arXiv subjects

Ali Mostafazadeh

Publications and source records attributed to Ali Mostafazadeh.

At least 73 records · Page 4Linked to original sources

arXiv2016

Dynamical Theory of Scattering, Exact Unidirectional Invisibility, and Truncated $\mathfrak{z}\,e^{-2ik_0x}$ potential

The dynamical formulation of time-independent scattering theory that is developed in [Ann. Phys. (NY) 341, 77-85 (2014)] offers simple formulas for the reflection and transmission amplitudes of finite-range potentials in terms of the solution of an initial-value differential equation. We prove a theorem that simplifies the application of this result and use it to give a complete characterization of the invisible configurations of the truncated $\mathfrak{z}\,e^{-2ik_0 x}$ potential to a closed interval, $[0,L]$, with $k_0$ being a positive integer multiple of $π/L$. This reveals a large class of exact unidirectionally and bidirectionally invisible configurations of this potential. The former arise for particular values of $\mathfrak{z}$ that are given by certain zeros of Bessel functions. The latter occur when the wavenumber $k$ is an integer multiple of $π/L$ but not of $k_0$. We discuss the optical realizations of these configurations and explore spectral singularities of this potential.

quant-ph↗

arXiv2016

Unidirectional Invisibility and Nonreciprocal Transmission in Two and Three Dimensions

We explore the phenomenon of unidirectional invisibility in two dimensions, examine its optical realizations, and discuss its three-dimensional generalization. In particular we construct an infinite class of unidirectionally invisible optical potentials that describe the scattering of normally incident transverse electric waves by an infinite planar slab with refractive-index modulations along both the normal directions to the electric field. A by-product of this investigation is a demonstration of nonreciprocal transmission in two dimensions. To elucidate this phenomenon we state and prove a general reciprocity theorem that applies to quantum scattering theory of real and complex potentials in two and three dimensions.

quant-ph↗

arXiv2016

Point Interactions, Metamaterials, and PT-Symmetry

We express the boundary conditions for TE and TM waves at the interfaces of an infinite planar slab of homogeneous metamaterial as certain point interactions and use them to compute the transfer matrix of the system. This allows us to demonstrate the omnidirectional reflectionlessness of Veselago's slab for waves of arbitrary wavelength, reveal the translational and reflection symmetry of this slab, explore the laser threshold condition and coherent perfect absorption for active negative-index metamaterials, introduce a point interaction modeling phase-conjugation, determine the corresponding antilinear transfer matrix, and offer a simple proof of the equivalence of Veselago's slab with a pair of parallel phase-conjugating plates. We also study the connection between certain optical setups involving metamaterials and a class of PT-symmetric quantum systems defined on wedge-shape contours in the complex plane. This provides a physical interpretation for the latter.

physics.optics↗

arXiv2016

Transfer Matrix Formulation of Scattering Theory in Two and Three Dimensions

In one dimension one can dissect a scattering potential $ v(x) $ into pieces $ v_i(x) $ and use the notion of the transfer matrix to determine the scattering content of $ v(x) $ from that of $ v_i(x) $. This observation has numerous practical applications in different areas of physics. The problem of finding an analogous procedure in dimensions larger than one has been an important open problem for decades. We give a complete solution for this problem and discuss some of its applications. In particular we derive an exact expression for the scattering amplitude of the delta-function potential in two and three dimensions and a potential describing a slab laser with a surface line defect. We show that the presence of the defect makes the slab begin lasing for arbitrarily small gain coefficients.

