Project B3 • Frequency combs

Principal investigators

Project summary

A frequency comb is an optical signal consisting of a multitude of equidistantly spaced optical frequencies. Since their discovery by Theodor Hänsch, awarded with the Nobel prize in 2005, their tremendous applicability was demonstrated in e.g. optical metrology, frequency metrology and optical communications. In this project, we aim to find a mathematical and electrotechnical framework for the generation of frequency combs with several key characteristics.

Particularly interesting comb sources are nonlinear Kerr microresonators, cf. Figure 1. These chip-scale devices allow for the generation of frequency combs in setups with a footprint comparable to a matchbox, enabling their use in small and mobile objects such as autonomous cars, servers in data-centres and handhelds. In proof-of-principle experiments, we demonstrated the application of microresonator frequency combs in optical communications at data transmission rates of more than 50 Tbits/s through a single optical fiber [Mea16, MPea17]. Furthermore, we performed optical distance measurements and achieved a sub-µm resolution at record sampling rates of 100 MHz [Gea17, Tea18].

To enable this technology for a widespread use, key characteristics such as energy efficiency and potential for mass production need to be investigated. We address these challenges in an interdisciplinary team consisting of mathematicians and engineers by phrasing practical demands and answering these with mathematical techniques. We focus on the optical power conversion efficiency of a driving optical field inside the resonator and the emerging frequency comb. Furthermore, we explore the potential of different material platforms. Here, especially Silicon is of interest due to its widespread use in the semiconductor industry.

From the mathematical point of view, the field inside the resonator is approximated by a complex-valued function \(a(t,x)\), which depends on time \(t\) and intracavity position \(x\), and satisfies the Lugiato–Lefever–equation (LLE) \[\frac{\partial a(t,x)}{\partial t}=\left[-1-\text{i}\zeta+\text{i}d\frac{\partial^2}{\partial x^2}+\text{i}|a(t,x)|^2\right]a(t,x)+f, \quad a(t,x+2\pi)=a(t,x).\] The LLE is a damped and forced nonlinear Schrödinger equation with parameters given by forcing \(f\) (amplitude of the driving field), detuning \(\zeta\) (relative offset between frequency of the driving field and a resonance frequency of the microresonator) and second-order dispersion \(d\). When pumping the microresonator with an incoming single-frequency field, the Kerr-nonlinear term in the LLE gives rise to new spectral components, resulting in a frequency comb. Of most interest are soliton frequency comb states. These are stationary strongly localized solutions in space, which feature a broad spectrum in the frequency domain, cf. Figure 2.

New dynamical models for frequency combs in silicon-based microresonator

From the practical point of view, Kerr comb generation in silicon based microresonators is promising due to the high nonlinear Kerr effect of silicon. In addition, silicon is a well established material for photonic integrated circuits. However, the effects of two photon absorption (TPA) and free carrier absorption (FCA) work against comb formations. To investigate the possibility of comb formations, we formulated a modified variant of the LLE model taking into account TPA and FCA in silicon based microresonator. In [GMR22], existence of frequency combs in presence of TPA was shown, in [Tro20], the modulation instability occurs for certain TPA, FCA, and pump parameters. Continue reading. Collapse content.

TPA leads to a nonlinear damping, represented by the parameter $r > 0$. Additionally, free carriers generated by TPA will impact the refractive index via the free carrier dispersion parameter $\mu$. The density $N$ of the free carriers is governed by a first order ODE coupled to the modified LLE. The resulting dynamical models are described as: \begin{align*} \frac{\partial a(t,x)}{\partial t} &= \left[-1- \mathrm{i} \zeta - \sigma \left(1 + \mathrm{i} \mu \right)N(x-tv, t) + \mathrm{i} d \frac{\partial^2}{\partial x^2} + (\mathrm{i} - r)|a(t,x)|^2 \right] a(t,x) + f,\\ \frac{\partial N(t,x)}{\partial t} &= r|a(x+tv, t)|^4-\frac{N(t,x)}{\tau}, \end{align*} where the positive quantities $\sigma$ and $\tau$ represent FCA cross-section and effective FCA lifetime. Since the field amplitude $a$ inside the microresonator is described in a reference frame moving with group velocity $v$ while the free carriers do not move, the offset ±$tv$ between the two reference frames is visible in the coupling of the two differential equations. In our investigations, the main findings are:

