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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 16 — Aug. 11, 2014
  • pp: 19277–19283
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Sub-kHz lasing of a CaF2 whispering gallery mode resonator stabilized fiber ring laser

M. C. Collodo, F. Sedlmeir, B. Sprenger, S. Svitlov, L. J. Wang, and H. G. L. Schwefel  »View Author Affiliations


Optics Express, Vol. 22, Issue 16, pp. 19277-19283 (2014)
http://dx.doi.org/10.1364/OE.22.019277


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Abstract

We utilize a high quality calcium fluoride whispering-gallery-mode resonator to passively stabilize a simple erbium doped fiber ring laser with an emission frequency of 196THz (wavelength 1530nm) to an instantaneous linewidth below 650Hz. This corresponds to a relative stability of 3.3 × 10−12 over 16μs. In order to characterize the linewidth we use two identical self-built lasers and a commercial laser to determine the individual lasing linewidth via the three-cornered-hat method. We further show that the lasers are finely tunable throughout the erbium gain region.

© 2014 Optical Society of America

1. Introduction

Before reaching the fundamental thermal noise floor limit [7

7. A. B. Matsko, A. A. Savchenkov, N. Yu, and L. Maleki, “Whispering-gallery-mode resonators as frequency references. I. Fundamental limitations,” J. Opt. Soc. B 24, 1324–1335 (2007). [CrossRef]

, 10

10. J. Alnis, A. Schliesser, C. Y. Wang, J. Hofer, T. J. Kippenberg, and T. W. Hänsch, “Thermal-noise-limited crystalline whispering-gallery-mode resonator for laser stabilization,” Phys. Rev. A 84, 011804 (2011). [CrossRef]

12

12. A. Chijioke, Q.-F. Chen, A. Y. Nevsky, and S. Schiller, “Thermal noise of whispering-gallery resonators,” Phys. Rev. A 85, 053814 (2012). [CrossRef]

], the fractional stability of a monolithic reference cavity is mainly determined by its deformation due to mechanical vibration or thermal effects, whose influence scales with the cavity’s dimensions [13

13. B. Sprenger, H. G. L. Schwefel, Z. H. Lu, S. Svitlov, and L. J. Wang, “CaF2 whispering-gallery-mode-resonator stabilized-narrow-linewidth laser,” Opt. Lett. 35, 2870–2872 (2010). [CrossRef] [PubMed]

]. Recently, there has been progress in reducing the effect of thermal noise by utilizing the birefringence of magnesium fluoride [14

14. L. M. Baumgartel, R. J. Thompson, and N. Yu, “Frequency stability of a dual-mode whispering gallery mode optical reference cavity,” Opt. Express 20, 29798–29806 (2012). [CrossRef]

, 15

15. I. Fescenko, J. Alnis, A. Schliesser, C. Y. Wang, T. J. Kippenberg, and T. W. Hänsch, “Dual-mode temperature compensation technique for laser stabilization to a crystalline whispering gallery mode resonator,” Opt. Express 20, 19185–19193 (2012). [CrossRef] [PubMed]

] or by increasing the resonator size [16

16. J. Li, H. Lee, and K. J. Vahala, “Microwave synthesizer using an on-chip Brillouin oscillator,” Nat. Commun. 4, 2097 (2013).

, 17

17. H. Lee, M.-G. Suh, T. Chen, J. Li, S. A. Diddams, and K. J. Vahala, “Spiral resonators for on-chip laser frequency stabilization,” Nat. Commun. 4, 2468 (2013). [CrossRef]

]. Therefore, and due to their compact sizes, WGM resonators are eminently suitable as a frequency reference [18

18. Y. K. Chembo, L. M. Baumgartel, and N. Yu, “Toward whispering-gallery-mode disk resonators for metrological applications,” SPIE Newsroom (2012). [CrossRef]

, 19

19. J. Li, H. Lee, and K. J. Vahala, “Low-noise brillouin laser on a chip at 1064 nm,” Opt. Lett. 39, 287–290 (2014). [CrossRef] [PubMed]

].

