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.     

Design of retrodiffraction gratings for polarization-insensitive and polarization-sensitive characteristics by using the Taguchi method

We present the design of retrodiffraction gratings that utilize total internal reflection (TIR) in a lamellar configuration to achieve high performance for both TE and TM polarized light and polarization-sensitive performance for gratings behaving as polarizer filters; the design was based on rigorous coupled wave analysis (RCWA) and the Taguchi method. The components can thus be fabricated from a single dielectric material and do not have to be coated with a metallic or dielectric film layer to enhance the reflectance. The effects of the structural and optical parameters of lamellar gratings were investigated, and the TIR gratings in a lamellar configuration were structurally and optically optimized in terms of the signal-to-noise ratio (S/N) and a statistical analysis of variance (ANOVA) of the refractive index, grating period, filling factor, and grating depth as control factors and the estimated efficiency by RCWA as a noise factor. For more accurate robustness, a two-step optimization process was used for each purpose. For TIR gratings designed to perform similarly for TE and TM incident polarization, the -1st-order efficiencies were estimated to be up to 92.0% and 88.5% for TE and TM polarization, respectively. Also, for the TIR gratings designed to achieve polarization-sensitive performance when behaving as a polarizer filters, the -1st-order diffraction efficiencies for TE and TM polarization were estimated to be up to 95.5% and 2.7%, respectively. From these analysis results, it was confirmed that the Taguchi method shows feasibility for an optimization approach to a technique for designing optical devices.     

Optimum parallel-face slanted surface-relief gratings

Using a combination of rigorous coupled-wave analysis and simulated annealing, parallel-face slanted surface-relief gratings (PFSSRGs) are optimized. For substrate-mode optical interconnects, profiles are presented for both polymer and silicon PFSSRGs for both TE and TM polarizations at normal incidence with grating periods designed to give a 45 degrees output angle in the negative-first forward-diffracted order. The resulting diffraction efficiencies range from 70% to 99%, with a majority of the optimized profiles yielding over 90%. Optimized polymer profiles for TE and TM polarizations exhibit similar high diffraction efficiencies, but the TM profiles generally require greater groove depths. Silicon profiles optimized for TM polarization have greater diffraction efficiencies than those for TE polarization. Profiles that can feasibly be fabricated are identified, and sensitivities to groove depth, filling factor, slant angle, and incident angle are shown to be modest.     

Measurement of duty cycles of metal grating masks formed on dielectric substrates

A nondestructive method for measuring the duty cycles of metal grating masks formed on top of dielectric substrates is proposed. For a near-normal angle of incidence, the zeroth diffracted order transmission efficiency curves for both TE and TM polarized probe lights, as a function of duty cycles, behave linearly in the duty cycle ranging from 0 to 1. By comparing the measured efficiencies, or the ratio of zeroth-order transmission efficiency for TM polarization to that for TE polarization, with that of the rigorous-coupled wave analysis (RCWA) method for a fixed grating period and depth, one can determine the duty cycle of the grating. By selecting the probe light appropriately, the measurement errors originating from deviations of the incident angle and grating depth can be negligible. This method is applicable for all metal gratings, which are not easy to measure nondestructively due to fine grooves smaller than the wavelength. This method is simple, accurate, nondestructive, and low-cost. The results of experimental verification are presented and show excellent agreement with scanning electron microscope images.     

Design of transmission blazed binary gratings for optical limiting with the form-birefringence theory

The structure of transmission blazed binary gratings for optical limiting is designed with the form-birefringence theory. This kind of grating has subwavelength features, can imitate the transmission blazed grating effectively, and has higher efficiencies than a transmission blazed grating with a subwavelength structure. The diffraction efficiencies are calculated and analyzed. For the normal incident light with 10.6 microm wavelength, the transmissivities for the designed grating at 0 degrees deviation angle for TE and TM polarizations are 0.05% and 5.09%, respectively, which are basically identical to the results of the finite-difference time-domain method. The diffraction efficiencies of the first transmitted order for TE and TM polarizations are 93.95% and 83.88%, respectively.     

Polarization dependence of diffraction gratings that have total internal reflection facets

The total internal reflection (TIR) grating is an integrated optical diffraction grating designed to achieve high efficiency for the retrodiffracted order by use of total internal reflection twice within a groove of the grating rather than by use of metalized grooves. Numerical calculations are presented for both TE and TM polarizations of incident light. When the TIR grating was used in the -mth-order Littrow mount with m > 13, the diffraction efficiency was found to decrease linearly with 1/m. The polarization dependence of the retrodiffraction efficiency exceeds 3 dB for TIR gratings formed in silica glass (n = 1.5) but is very small for gratings with InP-based technology (n = 3.2).     

