## Comparative analysis of spectral coherence in microresonator frequency combs |

Optics Express, Vol. 22, Issue 4, pp. 4678-4691 (2014)

http://dx.doi.org/10.1364/OE.22.004678

Acrobat PDF (8151 KB)

### 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

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_{2}[8

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_{2}[9

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^{5}–10

^{6}around 1.5

*μ*m) but the nonlinear coefficients are sufficiently high to yield frequency combs, in some cases with thousands of modes and a bandwidth spanning tens of terahertz [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]

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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. 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. 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**, 770–776 (2011). [CrossRef]

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]

**5**, 770–776 (2011). [CrossRef]

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]

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]

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. 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]

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]

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]

39. A. B. Matsko, W. Liang, A. A. Savchenko, and L. Maleki, “Chaotic dynamics of frequency combs generated with continuously pumped nonlinear microresonators,” Opt. Lett. **38**, 525–527 (2013). [CrossRef] [PubMed]

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]

32. A. B. Matsko, A. A. Savchenko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Mode-locked Kerr frequency combs,” Opt. Lett. **36**, 2845–2847 (2011). [CrossRef] [PubMed]

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

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]

31. Y. K. Chembo and N. Yu, “Modal expansion approach to optical-frequency-comb generation with monolithic whispering-gallery-mode resonators,” Phys. Rev. A **82**, 033801 (2010). [CrossRef]

33. A. B. Matsko, A. A. Savchenko, V. S. Ilchenko, D. Seidel, and L. Maleki, “Hard and soft excitation regimes of Kerr frequency combs,” Phys. Rev. A **85**, 023830 (2012). [CrossRef]

39. A. B. Matsko, W. Liang, A. A. Savchenko, and L. Maleki, “Chaotic dynamics of frequency combs generated with continuously pumped nonlinear microresonators,” Opt. Lett. **38**, 525–527 (2013). [CrossRef] [PubMed]

32. A. B. Matsko, A. A. Savchenko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Mode-locked Kerr frequency combs,” Opt. Lett. **36**, 2845–2847 (2011). [CrossRef] [PubMed]

34. Y. K. Chembo and C. R. Menyuk, “Spatiotemporal Lugiato-Lefever formalism for Kerr-comb generation in whispering gallery-mode resonators,” Phys. Rev. A **87**, 053852 (2013). [CrossRef]

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]

41. L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett. **58**, 2209–2211 (1987). [CrossRef] [PubMed]

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]

43. F. Leo, S. Coen, P. Kockaert, S. P. Goza, P. Emplit, and M. Haelterman, “Temporal cavity solitons in one-dimensional Kerr media as bits in an all-optical buffer,” Nat. Photonics **4**, 471–476 (2010). [CrossRef]

44. H. Lajunen, V. Torres-Company, J. Lancis, E. Silvestre, and P. Andres, “Pulse-by-pulse method to characterize partially coherent pulse propagation in instantaneous nonlinear media,” Opt. Express **18**, 14979–14991 (2010). [CrossRef] [PubMed]

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. 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]

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

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

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

**5**, 770–776 (2011). [CrossRef]

**84**, 053833 (2011). [CrossRef]

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. 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]

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

41. L. A. Lugiato and R. Lefever, “Spatial dissipative structures in passive optical systems,” Phys. Rev. Lett. **58**, 2209–2211 (1987). [CrossRef] [PubMed]

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]

48. F. Leo, L. Gelens, P. Emplit, M. Haelterman, and S. Coen, “Dynamics of one-dimensional Kerr cavity solitons,” Opt. Express **21**, 9180–9191 (2013). [CrossRef] [PubMed]

32. A. B. Matsko, A. A. Savchenko, W. Liang, V. S. Ilchenko, D. Seidel, and L. Maleki, “Mode-locked Kerr frequency combs,” Opt. Lett. **36**, 2845–2847 (2011). [CrossRef] [PubMed]

34. Y. K. Chembo and C. R. Menyuk, “Spatiotemporal Lugiato-Lefever formalism for Kerr-comb generation in whispering gallery-mode resonators,” Phys. Rev. A **87**, 053852 (2013). [CrossRef]

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]

**36**, 2845–2847 (2011). [CrossRef] [PubMed]

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

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

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]

*E*(

*t*,

*τ*) =

*a*

_{(−)}exp(−

*i*Ω

*τ*) +

*a*

_{0}+

*a*

_{(+)}exp(

*i*Ω

*τ*) [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. 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]

