Inverse design is an application of machine learning to device design, giving the computer maximal latitude in generating novel structures, learning from their performance, and optimizing them to suit the designer’s needs. Gradient-based optimizers, augmented by the adjoint method to efficiently compute the gradient, are particularly attractive for this approach and have proven highly successful with finite-element and finite-difference physics simulators. Here, we extend adjoint optimization to the transfer matrix method, an accurate and efficient simulator for a wide variety of quasi-1D physical phenomena. We leverage this versatility to develop a physics-agnostic inverse design framework and apply it to three distinct problems, each presenting a substantial challenge for conventional design methods: optics, designing a multivariate optical element for compressive sensing; acoustics, designing a high-performance anti-sonar submarine coating; and quantum mechanics, designing a tunable double-bandpass electron energy filter.

Inverse design1–3 is a large family of techniques that use machine optimization to achieve desired functionalities with minimal human intervention. The design process can be cast as an optimization problem
minϕL(ϕ,e,e*)subject to f(ϕ,e,e*)=0,
where L is the cost function and encodes the design objectives, including any fabrication constraints; ϕ are the optimization parameters; e are a set of complex fields describing the response of the device; and f describes the physics of the system. The computer minimizes L by adjusting ϕ based on the feedback it receives from simulations. This mimics the training of machine learning models, with a large number of simulations taking the place of pre-existing datasets. The results are rarely intuitive but typically perform better than rationally designed equivalents.2,3

The minimization is often carried out by gradient-based optimization algorithms such as gradient descent (GD) or quasi-Newton methods4,5 that use the derivative dL/dϕ to update ϕ. The most common strategy for total derivative estimation remains the finite difference method (FDM), with autodifferentiation often used to compute the requisite partial derivatives.6 With this technique, the computational complexity of optimizations scales with the number of parameters M, severely limiting the amount of tunable variables.

The adjoint method7 provides a much more efficient alternative to compute dL/dϕ,
dLdϕ=Lϕ+2ReeTfϕ,
(1)
where the adjoint fields e′ are the solution to
feTe+Le=0
(2)
(see the supplementary material for the derivation). Together, Eqs. (1) and (2) yield dL/dϕ with computational complexity equivalent to two simulations, regardless of M.

The adjoint method evolved out of control theoretic techniques8 and can be seen as a generalization of the back-propagation used to train neural networks. Since the 1970s,9 it has been applied to an ever-expanding set of temporal and spatial optimization problems and is now a staple of gradient-based inverse design in photonics,4,5,10–12 acoustics,13–15 and mechanics.16–18 It is typically used alongside finite-difference (FDFD),2 finite-element (FEM),19 or boundary element (BEM)20 simulation engines.

However, the integration of the adjoint method with many specialized simulation engines, including the transfer matrix method (TMM), has lagged. For a vast number of layer-like quasi-1D systems, the TMM has a number of potential advantages—namely, it is exact as long as the approximations used to derive the matrices hold21 and can efficiently simulate structures with highly heterogeneous geometries or properties without computationally costly fine-grained meshes.22 The TMM has already seen use in optimization problems using FDM-based algorithms23 or analytic techniques such as the needle method for optics.24,25 However, FDM gradient computation is extremely computationally costly, and analytic derivatives become impractical for complex cost functions.26 For certain physics- and geometry-specific TMM methods, such as generalized Mie theory, adjoint-based gradient descent has very recently been implemented,27,28 but a general, physics-agnostic framework is lacking.

In this article, we introduce TMM-based adjoint optimization (TMM-AO), a versatile inverse design framework for quasi-1D systems. Compatible with both gradient descent and quasi-Newton optimizers, TMM-AO offers a physics-agnostic approach that we showcase through examples in optics, acoustics, and quantum mechanics. Beyond these examples, TMM-AO can be applied to any system for which transfer matrices exist, including more complex geometries like arrays of spheres,28 structural mechanics,29 microwave circuits,30 nonlinear optics,31 and 2/3D time-domain dynamics.32 

To illustrate the use of transfer matrices, we will take as an example a multilayer system, but we note that in addition to a film of material, a “layer” may be a sphere, a lumped-element microwave component, an equipotential region, or many other objects with well-defined transfer functions.

Consider a stack of N − 1 layers, each infinite in x̂ and ŷ but finite in ẑ. A steady-state simulation of such a structure consists of finding the state vector ei, i ∈ 0, 1, …, N in each layer, with e0 lying in the incidence medium and eN in the transmission medium.

These state vectors can be any set of quantities that completely describe the nonuniform fields in a layer, such as the amplitude of counter-propagating plane waves for optics or the pressure and velocity for acoustics.