• Modulation instability occurs provided $r$ and $\sigma\tau$ stay below a critical threshold and simultaneously the pump power exceeds a critical value for sideband excitation, see Figure 3(a)
• Modulation instability is necessary for soliton comb formation - but not sufficient. Thus, dynamically increased detuning will drive the resonator into a final single soliton comb.

Based on these results, we have started working towards an experimental demonstration of Kerr comb generation in silicon based microresonators, that are equipped with a reverse-biased p-i-n junction to sweep out two-photon generated free carriers from the waveguide core, see Figure 3(b).

Generation of frequency combs by two-mode pumping

In applications, generation of frequency combs by pumping two (instead of one) resonant modes of the ring resonator can be advantageous with respect to quality measures such as optical power conversion efficiency and bandwidth. In [GJKR22], [GPKR22] we investigate this novel pumping scheme modeled by the extended LLE: $$\frac{\partial u(t,x)}{\partial t} = \left[-1- \mathrm{i}\zeta + \mathrm{i}d \frac{\partial^2}{\partial x^2} + \mathrm{i} |u(t,x)|^2 \right] u(t,x) + f_0 + f_1 \mathrm{e}^{ \mathrm{i}(k_1 x - \nu_1 t)}.$$ Here the parameters $f_0, f_1$ model the power of the two lasers sources with total power $f_0^2+f_1^2 = f^2$, $k_1$ is the index of the second pumped mode and $\nu_1$ represents the frequency difference between the second laser source and the corresponding resonant mode $k_1$. Continue reading. Collapse content.

Using analytical and numerical continuation and bifurcation methods, cf. Fig. 4, we were able to prove the existence of traveling wave solutions to the extended LLE.

Moreover, we numerically discovered that a symmetrical power distribution between the two lasers ($50 \%$ in each mode) simultaneously optimizes several important performance metrics of the Kerr-comb, see Fig. 5.

Applications of Kerr frequency combs

Apart from theoretical investigations and fundamental explorations of Kerr comb dynamics, we further demonstrated the viability of Kerr frequency combs in different application fields. In [MPea17], [MPea20], Kerr soliton combs were studied in the context of optical communications and experiments demonstrated their use for high-speed data transmission. Further applications include optical ranging [Tea18], and ultra-broadband photonic-electronic signal processing [Fea21], [Dea22]. Here, one of our main findings was that Kerr combs enable ultra-broadband analog-to-digital conversion with record-high acquisition bandwidth (320 GHz) and optical waveform measurement with a record-high bandwidth exceeding 600 GHz [Fea21], [Fea22], [Dea22]. Continue reading. Collapse content.

Continuation and bifurcation for the Lugiato-Lefever equation

In order to find soliton states analytically, several approaches for the stationary equation $$ -da'' + (\zeta - \mathrm{i}) a - |a|^2 a + \mathrm{i} f = 0 $$ were used [MR17], [GR20]. Further, in [Gea19] we developed a heuristic to find the most localized solitons by numerical path continuation. Continue reading. Collapse content.

The first approach is continuation for \(d>0\). We consider a reformulated version of the equation \[-du''+(\zeta-\epsilon\text{i})u-|u|^2u+\text{i}f=0\] on the real line. For \(\epsilon=f=0\) the soliton solution can be calculated explicitly, and then analytically continued into the regime where \(\epsilon>0\) and \(f>0\). This approach can also be used numerically as these solutions serve as approximations on a bounded interval. Scaling them back leads to a map of parameters where we can expect soliton solutions.

