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.     

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.     

Transmittance analysis of diffraction phase grating

In order to accurately analyze and design the transmittance characteristic of a diffraction phase grating, the validity of both the scalar diffraction theory and the effective medium theory is quantitatively evaluated by the comparison of diffraction efficiencies predicted from both simplified theories to exact results calculated by the rigorous vector electromagnetic theory. The effect of surface profile parameters, including the normalized period, the normalized depth, and the fill factor for the precision of the simplified methods is determined at normal incidence. It is found that, in general, when the normalized period is more than four wavelengths of the incident light, the scalar diffraction theory is useful to estimate the transmittance of the phase grating. When the fill factor approaches 0.5, the error of the scalar method is minimized, and the scalar theory is accurate even at the grating period of two wavelengths. The transmittance characteristic as a function of the normalized period is strongly influenced by the grating duty cycle, but the diffraction performance on the normalized depth is independent of the fill factor of the grating. Additionally, the effective medium theory is accurate for evaluating the diffraction efficiency within an error of less than around 1% when no higher-order diffraction waves appear and only the zero-order waves exist. The precision of the effective medium theory for calculating transmittance properties as a function of the normalized period, the normalized groove depth, and the polarization state of incident light is insensitive to the fill factor of the phase grating.     

Examination of the +1, −1 Surface Plasmon Mini-gap on a Gold Grating

Abstract

An energy gap in the excitation of surface plasmons is found for light at normal incidence to a gold grating. This gap occurs at the crossing of the plus and minus first order surface plasmons. It arises directly as a consequence of distortion of the grating from sinusoidality, the first harmonic of the grating providing coupling between the plus and minus one orders. Experiments have been performed using both wavelength scans, where at a fixed angle of incidence the wavelength of excitation is varied, and angle of incidence scans, where for a fixed wavelength the angle of incidence is varied a few degrees either side of normal to the grating. By fitting the angular dependent reflectivity scans using grating modelling theory the gold grating is characterized at all wavelengths. This then allows a detailed comparison of the theoretical dispersion curve with that obtained experimentally. The agreement for both p-polarized light (for angle dependence with the plane of incidence normal to the grating grooves) and for s-polarized light (angle dependence with the plane of incidence perpendicular to the grating grooves) is excellent. An apparent momentum gap in the lower energy branch of the dispersion curve, attributed to the loss of coupling strength, is found to move to the upper branch if the grating profile is inverted.     

A Simple Model for a Microrough Diffraction Grating That Predicts Diffuse Light Bands

Abstract

A real diffraction grating is simulated by a plane boundary with a coordinate-dependent surface impedance. This surface impedance is constructed as the sum of two complex functions: a perfectly periodic function representing a grating without defects, plus a randomly varying, spatially localized function that represents the microroughness or defects of a real grating. This alternative scattering problem is solved in a rigorous electromagnetic manner for two polarizations of the incident field. Our results show that when the magnetic field is parallel to the grooves the scattered light pattern exhibits intensity maxima. The behaviour of these peaks reproduces that reported for the diffuse light bands observed in the spectrum of a real grating.     

Estimation method for the light extraction efficiency of light-emitting elements with a rigorous grating diffraction theory

The light extraction efficiency of a light-emitting element with microstructured surface is analyzed with a rigorous grating diffraction theory. The grating theory reveals an improvement of extraction efficiency due to diffraction of light by the surface microstructure. The simulation results show that the improvement of extraction efficiency is due mainly to the reflected diffraction rather than to the transmitted diffraction. A part of total-internal-reflection light is diffracted into directions at less than the critical angle. Extraction efficiency is improved by multiple reflection and diffraction of light in a high-refractive-index layer. We propose a simple design method for an efficient surface microstructure from the viewpoint of reflected diffraction.     

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.     

Nonpolarizing guided-mode resonant grating filter for oblique incidence

A new type of guided-mode resonant grating filter is described. The filter is independent of polarization state for oblique incidence. The filter has a crossed grating structure, and the plane of incidence on the filter contains the symmetric axis of the grating structure. Theoretical considerations and numerical calculations using two-dimensional rigorous coupled-wave analysis show that a rhombic lattice structure is suitable to such filters. In this configuration an incident light wave is diffracted into the waveguide and is divided into two propagation modes whose directions are symmetric with respect to the plane of incidence. In particular, when the propagation directions of the two modes are perpendicular to each other, the fill factor of grating structure can be approximately 50%. The filter was designed for an incident angle of 45 degrees. Tolerances of setting errors and fabrication errors for this filter were estimated by numerical calculations.     