The state vectors in adjacent layers are connected by transfer matrices determined by the physics in each layer,
ej=Tj+1,jej+1.
(3)
The states in any two layers can then be related by
ej=Ti,jei=Tj+1,jTj+2,j+1Ti,i1ei.
(4)
In particular, the fields in the incidence and transmission media may be related by
e0=TN,0eN.
(5)
Making Eq. (5) a well-posed problem, thereby allowing the fields to be found, requires exactly half of the elements of e0 and eN to be constrained. The form of these constraints depends on the physics described by the transfer matrices. To create a physics-agnostic TMM framework, we define the global boundary matrices A and B and the global source vector c, such that
Ae0+BeN=c,
(6)
where A, B, and c are known quantities that, when properly chosen, encode the constraints on e0 and eN. With these definitions, the physics equation f = 0 required by the adjoint method can be written as
f=A00B1T1,000::.::001TN,(N1)e0e1:eNc0:0=0,
(7)
where 1 is the identity matrix, 0 is the zero matrix, and 0 is the zero vector. This equation can be solved efficiently by first finding eN via
ATN,0+BeN=c,
(8)
and then iterating Eq. (3) to find all other state vectors, completing the forward simulation.

We note here a significant advantage of TMM over FDFD/FEM: To simulate a structure with highly heterogeneous length scales (such as radically different layer sizes, local wavelengths or decay lengths, etc.), with FDFD/FEM, one has to use either a uniform mesh that is sufficiently fine for the smallest anticipated length scale, or an adaptive mesh that requires periodic re-meshing during optimization.22 Both options are computationally intensive. In TMM-AO, the dynamics of any layer, regardless of the local length scale, is fully described by a single state vector. For such layer-like systems, TMM-AO will always be faster than FDFD/FEM-based approaches (see the supplementary material).

To calculate the gradient dL/dϕ, we first find the adjoint fields e′ with a backward simulation using Eq. (2). Thanks to the linearity of TMM, the structure of the adjoint system bears striking resemblance to that of the forward system,
AT100T1,0T0::.:BT0TN,(N1)Te0e1:eN+L0L1:LN=0,
(9)
where
Lj=Lej.
(10)
Typically, when the concern is the input/output of the whole device, L will only explicitly depend upon the incident and transmitted fields, e0 and eN, in which case Lj = 0 ∀j ≠ 0, N, though this is by no means guaranteed. As with Eq. (7), Eq. (9) can be solved as a single matrix equation. However, this is very memory inefficient. Leveraging the sparsity of Eq. (9), a more efficient iterative algorithm may be derived. In particular, Eq. (9) is equivalent to
BTe0+TN,(N1)TeNLN=0,
(11)
ATe0e1L0=0,
(12)
ej+1=Tj,(j1)TejLj=0.
(13)
Equations (11) and (12) may be further combined with Eq. (13) to yield
ATN,0+BTe0=LNj=0N1TN,jTLj.
(14)
Using Eq. (14), e0 may be found, followed by e1 via Eq. (12), and finally, ej, j ≠ 0, 1, using Eq. (13). The remaining ingredients needed to compute dL/dϕ are ∂L/ϕ and f/ϕ. The explicit form of the cost function L is known to the designer and, therefore, ∂L/ϕ can typically be found analytically. Suppose there are M optimization parameters ϕm, and each transfer matrix can depend on any subset of these parameters. For maximal generality, it will be assumed that the global boundary matrices A and B, as well as the source vector c, also depend upon the optimization parameters, although in practice this is very uncommon. The partial Jacobian f/ϕ is then
Aϕ1e0+Bϕ1eNcϕ1AϕMe0+BϕMeNcϕMT1,0ϕ1e1T1,0ϕMe1::TN,(N1)ϕ1eNTN,(N1)ϕMeN.
(15)
With this matrix, we can finally use Eq. (1) to determine the full gradient
dLdϕm=Lϕm+2Ree0TAϕme0+BϕmeNcϕm+j=1NejTTj,(j1)ϕmej.
(16)
Equations (12)(14) and (16) are the key results that serve as the foundation of TMM-AO.

The TMM-AO algorithm (Fig. 1) is now complete: at each iteration, the state vectors can be found from Eqs. (3) and (8), the adjoint vectors from Eqs. (12)(14), and the gradient from Eq. (16). The explicit forms of the transfer matrices depend on the physics of the problem, but the algorithm itself is completely general. Details of the implementation can be found in the code repository.33 We emphasize again that only two simulations are needed per iteration, far fewer than existing FDM-based approaches.23 Even compared to the analytic derivatives required by, e.g., the needle method or automatic differentiation,34 the adjoint gradient exhibits superior scaling (see the supplementary material).

FIG. 1.

Inverse design using transfer-matrix-based adjoint optimization (TMM-AO). The highlighted section is the focus of this paper. An optimizer provides the current structure, which is used by the appropriate physics package to generate transfer matrices. A physics-agnostic simulator then performs the forward and backward simulations, which are used to compute the new cost function and gradient that are returned to the optimizer.