Here we report the setup and characterization of a free running widely tunabe fiber ring laser providing lasing linewidths below 1kHz. This is achieved by resonantly filtering the broad emission spectrum of an erbium doped fiber via the narrow-linewidth modes of a WGM resonator [20

20. K. Kieu and M. Mansuripur, “Active Q switching of a fiber laser with a microsphere resonator,” Opt. Lett. 31, 3568–3570 (2006). [CrossRef] [PubMed]

, 21

21. K. Kieu and M. Mansuripur, “Fiber laser using a microsphere resonator as a feedback element,” Opt. Lett. 32, 244–246 (2007). [CrossRef] [PubMed]

]. Only these narrow modes can pass the resonator and achieve gain in the following round trip. This establishes a narrow linewidth lasing behavior which can be tuned throughout the gain region. In order to quantify the lasing linewidth two identical systems were built which allow for the first time to determine the individual linewidth of the setups.

2. Experimental setup

Fig. 1 (a) Sketch of one of the whispering gallery mode resonator filtered lasers. The active medium is an erbium doped fiber with an emission spectrum in the telecommunication C-band. Broad shaping of the emission spectrum is achieved by a bandpass filter. Passive filtering of the conventional telecom fiber loop lasing modes is provided by a WGM resonator. Precise tunable single mode lasing is obtained without further active stabilization techniques. (b) Photograph of the WGM resonator sandwiched by the two prisms.

As our main task is the linewidth measurement, the tunability of the lasing frequency is paramount. With a broadband (1nm FWHM) optical bandpass filter a coarse tuning of the lasing mode’s wavelength over the whole emission spectrum of the erbium doped fiber in steps of the WGM resonator’s free spectral range (FSR) (∼20GHz) is possible due to mode competition. Further fine tuning can be achieved via temperature control of the resonator. Carefully combining both methods yields the possibility to continuously tune over more than a FSR.

In our experiment we observed a refinement from a multimodal lasing behavior spanning over 50GHz to a sub-kHz lasing line by including the WGM filter. Furthermore, in comparison with the cold cavity linewidth of the resonator (sub-MHz) the active lasing reduces the linewidth and improves the Q by three orders of magnitude. In order to verify these results, we modeled our ring laser setup analytically following and extending the approach by Wang [23

23. L. Wang, “Causal ‘all-pass’ filters and Kramers-Kronig relations,” Opt. Comm. 213, 27–32 (2002). [CrossRef]

]. We modeled the filtering mechanism introduced by the WGM resonator and implemented this model in an iterative numerical simulation, taking into account gain saturation. The experimentally observed characteristics could be reproduced by appropriate choice of parameters, the most crucial being the fiber cavity’s and the WGM resonator’s Q factors and the saturated intra-cavity power. The laser’s emission spectrum narrows with increasing intracavity power, which agrees well with a fully analytic approach [24

24. M. Eichhorn and M. Pollnau, “The Q-factor of a continuous-wave laser,” CLEO: Science and Innovations (2012).

]. Our simulation does not cover further aspects regarding the WGM resonator’s instability due to an increased lasing power, and is therefore not able to determine the optimal intracavity power.

3. Measurement procedure

As the frequency stability of the individual lasers cannot be measured directly, two identical systems were built. The beat note generated by mixing the emission spectra of the two lasers allows us to reconstruct the frequency stability of the combined system. For a later analysis we also recorded the beat note of both self-built systems with a commercial external cavity stabilized diode laser (Toptica DL pro design). We recorded 50ms (sampling rate is 200 megasamples/second, with 8 bit vertical resolution) of the beat note traces of the two approximately 10MHz detuned WGLs. First we analyze the phase noise of the system. By modeling the time-domain beat signal as V(t) = (V0 + ε(t))cos(2πf0t + φ(t)), where f0 is the carrier frequency, φ(t) the time-dependent phase noise and ε(t) the time-dependent amplitude noise, we can perform a demodulation to extract both the phase noise and amplitude noise. For this purpose we define the two quadratures I = V(t) · cos(2πf0t) and Q = V(t) · sin(2πf0t). The resulting terms are filtered from all the high frequency components of the quadratures and the time-domain phase noise can be written as φ(t)=arctanIQ [25

25. A. H. Safavi-Naeini, private communications.

]. In Fig. 2(a) we can see the frequency-domain phase noise extracted from the measured beat note traces. It clearly follows the usual f−2 Lorentzian behavior.

