INTERFEROMETER AND METHOD OF DESIGNING AN INTERFEROMETER

Abstract

A universal interferometer (100) for coupling modes of electromagnetic radiation according to a transformation has N inputs and N outputs for inputting and outputting N modes of electromagnetic radiation into and from the interferometer. Waveguides (101, 102, 103, 104, 105) pass through the interferometer to connect the N inputs to the N outputs and to carry the N modes of electromagnetic radiation. The waveguides provide crossing points between pairs of waveguides and a reconfigurable beam splitter (107) implements a reconfigurable reflectivity and a reconfigurable phase shift at each crossing point. The waveguides and crossing points are arranged such that each of the N modes of electromagnetic radiation is capable of coupling with each of the other modes of electromagnetic radiation at respective reconfigurable beam splitters. The couplings between modes at the reconfigurable beam splitters are configured such that the interferometer implements a transformation of the N modes between the N inputs and the N outputs.

Claims

1. A method of designing an interferometer for coupling a plurality of modes of electromagnetic radiation, the method comprising: for an interferometer comprising: N inputs for inputting N modes of electromagnetic radiation into the interferometer, N outputs for outputting N modes of electromagnetic radiation from the interferometer, and a plurality of waveguides arranged to pass through the interferometer to connect the N inputs to the N outputs and for carrying the N modes of electromagnetic radiation through the interferometer, wherein: N is a natural number, the plurality of waveguides are arranged to provide a plurality of crossing points between pairs of the plurality of waveguides such that at each crossing point the two modes of electromagnetic radiation carried by the two respective waveguides are capable of coupling with each other, wherein the plurality of waveguides and the plurality of crossing points are arranged such that each of the N modes of electromagnetic radiation is capable of coupling with each of the other modes of electromagnetic radiation at respective crossing points, and the plurality of waveguides are arranged such that the plurality of crossing points are arranged into N groups along the plurality of waveguides from the inputs to the outputs through the interferometer, wherein each group contains the maximum number of possible crossing points between pairs of waveguides, and wherein the crossing points in each group involve pairs of adjacent waveguides whose paths were not crossed in the previous group of crossing points: receiving a unitary matrix describing a desired transformation to be performed by the interferometer; operating on the unitary matrix with a plurality of transformation matrices to decompose the unitary matrix into a diagonal matrix, wherein the transformation matrices used to decompose the unitary matrix each represent the coupling between a pair of modes at a crossing point of a pair of waveguides, and wherein the transformation matrices are arranged to operate on the unitary matrix in an order that matches a sequence in which the crossing points in the interferometer may be arranged; determining, using the transformation matrices, the coupling for the pair of modes at each of the plurality of crossing points for use in designing the interferometer.

2. A method as claimed in claim 1, wherein the modes of electromagnetic radiation each have a wavelength of between 700 nm and 1600 nm.

3. A method as claimed in claim 1, wherein N is greater than 3.

4. A method as claimed in claim 1, wherein each of the transformation matrices comprises one or more elements that are representative of a reflectivity and a phase shift of the coupling between the pair of modes at the respective crossing point, and the method comprises: determining a reflectivity coefficient and a phase shift coefficient from each of the transformation matrices; and determining, using the reflectivity coefficient and the phase shift coefficient, the reflectivity and phase shift for the coupling of each of the pairs of modes at the respective crossing points for use in designing the interferometer.

5. A method as claimed in claim 4, wherein a beam splitter is arranged at each of one or more of the crossing points, and the method comprises configuring each beam splitter to couple the respective pair of modes according to the determined reflectivity coefficient and the determined phase shift coefficient at each of the plurality of crossing points.

6. A method as claimed in claim 5, wherein each of the beam splitters is adjustable.

7. A method as claimed in claim 1, comprising defining a unitary matrix describing the desired transformation to be performed by the interferometer.

8. A method as claimed in claim 1, wherein the unitary matrix is defined by an NN unitary matrix that describes the transformation of the annihilation operators of the N modes of the interferometer that the unitary matrix represents, wherein the annihilation operators and the unitary matrix satisfy the equation: .sub.out=.Math..sub.in, where .sub.out=(.sub.out,1, .sub.out,2, . . . , .sub.out,N) and .sub.in=(.sub.in,1, .sub.in,2, . . . , .sub.in,N) are column vectors representing the annihilation operators of all the N input modes and all the N output modes respectively.

9. A method as claimed in claim 1, wherein each transformation matrix is defined as an NN matrix T.sub.m,n(,), which is identity except for the (m,m), (m,n), (n,m) and (n,n) elements, that form a 22 sub-matrix that performs the transformation: $\left[\begin{array}{c}{\hat{a}}_{m,\mathrm{out}}\\ {\hat{a}}_{n,\mathrm{out}}\end{array}\right]=\left[\begin{array}{cc}{e}^{i.Math..Math.}.Math..Math.\cos .Math..Math.& -\sin .Math..Math.\\ e^{i.Math..Math.}.Math..Math.\sin .Math..Math.& \cos .Math..Math.\end{array}\right]\left[\begin{array}{c}{\hat{a}}_{m,\mathrm{in}}\\ {\hat{a}}_{n,\mathrm{in}}\end{array}\right],$ wherein each transformation matrix describing the coupling between modes m and n is defined as a reflectivity of cos , and a phase shift of its at input m, wherein m<n.

