Phase constraint for the waves diffracted by lossless symmetrical gratings at Littrow mount

The energy conservation of grating diffraction is analyzed in a particular condition of incidence in which two incident waves reach a symmetrical grating from the two sides of the grating normal at the first-order Littrow mounting. In such a situation the incident waves generate an interference pattern with the same period as the grating. Thus in each direction of diffraction, interference occurs between two consecutive diffractive orders of the symmetrical incident waves. By applying only energy conservation and the geometrical symmetry of the grating profile to this problem it is possible to establish a general constraint for the phases and amplitudes of the diffracted orders of the same incident wave. Experimental and theoretical results are presented confirming the obtained relations.     

Nonparaxial scalar treatment of sinusoidal phase gratings

Scalar diffraction theory is frequently considered inadequate for predicting diffraction efficiencies for grating applications where lambda/d>0.1. It has also been stated that scalar theory imposes energy upon the evanescent diffracted orders. These notions, as well as several other common misconceptions, are driven more by an unnecessary paraxial approximation in the traditional Fourier treatment of scalar diffraction theory than by the scalar limitation. By scaling the spatial variables by the wavelength, we have previously shown that diffracted radiance is shift invariant in direction cosine space. Thus simple Fourier techniques can now be used to predict a variety of wide-angle (nonparaxial) diffraction grating effects. These include (1) the redistribution of energy from the evanescent orders to the propagating ones, (2) the angular broadening (and apparent shifting) of wide-angle diffracted orders, and (3) nonparaxial diffraction efficiencies predicted with an accuracy usually thought to require rigorous electromagnetic theory.     

Improving resolution in microscopic holography by computationally fusing multiple, obliquely illuminated object waves in the Fourier domain

We present a computational method to increase the effective NA of a holographic microscopy system operating in air. Our optical system employs a reflection Mach-Zender architecture and computational reconstruction of the full complex (phase and amplitude) wavefront. Based on fundamental diffraction principles, different angles of incident illumination result in different diffracted orders of the object wave being imaged. We record, store, and computationally recombine these object waves to expand the spatial frequency response. Experimental results demonstrate an improvement in the effective NA of our system from 0.59 to 0.78.     

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.     

Rigorous theory of the diffraction of Gaussian beams by finite gratings: TE polarization

A rigorous modal theory for the diffraction of Gaussian beams from N equally spaced slits (finite grating) in a planar perfectly conducting thin screen is presented. The case of normal incidence and TE polarization state is considered; i.e., the electric field is parallel to the slits. The characteristics of the far-field diffraction patterns, the transmission coefficient, and the normally diffracted energy as a function of several optogeometrical parameters are analyzed within the so-called vectorial region, where the polarization effects are important. The diffraction pattern of an aperiodic grating is also considered. In addition, one diffraction property known to be valid in the scalar region is generalized to the vectorial region: the existence of constant-intensity angles in the far field when the incident beam wave is scanned along the N slits. The classical grating equation is tested for incident Gaussian beams under several conditions.     

Three-dimensional electromagnetic diffraction of a Gaussian beam by a perfectly conducting half-plane

Formulas are derived for the diffraction of a three-dimensional electromagnetic Gaussian beam by a perfectly conducting half-plane. The beam can be incident from any direction, and the main component of the electric field can point in any direction on the plane of the beam waist. The center of the beam waist is on the edge of the half-plane. The incident beam is constructed as a superposition of plane waves, and the total diffracted field is obtained from a superposition of the diffracted fields that are due to each plane wave. Physical constraints that limit the size and direction of the beam relative to the half-plane are described and incorporated into the theory. The scattered field in the far zone is obtained by asymptotic evaluation of the general formulas. Graphical results for the near-field as well as far-field patterns are presented and discussed.     

Design of binary diffraction gratings of liquid crystals in a linearly graded phase model

We report on a linearly graded phase model for the design of binary diffraction gratings of liquid crystals (LCs) associated with the periodic interfacial effect. The binary nature of the LC grating is produced by use of periodic striped domains in an alternating homeotropic and hybrid geometry. In our graded phase model the diffraction patterns and the diffracted intensities of the LC binary grating is primarily governed by three length scales: the cell thickness and two distortion parameters scaled by the grating period at two domain boundaries. The experimental data agree well with theoretical predictions made in our linearly graded phase model as well as the elastic continuum theory.     