Fig. 2 (a) Computationally determined phase noise including a fit of a Lorentzian at a central frequency of f0 = 9.468MHz. (b) Extracted oscillation frequencies of the beat notes of the three different lasers beat a the same time.

For the computation of the beat note’s frequency, contiguous, non overlapping basic time intervals of 1μs length are chosen from the trace in order to obtain a frequency stream. This basic time interval is chosen such that it covers at least ten periods of the beat note signal. We perform a nonlinear least-square fit of a sine function over these separate time domain traces. The only fitting parameters in the used model are frequency, amplitude, offset and phase, which are all assumed to be constant over the basic 1μs fit interval. The resulting time resolved oscillation frequency values of all three measured beat note traces are depicted in Fig. 2(b). We prefer this method in comparison to a conventional frequency counter because of the higher flexibility and robustness in case of a low signal to noise ratio. Furthermore, we avoid ambiguity in the interpretation of the resulting values and we are free to compute different types of variances [26

26. E. Rubiola, “On the measurement of frequency and of its sample variance with high-resolution counters,” Rev. Sci. Instr. 76, 054703 (2005). [CrossRef]

]. In order to determine the frequency stability of our lasers with respect to the averaging time the Allan deviation [27

27. D. Allan, H. Hellwig, P. Kartaschoff, J. Vanier, J. Vig, G. Winkler, and N. Yannoni, “Standard terminology for fundamental frequency and time metrology,” in “Frequency Control Symposium, 1988., Proceedings of the 42nd Annual,” (1988), pp. 419–425.

, 28

28. J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGunigal, J. A. Mullen, W. L. Smith, R. L. Sydnor, R. F. C. Vessot, and G. M. R. Winkler, “Characterization of frequency stability,” IEEE T. Instrum. Meas. IM-20, 105–120 (1971). [CrossRef]

]
σ2=12(M1)i=1M1(fifi+1)2
(1)
is used, where σ is the Allan deviation of the lasing frequency, fi denotes the value of the lasing frequency referring to the i-th basic time interval and M is the total amount of basic time intervals. Pre-averaging of the frequency stream over multiple basic time intervals allows us to compute Allan deviations for different averaging time scales.

4. Results

Fig. 3 Allan deviation values (corresponding to lasing linewidth in Hz) and relative stabilities (corresponding to lasing Q factor) for the whispering gallery lasers (WGL). (a) Direct evaluation of the beat note signal between WGL1 and WGL2 reports the combination of both noise sources. (b) Individual noise components were obtained via the three-cornered hat method, a possible correlation was taken into account.

In order to extract the individual frequency stabilities we add a third lasing system, namely a commercial Toptica DLpro design laser and perform a three-cornered hat measurement [29

29. J. Gray and D. Allan, “A method for estimating the frequency stability of an individual oscillator,” in “28th Annual Symposium on Frequency Control. 1974,” (1974), pp. 243–246.

,30

30. E. de Carlos López and J. M. López Romero, “Frequency stability estimation of semiconductor lasers using the three-cornered hat method,” (2006).

]. The individual linewidth is determined by recording the beat notes from the three possible combinations of laser pairs simultaneously and solving for the single laser variances
2σWGL12=σWGL1+WGL22+σWGL1+DLpro2σWGL2+DLpro2
(2)
(and permutations of this equation). This holds only if no correlation in the lasing frequency characteristics is predominant. For a more incontestable approach taking into account possible correlations, a correlation removal algorithm by Premoli and Tavella [31

31. A. Premoli and P. Tavella, “A revisited three-cornered hat method for estimating frequency standard instability,” IEEE Trans. Instrum. Meas. 42, 7–13 (1993). [CrossRef]

] has been used with WGL1 as a reference (choice of reference is not significant). The individual frequency stabilities, revised in this manner, are shown in Fig. 3(b).