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.     

Effective grating theory for resonance domain surface-relief diffraction gratings

An effective grating model, which generalizes effective-medium theory to the case of resonance domain surface-relief gratings, is presented. In addition to the zero order, it takes into account the first diffraction order, which obeys the Bragg condition. Modeling the surface-relief grating as an effective grating with two diffraction orders provides closed-form analytical relationships between efficiency and grating parameters. The aspect ratio, the grating period, and the required incidence angle that would lead to high diffraction efficiencies are predicted for TE and TM polarization and verified by rigorous numerical calculations.     

Light diffraction from rough gratings

Two holographic diffraction gratings with very similar parameters, designed for maximum performance inTM polarization, are investigated, their diffraction efficiency measured, an electron microscopic picture of their surface observed, and numerical simulation of light diffraction done with the surface roughness taken into account. It is demonstrated numerically that a small-scale roughness imposed on the grating surface could increase significantly both scattering and absorption from the surface, this influence being greater in the TM case. A very good coincidence between the numerical and experimental data is obtained in TE polarization, whereas in TM polarization only a qualitative agreement exists.     

Optimization of sawtooth surface-relief gratings: effects of substrate refractive index and polarization

The effect of the refractive index of the substrate together with the incident polarization on the optimization of sawtooth surface-relief gratings (SRGs) is investigated. The global optimum diffraction efficiencies of the -1st forward-diffracted order of sawtooth SRGs are 63.3% occurring at n2=1.47 for TE polarization and 73.8% occurring at n2=2.88 for TM polarization. Incident TE polarization has higher optimum diffraction efficiency than TM polarization for all n2<1.85. In contrast, TM polarization has higher optimum diffraction efficiency than TE polarization for all n2>1.85. A polymer (n2=1.5) optimum sawtooth SRG exhibits 62.6% efficiency for TE polarization. A silicon (n2=3.475) optimum sawtooth SRG exhibits 68.6% efficiency for TM polarization. These sawtooth SRGs are compared to right-angle-face trapezoidal SRGs. It is found that the optimum profiles of right-angle-face trapezoidal SRGs have only very slightly increased efficiencies over sawtooth SRGs (0.04% for TE and 0.55% for TM).     

光栅衍射特性的耦合波分析、计算与讨论

本文从麦克斯韦方程组及电磁场边界条件出发,推导了广泛应用于各类光栅衍射问题的矢量分析方法--严格的耦合波分析方法.针对光栅的衍射特性,编写了基于严格的耦合波分析方法的计算程序,并以TE模情形为例对光栅的衍射效率和收敛性作了数值计算.结果表明,当谐波数不断增加,即便对于厚光栅(d/λ>10)情形,光栅的衍射效率仍将收敛于某一确定值.     

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.     

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.     

Volume holographic grating couplers: rigorous analysis by use of the finite-difference frequency-domain method

Two configurations of volume holographic grating couplers are rigorously analyzed by means of the finite-difference frequency-domain method (FDFD) for both TE and TM polarizations and for 0- and 45-deg output coupling. The two configurations depend on the position of the grating coupler, which can be placed either in the film or in the cover waveguide region. The FDFD results are compared with those obtained by the rigorous coupled-wave analysis in conjunction with the leaky-mode approach (RCWA-LM). Because the FDFD method is a rigorous solution of the Maxwell equations, it simulates the VHGC configuration and takes into account the waveguide-coupler discontinuity effects as well as the multimode excitation and interference effects, all of which are neglected by the traditional RCWA-LM.     

Optimization of anisotropically etched silicon surface-relief gratings for substrate-mode optical interconnects

The optimum profiles of right-angle-face anisotropically etched silicon surface-relief gratings illuminated at normal incidence for substrate-mode optical interconnects are determined for TE, TM, and random linear (RL) polarizations. A simulated annealing algorithm in conjunction with the rigorous coupled-wave analysis is used. The optimum diffraction efficiencies of the -1 forward-diffracted order are 37.3%, 67.1%, and 51.2% for TE-, TM-, and RL-polarization-optimized profiles, respectively. Also, the sensitivities to grating thickness, slant angle, and incident angle of the optimized profiles are presented.     

