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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 4 — Feb. 24, 2014
  • pp: 4678–4691
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Comparative analysis of spectral coherence in microresonator frequency combs

Victor Torres-Company, David Castelló-Lurbe, and Enrique Silvestre  »View Author Affiliations


Optics Express, Vol. 22, Issue 4, pp. 4678-4691 (2014)
http://dx.doi.org/10.1364/OE.22.004678


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Abstract

Microresonator combs exploit parametric oscillation and nonlinear mixing in an ultrahigh-Q cavity. This new comb generator offers unique potential for chip integration and access to high repetition rates. However, time-domain studies reveal an intricate spectral coherence behavior in this type of platform. In particular, coherent, partially coherent or incoherent combs have been observed using the same microresonator under different pumping conditions. In this work, we provide a numerical analysis of the coherence dynamics that supports the above experimental findings and verify particular design rules to achieve spectrally coherent microresonator combs. A particular emphasis is placed in understanding the differences between so-called Type I and Type II combs.

© 2014 Optical Society of America

1. Introduction

An optical frequency comb is a laser source with a spectrum composed by a set of evenly spaced components that maintain the phase coherence across the whole bandwidth. Thanks to the self-referencing technique for femtosecond mode-locked lasers [1

1. D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, and S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000). [CrossRef] [PubMed]

, 2

2. S. A. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, and T. W. Hänsch, “Direct link between microwave and optical frequencies with a 300 THz femtosecond laser comb,” Phys. Rev. Lett. 84, 5102–5105 (2000). [CrossRef] [PubMed]

], it is now possible to synthesize/measure absolute optical frequencies with a performance level that before was only achievable by specialized laboratories [3

3. T. W. Hänsch, “Nobel Lecture: Passion for precision,” Rev. Mod. Phys. 78, 1297–1309 (2006). [CrossRef]

]. This has led to a revolution in different fields, ranging from optical clocks to precision spectroscopy [4

4. N. R. Newbury, “Searching for applications with a fine-tooth comb,” Nature Photon. 5, 186–188 (2011). [CrossRef]

]. These applications mainly use mode-locked lasers, whose repetition rates are typically less than 10 GHz. There are other applications (such as optical arbitrary waveform generation, coherent optical communications or radio-frequency photonics) that do not require self-referencing but would benefit from higher repetition rates and smaller sized combs [5

5. V. Torres-Company and A. M. Weiner, “Optical frequency comb technology for ultra-broadband radio-frequency photonics,” Laser and Photon. Rev. (in press, 2013). DOI [CrossRef] .

]. Recently, a novel platform has emerged with the prospect to achieve these two features simultaneously: the microresonator frequency comb [6

6. T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011). [CrossRef] [PubMed]

].

However, before realizing the full potential of this novel platform one must ensure that the requirement of full coherence across the bandwidth is indeed satisfied. Strikingly, recent experiments by Weiner and colleagues [24

24. F. Ferdous, H. X Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011). [CrossRef]

] and Papp and Diddams [25

25. S. B. Papp and S. A. Diddams, “Spectral and temporal characterization of a fused-quartz-microresonator optical frequency comb,” Phys. Rev. A 84, 053833 (2011). [CrossRef]

] revealed that this is not always the case. They focused on getting transform-limited pulses from microresonator combs by line-by-line pulse shaping [26

26. S. T. Cundiff and A. M. Weiner, “Optical arbitrary waveform generation,” Nat. Photonics 4, 760–766 (2010). [CrossRef]

]. Following the nomenclature of [24

24. F. Ferdous, H. X Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011). [CrossRef]

], there was a clear distinction between the noise performance exhibited by combs where the spectral distance between the CW pump laser and the first oscillating frequencies in the comb corresponded to one (Type I) or more (Type II) cavity’s FSRs (see Fig. 1) [24

24. F. Ferdous, H. X Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011). [CrossRef]

, 27

27. F. Ferdous, H. X. Miao, P. H. Wang, D. E. Leaird, K. Srinivasan, L. Chen, V. Aksyuk, and A. M. Weiner, “Probing coherence in microcavity frequency combs via optical pulse shaping,” Opt. Express 20, 21033–21043 (2012). [CrossRef] [PubMed]

29

29. T. Herr, K. Hartinger, J. Riemesberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr-frequency combs in microresonators,” Nat. Photonics 6, 480–487 (2012). [CrossRef]