Rectangular surface-relief transmission gratings with a very large first-order diffraction efficiency (~95%) for unpolarized light

Computer optimization shows that the first-order diffraction efficiency of a lossless-transmission surface-relief grating with a rectangular surface profile can be made very large (~95%) simultaneously for light of TE and TM polarizations incident near the Bragg angle by the proper choice of the fill factor. The case for visible light incident close to the Bragg angle on unslanted gratings with periodicities corresponding to Bragg angles of 30 degrees , 37.5 degrees , and 45 degrees is presented. The refractive index of the grating material was chosen in the range between 1.2 and 2.     

High transmission efficiency for surface plasmon resonance by use of a dielectric grating

The efficiency of the transmission of surface plasmon waves by use of a dielectric diffraction grating is discussed. The Kretschmann device allows us to obtain a surface plasmon resonance that consists of an absorption peak in the reflection spectrum. When surface plasmon resonance occurs, the TM-polarization mode of the incident electromagnetic wave is neither transmitted nor reflected. The procedure to transform an 4bsorption peak into a transmission peak is described. Transmittivity of 68% is obtained for a simple structure that consists of a thin-film layer of Ag coated on a volume diffraction grating and embedded between two dielectric media. The results presented herein were obtained by numerical simulations that were carried out by use of an algorithm based on the rigorous coupled-wave theory.     

Distribution of electric field and energy flux around the cracks on the surfaces of Nd-doped phosphate glasses

We simulate and calculate numerically the electromagnetic field and energy flux around a surface crack of an Nd-doped phosphate laser glass by using the finite-difference time-domain method. Because of a strong interference between the incident wave and the total internal reflections from the crack and the glass surface, the electric field is redistributed and enhanced. The results show that the electric-field distribution and corresponding energy flux component depend sensitively on the light polarization and crack geometry, such as orientation and depth. The polarization of the incident laser beam relative to the crack surfaces will determine the profile of the electric field around the crack. Under TE wave incidence, the energy flux peak is always inside the glass. But under TM wave incidence, the energy flux peak will be located inside the glass or inside the air gap. For both incident modes, the light intensification factor increases with the crack depth, especially for energy flux along the surface. Because cracks on the polished surfaces are the same as the roots extending down, the probability for much larger intensification occurring is high. The results suggest that the surface laser-damage threshold of Nd-doped phosphate may decrease dramatically with subsurface damage.     

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.     

Linear focusing by a plane grating with curved grooves

We study the field diffracted by a plane grating with curved (parabolic) grooves. We will demonstrate that when a monochromatic plane wave is incident on a grating with parabolic grooves the diffracted field has a focal line whose position depends on the curvature radius of the parabolas and the incidence angle of the light onto the grating. The effect described has potential applications in grating-based devices for focusing light without requiring any additional optics.     

Comparison of the calculated and the measured efficiencies of a normal-incidence grating in the 125-225- A wavelength range

The efficiency of a diffraction grating was measured near normal incidence in the 125-225-A wavelength range with synchrotron radiation. The grating pattern had 2400 grooves/mm and was recorded on a concave fused-silica blank by a holographic technique. The grooves were shaped by ion-beam etching to provide a facet with a blaze angle of 2.5 degrees as determined by atomic force microscopy. Because of the characteristics of the etching process the groove profile was approximately triangular, with the other facet inclined at an angle of 5.5 degrees to the surface. The measured efficiency was compared with the efficiency calculated by a computer program, small enough to run on a personal computer, that solved the periodic boundary-value problem corresponding to electromagnetic radiation incident on a diffraction grating with finite conductivity. The calculation was based on the nominal groove profile that was determined by atomic force microscopy. The measured and the calculated efficiencies were in good agreement. This investigation indicates that the diffraction efficiency of a normal-incidence grating can be calculated in the soft-x-ray region with a personal computer.     

