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Adjoint method and inverse design for nonlinear nanophotonic devices
Tyler W. Hughes, Momchil Minkov, Ian A. D. Williamson, Shanhui Fan
TL;DR
The paper develops an adjoint method for gradient optimization of nonlinear photonic devices. Applied to Kerr-material switches, the method produces compact devices that route light differently in linear and nonlinear regimes.
Problem
The paper addresses the need to broaden inverse design and gradient optimization to nonlinear photonic devices.
Method
The authors extend the adjoint variable method to electromagnetic systems with Kerr nonlinearity, enabling gradients for arbitrarily many design parameters with little marginal cost.
Results
The designed switches route light between ports according to input power, achieving 98.2% versus 3.1% transmission in one switch and 81.8%/5.9% versus 6.1%/80.8% in another.
Takeaways & Limitations
The approach broadens inverse design for producing novel nonlinear devices and may apply to saturable materials, frequency mixing, optical neural networks, and optical limiters.
Takeaways & Limitations
The demonstrated formulation assumes no explicit dependence of the nonlinearity on the design variable and confines nonlinearity to specified design regions.
Abstract
from arXiv · showhide
The development of inverse design, where computational optimization techniques are used to design devices based on certain specifications, has led to the discovery of many compact, non-intuitive structures with superior performance. Among various methods, large-scale, gradient-based optimization techniques have been one of the most important ways to design a structure containing a vast number of degrees of freedom. These techniques are made possible by the adjoint method, in which the gradient of an objective function with respect to all design degrees of freedom can be computed using only two full-field simulations. However, this approach has so far mostly been applied to linear photonic devices. Here, we present an extension of this method to modeling nonlinear devices in the frequency domain, with the nonlinear response directly included in the gradient computation. As illustrations, we use the method to devise compact photonic switches in a Kerr nonlinear material, in which low-power and high-power pulses are routed in different directions. Our technique may lead to the development of novel compact nonlinear photonic devices.
NONLINEAR ADJOINT METHOD
The nonlinear adjoint formulation computes objective gradients without explicitly differentiating the nonlinear field solution for every design variable. It requires one additional linear adjoint solve, preserving the efficiency needed for large-scale gradient optimization.
- Formulation: The method optimizes real design variables ϕ to maximize a real objective L that depends on the complex field e and its conjugate.The field is constrained by a nonlinear equation, and e and e* are treated as independent variables for differentiation.
- Gradient derivation: Differentiating the governing equation produces coupled expressions for de/dϕ and de*/dϕ.These derivatives are then reorganized so the gradient can be obtained through an adjoint field rather than solving separately for each parameter.
- Adjoint solve: The adjoint field e^aj is defined as the solution of an additional linear system analogous to the linear adjoint method.This construction replaces repeated forward sensitivities with one adjoint solve.
- Computational scaling: Since e^aj is solved only once regardless of the number of parameters, gradients for arbitrarily many free parameters have little marginal cost.For multiple parameters, ∂f/∂ϕ becomes a matrix while the adjoint solve remains unchanged.
APPLICATION TO KERR NONLINEARITY
The formalism is applied to frequency-domain Maxwell systems with Kerr nonlinearity, where the forward field is nonlinear but the adjoint problem remains linear. The nonlinear fields determine the effective permittivity used in that adjoint problem.
- Linear system: The linear frequency-domain Maxwell system uses electric fields e, relative permittivity ϵr, and a source vector proportional to the electric current.The permittivity distribution serves as the design variable in the linear formulation.
- Kerr model: Kerr nonlinearity introduces an intensity-dependent permittivity through the nonlinear susceptibility χ^(3).Other nonlinear terms can also be incorporated into the general formalism.
- Adjoint construction: The nonlinear adjoint field is computed using a linear region whose effective permittivity depends on the nonlinear fields.This relationship is illustrated in the nonlinear adjoint construction in Figure 1.
- Adjoint system: The nonlinear forward field is obtained from a nonlinear equation, while the adjoint problem is a linear system whose source depends on the nonlinear solution.The adjoint system is twice the size of the corresponding linear problem but has a similar form.
- Gradient evaluation: The gradient with respect to ϵr is evaluated after computing the adjoint field, as in the linear case.The presented derivation does not assume explicit dependence of the nonlinearity on the design variable, although the formalism can be extended to it.
INVERSE DESIGN OF OPTICAL SWITCHES
The frequency-domain nonlinear adjoint method is used to inverse-design compact Kerr-nonlinear optical switches, with power-dependent routing demonstrated in 1→1 and 1→2 devices.
