We present a general framework for inverse design of nanopatterned surfaces that maximize spatially averaged surface-enhanced Raman (SERS) spectra from molecules distributed randomly throughout a material or fluid, building upon a recently proposed trace formulation for optimizing incoherent emission. This leads to radically different designs than optimizing SERS emission at a single known location, as we illustrate using several 2D design problems addressing effects of hot-spot density, angular selectivity, and nonlinear damage. We obtain optimized structures that perform about 4 times better than coating with optimized spheres or bowtie structures and about 20 times better when the nonlinear damage effects are included.
Incoherent light is ubiquitous, yet designing optical devices that can handle its random nature is very challenging, since directly averaging over many incoherent incident beams can require a huge number of scattering calculations. We show how to instead solve this problem with a reciprocity technique which leads to three orders of magnitude speedup: one Maxwell solve (using any numerical technique) instead of thousands. This improvement enables us to perform efficient inverse design, large scale optimization of the metasurface for applications such as light collimators and concentrators. We show the impact of the angular distribution of incident light on the resulting performance, and show especially promising designs for the case of “annular” beams distributed only over nonzero angles.
We present a general analysis for finding and characterizing nonlinear exceptional point (EP) lasers above threshold. Using coupled mode theory and the steady-state nonlinear Maxwell-Bloch equations, we show that, for a system of coupled slabs, a nonlinear EP is obtained for a given ratio between the external pumps in each resonator, and that it is associated with a kink in the output power and lasing frequency. Through numerical linear stability analysis, we confirm that the EP laser can be stable for a large enough inversion population relaxation rate. We further show that the EP laser can be characterized by scattering a weak signal off the lasing cavity, so that the scattering frequency spectrum exhibits a quartic divergence.
We introduce a methodology for large-scale optimization of non-Fourier thermal transport in nanostructures, based upon the forward and adjoint phonon Boltzmann transport equation (BTE) and density-based topology optimization. To this end, we also develop the transmission interpolation model (TIM), an interface-based method that allows for smooth interpolation between void and solid regions. We first use our approach to tailor the effective thermal conductivity tensor of a periodic nanomaterial; then, we maximize classical phonon size effects under constrained diffusive transport, obtaining more than a four-fold degradation in the thermal conductivity with respect to commonly-employed configurations. Our method enables systematic optimization of materials for heat management and conversion, and, more broadly, the design of devices where diffusive transport is not valid.
We present a method for the end-to-end optimization of computational imaging systems that reconstruct targets using compressed sensing. Using an adjoint analysis of the Karush-Kuhn-Tucker conditions, we incorporate a fully iterative compressed sensing algorithm that solves an L_{1}-regularized minimization problem, nested within the end-to-end optimization pipeline. We apply this method to jointly optimize the optical and computational parameters of metasurface-based imaging systems for underdetermined recovery problems. This allows us to investigate the interplay of nanoscale optics with the design goals of compressed sensing imaging systems. Our optimized metasurface imaging systems are robust to noise, significantly improving over random scattering surfaces and approaching the ideal compressed sensing performance of a Gaussian matrix.
Spatially incoherent light sources, such as spontaneously emitting atoms, naively require Maxwell's equations to be solved many times to obtain the total emission, which becomes computationally intractable in conjunction with large-scale optimization (inverse design). We present a trace formulation of incoherent emission that can be efficiently combined with inverse design, even for topology optimization over thousands of design degrees of freedom. Our formulation includes previous reciprocity-based approaches, limited to a few output channels (e.g. normal emission), as special cases, but generalizes to a continuum of emission directions by exploiting the low-rank structure of emission problems. We present several examples of incoherent-emission topology optimization, including tailoring the geometry of fluorescent particles, a periodically emitting surface, and a structure emitting into a waveguide mode, as well as discussing future applications to problems such as Raman sensing and cathodoluminescence.
We present a “physics-enhanced deep-surrogate” (“PEDS”) approach towards developing fast surrogate models for complex physical systems described by partial differential equations (PDEs) and similar models: we show how to combine a low-fidelity “coarse” solver with a neural network that generates “coarsified” inputs, trained end-to-end to globally match the output of an expensive high-fidelity numerical solver. In this way, by incorporating limited physical knowledge in the form of the low-fidelity model, we find that a PEDS surrogate can be trained with at least ~10× less data than a “black-box” neural network for the same accuracy. Asymptotically, PEDS appears to learn with a steeper power law than black-box surrogates, and benefits even further when combined with active learning. We demonstrate feasibility and benefit of the proposed approach by using an example problem in electromagnetic scattering that appears in the design of optical metamaterials.
