arXiv subjects

Farhang Loran

Publications and source records attributed to Farhang Loran.

At least 19 recordsLinked to original sources

arXiv2026

The Photon Gas in Classical Mechanics: A Statistical-Mechanical Treatment of Classical Field Theory

The thermodynamics of cavity radiation is usually introduced through the quantum theory of a photon gas. In contrast, we investigate which features of thermal radiation follow from classical electrodynamics alone. We show, using two complementary approaches, that the proportionalities between energy density, radiation pressure, and energy flux arise without invoking photons or quantum statistics. The first approach is based on explicit solutions of Maxwell's equations and spatio-temporal averaging of electromagnetic waves. The second approach develops the classical statistical mechanics of the electromagnetic field from first principles by constructing the canonical Hamiltonian, implementing the gauge constraints, formulating the partition function as a functional integral, and evaluating the corresponding field correlation functions. By combining wave analysis, statistical mechanics, constrained Hamiltonian dynamics, and functional methods within a single classical framework, this work provides a coherent classical formulation of electromagnetic radiation thermodynamics.

physics.class-ph↗

arXiv2026

Exactly Solvable Diffraction-Grating Scattering Problems for TE and TM waves

In J.~Phys.~A {\bf 31}, 3493 (1998), Berry provides an analytic solution of the problem of determining the diffracted beam intensities for atoms incident upon a grating given by the potential, $v(x,y):=iV_0(e^{i\mathfrak{K}\,y}-1)$ for $0\leq x\leq\ell$ and $v(x,y):=0$ otherwise, where $V_0, \mathfrak{K}$, and $\ell$ are positive real parameters. This result, which relies on the paraxial approximation, readily applies to the diffraction of transverse electric (TE) waves by the nonmagnetic optical grating given by the relative permittivity, $\hat\varepsilon(x,y):=1-v(x,y)/k^2$, where $k$ is the incident wavenumber. We show that Berry's grating belongs to a larger class of possibly magnetic diffraction gratings whose scattering problem for both TE and transverse magnetic (TM) waves are exactly solvable. Our analysis makes use of a recently developed dynamical formulation of stationary scattering that is based on the idea of mapping the scattering problem to the quantum dynamics generated by an effective non-Hermitian Hamiltonian operator. {We use this formulation to derive explicit analytic expressions for the diffracted beam amplitudes that are valid beyond paraxial approximation.} As a concrete example, we offer the exact solution of the scattering problem for TE and TM waves incident upon a generalization of Berry's grating.

physics.optics↗

arXiv2026

Scattering of TE and TM waves by inhomogeneities of a 2D material, low-frequency behavior of the scattering amplitude, and low-frequency invisibility

The propagation of the transverse electric (TE) and transverse magnetic (TM) waves in an effectively two-dimensional (2D) isotropic medium is described by Bergmann's equation of acoustics. We develop a dynamical formulation of the stationary scattering of these waves and explore its application in the study of the low-frequency behavior of the scattering data. Specifically, we introduce a suitable notion of fundamental transfer matrix for TE and TM waves in 2D. This is an integral operator $\widehat{\mathbf{M}}$ that carries the information about the scattering properties of the medium and admits a Dyson series expansion involving a non-Hermitian Hamiltonian operator. For situations where the inhomogeneities of the medium are confined to a layer of thickness $\ell$, we use the Dyson series for $\widehat{\mathbf{M}}$ to construct the series expansion of the scattering amplitude in powers of $k\ell$, where $k$ is the incident wavenumber. We derive analytic expressions for the leading- and next-to-leading-order terms of this series, verify the effectiveness of their application to a class of exactly solvable models, and use them to study low-frequency invisibility. In particular, we develop a low-frequency cloaking scheme which is applicable for both TE and TM waves. Our results have immediate applications in the study of low-frequency scattering of acoustic waves in a 2D fluid as these waves are also described by Bergmann's equation.

math-ph↗

arXiv2025

Does a Curved Mirror Honestly Reflect Your Identity? A Study of Multipole Images in Front of a Grounded Sphere

The method of image charges is a powerful and elegant technique in electrostatics, commonly used to determine the electric field generated by point charges near conductors of various shapes. While standard problems focus on single charges interacting with conductors, the behavior of multipoles in such configurations has received comparatively less attention, particularly beyond the well-studied case of a flat plane. In this paper, we explore the image formation of electric dipoles and quadrupoles near a conducting sphere and uncover a wonderful result: the image of a given multipole is not necessarily of the same type. Instead, alongside the expected multipole image, the resulting image configuration also includes lower-order multipole contributions. This finding broadens the understanding of electrostatic images and offers new insights into the interaction of multipoles with conducting boundaries.

