Curved Grating Spectrometer and Wavelength Multiplexer or Demultiplexer with Very High Wavelength Resolution
20170102270 ยท 2017-04-13
Assignee
Inventors
Cpc classification
G02B5/1861
PHYSICS
G02B6/2938
PHYSICS
G02B6/29326
PHYSICS
International classification
Abstract
The present application discloses a system comprising a compact curved grating (CCG) and its associated compact curved grating spectrometer (COGS) or compact curved grating wavelength multiplexer/demultiplexer (WMDM) module and a method for making the same. The system is capable of achieving a very small (resolution vs. size) RS factor. The location of the entrance slit and detector can be adjusted in order to have the best performance for a particular design goal. The initial groove spacing is calculated using a prescribed formula dependent on operation wavelength. The location of the grooves is calculated based on two conditions. The first one being that the path-difference between adjacent grooves should be an integral multiple of the wavelength in the medium to achieve aberration-free grating focusing at the detector or a first anchor output slit even with large beam diffraction angle from the entrance slit or input slit, the second one being specific for a particular design goal of a curved-grating spectrometer.
Claims
1.-17. (canceled)
19. A wavelength multiplexer/demultiplexer/spectrometer or compact curved grating spectrometer using discrete optical components or with integration possibility as a wavelength dispersion element in a photonic integrated circuit, enabling dispersion of light spectra around a wavelength .sub.BI1, the wavelength multiplexer/demultiplexer/spectrometer comprising: at least one input slit; a plurality of output slits; and a curved grating, the curved grating configured for processing the spectra compositions of the optical beam including a plurality of grooves, the position of each groove being adjustable for controlling a performance of the wavelength multiplexer/demutiplexer/spectrometer, and the position of the input slit and each of the output slits being adjustable for controlling a performance of the wavelength multiplexer/demutiplexer/spectrometer, wherein the input slit allows an entry of the optical beam into the wavelength multiplexer/demutiplexer/spectrometer, a location of the input slit being adjustable, and further the location of the input slit X.sub.I1 specified by a first input angle .sub.I1 that is sustained between the line joining the input slit to the grating center and a normal line to the grating center, and a first input distance S.sub.I1 from the grating center to the input slit, further wherein a first output slit for allowing the exiting of a first output optical beam having a first anchor output wavelength .sub.I1-O1A, a location of the first anchor output slit being adjustable, and further the location of the first anchor output slit specified by a first output angle .sub.O1A that is sustained between the line joining the first output slit to the grating center and a normal line to the grating center, and a first output distance S.sub.O1A from the grating center to the first anchor output slit, further wherein a medium in which the light propagates in having an effective refractive index of propagation n.sub.gr, wherein in the case of free space, n.sub.gr is the material refractive index, and in the case of a planar waveguide, n.sub.gr is the effective refractive index of propagation within the planar waveguide, further wherein a position of the i.sup.th groove is specified by its x-y coordinates X.sub.i=(x.sub.i, y.sub.i), the x-y coordinates are specified with respect to the grating center and the input slit, for which the grating center has the coordinate X.sub.0=(0, 0) and the input slit has the coordinate X.sub.I1=(S.sub.I1*Sin(.sub.I1), S.sub.I1*Cos(.sub.I1)), wherein the given value of the input circle radius R where R is related to the input slit position by S.sub.I1=R*Cos(.sub.I1), around the grating center at X.sub.0=(0, 0), two initial grating teeth are chosen to be located at a distance d apart from each other so that they are placed at locations
X.sub.1=(d/2,R(R.sup.2(d/2).sup.2).sup.1/2)
and
X.sub.1=(d/2,R(R.sup.2(d/2).sup.2).sup.1/2), where when given the first anchor output wavelength .sub.I1-O1A, the distance d is to be determined as follows: choosing a grating order and denoting the order by an integer m, and obtaining the grating parameter d from
d*(Sin(.sub.O1A)+Sin(.sub.I1))=m*.sub.I1-O1A/n.sub.gr, further wherein the locations of all other grooves are given by computing the coordinate of each groove with the i.sup.th groove's coordinate X.sub.i given by the following two conditions: further wherein the locations of all other grooves are given by computing the coordinate of each groove with the i.sup.th groove's coordinate X.sub.i given by a first condition:
Sgn(ij a)*([D.sub.1(.sub.I1,S.sub.I1,X.sub.i)+D.sub.2(.sub.O1A,S.sub.O1A,X.sub.i)][D.sub.1(.sub.O1,S.sub.I1,X.sub.ja)+D.sub.2(.sub.O1A,S.sub.O1A,X.sub.ja)])=m*.sub.I1-O1A/n.sub.gr, hereby referred to as Eq. (1), wherein D.sub.1(.sub.I1,S.sub.I1, X.sub.i) is the distance from X.sub.i to the first input slit location X.sub.I1 specified by .sub.I1 and S.sub.I1, D.sub.2(.sub.O1A,S.sub.O1A,X.sub.i) is the distance from X.sub.i to the first anchor output slit location specified by .sub.O1A and S.sub.O1A, and the position of groove ja, X.sub.ja is typically already known, and a second condition such that a function f is equal to a numerical constant, functionally expressed as:
f(X.sub.i)=constant where the above constant can be depending on other design parameters such as the input slit and output slit positions or the positions of the adjacent grooves (e.g. .sub.I1,S.sub.I1,.sub.O1,S.sub.O1, .sub.I1-O1, m, n.sub.gr, {X.sub.j}) that are already known and treated as part of the constant, wherein the positions {X.sub.j} represent the positions of some grating teeth that are already known. wherein the second constraint is further given by choosing the function f so that:
[D.sub.1(.sub.I1,S.sub.I1,X.sub.i)+D.sub.3(.sub.O2A,S.sub.O2A,X.sub.i)][D.sub.1(.sub.I1,S.sub.I1,X.sub.i1)+D.sub.3(.sub.O2A,S.sub.O2A,X.sub.i1)]=m*.sub.I1-O2A/n.sub.gr, hereby referred to as Eq. (2), wherein D.sub.3(.sub.O2A,S.sub.O2A,X.sub.i) is the distance from the i-th groove located at X.sub.i to the second anchor output slit specified by a third angle .sub.O2A that is sustained between the line joining the second output slit to the grating center and a normal line of the grating center, and a second output distance S.sub.O2A from the grating center to the second output slit, wavelength .sub.I1-O2A is a second wavelength that is the wavelength for the second output slit given by:
d*(Sin(.sub.O2A)+Sin(.sub.I1))=m*.sub.I1-O2A/n.sub.gr, and by solving Eq.(1) and Eq.(2) for the x-coordinate x.sub.i and y-coordinate y.sub.i of the i.sup.th groove at X.sub.i=(x.sub.i, y.sub.i), exact locations of other grooves X.sub.i's are obtained, further wherein for more than one of the plurality of the output waveguides, the waveguide has a first tapering region forming the output mouth that tapered from the entrance mouth width to near or smaller than a waveguide width that supports only the fundamental mode, further wherein somewhere after the first tapering region is a first straight waveguide that can have zero length or afinite length.
20. The wavelength multiplexer/demultiplexer/spectrometer as recited in claim 19, wherein one or both of the two anchor output slits are not physically present, and are only used for the purpose of generating the grating teeth.
21. A second device wavelength multiplexer/demultiplexer/spectrometer with a grating that is basically the same design as recited in claim 19, wherein two gratings are the same design if a set of grooves involved in reflecting more than 50% of the total power reflected by the grating towards the same output slit location has each of its groove's S satisfying S<.sub.I1-O1A/(n.sub.gr), where S=(x.sup.2+y.sup.2).sup.0.5 with x being the spatial deviation of grating groove position in the second device from the designed position of claim 19 in the x-coordinate and y being the spatial deviation of grating groove position in the second device from the designed position of claim 19 in the y-coordinate.
22. The wavelength multiplexer/demultiplexer/spectrometer as recited in claim 19, wherein the equation: d*(Sin(.sub.O1A)+Sin(.sub.I1))=m*.sub.I1-O1A/n.sub.gr, is replaced by:
[D.sub.1(.sub.I1,S.sub.I1,X.sub.1)+D.sub.2(.sub.O1A,S.sub.O1A,X.sub.1)][D.sub.1(.sub.I1,S.sub.I1,X.sub.1)+D.sub.2(.sub.O1A,S.sub.O1A,X.sub.1)]=m*.sub.I1-O1A/n.sub.grI1-O1A Further the equation: d*(Sin(.sub.O2A)+Sin(.sub.I1))=m*.sub.I1-O2A/n.sub.gr, is replaced by:
[D.sub.1(.sub.I1,S.sub.I1,X.sub.1)+D.sub.2(.sub.O2A,S.sub.O2A,X.sub.1)][D.sub.1(.sub.I1,S.sub.I1,X.sub.1)+D.sub.2(.sub.O2A,S.sub.O2A,X.sub.1)]=m*.sub.I1-O2A/n.sub.grI1-O2A.
23. A wavelength multiplexer/demultiplexer/spectrometer or compact curved grating spectrometer using discrete optical components or with integration possibility as a wavelength dispersion element in a photonic integrated circuit, enabling dispersion of light spectra around a wavelength .sub.BI1, the wavelength multiplexer/demultiplexer/spectrometer comprising: at least one input slit; a plurality of output slits; and a curved grating, the curved grating configured for processing the spectra compositions of the optical beam including a plurality of grooves, the position of each groove being adjustable for controlling a performance of the wavelength multiplexer/demutiplexer/spectrometer, and the position of the input slit and each of the output slits being adjustable for controlling a performance of the wavelength multiplexer/demutiplexer/spectrometer, wherein the input slit allows an entry of the optical beam into the wavelength multiplexer/demutiplexer/spectrometer, a location of the input slit being adjustable, and further the location of the input slit X.sub.I1 specified by a first input angle .sub.I1 that is sustained between the line joining the input slit to the grating center and a normal line to the grating center, and a first input distance S.sub.I1 from the grating center to the input slit, further wherein a first output slit for allowing the exiting of a first output optical beam having a first anchor output wavelength .sub.I1-O1A, a location of the first anchor output slit being adjustable, and further the location of the first anchor output slit specified by a first output angle .sub.O1A that is sustained between the line joining the first output slit to the grating center and a normal line to the grating center, and a first output distance S.sub.O1A from the grating center to the first anchor output slit, further wherein a medium in which the light propagates in having an effective refractive index of propagation n.sub.gr, wherein in the case of free space, n.sub.gr is the material refractive index, and in the case of a planar waveguide, n.sub.gr is the effective refractive index of propagation within the planar waveguide, further wherein a position of the i.sup.th groove is specified by its x-y coordinates X.sub.i(x.sub.i, y.sub.i), the x-y coordinates are specified with respect to the grating center and the input slit, for which the grating center has the coordinate X.sub.0=(0, 0) and the input slit has the coordinate X.sub.I1=(S.sub.I1*Sin(.sub.I1), S.sub.I1*Cos(.sub.I1)), wherein the given value of the input circle radius R where R is related to the input slit position by S.sub.I1=R*Cos(.sub.I1), around the grating center at X.sub.0=(0, 0), two initial grating teeth are chosen to be located at a distance d apart from each other so that they are placed at locations
X.sub.1=(d/2,R(R.sup.2(d/2).sup.2).sup.1/2)
and
X.sub.1=(d/2,R(R.sup.2(d/2).sup.2).sup.1/2), where when given the first anchor output wavelength .sub.I1-O1A, the distance d is to be determined as follows: choosing a grating order and denoting the order by an integer m, and obtaining the grating parameter d from
d*(Sin(.sub.O1A)+Sin(.sub.I1))=m*.sub.I1-O1A/n.sub.gr, further wherein the locations of all other grooves are given by computing the coordinate of each groove with the i.sup.th groove's coordinate X.sub.i given by the following two conditions: further wherein the locations of all other grooves are given by computing the coordinate of each groove with the i.sup.th groove's coordinate X.sub.i given by a first condition:
Sgn(ija)*([D.sub.1(.sub.I1,S.sub.I1,X.sub.i)+D.sub.2(.sub.O1A,S.sub.O1A,X.sub.i)][D.sub.1(.sub.I1,S.sub.I1,X.sub.ja)+D.sub.2(.sub.O1A,S.sub.O1A,X.sub.ja)])=m*.sub.I1-O1A/n.sub.gr, hereby referred to as Eq. (1), wherein D.sub.1(.sub.I1,S.sub.I1, X.sub.i) is the distance from X.sub.i to the first input slit location X.sub.I1 specified by .sub.I1 and S.sub.I1, D.sub.2(.sub.O1A,S.sub.O1A,X.sub.i) is the distance from X.sub.i to the first anchor output slit location specified by .sub.O1A and S.sub.O1A, and the position of groove ja, X.sub.ja is typically already known, and a second condition such that a function f is equal to a numerical constant, functionally expressed as:
f(X.sub.i)=constant where the above constant can be depending on other design parameters such as the input slit and output slit positions or the positions of the adjacent grooves (e.g. .sub.I1,S.sub.I1,.sub.O1,S.sub.O1, .sub.I1-O1, m, n.sub.gr, {X.sub.j}) that are already known and treated as part of the constant, wherein the positions {X.sub.j} represent the positions of some grating teeth that are already known, wherein the second constraint is further given by choosing the function f so that:
[D.sub.1(.sub.I1,S.sub.I1,X.sub.i)+D.sub.3(.sub.O2A,S.sub.O2A,X.sub.i)][D.sub.1(.sub.I1,S.sub.I1,X.sub.i1)+D.sub.3(.sub.O2A,S.sub.O2A,X.sub.i1)]=m*.sub.I1-O2A/n.sub.gr, hereby referred to as Eq. (2), wherein D.sub.3(.sub.O2A,S.sub.O2A,X.sub.i) is the distance from the i-th groove located at X.sub.i to the second anchor output slit specified by a third angle .sub.O2A that is sustained between the line joining the second output slit to the grating center and a normal line of the grating center, and a second output distance S.sub.O1A from the grating center to the second output slit, wavelength .sub.I1-O2A is a second wavelength that is the wavelength for the second output slit given by:
d*(Sin(.sub.O2A)+Sin(.sub.I1))=m*.sub.I1-O2A/n.sub.gr, and by solving Eq.(1) and Eq.(2) for the x-coordinate x.sub.i and y-coordinate y.sub.i of the i.sup.th groove at X.sub.1=(x.sub.1, y.sub.i), exact locations of other grooves X.sub.i's are obtained, wherein two anchor output slit positions X.sub.O1A and X.sub.O2A, and the input slit position X.sub.I1 are descried by two lines, a first line called anchor-output-slits line L.sub.AOS, joins X.sub.O1A and X.sub.O2A with a midpoint at A.sub.AOSM, and a second line called input to anchor-Output-slits-midpoint line L.sub.IM, joins A.sub.AOSM and X.sub.I1, wherein a line joining the input X.sub.I1 to the grating center is called line L.sub.I1, the angle between a line at 90 degrees to line L.sub.I1 and the line L.sub.IM is IM, which takes on a value of 0 when line L.sub.I1 and line L.sub.IM are perpendicular to each other, and take on a positive value when line L.sub.IM is rotated about the input slit point X.sub.I1 in a direction to bring the point X.sub.L(O1A,O2A)M closer in its distance to the grating center, wherein a line joining the midpoint A.sub.AOSM to the grating center is called line L.sub.GM, the angle between a line at 90 degrees to line L.sub.GM and the line L.sub.IM is .sub.GM, which takes on a value of 0 when line L.sub.IM and line L.sub.GM are perpendicular to each other, and take on a positive value when line L.sub.IM is rotated about the input slit point X.sub.I1 in a direction to bring the point A.sub.AOSM closer in its distance to the grating center; and wherein the angle between line L.sub.IM and the line L.sub.AOS is .sub.AM, which takes on a value of 0 when line L.sub.IM and line L.sub.AOS are parallel to each other, and take on a positive value when line L.sub.AOS is rotated about its midpoint X.sub.AOSM in a direction that brings the furthest end of line L.sub.AOS from X.sub.I1 closer in its distance to the grating center, the two anchor output slit positions X.sub.O1A and X.sub.O2A are placed such that .sub.IM is within +45 and 45 and .sub.AM is within +45 and 45.
24. The wavelength multiplexer/demultiplexer/spectrometer as recited in claim 23 wherein the two anchor output slit positions X.sub.O1A and X.sub.O2A are placed such that .sub.GM is within +45 and 45 and .sub.AM is within +45 and 45.
25. The wavelength multiplexer/demultiplexer/spectrometer as recited in claim 23 wherein the two anchor output slit positions X.sub.O1A and X.sub.O2A are placed such that .sub.IM is within +30 and 30 and .sub.AM is within +30 and 30.
26. The wavelength multiplexer/demultiplexer/spectrometer as recited in claim 23 wherein the two anchor output slit positions X.sub.O1A and X.sub.O2A are placed such that .sub.GM is within +30 and 30 and .sub.AM is within +30 and 30.
27. The wavelength multiplexer/demultiplexer/spectrometer as recited in claim 23 wherein the two anchor output slit positions X.sub.O1A and X.sub.O2A are placed such that .sub.IM is within +15 and 15 and .sub.AM is within +15 and 15.
28. The wavelength multiplexer/demultiplexer/spectrometer as recited in claim 23 wherein the two anchor output slit positions X.sub.O1A and X.sub.O2A are placed such that .sub.GM is within +15 and 15 and .sub.AM is within +15 and 15.
29. The wavelength multiplexer/demultiplexer/spectrometer as recited in claim 23, wherein one or both of the two anchor output slits are not physically present, and are only used for the purpose of generating the grating teeth.
30. A second device wavelength multiplexer/demultiplexer/spectrometer with a grating that is basically the same design as recited in claim 23, wherein two gratings are the same design if a set of grooves involved in reflecting more than 50% of the total power reflected by the grating towards the same output slit location has each of its groove's S satisfying S<.sub.I1-O1A/(n.sub.gr), where S=(x.sup.2+y.sup.2).sup.0.5 with x being the spatial deviation of grating groove position in the second device from the designed position of claim 23 in the x-coordinate and y being the spatial deviation of grating groove position in the second device from the designed position of claim 23 in the y-coordinate.
31. The wavelength multiplexer/demultiplexer/spectrometer as recited in claim 23, wherein the equation: d*(Sin(.sub.O1A)+Sin(.sub.I1))=m*.sub.I1-O1A/n.sub.gr, is the equation:
[D.sub.1(.sub.I1,S.sub.I1,X.sub.1)+D.sub.2(.sub.O1A,S.sub.O1A,X.sub.1)][D.sub.1(.sub.I1,S.sub.I1,X.sub.1)+D.sub.2(.sub.O1A,S.sub.O1A,X.sub.1)]=m*.sub.I1-O1A/n.sub.grI1-O1A Further the equation: d*(Sin(.sub.O2A)+Sin(.sub.I1))=m*.sub.I1-O2A/n.sub.gr, is the equation:
[D.sub.1(.sub.I1,S.sub.I1,X.sub.1)+D.sub.2(.sub.O2A,S.sub.O2A,X.sub.1)][(.sub.I1,S.sub.I1,X.sub.1)+D.sub.2(.sub.O2A,S.sub.O2A,X.sub.1)]=m*.sub.I1-O2A/n.sub.grI1-O2A.
32. A wavelength multiplexer/demultiplexer/spectrometer or compact curved grating spectrometer using discrete optical components or with integration possibility as a wavelength dispersion element in a photonic integrated circuit, enabling dispersion of light spectra around a wavelength .sub.BI1, the wavelength multiplexer/demultiplexer/spectrometer comprising: at least one input slit; a plurality of output slits; and a curved grating, the curved grating configured for processing the spectra compositions of the optical beam including a plurality of grooves, the position of each groove being adjustable for controlling a performance of the wavelength multiplexer/demutiplexer/spectrometer, and the position of the input slit and each of the output slits being adjustable for controlling a performance of the wavelength multiplexer/demutiplexer/spectrometer, wherein the input slit allows an entry of the optical beam into the wavelength multiplexer/demutiplexer/spectrometer, a location of the input slit being adjustable, and further the location of the input slit X.sub.I1 specified by a first input angle .sub.I1 that is sustained between the line joining the input slit to the grating center and a normal line to the grating center, and a first input distance S.sub.I1 from the grating center to the input slit, further wherein a first output slit for allowing the exiting of a first output optical beam having a first anchor output wavelength .sub.I1-O1A, a location of the first anchor output slit being adjustable, and further the location of the first anchor output slit specified by a first output angle .sub.O1A that is sustained between the line joining the first output slit to the grating center and a normal line to the grating center, and a first output distance S.sub.O1A from the grating center to the first anchor output slit, further wherein a medium in which the light propagates in having an effective refractive index of propagation n.sub.gr, in the case of free space, n.sub.gr is the material refractive index. In the case of a planar waveguide, n.sub.gr is the effective refractive index of propagation within the planar waveguide, further wherein a position of the i.sup.th groove is specified by its x-y coordinates X.sub.i=(x.sub.i, y.sub.i), the x-y coordinates are specified with respect to the grating center and the input slit, for which the grating center has the coordinate X.sub.0=(0, 0) and the input slit has the coordinate X.sub.I1=(S.sub.I1*Sin(.sub.I1), S.sub.I1*Cos(.sub.I1)), wherein the given value of the input circle radius R where R is related to the input slit position by S.sub.I1=R*Cos(.sub.I1), around the grating center at X.sub.0=(0, 0), two initial grating teeth are chosen to be located at a distance d apart from each other so that they are placed at locations
X.sub.1=(d/2,R(R.sup.2(d/2).sup.2).sup.1/2)
and
X.sub.1=(d/2,R(R.sup.2(d/2).sup.2).sup.1/2), where when given the first anchor output wavelength .sub.I1-O1A, the distance d is to be determined as follows: choosing a grating order and denoting the order by an integer m, and obtaining the grating parameter d from
d*(Sin(.sub.O1A)+Sin(.sub.I1))=m*.sub.I1-O1A/n.sub.gr, further wherein the locations of all other grooves are given by computing the coordinate of each groove with the i.sup.th groove's coordinate X.sub.i given by the following two conditions: further wherein the locations of all other grooves are given by computing the coordinate of each groove with the i.sup.th groove's coordinate X.sub.i given by a first condition:
Sgn(ija)*([D.sub.1(.sub.I1,S.sub.I1,X.sub.i)+D.sub.2(.sub.O1A,S.sub.O1A,X.sub.i)][D.sub.1(.sub.I1,S.sub.I1,X.sub.ja)+D.sub.2(.sub.O1A,S.sub.O1A,X.sub.ja)])=m*.sub.I1-O1A/n.sub.gr, hereby referred to as Eq. (1), wherein D.sub.1(.sub.I1,S.sub.I1, X.sub.i) is the distance from X.sub.i to the first input slit location X.sub.I1 specified by .sub.I1 and S.sub.I1, D.sub.2(.sub.O1A,S.sub.O1A,X.sub.i) is the distance from X.sub.i to the first anchor output slit location specified by .sub.O1A and S.sub.O1A, and the position of groove ja, X.sub.ja is typically already known, and a second condition such that a function f is equal to a numerical constant, functionally expressed as:
f(X.sub.i)=constant where the above constant can be depending on other design parameters such as the input slit and output slit positions or the positions of the adjacent grooves (e.g. .sub.I1,S.sub.I1,.sub.O1,S.sub.O1, .sub.I1-O1, m, n.sub.gr, {X.sub.j}) that are already known and treated as part of the constant, wherein the positions {X.sub.j} represent the positions of some grating teeth that are already known, wherein the second constraint is further given by choosing the function f so that:
[D.sub.1(.sub.I1,S.sub.I1,X.sub.i)+D.sub.3(.sub.O2A,S.sub.O2A,X.sub.i)][D.sub.1(.sub.I1,S.sub.I1,X.sub.i1)+D.sub.3(.sub.O2A,S.sub.O2A,X.sub.i1)]=m*.sub.I1-O2A/n.sub.gr, hereby referred to as Eq. (2), wherein D.sub.3(.sub.O2A,S.sub.O2A,X.sub.i) is the distance from the i-th groove located at X.sub.i to the second anchor output slit specified by a third angle .sub.O2A that is sustained between the line joining the second output slit to the grating center and a normal line of the grating center, and a second output distance S.sub.O2A from the grating center to the second output slit, wavelength .sub.I1-O2A is a second wavelength that is the wavelength for the second output slit given by:
d*(Sin(.sub.O2A)+Sin(.sub.I1))=m*.sub.I1-O2A/n.sub.gr, and by solving Eq.(1) and Eq.(2) for the x-coordinate x.sub.i and y-coordinate y.sub.i of the i.sup.th groove at X.sub.i=(x.sub.i, y.sub.i), exact locations of other grooves X.sub.i's are obtained, wherein the plurality of more than one output slits with location positions at X.sub.O1, . . . X.sub.Ok wherein X.sub.Ok is determined as follows: defining a point P to have position given by X.sub.L(i,i1)I1-OkP=(x.sub.L(i,i1)I1-OkP, y.sub.L(i,i1)I1-OkP), where x.sub.L(i,i1)I1-OkP=Sin(.sub.L(i,i1)I1-OkP) and y.sub.L(i,i1)I1-OkP=S.sub.L(i,i1)I1-OkP*Cos(.sub.L(i,i1)I1-OkP), for which the following equation is satisfied based on the grating grooves of number i and number (i1):
[D.sub.1(.sub.I1,S.sub.I1,X.sub.i)+D.sub.2(.sub.L(i,i1)I1-OkP,S.sub.L(i,i1)I1-OkP,X.sub.i)][D.sub.1(.sub.I1,S.sub.I1,X.sub.i1)+D.sub.2(.sub.L(i,i1)I1-OkP,S.sub.L(i,i1)I1-OkP,X.sub.i1)]=m*.sub.I1-Ok/n.sub.grI1-Ok wherein the locus of all position point P forms a line L.sub.(i,i1)I1-Ok, wherein D.sub.2(.sub.L(i,i1)I1-OkP,S.sub.L(i,i1)I1-OkP,X.sub.i) is the distance from X.sub.i to the point P on line L.sub.(i,i1)I1-Ok c(i,i1)L, D.sub.1(.sub.I1,S.sub.I1,X.sub.i) is the distance from X.sub.i to the first input slit at X.sub.I1. Line L.sub.(i,i1)I1-Ok is generated when S.sub.L(i,i1)I1-OkP 19-189kP(i,i1)S increases from an initial small value to a value larger than the estimated position of S.sub.Ok; wherein the second line, Line L.sub.(j,j-1)I1-Ok is generated with grating grooves of number j and (j1) for which a point Q having position given by X.sub.L(j,j-1)I1-OkQ=(x.sub.L(j,j-1)I1-OkQ, y.sub.L(j,j-1)I1-OkQ, where X.sub.L(j,j-1)I1-OkQ=S.sub.L(j,j-1)I1-OkQ*Sin(.sub.L(j,j-1)I1-OkQ) and y.sub.L(j,j-1)I1-OkQ=S.sub.L(j,j-1)I1-OkQ*Cos(.sub.L(j,j-1)I1-OkQ), for which the following equation is satisfied based on the grating grooves of number j and number (j1):
[D.sub.1(.sub.I1,S.sub.I1,X.sub.j)+D.sub.2(.sub.L(j,j-1)I1-OkQ,S.sub.L(j,j-1)I1-OkQ,X.sub.j)][D.sub.1(.sub.I1,S.sub.I1,X.sub.j-1)+D.sub.2(.sub.I1,S.sub.I1X.sub.j-1)]=m*.sub.I1-Ok/n.sub.grI1-Ok wherein the locus of all position point Q forms a line L.sub.(j,j-1)I1-Ok, wherein the grating groove pairs (j,j1) for grating grove at X.sub.j and X.sub.j-1 are chosen to lie on the opposite side of the grating center from that of grating groove pair (i,i1) for grating grove at X.sub.i and X.sub.i+1, and the location of X.sub.Ok of output slit k that shall receive beam spectral component at wavelength .sub.I1-Ok is then chosen to be a point near the point X.sub.Okest, called the estimated output location, where the point X.sub.Okes is obtained by a function V=V({X.sub.Ok(i,i1;j,j-1)}) that is dependent on all the vectors X.sub.Ok(i,i1;j,j-1) generated by a selected set of the grating groove pairs with different values of i,i1 or j,j1, such that:
X.sub.Okest=V({X.sub.Ok(i,i1;j,j-1)}) wherein near is within three times the beam diameter generated by the input beam at X.sub.Okest defined by the full-width half-maximum of the beam intensity width, or three times the width W.sub.Ok of the slit at X.sub.Ok, whichever is larger.
