Effect of the Dispersion Orders on the Widths of the Coexistence Domain and Combs Spectra of Bright and Dark Solitons in Microresonators

Kangmafo Miendjim, Willy André; Kammegne, Caouis; Tamo Tatietse, Thomas; Talla Mbé, Jimmi Hervé

doi:https://doi.org/10.1155/2023/5537645

International Journal of Optics

On this page

Abstract Introduction Conclusion Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 5537645 | https://doi.org/10.1155/2023/5537645

Effect of the Dispersion Orders on the Widths of the Coexistence Domain and Combs Spectra of Bright and Dark Solitons in Microresonators

Willy André Kangmafo Miendjim,¹Caouis Kammegne,¹Thomas Tamo Tatietse,²and Jimmi Hervé Talla Mbé¹

Academic Editor: Mustapha Tlidi

Received14 Jun 2023

Revised11 Nov 2023

Accepted20 Nov 2023

Published08 Dec 2023

Abstract

Using the Lugiato–Lefever model, we base on the hysteresis approach to analyze the coexistence domain of bright and dark solitons in the zero, normal, and anomalous dispersion regimes. Our results also highlight that the fourth-order dispersion term affects the width of the frequency combs of both dark and bright solitons. It also allows the appearance of dispersive waves on the soliton spectra that disappear for high values of the fourth-order dispersion followed by the soliton destabilization into harmonic oscillations and oscillation packages.

1. Introduction

Optical frequency combs are among the discoveries that have most revolutionized the world of very high-precision metrology. They have become almost ubiquitous tools for optical analysis, biomedical imaging, optical communication, quantum information, and more [1–7]. Before the exploration of the optical Kerr effect in whispering-gallery-mode resonators, optical frequency combs were developed from mode-locked lasers. Nowadays, optical frequency combs have also found much border sources such as continuous lasers [8], optical fibers [9], optoelectronic oscillators [10, 11], and more [1, 7, 12]. However, in 2004, two independent teams generated optical frequency combs for the first time through amorphous [13] and crystalline [14] whispering-gallery-mode resonators.

Among the resonator parameters that explain the generation of Kerr frequency combs, chromatic dispersion plays a major role. Indeed, it strongly affects the pumping power required for combs generation, the range of the generated frequency combs, the shape of the power spectrum envelope, and even the repetition rate of the generated pulse in the resonator [15, 16]. The generation of Kerr optical frequency combs was extensively studied considering zero dispersion regime [17, 18] and high-order dispersion regimes that strongly affect the dynamics of Kerr frequency combs [19–21]. For example, dark solitons in the anomalous group velocity dispersion regime are only obtained when considering the fourth-order dispersion term [20, 22–24]. Besides, two branches of dark solitons having different polarization states and power peaks may coexist in normal dispersion optical resonators subject to a coherent optical injection [25]. Furthermore, it was reported that the third-order dispersion term allows the coexistence of dark and bright solitons in the cavity for a pump power taken around a critical value of the pump [23, 26, 27]. It was also shown that one of the modifications brought by higher order dispersion terms is the generation of dispersive waves [15] which can originate from the oscillation of the tails of the Kerr solitons induced by the third-order dispersion (TOD) [28]. It has been shown experimentally that the emission of a dispersive wave widens the spectral bandwidth of the frequency combs to two-thirds of an octave [29].

In this paper, we exploit the hysteresis curves to determine the areas in parameter space where solitons coexist. Besides, the broadening of the frequency combs spectral bandwidth under the effect of the fourth-order dispersion (FOD) term is also studied. It could be important if one can harness the range of the system parameters notably the pump, delimiting the coexistence of solitons. Moreover, the effect of higher order dispersion terms on the frequency combs and dispersive waves could bring a general benefit for telecommunication applications and high-frequency metrology [29].

This paper is structured as follows. In Section 2, we present the Lugiato–Lefever equation (LLE) model; then, in Section 3, we highlight the parameter space region where dark and bright solitons coexist in the case of zero, normal, and anomalous group velocity dispersion regimes taking into account the third-order dispersion (TOD). In Section 4, we characterize the effects of fourth-order dispersion (FOD) on the width of the Kerr combs spectra. The paper ends with a conclusion.

2. The Model

The deterministic dynamics of Kerr frequency combs can be accurately studied using the Lugiato–Lefever equation (LLE) [30, 31]. This equation governs the spatiotemporal evolution of the laser field inside the cavity of the whispering-gallery-mode resonators [32, 33]. The scaled form of this equation is as follows [23, 34]:where is the total field inside the whispering-gallery-mode resonator, is the azimuthal angle along the circumference of the cavity, is the normalized time, is the dimensionless frequency detuning, and represents the external pump amplitude. are the order dispersion parameters. The parameter represents the second-order dispersion parameter of the resonator, defined as normal for and anomalous for . is the third-order dispersion (TOD) parameter and stands for the fourth-order dispersion (FOD) parameter. The numerical method used to integrate the LLE will be the split-step Fourier algorithm [35].