10. A method as claimed in claim 9, wherein the unitary matrix .Math. is decomposed into a product of a plurality of T.sub.m,n matrices that satisfy $\hat{U}=\hat{D}.Math.\underset{\left(m,n\right)S}{}.Math.{\hat{T}}_{m,n}$ wherein S defines the ordered sequence in which the {circumflex over (T)}.sub.m,n matrices are to be applied to the unitary matrix .Math. in turn and {circumflex over (D)} is the resultant diagonal matrix, and wherein the interferometer is designed with a plurality of beam splitters and phase shifters arranged in the ordered configuration determined by S, and wherein the beam splitters and phase shifters are each configured to have the values of and as determined by the respective transformation matrix.

11. A method as claimed in claim 1, comprising operating on the unitary matrix with the plurality of transformation matrices, wherein the operation of each transformation matrix on the unitary matrix nulls a respective different non-diagonal element of the unitary matrix.

12. A method as claimed in claim 1, comprising operating on the unitary matrix with the plurality of transformation matrices, wherein the operation of each transformation matrix on the unitary matrix nulls the elements of the lower or upper triangle of the unitary matrix to decompose the unitary matrix.

13. A method as claimed in claim 12, wherein the elements of the lower or upper triangle of the unitary matrix are nulled in an order such that a triangle of increasing size of nulled elements is formed until the whole of the lower or upper triangle has been nulled.

14. A method as claimed in claim 11, wherein the initial element nulled is the bottom left hand corner element in the lower triangle or the top right hand corner element in the upper triangle of the unitary matrix.

15. A method as claimed in claim 11, wherein when the initial element nulled is the bottom left hand corner element in the lower triangle of the unitary matrix, the order in which the elements of the unitary matrix are nulled, for an NN unitary matrix .Math. is: for i from 1 to N1: when i is odd: looping over j from 0 to i1: null element (Nj,ij); when i is even: looping over j from 1 to i: null element (N+ji,j).

16. A method as claimed in claim 11, wherein when the initial element nulled is the bottom left hand corner element in the lower triangle of the unitary matrix, the order in which the transformation matrices are applied to the unitary matrix, for an NN unitary matrix .Math. is: for i from 1 to N1: when i is odd: looping over j from 0 to i1: multiply .Math. (updated from the previous step) from the right with a {circumflex over (T)}.sub.ij,ij+1.sup.1 matrix; when i is even: looping over j from 1 to i: multiply .Math. (updated from the previous step) from the left with a {circumflex over (T)}.sub.N+ji1,N+ji, matrix.

17. A method as claimed in claim 1, comprising outputting the determined couplings for the pairs of modes at each of the plurality of crossing points.

18. A method as claimed in claim 1, comprising designing and manufacturing the interferometer using the determined couplings for the pair of modes at each of the plurality of crossing points.

19. A non-transitory computer readable storage medium storing computer software code which when executing on a data processing system performs a method of designing an interferometer for coupling a plurality of modes of electromagnetic radiation, the method comprising: for an interferometer comprising: N inputs for inputting N modes of electromagnetic radiation into the interferometer, N outputs for outputting N modes of electromagnetic radiation from the interferometer, and a plurality of waveguides arranged to pass through the interferometer to connect the N inputs to the N outputs and for carrying the N modes of electromagnetic radiation through the interferometer, wherein: N is a natural number, the plurality of waveguides are arranged to provide a plurality of crossing points between pairs of the plurality of waveguides such that at each crossing point the two modes of electromagnetic radiation carried by the two respective waveguides are capable of coupling with each other, wherein the plurality of waveguides and the plurality of crossing points are arranged such that each of the N modes of electromagnetic radiation is capable of coupling with each of the other modes of electromagnetic radiation at respective crossing points, and the plurality of waveguides are arranged such that the plurality of crossing points are arranged into N groups along the plurality of waveguides from the inputs to the outputs through the interferometer, wherein each group contains the maximum number of possible crossing points between pairs of waveguides, and wherein the crossing points in each group involve pairs of adjacent waveguides whose paths were not crossed in the previous group of crossing points: receiving a unitary matrix describing a desired transformation to be performed by the interferometer; operating on the unitary matrix with a plurality of transformation matrices to decompose the unitary matrix into a diagonal matrix, wherein the transformation matrices used to decompose the unitary matrix each represent the coupling between a pair of modes at a crossing point of a pair of waveguides, and wherein the transformation matrices are arranged to operate on the unitary matrix in an order that matches a sequence in which the crossing points in the interferometer may be arranged; determining, using the transformation matrices, the coupling for the pair of modes at each of the plurality of crossing points for use in designing the interferometer.

Description

[0138] Certain preferred embodiments of the invention will now be described, by way of example only, with reference to the accompanying drawings in which:

[0139] FIG. 1 shows a schematic of a layout of an interferometer;

[0140] FIG. 2 shows a schematic diagram of the layout of an interferometer according to an embodiment of the invention;

[0141] FIGS. 3a-3e show the steps of decomposing a unitary matrix and the corresponding couplings that are introduced between the modes of an interferometer according to an embodiment of the present invention;

[0142] FIG. 4 shows the order in which the matrix elements of the unitary matrix are nulled according to an embodiment of the present invention;

[0143] FIG. 5 shows a schematic diagram of the layout of an interferometer according to another embodiment of the invention; and

[0144] FIGS. 6a-6c show the order in which the matrix elements of the unitary matrix are nulled according to another embodiment of the present invention.