quant-ph↗

arXiv2015

Perturbative Unidirectional Invisibility

We outline a general perturbative method of evaluating scattering features of finite-range complex potentials and use it to examine complex perturbations of a rectangular barrier potential. In optics, these correspond to modulated refractive index profiles of the form $n(x)=n_0+f(x)$, where $n_0$ is real, $f(x)$ is complex-valued, and $|f(x)|\ll1\leq n_0$. We give a comprehensive description of the phenomenon of unidirectional invisibility for such media, proving five general theorems on its realization in ${\cal PT}$-symmetric and non-${\cal PT}$-symmetric material. In particular, we establish the impossibility of unidirectional invisibility for ${\cal PT}$-symmetric samples whose refractive index has a constant real part and show how a simple scaling transformation of a unidirectionally invisible ${\cal PT}$-symmetric index profile with $n_0=1$ may be used to generate a hierarchy of unidirectionally invisible ${\cal PT}$-symmetric index profiles with $n_0>1$. The results pertaining unidirectional invisibility for $n_0>1$ open up the way for the experimental studies of this phenomenon in a variety of active material. As an application of our general results, we show that a medium with $n(x)=n_0+ζe^{iK x}$, $ζ$ and $K$ real, and $|ζ|\ll 1$ can support unidirectional invisibility only for $n_0=1$. We then construct unidirectionally invisible index profiles of the form $n(x)=n_0+\sum_\ell z_\ell e^{iK_\ell x}$, with $z_\ell$ complex, $K_\ell$ real, $|z_\ell|\ll 1$, and $n_0>1$.

quant-ph↗

arXiv2015

Active Invisibility Cloaks in One Dimension

We outline a general method of constructing finite-range cloaking potentials which render a given finite-range real or complex potential $v(x)$ unidirectionally reflectionless or invisible at a wavenumber $k_0$ of our choice. We give explicit analytic expressions for three classes of cloaking potentials which achieve this goal while preserving some or all of the other scattering properties of $v(x)$. The cloaking potentials we construct are the sum of up to three constituent unidirectionally invisible potentials. We also discuss their application in making $v(x)$ bidirectionally invisible at $k_0$, and demonstrate the application of our method to obtain anti-reflection and invisibility cloaks for a Bragg reflector.

physics.class-ph↗

arXiv2015

Physics of Spectral Singularities

Spectral singularities are certain points of the continuous spectrum of generic complex scattering potentials. We review the recent developments leading to the discovery of their physical meaning, consequences, and generalizations. In particular, we give a simple definition of spectral singularities, provide a general introduction to spectral consequences of ${\cal PT}$-symmetry (clarifying some of the controversies surrounding this subject), outline the main ideas and constructions used in the pseudo-Hermitian representation of quantum mechanics, and discuss how spectral singularities entered in the physics literature as obstructions to these constructions. We then review the transfer matrix formulation of scattering theory and the application of complex scattering potentials in optics. These allow us to elucidate the physical content of spectral singularities and describe their optical realizations. Finally, we survey some of the most important results obtained in the subject, drawing special attention to the remarkable fact that the condition of the existence of linear and nonlinear optical spectral singularities yield simple mathematical derivations of some of the basic results of laser physics, namely the laser threshold condition and the linear dependence of the laser output intensity on the gain coefficient.

quant-ph↗

arXiv2015

Composition of Transfer Matrices for Potentials with Overlapping Support

For a pair of real or complex scattering potentials $v_j:\mathbb{R}\to\mathbb{C}$ ($j=1,2$) with support $I_j$ and transfer matrix $M_j$, the transfer matrix of $v_1+v_2$ is given by the product $M_2 M_1$ provided that $I_1$ lies to the left of $I_2$. We explore the prospects of generalizing this composition rule for the cases that $I_1$ and $I_2$ have a small intersection. In particular, we show that if $I_1$ and $I_2$ intersect in a finite closed interval of length $\ell$ in which both the potentials are analytic, then the lowest order correction to the above composition rule is proportional to $\ell^5$. This correction is of the order of $\ell^3$, if $v_1$ and $v_2$ are respectively analytic throughout this interval except at $x=\ell$ and $x=0$. We use these results to explore the superposition of a pair of unidirectionally invisible potentials with overlapping support.