]. The former class allows for getting transform-limited pulses after phase compensation: an indirect indication of spectrally coherent behavior. For Type II combs and for certain pump setting conditions, it was not possible to achieve transform-limited pulses. This led to a degradation in the peak to background ratio in the measured autocorrelation trace when compared to the corresponding digitally compressed one [24

24. F. Ferdous, H. X Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011). [CrossRef]

,25

25. S. B. Papp and S. A. Diddams, “Spectral and temporal characterization of a fused-quartz-microresonator optical frequency comb,” Phys. Rev. A 84, 053833 (2011). [CrossRef]

]. The explanation for this phenomenon was that the spectral phases of the microresonator comb were fluctuating randomly over a certain range (see supplementary information from [24

24. F. Ferdous, H. X Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011). [CrossRef]

]). Different groups have reported similar observations. Recent multi-heterodyne experiments showed that Type II combs operating in the partially coherent regime do not have a spectrum formed by evenly spaced frequency components, leading to decreased noise performance as inferred by the broadening of the RF beat note of the microresonator comb [29

29. T. Herr, K. Hartinger, J. Riemesberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr-frequency combs in microresonators,” Nat. Photonics 6, 480–487 (2012). [CrossRef]

]. An important conclusion is that neither the formation dynamics nor the noise behavior is exclusive of a particular material or geometry [29

29. T. Herr, K. Hartinger, J. Riemesberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr-frequency combs in microresonators,” Nat. Photonics 6, 480–487 (2012). [CrossRef]

]. In [25

25. S. B. Papp and S. A. Diddams, “Spectral and temporal characterization of a fused-quartz-microresonator optical frequency comb,” Phys. Rev. A 84, 053833 (2011). [CrossRef]

, 28

28. P. H. Wang, F. Ferdous, H. X Miao, J. Wang, D. E. Leaird, K. Srinivasan, L. Chen, V. Aksyuk, and A. M. Weiner, “Observation of correlation between route to formation, coherence, noise, and communication performance of Kerr combs,” Opt. Express 20, 29284–29295 (2012). [CrossRef]

], it was observed that when the RF beat note appears broadened, the comb light couldn’t be temporally compressed. This has a strong implication in optical communications because this apparent lack of spectral coherence leads to degradation in the bit error rate performance when the comb is used as a multiwavelength transmitter [28

28. P. H. Wang, F. Ferdous, H. X Miao, J. Wang, D. E. Leaird, K. Srinivasan, L. Chen, V. Aksyuk, and A. M. Weiner, “Observation of correlation between route to formation, coherence, noise, and communication performance of Kerr combs,” Opt. Express 20, 29284–29295 (2012). [CrossRef]

].

Fig. 1 Modulation instability (MI) is the underlying phenomenon that leads to parametric oscillation in microresonator frequency combs. In Type II combs, the MI sidebands generated by the CW pump lead to parametric growth of frequencies that are several FSRs away. These frequency components grow in power and nonlinear mix with the pump. This microresonator may lead to a spectrally partially coherent comb [29]. However, the comb dynamics can be altered by actively manipulating the CW pump settings in the course of comb formation. This may lead to the formation of stable combs [33, 39] and cavity solitons [42]. We show that these solutions are indeed stable and spectrally coherent, but they strongly depend on the particular initial conditions of the system. For Type I combs, the first oscillating modes are beside the pump and display a spectrally coherent behavior. These solutions are robust to the noise conditions.

Regarding coherence studies in microresonator combs, Erkintalo and Coen [45

45. M. Erkintalo and S. Coen, “Coherence properties of Kerr frequency combs,” Opt. Lett. 39, 283–286 (2014). [CrossRef]

] have analyzed numerically the first-order-degree of spectral coherence when the comb is operating under different regimes, which are linked to the solutions of the Lugiato-Lefever equation [38

38. S. Coen and M. Erkintalo, “Universal scaling laws of Kerr frequency combs,” Opt. Lett. 38, 1790–1792 (2013). [CrossRef] [PubMed]

]. They find that stable and coherent spectra can be obtained in Type II combs at either the onset of modulation instability or when cavity solitons are formed [45

45. M. Erkintalo and S. Coen, “Coherence properties of Kerr frequency combs,” Opt. Lett. 39, 283–286 (2014). [CrossRef]

]. Here, we complement this study by analyzing the connection between spectral coherence and the dynamics of Type I versus Type II microresonator combs. In line with recent observations [24

24. F. Ferdous, H. X Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011). [CrossRef]

,25

25. S. B. Papp and S. A. Diddams, “Spectral and temporal characterization of a fused-quartz-microresonator optical frequency comb,” Phys. Rev. A 84, 053833 (2011). [CrossRef]