[0145] There are many applications in which we might want to interfere modes of light, such as in a photonics chip, radio-frequency interferometers and quantum systems that undergo controllable unitary or linear evolution described by beam splitter-like operations.

[0146] This particular embodiment of the present invention may be useful for interfering modes of light in an NN interferometer, where N is the number of modes of light, to produce an arbitrary interference pattern. As will become apparent, this embodiment of the invention provides a means for determining the reflection coefficients and phase shifts of an interferometer for creating an arbitrary interference pattern.

[0147] FIG. 2 shows a schematic diagram of the layout of an interferometer 100 that is arranged to interfere the paths 101, 102, 103, 104 and 105 of five modes of light along the length of the interferometer. FIG. 2 illustrates an embodiment of this invention for an interferometer with N=5 modes of light for purposes of clarity and simplicity. In practice N may be much larger, for example, approximately one hundred, or orders of magnitude greater.

[0148] The interferometer in FIG. 2 includes five inputs and five outputs, the inputs matching one to one with the outputs via five paths through the interferometer. The five paths are arranged to carry five respective modes of light through the interferometer along which each mode of light passes through a series of beam splitters. The paths 101, 102, 103, 104 and 105 are arranged to cross each other in the beam splitter 108 at crossing points between two paths such that each mode of light at the input is systematically interfered with every other mode of light by means of the beam splitters. Thus it will be seen that the crossing points are arranged into five groups 106 along the waveguide from the inputs to the outputs. The paths and the beam splitters are arranged in the topography shown in FIG. 2. For example, a beam splitter 108 is located at position 107, where two paths, 104 and 105, cross and their respective modes are interfered.

[0149] A phase shift is applied before each beam splitter 108 by means of a phase-shifting device to at least one mode of light. A minimum of N(N1)/2 beam splitters are required to interfere all the modes of light in an NN interferometer; 10 beam splitters are used in the 55 interferometer illustrated in FIG. 2.

[0150] By selecting the reflectivity coefficient of each of the beam splitters 108, and selecting the phase shifts applied to the modes of light before each beam splitter 108, the interferometer can be used to create any arbitrary interference pattern.

[0151] Using an embodiment of the invention, a desired interference pattern can be converted into a design for the arrangement of: the order of interference of N modes of light in an interferometer; the reflection coefficient applied at each interference point; and the phase shift applied before at the interference point to one or both of the modes of light.

[0152] A beam splitter having reflectivity cos ([0,2]) on modes labelled m and n (m<n) at input m, with a phase shift, at input m, can be written as an NN matrix {circumflex over (T)}.sub.m,n(,) which is the identity matrix except for the (m, m), (m, n), (n, m), (n, n) elements, which form a 22 submatrix effecting the following transformation:

[00007]$\left[\begin{array}{c}{\hat{a}}_{m,\mathrm{out}}\\ {\hat{a}}_{n,\mathrm{out}}\end{array}\right]=\left[\begin{array}{cc}{e}^{i.Math..Math.}.Math..Math.\cos .Math..Math.& -\sin .Math..Math.\\ e^{i.Math..Math.}.Math..Math.\sin .Math..Math.& \cos .Math..Math.\end{array}\right]\left[\begin{array}{c}{\hat{a}}_{m,\mathrm{in}}\\ {\hat{a}}_{n,\mathrm{in}}\end{array}\right]$

[0153] A property of the {circumflex over (T)}.sub.m,n(,) matrices is that for any .Math., there are specific values of and that null the (m,n) element of matrix .Math.{circumflex over (T)}.sub.m,n(,). For notational simplicity, the explicit dependence of the {circumflex over (T)}.sub.m,n(,) matrices on and is omitted in the below description.

[0154] A further matrix used in the decomposition are matrices represented by {circumflex over (T)}.sub.m,n.sup.1(,). A matrix {circumflex over (T)}.sub.m,n.sup.1(, ) is the inverse of a matrix {circumflex over (T)}.sub.m,n(, ). {circumflex over (T)}.sub.m,n.sup.1(, ) is identity except for the (m, m), (m, n), (n, m), (n, n) elements, which form a 22 submatrix effecting the following transformation:

[00008]$\left[\begin{array}{c}{\hat{a}}_{m,\mathrm{out}}\\ {\hat{a}}_{n,\mathrm{out}}\end{array}\right]=\left[\begin{array}{cc}{e}^{i.Math..Math.}.Math..Math.\cos .Math..Math.& e^{-i.Math..Math.}.Math..Math.\sin .Math..Math.\\ -\sin .Math..Math.& \cos .Math..Math.\end{array}\right]\left[\begin{array}{c}{\hat{a}}_{m,\mathrm{in}}\\ {\hat{a}}_{n,\mathrm{in}}\end{array}\right]$

[0155] The {circumflex over (T)}.sub.m,n.sup.1 matrices can null the (m,n) element of the matrix {circumflex over (T)}.sub.m,n.sup.1(, ).Math.. As with the {circumflex over (T)}.sub.m,n matrices, the explicit dependence on and in the {circumflex over (T)}.sub.m,n.sup.1(, ) matrices will be omitted for simplicity.

[0156] {circumflex over (T)}.sub.m,n.sup.1 matrices represent a physical implementation of a beam splitter of reflectivity cos with a phase shift at input m (as opposed to output m for the {circumflex over (T)}.sub.m,n matrices). The inverse of any {circumflex over (T)}.sub.m,n matrix is a {circumflex over (T)}.sub.m,n.sup.1 matrix.

