METHOD AND DEVICE FOR PRODUCING A REFERENCE FREQUENCY

Abstract

The invention relates to a method for producing a reference frequency f. According to the invention, the use of a first optical resonator (3a; 24) and of a second optical resonator (25) is provided, wherein the first resonator (3a; 24) has a first resonator mode having a first frequency f1 and the second resonator (25) has a second resonator mode having a second frequency f2, wherein the frequencies of the two resonator modes are functions of an operating parameter BP and assume the values f1 and f2at a specified value BP.sub.0 of the operating parameter such that f1(BP.sub.0)=f1 and f2(BP.sub.0)=f2 apply, wherein the resonators (3a; 24, 25) are designed in such a way that the respective first derivatives of the frequencies f1(BP), f2BP) with respect to BP or at least respective difference quotients around BP.sub.0 correspond within a deviation of at most 0.1%, wherein light of the first frequency f1 is stabilized to the first frequency f1 by means of the first resonator and light of the second frequency f2 is stabilized to the second frequency f2 by means of the second resonator, and wherein the difference between the stabilized frequencies f1 and f2, f=|f1f2|, is determined in order to obtain the stabilized reference frequency f.

Claims

1. A method for producing a reference frequency f using a first optical resonator and a second optical resonator, wherein the first resonator has a first resonator mode with a first frequency f1 and the second resonator has a second resonator mode with a second frequency f2, wherein the frequencies of the two resonator modes are functions of an operating parameter BP, and take the values f1 and f2 at a set value BP.sub.0 of the operating parameter, so that f1(BP.sub.0)=f1 and f2(BP.sub.0)=f2, wherein the resonators are designed so that the first derivative of the frequencies f1(BP), f2(BP) with respect to BP, or at least a difference ratio around BP.sub.0, corresponds up to a deviation of a maximum of 0.1%, wherein light of the first frequency f1 is stabilized at the first frequency f1 by the first resonator, and light of the second frequency f2 is stabilized at the second frequency f2 by the second resonator, and wherein the difference between the stabilized frequencies f1 and f2, f=|f1f2|, is determined in order to obtain the stabilized reference frequency f.

2. The method as in claim 1, wherein the first optical resonator has a resonator length L.sub.1 and a linear temperature coefficient .sub.1 and the second optical resonator has a resonator length L.sub.2 and a linear temperature coefficient .sub.2, wherein the resonators are designed so that m.sub.1*.sub.1*L.sub.2*n.sub.2=m.sub.2*.sub.2*L.sub.1*n.sub.1 up to a deviation of a maximum of 0.1%, with m.sub.1, m.sub.2 being whole numbers, which correspond to the wavelength numbers of the first or second resonator mode in the first or second resonator, and n.sub.1, n.sub.2 correspond to the refractive indices for the first resonator mode in the first resonator and the second resonator mode in the second resonator, or that coupled modes are present in the first and second resonator, and a mode spectrum that is split because of the coupling contains the first resonator mode and the second resonator mode.

3. The method as in claim 1, wherein the first resonator simultaneously also forms the second resonator and is identical to it.

4. The method as in claim 3, wherein the mode coupling is produced by an at least partially reflecting element.

5. The method as in claim 1, wherein the mode coupling is produced by evanescent coupling of the first resonator to the second resonator.

6. The method as in claim 1, wherein the two resonator modes can each be described by a longitudinal index and two transversal indices, wherein the first resonator mode and the second resonator mode have the same longitudinal index and at least one different transversal index.

7. The method as in claim 1, wherein light of a third frequency f3 is produced and is stabilized by means of a resonator, wherein f3 has a greater dependence on the operating parameter than f, that a comparison frequency f4 is given by
f4=|f3f1| or f4=|f3f2| and that the ratio f4/f or the difference f4f is determined and is used to control operating parameter regulating means, which are provided to regulate the operating parameter of the first resonator and/or second resonator.

8. The method as in claim 7, wherein the first resonator and/or the second resonator or a third resonator is used to stabilize the light of the third frequency f3.

9. The method as in claim 7, wherein the light of the third frequency f3 is formed by a comb mode of a frequency comb.

10. The method as in claim 1, wherein a Fabry-Prot resonator is used as the first resonator and/or second resonator.

11. The method as in claim 1, wherein an optical ring resonator is used as first resonator and/or as second resonator.

12. The method as in claim 1, wherein an optical resonator made as a waveguide on an optical chip is used as first resonator and/or as second resonator.

13. A device for production of a reference frequency f, wherein a first optical resonator, which has a first resonator mode with a first frequency f1, and a second optical resonator, which has a second resonator mode with a second frequency f2, are provided, wherein the frequencies of the two resonator modes are functions of an operating parameter BP, and take on the values f1 and f2 at a set value BP.sub.0 of the operating parameter, so that f1(BP.sub.0)=f1 and f2(BP.sub.0)=f2, wherein the resonators are designed so that the first derivative of the frequencies f1(BP), f2(BP) with respect to BP, or at least one difference ratio around BP.sub.0, corresponds up to a deviation of a maximum of 0.1%, the device further comprising first light producing means to produce light of the first frequency f1 and second light producing means to produce light of the second frequency f2, wherein the first light producing means and the second light producing means comprise at least one laser, the device further comprising first stabilization means, to stabilize the first frequency f1, and second stabilization means to stabilize the second frequency f2, wherein determination means are provided in order to determine the difference between the stabilized frequencies f1 and f2, f=|f1f|, and to maintain the stabilized reference frequency f.

14. The device as in claim 13, wherein the first optical resonator has a resonator length L.sub.1 and a linear temperature coefficient .sub.1 and the second optical resonator has a resonator length L2 and a linear temperature coefficient .sub.2, wherein the resonators are designed so that m.sub.1*.sub.1*L.sub.2*n.sub.2=m.sub.2*.sub.2*L.sub.1*n.sub.1 up to a deviation of a maximum of 0.1%, with m.sub.1, m.sub.2 being whole numbers, which correspond to the wavelength number of the first or second resonator mode in the first or second resonator, and n.sub.1, n.sub.2 correspond to the refractive indices for the first resonator mode in the first resonator and the second resonator mode in the second resonator, or that coupled modes are present in the first and second resonator, and a mode spectrum that is split because of the coupling contains the first resonator mode and the second resonator mode.