33. The wavelength multiplexer/demultiplexer/spectrometer as recited in claim 32, wherein the location of X.sub.Ok of output slit k that shall receive beam spectral component at wavelength .sub.I1-Ok is then chosen to be a point very near the point X.sub.Okest, called the estimated output location, where the point X.sub.Okes is obtained by a function V=V({X.sub.Ok(i,i1;j,j-1)}) that is dependent on all the vectors X.sub.Ok(i,i1;j,j-1) generated by a selected set of the grating groove pairs with different values of i,i1 or j,j1, so that:
X.sub.Okest=V({X.sub.Ok(i,i1;j,j-1)}) wherein very near is within half the beam diameter generated by the input beam at X.sub.Okest defined by the full-width half-maximum of the beam intensity width, or half the width W.sub.Ok of the slit at X.sub.Ok, whichever is larger.
34. The wavelength multiplexer/demultiplexer/spectrometer as recited in claim 32, wherein the location of X.sub.Ok of output slit k that shall receive beam spectral component at wavelength .sub.I1-OK is then chosen to be a point at the point X.sub.Okest, called the estimated output location, where the point X.sub.Okes is obtained by a function V=V({X.sub.Ok(i,i1;j,j-1)}) that is dependent on all the vectors X.sub.Ok(i,i1;j,j-1) generated by a selected set of the grating groove pairs with different values of i,i1 or j,j1, so that:
X.sub.Okest=V({X.sub.Ok(i,i1;j,j-1)}) wherein at is within 10% of the beam diameter generated by the input beam at X.sub.Okest defined by the full-width half-maximum of the beam intensity width, or 10% of the width W.sub.Ok of the slit at X.sub.Ok, whichever is larger.
35. The wavelength multiplexer/demultiplexer/spectrometer as recited in claim 32, wherein V=V({X.sub.Ok(i,i1;j,j-1)}) involves a weighted average of the intersecting points between pairs of lines in which the averaging is weighted by multiplying the solution X.sub.Ok(i,i1;j,j-1) with the total input beam power or its positive power exponential that reaches both grating groove i,i1 pair and groove j,j1 pair, and by vectorially summing up all vectors obtained after such multiplications computed from a set of vectors X.sub.Ok(i,i1;j,j-1) obtained from a selected set of grooves, and divided by the sum of the total input beam power used in the weighting multiplications, then gives the x and y coordinate values x.sub.Okest, and y.sub.Okest, for obtaining the estimated output slit location X.sub.Okest=(x.sub.Okest, y.sub.Okest), wherein P.sub.I1(i,i1;j,j-1) is the total input beam power that falls on the surfaces of both groove i,i1 pair and groove j,j1 pair from input slit SL.sub.I1 due to beam spatial diffraction from the input slit SL.sub.I1, then X.sub.Okest is given by:
X.sub.Okest=V({X.sub.Ok(i,i1;j,j-1)})=[Sum({i,i1;j,j1})([P.sub.I1(i,i1;j,j-1)].sup.NX.sub.Ok(i,i1;j,j-1))]/[Sum({i,i1;j,j1})(P.sub.I1(i,i1;j,j-1))], where Sum({i,i1;j,j1}) denotes sum over the range of all the i,i1 and j,j1 pairs in the set {i,i1;j,j1} defined above, and N in Eq. 0 is taking P to the power of N, where N is a positive real number larger than 0.
36. The wavelength multiplexer/demultiplexer/spectrometer as recited in claim 32, wherein V=V({X.sub.Ok(i,i1;j,j-1)}) involves a weighted average of the intersecting points between pairs of lines in which the averaging is weighted by multiplying the solution X.sub.Ok(i,i1;j,j-1) with a function that depends on total input beam power that reaches both grating groove i,i1 pair and groove j,j1 pair, and vectorially summing up all vectors obtained after such multiplications computed from a set of vectors X.sub.Ok(i,i1;j,j-1) obtained from a selected set of grooves, and divided by the sum of the total input beam power used in the weighting multiplications, then gives the x and y coordinate values x.sub.Okest, and y.sub.Okest, for obtaining the estimated output slit location X.sub.Okest=(x.sub.Okest, y.sub.Okest), wherein P.sub.I1(i,i1;j,j-1) is the total input beam power that falls on the surfaces of both groove i,i1 pair and groove j,j1 pair from input slit SL.sub.I1 due to beam spatial diffraction from the input slit SL.sub.I1, then X.sub.Okest is given by:
X.sub.Okest=V({X.sub.Ok(i,i1;j,j-1)})=[Sum({i,i1;j,j1})(f[P.sub.I1(i,i1;j,j-1)]X.sub.Ok(i,i1;j,j-1))]/[Sum({i,i1;j,j1})P.sub.I1(i,i1;j,j-1))], and where f[P.sub.I1(i,i1;j,j-1)] is any mathematical function of P.sub.I1(i,i1;j,j-1).
37. The wavelength multiplexer/demultiplexer/spectrometer as recited in claim 32, wherein one or both of the two anchor output slits are not physically present, and are only used for the purpose of generating the grating teeth.
38. A second device wavelength multiplexer/demultiplexer/spectrometer with a grating that is basically the same design as recited in claim 32, wherein two gratings are the same design if a set of grooves involved in reflecting more than 50% of the total power reflected by the grating towards the same output slit location has each of its groove's S satisfying S<.sub.I1-O1A/(n.sub.gr), where S=(x.sup.2+y.sup.2).sup.0.5 with x being the spatial deviation of grating groove position in the second device from the designed position of claim 32 in the x-coordinate and y being the spatial deviation of grating groove position in the second device from the designed position of claim 32 in the y-coordinate.
39. The wavelength multiplexer/demultiplexer/spectrometer as recited in claim 32, wherein the equation: d*(Sin(.sub.O1A)+Sin(.sub.I1))=m*.sub.I1-O1A/n.sub.gr, is the equation:
[D.sub.1(.sub.I1,S.sub.I1,X.sub.1)+D.sub.2(.sub.O1A,S.sub.O1A,X.sub.1)][D.sub.1(.sub.I1,S.sub.I1,X.sub.1)+D.sub.2(.sub.O1A,S.sub.O1A,X.sub.1)]=m*.sub.I1-O1A/n.sub.grI1-O1A Further the equation: d*(Sin(.sub.O2A)+Sin(.sub.I1))=m*.sub.I1-O2A/n.sub.gr, is the equation:
[D.sub.1(.sub.I1,S.sub.I1,X.sub.1)+D.sub.2(.sub.O2A,S.sub.O2A,X.sub.1)][D.sub.1(.sub.I1,S.sub.I1,X.sub.1)+D.sub.2(.sub.O2A,S.sub.O2A,X.sub.1)]=m*.sub.I1-O2A/n.sub.grI1-O2A.
Description
BRIEF DESCRIPTION OF THE DRAWINGS
[0082] The preferred embodiments of the invention will hereinafter be described in conjunction with the appended drawings provided to illustrate and not to limit the invention, wherein like designations denote like elements, and in which
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DETAILED DESCRIPTION
[0114] The present invention discloses a system comprising a compact curved grating (CCG), its associated compact curved grating spectrometer (CCGS) or wavelength Mux/deMux (WMDM) module and a method for making the same. The system is capable of achieving very small (resolution vs. size/area) RS.sub.SM/RA.sub.SM factor thereby enabling high resolution at small size. It is also capable of achieving high adjacent-wavelength power extinction ratio .sub.ace(.sub.SM), and high spectral output efficiency .sub.eff(.sub.SM) at the detecting wavelength .sub.SM.
[0115] The uses of CCGS or WMDM module include an isolated optical spectrometer or wavelength Mux/deMux using discrete optical components such as slits, grating, spectrometer or wavelength Mux/deMux casing, photodetector, photodectector array, or motor drive. The CCGS or WMDM module could also be used as a wavelength dispersion element in a photonic integrated circuit or electronic-photonic integrated circuit.
[0116] The integrated circuit can be based on various materials including but not limited to glass (silica) waveguide, semiconductor waveguide, polymer waveguide, or any other type of optical waveguiding devices. Semiconductor waveguides include silicon or compound semiconductor waveguides such as III-V (GaAs, InP, InGaAsP, InAlAsP etc). The wavelength dispersion element based on the CCGS or WMDM module in the photonic integrated circuit can be integrated with optical detector, laser, amplifier, waveguide, modulator, splitter, multimode interference devices, other wavelength filters, array-waveguide-based devices, and other photonic devices, materials, or components to achieve a multi-component photonic integrated circuit with useful functionalities. The CCG explained below is a High-Resolution Compact Curved Grating (HR-CCG) that tries to alleviate the disadvantages associated with prior art mentioned earlier, by providing a high resolution in a small (compact) module with high spectral filtering performances such as high adjacent-wavelength power extinction and high spectral power output efficiency.
[0117] The present invention discloses a device that will have a wide range of utilities and can be used as a device in optical spectrometers, wavelength channel multiplexers, wavelength channel demultiplexers, wavelength or frequency filters, wavelength combiners, wavelength splitters, optical spectrum analyzers, wavelength detectors, spectra dispersion devices, optical arbitrary waveform generators, optical dispersion compensators, optical signal processors, and optical wavelength-domain or frequency-domain processors, for combining, filtering, analyzing, processing, or detecting the spectral compositions of an input optical beam or plurality of input beams, with one or plurality of output beams, and methods of making the same.
[0118] We have improved on the current Rowland design, enabling curved-grating spectrometer with 10-100 times smaller linear size (or 100-10,000 time smaller area) using our HR-CCG with large-angle aberration-corrected design. The typical Rowland design can only reach a useful divergence angle .sub.div of 4 DEG, beyond which serous aberration in the refocusing beam will occur to limit wavelength resolution. In
[0119] We have used discrete time solution of vectorial Maxwell's equations to simulate the HR-CCG design, which verified the high resolution nature of our grating as predicted by the ray-tracing method. Referring to
[0120] Referring to
[0121] Referring to
[0122] Referring to
[0123] Referring to
[0124] The planar waveguiding region referred to above is basically the region in which wave or beam propagates between the input slit and the grating or the grating and the output slit and is basically the grating-propagating region referred to as GPR 1020 in
[0125] Above illustrate the various general geometrical configurations of the present invention for the purpose of illustration and not limitation. It would be obvious to those skilled in the art to combine, separate, or utilize the components in these illustrative configurations in various ways.
FURTHER DEFINITIONS OF COMMON TERMINOLOGIES
[0126] The subsection below further defines the common terminologies useful for describing the present invention.
[0127] It is known in the art that a relatively compact optical spectrometer can be achieved using a curved grating. The schematics of such a curved grating spectrometer device 1000 is shown in
Input Region Specification, Grating Center Circle Normal Line, and Grating-Center-to-Input-Slit Line.
[0128] A first input optical beam B.sub.I1 1101 entering a first input slit SL.sub.I1 1201, where the subscript I1 is the label for input slit 1 or the first input slit. The width of the input slit is specified by a first input slit width W.sub.I1 1291W. The location of the center point PX.sub.I1 1291O of the first input slit SL.sub.I1 1201 is specified by a first input angle .sub.I1 1271 that is an angle sustained between the line L.sub.I1 1251 joining the center point PX.sub.I1 1291O of the input slit SL.sub.I1 1201 to the grating center CGC 1050 (called grating-center to input-slit line), and a normal line to a circle described below at the grating center (called grating-center circle normal line) L.sub.GCCN 1050N. The input slit location is further specified by a first input distance S.sub.I1 1261 from the grating center CGC 1050 to the center point PX.sub.I1 1291O of the first input slit SL.sub.I1 1201. The first input angle .sub.I1 1271 is zero when line L.sub.I1 1251 is parallel to grating-center circle normal line L.sub.GCCN 1050N. The angle .sub.I1 1271 takes on positive value when the line L.sub.I1 1251 is rotated counter-clockwise (CCW) about the grating center CGC 1050 from this zero-angle position, and takes on negative value when it is rotated clockwise (CW).
[0129] More specifically, as shown in
[0130] In the situation in which there are plurality of input slits, there is in general no requirement that these input slits be situated on this input circle, though preferably, their input angles will all be defined with respect to the same grating-center circle normal line L.sub.GCCN 1050N.
[0131] Optical Axis Definition.
[0132] As shown by
[0133] Let x.sub.I measures the perpendicular distance from the optical axis point P.sub.ZBI1 1151O to a point P.sub.ZBI1(x.sub.P) 1161 for which x.sub.P>0 on left side of the optical axis and x.sub.P<0 on right side of the optical axis. More specifically, left means 9 O'clock side of the direct of propagation and right is 3 O'clock side if the direction of propagation is the 12 O'clock direction. Then I(x.sub.P,Z.sub.BI1) 1151I as a function of x.sub.P denotes the transverse or lateral intensity profile of the input beam at Z.sub.BI1 1151D.
[0134] Input-Slit-Width Angle and Input-Slit-Mouth Line.
[0135] Let the width of the input slit W.sub.I1 1291W spans an angle .sub.I1 1281 at the grating center (called angle sustained by input slit width or input-slit-width angle) so that the CCW side (or left side; with front facing the slit) of the edge of the slit is at angle .sub.I1+LI.sub.1/2 and the CW side (or right side) of the slit is at .sub.I1.sub.RI1/2 where .sub.LI1/2 1281L and .sub.RI1/2 1281R are the angular span to the left and the right of input angle .sub.I1 for the input slit. Here, the direction from the curved grating center CGC 1050 to the input slit SL.sub.O1 1201 is the front direction or 12 O'clock direction and the left/right side is the counter-clockwise/clockwise side towards the 9/3 O'clock direction. This will be the sign conventions for all the angles below that are referencing the curved grating center CGC 1050 as the pivot of rotation.
[0136] The left edge of the first input slit at angle .sub.I1+.sub.LI1/2 is denoted as point PX.sub.LI1 1291L. The location of the right edge of the first input slit at angle .sub.I1.sub.RI1/2 is denoted as point PX.sub.RI1 1291R. The points PX.sub.I1 1291O, PX.sub.LI1 1291L, and PX.sub.RI1 1291R is an approximate straight line and point PX.sub.I1 1291O is at the middle of the line joining point PX.sub.LI1 1291L and point PX.sub.RI1 1291R, called the first input slit mouth line L.sub.(PXLI1-PXRI1). Thus, typically the left and right sides of the above angular spans are equal so that .sub.LI1=.sub.RI1=.sub.I1.
[0137] The spatial x-y coordinates for the points at the input slits are: for PX.sub.I1 1291O its x-y coordinates are denoted by X.sub.I1 1291OC; for PX.sub.LI1 1291L its x-y coordinates are denoted by X.sub.LI1 1291LC; for PX.sub.RI1 1291R its x-y coordinates are denoted by X.sub.RI1 1291RC. The width of the input slit W.sub.I1 is then given by W.sub.I1=|X.sub.LI1X.sub.RI1| 1291W.
[0138] Typically line L.sub.(PXLI1-PXRI1) is designed so that it is perpendicular to the input line L.sub.I1 1251, though it is not always so.
[0139] While the input-slit-mouth line L.sub.(PXLI1-PXRI1) may or may not be perpendicular to the input line L.sub.I1 1251, the launching of the input beam B.sub.I1 1101 is done in a way so that its optical axis of propagation given by line L.sub.BI1 1151OL in
[0140] Relation Between Input-Center Ray, Input Optical Axis, and Grating-Center-to-Input-Slit Line, and Plurality of Input Slits.
[0141] Note that input center ray L.sub.I1-0 1620O and the input beam optical axis L.sub.BI1 1151OL are normally close to coinciding with each other. Also the input beam optical axis L.sub.BI1 1151OL and grating-center-to-input-slit line L.sub.I1 1251 are normally close to coinciding with each other. Thus, the point PX.sub.I1 1291O where the grating-center-to-input-slit line L.sub.I1 1251 meets the input slit at coordinate X.sub.I1 1291OC is normally close to the same point as the point where the input beam optical axis L.sub.BI1 1151OL meets the input slit at coordinate X.sub.BI1 11810.
[0142] The x-y coordinates for the various input beam points at the input slit location are given as follows: the x-y coordinate for the optical axis L.sub.BI1 1151OL at the input slit location is denoted by X.sub.BI1 1181OC; the x-y coordinate for the line L.sub.RBI1-IP % 1151RL at the input slit location is denoted by X.sub.RBI1-IP % 1181RC; the x-y coordinate for the line L.sub.LBI1-IP % 1151LL at the input slit location is denoted by X.sub.LBI1-IP % 1181LC. The beam width at the input slit location is denoted by W.sub.BI1-IP %=|X.sub.LBI1-IP %X.sub.RBI1-IP % 11181W.
[0143] In general, there may be more than one input slit and the second input slit will be labeled as SL.sub.I2 1202. Likewise, input slit n will be labeled as SL.sub.In 120n. All the other geometrical parameters will also follow the same numbering system as this numbering system. For example, the input line for the second input slit will be labeled as line L.sub.I2 1282. Likewise, input line for the input slit n will be labeled as L.sub.In 128n (i.e. by changing the last digit to correspond to the slit number) etc. This labeling system could illustrate i from i=0 to at most up to i=9 or 9 (i.e. up to |i|=9, where |x| means taking the absolute value of the number x). However, those skilled in the art will know how to extend it further to groove number |i|>9 if needed.
[0144] As the input beam exit input slit SL.sub.I1 1201, due to optical diffraction, the input beam width will also become larger giving rise to a spatially diverging beam width when the beam propagates towards the grating. The angle along which the beam diverges is called the beam's divergence angle as already discussed above. As explained there, the definition of interest for the divergence angle will depend on the applications involved.
[0145] Output Region Specification.
[0146] The curvature of the grating helps to refocus the diverging beam from the first input slit SL.sub.I1 1201 to a first output slit SL.sub.O1 1401 with a first output slit width W.sub.O1 1491W, where the subscript O1 is the label for output slit 1 or the first output slit. Note that the term output slit is also referred to as exit slit below so that the terms output slit and exit slit will be used totally interchangeably. Similarly, the terms input slit and entrance slit will be used totally interchangeably.
[0147] As shown in
[0148] Output-Slit-Width Angle and Output-Slit-Mouth Line.
[0149] Let the width of the output slit W.sub.O1 1491W spans an angle .sub.O1 1481 (called angle sustained by output slit width or output-slit-width angle) at the grating center so that the CCW side (or left side; with front facing the slit) of the edge of the slit is at angle .sub.O1+.sub.LO1/2 and the CW side (or right side) of the slit is at .sub.O1R.sub.O1/2 where L.sub.O1/2 1481L and R.sub.O1/2 1481R are the angular span to the left and the right of output angle .sub.O1 for the output slit. Here, the direction from the curved grating center CGC 1050 to the output slit SL.sub.O1 1401 is the front direction or 12 O'clock direction and the left/right side is the counter-clockwise/clockwise side towards the 9/3 O'clock direction. This will be the sign conventions for all the angles below that are referencing the curved grating center CGC 1050 as the pivot of rotation.
[0150] The left edge of the first output slit at angle .sub.O1+L.sub.O1/2 is denoted as point PX.sub.LO1 1491L. The location of the right edge of the first input slit at angle .sub.O1R.sub.O1/2 is denoted as point PX.sub.LO1 1491R. The points PX.sub.O1 1491O, PX.sub.LO1 1491L, and PX.sub.RO1 1491R form a straight line and point PX.sub.O1 1491O is at the middle of the line joining point PX.sub.LO1 1491L and point PX.sub.RO1 1491R, called the first output-slit-mouth line L.sub.(PXLO1-PXRO1). Thus, the left and right sides of the above angular spans are equal so that .sub.LO1=.sub.RO1=.sub.O1.
[0151] The spatial x-y coordinates for the points at the output slits are: for PX.sub.O1 1491O its x-y coordinates are denoted by X.sub.O1 1491OC; for PX.sub.LO1 1491L its x-y coordinates are denoted by X.sub.LO1 1491LC; for PX.sub.RO1 1491R its x-y coordinates are denoted by X.sub.RO1 1491RC. The width of the output slit W.sub.O1 is then given by W.sub.O1=|X.sub.LO1X.sub.RO1| 11491W.
[0152] Typically line L.sub.(PXLO1-PXRO1) is designed so that it is perpendicular to the output line L.sub.O1 1451, though it is not always so.
[0153] While the first output-slit-mouth line L.sub.(PXLO1-PXRO1) may or may not be perpendicular to the output line L.sub.O1 1451, the receiving of the output beam B.sub.I1-O1 1301 is done in a way so that its optical axis of propagation given by line L.sub.BI1-O1 1351OL in
[0154] In general, there may be more than one output slit and the second output slit will be labeled as SL.sub.O2 1402. Likewise, output slit n will be labeled as SL.sub.On 140n. All the other geometrical parameters will also follow the same numbering system as this numbering system. For example, the output line for the second output slit will be labeled as line L.sub.O2 1482. Likewise, output line for the output slit n will be labeled as L.sub.On 148n (i.e. by changing the last digit to correspond to the slit number) etc.
[0155] Output Geometrical Spectral Width.
[0156] Let .sub.LI1-O1/2 1321L be the deviation from the output center wavelength .sub.I1-O1 1321 to the left such that the new wavelength .sub.I1-O1+.sub.LI1-O1/2 will give output angle .sub.O1+.sub.LO1/2 or .sub.I1-Out(.sub.I1-O1+.sub.LI1-O1/2)=.sub.O1+.sub.LO1/2. Hence, the equation .sub.I1-Out(.sub.I1-O1+.sub.LI1-O1/2)=.sub.O1+.sub.LO1/2 can also be used to define L.sub.I1-O1/2 1321L. Likewise, Let .sub.RI1-O1/2 1321R be the deviation from the output center wavelength .sub.I1-O1 1321 to the right, which is then given by .sub.I1-Out(.sub.I1-O1.sub.RI1-O1/2)=.sub.O1.sub.RO1/2. Note that .sub.LI1-O1/2 1321 LRes or .sub.RI1-O1/2 1321RRes may take on a positive or a negative value. The right edge wavelength is .sub.RI1-O1=.sub.I1-O1+R.sub.I1-O1/2 1321R. The left edge wavelength is .sub.LI1-O1=.sub.I1-O1+L.sub.LI1-O1/2 1321L. The total spectral deviation is
[0157] Adding the left and right wavelength deviations that span the range of angle that covers the output slit width then gives the output geometrical spectral width or output geometrical resolution .sub.I1-O1 (in wavelength) 1381GRes at output slit SL.sub.O1 1401 for the beam from input slit SL.sub.I1 1201, where .sub.I1-O1=|L.sub.I1-O1/2|+|.sub.RI1-O1/2| 1381GRes. This spectral width .sub.I1-O1 1381GRes will be referred to as the first output slit geometrical spectral width for beam from the first input slit or simply as first output slit geometrical spectral width when the context is clear which input slit the beam comes from.
[0158] Output Spectral Resolution Bandwidth and Output Power Spectrum.
[0159] In an ideal situation, the output power spectrum PS.sub.I1-O1() 1331 received/passed/detected by the first output slit would be from the input beam spectrum PS.sub.I1() 1131 between wavelengths .sub.RI1-O1=X.sub.I1-O1.sub.RI1-O1/2 1321R and .sub.LI1-O1=X.sub.I1-O1+.sub.LI1-O1/2 1321L around the center wavelength .sub.I1-O1 1321, where .sub.LI1-O1/2 1321 LRes and .sub.RI1-O1/2 1321RRes correspond to the angular deviations to the left and right side of the diffraction angle of this center wavelength 1321. Adding the left and right wavelength deviations that span the range of angle that covers the output slit width then gives the output geometrical spectral width or output geometrical resolution .sub.I1-O1 (in wavelength) 1381GRes at output slit SL.sub.O1 1401 for the beam from input slit SL.sub.I1 1201, where .sub.I1-O1=|.sub.LI1-O1/2|+|R.sub.I1-O1/2| 1381GRes.
[0160] The spectral width of the actual output power spectrum is called the spectral resolution bandwidth denoted as .sub.Res-I1-O1 1381Res. It is defined more precisely below. In the ideal situation, .sub.Res-I1-O1 1381Res is basically equal to the output geometrical spectral width .sub.I1-O1 1381GRes or .sub.Res-I1-O1=.sub.I1-O1.
[0161] However, in practice, due to spatial aberration of the output beam, the actual spectral resolution bandwidth denoted as .sub.Res-I1-O1 1381Res that take into account the spatial spread of the output beam width and spatial distortion of the beam when it focuses at the output slit 1401 location is larger than the ideal situation determined by geometry so that actual spectral resolution bandwidth .sub.Res-I1-O1 1381Res is larger than the output geometrical spectral width .sub.I1-O1 1381GRes or .sub.Res-I1-O1>.sub.I1-O1.
[0162] Output Power Spectrum and Spectral Power Output Efficiency.
[0163] The output power spectrum after the beam from input slit SL.sub.I1 1201 goes through output slit SL.sub.O1 1401 is denoted by PS.sub.I1-O1() 1331. It can be expressed in terms of the input beam power spectrum by: PS.sub.I1-O1()=.sub.effI1-O1()*PS.sub.I1(), where .sub.effI1-O1(), with a value between 0 and 1, is the efficiency factor for receiving/passing/detecting the input beam spectrum at the wavelength , called the spectral power output efficiency at wavelength .
[0164] Output Power.
[0165] The optical power received/passed/detected by the first output slit SL.sub.O1 1401 called the output power for a beam going from input slit SL.sub.I1 1201 to output slit SL.sub.O1 1401 over a small spectral bandwidth centered at wavelength .sub.A (small comparing to the spectral bandwidth of PS.sub.I1-O1() at .sub.A or more precisely, small enough so that PS.sub.I1-O1() at .sub.A does not change much over the wavelength bandwidth ) for the beam from input slit SL.sub.I1 be P.sub.I1-O1(.sub.A; ). P.sub.I1-O1(.sub.A; ) is then given by PS.sub.I1-O1(.sub.A)* and is thus related to the spectral density of the input beam PS.sub.I1(.sub.A) by:
P.sub.I1-O1(.sub.A;)=PS.sub.I1-O1(.sub.A)*=.sub.effI1-O1(.sub.A)*PS.sub.I1(.sub.A)*.(9)
In the situation that is large, Eq. (9) should be more precisely converted to an integration of PS.sub.I1-O1() with respect to wavelength over wavelength bandwidth centered at wavelength =.sub.A given by:
[0166] Total Output Power.
[0167] The total optical power received/passed/detected by the first output slit called the total output power for a beam going from input slit SL.sub.I1 1201 to output slit SL.sub.O1 1401 is then given by P.sub.I1-O1(; ) above, with the wavelength given by the output center wavelength .sub.I1-O1 1321 and the spectral width given by the spectral resolution bandwidth .sub.Res-I1-O1 1381Res or more precisely integrated over the entire wavelength, and is denoted as P.sub.I1-O1. That is, it is approximately given by:
P.sub.I1-O1=P.sub.I1-O1(.sub.I1-O1;.sub.Res-I1-O1)=.sub.effI1-O1(.sub.I1-O1)*PS.sub.I1(.sub.I1-O1)*.sub.Res-I1-O1.(11)
[0168] Or by more precisely by the integration below:
[0169] Light through the first output slit SL.sub.O1 1401 is then detected by a first photodetector Det.sub.O1 1311.
[0170] Definition of Input Beam's Divergence-Diffraction Angle at Integrated Power Point.
[0171] Let us denote the input beam's full divergence angle due to diffraction from the input slit by .sub.dvdf-BI1-IP % 1141, which is defined by the angle made between the two lines traced out by the beam intensity locus points P.sub.RBI1-IP % 1151R (coordinate at X.sub.RBI1-IP % 1151RC) and P.sub.LBI1-IP % 1151L (coordinate at X.sub.LBI1-IP % 1151LC) on both sides of the beam, where P.sub.RBI1-IP % 1151R, and P.sub.LBI1-IP % 1151L are the locations of the intensity points such that the integrated power of the beam from the beam's intensity peak to each of the intensity point is IP/2 percent (IP/2%), where IP % is given by the parameter in the subscript of .sub.dvdf-BI1-IP % 1141. Adding up both the left and right sides will give the percentage of the integrated optical power (IP %) between points P.sub.RBI1-IP % 1151R and P.sub.LBI1-IP % 1151L. Thus IP % denotes the percentage of the integrated power of the beam between points P.sub.RBI1-IP % 1151R and P.sub.LBI1-IP % 1151L of the beam perpendicular to the beam axis of propagation. This angular of divergence of the input beam is due to optical beam diffraction effect from a small finite aperture formed by the input slit width, as is also called the beam's diffraction angle. Thus, we will refer to it as divergence-diffraction angle. It should not be confused with another diffraction effect which is the diffraction of the input beam to different output angles by the curved grating CG 1010. In that case it is called the grating diffraction angle. The angle .sub.dvdf-BI1-IP % 1141 defined here will be called the Input beam's divergence-diffraction angle at IP % integrated power point. It is labeled with subscript dvdf referring to divergence-diffraction.
