Aberrations of diffracted wave fields. II. Diffraction gratings

The Rayleigh-Sommerfeld theory is applied to diffraction of a spherical wave by a grating. The grating equation is obtained from the aberration-free diffraction pattern, and its aberrations are shown to be the same as the conventional aberrations obtained by using Fermat's principle. These aberrations are shown to be not associated with the diffraction process. Moreover, it is shown that the irradiance distribution of a certain diffraction order is the Fraunhofer diffraction pattern of the grating aperture as a whole aberrated by the aberration of that order.     

Evolution process from ghost diffraction to ghost imaging in a lensless imaging system

Ghost diffraction and ghost imaging are investigated in a lensless imaging system. The evolution process from ghost diffraction to ghost imaging is discussed when the object is moved far away from the source in the test arm. The relation of ghost diffraction and imaging is also studied, and it is found that the visibility of ghost imaging is always better than that of ghost diffraction.     

Diffraction,Asymptotics and Catastrophes

First the method of stationary phase is used to develop various semiuniform asymptotic techniques that can be used to calculate the field in regions in which diffraction catastrophes occur. Three types of catastrophes are treated, namely those that take place at a shadow boundary, at a caustic and at a cusp. Then the diffraction of a spherical wave by a circular aperture is considered, and it is shown how the use of standard diffraction theories leads to diffraction integrals that can be evaluated by means of the semi-uniform asymptotic techniques developed for the three diffraction catastrophes mentioned above. Finally, it is pointed out how the asymptotic results provide new physical insight into the diffraction process, as well as offering great savings in computing time.     

J. H. Kulick, J. M. Jarem, R. G. Lindquist, S. T. Kowel, and M. W. Friends are with the Department of Electrical and Computer Engineering, University of Alabama in Huntsville, Huntsville, Alabama 35899

The development and modeling of a liquid-crystal phase grating for real-time diffractive three-dimensional displays are discussed. The system being developed, which is called the ICVision system, utilizes a number of ideas that will result in a rugged, low-power three-dimensional display offering both vertical and horizontal parallax and eventually full color. Fringing fields created between interdigitated electrodes formed on top of VLSI die will induce a diffraction pattern in a thin layer of liquid crystal that will cover the die. A detailed electrostatic and diffraction analysis of liquid-crystal phase-grating regions that will make up the final display is given here. The electrostatic analysis is developed by use of the method of moments. The diffraction analysis is developed by use of rigorous coupled-wave diffraction theory. The numerical results obtrained from the mathematical model are compared with experimental diffraction results from preliminary LCD cells that have been assembled as prototype ICVision devices.     

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.     

Formation of anisotropic diffraction gratings in a polymer-dispersed liquid crystal by polarization modulation using a spatial light modulator

Anisotropic diffraction gratings based on a holographic polymer-dispersed liquid crystal (HPDLC) are realized by interferometric exposure using a spatial light modulator (SLM). The SLM is used in the HPDLC grating formation for anisotropic holographic recordings of two-dimensional polarization states for an incident light beam. The diffraction efficiency for P-polarization and the distinctive ratio of diffraction efficiency in P-polarization to that in S-polarization increases with the signal level applied to the SLM. The resulting volume gratings exhibit diffraction efficiency of more than 60% and a distinctive ratio of diffraction over 100. The microscopic origin of the anisotropic property is investigated by an optical polarizing microscope. The novel characteristics of the anisotropic diffraction properties of HPDLC are applied to an image reconstruction technique.     

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.     

Grazing Diffraction by Volume Phase Gratings

Light diffraction by a volume phase grating with slanted fringes is considered. An analytical solution of the system of second-order coupled wave equations is obtained and applied to the problems of grazing diffraction by a transmission grating. It is shown that a strong backward diffracted wave arises, demonstrating that the grating can behave simultaneously as a transmission and a reflection grating. The diffraction efficiency balance and angular selectivity are analysed.     

