Shift and shape of grating diffracted beams

Abstract

The grating diffraction of beams is theoretically investigated by applying an electromagnetic method (the Integral Equation System Method with Parametrization of the grating profile = IESMP) to their plane wave components. For the first time, explicit values for the displacement of grating diffracted Gaussian beams are calculated with this method. For total reflection this displacement of beams is known as the Goos-Hänchen shift. A maximum shift of 36 μm has been found for the investigated sinusoidal grating near an anomaly which is much greater than the known Goos–Hänchen shift of about 1 μm for the total reflection case. The replacement of the angular spectrum of plane waves with constant wavelength by a wavelength spectrum of plane waves of constant direction allows an analogous treatment of short-time pulses. Surprisingly, the above anomaly causes a maximum temporal shift of 80 fs for the pulse diffraction. These temporal shifts and additional effects like pulse deformations can influence ultra short-time pulse experiments. Furthermore, the behaviour of temporally and spatially Gaussian shaped light pulses (TSG pulses) by grating diffraction are studied considering the diffraction of an angular and wavelength dependent spectrum of plane waves. The diffraction of a short TSG pulse at the above grating deforms the pulse and creates an additional smaller satellite pulse. All described effects occur only at positions of the space–time complex filtering function in the angular-wavelength frequency space with high gradient of the phase.     

Resonance radiation of electromagnetic waves by a diffraction grating filled with metamaterial

We consider the diffraction of a plane inhomogeneous H-polarized wave, generated by a modulated sheet electron beam, on a periodic reflective grating with rectangular grooves filled by a metamaterial with the effective permittivity characterized by a frequency dispersion. The problem is solved using the method of reexpansion. The phenomenon of resonance radiation of electromagnetic waves by the grating is discovered and the conditions for its manifestation are established.     

Polarization of holographic grating diffraction. I. General theory

The full polarization property of volume holographic grating diffraction is investigated theoretically. With a simple volume grating model, the diffracted fields and Mueller matrices are first derived from Maxwell's equations by using the Green's function algorithms. The formalism is derived for the general case that the diffraction beam and the grating wave vector are not in the plane of incidence, where s waves and p waves are not decoupled. The derived photon-momentum relations determine the Bragg angle selectivity. The parameters of diffraction strength related to the hologram-writing process and material are defined and are not necessarily small in general. The diffracted-beam profiles are analytically calculated by using the known grating shape function. This theory has provided a fundamental understanding of the polarization phenomena of a real holographic diffraction grating device. The derived algorithm would provide a simulation-analysis tool for the engineering design of real holographic beam combiner/splitter devices.     

On diffraction of waves by the infinite grating of circular dielectric cylinders at oblique incidence: Floquet representation

Abstract

We investigate the diffraction of plane E-polarized electromagnetic waves by an infinite array of infinitely long circular dielectric cylinders for the most generalized case of oblique incidence. Employing the Sommerfeld integral representation of the Hankel function, and exploiting Poisson's summation formula in separation-of-variables solution, we have acquired a new representation for the diffraction of waves by the infinite grating of the circular dielectric cylinders at oblique incidence. The exact solutions for the external electric and magnetic field intensities have been derived in terms of the diffraction angles of the grating which are obtained by solving the grating equation. In addition, the transmitted and reflected fields of the infinite grating have been presented in terms of propagating and evanescent Floquet modes.     

Some Properties of Fresnel Images of a Square Wave Amplitude Grating

The self-imaging phenomenon of a square wave binary amplitude grating is studied. The intensity distributions in the Fresnel diffraction image planes lying at distances z = (M -1/4)d ²/u from the grating illuminated by a plane wave are calculated. It is found that these images are very similar to the object structure. As a consequence, some potential applications are proposed.     

Broadband quasi-total autocollimation cross-polarization wave transformation

The problem of diffraction of plane E-and H-polarized waves on a chiral layer supported on a periodic reflective grating composed of rectangular metal rods is solved using the methods of reexpansion and generalized scattering matrices. The diffraction on this structure is accompanied by quasi-total autocollimation cross-polarization wave transformation in a broad frequency band.     

Diffraction of gaussian beams by plane apertures: A different approach

Abstract

We first develop an integral equation formalism to solve the boundary value problems of the paraxial scalar wave equation when data, of the Dirichlet or Neumann type, are given on a closed surface S, in particular when S is a plane. Then, we show how this formalism can be used to analyse the scattering by perfectly conducting planes of paraxial waves, specially of Gaussian beams, and how it may be applied to paraxial wave diffraction by plane apertures. Finally, the diffraction of Gaussian beams by slits, rectangular and circular apertures is investigated.     

On the Phenomenology of Self-organized Quasi-phase-matched Second-harmonic Generation in Optical Fibres

Abstract

A simple phenomenological theory of second-harmonic generation in centrosymmetric media is presented in this analytical paper. It is shown that in the medium without memory the energy transfer between interacting waves is negligibly small. For the medium with an optical memory (like phosphorus-germanium-doped glass fibres), the process of formation of χ ⁽²⁾ grating, owing to the third-order optical rectification and the electric-field-induced deprivation of centrosymmetric structure, is described from the phenomenological point of view. Second harmonic generation at periodical χ ⁽²⁾ grating is studied. In the case of resonant interaction, second-harmonic radiation can be systematically amplified. For the non-monochromatic and pulse-form radiations the effective length of the medium is limited largely due to the Kerr nonlinearities. The estimated effective fibre length is in good agreement with previous experiments.     