[0172] Definition of Input Beam's Right/Left Half-Divergence-Diffraction Angle at Integrated Power Point.
[0173] The locus points P.sub.RBI1-IP % forming the line L.sub.RBI1-IP % 1151RL hit the grating surface at point IG.sub.RBI1-IP % 1171R and the locus points P.sub.LBI1-IP % forming the line L.sub.LBI1-IP % 1151LL hit the grating surface at point IG.sub.LBI1-IP % 1171L. The angle sustained by the line L.sub.RBI1-IP % 1151RL and the optical axis L.sub.BI1 1151OL is the right half-divergence-diffraction angle at IP % integrated power point .sub.Rdvdf-BI1-IP % 1141R. The angle sustained by the line L.sub.LBI1-IP % 1151LL and the optical axis L.sub.BI1 1151OL is the left half-divergence-diffraction angle at IP % integrated power point .sub.Ldvdf-BI1-IP % 1141L. The two angles are typically equal to each other though not always so and they added up to be equal to the divergence-diffraction angle .sub.dvdf-BI1-IP % 1141 so that: .sub.dvdf-BI1-IP %=.sub.Rdvdf-BI1-IP+.sub.Ldvdf-BI1-IP %. When the context is clear, these angles will be referred to simply as the input beam divergence angles.
[0174] Definition of Beam Width at IP % Integrated Power.
[0175] The distance between two points X.sub.RBI1-IP % and X.sub.LBI1-IP % given by: |X.sub.RBI1-IP %X.sub.LBI1-IP %| and denoted as W.sub.BI1-IP %(Z.sub.BI1) 1151W (with W.sub.BI1-IP %(Z.sub.BI1)=|X.sub.RBI1-IP %X.sub.LBI1-IP %|) is referred to as the beam's width encompassing (or at) IP % of the integrated center beam power.
[0176] For example with IP %=50%, .sub.dvdf-BI1-50% is the divergence angle defined by the angle made between the two lines traced out by the beam intensity locus points P.sub.RBI1-50% 1151R (coordinate at X.sub.RBI1-50% 1151RC) and P.sub.LBI1-50% 1151L (coordinate at X.sub.LBI1-50% 1151LC) on both sides of the beam that each encompasses 25% of the power of the beam from the beam's optical axis to each of the intensity point at P.sub.LZBI1-50% or P.sub.RZBI1-50% so that [.sub.0.sup.X.sup.
[0177] Relation to the Usual Angle of Beam Divergence at Lie Intensity Point.
[0178] It is useful to relate this beam width and beam divergence angle definitions to the more commonly used definitions based on Gaussian beam approximation and description. As is known to those skilled in the art, for the purpose of estimation and description, often an optical beam such as the input beam can be approximated or fitted with a Gaussian beam intensity profile I(x,Z.sub.BI1)=I(0,Z.sub.BI1)*Exp(x.sup.2). Let the fitted Gaussian beam width W.sub.BI1-1=1/e be defined by 1/e points of the Gaussian beam intensity profile, which is the points X.sub.LB-I=1/e and X.sub.RB-I=1/e at which I(x.sub.LB-I=1/e,Z.sub.BI1)=(1/e)*I(0,Z.sub.BI1) and I(x.sub.RB-I=1/e,Z.sub.BI1)=(1/e)*I(0,Z.sub.BI1) so that W.sub.BI1-1=1/e=X.sub.LB-1=1/eX.sub.RB-I=1/e. It can be shown that W.sub.BI1-1=1/e is approximately equal to (1/2.3) of W.sub.BI1-IP %=95% so that W.sub.BI1-1=1/e(W.sub.BI1-95%/2.3). The term angle of beam divergence or also called angle of beam diffraction in the literature commonly referred to is defined as the angle made between the two lines defined by the locus points traced out by x.sub.LB-I=1/e and X.sub.RB-I=1/e, and will be denoted by .sub.dvdf-BI1-I=1/e. It will be referred to as Input beam's divergence angle at 1/e intensity point.
[0179] In order not to be confused with angle of grating diffraction, we prefer to refer to this angle of beam width spreading as angle of beam divergence-diffraction or simply as angle of beam divergence. It is important to note that for our applications, we need to deviate from this common terminology and there is no one single angle of beam divergence. The angle of beam divergence of interest is depending on design needs and the angle .sub.dvdf-BI1-IP % 1141 as defined above describes the x percentage of integrated center beam power (with IP %=x %) being encompassed by the angles. It will be used to describe the various angles of divergence of interest below as they will be more specifically relevant and useful, instead of that more commonly used in the literature based on Gaussian beam approximation and description.
[0180] Definition of the Beam Waist Under Gaussian Beam Approximation and the Divergence Angle at 1/e Intensity Point of the Beam.
[0181] If we approximate or decompose the input beam in terms of the Gaussian beam at the fundamental mode, we can let W.sub.BI1-I=1/e(x=0,Z.sub.BI1=0) be the Gaussian beam waist at the input slit mouth (i.e. at Z.sub.BI1=0) at which the wavefront is flat. Note that the location along the optical axis (i.e. the Z.sub.BI1 defined above) at which the Gaussian beam or any optical beam has a flat wavefront is referred to in the art as the beam waist of the beam propagation.
[0182] If we approximate or decompose the input beam in terms of the Gaussian beam at the fundamental mode, the first input beam angle of divergence at 1/e intensity points denoted by .sub.dvdf-BI1-I=1/e can be given approximately in terms of its 1/e Gaussian beam's approximated beam waist W.sub.BI1-I=1/e(x=0,Z.sub.BI1=0). It is .sub.dvdf-BI1-I=1/e=2*(.sub.BI1/(Pi*W.sub.BI1-I=1/e(x=0,Z.sub.BI1=0))) (in Radian), where .sub.BI1 1121 is the first input-beam center wavelength shown in
[0183] Output Beam's Propagation and Optical Axis.
[0184] Let Z.sub.BI1-O1 1351D measures the distance of propagation along the optical axis L.sub.BI1-O1 1351OL of the output beam from the curved grating CG 1010 to output slit SL.sub.O1 1201 so that Z.sub.BI1-O1=0 at the curved grating surface and Z.sub.BI1-O1>0 when the beam propagates towards the output slit. The output beam is obtained from the input beam from input slit SL.sub.I1 reflected and diffracted by the grating. The optical axis L.sub.BI1-O1 1351OL hits the grating surface at point OG.sub.BI1-O1 1371O at which Z.sub.BI1-O1=0. A point on the optical axis at z.sub.I=Z.sub.BI1-O1 is called point P.sub.ZBI1-O1 1351. The x-y coordinates for that point is referred to as X.sub.ZBI1-O1 1351C. The optical axis line for the output beam is made up by the locus points traced out by points P.sub.ZBI1-O1 1351 or coordinates X.sub.ZBI1-O1 1351C. This optical axis L.sub.BI1-O1 1351OL is referred to as the output beam's convergence-focusing optical axis.
[0185] In the typical application, point OG.sub.BI1-O1 1371O is assumed to coincide with point IG.sub.BI1 1171O for the input beam. The optical axis is typically defined by the locus points traced out by the peak intensity of the fundamental mode of the input beam. Let x.sub.I measures the perpendicular distance from the optical axis point P.sub.ZBI1-O1 1351 to a point P.sub.LZBI1-O1(x.sub.P) 1361 for which x.sub.P>0 on left side of the optical axis and x.sub.P<0 on the right side. More specifically, left means 9 O'clock side of the direct of propagation and right is 3 O'clock side if the direction of propagation is the 12 O'clock direction. Then I(x.sub.P,Z.sub.BI1-O1) as a function of x.sub.P denotes the transverse or lateral intensity profile of the output beam at Z.sub.BI1-O1 1351D.
[0186] Output Beam's Full Convergence-Focusing Angle.
[0187] Assuming the input beam is a monochromatic source or a light beam with a very narrow spectral width at wavelength .sub.I1-O1. Then after the monochromatic input beam is diffracted by the curved grating, the grating's surface curvature and diffraction properties will act together to focus the output beam with a focusing angle that is converging the beam size to a small focused spot at the output slit. The angle of focusing will be called the convergence-focusing angle. Let us denote this output beam's full convergence-focusing angle by .sub.cvfo-BI1-O1-IP % 1341, which is defined by the angle made between the two lines traced out by the beam intensity locus points P.sub.RBI1-O1-IP % 1351R (coordinate at X.sub.RBI1-O1-IP % 1351RC) and P.sub.LBI1-O1-IP % 1351L (coordinate at X.sub.LBI1-O1-IP % 1351LC) on both sides of the beam, where P.sub.RBI1-O1-IP % 1351R, and P.sub.LBI1-O1-IP % 1351L are the locations of the intensity points such that the integrated power of the beam from the beam's intensity peak to each of the intensity point is IP/2 percent (IP/2%), where IP % is given by the parameter in the subscript of .sub.cvfo-BI1-O1-IP % 1341. Adding up both the left and right sides will give the percentage of the integrated optical power (IP %) between points P.sub.RBI1-O1-IP % 1351R and P.sub.LBI1-O1-IP % 1351L. Thus IP % denotes the percentage of the integrated power of the beam between points P.sub.RBI1-O1-IP % 1351R and P.sub.LBI1-O1-IP % 1351L of the beam perpendicular to the beam's optical axis of propagation. This angular of divergence of the input beam is due to optical beam diffraction effect from a small finite aperture formed by the input slit width, as is also called the beam's diffraction angle. Thus, we will refer to it as convergence-focusing angle. The angle .sub.cvfo-BI1-O1-IP % 1341 defined here will be called the Out beam's convergence-diffraction angle at IP % integrated power point. It is labeled with subscript cvfo referring to convergence-focusing.
[0188] Meeting of the Output Beam and Input Beam at the Grating Surface.
[0189] The locus points x.sub.LBI1-O1-IP % 1351LC is assumed to hit the grating surface at point OG.sub.LBI1-O1-IP % 1371L and the locus points X.sub.RBI1-O1-IP % 1351RC hits the grating surface at point OG.sub.RBI1-O1-IP % 1371R. Above, we already describe the optical axis L.sub.BI1-O1 1151OL traced out by locus points x.sub.BI1-O1 1151OC is assumed to hit the grating surface at point OG.sub.BI1-O1 1371O. In the typical application, point OG.sub.LBI1-O1-IP % 1371L is assumed to coincide with point IG.sub.LBI1-IP % 1171L for the input beam and OG.sub.RBI1-O1-IP % 1371R is assumed to coincide with point IG.sub.RBI1-IP % 1171R for the input beam.
[0190] The coordinates for all these points at the grating surfaces are: the coordinate for OG.sub.BI1-O1 1371O is X.sub.GBI1-O1 1371OC; the coordinate for OG.sub.RBI1-O1-IP % 1371R is X.sub.GRBI1-O1-IP % 1371RC; the coordinate for OG.sub.LBI1-O1-IP % 1371L is X.sub.GLBI1-O1-IP % 1371LC; the coordinate for IG.sub.BI1-IP % 1171 is X.sub.GBI1-IP % 1171C; the coordinate for IG.sub.RBI1-IP % 1171R is X.sub.GRBI1-IP % 1171RC; the coordinate for IG.sub.LBI1-IP % 1171L is X.sub.GLBI1-IP % 1171LC; the coordinate for CGC 1050 is X.sub.0.
[0191] Definition of Beam Width at IP % Integrated Power.
[0192] The distance between two points X.sub.RBI1-O1-IP % 1351RC and X.sub.LBI1-O1-IP % 1351LC given by: |X.sub.RBI1-O1-IP %X.sub.LBI1-O1-IP %| and denoted as W.sub.BI1-O1-IP %(Z.sub.BI1-O1) 1351W (with W.sub.BI1-O1-IP %(Z.sub.BI1-O1)=|X.sub.RBI1-O1-IP %X.sub.LBI1-O1-IP %|) is referred to as the beam's width encompassing (or at) IP % of the integrated center beam power.
[0193] Definition of Output Beam's Right/Left Half-Convergence-Focusing Angle at Integrated Power Point.
[0194] The locus points P.sub.RBI1-O1-IP % forming the line L.sub.RBI1-O1-IP % 1351RL hit the grating surface at point OG.sub.RBI1-O1-IP % 1371R and the locus points P.sub.LBI1-O1-IP % forming the line L.sub.LBI1-O1-IP % 1351LL hit the grating surface at point OG.sub.LBI1-O1-IP % 1371L. The angle sustained by the line L.sub.RBI1-O1-IP % 1351RL and the optical axis L.sub.BI1-O1 1351OL is the right half-convergence-focusing angle at IP % integrated power point .sub.Rcvfo-BI1-O1-IP % 1341R. The angle sustained by the line L.sub.LBI1-O1-IP % 1351LL and the optical axis L.sub.BI1-O1 1351OL is the left half-convergence-focusing angle at IP % integrated power point .sub.Lcvfo-BI1-O1-IP % 1341L. The two angles are typically equal to each other though not always so and they added up to be equal to the divergence-diffraction angle .sub.cvfo-BI1-O1-IP % 1341 so that: .sub.cvfo-BI1-O1-IP %=.sub.Rcvfo-BI1-O1-IP %+.sub.Lcvfo-BI1-O1-IP %. When the context is clear, these angles will be referred to simply as the output beam convergence angles.
[0195] Relation Between Output-Center Ray, Output Optical Axis, and Grating-Center-to-Output-Slit Line, and Plurality of Output Slits.
[0196] In the situation in which the input is a monochromatic beam at wavelength .sub.I1-O1 1321, the output beam after reflection and diffraction from the grating is a well-defined beam and we can refer to various output-beam related variables including output-center ray, output optical axis etc. Note that output-center ray L.sub.O1-0 1820O and the output beam optical axis L.sub.BI1-O1 1351OL are normally close to coinciding with each other. Also the output beam optical axis L.sub.BI1-O1 1351OL and grating-center-to-Output-slit line L.sub.O1 1451 are normally close to coinciding with each other.
[0197] The x-y coordinates for the various output beam points at the output slit location are given as follows: the x-y coordinate for the optical axis L.sub.BI1-O1 1351OL at the output slit location is denoted by X.sub.BI1-O1 1381OC; the x-y coordinate for the line L.sub.RBI1-O1-IP % 1351RL at the output slit location is denoted by X.sub.RBI1-O1-IP % 1381RC; the x-y coordinate for the line L.sub.LBI1-O1-IP % 1351LL at the output slit location is denoted by X.sub.LBI1-O1-IP % 1381LC. The beam width at the output slit location (called output-slit beam width) is denoted by W.sub.BI1-O1-IP %=|X.sub.LBI1-O1-IP %X.sub.RBI1-O1-IP %| 1381W.
[0198] Thus, the point PX.sub.O1 1491O where the grating-center output slit line L.sub.O1 1451 meets the output slit at coordinate X.sub.O1 1491OC is normally close to the same point as the point where the output beam optical axis L.sub.BI1-O1 1351OL meets the output slit at coordinate X.sub.BI1-O1 13810C.
[0199] Output Beam's Focused Beam Waist Location and Beam Waist Width.
[0200] As noted above, if the input beam is a monochromatic beam at wavelength .sub.I1-O1 1321, then the output beam typically will achieve a minimal beam width at or around some spatial location. We shall call that the output focused beam waist location X.sub.BI1-O1 1391OC. The beam waist width at that location is denoted by W.sub.BI1-O1-IP % 1391W and is called the beam waist width of the output beam.
[0201] The x-y coordinates for the various output beam points at this beam waist location are given as follows: the x-y coordinate for the optical axis L.sub.BI1-O1 1351OL at the output beam waist location is denoted by X.sub.BI1-O1 1391OC; the x-y coordinate for the line L.sub.RBI1-O1-IP % 1351RL at the output beam waist location is denoted by X.sub.RBI1-O1-IP % 1391RC; the x-y coordinate for the line L.sub.LBI1-O1-IP % 1351LL at the output beam waist location is denoted by X.sub.LBI1-O1-IP % 1391LC. The beam width at the input slit location is denoted by W.sub.BI1-IP %=|X.sub.LBI1-IP %X.sub.RBI1-IP %| 1181W.
[0202] Output Beam's Full Divergence-Diffraction Angle.
[0203] It is also useful to think of launching an optical beam from the output slit instead of the input slit and see how its angle of propagation diffracts. Let us called that the reversed output beam B.sub.OR1 1301R, as shown in
[0204] This divergence-diffraction of the output beam from output slit SL.sub.O1 is shown in
[0205] Optical axis definition. As shown by
[0206] Let x.sub.I measures the perpendicular distance from the optical axis point P.sub.ZBO1 1551O to a point P.sub.ZBO1(x.sub.P) 1561 for which x.sub.P>0 on left side of the optical axis and x.sub.P<0 on right side of the optical axis. More specifically, left means 9 O'clock side of the direct of propagation and right is 3 O'clock side if the direction of propagation is the 12 O'clock direction. Then I(x.sub.P,Z.sub.BO1) 1551I as a function of x.sub.P denotes the transverse or lateral intensity profile of the input beam at Z.sub.BO1 1551D.
[0207] Definition of Output Beam's Right/Left Half-Divergence-Diffraction Angle at Integrated Power Point.
[0208] The locus points P.sub.RBO1-IP % 1551R forming the line L.sub.RBO1-IP % 1551RL hit the grating surface at point IG.sub.RBO1-IP % 1571R and the locus points P.sub.LBO1-IP % 1551L forming the line L.sub.LBO1-IP % 1551LL hit the grating surface at point IG.sub.LBO1-IP % 171 1571L. The angle sustained by the line L.sub.RBO1-IP % 1551RL and the optical axis L.sub.BO1 1551OL is the right half-divergence-diffraction angle at IP % integrated power point .sub.Rdvdf-BO1-IP % 1541R. The angle sustained by the line L.sub.LBO1-IP % 1551LL and the optical axis L.sub.BO1 1551OL is the left half-divergence-diffraction angle at IP % integrated power point .sub.Ldvdf-BO1-IP % 1541L. The two angles are typically equal to each other though not always so and they added up to be equal to the divergence-diffraction angle .sub.dvdf-BO1-IP % 1541 so that: .sub.dvdf-BO1-IP %=.sub.Rdvdf-BO1-IP %+.sub.Ldvdf-BO1-IP %. When the context is clear, these angles will be referred to simply as the output beam divergence angles.
[0209] The x-y coordinates for the various output beam points at the output slit location are given as follows: the x-y coordinate for the optical axis L.sub.BO1 1551OL at the output slit location is denoted by X.sub.BO1 1581OC; the x-y coordinate for the line L.sub.RBO1-IP % 1581RL at the output slit location is denoted by X.sub.RBO1-IP % 1581RC; the x-y coordinate for the line L.sub.LBO1-IP % 1551LL at the output slit location is denoted by X.sub.LBO1-IP % 1581LC. The beam width at the output slit location is denoted by W.sub.BO1-IP % 1581W and is called the beam waist width of the slit.
[0210] Factors Affecting the Output Spectral Resolution Bandwidth.
[0211] As is well known to those skilled in the art, the resolution of the spectrometer increases with decreasing first input slit width W.sub.I1 1291W. The imaging through the curved grating requires the width of the output slit, such as the width of the first output slit W.sub.O1 1491W, to be chosen appropriately to minimize the wavelength resolution there. Let this wavelength resolution at output slit SL.sub.O1 be denoted by .sub.Res-I1-O1 (called the first output slit spectral resolution bandwidth for beam from the first input slit or simply as first output slit spectral resolution bandwidth when the context is clear which input slit the beam comes from).
[0212] It is important to note that .sub.Res-I1-O1 is dependent on: (1) the input slit width, (2) the output slit width, and (3) the grating groove design (the spatial aberration of the focusing beam can be caused by the curved grating's groove design). For the typical curved grating design, especially the usual Rowland curved grating design, the input and output slit widths, W.sub.I1 1291W and W.sub.O1 1491W are about equal.
[0213] Spectral Resolution Bandwidth.
[0214] Let P.sub.MI1-O1(.sub.I1) be the output optical power at output slit SL.sub.O1 when the input beam from input slit SL.sub.I1 is a monochromatic beam (denoted by the subscript M in P.sub.MI1-O1(.sub.I1)) with wavelength .sub.I1. Let .sub.I1-Ppk be the peak wavelength at which P.sub.MI1-O1() is maximum in value when =.sub.I1-Ppk. Let .sub.Lg-I1-P=0.5 be the long wavelength side with respect to .sub.I1-IPk with .sub.Lg-I1-P=0.5>.sub.I1-pk at which P.sub.MI1-O1(.sub.Lg-I1-P=0.5) drops to 0.5=50% of its peak power so that P.sub.MI1-O1(.sub.Lg-I1-P=0.5)=0.5*P.sub.MI1-O1(.sub.I1-pk). Let .sub.St-I1-P=0.5 be the short wavelength side with respect to .sub.I1-Ppk with .sub.St-I1-P=0.5<.sub.I1-pk at which P.sub.MI1-O1(.sub.I1) drops to 0.5=50% of its peak power so that P.sub.MI1-O1(.sub.St-I1-P=0.5)=0.5*P.sub.MI1-O1(.sub.I1-pk). As shown in
[0215] Input and Output Waveguides as Input and Output Slits.
[0216] When there is a channel waveguide defining the input or output slit as shown by
[0217] The output slit SL.sub.O1 1401 location is typically (though not always as explained below) approximately defined by a waveguide mouth acting as an output slit and will be denoted in the same way as the slit as SL.sub.O1 1401 beyond which (towards the grating direction) the channel waveguide terminates and a planar waveguiding region, region GPR 1020, starts. The physical width of the output waveguide mouth at the termination point is waveguide mouth width MW.sub.O1 1491MW, defined by the width of its waveguide core WGW.sub.O1 1991W at the waveguide mouth location. In general, this width MW.sub.O1 1491MW may not be the same as the slit width W.sub.O1 1491W, but is typically quite close to W.sub.O1 1491W for the case where the refractive index contrast between the waveguide core and two waveguide cladding regions to the right and left of the waveguide core is large. The coordinate location of the middle of the output waveguide mouth MSL.sub.O1 1401M is MX.sub.O1 1491MOC.
[0218] Let the cross-section of the input channel waveguide WG.sub.I1 1701 along A-A of
[0219] Let the cross-section of the output channel waveguide WG.sub.O1 1901 along B-B of
[0220] Let the cross-section of the planar waveguiding region, region GPR 1020's, along C-C of
[0221] In many situations (though not always), the channel and planar waveguiding regions are approximately equal so that n.sub.pIco 1040O is approximately equal to n.sub.chcO1l 1711O, n.sub.TpIcd 1040T is approximately equal to n.sub.TchcdI1 1711T, and n.sub.BpIcd 1040B is approximately equal to n.sub.BchcdI1 1711B. Also, typically (though not always) the refractive indices for all the input channel waveguides and output channel waveguide are approximately equal to each other.
[0222] The waveguide at close to the mouth can take on shape of constant width or can be tapering in width with linear shape or an arbitrary curvilinear shape, including different types such as up taper (becoming wider at the waveguide exit mouth towards the grating), down taper (becoming narrower at the waveguide exit mouth towards the grating) as shown in
[0223] As is known to those skilled in the art, a beam in a waveguide is clearly guided when the phase front of the beam is a plane or flat phase front. If the beam is clearly guided all the way to the physical input mouth location MX.sub.I1 1291MOC of the input channel waveguide mouth MSL.sub.I1 1201M (where the waveguide terminates), then the physical mouth location MX.sub.I1 1291MOC of the input waveguide mouth MSL.sub.I1 1201M becomes the slit location X.sub.I1 1291OC. At this location, the input beam will have a plane phase front with a certain beam width MW.sub.BI1-IP % 1181MW, then the beam width MW.sub.BI1-IP % 1181MW of the input waveguide mouth MSL.sub.I1 1201M becomes the equivalent beam width W.sub.BI1-IP % 1181W of an equivalent slit SL.sub.I1 1201 at X.sub.I1 1291OC with slit width W.sub.I1 1291W.
[0224] After the beam propagates out of the waveguide mouth MSL.sub.I1 1201M, its phase front will begin to become curved in the lateral direction (i.e. for direction within the plane of the planar waveguide) when the beam propagates into the planar waveguiding region, region GPR 1020, due to optical diffraction for a freely propagating (i.e. unguided) beam (the beam becomes unguided in the direction within the plane (i.e. the x-y plane for the x-y-z Cartesian coordinate system defined above) though it is still guided in the direction perpendicular to the plane (i.e. the z direction) of the planar waveguide). This location for the beam is also called the beam waist of the beam as is known to those skilled in the art.
[0225] As is known to those skilled in the art, a beam in an input tapering waveguide region TWG.sub.I1 1701T may not be clearly guided when the taper is fast or when the waveguide width changes rapidly within a short distance, as in that case the phase front of the beam would not be a plane or flat phase front and would attain a beam phase front radius of curvature that is finite (plane or flat phase front corresponds to an infinite radius of curvature). In that case, based on the wave propagation in the planar waveguiding region, region GPR 1020, as is well known to those skilled in the art, it is possible to approximately fit the propagating wave (in wavefront and intensity profile) in the grating planar waveguiding region GPR 1020 (the wave-fronts indicated with X in case (iii) shown in
[0226] Output Case
[0227] Similarly for the output waveguide, except that we shall think of each guided mode of the output waveguide and think of the reverse process in which a beam is launched into a particular mode of the output waveguide and the mode is propagating towards the grating, like the input waveguide case discussed. Then the various definitions for the input waveguide can be used for the output waveguides as well.
[0228] If the beam is clearly guided all the way to the physical output mouth location MX.sub.O1 1491MOC of the output channel waveguide mouth MSL.sub.O1 1401M (where the waveguide terminates), then the physical mouth location MX.sub.O1 1491MOC of the output waveguide mouth MSL.sub.O1 1401M becomes the slit location X.sub.O1 1491OC. At this location, the output beam will have a plane phase front with a certain beam width MW.sub.BO1-IP % 1581MW, then the beam width MW.sub.BO1-IP % 1581MW of the output waveguide mouth MSL.sub.O1 1401M becomes the equivalent beam width W.sub.BO1-IP % 1581W of an equivalent slit SL.sub.O1 1401 at X.sub.O1 1491OC with slit width W.sub.O1 1491W.
[0229] After the beam propagates out of the waveguide mouth MSL.sub.O1 1401M, its phase front will begin to become curved in the lateral direction (i.e. for direction within the plane of the planar waveguide) when the beam propagates into the planar waveguiding region, region GPR 1020, due to optical diffraction for a freely propagating (i.e. unguided) beam (the beam becomes unguided in the direction within the plane (i.e. the x-y plane for the x-y-z Cartesian coordinate system defined above) though it is still guided in the direction perpendicular to the plane (i.e. the z direction) of the planar waveguide). This location for the beam is also called the beam waist of the beam as is known to those skilled in the art.
[0230] As is known to those skilled in the art, a beam in an output tapering waveguide region TWG.sub.O1 1901T may not be clearly guided when the taper is fast or when the waveguide width changes rapidly within a short distance, as in that case the phase front of the beam would not be a plane or flat phase front and would attain a beam phase front radius of curvature that is finite (plane or flat phase front corresponds to an infinite radius of curvature). In that case, based on the wave propagation in the planar waveguiding region, region GPR 1020, as is well known to those skilled in the art, it is possible to approximately fit the propagating wave (in wavefront and intensity profile) in the grating planar waveguiding region GPR 1020 (the wave-fronts indicated with X in case (iv) shown in
[0231] Equivalent Input or Output Slit Location and Slit Width for Channel Waveguiding Case.
[0232] The equivalence between the input or output slit and input and output waveguide parameters would be of interest when the location of the input or output slit is important and the diffraction angle of the beam is important. The width of the input or output slit mainly affects the beam diffraction angle. There are situations in which the detailed intensity profile of the input beam can affect certain performances of the spectrometer at the output slit. In those situations, the waveguide may generate different intensity profile that cannot be well matched by a uniform intensity across the slit for the purpose involved. This invention includes the use of slit as well as the use of waveguide to generate their respective input intensity profiles. When a waveguide is referred to, the waveguide includes waveguides with straight waveguide mouth, tapering waveguide mouth, and waveguide mouth with arbitrary refractive index variation of the core and cladding materials at the waveguide mouth.