The second approach addresses bifurcation with respect to the parameter \(\zeta\), cf. [MR17], and preprint [Mea16]. At certain values, non-trivial solutions bifurcate from the trivial curve, cf. Figure 6.

In our investigations, we observe the following heuristic:
For \(d>0\) the most localized solitons are at the first turning point of the last bifurcating branch (Label A).
For \(d<0\) the most localized solitons are at the second turning point of the first bifurcating branch (Label B).
Using this heuristic, we can identify parameter regimes where the most localized soliton solutions are found. Defining quality measures such as combwidth and power conversion efficiency, this enables us to determine a universal conversion efficiency map, cf. Figure 4, which is essential for designing microresonators embedded in integrated photonic systems, cf. [Gea19].

Within our project we also consider numerical time integration of the LLE via a Strang-Splitting between its nonlinear and dispersive part; cf. [JMS17]. One of our observations was that time integration follows the bifurcation branches when the detuning is swept over time.

Publications

1. L. Bengel and B. de Rijk. Existence and stability of soliton-based frequency combs in the Lugiato–Lefever equation. Phys. D, 483:134922, December 2025. URL https://doi.org/10.1016/j.physd.2025.134922. [preprint] [files] [bibtex]

2. M. Haragus, M. A. Johnson, W. R. Perkins, and B. de Rijk. Nonlinear subharmonic dynamics of spectrally stable Lugiato–Lefever periodic waves. Comm. Math. Phys., 405(10):227, 31, October 2024. URL https://doi.org/10.1007/s00220-024-05104-5. [preprint] [bibtex]

3. L. Bengel, D. Pelinovsky, and W. Reichel. Pinning in the extended Lugiato–Lefever equation. SIAM J. Math. Anal., 56(3):3679–3702, June 2024. URL https://doi.org/10.1137/23M1550700. [preprint] [files] [bibtex]

4. L. Bengel. Stability of solitary wave solutions in the Lugiato–Lefever equation. Z. Angew. Math. Phys., 75:130, 19pp., June 2024. URL https://doi.org/10.1007/s00033-024-02273-0. [preprint] [bibtex]

5. E. Gasmi, H. Peng, C. Koos, and W. Reichel. Bandwidth and conversion-efficiency analysis of Kerr soliton combs in dual-pumped resonators with anomalous dispersion. Phys. Rev. A, 108(2):023505, August 2023. URL https://doi.org/10.1103/PhysRevA.108.023505. [preprint] [bibtex]

6. E. Gasmi, T. Jahnke, M. Kirn, and W. Reichel. Global continua of solutions to the Lugiato–Lefever model for frequency combs obtained by two-mode pumping. Z. Angew. Math. Phys., 74:168, 31pp., July 2023. URL https://doi.org/10.1007/s00033-023-02060-3. [preprint] [files] [bibtex]

7. J. Gärtner, R. Mandel, and W. Reichel. The Lugiato–Lefever equation with nonlinear damping caused by two photon absorption. J. Dyn. Diff. Equat., 34(3):2201–2227, September 2022. URL https://doi.org/10.1007/s10884-021-09943-x. [preprint] [bibtex]

8. J. Gärtner and W. Reichel. Soliton solutions for the Lugiato–Lefever equation by analytical and numerical continuation methods. In W. Dörfler, M. Hochbruck, D. Hundertmark, W. Reichel, A. Rieder, R. Schnaubelt, and B. Schörkhuber, editors, Mathematics of Wave Phenomena, Trends in Mathematics, pages 179–195, October 2020. Birkhäuser Basel. [bibtex]

9. P. Marin-Palomo, J. N. Kemal, T. J. Kippenberg, W. Freude, S. Randel, and C. Koos. Performance of chip-scale optical frequency comb generators in coherent WDM communications. Opt. Express, 28(9):12897–12910, April 2020. URL https://doi.org/10.1364/OE.380413. [bibtex]