The equilibria of the system are obtained by cancelling any variation with respect to the time and space in equation (1). The study of the stability of the system done using the homogeneous solution in steady state yieldswhere is the homogeneous steady-state solution. For the values of , equation (2) grows monotonically for ; only one stable equilibrium point exists and therefore bright and dark solitons are not expected. For , equation (2) has three real solutions , , and , satisfying , with and being stable, while is unstable [20, 35]. The subscripts , , and stand for the bottom, medium, and high, respectively. In this case, bright and dark solitons are expected for pumps around a critical value given by [36]

3. Determination of the Coexistence Domain of Stable Localized Structures Taking into Account the TOD

In this section, we study the coexistence domain of localized structures (dark and bright solitons) in the cavity by considering the third-order dispersion term (TOD). The width of the coexistence domain is evaluated using the mean energy of the field inside the whispering-gallery-mode resonator when stable bright and dark solitons are formed inside the cavity [26]:where is the time scale of the resonator roundtrip.

3.1. Coexistence Domain of Stable Localized Structures When the Group Velocity Dispersion Is Zero

It has been shown in [23] that for an appropriate choice of , bright and dark solitons coexist for pump power values taken in the vicinity of a critical pump power value (equation (3)). To delimit the coexistence domain, in Figure 1, we plot the bifurcation diagrams of the mean energy solutions (equation (4)) as a function of the external pump amplitude for , , and . Figures 1(a) and 1(b) show the bifurcation diagrams for increasing and decreasing , respectively. The red lines of low energy represent bright solitons, the black lines of high energy characterize dark solitons, and the blue lines are coherent states. As increases, we observe in the cavity for , the transition from a bright soliton to a dark soliton (Figure 1(a)). For , the blue line represents the high energy stable equilibrium and also highlights a threshold power beyond which there is no dark soliton in the cavity. Similarly, as decreases, for , a transition occurs from a dark soliton to a bright soliton (Figure 1(b)). For , the blue line represents the stable low energy equilibrium , which highlights the threshold power below which bright soliton does not exist in the cavity. Superposing Figures 1(a) and 1(b), a hysteresis zone for is observed around a critical value where bright and dark solitons coexist in the cavity (Figure 1(c)). This gives the range of pump powers where bright and dark solitons coexist for zero GVD . The plots of some bright and dark solitons as well as their frequency combs using parameters of Figure 1 are given in Figure 2(a). In the next subsection, we will also study the coexistence domain of the localized structures in the normal and anomalous GVD regimes and take into account the TOD.

(a)

(b)

(c)

3.2. Coexistence Domain of Stable Localized Structures in the Case of Normal and Anomalous Dispersion Regimes

In the case of normal GVD , the bifurcations for increasing and decreasing values of can be also used to evaluate the values where these structures can coexist as shown in Figures 3(a) and 3(b), respectively. Figure 3(c) is a superposition of the two and witnesses such coexistence. The coexistence of these solitons is illustrated in Figure 2(b) taking . The red lines characterize bright solitons, while the black ones represent dark solitons. The blue lines are low and high energy coherent states that occur for and , respectively. Indeed, in the normal GVD regime, the threshold pump amplitudes required for the generation of Kerr solitons are higher than those in the zero GVD regime. We highlight a range of values of but not exactly around the critical where bright and dark solitons coexist (Figure 3(c)) versus when (Figure 1(c)). Moreover, the width of the coexistence domain is larger (0.14) than in the case of zero GVD (0.03). It is also important to note that these solitons are obtained for a detuning which is higher than the detuning for which solitons are recorded in the case of zero GVD [23].

(a)

(b)

(c)

In the case of anomalous GVD , the bifurcation diagram of stable stationary solutions as a function of the pump amplitude is shown in Figure 4. The red, black, and blue lines also characterize the bright and dark solitons and the stable coherent states, respectively. Samples of bright and dark solitons as well as their frequency combs are shown in Figure 2(c) for the same value of . Note that the fourth-order term is inserted to favour the formation of bright solitons [15]. There is also a hysteresis zone occurring in higher range of ( and around the critical pump ) where bright and dark solitons coexist with a width of the coexistence domain equal to 0.096. This width is reduced compared to 0.14 found in the case of normal GVD.