[0233] For the case of an input or output waveguide that is a strongly guiding channel waveguide in the x-y plane (i.e. a waveguide with high refractive index contrast between the waveguide core region and the waveguide cladding region in the plane of the planar waveguide), if the beam is clearly guiding at the mouth, then the slit location is well approximated by the physical location of the mouth and the two edges of the slit are well approximated by the two edges of the waveguide mouth.
[0234] However, if the beam is not clearly guiding, not strongly guiding, or not guiding at the mouth (e.g. when its phase front is curved), then the input or output slit location and slit width are taken to be those that will produce a propagating beam that can well approximate the actual beam from the waveguide mouth in terms of the beam width, beam diffraction angle, and beam intensity variation.
[0235] In either case, the slit location is well approximated by the virtual beam waist location and the slit width is well approximated by the width of the slit that will give a beam at the slit that matches the beam at the virtual beam waist in terms of beam intensity width and beam intensity profile. In the case the beam is clearly guiding, the virtual beam waist is then given by the actual beam waist at the waveguide mouth.
[0236] The terms input slit and input waveguide mouth, input slit width and input waveguide mouth width are used interchangeably. When there is an ambiguity, the input waveguide mouth location and width are understood to be the location and width of the equivalent input slit as defined above and not the physical location and width of the actual physical waveguide mouth involved.
[0237] In short, for the input it is typically a good approximation to take the equivalent slit location X.sub.I1 1291OC for an equivalent input slit SL.sub.I1 1201 as the virtual beam waist location TWX.sub.I1 1291TWOC for an input waveguide TWG.sub.I1 1701T, and take the equivalent slit physical width W.sub.I1 1291W for an equivalent input slit SL.sub.I1 1201 as the slit physical width that will produce the width of a beam waist W.sub.BI1-IP % 1181W that is about equal to the virtual beam waist width TWW.sub.BI1-IP % 1181TWW (i.e. W.sub.BI1-IP % 1181WTWW.sub.B1-IP % 1181TWW).
[0238] For the output it is also typically a good approximation to take the equivalent slit location X.sub.O1 1491TOC for an equivalent output slit SL.sub.01 1401 as the virtual beam waist location TWX.sub.O1 1491TMOC for an output waveguide TWG.sub.01 1901T, and take the equivalent slit physical width W.sub.O1 1491W for an equivalent output slit SL.sub.01 1401 as the slit physical width that will produce the width of a beam waist W.sub.BO1-IP % 1581W that is about equal to the virtual beam waist width TWW.sub.BO1-IP % 1581TWW (i.e. W.sub.BO1-IP % 1581WTWW.sub.BO1-IP % 1581TWW).
[0239] As is well known to those skilled in the art. The above approximation methods are accurate to better than plus-minus 50%. Mouth width design that falls within plus-minus 50% of the design width obtained using the above descriptions gives similar functionalities and is incorporated as inclusive in the embodiments of the present invention.
[0240] Optimal Output Slit/Waveguide Design.
[0241] Typically, a good optimal output slit design is to first find the output focused beam waist location X.sub.BI1-O1 1391OC and place the output slit SL.sub.O1 1401 location X.sub.O1 1491OC close to it (i.e. X.sub.O1 1491OCX.sub.BI1-O1 1391OC), and the slit width is such that the beam waist width of the slit W.sub.BO1-IP % 1581W matches beam waist width of the output beam (given a monochromatic input beam) W.sub.BI1-O1-IP % 1391W (i.e. W.sub.BO1-IP % 1581WW.sub.BI1-O1-IP % 1391W).
[0242] Below, we will not specifically distinguish or label waveguide situations, and will simply use the slit location and width, with the understanding that they can represent the waveguide situations as well based on the equivalent width and position approach discussed above.
A Method of Generating the Grating
[0243] First, referring back to
[0244] Second, the location PX.sub.O1 1491O of output slit SL.sub.O1 (or waveguide or photodetector) can be adjusted in order to have the best performance for a particular design goal. Thus, the location PX.sub.O1 1491O of output slit SL.sub.O1 (or waveguide or photodetector) 1401, specified by the output angle .sub.O1 1471 with respect to the grating-center circle normal line L.sub.GCCN 1050N and the distance S.sub.O1 1461 from the grating center is not necessarily on the same circle where entrance slit or input slit (or waveguide) 1201 is located.
[0245] Third, the relation between .sub.I1 1271, .sub.O1 1471, and the initial groove parameter d is given by the approximate grating formula (valid at far field) as is known to those skilled in the art,
d*(Sin(.sub.O1)+Sin(.sub.I1))=m*.sub.I1-O1/n.sub.gr(13),
[0246] where m is the diffraction order and .sub.I1-O1 1321 is the wavelength that will be diffracted by the grating to the output slit SL.sub.01 1401. The medium in which the light propagates in can be air or a material medium with an effective refractive index of propagation n. In the case of free space, n is the material refractive index. In the case of a planar waveguide, n.sub.gr is the effective refractive index of propagation within the planar waveguide.
[0247] Note that Eq. (13) primarily gives the output wavelength .sub.I1-O1/n.sub.gr which is the wavelength diffracted to the output slit SL.sub.O1 1401 from input slit SL.sub.I1 1201 in the material of refractive index n, when given the grating order m, based on a grating parameter d that can be interpreted as the approximate distance between any two adjacent groove at the grating center.
[0248] Fourth, a choice of the initial groove positions are made. In a preferred embodiment, they are X.sub.1=(d/2, 0) and X.sub.1=(d/2, 0), or alternatively X.sub.1=(d/2, R(R.sup.2(d/2).sup.2).sup.1/2) and X.sub.1=(d/2, R(R.sup.2(d/2).sup.2).sup.1/2) (so that X.sub.1 and X.sub.1 will be exactly on the input circle IC 1080). With these position vectors, the two initial grooves are located on line forming the tangent L.sub.GCT 1050T to the grating center curve L.sub.GCC 1050CV that is approximately perpendicular to the grating-center circle normal line L.sub.GCCN 1050N and have the initial groove spacing of d at the grating center.
[0249] In another embodiment, they are X.sub.0=(0,0), X.sub.1=(d, R(R.sub.2d.sup.2).sup.1/2) and X.sub.1=(d, R(R.sup.2d.sup.2).sup.1/2). With these position vectors, three initial grooves are located on a circle radius R, have the initial groove spacing of d at the grating center, and has a tangent L.sub.GCT 1050T to the grating-center curve L.sub.GCC 1050CV that is approximately perpendicular to the grating-center circle normal line L.sub.GCCN 1050N and have the initial groove spacing of d at the grating center. While this embodiment is an acceptable alternative, it is not generally a preferred alternative as it is particularly good only if the output slit is located close to the input circle IC 1080 or the Rowland circle with a radius R/2 (the three grooves will give a focus of the output beam at close to the input circle. While other grooves to be generated will give focus at any output slit point off the input circle, there will be aberration (i.e. deviation of the focusing points) from these initial three grooves that will give focus only at close to the input circle).
[0250] Fifth, the locations of other grooves X.sub.i's are obtained by two conditions. The first of these conditions being that the path-difference between adjacent grooves should be an integral multiple of the wavelength in the medium. The first condition can be expressed mathematically by:
Sgn(ija)*([D.sub.1(.sub.1,S.sub.I1,X.sub.i)+D.sub.2(.sub.O1A,S.sub.O1A,X.sub.i)][D.sub.1(.sub.I1,S.sub.I1,X.sub.ja)+D.sub.2(.sub.O1A,S.sub.O1A,X.sub.ja)])=m*.sub.I1-O1A/n.sub.gr,(14)
[0251] Where as shown in
[0252] The position of groove ja, X.sub.ja is typically already known. For an illustration and not limitation, if the grooves close to the grating center are already known, then groove ja is taken to be a groove adjacent to groove i so that X.sub.ja=X.sub.i+1 for i>0 (so ja=+|i1|=i1 is the previous groove close to i=0 that is already solved) and X.sub.ja=X.sub.i+1 for i<0 (so ja=|i1|=i+1 is the previous groove close to i=0 that is already solved). This is only an illustration as there can be situations, for example, the initial grooves may not be at the grating center. Sgn(ija) takes on value +1 or 1. Sgn(ija) is +1 if i>ja, and 1 if i<ja. This mathematical expression is numerically exact for the optical path difference requirement in the diffraction grating and is actively adjusted for every groove on HR-CCG.
[0253] The second of these conditions being specific for a particular design goal of a curved-grating spectrometer. The second condition in general can be mathematically expressed as
f(X.sub.i)=constant(15)
[0254] where Eq. (15) can be depending on other design parameters such as the input slit and output slit positions or the positions of the adjacent grooves (e.g. .sub.I1,S.sub.I1,.sub.O1,S.sub.O1, .sub.I1-O1, m, n.sub.gr, {X.sub.i}) that are already known and hence can be treated as part of the constant. (e.g. .sub.I1,S.sub.I1,.sub.O1,S.sub.O1, .sub.I1-O1, m, n.sub.gr, {X.sub.i}) that are already known and hence can be treated as part of the constant. The positions {X.sub.i} represent the location positions of some grating teeth that are already known. The functional variable involved is X.sub.i which is the variable to be solved. Specific examples of the second condition are described later in the application. The unknown variables in both equations Eq. (14) and Eq. (15) are x- and y-coordinates of the location vector X.sub.i of the i-th groove. For a given input-slit (or input-waveguide) location (.sub.I1, S.sub.I1), output slit (or waveguide or photodetector) location (.sub.O1, S.sub.O1), and the previous, i.e., ja-th, groove position X.sub.ja, X.sub.i is completely specified by equations Eq. (14) and Eq. (15) for a given wavelength .sub.I1-O1 to output slit SL.sub.O1, effective refractive index of propagation n.sub.gr, and the diffraction order m.
[0255] The above two equations Eq. (14) and Eq. (15) are needed to solve for the two unknown numbers in X.sub.i=(x.sub.i, y.sub.i), namely x-coordinate x.sub.i and y-coordinate y.sub.i of the i.sup.th groove. These two equations are solved analytically, numerically, or computationally for the values of X.sub.i=(x.sub.i, y.sub.i) using equations solving methods that are already known to those skilled in the art. The groove positions X.sub.i starting from i=0, 1, 2 . . . or i=0, 1, 2 . . . are iteratively solved with the groove location of the preceding groove X.sub.ja already solved or specified starting from the location of initial groove X.sub.0=(0,0) or X.sub.1 or X.sub.1, or any other initial groove position, whichever is applicable.
[0256] The last of the HR-CCG specification, namely the Fifth step, ensures that every ray from each groove focuses to a single point. This ensures the rays from HR-CCG will focus at output slit SL.sub.O1 with minimal spatial focusing aberration, and therefore enabling a small focused spot size at the output slit.
[0257] Other Choices of Initial Grooves.
[0258] Note that Eq.(13) is an approximate formula assuming the distance between two adjacent grooves at near the grating center is approximately d and the line joining the two grooves are perpendicular to the designated grating normal line. It becomes exact only with far-field approximation, which is valid only when S.sub.I1 1261 and S.sub.O1 1461 are much larger than d or when the grating is large in size comparing to d.
[0259] The more exact form is the same as Eq. (14) by taking two adjacent grooves at the grating center such as i=1 and ja=0 groove giving:
[D.sub.1(.sub.I1,S.sub.I1,X.sub.1)+D.sub.2(.sub.O1,S.sub.O1,X.sub.1)][D.sub.1(.sub.I1,S.sub.I1,X.sub.0)+D.sub.2(.sub.O1,S.sub.O1,X.sub.0)]=m*.sub.I1-O1/n.sub.gr,(16)
Specifically, Eq.(13) can be replaced by Eq. (16) with the distance between X.sub.0 and X.sub.1 set to be equal to approximately equal to d (e.g. X.sub.0=(0,0), X.sub.1=(d, 0), or alternatively X.sub.0=(0,0), X.sub.1=(d, R(R.sup.2d.sup.2).sup.1/2)), and with the input slit location (i.e. .sub.I1,S.sub.I1) given and the output slit location (i.e. .sub.O1,S.sub.O1) given, Eq.(16) can be used to solve for .sub.I1-O1 (with m chosen and n.sub.gr given), or alternatively when given .sub.I1-O1 can be used to solve for .sub.O1, (with m chosen and n.sub.gr given, and S.sub.O1 given (or S.sub.O1 taken to be long such as infinity)). These will be just like what Eq. (13) would do but without the need for far field approximation. Note that alternatively we can symmetrically choose the center two grooves to be at X.sub.1 and X.sub.1 with X.sub.1 and X.sub.1 spaced at d/2 away from the origin X.sub.0 (i.e. with X.sub.1=(d/2, 0) and X.sub.1=(d/2, 0) or alternatively, X.sub.1=(d/2, R(R.sup.2(d/2).sup.2).sup.1/2) and X.sub.1=(d/2, R(R.sup.2(d/2).sup.2).sup.1/2). In that case X.sub.0 will not be used as a groove location though it is the center (or origin) of the coordinate center.
[0260] It is easy to show that if X.sub.0 and X.sub.1 are given as above (e.g. X.sub.0=(0,0), X.sub.i=(d, 0)), then approximately: D.sub.1(.sub.I1,S.sub.I1, X.sub.1)D.sub.1(.sub.I1,S.sub.I1, X.sub.0)d*Sin(.sub.I1) and approximately: D.sub.2(.sub.O1,S.sub.O1, X.sub.1)D.sub.2(.sub.O1,S.sub.O1, X.sub.0)d*Sin(.sub.O1). Thus Eq.(16) can be reduced to Eq. (13) that d*(Sin(.sub.O1)+Sin(.sub.I1))=m*.sub.I1-O1/n.sub.gr.
[0261] Thus, the use of Eq.(13) is not for the purpose of grating structural design limitation as various other equations can be used to achieve similar goal. In particular, it shall not be used to limit the grating design as the grating design and performances are largely defined by majority of the rest of the grating grooves. Its use is merely to estimate the output wavelength .sub.I1-O1/n.sub.gr which is the wavelength in the planar waveguiding material diffracted to the output slit SL.sub.O1 1401 from input slit SL.sub.I1 1201 in the material of refractive index n.sub.gr, when given the grating order m. While a target grating parameter d that can be interpreted as the approximate distance between any two adjacent groove at the grating center is useful for the estimation, it is not an essential result.
[0262] To put it simply, there is only one initial groove location needed to generate the other grating grooves utilizing Eqs. (14) and (15). The specification of two initial grooves such as the one given by X.sub.1=(d/2, 0) and X.sub.1=(d/2, 0) (or alternatively X.sub.1=(d/2, R(R.sup.2(d/2).sup.2).sub.1/2) and X.sub.1=(d/2, R(R.sup.2(d/2).sup.2).sup.1/2)), ensures that the grating-center circle normal line L.sub.GCCN 1050N will indeed be approximately perpendicular to the tangent line L.sub.GCT 1050T to the grating-center curve L.sub.GCC 1050CV. That means the actual grating center tangent normal line L.sub.GCTN 1050TN (that by definition perpendicular to the tangent L.sub.GCT 1050T) is coincided with the grating-center circle normal line L.sub.GCCN 1050N. Otherwise they do not coincide. The grating-center circle normal line L.sub.GCCN 1050N is what the initial input slit angle .sub.I1 1271 and output slit angle .sub.O1 1471 are defined based on. However, if the new actual grating center tangent normal line L.sub.GCTN 1050TN does not coincide with the grating-center circle normal line L.sub.GCCN 1050N, then by right the input angle and output angle in Eq. (14) for example, shall be given by .sub.I1 1271TNA and .sub.O1 1471TNA instead of by .sub.I1 1271 and .sub.O1 1471 (.sub.I1 1271TNA and .sub.O1 1471TNA are angles based on the grating center tangent normal line L.sub.GCTN 1050TN).
[0263] The specification of three initial grooves is thus overdone but is some time used. When the third initial groove is specified rather than having the third groove computed using Eqs. (14) and (15), it sometime may generate small amount of aberration as it may not be consistent with the rest of the grating groove that are generated using Eqs. (14) and (15).
[0264] Other Alternatives to the Third Step.
[0265] The essential function of Third Step is to pick or designate output free-space wavelength .sub.I1-O1 1321 and grating order m, and when possible, also make sure the grating-center curve L.sub.GCC 1050CV for the initial few grooves has a tangent L.sub.GCT 1050T that is perpendicular to the grating-center circle normal line L.sub.GCCN 1050N. The .sub.I1-O1 1321 and grating order m can then be used in step Five to generate all other grooves by starting from just one grating center groove.
[0266] In an embodiment, for the purpose of illustration and not limitation, we can specify the grating center groove X.sub.0 1600O as the only initial groove, and .sub.I1-O1 1321 and grating order m are arbitrarily chosen. In that case, it is convenient to let X.sub.0 1600O to be the coordinate origin X.sub.0=(0,0). The grooves positions are thus essentially all given by step Five, which then ensures that the grating design resulted will give diffract light to output slit SL.sub.O1 1401 from input slit SL.sub.I1 1201 at wavelength .sub.I1-O1 1321 with aberration-free focusing at output slit SL.sub.O1 1401, even for a large grating angle span. The actual distances or spacing between two adjacent grooves generated is automatically determined and we may label it as d at near the grating center, which may be substantially deviating from d at far from the grating center.
[0267] For example, X.sub.0 1600O can serve as the previous groove position for X.sub.1 1601P and X.sub.1 1601N and the rest of the grating grooves can be generated from there by using Eq. (14) and (15) in the Fifth Step. For example, if Eq.(15) is the constant arc case, then the arc length choice becomes a variable of Eq. (15), which will result in certain distance d between X.sub.0 and X.sub.1. This procedure, while can be adopted, however, does not ensure the grating-center curve L.sub.GCC 1050CV for the initial few grooves has a tangent L.sub.GCT 1050T that is perpendicular to the grating-center circle normal line L.sub.GCCN 1050N.
[0268] Flexibility in Choosing Initial Grooves.
[0269] There are various other possible specifications of the initial groove positions that can be made. For example, instead if three initial grooves or one initial groove, the initial grooves can also be chosen to be only two grooves at X.sub.1=(d/2, R(R.sup.2(d/2).sup.2).sup.1/2) and X.sub.1=(d/2, R(R.sup.2(d/2).sup.2).sup.1/2) and there is no groove at X.sub.0. Alternatively, we can also specify X.sub.1=(d/2, 0) and X.sub.1=(d/2, 0) and there is no groove at X.sub.0, which is just a shift in the y coordinate by the amount R(R.sup.2(d/2).sup.2).sup.1/2 and is mounting to a redefinition of the distance to the input or the output slits. This procedure does ensure the grating-center curve L.sub.GCC 1050CV for the initial few grooves has a tangent L.sub.GCT 1050T that is perpendicular to the grating-center circle normal line L.sub.GCCN 1050N.
[0270] Arbitrary Reference for the Input and Output Angles.
[0271] It is important to note that in the general situation where Eq.(13) (or the more exact Eq. (16)) is not used to generate the grating with either the initial two points or three points as described above, then after the grating is generated, there is no guarantee that the grating-center circle normal line L.sub.GCCN 1050N is perpendicular to the grating-center tangent line L.sub.GCT 1050T to the resulted grating-center curve L.sub.GCC 1050CV at the grating center CGC 1050.
[0272] If the grating generated produces a tangent line L.sub.GCT 1050T that is not perpendicular to grating-center circle normal line L.sub.GCCN 1050N (which is used to define angle .sub.I1 1271), it will just amount to an offset of the angle .sub.I1 1271 with respect to an actual normal line that is actually perpendicular to the tangent line L.sub.GCT 1050T. That means the normal line L.sub.GCCN 1050N used is rotated by an angle, say .sub.I1, from this actual normal line. With respect to this actual normal line, the value of the input angle .sub.I1 will just be given in terms of .sub.I1 by .sub.I1=.sub.I1+.sub.I1.
[0273] In some cases, the initial grooves generated may not fit the equations (e.g. Eqs. (14) and (15)) imposed by step Five above (e.g. for the case of three initial grooves on circle of radius R, they give focusing only at near the input circle IC 1080 or the Rowland circle with a radius R/2) and hence may even give a distortion or deviation from the focusing properties of other grooves (e.g. that could be giving focusing at off the input circle).
[0274] Thus, the way the initial grating grooves are chosen shall not be used to limit the grating design as the grating design and performances are largely defined by majority of the rest of the grating grooves and not necessarily the initial few grooves. Thus, variations are allowed in the positions of the grating groove as long as they fall within the domain of grating-groove variation applicability discussed next.
[0275] Domain of Grating-Groove Variation Applicability.
[0276] As is known to those skilled in the art, the grating performances are depending on the collective results of diffraction and wave interference from the majority of the grating grooves. They are not depending on just a few grating grooves. They are also not too sensitive to the grating grooves being moved spatially by an amount S less than about of an optical wavelength in the material given by .sub.I1-O1/(2*n.sub.gr), where S=(x.sub.2y.sup.2).sup.0.5 with x being the spatial deviation from the designed position in the x-coordinate and y being the spatial deviation from the designed position in the y-coordinate. If the design of a grating groove position in accordance to an embodiment of the present invention is X.sub.jDn=(x.sub.jDn, y.sub.jDn) and another design or implementation or realization of the grating groove is at X.sub.jIm=(x.sub.jIm, y.sub.jIm), then x=|x.sub.jDNx.sub.jIm| and y=|y.sub.jDNy.sub.jIm|. Moreover, two gratings or grating designs or grating implementations or grating realizations can achieve similar output spectral filtering performances for about half or more than half of the filtered spectrum if at least for a collection of grating grooves that are involved in reflecting more than 50% of the total power reflected by the grating towards the same output slit location, they are similar in their groove positions to each other in both gratings. Similar grating groove position means S<.sub.I1-O1/(2*n.sub.gr).
[0277] While the five steps above is a method of generating the set of positions for all the grating grooves in accordance with an embodiment of the present invention, there are other method that could generate the set of positions for all the grating grooves.
[0278] Thus, the grating performances will be similar as long as for this collection of the grating grooves (that are involved in reflecting more than 50% of the total power reflected by the grating towards the same output slit location), the deviation denoted by S of each grating groove position from the designed values is less than about half of the wavelength in the material so that S<.sub.I1-O1/(2*n.sub.gr).
[0279] Obviously, smaller deviation (e.g. S<.sub.I1-O1/(4*n.sub.gr) or S<.sub.I1-O1/(10*n.sub.gr) or a larger set of grooves involved (e.g. the set of grooves involve in over 70% of the grating total power reflection instead of 50%, or the set of grooves involve in over 90% of the grating total power reflection instead of 50%) will ensure even closer performances to the desired design. These allowed deviations (e.g. a set of grooves involved in reflecting more than 50% of the total power reflected by the grating towards the same output slit location has each of its groove's S satisfying S<.sub.I1-O1/(2*n.sub.gr)) describe the maximum deviations allowed. When two gratings meet such conditions, we will consider them to be the same design within the allowances of design variations for the purpose of this invention. The minimal of which is given by same design condition (A): two gratings are considered the same design if a set of grooves involved in reflecting more than 50% of the total power reflected by the grating towards the same output slit location has each of its groove's S satisfying S<.sub.I1-O1/(2*n.sub.gr); a tighter one is given by same design condition (B): two gratings are considered the same design if a set of grooves involved in reflecting more than 50% of the total power reflected by the grating towards the same output slit location has each of its groove's S satisfying S<.sub.I1-O1/(4*n.sub.gr); another tighter one is given by same design condition (C): two gratings are considered the same design if a set of grooves involve in over 70% if a set of grooves involved in reflecting more than 70% of the total power reflected by the grating towards the same output slit location has each of its groove's S satisfying S<.sub.I1-O1/(10*n.sub.gr). As yet another tighter one is given by same design condition (D): two gratings are considered the same design if a set of grooves involved in reflecting more than 90% of the total power reflected by the grating towards the same output slit location has each of its groove's S satisfying S<.sub.I1-O1/(10*n.sub.gr). The applicability of which is depending on grating applications. For example, for the usual spectral analysis application, same-design-conditions (A) and (B) is applicable, for the DWDM (dense wavelength division multiplexing) wavelength channel filtering applications in fiber-optic communications, same-design-conditions (C) and (D) are applicable,
[0280] In terms of the performance of the grating design, the grating groove positions are what are important and not the method of generating those positions. Thus, in terms of the specification of the grating structure in accordance with an embodiment of the present invention, the method of obtaining the grating groove position is for the purpose of illustration and not limitation. In terms of the method of obtaining the grating structure in accordance to another embodiment of the present invention, the steps of obtaining the grating structure are important.
[0281] Independence on the Choice of Initial Groove as Coordinate Origin.
[0282] It is important to note that the choice of X.sub.0=(0,0) to be the coordinate origin or not should not materially alter the grating design as the choice of reference coordinate system is arbitrary. The choice of a reference coordinate system is only used as a language or reference to describe the locations of all the groove points. As is well known to those skill in the art, other coordinate systems and references can be used as there is always a way to describe the same set of groove points in terms of another coordinate system.
Constant Angle Geometry
[0283] An exemplary embodiment of HR-CCG specified above is shown in
[0284] where X.sub.I1=(S.sub.I1*Sin(.sub.I1), S.sub.I1*Cos(.sub.I1)) 13-1291OC is the position vector of input or entrance slit SL.sub.I1 13-1201, X.sub.O1=(S.sub.O1*Sin(.sub.O1), S.sub.O1*Cos(.sub.O1)) 13-1491OC is the position vector of output slit (or waveguide or photodetector) SL.sub.O1 13-1401, X.sub.i 13-160|i|P/N (N if i<0 and P if i>0) is the position of a groove i, and .sub.i 13-163|i|P/N (163|i|N if i<0 or 1603|i|P (if i>0) is the difference in angular position between successive i.sup.th and ja.sup.th grooves (e.g. ja is the groove that precedes groove i (closer to grating center) so that |ja|=(|i|1)). Position X.sub.ja is typically already known or solved. In Eq. 17, operator .Math. means the inner product in vector analysis and defined as A.Math.B|A|*|B|*Cos(). The vertical bar | indicates taking the absolute value or the length of a vector. Equivalently, Eq.(17) can be written as:
[0285] Because |.sub.i| is constant for all grooves, it is same as the angular position difference between the center groove at X.sub.0 13-1601 and the first groove at X.sub.1 13-1601P, i.e.
[0286] In this particular case, the position of entrance slit or input slit (or waveguide) SL.sub.I1 13-1201 exit slit or output slit (or waveguide) SL.sub.O1 13-1401 and the angular spacing between the grooves are X.sub.I1=(23.49, 16.45) 13-1291OC, X.sub.O1=(17.26, 33.46) 13-1491OC, and .sub.i=.sub.1=4.13 13-1611P. In this example, wave-front of the diverging input beam propagating toward the curved grating is sliced into a set of narrow beams with angular extension by the curved-grating. Each beam with angular extension undergoes reflective diffraction by each groove. At a particular wavelength, grating diffraction at a particular groove is equivalent to redirecting a particular narrow beam to an output slit (or waveguide or photodetector) SL.sub.O1 13-1401 location with .sub.O1 13-1471, basically due to constructive interference of electro-magnetic waves reflected from adjacent grating grooves. In geometrical optics, it is regarded as due to constructive interference of rays or beams reflected from adjacent grating grooves. The geometrical optic picture is physically less precise but gives reasonable results in predicting the direction of the output beam due to grating diffraction. The position vectors X.sub.i's calculated from Eq. (2) and Eq. (4) are listed in the Table 2. As shown in
TABLE-US-00003 X.sub.10 (23.17, 15.28) X.sub.9 (22.24, 12.89) X.sub.8 (20.97, 10.60) X.sub.7 (19.36, 8.43) X.sub.6 (17.42, 6.44) X.sub.5 (15.17, 4.65) X.sub.4 (12.62, 3.10) X.sub.3 (9.80, 1.83) X.sub.2 (6.74, 0.87) X.sub.1 (3.60, 0.14) X.sub.0 (0.00, 0.00) X.sub.1 (3.60, 0.14) X.sub.2 (7.30, 0.70) X.sub.3 (11.06,1.70).sup. X.sub.4 (14.83, 3.13) X.sub.5 (18.57, 5.02) X.sub.6 (22.22, 7.36) X.sub.7 (25.73, 10.16) X.sub.8 (29.06, 13.39) X.sub.9 (32.16, 17.06) X.sub.10 (34.98, 21.15)
[0287] The above example has been used for illustration purposes only and should not be construed in any way as limiting the scope of the invention.