math-ph↗

arXiv2015

Lasing Threshold Condition for Oblique TE and TM Modes, Spectral Singularities, and Coherent Perfect Absorption

We study spectral singularities and their application in determining the threshold gain coefficient $g^{(E/M)}$ for oblique transverse electric/magnetic (TE/TM) modes of an infinite planar slab of homogenous optically active material. We show that $g^{(E)}$ is a monotonically decreasing function of the incidence angle $θ$ (measured with respect to the normal direction to the slab), while $g^{(M)}$ has a single maximum, $θ_c$, where it takes an extremely large value. We identify $θ_c$ with the Brewster's angle and show that $g^{(E)}$ and $g^{(M)}$ coincide for $θ=0$ (normal incidence), tend to zero as $θ\to 90^\circ$, and satisfy $g^{(E)} θ_c$ is smaller inside the gain region than outside it. The converse is true for the TM waves with $θ<θ_c$ and all TE waves.

quant-ph↗

arXiv2014

Unidirectionally Invisible Potentials as Local Building Blocks of all Scattering Potentials

We give a complete solution of the problem of constructing a scattering potential v(x) that possesses scattering properties of one's choice at an arbitrary prescribed wavenumber. Our solution involves expressing v(x) as the sum of at most six unidirectionally invisible finite-range potentials for which we give explicit formulas. Our results can be employed for designing optical potentials. We discuss its application in modeling threshold lasers, coherent perfect absorbers, and bidirectionally and unidirectionally reflectionless absorbers, amplifiers, and phase shifters.

quant-ph↗

arXiv2014

Generalized Unitarity and Reciprocity Relations for PT-symmetric Scattering Potentials

We derive certain identities satisfied by the left/right-reflection and transmission amplitudes, $R^{l/r}(k)$ and $T(k)$, of general ${\cal PT}$-symmetric scattering potentials. We use these identities to give a general proof of the relations, $|T(-k)|=|T(k)|$ and $|R^r(-k)|=|R^l(k)|$, conjectured in [Z. Ahmed, J. Phys. A 45 (2012) 032004], establish the generalized unitarity relation: $R^{l/r}(k)R^{l/r}(-k)+|T(k)|^2=1$, and show that it is a common property of both real and complex ${\cal PT}$-symmetric potentials. The same holds for $T(-k)=T(k)^*$ and $|R^r(-k)|=|R^l(k)|$.

quant-ph↗

arXiv2014

Spectral Singularities and CPA-Laser Action in a Weakly Nonlinear PT-Symmetric Bilayer Slab

We study optical spectral singularities of a weakly nonlinear PT-symmetric bilinear planar slab of optically active material. In particular, we derive the lasing threshold condition and calculate the laser output intensity. These reveal the following unexpected features of the system: 1. For the case that the real part of the refractive index $η$ of the layers are equal to unity, the presence of the lossy layer decreases the threshold gain; 2. For the more commonly encountered situations when $η-1$ is much larger than the magnitude of the imaginary part of the refractive index, the threshold gain coefficient is a function of $η$ that has a local minimum. The latter is in sharp contrast to the threshold gain coefficient of a homogeneous slab of gain material which is a decreasing function of $η$. We use these results to comment on the effect of nonlinearity on the prospects of using this system as a CPA-laser.

quant-ph↗

arXiv2014

Adiabatic Approximation, Semiclassical Scattering, and Unidirectional Invisibility

The transfer matrix of a possibly complex and energy-dependent scattering potential can be identified with the $S$-matrix of a two-level time-dependent non-Hermitian Hamiltonian H(t). We show that the application of the adiabatic approximation to H(t) corresponds to the semiclassical description of the original scattering problem. In particular, the geometric part of the phase of the evolving eigenvectors of H(t) gives the pre-exponential factor of the WKB wave functions. We use these observations to give an explicit semiclassical expression for the transfer matrix. This allows for a detailed study of the semiclassical unidirectional reflectionlessness and invisibility. We examine concrete realizations of the latter in the realm of optics.

