Many key functionalities of optical frequency combs, such as self-referencing and broad spectral access, rely on coherent supercontinuum generation (SCG). While nanophotonic waveguides have emerged as a compact and power-efficient platform for SCG, their geometric degrees of freedom have not been fully utilized due to the underlying complex nonlinear and noise-dependent stochastic physics. Here, we introduce inverse design to unlock free-form waveguides for coherent SCG. The efficacy of our design is numerically and experimentally demonstrated on Si3N4 waveguides, producing flat and coherent spectra from visible to mid-infrared wavelengths. Our work has direct applications in developing chip-based broadband light sources for spectroscopy, metrology, and sensing across multiple spectral regimes.

Formed by equally spaced spectral lines with fixed phase relations, optical frequency combs (OFCs) have received recent intense interest for their revolutionary impacts on timing, sensing, and spectroscopy.1 These diverse applications demand OFCs operated at different wavelengths, from visible bands for astronomical calibration2 to mid-infrared (mid-IR) bands for trace-gas detection.3 Such spectral windows can be unlocked through supercontinuum generation (SCG), which spectrally broadens narrowband comb generators via third-order nonlinear optical processes. Indeed, SCG has become an essential component in many OFCs to realize octave-spanning spectra for f − 2f self-referencing.4 Over the past decade, the platforms for SCG have evolved from microstructured fibers5 to nanophotonic waveguides,6 with increased optical confinement and Kerr nonlinearity. A wide selection of waveguide materials, such as Si,7,8 SiO2,9 Si3N4,10,11 LiNbO3,12,13 AlN,14–16 Ta2O5,17–19 and AlGaAs,20,21 have provided a large transparency window for multi-octave SCG from ultraviolet to mid-infrared wavelengths.22 Along with lithography-based geometric dispersion engineering, more complex geometries for SCG have been studied, such as non-uniform waveguides23–26 and waveguide arrays.27 

In addition to the spectral broadening, the parasitic amplification of noises during SCG could destroy the comb structure. Despite the established theoretical model, designing coherent SCG is usually empirical (e.g., using short pulses and short waveguides) due to the underlying nonlinear and stochastic processes induced by input noise.5,28 Recently, inverse design has shown the potential to expedite SCG design by generating waveguide geometries from the target spectra.29,30 However, incorporating coherence as an objective for optimization remains challenging and time-consuming since most inverse-design algorithms are not compatible with stochastic processes. As a result, geometric optimization of waveguides31,32 and fibers33 for coherent SCG have only been tested with few degrees of freedom.

In this work, we introduce an efficient protocol for the inverse design of coherent SCG. The waveguide widths along the propagation direction are freely varied, which impacts both the local dispersion and nonlinear coefficient. Utilizing the particle swarm optimization (PSO) algorithm, multiple objectives, including power, spectral flatness, and coherence, can be simultaneously optimized according to preset preference levels. Our approach can also be parallelized to boost computational efficiency. The advantages of the inverse-designed free-form waveguides over uniform waveguides are numerically and experimentally verified on Si3N4 platforms. In addition, an automated algorithm is developed for the optimal strategy to create self-referenced OFCs.

The waveguide geometry and computational flow are shown in Fig. 1. The free-form waveguide comprises multiple straight segments with distinct lengths and widths, which are connected by tapered segments for adiabatic transitions. Varying the waveguide width from 0.8 to 10 µm, with a 100 µm transition length, yields ∼99% transmission of the TE polarization fundamental mode at the wavelength of 1560 nm. To ensure a conservative estimate, we set the maximum gradient of the tapered transition structure to 25 nm/μm (gradient = half the width change/length). To compute the dispersion and effective mode area for a given cross section, a vector finite difference mode solver is employed.34 Given the significant alterations in waveguide geometry, potentially resulting in mode leakage into the surrounding silica, we include the imaginary part of the silica index.35,36 Since our simulations primarily target wavelengths below 4 µm, we disregard the imaginary part of the silicon nitride index, considering its transparent window to be 0.4–4.5 µm.6,37 To expedite the computations, a database is established to store these waveguide properties for subsequent simulations.