The more stable WGM resonator laser reaches a relative stability of (1.67 ± 1.60) × 10−12 for an averaging time of 16μs. Equivalently, this corresponds to a lasing linewidth of (328 ± 314) Hz at the laser’s emission frequency of 196THz. In a conservative estimate we report a relative stability of 3.3 × 10−12 and a linewidth of 650Hz, respectively. These values agree well with the results of the prior estimation using the combined frequency stability directly. Compared to the phase noise considerations presented in Fig. 2(a) we find a noise stability which is one order of magnitude worse.

5. Conclusions

Our evaluation method for the lasing stability is based on the analysis of the digitized time domain beat note traces and avoids the standard frequency counter approach. The reason for the different performances of the phase noise considerations and the Allan deviation lies in the fact that the latter takes into account the instantaneous frequency components which are determined from the beat trace, whereas the former assumes a perfect theoretical carrier frequency. Thus the Allan deviation gives a more complete noise measure.

Acknowledgments

The authors would like to thank Dmitry V. Strekalov and Josef U. Fürst for stimulating discussions and Gerd Leuchs for his support. We acknowledge support by the Deutsche Forschungsgemeinschaft and the Friedrich-Alexander-Universität Erlangen-Nürnberg within the funding program Open Access Publishing.

References and links

1.

S. N. Lea, “Limits to time variation of fundamental constants from comparisons of atomic frequency standards,” Rep. Prog. Phys. 70, 1473–1523 (2007). [CrossRef]

2.

M. Baaske and F. Vollmer, “Optical resonator biosensors: Molecular diagnostic and nanoparticle detection on an integrated platform,” ChemPhysChem 13, 427–436 (2012). [CrossRef] [PubMed]

3.

G. N. Conti, S. Berneschi, A. Barucci, F. Cosi, S. Soria, and C. Trono, “Fiber ring laser for intracavity sensing using a whispering-gallery-mode resonator,” Opt. Lett. 37, 2697–2699 (2012). [CrossRef]

4.

K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003). [CrossRef] [PubMed]

5.

I. S. Grudinin, A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Ultra high Q crystalline microcavities,” Opt. Comm. 265, 33–38 (2006). [CrossRef]

6.

A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, and L. Maleki, “Optical resonators with ten million finesse,” Opt. Express 15, 6768–6773 (2007). [CrossRef] [PubMed]

7.

A. B. Matsko, A. A. Savchenkov, N. Yu, and L. Maleki, “Whispering-gallery-mode resonators as frequency references. I. Fundamental limitations,” J. Opt. Soc. B 24, 1324–1335 (2007). [CrossRef]

8.

W. Liang, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “Whispering-gallery-mode-resonator-based ultranarrow linewidth external-cavity semiconductor laser,” Opt. Lett. 35, 2822–2824 (2010). [CrossRef] [PubMed]

9.

E. Dale, W. Liang, D. Eliyahu, A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, D. Seidel, and L. Maleki, “On phase noise of self-injection locked semiconductor lasers,” Proc. SPIE 8960, 89600X (2014).

10.

J. Alnis, A. Schliesser, C. Y. Wang, J. Hofer, T. J. Kippenberg, and T. W. Hänsch, “Thermal-noise-limited crystalline whispering-gallery-mode resonator for laser stabilization,” Phys. Rev. A 84, 011804 (2011). [CrossRef]

11.

K. Numata, A. Kemery, and J. Camp, “Thermal-Noise limit in the frequency stabilization of lasers with rigid cavities,” Phys. Rev. Lett. 93, 250602 (2004). [CrossRef]

12.

