Angular responses of the first diffracted order in over-modulated volume diffraction gratings

Abstract

Kogelnik's coupled-wave (CW) theory has been used for decades to predict the diffraction efficiency of volume diffraction gratings. Although this theory has been applied with success to volume diffraction gratings recorded under a great variety of experimental conditions, its predictions deviate from the actual behaviour whenever the hologram is thin or the refractive index is high. In these cases, it is necessary to use a more general CW theory or the rigorous coupled-wave (RCW) theory. Both of these theories allow for more than two orders to propagate inside the hologram. The difference between them is that in the CW theory the second derivatives that appear in the coupled equations are disregarded. The RCW theory does not incorporate any approximation and thus, since it is rigorous, permits judging the accuracy of the approximations included in Kogelnik's CW theory and the more general CW theory. In this article a comparison between the predictions of the three theories for phase transmission diffraction gratings is carried out. Over-modulated diffraction gratings are also recorded in photographic emulsions in order to study the applicability of Kogelnik's theory in this case. Good agreement between theory and experiment is found for both Kogelnik's theory and the RCW theory formulations in the particular experimental cases studied.     

A treatment of the general volume holographic grating as an array of parallel stacked mirrors

An alternative model to Kogelnik's coupled wave theory of the volume holographic grating is developed in terms of an infinite array of parallel stacked mirrors. The model is based on summing the individual Fresnel reflections from an infinite number of infinitesimal discontinuities in the permittivity profile. The resulting first-order coupled partial differential equations are solved in a rotated frame of reference in order to derive analytical expressions for the diffraction efficiency of the general slanted grating at an arbitrary angle of incidence. The model has been tested using computational solutions of the Helmholtz equation for the unslanted reflection grating. For index modulations characteristic of modern silver halide and photopolymer materials used in display and optical element holography the new model shows excellent agreement with the numerical results. Kogelnik's model also provides good agreement as long as the dephasing parameter is not too large. The model has been tested against Kogelnik's theory for a variety of cases with finite fringe slant with good agreement for typical index modulations. A further advantage of the new model is that colour holographic gratings may be treated at and away from Bragg resonance. Numerical and analytical results are presented concerning the diffractive efficiency of two- and three-colour holographic gratings.     

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.     

High-efficiency volume holograms recording on acrylamide and N,N′methylene-bis-acrylamide photopolymer with pulsed laser

In order to achieve a better understanding of the mechanisms of hologram formation and higher diffraction efficiencies in volume gratings stored in acrylamide based photopolymers, a crosslinker (N,N′methylene-bis-acrylamide) has been incorporated in the photopolymer to record holograms by pulsed laser exposure. The presence of this component increases the polymerization rate and refractive index modulation. The recording was performed using a holographic copying process. The original was a grating of 1000?lines/mm processed using silver halide sensitized gelatin. First, the effect of the pulse fluence was investigated. When the pulse fluence was optimized, the results obtained using the new composition of material were compared with those using the composition without a crosslinker. Using a pulsed laser at 532?nm the photopolymer without crosslinker presented diffraction efficiencies slightly less than 60%. On the other hand, when the crosslinker was introduced in the photopolymer composition, the diffraction efficiencies achieved were higher than 85%. The non-linearity of the material's response was also studied comparing the energetic sensitivity, diffraction efficiencies and index modulation of gratings recorded with pulsed and continuous laser exposure. This study was performed fitting the angular scan of each grating using Kogelnik's theory.     

Finite-number-of-periods holographic gratings with finite-width incident beams: analysis using the finite-difference frequency-domain method

The effects of finite number of periods (FNP) and finite incident beams on the diffraction efficiencies of holographic gratings are investigated by the finite-difference frequency-domain (FDFD) method. Gratings comprising 20, 15, 10, 5, and 3 periods illuminated by TE and TM incident light with various beam sizes are analyzed with the FDFD method and compared with the rigorous coupled-wave analysis (RCWA). Both unslanted and slanted gratings are treated in transmission as well as in reflection configurations. In general, the effect of the FNP is a decrease in the diffraction efficiency with a decrease in the number of periods of the grating. Similarly, a decrease in incident-beam width causes a decrease in the diffraction efficiency. Exceptions appear in off-Bragg incidence in which a smaller beam width could result in higher diffraction efficiency. For beam widths greater than 10 grating periods and for gratings with more than 20 periods in width, the diffraction efficiencies slowly converge to the values predicted by the RCWA (infinite incident beam and infinite-number-of-periods grating) for both TE and TM polarizations. Furthermore, the effects of FNP holographic gratings on their diffraction performance are found to be comparable to their counterparts of FNP surface-relief gratings.     