15. The device as in claim 13, wherein the first stabilization means comprises first modulation means to modulate side bands on the light of the first frequency f1, and first demodulation means with a first detector to produce a first error signal by means of the modulated light of the first frequency f1 that is reflected back or transmitted to the first detector, and first regulating means to regulate, by means of the first error signal, the first light producing means so that the first frequency f1 becomes stabilized, and that the second stabilization means comprises second modulation means to modulate side bands on the light of the second frequency f2, and second demodulation means with a second detector to produce a second error signal by means of the modulated light of the second frequency f2 that is reflected back or transmitted to the second detector, and second regulating means to regulate, by means of the second error signal, the second light producing means so that the second frequency f2 becomes stabilized.

16. The device as in claim 13, wherein the first resonator simultaneously also forms the second resonator and is identical to it.

17. The device as in claim 16, wherein an at least partially reflecting element is provided to produce the mode coupling.

18. The device as in claim 13, wherein the first regulator is evanescently coupled to the second resonator to produce the mode coupling.

19. The device as in claim 13, wherein the two resonator modes can each be described by a longitudinal index and two transversal indices, wherein the first resonator mode and the second resonator mode have the same longitudinal index and at least one different transversal index.

20. The device as in claim 13, wherein third light producing means are provided to produce light of a third frequency f3 along with a resonator for stabilization, wherein f3 has a greater dependence on the operating parameter than f, a comparison frequency f4 is given by
f4=|f3f1| or f4=|f3f2|, wherein additional determination means are provided to determine the ratio f4/f or the difference f4 f, and that the operating parameter regulating means are provided to control the operating parameter of the first resonator and/or of the second resonator as a function of ratio f4/f or the difference f4f.

21. The device as in claim 20, wherein the resonator for stabilizing the light of the third frequency f3 is the first resonator and/or the second resonator or a third resonator.

22. The device as in claim 20, wherein the third light producing means comprises a frequency comb to form the light of the third frequency f3 as a comb mode of the frequency comb.

23. The device as in claim 13, wherein the first resonator and/or the second resonator is a Fabry-Prot resonator.

24. The device as in claims 13, wherein the first resonator and/or the second resonator is an optical ring resonator.

25. The device as in claim 13, wherein the first resonator and/or the second resonator is made as a waveguide on an optical chip.

26. The method of claim 1 wherein the operating parameter comprises a temperature.

27. The method of claim 7 wherein the operating parameter comprises a temperature.

28. The method of claim 7 wherein the operating parameter regulating means comprises a temperature regulating means and the operating parameter comprises a temperature.

29. The device of claim 13 wherein the operating parameter BP comprises a temperature.

30. The device of claim 13 wherein the first and second light producing means together comprise at least one laser.

31. The device as in claim 20 wherein the operating parameter comprises a temperature.

Description

BRIEF DESCRIPTION OF THE FIGURES

[0040] The invention will now be explained in more detail by means of embodiment examples. The drawings are exemplary and are intended to represent the ideas of the invention, but not to limit it in any way or even to reproduce it conclusively.

[0041] Here:

[0042] FIG. 1 shows a schematic representation of an optical resonator of the Fabry-Prot type (also called cavity)

[0043] FIG. 2a shows the change of the frequency dependences with length L of the resonator in FIG. 1;

[0044] FIG. 2b shows an enlarged detail view of a portion of FIG. 2a;

[0045] FIG. 3 shows a schematic representation of an embodiment of a device according to the invention for producing a reference frequency, wherein the frequency of an oscillator is stabilized at the reference frequency;

[0046] FIG. 4 shows a schematic representation of another embodiment of the device according to the invention, wherein the length of the resonator is stabilized by means of the temperature at the ratio of two frequency differences between three optical modes;

[0047] FIG. 5 shows a schematic representation of another embodiment of the device according to the invention, in which the length of the resonator is stabilized by means of the temperature at the difference between the optical frequency of one mode of the resonator and a mode of a frequency comb, wherein the tooth spacing in the frequency comb is determined by the reference frequency;

[0048] FIGS. 6a) through d) show schematic representations of the relevant frequencies for the application in FIG. 5;

[0049] FIGS. 7a) through d) show the effect of deviations of resonator properties from their ideal values;

[0050] FIG. 8 shows another embodiment of the device according to the invention, wherein three ring resonators are used;

[0051] FIG. 9 shows another embodiment of the device according to the invention, analogous to FIG. 8, but where the first and the second ring resonators are coupled;

[0052] FIG. 10a shows a diagrammatic illustration of the frequencies of the modes of the individual ring resonators in FIG. 8 and the coupled ring resonators in FIG. 9 as a function of the temperature (given with respect to a working point temperature);

[0053] FIG. 10b shows a diagrammatic illustration of the reference frequency in the case of the coupled ring resonators in FIG. 9 as a function of the temperature (given with respect to an operating point temperature).

WAYS OF IMPLEMENTING THE INVENTION

[0054] FIG. 1 shows a schematic representation of a Fabry-Prot resonator 3a, which can be used according to the invention to produce a stable reference frequency. In such an optical resonator 3a, the frequency of an optical mode is determined by radii of curvature R1, R2 of a first mirror 1 and a second mirror 2 and by a length L of the resonator 3a. The frequency f of a mode of the resonator 3a in vacuum is

[00001] $f = \frac{c}{2 .Math. L} [l + \frac{1 + m + n}{} .Math. \cos^{- 1} (\sqrt{1 - \frac{L}{R_{1}}} .Math. \sqrt{1 - \frac{L}{R_{2}}})]$

[0055] The length L of the resonator is determined by the dimension of a mount 3 of the mirrors 1, 2. The velocity of light c determines the round trip time of a light particle or photon in resonator 3a. Said round trip time is further determined by the transversal mode indices m and n, which are part of the whole, positive numbers, including 0. The Hermite-Gauss basis for the resonator modes was chosen for the above formula. The principle represented herein is, however, basis-independent and can likewise be described in any other complete basis.