Constant Angle with One or Plurality of Outputs on Input Circle or Rowland Circle
[0288] Due to the imaging property of curved grating surfaces as shown by Rowland, when the spectrometer is constructed so that the input slit is on a circle called the input circle IC 1080 or the Rowland circle with a radius R/2 which is half the radius of curvature of the grating-center curve, the output focusing point for a particular wavelength will be approximately on the input circle or Rowland circle.
[0289] The current invention includes the situations in which one or more than one (i.e. plurality) of output slits are placed along an output plane, each slit placed along a particular output angle to detect a particular wavelength of light, and light at that wavelength is focused (i.e. is achieving its minimal beam width) at the location of that output slit. The output plane is called the focusing field.
[0290] When the first output slit is placed on the input circle IC 1080 or the Rowland circle with a radius R/2, as noted about, the property of the imaging of the curved grating surface will result in an output plane or focusing field that is curved and in fact approximately along the input circle or Rowland circle.
[0291] This in another alternative embodiment, High-Resolution Compact Curved Grating has a constant angle and output slit (or waveguide or photodetector) SL.sub.O1 present on the circle of radius R/2. In this embodiment, each groove surface has an angular extension ().sub.i from entrance slit or input slit (or waveguide) SL.sub.I1. In this example, the angular extensions ().sub.i are kept constant for all grooves. In addition, both entrance slit or input slit (or waveguide) SL.sub.I1 and output slits (or waveguide or photodetector) SL.sub.O1 are located on or near a circle of radius R/2, where R is the radius of a circle formed by three initial groove locations X.sub.0, X.sub.1, and X.sub.1.
Constant Angle with One or Plurality of Outputs on Arbitrary Locations
[0292] In as yet another alternate embodiment, High-Resolution Compact Curved Grating has a constant angle with output slit (or waveguide or photodetector) SL.sub.O1 present at an arbitrary location, as shown by
Various Curvilinear Shapes for the Groove Surfaces.
[0293] There are two commonly used shapes of grooves in the grating used in the free-space spectrometer. They are straight line and sinusoidal shape. These two shapes are widely used because of ease of manufacturing process. In an embodiment of the present invention, for a curved-grating, ideal shape of reflecting surface is not a straight line, but a curved shape that can image entrance slit or input slit (or input waveguide mouth) SL.sub.I1 1205 at output slit (or output waveguide mouth or photodetector) SL.sub.O1 1405 location. Ideal aberration-free curved mirror is an ellipse with its focal point located at source and image. Therefore, as shown in
[0294] This, in another embodiment to be described next, elliptical shape is used for each groove as just described above.
Constant Arc Case
[0295] The geometrical specification of the HR-CCG with constant arc-length (the Constant-Arc Case) and output slit (or waveguide or photodetector) SL.sub.O1 is as described below.
[0296] First, refer to
[0297] Second, the location of output slit (or waveguide or photodetector) SL.sub.O1 1401 can be adjusted in order to have the best performance for a particular design goal. Thus, the location of output slit (or waveguide or photodetector) X.sub.O1 1491OC, specified by the angle .sub.O1 1471 with respect to the grating-center circle normal line L.sub.GCCN 1050N and the distance S.sub.O1 of 1461 from the grating center CGC 1050 is not necessarily on the same circle where entrance slit or input slit (or waveguide) SL.sub.I1 1201 is located.
[0298] Third, the relation between .sub.I1 1271, .sub.O1 1471, and the initial groove spacing d is given by the grating formula d*(Sin(.sub.O1)+Sin(.sub.I1))=m*.sub.I1-O1/n.sub.gr where m is the diffraction order, n.sub.gr is the effective refractive index of propagation of the medium, and .sub.I1-O1 is an operation wavelength.
[0299] Fourth, initial groove positions are X.sub.1=(d, 0) 1601P and X.sub.1=(d, 0) 1601N, or alternatively X.sub.1=(d/2, R(R.sup.2(d/2).sup.2).sup.1/2) and X.sub.1=(d/2, R(R.sup.2(d/2).sup.2).sup.1/2) (so that X.sub.1 and X.sub.1 will be exactly on the input circle IC 1080). With these position vectors, two initial grooves are located on a circle of radius R and have the initial groove spacing of d at the grating center. This circle segment of radius R at the grating center then forms the grating-center curve.
[0300] Fifth, the location of other grooves X.sub.i's 160|i|P/N (N if i<0 and P if i>0) are obtained by the following two conditions. The first condition being the path-difference between adjacent grooves should be an integral multiple of the wavelength in the medium, which is mathematically expressed as
[D.sub.1(.sub.I1,S.sub.I1,X.sub.1)+D.sub.2(.sub.O1,S.sub.O1,X.sub.1)][D.sub.1(.sub.I1,S.sub.I1,X.sub.0)+D.sub.2(.sub.O1,S.sub.O1,X.sub.0)]=m*.sub.I1-O1/n.sub.gr, (20)
[0301] The second condition being the arc lengths of all the grooves are the same throughout the HR-CCG. This second condition can be mathematically expressed as:
where S.sub.i is the arc-length of the i.sup.th groove. Suppose X.sub.i1 is already known, this equation requires the knowledge of X.sub.i+1, which is still unknown. However, with the constraint of the fact that each X.sub.i is located at the center of the groove, the above expression can be reduced to the following expression without X.sub.i+1.
[0302] Or equivalently:
where X.sub.ja is the position of a groove ja that is adjacent to groove i (e.g., ja can be i1 or i+1). Typically X.sub.ja is already known.
Constant Arc with Output Slits on Rowland Circle
[0303] For the purpose of illustration and not limitation,
TABLE-US-00004 X.sub.15 (55.43, 23.48) X.sub.14 (52.90, 20.32) X.sub.13 (50.07, 17.38) X.sub.12 (46.98, 14.68) X.sub.11 (43.67, 12.21) X.sub.10 (40.17, 9.98) X.sub.9 (36.52, 7.99) X.sub.8 (32.74, 6.24) X.sub.7 (28.84, 4.72) X.sub.6 (24.86, 3.43) X.sub.5 (20.81, 2.35) X.sub.6 (16.71, 1.48) X.sub.5 (20.81, 2.35) X.sub.4 (16.71, 1.48) X.sub.3 (12.57, 0.82) X.sub.2 (8.39, 0.36) X.sub.1 (4.20, 0.09) X.sub.0 (0.00, 0.00) X.sub.1 (4.20, 0.09) X.sub.2 (8.39, 0.35) X.sub.3 (12.57, 0.77) X.sub.4 (16.73, 1.34) X.sub.5 (20.86, 2.07) X.sub.6 (24.97, 2.94) X.sub.7 (29.04, 3.96) X.sub.8 (33.07, 5.10) X.sub.9 (37.06, 6.37) X.sub.10 (41.01, 7.76) X.sub.11 (44.91, 9.28) X.sub.12 (48.77, 10.90) X.sub.13 (52.57, 12.63) X.sub.14 (56.33, 14.47)
[0304] The above example has been for illustration purposes only and should not in any way be limiting the scope of the above-described embodiment or invention as a whole.
[0305] The performance of the HR-CCG with the constant arc-length and output slit (or waveguide or photodetector) on a tangent circle is compared with a Rowland design with the same parameters such as .sub.I1, S.sub.I1, O1, S.sub.O1, R, m, d, and .sub.I1-O1. It is a direct comparison of a Rowland curved-grating spectrometer described in
[0306] In another embodiment to be described next, elliptical shape is used for each groove, and the length of this elliptical shape in each groove is kept constant (constant arc). Center positions of the grooves X.sub.i's in this example are determined so that the length of each elliptical groove is the same.
Constant Arc with Outputs Near a Straight Line or Near Flat Field Output Case
[0307] Due to the imaging property of curved grating surfaces as shown by Rowland, when the spectrometer is constructed so that the input slit is on a circle called the input circle IC 1080 or the Rowland circle with a radius R/2 that has a radius which is half the radius of curvature of the grating-center curve, the output focusing point for a particular wavelength will be approximately on the input circle or Rowland circle.
[0308] The current invention includes the situations in which one or more than one (i.e. plurality) of output slits are placed along an output plane, each slit placed along a particular output angle to detect a particular wavelength of light, and light at that wavelength is focused (i.e. is achieving its minimal beam width) at the location of that output slit. The output plane is called the focusing field.
[0309] When the first output slit is placed on the input circle or Rowland circle, as noted above, the property of the imaging of the curved grating surface will result in an output plane or focusing field that is curved and in fact approximately along the input circle IC 1080 or Rowland circle. In some application situations, such curved output plane or output focusing field is undesirable.
[0310] In some application situations, it is desirable that the output plane is closed to being a straight line. In that situation we will refer to the spectrometer as a Flat Field spectrometer.
[0311] In another embodiment, High-Resolution Compact Curved Grating has a constant arc with the first output slit (or waveguide or photodetector) being present not on the input circle IC 1080 or Rowland circle but along a straight line that passes through the input slit on one end and the output slit on the other end, and the line is approximately perpendicular to the grating center circle normal line L.sub.GCCN 1050N. With reference to
[0312] In as yet another embodiment, the grating is the constant arc design mentioned above. Furthermore, there are plurality of output slits, in which each output slit is placed along a particular output angle to detect a particular wavelength of light, and the plurality of the output slits are placed close to the input-slit CPD line.
[0313] It can be shown that for grating with constant arc design, for an output slit placed at particular output angle to detect a particular wavelength of light, light at that wavelength will be focused near the input-slit CPD line. This is a useful property of the constant arc design.
[0314] Such flat field output is useful for example, when the input waveguide or slit is small and the input beam diffraction divergence angle is large. This also means the converging or focusing beam onto the output waveguides or slits will also have large angle. The flat field means that the output waveguide mouth (or slit) of one channel will not block the large angle converging beam that reach the adjacent output channel waveguide (or slit).
Constant Arc with Arbitrary Output Locations
[0315] In another alternate embodiment, High-Resolution Compact Curved Grating has a constant arc with an output slit (or waveguide or photodetector) SL.sub.O1 or plurality of output slits present at arbitrary locations.
Grating Groove Near or on Outer Rowland Circle
[0316] In as yet another alternate embodiment, High-Resolution Compact Curved Grating has grooves lying on or near the circle of radius R (the near-Outer-Rowland Case) where R is the radius of a circle formed by three initial groove locations X.sub.0, X.sub.1, and X.sub.2. The output slit SL.sub.O1 can be located at anywhere, including but not limited to the input circle IC 1080 or Rowland circle, as depicted in
Sgn(ija)*([D.sub.1(.sub.I1,S.sub.I1,X.sub.i)+D.sub.2(.sub.O1,S.sub.O1,X.sub.i)][D.sub.1(.sub.I1,S.sub.I1,X.sub.ja)+D.sub.2(.sub.O1,S.sub.O1,X.sub.ja)])=m*.sub.I1-O1/n.sub.gr, (24A)
where the groove i is next to some groove ja. The position of groove ja, X.sub.ja is typically already known. For an illustration and not limitation, if the grooves close to the grating center are already known, then groove ja is taken to be a groove adjacent to groove i so that X.sub.ja=X.sub.i1 for i>0 (so ja=+|i1|=i1 is the previous groove close to i=0 that is already solved) and X.sub.ja=X.sub.1+1 for i<0 (so ja=|i1|=i+1 is the previous groove close to i=0 that is already solved). This is only an illustration as there can be situations, for example, the initial grooves may not be at the grating center. Sgn(ija) takes on value +1 or 1. Sgn(ija) is +1 if i>ja, and 1 if i<ja.
[0317] Secondly, the angular locations of the grooves are chosen so that each groove is located at or near the circle of radius R throughout the HR-CCG, where R is the radius of a circle formed by three initial groove locations X.sub.0, X.sub.1, and X.sub.1.
[0318] In another alternate embodiment, the High-Resolution Compact Curved Grating with grooves on or near the circle of radius R or the outer input circle 1070 (the near-Rowland case) has one or plurality of output slits (or waveguides or photodetectors) present at arbitrary locations.
Grating Groove Near or on Outer Elliptical Curve
[0319] In as yet another alternate embodiment, High-Resolution Compact Curved Grating has grooves lying on or near an elliptical curve that has a radius of curvature R (the near-Ellipse Case) at or near the grating center, where R/2 is the radius of the input circle 1080 or Rowland circle that passes through the grating center and the input slit SL.sub.I1. The elliptical curve is also part of an ellipse with two foci of the ellipse at the input slit and the image point of the input slit. The image point of the input slit is obtained by using the grating-center circle normal line L.sub.GCCN 1050N as the plane of reflection for the input slit point.
[0320] The output slit SL.sub.O1 can be located on anywhere, including but not limited to the input circle 1080 or Rowland circle (like
Sgn(ija)*([D.sub.1(.sub.I1,S.sub.I1,X.sub.i)+D.sub.2(.sub.O1,S.sub.O1,X.sub.1)][D.sub.1(.sub.I1,S.sub.I1,X.sub.ja)+D.sub.2(.sub.O1,S.sub.O1,X.sub.ja)])=m*.sub.I1-O1/n.sub.gr, (24B)
where the groove i is next to some groove ja. The position of groove ja, X.sub.ja is typically already known. For an illustration and not limitation, if the grooves close to the grating center are already known, then groove ja is taken to be a groove adjacent to groove i so that X.sub.ja=X.sub.11 for i>0 (so ja=+|i1|=i1 is the previous groove close to i=0 that is already solved) and X.sub.ja=X.sub.i+1 for i<0 (so ja=|i1|=i+1 is the previous groove close to i=0 that is already solved). This is only an illustration as there can be situations, for example, the initial grooves may not be at the grating center. Sgn(i-ja) takes on value +1 or 1. Sgn(ija) is +1 if i>ja, and 1 if i<ja.
[0321] Secondly, the angular locations of the grooves are chosen so that each groove lies on or near the elliptical curve obtained above throughout the HR-CCG, where R is the radius of a circle formed by three initial groove locations X.sub.0, X.sub.1, and X.sub.1.
[0322] In another alternate embodiment, the High-Resolution Compact Curved Grating with grooves on or near an ellipse (the near-Ellipse case) has one or plurality of output slits (or waveguides or photodetectors) present at arbitrary locations.
Broadband Two Anchor Wavelengths Case
[0323]
[0324] First, the location of the entrance slit or input slit (or waveguide) 16-1201 is adjustable in order to have the best performance for a particular design goal. Thus, the location X.sub.I1 16-1291OC of an entrance slit or input slit (or waveguide) 16-1201 is specified by angle .sub.I1 with respect to a grating-center circle normal line L.sub.GCCN 16-1050N and the distance S.sub.I1 from curved grating center CGC 1050.
[0325] Second, the location of the output slit (or waveguide or photodetector) for two different wavelengths .sub.I1-O1A and .sub.I1-O2A is adjustable in order to have the best performance for a particular design goal. The location X.sub.O1A 16-1491AOC of the first output slit (or waveguide or photodetector) SL.sub.O1A 16-1401A for wavelength .sub.I1-O1A is specified by the angle O1A 16-1471A with respect to a grating-center circle normal line L.sub.GCCN 16-1050N and the distance S.sub.O1A 16-1461A from the curved grating center CGC 1050.
[0326] The location X.sub.O1B 16-1491BOC of the second output slit (or waveguide or photodetector) SL.sub.01B 16-1401B for wavelength .sub.I1-O1B is specified by the angle .sub.O1B with respect to a grating-center circle normal line L.sub.GCCN 1050N and the distance S.sub.O1B from the curved grating center CGC 1050.
[0327] Note that output slit SL.sub.01A 16-1401A, and output slit SL.sub.02A 16-1402A are not necessarily on the same circle where entrance slit or input slit (or waveguide) SL.sub.I1 16-1201 is located.
[0328] Third, the relation between .sub.I1, .sub.O1A, .sub.O2A and the initial groove spacing d is given by the grating formula,
d*Sin(.sub.O1A)+Sin(.sub.O1A)=m*.sub.I1-O1A/n.sub.gr(25)
d*Sin(.sub.O2A)+Sin(.sub.I1))=m*.sub.I1-O2A/n.sub.gr(26)
where m is the diffraction order and n is the effective refractive index of propagation of the medium. For example for the purpose of illustration but not limitation, when given the locations of the two anchor output slits, .sub.O1A, .sub.O2A are known, and with m, n.sub.gr, and d chosen, Eqs. (25) and (26) can be used to solve for .sub.I1-O1A and .sub.I1-O2A. The values for .sub.I1-O1A and .sub.I1-O2A can then be used in the next step (forth step) to obtain all the grating groove positions starting from a groove at X.sub.0.
[0329] Fourth, locations of other grooves X.sub.i's are obtained by two conditions. The first of these conditions being that the path-difference between adjacent grooves should be an integral multiple of the wavelength .sub.I1-O1A in the medium. The first condition can be expressed mathematically by:
Sgn(ija)*([D.sub.1(.sub.I1,S.sub.I1,X.sub.i)+D.sub.2(.sub.O1A,S.sub.O1A,X.sub.i)][D.sub.1(.sub.I1,S.sub.I1,X.sub.ja)+D.sub.2(.sub.O1A,S.sub.O1A,X.sub.ja)])=m*.sub.I1-O1A/n.sub.gr, (27)
where the groove i is next to some groove ja. The position of groove ja, X.sub.ja is typically already known. For an illustration and not limitation, if the grooves close to the grating center are already known, then groove ja is taken to be a groove adjacent to groove i so that X.sub.ja=X.sub.i1 for i>0 (so ja=+|i1|=i1 is the previous groove close to i=0 that is already solved) and X.sub.ja=X.sub.i+1 for i<0 (so ja=|i1|=i+1 is the previous groove close to i=0 that is already solved). This is only an illustration as there can be situations, for example, the initial grooves may not be at the grating center. Sgn(ija) takes on value +1 or 1. Sgn(ija) is +1 if i>ja, and 1 if i<ja.
[0330] D.sub.1(.sub.I1,S.sub.I1,X.sub.i) is the distance from the i-th groove located at X.sub.i to entrance slit or input slit (or waveguide) location X.sub.I1 16-1291OC specified by ell and S.sub.I1, D.sub.2(.sub.O1A,S.sub.O1A,X.sub.i) is the distance from the i-th groove located at X.sub.i to output slit (or waveguide or photodetector) location X.sub.O1A 16-1491AOC specified by .sub.O1A and S.sub.O1A, m is the diffraction order, and n is the effective refractive index of propagation of the medium. This mathematical expression is numerically exact for the optical path difference requirement in the diffraction grating and is actively adjusted for every groove on HR-CCG.
[0331] The second of these conditions being that the path-difference between adjacent grooves should be an integral multiple of the wavelength .sub.I1-O2A in the medium. The second condition can be expressed mathematically by:
Sgn(ija)*([D.sub.1(.sub.I1,S.sub.I1,X.sub.i)+D.sub.3(.sub.O2A,S.sub.O2A,X.sub.i)][D.sub.1(.sub.I1,S.sub.I1,X.sub.ja)+D.sub.3(.sub.O2A,S.sub.O2A,X.sub.ja)])=M*.sub.I1-O2A/n.sub.gr,(28)
where the groove i is next to some groove ja. The position of groove ja, X.sub.ja is typically already known. For an illustration and not limitation, if the grooves close to the grating center are already known, then groove ja is taken to be a groove adjacent to groove i so that X.sub.ja=X.sub.i1 for i>0 (so ja=+|i1|=i1 is the previous groove close to i=0 that is already solved) and X.sub.ja=X.sub.i+1 for i<0 (so ja=|i1|=i+1 is the previous groove close to i=0 that is already solved). This is only an illustration as there can be situations, for example, the initial grooves may not be at the grating center. Sgn(ija) takes on value +1 or 1. Sgn(ija) is +1 if i>ja, and 1 if i<ja.
[0332] D.sub.3(.sub.O2A,S.sub.O2A,X.sub.i) is the distance from the i-th groove located at X.sub.i to output slit (or waveguide or photodetector) location X.sub.O2A 16-1492AOC specified by .sub.O2A 16-1472A and S.sub.O2A 16-1461A. This mathematical expression is numerically exact for the optical path difference requirement in the diffraction grating and is actively adjusted for every groove on HR-CCG. Solving Equations (27) and (28) together, exact locations of other grooves X.sub.i's can be obtained.
Broadband Two Anchor Wavelengths Case with Input Slit and Two Output Slits on a Same Circle with Circle Center Near Grating Center
[0333]
[0334] In an application, such relatively equidistance locations from the grating for the output slits is useful for example, when the input waveguide or slit is small and the input beam diffraction divergence angle is large. This also means the converging or focusing beam onto the output waveguides or slits will also have large angle. The equidistance means that the output waveguide mouth (or slit) of one channel will not block the large angle converging beam that reach the adjacent output channel waveguide (or slit).
[0335] In this case, optionally, in some applications one can place a curved reflecting mirror passing through points X.sub.O1A 17-1491AOC and X.sub.O2A 17-1492AOC, with the nominal direction of the curve at X.sub.O1A 17-1491AOC and X.sub.O2A 17-1492AOC pointing towards the grating center CGC 17-1050. As light at wavelength .sub.I1-O1A enters entrance slit or input slit (or waveguide) SL.sub.I1 17-1201 and is diffracted by the grating towards output slit SL.sub.O1A 17-1401A, it will hit the mirror and be reflected back by the mirror along the same path back to the grating and further focusing back to the entrance slit or input slit (or waveguide) SL.sub.I1 17-1201. Similarly, as light at wavelength .sub.I1-O2A enters entrance slit or input slit (or waveguide) SL.sub.I1 17-1201 and is diffracted by the grating towards output slit SL.sub.O2A 17-1402A, it will hit the mirror and be reflected back by the mirror along the same path back to the grating and further focusing back to the entrance slit or input slit (or waveguide) SL.sub.I1 17-1201. An application that may make use of this property is when one want to reflected the beam back to the input slit location such as to form an optical cavity in the case of a laser, or to increase the path length of the beam in certain compact spectrometer applications.
Broadband Two Anchor Wavelengths Case with Input Slit and Three or Plurality of the Output Slits Near a Straight Line
[0336] In another embodiment, also illustrated by
[0337] The three output slit (or waveguide) locations X.sub.O1A 17-1491AOC, X.sub.O2A 17-1492AOC, X.sub.O3 17-14930C, can be made to be located on or near a circle of radius R.sub.out, called the radius of curvature of the output plane or output-plane radius. This circle of radius R.sub.out, can have a center of curvature either closed to or away from the grating center.
[0338] In one embodiment, S.sub.O1A and S.sub.O2A are chosen so that the locations X.sub.O1A 17-1491AOC, X.sub.O2A 17-1492AOC, X.sub.O3 17-14930C lie on a near straight line and R.sub.out is large.
[0339] In an application, such relatively equidistance locations from the grating and flat field (near straight-line) arrangement for the output slits is useful for example, when the input waveguide or slit is small and the input beam diffraction divergence angle is large. This also means the converging or focusing beam onto the output waveguides or slits will also have large angle. The flat field means that the output waveguide mouth (or slit) of one channel will not block the large angle converging beam that reach the adjacent output channel waveguide (or slit).
[0340] Optionally, in some applications, the three focal points can form a nearly flat surface of reflection if a mirror surface 17-905 is placed across the output slit locations, so that the focused beams with plane wavefront at their focal points will be reflected directly back to the entrance slit SL.sub.I1 17-1201. In that case, each of the reflected beam will trace back its own original physical beam propagation and hence will achieve maximum reflection back into the input slit or waveguide SL.sub.I1 17-1201. An application that may make use of this property is when one want to reflected the beam back to the input slit location such as to form an optical cavity in the case of a laser, or to increase the path length of the beam in certain compact spectrometer applications.
Broadband Two Anchor Wavelengths Case with Input Slit and Plurality of Output Slits on a Same Circle and Mirrors Near Output Slit with Curved Mirror Surface to Match Beam's Phase Front Curvature
[0341] As shown in
Broadband Two Anchor Wavelengths Multiple Outputs and Inputs Case
[0342] An optical grating spectrometer device that can be used in various devices include but not limited to a wavelength multiplexer, wavelength demultiplexer, optical spectra processing device, or a device in an optical spectrometers, wavelength channel multiplexers, wavelength channel demultiplexers, wavelength or frequency filters, wavelength combiners, wavelength splitters, optical spectrum analyzers, wavelength detectors, spectra dispersion devices, optical arbitrary waveform generators, optical dispersion compensators, optical signal processors, and optical wavelength-domain or frequency-domain processors, for combining, filtering, analyzing, processing, or detecting the spectral compositions of an input optical beam or plurality of input beams, with one or plurality of output beams.
[0343]
[0344] Specifically, the optical grating spectrometer device 18-1000 enabling a processing of light spectra in a range of operation wavelengths centered at wavelength .sub.c. As shown by
[0345] The optical grating spectrometer device 18-1000 also comprises at least an output slit SL.sub.O1 18-1401 (at position X.sub.O1 18-1491OC) or a plurality of output slits, say No number of slits given by {SL.sub.O1 18-1401 (at position X.sub.O1 18-1491OC), SL.sub.O2 18-1402 (at position X.sub.O2 18-14920C), SL.sub.h3 18-140h (at position X.sub.O3 18-1493OC), . . . , and SL.sub.ONo 18-140No (at position X.sub.ONo 18-149NoOC)}, and a curved grating CG 18-1010. The curved grating CG 18-1010 for processing the spectral compositions of at least an optical beam B.sub.I1 18-1101 that goes through slit SL.sub.I1 18-1201. There may be other optical beams Bit 18-1102 that goes through slit SL.sub.I2 18-1202, B.sub.Ik 18-110k that goes through slit SL.sub.Ik 18-120k, . . . , and B.sub.IN 18-110N that goes through slit SL.sub.IN 18-120N etc. The grating CG 18-1010 comprises a plurality of grooves at positions X.sub.2, X.sub.1, X.sub.0, X.sub.1, X.sub.2 . . . , the position of each groove being adjustable for controlling a performance of the grating spectrometer, the position of each of the input slits being adjustable for controlling a performance of the of the grating spectrometer, and the position of each of the output slits being adjustable for controlling a performance of the grating spectrometer, are determined as follows:
[0346] First: for the purpose of describing the design of the optical grating spectrometer device, a Cartesian coordinate system is set up with vector X=(x,y,z) denoting a spatial point in the coordinate system for which the real number x is the x-coordinate, y is the y-coordinate, and z is the z-coordinate for the vector. The coordinate origin is at vector (0,0,0). The optical beam propagating in the grating system is assumed to be propagating approximately parallel to the two-dimensional x-y plane at z=0. In this plane, it is sufficient to describe the x and y coordinates, and the vector in this plane will be denoted by X=(x,y) with the origin at (0,0).
[0347] In the case of applications to a wavelength multiplexer/demutiplexer/spectrometer/spectra-processing-device with freely propagating optics, the grating grooves are a set of planes approximately perpendicular to x-y plane at z=0. The spacing and relative locations of these planes with respect to each other at z=0 can be described by a function that depends on the x and y coordinates. The angular deviation of the optical beam diffracted by the grating is largely in a direction parallel to the x-y plane at z=0.
[0348] As is well known to those skilled in the art, the grating still can have a curvilinear surface as a function of z in the z direction but it will be to perform only the regular function of focusing and imaging the optical beam in the z direction and will not have the function of spreading the optical beam in different spatial directions for different frequency components due to the grating grooves. In one embodiment, this focusing function in the z-direction is achieved via curvilinear surface that has an elliptical shape in the z direction with two foci, one focus at the location of the input slit and the other focus at the location of the output slit. In another embodiment, this focusing ellipse is approximated by a circle whose radius of curvature matches the radius of curvature of the said elliptical curve around the z=0 region where most of the energy of the beam is hitting the grating. Thus, the wavelength dispersion functionalities of the grating grooves that depend only on the relative positions and spacing of the parallel lines of the grating grooves with respect to each other in the x and y directions can be described by functions in the x-y plane with two-dimensional coordinates denoted by vector X=(x,y). When the x-y positions of these grating grooves are joined with a line (say at the z=0 plane), they will form a curvilinear line in the two-dimensional x-y plane that describes the grating surface (at near z=0 plane).
[0349] When this device is applied as a device in an integrated optic circuit or electronic-photonic integrated circuit, the optical beam in the optical grating spectrometer device will be confined within a planar waveguide with its plane parallel to the z=0 plane. As is known to those skilled in the art, a planar waveguide has a layer or set of layers of materials made up of materials with relatively high refractive indices forming a two-dimensional waveguide core with refractive index now 18-1040O (see
[0350] Hence, for the purpose of illustration and not limitations, all the grating groove positions and all the positions for the input slits or output slits are described by a two-dimensional coordinate system that depends only on the x and y coordinates.
