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Department: search.filters.department.Department of MathematicsSubject: search.filters.subject.Optics

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Now showing 1 - 10 of 30
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    PublicationOpen Access
    Active invisibility cloaks in one dimension
    (American Physical Society (APS), 2015) Department of Mathematics; Department of Physics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; Department of Physics; College of Sciences; 4231
    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 wave number, 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 discuss their utility in making v(x) bidirectionally invisible at k(0) and demonstrate the application of our method to obtain antireflection and invisibility cloaks for a Bragg reflector.
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    PublicationOpen Access
    Addendum to 'Unidirectionally invisible potentials as local building blocks of all scattering potentials'
    (American Physical Society (APS), 2014) Department of Mathematics; Department of Physics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; Department of Physics; College of Sciences; 4231
    In [Phys. Rev. A 90, 023833 (2014)], we offer a solution to the problem of constructing a scattering potential v(x) which possesses scattering properties of one's choice at an arbitrarily prescribed wave number. This solution involves expressing v(x) as the sum of n <= 6 finite-range unidirectionally invisible potentials. We improve this result by reducing the upper bound on n from 6 to 4. In particular, we show that we can construct v(x) as the sum of up to n = 3 finite-range unidirectionally invisible potentials, unless if it is required to be bidirectionally reflectionless.
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    PublicationOpen Access
    Adiabatic series expansion and higher-order semiclassical approximations in scattering theory
    (Institute of Physics (IOP) Publishing, 2014) Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231
    The scattering properties of any complex scattering potential, nu: R -> C, can be obtained from the dynamics of a particular non-unitary two-level quantum system. S-nu. The application of the adiabatic approximation to S-nu 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 S-nu. This results in a high-energy approximation scheme that unlike the semiclassical approximation can be applied for potentials with large derivatives.
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    PublicationOpen Access
    Blowing up light: a nonlinear amplification scheme for electromagnetic waves
    (Optical Society of America (OSA), 2018) Department of Mathematics; Department of Physics; Mostafazadeh, Ali; Ghaemidizicheh, Hamed; Hajizadeh, Sasan; Faculty Member; PhD Student; Department of Mathematics; Department of Physics; Graduate School of Sciences and Engineering; 4231; N/A; N/A
    We use the blow-up solutions of nonlinear Helmholtz equations to introduce a nonlinear resonance effect that is capable of amplifying electromagnetic waves of a particular intensity. To achieve this, we propose a scattering setup consisting of a Kerr slab with a negative (defocusing) Kerr constant placed to the left of a linear slab in such a way that a left-incident coherent transverse electric wave with a specific incidence angle and intensity realizes a blow-up solution of the corresponding Helmholtz equation whenever its wavenumber k takes a certain critical value, k(*). For k = k(*), the solution blows up at the right-hand boundary of the Kerr slab. For k < k(*), the setup defines a scattering system with a transmission coefficient that diverges as (k - k(*))(-4) for k -> k(*). By tuning the distance between the slabs, we can use this setup to amplify coherent waves with a wavelength in an extremely narrow spectral band. For nearby wavelengths, the setup serves as a filter. Our analysis makes use of a nonlinear generalization of the transfer matrix of the scattering theory as well as properties of unidirectionally invisible potentials.
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    PublicationOpen Access
    Class of exactly solvable scattering potentials in two dimensions, entangled-state pair generation, and a grazing-angle resonance effect
    (American Physical Society (APS), 2017) Loran, Farhang; Department of Mathematics; Department of Physics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; Department of Physics; College of Sciences; 4231
    We provide an exact solution of the scattering problem for the potentials of the form v(x,y) = chi(a)(x)[v(0)(x) + v(1)(x)e(i alpha y)], where chi(a)(x) := 1 for x is an element of [0,a], chi(a)(x) := 0 for x is an element of [0,a], v(j)(x) are real or complex-valued functions, chi(a)(x)v(0)(x) is an exactly solvable scattering potential in one dimension, and alpha is a positive real parameter. If alpha exceeds the wave number k of the incident wave, the scattered wave does not depend on the choice of v(1)(x). In particular, v(x,y) is invisible if v(0)(x) = 0 and k< alpha. For k > alpha and v(1)(x) = 0, the scattered wave consists of a finite number of coherent plane-wave pairs Psi(+/-)(n) with wave vector: k(n) = (+/-root k(2)- [n alpha](2),n alpha), where n = 0,1,2, . . .< k/alpha. This generalizes to the scattering of wave packets and suggests means for generating quantum states with a quantized component of momentum and pairs of states with an entangled momentum. We examine a realization of these potentials in terms of certain optical slabs. If k = N alpha for some positive integer N, Psi(+/-)(N) coalesce and their amplitude diverge. If k exceeds N alpha slightly, Psi(+/-)(N) have a much larger amplitude than Psi(+/-)(n) with n < N. This marks a resonance effect that arises for the scattered waves whose wave vector makes a small angle with the faces of the slab.