[0056] The longitudinal mode index is given by 1, also part of the whole positive numbers. The frequency difference between two modes with longitudinal indices l1 and l2 and transversal indices m1=m2=n1=n2=0 is given by

[00002] $.Math. .Math. f (0, 0) = \frac{c}{2 .Math. L} .Math. (l_{2} - l_{1})$

[0057] 19 and thus is always dependent of the length L of the resonator 3a. According to the invention, the frequency difference f between two modes with the same longitudinal index 1, but different transversal indices, for example m1=0, n1=0 and m2=1, n2=0, can be used as reference frequency, since it proves to be extremely stable, as further explained below. The resulting frequency difference f is

[00003] $.Math. .Math. f = \frac{c}{2 .Math. L} .Math. \frac{m + n}{} .Math. \cos^{- 1} (\sqrt{1 - \frac{L}{R_{1}}} .Math. \sqrt{1 - \frac{L}{R_{2}}})$

[0058] Through the non-monotonic behavior of the arccos function it is possible to find combinations of L, R4 and R2 for which the dependence of the frequency on length has reversal points and even deviates from the reference value only in the third order.

[0059] This can be seen in FIG. 2a. Here the first derivative of the frequency difference or the reference frequency f with respect to the resonator length L is shown for different combinations of R1 and R2. In FIG. 2a, the resonator length L is normalized to the maximum length L.sub.max=R1+R2 for all functions. The derivative is in turn normalized to the local frequency difference or the local reference frequency f at the length L/L.sub.max. The functions shown in FIG. 2a correspond to the following configurations of resonator 3a mirror 1 flat, mirror 2 curved (planoconcave, dashed line); mirror 1 and mirror 2 having the same radius of curvature (symmetric, dotted line); mirror 1 with radius R1, mirror 2 with radius R2=*R1 (optimum, solid line). The factor is further explained below.

[0060] For comparison, FIG. 2a also shows the behavior of the normalized derivative for the frequency of a single optical mode with m=n=0 (dot-dash line).

[0061] The region around L/L.sub.max0.8 is shown in detail in FIG. 2b. Here one can clearly see that the derivatives for both the symmetric and the planoconcave geometry cross through zero, while the derivative for the optimum case even has quadratic behavior.

[0062] The optimum case can be achieved by setting the ratio of the radii of curvature to be 1.7048. The most stable point then lies at an optimum resonator length of L.sub.opt 2.0428*R1. The quantity L.sub.opt can be found by solving the transcendental equation

[00004] $L_{opt} = \frac{\tan .Math. .Math. L_{opt}}{1 + \tan .Math. .Math. L_{opt}} .$

[0063] The optimum ratio of the radii of curvature R1, R2 can then be expressed as

[00005] $= \frac{L_{opt} (1 - L_{opt})}{1 - L_{opt} - \cos^{2} .Math. L_{opt}} .$

[0064] The zero-crossing length for symmetric and planoconcave resonators 3a can thus be found by solving the equation

[00006] $\frac{1}{2} .Math. \sqrt{\frac{L_{x}}{1 - L_{x}}} = \cos^{- 1} (1 - L_{x})$

with L.sub.x=L/L.sub.max, and is 0.8446*L.sub.max.

[0065] Here it should be noted that all resonators 3a with R2/R1< have at least one zero crossing in the first derivative. With favorable combinations of radii R1, R2 and length L, a stable frequency difference f between the selected modes can thus be determined. Such a resonator 3a can thus serve as frequency reference that is insensitive, for example, to changes of length caused by vibrations.

[0066] Factors that affect both the length L and the radii of curvature R1, R2 of the mirrors 1, 2 have a more noticeable effect on the reference frequency f. The most important factor is the temperature, especially in a vacuum. It can be established by expansion of the expression for the frequency difference

[00007] $.Math. .Math. f (T) = \frac{c}{2 .Math. L (T)} .Math. \frac{m + n}{} .Math. \cos^{- 1} (\sqrt{1 - \frac{L (T)}{R_{1} (T)}} .Math. \sqrt{1 - \frac{L (T)}{R_{2} (T)}}) .Math. .Math. with$ $L (T) = L (1 +_{S, 1} .Math. T + \frac{1}{2} .Math._{S, 2}^{2} .Math. T^{2}) .Math. .Math. and$ $R_{q} (T) = R_{q} (1 +_{R_{q}, 1} .Math. T + \frac{1}{2} .Math._{R_{q}, 2}^{2} .Math. T^{2})$

(with q=1 or q=2). Only linear and quadratic terms of the temperature dependence, which in practically all cases are dominant, were given here. Here .sub.S,1 and .sub.S,2 are the linear and quadratic coefficients of expansion of the mount, while .sub.R.sub.q.sub.,1 and .sub.R.sub.q.sub.,2 are the coefficients of expansion of the mirrors. T is the temperature difference at an operating point temperature at which the length is L.

[0067] For the sake of simplicity, it is assumed below that the two mirrors 1, 2 consist of the same material, so that .sub.R.sub.1.sub.,1=.sub.R.sub.2.sub.,1=.sub.R,1 and .sub.R.sub.1.sub.,2=.sub.R.sub.2.sub.,2=.sub.R,2. At this point it should be noted that the temperature dependence can be reduced still further by an appropriate choice of different coefficientsfor example by the selection of different materialsfor the two mirrors 1, 2.

[0068] Since, as explained above, small changes of length are negligible, the reference frequency f(T) mainly changes because of the thermal expansion of the mirrors 1, 2. The mirrors 1, 2 can be made of a material that, at a certain temperature, preferably the operating point temperature, has a zero crossing of the linear temperature coefficient, in other words, .sub.R,1=0, so that only a quadratic temperature dependence exists. Said materials are routinely used in optics, for example ULE glass at room temperature or silicon at 124 K.

[0069] However, if the mount 3 consists of a material with a nonvanishing linear thermal coefficient, the frequency difference between transversal ground modes f(T, 0, 0) and their optical frequencies will have a corresponding thermal dependence.

[0070] Said sensitivity is utilized for self-stabilization of the resonator in the embodiment examples in FIGS. 4 and 5 that are described below.