*a*

_{(−)}and

*a*

_{(+)}are much smaller than

*a*

_{0}, and this last term satisfies the optical bistability equation

*P*

_{0}= |

*a*

_{0}|

^{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

**17**, 745–747 (1992). [CrossRef] [PubMed]

*κ*(Ω) =

*δ*

_{0}−

*Lβ*

_{2}Ω

^{2}/2 − 2

*γLP*

_{0}equals zero. This occurs at the angular frequency [46

**17**, 745–747 (1992). [CrossRef] [PubMed]

*a*

_{(−)}and

*a*

_{(+)}will experience net gain as long as

*P*

_{0}>

*P*

_{0,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, where critical coupling is assumed. Parametric oscillation in microresonator combs has been previously studied [30

**71**, 033804 (2005). [CrossRef]

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]

*Q*

^{2}dependence in the required threshold pump power (see e.g. Eq. (10) in [30

**71**, 033804 (2005). [CrossRef]

*δ*

_{0}= 0, the pump power threshold becomes

*P*

_{in,th}= 2

*α*

^{2}/(

*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]

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

**5**, 770–776 (2011). [CrossRef]

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]

**6**, 480–487 (2012). [CrossRef]

**5**, 770–776 (2011). [CrossRef]

**5**, 770–776 (2011). [CrossRef]

**20**, 21033–21043 (2012). [CrossRef] [PubMed]

**20**, 29284–29295 (2012). [CrossRef]

_{m}= 2

*π*FSR, imposes the dispersion of the cavity to satisfy 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,

*P*

_{0,th}, we get

*β*

_{2}= −

*α*/(

*Lπ*

^{2}FSR

^{2}) (in close agreement to what is found in [29

**6**, 480–487 (2012). [CrossRef]

*P*

_{0}is not necessarily the intracavity power at threshold.

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

## 3. Numerical results

**39**, 283–286 (2014). [CrossRef]

*Ẽ*(

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

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]

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]

*E*(

*t*,

*τ*) is calculated within the temporal window −

*t*/2 ≤

_{R}*τ*≤

*t*/2, and the steps taken in the variable

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

*t*

_{2}−

*t*

_{1}and 1000 different instants

*t*

_{1}of the evolution time. As demonstrated in [45

**39**, 283–286 (2014). [CrossRef]

*t*

_{ph}=

*t*/(2

_{R}*α*), 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

*t*

_{2}−

*t*

_{1}to be more than one order of magnitude longer than

*t*

_{ph}.

### 3.1. Example A: Type II microresonator comb

*L*= 2

*πr*with

*r*= 100

*μ*m,

*α*=

*θ*= 0.003,

*γ*= 1000 (W·km)

^{−1},

*β*

_{2}= −48.5 ps

^{2}/km,

*β*

_{3}= 0.131 ps

^{3}/km and

*β*

_{4}= 0.0025 ps

^{4}/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. 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]

*δ*

_{0}≥ −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]

**6**, 480–487 (2012). [CrossRef]

_{m}given by Eq. (2) and are ±2

*π*× 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 ±2Ω

_{m}and ±3Ω

_{m}, 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]

**6**, 480–487 (2012). [CrossRef]

**20**, 21033–21043 (2012). [CrossRef] [PubMed]

**6**, 480–487 (2012). [CrossRef]

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]

*t*is calculated over long time distances. As explained before, the coherence is calculated for pairs of spectral waveforms separated

*t*

_{2}−

*t*

_{1}> 10

*t*

_{ph}. 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

*δ*

_{0}= −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,

*δ*

_{0}= −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,

*δ*

_{0}= 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

**39**, 283–286 (2014). [CrossRef]

### 3.2. Example B: Type I microresonator comb

*r*= 10

*μ*m,

*α*=

*θ*= 0.001,

*γ*= 1100 (W·km)

^{−1}and

*β*

_{2}= −623.4 ps

^{2}/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 10

*t*

_{ph}. The average spectral envelopes and degrees of coherence are displayed in Figs. 4(a)–4(d) for different pump powers but keeping

*δ*

_{0}= 0.

