Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Physics A Mathematical and General

Diffraction by volume gratings with imaginary potentials

Michael Berry, Duncan H J O’Dell

Published Feb 27, 1998• 129 citations
Publisher

Abstract

A medium described by an imaginary potential (or imaginary (refractive index) ), that varies sinusoidally in one direction, acts as a volume grating for plane waves incident on it obliquely or normally. Two peculiar features are identified. First, if the potential is weak, so that there are only two significant diffracted beams near the Bragg angle, and three for normal incidence, diffraction is strongly affected by degeneracies of the non-Hermitian matrix generating the `Bloch waves' in the grating; the effect of these degeneracies is very different from that of the Hermitian degeneracies for transparent gratings. Second, if the potential is strong and the grating thick, the asymptotic distribution of intensities among the diffracted beams (momentum distribution) is a rather narrow Gaussian, and dominated by a single set of complex rays; this is very different from the semiclassical limit for transparent gratings, where the rays form families of caustics proliferating with thickness, and with a wider momentum distribution.

DOI
10.1088/0305-4470/31/8/019

Related papers

Lop-sided diffraction by absorbing crystals

Michael Berry

Diffraction of atoms by a particular absorbing `crystal of light', that has been studied experimentally, is described by the complex potential . The diffracted beam intensities can be calculated exactly, as a function of (dimensionless) potential strength , angle of incidence , and crystal thickness . Only the beams have nonzero intensity; this `lop-sidedness' is a dramatic violation of Friedel's law. The n th beam is strong at Bragg angles , and the peaks get sharper with increasing ; they correspond to degeneracies of the (non-Hermitian) governing matrix. For normal incidence , the are periodic in , and as n increases they approach a self-similar function of .

Journal of Physics A Mathematical and General•Apr 17, 1998•84 citations

The diraction of atoms by light

Duncan H J O’Dell, H. H. Wills

The diffraction of a coherent, stationary, beam of atoms from a standing wave of light is considered. Particular attention is paid to long interaction times for which the motion of the atoms inside the light field must be taken into account. This is the regime of dynamical diffraction, and is treated within the framework of a set of differential difference equations due to Raman and Nath. The popular Raman-Nath approximation, in which the kinetic energy imparted to the atoms by the light wave is ignored, is no longer valid and the full equations must be solved. When the interaction between the atoms and the light is very weak then few beams are produced. To describe spontaneous emission from the atom one can invoke a complex potential as a model for dissipation. If the atom beam is obliquely incident one finds anomalously high transmission close to the Bragg angles which is the physical manifestation of degeneracies in non-Hermitian matrices. The classical limit is approached when the atom-light interaction is strong, producing many beams. The classical dynamics (geometrical rays) reveal that the farfield intensity pattern is dominated by caustics. These proliferate with increasing interaction distance. Two methods are presented for analytically treating the semiclassical limit of the wave mechanics. The first begins with W.K.B. solutions to the continuised Raman-Nath equations. Mapping onto parabolic cylinder functions removes the spurious divergences of the W.K.B. solutions, giving eigenvectors which are uniformly valid for all scattering angles and may be summed to give the wavefunction. The second method transforms the original W.K.B. eigensum, using the Poisson summation formula, giving a new sum, each term of which has classical significance. This method shows that wave theory decorates the classical caustics with an Airy function intensity profile, as predicted by catastrophe theory.

•Jan 1, 1998•1 citation

Fresnel diffraction is a basic quantum mechanical effect

Mohammad Taghi Tavassoly

Using the Fresnel–Kirchhoff integral, we express the state of the light that diffracts from an aperture to a point, by superposition of two or more waves. The amplitudes of the waves are determined by the distances of the point from the ray optics borders of the emerging beams. The interference approach to the Fresnel diffraction leads to a universal formulation of the normalized intensity distribution on the diffraction pattern that determines the state of the incident light, the parameters of the aperture, and its distance from the observation plane, uniquely. In the photon approach, according to the uncertainty principle, confining the position of a photon to an aperture changes the probable propagation direction of the photon in an interval that leads to the diffraction of the photon or light. Thus, the normalized intensity distribution on the diffraction pattern is the probability distribution of a photon. To this probability distribution, a probability amplitude distribution is associated that diffracts similar to a wave in Fresnel diffraction. Applying the introduced approach to the study of the diffraction, from different simple apertures and phase steps, we deduce general behaviors of the Fresnel diffraction and show it is involved in all aspects of light phenomena.