A. Chijioke, Q.-F. Chen, A. Y. Nevsky, and S. Schiller, “Thermal noise of whispering-gallery resonators,” Phys. Rev. A 85, 053814 (2012). [CrossRef]

13.

B. Sprenger, H. G. L. Schwefel, Z. H. Lu, S. Svitlov, and L. J. Wang, “CaF2 whispering-gallery-mode-resonator stabilized-narrow-linewidth laser,” Opt. Lett. 35, 2870–2872 (2010). [CrossRef] [PubMed]

14.

L. M. Baumgartel, R. J. Thompson, and N. Yu, “Frequency stability of a dual-mode whispering gallery mode optical reference cavity,” Opt. Express 20, 29798–29806 (2012). [CrossRef]

15.

I. Fescenko, J. Alnis, A. Schliesser, C. Y. Wang, T. J. Kippenberg, and T. W. Hänsch, “Dual-mode temperature compensation technique for laser stabilization to a crystalline whispering gallery mode resonator,” Opt. Express 20, 19185–19193 (2012). [CrossRef] [PubMed]

16.

J. Li, H. Lee, and K. J. Vahala, “Microwave synthesizer using an on-chip Brillouin oscillator,” Nat. Commun. 4, 2097 (2013).

17.

H. Lee, M.-G. Suh, T. Chen, J. Li, S. A. Diddams, and K. J. Vahala, “Spiral resonators for on-chip laser frequency stabilization,” Nat. Commun. 4, 2468 (2013). [CrossRef]

18.

Y. K. Chembo, L. M. Baumgartel, and N. Yu, “Toward whispering-gallery-mode disk resonators for metrological applications,” SPIE Newsroom (2012). [CrossRef]

19.

J. Li, H. Lee, and K. J. Vahala, “Low-noise brillouin laser on a chip at 1064 nm,” Opt. Lett. 39, 287–290 (2014). [CrossRef] [PubMed]

20.

K. Kieu and M. Mansuripur, “Active Q switching of a fiber laser with a microsphere resonator,” Opt. Lett. 31, 3568–3570 (2006). [CrossRef] [PubMed]

21.

K. Kieu and M. Mansuripur, “Fiber laser using a microsphere resonator as a feedback element,” Opt. Lett. 32, 244–246 (2007). [CrossRef] [PubMed]

22.

B. Sprenger, H. G. L. Schwefel, and L. J. Wang, “Whispering-gallery-mode-resonator-stabilized narrow-linewidth fiber loop laser,” Opt. Lett. 34, 3370–3372 (2009). [CrossRef] [PubMed]

23.

L. Wang, “Causal ‘all-pass’ filters and Kramers-Kronig relations,” Opt. Comm. 213, 27–32 (2002). [CrossRef]

24.

M. Eichhorn and M. Pollnau, “The Q-factor of a continuous-wave laser,” CLEO: Science and Innovations (2012).

25.

A. H. Safavi-Naeini, private communications.

26.

E. Rubiola, “On the measurement of frequency and of its sample variance with high-resolution counters,” Rev. Sci. Instr. 76, 054703 (2005). [CrossRef]

27.

D. Allan, H. Hellwig, P. Kartaschoff, J. Vanier, J. Vig, G. Winkler, and N. Yannoni, “Standard terminology for fundamental frequency and time metrology,” in “Frequency Control Symposium, 1988., Proceedings of the 42nd Annual,” (1988), pp. 419–425.

28.

J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGunigal, J. A. Mullen, W. L. Smith, R. L. Sydnor, R. F. C. Vessot, and G. M. R. Winkler, “Characterization of frequency stability,” IEEE T. Instrum. Meas. IM-20, 105–120 (1971). [CrossRef]

29.

J. Gray and D. Allan, “A method for estimating the frequency stability of an individual oscillator,” in “28th Annual Symposium on Frequency Control. 1974,” (1974), pp. 243–246.

30.

E. de Carlos López and J. M. López Romero, “Frequency stability estimation of semiconductor lasers using the three-cornered hat method,” (2006).