FIG. 1.

Inverse design using transfer-matrix-based adjoint optimization (TMM-AO). The highlighted section is the focus of this paper. An optimizer provides the current structure, which is used by the appropriate physics package to generate transfer matrices. A physics-agnostic simulator then performs the forward and backward simulations, which are used to compute the new cost function and gradient that are returned to the optimizer.

Close modal

As a demonstration, we use TMM-AO to inverse design optical, acoustic, and quantum devices that pose significant challenges to rational design using transfer matrices built from Maxwell’s equations, the displacement potential wave equation, and the Schrödinger equation, respectively. These examples utilize TMM-AO in a number of diverse situations: with optics, we seek to exactly match a target transmission spectrum; with acoustics, we seek to minimize full-spectrum-integrated reflection; and with quantum, we optimize the voltage levels of an electrode array to create a tunable electron energy filter.

We use L-BFGS-B,35 an efficient quasi-Newton optimizer, for moderate-scale problems commonly used in both inverse design and neural network training.36 There are many other optimizers, refined for use in training larger neural networks, that may be used with TMM-AO; we compare a number of these algorithms from the PyTorch library in the supplementary material. See the supplementary material also for details of the cost functions, transfer matrices, and state vectors, as well as additional examples.

Multivariate optical computing (MOC) is an emerging technique that can extract the concentration of a specific analyte despite strong interference from other chemicals, with only a few single-point detectors.6,37 It relies on sets of quantitatively precise optical filters called multivariate optical elements (MOEs) that transmit or reflect a particular spectral response that varies linearly with the analyte concentration38 [Fig. 2(a)]. So far, such MOEs have been designed primarily using FDM-based quasi-Newton optimization;23 this is, however, a very computationally demanding process.39 

FIG. 2.

TMM-AO inverse design results. (a)–(d) MOE for multivariate optical computing, (e)–(h) anti-sonar coatings for submarines, and (i)–(l) electron energy filter. (a), (e), and (i) Intended applications. (b), (f), and (j) Cost function during optimization and design at selected iterations. (c), (g), and (k) Final designs and electric field intensity, local absorption, and wavefunction magnitude therein. (d) Transmission of the final MOE design, compared to the target profile. (h) Echo reduction of the final design compared with solid Alberich tile and periodic multilayer, with the target frequency band shaded. (l) Performance of the energy filter with target passbands shaded.

FIG. 2.

TMM-AO inverse design results. (a)–(d) MOE for multivariate optical computing, (e)–(h) anti-sonar coatings for submarines, and (i)–(l) electron energy filter. (a), (e), and (i) Intended applications. (b), (f), and (j) Cost function during optimization and design at selected iterations. (c), (g), and (k) Final designs and electric field intensity, local absorption, and wavefunction magnitude therein. (d) Transmission of the final MOE design, compared to the target profile. (h) Echo reduction of the final design compared with solid Alberich tile and periodic multilayer, with the target frequency band shaded. (l) Performance of the energy filter with target passbands shaded.

Close modal

We take as our design objective the target spectrum for methane sensing in petroleum mixtures from a recent study.6,39 The authors’ design consists of a 5 µm, 5-layer, Si/SiO2 multispectral filter on one side of a glass substrate and a 5 µm, 16-layer bandpass filter on the other. Using TMM-AO and random initial conditions, we combine both filters in a remarkably thinner and simpler 3.3 µm, 11-layer stack on just one side of the substrate [Figs. 2(b)2(d)]. This device can be readily fabricated with standard coating techniques.39 

Here, we vary the layer thicknesses of a binary stack of Si and SiO2, with air as the incidence medium and BK7 optical glass as the transmission medium.40,41 The final structure is highly unintuitive and creates a variety of resonances throughout the depth of the device, with stronger resonances at wavelengths with higher transmission [Fig. 2(d)]. The end result is a transmission profile that closely mirrors the target, despite the much thinner and simpler design [Fig. 2(d)]. While this MOE is designed for normal incidence, it remains operational at moderate incident angles (supplementary material Fig. 1). With the reduced computational burden provided by the adjoint method, we expect that TMM-AO will significantly expedite the design of MOEs and expand their adoption.

To provide further examples and to demonstrate cost functions that depend upon more than just transmission, we provide three additional optics examples in supplementary material, Fig. 2: a second MOE operating in the visible range,23 a polarization-independent mid-IR beamsplitter, and a polarization-dependent dichroic beamsplitter, a device very difficult to rationally design.

