Fresnel and Fraunhofer diffraction of a Gaussian laser beam by fork-shaped gratings

Expressions describing the vortex beams that are generated by the process of Fresnel diffraction of a Gaussian beam incident out of waist on fork-shaped gratings of arbitrary integer charge p, and vortex spots in the case of Fraunhofer diffraction by these gratings, are deduced. The common general transmission function of the gratings is defined and specialized for the cases of amplitude holograms, binary amplitude gratings, and their phase versions. Optical vortex beams, or carriers of phase singularity with charges mp and -mp, are the higher negative and positive diffraction-order beams. The radial part of their wave amplitudes is described by the product of the mpth-order Gauss-doughnut function and a Kummer function, or by the first-order Gauss-doughnut function and the difference of two modified Bessel functions whose orders do not match the singularity charge value. The wave amplitude and the intensity distributions are discussed for the near and far fields in the focal plane of a convergent lens, as well as the specialization of the results when the grating charge p=0; i.e., the grating turns from forked into rectilinear. The analytical expressions for the vortex radii are also discussed.     

Lamellar Gratings as Polarization Components for Specularly Reflected Beams

Abstract

High spatial frequency lamellar gratings are shown to function as phase compensators, quarter-wave and half-wave retarders, and polarization rotators that operate on specularly reflected (zeroth-order) beams. These gratings are designed using rigorous coupled-wave and modal grating diffraction theories. Controlling the geometrical parameters of these gratings allows for engineering the phase retardation and polarization conversion introduced to a reflected beam. Fabrication and operational tolerances for these elements are discussed. Wavelength and polar angle of incidence variation affect the performance of these elements more strongly than variations in other geometrical and operational parameters.     

Optics for Neutral Atomic Beams: Reflection and Diffraction of Sodium Atoms by Evanescent Laser Light Fields

Abstract

Realization of the analogues of optical components for the manipulation of atomic de Broglie waves has recently become possible through the use of laser-based techniques. The strong fields generated by laser light have already been used to reflect atoms using travelling evanescent waves, and to diffract atomic beams through small angles using transmission gratings formed by a standing laser light wave in vacuo. Present experiments aim to combine these techniques to produce a reflection diffraction grating formed by a standing evanescent laser-light field, and to exploit the larger diffraction angles made possible theoretically by such a scheme.     

Pitch-modulated phase grating and its application to displacement encoder

Abstract

A binary phase grating with modulated pitch is investigated for a simple displacement encoder. The grating consists of a binary phase grating to eliminate the zeroth-order diffraction, and the pitch of the grating is modulated to compensate the higher harmonics of the encoder displacement signal. Therefore, an undistorted sinusoidal signal as a function of displacement is obtained by simply superimposing a conventional binary grating on the pitch-modulated phase gratings for any air gap between the gratings. The characteristics of the proposed gratings and the encoder signal are investigated by the Fresnel diffraction theory. The proposed grating has been fabricated lithographically, and the signal was examined experimentally. Considering these results, the proposed technique can suppress interpolation error and will be useful for an encoder in precision machining.     

Dynamic Control of Light Direction Enabled by Stimuli‐Responsive Liquid Crystal Gratings

The ability to control light direction with tailored precision via facile means is long‐desired in science and industry. With the advances in optics, a periodic structure called diffraction grating gains prominence and renders a more flexible control over light propagation when compared to prisms. Today, diffraction gratings are common components in wavelength division multiplexing devices, monochromators, lasers, spectrometers, media storage, beam steering, and many other applications. Next‐generation optical devices, however, demand nonmechanical, full and remote control, besides generating higher than 1D diffraction patterns with as few optical elements as possible. Liquid crystals (LCs) are great candidates for light control since they can form various patterns under different stimuli, including periodic structures capable of behaving as diffraction gratings. The characteristics of such gratings depend on several physical properties of the LCs such as film thickness, periodicity, and molecular orientation, all resulting from the internal constraints of the sample, and all of these are easily controllable. In this review, the authors summarize the research and development on stimuli‐controllable diffraction gratings and beam steering using LCs as the active optical materials. Dynamic gratings fabricated by applying external field forces or surface treatments and made of chiral and nonchiral LCs with and without polymer networks are described. LC gratings capable of switching under external stimuli such as light, electric and magnetic fields, heat, and chemical composition are discussed. The focus is on the materials, designs, applications, and future prospects of diffraction gratings using LC materials as active layers.     

Theoretical Analysis of the Lau Effect with a Sinusoidal Phase Grating

Abstract

By means of both partial coherence theory and diffraction theory, the Lau effect with a sinusoidal phase grating in a double-grating configuration is analysed. The optical intensity distribution at the observation plane is obtained. This distribution is related to the modulation of the phase grating and the separation of the two gratings. The results are somewhat different from the case of double amplitude gratings.     