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    PublicationOpen Access
    Comment on “Identical motion in classical and quantum mechanics”
    (American Physical Society (APS), 1999) Department of Mathematics; Department of Physics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; Department of Physics; College of Sciences; 4231
    Makowski and Konkel [Phys. Rev. A 58, 4975 (1998)] have obtained certain classes of potentials which lead to identical classical and quantum Hamilton-Jacobi equations. We obtain the most general form of these potentials.
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    PublicationOpen Access
    Comment on “Quartic anharmonic oscillator and non-Hermiticity”
    (American Physical Society (APS), 2005) Department of Mathematics; Department of Physics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; Department of Physics; College of Sciences; 4231
    The analysis of the paper by J.-L. Chen et al. [Phys. Rev. A 67, 012101 (2003)] suffers from a conceptual error. Its results contradict some well-established mathematical facts and are incorrect.
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    PublicationMetadata only
    Comment on "Scattering of light by a parity-time-symmetric dipole beyond the first Born approximation"
    (American Physical Society (APS), 2022) Loran, Farhang; Seymen, Sema; Turgut, O. Teoman; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231
    In the work by Rebouças and Brandão [Phys. Rev. A 104, 063514 (2021)2469-992610.1103/PhysRevA.104.063514] the authors compute the scattering amplitude for a PT-symmetric double-δ-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. 
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    PublicationOpen Access
    Exactness of the Born approximation and broadband unidirectional invisibility in two dimensions
    (American Physical Society (APS), 2019) Loran, Farhang; Department of Mathematics; Department of Physics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; Department of Physics; College of Sciences; 4231
    Achieving exact unidirectional invisibility in a finite frequency band has been an outstanding problem for many years. We offer a simple solution to this problem in two dimensions that is based on our solution to another more basic open problem of scattering theory, namely, finding potentials v(x, y) whose scattering problem is exactly solvable via the first Born approximation. Specifically, we find a simple condition under which the first Born approximation gives the exact expression for the scattering amplitude whenever the wave number for the incident wave is not greater than a given critical value alpha. Because this condition only restricts the y dependence of v(x, y), we can use it to determine classes of such potentials that have certain desirable scattering features. This leads to a partial inverse scattering scheme that we employ to achieve perfect (nonapproximate) broadband unidirectional invisibility in two dimensions. We discuss an optical realization of the latter by identifying a class of two-dimensional isotropic active media that do not scatter incident transverse waves with wave numbers in the range (alpha/root 2, alpha] and source located at x = infinity while scattering the same waves if their source is relocated to x = -infinity.
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    PublicationOpen Access
    Fundamental transfer matrix and dynamical formulation of stationary scattering in two and three dimensions
    (American Physical Society (APS), 2021) Loran, Farhang; Department of Mathematics; Department of Physics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; Department of Physics; College of Sciences; 4231
    We offer a consistent dynamical formulation of stationary scattering in two and three dimensions (2D and 3D) that is based on a suitable multidimensional generalization of the transfer matrix. This is a linear operator acting in an infinite-dimensional function space which we can represent as a 2 x 2 matrix with operator entries. This operator encodes the information about the scattering properties of the potential and enjoys an analog of the composition property of its one-dimensional ancestor. Our results improve an earlier attempt in this direction [Phys. Rev. A 93, 042707 (2016)] by elucidating the role of the evanescent waves. We show that a proper formulation of this approach requires the introduction of a pair of intertwined transfer matrices, each related to the time-evolution operator for an effective nonunitary quantum system. We study the application of our findings in the treatment of the scattering problem for delta-function potentials in 2D and 3D and clarify its implicit regularization property which circumvents the singular terms appearing in the standard treatments of these potentials. We also discuss the utility of our approach in characterizing invisible (scattering-free) potentials and potentials for which the first Born approximation provides the exact expression for the scattering amplitude.