Publications with Fulltext

Permanent URI for this collectionhttps://hdl.handle.net/20.500.14288/6

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

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Now showing 1 - 10 of 25
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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.
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    PublicationOpen Access
    Transfer matrices as nonunitary S matrices, multimode unidirectional invisibility, and perturbative inverse scattering
    (American Physical Society (APS), 2014) Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231
    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(tau). If v is real valued, H(tau) is pseudo-Hermitian and its exceptional points correspond to the classical turning points of v. Applying time-dependent perturbation theory to H(tau) 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-order 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.
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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
    Perfect broadband invisibility in isotropic media with gain and loss
    (The Optical Society (OSA) Publishing, 2017) Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; College of Sciences; 4231
    We offer a simple route to perfect omnidirectional invisibility in a spectral band of desired width. Our approach is based on the observation that in two dimensions a complex potential v(x; y) is invisible for incident plane waves with a wavenumber not exceeding a pre-assigned value a, provided that its Fourier transform with respect to y, which we denote by v (x; R-y), vanishes for R-y <= 2a. We can fulfill this condition for potentials modeling the permittivity profile of an optical slab. Such a slab is perfectly invisible for any transverse electric wave whose wavenumber is in the range [0; a]. Our results also apply to transverse magnetic waves propagating in a medium with a relative permittivity epsilon (x; y) that is a smooth bounded function with a positive real part.
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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
    Generalized unitarity relation for linear scattering systems in one dimension
    (Springer, 2019) Department of Physics; Department of Mathematics; Mostafazadeh, Ali; Faculty Member; Department of Physics; Department of Mathematics; College of Sciences; 4231
    We derive a generalized unitarity relation for an arbitrary linear scattering system that may violate unitarity, time-reversal invariance, PT - symmetry, and transmission reciprocity.
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    PublicationOpen Access
    Spectral singularities and whispering gallery modes of a cylindrical gain medium
    (American Physical Society (APS), 2013) Department of Mathematics; Mostafazadeh, Ali; Sarısaman, Mustafa; Faculty Member; Researcher; Department of Mathematics; College of Sciences; 4231; N/A
    Complex scattering potentials can admit scattering states that behave exactly like a zero-width resonance. Their energy is what mathematicians call a spectral singularity. This phenomenon admits optical realizations in the form of lasing at the threshold gain, and its time-reversal is responsible for antilasing. We study spectral singularities and whispering gallery modes (WGMs) of a cylindrical gain medium. In particular, we introduce a class of WGMs that support a spectral singularity and, as a result, have a divergent quality factor. These singular gallery modes (SGMs) are excited only if the system has a positive gain coefficient, but typically the required gain is extremely small. More importantly, given any amount of gain, there are SGMs requiring smaller gain than this amount. This means that, in principle, the system lacks a lasing threshold. Furthermore, the abundance of these modes allows for configurations where a particular value of the gain coefficient yields an effective excitation of two distant SGMs. This induces lasing at two different wavelengths.
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    PublicationOpen Access
    Optimal control of molecular motion expressed through quantum fluid dynamics
    (American Physical Society (APS), 2000) Dey, B. K.; Rabitz, H.; Department of Mathematics; Aşkar, Attila; Faculty Member; Department of Mathematics; College of Sciences; 178822
    A quantum fluid-dynamic (QFD) control formulation is presented for optimally manipulating atomic and molecular systems. In QFD the control quantum system is expressed in terms of the probability density rho and the quantum current j. This choice of variables is motivated by the generally expected slowly varying spatial-temporal dependence of the fluid-dynamical variables. The QFD approach is illustrated for manipulation of the ground electronic state dynamics of HCl induced by an external electric field.
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    PublicationOpen Access
    Spectral singularities and tunable slab lasers with 2D material coating
    (Optical Society of America (OSA), 2020) Ghaemi-Dizicheh, Hamed; Sarısaman, Mustafa; Department of Mathematics; Department of Physics; Mostafazadeh, Ali; Faculty Member; Department of Mathematics; Department of Physics; College of Sciences; 4231
    We investigate linear and nonlinear spectral singularities in the transverse electric and transverse magnetic modes of a slab laser consisting of an active planar slab sandwiched between a pair of graphene or Weyl semimetal thin sheets. The requirement of the presence of linear spectral singularities gives the laser threshold condition while the existence of nonlinear spectral singularities due to an induced weak Kerr nonlinearity allows for computing the laser output intensity in the vicinity of the threshold. The presence of the graphene and Weyl semimetal sheets introduces additional physical parameters that we can use to tune the output intensity of the laser. We provide a comprehensive study of this phenomenon and report peculiarities of lasing in the transverse magnetic (TM) modes of the slab with Weyl semimetal coatings. In particular, we reveal the existence of a critical angle such that no lasing seems possible for TM modes of the slab with the smaller emission angle. Our results suggest that for TM modes with an emission angle slightly exceeding the critical angle, the laser output intensity becomes highly sensitive to the physical parameters of the coating.
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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.