4. Effect of the FOD on the Width of the Kerr Combs Spectrum

In this section, we investigate the effect of the FOD on the width of the Kerr combs spectrum in the case of normal and anomalous GVD. These Kerr comb spectra are the frequency characteristics of Kerr solitons formed by switching waves [23]. Recall that the addition of the FOD affects the bandwidth of Kerr optical frequency combs [15, 29]. Here, we vary the fourth-order dispersion term and analyze the effect on the combs spectrum in the cases of normal and anomalous GVD regimes.

4.1. Effect of the FOD on the Width of the Kerr Combs Spectrum in the Case of Normal GVD Regime

We present in Figure 5 the bright solitons obtained with the normal GVD and the corresponding frequency combs for different values of . Note that the simulations are done for , , , and . It can be seen that as the value of increases, the soliton combs lose their symmetry and widen towards low frequencies. For values within , there is a generation of a dispersive wave of low power on the side where the spectrum spreads (see Figures 5, , and ). This is in agreement with the work of Bao et al. [15], where it is shown that one of the modifications brought by the higher order dispersion terms is the generation of a dispersive wave. As is increased, the dispersive wave gains in power as well as it approaches the central mode of the frequency combs (Figures 5, , and ). Conversely, the frequency combs spread out more and more towards the dispersive wave until they merge. For values of , the soliton destabilizes and gives harmonic oscillations with distinctive peaks in the spectrum (Figures 5(e) and ).

In Figure 6, we present the dark solitons and the corresponding spectra for different values of in the case of normal GVD. As with bright solitons, we observe a broadening of the spectrum with the generation of a dispersive wave for (see Figure 6) far from the pump mode with low power. But, unlike bright solitons, the increase in the value of directly destabilizes the soliton into an oscillating pattern similar to an amplitude modulation (see Figures 6 and ()).

4.2. Effect of the FOD on the Width of the Kerr Combs Spectrum in the Case of Anomalous GVD Regime

For different values of in the case of the anomalous GVD, we illustrate bright solitons and their combs (Figure 7). As in the case of normal GVD, other parameters are fixed as follows: , , , and .

Similarly, increasing the value of also causes the combs spectrum to broaden and progressively become asymmetric. Likewise, the dispersive waves appear in the bright soliton spectrum from the value of . We also observe the shift of the dispersive wave towards the central mode and for , the soliton destabilizes into oscillation packages characterized by nonuniform packages in the spectrum that are almost symmetric from the central frequency (Figures 7 and ). In the case of anomalous GVD, the dispersive wave in the frequency combs of the dark solitons was not observed (not shown) indicating that these solitons are more sensitive to a slight change of the fourth-order dispersion term.

5. Conclusion

In this paper, we have studied the effects of higher order dispersion terms on the coexistence of solitons and optical frequency combs generated by whispering-gallery-mode microresonators. By playing with the values of the chromatic dispersion parameters and their signs, we have shown that they strongly influence the dynamics of Kerr frequency combs (the type of pulse, their stability, the shape of the frequency combs envelope, and the bandwidth of the frequency combs). In the case of zero and anomalous GVD regimes, we have also highlighted that the pump power ranges where bright and dark solitons coexist are found around the critical pump (the critical pump depends only on the detuning ).

Similar to the results obtained by Coen et al. [31], it has been noticed a broadening of the frequency combs spectra accompanied by the generation of dispersive waves. This is caused by the FOD in the case of the normal GVD (for both bright and dark solitons), but only with bright solitons in the case of anomalous GVD. Besides, the dispersive waves have been found abundantly with bright solitons, that is with a larger range of the FOD, while they are selective with dark solitons, that is with a specific value of the FOD. This highlights the possibility of controlling the width of the Kerr frequency combs with the fourth-order dispersion term. Future works will address the influence of thermo-optic effects on the widths of the hysteresis and the frequency combs spectra in microresonators [37].

Data Availability

The data used to support the funding of this study are available from the corresponding authors upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