10. J. Gärtner, P. Trocha, R. Mandel, C. Koos, T. Jahnke, and W. Reichel. Bandwidth and conversion efficiency analysis of dissipative Kerr soliton frequency combs based on bifurcation theory. Phys. Rev. A, 100(3):033819, September 2019. URL https://doi.org/10.1103/PhysRevA.100.033819. [preprint] [bibtex]

11. R. Mandel. Global secondary bifurcation, symmetry breaking and period-doubling. Topol. Methods Nonlinear Anal., 53(2):779–800, June 2019. URL https://doi.org/10.12775/TMNA.2019.025. [preprint] [bibtex]

12. P. Trocha, M. Karpov, D. Ganin, M. H. P. Pfeiffer, A. Kordts, S. Wolf, J. Krockenberger, P. Marin-Palomo, C. Weimann, S. Randel, W. Freude, T. J. Kippenberg, and C. Koos. Ultrafast optical ranging using microresonator soliton frequency combs. Science, 359(6378):887–891, February 2018. URL https://doi.org/10.1126/science.aao3924. [bibtex]

13. T. Jahnke, M. Mikl, and R. Schnaubelt. Strang splitting for a semilinear Schrödinger equation with damping and forcing. J. Math. Anal. Appl., 455(2):1051–1071, November 2017. URL https://doi.org/10.1016/j.jmaa.2017.06.004. [preprint] [bibtex]

14. P. Marin-Palomo, J. N. Kemal, M. Karpov, A. Kordts, J. Pfeifle, M. H. P. Pfeiffer, P. Trocha, S. Wolf, V. Brasch, M. H. Anderson, R. Rosenberger, K. Vijayan, W. Freude, T. J. Kippenberg, and C. Koos. Microresonator-based solitons for massively parallel coherent optical communications. Nature, 546:274–279, June 2017. URL https://doi.org/10.1038/nature22387. [bibtex]

15. D. Ganin, P. Trocha, M. Pfeiffer, M. Karpov, A. Kordts, J. Krockenberger, P. Marin-Palomo, S. Wolf, S. Randel, W. Freude, T. J. Kippenberg, and C. Koos. Ultrafast dual-comb distance metrology using dissipative Kerr solitons. In Conference on Lasers and Electro-Optics, STh4L.6, May 2017. Optical Society of America. [bibtex]

16. R. Mandel and W. Reichel. A priori bounds and global bifurcation results for frequency combs modeled by the Lugiato-Lefever equation. SIAM J. Appl. Math., 77(1):315–345, February 2017. URL https://doi.org/10.1137/16M1066221. [preprint] [bibtex]

17. P. Marin, J. N. Kemal, P. Trocha, S. Wolf, A. Kordts, M. Karpov, M. H. P. Pfeiffer, V. Brasch, W. Freude, and C. Koos. 34.6 Tbit/s WDM transmission using soliton Kerr frequency combs as optical source and local oscillator. In European Conference on Optical Communication (ECOC), pages 415–417, December 2016. [bibtex]

18. P. Marin, J. Pfeifle, M. Karpov, P. Trocha, R. Rosenberger, K. Vijayan, S. Wolf, J. N. Kemal, A. Kordts, M. Pfeiffer, V. Brasch, W. Freude, T. Kippenberg, and C. Koos. 50 Tbit/s massively parallel WDM transmission in C and L band using interleaved cavity-soliton Kerr combs. In Conference on Lasers and Electro-Optics, STu1G.1, June 2016. [preprint] [bibtex]

Preprints

19. J. Alexopoulos. Nonlinear dynamics of periodic Lugiato–Lefever waves against sums of co-periodic and localized perturbations. CRC 1173 Preprint 2025/17, Karlsruhe Institute of Technology, May 2025. [bibtex]

20. L. Bengel and B. de Rijk. Multiple front and pulse solutions in spatially periodic systems. CRC 1173 Preprint 2025/5, Karlsruhe Institute of Technology, February 2025. [files] [bibtex]