[0157] A given N-mode interferometer can be represented by an NN unitary matrix .Math. that describes the transformation of the annihilation operators of the modes of light by means of the equation

.sub.out=.Math..sub.in

where .sub.out=(.sub.out,1, .sub.out,2, . . . , .sub.out,N) and .sub.in=(.sub.in,1, .sub.in,2, . . . , .sub.in,N) are vectors representing the annihilation operators of all the input and output modes, i.e. .Math. describes the desired interference pattern to be achieved by the interferometer. Thus first a desired matrix .Math. is defined to reflect the desired interference pattern.

[0158] The decomposition of the unitary matrix .Math. will now be described for a general NN interferometer having N inputs and N outputs. Thereafter, this method will then be applied to the specific example shown in FIGS. 3a-3e. FIGS. 3a-3e show the steps of decomposing the unitary matrix .Math. and the corresponding couplings that are introduced between the modes 101, 102, 103, 104, 105 of the interferometer 100, 203.

[0159] Once the matrix .Math. has been defined, the reflectivity coefficient of each of the beam splitters and the phase shift applied at each phase shifter can be calculated by decomposing .Math.. Decomposing .Math. into a series of simpler matrices, where each matrix represents a single 22 beam splitter, allows an NN interferometer to be described with at most N(N1)/2 beam splitters with one or two phase shifters associated with each beam splitter.

[0160] The decomposition finds, for a given N, that the U matrix can be written as

[00009]$\hat{U}=\hat{D}\left(\underset{\left(m,n\right)S}{}.Math.{\hat{T}}_{m,n}\right)$

where the and values of the {circumflex over (T)}.sub.m,n.sup.1 matrices depend on .Math., is the product operator, S defines a specific ordered sequence of two-mode transformations and {circumflex over (D)} is a diagonal matric with complex elements with modulus equal to one on the diagonal.

[0161] An interferometer composed of beam splitters and phase shifters in the configuration defined by S, with values defined by the and in the {circumflex over (T)}.sub.m,n matrices, will implement transformation .Math.. {circumflex over (D)} can be implemented in an interferometer by phase shifts on all individual modes at the output of the interferometer.

[0162] The unitary matrix decomposition procedure is implemented by consecutively nulling elements of .Math. using {circumflex over (T)}.sub.m,n and {circumflex over (T)}.sub.m,n.sup.1 matrices. FIG. 4 shows the order in which the matrix elements are nulled by the triangle 301.

[0163] The numerical value in each element (given in Roman numerals) of the triangle 301 indicates the ordering of the nulling is this embodiment of the invention. The first element to be nulled is at the bottom left of the matrix. The following elements are then nulled in consecutive diagonals. An underlined element in FIG. 4 located in row i of the matrix is nulled with a {circumflex over (T)}.sub.i1,i. An element which is not underlined in FIG. 4 is nulled with a {circumflex over (T)}.sub.i,i+1.sup.1 matrix.

[0164] For each row i from 1 to N1, if i is odd then for each column, j, between 0 and i1 find a {circumflex over (T)}.sub.ij,ij+1.sup.1 matrix that nulls element (Nj, ij) of .Math.. After each null operation for an odd i, .Math. is updated such that .Math.=.Math.{circumflex over (T)}.sub.ij,ij+1.sup.1If i is even then for each j from 1 to i find a {circumflex over (T)}.sub.N+ji1,N+ji matrix that nulls element (N+ji,j) of .Math.. After each null operation for an even i, .Math. is updated such that .Math.={circumflex over (T)}.sub.N+ji1,N+ji.Math..

[0165] The decomposition yields an equation

[00010]$\left(\underset{\left(m,n\right)S_L}{}.Math.{\hat{T}}_{m,n}\right).Math.\hat{U}\left(\underset{\left(m,n\right)S_R}{}.Math.{\hat{T}}_{m,n}^{-1}\right)=\hat{D}$

where S.sub.L and S.sub.R are the respective orderings of the (m, n) indices for the {circumflex over (T)}.sub.m,n or {circumflex over (T)}.sub.m,n.sup.1matrices yielded by our decomposition. This can be rewritten as

[00011]$\begin{array}{cc}\hat{U}=\left(\underset{\left(m,n\right)S_L^T}{}.Math.{\hat{T}}_{m,n}^{-1}\right).Math.\hat{D}\left(\underset{\left(m,n\right)S_R^T}{}.Math.{\hat{T}}_{m,n}\right)& \left(1\right)\end{array}$

[0166] If {circumflex over (D)} consists of single-mode phase shifts, then for any {circumflex over (T)}.sub.m,n.sup.1 matrix one can find a {circumflex over (D)} matrix of single mode phase-shifts and a D.sub.m?, matrix such that {circumflex over (T)}.sub.m,n.sup.1{circumflex over (D)}={circumflex over (D)}{circumflex over (T)}.sub.m,n, hence Equation (1) can be rewritten as

[00012]$\hat{U}=\hat{D}\left(\underset{\left(m,n\right)S}{}.Math.{\hat{T}}_{m,n}\right)$