*P*

_{in}= 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

**5**, 770–776 (2011). [CrossRef]

*Ẽ*

_{1}(

*t*,

*ω*) and

*Ẽ*

_{2}(

*t*,

*ω*) are calculated at a fixed instant time

*t*for different random seeds. We note that this magnitude is conceptually closer to the one considered when evaluating the coherence properties in supercontinuum fiber 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. J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fibers,” Rev. Mod. Phys. **78**, 1135–1184 (2006). [CrossRef]

*t*= 10000

*t*

_{ph}. As in the previous case, we compute 1000 random seeds to calculate the average in Eq. (7). All the spectra are superposed (gray curves) and the average spectrum is shown in pink solid line. We observe that the spectral envelope of the comb remains constant in amplitude and is almost identical to the one calculated by averaging, for a single random seed, over multiple roundtrip times [pink solid line in Fig. 4(b) here displayed as dashed yellow]. However the degree of coherence as defined by Eq. (7), shown in dashed blue curve under the spectrum, indicates a highly incoherent behavior for all the frequency components except for the pump. What occurs is that the pulses achieve the same temporal profile as in Fig. 4(b) but appear randomly delayed within the cavity period for different random seeds. This leads to identical spectra with a linear spectral phase ramp with random slope. When the delay is compensated

*offline*, we observe a substantial increase in the spectral coherence as defined by Eq. (7) (blue solid line). This curve indeed matches the degree of coherence as calculated by Eq. (6), which is displayed for completeness in red dashed line in Fig. 5.

## 4. Stability and coherence of temporal cavity solitons

33. A. B. Matsko, A. A. Savchenko, V. S. Ilchenko, D. Seidel, and L. Maleki, “Hard and soft excitation regimes of Kerr frequency combs,” Phys. Rev. A **85**, 023830 (2012). [CrossRef]

**88**, 023819 (2013). [CrossRef]

33. A. B. Matsko, A. A. Savchenko, V. S. Ilchenko, D. Seidel, and L. Maleki, “Hard and soft excitation regimes of Kerr frequency combs,” Phys. Rev. A **85**, 023830 (2012). [CrossRef]

**88**, 023819 (2013). [CrossRef]

39. A. B. Matsko, W. Liang, A. A. Savchenko, and L. Maleki, “Chaotic dynamics of frequency combs generated with continuously pumped nonlinear microresonators,” Opt. Lett. **38**, 525–527 (2013). [CrossRef] [PubMed]

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]

**6**, 480–487 (2012). [CrossRef]

56. K. Saha, Y. Okawachi, B. Shim, J. S. Levy, R. Salem, A. R. Johnson, M. A. Foster, M. R. E. Lamont, M. Lipson, and A. L. Gaeta, “Modelocking and femtosecond pulse generation in chip-based frequency combs,” Opt. Express **21**, 1335–1343 (2013). [CrossRef] [PubMed]

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]

**38**, 1790–1792 (2013). [CrossRef] [PubMed]

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]

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]

**38**, 1790–1792 (2013). [CrossRef] [PubMed]

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]

**39**, 283–286 (2014). [CrossRef]

**8**, 145–152 (2014). [CrossRef]

*δ*

_{0}= −0.025 for a time duration corresponding to 120

*t*

_{ph}. The comb remains here in the stable MI stage [as indicated by Eq. (4)]. We then sweep linearly in time the detuning for 120

*t*

_{ph}more until it reaches

*δ*

_{0}= 0.070. We note that this value lies within the range 3(

*P*

_{in}

*γLα*/4)

^{1/3}≤

*δ*

_{0}≤

*π*

^{2}

*P*

_{in}

*γL*/(8

*α*), where CSs are expected to form [38

**38**, 1790–1792 (2013). [CrossRef] [PubMed]

**38**, 3478–3481 (2013). [CrossRef] [PubMed]

*t*

_{ph}from each other. Snapshots in time and frequency at the relevant regimes are shown in Fig. 7. Our analysis of the spectral coherence reveals that the cavity pulse is highly coherent [Fig. 6(d)], confirming what has been recently found by analyzing the LLE solutions [45

**39**, 283–286 (2014). [CrossRef]

**8**, 145–152 (2014). [CrossRef]

**8**, 145–152 (2014). [CrossRef]

**39**, 283–286 (2014). [CrossRef]

**8**, 145–152 (2014). [CrossRef]

**8**, 145–152 (2014). [CrossRef]

**6**, 480–487 (2012). [CrossRef]

## 5. Conclusions and discussion

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]

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]

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]

## Acknowledgments

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56. | K. Saha, Y. Okawachi, B. Shim, J. S. Levy, R. Salem, A. R. Johnson, M. A. Foster, M. R. E. Lamont, M. Lipson, and A. L. Gaeta, “Modelocking and femtosecond pulse generation in chip-based frequency combs,” Opt. Express |

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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. |

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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

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