The Structure of Fraunhofer Diffraction Patterns of Apertures with N-fold Rotational Symmetry

The moment theorem is used to show that the innermost part of the Fraunhofer diffraction pattern of any real aperture with higher than two-fold rotational symmetry is rotationally invariant. Then a formalism is presented in which aperture transmission-functions are represented by series of Zernike circle polynomials and diffracted field-amplitudes by series of Bessel functions, from which it is easily shown that the diffraction patterns of such apertures consist of regions, contained between well-defined values of the radius, whose rotational symmetries are integral multiples of that of the aperture. The central region, extending from = 0 to , N ( measures the diffraction angle, and N is the degree of rotational symmetry of the aperture) is rotationally invariant, and successive circumjacent regions have progressively higher rotational symmetries. The diffraction patterns of sectoral apertures and of rings of pinholes are derived and shown to exemplify these general conclusions. Finally it is shown how the diffraction patterns of some apertures (‘chiral apertures’) with rotational symmetries but no mirror symmetry can be deduced from the diffraction pattern of a related aperture with mirror symmetries, to which a chiral perturbation is applied.     

Theoretical description of fiber Bragg reflectors prepared by Fresnel diffraction images

The theory of Fresnel diffraction images is applied to Bragg-grating formation in a germanium-doped silica fiber. Fresnel diffraction images arise from the near-field diffraction at a periodic mask. The diffraction images are calculated as a function of the propagation distance for several mask configurations. The average of the diffraction-image intensities is calculated for a single longitudinal repetitive interval, and it is shown that the period of the resulting average intensity field is twice that of the original mask period. In some cases the periodic mask can be predicted for a desired average intensity field by calculation of the magnitude of its Fourier coefficients.     

Diffraction characteristics of 10 fs laser pulses passing through an aperture

Results are presented on experimental and theoretical work performed to compare diffraction phenomena for ultrashort 10 fs pulses and continuous-wave propagation modes illuminating different-sized pinholes and slits. Results demonstrate that 10 fs pulses do not produce high-frequency diffraction like that produced with continuous-wave illumination. The diffraction through a 1 mm pinhole of temporally stretched pulses obtained by using fused silica plates whose frequency spectrum remains the same is compared with those of 10 fs pulses. The overall diffraction intensity profiles are, however, nearly identical in this case. The simulations of diffraction patterns for 100 fs, 10 fs, and 1 fs incident pulse were compared theoretically for different aperture sizes and frequencies. Calculations indicate that the lack of high-frequency diffraction for the mode-locked case is due to the broadband nature of the ultrashort laser pulses; i.e., the distribution of the frequency contained in the pulse ends up washing out when objects are illuminated with pulses of broad frequency content. The results of this work have important application in biomedical imaging and remote imaging applications, to name only a few.     

Diffracted radiance: a fundamental quantity in nonparaxial scalar diffraction theory

Most authors include a paraxial (small-angle) limitation in their discussion of diffracted wave fields. This paraxial limitation severely limits the conditions under which diffraction behavior is adequately described. A linear systems approach to modeling nonparaxial scalar diffraction theory is developed by normalization of the spatial variables by the wavelength of light and by recognition that the reciprocal variables in Fourier transform space are the direction cosines of the propagation vectors of the resulting angular spectrum of plane waves. It is then shown that wide-angle scalar diffraction phenomena are shift invariant with respect to changes in the incident angle only in direction cosine space. Furthermore, it is the diffracted radiance (not the intensity or the irradiance) that is shift invariant in direction cosine space. This realization greatly extends the range of parameters over which simple Fourier techniques can be used to make accurate calculations concerning wide-angle diffraction phenomena. Diffraction-grating behavior and surface-scattering effects are two diffraction phenomena that are not limited to the paraxial region and benefit greatly from this new development.     

Efficient optimization of diffractive optical elements based on rigorous diffraction models.

An efficient optimization strategy for the design of diffractive optical elements that is based on rigorous diffraction theory is described. The optimization algorithm combines diffraction models of different degrees of accuracy and computational complexity. A fast design algorithm for diffractive optical elements is used to yield estimates of the optimum surface profile based on paraxial diffraction theory. These estimates are subsequently evaluated with a rigorous diffraction model. This scheme allows one to minimize the need to compute diffraction effects rigorously, while providing accurate design. We discuss potential applications of this scheme as well as details of an implementation based on a modified Gerchberg-Saxton algorithm and the finite-difference time-domain method. Illustrative examples are provided in which we use the algorithm to design Fourier array illuminators.     

Integral method for echelles covered with lossless or absorbing thin dielectric layers

We make a generalization of the integral method in the electromagnetic theory of gratings to study diffraction by echelles covered with dielectric lossless or absorbing layers. Numerical examples are given that show that, as in the resonance domain, the diffraction efficiency is more complicated than being a simple product of lossless diffraction efficiency curves and plane surface reflectivity.     