[0071] In the above description it was always assumed that the optical resonator 3a is situated in an evacuated container. The invention can, however, be made functional even in a light-permeable medium like air. For this, the dependence of the velocity of light on refractive index and its dependence on pressure P, temperature T, and other ambient factors X must be introduced into all of the formulas by means of the substitution c.fwdarw.c/n(T,P,X). In this expansion, ambient effects that are dependent on the temperature, for example pressure P in an airtight resonator 3a, can be compensated by slight modification of the parameters. Temperature-independent parameters, for example a contamination of the medium, cannot be compensated and lead to a frequency shift.

[0072] FIG. 3 schematically shows an embodiment example of a device according to the invention for producing the reference frequency f, wherein the frequency of a regulatable oscillator 7 is stabilized at the reference frequency f. A laser 4 produces a linearly polarized optical wave with low bandwidth. It is directed to an electro-optical modulator 4a, which produces sidebands on the optical wave. The modulated optical wave is directed through a polarized beam splitter 4b and a quarter-wave optical retarder 4c to the optical resonator. The latter is again shown as a Fabry-Prot resonator 3a in FIG. 3, and also in FIGS. 4 and 5, but basically any other type of resonator, in particular a ring resonator, could also be used.

[0073] The polarization optics 4b and 4c ensure that the light reflected by resonator 3a exits the beam splitter 4b in the direction of a demodulator 4d. The sidebands enable, by means of the demodulator 4d, an error signal to be produced, by means of which the laser frequency can be stabilized at the frequency f1 of a first resonator mode, which is a transversal ground mode M1 (with mode indices l1, m1=0, n1=0) of the optical resonator 3a, which is indicated in FIG. 3 by the dashed arrow pointing to laser 4. Said stabilization behavior is known as Pound-Drever-Hall behavior, see Reference 6 [Dreyer, R. W. P., Hall, J. L., Kowalski, F. V., Hough, J., Ford, G. M., Munley, A. J., & Ward, H; Laser phase and frequency stabilization using an optical resonator; Applied Physics B, 31(2), 97-105 (1983)].

[0074] A part of the laser beam of the laser 4 is now directed to an acousto-optical frequency shifter 6 by means of a beam splitter 5. Said beam is directed through a second electro-optical modulator 6a and guided through a second polarizing beam splitter 6b and a second quarter-wave optical retarder 6c to the optical resonator 3a. The purpose of this second beam is to stimulate a second resonator mode in the form of a higher transversal mode M2 with the same longitudinal index l2=l1 (for example with indices m2=1, n2=0) and the frequency f2. For this, the beam profile must be matched to that of the desired second resonator mode. This can be achieved, for example, via a structured phase plate 6e, as often takes place in optics and quantum optics.

[0075] Said beam, after its re-exit or reflection by the resonator 3a, is directed through the polarization optics 6b, 6c to a second demodulator 6d. The second electro-optical modulator 6a can be driven at a different frequency from the electro-optical modulator 4a in order to enable clean demodulation at the second modulator 6d. The error signal here is ascertained directly by the Pound-Drever-Hall technique: The frequency f2 is given by f1+f. However, the laser beam has the frequency f1+f(7), wherein f(7) is the frequency of oscillator 7. If f(7)f, then the Pound-Drever-Hall measurement outputs an error signal. The error signal thus arises through the difference between f1+f(7) and f2. Therefore, one can say in simpler terms that the resulting error signal corresponds to the difference between the oscillator frequency f(7) and f, and the frequency f(7) of the oscillator 7 is stabilized to the frequency differences between the two stimulated modes of the resonator 3a, thus to the reference frequency f.

[0076] A temperature regulator 8 can minimize the already quite low sensitivity of the structure even further. Such regulators are commercially available for stabilization up to variations around 10.sup.3 K.

[0077] The resonator 3a itself can be used to achieve a still higher frequency accuracy or stability of the reference frequency f. An example of such a system is shown in the embodiment example in FIG. 4. Here an additional partial beam is guided from beam splitter 5 to an additional acousto-optical modulator 10. Said modulator is driven at a frequency that is produced directly by oscillator 7 by means of a suitable fixed frequency multiplier 9, which multiplies the incoming frequency by a factor a.sub.f. The acousto-optical modulator 10 thus produces a beam that, at the desired operating point, is exactly resonant with an additional longitudinal third resonator mode M3 (with indices l3l1, n=0, m=0) having a frequency f3. By means of an additional electro-optical modulator 10a, which is connected after the acousto-optical modulator 10, sidebands are produced on the optical wavesimilar to the function of the electro-optical modulator 4a described above.

[0078] Since the frequency spacing f4 between the frequency f3 of said third resonator mode and the laser frequency or the frequency f1 of the first resonator mode has a far greater dependence on temperature than the frequency spacing f of the second frequency f2 of the higher transversal mode M2 of the laser frequency f1, a temperature deviation will lead to the resonance condition for f3 no longer being satisfied. This deviation is registered by a third demodulator 10d. Similar to what was described above, the error signal can be ascertained as the difference between f1+f*a.sub.f and f3. Here, f3 is the frequency of the resonator mode M3 at the set value of the temperature. More simply expressed, the error signal thus corresponds to the difference between f4 and f*a.sub.f. The error signal can now be sent to a heating current source 12 in order to control it.

[0079] In the example shown, optical circulators 11 and additional polarization optics 13 are used to direct the different beams to the desired demodulators 4d, 10d. The absolute frequency deviation f4 is proportional to the difference of the longitudinal indices l3-l1. Said difference is limited to about 1 GHz in this example by the modulation frequency of the acousto-optical modulator 10, but it can be higher by orders of magnitude in other embodiments. The latter can be achieved, for example, by the different light frequencies f1, f3 and possibly f2 being produced by different laser sources.

[0080] In practice, the mode index 11 can be a number in the range from about 10.sup.1 to 10.sup.5, since the optical wavelength is 1 m in order of magnitude. The frequency f1 is thus a few hundred THz. Said frequency f1 is clearly greater than f and f4. In order to be able to choose f4 to be correspondingly great as well and thus to enable a considerably more accurate self-stabilization of resonator 3a, an optical frequency comb 15 can be used; see FIG. 5.