Journal of the Optical Society of America B•May 14, 2021•8 citations

Self-diffraction in population gratings

H. L. Fragnito, Silvania F. Pereira, Alvin E. Kiel

We present a nonperturbative theory of self-diffraction in thick, pseudostationary population gratings induced by the interference of two nearly collinear incident laser beams of arbitrary relative and absolute intensities. At large saturation parameters, the grating becomes nonsinusoidal, and high-order diffraction occurs efficiently. We obtain an exact expression for the Fourier coefficients of the grating that significantly simplifies the interpretation of the various coherent coupling mechanisms and permits efficient computer simulations. In the case of small depletion of the pump waves, we obtain simple analytic expressions for the scattered intensity into any order of self-diffraction. We present our results in a manner that should aid an experimenter in choosing the optical thickness of the unsaturated medium and the sum and ratio of the pump intensities to obtain optimum diffraction intensity in a given order.

Journal of the Optical Society of America B•Aug 1, 1987•25 citations

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

Farhang Loran, Ali Mostafazadeh

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.opticsCross-listedarXiv:2607.13287•Jul 16, 2026

Atomic diffraction by a thin phase grating

Carsten Henkel, J.-Y. Courtois, Alain Aspect

We present a semiclassical perturbation method for the description of atomic diffraction by a weakly modulated potential. It proceeds in a way similar to the treatment of light diffraction by a thin phase grating, and consists in calculating the atomic wavefunction by means of action integrals along the classical trajectories of the atoms in the absence of the modulated part of the potential. The capabilities and the validity condition of the method are illustrated on the well-known case of atomic diffraction by a Gaussian standing wave. We prove that in this situation the perturbation method is equivalent to the Raman-Nath approximation, and we point out that the usually-considered Raman-Nath validity condition can lead to inaccuracies in the evaluation of the phases of the diffraction amplitudes. The method is also applied to the case of an evanescent wave reflection grating, and an analytical expression for the diffraction pattern at any incidence angle is obtained for the first time. Finally, the application of the method to other situations is briefly discussed.