Diffractive light attenuators with variable transmission

Abstract

A new application of diffractive optical elements (DOEs) for continuous or multistage adjustment of optical radiation intensity is described. The diffractive attenuators are linear or circular gratings (amplitude or phase) with constant period and diffraction efficiency that varies across the grating. The zero order of diffraction is used as the output and transmitted through the grating without angular deviation. The diffractive attenuators, in distinction to conventional analogues, allow one to change the intensity of the light beam according to predetermined function and have no limitations for power of the regulated light beam. These elements can be used in optical systems as a beam splitter with adjusted splitting coefficient. The experimental results on a circular diffractive attenuator fabricated by direct laser writing on a chromium film are presented. The range of transmission variation was 20 times within a 340° angle of attenuator turn. The possibility to use a phase diffractive attenuator as a light radiation modulator for a powerful technological laser is discussed.     

Stabilization of volume gratings recorded in polyvinyl alcohol-acrylamide photopolymers with diffraction efficiencies higher than 90%

Abstract

Obtaining holographic volume gratings with high diffraction efficiencies that can be used under white light has been a serious problem for the polyvinyl alcohol-acrylamide-based photopolymers developed by other researchers. In this paper we propose to eliminate the residual monomer in order to stabilize the holographic gratings. The residual dye and residual monomer are the main problems in achieving high diffraction efficiencies stable under white light. In order to polymerize the residual monomer we illuminate the gratings with coherent green light and incoherent white light and we heat the grating at 80°C for different times. We also study the conservation of gratings dried in critical conditions of humidity and temperature. After stabilization the diffraction efficiencies achieved were clearly higher than 90%.     

Off-Bragg analysis of the diffraction efficiency of reflection photorefractive holograms

Formulas are given for the calculation of diffraction efficiency of reflection-type gratings recorded in a photorefractive medium. The analysis incorporates the coupled-wave theory that was developed for photorefractive hologram gratings. This analysis takes into account grating slant with respect to the medium surface, light absorption during reconstruction, any incident angle of the reference beam, and any photorefractive phase shift. General solutions for signal and reference wave functions are given in a closed-form expression by use of a hypergeometric function. The optimum media parameters and recording conditions for high diffraction efficiency are obtained by the derived formulas. The diffraction properties for off-Bragg conditions are also discussed.     

Beam Splitter Model of Two-beam Coupling in Photorefractive Materials

Abstract

The phenomenon of two-beam coupling in photorefractive crystals results from the interaction of two beams with the index grating written by those two beams. In this paper we show that the process can be understood by considering the photorefractive crystal to be a nonlinear beam splitter, with reflectivity and transmission coefficients dependent upon the relative intensities of the two beams. The relative phase relationships upon reflection, determined by the Fresnel equations, as well as the phase shift between the intensity and index grating, are the sources of the asymmetric coupling. A simple derivation of coupled mode equations describing two-beam coupling based upon the beam splitter model is proffered.     

Resonant atomic diffraction: real versus imaginary potentials

In 1997, Zeilinger's group claimed to have observed atomic diffraction from a purely imaginary potential, that is an absorptive grating, made with resonant light waves. In a 1990 summary of their previous research, Kazantsev et al. stated that the diffraction of atoms from resonant light waves is due to a real potential (a resonant phase grating). We resolve this conflict and argue that absorptive gratings of light do not exist for atoms. Analysis of our metastable argon experiments with resonant 801.7?nm light supports this view. We discuss why these issues are important for understanding transverse momentum conservation in diffraction processes.     

Azimuth-angle-dependent reflectivity data from metallic gratings

Abstract

In this work, we have used a new technique to characterize optically a periodically modulated metal–dielectric interface involving the measurement of the azimuthal-angle-dependent specular reflectivity. This method presents advantages over the conventional polar angle scan experiment since it requires no moving signal detector. The data recorded has been fitted to a conical version of Chandezon's differential formalism using a single set of parameters describing the grating profile and metal permittivity. The fitted grating profile is in good agreement with the form found by the use of an atomic force microscope. This work therefore provides a new procedure for characterizing gratings as well as a convincing test of conical diffraction theory.     

Amplitude modulation and beam-steering properties of active binary phase gratings with reconfigurable absorption areas

By combining a binary phase grating and materials with a controllable absorption, it is shown theoretically that it is possible to modulate the light in a zero-order diffraction beam and that a high contrast level for the beam modulation can be obtained. The intensity of higher diffraction orders also changes, but it is calculated that high contrasts cannot be achieved for these higher-order beams with the active gratings that we examine. This specific modulator design that we use can be applied both for transmitted and for reflected light. Using the same ideas, one may build a beam-deflection device by varying the period of the grating by selectively changing the absorption levels in the grating. The deflection efficiency of the device can be improved compared with other designs by use of a grating with a reduced intensity of the nondeflected zero-order beam.     

Optical beam deflection by phase-controllable dual-period grating employing liquid crystal

A dual-period grating is proposed that can change the period and the phase difference in the grating structure, allowing the switchable diffraction allocation of transmitted light. Liquid crystal is assumed to be a variable-refractive-index medium confined in the grating grooves. The distribution of the transmitted diffraction efficiencies is analysed using a rigorous diffraction analysis for dual-period gratings and this reveals that the efficiencies can be maximized among the zeroth-, first-, and second-order waves successively according to the refractive index change in the liquid crystal. The analytical model provides insight into the operation of the device and emphasizes its potential application as an optical beam deflector with large deflection angles.     