T-matrix Approach for Calculating the Electromagnetic Fields Diffracted by a Corrugated,Anisotropic Grating

Abstract

The diffraction of electromagnetic plane waves incident on a corrugated grating made of a uniaxial dielectric material is studied using a fully vectorial treatment based on a T-matrix approach and the Rayleigh hypothesis. The optic axis is considered parallel to the mean surface of the periodic interface. The theory is exemplified numerically for the case of sinusoidal gratings made of sodium nitrate.     

Acoustic Emission Surface Responses from Mixed-Mode Crackings

An analytical expression for the relationship between acoustic emission (AE) surface motion and displacement discontinuity of a finite crack extending in mixed mode was investigated based on an analogy to earthquake problems in seismology and crackings in fracture mechanics. The displacement field in an infinite body was expressed in terms displacement discontinuity for a growing plane crack with arbitrary shape and orientation. The complete field response, including far-field, near-field, and intermediate-field motion has been studied. The displacement field in an infinite body was converted into the normal surface motion at the sensor site at the boundary of a half-space through local modification, considering the first reflection and mode conversion of the incident P and S waves at the traction-free boundary. The surface motion obtained is valid for finite crack propagation, while an algebraic equation for microcracking is given as a special case for illustration. Using these results, we will be able to analyze the fracture mode and relative location and orientation of sensor and crack plane according to the waveform obtained.     

Optical helicity of interfering waves

Helicity is a property of light which is familiar from particle physics but less well-known in optics. In this paper we recall the explicit form taken by the helicity of light within classical electromagnetic theory and reflect upon some of its remarkable characteristics. The helicity of light is related to, but is distinct from, the spin of light. To emphasise this fact, we draw a simple analogy between the helicity of light and electric charge and between the spin of light and electric current. We illustrate this and other observations by examining various superpositions of plane waves explicitly.     

An Approximate Linear Relationship for the Optical Response of Turbid Media

The optical response of a homogeneous turbid medium is often expressed analytically with the aid of the simplified theory of Kubelka and Munk. The equations derived from this theory yield the response A as an explicit function of absorption Kand scatter S. In practice, however, an inverted form is frequently needed, which would display absorption as an explicit function of A and S. No closed-form rigorous inversion of the response function A(S, K) is available, though approximate solutions exist. All direct inversions, analytical or empirical, are highly non-linear, even though for measurement and instrumentation purposes a linear characteristic is highly desirable. This paper shows that this can be achieved by using a logarithmic transform for the transmittance and a reciprocal one for the reflectance mode. Graphs are given for the coefficients of the linear approximation applicable over a wide range of the parameters S and K ; also shown is the mean error incurred in this range; it is nowhere in excess of a few per cent. The values displayed were obtained numerically using the least-square method of linear approximation. It is expected that the linearized form will be useful in many areas of applied densitometry. The advantages should apply not only to instrument design and measurement procedures, but also to theoretical studies aimed at practical situations.     

Reformulation of the fourier modal method for surface-relief gratings made with anisotropic materials

Abstract

By the use of the correct Fourier factorization rules presented in an earlier paper by Li the Fourier modal method (FMM) for anisotropic surface-relief gratings is reformulated. The newly formulated FMM converges much faster than the old formulation when the permittivity contrast in the grating groove region is large. Highly conducting lamellar gratings coated with anisotropic materials can now be analysed easily. In addition, a simple set of criteria are given for determining the energy propagation directions of the plane waves associated with the real solutions of the Booker quartic.     

Improvement of PUFEM for the numerical solution of high‐frequency elastic wave scattering on unstructured triangular mesh grids

The purpose of this paper, which builds on previous work (Int. J. Numer. Meth. Engng 2009; 77 :1646–1669), is to improve a numerical scheme based on the partition of unity finite element method (PUFEM) for the solution of the time harmonic elastic wave equations. The approach consists to approximate the displacement field by the standard finite element shape functions, enriched locally by superimposing pressure (P) and shear (S) plane waves. The aim is to accurately model two‐dimensional elastic wave problems on relatively coarse mesh grids, capable of containing many wavelengths per nodal spacing, for wide ranges of frequencies. This allows us to relax the traditional requirement of about 10 nodal points per S wavelength. In this work, an exact integration scheme for the linear triangular finite element is developed to evaluate the oscillatory integrals arising from the use of the PUFEM. The main contribution here consists in developing an explicit closed‐form solution for two‐dimensional wave‐based integrals, when the phase variation is linear in the local coordinate element system. The evaluation of the element mass matrix is performed from appropriate edge integrals. All other element matrices, obtained by adequate splitting of the element stress tensor matrix, are simply deduced from the element mass matrix entries. The results show clearly that the proposed integration scheme evaluates accurately the entries of the global matrix with drastic reduction of the computational time. Numerical tests dealing with the scattering of S elastic plane waves by a circular rigid body show that, for the same discretization level, it is possible to improve the accuracy by using large elements associated with high numbers of approximating plane waves rather than using small elements with less plane waves. However, this increases the conditioning and the fill‐in of the global matrix. At high frequency, it is even possible to push the number of degrees of freedom per S wavelength under 2 and still achieve good accuracy. Finally, some remarks on the choice of the numbers of P and S plane waves leading to better accuracy and conditioning are discussed. Copyright © 2010 John Wiley & Sons, Ltd.     