FIG. 1.

Working flow of the inverse design algorithm. The algorithm contains four main parts, namely, objectives, restrictions, simulation solver, and repository. The right panel shows the geometry of the free-form waveguide, with its cross-sectional modal profiles and dispersion profiles varying at different locations.

FIG. 1.

Working flow of the inverse design algorithm. The algorithm contains four main parts, namely, objectives, restrictions, simulation solver, and repository. The right panel shows the geometry of the free-form waveguide, with its cross-sectional modal profiles and dispersion profiles varying at different locations.

Close modal
The SCG solver is based on the generalized nonlinear Schrödinger equation (GNLSE),38,39 selected for its capability to model broadband spectra,
Ã(z,ω)exp(D̂(ω)z)z=in2n0(ω)ωcneff(ω)Aeff1/4(ω)exp(D̂(ω)z)FĀ(z,T)|Ā(z,T)|2.
(1)
The dispersion operator D̂(ω) and the function Ā(z,T) are defined as follows:
Ā(z,T)=F1Ã(z,ω)Aeff1/4(ω),
(2)
D̂(ω)=iβ(ω)βω0β1ω0ωω0α(ω)/2,
(3)
where Ã(z,ω) represents the complex spectral envelope, α(ω) denotes the linear loss, β1 corresponds to the first-order dispersion, Aeff is the effective mode area, n2 refers to the nonlinear index, n0 represents the linear refractive index, neff signifies the effective index, and c is the speed of light in vacuum. We solve Eq. (1) using ordinary differential equation (ODE) integrators in MATLAB.40 The propagation step set for ODE is 1 µm to ensure convergence, with the actual computed step sizes auto-adapted accordingly. Using data retrieved from the database, the SCG solver generates an output spectrum in ∼10 s to 1 min.
We present the inverse design process as an optimization problem, focusing on three figures of merit (FOMs): power, flatness, and coherence within a specific spectral window (ω1 to ω2). The FOMs for power and flatness are represented by the following equations:
FOM1=ω1ω2c(ω)log|Ãω|2dω,
(4)
FOM2=ω1ω2log|Ã(ω)|2log|Ã(ω)|2̄2log|Ã(ω)|2̄2(ω2ω1)dω.
(5)
Here, c(ω) represents a weight function to adjust the significance of specific wavelengths. By default, c(ω) is set to a constant value, reflecting our emphasis on the overall waveband rather than individual wavelengths in our paper. To avoid the generation of large spikes on the spectrum, we use the logarithm of the power spectral density (log|Ã(ω)|2) rather than the absolute power levels. The coherence is evaluated by simply comparing the difference between two realizations (Ã1 and Ã2) that are independently simulated with and without noise seedings added to the input pulse,
FOM3=ω1ω2log|Ã1ω|2log|Ã2ω|22dω.
(6)
We find that FOM3 is sufficient for the optimization process, which is further verified using coherence calculated from more independent realizations (see Sec. III).

To minimize these FOMs simultaneously, we employ the PSO algorithm41 and store the optimal solutions in a repository using Pareto dominance.42, FOM preference guidance is introduced to ensure that the population evolves toward FOMs with higher priority in each iteration, thus avoiding unnecessary solutions on the edges of the solution set. Constraints, such as the total length of the waveguide and minimum feature size during fabrication, are also included. Finally, the solutions in the repository are refined to reveal the optimal results. In contrast to genetic algorithms,31,33 the PSO algorithm is insensitive to initial values. The numerous local minima of the GNLSE can be avoided since the PSO actively exchanges local and global information. Notably, PSO is compatible with parallel computation, which can expedite the optimization. For example, on a regular desktop PC, five simulation solvers can perform 5000 to 10 000 computations (computations = particle size · iteration number) in 10 h. In our observations, with 80 particles, the optimized results exhibit slight changes after 200 iterations.