,27

27. F. Ferdous, H. X. Miao, P. H. Wang, D. E. Leaird, K. Srinivasan, L. Chen, V. Aksyuk, and A. M. Weiner, “Probing coherence in microcavity frequency combs via optical pulse shaping,” Opt. Express 20, 21033–21043 (2012). [CrossRef] [PubMed]

,28

28. P. H. Wang, F. Ferdous, H. X Miao, J. Wang, D. E. Leaird, K. Srinivasan, L. Chen, V. Aksyuk, and A. M. Weiner, “Observation of correlation between route to formation, coherence, noise, and communication performance of Kerr combs,” Opt. Express 20, 29284–29295 (2012). [CrossRef]

], we also find that Type II combs are coherent as long as the oscillating modes remain incapable to provide net MI gain by themselves. Otherwise, we observe a degradation of the coherence, but only in particular regions of the spectrum. We also find that cavity solitons are stable and spectrally coherent but these solutions are susceptible to the vacuum fluctuations that drive the dynamics of the comb. The most important observation of our work is that Type I combs emerge in a natural manner and are indeed spectrally coherent, regardless of the initial seed conditions. This work highlights the relevance of reporting stability, shot-to-shot fluctuations, spectral coherence and repeatability in microresonator comb experiments.

2. Parametric oscillation revisited in the framework of the Lugiato-Lefever equation

The LLE reads as
tRE(t,τ)t=[αiδ0+iLk2βkk!(iτ)k+iγL|E(t,τ)|2]E(t,τ)+θEin.
(1)

Parametric oscillation in microresonators is analyzed by probing the LLE with the ansatz E(t, τ) = a(−) exp(−iΩτ) + a0 + a(+) exp(iΩτ) [36

36. T. Hansson, D. Modotto, and S. Wabnitz, “Dynamics of the modulation instability in microresonator frequency combs,” Phys. Rev. A 88, 023819 (2013). [CrossRef]

, 46

46. M. Haelterman, S. Trillo, and S. Wabnitz, “Additive-modulation-instability ring laser in the normal dispersion regime of a fiber,” Opt. Lett. 17, 745–747 (1992). [CrossRef] [PubMed]

], where a(−) and a(+) are much smaller than a0, and this last term satisfies the optical bistability equation a0[α+i(δ0γLP0)]=θEin, where P0 = |a0|2. It is easy to show that in a first-order-dispersion approximation the LLE presents exponentially growing solutions for a(−) and a(+), proportional to exp[Λ(Ω)t] [46

46. M. Haelterman, S. Trillo, and S. Wabnitz, “Additive-modulation-instability ring laser in the normal dispersion regime of a fiber,” Opt. Lett. 17, 745–747 (1992). [CrossRef] [PubMed]

]. The gain coefficient is Λ(Ω)=α+(γLP0)2(Δκ(Ω))2 and is maximum when the phase-mismatch Δκ(Ω) = δ02Ω2/2 − 2γLP0 equals zero. This occurs at the angular frequency [46

46. M. Haelterman, S. Trillo, and S. Wabnitz, “Additive-modulation-instability ring laser in the normal dispersion regime of a fiber,” Opt. Lett. 17, 745–747 (1992). [CrossRef] [PubMed]

]
Ωm2=2Lβ2(δ02γLP0).
(2)
At this frequency, the modes a(−) and a(+) will experience net gain as long as P0 > P0,th = α/(γL), which defines the required power threshold for parametric oscillation. Considering the optical bistability condition, the above threshold for the intracavity power provides the CW pump power needed to achieve parametric oscillation,
Pin,th=1γL[α2+(δ0α)2],
(3)
where critical coupling is assumed. Parametric oscillation in microresonator combs has been previously studied [30

30. A. B. Matsko, A. A. Savchenko, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Optical hyperparametric oscillations in a whispering-gallery-mode resonator: Threshold and phase diffusion,” Phys. Rev. A 71, 033804 (2005). [CrossRef]

, 40

40. T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93, 083904 (2004). [CrossRef]

]. It is interesting to note that the 1/Q2 dependence in the required threshold pump power (see e.g. Eq. (10) in [30

30. A. B. Matsko, A. A. Savchenko, D. Strekalov, V. S. Ilchenko, and L. Maleki, “Optical hyperparametric oscillations in a whispering-gallery-mode resonator: Threshold and phase diffusion,” Phys. Rev. A 71, 033804 (2005). [CrossRef]