Polarization-independent wideband mixed metal dielectric reflective gratings

A polarization-independent wideband mixed metal dielectric grating with high efficiency of the -1st order is analyzed and designed in Littrow mounting. The mixed metal dielectric grating consists of a rectangular-groove transmission dielectric grating on the top layer and a highly reflective mirror composed of a connecting layer and a metal film. Simplified modal analysis is carried out, and it shows that when the phase difference accumulated by the two propagating modes is odd multiples of π/2, the diffraction efficiency of the -1st order will be high. Selecting grating depth and duty cycle for satisfying the phase difference condition for both TE (electric field parallel to grooves) and TM (magnetic field parallel to grooves) polarizations, a polarization-independent high-efficiency grating can be designed. Using rigorous coupled-wave analysis and a simulated annealing algorithm, geometric parameters of the reflective grating are exactly obtained. The optimized grating for operation around a wavelength of 800 nm exhibits diffraction efficiencies higher than 90% for both TE and TM polarizations over a 120 nm wavelength bandwidth. The simplified modal analysis can be applied in other types of reflective gratings if the top layer is a dielectric transmission grating.     

The Properties of Holographic Overlap Gratings

The analysis of a holographic beam expander which can, under perfect lossless conditions, convert a plane input beam into a plane diffracted beam, with a specified diffraction efficiency and beam width expansion ratio, is presented. Using solutions of the coupled-wave differential equations of [2], the effects of average loss, and both constant and spatially dependent dephasing are studied. These solutions are also used to compare the properties of overlap gratings with Kogelnik's [1] results for infinite gratings, and to extend the analysis of [4] to lossy, dephased cases. The maximum possible diffraction efficiency of an absorption overlap grating is shown to depend on its shape and also to lie between the values for Kogelnik's reflection and transmission cases.     

Simple measuring technique for the diffraction efficiency of slanted volume gratings at various wavelengths

To measure diffraction efficiencies of gratings as a function of wavelength, it is necessary to have quasi-monochromatic light sources of various wavelengths. We propose a method to measure the wavelength dependence of the grating diffraction efficiency by using a quasi-monochromatic light source. This method of estimating the real diffraction characteristics of the gratings for various wavelengths is very useful and simple. First the diffraction efficiency of the grating as a function of various incident-beam angles of monochromatic light is measured, then, using these data, we can obtain the diffraction efficiencies for various wavelengths of the same incident angle of light by virtue of a mathematical-conversion method. The mathematical-conversion results for two laminated differently slanted angle gratings of the same volume grating period are in good agreement with the experimental ones.     

Comparison of beam propagation method and rigorous coupled-wave analysis for single and multiplexed volume gratings

Multiplexed holographic structures have been suggested to provide large capacity and parallel access as three-dimensional storage media. One of the most widely used techniques in the literature for analyzing such structures has been the coupled-wave analysis and its variations. Another approach that is becoming increasingly popular because of the ease with which it can be implemented is the beam propagation method (BPM). The BPM is quantitatively compared with the rigorous coupled-wave analysis for the cases of single and multiplexed gratings. Normal and off-normal incidence as well as TE and TM polarizations are considered for single (slanted and unslanted) and multiplexed gratings. It is shown that the BPM, even in its most rudimentary form, is a powerful and accurate calculational method that is especially suited for analyzing the many multiplexed grating diffraction problem.     

Development of multilayer laminar-type diffraction gratings to achieve high diffraction efficiencies in the 1-8 keV energy region

W/C and Co/SiO(2) multilayer gratings have been fabricated by depositing a multilayer coating on the surface of laminar-type holographic master gratings. The diffraction efficiency was measured by reflectometers in the energy region of 0.6-8.0 keV at synchrotron radiation facilities as well as with an x-ray diffractometer at 8.05 keV. The Co/SiO(2) and W/C multilayer gratings showed peak diffraction efficiencies of 0.47 and 0.38 at 6.0 and 8.0 keV, respectively. To our knowledge, the peak efficiency of the W/C multilayer grating is the highest measured with hard x rays. The diffraction efficiency of the Co/SiO(2) multilayer gratings was higher than that of the W/C multilayer grating in the energy range of 2.5-6.0 keV. However, it decreased significantly in the energy above the K absorption edge of Co (7.71 keV). For the Co/SiO(2) multilayer grating, the measured diffraction efficiencies agreed with the calculated curves assuming a rms roughness of approximately 1 nm.