physics.class-ph↗

arXiv2025

Reciprocity Theorem and Fundamental Transfer Matrix

Stationary potential scattering admits a formulation in terms of the quantum dynamics generated by a non-Hermitian effective Hamiltonian. We use this formulation to give a proof of the reciprocity theorem in two and three dimensions that does not rely on the properties of the scattering operator, Green's functions, or Green's identities. In particular, we identify reciprocity with an operator identity satisfied by an integral operator $\widehat{\mathbf{M}}$, called the fundamental transfer matrix. This is a multi-dimensional generalization of the transfer matrix $\mathbf{M}$ of potential scattering in one dimension that stores the information about the scattering amplitude of the potential. We use the property of $\widehat{\mathbf{M}}$ that is responsible for reciprocity to identify the analog of the relation, $\det{\mathbf{M}}=1$, in two and three dimensions, and establish a generic anti-pseudo-Hermiticity of the scattering operator. Our results apply for both real and complex potentials.

quant-ph↗

arXiv2025

Low-Frequency Scattering of TE and TM Waves by an Inhomogeneous Medium with Planar Symmetry

Stationary scattering of TE and TM waves propagating in an isotropic medium with planar symmetry is described by Bergmann's equation in one dimension. This is a generalization of Helmholtz equation which allows for developing transfer matrix methods to deal with the corresponding scattering problems. We use a dynamical formulation of stationary scattering to study the low-frequency scattering of these waves when the inhomogeneities of the medium causing the scattering are confined to a planar slab. This formulation relies on the construction of an effective two-level non-Hermitian quantum system whose time-evolution operator determines the transfer matrix. We use it to construct the low-frequency expansions of the transfer matrix and the reflection and transmission coefficients of the medium, introduce a generalization of Brewster's angle for inhomogeneous slabs at low frequencies, and derive analytic conditions for transparency and reflectionlessness of PT-symmetric and non-PT-symmetric slabs at these frequencies. We also discuss the application of this method to deal with the low-frequency scattering of TE and TM waves when the carrier medium occupies a half-space and the waves satisfy boundary conditions with planar symmetry at the boundary of the half-space. Because acoustic waves propagating in a compressible fluid with planar symmetry are also described by Bergmann's equation, our results apply to the low-frequency scattering of these waves.

physics.optics↗

arXiv2025

Dynamical formulation of low-frequency scattering in two and three dimensions

The transfer matrix of scattering theory in one dimension can be expressed in terms of the time-evolution operator for an effective non-unitary quantum system. In particular, it admits a Dyson series expansion which turns out to facilitate the construction of the low-frequency series expansion of the scattering data. In two and three dimensions, there is a similar formulation of stationary scattering where the scattering properties of the scatterer are extracted from the evolution operator for a corresponding effective quantum system. We explore the utility of this approach to scattering theory in the study of the scattering of low-frequency time-harmonic scalar waves, $e^{-iωt}ψ(\mathbf{r})$, with $ψ(\mathbf{r})$ satisfying the Helmholtz equation, $[\nabla^2+k^2\hat\varepsilon(\mathbf{r};k)]ψ(\mathbf{r})=0$, $ω$ and $k$ being respectively the angular frequency and wavenumber of the incident wave, and $\hat\varepsilon(\mathbf{r};k)$ denoting the relative permittivity of the carrier medium which in general takes complex values. We obtain explicit formulas for low-frequency scattering amplitude, examine their effectiveness in the study of a class of exactly solvable scattering problems, and outline their application in devising a low-frequency cloaking scheme.

quant-ph↗

arXiv2024

Scattering of TE and TM waves and quantum dynamics generated by non-Hermitian Hamiltonians

The study of the scattering of electromagnetic waves by a linear isotropic medium with planar symmetry can be reduced to that of their TE and TM modes. For situations where the medium consists of parallel homogeneous slabs, one may use the standard transfer matrix technique to address the scattering problem for these modes. We extend the utility of this technique to inhomogeneous permittivity and permeability profiles by proposing a dynamical formulation of the scattering of TE and TM waves in which the transfer matrix for the medium is given in terms of the evolution operator for an effective non-unitary quantum system. This leads to a system of dynamical equations for the reflection and transmission amplitudes. Decoupling these equations we reduce the solution of the scattering problem for TE and TM modes to that of an initial-value problem for a Riccati equation. We discuss the application of this observation in identifying media that do not reflect TE or TM waves with given wavenumber and incidence angle.

quant-ph↗

arXiv2024

Can N-th Order Born Approximation Be Exact?