[0081] In this embodiment example, the modes of the frequency comb 15 are directly whole number proportional to the frequency afforded by a suitable fixed frequency divider 14. Such a frequency comb 15 is commercially available. In the embodiment example in FIG. 5, the frequency of oscillator 7 is directed through the frequency divider 14 to the frequency comb 15 and sets the tooth spacing between the comb modes there. A part of the laser light is now guided by the beam splitter optics 5 to a photodiode 16, at which light from frequency comb 15 also arrives. Beat frequencies arise here due to the mixing of the optical frequencies from frequency comb 15 and laser 4. One of these beats can be isolated by means of a suitable filter. At a frequency comparator 17a (usually a phase regulator loop), said beat frequency can be compared to that of the stabilized oscillator 7. For this, the frequency of the stabilized oscillator 7 in general must again be adjusted by a fixed additional frequency multiplier 17 to the beat frequency. The resulting deviation signal canas in the previous embodiment example of FIG. 4be directed to a heating current source 12 in order to stabilize the length of the resonator 3a with high accuracy.

[0082] The effect of a temperature change is shown in FIG. 6. Here, the lines a) and b) show the modes of the optical resonator 3a at the set temperature, while the effect of the temperature change is represented in lines c) and d). A temperature change changes the optical frequency f1 of the transversal ground mode, f(l1, 0, 0).fwdarw.f(l1, 0, 0)(T), the frequency difference from this mode to the next transversal ground mode, f(0,0).fwdarw.f(0, 0)(T), and also the reference frequency f.fwdarw.f(1+). However, the relative change of the reference frequency is considerably smaller than the relative change of the two other said frequencies. For the application in FIG. 4, this means that while the temperature has little effect on the reference frequency f, it has a clearly measurable effect on the ratio of the two frequencies, in other words

[00008] $1 + 1, but .Math. .Math. .Math. \frac{.Math. .Math. f (0, 0)}{.Math. .Math. f} - \frac{.Math. .Math. f (0, 0) .Math. (T)}{.Math. .Math. f (1 +)} .Math. 0.$

[0083] In the application in FIG. 5, the change of the reference figure is transferred multiplicatively to the tooth spacing of the frequency comb 15. The original comparison mode of the frequency comb 15 with comb mode index m.sub.k and mode spacing f.sub.k has a frequency m.sub.kf.sub.k and is shifted to the frequency m.sub.kf.sub.k(1+) by a change of temperature or length. The original spacing of the selected transversal ground mode of the resonator 3a to the nearest comb mode f=f(l1, 0, 0)m.sub.kf.sub.k thus changes to f (T)=f(l1, 0, 0)(T)m.sub.kf.sub.k(1). Here, too, the reference frequency remains close to its original value, while a measurable shift of the frequency spacing |f(T)f|>>0 can arise.

[0084] That is, f1 is given by the frequency of the mode ml to which the laser 4 is adjusted; f2 is given by the frequency of a higher transversal mode m2, to which the frequency of the laser 4, in addition to the frequency of the acousto-optical frequency shifter 6, is adjusted; f3 is the frequency of the comb mode in the vicinity of f1, given by m.sub.kf.sub.k; f4 is the frequency difference f3f1.

[0085] The self regulation strategies will now be explained further by means of a numerical example.

[0086] The accuracy of the stabilization is dependent on the frequency resolution capacity of the system. Generally, the resonance frequency of a mode in an optical resonator 3a can be determined with an accuracy of

[00009] $_{f} = \frac{c}{2 .Math. L} .Math. \frac{1}{4 .Math. F} .Math. \sqrt{\frac{.Math. .Math. c}{.Math. .Math. P}} .Math. \frac{1}{\sqrt{t}} .$

Here, t is the measurement time, h is the reduced Planck constant, is the wavelength, and P is the optical power. F specifies the finesse of the resonator, which is given by the quality of the mirrors.

[0087] Realistic values of these parameters are P=100 W and F=100,000. A higher optical power can indeed improve the frequency resolution, but if the values are too large, it leads to heating of the mirrors due to absorption. Said effect is negligible at 100 W. For the following numerical examples, the value =1.55 m is selected, since this is a common wavelength in optical telecommunications. It is additionally assumed that the resonatoras in Reference 3 [Hagemann, C., et al.; Ultrastable laser with average fractional frequency drift rate below 510.sup.19/s; Optics Letters (39) 17, 5102-5105 (2014)]has a length of L=21 cm. For a measurement duration of one second, a frequency resolution of .sub.f(min)=0.064 mHz can thus be achieved. The resolution for the difference between two frequencies can then be estimated with f(min)=.sub.f(min)2=0.09 mHz.

[0088] In the embodiment example of FIG. 5, the mount 3 is made, for example, of aluminum, with a linear coefficient of expansion of 23 ppm/K, while the mirrors 1, 2 are made, for example, of ULE glass. Resonator 3a is at the zero-crossing temperature of the linear coefficient of expansion of the mirrors (22 C.) and in a vacuum. The mirrors 1, 2 have the radii of curvature described above, R2=R1 and R1=L/L.sub.opt. If the method in FIG. 4 is employed and l2l1=1 is valid, then there will be a frequency difference of f(0,0)=714 MHz between two adjacent longitudinal ground modes, while the difference between the first longitudinal ground mode and the second transversal mode will be f=464 MHz. Thus, the combined frequency resolution corresponds to a temperature change of about 10 nK. This in turn corresponds to a relative frequency change of the reference frequency of 210.sup.25 per second, which is less than the current best mark of Reference 3 by 6 orders of magnitude. The relative change of length of the resonator 3 in this example is 2.310.sup.13, and this value is 3 orders of magnitude higher than the current smallest thermal fluctuations of length from Reference 3.

[0089] In this regard, one should also note the following: If the value of 10.sup.16 from Reference 3 is assumed as the smallest possible value for thermal length fluctuations, then the frequency shift corresponding to such a change of length, which is measured by the method in FIG. 5 with frequency comb stabilization, is still 19.5 mHz. The change would therefore be solvable with the assumed values and thus regulatable. In this case, the relative change of the reference frequency would be less than 10.sup.31. It should also be stressed at this point that long-term shifts, as described in Reference 3, are suppressed to the greatest possible extent by the proposed self-correction.

