@article{RodriguezJo07-iir-eprint,
author = {Alejandro Rodriguez and Steven G. Johnson},
title = {Efficient generation of correlated random numbers using {Chebyshev}-optimal magnitude-only {IIR} filters},
journal = {arXiv.org e-Print archive},
eprint = {arXiv:physics/0703152},
year = 2007,
month = {March},
abstract = {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.}
}
@article{McCauleyRo11-repulsion-eprint,
author = {Alexander P. McCauley and Alejandro W. Rodriguez and M. T. Homer Reid and Steven G. Johnson},
title = {{Casimir} repulsion beyond the dipole regime},
journal = {arXiv.org e-Print archive},
year = 2011,
eprint = {arXiv:1105.0404},
month = {May},
abstract = {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.}
}
@article{MillerQi13-eprint,
author = {Owen D. Miller and Wenjun Qiu and John D. Joannopoulos and Steven G. Johnson},
title = {Comment on `{A} self-assembled three-dimensional cloak in the visible' in {\it {S}cientific {R}eports} {\bf 3}, 2328},
journal = {arXiv.org e-Print archive},
year = 2013,
eprint = {arXiv:1310.1503},
month = {October},
abstract = {M\"uhlig 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.}
}
@article{Johnson06-bump,
author = {Steven G. Johnson},
title = {Saddle-point integration of {$C_\infty$} ``bump'' functions},
journal = {arXiv.org e-Print archive},
year = 2015,
eprint = {arXiv:1508.04376},
month = {August},
abstract = {This technical note describes the application of saddle-point integration to the asymptotic Fourier analysis of the well-known $C_{\infty}$ ``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.}
}
@article{MillerHs15-bw,
author = {Owen D. Miller and Chia Wei Hsu and Emma Anquillare and John D. Joannopoulos and Marin Solja{\v{c}}i{\'{c}} and Steven G. Johnson},
title = {Power--bandwidth limitations of an optical resonance},
journal = {arXiv.org e-Print archive},
year = 2015,
eprint = {arXiv:1510.06902},
month = {October},
abstract = {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.}
}
@article{HernandezPi17,
author = {Felipe Hern{\'a}ndez and Adi Pick and Steven G. Johnson},
title = {Scalable computation of {J}ordan chains},
journal = {arXiv.org e-Print archive},
year = 2017,
eprint = {arXiv:1704.05837},
month = {April},
abstract = {We present an algorithm to compute the Jordan chain of a nearly defective matrix with a $2\times2$ 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 $O(\varepsilon)$ error, with $\varepsilon$ 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 $212^2\times212^2$. An extension of this algorithm is also presented which can achieve higher order convergence [$O(\varepsilon^2)$] when the matrix derivative is known.}
}
@article{ReidMi17,
author = {M. T. Homer Reid and O. D. Miller and A. G. Polimeridis and A. W. Rodriguez and E. M. Tomlinson and Steven G. Johnson},
title = {Photon torpedoes and {Rytov} pinwheels: Integral-equation modeling of non-equilibrium fluctuation-induced forces and torques on nanoparticles},
journal = {arXiv.org e-Print archive},
year = 2017,
eprint = {arXiv:1708.01985},
month = {August},
abstract = {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 \textit{sign} of the induced torque on the temperature of the emitter.}
}
@article{Johnson19,
author = {Steven G. Johnson},
title = {Accurate solar-power integration: Solar-weighted {G}aussian quadrature},
journal = {arXiv.org e-Print archive},
year = 2019,
eprint = {arXiv:1912.06870},
month = {December},
abstract = {In this technical note, we explain how to construct Gaussian quadrature rules for efficiently and accurately computing integrals of the form $\int S(\lambda)f(\lambda)d\lambda$ where $S(\lambda)$ is the solar irradiance function tabulated in the ASTM standard and $f(\lambda)$ is an arbitary application-specific smooth function. This allows the integral to be computed accurately with a relatively small number of $f(\lambda)$ evaluations despite the fact that $S(\lambda)$ 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 $f(\lambda)$ is often a computationally expensive function such as an absorbance calculated by solving Maxwell's equations.}
}
@article{Oskooi20,
author = {Ardavan Oskooi and Christopher Hogan and Alec M. Hammond and M.T. Homer Reid and Steven G. Johnson},
title = {Factorized machine mearning for performance modeling of massively parallel heterogeneous physical simulations},
journal = {arXiv.org e-Print archive},
year = 2020,
eprint = {arXiv:2003.04287},
month = {March},
abstract = {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.}
}
@article{CerjanOs20,
author = {Alexander Cerjan and Ardavan Oskooi and Song-Liang Chua and Steven G. Johnson},
title = {Modeling lasers and saturable absorbers via multilevel atomic media in the \textit{Meep} {FDTD} software: Theory and implementation},
journal = {arXiv.org e-Print archive},
year = 2020,
eprint = {arXiv:2007.09329},
month = {July},
abstract = {This technical note describes the physical model, numerical implementation, and validation of multilevel atomic media for lasers and saturable absorbers in \emph{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 \emph{Meep} implementation is validated using a separate FDTD density-matrix model as well as a frequency-domain solver based on steady-state \textit{ab-initio} laser theory (SALT).}
}
@article{XuJo21,
author = {Guanpeng Xu and Steven G. Johnson},
title = {Modified discrete {Laguerre} polynomials for efficient computation of exponentially bounded {Matsubara} sums},
journal = {arXiv.org e-Print archive},
year = 2021,
eprint = {arXiv:2101.01655},
month = {January},
abstract = {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 $F(0)/2+F(h)+F(2h)+\cdots$ 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 $F$.}