We first compare the performance of free-form waveguides to that of uniform waveguides. Here, we select Si3N4 waveguides and optimize FOMs regarding intensity, flatness, and coherence within both the visible (400–760 nm) and mid-IR (2500–4000 nm) bands. These Si3N4 waveguides are designed with air cladding on a silica substrate, with a fixed length of 4 mm and a thickness of 740 nm. The input pulse used in our simulation is a sech-shaped pulse with a central wavelength of 1560 nm, peak power of 1.5 kW, and pulse width of 90 fs. During optimization, we consider a width ranging from 800 to 9000 nm, with tapered segments featuring a gradient scale of 2.5–25 nm/μm. Only TE fundamental modes are considered hereafter. The results of SCG are shown in Fig. 2(a). For the uniform waveguide (1 segment), the produced spectra are relatively narrow and exhibit disjointed coverage in the visible, near-IR, and mid-IR regions. In contrast, the free-form waveguides produce much broader and flatter spectra. For instance, a waveguide with 29 segments generates 700 times and 20 times higher power than the uniform waveguide in the visible and mid-IR bands, respectively. Figure 2 shows that as the number of segments increases, the geometries of the waveguides do not converge to the same design. It is because the length of each segment should be greater than the nonlinear length defined by (γP)−1 to have a significant effect on the spectrum. In our specific case, this minimum length extends to hundreds of micrometers, indicating a maximum number of segments around 30. Once this limit is reached, further increasing the number of segments would reduce computational efficiency.

FIG. 2.

SCG in uniform and free-form waveguides. (a) Top-view geometries of the waveguides. The scale bar of the vertical axis is to indicate the width of the waveguides. The number of segments for each waveguide is indicated at the input port. (b) Simulated output spectra from the waveguides in panel (a). The spectrum of the input pulse is indicated in gray. The coherence of the spectra is also plotted in green. The target spectral window is indicated by the dotted lines.

FIG. 2.

SCG in uniform and free-form waveguides. (a) Top-view geometries of the waveguides. The scale bar of the vertical axis is to indicate the width of the waveguides. The number of segments for each waveguide is indicated at the input port. (b) Simulated output spectra from the waveguides in panel (a). The spectrum of the input pulse is indicated in gray. The coherence of the spectra is also plotted in green. The target spectral window is indicated by the dotted lines.

Close modal
We further calculate the first-order coherence g12(1) of the spectra based on
g12(1)(ω)=Ã1(ω)Ã2(ω)Ã1(ω)2Ã2(ω)2,
(7)
where Ã1 and Ã2 represent two realizations with different random noise seedings.5 The averaged coherence is determined using 50 independently simulated spectra. Notably, the SCG in the free-form waveguides manifests near-unity coherence throughout the spectral range under investigation, even in regions of the spectrum where the signal is below the scale shown but surpasses the noise floor, showing significant advantages over spectra generated from the uniform waveguide.

Although the inverse design is usually considered a black box, understanding the key dynamics that leverage the performance of free-form waveguides is important. As an example, we investigate SCG in a waveguide consisting of 29 segments (Fig. 3). Based on the evolution of the FOMs, the SCG can be classified into five stages. In the initial stage, the waveguide width gradually changes from 800 to 5600 nm over a length of 1.687 mm, leading to pulse compression due to self-phase modulation. Subsequently, the waveguide narrows down to 800 nm, facilitating the generation of a higher-order soliton with a soliton number exceeding 6. In the second stage, soliton fission occurs and significantly broadens the spectrum. Subsequently, dispersive waves start to form in the visible and mid-IR bands. Modulational instability is effectively suppressed due to continuous tuning of the wavelength of the parametric gain through the tapered structure. These phenomena align well with the physical mechanisms commonly employed in the design of coherent SCG.5 

FIG. 3.

Dynamics of SCG in free-form waveguides. (a) Top-view of the waveguide geometry. Insets show the dispersion profiles at three different locations. AGVD: anomalous group velocity dispersion. NGVD: normal group velocity dispersion. (b) Simulated evolution of FOMs. (c) Simulated temporal evolution. (d) Simulated spectral evolution. DW: dispersive wave; SF: soliton fission; SPM: self-phase modulation; DSP: dispersive effect; XPM: cross-phase modulation.