[0167] This decomposition is illustrated in FIGS. 3a-3e for a 55 interferometer. The decomposition starts with a random 55 unitary matrix a 201, and a blank interferometer, 203, shown in FIG. 3a. Elements in the random 55 unitary matrix a 201, are represented by asterisk symbols. The bottom left element of .Math., 201, is the first one nulled, with a TV, which causes the first two columns of .Math. to mix. This corresponds to adding the top-left beam splitter in the interferometer, i.e. to couple the top two modes 101, 102 in the interferometer 203. Successive elements of .Math., 201, are nulled in the order given by the triangle 301 in FIG. 4. This is shown in FIGS. 3b-3e, with a sequence of {circumflex over (T)}.sub.m,n and {circumflex over (T)}.sub.m,n.sup.1 matrices 205 which operate on .Math., 201, to null the respective elements. In FIG. 3e, .Math., 201, is a lower triangular matrix, which by virtue of its unitarity is diagonal. The sequence of {circumflex over (T)}.sub.m,n and {circumflex over (T)}.sub.m,n.sup.1 matrices 205 and .Math., 201, given in FIG. 3e can be rewritten as .Math.=DT.sub.3,4T.sub.4,5T.sub.1,2T.sub.2,3T.sub.3,4T.sub.4,5T.sub.1,2T.sub.2,3T.sub.3,4T.sub.1,2.

[0168] Alternative embodiments of the present invention may be presented by relabeling modes in the interferometer or by decomposing the elements in .Math. in a different order to that presented in FIG. 4. One such embodiment is described as follows.

[0169] FIG. 5 shows a schematic diagram of the layout of the interferometer 400 that is arranged to interfere the paths 1, 2, 3, 4 and 5 of five modes of light along the length of the interferometer. FIG. 5 illustrates an embodiment of this invention for an interferometer with N=5 modes of light for purposes of clarity and simplicity. In practice N may be much larger, for example, approximately one hundred, or orders of magnitude greater.

[0170] The interferometer in FIG. 5 includes five inputs and five outputs, the inputs matching one to one with the outputs via five paths through the interferometer. The five paths are arranged to carry five respective modes of light through the interferometer along which each mode of light passes through a series of beam splitters. The paths 1, 2, 3, 4 and 5 are arranged to cross each other in the beam splitter at crossing points between two paths such that each mode of light is systematically interfered with every other mode of light by means of the beam splitters. The paths and the beam splitters are arranged in the topography shown in FIG. 5. For example, a beam splitter is located at position 405, where two paths 1, 4 cross and their respective modes are interfered.

[0171] The dashed ovals 403 help indicate the sequence of beam splitters that correspond to the ordering identified in an embodiment of the invention. The {circumflex over (T)}.sub.m,n matrices, 401, indicate which {circumflex over (T)}.sub.m,n matrix corresponds to which crossing point.

[0172] A phase shift is applied before each beam splitter by means of a phase-shifting device to at least one mode of light. A minimum of N(N1)/2 beam splitters are required to interfere all the modes of light in an NN interferometer; 10 beam splitters are used in the 55 interferometer illustrated in FIG. 5.

[0173] By selecting the reflectivity coefficient of each of the beam splitters, and selecting the phase shifts applied to the modes of light at before each beam splitter, the interferometer can be used to create any arbitrary interference pattern.

[0174] Using an embodiment of the invention, a desired interference pattern can be converted into a design for the arrangement of: the order of interference of N modes of light in an interferometer; the reflection coefficient applied at each interference point; and the phase shift applied before at the interference point to one or both of the modes of light.

[0175] A beam splitter having reflectivity cos ([0,2]) on modes labelled m and n (m<n) at output m, with a phase shift, , at output m can be written as an NN matrix {circumflex over (T)}.sub.m,n(, ) which is the identity matrix except for the (m, m), (m, n), (n, m), (n, n) elements, which form a 22 submatrix effecting the following transformation:

[00013]$\left[\begin{array}{c}{\hat{a}}_{m,\mathrm{out}}\\ {\hat{a}}_{n,\mathrm{out}}\end{array}\right]=\left[\begin{array}{cc}{e}^{i.Math..Math.}.Math..Math.\cos .Math..Math.& -e^{i.Math..Math.}.Math..Math.\sin .Math..Math.\\ \sin .Math..Math.& \cos .Math..Math.\end{array}\right]\left[\begin{array}{c}{\hat{a}}_{m,\mathrm{in}}\\ {\hat{a}}_{n,\mathrm{in}}\end{array}\right]$

[0176] A property of the {circumflex over (T)}.sub.m,n(, ) matrices is that for any .Math., there are specific values of and that null the (m,n) element of matrix .Math.{circumflex over (T)}.sub.m,n(, ). For notational simplicity, the explicit dependence of the {circumflex over (T)}.sub.m,n(, ) matrices on and is omitted in the below description.

[0177] A further matrix used in the decomposition are matrices represented by {circumflex over (T)}.sub.m,n.sup.(, ). A matrix {circumflex over (T)}.sub.m,n.sup.1(, ) is the inverse of a matrix {circumflex over (T)}.sub.m,n(, ). {circumflex over (T)}.sub.m,n.sup.1(, ) are identity except for the (m, m), (m, n), (n, m), (n, n) elements, which form a 22 submatrix effecting the following transformation:

[00014]$\left[\begin{array}{c}{\hat{a}}_{m,\mathrm{out}}\\ {\hat{a}}_{n,\mathrm{out}}\end{array}\right]=\left[\begin{array}{cc}{e}^{i.Math..Math.}.Math..Math.\cos .Math..Math.& \sin .Math..Math.\\ -e^{i.Math..Math.}.Math..Math.\sin .Math..Math.& \cos .Math..Math.\end{array}\right]\left[\begin{array}{c}{\hat{a}}_{m,\mathrm{in}}\\ {\hat{a}}_{n,\mathrm{in}}\end{array}\right]$

[0178] The {circumflex over (T)}.sub.m,n.sup.1 matrices can null the (m,n) element of the matrix {circumflex over (T)}.sub.m,n.sup.1(, ).Math.. As with the {circumflex over (T)}.sub.m,n matrices, the explicit dependence on and in the {circumflex over (T)}.sub.m,n.sup.1(, ) matrices will be omitted for simplicity.