[0090] The calculations presented above start from optimum values. The illustrations in FIG. 7 show the effect of deviations from the optimum parameters. Large variations of 1 K in the temperature and 1 m in the length of resonator 3a are assumed in order to illustrate their effects. The curves show the order of magnitude of the absolute value of the average relative deviation of f that results from these variations of length and temperature. Not all curves are shown. FIG. 7a) shows the effect of deviations of the temperature (T) and the length (L) from their ideal values around the operating point. Here, one can see that the assumed variation at the ideal values have a very small effect. The effect increases if there are deviations from the ideal values, but for increasing temperature deviations, the importance of length deviations becomes increasingly smaller.

[0091] FIGS. 7b), c) and d) show the effect of deviations in length and radius of curvature of a mirror for three different temperature deviations. Here again, one can see that small deviations of length have a small effect, but the system is relatively sensitive to deviations in radius of curvature. Nevertheless, even a nonideal resonator 3a can be made variation-robust by an adjustment of the temperature. For comparison, for said variation ranges, the frequency of the optical mode of a resonator 3a that is made entirely of ULE glass exhibits values on the order of magnitude of 10.sup.21 due to temperature variation and 10.sup.13 due to length variations.

[0092] Of course, the methods for self-stabilization can also be used for nonideal optical resonators, but the reference frequency f will not exhibit the best possible stability.

[0093] Finally, embodiment examples that can easily be integrated into microsystems that are suitable for mass production will now be presented. Modern communications and data processing systems increasingly contain photonic components, which consist of optical waveguides on chips. An optical waveguide is in general a structure that consists of a core with a higher refractive index than the medium surrounding it. As a result, light has propagating modes in the waveguide that can be roughly understood via an image of the total internal reflection. Meanwhile, a large number of chip-integrated laser sources, modulators, and detectors have been developed. With these components, chip-based optical resonators can be used as frequency references in a fully integrated photonic system, or a stable reference frequency f can be produced in this way in accordance with the invention.

[0094] There are various types of chip-based resonators. For example, it is also possible to modify the methods described above for waveguide geometries with integrated Bragg mirrors, so that in the end, a resonator that corresponds to the Fabry-Prot type is produced. Another possible type of waveguide resonator is the whispering gallery mode resonator, also called a ring resonator, which is very attractive for use, since it can have very high Q factors, see, for example, Reference 4 [D. T. Spencer, J. F. Bauters, M. J. R. Heck, and J. E. Bowers; Integrated waveguide coupled Si3N4 resonators in the ultrahigh-Q regime; Optica, Vol. 1, No. 3, p. 153, Sep. 20, (2014)]. This type of resonator in general consists of a closed waveguide, which can, for example, be circular, elliptical, or stadium-shaped, but in principle can take any closed form. For the sake of simplicity, round resonators are assumed in the following embodiment examples, but the description is applicable to any desired geometry by replacing the resonator length.

[0095] Light is guided into and out of the ring resonators by evanescent coupling. The resonance frequencies can be calculated via the extent and the effective refractive index of the propagating modes. However, there are no analytical methods to calculate the mode spectrum for these systems. For this reason, numerical minimizing methods must be employed. Nevertheless, even here ranges can be found for which the frequency difference between two modes is minimally dependent on the temperature.

[0096] The light modes of a waveguide have a portion in the medium surrounding the core, and said portion is mode-dependent. The fact that even here stable mode pairs can exist thus stems from the fact that different modes thereby have different effective refractive indices, which also have different temperature dependences. Thus, mode pairs for which the change of the resonance frequencies is nearly identical for small operating parameter variations, in particular temperature variations, can also be produced here.

[0097] The mode pairs can be produced in a number of waveguide arrangements. For example, all three required modes can be produced in a single ring resonator (not shown). Alternatively, the modes can be situated in three different resonators 24, 25, 26, through which greater freedoms in determining the mode properties result, see FIG. 8. Another variation is the production of a pair 35 of ring resonators that are coupled to each other, see FIG. 9. Through the coupling, a mode pair is produced whose frequency spacing can be finely determined by the fabrication parameters (radii and spacing between the rings), see, for example, Reference 5 [Zhang, Z., Dainese, M., Wosinski, L., & Qiu, M.; Resonance-splitting and enhanced notch depth in SOI ring resonators with mutual mode coupling; Optics Express, 16(7), 4621-4630 (2008)].

[0098] The resonance frequency in a circular cavity resonator is given by

[00010] $f_{r} (T) = \frac{mc}{2 .Math. .Math. .Math. rn (1 + .Math. .Math. T)} \frac{mc}{2 .Math. .Math. .Math. rn} .Math. (1 - .Math. .Math. T) .$

[0099] Here, m is the number of wavelengths in the resonator, n is the refractive index, and is the linear temperature coefficient. Quantity T again gives the deviation of the temperature from the desired set value or the operating point temperature. Small temperature changes are assumed, so that quadratic deviations can be neglected. The temperature coefficient then describes all effects that influence the propagation of the modes. These are, for example, thermal expansion, which affects the radius r of the ring and the dimensions of the waveguide, and the thermal dependence of the refractive index n of the core and jacket. In general, the dimensional dependence for all modes will be the same, but the change of the refractive index n for each mode is different. Thus, a slightly different coefficient results for each mode. The frequency difference between two modes is then given by

[00011] $.Math. .Math. f_{r} (T) = \frac{c}{2 .Math.} .Math. (\frac{m_{2} (1 -_{2} .Math. T)}{r_{2} .Math. n_{2}} - \frac{m_{1} (1 -_{1} .Math. T)}{r_{1} .Math. n_{1}}),$

wherein the subscripts 1 and 2 represent the relevant mode. The zero crossing of the first derivative with the condition m.sub.2.sub.2r.sub.1n.sub.1=m.sub.1.sub.1r.sub.2n.sub.z is found via the above expression. The required ratio between mode indexes, radii, and refractive indexes, as well as their thermal dependences can be found via these two expressions. Detailed tests showed that this condition can be relaxed somewhat in order to achieve satisfactory results, namely to the design condition m.sub.1*.sub.1*L.sub.2*n.sub.2=m.sub.2*.sub.2*L.sub.1*n.sub.10.1%.