FIG. 3.

Dynamics of SCG in free-form waveguides. (a) Top-view of the waveguide geometry. Insets show the dispersion profiles at three different locations. AGVD: anomalous group velocity dispersion. NGVD: normal group velocity dispersion. (b) Simulated evolution of FOMs. (c) Simulated temporal evolution. (d) Simulated spectral evolution. DW: dispersive wave; SF: soliton fission; SPM: self-phase modulation; DSP: dispersive effect; XPM: cross-phase modulation.

Close modal

However, the insertion of a wide waveguide after soliton fission has minimal impact on the FOMs. Indeed, its impacts are more pronounced in the time domain, as the split pulses continue to propagate at speeds determined by their respective group velocities and are relocated at the end of this stage. It ensures substantial temporal overlap between these pulses in the final stage, which further broadens the dispersive waves due to cross-phase modulation. This wave-trapping stage is critical for free-form waveguides to outperform uniform waveguides, and the required precise control of the dispersion to realize pulse relocation remains a challenge for forward design.

In this section, we present some exemplary designs generated by our algorithm. The first example targets broadband visible OFCs within the wavelength range between 400 and 760 nm (394 to 749 THz). The inverse design is performed for a comb generator operating at 1560 nm with a pulse width of 90 fs and Si3N4 waveguides with a length of 4 mm and a thickness of 740 nm [see Fig. 4(a)]. These waveguides are designed for different pulse energies ranging from 82 to 205 pJ. The overall width of the waveguide decreases with lower-power input pulses, indicating a higher nonlinear coefficient required for spectral broadening. The power of the comb lines in the visible band increases with higher input power levels, yet every spectrum is capable of delivering coherent combs spanning a few hundred terahertz with power variations within 5 dB.

FIG. 4.

Free-form waveguides for coherent SCG in the visible and mid-IR bands. (a) Waveguide geometries and (b) optical spectra optimized for coherent SCG in the visible band. (c) Waveguide geometries and (d) optical spectra optimized for coherent SCG in the mid-IR band. The input pulse energies are indicated at the input port. The spectra of the input pulse and the coherence are overlaid as gray and green traces, respectively. The target spectral window is indicated by dotted lines.

FIG. 4.

Free-form waveguides for coherent SCG in the visible and mid-IR bands. (a) Waveguide geometries and (b) optical spectra optimized for coherent SCG in the visible band. (c) Waveguide geometries and (d) optical spectra optimized for coherent SCG in the mid-IR band. The input pulse energies are indicated at the input port. The spectra of the input pulse and the coherence are overlaid as gray and green traces, respectively. The target spectral window is indicated by dotted lines.

Close modal

We then perform tests in the mid-IR bands (2500–4000 nm) with the same settings of the comb generator and waveguide constraints. Excellent coherence and flatness are achieved for all input power levels, with spectral variations below 5 dB achieved over a 1500 nm wavelength span [Fig. 4(d)]. This result has surpassed previous demonstrations in uniform11 and arrayed27 Si3N4 waveguides. Notably, our primary focus lies in the spectral band rather than individual wavelengths. Therefore, fabrication inaccuracies leading to waveguide width shifts of up to 100 nm may cause slight shifts but maintain maximum spectral variations within 3 dB.