This note is intended as a brief introduction to the theory and practice of perfectly matched layer (PML) absorbing boundaries for wave equations, originally developed for MIT courses 18.369 and 18.336. It focuses on the complex stretched-coordinate viewpoint, and also discusses the limitations of PML.
Inverse design arises in a variety of areas in engineering such as acoustic, mechanics, thermal/electronic transport, electromagnetism, and optics. Topology optimization is a major form of inverse design, where we optimize a designed geometry to achieve targeted properties and the geometry is parameterized by a density function. This optimization is challenging, because it has a very high dimensionality and is usually constrained by partial differential equations (PDEs) and additional inequalities. Here, we propose a new deep learning method – physics-informed neural networks with hard constraints (hPINNs) – for solving topology optimization. hPINN leverages the recent development of PINNs for solving PDEs, and thus does not rely on any numerical PDE solver. However, all the constraints in PINNs are soft constraints, and hence we impose hard constraints by using the penalty method and the augmented Lagrangian method. We demonstrate the effectiveness of hPINN for a holography problem in optics and a fluid problem of Stokes flow. We achieve the same objective as conventional PDE-constrained optimization methods based on adjoint methods and numerical PDE solvers, but find that the design obtained from hPINN is often simpler and smoother for problems whose solution is not unique. Moreover, the implementation of inverse design with hPINN can be easier than that of conventional methods.
We develop a new type of orthogonal polynomial, the modified discrete Laguerre (MDL) polynomials, designed to accelerate the computation of bosonic Matsubara sums in statistical physics. The MDL polynomials lead to a rapidly convergent Gaussian “quadrature” scheme for Matsubara sums, and more generally for any sum F(0)/2+F(h)+F(2h)+... of exponentially decaying summands F(nh)=f(nh)e^{-nhs} where hs>0. We demonstrate this technique for computation of finite-temperature Casimir forces arising from quantum field theory, where evaluation of the summand F requires expensive electromagnetic simulations. A key advantage of our scheme, compared to previous methods, is that the convergence rate is nearly independent of the spacing h (proportional to the thermodynamic temperature). We also prove convergence for any polynomially decaying F.
This technical note describes the physical model, numerical implementation, and validation of multilevel atomic media for lasers and saturable absorbers in Meep: a free/open-source finite-difference time-domain (FDTD) software package for electromagnetics simulation. Simulating multilevel media in the time domain involves coupling rate equations for the populations of electronic energy levels with Maxwell's equations via a generalization of the Maxwell–Bloch equations. We describe the underlying equations and their implementation using a second-order discretization scheme, and also demonstrate their equivalence to a quantum density-matrix model. The Meep implementation is validated using a separate FDTD density-matrix model as well as a frequency-domain solver based on steady-state ab-initio laser theory (SALT).
We demonstrate neural-network runtime prediction for complex, many-parameter, massively parallel, heterogeneous-physics simulations running on cloud-based MPI clusters. Because individual simulations are so expensive, it is crucial to train the network on a limited dataset despite the potentially large input space of the physics at each point in the spatial domain. We achieve this using a two-part strategy. First, we perform data-driven static load balancing using regression coefficients extracted from small simulations, which both improves parallel performance and reduces the dependency of the runtime on the precise spatial layout of the heterogeneous physics. Second, we divide the execution time of these load-balanced simulations into computation and communication, factoring crude asymptotic scalings out of each term, and training neural nets for the remaining factor coefficients. This strategy is implemented for Meep, a popular and complex open-source electrodynamics simulation package, and are validated for heterogeneous simulations drawn from published engineering models.
In this technical note, we explain how to construct Gaussian quadrature rules for efficiently and accurately computing integrals of the form ∫S(λ)f(λ)dλ where S(λ) is the solar irradiance function tabulated in the ASTM standard and f(λ) is an arbitary application-specific smooth function. This allows the integral to be computed accurately with a relatively small number of f(λ) evaluations despite the fact that S(λ) is non-smooth and wildly oscillatory. Julia software is provided to compute solar-weighted quadrature rules for an arbitrary bandwidth or number of points. We expect that this technique will be useful in solar-energy calculations, where f(λ) is often a computationally expensive function such as an absorbance calculated by solving Maxwell's equations.