Formation of the micro-sized grains under influence of the nanosecond laser pulses

The occurrence of a periodic photo-induced grating having about 3.2-μm period and modulation heights up to 150 nm is discovered here under the irradiation of 10-ns 1,064-nm laser pulses with Ag nano-particles deposited on ITO substrates. It is shown that the main role is played by the ITO resistance. This can be explained as the result of interaction between two coherent waves. The first wave is engendered by the fundamental laser and the second one is due to the two-photon absorption phenomenon. It is important mentioning that the observed photo-induced grating is obtained only from a treatment by single 10-ns laser pulses with an effective pump power below 0.7 GW/cm². Since the laser power densities are markedly low and the pulse durations larger than those of the femtosecond laser, the origin of this result is expected to be ascribed to the plasmon excitation leading to further thermoionization.     

Internal reflections in bleached reflection holograms

Bleached reflection holograms produced by two plane waves in Agfa 8E56 photographic emulsion and recorded in an index-matching liquid tank at 514 nm are studied. Replay at 514 nm is both in the tank and in air, boundary reflections giving rise to multiple-output beams in the latter case. The intensities in the various output beams are measured as a function of the angle of incidence of the input beam. A simple theory based on two-wave grating diffraction and the Fresnel boundary coefficients is formulated and shown to agree with good approximation with the observed intensities of all significant output beams.     

Influence of processing conditions on geometrical features of laser claddings obtained by powder injection

The influence of processing conditions on the geometrical aspects of laser claddings is studied. Attention is drawn to the following laser-processing parameters: laser-power interaction time, surface of the irradiated zone, powder-feed rate. Different Ni-Cr-Co-Fe base powders are deposited on an austenitic stainless steel. It is observed that the mass of powder participating in the clad layer — which is proportional to the effective cross-sectionS of the layer — increases linearly with irradiance and interaction time. However, threshold values are detected; they are due to the existence of a minimum energy value required to melt the powder particles. The influence of the powder-feed rate, F, is more complicated. For low values,S is directly proportional toF, while beyond a critical value,F ^*, the powder injected in the laser beam acts as a screen, the opacity of which increases due to the multiple scattering phenomenon; energy absorption is more and more efficient and process efficiency is enhanced. However, this increase is limited to a maximum value, since the energy delivered by the laser source is also limited and since cladding requires melting of the particles. A simple explanation is proposed, based on mass- and energy-transfer governing equations.     

A Self-aligning Interferometer Suitable for White or Monochromatic Light Consisting Solely of a Mirror and a Reflection Grating

Abstract

A simple interferometer constructed from a mirror and a reflection grating is described. This interferometer can operate with monochromatic or wideband light, and versions of it may be constructed to suit radiations other than light, such as sound waves or matter waves. A most unusual feature of the interferometer is its high tolerance to large misalignments of its optical elements.     

Properties of Diffraction Grating Anomalies

Values provided by three different theories of the diffraction by gratings are compared with measurements made at millimetre wavelengths. The results based on the integral equation method of Pavageau and Bousquet give the best agreement with the experimentally-determined shapes of Wood anomalies. The wavelength dependence of P polarization anomalies is considered. Some properties of first-order P polarization resonance anomalies are demonstrated. The existence of second-order P resonances is shown. Second, third and fourth-order S polarization resonance anomalies have been found to occur for sinusoidal profile gratings at groove depths given by a simple formula. These anomalies cause large changes in grating efficiency to occur in very short wavelength intervals, and so might prove useful in spectroscopy.     

Travelling and evanescent parts of the optical near field

Abstract

The optical near field of a localized source has been studied by means of the angular spectrum representation of the electromagnetic Green's tensor. This Green's tensor can be expressed in terms of four auxiliary functions, which depend on the field point through the dimensionless radial distance q to the source, or origin of coordinates, and the polar angle ρ with the z axis. Each function separates into a part containing travelling (radiative) waves and a part which is a superposition of evanescent (decaying) waves. We have derived series expansions in q of the four functions, both for the travelling and for the evanescent parts. It is shown that the travelling waves are finite at the origin of coordinates, and that all singular behaviour of the radiation field is governed by the evanescent waves. It is illustrated numerically that the series expansions are applicable up to about five wavelengths from the origin. In order to extend the range to also cover larger values of q, we have derived series expansions of the auxiliary functions which converge rapidly near the x-y plane, and a full asymptotic expansion with the z coordinate as the large variable. Finally, from the properties of the Taylor coefficients we have derived simple new integral representations for the auxiliary functions.