[0179] {circumflex over (T)}.sub.m,n.sup.1 matrices represent a physical implementation of a beam splitter of reflectivity cos with a phase shift at input m (as opposed to output m for the {circumflex over (T)}.sub.m,n matrices). The inverse of any {circumflex over (T)}.sub.m,n matrix is a {circumflex over (T)}.sub.m,n.sup.1 matrix.

[0180] A given N-mode interferometer can be represented by an NN unitary matrix that describes the transformation of the annihilation operators of the modes of light by means of the equation

.sub.out=.Math..sub.in

where .sub.out=(.sub.out,1, .sub.out,2, . . . , .sub.out,N) and .sub.in=(.sub.in,1, .sub.in,2, . . . , .sub.in,N) are vectors representing the annihilation operators of all the input and output modes, i.e. .Math. describes the desired interference pattern to be achieved by the interferometer. Thus first a desired matrix .Math. is defined to reflect the desired interference pattern.

[0181] The decomposition of the unitary matrix .Math. will now be described for a general NN interferometer having N inputs and N outputs. Thereafter, this method will then be applied to the specific example shown in FIG. 5.

[0182] Once the matrix .Math. has been defined the reflectivity coefficient of each of the beam splitters and the phase shift applied at each phase shifter, can be calculated by decomposing a matrix .Math.. Decomposing .Math. into a series of simpler matrices, where each matrix represents a single 22 beam splitter, allows an N x N interferometer to be described with at most N(N1)/2 beam splitters with one or two phase shifters associated with each beam splitter.

[0183] The decomposition finds, for a given N, a fixed sequence, S, for which any NN .Math. matrix can be written as

[00015]$\hat{U}=\hat{D}\left(\underset{\left(m,n\right)S}{}.Math.{\hat{T}}_{m,n}^{-1}\right),$

where the and values of the {circumflex over (T)}.sub.m,n.sup.1 matrices depend on .Math. and is the product operator.

[0184] The decomposition begins by nulling element (N(N+1)/2, (N1)/2) of .Math. if N is odd, or element (N/2+1,N/2) of .Math. if N is even. The selected element is nulled by multiplying it by a {circumflex over (T)}.sub.m,n matrix on the right with the values of and selected so as to null the element.

[0185] To null the remainder of the elements below the diagonal of .Math. the following operations are applied. The operation chosen at each iteration is dependent upon the coordinates, (i, j), of the last nulled element of .Math..

[0186] If iN/2+1 and i1=j, then the greatest n for which the (n, j) element has been nulled is found. Element (n+1, j) is nulled by multiplying .Math. (and any previously included {circumflex over (T)}.sub.m,n and {circumflex over (T)}.sub.m,n.sup.1 matrices) from the left by a {circumflex over (T)}.sub.j,n+1.sup.1 matrix.

[0187] Otherwise, if j>(N1)/2 and j+1=i, the smallest n is then found for which the (i, n) element has been nulled. Null element (i, n1) by multiplying .Math. (and any previously included {circumflex over (T)}.sub.m,n and {circumflex over (T)}.sub.m,n.sup.1 matrices) from the right by a {circumflex over (T)}.sub.n1,i matrix.

[0188] Otherwise, if (i1,j) has not been nulled, then it is nulled by multiplying .Math. (and any previously included {circumflex over (T)}.sub.m,n and {circumflex over (T)}.sub.m,n.sup.1 matrices) from the right by a {circumflex over (T)}.sub.j,i1 matrix.

[0189] If none of the above conditions are fulfilled, the matrix (i,j+1) is nulled by multiplying .Math. (and any previously included {circumflex over (T)}.sub.m,n and {circumflex over (T)}.sub.m,n.sup.1 matrices) from the left by a {circumflex over (T)}.sub.j,n+1.sup.1 matrix.

[0190] The (i,j) coordinates of the nulled element are then used to determine the next coordinate to null, using the selection criteria above.

[0191] FIGS. 6a-6c show the order of nulling the matrix elements for an N=5 matrix according to another embodiment of the invention. The matrix 501 shown in FIG. 6a has already been partially-nulled to form a triangle of nulled elements below the diagonal, using the steps outlined above. Referring to FIG. 6b, matrix 503 shows the next steps of nulling the matrix. The element immediately below the left-most column 51 of the nulled triangle is nulled first. The element immediately below the second left-most column 52 of the nulled triangle is nulled second. The element immediately below the third left-most column 53 of the nulled triangle is nulled third.

[0192] The element immediately below the fourth left-most column 54 of the nulled triangle is nulled fourth. In this example the elements immediately below all the columns of the nulled-triangle in matrix 501 (shown in FIG. 6a) have now also been nulled.