Anti-sonar coatings are employed on all modern military submarines to reduce their sonar cross section [Fig. 2(e)]. The currently deployed state-of-the-art are porous rubber composites known as Alberich tiles.42 The figure-of-merit for such coatings is the echo reduction,43 
ER(f)=10log10Pr(f)Pi(f),
(17)
where Pr(f) is the reflected acoustic power at frequency f and Pi(f) is the incident acoustic power. It is also useful to define the integrated full-spectrum echo reduction,
ER=10log10Pr(f)dfPi(f)df.
(18)

We focus on the 1–10 kHz band, covering most intermediate-range military sonars.44 The performance of modern military tiles is not publicly disclosed, but the closest commercially available equivalents45 boast an integrated ER of about −7.1 dB for 10 cm of tile. Using TMM-AO, we can radically improve ER to −15.4 dB without increasing the total thickness by strategically adding spacer layers composed of polyurethane (PU), a common material for marine applications that bonds well to rubber46,47 [Figs. 2(f)2(h)].

Here, we vary the layer thicknesses of a binary stack of Alberich tile (using an averaged density and modulus to account for porosity) and PU, with seawater48 as the incidence medium and a 5 cm steel substrate49 followed by air50 in the transmission region, representing the hull and interior of a submarine.

As evidenced by the local absorption throughout the stack [Fig. 2(g), compared with supplementary material, Fig. 3], the TMM-AO design utilizes the full depth of the coating more effectively to dissipate acoustic energy in the design window. The result is a 10-layer device with much greater ER (−15.4 dB) than either a solid Alberich tile (−7.1 dB) or an evenly spaced periodic multilayer (−9.7 dB) of the same total thickness.

We note that this is not simply an anti-reflection coating, which typically maximizes transmission. Very little energy is transmitted (supplementary material Fig. 4), meaning the high ER of the final design is derived almost exclusively from enhanced absorption—something difficult to achieve with rational design. TMM-AO consistently generates distinct, high-performance structures given random initial conditions, as can be seen in the statistics of 180 trials of various numbers of layers (supplementary material Fig. 5).

Electrons or ions with well-defined energies are indispensable for studying a wide variety of quantum phenomena, from coherence times51 to gravitational and Aharonov–Bohm effects.52 Energy filters capable of isolating particles within one or more narrow, tunable bands are, therefore, highly desirable. Such filtering is typically achieved by magnetic prisms,53 but this requires a device large enough to separate and recombine the beams and is not easily tunable when multiple passbands are needed.

Instead, we envision an in-line energy filter composed of a microscale sequence of uniformly spaced electrodes held at specific electric potentials [Fig. 2(i)]. By changing the voltage on each electrode, we may control the transmission spectrum at will by directly acting on the particles’ wavefunctions.54 We target a double-bandpass filter for cold electrons around 1 K.55 Rational designs for such a device do not exist to the best of our knowledge.

The potential profile obtained by TMM-AO is highly unintuitive but creates precise resonances that enhance passband transmission while blocking undesirable energies, as shown in the electron wavefunction magnitude throughout the device [Fig. 2(k)]. The result is >90% transmission within two well-defined passbands and negligible transmission outside [Fig. 2(l)]. Moreover, we can change the locations of the passbands and even merge them using the same electrodes by simply applying different inverse-designed potential profiles (supplementary material Fig. 6). We note that fabricating such a device would be challenging but not out of reach with modern lithography and deep-etch technologies.56 

We have presented TMM-AO, a versatile machine optimization framework to efficiently design quasi-1D systems. It is compatible with any optimizer based on first-order gradients and is applicable to a wide range of physics characterized by transfer matrices. Beyond our demonstrations, transfer matrix simulation methods have been developed for other types of linear29,30 and nonlinear31,57 systems and even for higher dimensions.32,58 TMM-AO can be readily extended to any such system describable with transfer matrices.

The TMM-AO algorithm is also straightforward to use. It has been designed to be accessible to scientists and engineers unfamiliar with machine learning and optimization, providing an easy-to-understand framework that enables the user to quickly adapt the algorithm to new kinds of physics. With such broad applicability, TMM-AO has the potential to become a mainstay in inverse design.

See the supplementary material for details on the derivation of the adjoint gradient; a comparison of the simulation, gradient computation, and optimization algorithms used in TMM-AO with alternatives; and the explicit form of the transfer matrices used in this work, as well as those figures referenced in the main text.

The authors have no conflicts to disclose.

Nathaniel Morrison: Conceptualization (equal); Software (equal); Writing – original draft (equal); Writing – review & editing (equal). Shuaiwei Pan: Conceptualization (equal); Writing – review & editing (equal). Eric Y. Ma: Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

The tmmao module is available at https://github.com/Ma-Lab-Cal/tmmao.33 All the data used in generating the results in the main text are available at https://github.com/Ma-Lab-Cal/tmmao-data.59 The data used to generate the figures in the supplementary material are available upon reasonable request.

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