For the scattering of scalar waves in two and three dimensions and electromagnetic waves in three dimensions, we identify a condition on the scattering interaction under which the $N$-th order Born approximation gives the exact solution of the scattering problem for some $N\geq 1$.

quant-ph↗

arXiv2024

Exactness of the first Born approximation in electromagnetic scattering

For the scattering of plane electromagnetic waves by a general possibly anisotropic stationary linear medium in three dimensions, we give a condition on the permittivity and permeability tensors of the medium under which the first Born approximation yields the exact expression for the scattered wave whenever the incident wavenumber $k$ does not exceed a pre-assigned value $α$. We also show that under this condition the medium is omnidirectionally invisible for $k\leq α/2$, i.e., it displays broadband invisibility regardless of the polarization of the incident wave.

math-ph↗

arXiv2023

Broadband directional invisibility

The discovery of unidirectional invisibility and its broadband realization in optical media satisfying spatial Kramers-Kronig relations are important landmarks of non-Hermitian photonics. We offer a precise characterization of a higher-dimensional generalization of this effect and find sufficient conditions for its realization in the scattering of scalar waves in two and three dimensions and electromagnetic waves in three dimensions. More specifically, given a positive real number $α$ and a continuum of unit vectors $Ω$, we provide explicit conditions on the interaction potential (or the permittivity and permeability tensors of the scattering medium in the case of electromagnetic scattering) under which it displays perfect (non-approximate) invisibility whenever the incident wavenumber $k$ does not exceed $α$ (i.e., $k\in(0,α]$) and the direction of the incident wave vector ranges over $Ω$. A distinctive feature of our approach is that it allows for the construction of potentials and linear dielectric media that display perfect directional invisibility in a finite frequency domain.

physics.class-ph↗

arXiv2023

Introducing a general method for solving electromagnetic radiation problem in an arbitrary linear medium

Numerical transfer matrices have been widely used in the study of wave propagation and scattering. These may be viewed as descretizations of a recently introduced fundamental notion of transfer matrix which admits a representation in terms of the evolution operator for an effective non-unitary quantum system. We use the fundamental transfer matrix to develop a general method for the solution of the problem of radiation of an oscillating source in an arbitrary, possibly non-homogenous, anisotropic, and active or lossy linear medium. This allows us to obtain an analytic solution of this problem for an oscillating source located in the vicinity of a planar collection of possibly anisotropic and active/lossy point scatterers such as those modeling a two-dimensional photonic crystal.

physics.class-ph↗

arXiv2022

Fundamental transfer matrix for electromagnetic waves, scattering by a planar collection of point scatterers, and anti-PT-symmetry

We develop a fundamental transfer-matrix formulation of the scattering of electromagnetic (EM) waves that incorporates the contribution of the evanescent waves and applies to general stationary linear media which need not be isotropic, homogenous, or passive. Unlike the traditional transfer matrices whose definition involves slicing the medium, the fundamental transfer matrix is a linear operator acting in an infinite-dimensional function space. It is given in terms of the evolution operator for a non-unitary quantum system and has the benefit of allowing for analytic calculations. In this respect it is the only available alternative to the standard Green's-function approaches to EM scattering. We use it to offer an exact solution of the outstanding EM scattering problem for an arbitrary finite collection of possibly anisotropic nonmagnetic point scatterers lying on a plane. In particular, we provide a comprehensive treatment of doublets consisting of pairs of isotropic point scatterers and study their spectral singularities. We show that identical and $\mathcal{P}\mathcal{T}$-symmetric doublets do not admit spectral singularities and cannot function as a laser unless the real part of their permittivity equals that of vacuum. This restriction does not apply to doublets displaying anti-$\mathcal{P}\mathcal{T}$-symmetry. We determine the lasing threshold for a generic anti-$\mathcal{P}\mathcal{T}$-symmetric doublet and show that it possesses a continuous lasing spectrum.

physics.optics↗

arXiv2022

Propagating-wave approximation in two-dimensional potential scattering

We introduce a nonperturbative approximation scheme for performing scattering calculations in two dimensions that involves neglecting the contribution of the evanescent waves to the scattering amplitude. This corresponds to replacing the interaction potential $v$ with an associated energy-dependent nonlocal potential ${\mathscr{V}}_k$ that does not couple to the evanescent waves. The scattering solutions $ψ(\mathbf{r})$ of the Schrödinger equation, $(-\nabla^2+{\mathscr{V}}_k)ψ(\mathbf{r})=k^2ψ(\mathbf{r})$, has the remarkable property that their Fourier transform $\tildeψ(\mathbf{p})$ vanishes unless $\mathbf{p}$ corresponds to the momentum of a classical particle whose magnitude equals $k$. We construct a transfer matrix for this class of nonlocal potentials and explore its representation in terms of the evolution operator for an effective non-unitary quantum system. We show that the above approximation reduces to the first Born approximation for weak potentials, and similarly to the semiclassical approximation, becomes valid at high energies. Furthermore, we identify an infinite class of complex potentials for which this approximation scheme is exact. We also discuss the appealing practical and mathematical aspects of this scheme.

