Optimized Multimode Edge-Coupler Designs

• Multimode-inspired edge-coupler designs are optimization-driven methods that leverage adjoint-based topology optimization to achieve efficient fiber-to-chip mode coupling over multiple wavelengths.
• The design framework systematically parameterizes a localized photonic interface, optimizing the dielectric profile to ensure critical coupling and minimal reflection across broadband spectra.
• These designs facilitate compact photonic interfaces for nonlinear frequency conversion and multiplexed applications while maintaining fabrication tolerance and robust multi-mode performance.

Multimode-inspired edge-coupler designs refer to the application of adjoint-based topology optimization methodologies—originally developed for compact multimode cavity–waveguide couplers—to the problem of efficiently interfacing fiber or free-space inputs with on-chip photonic waveguides. The approach systematically parameterizes a localized design region at the photonic interface, optimizes the dielectric profile to maximize desired modal overlap and energy transfer across several frequencies, and enforces fabrication constraints for robust, multi-wavelength operation. This framework, as advanced by Jin et al. (Jin et al., 2018), enables the realization of compact devices capable of critical or near-critical coupling over broad bands and multiple modes relevant for nonlinear and multiplexed photonic chip applications.

1. Topology Optimization Framework

Adjoint-based large-scale topology optimization forms the core methodology for multimode-inspired edge-coupler design. The design region $\Omega_c$ is discretized on a uniform grid, and a continuous density field $\rho(\mathbf{r}) \in [0,1]$ defines the local permittivity:

$$\varepsilon(\mathbf{r}) = \varepsilon_{\text{sub}} + \rho(\mathbf{r}) \left( \varepsilon_{\text{st}} - \varepsilon_{\text{sub}} \right)$$

where $\varepsilon_{\text{sub}}$ and $\varepsilon_{\text{st}}$ denote substrate and structure permittivities, respectively. During optimization, $\rho(\mathbf{r})$ is driven in the interval $[0,1]$; post-convergence, a projection or density filter enforces fabricable binary features, with minimum wall thickness set by process capabilities (e.g., $\gtrsim 120\,\text{nm}$ for GaP).

The optimization is governed by multi-modal objectives. For $N$ targeted frequencies or cavity modes, quality factors are defined:

$$Q_{i,\text{r}} = \frac{\omega_i U_i}{P_{i,\text{r}}}, \quad Q_{i,\text{c}} = \frac{\omega_i U_i}{P_{i,\text{c}}}$$

with $P_{i,\text{r}}$ the radiated power and $P_{i,\text{c}}$ the power coupled into the desired port (waveguide or output mode). Objective functions include rate-matching penalization,

$$\mathcal{F}(\rho) = \max_{i=1..N} \left[ Q_{i,\text{c}}(\rho) - \xi_i Q_{i,\text{r}}(\rho) \right]^2$$

subject to constraints on radiative $Q$ degradation, or equivalent energy-maximization formulations for unidirectional couplers.

2. Objective Formulation and Modal Overlap

The forward electromagnetic problem for each frequency $\omega_i$ is posed as:

$$\left[ \nabla \times \mu^{-1} \nabla \times - \omega_i^2 \varepsilon(\mathbf{r}) \right] \mathbf{E}_i(\mathbf{r}) = i \omega_i \mathbf{J}_i(\mathbf{r})$$

where $\mathbf{J}_i$ are adjoint current sources seeded from cavity eigenmodes. Stored energy in the cavity,

$$U_i = \frac{1}{2} \int_{V_{\text{cavity}}} \varepsilon(\mathbf{r}) \left| \mathbf{E}_i(\mathbf{r}) \right|^2 dV$$

and power flux into ports,

$$P_{i,k} = \frac{1}{2} \int_{\Sigma_k} \text{Re}[ \mathbf{E}_i^* \times \mathbf{H}_i ] \cdot \hat{n} \, dS$$

are computed for quantifying the coupling efficiency.

Edge coupler objectives directly maximize the waveguide mode power,

$$\mathcal{F}_{\text{edge}}(\rho) = \frac{P_{\text{wg}}(\rho)}{P_{\text{in}}} = \frac{ |\int_{\Sigma_{\text{wg}}} \mathbf{E}^*_{\text{wg}} \times \mathbf{H}(\rho)\, dS|^2 }{ P_{\text{in}} }$$

while minimizing spurious reflection or mode mismatch.

For nonlinear frequency-conversion designs, the figure of merit incorporates spatial overlap integrals—e.g., for second-harmonic or sum-frequency generation:

$$\mathrm{FOM} = |\beta|^2 \prod_{i=1,2,s} \frac{Q_{i,\text{r}}}{2 + Q_{i,\text{r}}/Q_{i,\text{c}} + Q_{i,\text{c}}/Q_{i,\text{r}} }$$

with peak performance achieved when $Q_{\text{c}} = Q_{\text{r}}$ for all modes. The overlap parameter $\beta$ is computed as

$$\beta = \frac{ \int_{V_{\text{cavity}}} \chi^{(2)}(\mathbf{r}) E_1(\mathbf{r}) E_2(\mathbf{r}) E_s^*(\mathbf{r})\, dV }{ \sqrt{ U_1 U_2 U_s } }$$