]) can alternatively be obtained in the framework of the LLE. In particular, if we take Eq. (3) and consider δ0 = 0, the pump power threshold becomes Pin,th = 2α2/(γL), in agreement with [49

49. Y. K. Chembo, D. V. Strekalov, and N. Yu, “Spectrum and dynamics of optical frequency combs generated with monolithic whispering gallery mode resonators,” Phys. Rev. Lett. 104, 103902 (2010). [CrossRef] [PubMed]

]. Note that this threshold is different from the absolute parametric threshold, which is given for a fixed pump power when the detuning satisfies [38

38. S. Coen and M. Erkintalo, “Universal scaling laws of Kerr frequency combs,” Opt. Lett. 38, 1790–1792 (2013). [CrossRef] [PubMed]

]
δ0αPinγLα2.
(4)

As firstly observed in [24

24. F. Ferdous, H. X Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011). [CrossRef]

, 27

27. F. Ferdous, H. X. Miao, P. H. Wang, D. E. Leaird, K. Srinivasan, L. Chen, V. Aksyuk, and A. M. Weiner, “Probing coherence in microcavity frequency combs via optical pulse shaping,” Opt. Express 20, 21033–21043 (2012). [CrossRef] [PubMed]

29

29. T. Herr, K. Hartinger, J. Riemesberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr-frequency combs in microresonators,” Nat. Photonics 6, 480–487 (2012). [CrossRef]

], microresonator combs whose first oscillating frequencies appear one FSR away from the pump appear to be stable and admit compressibility to the transform-limited duration [24

24. F. Ferdous, H. X Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011). [CrossRef]

]. Thus, it is important to assess the design rules that lead to this type of microresonator comb (so-called Type I [24

24. F. Ferdous, H. X Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011). [CrossRef]

,27

27. F. Ferdous, H. X. Miao, P. H. Wang, D. E. Leaird, K. Srinivasan, L. Chen, V. Aksyuk, and A. M. Weiner, “Probing coherence in microcavity frequency combs via optical pulse shaping,” Opt. Express 20, 21033–21043 (2012). [CrossRef] [PubMed]

,28

28. P. H. Wang, F. Ferdous, H. X Miao, J. Wang, D. E. Leaird, K. Srinivasan, L. Chen, V. Aksyuk, and A. M. Weiner, “Observation of correlation between route to formation, coherence, noise, and communication performance of Kerr combs,” Opt. Express 20, 29284–29295 (2012). [CrossRef]

]). From Eq. (2), the Type I condition, i.e. Ωm = 2πFSR, imposes the dispersion of the cavity to satisfy
β2=(δ02γLP0)2Lπ2FSR2.
(5)
This indicates that, depending on the cavity detuning, either normal or anomalous dispersion may lead to Type I combs. For zero detuning and considering the intracavity power at threshold, P0,th, we get β2 = −α/(2FSR2) (in close agreement to what is found in [29

29. T. Herr, K. Hartinger, J. Riemesberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr-frequency combs in microresonators,” Nat. Photonics 6, 480–487 (2012). [CrossRef]

]), which means that anomalous dispersion is required. Equation (5) can be considered as a generalization for the Type I design rule, where P0 is not necessarily the intracavity power at threshold.

In the following sections, we shall verify that spectrally coherent combs are achievable for Type I combs. For completeness, these results are benchmarked to the coherence properties of Type II combs whose detuning is just above the minimum defined by Eq. (4) [45

45. M. Erkintalo and S. Coen, “Coherence properties of Kerr frequency combs,” Opt. Lett. 39, 283–286 (2014). [CrossRef]

], or temporal cavity solitons.

3. Numerical results

The spectral coherence of microresonator combs is evaluated by means of the following figure of merit [45

45. M. Erkintalo and S. Coen, “Coherence properties of Kerr frequency combs,” Opt. Lett. 39, 283–286 (2014). [CrossRef]

]
|g(ω;t1,t2)|=|E˜*(t1,ω)E˜(t2,ω)||E˜(ω)|2.
(6)
Here, the complex field (t, ω) is the Fourier transform of E(t, τ) with respect to τ. The frequency dependence of the two-time correlation function has been widely used to assess the noise performance of different supercontinuum sources [50

50. J. M. Dudley and S. Coen, “Coherence properties of supercontinuum spectra generated in photonic crystal and tapered fibers,” Opt. Lett. 27, 1180–1182 (2002). [CrossRef]

, 51

51. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fibers,” Rev. Mod. Phys. 78, 1135–1184 (2006). [CrossRef]

