Preprints — Nanostructures and Computation Group

(See also our published papers and our home page.)
C. Pérez-Arancibia, R. Pestourie, and S. G. Johnson, “Sideways adiabaticity: Beyond ray optics for slowly varying metasurfaces,” e-Print archive, arXiv:1806.05330, June 2018. [ bib | arXiv ]
Optical metasurfaces (subwavelength-patterned surfaces typically described by variable effective surface impedances) are typically modeled by an approximation akin to ray optics: the reflection or transmission of an incident wave at each point of the surface is computed as if the surface were locally uniform, and then the total field is obtained by summing all of these local scattered fields via a Huygens principle. (Similar approximations are found in scalar diffraction theory and in ray optics for curved surfaces.) In this paper, we develop a precise theory of such approximations for variable-impedance surfaces. Not only do we obtain a type of adiabatic theorem showing that the zeroth-order' locally uniform approximation converges in the limit as the surface varies more and more slowly, including a way to quantify the rate of convergence, but we also obtain an infinite series of higher-order corrections. These corrections, which can be computed to any desired order by performing integral operations on the surface fields, allow rapidly varying surfaces to be modeled with arbitrary accuracy, and also allow one to validate designs based on the zeroth-order approximation (which is often surprisingly accurate) without resorting to expensive brute-force Maxwell solvers. We show that our formulation works arbitrarily close to the surface, and can even compute coupling to guided modes, whereas in the far-field limit our zeroth-order result simplifies to an expression similar to what has been used by other authors.

D. M. Kita, J. Michon, S. G. Johnson, and J. Hu, “Are slot and sub-wavelength grating waveguides better than strip waveguides for sensing?,” e-Print archive, arXiv:1805.03321, May 2018. [ bib | arXiv ]
The unique ability of slot and sub-wavelength grating (SWG) waveguides to confine light outside of the waveguide core material has attracted significant interest in their application to chemical and biological sensing. However, high sensitivity to sidewall roughness induced scattering loss in these structures compared to strip waveguides casts doubt on their efficacy. In this article, we seek to settle the controversy by quantitatively comparing the sensing performance of various waveguide geometries through figures of merit that we derive for each mode of sensing. These methods take into account both modal confinement and roughness scattering loss, the latter of which is computed using a volume-current (Green's-function) method with a first Born approximation. For devices based on the standard 220 nm silicon-on-insulator (SOI) platform whose propagation loss is predominantly limited by random line-edge sidewall roughness scattering, our model predicts that properly engineered TM-polarized strip waveguides claim the best performance for refractometry and absorption spectroscopy, while optimized slot waveguides demonstrate >5× performance enhancement over the other waveguide geometries for waveguide-enhanced Raman spectroscopy.

H. Shim, L. Fan, S. G. Johnson, and O. D. Miller, “Fundamental limits to near-field optical response, over any bandwidth,” arXiv, arXiv:1805.02140, May 2018. [ bib | arXiv ]
We develop an analytical framework to derive upper bounds to light-matter interactions in the optical near field, where applications ranging from spontaneous-emission amplification to greater-than-blackbody heat transfer show transformative potential. Our framework connects the classic complex-analytic properties of causal fields with newly developed energy-conservation principles, resulting in a new class of power-bandwidth limits. We show that at specific frequency and bandwidth combinations, the bounds can be closely approached by canonical plasmonic geometries, with the opportunity for new designs to emerge away from those frequency ranges. Embedded in the bounds is a material “figure of merit,” which determines the maximum response of any material (metal/dielectric, bulk/2D, etc.), for any frequency and bandwidth. We focus on spontaneous-emission enhancements as encoded in the local density of states (LDOS), and anticipate extensions to Casimir forces, nonlinear Raman scattering, engineered Lamb shifts, and more.