S. A. Diddams, K. Vahala, and T. Udem, “Optical frequency combs: coherently uniting the electromagnetic spectrum,” Science, vol. 369, no. 6501, Article ID eaay3676, 2020.
View at: Publisher Site | Google Scholar
S. Zhang, T. Bi, G. N. Ghalanos, N. P. Moroney, L. Del Bino, and P. Del’Haye, “Dark-bright soliton bound states in a microresonator,” Physical Review Letters, vol. 128, no. 3, Article ID 033901, 2022.
View at: Publisher Site | Google Scholar
L. Bahloul, L. Cherbi, A. Hariz, and M. Tlidi, “Temporal localized structures in photonic crystal fibre resonators and their spontaneous symmetry-breaking instability,” Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol. 372, Article ID 20140020, 2014.
View at: Publisher Site | Google Scholar
B. Kostet, S. S. Gopalakrishnan, E. Averlant, Y. Soupart, K. Panajotov, and M. Tlidi, “Vectorial dark dissipative solitons in Kerr resonators,” OSA Continuum, vol. 4, no. 5, pp. 1564–1570, 2021.
View at: Publisher Site | Google Scholar
T. Herr, V. Brasch, J. D. Jost et al., “Temporal solitons in optical microresonators,” Nature Photonics, vol. 8, no. 2, pp. 145–152, 2014.
View at: Publisher Site | Google Scholar
C. Y. Wang, T. Herr, P. Del’Haye et al., “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-q toroid microcavity,” Nature Communications, vol. 4, no. 1345, pp. 1–7, 2013.
View at: Google Scholar
L. Chang, S. Liu, and J. E. Bowers, “Integrated optical frequency comb technologies,” Nature Photonics, vol. 16, no. 2, pp. 95–108, 2022.
View at: Publisher Site | Google Scholar
D. A. Long, A. J. Fleisher, K. O. Douglass et al., “Multiheterodyne spectroscopy with optical frequency combs generated from a continuous-wave laser,” Optics Letters, vol. 39, no. 9, pp. 2688–2690, 2014.
View at: Publisher Site | Google Scholar
F. C. Cruz, “Optical frequency combs generated by fourwave mixing in optical fibers for astrophysical spectrometer calibration and metrology,” Optics Express, vol. 16, no. 17, Article ID 13267, 2008.
View at: Publisher Site | Google Scholar
R. M. Kouayep, J. H. Talla Mbé, and P. Woafo, “Power spectrum analysis of time-delayed optoelectronic oscillators with wide and narrow band nonlinear flters,” Optical and Quantum Electronics, vol. 55, no. 62, pp. 1–19, 2022.
View at: Google Scholar
Y. Cheng and J. Yan, “20 GHz self-oscillating parametric optical frequency comb generator using an electroabsorption modulated laser based dual-loop optoelectronic oscillator,” IET Optoelectronics, vol. 14, no. 6, pp. 417–421, 2020.
View at: Publisher Site | Google Scholar
T. Fortier and E. Baumann, “20 years of developments in optical frequency comb technology and applications,” Communications Physics, vol. 2, no. 1, p. 153, 2019.
View at: Publisher Site | Google Scholar
T. Kippenberg, S. Spillane, and K. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-q toroid microcavity,” Physical Review Letters, vol. 93, no. 8, Article ID 083904, 2004.
View at: Publisher Site | Google Scholar
V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Nonlinear optics and crystalline whispering gallery mode cavities,” Physical Review Letters, vol. 92, no. 4, Article ID 043903, 2004.
View at: Publisher Site | Google Scholar
C. Bao, H. Taheri, L. Zhang et al., “High-order dispersion in kerr comb oscillators,” Journal of the Optical Society of America B, vol. 34, no. 4, pp. 715–725, 2017.
View at: Publisher Site | Google Scholar
R. D. D. Bitha and A. M. Dikandé, “Elliptic-type soliton combs in optical ring microresonators,” Physical Review A, vol. 97, no. 3, Article ID 033813, 2017.
View at: Google Scholar
M. H. Anderson, W. Weng, G. Lihachev, A. Tikan, J. Liu, and T. J. Kippenberg, “Zero dispersion kerr solitons in optical microresonators,” Nature Communications, vol. 13, no. 1, p. 4764, 2022.
View at: Publisher Site | Google Scholar
S. Zhang, T. Bi, and P. Del’Haye, “Microresonator soliton frequency combs in the zero-dispersion regime,” 2022, https://arxiv.org/abs/2204.02383.
View at: Google Scholar
P. Parra-Rivas, S. Hetzel, Y. V. Kartashov et al., “Quartic kerr cavity combs: bright and dark solitons,” Optics Letters, vol. 47, no. 10, pp. 2438–2441, 2022.
View at: Publisher Site | Google Scholar
M. Tlidi and L. Gelens, “High-order dispersion stabilizes dark dissipative solitons in all-fiber cavities,” Optics Letters, vol. 35, no. 3, pp. 306–308, 2010.
View at: Publisher Site | Google Scholar