]. Different figures of merit, e.g. the two-frequency correlation function [52

52. G. Genty, M Surakka, J. Turunen, and A. T. Friberg, “Second-order coherence of supercontinuum light,” Opt. Lett. 35, 3057–3059 (2010). [CrossRef] [PubMed]

] or higher-order correlations [53

53. T. Godin, B. Wetzel, T. Sylvestre, L. Larger, A. Kudlinski, A. Mussot, A. Ben Salem, M. Zghal, G. Genty, F. Dias, and J. M. Dudley, “Real time noise and wavelength correlations in octave-spanning supercontinuum generation,” Opt. Express 21, 18452–18460 (2013). [CrossRef] [PubMed]

] can be studied too, but this is a topic beyond the scope of this work.

The brackets above denote ensemble averaging. In practice, we solve Eq. (1) starting from an empty ring. The waveform E(t, τ) is calculated within the temporal window −tR/2 ≤ τtR/2, and the steps taken in the variable t correspond to a single cavity roundtrip time. Before every step, we load the CW pump with statistically independent noise consisting of one photon per spectral bin with random phase [50

50. J. M. Dudley and S. Coen, “Coherence properties of supercontinuum spectra generated in photonic crystal and tapered fibers,” Opt. Lett. 27, 1180–1182 (2002). [CrossRef]

, 51

51. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fibers,” Rev. Mod. Phys. 78, 1135–1184 (2006). [CrossRef]

]. We calculate the complex degree of coherence at a fixed time difference t2t1 and 1000 different instants t1 of the evolution time. As demonstrated in [45

45. M. Erkintalo and S. Coen, “Coherence properties of Kerr frequency combs,” Opt. Lett. 39, 283–286 (2014). [CrossRef]

], for time differences shorter or in the order of the photon lifetime in the cavity, tph = tR/(2α), the light source may display coherent behavior even in a regime where the comb is inherently unstable. In order to avoid these artifacts, we choose t2t1 to be more than one order of magnitude longer than tph.

3.1. Example A: Type II microresonator comb

In our first example, the microresonator is designed with FSR = 226 GHz, L = 2πr with r = 100 μm, α = θ = 0.003, γ = 1000 (W·km)−1, β2 = −48.5 ps2/km, β3 = 0.131 ps3/km and β4 = 0.0025 ps4/km. These parameters could be realistically obtained with a silicon nitride microresonator and are similar to the ones reported in [13

13. Y. Okawachi, K. Saha, J. S. Levy, Y. H. Wen, M. Lipson, and A. L. Gaeta, “Octave-spanning frequency comb generation in a silicon nitride chip,” Opt. Lett. 36, 3398–3400 (2011). [CrossRef] [PubMed]

,35

35. S. Coen, H. G. Randle, T. Sylvestre, and M. Erkintalo, “Modeling of octave-spanning Kerr frequency combs using a generalized mean-field Lugiato-Lefever model,” Opt. Lett. 38, 37–39 (2013). [CrossRef] [PubMed]

]. We consider a CW pump power of 1.5 W. From Eq. (4), the required detuning for parametric oscillation is δ0 ≥ −0.028. In Fig. 2 we plot the dynamics of the microresonator comb at different detuning values satisfying the above inequality. For values close to the threshold [see Fig. 2(a)], only a few discrete frequency components are generated. In a first step, the pump generates two new frequencies through degenerate four-wave mixing [6

6. T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011). [CrossRef] [PubMed]

, 29

29. T. Herr, K. Hartinger, J. Riemesberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr-frequency combs in microresonators,” Nat. Photonics 6, 480–487 (2012). [CrossRef]

], that is the fundamental interaction behind the modulation instability process [54

54. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

]. The position of these lines corresponds to the frequencies ±Ωm given by Eq. (2) and are ±2π × 6.7 THz in this case, much higher than the cavity’s FSR. In a second step, these three frequencies interact through a nondenegerate and stimulated four-wave mixing process. Consequently, new frequencies appear at ±2Ωm and ±3Ωm, keeping in this way the equidistance between lines [6

6. T. J. Kippenberg, R. Holzwarth, and S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011). [CrossRef] [PubMed]

, 29

29. T. Herr, K. Hartinger, J. Riemesberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr-frequency combs in microresonators,” Nat. Photonics 6, 480–487 (2012). [CrossRef]

]. We observe that the comb evolves towards a stable steady state after several roundtrips. Pulses are formed as soon as the new frequencies emerge. Figure 2(b) displays the comb dynamics for the case in which the CW pump is closer to resonance. Here there is an initial state similar to the case in Fig. 2(a), but the first oscillating modes acquire sufficient power to stimulate the growth of frequency components located between them and the pump through a degenerate four-wave-mixing process [27