}
@article{JohnsonPMLnotes21,
author = {Steven G. Johnson},
title = {Notes on Perfectly Matched Layers ({PML}s)},
journal = {arXiv.org e-Print archive},
year = 2021,
month = {August},
eprint = {arXiv:2108.05348},
abstract = {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.}
}
@article{SunSha25,
author = {Mengdi Sun and Ata Shakeri and Arvin Keshvari and Dimitrios Giannakopoulos and Qing Wang and Wei Ting Chen and Steven G. Johnson and Zin Lin},
title = {Scalable freeform optimization of wide-aperture {3D} metalenses by zoned discrete axisymmetry},
journal = {arXiv.org e-Print archive},
year = 2025,
month = {January},
eprint = {arXiv:2501.07979},
abstract = { We introduce a novel framework for design and optimization of 3D freeform metalenses that attains nearly linear scaling of computational cost with diameter, by breaking the lens into a sequence of radial ``zones'' with $n$-fold discrete axisymmetry, where n increases with radius. This allows vastly more design freedom than imposing continuous axisymmetry, while avoiding the compromises of the locally periodic approximation (LPA) or scalar diffraction theory. Using a GPU-accelerated finite-difference time-domain (FDTD) solver in cylindrical coordinates, we perform full-wave simulation and topology optimization within each supra-wavelength zone. We validate our approach by designing millimeter and centimeter-scale, poly-achromatic, 3D freeform metalenses which outperform the state of the art. By demonstrating the scalability and resulting optical performance enabled by our ``zoned discrete axisymmetry'' (ZDA) and supra-wavelength domain decomposition, we highlight the potential of our framework to advance large-scale meta-optics and next-generation photonic technologies.}
}
@book{MatrixCalc,
author = {Paige Bright and Alan Edelman and Steven G. Johnson},
title = {Matrix Calculus (for Machine Learning and Beyond)},
publisher = {arXiv.org e-Print archive},
year = 2025,
eprint = {arXiv:2501.14787},
month = {January},
abstract = {This course, intended for undergraduates familiar with elementary calculus and linear algebra, introduces the extension of differential calculus to functions on more general vector spaces, such as functions that take as input a matrix and return a matrix inverse or factorization, derivatives of ODE solutions, and even stochastic derivatives of random functions. It emphasizes practical computational applications, such as large-scale optimization and machine learning, where derivatives must be re-imagined in order to be propagated through complicated calculations. The class also discusses efficiency concerns leading to ``adjoint'' or ``reverse-mode'' differentiation (a.k.a. ``backpropagation''), and gives a gentle introduction to modern automatic differentiation (AD) techniques. }
}
@article{ChenCh25,
author = {Mo Chen and Ka Fai Chan and Alec M. Hammond and Chi Hou Chan and Steven G. Johnson},
title = {Inverse design of 3D-printable metalenses with complementary dispersion for terahertz imaging},
journal = {arXiv.org e-Print archive},
year = 2025,
eprint = {arXiv:2502.10520},
month = {February},
abstract = {This study formulates a volumetric inverse-design methodology to generate a pair of complementary focusing metalenses for terahertz imaging: the two lenses exhibit equal and opposite shifts in focal length with frequency. An asymmetry arises, where we find a focal length that decreases with frequency to be more challenging to achieve (without material dispersion) given fabrication constraints, but it is still possible. We employ topology optimization, coupled with manufacturing constraints, to explore fully freeform designs compatible with 3D printing. Formulating an optimization problem that quantifies the goal of maximal complementary focal shifts, while remaining differentiable and tractable, requires a carefully selected sequence of constraints and approximations. }
}
@article{ChaoHa25,
author = {Pengning Chao and Ian M. Hammond and Steven G. Johnson},
title = {Sum-of-squares bounds on surface-enhanced {Raman} scattering},
journal = {arXiv.org e-Print archive},
year = 2025,
eprint = {arXiv:2502.09821},
month = {February},
abstract = {Surface-enhanced Raman scattering (SERS) is a critical tool for chemical sensing and spectroscopy, and a key question is how to optimally design nanostructures for maximizing SERS.
We present fundamental limits on spatially-averaged SERS via periodic metasurfaces, derived using sum-of-squares (SOS) programming. This work represents the first use of SOS techniques to optics, overcoming difficulties that prior bounding techniques have with regards to non-linear photonic processes with higher order figures of merit. Our bounds on the $\int \Vert \mathbf{E} \Vert^4$ SERS enhancement factor for 2D examples demonstrate remarkable tightness when compared with inverse-designed dielectric and metallic structures for both electrical field out-of-plane ($E_z$) and in-plane ($H_z$) polarizations. We show that delocalized high-Q guided modes can achieve significant, theoretically diverging SERS enhancement even in the presence of material loss. For metallic structures, we demonstrate a fundamental performance limitation for $E_z$ polarized drive fields due to surface plasmon excitation restrictions. By varying the separation between Raman-active molecules and the metasurface design region, we also find material-dependent bounds on the maximum strength of field singularities. Our results offer insights into optimal metasurface design strategies for enhancing light-matter interactions, and our methodology may be adapted to the study of other nonlinear photonics design problems.}
}
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