Y. Liu, L. Fan, E. Lee, N. Fang, S. G. Johnson, and O. D. Miller, “Optimality of optical forces and torques on nanoparticles via illumination/scattering channels,” e-Print archive, arXiv:1805.11471, May 2018. [ bib | arXiv ]
A universal property of resonant subwavelength scatterers is that their optical cross-sections are proportional to a square wavelength, λ2, regardless of whether they are plasmonic nanoparticles, two-level quantum systems, or RF antennas. The maximum cross-section is an intrinsic property of the incident field: plane waves, with infinite power, can be decomposed into multipolar orders with finite powers proportional to λ2. In this Letter, we identify λ2/c and λ3/c as analogous force and torque constants, derived within a more general quadratic scattering-channel framework for upper bounds to optical force and torque for any illumination field. This framework also simplifies the reverse problem: computing optimal “holographic” incident beams, for a fixed collection of scatterers. We analyze structures and incident fields that approach the bounds, which for nonspherical, wavelength-scale bodies show a rich interplay between scattering channels. This framework should enable optimal mechanical control of nanoparticles with light.

M. Benzaouia, G. Tokic, O. D. Miller, D. K. P. Yue, and S. G. Johnson, “From solar cells to ocean buoys: Wide-bandwidth limits to absorption by metaparticle arrays,” e-Print archive, arXiv:1804.00600, April 2018. [ bib | arXiv ]
In this paper, we develop an approximate wide-bandwidth upper bound to the absorption enhancement in arrays of metaparticles, applicable to general wave-scattering problems and motivated here by ocean-buoy energy extraction. We show that general limits, including the well-known Yablonovitch result in solar cells, arise from reciprocity conditions. The use of reciprocity in the stochastic regime leads us to a diffusion model from which we derive our main result: an analytical prediction of optimal array absorption that closely matches exact simulations for both random and optimized arrays. This result also enables us to propose and quantify approaches to increase performance through careful particle design and/or using external reflectors.

M. T. H. Reid, O. D. Miller, A. G. Polimeridis, A. W. Rodriguez, E. M. Tomlinson, and S. G. Johnson, “Photon torpedoes and Rytov pinwheels: Integral-equation modeling of non-equilibrium fluctuation-induced forces and torques on nanoparticles,” e-Print archive, arXiv:1708.01985, August 2017. [ bib | arXiv ]
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.

F. Hernández, A. Pick, and S. G. Johnson, “Scalable computation of Jordan chains,” e-Print archive, arXiv:1704.05837, April 2017. [ bib | arXiv ]
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 2122×2122. An extension of this algorithm is also presented which can achieve higher order convergence [O2)] when the matrix derivative is known.

O. D. Miller, C. W. Hsu, E. Anquillare, J. D. Joannopoulos, M. Soljačić, and S. G. Johnson, “Power-bandwidth limitations of an optical resonance,” e-Print archive, arXiv:1510.06902, October 2015. [ bib | arXiv ]
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.

S. G. Johnson, “Saddle-point integration of C “bump” functions,” e-Print archive, arXiv:1508.04376, August 2015. [ bib | arXiv ]
This technical note describes the application of saddle-point integration to the asymptotic Fourier analysis of the well-known C “bump” function exp[-(1-x2)-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.

O. D. Miller, W. Qiu, J. D. Joannopoulos, and S. G. Johnson, “Comment on `A self-assembled three-dimensional cloak in the visible' in scientific reports 3, 2328,” e-Print archive, arXiv:1310.1503, October 2013. [ bib | arXiv ]
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.

A. P. McCauley, A. W. Rodriguez, M. T. H. Reid, and S. G. Johnson, “Casimir repulsion beyond the dipole regime,” e-Print archive, arXiv:1105.0404, May 2011. [ bib | arXiv ]
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 103. We conclude that this enhancement, although quite dramatic, is still too small to yield detectable repulsive Casimir forces.

A. Rodriguez and S. G. Johnson, “Efficient generation of correlated random numbers using Chebyshev-optimal magnitude-only IIR filters,” e-Print archive, arXiv:physics/0703152, March 2007. [ bib | arXiv ]
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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See also our published papers and our home page.