[0351] After setting up a two-dimensional coordinate system described by vector X=(x,y) with the coordinate origin at (0,0), the input/output slit positions and the positions of the grating grooves are specified for an optical grating spectrometer device 18-1000 with a curved grating CG 18-1010 as described below. The center for the curved grating called the curved grating center CGC 18-1050 is chosen to be situated at X.sub.0 that is set also to be at the coordinate origin so that X.sub.0=(0,0) 18-1600O. In this embodiment, the coordinate axes are set up so that the y direction is parallel to the grating-center circle normal line L.sub.GCCN 18-1050N and the x=0 line coincides with the grating-center circle normal line L.sub.GCCN 18-1050N and passes through the grating center GC 18-1050 at X.sub.0. Thus, the grating-center circle normal line L.sub.GCCN 18-1050N coincides with the line that gives the y-axis.
[0352] There is a first input slit SL.sub.I1 18-1201 for allowing an entry of an input optical beam B.sub.I1 18-1101 into device 18-1000, a location of the first input slit being adjustable, and further the location of the first input slit is specified by a first input angle .sub.I1 18-1271 that is sustained between the line joining the first input slit location X.sub.I1 18-1291OC to the grating center CGC 18-1050 and the grating-center circle normal line L.sub.GCCN 18-1050N, and further specified by a first input distance S.sub.In1 18-1261 from the curved grating center CGC 18-1050 to the first input slit location X.sub.I1 18-1291OC. The width of the first input slit being adjustable, and further the width of the first input slit being specified by a first input slit width W.sub.I1 18-1291W. The angle .sub.I1 18-1271 is zero when the line joining the input slit to the grating center is parallel to grating-center circle normal line L.sub.GCCN 18-1050N, takes on positive value when it is rotated about the curved grating center CGC 18-1050 from this zero-angle position towards the negative x direction, and takes on negative value when it is rotated about the grating center from this zero-angle position towards the positive x direction from this zero-angle position. This will be the sign conventions for all the angles below that are referenced to the grating center CGC 18-1050 as the pivot of rotation. In terms of the Cartesian coordinates expressed in angle .sub.I1 18-1271 and distance S.sub.I1 18-1261, the input slit SL.sub.I1 18-1201 is situated at X.sub.I1=(S.sub.I1*Sin(.sub.I1), S.sub.I1*Cos(.sub.I1)) 18-1291OC.
[0353] Second: a first output slit called the first anchor output slit SL.sub.O1A 18-1401A for allowing the exiting of a first anchor output optical beam B.sub.O1A 18-1301A, is specified. As will be clear below, the output slit of interest SL.sub.O1A 18-1401A is referred to as an anchor output slit as the position of this output slit SL.sub.O1A 18-1401A will be used to obtain the grating groove positions and hence will serve as an anchor for the groove positions generation. The location of the first anchor output slit X.sub.O1A 18-1491AOC being adjustable, and further the location of the first anchor output slit X.sub.O1A 18-1491AOC specified by a first anchor output angle .sub.O1A 18-1471A that is sustained between the line joining the first anchor output slit location X.sub.O1A 18-1491AOC to the grating center CGC 18-1050 and the grating-center circle normal line L.sub.GCCN 18-1050N, and further specified by a first anchor output distance S.sub.O1A 18-1461A from the grating center CGC 18-1050 to the first anchor output slit location X.sub.O1A 18-1491AOC. In the subscripts for .sub.O1A 18-1471A and S.sub.O1A 18-1461A, O1A refers to the first anchor output. The coordinate of the first anchor output slit is denoted by X.sub.O1A 18-1491AOC and is given by X.sub.O1A=(S.sub.O1A*Sin(.sub.O1A), S.sub.O1A*Cos(.sub.O1A)) 18-1491AOC. The width of the first anchor output slit being adjustable, and further the width of the first anchor output slit is being specified by a first anchor output slit width W.sub.O1A 18-1491W. Thus, the location of output slit is not necessarily on the Rowland circle; and
[0354] Third: the medium in the grating diffraction region between any of the input slit and the grating or any of the output slit and the grating, has an effective propagating refractive index around n.sub.gr 18-1040 for the input optical beam with spectra compositions around wavelength .sub.BI1 18-1121, as shown in
[0355] A grating order is chosen and denoted by m, which is an integer (can be positive or negative). A particular optical wavelength of the spectral component of the input beam is chosen to be diffracted to the first anchor output slit SL.sub.O1A 18-1201A. The free-space wavelength for this spectral component is denoted by .sub.I1-O1A 18-1321A and its frequency is f.sub.I1-O1A=(c/.sub.I1-O1A), where c is the speed of light in vacuum. The value of .sub.I1-O1A 18-1321A (see
[0356] Note that this anchor output slit may be one of the No output slits specified as {SL.sub.O1, . . . , SL.sub.ONo} or it may be addition to the No output slits specified as {SL.sub.O1, . . . , SL.sub.Ono}. Such anchor output slits are defined for the purpose of discussion such as to define how the grating is generated, and not limitation in that in the actual implementation, this middle-wavelength output slit does not have to actually physically exist. In any case, it is specifically specified as the anchor slit and it carries a subscript A to distinguish it from the other output slits whose positions will be determined differently from such anchor output slits.
[0357] Fourth: The position of the i.sup.th groove is specified by its x-y coordinates X.sub.i=(x.sub.i, y.sub.i) 18-160|i|P/N. Below, 18-160|i|P/N shall be taken as to mean it is given by 18-160|i|N if i<0, 18-160|i|P if i>0, and 18-1600O if i=0. The x-y coordinates are specified with respect to the grating center X.sub.0 18-1600O and the input slit X.sub.I1 18-1291OC. The angle .sub.grI1-i 18-161|i|P/N is the angle made by the line joining the input slit location X.sub.I1 18-1291OC to the groove location X.sub.i 18-160|i|P/N and the line joining the input slit location X.sub.I1 18-1291OC to the grating center X.sub.0 18-1600O. Below, 18-161|i|P/N shall be taken as to mean it is given by 18-161|i|N if i<0 and 18-161|i|P if i>0. 18-161|i|P/N defined above can be used to give the divergence angle span of the input beam intercepted by the grating from the grating groove at i=0 up to groove number i, and may be referred to as the input-ray angle at groove i. The value of .sub.grI1-i 18-161|i|P/N is zero when X.sub.i=X.sub.0, is positive when i>0, and is negative when i<0.
[0358] Fifth, the locations of all other grooves are given by computing the coordinate of each groove with the i.sup.th groove's coordinate X.sub.i 18-160|i|P/N given by the following two conditions. The first of these conditions being that the path-difference between adjacent grooves should be an integral multiple of the wavelength in the medium. The first condition can be expressed mathematically by:
Sgn(ija)*([D.sub.1(.sub.I1,S.sub.I1,X.sub.i)+D.sub.2(.sub.O1A,S.sub.O1,X.sub.i)][D.sub.1(.sub.I1,S.sub.I1,X.sub.ja)+D.sub.2(.sub.O1A,S.sub.O1A,X.sub.ja)])=m*.sub.I1-O1A/n.sub.grI1-O1A,(29),
where D.sub.1(.sub.I1,S.sub.I1,X.sub.i) is the physical distance from an i-th groove located at X.sub.i 18-160|i|P/N to the input slit's (or input waveguide's) SL.sub.I1 18-1201 position X.sub.I1 18-1291OC specified by .sub.I1 18-1271 and S.sub.I1 18-1261, D.sub.2(.sub.O1A,S.sub.O1A,X.sub.i) is the distance from the i-th groove located at X.sub.i 18-160|i|P/N to the first anchor output slit's (or waveguide's or photodetector's) SL.sub.O1A 18-1401A position X.sub.O1A 18-1491AOC specified by .sub.O1A 18-1471A and S.sub.O1A 18-1461A, m is the diffraction order, and n.sub.grI1-O1A 18-1041A is the averaged effective refractive index of propagation of the medium at wavelength .sub.I1-O1A 18-1321A. Note that some time we would refer to just a general planar-waveguiding refractive index n.sub.gr 18-1040 for the grating region. When there is substantial refractive index variation as a function of the wavelength, the denotation of n.sub.gr 18-1040 by n.sub.grI1-O1A 18-1041A or n.sub.grI1-O2A 18-1042A is just to make the statement more precise as is well known to those skilled in the art. Hence, they are used interchangeably and n.sub.gr 18-1040 is often used for simplicity and is not meant to limit the scope of the present invention. Eq. (29) ensures that the free-space wavelength .sub.I1-O1A 18-1322A in Eq.(29) is the wavelength that will diffract to this first anchor output slit SL.sub.O1A 18-1401A from input slit SL.sub.I1, and is called the first anchor output wavelength.
[0359] In Eq. (29), groove ja is taken to be a groove adjacent to groove i, and Sgn(ija) takes on value +1 or 1. Sgn(ija) is +1 if i>ja, and 1 if i<ja. The position of groove ja, X.sub.ja is typically already known. For an illustration and not limitation, if the grooves close to the grating center are already known, then X.sub.ja=X.sub.i1 for i>0 (so ja=+|i1|=i1 is the previous groove close to i=0 that is already solved) and X.sub.ja=X.sub.i+1 for i<0 (so ja=|i1|=i+1 is the previous groove close to i=0 that is already solved). Since the groove position X.sub.0 at the grating center with i=0 is given, this will give one of the two equations needed to generate groove positions X.sub.1 and X.sub.1 with i=1 and i=1, and one of the two equations needed to generate all other grooves similarly. This is only an illustration as there can be situations. For example, the initial grooves may not be at the grating center at X.sub.0 and may be at other groove locations. The mathematical expression given by Eq. 29 is numerically exact for the optical path difference requirement in the diffraction grating and is actively adjusted for every groove on HR-CCG.
[0360] The second of these conditions being specific for a particular design goal of a curved-grating spectrometer. The second condition in general can be mathematically expressed as
f(X.sub.i)=constant(30)
where in Eq. (30), the function f or the constant on the right-hand-side of Eq. (30), can be depending on other design parameters such as the input slit and output slit positions or the positions of the adjacent grooves and other parameters (e.g. .sub.I1,S.sub.I1,.sub.O1A,S.sub.O1A, m, n.sub.grI1-O1A, X.sub.ja) that are already known. The functional variable involved is X.sub.i 18-160|i|P/N which is the variable to be solved. Specific examples of the second condition are described later in the application. Eq. (30) will give the second of the two equations needed to generate all the groove positions.
[0361] The unknown variables in both equations Eq. (29) and Eq. (30) are x- and y-coordinates of the location vector X.sub.i 18-160|i|P/N of the i-th groove X.sub.i=(x.sub.i, y.sub.i). For a given input-slit (or input-waveguide) location X.sub.I1 18-1291OC given by .sub.I1 18-1271 and S.sub.I1 18-1261, anchor output slit (or waveguide or photodetector) location X.sub.O1A 18-1491AOC given by .sub.O1A 18-1471A and S.sub.O1A 18-1461A, and the previous ja-th groove position X.sub.ja, the positional vector for the i-th groove X.sub.i 18-160|i|P/N is completely specified by equations Eq. (29) and Eq. (30) for a given wavelength .sub.I1-O1A 18-1321A to output slit SL.sub.O1A 18-1201A, effective refractive index of propagation n.sub.gr-I1-O1A, and the diffraction order m.
[0362] The above two equations Eq. (29) and Eq. (30) are needed to solve for the two unknown numbers in X.sub.i=(x.sub.i, y.sub.i), namely x-coordinate x.sub.i and y-coordinate y.sub.i of the i.sup.th groove. These two equations are solved analytically, numerically, or computationally for the values of X.sub.i=(x.sub.i, y.sub.i) 18-160|i|P/N using equations solving methods that are known to those skilled in the art. The groove positions X.sub.i starting from i=0, 1, 2 . . . or i=0, 1, 2 . . . are iteratively solved with the groove location of the preceding groove X.sub.ja already solved or specified starting from the location of initial grooves X.sub.1 or X.sub.1, or any other initial groove positions, such as X.sub.0=(0,0), whichever is applicable.
[0363] This fifth step of the HR-CCG specifications ensures that the rays from all the grooves basically converge to a single point at the first anchor output location X.sub.O1A 18-1491AOC. This ensures the rays from HR-CCG will focus well at output slit SL.sub.O1 18-1401 with minimal spatial focusing aberration, and therefore enabling a small focused spot size at the output slit.
[0364] Sixth: In an alternative embodiment, the second constraint is further given by choosing the function f so that:
Sgn(ija)*([D(.sub.I1,S.sub.I1,X.sub.i)+D.sub.2(.sub.O2A,S.sub.O2A,X.sub.i)][D.sub.1(.sub.I1,S.sub.I1,X.sub.ja)+D.sub.2(.sub.O2A,S.sub.O2A,X.sub.ja)])=m*.sub.I1-O2A/n.sub.gr-I1-O2A,(31)
where D.sub.2(.sub.O2A,S.sub.O2A,X.sub.i) is the distance from X.sub.i 18-160|i|P/N to a second anchor output slit SL.sub.O2A 18-1402A at position X.sub.O2A 18-1492AOC, whose location is specified by a second anchor output angle .sub.O2A 18-1471A that is sustained between the line joining the second anchor output slit SL.sub.O2A 18-1402A position X.sub.O2A 18-1492AOC to the grating center CGC 18-1050 at X.sub.0 and the grating-center circle normal line L.sub.GCCN 18-1050N, and a second anchor output distance S.sub.O2A 18-1461A from the grating center CGC 18-1050 at X.sub.0 to the second anchor output slit SL.sub.O2A 18-1402A position X.sub.O2A 18-1492AOC. Note for example that f in Eq. (30) is the left hand side of Eq. 31 and the constant in Eq. (30) is the right hand side of Eq. (31).
[0365] The coordinate of the second anchor output slit SL.sub.O2A 18-1402A is denoted by: X.sub.O2A 18-1492AOC (32A)
and is given by
X.sub.O2A=(S.sub.O2A*Sin(.sub.O2A),S.sub.O2A*Cos(.sub.O2A)).(32B)
[0366] The width of the second anchor output slit SL.sub.O2A 18-1402A being adjustable, and further the width of the second anchor output slit specified by a second anchor output slit width W.sub.O2A 18-1492AW. The free-space wavelength .sub.I1-O2A 18-1322A in Eq.(31) is the wavelength that will diffract to this second anchor output slit SL.sub.O2A 18-1402A from input slit SL.sub.I1, and is called the second anchor output wavelength.
[0367] Note that this second anchor output slit may be one of the No output slits specified as {SL.sub.O1, . . . , SL.sub.ONo} or it may be addition to the No output slits specified as {SL.sub.O1, . . . , SL.sub.ONo}. Such anchor output slits are defined for the purpose of discussion such as to define how the grating is generated, and not limitation in that in the actual implementation, this middle-wavelength output slit does not have to actually physically exist. In any case, it is specifically specified as the anchor slit and it carries a subscript A to distinguish it from the other output slits whose positions will be determined differently from such anchor output slits.
[0368] In Eq. (31), groove ja is taken to be a groove adjacent to groove i, and Sgn(ija) takes on value +1 or 1. Sgn(ija) is +1 if i>ja, and 1 if i<ja. The position of groove ja, X.sub.ja is typically already known. For an illustration and not limitation, if the grooves close to the grating center are already known, then X.sub.ja=X.sub.i1 for i>0 (so ja=+|i1|=i1 is the previous groove close to i=0 that is already solved) and X.sub.ja=X.sub.i+1 for i<0 (so ja=|i1|=i+1 is the previous groove close to i=0 that is already solved).
[0369] Eq. (31) imposes that spectral energy in the input beam at a second anchor output wavelength .sub.I1-O2A will be diffracted by the grating to the second anchor output slit direction at second anchor output angle .sub.O2A. The wavelength .sub.I1-O2A 8-1322A is at this point unknown and has to be solved and there are various ways to do so.
[0370] Arbitrariness in the Generation of the Initial Grooves
[0371] (Preferred embodiment: two-groove case with exact solution and with .sub.I1-O1A first chosen) In another preferred embodiment, the initial set of grooves is made in the following ways: an initial two groove positions are set at:
X.sub.1=(d/2,0)(33A)
and
X.sub.1=(d/2,0)(34A)
and there is no groove at X.sub.0. Alternatively, the two grooves can be:
X.sub.0=(0,0)(33B)
and
X.sub.1=(d,0)(34B)
or alternatively, the two grooves can also be:
X.sub.0=(0,0)(33C)
and
X.sub.1=(d,0)(34C)
or alternatively, the two grooves can also be:
X.sub.1=(d/2,R(R.sup.2(d/2).sup.2).sup.1/2)(33D)
and
X.sub.1=(d/2,R(R.sup.2(d/2).sup.2).sup.1/2(34D)
or alternatively, the two grooves can also be:
X.sub.0=(0,0)(33E)
and
X.sub.1=(d,R(R.sup.2d.sup.2).sup.1/2)(34E)
or alternatively, the two grooves can also be:
X.sub.0=(0,0)(33F)
and
X.sub.1=(d,R(R.sup.2d.sup.2).sup.1/2)(34F)
[0372] In one embodiment, the parameter d being adjustable such that these two points gives for the chosen wavelength .sub.I1-O1A 18-1321A and output slit location given by .sub.O1A,S.sub.O1A, the following equation is satisfied:
d*(Sin(.sub.O1A)+Sin(.sub.I1))=m*.sub.I1-O2A/n.sub.grI1-O1A,(35A)
In the case of the alternative embodiment given in the sixth step (Eq. 31) above, the wavelength .sub.I1-O2A 18-1322A is further solved by requiring that:
d*(Sin(.sub.O2A)+Sin(.sub.I1))=m*.sub.I1-O2A/n.sub.grI1-O2A,(35B)
with n.sub.grI1-O2A 18-1042A being the refractive index of the grating diffraction region at the free-space wavelength .sub.I1-O2A 18-1322A.
[0373] In another embodiment for the case involving X.sub.1 and X.sub.1 as the initial grooves, the parameter d being adjustable such that these two points gives for the chosen wavelength .sub.I1-O1A 18-1321A and output slit location given by .sub.O1A,S.sub.O1A, the following equation is satisfied:
[D.sub.1(.sub.I1,S.sub.I1,X.sub.1)+D.sub.2(.sub.O1A,S.sub.O1A,X.sub.1)][D.sub.1(.sub.I1,S.sub.I1,X.sub.1)+D.sub.2(.sub.O1A,S.sub.O1A,X.sub.1)]=m*.sub.I1-O1A/n.sub.grI1-O1A(36A)
In the case of the alternative embodiment given in the sixth step (Eq. 31) above, the wavelength .sub.I1-O2A 18-1322A is further solved by requiring that:
[D.sub.1(.sub.I1,S.sub.I1,X.sub.1)+D.sub.2(.sub.O2A,S.sub.O2A,X.sub.1)][D.sub.1(.sub.I1,S.sub.I1,X.sub.1)+D.sub.2(.sub.O2A,S.sub.O2A,X.sub.1)]=m*.sub.I1-O2A/n.sub.grI1-O2A (36B)
[0374] In another embodiment for the case involving X.sub.0 and either X.sub.1 or X.sub.1 as the initial grooves, the parameter d being adjustable such that these two points gives for the chosen wavelength .sub.I1-O1A 18-1321A and output slit location given by .sub.O1A,S.sub.O1A, the following equation is satisfied:
j0([D.sub.1(.sub.I1,S.sub.I1,X.sub.j0)+D.sub.2(.sub.O1A,S.sub.O1A,X.sub.j0)][D.sub.1(.sub.I1,S.sub.I1,X.sub.0)+D.sub.2(.sub.O1A,S.sub.O1A,X.sub.0)])=m(.sub.I1-O1A/n.sub.grI1-O1A(36C)
j0*([D.sub.1(.sub.I1,S.sub.I1,X.sub.j0)+D.sub.2(.sub.O2A,S.sub.O2A,X.sub.j0)][D.sub.1(.sub.I1,S.sub.I1,X.sub.0)+D.sub.2(.sub.O2A,S.sub.O2A,X.sub.0)])=m*.sub.I1-O2A/n.sub.grI1-O2A (36D)
where the X.sub.j0 is the grating groove position adjacent to X.sub.0 (i.e. j0=1 or 1). The left multiplication by j0 is just to ensure the sign on the left side of the equation comes out correct.
[0375] (Three initial grooves with .sub.I1-O1A first chosen) In as yet another embodiment, with .sub.I1-O1A first chosen, the initial grooves are generated by taking the following three grooves as initial grooves:
X.sub.0=(0,0)(37A),
X.sub.1=(d,R(R.sub.2d.sub.2).sup.1/2),(37B)
and
X.sub.1=(d,R(R.sup.2d.sup.2).sup.1/2)(37C)
Note that these grooves that are on the outer input circle 18-1070 of radius R.
The parameter d being adjustable such that for the chosen wavelength .sub.I1-O1A 18-1321A and output slit angles given by .sub.O1A and .sub.O2A, either the exact Eqs.(38A) and (38B) below, or the approximate Eqs.(35A) and (35B) or other similar approximate equations as Eqs.(35A) and (35B) are obeyed, which then determines the free-space wavelength .sub.I1-O2A 18-1322A.
j0*([D.sub.1(.sub.I1,S.sub.I1,X.sub.j0)+D.sub.2(.sub.O1A,S.sub.O1A,X.sub.j0)][D.sub.1(.sub.I1,S.sub.I1,X.sub.0)+D.sub.2(.sub.O1A,S.sub.O1A,X.sub.0)])=m*.sub.I1-O1A/n.sub.grI1-O1A (38A)
j0*[D.sub.1(.sub.I1,S.sub.I1,X.sub.j0)+D.sub.2(.sub.O2A,S.sub.O2A,X.sub.j0)][D.sub.1(.sub.I1,S.sub.I1,X.sub.0)+D.sub.2(.sub.O2A,S.sub.O2A,X.sub.0)])=m*.sub.I1-O2A/n.sub.grI1-O2A (38B)
where the X.sub.j0 is the grating groove position adjacent to X.sub.0 (i.e. j0=1 or 1). The left multiplication by j0 is just to ensure the sign on the left side of the equation comes out correct. Note the location of X.sub.j0 is dependent on and varies with d as specified by Eqs. (37A/B/C). Note in general, choosing j0=1 will give a slightly different result for .sub.I1-O2A 18-1322A than if one chooses j0=1. This is because having 3 initial grooves is redundant and may not be totally consistent with the exact solution for the grating grooves for output slits SL.sub.O2A or SL.sub.O2A. Thus, one shall only pick either j0=1 or j0=1 to solve for .sub.I1-O1A and .sub.I1-O2A.
[0376] (d (instead of .sub.I1-O1A) and Outputs first chosen case for alternative embodiment) In other embodiments, for the above two embodiments involving placement of initial grooves with parameter d (e.g. Eqs. 33A/B/C/D/E/F and 34A/B/C/D/E/F, or Eqs. 37A/B/C), the parameter d is first chosen and (instead of to be first chosen), output slit locations are also given (by .sub.O1A and S.sub.O1A and .sub.O2A and S.sub.O2A). In the case of the alternative embodiment given in the sixth step (Eq. 31) above, .sub.I1-O1A and .sub.I1-O2A are then solved to satisfy either the exact Eqs.(36A) and (36B) (or Eqs.(36C) and (36D)) for the case of Eqs. 33A/B/C/D/E/F and 34A/B/C/D/E/F (or Eqs.(38A) and (38B) for the case of Eqs. 37A/B/C), or the approximate Eqs.(35A) and (35B) or other similar approximate equations as Eqs.(35A) and (35B).
[0377] (Another Preferred Embodiment: d, .sub.I1-O1A, and (S.sub.O1A, S.sub.O2A) first chosen case) In another also preferred embodiments, for the above two embodiments involving placement of initial grooves with parameter d (e.g. Eqs. 33A/B/C/D/E/F and 34A/B/C/D/E/F, or Eqs. 37A/B/C), the parameter d is first chosen, .sub.I1-O1A is chosen, and output slit distances are also given (by S.sub.O1A and S.sub.O2A). Then .sub.O1A and .sub.O2A are solved to satisfy either the exact Eqs.(36A) and (36B) (or Eqs.(36C) and (36D)) for the case of Eqs. 33A/B/C/D/E/F and 34A/B/C/D/E/F (or Eqs.(38A) and (38B) for the case of Eqs. 37A/B/C), or the approximate Eqs.(35A) and (35B) or other similar approximate equations as Eqs.(35A) and (35B).
[0378] In the case of the grating generated by Eq. (29) and Eq. (31) (or Eq. (29) and Eq. (30)), the above examples shows various ways to obtain the value for .sub.I1-O2A or .sub.O2A for use in Eq. (31) that ensures that for the grating grooves so generated, the tangent to the grating-center curve passing through the few grooves closest to the grating center is perpendicular to the grating-center circle normal line L.sub.GCCN 18-1050N at the grating center, and the few grooves basically obey the exact Eqs.(36A) and (36B) (or Eqs.(36C) and (36D)) for the case of Eqs. 33A/B/C/D/E/F and 34A/B/C/D/E/F (or Eqs.(38A) and (38B) for the case of Eqs. 37A/B/C), or the approximate Eqs.(35A) and (35B) or other similar approximate equations as Eqs.(35A) and (35B). It is done by construction, requiring that the few grooves approximately lie on a circle of radius R with such property.
[0379] (One initial groove with .sub.I1-O1A first chosen) In another embodiment, with .sub.I1-O1A first chosen, the initial grooves are generated by taking the groove X.sub.0 as the only groove in the initial set of grooves.
X.sub.0=(0,0)(39)
In this case, there is no constraint on X.sub.1 or X.sub.1. A grating still can be generated based on Eq. (29) and Eq. (31) (or Eq. (29) and Eq. (30)), but there is no guarantee that the actual grating center tangent normal line L.sub.GCTN 1050TN (that by definition perpendicular to the tangent L.sub.GCT 1050T) will coincided with the grating-center circle normal line L.sub.GCCN 1050. As discussed above, when that happens, it just amount to a redefinition of the angles for the input and output slits, and the actual grating center tangent normal line L.sub.GCTN 1050TN shall take on the role of the grating-center circle normal line L.sub.GCCN 1050 instead. To put it in another way, it is worth noting that an arbitrary choices of the values of .sub.I1-O1A and .sub.I1-O2A given all other parameters like .sub.O1A and .sub.O2A etc or the values of .sub.O1A and .sub.O2A given all other parameters like .sub.I1-O1A and .sub.I1-O2A etc, that deviate from the values given by the above procedure (e.g. by picking X.sub.0 as the only initial point) will still enable the grating grooves to be generated based on Eq. (29) and Eq. (31) (or Eq. (29) and Eq. (30)), however, the tangent to the grating-center curve joining the grooves closest to the grating center will no longer be perpendicular to the grating-center circle normal line L.sub.GCCN 18-1050N. This simply results in a re-orientation of the true grating center normal line so that a new grating-center normal line that is perpendicular to the tangent to the grating-center curve joining the grooves shall be used to measure the input slit angle and output slit angle and hence altering their angles to new values. Under this new grating-center normal line, the grating so generated would be equivalent to the case in which the value of .sub.I1-O2A is given by one of the above procedures but with the input slit angle and output slit angle altered to their new values.
Broadband Two Anchor Wavelengths with Multiple Outputs Slits
[0380]
[0381] Specifically, this section refer to
[0382] Let the output angle for wavelength .sub.BI1-M 19-1121M after diffraction from the grating be .sub.I1-OM 19-147M. For a potential output slit, called the middle-wavelength output slit SL.sub.OM 19-140M at .sub.I1-OM 19-147M and location .sub.OM 19-149M, the output wavelength from this output slit for the beam from input slit SL.sub.I1 will be labelled as .sub.I1-OM 19-132M, and .sub.I1-OM=.sub.BI1-M. These are defined for the purpose of discussion such as to define how the grating is generated, and not limitation in that in the actual implementation, this middle-wavelength output slit does not have to actually physically exist.
[0383] In an embodiment of the present invention .sub.I1-OM 19-132M is placed between the first anchor output wavelength .sub.I1-O1A 19-1321A and the second anchor output wavelength .sub.I1-O2A 19-1322A as it would be advantages because the spatial focusing aberrations at the first anchor output slit SL.sub.O1A 19-1401A and the second anchor output slit SL.sub.O2A 19-1402A are essentially zero, giving the highest spectral resolution at .sub.I1-O1A 19-1321A and .sub.I1-O2A 19-1322A. That means the spectral resolution for a wavelength .sub.I1-Om 19-132m (where m is an integer labeling the channel number) around the designed middle wavelength .sub.I1-OM 19-132M, that is between .sub.I1-O1A and .sub.I1-O2A will also be minimized for which the output slit SL.sub.Ok 19-140k receiving that wavelength will have an output slit angle .sub.I1-Ok 19-147k that is between .sub.I1-O1A 19-1471A and .sub.I1-O2A 19-1472A, and thus it is spatially located in-between the anchor output slits SL.sub.O1A 19-1401A and SL.sub.O2A 19-1402A and generally near a line joining the anchor output slits SL.sub.O1A 19-1401A and SL.sub.O2A 19-1402A.