quant-ph↗

arXiv2022

Existence of the transfer matrix for a class of nonlocal potentials in two dimensions

Evanescent waves are waves that decay or grow exponentially in regions of the space void of interaction. In potential scattering defined by the Schrödinger equation, $(-\nabla^2+v)ψ=k^2ψ$ for a local potential $v$, they arise in dimensions greater than one and are present regardless of the details of $v$. The approximation in which one ignores the contributions of the evanescent waves to the scattering process corresponds to replacing $v$ with a certain energy-dependent nonlocal potential $\hat{\mathscr{V}}_k$. We present a dynamical formulation of the stationary scattering for $\hat{\mathscr{V}}_k$ in two dimensions, where the scattering data are related to the dynamics of a quantum system having a non-self-adjoint, unbounded, and nonstationary Hamiltonian operator. The evolution operator for this system determines a two-dimensional analog of the transfer matrix of stationary scattering in one dimension which contains the information about the scattering properties of the potential. Under rather general conditions on $v$, we establish the strong convergence of the Dyson series expansion of the evolution operator and prove the existence of the transfer matrix for $\hat{\mathscr{V}}_k$ as a densely-defined operator acting in $\mathbb{C}^2\otimes L^2(-k,k)$.

math-ph↗

arXiv2022

Renormalization of multi-delta-function point scatterers in two and three dimensions, the coincidence-limit problem, and its resolution

In two and three dimensions, the standard treatment of the scattering problem for a multi-delta-function potential, $v(\mathbf{r})=\sum_{n=1}^N\mathfrak{z}_nδ(\mathbf{r}-\mathbf{a}_n)$, leads to divergent terms. Regularization of these terms and renormalization of the coupling constants $\mathfrak{z}_n$ give rise to a finite expression for the scattering amplitude of this potential, but this expression has an important short-coming; in the limit where the centers $\mathbf{a}_n$ of the delta functions coincide, it does not reproduce the formula for the scattering amplitude of a single-delta-function potential, i.e., it seems to have a wrong coincidence limit. We provide a critical assessment of the standard treatment of these potentials and offer a resolution of its coincidence-limit problem. This reveals some previously unnoticed features of this treatment. For example, it turns out that the standard treatment is incapable of determining the dependence of the scattering amplitude on the distances between the centers of the delta functions. This is in sharp contrast to the treatment of this problem offered by a recently proposed dynamical formulation of stationary scattering. For cases where the centers of the delta functions lie on a straight line, this formulation avoids singularities of the standard approach and yields an expression for the scattering amplitude which has the correct coincidence limit.

quant-ph↗

arXiv2022

Singularity-free treatment of delta-function point scatterers in two dimensions and its conceptual implications

In two dimensions, the standard treatment of the scattering problem for a delta-function potential, $v(\mathbf{r})=\mathfrak{z}\,δ(\mathbf{r})$, leads to a logarithmic singularity which is subsequently removed by a renormalization of the coupling constant $\mathfrak{z}$. Recently, we have developed a dynamical formulation of stationary scattering (DFSS) which offers a singularity-free treatment of this potential. We elucidate the basic mechanism responsible for the implicit regularization property of DFSS that makes it avoid the logarithmic singularity one encounters in the standard approach to this problem. We provide an alternative interpretation of this singularity showing that it arises, because the standard treatment of the problem takes into account contributions to the scattered wave whose momentum is parallel to the detectors' screen. The renormalization schemes used for removing this singularity has the effect of subtracting these unphysical contributions, while DFSS has a built-in mechanics that achieves this goal.

quant-ph↗

arXiv2022

Comment on "Scattering of light by a parity-time-symmetric dipole beyond the first Born approximation"

In [J. A. Rebouças and P. A. Brandão, Phys. Rev. A 104, 063514 (2021)] the authors compute the scattering amplitude for a $\mathcal{P}\mathcal{T}$-symmetric double-delta-function potential in three dimensions by invoking the far-zone approximation and summing the resulting Born series. We show that the analysis of this paper suffers from a basic error. Therefore its results are inconclusive. We give an exact closed-form expression for the scattering amplitude of this potential.

quant-ph↗