3. Key Device Realizations

Device results exemplifying this framework include multimode couplers for second-harmonic generation (SHG), nondegenerate sum-frequency generation (SFG), and 6-mode comb coupling:

Process	λ targets (nm)	Coupler region (μm × μm)	Radiative $Q^0_{\text{r}}$	$Q_{\text{r}}$ after optimization	$|T+R|$ at resonance	Remarks
SHG	{1500, 750}	3.75 × 1.5	{1.4e3, 4.6e3}	{4.1e3, 1.0e4}	<2%	Critical coupling
SFG	{1500, 907, 565}	5.4 × 2.0	{640, 5.3e4, 3.2e4}	{1.4e3, 9e4, 1e5}	<1%	Three-mode critical
Comb	f_i={0.667,…,1.157}	4.5 × 4.5	$\sim$10^4–10^5	—	2–13%	6-mode critical

All designs achieved near-critical coupling $Q_{\text{c}} \approx Q_{\text{r}}$ at each target and minimal feature sizes as small as 15 nm (comb) and 120 nm (SHG/SFG). Computational cost scales linearly with the number of target frequencies ($\sim$200–300 forward+adjoint solves per device).

4. Practical Generalization to Edge Coupling

The adjoint optimization protocol is readily generalized from waveguide–cavity interfaces to photonic chip edge couplers. Adjustments include:

• Mode source placed on the fiber/collimated input boundary.
• Boundary conditions with modal expansion and PML for absorption.
• Maximization of the on-chip fundamental waveguide mode flux; overlap integration with the desired output mode.
• Constraints on minimum feature size, geometry connectivity, and maximal allowed reflection coefficient.
• Tolerance and robustness assessed by Monte Carlo simulations over random $\pm 10\,\text{nm}$ perturbations in $\rho(\mathbf{r})$.

A plausible implication is the capacity for these edge coupler designs to be adapted for broad-band, multi-mode fiber-to-chip links for multiplexed or nonlinear signal processing applications, with fabrication-tolerant features.

5. Algorithmic Workflow and Implementation

The comprehensive recipe for implementation comprises:

1. Bare Cavity Characterization: Compute bare eigenmodes $E^0_i$, radiative $Q^0_{i,\text{r}}$.
2. Design Region Initialization: Discretize $\Omega_c$, choose $\varepsilon_{\text{sub}}$, $\varepsilon_{\text{st}}$.
3. Target Mode/Parameter Selection: Specify frequencies $\omega_i$, coupling ratios $\xi_i$.
4. Initialization: Set initial $\rho(\mathbf{r})$ (e.g., uniform or ribbon).
5. Adjoint Optimization Loop:
   • For each $\omega_i$, solve forward Maxwell's equations for $E_i(\mathbf{r})$, $H_i(\mathbf{r})$.
   • Calculate stored energy $U_i$, fluxes $P_{i,\text{c}}$, $P_{i,\text{r}}$, $Q$ factors.
   • Evaluate objective $\mathcal{F}$ or $\mathcal{F}'$, and constraints $\mathcal{G}_i$.
   • Build adjoint sources from $\partial \mathcal{F} / \partial E_i$.
   • Solve adjoint problems; accumulate gradients via overlap integrals.
   • Update $\rho(\mathbf{r})$ using projected gradient or the Method of Moving Asymptotes.
   • Filter and project density to maintain desired feature sizes and binarity.
6. Post-processing: Convert to binary permittivity, verify with full-resimulation.

Performance monitoring includes N (number of frequencies), grid resolution ($\geq 13$ pixels per $\lambda$ in high $\varepsilon$ regions), minimal feature size radius, regularization weights, and convergence to $\Delta \mathcal{F} < 10^{-3}$ or $\xi_i$ within $\pm 5\%$ for all modes.

Expected footprints are typically few $\times (\lambda_{\max})^2$, e.g., $4\,\mu\text{m} \times 4\,\mu\text{m}$ for $\lambda_{\max} = 1.5\,\mu\text{m}$, with critical coupling ($|T+R| < 5\%$) at all $\lambda$ for resolved designs (Jin et al., 2018).

6. Trade-offs, Fabrication Issues, and Robustness

Increasing the number of coupled modes or spectral bandwidth requires greater footprint and finer structural features, decreasing fabrication tolerance. Designs targeting fewer modes, such as SHG or SFG, accommodate more aggressive smoothing and $\geq 120\,\text{nm}$ features, beneficial for robustness. Regularization, density filtering, and binarization promote tolerance to process variability, while systematic Monte Carlo perturbation studies quantify the statistical performance impact.

7. Conceptual Schematics and Physical Significance

A conceptual layout comprises a multimode cavity or a fiber source (edge) interfaced with a single waveguide via a topology-optimized compact region. A typical schematic shows the design area bridging chip edge or cavity to waveguide, capable of matching multiple resonant frequencies with critical coupling.

Side-view cross-sections illustrate the adiabatic transformation from an incident fiber mode through a quasi-periodic metastructure into the guided on-chip mode, underlining the physical principle behind efficient mode conversion.

Application of these design principles in edge couplers fundamentally advances the miniaturization, multi-mode operation, and fabrication-tolerance of silicon photonic interconnects and nonlinear device interfaces (Jin et al., 2018).