[0384] The spatial closeness of slit SL.sub.Ok 19-140k to the aberration free points at slits SL.sub.O1A 19-1401A and SL.sub.O2A 19-1402A makes the focusing at SL.sub.Ok 19-140m also nearly aberration free and hence high spectral resolution can be achieved at output wavelength .sub.I1-Ok 19-132k.
[0385] Furthermore, if 19-1000 is designed for input beam from slit SL.sub.I1 19-1201 that is expected to have a spectral span from .sub.BI1-M 19-1121M to a wavelength .sub.LBI1-M-X % 19-1121(X %) ML for beams at output angle .sub.LI1-OM-X % 19-147M(X %) L where .sub.LI1-OM-X % is to the left (assuming the front is facing the output slit looking from the grating center) of the output angle .sub.I1-OM 19-147M for wavelength .sub.I1-OM 19-132M that encompasses X % of the total spectral energy on one side of .sub.I1-OM 19-132M from .sub.I1-OM to X.sub.LBI1-M-X %, and a spectral span from X.sub.BI1-M 19-1121M to a wavelength .sub.RBI1-M-Y % 19-1121(Y %) MR for beams at output angle .sub.RI1-OM-Y % 19-147M(Y %) R where .sub.RI1-OM-Y % 19-147M(Y %) R is to the right of the output angle .sub.I1-OM 19-147M for wavelength .sub.I1-OM 19-132M that encompasses Y % of the total spectral energy on one side of .sub.I1-OM 19-132M from .sub.I1-OM to .sub.RBI1-M-Y % 19-1121 (Y %) MR.
[0386] For output slit SL.sub.LOM-X % 19-140M(X %) L at L.sub.I1-OM-X % 19-147M(X %) L, the output wavelength from this output slit for the beam from input slit SL.sub.I1 will be labelled as .sub.LI1-OM-S % 19-132M(X %) L, and .sub.LI1-OM-X %=.sub.LBI1-M-X %. Likewise, for output slit SL.sub.ROM-Y % 19-140M(Y %) R at .sub.RI1-OM-Y % 19-147M(Y %) R, the output wavelength from this output slit for the beam from input slit SL.sub.I1 will be labelled as .sub.RI1-OM-Y % 19-132M(Y %) R, and .sub.RI1-OM-Y %=.sub.RBI1-M-Y %.
[0387] For some applications that is designed to process a beam spectral width .sub.BI1-M 19-1121MSW (see
[0388] For some applications that is designed to process a beam spectral width .sub.BI1-M 19-1121MSW (see
[0389] In other applications that is designed to process a relatively wide spectral width (e.g spectral width .sub.I1-M 18-1121MSW with .sub.I1-M>0.1% of .sub.I1-M 18-1121M), it is typically advantages to place the anchor outputs at around 25% of the energy or more at both sides.
[0390] In as yet another applications that is designed to process a relatively wide spectral width (e.g spectral width .sub.I1-M 18-1121MSW with .sub.I1-M>1% of .sub.I1-M 18-1121M), it is typically advantages to place the anchor outputs at around 40% of the energy or more at both sides so that .sub.I1-O1A<.sub.I1-L40% and .sub.I1-O2A>.sub.I1-R40% (for the situation .sub.I1-O1A<.sub.I1-O2A and .sub.I1-L40%<.sub.I1-R40%; for .sub.I1-L40%>.sub.I1-R40% just exchange .sub.I1-L40% and .sub.I1-R40%) or .sub.I1-O2A<.sub.I1-L40% and .sub.I1-O1A>.sub.I1-R40% (for the situation .sub.I1-O2A<.sub.I1-O1A and .sub.I1-L40%<.sub.I1-R40%; for .sub.I1-L40%>.sub.I1-R40% just exchange .sub.I1-L40% and .sub.I1-R40%).
Plurality of Output Slits
[0391] As shown in
x.sub.Ok=S.sub.Ok*Sin(.sub.Ok)(40A)
and its y-coordinate:
y.sub.Ok=S.sub.Ok*Cos(.sub.Ok).(40B)
The width of the k.sup.th output slit being adjustable, and further the width the k.sup.th output slit specified by a k.sup.th output slit width W.sub.Ok 19-149kW.
Wavelengths and Angles of Output Slits
[0392] The output slit wavelength .sub.I1-Ok 19-132k is given by:
d*(Sin(.sub.Ok)+Sin(.sub.I1))=m*.sub.I1-Ok/n.sub.grI1-Ok,Eq. (41)
which will ensure that spectral energy in the input beam at output wavelength .sub.I1-Ok 19-132k will be diffracted by the grating to the k.sup.th output slit direction at output angle .sub.Ok 19-147k. As noted before, equation like Eq. (41) is an approximate form. Alternatively, the exact form below:
j0*([D.sub.1(.sub.I1,S.sub.I1,X.sub.j0)+D.sub.2(.sub.Ok,S.sub.Ok,X.sub.j0)][D.sub.1(.sub.I1S.sub.I1,X.sub.0)+D.sub.2(.sub.Ok,S.sub.Ok,X.sub.0)])=m*.sub.I1-Ok/n.sub.grI1-Ok(42)
where the X.sub.j0 is the grating groove position adjacent to X.sub.0 (i.e. j0=1 or 1). The left multiplication by j0 is just to ensure the sign on the left side of the equation comes out correct. Eq. (42) can be used to obtain wavelength .sub.I1-Ok 19-132k. Note that Eq. (42) is depending on two grating center grooves (X.sub.0 and X.sub.j0) whose positions are already solved (this is equivalent to knowing d in Eq. (41)) and .sub.Ok 19-147k, S.sub.Ok 19-146k are given by the position of the output slit k X.sub.Ok 19-149kOC. Hence the only unknown in Eq. (42) is .sub.I1-Ok 19-132k, which can then be solved numerically for the value that satisfies Eq. (42). As shown by the approximate form Eq.(41), .sub.I1-Ok 19-132k is basically determined by the output angle .sub.Ok 19-147k and is only weakly depending S.sub.Ok 19-146k, which is needed only when one uses the more exact form of Eq.(41). Thus, for every angle .sub.Ok 19-147k, there is an output wavelength .sub.I1-Ok 19-132k for that angle. Thus, one has a choice to either pick the wavelength of the output slit and find its angle or pick the angle of the output slit and obtain its wavelength later.
[0393] Below, we will discuss for an output slit SL.sub.Ok 19-140k with a given intended output wavelength .sub.I1-Ok 19-132k, how one would get its preferred location X.sub.Ok 19-149kOC.
Determining the Preferred Position for the Output Slit k
[0394]
[0395] Specifically, as shown in
[D.sub.1(.sub.O1,D.sub.I1,X.sub.i)+D.sub.2(.sub.L(i,i1)I1-OkP,S.sub.L(i,i1)I1-OkP,X.sub.i)][D.sub.1(.sub.I1,S.sub.I1,X.sub.i1)+D.sub.2(.sub.L(i,i1)I1-OkP,S.sub.L(i,i1)I1-OkP,X.sub.i1)]=m*.sub.I1-Ok/n.sub.grI1-Ok(43)
wherein D.sub.2(.sub.L(i,i1)I1-OkP,S.sub.L(i,i1)I1-OkP,X.sub.i) is the distance from X.sub.i 19-160|i|P/N to the point P 19-189kP on line L.sub.(i,i1)I1-Ok c(i,i1)L, D.sub.1(.sub.I1,S.sub.I1,X.sub.i) is the distance from X.sub.i 19-160|i|P/N to the first input slit at X.sub.I1. Line L.sub.(i,i1)I1-Ok 19-189k(i,i1)L is generated when S.sub.L(i,i1)I1-OkP 19-189kP(i,i1)S increases from an initial small value to a value larger than the estimated position of S.sub.Ok 19-146k which is typically close to the value of S.sub.O1A 19-1461A or S.sub.O2A 19-1462A that are already known. There are many ways to parameterize line L.sub.(i,i1)I1-Ok 19-189k(i,i1)L. The locus of points tracing out line L.sub.(i,i1)I1-Ok 19-189k(i,i1)L is independent on parameterization and is solely dependent on the above equation that completely defines the locus of points tracing out line L.sub.(i,i1)I1-Ok 19-189k(i,i1)L.
[0396] The second line, called Line L.sub.(j,j-1)I1-Ok 19-189k(j,j1)L being parameterized by an angle .sub.L(j,j-1)I1-OkQ 19-189kQ(j,j1)D and a distance S.sub.L(j,j-1)I1-OkQ 19-189kQ(j,j1)S. The angle .sub.L(j,j-1)I1-OkQ 19-189 kQ(j,j1)D is sustained between the line joining a point Q (along line L.sub.(j,j-1)I1-Ok 19-189k(j,j1)L) to the grating center CGC 19-1050 and the grating-center circle normal line L.sub.GCCN 19-1050N. The distance S.sub.L(j,j-1)I1-OkQ 19-189kQ(j,j1)S from the grating center CGC 19-1050 to the same point Q (along L.sub.(j,j-1)I1-Ok 19-189k(j,j1)L). Equivalently Line L.sub.(j,j-1)I1-Ok 19-189k(j,j1)L is parameterized by the coordinates X.sub.L(j,j-1)I1-OkQ=(x.sub.L(j,j-1)I1-OkQ, y.sub.L(j,j-1)I1-OkQ) 19-189kQ(i,i1)C where x.sub.L(j,j-1)I1-OkQ=S.sub.L(j,j-1)I1-OkQ*Sin(.sub.L(j,j-1)I1-OkQ) 19-189kQ(i,i1)Cx and y.sub.L(j,j-1)I1-OkQ=S.sub.L(j,j-1)I1-OkQ*Cos(.sub.L(j,j-1)I1-OkQ) 19-189kQ(i,i1)Cy, for which the following equation is satisfied based on the grating grooves of number j and (j1):
[D.sub.1(.sub.I1,S.sub.I1,X.sub.j)+D.sub.2(.sub.L(j,j-1)I1-OkQ,S.sub.L(j,j-1)I1-OkQ,X.sub.j)][D.sub.1(.sub.I1,S.sub.I1,X.sub.j-1)+D.sub.2(.sub.L(j,j-1)I1-OkQ,S.sub.L(j,j-1)I1-OkQ,X.sub.j-1)]=m*.sub.I1-Ok/n.sub.grI1-Ok(44)
wherein D.sub.2(.sub.L(j,j-1)I1-OkQ, S.sub.L(j,j-1)I1-OkQ, X.sub.j) is the distance from X.sub.j 19-160|j|P/N to a point Q with coordinate X.sub.L(j,j-1)I1-OkQ 19-189kQ(j,j1)C on line L.sub.(j,j-1)I1-Ok 19-189k(j,j1)L, D.sub.1(.sub.I1,S.sub.I1, X.sub.j) is the distance from X.sub.j 19-160|j|P/N to the first input slit at X.sub.I1 19-1201. The first line is generated when S.sub.L(j,j-1)I1-OkQ 19-189kQ(i,i1)S increases from an initial small value to a value larger than the estimated position of S.sub.Ok 19-146k, which is typically close to the value of S.sub.O1A 19-1461A or S.sub.O2A 19-1462A that are already known. There are many ways to parameterize line L.sub.(j,j-1)I1-Ok 19-189k(j,j1)L. The locus of points tracing out L.sub.(j,j-1)I1-Ok 19-189k(j,j1)L is independent on parameterization and is solely dependent on the above equation that completely defines the locus of points tracing out Line-j.
Determining the Preferred Location for the Output Slit k Via Averaging Intersecting Points.
[0397] As an exemplary embodiment, it is typically advantages to choose two pairs of grating groove, each from one side of the grating center. In that case, let the grating groove pairs (j,j1) for grating grove at X.sub.i and X.sub.j-1 be chosen to lie on the opposite side of the grating center from that of grating groove pair (i,i1) for grating grove at X.sub.i and X.sub.i1. For example if j1>0, then i<0 and if i1>0, then j<0, and i=0 gives the position at the grating center. The intersecting point between line L.sub.(i,i1)I1-Ok and line L.sub.(j,j-1)I1-Ok then gives the coordinate X.sub.Ok(i,i1;j,j-1). The intersecting point between line L.sub.(i,i1)I1-Ok 19-189k(i,i1)L and line L.sub.(j,j-1)I1-Ok 19-189k(j,j1)L then gives the coordinate .sub.I1-Ok(i,i1;j,j-1) 19-149kOC(i,i1;j,j1). Below, we will replace subscript I1-Ok by Ok in X.sub.I1-Ok(i,i1;j,j-1) so .sub.I1-Ok(i,i1;j,j-1)=X.sub.Ok(i,i1;j,j-1) for simplicity with the understanding that it is generated for beam from input slit I1. Likewise we will also do the replacement for other .sub.I1-OK variables. The location of X.sub.Ok 19-149kOC of output slit k that shall receive beam spectral component at wavelength .sub.I1-OK is then chosen to be a point either at, near, or very near the point X.sub.Okest 19-149kOCest, called the estimated output location, where the point X.sub.Okest 19-149kOCest is obtained by a function V=V({X.sub.Ok(i,i1;j,j-1)}) that is dependent on all the vectors X.sub.Ok(i,i1;j,j-1) 19-149kOC(i,i1;j,j1) generated by a selected set of the grating groove pairs with different values of i,i1 or j,j1, so that:
X.sub.Okest=V({X.sub.Ok(i,i1;j,j-1)})(45)
[0398] By near means the placement is within three times the beam diameter generated by the input beam at X.sub.Okest 19-149kOCest defined by the full-width half-maximum of the beam intensity width, or three times the width W.sub.Ok 19-149kW of the slit at X.sub.Ok 19-149kOC, whichever is larger. By very near means the placement is within half the beam diameter generated by the input beam at X.sub.Okest 19-149kOCest defined by the full-width half-maximum of the beam intensity width, or half the width W.sub.Ok 19-149kW of the slit at X.sub.Ok 19-149kOC, whichever is larger. By at means the placement is within 10% of the beam diameter generated by the input beam at X.sub.Okest 19-149kOCest defined by the full-width half-maximum of the beam intensity width, or 10% of the width W.sub.Ok 19-149kW of the slit at X.sub.Ok 19-149kOC, whichever is larger.
Determining the Preferred Location for the Output Slit k Via Averaging Intersecting Points from Rays Near Grating Center
[0399] In an exemplary embodiment, the estimated output slit location X.sub.Okest 19-149kOCest is given by one of X.sub.Ok(i,i1;j,j-1) 19-149kOC(i,i1;j,j1) in which the grooves i,j are close to the grating center CGC 19-1050 within the angle at plus and minus 30 from the grating center CGC 19-1050 for which the angle is measured with respect to the line joining the input slit to the grating center and pivoted at the input slit. That is let .sub.grI1-i 19-161|i|P/N be the angle made by two lines pivoted (or joint) at the input slit location: the line joining the input slit location X.sub.I1 19-1291OC to the groove location X.sub.i 19-160|i|P/N (called line L.sub.I1-i 19-162|i|P/N or input light ray to groove i) and the line joining the input slit location X.sub.I1 19-1291OC to the groove at grating center X.sub.0 19-1600O (called line L.sub.I1 19-1251 or grating-center to input-slit line). Then the above requires that .sub.grI1-i<30.
Determining the Preferred Location for the Output Slit k Via an Weighted Average of the Intersecting Points by Multiplying with Beam Power
[0400] In an exemplary embodiment of the averaging function V=V({X.sub.Ok(i,i1;j,j-1)}) shown in
X.sub.Okest=V({X.sub.Ok(i,i1;j,j-1)})=[SUM({i,i1;j,j1})([P.sub.I1(i,i1;j,j-1)].sup.NX.sub.Ok(i,i1;j,j-1))]/[SUM({i,i1;j,j1})(P.sub.I1(i,i1;j,j-1))].(46A)
where Sum({i,i1;j,j1}) denotes sum over the range of all the i,i1 and j,j1 pairs in the set {i,i1;j,j1} defined above, and N in Eq. (46A) is taking P to the power of N, where N is a positive real number larger than 0. In an exemplary embodiment, N=1. Alternatively, it can be multiplied by any function of the power:
X.sub.Okest=V({X.sub.Ok(i,i1;j,j-1)})=[Sum({i,i1;j,j1})(f[P.sub.I1(i,i1;j,j-1)]X.sub.Ok(i,i1;j,j-1))]/[Sum({i,i1;j,j1})(P.sub.I1(i,i1;j,j-1))].(46B)
where f[P.sub.I1(i,i1;j,j-1)] is any mathematical function of P.sub.I1(i1;j,j-1).
[0401] Note that one or both of the two anchor output slits may not be physically present. In that case, their positions are still designated but are only used for the purpose of generating the grating teeth, and are not to be used for forming physical output slit locations or output waveguide locations.
Determining the Preferred Location for the Output Slit k Via an Weighted Average of the Intersecting Points from Rays Originated Symmetrically from Grating Center
[0402] In an exemplary embodiment, the estimated output slit location X.sub.Okest 19-149kOCest is given by the function V=V({X.sub.Ok(i,i1;j,j-1)}) above with j,j1 set to the groove i,(i1) wherein groove i is opposite to groove i and (i1) is opposite to groove i1 with respect to the curved grating center CGC 19-1050. The sum is then taken over all pairs of grooves (i, i1) of the grating.
Generating Multiple Input Silts/Waveguides for Multiple Output and Input Slits/Waveguides Case
[0403] In another exemplary embodiment for the case with multiple input slits/waveguides beside multiple output slits/waveguides, input slit/waveguide SL.sub.I1 is first used to generate all the output slits using the method described above. Once the output slit/waveguide positions are determined using input slit/waveguide SL.sub.I1, one can then use one of the output slit/waveguide, including either one of the two anchor slits/waveguides acting as new input slit and SL.sub.I1 acting as one of the new anchor output slit and then designate another input slit location as the second new anchor output slit. The other plurality of multiple input slit locations are then generated just like the way we generate the multiple output slits described above.
Two Anchor Outputs Slits In-Line with Input Slit
[0404]
[0405] Specifically, as shown in
Two Anchor Outputs Slits on a Line Rotated from In-Line with Input Slit
[0406]
[0407] Specifically, as shown in
[0408] The midpoint of L.sub.(O1A,O2A) is denoted as X.sub.L(O1A,O2A)M 19-1400AM or referred to as anchor-Output-slits-midpoint. X.sub.L(O1A,O2A)M is also called X.sub.AOSM 19-1400AM. The line joining the input slit location X.sub.I1 to X.sub.L(O1A,O2A)M called line L.sub.(I1-(O1A,O2A)M) 19-1400IML (or as line L.sub.IM so L.sub.IM=L.sub.(I1-(O1A,O2A)M)) referred to as input to anchor-output-slits-midpoint line 19-1400IML.
[0409] The line joining the input slit to the grating center at X.sub.0 is called line L.sub.I1 19-1251 or grating-center to input-slit line. The line joining the grating center at X.sub.0 19-1600O to the anchor-Output-slits-midpoint X.sub.L(O1A,O2A)M is called line L.sub.(GC-(O1A,O2A)M)) 19-1400GML or grating-center to anchor-Output-slits-midpoint line (denoted as L.sub.GM=L.sub.(GC-(O1A,O2A)M)).
[0410] The angle between line L.sub.I1 plus 90 degrees (shown as Line A) and line L.sub.IM is .sub.IM 19-1470IMD, which takes on a value of 0 when line L.sub.I1 and line L.sub.IM are perpendicular to each other, and take on a positive value when line L.sub.IM is rotated about the input slit point X.sub.I1 19-1291OC in a direction to bring the point X.sub.AOSM=X.sub.L(O1A,O2A)M closer in its distance to the grating center.
[0411] The angle .sub.IM 19-1470IMD being adjustable for controlling a performance of the optical gratin spectrometer.
[0412] The angle between line L.sub.GM plus 90 degrees (shown as Line B) and line L.sub.IM is .sub.GM 19-1470GMD, which takes on a value of 0 when line L.sub.GM and line L.sub.IM are perpendicular to each other, and take on a positive value when line L.sub.IM is rotated about the input slit point X.sub.I1 19-1201 in a direction to bring the point A.sub.AOSM=X.sub.L(O1A,O2A)M closer in its distance to the grating center. Note that .sub.GM and .sub.IM are not independent and are geometrically related through or constrained by .sub.I1 19-1271, .sub.O1A 19-1471A, and .sub.O2A 19-1472A.
[0413] The angle made between line L.sub.IM and line L.sub.AOS is .sub.AM 19-1470AMD which takes on a zero value when X.sub.I1, X.sub.O1A, and X.sub.O2A are all in a straight line (i.e when line L.sub.IM and line L.sub.AOS are parallel to each other) and take on a positive value when line L.sub.AOS is rotated about its midpoint at A.sub.AOSM in a direction that brings the furthest end of line L.sub.AOS from the input slit point X.sub.I1 19-1291OC closer in its distance to the grating center at X.sub.0. The furthest end of line L.sub.AOS is the end point of L.sub.AOS that is furthest away from X.sub.I1.
[0414] The angle .sub.AM 19-1470AMD being adjustable for controlling a performance of the optical grating spectrometer device 1000.
Two Anchor Outputs Slits on a Line Rotated from In-Line with Input Slit by 45 Degree and Rotated from Perpendicular with Grating-Center to Input-Slit Line by 45 Degree
[0415] The larger the angles .sub.IM and .sub.AM are, typically the worse the aberration for the channels in-between the anchor slits. High grating resolution requires the input slit size to be small, which will result in a large beam divergence angle from the input slit to the grating and a large beam convergence angle from the grating to the output slit. If two output slits are closely spaced, and if one slit is has a further distance from the grating center than the other slit (we will call this as behind the other slit), then the convergence beam to the one behind may hit the mouth of the one in front, which means some energy will go into the slit in front. This result in adjacent channel cross talk as is known those skilled in the art. That is if we regard each slit as one wavelength channel. This can be avoided if all the output slits are about equal distance from the grating center. That means .sub.GM is around zero so the line joining the two anchor output slits is perpendicular to the line joining the grating center to the midpoint between the two anchor output slits. An angle for .sub.GM up to plus and minus 45 degrees may still be alright to reduce adjacent channel cross talks, depending on the distance between two adjacent output slits and the angle of convergence for the beam from the grating to the output slit.
[0416] The angle .sub.IM is another angle of measurement that has a value close to .sub.GM if the input slit angle is small (less than plus and minus 45 degrees). The angle .sub.IM is around zero when the line joining the two anchor output slits is perpendicular to the line joining the grating center to the input slits. When the input slit angle is relatively small (less than plus and minus 45 degrees), we can also impose similar requirement on IM as the above requirement for .sub.GM.
[0417] Thus, for many applications, it is preferred that the anchor output slits at X.sub.O1A 19-1401A and X.sub.O2A 19-1402A are located such that .sub.IM is within +45 and 45 and the anchor output slits at X.sub.O1A 19-1401A and X.sub.O2A 19-1402A are located such that .sub.AM is within +45 and 45.
Two Anchor Outputs Slits on a Line Rotated from In-Line with Input Slit by 45 Degree and Rotated from Perpendicular with Grating-Center to Anchor-Output-Slits-Midpoint Line by 45 Degree
[0418] The larger the angles .sub.GM and .sub.AM are, typically the worse the aberration for the channels in-between the anchor slits. High grating resolution requires the input slit size to be small, which will result in a large beam divergence angle from the input slit to the grating and a large beam convergence angle from the grating to the output slit. If two output slits are closely spaced, and if one slit is has a further distance from the grating center than the other slit (we will call this as behind the other slit), then the convergence beam to the one behind may hit the mouth of the one in front, which means some energy will go into the slit in front. This result in adjacent channel cross talk as is known those skilled in the art. That is if we regard each slit as one wavelength channel. This can be avoided if all the output slits are about equal distance from the grating center. That means .sub.GM is around zero so the line joining the two anchor output slits is perpendicular to the line joining the grating center to the midpoint between the two anchor output slits. An angle for .sub.GM up to plus and minus 45 degrees may still be alright to reduce adjacent channel cross talks, depending on the distance between two adjacent output slits and the angle of convergence for the beam from the grating to the output slit.
[0419] Thus, for many applications, it is preferred that the anchor output slits at X.sub.O1A 19-1401A and X.sub.O2A 19-1402A are located such that .sub.GM is within +45 and 45 and the anchor output slits at X.sub.O1A 19-1401A and X.sub.O2A 19-1402A are located such that .sub.AM is within +45 and 45.
Input Angle within 45 Degree
[0420] In another embodiment, the output slits of wherein the input angle .sub.I1 19-1271 is less than 45 and the location of at least one of the output slits is within the area bounded by the input circle IC 19-1080 with a radius R/2.
Two Anchor Outputs Slits on a Line Rotated from In-Line with Input Slit by 30 Degree and Rotated from Perpendicular with Grating-Center to Input-Slit Line by 30 Degree
[0421] The larger the angles .sub.IM and .sub.AM are, typically the worse the aberration for the channels in-between the anchor slits. An angle for .sub.IM up to plus and minus 30 degrees will further reduce adjacent channel cross talks (comparing to plus and minus 45 degrees), especially when the output beam convergence angle is large.
[0422] Thus, for some more stringent applications such as in certain optical networks, it is preferred that the anchor output slits at X.sub.O1A 19-1401A and X.sub.O2A 19-1402A are located such that .sub.IM is within +30 and 30 and the anchor output slits at X.sub.O1A 19-1401A and X.sub.O2A 19-1402A are located such that .sub.AM is within +30 and 30
Two Anchor Outputs Slits on a Line Rotated from In-Line with Input Slit by 30 Degree and Rotated from Perpendicular with Grating-Center to Anchor-Output-Slits-Midpoint Line by 30 Degree
[0423] The larger the angles .sub.GM and .sub.AM are, typically the worse the aberration for the channels in-between the anchor slits. An angle for .sub.GM up to plus and minus 30 degrees will further reduce adjacent channel cross talks (comparing to plus and minus 45 degrees), especially when the output beam convergence angle is large.
Thus, for some more stringent applications such as in certain optical networks, it is preferred that the anchor output slits at X.sub.O1A 19-1401A and X.sub.O2A 19-1402A are located such that .sub.GM is within +30 and 30 and the anchor output slits at X.sub.O1A 19-1401A and X.sub.O2A 19-1402A are located such that .sub.AM is within 30 and 30.
Two Anchor Outputs Slits on a Line Rotated from In-Line with Input Slit by 15 Degree and Rotated from Perpendicular with Grating-Center to Input-Slit Line by 15 Degree
[0424] The larger the angles .sub.IM and .sub.AM are, typically the worse the aberration for the channels in-between the anchor slits. An angle for .sub.IM up to plus and minus 15 degrees will further reduce adjacent channel cross talks (comparing to plus and minus 30 degrees), especially when the output beam convergence angle is large.
[0425] Thus, for some even more stringent applications such as in certain stringent optical networks, it is preferred that the anchor output slits at X.sub.O1A 19-1401A and X.sub.O2A 19-1402A are located such that .sub.IM is within +15 and 15 and the anchor output slits at X.sub.O1A 19-1401A and X.sub.O2A 19-1402A are located such that .sub.AM is within +15 and 15.
Two Anchor Outputs Slits on a Line Rotated from In-Line with Input Slit by 15 Degree and Rotated from Perpendicular with Grating-Center to Anchor-Output-Slits-Midpoint Line by 15 Degree
[0426] The larger the angles .sub.GM and .sub.AM are, typically the worse the aberration for the channels in-between the anchor slits. An angle for .sub.GM up to plus and minus 15 degrees will further reduce adjacent channel cross talks (comparing to plus and minus 30 degrees), especially when the output beam convergence angle is large.
[0427] Thus, for some even more stringent applications such as in certain stringent optical networks, it is preferred that the anchor output slits at X.sub.O1A 19-1401A and X.sub.O2A 19-1402A are located such that .sub.GM is within +15 and 15 and the anchor output slits at X.sub.O1A 19-1401A and X.sub.O2A 19-1402A are located such that .sub.AM is within 15 and 15.