21. L. Bengel and W. Reichel. Analysis of Kerr frequency combs in microresonators with two-photon absorption and free-carriers. CRC 1173 Preprint 2024/2, Karlsruhe Institute of Technology, February 2024. [files] [bibtex]

22. P. Maier, Y. Chen, Y. Xu, Y. a. B. M. Bao, D. Geskus, R. Dekker, J. Liu, P.-I. Dietrich, H. Peng, S. Randel, W. Freude, T. J. Kippenberg, and C. Koos. Sub-kHz-linewidth external-cavity laser (ECL) with Si${}_3$N${}_4$ resonator used as a tunable pump for a Kerr frequency comb. Preprint, Karlsruhe Institute of Technology (KIT), November 2022. [bibtex]

23. T. Jahnke and B. Stein. Stochastic Galerkin-collocation splitting for PDEs with random parameters. CRC 1173 Preprint 2018/28, Karlsruhe Institute of Technology, October 2018. [bibtex]

Theses

24. L. Bengel. On the existence and stability of nonlinear waves in Lugiato–Lefever models and in systems with periodic coefficients. PhD thesis, Karlsruhe Institute of Technology (KIT), October 2025. [bibtex]

25. E. Gasmi. On the Lugiato–Lefever model for frequency combs in a dual-pumped ring resonator with an appendix on band structures for periodic fractional Schrödinger operators. PhD thesis, Karlsruhe Institute of Technology (KIT), November 2022. [bibtex]

26. P. Marin-Palomo. Chip-scale optical frequency comb sources for terabit communications. PhD thesis, Karlsruhe Institute of Technology (KIT), February 2021. [bibtex]

27. P. Trocha. Kerr-nonlinear microresonators and frequency combs: modelling, design and applications. PhD thesis, Karlsruhe Institute of Technology (KIT), November 2020. [bibtex]

28. J. Gärtner. Continuation and bifurcation of frequency combs modeled by the Lugiato–Lefever equation. PhD thesis, Karlsruhe Institute of Technology (KIT), February 2019. [bibtex]

29. J. Pfeifle. Terabit-rate transmission using optical frequency comb sources. PhD thesis, Karlsruhe Institute of Technology (KIT), July 2016. [bibtex]

Other references

30. D. Drayss, D. Fang, C. Füllner, G. Likhachev, T. Henauer, Y. Chen, H. Peng, P. Marin-Palomo, T. Zwick, W. Freude, T. J. Kippenberg, S. Randel, and C. Koos. Slice-less optical arbitrary waveform measurement (OAWM) in a bandwidth of more than 600 GHz. In 2022 Optical Fiber Communications Conference and Exhibition (OFC), pages 1–3, March 2022. [bibtex]

31. D. Fang, A. Zazzi, J. Müller, D. Drayss, C. Füllner, P. Marin-Palomo, A. Tabatabaei Mashayekh, A. Dipta Das, M. Weizel, S. Gudyriev, W. Freude, S. Randel, J. C. Scheytt, J. Witzens, and C. Koos. Optical arbitrary waveform measurement (OAWM) using silicon photonic slicing filters. J. Light. Technol., 40(6):1705–1717, March 2022. URL https://doi.org/10.1109/JLT.2021.3130764. [bibtex]

32. D. Fang, D. Drayss, G. Lihachev, P. Marin-Palomo, H. Peng, C. Füllner, A. Kuzmin, J. Liu, R. Wang, V. Snigirev, A. Lukashchuk, M. Zhang, P. Kharel, J. Witzens, C. Scheytt, W. Freude, S. Randel, T. J. Kippenberg, and C. Koos. 320 GHz analog-to-digital converter exploiting Kerr soliton combs and photonic-electronic spectral stitching. In 2021 European Conference on Optical Communication (ECOC), pages 1–4, September 2021. [bibtex]