We present new theoretical tools, based on fluctuational electrodynamics and the integral-equation approach to computational electromagnetism, for numerical modeling of forces and torques on bodies of complex shapes and materials due to emission of thermal radiation out of thermal equilibrium. This extends our recently-developed fluctuating-surface-current (FSC) and fluctuating-volume-current (FVC) techniques for radiative heat transfer to the computation of non-equilibrium fluctuation-induced forces and torques; as we show, the extension is non-trivial due to the greater computational cost of modeling radiative momentum transfer, including new singularities that must be carefully neutralized. We introduce a new analytical cancellation technique that addresses these challenges and allows, for the first time, accurate and efficient prediction of non-equilibrium forces and torques on bodies of essentially arbitrary shapes—including asymmetric and chiral particles—and complex material properties, including continuously-varying and anisotropic dielectrics. We validate our approach by showing that it reproduces known results, then present new numerical predictions of non-equilibrium self-propulsion, self-rotation, and momentum-transfer phenomena in complex geometries that would be difficult or impossible to study with existing methods. Our findings indicate that the fluctuation-induced dynamics of micron-size room-temperature bodies in cold environments involve microscopic length scales but macroscopic time scales, with typical linear and angular velocities on the order of microns/second and radians/second; For a micron-scale gear driven by thermal radiation from a nearby chiral emitter, we find a strong and non-monotonic dependence of the magnitude and even the sign of the induced torque on the temperature of the emitter.
We present an algorithm to compute the Jordan chain of a nearly defective matrix with a 2×2 Jordan block. The algorithm is based on a inverse-iteration procedure and only needs information about the invariant subspace corresponding to the Jordan chain, making it suitable for use with large matrices arising in applications, in contrast with existing algorithms which rely on an SVD. The algorithm produces the eigenvector and Jordan vector with O(ε) error, with ε being the distance of the given matrix to an exactly defective matrix. As an example, we demonstrate the use of this algorithm in a problem arising from electromagnetism, in which the matrix has size 212^{2}×212^{2}. An extension of this algorithm is also presented which can achieve higher order convergence [O(ε^{2})] when the matrix derivative is known.
We present shape-independent upper limits to the power–bandwidth product for a single resonance in an optical scatterer, with the bound depending only on the material susceptibility. We show that quasistatic metallic scatterers can nearly reach the limits, and we apply our approach to the problem of designing N independent, subwavelength scatterers to achieve flat, broadband response even if they individually exhibit narrow resonant peaks.
This technical note describes the application of saddle-point integration to the asymptotic Fourier analysis of the well-known C_{∞} “bump” function exp[-(1-x^{2})^{-1}], deriving both the asymptotic decay rate k^{-3/4} exp(-sqrt(k)) of the Fourier transform F(k) and the exact coefficient. The result is checked against brute-force numerical integration and is extended to generalizations of this bump function.
Mühlig et. al. propose and fabricate a “cloak” comprised of nano-particles on the surface of a sub-wavelength silica sphere. However, the coating only reduces the scattered fields. This is achieved by increased absorption, such that total extinction increases at all wavelengths. An object creating a large shadow is generally not considered to be cloaked; functionally, in contrast to the relatively few structures that can reduce total extinction, there are many that can reduce scattering alone.
We extend a previous result [Phys. Rev. Lett. 105, 090403 (2010)] on Casimir repulsion between a plate with a hole and a cylinder centered above it to geometries in which the central object can no longer be treated as a point dipole. We show through numerical calculations that as the distance between the plate and central object decreases, there is an intermediate regime in which the repulsive force increases dramatically. Beyond this, the force rapidly switches over to attraction as the separation decreases further to zero, in line with the proximity force approximation. We demonstrate that this effect can be understood as a competition between an increased repulsion due to a larger polarizability of the central object interacting with increased fringing fields near the edge of the plate, and attractive forces due primarily to the nonzero thickness of the plate. In comparison with our previous work, we find that using the same plate geometry but replacing the single cylinder with a ring of cylinders, or more generally an extended uniaxial conductor, the repulsive force can be enhanced by a factor of approximately 10^{3}. We conclude that this enhancement, although quite dramatic, is still too small to yield detectable repulsive Casimir forces.
We compare several methods for the efficient generation of correlated random sequences (colored noise) by filtering white noise to achieve a desired correlation spectrum. We argue that a class of IIR filter-design techniques developed in the 1970s, which obtain the global Chebyshev-optimum minimum-phase filter with a desired magnitude and arbitrary phase, are uniquely suited for this problem but have seldom been used. The short filters that result from such techniques are crucial for applications of colored noise in physical simulations involving random processes, for which many long random sequences must be generated and computational time and memory are at a premium.
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