27. F. Ferdous, H. X. Miao, P. H. Wang, D. E. Leaird, K. Srinivasan, L. Chen, V. Aksyuk, and A. M. Weiner, “Probing coherence in microcavity frequency combs via optical pulse shaping,” Opt. Express 20, 21033–21043 (2012). [CrossRef] [PubMed]

,29

29. T. Herr, K. Hartinger, J. Riemesberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr-frequency combs in microresonators,” Nat. Photonics 6, 480–487 (2012). [CrossRef]

,55

55. M. R. E. Lamont, Y. Okawachi, and A. L. Gaeta, “Route to stabilized ultrabroadband microresonator-based frequency combs,” Opt. Lett. 38, 3478–3481 (2013). [CrossRef] [PubMed]

]. These new lines generate more frequency components by non-degenerate and stimulated four-wave-mixing processes. In the time domain, the mixing leads to strong intensity variations. Finally, in the case in which the comb is initiated by pumping on resonance, Fig. 2(c), a broader microresonator comb is rapidly obtained, and the spectrum fills quickly all the FSRs between the pump and the first oscillating modes. However, the waveform does not approach a steady state and there are strong spectral and intensity variations from one roundtrip to the next.

Fig. 2 Dynamics of a Type II microresonator comb in spectral domain (top row) and time domain (bottom row) for different detuning conditions. Corresponding average spectra, shot-to-shot fluctuations and spectral coherence are displayed in Fig. 3.

We have calculated the spectral coherence for each example above as per Eq. (6) and the results are presented in Fig. 3. For each detuning, the evolution of the waveform over the variable t is calculated over long time distances. As explained before, the coherence is calculated for pairs of spectral waveforms separated t2t1 > 10tph. In order to get conclusive statistics, we compute 1000 consecutive pairs. The evolution of the field over time is shown on the left column in Fig. 3. On the right column, the field realizations are superimposed and the average spectrum corresponding to the comb envelope is calculated and shown in pink. For δ0 = −0.014 in Fig. 3(a), the fluctuations appear only at the background level and the comb displays a spectrally coherent behavior. The MI lines remain highly coherent and stable upon evolution. For detuning values closer to resonance, δ0 = −0.007 in Fig. 3(b), the spectral regions around the first oscillating modes remain coherent, however the new comb lines that arise in between are partially coherent. One can indeed observe stronger amplitude fluctuations in these spectral regions and conclude that the comb is spectrally partially coherent. Finally, right on resonance, δ0 = 0 in Fig. 3(c), the spectral envelope of the combs is much smoother but there appear large spectral fluctuations from shot to shot that lead to a degradation of the spectral coherence across the whole bandwidth. These findings are in agreement with the analysis carried out in [45

45. M. Erkintalo and S. Coen, “Coherence properties of Kerr frequency combs,” Opt. Lett. 39, 283–286 (2014). [CrossRef]

] based on the stability of the LLE solutions.

Fig. 3 Spectral coherence and shot-to-shot fluctuations for the microresonator combs in Fig. 2.

3.2. Example B: Type I microresonator comb

We now consider a slightly different microresonator, with design parameters FSR = 2.41 THz, r = 10 μm, α = θ = 0.001, γ = 1100 (W·km)−1 and β2 = −623.4 ps2/km. These parameters are chosen so that Eq. (5) is satisfied at resonance for a CW pump power of 0.2 W. The field evolution is calculated in the same manner as before. Likewise, the complex degree of coherence is computed for a fixed time difference greater than 10tph. The average spectral envelopes and degrees of coherence are displayed in Figs. 4(a)–4(d) for different pump powers but keeping δ0 = 0.

Fig. 4 (a)–(d) Average envelope spectra, degree of coherence and intensity profile for a Type I microresonator comb at different pump power leves. (e) MI gain bandwith shift for different power levels. At 0.2 W the maximum gain coincides with the FSR of the microresonator cavity.