Beyond the straightforward optimization problem presented above, our algorithm also allows for the design of waveguides to enable f − 2f self-referencing of OFCs. This task does not have a fixed wavelength to optimize, as it only requires the comb to be coherently broadened to an octave, such that the low-frequency (ωo) components can be doubled to beat with the high-frequency components (2ωo). The frequency of the beatnote corresponds to the carrier-envelope-offset frequency of the comb. To maximize the signal-to-noise ratio of the beatnote, we first search for the optimal frequency using the following criterion:
ωo=minlog|Aω|2+log(A2ω2).
(8)
The FOMs for this problem are given by
FOM1=log|Aωo|2,
(9)
FOM2=log|A2ωo|2,
(10)
FOM3=ωoεωo+εlog|A1ω|2log|A2ω|22dω2ωoε2ωo+εlog|A1ω|2log|A2ω|22dω,
(11)
which, respectively, measure the intensities of the spectral components and their coherence. A1 and A2 are two different realizations of the spectra with different noise inputs, and ɛ denotes the phase-matching bandwidth of the doubler. We perform calculations on air-cladded Si3N4 waveguides with a fixed length of 10 mm and a thickness of 740 nm, and the results are shown in Fig. 5. The optimal frequencies for self-referencing are dynamically adapted to lower pulse energies. For pulse energy above 50 pJ, the lower frequency is chosen to be at the pump frequency of around 200 THz. However, for pulse energy of 31 pJ, no dispersive waves are generated at 400 THz, and the optimal frequencies are shifted to around 125 THz. Note that when the input pulse energy is below 20 pJ, it is not possible to realize octave-spanning OFCs using this comb generator. The power threshold is insufficient to induce soliton fission, with our free-form waveguide primarily modulating light post-soliton fission. It should be noted that our simulation using 150 fs in the three examples above can also achieve coherent and objective-satisfying SCG, highlighting the flexibility of our design tool with longer pump pulses.
FIG. 5.

Free-form waveguides for self-referencing of OFCs. (a) Optimized waveguide geometries for different input pulse energies. The input pulse energies are indicated at the input port. (b) Simulated output spectra from the waveguides in panel (a). The spectra of the input pulse are indicated in gray. The coherence of the spectra is also plotted in green. The arrows indicate the optimal wavelengths for realizing f − 2f self-referencing.

FIG. 5.

Free-form waveguides for self-referencing of OFCs. (a) Optimized waveguide geometries for different input pulse energies. The input pulse energies are indicated at the input port. (b) Simulated output spectra from the waveguides in panel (a). The spectra of the input pulse are indicated in gray. The coherence of the spectra is also plotted in green. The arrows indicate the optimal wavelengths for realizing f − 2f self-referencing.

Close modal

We fabricate Si3N4 waveguides to validate the efficacy of our design. The geometry of the 740 nm thick air-clad Si3N4 waveguide is shown in Fig. 6(a), which has been optimized for generating ultra-broadband OFCs from the visible to mid-IR wavelengths. The patterns are precisely defined through electron-beam lithography and are transferred to the Si3N4 layer via reactive ion etching. The input pulse is derived from an erbium-doped-fiber mode-locked laser with a 50 fs pulse duration and a 100-MHz repetition rate. It is coupled to the waveguide through a lensed fiber, which is customized alongside the laser to output unchirped pulses, with a 5 dB coupling loss per facet. The output from the waveguide is collimated by a parabolic reflector and then coupled into optical fibers by another parabolic reflector [Fig. 6(b)]. This effectively overcomes the chromatic and spherical aberrations associated with free-space optical paths. The spectra shown in Fig. 6(c) are acquired at three different pulse energy levels, which are analyzed by three optical spectral analyzers operating at different wavelength ranges. For a 135 pJ input pulse, the spectrum extends to an octave, with substantial dispersive waves generated at the visible wavelength. When the pulse energy is increased to 540 pJ, a broad spectrum from 550 to 2700 nm emerges with intensity variations within 20 dB. In Fig. 6(c), we observe good agreement between simulation and experiment, despite some discrepancies, which may arise from the difference between the actual dispersion of the waveguide and parameters used in simulation. In the experiments, some input light may couple into higher-order modes, resulting in lower actual coupled power compared to the measurements. Furthermore, the roughness arising from the fabrication process can lead to nonlinear coupling between multi-modes, which further affects the spectra. Beyond 2800 nm, all the simulated results predict a larger spectral density, likely due to increased scattered losses in longer wavelengths and OH absorption in the silica substrate.43 We then filter the spectrum near 650 nm for photodetection. A clean beatnote at 100 MHz is observed [Fig. 6(d)], which corresponds to the repetition rate of the mode-locked laser. Moreover, we further measure the relative intensity noise (RIN) for the 540 pJ results shown in Fig. 6(c). As shown in Fig. 6(e), the noise increases by up to 10 dB after SCG compared to that of the pump laser, indicating that the noise introduced below 1000 nm is not excessive.44,45

FIG. 6.