[0100] The relationship between mode indexes and radii is limited by the light frequency, which, for example, for telecommunications applications, lies in the range around 195 THz. In addition, for a given waveguide technology, one must keep in mind the dependence of the losses on resonator radius, so that the choice of the radii can be limited. This is because waveguides have endless transmission. The longer the waveguide, the more light goes lost. Conversely, small ring radii also lead to losses because of the greater curvature. Because of this, there is a range of a particularly useful radii.

[0101] Moreover, both the refractive indices n.sub.1, n.sub.2 and the thermal coefficients .sub.1, .sub.2 are dependent on the waveguide dimensions and therefore need to be optimized together to the target value. In principle, a desired frequency difference and an appropriate size reduction can therefore be chosen for the radii r.sub.1, r.sub.2. Together with the achievable range of the refractive indices n.sub.1, n.sub.2, this results in boundaries for the selectable mode indices. Accordingly, the refractive indices n.sub.1, n.sub.2 and temperature coefficients .sub.1, .sub.2 can be selected for the desired frequency difference f. For example, it can be seen from Reference 4 that differences in the refractive index between 0% and 0.15% are absolutely achievable by means of dimensioning the waveguide. The largest corresponding difference between the temperature dependences of the two modes can be calculated to be 0.8%. For modes in a single ring (r.sub.1=r.sub.2), stable mode pairs with a smallest frequency distance of 11 GHz at a radius of 5 mm result from this. The frequency difference can be finely adjusted via the radius. In the case of two different rings, the frequency difference can be more finely and more freely adjusted via the different radii. The stable frequency is extremely sensitive to the radii of the rings both for one and for two rings. Thus, a deviation in radius by 0.002 percent (0.02 per mil) can give rise to a deviation of up to two percent in the frequency. Therefore, in practice, the manufacturing process must be adjusted very precisely to a starting frequency. However, in the electronics sphere, frequency conversion is routine and therefore a known frequency deviation can at least be corrected here and the stability can nevertheless be used. Moreover, for many applications, a stable, accurately known frequency is sufficient.

[0102] Because the mode indices are whole numbers, the above described design condition will not be exactly satisfied in practice, since the dimensions, refractive indices, and thermal dependences are subject to certain manufacturing variations. Two effects that have been neglected up to now can be used to still produce a zero crossing of the temperature dependence: First, the modes are subject to different dispersion relations, so that the refractive indices (at constant temperature) can be nicely modified, even though in steps, through the choice of the mode indices m.sub.1 and m.sub.2. Second, the modes are also subject to higher order thermal coefficients, so that the values .sub.1 and .sub.2 can be finely set by changing the operating point temperature.

[0103] Another variation for producing a stable mode pair is the coupling of two modes. Said coupling can be produced in a single ring by a reflecting element or can be produced via evanescent coupling of two rings (see Reference 5).

[0104] In a single ring, the variants of a three-dimensional mode, which propagate clockwise and counterclockwise in the ring, are coupled through this. Since the three-dimensional modes are nominally identical, their thermal coefficients will also be nearly the same, so that a high stability can be expected. The strength of the coupling, which in this case is exactly the reference frequency f, can be determined via the reflectivity. However, a single reflecting element (as used in Reference 5) will in general lead to scattering losses, since it causes a non-adiabatic change of the mode parameters and thus a scattering into free modes in the jacket.

[0105] These losses can be largely avoided through the use of two rings, since the coupling can arise by means of evanescence between the modes of the rings, so that the propagation parameters only change slowly along the rings. This application variant is well suited for stable frequencies f in the range around 250 MHz. This value results from the fact that the line widths of good ring resonators with diameters around 1 cm lie in the range of 10 MHz, while the spacing between modes with said dimensioning is 6.5 GHz (Reference 4). The stable frequency difference f is thus clearly larger than the line width, so that the modes can be easily resolved. On the other hand, it is clearly smaller than the mode spacing in one ring, so that the overlap with the next mode is vanishingly small. The strength of coupling between the rings is exponentially dependent on the minimum spacing of the rings (which do not have to be concentric), and thus can be selected. At the point where the modes of the two individual rings have the same frequency f.sub.0, two modes, which are split by the coupling 2 g, arise due to the coupling. The frequency spacing f.sub.K(T) between the two modes for rings with thermal coefficients .sub.1 and .sub.2 is given by

[00012] $.Math. .Math. f_{g} (T) = 2 .Math. g .Math. \sqrt{1 + {(f_{0} .Math. \frac{_{1} -_{2}}{2 .Math. g} .Math. T)}^{2}} 2 .Math. g + \frac{1}{4 .Math. g} .Math. {((_{1} -_{2}) .Math. f_{0} .Math. T)}^{2},$

which is thus temperature-insensitive to resonance up to a first order of magnitude. Several small corrections were neglected here, for example a small shift of temperature

[00013] $({(\frac{2 .Math. g}{f_{0}})}^{2} .Math. \frac{\sqrt{_{1} .Math._{2}}}{{(_{1} -_{2})}^{2}}),$

which for the parameters used here is much smaller than 1 Kelvin, as well as a further shift due to the final width and asymmetry of the resonance lines, and a shift due to a small temperature dependence of the strength of coupling. In practice, these shifts must be determined by measurement. Moreover, in general, the resonance frequencies of the two rings will not be identical at the desired operating point temperature. Through the difference in the thermal coefficients .sub.1 and .sub.2, the rings can, however, be brought into resonance. However, since the quadratic term of the temperature dependence is also dependent on said difference, the sensitivity can be reconciled against the susceptibility to detuning. For example, for a difference of the temperature coefficients of 0.2%, a thermal tuning of about 15 K is necessary to achieve the resonance condition. Nonetheless, this variant can then be used to produce a stable reference frequency and another mode of the system, or a third ring can serve as temperature-dependent element for long-term stabilization according to the invention. It should be noted at this point that the same effect can also be achieved with coupled Fabry-Prot resonators.