Spanning Angle of the Grating Teeth
Smaller Input Slit Width Case
[0428]
[0429] Specifically, as shown in
[0430] Let the grating input left angular span be defined as .sub.grI1L 19-1651L where .sub.grI1L=|.sub.grI1-MP|, where i=MP is the maximum groove number to left side of the grating with i>0 and .sub.grI1-MP 18-161|MP|P is its input-groove angle (also called input-ray angle at groove MP). Let the grating input right angular span be defined as .sub.grI1R 19-1651R where .sub.grI1R=|.sub.grI1-MN|, j=MN is the maximum groove number to the right side of the grating with j<0 and .sub.grI1-MN 19-161|MN|N is its input-groove angle (also called input-ray angle at groove MN). Then the grating input total angular span .sub.grI1 19-1651 is defined as .sub.grI1=.sub.grI1L+.sub.grI1R=|.sub.grI1-MP|+|.sub.grI1-MN|.
The grating is to be designed with a large enough angular span .sub.grI1 with .sub.grI1>.sub.dvdf-BI1-95%, where larger than .sub.dvdf-BI1-95% 19-1141 means that over 95% of the beam energy from the input beam is intercepted by the grating and all the rays from the input slit, after reflecting from the grating, will converge to the first anchor output slit at around a single point at wavelength .sub.I1-O1A and also converge to the second anchor output slit at around a single point at wavelength .sub.I1-O2A, thereby resulting in minimal beam focusing aberrations at the two anchor output slits. Otherwise if .sub.grI1<.sub.dvdf-BI1-95%, then the beam after reflecting from the grating will have not just power loss but smaller beam converging angle to the output slits, which means it will not be able to focus to as small a spot size than if the beam converging angle is larger. This will result a loss in spectral resolution for the optical grating spectrometer as well.
Medium Input Slit Width Case
[0431] As shown in
The grating is to be designed with a large enough angular span .sub.grI1 with .sub.grI1>.sub.dvdf-BI1-90%, where larger than .sub.dvdf-BI1-90% 19-1141 means that over 90% of the beam energy from the input beam is intercepted by the grating and all the rays from the input slit, after reflecting from the grating, will converge to the first anchor output slit at around a single point at wavelength .sub.I1-O1A and also converge to the second anchor output slit at around a single point at wavelength .sub.I1-O2A, thereby resulting in minimal beam focusing aberrations at the two anchor output slits. Otherwise if .sub.grI1<.sub.dvdf-BI1-90%, then the beam after reflecting from the grating will have not just power loss but smaller beam converging angle to the output slits, which means it will not be able to focus to as small a spot size than if the beam converging angle is larger. This will result a loss in spectral resolution for the optical grating spectrometer as well.
Large Input Slit Width Case
[0432] As shown in
The grating is to be designed with a large enough angular span grI1 with .sub.grI1>.sub.dvdf-BI1-80%, where larger than .sub.dvdf-BI1-80% 19-1141 means that over 80% of the beam energy from the input beam is intercepted by the grating and all the rays from the input slit, after reflecting from the grating, will converge to the first anchor output slit at around a single point at wavelength .sub.I1-O1A and also converge to the second anchor output slit at around a single point at wavelength .sub.I1-O2A, thereby resulting in minimal beam focusing aberrations at the two anchor output slits. Otherwise if .sub.grI1<.sub.dvdf-BI1-80%, then the beam after reflecting from the grating will have not just power loss but smaller beam converging angle to the output slits, which means it will not be able to focus to as small a spot size than if the beam converging angle is larger. This will result a loss in spectral resolution for the optical grating spectrometer as well.
Very Large Input Slit Width Case
[0433] As shown in
The grating is to be designed with a large enough angular span grI1 with .sub.grI1>.sub.dvdf-BI1-70%, where larger than .sub.dvdf-BI1-70% 19-1141 means that over 70% of the beam energy from the input beam is intercepted by the grating and all the rays from the input slit, after reflecting from the grating, will converge to the first anchor output slit at around a single point at wavelength .sub.I1-O1A and also converge to the second anchor output slit at around a single point at wavelength .sub.I1-O2A, thereby resulting in minimal beam focusing aberrations at the two anchor output slits. Otherwise if .sub.grI1<.sub.dvdf-BI1-70%, then the beam after reflecting from the grating will have not just power loss but smaller beam converging angle to the output slits, which means it will not be able to focus to as small a spot size than if the beam converging angle is larger. This will result a loss in spectral resolution for the optical grating spectrometer as well.
Grating Angular Span and Output Slit Width
[0434]
[0435] Specifically, as shown in
[0436] Let the output beam's full convergence focusing angle at output slit be .sub.cvfo-BI1-Ok-50% 19-134k for a beam entering the output slit k with slit width W.sub.Ok 19-149kW from an input beam reflected and diffracted from the grating, defined by the angle between the two lines traced out by the beam intensity points on both sides of the beam that each encompasses 25% of the power of the beam from the beam's intensity peak to each of the two intensity points.
[0437] The grating is to be designed with a large enough angular span .sub.grI1 with .sub.grI1>.sub.dvdf-BI1-90%, where larger than .sub.dvdf-BI1-90% 19-1141 means that over 45% of the beam energy from the input beam is intercepted by the grating and all the rays from the input slit, after reflecting from the grating, will converge to the first anchor output slit at around a single point at .sub.I1-O1A and also converge to the second anchor output slit at around a single point at wavelengths .sub.I1-O2A, thereby resulting in minimal beam focusing aberrations at the two anchor output slits. This will enable high spectral resolution.
[0438] Furthermore the output slit width W.sub.Ok are designed so that .sub.dvdf-BO1-90% 19-1541 is about equal to .sub.cvfo-BI1-Ok-90% 19-134k and differs from .sub.cvfo-BI1-Ok-90% 19-134k by no more than plus and minus 50% of the value of .sub.cvfo-BI1-Ok-00% 19-134k in one aspect of the embodiment to achieve high spectral resolution and low optical loss for the output beam.
[0439] In another aspect of the embodiment, the grating input total angular span .sub.grI1 19-1651 and the output slit width W.sub.Ok are designed so that .sub.dvdf-BO1-50% 19-1541 is about equal to .sub.cvfo-BI1-Ok-50% 19-134k and differs from .sub.cvfo-BI1-Ok-50% 19-134k by no more than plus and minus 25% of the value of .sub.cvfo-BI1-Ok-50% 19-134k to achieve high spectral resolution and medium-low optical loss for the output beam.
[0440] In as yet another aspect of the embodiment, the grating input total angular span .sub.grI1 19-1651 and the output slit width W.sub.Ok are designed so that .sub.dvdf-BO1-50% 19-1541 is about equal to .sub.cvfo-BI1-Ok-50% 19-134k and differs from .sub.cvfo-BI1-Ok-50% 19-134k by no more than plus and minus 10% of the value of .sub.cvfo-BI1-Ok-50% 19-134k to achieve high spectral resolution and ultra-low optical loss for the output beam.
Output Slit Width
[0441]
[0442] Specifically, as shown in
W.sub.Okest=W.sub.I1*S.sub.Okest/S.sub.ROk(47)
and then W.sub.Ok 19-149 kW shall differs W.sub.Okest 19-149kestW by no more than plus and minus 10% of the value of W.sub.Okest 19-149kestW. In the above Eq. 47, S.sub.Okest 19-146kest is the distance from the grating center X.sub.0 to X.sub.Ok 19-149kOC along the angle .sub.Ok 19-147k for the case of W.sub.Okest 19-149kWest, and S.sub.ROk 19-146kR is the distance from the grating center X.sub.0 to the input circle IC 19-1080 or the Rowland circle with a radius R/2 along the angle .sub.Ok 19-1471 (for the case of W.sub.Ok 19-149kW), or along the angle .sub.O1A (for the case of W.sub.O1A), or along the angle .sub.O2A (for the case of W.sub.O2A).
[0443] W.sub.Ok can also be directly given approximately by having its value differ by no more than 10% of the value W.sub.Ok given below:
W.sub.Ok=W.sub.I1*S.sub.Ok/S.sub.ROk(48)
Two Anchor Outputs with Multiple Output Waveguides Designed to Reduce Adjacent Channel Cross Talks
[0444]
[0445] Specifically, the design of the grating, input slits placements and output slits placements are only part of the requirement to obtain high adjacent channel extinction ratio. The way the multiple waveguides at the output slit locations are placed and channel out and the absorption of unwanted back scattered light from other photonic device components is also important.
[0446] It also shows multiple outputs with output waveguides WG.sub.O1 20-1901, WG.sub.O2 20-1902, . . . WG.sub.Ok 20-190k. The corresponding waveguide mouth, for example for output waveguide WG.sub.Ok 20-190k is MSL.sub.Ok 20-140kM and the physical width of the waveguide mouth is waveguide mouth width MW.sub.Ok 20-149kMW usually defined by the width of its waveguide core. The coordinate location of the middle of the output waveguide mouth MSL.sub.Ok 20-140kM is MX.sub.Ok 20-149kMOC.
[0447] Further, an input and output waveguide at close to the mouth can take on shape of constant width or can be tapering in width with linear shape or an arbitrary curvilinear shape as shown in
[0448] For the input waveguide mouth at input 1, the tapering region is referred to as region TWG.sub.I1 20-1701T. The tapered input waveguide can be characterized by a virtual beam waist width given by TWW.sub.BI1-IP % 20-1181TWW, and virtual beam waist location given by TWX.sub.I1 20-1291TWOC (see (iii) in
[0449] For the output waveguide mouth at output k, the tapering region referred to as region TWG.sub.Ok 20-190kT. The tapered output waveguide can be characterized by a virtual beam waist width given by TWW.sub.BOk-IP % 20-158kTWW, and virtual beam waist location given by TWX.sub.Ok 20-149kTWOC (see (iv) in
[0450] The tapering mouth region may taper the waveguide width in a linear fashion (linear shape), or parabolic shape, or arbitrary curvilinear shape.
[0451] Let the cutoff waveguide width that supports only the up to mode v as is known to those skilled in the art be width WGW.sub.Okv 20-199kv for waveguide WG.sub.Ok 20-190k. In a preferred embodiment as illustrated by device 20-1000, the output waveguide has a tapering mouth region that rapidly tapered from the entrance mouth width MW.sub.Ok 20-149kMW to near or smaller than the waveguide width WGW.sub.Ok0 20-199k0 that supports only the fundamental mode referred to as mode-0 (i.e. with v=0) or more precisely the waveguide width that cutoff the propagation of mode 1. Near means within 50% of the value of WGW.sub.Ok0 20-199k0. This tapering region reduces the wave coupling between adjacent waveguides by separating the distance between the waveguides and hence reduces the adjacent channel crosstalk or increases the adjacent channel extinction. This is then followed by an optional section of straight waveguide SWG.sub.Ok 20-190kS with a waveguide width SWGW.sub.Ok 20-190kSW. Thus, this straight waveguide can have zero length (if the waveguide is absent) or finite length. In an embodiment width SWGW.sub.Ok 20-190kSW is near or smaller than the fundamental mode width WGW.sub.Ok0 20-199k0. After that is a section of bending region called region with waveguide B1WG.sub.Ok 20-190kB1 and fanning out region called region with waveguide FWG.sub.Ok 20-190kF in which the waveguides are further separated from each other in a radial-like fashion by bending slightly and then fan out radially as illustrated (see
[0452] After that is another bending of the waveguide called bending waveguide B2WG.sub.Ok 20-190kB2 so that the fanning out waveguide is joined to a section of parallel propagating waveguide called P1WG.sub.Ok 20-196k. In an exemplary embodiment, as an option, the locations where the fanning out waveguides end and begin to bend as bending waveguides B2WG.sub.Ok 20-190kB2 form an approximate circle called fanning out waveguide circle FWGC 20-1090 as shown in
[0453] At the parallel propagating waveguide P1WG.sub.Ok 20-196k region, the waveguides, after being separated by a distance P1WGD.sub.Ok(k+1) 20-196k(k+1)D (between waveguide k and waveguide k+1), is propagated almost parallel to each other. In this region, to reduce loss, the waveguide width P1WGW.sub.Ok 20-196kW is tapered out to larger than the fundamental mode width WGW.sub.Ok0 20-199k0 via a tapering region T1P1WG.sub.Ok 20-196kT1. Close to the end of P1WG.sub.Ok 20-196k, the waveguide is tapered back to near or smaller than the fundamental mode width WGW.sub.Ok0 20-199k0 via another tapering region T2P1WG.sub.Ok 20-196kT2. After that is a section of waveguide bending region called region B2P1WG.sub.Ok0 20-196kB1. In an embodiment, in this region, each waveguide undergoes a substantial bending such as close to a 90 bend. In other embodiment, the bending may be less substantial.
[0454] The above B2WG.sub.Ok 20-190kB2 waveguide has a waveguide width B2WGW.sub.Ok 20-190kB2W. In an embodiment width B2WGW.sub.Ok 20-190kB2W is near or smaller than the fundamental mode width WGW.sub.Ok0 20-199k0. B1P1WG.sub.Ok 20-196kB1 has a waveguide width B1P1WGW.sub.Ok 20-196kB1W. In an embodiment width B1P1WGW.sub.Ok 20-196kB1W is near or smaller than the fundamental mode width WGW.sub.Ok0 20-199k0.
[0455] After that is another section of parallel propagating waveguide called P2WG.sub.Ok 20-197k at which the waveguides, after being separated by a distance (between waveguide k and waveguide k+1) P2WGD.sub.Ok(k+1) 20-197k(k+1)D, is propagated almost parallel to each other. In this region, to reduce loss, the waveguide width P2WGW.sub.Ok 20-197kW is tapered out to larger than the fundamental mode width WGW.sub.Ok0 20-199k0 via a tapering region T1P2WG.sub.Ok 20-197kT1. Close to the end of P2WG.sub.Ok 20-197k, optionally the waveguide may be tapered back to near the fundamental mode width WGW.sub.Ok0 20-199k0 via another tapering waveguide T2P2WG.sub.Ok 20-197kT2. Thus, this tapering waveguide can have zero length (if the waveguide is absent) or finite length.
[0456] The substantial bending at B1WG.sub.Ok 20-190kB, B2WG.sub.Ok 20-190kB2, and B1P1WG.sub.Ok0 20-196kB1 regions can help to shred what is called the higher order modes such as the first order mode, which will make the output spectrum more pure such as reducing adjacent channel cross talk and kinks in the output spectrum. Thus the bending part serves as a mode filter for higher order mode and passes mainly the fundamental mode.
[0457] Optionally, the input waveguide(s) may use the same waveguide tapering, bending, or fanning out scheme, similar to that described above for the output waveguides as illustrated in
Two Anchor Outputs with Multiple Output Waveguides and Absorber or Structures to Reduce Light Reflection into Output Waveguides
[0458]
[0459] Specifically, in order to reduce back scattering for wave that propagates to outside the grating planar-waveguiding region GPR 20-1020, as shown by
[0460] Example of wave dissipating structure include tapered structures that send the wave into the region above the planar waveguide occupied by certain materials (e.g dielectric material or air) or the region below the planar waveguide occupied by certain materials (e.g. dielectric material or the substrate). An example of wave dissipating structure is simply via zigzagging the edges of the GPR 20-1020 region. An example of the zigzagging is a teeth-like structure with teeth spacing (called dissipating structure teach spacing DSTS 20-1981TS) and teeth length (called dissipating structure teach length DSTL 20-1981TL). In an exemplary embodiment DSTS=50 and DSTL=100 nm. In general, DSTS is smaller than an optical wavelength in the material, and DSTL is larger than 0.25 of an optical wavelength in the material. The zigzagging can also take on other curvilinear shapes such as square shape, sinusoidal shape, triangular shape etc as long as it dissipate additional optical energy that propagates towards it as shown by beam BP 1101P in
[0461] In summary, tapering region TWG.sub.Ok 20-190kT and fanning-out region FWG.sub.Ok 20-190kF both act to decouple from adjacent waveguide and can be called waveguide mode decoupling region, which will increase adjacent channel extinction. Next region BWG.sub.Ok 20-190kB and region B2WG.sub.Ok 20-190kB2 both involving waveguide bending, will help to shred higher-order modes and can be called waveguide mode filter region, which will help to reduce side modes in the output spectrum and increase the adjacent channel extinction as well. In alternative embodiments, one or more of the four regions TWG.sub.O1 20-1901T, FWG.sub.O1 20-1901F, BWG.sub.O1 20-1901B, and region B2WG.sub.O1 20-1901B2, may be used alone or in combinations to reach the purposes discussed above. Thus, they do not all have to be used together or in the sequences discussed above though what is discussed above would be the preferred embodiment.
Another Description of Grating Generating Method
[0462] As an exemplary embodiment, the wavelength multiplexer/demultiplexer/spectrometer or compact curved grating spectrometer using discrete optical components or with integration possibility as a wavelength dispersion element in a photonic integrated circuit, enabling dispersion of light spectra around a wavelength .sub.BI1. The wavelength multiplexer/demultiplexer/spectrometer comprising:
[0463] at least one input slit;
[0464] a plurality of output slits; and
[0465] a curved grating, the curved grating configured for processing the spectra compositions of the optical beam including a plurality of grooves, the position of each groove being adjustable for controlling a performance of the wavelength multiplexer/demutiplexer/spectrometer, and the position of the input slit and each of the output slits being adjustable for controlling a performance of the wavelength multiplexer/demutiplexer/spectrometer,
[0466] wherein the input slit allows an entry of the optical beam into the wavelength multiplexer/demutiplexer/spectrometer, a location of the input slit being adjustable, and further the location of the input slit X.sub.I1 specified by a first input angle .sub.I1 that is sustained between the line joining the input slit to the grating center and a normal line to the grating center, and a first input distance S.sub.I1 from the grating center to the input slit.
[0467] further wherein a first output slit for allowing the exiting of a first output optical beam having a first anchor output wavelength .sub.I1-O1A, a location of the first anchor output slit being adjustable, and further the location of the first anchor output slit specified by a first output angle .sub.O1A that is sustained between the line joining the first output slit to the grating center and a normal line to the grating center, and a first output distance S.sub.O1A from the grating center to the first anchor output slit,
[0468] further wherein a medium in which the light propagates in having an effective refractive index of propagation n.sub.gr. In the case of free space, n.sub.gr is the material refractive index. In the case of a planar waveguide, n.sub.gr is the effective refractive index of propagation within the planar waveguide,
[0469] further wherein a position of the i.sup.th groove is specified by its x-y coordinates X.sub.i=(x.sub.i, y.sub.i), the x-y coordinates are specified with respect to the grating center and the input slit, for which the grating center has the coordinate X.sub.0=(0, 0) and the input slit has the coordinate X.sub.I1=(S.sub.I1*Sin(.sub.I1), S.sub.I1*Cos(.sub.I1)).
[0470] With the given value of the input circle radius R where R is related to the input slit position by S.sub.I1=R*Cos(.sub.I1), around the grating center at X.sub.0=(0, 0), two initial grating teeth are chosen to be located at a distance d apart from each other so that they are placed at locations:
X.sub.1=(d/2,R(R.sup.2(d/2).sup.2).sup.1/2)(49A)
and
X.sub.1=(d/2,R(R.sup.2(d/2).sup.2).sup.1/2)(49B)
where when given the first anchor output wavelength .sub.I1-O1A, the distance d is to be determined as follows:
[0471] A grating order is chosen and denoted by an integer m. Then the grating parameter d is obtained approximately from
d*Sin(.sub.O1A)+Sin(.sub.I1))=m*.sub.I1-O1A/n.sub.gr,(49C)
[0472] further wherein the locations of all other grooves are given by computing the coordinate of each groove with the i.sup.th groove's coordinate X.sub.i given by the following two conditions:
The first condition is
Sgn(iJa)*([D.sub.1(.sub.I1,S.sub.I1,X.sub.i)+D.sub.2(.sup.O1A,S.sub.O1A,X.sub.i)][D.sub.1(.sub.I1,S.sub.I1,X.sub.ja)+D.sub.2(.sub.O1A,S.sub.O1A,X.sub.ja)])=m*.sub.I1-O1A/n.sub.gr, (50)
wherein D.sub.1(.sub.I1,S.sub.I1, X.sub.i) is the distance from X.sub.i to the first input slit location X.sub.I1 specified by .sub.I1 and S.sub.I1, D.sub.2(.sub.O1A,S.sub.O1A,X.sub.i) is the distance from X.sub.i to the first anchor output slit location specified by .sub.O1A and S.sub.O1A. The position of groove ja, X.sub.ja is typically already known. For an illustration and not limitation, if the grooves close to the grating center are already known, then groove ja is taken to be a groove adjacent to groove i so that X.sub.ja=X.sub.i1 for i>0 (so ja=+|i1|=i1 is the previous groove close to i=0 that is already solved) and X.sub.ja=X.sub.1+1 for i<0 (so ja=|i1|=i+1 is the previous groove close to i=0 that is already solved). Sgn(ija) takes on value +1 or 1. Sgn(ija) is +1 if i>ja, and 1 if i<ja. The second condition is such that a function f is equal to a numerical constant, functionally expressed as:
f(X.sub.i)=constant(51)
where the above constant can be depending on other design parameters such as the input slit and output slit positions or the positions of the adjacent grooves (e.g. .sub.I1,S.sub.I1,.sub.O1,S.sub.O1, .sub.I1-O1, m, n.sub.gr, {X.sub.i}) that are already known and hence can be treated as part of the constant. The positions {X.sub.i} represent the positions of some grating teeth that are already known.
[0473] The unknown variables in both equations Eq. (50) and Eq. (51) are x- and y-coordinates of the location vector X.sub.i of the i-th groove. For a given input-slit (or input-waveguide) location (.sub.I1, S.sub.I1), output slit (or waveguide or photodetector) location (.sub.O1, S.sub.O1), and the previous, i.e., ja-th, groove position X.sub.ja, X.sub.i is completely specified by equations Eq. (50) and Eq. (51) for a given wavelength .sub.I1-O1 to output slit SL.sub.O1, effective refractive index of propagation n.sub.gr, and the diffraction order m,
[0474] wherein the second constraint is further given by choosing the function f so that:
[D.sub.1(.sub.I1,S.sub.I1,X.sub.i)+D.sub.3(.sub.O2A,S.sub.O2A,X.sub.i)][D.sub.1(.sub.I1,S.sub.I1,X.sub.i1)+D.sub.3(.sub.O2A,S.sub.O2A,X.sub.i1)]=m*.sub.I1-O2A/n.sub.gr,(52)
[0475] wherein D.sub.3(.sub.O2A,S.sub.O2A,X.sub.i) is the distance from the i-th groove located at X.sub.i to the second anchor output slit specified by a third angle .sub.O2A that is sustained between the line joining the second output slit to the grating center and a normal line of the grating center, and a second output distance S.sub.O2A from the grating center to the second output slit, wavelength .sub.I1-O2A is a second wavelength that is the wavelength for the second output slit given by:
d*Sin(.sub.O2A)+Sin(.sub.I1))=m*.sub.I1-O2A/n.sub.gr,(53)
and by solving (50) and (52) for the x-coordinate x.sub.i and y-coordinate y.sub.i of the i.sup.th groove at X.sub.i=(x.sub.i, y.sub.i), exact locations of other grooves X.sub.i's are obtained.
[0476] In another embodiment, Eq. (49C): d*Sin(.sub.O1A)+Sin(.sub.I1))=m*.sub.I1-O1A/n.sub.gr, is replaced by the more accurate:
[D.sub.1(.sub.I1,S.sub.I1,X.sub.1)+D.sub.2(.sub.O1A,S.sub.O1A,X.sub.1)][D.sub.1(.sub.I1,S.sub.I1,X.sub.1)+D.sub.2(.sub.O1A,S.sub.O1A,X.sub.1)]=m*.sub.I1-O1A/n.sub.grI1-O1A (54)
further, Eq. (53): d*Sin(.sub.O2A)+Sin(.sub.I1))=m*.sub.I1-O2A/n.sub.gr, is replaced by the more accurate:
[D.sub.1(.sub.I1,S.sub.I1,X.sub.1)+D.sub.2(.sub.O2A,S.sub.O2A,X.sub.1)][D.sub.1(.sub.I1,S.sub.I1,X.sub.1)+D.sub.2(.sub.O2A,S.sub.O2A,X.sub.1)]=m*.sub.I1-O2A/n.sub.grI1-O2A (55)
Note the locations of X.sub.1 and X.sub.1 are dependent on and varies with d as specified by Eqs. (49A) and (49B).
Independence on Geometry Generating Method and Design Tolerances
[0477] As is known to those skilled in the art, the grating performances are depending on the collected results of diffraction and wave interference from the majority of the grating grooves. They are not depending on just a few grating grooves. They are also not too sensitive to the grating grooves being moved spatially by an amount S less than about an optical wavelength in the material given by .sub.I1-O1A/(n.sub.grI1-O1A) for output slit SL.sub.O1A for example, where
S=(x.sup.2+y.sup.2).sup.0.5,(56)
with x being the spatial deviation of grating groove position in another grating design from the designed position (in accordance with an embodiment of this invention) in the x-coordinate and y being the spatial deviation of grating groove position in another grating design from the designed position (in accordance with an embodiment of this invention) in the y-coordinate. If the design of a grating groove position in accordance with an embodiment of this invention is X.sub.jDn=(x.sub.jDn, y.sub.jDn) and another design or implementation or realization of the grating groove is at X.sub.jIm=(x.sub.jIm, y.sub.jIm), then x=|x.sub.jDnx.sub.jIm| and y=|y.sub.jDny.sub.jIm|. Moreover, two gratings or grating designs or grating implementations or grating realizations can achieve similar output spectral filtering performances for about half or more than half of the filtered spectrum if at least for a collection of grating grooves that are involved in reflecting more than 50% of the total power reflected by the grating towards the same output slit location, they are similar in their groove positions to each other in both gratings. Similar grating groove position means:
S<.sub.I1-O1A/(n.sub.grI1-O1A).(57)
[0478] While the steps above is a method of generating the set of positions for all the grating grooves in accordance with an embodiment of the present invention, there are other methods that could generate the same set of positions for all the grating grooves.
[0479] Thus, the grating performances will be similar as long as for this collection of the grating grooves (that are involved in reflecting more than 50% of the total power reflected by the grating towards the same output slit location), the deviation denoted by S of each grating groove position from the designed values is less than about half of the wavelength in the material so that S<.sub.I1-O1A/(n.sub.grI1-O1A).
[0480] Obviously, smaller deviation (e.g. S<.sub.I1-O1A/(2*n.sub.grI1-O1A) or S<.sub.I1-O1A/(10*n.sub.grI1-O1A) or a larger set of grooves involved (e.g. the set of grooves involve in over 70% of the grating total power reflection instead of 50%, or the set of grooves involve in over 90% of the grating total power reflection instead of 50%) will ensure even closer performances to the desired design. These allowed deviations (e.g. a set of grooves involved in reflecting more than 50% of the total power reflected by the grating towards the same output slit location has each of its groove's S satisfying S<.sub.I1-O1A/(n.sub.grI1-O1A)) describe the maximum deviations allowed. When two gratings meet such conditions, we will consider them to be the same design within the allowances of design variations for the purpose of this invention. The minimal of which is given by same-design-condition (A): two gratings are considered basically the same design if a set of grooves involved in reflecting more than 50% of the total power reflected by the grating towards the same output slit location has each of its groove's S satisfying S<.sub.I1-O1A/(n.sub.grI1-O1A); a tighter one is given by same-design-condition (B): two gratings are considered essentially the same design if a set of grooves involved in reflecting more than 50% of the total power reflected by the grating towards the same output slit location has each of its groove's S satisfying S<.sub.I1-O1A/(2*n.sub.grI1-O1A); (C): two gratings are considered highly the same design if a set of grooves involved in reflecting more than 50% of the total power reflected by the grating towards the same output slit location has each of its groove's S satisfying S<.sub.I1-O1A/(4*n.sub.grI1-O1A); another tighter one is given by same-design-condition (D): two gratings are considered strongly the same design if a set of grooves involve in over 70% if a set of grooves involved in reflecting more than 50% of the total power reflected by the grating towards the same output slit location has each of its groove's S satisfying S<.sub.I1-O1A/(10*n.sub.grI1-O1A). As yet another tighter one is given by same-design-condition (E): two gratings are considered very strongly the same design if a set of grooves involved in reflecting more than 90% of the total power reflected by the grating towards the same output slit location has each of its groove's S satisfying S<.sub.I1-O1A/(10*n.sub.grI1-O1A). The applicability of which is depending on grating applications. For example, for the usual spectral analysis application, same-design-conditions (A) and (B) is applicable, for the DWDM (dense wavelength division multiplexing) wavelength channel filtering applications in fiber-optic communications, same-design-conditions (C), (D), and (E) are applicable.