Vector diffraction analysis of optical disk readout

The optical disk readout signals from ROM disks are presented by use of a rigorous three-dimensional vector diffraction method. The optical disk is modeled as a crossed metal grating without restriction on the form of the information marks, and the permittivity of the metal is taken into account. The diffracted field from the disk is obtained by means of decomposing the focused incident beam into a spectrum of plane waves and then calculating the diffracted plane waves for each respective incident component. The readout signal is obtained by integration of the energy-flux density of the diffracted field according to the detection scheme of the optical disk system. A typical digital versatile disk (DVD) system is applied with this theory, and the result is far from that of scalar diffraction theory.     

Polarization properties of inversely twisted nematic liquid-crystal gratings

Based on the Jones matrix representation of twisted nematic liquid crystals (LC's), we have carried out a theoretical analysis of the polarization properties of inversely twisted nematic (ITN) LC gratings. Some interesting polarization behaviors are expected in the ITN LC grating. When a linearly polarized light parallel or perpendicular to the grating direction is incident on the ITN LC grating, the diffracted light in the 0th order is linearly polarized with the same polarization direction of incident light, while the diffracted light in high orders is linearly polarized perpendicular to that of incident light. Using a multirubbing alignment technique, we have practically prepared an ITN LC grating with ?45 degrees inversely twisted structures. The experimental investigations of the optical characteristics of the ITN LC grating demonstrate agreement with theoretical expectations.     

Effect of spurious diffraction orders in arbitrary multifoci patterns produced via phase-only holograms

We analyze the effect of spurious diffraction orders when generating functional multifoci patterns produced by illuminating a phase-only hologram with a single Gaussian beam. Using a practical device for encoding a hologram generates an undesirable zero order and high-diffraction orders at the Fourier plane. This translates to the fact that a significant fraction of the incident light does not necessarily convert to functional multifoci patterns. In most applications, the zero order can be avoided by generating foci patterns shifted off the optical axis, which further increases the amount of light distributed to spurious high-diffraction orders owing to the reduction of light directed to the desired foci pattern. We analyze the amount of light dispersed to spurious orders and show that these unwanted orders can be a major limiting factor for most applications based on arbitrary multifoci patterns.     

Fringe pattern of the field diffracted by axicons

The far-field intensity pattern of laser beams diffracted by axicons is extensively characterized both theoretically and experimentally. The regular structure of the pattern, consisting of high-contrast fringes, is explained. The experimental results have been interpreted by representing the diffracted field as generated by an extended virtual source shaped as a circle centered on the optical axis of the incident laser beam. The simulations include modifications to the diffraction pattern arising from the laser radiation diffraction limit at the axicon tip, and they reproduce well the measured intensity profile at different distances from the axicon.     

Propagation characteristics of a diffracted M2 beam

The propagation characteristics of a beam diffracted by a circular aperture are investigated. The beam-quality factor M2 defined by an 86.5% power-content radius is given theoretically and experimentally as a function of the truncation ratio. It is found that the theoretical limit of M2 is 2.37 times as great as that of an incident beam as the truncation ratio approaches 0. For a weakly diffracted beam a simple formula giving M2 is derived. Although M2 does not increase much with diffraction, the influence of diffraction should be taken into account in beam brightness.     

Computer Reconstruction of a Random Rough Surface in the Presence of Higher Diffraction Orders

Abstract

The profile of a random rough surface (RRS), whose mean roughness R_a is greater than the light wavelength, is visualized by computer processing. The surface is presented as a sum of sinusoidal gratings. The light diffracted from this surface field is registered by a photodiode array. The second and third diffraction orders from each grating are taken into account in computer processing of the diffracted field and the mixing field–the field obtained at the mixing of the reference and the diffraction fields. The criterion for taking into account higher diffraction orders is the asymmetry of the diffraction pattern to the left and to the right relative to the central peak (the field of zero diffraction orders obtained from each grating) The number of the diffraction orders higher than the first is defined from the average intensity distribution between the central peak and the diffraction orders to the left and to the right at arbitrary light wavelength. The surface profile is reconstructed by a computer program and the mean roughness R_a is calculated. The obtained value of R_a is in satisfactory agreement with that measured by the contact pin method.     

Comprehensive focusing analysis of various Fresnel zone plates

A series-form expression for the individual diffracted field of a general annular ring is derived from the Rayleigh-Sommerfeld diffraction integral. It can be used for the accurate and fast simulation of any diffractive focusing element composed of concentric transparent rings. We present a comprehensive analysis, based on the leading term and the linear superposition principle, of the focusing performances of various Fresnel zone plates. Many problems, such as the equivalent aperture function, the diffraction efficiency, the focal spot pattern, the suppression of higher orders and the appearance of "fractional orders," and the explanation for the appearance of Fraunhofer diffraction patterns, are analytically investigated in detail. Because of the great similarity between Fresnel zone plates and multilevel diffractive lenses, most of the obtained results are also applicable to multilevel diffractive lenses.     