[0193] The next steps are illustrated in matrix 505 of FIG. 6c. The element immediately to the left of the bottom row 55 of the nulled triangle is nulled first. The element immediately to the left of the second bottom row 56 of the nulled triangle is nulled second. The element immediately to the left of the third bottom row 57 of the nulled triangle is nulled third. The element immediately to the left of the fourth bottom row 57 of the nulled triangle is nulled fourth. The element immediately to the left of the fifth bottom row 58 of the nulled triangle is nulled fifth. The element immediately to the left of the sixth bottom row 59 of the nulled triangle is nulled sixth. The nulled triangle is thereby expanded. The steps can be continued to null further rows and columns of an NN matrix.

[0194] The decomposition yields an equation

[00016]$\left(\underset{\left(m,n\right)S_L}{}.Math.{\hat{T}}_{m,n}^{-1}\right).Math.\hat{U}\left(\underset{\left(m,n\right)S_R}{}.Math.{\hat{T}}_{m,n}\right)=\hat{D},$

where S.sub.L and S.sub.R correspond to the sequences of the beam splitter matrices used the left and right sides of .Math., respectively, and {circumflex over (D)} is a diagonal matrix with complex diagonal elements with modulus equal to one. This is equivalent to

[00017]$\begin{array}{cc}\hat{U}=\left(\underset{\left(m,n\right)S_R}{}.Math.{\hat{T}}_{m,n}\right)=\left(\underset{\left(m,n\right)S_L^T}{}.Math.{\hat{T}}_{m,n}\right).Math.\hat{D}.& \left(2\right)\end{array}$

Equation (2) can be rewritten as

[00018]$\begin{array}{cc}\hat{U}=\left(\underset{\left(m,n\right)S_L^T}{}.Math.{\hat{T}}_{m,n}\right).Math.\hat{D}\left(\underset{\left(m,n\right)S_R^T}{}.Math.{\hat{T}}_{m,n}^{-1}\right).& \left(3\right)\end{array}$

{circumflex over (T)}.sub.m,n.sup.1 matrices are physically implemented by a phase shift on one mode followed by a two-mode beam splitter and the {circumflex over (T)}.sub.m,n matrices are physically implemented by a two-mode beam splitter followed by a phase shift on one mode. {circumflex over (D)} physically corresponds to additional single-mode phases on all the modes. Hence, the interferometer design given by Equation (3) can be implemented with a phase shift in between each beam splitter and at the input and output of the interferometer.

[0195] In an alternative embodiment, the interferometer may be physically implemented using phase shifts at only one input of each of the beam splitters. To determine the design for the interferometer with phase shifts at only one input of each of the beam splitters, the steps noted above to find Equation (2) should initially be followed.

[0196] It is first noted that for any given (m, n) and {circumflex over (T)}.sub.m,n matrix, and for any diagonal {circumflex over (D)} matrix, the (m, n) element of {circumflex over (T)}.sub.m,n{circumflex over (D)} can be nulled by multiplying from the right by the appropriate {circumflex over (T)}.sub.m,n matrix (where the prime symbol indicates a {circumflex over (T)}.sub.m,n matrix with different values for and ), such that

{circumflex over (T)}.sub.m,n{circumflex over (D)}{circumflex over (T)}.sub.m,n={circumflex over (D)},

where {circumflex over (D)} is a diagonal matrix. It follows that

{circumflex over (T)}.sub.m,n{circumflex over (D)}={circumflex over (D)}={circumflex over (D)}{circumflex over (T)}.sub.m,n.sup.1. (4)

[0197] Combining Equations (2) and (4) yields

[00019]$\hat{U}\left(\underset{\left(m,n\right)S_R}{}.Math.{\hat{T}}_{m,n}\right)={\hat{D}}^{}\left(\underset{\left(m,n\right)S_L^T}{}.Math.{\hat{T}}_{m,n}^{-1}\right),$

therefore

[00020]$\begin{array}{cc}\hat{U}={\hat{D}}^{}\left(\underset{\left(m,n\right)S_L^T.Math.S_R^T}{}.Math.{\hat{T}}_{m,n}^{-1}\right),& \left(5\right)\end{array}$

where , the union operator, preserves the ordering of S.sub.L.sup.T and S.sub.R.sup.T. Equation (5) shows the completed decomposition for determining the reflection coefficients and phase shifts of the interferometer to achieve an arbitrary interference pattern. Hence Equation (5) yields the design for an interferometer with phase shifts at only one input of each of the beam splitters.

[0198] In some applications, phase shifts at the output of an interferometer are irrelevant.

[0199] In these cases {circumflex over (D)} is physically irrelevant and phase shifts are only required at one input of every beam splitter in the interferometer.

[0200] In other applications phase shifts are used at the output of an interferometer. In these cases {circumflex over (D)} is physically relevant. This means phase shifts are applied at one input of every beam splitter in the interferometer and after the final stage of beam splitters of the interferometer for each mode of light

[0201] To show how the decomposition corresponds to a physical implementation of an interferometer, Equation (5) is applied to an odd-numbered N, which yields

[00021]$\begin{array}{cc}\hat{U}={\hat{D}}^{}.Math.{\hat{T}}_{\left(\left(N+3\right).Math.\text{/}.Math.2,\left(N-1\right).Math.\text{/}.Math.2\right)}^{-1}.Math..Math.\mathrm{.Math.}.Math..Math.\left(\underset{n=2}{\overset{N-2}{.Math.}}.Math..Math.{\hat{T}}_{\left(n,N-1\right)}\right).Math.\left(\underset{n=1}{\overset{N-1}{.Math.}}.Math..Math.{\hat{T}}_{\left(n,N\right)}\right).Math.\left(\underset{n=2}{\overset{N-1}{.Math.}}.Math..Math.{\hat{T}}_{\left(1,n\right)}^{-1}\right).Math.\left(\underset{n=3}{\overset{N-2}{.Math.}}.Math..Math.{\hat{T}}_{\left(2,n\right)}^{-1}\right).Math..Math.\mathrm{.Math.}.Math..Math.{\hat{T}}_{\left(\left(N-1\right).Math.\text{/}.Math.2,\left(N+1\right).Math.\text{/}.Math.2\right)}^{-1},& \left(6\right)\end{array}$

where the {circumflex over (T)}.sub.m,n.sup.1 matrices on the first line of Equation (6) are ordered according to S.sub.L.sup.t and the {circumflex over (T)}.sub.m,n.sup.1 matrices on the second line are ordered according to S.sub.R.sup.T.