[0106] A variant with three different resonators is shown in FIG. 8. Here, the light from three different modulated laser light sourcesa first source 18 to produce light of frequency f1, a second source 19 to produce light of frequency f2 and a third source 20 to produce light of frequency f3is guided in three waveguidesa first waveguide 21, a second waveguide 22, and a third waveguide 23. The modes of these waveguides overlap slightly with those of three ring resonatorsa first ring resonator 24, a second ring resonator 25, and a third ring resonator 26and in this way couple evanescent light in their selected resonance modes. The transmitted light is collected by the relevant detectorsa first detector 27, a second detector 28, and a third detector 29. Here, too, the frequency of each laser 18, 19, 20 can be kept at the desired resonance in the relevant ring 24, 25, 26 by means of the side bands produced by the modulation, so that the frequencies f1, f2, f3 are stabilized. Again the Pound-Drever-Hall technique (Reference 6) is used for this. The modulation of the lasers 18, 19, 20, the demodulation of the detector signals, and the regulation of the laser beam are carried out in the integrated electronic modules 37. The deviation of the laser frequency from the resonant mode frequency, which can be seen from the demodulation, is carried out by a correction of the laser beam for each laser light source 18, 19, 20 individually.

[0107] Moreover, the light from the laser light sources 18 and 19 is guided by means of integrated beam splittersa first beam splitter 30 and a second beam splitter 31to another, fourth detector 32, at which the beat frequency f between the two mode frequencies f1, f2 in rings 24 and 25 is measured. The same technique can be used for the laser light sources 19 and 20 via the second beam splitter 31 and a third beam splitter 33 in order to measure, at a fifth detector 34, the beat f4 between the mode frequencies f2 and f3 of rings 25 and 26.

[0108] Because of the described design of the resonators 24, 25, the beat frequency f at the fourth detector 32 is highly stable, while the beat frequency f4 at detector 34 is sensitive to disruptions. A part of the light, which carries the stable beat frequency f, can be sent on to an optical output 36 as an optical signal of the chip and can be distributed via optical fibers or as a free beam to other devices.

[0109] Moreover, the measured beat frequency can, just like in the preceding description, serve for self-stabilization of the device or the reference frequency f: The beat frequencies f and f4 from detectors 32 and 34 can again be compared at an electronic unit 38, and changes of preset values can be used for regulation of a heating current source. Thus, the temperature of the chip and thus the clock frequency can be precisely stabilized.

[0110] Preferably, this needs to take place considerably slower than the correction of the laser beams. In practice, this condition is easily satisfied, since the thermal regulation at best can take place on a millisecond scale, while the laser beam can be corrected in less than one microsecond.

[0111] Last, a variant with coupled resonators is shown in FIG. 9. All elements keep their function as in FIG. 8 up to rings 25 and 26, which are replaced by the coupled ring resonator pair 35. In this drawing, the laser modulation and regulation loops, as well as the temperature regulation, were left out for clarity, but they likewise have the same functions here as in FIG. 8. The modes from the sources 18 and 19 couple via the waveguides 21 and 22 here, each to a mode of the stable mode pair in ring resonator pair 35. FIGS. 10a and 10b serve to illustrate this.

[0112] FIG. 10a shows the crossing between two modes of the coupled ring resonator pair 35. The dashed lines represent the frequencies of the modes of the individual resonators of the ring resonator pair 35, while the solid lines show the frequencies of the modes of the coupled system. Through the different thermal coefficients, the frequencies of the individual rings intersect. The coupling leads the resonance to be nearly parallel to the frequencies of the coupled system.

[0113] FIG. 10b shows the course of the frequency difference f.sub.g(T) for a difference of 0.2% between the thermal coefficients and a coupling frequency of 2 g=250 MHz. It should be noted here that the chip-integrated resonator, of course, is also suitable for the frequency comb-based stabilization (described by means of FIGS. 5 and 6), and thus can be used for integrated stabilization of chip-based frequency combs.

[0114] The preceding descriptions always started from basically unstable light sources. If a light source with low frequency variations about a frequency f1 is available, the principle of the invention can also be used to carry said stability to the difference frequency f2f1. In doing so, the variations of the light source frequency are additionally suppressed by the lower (for example, quadratic or cubic) dependence of the frequency difference.

[0115] In conclusion, the following should be quite generally noted for material choice: Usually, silicon oxide, titanium oxide, silicon or silicon nitride are used as materials for waveguide systems at wavelengths 1.5 m. Any material that is transparent in the desired wavelength range and that can be milled and polished in the correct shape (or otherwise produced in the correct shape), can be used as the mirror substrate. Quartz glass and silicon are useful for the indicated 1.5 m wavelength. Metals, crystals (for example silicon or quartz glass) or ceramics can be used as the mount (see mount 3 in FIG. 1).

REFERENCE NUMBER LIST

[0116] 1 First mirror [0117] 2 Second mirror [0118] 3 Mount [0119] 3a Fabry-Prot resonator [0120] 4 Laser [0121] 4a Electro-optical modulator [0122] 4b Polarizing beam splitter [0123] 4c Quarterwave retardation optics [0124] 4d Demodulator [0125] 5 Beam splitter [0126] 6 Acousto-optical frequency shifter [0127] 6a Second electro-optical modulator [0128] 6b Second polarizing beam splitter [0129] 6c Second quarter-wave retardation optics [0130] 6d Second demodulator [0131] 6e Structured phase plate [0132] 7 Oscillator [0133] 8 Temperature regulator [0134] 9 Frequency multiplier [0135] 10 Additional acousto-optical modulator [0136] 10a Additional electro-optical modulator [0137] 10d Third demodulator [0138] 11 Optical circulator [0139] 12 Heating current source [0140] 13 Additional polarization optics [0141] 14 Frequency divider [0142] 15 Frequency comb [0143] 16 Photodiode [0144] 17 Additional frequency multiplier [0145] 17a Frequency comparator [0146] 18 First laser light source [0147] 19 Second laser light source [0148] 20 Third laser light source [0149] 21 First waveguide [0150] 22 Second waveguide [0151] 23 Third waveguide [0152] 24 First ring resonator [0153] 25 Second ring resonator [0154] 26 Third ring resonator [0155] 27 First detector [0156] 28 Second detector [0157] 29 Third detector [0158] 30 First beam splitter [0159] 31 Second beam splitter [0160] 32 Beam detector [0161] 33 Third beam splitter [0162] 34 Fifth detector [0163] 35 Coupled ring resonator pair [0164] 36 Optical output [0165] 37 Electronic module [0166] 38 Electronic unit

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