The comb is always highly coherent in this case. The broadest comb envelope corresponds to the case in which the MI gain peak matches the FSR of the cold cavity, as Fig. 4(e) indicates. Under the spectral envelope figures, we plot the corresponding intensity profile (temporally shifted for clarity) and compare it to the transform-limited case. For the optimum case in which Pin = 0.2 W, the pulse is very close to the transform limit. For higher powers [Figs. 4(c) and 4(d)], the pulse deviates from the optimally compressed case, yet the degree of coherence is 1. This means that the spectral phase is not uniform but high-quality ultrashort pulses can be achieved with the aid of a line-by-line pulse-shaping device [24

24. F. Ferdous, H. X Miao, D. E. Leaird, K. Srinivasan, J. Wang, L. Chen, L. T. Varghese, and A. M. Weiner, “Spectral line-by-line pulse shaping of on-chip microresonator frequency combs,” Nat. Photonics 5, 770–776 (2011). [CrossRef]

].

Fig. 5 Analysis of the universality of the stable solutions for Type I coherent combs. (Top) The different realizations calculated at a fixed instant time for different random seeds. (Bottom) the degree of coherence calculated at fixed time t over multiple noise seeds (blue dash curve) is however substantially different when compared to the one calculated at multiple instant times for a fixed seed (red dash curve). After compensating for a linear spectral phase ad hoc, the degree of coherence calculated at a fixed time for various noise seeds (blue solid line) is identical to the degree of coherence calculated for a fixed seed and various instant times.

4. Stability and coherence of temporal cavity solitons

Fig. 6 Temporal cavity soliton formation in microresonators. (a) Dynamic evolution of detuning. Corresponding (b) spectral and (c) time domain evolutions. Note the difference in time scales at different stages. The detuning is changed dynamically in the course of the CS formation. In the first stage, it is kept constant. The points marked as A and C are indicated by the dashed white lines in (b) and (c). Once the cavity soliton is formed, the waveform appears stable and coherent (d). The average intensity indicates a pulse with 25 fs duration (e).
Fig. 7 Analysis of cavity soliton formation under different noise conditions. The sweep and other parameters are identical to those in Fig. 6. The points A–C are indicated in Fig. 6(a).

Next, we are interested in the robustness of this solution for different noise seeds, for which we repeat the above simulation considering the same microresonator parameters, including the sweep and pump power. The only difference now is the set of independent random noise seeds accompanying the CW pump every roundtrip, which is otherwise inaccessible by any experimental means. Interestingly, the system achieves a different steady state consisting of two temporal cavity solitons. This waveform also has a high degree of spectral coherence. Figure 7 compares the relevant waveforms in time and frequency domains obtained at particular instant times for the two sets of noise considered. In the initial MI stage, the waveforms are almost identical, simply shifted in time. However, the waveform achieved in the chaotic stage [indicated as time B in Fig. 7] is different. When the system achieves the steady state, the exact waveform depends on the set of noise seeds accompanying the pump every roundtrip. In the case presented in Fig. 7(b) it shows two solitons but in other runs the system evolves towards 1, 2, 3, 4, 5 solitons or simply collapses into a continuous wave. We know that the conclusion depends on the particular ramp programmed. Whether the system evolves to a single soliton regardless the noise conditions for an optimal ramp choise is an open question.

Such a fine sensitivity of CSs to the particular noise conditions has been observed experimentally [42

42. T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014). [CrossRef]

] and discussed in [42

42. T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014). [CrossRef]

, 45

45. M. Erkintalo and S. Coen, “Coherence properties of Kerr frequency combs,” Opt. Lett. 39, 283–286 (2014). [CrossRef]

]. An expression for the maximum number of stationary (non interacting) CSs is provided in [42

42. T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014). [CrossRef]

]. It is important to note [42

42. T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kondratiev, M. L. Gorodetsky, and T. J. Kippenberg, “Temporal solitons in optical microresonators,” Nat. Photonics 8, 145–152 (2014). [CrossRef]

] that the condition to get a maximum number of solitons equal to 1 coincides with the design rule for Type I microresonators [29

29. T. Herr, K. Hartinger, J. Riemesberger, C. Y. Wang, E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J. Kippenberg, “Universal formation dynamics and noise of Kerr-frequency combs in microresonators,” Nat. Photonics 6, 480–487 (2012). [CrossRef]

].

5. Conclusions and discussion

We have analyzed the spectral coherence of microring resonator combs. There is a strikingly different behavior between Type I and Type II combs. In the soft excitation regime, Type II combs are spectrally coherent just at the onset of parametric oscillation, where only the primary lines oscillate and mix with the CW pump. The spectral coherence is severely degraded when the spectral gap between these lines and the CW pump fills in.

This type of combs may however admit the formation of temporal cavity solitons. This requires operating the microresonator in the hard excitation regime by, e.g., realizing a proper detuning of the CW pump in the course of comb formation. We showed that CSs are spectrally coherent and stable, but their formation is very sensitive to vacuum fluctuations. On the contrary, when the microresonator is designed to provide Type I combs, the system always approaches a steady state, stable, spectrally coherent solution regardless of the noise conditions.