Experimental characterization of SCG in free-form waveguides. (a) The designed waveguide geometry and the scanning electron microscope images of the fabricated waveguide. (b) Experimental setup for the characterization of broadband SCG. (c) Measured optical spectra (colored shadings) and simulated optical spectra (gray lines) for input pulse energies of 135, 270, and 540 pJ. (d) Measured radiofrequency beatnote of the optical spectra indicated by the shading in the bottom panel of (c). RBW: resolution bandwidth. (e) Measured RIN of the source and output spectra for input pulse energies of 540 pJ at wavelengths from 300 to 1000 nm.

FIG. 6.

Experimental characterization of SCG in free-form waveguides. (a) The designed waveguide geometry and the scanning electron microscope images of the fabricated waveguide. (b) Experimental setup for the characterization of broadband SCG. (c) Measured optical spectra (colored shadings) and simulated optical spectra (gray lines) for input pulse energies of 135, 270, and 540 pJ. (d) Measured radiofrequency beatnote of the optical spectra indicated by the shading in the bottom panel of (c). RBW: resolution bandwidth. (e) Measured RIN of the source and output spectra for input pulse energies of 540 pJ at wavelengths from 300 to 1000 nm.

Close modal

We have demonstrated the inverse design of coherent SCG in free-form nanophotonic waveguides despite nonlinear and stochastic processes. Unlike previous inverse design methods for SCG, our approach prioritizes coherence as an FOM in optimization and enables parallel computation, significantly reducing time consumption. This results in broadband spectra that sustain OFC functionalities. Several future improvements can further enhance the advantage of our free-form waveguides over uniform waveguides. First, increasing the length of the waveguide provides a larger parameter space for the design. Indeed, recent advances in fabrication technology have enabled low-loss, meter-long nonlinear waveguides.46,47 Second, other nonlinear processes, e.g., Raman and Pockels effect, can contribute to further extending the spectral coverage. For example, LiNbO3-on-insulator and AlN waveguides are capable of generating OFC into the visible or ultraviolet bands via second-order nonlinearity.12–16 Furthermore, the complex spatiotemporal dynamics in multimode waveguides could be harnessed as another degree of freedom.48,49 The resulting increased computational load is expected to be resolved by computationally efficient data-driven methods, such as deep neural networks.30,50 We are also anticipating more functional components to be involved in the design, including periodic gratings,51 coupled waveguides,27 and resonant structures.52,53 These highly flexible devices have the potential to create diverse light sources for a wide range of applications, such as optical coherence tomography,54–57 ultrashort pulse generation,58,59 and optical communications.60 

This work was supported by the National Key R&D Plan of China (Grant No. 2021YFB2800601), the National Natural Science Foundation of China (Grant No. 92150108), the Beijing Natural Science Foundation (Grant No. Z210004), the High-performance Computing Platform of Peking University, and the micro/nano Fabrication Laboratory of Synergetic Extreme Condition User Facility (SECUF).

The authors have no conflicts to disclose.

C.-Y.L. and Y.L. contributed equally to this work.

Chia-Yi Lee: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Methodology (equal); Software (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Yanwu Liu: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Resources (equal); Validation (equal); Writing – review & editing (equal). Yinke Cheng: Resources (equal); Validation (equal); Writing – review & editing (equal). Chenghao Lao: Resources (equal); Validation (equal); Writing – review & editing (equal). Qi-Fan Yang: Conceptualization (equal); Funding acquisition (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal).

The code and data used to produce the plots within this work will be released on the repository Figshare upon publication of this preprint.

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