arXiv:quant-ph/0408154•Aug 25, 2004•0 citations
A medium described by an imaginary potential (or imaginary (refractive index) ), that varies sinusoidally in one direction, acts as a volume grating for plane waves incident on it obliquely or normally. Two peculiar features are identified. First, if the potential is weak, so that there are only two significant diffracted beams near the Bragg angle, and three for normal incidence, diffraction is strongly affected by degeneracies of the non-Hermitian matrix generating the `Bloch waves' in the grating; the effect of these degeneracies is very different from that of the Hermitian degeneracies for transparent gratings. Second, if the potential is strong and the grating thick, the asymptotic distribution of intensities among the diffracted beams (momentum distribution) is a rather narrow Gaussian, and dominated by a single set of complex rays; this is very different from the semiclassical limit for transparent gratings, where the rays form families of caustics proliferating with thickness, and with a wider momentum distribution.
Diffraction of atoms by a particular absorbing `crystal of light', that has been studied experimentally, is described by the complex potential . The diffracted beam intensities can be calculated exactly, as a function of (dimensionless) potential strength , angle of incidence , and crystal thickness . Only the beams have nonzero intensity; this `lop-sidedness' is a dramatic violation of Friedel's law. The n th beam is strong at Bragg angles , and the peaks get sharper with increasing ; they correspond to degeneracies of the (non-Hermitian) governing matrix. For normal incidence , the are periodic in , and as n increases they approach a self-similar function of .
The diffraction of a coherent, stationary, beam of atoms from a standing wave of light is considered. Particular attention is paid to long interaction times for which the motion of the atoms inside the light field must be taken into account. This is the regime of dynamical diffraction, and is treated within the framework of a set of differential difference equations due to Raman and Nath. The popular Raman-Nath approximation, in which the kinetic energy imparted to the atoms by the light wave is ignored, is no longer valid and the full equations must be solved. When the interaction between the atoms and the light is very weak then few beams are produced. To describe spontaneous emission from the atom one can invoke a complex potential as a model for dissipation. If the atom beam is obliquely incident one finds anomalously high transmission close to the Bragg angles which is the physical manifestation of degeneracies in non-Hermitian matrices. The classical limit is approached when the atom-light interaction is strong, producing many beams. The classical dynamics (geometrical rays) reveal that the farfield intensity pattern is dominated by caustics. These proliferate with increasing interaction distance. Two methods are presented for analytically treating the semiclassical limit of the wave mechanics. The first begins with W.K.B. solutions to the continuised Raman-Nath equations. Mapping onto parabolic cylinder functions removes the spurious divergences of the W.K.B. solutions, giving eigenvectors which are uniformly valid for all scattering angles and may be summed to give the wavefunction. The second method transforms the original W.K.B. eigensum, using the Poisson summation formula, giving a new sum, each term of which has classical significance. This method shows that wave theory decorates the classical caustics with an Airy function intensity profile, as predicted by catastrophe theory.
Using the Fresnel–Kirchhoff integral, we express the state of the light that diffracts from an aperture to a point, by superposition of two or more waves. The amplitudes of the waves are determined by the distances of the point from the ray optics borders of the emerging beams. The interference approach to the Fresnel diffraction leads to a universal formulation of the normalized intensity distribution on the diffraction pattern that determines the state of the incident light, the parameters of the aperture, and its distance from the observation plane, uniquely. In the photon approach, according to the uncertainty principle, confining the position of a photon to an aperture changes the probable propagation direction of the photon in an interval that leads to the diffraction of the photon or light. Thus, the normalized intensity distribution on the diffraction pattern is the probability distribution of a photon. To this probability distribution, a probability amplitude distribution is associated that diffracts similar to a wave in Fresnel diffraction. Applying the introduced approach to the study of the diffraction, from different simple apertures and phase steps, we deduce general behaviors of the Fresnel diffraction and show it is involved in all aspects of light phenomena.
We present a nonperturbative theory of self-diffraction in thick, pseudostationary population gratings induced by the interference of two nearly collinear incident laser beams of arbitrary relative and absolute intensities. At large saturation parameters, the grating becomes nonsinusoidal, and high-order diffraction occurs efficiently. We obtain an exact expression for the Fourier coefficients of the grating that significantly simplifies the interpretation of the various coherent coupling mechanisms and permits efficient computer simulations. In the case of small depletion of the pump waves, we obtain simple analytic expressions for the scattered intensity into any order of self-diffraction. We present our results in a manner that should aid an experimenter in choosing the optical thickness of the unsaturated medium and the sum and ratio of the pump intensities to obtain optimum diffraction intensity in a given order.
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):=iV0(eiK y−1)v(x,y):=iV_0(e^{i\mathfrak{K}\,y}-1)v(x,y):=iV0​(eiKy−1) for 0≤x≤ℓ0\leq x\leq\ell0≤x≤ℓ and v(x,y):=0v(x,y):=0v(x,y):=0 otherwise, where V0,KV_0, \mathfrak{K}V0​,K, and ℓ\ellℓ 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, ε^(x,y):=1−v(x,y)/k2\hat\varepsilon(x,y):=1-v(x,y)/k^2ε^(x,y):=1−v(x,y)/k2, where kkk 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.
We present a semiclassical perturbation method for the description of atomic diffraction by a weakly modulated potential. It proceeds in a way similar to the treatment of light diffraction by a thin phase grating, and consists in calculating the atomic wavefunction by means of action integrals along the classical trajectories of the atoms in the absence of the modulated part of the potential. The capabilities and the validity condition of the method are illustrated on the well-known case of atomic diffraction by a Gaussian standing wave. We prove that in this situation the perturbation method is equivalent to the Raman-Nath approximation, and we point out that the usually-considered Raman-Nath validity condition can lead to inaccuracies in the evaluation of the phases of the diffraction amplitudes. The method is also applied to the case of an evanescent wave reflection grating, and an analytical expression for the diffraction pattern at any incidence angle is obtained for the first time. Finally, the application of the method to other situations is briefly discussed.