31.

A. Premoli and P. Tavella, “A revisited three-cornered hat method for estimating frequency standard instability,” IEEE Trans. Instrum. Meas. 42, 7–13 (1993). [CrossRef]

OCIS Codes
(060.2390) Fiber optics and optical communications : Fiber optics, infrared
(060.2410) Fiber optics and optical communications : Fibers, erbium
(140.3410) Lasers and laser optics : Laser resonators
(140.3560) Lasers and laser optics : Lasers, ring
(140.3425) Lasers and laser optics : Laser stabilization
(130.7408) Integrated optics : Wavelength filtering devices

ToC Category:
Lasers and Laser Optics

History
Original Manuscript: June 17, 2014
Revised Manuscript: July 22, 2014
Manuscript Accepted: July 24, 2014
Published: August 1, 2014

Citation
M. C. Collodo, F. Sedlmeir, B. Sprenger, S. Svitlov, L. J. Wang, and H. G. L. Schwefel, "Sub-kHz lasing of a CaF2 whispering gallery mode resonator stabilized fiber ring laser," Opt. Express 22, 19277-19283 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-16-19277


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References

  1. S. N. Lea, “Limits to time variation of fundamental constants from comparisons of atomic frequency standards,” Rep. Prog. Phys.70, 1473–1523 (2007). [CrossRef]
  2. M. Baaske and F. Vollmer, “Optical resonator biosensors: Molecular diagnostic and nanoparticle detection on an integrated platform,” ChemPhysChem13, 427–436 (2012). [CrossRef] [PubMed]
  3. G. N. Conti, S. Berneschi, A. Barucci, F. Cosi, S. Soria, and C. Trono, “Fiber ring laser for intracavity sensing using a whispering-gallery-mode resonator,” Opt. Lett.37, 2697–2699 (2012). [CrossRef]
  4. K. J. Vahala, “Optical microcavities,” Nature424, 839–846 (2003). [CrossRef] [PubMed]
  5. I. S. Grudinin, A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Ultra high Q crystalline microcavities,” Opt. Comm.265, 33–38 (2006). [CrossRef]
  6. A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, and L. Maleki, “Optical resonators with ten million finesse,” Opt. Express15, 6768–6773 (2007). [CrossRef] [PubMed]
  7. A. B. Matsko, A. A. Savchenkov, N. Yu, and L. Maleki, “Whispering-gallery-mode resonators as frequency references. I. Fundamental limitations,” J. Opt. Soc. B24, 1324–1335 (2007). [CrossRef]
  8. W. Liang, V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, D. Seidel, and L. Maleki, “Whispering-gallery-mode-resonator-based ultranarrow linewidth external-cavity semiconductor laser,” Opt. Lett.35, 2822–2824 (2010). [CrossRef] [PubMed]
  9. E. Dale, W. Liang, D. Eliyahu, A. A. Savchenkov, V. S. Ilchenko, A. B. Matsko, D. Seidel, and L. Maleki, “On phase noise of self-injection locked semiconductor lasers,” Proc. SPIE8960, 89600X (2014).
  10. J. Alnis, A. Schliesser, C. Y. Wang, J. Hofer, T. J. Kippenberg, and T. W. Hänsch, “Thermal-noise-limited crystalline whispering-gallery-mode resonator for laser stabilization,” Phys. Rev. A84, 011804 (2011). [CrossRef]
  11. K. Numata, A. Kemery, and J. Camp, “Thermal-Noise limit in the frequency stabilization of lasers with rigid cavities,” Phys. Rev. Lett.93, 250602 (2004). [CrossRef]
  12. A. Chijioke, Q.-F. Chen, A. Y. Nevsky, and S. Schiller, “Thermal noise of whispering-gallery resonators,” Phys. Rev. A85, 053814 (2012). [CrossRef]
