Adjoint method and inverse design for nonlinear nanophotonic devices

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.Source: NONLINEAR ADJOINT METHOD · pp. 1–2

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.Source: NONLINEAR ADJOINT METHOD · p. 2

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.Source: NONLINEAR ADJOINT METHOD · p. 2

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.Source: NONLINEAR ADJOINT METHOD · p. 2

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.Source: APPLICATION TO KERR NONLINEARITY · p. 2

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.Source: APPLICATION TO KERR NONLINEARITY · pp. 2–3

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.Source: Figure 1 · p. 3, APPLICATION TO KERR NONLINEARITY · p. 3

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.Source: APPLICATION TO KERR NONLINEARITY · p. 3

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.Source: APPLICATION TO KERR NONLINEARITY · pp. 2–3

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.Source: INVERSE DESIGN OF OPTICAL SWITCHES · p. 3

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.Source: INVERSE DESIGN OF OPTICAL SWITCHES · pp. 4–6

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.Source: INVERSE DESIGN OF OPTICAL SWITCHES · p. 5, Figure 3 · p. 5

• The extended adjoint method enables gradient optimization of electromagnetic systems with Kerr nonlinearity.Source: DISCUSSION · p. 6
• 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.Source: DISCUSSION · p. 6
• Potential applications include nonlinear elements for optical neural networks and compact optical limiters in photonic networks.Source: DISCUSSION · p. 6
• The authors made a software package implementing the discussed algorithms publicly available.Source: DISCUSSION · p. 6
• The paper broadens inverse design toward novel nonlinear photonic devices.Source: DISCUSSION · p. 6

• Table S1 lists parameters used in the optimization study for the 2-port and 3-port devices.Source: Table S1 · p. 8
• Figure S1 plots objective function versus optimization iteration for the 2-port and 3-port devices.Source: Figure S1 · p. 8

• The inverse-design demonstration assumes the nonlinear susceptibility is proportional to material density in the design region.Source: PERMITTIVITY-DEPENDENT NONLINEAR… · p. 8
• 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.Source: PERMITTIVITY-DEPENDENT NONLINEAR… · p. 8
• When relative permittivity is optimized, the nonlinear adjoint problem requires derivatives with respect to the fields and relative permittivity.Source: PERMITTIVITY-DEPENDENT NONLINEAR… · p. 9
• Because susceptibility depends on relative permittivity, the derivative with respect to relative permittivity has a more complicated form than when susceptibility is fixed.Source: PERMITTIVITY-DEPENDENT NONLINEAR… · p. 9
• The adjoint field retains its main-text form, but the sensitivity calculation must use the modified partial derivative with respect to relative permittivity.Source: PERMITTIVITY-DEPENDENT NONLINEAR… · p. 9

• Filtering and projection schemes are used to create realistic devices with larger minimum features and binarized permittivity distributions.Source: MAINTAINING MINIMUM FEATURE SIZE AND… · p. 9
• A design density ρ between 0 and 1 is low-pass filtered to smooth features below the chosen length scale R.Source: MAINTAINING MINIMUM FEATURE SIZE AND… · p. 9
• Projection converts the filtered density into a projected density for binarizing the structure.Source: MAINTAINING MINIMUM FEATURE SIZE AND… · p. 9
• The projection midpoint is controlled by η, while β controls projection strength; typical values are η = 0.5 and β around 100.Source: MAINTAINING MINIMUM FEATURE SIZE AND… · p. 9
• The final permittivity is set using the maximum permittivity ϵm.Source: MAINTAINING MINIMUM FEATURE SIZE AND… · p. 9
• 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.Source: MAINTAINING MINIMUM FEATURE SIZE AND… · p. 9
• 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.Source: Figure S2 · p. 10

• 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.Source: NONLINEAR INDEX SHIFT · p. 10
• The corresponding bandwidth-limited Gaussian pulse bandwidth is approximately 4.5 GHz.Source: NONLINEAR INDEX SHIFT · p. 10
• The final structures have maximum refractive index shifts below 5 × 10−3 and objective-function FWHM bandwidths above 10 GHz.Source: NONLINEAR INDEX SHIFT · p. 10
• 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.Source: NONLINEAR INDEX SHIFT · p. 10

• The supplementary transmission-versus-frequency results for both devices are shown in Fig. S3 in the linear regime.Source: TRANSMISSION SPECTRA · p. 10
• 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.Source: Figure S3 · p. 11
• The x-axis represents the frequency difference relative to the design frequency.Source: Figure S3 · p. 11