Analysis and simulations of two-wave mixing in photopolymer holographic recording media

Abstract

The process of two-wave mixing in photopolymer recording materials was investigated theoretically. The diffraction grating already forms during exposition and it may influence the original interference field distribution through diffraction of waves on the refractive index modulation. In order to show this, Kogelnik's coupled wave theory was extended to demonstrate the possibility of energy transfer from one recording wave to the other. The energy transfer and the intensity distribution during the recording process were systematically analysed depending on the boundary conditions. As a next step, the first harmonic model of the transmission grating recording, based on a simple material model, was implemented and solved. The ratio of the input intensities was found to be a crucial parameter and thus extensive simulations for various ratios of intensities were carried out. Modelling implies that the interference field and the refractive index grating just coincide for equal intensities. For intensities differing from unity they do not overlap themselves during the recording process. It has also turned out that the diffraction efficiency of the recorded grating drops against the case where the effects of two-wave mixing are not considered. The results of our analysis and simulation help give a better understanding of the physics of the recording process and proper adjustment of recording parameters in such applications as optical holography and holographic memories.     

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

Formulas for calculating the diffraction efficiency of gratings recorded in a photorefractive medium are given. The analysis uses coupled-wave theory for photorefractive hologram gratings and takes into account the photorefractive phase shift and fringe-bending effect. General solutions for diffracted (signal) and undiffracted (reference) waves are derived in a closed-form expression. By use of the derived formulas the diffraction efficiency for angle mismatch from the Bragg condition can easily be evaluated as compared with numerical-analysis methods. The diffraction efficiency is also quantified in terms of medium parameters and recording and reconstruction conditions.     

Diffraction properties of volume holographic gratings for an ultrashort pulsed beam with different polarization states readout

The diffraction properties of volume holographic gratings are studied when the gratings are illuminated by an ultrashort pulsed beam with different polarization states. The developed coupled wave theory of Kogelnik is used. Considering the dispersion effect of the grating media, solutions for the diffracted and transmitted intensities, diffraction efficiencies and the bandwidths of the gratings are given in transmission volume holographic gratings and reflection volume holographic gratings. The bandwidths of the gratings are reduced by the dispersion effect of the grating media. They also have different influences on the diffraction of an ultrashort pulsed beam with different polarization states. For different values of the ratio of the spectral bandwidth of the input pulse to that of the grating, the changes of the spectral and temporal distributions of the diffracted intensities, as well as the diffraction efficiencies of the gratings are shown.     

Coherent control of symmetric and asymmetric diffraction grating via relative phase

ABSTRACT

We present a theoretical model to realize the symmetric and asymmetric diffraction grating in a four-level atomic medium. The proposed atomic medium follows a double lambda configuration where four fields interact with it. We get control over symmetric and asymmetric behavior of the diffraction grating by manipulating the relative phase of the fields. Interestingly, the symmetric and asymmetric diffraction grating become prominent when the vortex beam is used instead of the plane wave. Enhanced first, second, and third-order diffraction gratings are achieved via the vortex beam. Further, we find control over asymmetric diffraction gratings by the relative phase of the fields. Coherent control of asymmetric diffraction grating in negative and positive diffracted angles is also achieved via the relative phase.     

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 Efficiency and Angular Selectivity of Volume Phase Holograms Recorded in Photorefractive Materials

The recording and replay of volume phase gratings in photorefractive crystals is investigated for both transmission and reflection geometries. Differential equations are derived and solved for a range of parameters including the length of the crystal, the magnitude, spatial distribution and phase angle of the refractive index modulation, the beam ratio at recording, and the angular range at replay. The recording process is assumed to reach a steady-state limit before replay with a weak probe beam. Solution of the repaly equations is mainly by numerical integration, although analytic solutions are derived for special cases. It is found that in certain cases the diffraction efficiency can be greatly increased by replaying the hologram at an angle different from the recording angle.     

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.     

The influence of the procedure on the dynamic range of bleached silver halide emulsions

Abstract

Bleached holograms are one of the most interesting techniques used to produce phase holograms of high quality on photographic emulsions. Of particular interest is the method of fixation-free rehalogenating bleaching. Using Kogelnik's coupled-wave theory we shall obtain information about the characteristics of the final holograms recorded in bleaching procedures. In this way, it is possible to analyse the particular mechanisms involved in the formation of the holograms. In this paper we study the influence of the different experimental conditions on the refractive index modulation created inside the hologram. It will be demonstrated that it is possible to maintain the values of diffraction efficiency over a wide range of exposures if the correct experimental conditions are chosen. On the other hand, it will also be demonstrated that the use of fixation-free rehalogenating bleaching techniques allows high-refractive-index modulations to be obtained. Moreover, it is possible to attain a theoretical diffraction efficiency of 100%, which is only limited by absorption and scattering.     

Generalized Curvilinear Diffraction Gratings: III . Euclidian diffraction gratings

This paper is the third in a series of papers on the diffraction and imaging properties of generalized curvilinear diffraction gratings. Here we study classes of plane uncrossed diffraction gratings which have the common property that their Fourier transform patterns consist of sharp-line caustics. These types of generalized gratings are referred to as ‘Euclidian diffraction gratings’, because the condition for the formation of line caustics is equivalent to the global vanishing of the gaussian curvature of the grating line pattern function. The analysis is applied to the laser material processing system discussed in our last paper [1].     