quant-ph↗

arXiv2014

Adiabatic Series Expansion and Higher-Order Semiclassical Approximations in Scattering Theory

The scattering properties of any complex scattering potential, v:R -> C, can be obtained from the dynamics of a particular non-unitary two-level quantum system S_v. The application of the adiabatic approximation to S_v yields a semiclassical treatment of the scattering problem. We examine the adiabatic series expansion for the evolution operator of S_v and use it to obtain corrections of arbitrary order to the semiclassical formula for the transfer matrix of v. This results in a high-energy approximation scheme that unlike the semiclassical approximation can be applied for potentials with large derivatives.

quant-ph↗

arXiv2014

Transfer Matrices as Non-Unitary S-Matrices, Multimode Unidirectional Invisibility, and Perturbative Inverse Scattering

We show that in one dimension the transfer matrix M of any scattering potential v coincides with the S-matrix of an associated time-dependent non-Hermitian 2 x 2 matrix Hamiltonian H(τ). If v is real-valued, H(τ) is pseudo-Hermitian and its exceptional points correspond to the classical turning points of v. Applying time-dependent perturbation theory to H(τ) we obtain a perturbative series expansion for M and use it to study the phenomenon of unidirectional invisibility. In particular, we establish the possibility of having multimode unidirectional invisibility with wavelength-dependent direction of invisibility and construct various physically realizable optical potentials possessing this property. We also offer a simple demonstration of the fact that the off-diagonal entries of the first Born approximation for M determine the form of the potential. This gives rise to a perturbative inverse scattering scheme that is particularly suitable for optical design. As a simple application of this scheme, we construct an infinite-range unidirectionally invisible potential.

quant-ph↗

arXiv2014

A Differential Integrability Condition for Two-Dimensional Hamiltonian Systems

We review, restate, and prove a result due to Kaushal and Korsch [Phys. Lett. A 276, 47 (2000)] on the complete integrability of two-dimensional Hamiltonian systems whose Hamiltonian satisfies a set of four linear second order partial differential equations. In particular, we show that a two-dimensional Hamiltonian system is completely integrable, if the Hamiltonian has the form $H=T+V$ where $V$ and $T$ are respectively harmonic functions of the generalized coordinates and the associated momenta.

math-ph↗

arXiv2013

A Dynamical Formulation of One-Dimensional Scattering Theory and Its Applications in Optics

We develop a dynamical formulation of one-dimensional scattering theory where the reflection and transmission amplitudes for a general, possibly complex and energy-dependent, scattering potential are given as solutions of a set of dynamical equations. By decoupling and partially integrating these equations, we reduce the scattering problem to a second order linear differential equation with universal initial conditions that is equivalent to an initial-value time-independent Schrodinger equation. We give explicit formulas for the reflection and transmission amplitudes in terms of the solution of either of these equations and employ them to outline an inverse-scattering method for constructing finite-range potentials with desirable scattering properties at any prescribed wavelength. In particular, we construct optical potentials displaying threshold lasing, anti-lasing, and unidirectional invisibility.

quant-ph↗

arXiv2013

Spectral Singularities in the Surface Modes of a Spherical Gain Medium

We study the surface modes of a homogeneous spherical gain medium and provide a comprehensive analytic treatment of a special class of these modes that support spectral singularities. Because the latter have a divergent quality factor, we call them the singular gallery modes. We show that they can be excited using arbitrarily small amounts of gain, and as a result, the system lacks a lasing threshold, effectively. This shows that we can realize spectral singularities in the surface modes of extremely small spherical samples with modest amounts of gain. We also examine the possibility of exciting singular gallery modes with different wavelengths using the same amount of gain. This corresponds to the situation where the system undergoes simultaneous lasing at different wavelengths.

quant-ph↗