- Design setup: The devices are optimized at a free-space wavelength of 2µm using FDFD discretization on two-dimensional structures.The design uses a fixed region with permittivity constrained between air and Al2S3, while nonlinearity is confined to the design region.
- Fabrication-oriented constraints: Low-pass filtering and projection produce binarized structures with large, smoothed features, while optimization can converge in only a few hundred iterations.The nonlinear refractive-index shifts remain below the stated Al2S3 damage threshold for sub-nanosecond pulses.
- 1→2 switch: The 1→2 switch routes light to the right port in the linear regime and to the bottom port in the nonlinear regime.The objective is normalized to a maximum value of 1 for perfect switching operation.
DISCUSSION
The work extends adjoint optimization to nonlinear photonic devices and identifies broader applications for designing compact nonlinear components.
- The extended adjoint method enables gradient optimization of electromagnetic systems with Kerr nonlinearity.
- The formalism can also address non-frequency-mixing nonlinearities such as saturable gain or absorption.The authors further describe a possible generalization to frequency-mixing problems.
- Potential applications include nonlinear elements for optical neural networks and compact optical limiters in photonic networks.
- The authors made a software package implementing the discussed algorithms publicly available.
- The paper broadens inverse design toward novel nonlinear photonic devices.
OPTIMIZATION DETAILS
The supplementary optimization details identify the parameter table and track objective-function values across iterations for the two demonstrated devices.
- Table S1 lists parameters used in the optimization study for the 2-port and 3-port devices.
- Figure S1 plots objective function versus optimization iteration for the 2-port and 3-port devices.
PERMITTIVITY-DEPENDENT NONLINEAR SUSCEPTIBILITY
The nonlinear susceptibility is modeled as dependent on the material distribution, requiring that dependence to be included when computing adjoint sensitivities.
- The inverse-design demonstration assumes the nonlinear susceptibility is proportional to material density in the design region.
- The susceptibility vector is expressed using the scalar nonlinear susceptibility and the relative permittivity vector.The maximum allowed relative permittivity corresponds to the material permittivity.
- When relative permittivity is optimized, the nonlinear adjoint problem requires derivatives with respect to the fields and relative permittivity.
- Because susceptibility depends on relative permittivity, the derivative with respect to relative permittivity has a more complicated form than when susceptibility is fixed.
- The adjoint field retains its main-text form, but the sensitivity calculation must use the modified partial derivative with respect to relative permittivity.
MAINTAINING MINIMUM FEATURE SIZE AND BINARIZATION
Filtering and projection transform a continuous design density into a smoothed, binarized permittivity distribution while preserving compatibility with adjoint sensitivity calculations.
- Filtering and projection schemes are used to create realistic devices with larger minimum features and binarized permittivity distributions.
- A design density ρ between 0 and 1 is low-pass filtered to smooth features below the chosen length scale R.
- Projection converts the filtered density into a projected density for binarizing the structure.
- The projection midpoint is controlled by η, while β controls projection strength; typical values are η = 0.5 and β around 100.
- The final permittivity is set using the maximum permittivity ϵm.
- In the main-text optimizations, filtering and binarization were applied only inside the design region and required minimal adjoint-sensitivity modifications.Derivatives through the projection and filtering stages were used to obtain sensitivities with respect to the underlying density ρ.
- Figure S2 illustrates the sequence from original density through filtering and projection to the final relative-permittivity distribution using R = 200nm, β = 100, and η = 0.5.
NONLINEAR INDEX SHIFT
The authors estimate the nonlinear index shift sustainable in Al2S3 and compare it with the shifts and bandwidths of the final structures to assess damage-free switching.
- Al2S3 has a nonlinear index n2 between 3 × 10−14 and 2 × 10−13 cm2/W, and its 2.5 J/cm2 damage threshold corresponds to 2.5 × 1010 W/cm2 for 100 ps pulses.
- The corresponding bandwidth-limited Gaussian pulse bandwidth is approximately 4.5 GHz.
- The final structures have maximum refractive index shifts below 5 × 10−3 and objective-function FWHM bandwidths above 10 GHz.
- These values suggest the structures can produce the desired switching effects without damage using roughly 100 ps pulses and input powers around 100 mW/µm.
TRANSMISSION SPECTRA
Figure S3 presents the linear-regime transmission spectra for the two devices, with frequency shown relative to the design frequency.
- The supplementary transmission-versus-frequency results for both devices are shown in Fig. S3 in the linear regime.
- Panel (a) shows transmission through the 2-port device, while panel (b) shows transmission through the right and bottom ports of the 3-port device.
- The x-axis represents the frequency difference relative to the design frequency.