Design and fabrication of a high-efficiency extreme-ultraviolet binary phase-only computer-generated hologram

As the development of extreme-ultraviolet (EUV) lithography progresses, interest grows in the extension of traditional optical components to the EUV regime. The strong absorption of EUV by most materials and its extremely short wavelength, however, make it very difficult to implement many components that are commonplace in the longer wavelength regimes. One such component is the diffractive optical element used, for example, in illumination systems to efficiently generate modified pupil fills. The fabrication and characterization of an EUV binary phase-only computer-generated hologram is demonstrated, allowing arbitrary far-field diffraction patterns to be generated. Based on reflective architecture, the fabricated device is extremely efficient. Based on an identically fabricated null hologram, the absolute efficiency into one diffracted order of 22% has been demonstrated. In the case where axially symmetric diffraction patterns are desired (both positive and negative diffraction orders can be used), the efficiency can be twice as high.     

Binary blazed reflection gratings

A reflection grating with a binary surface profile is presented that has high diffraction efficiency. The measured intensity for the + 1st diffracted order was 77%. The binary grating is composed of a minilattice with feature sizes comparable with the wavelength of the incident light. The overall structure is designed in such a way that it imitates a conventional blazed grating. The grating also has interesting polarization properties. The main part of the TE-polarized light is diffracted into the 1st diffracted order, and most of the TM-polarized light remains in the 0th diffracted order. The measurements of the grating are compared with rigorous diffraction theory and found to be in reasonable agreement.     

Spectral properties of multiorder diffractive lenses

Diffractive lenses have been traditionally designed with the first diffracted order. The spectral characteristics of diffractive lenses operating in higher diffracted orders differ significantly from the first-order case. Multiorder diffractive lenses offer a new degree of freedom in the design of broadband and multispectral optical systems that include diffractive optical elements. It is shown that blazing the surface-relief diffractive lens for higher diffraction orders enables the design of achromatic and apochromatic singlets. The wavelength-dependent optical transfer function and the associated Strehl ratio are derived for multiorder diffractive lenses. Experiments that illustrate lens performance in two spectral bands are described, and the results show excellent agreement with the theoretical predictions.     

Rib Waveguide Theory by the Spectral Index Method

The geometric theory of diffraction is a well known and very successful asymptotic technique in electromagnetics for the evaluation of scattered fields from objects that are large compared to the wavelength. Our recent work has produced approximate diffraction coefficients for diffracted rays from edges and corners in planar conducting or dielectric surfaces. These diffraction coefficients are tested here by comparing theoretical predictions with experimental measurements made at 30 GHz of the diffracted field from a conducting quarter plane. Very good agreement is found between this asymptotic approach and experimental results for a linearly polarised incident field, with the E field aligned with one of the edges of the quarter plane. The incident field direction is varied from 5° to 54° to the normal to the quarter plane and the total field is measured over a plane behind the target. Average agreement for intensity values is found to be around 95%.     

Design and construction of linear laser encoders that possess high tolerance of mechanical runout

A linearly diffracted laser encoder that has high tolerance of head-to-scale misalignment and a high signal-to-noise ratio is described. The preservation of parallelism between the incident and the diffracted beams, which can be attributed to a built-in folded 1x telescope, allows for the high alignment tolerance. It can be shown that, by coupling this newly developed circular polarization interferometer configuration with grating scale geometry optimization, one can eliminate the problems associated with signal distortion that arise from various efficiencies of the p- and the s-polarized light beams and obtain a high signal-to-noise ratio. Both theoretical and experimental results are presented to confirm the improved results and performance.     

Some properties of the optical resonances in a single subwavelength slit

Diffraction of TM-polarized waves by a slit in a thick screen of infinite conductivity is treated. The case of an arbitrary incident beam wave is considered. We study the resonances that appear when the wavelength of the incident beam wave is larger than the slit width, i.e., the subwavelength regime where a one-mode model for the slit can be considered. High anomalous values (resonances) of the transmission coefficient, the angular diffracted energy, and the magnetic field within the slit are analyzed. A simple linear relationship to determine the resonant wavelengths is proposed. We show that the transmission coefficient, the normal diffracted energy, and the magnetic field within the cavity are linear functions of the resonant wavelength and the thickness of the screen. Additionally and surprisingly, we reveal that under certain conditions the incident beam wave via the diffraction can give a suppressed light transmission; i.e., a minimum in the transmission is obtained where a maximum is expected.