We wish to emphasize that the above are not the only possibilities to achieve microresonator combs operating in a high-coherence state. As recent experiments indicate, stable microresonator combs can be obtained by pumping with a waveform composed of multiple CW waves (parametric seeding) [57

57. S. B. Papp, P. Del’Haye, and S. A. Diddams, “Parametric seeding of a microresonator optical frequency comb,” Opt. Express 21, 17615–17624 (2013). [CrossRef] [PubMed]

], or by placing the microring in a fiber cavity [58

58. M. Peccianti, A. Pasquazi, Y. Park, B. E. Little, S. T. Chu, D. J. Moss, and R. Morandotti, “Demonstration of a stable ultrafast laser based on a nonlinear microcavity,” Nature Commun. 3, 765 (2012). [CrossRef]

]. Other observations leading to low-noise states show features akin to injection locking between ensembles of comb modes [59

59. P. Del’Haye, S. B. Papp, and S. A. Diddams, “Self-injection locking and phase-locked states in microresonator-based optical frequency combs,” Phys. Rev. Lett. 112, 043905 (2014). [CrossRef]

]. Further numerical and experimental work is needed to understand the spectral coherence for these new mechanisms and provide general design rules for self-starting spectrally coherent microresonator combs.

Acknowledgments

Victor Torres acknowledges stimulating discussions with Prof. A. M. Weiner. This work has been partly funded by the Swedish Research Council (VR), by the Plan Nacional I+D+I under the research project TEC2008-05490, Ministerio de Ciencia e Innovación (Spain), and by the Generalitat Valenciana under the grant PROMETEO 2009-077. David Castelló-Lurbe gratefully acknowledges funding from the Generalitat Valenciana (VALi+d predoctoral contract).

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59.

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OCIS Codes
(030.1640) Coherence and statistical optics : Coherence
(320.7110) Ultrafast optics : Ultrafast nonlinear optics
(130.3990) Integrated optics : Micro-optical devices

ToC Category:
Coherence and Statistical Optics

History
Original Manuscript: December 26, 2013
Revised Manuscript: February 8, 2014
Manuscript Accepted: February 13, 2014
Published: February 20, 2014

Citation
Victor Torres-Company, David Castelló-Lurbe, and Enrique Silvestre, "Comparative analysis of spectral coherence in microresonator frequency combs," Opt. Express 22, 4678-4691 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-4-4678


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References

  1. D. J. Jones, S. A. Diddams, J. K. Ranka, A. Stentz, R. S. Windeler, J. L. Hall, S. T. Cundiff, “Carrier-envelope phase control of femtosecond mode-locked lasers and direct optical frequency synthesis,” Science 288, 635–639 (2000). [CrossRef] [PubMed]
  2. S. A. Diddams, D. J. Jones, J. Ye, S. T. Cundiff, J. L. Hall, J. K. Ranka, R. S. Windeler, R. Holzwarth, T. Udem, T. W. Hänsch, “Direct link between microwave and optical frequencies with a 300 THz femtosecond laser comb,” Phys. Rev. Lett. 84, 5102–5105 (2000). [CrossRef] [PubMed]
  3. T. W. Hänsch, “Nobel Lecture: Passion for precision,” Rev. Mod. Phys. 78, 1297–1309 (2006). [CrossRef]
  4. N. R. Newbury, “Searching for applications with a fine-tooth comb,” Nature Photon. 5, 186–188 (2011). [CrossRef]
  5. V. Torres-Company, A. M. Weiner, “Optical frequency comb technology for ultra-broadband radio-frequency photonics,” Laser and Photon. Rev. (in press, 2013). DOI . [CrossRef]
  6. T. J. Kippenberg, R. Holzwarth, S. A. Diddams, “Microresonator-based optical frequency combs,” Science 332, 555–559 (2011). [CrossRef] [PubMed]
  7. P. Del’Haye, A. Schliesser, O. Arcizet, T. Wilken, R. Holzwarth, T. J. Kippenberg, “Optical frequency comb generation from a monolithic microresonator,” Nature 450, 1214–1217 (2007). [CrossRef]
  8. A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, I. Solomatine, D. Seidel, L. Maleki, “Tunable optical frequency comb with a crystalline whispering gallery mode resonator,” Phys. Rev. Lett. 101, 093902 (2008). [CrossRef] [PubMed]
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