Beam splitting of low-contrast binary gratings under second Bragg angle incidence

Beam splitting of low-contrast rectangular gratings under second Bragg angle incidence is studied. The grating period is between lambda and 2lambda. The diffraction behaviors of the three transmitted propagating orders are illustrated by analyzing the first three propagating grating modes. From a simplified modal approach, the design conditions of gratings as a high-efficiency element with most of its energy concentrated in the -2nd transmitted order (~90%) and of gratings as a 1 x 2 beam splitter with a total efficiency over 90% are derived. The grating parameters for achieving exactly the splitting pattern by use of rigorous coupled-wave analysis verified the design method. A 1 x 3 beam splitter is also demonstrated. Moreover, the polarization-dependent diffraction behaviors are investigated, which suggest the possibility of designing polarization-selective elements under such a configuration. The proposed concept of using the second Bragg angle should be helpful for developing new grating-based devices.     

Finite-difference-time-domain analysis of finite-number-of-periods holographic and surface-relief gratings

The total-field-scattered-field formulation of the finite-difference time-domain method (FDTD) is used to analyze the diffraction of finite incident beams by finite-number-of-periods holographic and surface-relief gratings. Both second-order and fourth-order FDTD formulations are used with various averaging schemes to treat permittivity discontinuities and a comparative study is made with alternative numerical methods. The diffraction efficiencies for gratings of several periods and various beam sizes, for both TE and TM polarization cases, are calculated and the FDTD results are compared with the finite-difference frequency-domain (FDFD) method results in the case of holographic gratings, and with the boundary element method results in the case of surface-relief gratings. Furthermore, the convergence of the FDTD results to the rigorous coupled-wave analysis results is investigated as the number of grating periods and the incident beam size increase.     

Efficient holographic recording in novel azo-containing polymer

We report on periodic structures inscribed through holographic recording by both intensity light patterns and polarization patterns in novel azo-containing polymer poly[oxyethylene({6-[4-(biphenyl-4-azo)phenoxy]hexyl}imino) ethylene-carbonylimino(2-methyl-1,3-phenylene)iminocarbonyl] abbreviated as A-TDI. The kinetic of grating recording and diffraction efficiency of recorded gratings using polarized cw Ar⁺ laser light have been measured for s–s, p–p and s–p polarization configurations. The translation grating technique applied during the light self-diffraction process allowed for investigations of phase and amplitude gratings contributions to light diffraction in the studied polymer. Full reversibility of polarization (s–p) grating recording and observation of a half-period structure development during optical erasure process makes this material a good candidate for further studies as a potential material for dynamic holography purposes.     

Spatial frequency filtering in holographic image reconstruction

We show that reconstructing a hologram by its object beam leads to spatial filtering, namely, to the suppression of spatial frequency components that are perpendicular to the plane of the recording beams. We conduct experiments by rotating the transparencies of Ronchi gratings and crossed gratings and measuring the corresponding diffraction efficiencies. Good agreement is found with a theory based on image reconstruction with a beam that does not quite satisfy the Bragg condition.     

Extension du Concept D'espaces de Fourier à la Spectroscopie

This paper is the second in a series [1] of papers on the diffraction and imaging properties of generalized curvilinear diffraction gratings. In this article, a set of generalized ray equations is derived and applied to the calculation of the caustic properties of generalized gratings. No restriction is placed on the form of the ruling pattern or on the shape of the grating surface. For generalized uncrossed and off-axis plane gratings, the ruling pattern may be regarded as a contour map of some abstract surface. In this case, it is shown that the far-field caustic properties of the grating are determined by the regions of vanishing gaussian curvature on the abstract surface. As an application of the theory, a new method for laser material processing is discussed.     

Gratings in a conical diffraction mounting for an extreme-ultraviolet time-delay-compensated monochromator

The conical diffraction mounting in which the direction of incident light belongs to a plane parallel to the direction of the grooves has the unique property of maintaining high diffraction efficiency, even in the extreme-ultraviolet (EUV) region. This property is useful for designing high-throughput time-delay-compensated monochromators for the spectral selection of ultrashort EUV pulses as the high-order harmonics generated by the interaction between an ultrashort laser pulse and a gas jet. The time compensation allows one to exploit the femtosecond scale duration of the harmonics both to have high intensity and to reach an unprecedented temporal resolution for pump and probe experiments. Because two gratings have to be used for time compensation, the high diffraction efficiency becomes an essential requirement, which can be fulfilled by the conical diffraction mounting. Measurements recently accomplished at the Bending Magnet for Emission Absorption and Reflectivity (BEAR) beam line (ELETTRA Synchrotron, Trieste, Italy) for three gratings in the 10-90 nm region are reported here that show a peak efficiency of as much as 0.7 in the first order. A model computing the electromagnetic propagation and the grating efficiency, implemented and tested with the experimental data, permits the study and design of rather complex systems operating in the conical mounting. Basic physical principles and mathematical aspects of the model are discussed here.