A. G. Vladimirov, M. Tlidi, and M. Taki, “Dissipative soliton interaction in Kerr resonators with high-order dispersion,” Physical Review A, vol. 103, no. 6, Article ID 063505, 2021.
View at: Publisher Site | Google Scholar
M. Tlidi, L. Bahloul, L. Cherbi, A. Hariz, and S. Coulibaly, “Drift of dark cavity solitons in a photonic-crystal fiber resonator,” Physical Review A, vol. 88, no. 3, Article ID 035802, 2013.
View at: Publisher Site | Google Scholar
J. H. Talla Mbé and Y. K. Chembo, “Coexistence of bright and dark cavity solitons in microresonators with zero, normal, and anomalous group-velocity dispersion: a switching wave approach,” Journal of the Optical Society of America B, vol. 37, no. 11, p. A69, 2020.
View at: Publisher Site | Google Scholar
G. Wang, W. Zhang, K. Han, C. Lu, H. Zhang, and S. Fu, “Gete based modulator for the generation of soliton, soliton molecule and bright-dark soliton pair,” Infrared Physics and Technology, vol. 125, Article ID 104304, 2022.
View at: Publisher Site | Google Scholar
B. Kostet, Y. Soupart, K. Panajotov, and M. Tlidi, “Coexistence of dark vector soliton Kerr combs in normal dispersion resonators,” Physical Review A, vol. 104, no. 5, Article ID 053530, 2021.
View at: Publisher Site | Google Scholar
P. Parra-Rivas, D. Gomila, and L. Gelens, “Coexistence of stable dark-and bright-soliton kerr combs in normal-dispersion resonators,” Physical Review A, vol. 95, no. 5, Article ID 053863, 2017.
View at: Publisher Site | Google Scholar
A. Coillet, R. Henriet, K. Phan Huy et al., “Microwave photonics systems based on whispering-gallery-mode resonators,” Journal of Visualized Experiments, vol. 78, Article ID e50423, 2013.
View at: Publisher Site | Google Scholar
A. V. Cherenkov, V. E. Lobanov, and M. L. Gorodetsky, “Dissipative Kerr solitons and Cherenkov radiation in optical microresonators with third-order dispersion,” Physical Review A, vol. 95, no. 3, Article ID 033810, 2017.
View at: Publisher Site | Google Scholar
Y. Okawachi, M. R. Lamont, K. Luke et al., “Bandwidth shaping of microresonator-based frequency combs via dispersion engineering,” Optics Letters, vol. 39, no. 12, pp. 3535–3538, 2014.
View at: Publisher Site | Google Scholar
Y. K. Chembo, D. V. Strekalov, and N. Yu, “Spectrum and dynamics of optical frequency combs generated with monolithic whispering gallery mode resonators,” Physical Review Letters, vol. 104, no. 10, Article ID 103902, 2010.
View at: Publisher Site | Google Scholar
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,” Optics Letters, vol. 38, no. 1, pp. 37–39, 2013.
View at: Publisher Site | Google Scholar
Y. K. Chembo and C. R. Menyuk, “Spatiotemporal lugiato-lefever formalism for kerr-comb generation in whispering-gallery-mode resonators,” Physical Review A, vol. 87, no. 5, Article ID 053852, 2013.
View at: Publisher Site | Google Scholar
M. Liu, H. Huang, Z. Lu, Y. Wang, Y. Cai, and W. Zhao, “Dynamics of dark breathers and Raman-kerr frequency combs influenced by high-order dispersion,” Optics Express, vol. 29, no. 12, p. 18095, 2021.
View at: Publisher Site | Google Scholar
J. Wang, A.-G. Sheng, X. Huang, R.-Y. Li, and G.-Q. He, “Eigenvalue spectrum analysis for temporal signals of Kerr optical frequency combs based on nonlinear Fourier transform,” Chinese Physics B, vol. 29, no. 3, Article ID 034207, 2020.
View at: Publisher Site | Google Scholar
C. Godey, I. V. Balakireva, A. Coillet, and Y. K. Chembo, “Stability analysis of the spatiotemporal lugiato-lefever model for kerr optical frequency combs in the anomalous and normal dispersion regimes,” Physical Review A, vol. 89, no. 6, Article ID 063814, 2014.
View at: Publisher Site | Google Scholar
J. H. Talla Mbé, C. Milián, and Y. K. Chembo, “Existence and switching behavior of bright and dark kerr solitons in whispering-gallery mode resonators with zero group-velocity dispersion,” The European Physical Journal D, vol. 71, no. 7, p. 196, 2017.
View at: Publisher Site | Google Scholar
B. Azah Bei Cho, I. Ndifon Ngek, and A. M. Dikandé, “Soliton crystals in optical Kerr microresonators in the presence of thermo-optic effects,” Journal of Optics, vol. 24, no. 11, Article ID 115501, 2022.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2023 Willy André Kangmafo Miendjim et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

111

Downloads

173

Citations