[0202] The last term on the first line of Equation (6), .sub.n=1.sup.N1{circumflex over (T)}.sub.(n,N), is implemented in an interferometer by crossing mode N with every other mode in the centre of the interferometer circuit, in descending order of modes. This is shown by mode 5 in FIG. 5, which shows a N=5 interferometer where mode 5 interferes with the other modes in the order 4, 3, 2 then 1.

[0203] The first term on the second line of Equation (6), .sub.n=2.sup.N1{circumflex over (T)}.sub.(1,n).sup.1, corresponds to mode 1 crossing over every mode except N in the same order. Physically, this means that modes 1 and N are parallel to each other in the interferometer circuit, and modes 1 and N only cross over each other once they have crossed every other mode. This is shown by mode 1 in FIG. 5, which shows a N=5 interferometer where mode 1 interferes with the other modes in the order 4, 3, 2 then 5.

[0204] The penultimate term on the first line of Equation (6), .sub.n=2.sup.N2{circumflex over (T)}.sub.(n,N-1).sup.1, corresponds to mode N1 crossing over every mode in the same order except 1 and N. Physically, this means that mode N1 first crosses modes 1, then N, and then goes parallel to them to cross every other mode. This is illustrated by mode 4 in FIG. 5, which shows a N=5 interferometer where mode 4 interferes with the other modes in the order 1, 5, 3 then 2.

[0205] As will be appreciated by the skilled person, this reasoning can be applied to other terms in Equation (6) as required by the value of N, giving the crossing order between each of the N modes with each of the other N modes.

[0206] It will be appreciated by the skilled person that the ordering and labelling of the modes at the input, as shown by modes 101-105 in FIGS. 2 and 1-5 in FIG. 5, can be reordered or relabelled to achieve a desired order of modes at the output

[0207] Once the transformation matrices (the {circumflex over (T)}.sub.m,n matrices) have been determined, e.g. according to one of the above embodiments, e.g. using a computer, an interferometer can be designed and manufactured using the couplings determined from these transformation matrices. For example, in one embodiment a semi-conductor based integrated photonics circuit based interferometer is designed and manufactured having beam splitters arranged at each of the crossing points between the waveguides that carry the modes of light therethrough (e.g. in the arrangement shown in FIGS. 2-5, which are extendible to any N), with the beam splitters (e.g. Mach-Zehnder interferometers) being configured to have a reflectivity and phase shift as determined from the couplings.

[0208] Whilst the embodiments of the invention have been described with reference to the implementations shown in FIGS. 2 and 5, the skilled person will appreciate that phase shifts could be applied before, after or between the beam splitters.

[0209] In the embodiments discussed it will be seen that the nulling process may be started at two different elements in the matrix U; however the Applicant envisages that the nulling process may start at any suitable element.

[0210] An interferometer can thus be designed for N modes of light according to the desired interference pattern (as defined by the U matrix). The skilled person will appreciate that this could be applied to any wavelength of electromagnetic waves. The Applicants envisage that the interferometers can in particular be designed for photonic circuits.

[0211] While the embodiment described above with reference to FIG. 2 shows a preferred implementation, the Applicants envisage that a number of variations exist. For example, N could take any value over a several orders of magnitude from approximately 10.sup.1 to 10.sup.6 or greater, modes could be rearranged into any desired order, phase shifters could be placed between beam splitters, i.e. not integrated therewith, and interferometers manufactured to the design could be manufactured in any suitable and desired way, using any suitable material or materials, such as silica, silicon or lithium niobate.

[0212] In one example the interferometer is manufactured as a silica-on-silicon photonic chip designed to operate at near infrared wavelengths. The photonic chip is compatible with optical fibre technology and contains optical waveguides as well as thermo-optic phase shifters. The thermo-optic phase shifters comprise resistors placed just above the optical waveguide. The photonic chip also contains controllable beam splitters which are implemented by Mach-Zehnder interferometers that each contains a thermo-optic phase shifter. The phase shifters and variable beam splitters are computer-controlled.

[0213] The skilled person will appreciate that the above embodiments are a particularly efficient way of interfering N modes of light with one-another in a compact arrangement with a high balance of loss between the modes of light, using the minimum number of beam splitters.

[0214] Furthermore, as well as being used to implement an, e.g., universal transformation described by a unitary matrix, embodiments of the invention may be used to implement a linear transformation, described by a sub-matrix that is embedded into a unitary matrix.

[0215] (SIQS)The work leading to this invention has received funding from the European Union Seventh Framework Programme (FP7 2007-2013) under grant agreement no. 600645.

[0216] (MOQUACINO)The work leading to this invention has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7 2007-2013) under grant agreement no. 339918.