Complex-valued Fresnel-transform sampling

VanderLugt [Appl. Opt. 29, 3352 (1990)] presented sampling rates for the amplitude of Fresnel diffraction patterns. These apply to any plane in a coherent optical system. Although these sampling rates represent the amplitude of diffraction patterns accurately, they are not adequate for the retention of complete information in complex-valued Fresnel diffraction patterns. I show this by considering the ability to reconstruct the original input image through backward diffraction of the forward diffraction pattern of such an image. I then extend the VanderLugt sampling techniques such that reliable sampling of the phase of these Fresnel diffraction patterns can also be achieved. The analysis is restricted to lensless optical systems. The new sampling rates are tested with numerical computations of Fresnel diffraction patterns and rigorous scalar diffraction patterns in both forward and backward directions.     

Fraunhofer Diffraction by Koch Fractals

Abstract

Analytical expressions are derived for the complex amplitude in the Fraunhofer diffraction field of an arbitrary Koch fractal with a finite range of self-similarity. Results of the numerical evaluation for the intensity distribution of Fraunhofer diffraction patterns are compared with those obtained experimentally. It is shown that the diffraction pattern of the Koch fractal can be divided into two areas, a central fractal area and a periodic area, and that the former is surrounded by the latter. The existence of the periodic area is a consequence of the finite inner cut-off of the self-similarity of the object fractal. On the other hand, the outer cut-off gives rise to a small core area at the centre of the diffraction pattern.     

Average lattices and aperiodic structures

Statistically averaged lattices provide a common basis to understand the diffraction properties of structures displaying deviations from regular crystal structures. An average lattice is defined and examples are given in one and two dimensions along with their diffraction patterns. The absence of periodicity in reciprocal space corresponding to aperiodic structures is shown to arise out of different projected spacings that are irrationally related, when the grid points are projected along the chosen coordinate axes. It is shown that the projected length scales are important factors which determine the existence or absence of observable periodicity in the diffraction pattern more than the sequence of arrangement.     

Two-dimensionally-periodic diffractive optical elements: limitations of scalar analysis

The range of validity of the scalar diffraction analysis is quantified for the case of two-dimensionally-periodic diffractive optical elements (crossed gratings). Three canonical classes of two-dimensionally-periodic grating structures are analyzed by using the rigorous coupled-wave analysis as well as the scalar diffraction analysis. In all cases the scalar-analysis diffraction efficiencies are compared with the exact diffraction efficiencies. The error in using the scalar analysis is then determined as a function of the grating-period(s)-to-wavelength ratio(s), the minimum feature size, the grating depth, the refractive index of the grating, the incident polarization, and the number of phase levels. The three classes of two-dimensional (2-D) unit cells are as follows: (1) a rectangular pillar, (2) an elliptical pillar, and (3) an arbitrarily pixellated multilevel 2-D unit cell that is representative of more complicated diffractive optical elements such as computer-generated holograms. In all cases a normally incident electromagnetic plane wave is considered. It is shown that the error of the scalar diffraction analysis in the case of two-dimensionally-periodic diffractive optical elements is greater than that for the corresponding one-dimensionally-periodic counterparts. In addition, the accuracy of the scalar diffraction analysis degrades with increasing refractive index, grating thickness, and asymmetry of the 2-D unit cell and with decreasing grating-period-to-wavelength ratio and feature size.     

A fast method of calculating diffraction loss between two facing transducers

A fast method of calculating the diffraction loss between two facing circular ultrasonic transducers of unequal size is presented. This problem is directly applicable for minimization of diffraction loss in acoustic lens design. Graphs for amplitude and phase are presented that can be used to design lenses with the optimal transducer size for minimum diffraction loss. The theory is extended to include the diffraction loss determination in anisotropic materials. The results are in good agreement with previous experimental and theoretical results of the equal-transducer-size case. The effect of diffraction on pulsed excitation is also treated.     

Reconstruction in diffraction imaging

The problem of reconstruction in imaging systems that are modeled using the Helmholtz wave equation (diffraction imaging) is addressed. A spectral analysis of the available diffraction data is presented to help develop algorithms and constraints on a diffraction imaging system's parameters for accurate reconstruction of the desired image. Means of reducing the execution time of these algorithms and their relationship to currently available filtered backpropagation and unified Fourier reconstruction methods are also discussed.