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.     

Coupling of light into nanowire arrays and subsequent absorption

We present a theoretical study of the absorption of light in periodic arrays of InP nanowires. The absorption in the array depends strongly on the diameter and the length of the nanowires, as well as the period of the array. Nanowires of a length of just 2 microm are able, after an appropriate choice for the other parameters, to absorb more than 90% of the incident energy of TE and TM polarized light, with photon energies almost all the way down to the band gap energy and an incidence angle up to 50 degree. This high total absorption arises from a good coupling of the incident light into the nanowire array at the top interface between air and the array and absorption inside the array before the light reaches the interface between the nanowires and the substrate. We find that for a given photon energy there exists a critical nanowire diameter above which a dramatic increase in the absorption occurs. The critical diameter decreases for increasing photon energies, and is directly related to the dispersion of waveguiding modes in single isolated nanowires. A characterization showed that the absorption characteristics of the nanowire arrays are very promising for photovoltaic applications.     

Anomalous spatial modifications of beams diffracted by two-dimensional periodic media

By using a rigorous plane-wave representation, we examine the diffracted fields generated by a Gaussian beam incident onto the planar upper boundary of a 2D periodic structure. We first determine a geometric profile for every diffracted beam by neglecting the amplitude variation of its plane-wave spectrum. We then account for the spectral variation and show that, with respect to that geometric profile, every actual diffracted beam exhibits spatial modifications in the form of 2D lateral displacements, focal shifts, angular deviations, and beam-width alterations. These effects are relatively large if the incidence conditions tend to generate grating resonances. The magnitudes of the beam modifications are illustrated by using a canonic grating model that consists of a planar surface whose impedance varies sinusoidally along its two orthogonal directions. We also develop accurate analytical expressions for the spatial modifications by expressing the spectral amplitude functions in terms of Padé approximants. We thus find that the 2D spatial effects exhibit greater complexity and include features that are absent in previously reported cases involving 1D periodic surfaces.     

Pound-Drever-Hall error signals for the length control of three-port grating coupled cavities

Gratings enable light coupling into an optical cavity without transmission through any substrate. This concept reduces light absorption and substrate heating and was suggested for light coupling into the arm cavities of future gravitational wave detectors. One particularly interesting approach is based on all-reflective gratings with low diffraction efficiencies and three diffraction orders (three ports). However, it was discovered that, generally, three-port grating coupled cavities show an asymmetric resonance profile that results in asymmetric and low quality Pound-Drever-Hall error signals for cavity length control. We experimentally demonstrate that this problem is solved by the detection of light at both reflection ports of the cavity and the postprocessing of the two demodulated electronic signals.     

Phase and Amplitude Behaviour in Conical Diffraction Near the Resonant Excitation of Surface Plasmons

Abstract

The efficiencies and phase shifts of light specularly reflected by corrugated metallic gratings in conical mounting are studied by means of rigorous formalisms based on conformal and non-conformal transformations. The reflected field is separated into components co- and cross-polarized with respect to p- and s-polarized incident waves. The efficiency of these components and their relative phase shifts are calculated as functions of the angles of incidence in regions where coupling between light and surface plasmons occurs. The sensitivity of the efficiency of p–s conversion to changes in grating geometry is tested by comparing results obtained for cycloidal, sinusoidal and triangular profiles with different groove heights.     

Reduction of polarization-induced artifacts in grating-based spectrometers

An optical device that converts unpolarized light into a single polarization state is described. The device is based on a polarizing beam splitter that separates the two polarization directions. The beam splitter is combined with two pairs of equilateral prisms that are used to collimate the two beams in terms of both propagation and polarization directions. When it is used in combination with a blazed diffraction grating, this device is shown to effectively remove the polarization dependence of the first-order diffracted power. The device has an insertion loss of approximately 14% for purely s-polarized light. However, for unpolarized light incident upon the two gratings studied here, the increased throughput of the p-polarized component leads to an average relative gain in overall efficiency of 13%-19%, depending on the grating. In collimating the two polarization directions, the device may cause a reduction in spectral resolution for a rectangular entrance slit. As a result, the device is more likely to find use in spectrometers that have a circular aperture, such as that provided by an optical fiber.