  13. B. Sprenger, H. G. L. Schwefel, Z. H. Lu, S. Svitlov, and L. J. Wang, “CaF2 whispering-gallery-mode-resonator stabilized-narrow-linewidth laser,” Opt. Lett.35, 2870–2872 (2010). [CrossRef] [PubMed]
  14. L. M. Baumgartel, R. J. Thompson, and N. Yu, “Frequency stability of a dual-mode whispering gallery mode optical reference cavity,” Opt. Express20, 29798–29806 (2012). [CrossRef]
  15. I. Fescenko, J. Alnis, A. Schliesser, C. Y. Wang, T. J. Kippenberg, and T. W. Hänsch, “Dual-mode temperature compensation technique for laser stabilization to a crystalline whispering gallery mode resonator,” Opt. Express20, 19185–19193 (2012). [CrossRef] [PubMed]
  16. J. Li, H. Lee, and K. J. Vahala, “Microwave synthesizer using an on-chip Brillouin oscillator,” Nat. Commun.4, 2097 (2013).
  17. H. Lee, M.-G. Suh, T. Chen, J. Li, S. A. Diddams, and K. J. Vahala, “Spiral resonators for on-chip laser frequency stabilization,” Nat. Commun.4, 2468 (2013). [CrossRef]
  18. Y. K. Chembo, L. M. Baumgartel, and N. Yu, “Toward whispering-gallery-mode disk resonators for metrological applications,” SPIE Newsroom (2012). [CrossRef]
  19. J. Li, H. Lee, and K. J. Vahala, “Low-noise brillouin laser on a chip at 1064 nm,” Opt. Lett.39, 287–290 (2014). [CrossRef] [PubMed]
  20. K. Kieu and M. Mansuripur, “Active Q switching of a fiber laser with a microsphere resonator,” Opt. Lett.31, 3568–3570 (2006). [CrossRef] [PubMed]
  21. K. Kieu and M. Mansuripur, “Fiber laser using a microsphere resonator as a feedback element,” Opt. Lett.32, 244–246 (2007). [CrossRef] [PubMed]
  22. B. Sprenger, H. G. L. Schwefel, and L. J. Wang, “Whispering-gallery-mode-resonator-stabilized narrow-linewidth fiber loop laser,” Opt. Lett.34, 3370–3372 (2009). [CrossRef] [PubMed]
  23. L. Wang, “Causal ‘all-pass’ filters and Kramers-Kronig relations,” Opt. Comm.213, 27–32 (2002). [CrossRef]
  24. M. Eichhorn and M. Pollnau, “The Q-factor of a continuous-wave laser,” CLEO: Science and Innovations (2012).
  25. A. H. Safavi-Naeini, private communications.
  26. E. Rubiola, “On the measurement of frequency and of its sample variance with high-resolution counters,” Rev. Sci. Instr.76, 054703 (2005). [CrossRef]
  27. D. Allan, H. Hellwig, P. Kartaschoff, J. Vanier, J. Vig, G. Winkler, and N. Yannoni, “Standard terminology for fundamental frequency and time metrology,” in “Frequency Control Symposium, 1988., Proceedings of the 42nd Annual,” (1988), pp. 419–425.
  28. J. A. Barnes, A. R. Chi, L. S. Cutler, D. J. Healey, D. B. Leeson, T. E. McGunigal, J. A. Mullen, W. L. Smith, R. L. Sydnor, R. F. C. Vessot, and G. M. R. Winkler, “Characterization of frequency stability,” IEEE T. Instrum. Meas.IM-20, 105–120 (1971). [CrossRef]
  29. J. Gray and D. Allan, “A method for estimating the frequency stability of an individual oscillator,” in “28th Annual Symposium on Frequency Control. 1974,” (1974), pp. 243–246.
  30. E. de Carlos López and J. M. López Romero, “Frequency stability estimation of semiconductor lasers using the three-cornered hat method,” (2006).
  31. A. Premoli and P. Tavella, “A revisited three-cornered hat method for estimating frequency standard instability,” IEEE Trans. Instrum. Meas.42, 7–13 (1993). [CrossRef]

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