Analysis and observation of finite beam Bragg diffraction by a thick planar phase grating

A numerical-impulse-response technique for studying the propagation and diffraction of finite-width beams in planar phase gratings is described. It can account for both symmetric and asymmetric diffractions. The grating-length-to-beam-width ratio is shown to govern the extent of beam-profile distortion and selectivity sidelobe suppression. Trade-offs between diffraction efficiency and beam profile have also been demonstrated. Theoretical results have been verified by experimental observations in a planar waveguide geometry of diffracted beams that change from a single diffraction peak to multiple peaks as the grating-length-to-beam-width ratio increases.     

Polarizationally Non-sensitive Pseudo-holographic Computer Reconstruction of Random Rough Surface

Abstract

The function, describing a profile of a random rough surface (RRS) is expanded in a Fourier series, i.e. the surface is considered as a composition of sinusoidal gratings. The total diffracted optical field from this RRS is a sum of the fields due to all harmonic gratings, since Kirchhoff's condition for ‘locally flat surface’ is realized for each harmonic grating at a given light wavelength and at an appropriate choice of the basic grating period. The registered s and p components of the diffracted (+1 diffraction orders of each harmonic gratings), incident and mixed optic fields are separated with an optical analyser. These fields are experimentally measured and from these values the phase and the amplitude of each grating are determined. The profile of the surface is reconstructed for s and p polarization of the light scattered field, when the electric vector of the incident light concludes an arbitrary angle with the incidence plane. The mean roughness is determined in both cases. It is shown, that both reconstructions of the profile and the determination of the mean roughness are not dependent on the polarization of the incident light. The separation of the s and p components is of great importance at the two-dimensional reconstruction, when independent of incident light polarization (s or p), the scattered optical field is always depolarized. In this case the profile of the two-dimensional surface can be easily reconstructed with s or p component of the mixing and diffracted fields.     

Statistical Model for Noise Gratings Recorded in Volume Holograms

Abstract

A statistical model is used to study the formation of noise gratings in holograms and the mechanism that causes radiation scattering in them. The theoretical model presented describes noise scattering in terms of diffraction due to the high number of spurious gratings formed during the recording stage by the interference of a single reference beam with its own scattered radiation. Numerical results calculated from the theory showing the dependence of noise grating efficiency on exposure are compared with experimental results obtained with bleached emulsions, and good agreement is found.     

Polarization-dependent diffraction efficiency of a photorefractive volume grating and suppression of this efficiency

In terms of refractive-index ellipsoid of a uniaxial crystal, the relationship between the diffraction efficiency of a volume grating and the polarization state of a readout beam is theoretically analyzed. The direction of a refractive light beam and the corresponding refractive-index modulation will both be changed by a variation of the polarization state. In the polarization state of the readout beam, which may lead to a strong variation in the diffraction efficiency of the volume grating. This kind of polarization-dependent diffraction efficiency of a volume grating in an anisotropic crystal is extremely disadvantageous for some applications. A method to suppress the polarization-dependent diffraction efficiency by use of double volume gratings is presented, and experiments with LiNbO3:Fe crystal are also demonstrated. The experimental results indicate that this method can well suppress the polarization-dependent diffraction efficiency of a volume grating. Furthermore, the diffraction properties of the double volume gratings are almost independent of the polarization state of the readout beam. The relative values of the diffraction peaks are calculated on the basis of the relationship between index modulation and the state of polarization. The experimental values are in good agreement with the theoretical analyses.     

Diffraction of a one-dimensional phase grating in the deep Fresnel field

We analyze theoretically the diffraction of phase gratings in the deep Fresnel field on the basis of the theory of scalar diffraction and Green's theorem and present the general formula for the diffraction intensity of a one-dimensional sinusoidal phase grating. The numerical calculations show that in the deep Fresnel region the diffraction distribution can be described by designating three characteristic regions that are influenced by the parameters of the grating. The microlensing effect of the interface of the phase grating provides the corresponding explanation. Moreover, according to the viewpoint that the diffraction intensity distribution is the result of the interference of the diffraction orders of the grating, we find that the diffraction patterns, depending on the carved depth of the phase grating, are determined by the contributing diffraction orders, their relative power, and the quasi-Talbot effect of the phase grating, which results from the second meeting of the diffraction orders carrying most of the power of the total field, as in the case of the amplitude grating.     

Effect of Slab Interfaces on Diffraction of Visible Light by a Thick Volume Grating

Abstract

A theory of visible light diffraction by a volume grating is given for the case when reflection at slab interfaces cannot be neglected. A method of approximate solution of the second-order coupled-wave equations, applicable over a wide range of the grating vector orientation, is proposed. An appropriate boundary-value problem is discussed and solved in the two-wave approximation. The results are analytical and expressed in matrix notation suitable for both transmission and reflection diffraction gratings. Particular attention is given to the diffraction regime that arises in a slanted grating under the total internal reflection of the diffracted wave (TIRDW). Strong transmission anomalies occurring in the vicinity of the TIRDW threshold are established and analysed.