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.     

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.     

Fabrication and simulation of diffractive optical elements with superimposed antireflection subwavelength gratings

With the aim of reducing surface reflections and increasing the diffraction efficiency we investigated the superposition of subwavelength phase gratings onto blazed phase gratings. With direct-write electron-beam lithography bare blazed gratings and blazed gratings carrying subwavelength gratings were fabricated and their optical performances compared. For TE polarization the subwavelength-carrying gratings showed a maximum diffraction efficiency of 90.6%, whereas the corresponding maximum value for the bare grating was 86.3%. The experiment was simulated with rigorous diffraction 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.     

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.     

Deterministic design of binary phase-only blazed grating with subwavelength features under limitation on spatial resolution of fabrication technique

We propose a deterministic design method of a blazed grating consisting of a binary grating with subwavelength structures for a multilevel phase modulation. The feasible shapes of binary subwavelength microstructures are restricted to a few kinds of surface profiles by constraints in an actual fabrication technique. The relationship between the feasible shapes of binary subwavelength microstructures and their phase modulations can be calculated by an electromagnetic analysis and tabulated. Using the relationship, a deterministic design of a binary grating with subwavelength structures is simply realized. We have designed the binary blazed grating with subwavelength structures and investigated its performance. Its diffraction efficiency is in good agreement with that by a conventional statistical design method.     

Color separation of high-density dielectric rectangular grating in the Fresnel diffraction region

A high-density dielectric rectangular grating is designed for color separation in a Fresnel diffraction field. The Fresnel field distribution is analyzed and the optimization conditions for color separation are given. The process of the modes propagating and energy exchanging with the diffraction orders are expressed by modal method. The color separation for different polarizations can be realized. The energy efficiency is 96.3% at the 633 nm wavelength and 86.9% at the 488 mm wavelength for both TE polarizations, while the energy efficiency is theoretically 96.3% at the 633 nm wavelength for TE polarization and 90.6% at the 488 nm wavelength for TM polarization. The field distributions are scanned by the near-field scanning optical microscopy, and the efficiency is 71.2% for the 633 nm wavelength and 67.3% for the 488 nm wavelength for both TE polarizations experimentally.     

Narrowband, polarization-independent free-space wave notch filter

A two-dimensional-corrugated-slab-waveguide add/drop filter providing 100% resonant reflection at 1.55 microm wavelength for both TE and TM polarizations with identical FWHM is designed. The fabricated device exhibits a reflectivity spectrum of more than 95% peak reflection for both polarizations at 1.537 microm. The coupling scheme involves the TE0 guided mode only; it is made relatively tolerant by means of a double-sided crossed grating.     

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 efficiency of 200-nm-period critical-angle transmission gratings in the soft x-ray and extreme ultraviolet wavelength bands

We report on measurements of the diffraction efficiency of 200-nm-period freestanding blazed transmission gratings for wavelengths in the 0.96 to 19.4 nm range. These critical-angle transmission (CAT) gratings achieve highly efficient blazing over a broad band via total external reflection off the sidewalls of smooth, tens of nanometer thin ultrahigh aspect-ratio silicon grating bars and thus combine the advantages of blazed x-ray reflection gratings with those of more conventional x-ray transmission gratings. Prototype gratings with maximum depths of 3.2 and 6 μm were investigated at two different blaze angles. In these initial CAT gratings the grating bars are monolithically connected to a cross support mesh that only leaves less than half of the grating area unobstructed. Because of our initial fabrication approach, the support mesh bars feature a strongly trapezoidal cross section that leads to varying CAT grating depths and partial absorption of diffracted orders. While theory predicts broadband absolute diffraction efficiencies as high as 60% for ideal CAT gratings without a support mesh, experimental results show efficiencies in the range of ～50-100% of theoretical predictions when taking the effects of the support mesh into account. Future minimization of the support mesh therefore promises broadband CAT grating absolute diffraction efficiencies of 50% or higher.     

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.     

Preferential-order waveguide grating couplers: a comparative rigorous analysis using the finite-difference time-domain method

We rigorously analyze and compare preferential-order waveguide grating output couplers using the finite-difference time-domain method in the total-field/scattered-field formulation for TE and TM polarizations. Four kinds of preferential-order grating couplers are studied: volume holographic grating couplers, slanted parallelogrammic surface-relief grating couplers, double-corrugated surface-relief grating couplers, and reflecting-stack surface-relief grating couplers. The outcoupling efficiencies and branching ratios of the couplers, revealing their preferentiality, are calculated and compared with the rigorous coupled-wave analysis leaky-mode method. In addition, their performance is examined in terms of the main design parameters and the excitation wavelength.     

Efficiency of a grazing-incidence off-plane grating in the soft-x-ray region

Efficiency measurements of a grazing-incidence diffraction grating in the off-plane mount were performed using polarized synchrotron radiation. The grating had 5000 grooves/mm, an effective blaze angle of 14 degrees, and was gold coated. The efficiencies in the two polarization orientations (TM and TE) were measured in the 1.5-5.0 nm wavelength range and were compared with the efficiencies calculated using the PCGrate-SX code. The TM and TE efficiencies differ, offering the possibility of performing unique science studies of astrophysical, solar, and laboratory sources by exploiting the polarization sensitivity of the off-plane grating.     

Optical measurement of depth and duty cycle for binary diffraction gratings with subwavelength features

We describe a new and unique method for simultaneous determination of the groove depth and duty cycle of binary diffraction gratings. For a near-normal angle of incidence, the +1 and -1 diffracted orders will behave nearly the same as the duty cycle is varied for a fixed grating depth. The difference in their behavior, quantified as the ratio of their respective diffraction efficiencies, is compared to a look-up table generated by rigorous coupled-wave theory, and the duty cycle of the grating is thus obtained as a function of grating depth. Performing the same analysis for the orthogonal probe-light polarization results in a different functional dependence of the duty cycle on the grating depth. By use of both TE and TM polarizations, the depth and duty cycle for the grating are obtained by the intersection of the functions generated by the individual polarizations. These measurements can also be used to assess qualitatively both the uniformity of the grating and the symmetry of the grating profile. Comparison with scanning electron microscope images shows excellent agreement. This method is advantageous since it can be carried out rapidly, is accurate and repeatable, does not damage the sample, and uses low-cost, commonly available equipment. Since this method consists of only four fixed simple measurements, it is highly suitable for quality control in a manufacturing environment.     

Analysis of diffraction gratings by using an edge element method

Typically the grating problem is formulated for TE and TM polarizations by using, respectively, the electric and magnetic fields aligned with the grating wall and perpendicular to the plane of incidence, and this leads to a one-field-component problem. For some grating profiles such as metallic gratings with a triangular profile, the prediction of TM polarization by using a standard finite-element method experiences a slower convergence rate, and this reduces the accuracy of the computed results and also introduces a numerical polarization effect. This discrepancy cannot be seen as a simple numerical issue, since it has been observed for different types of numerical methods based on the classical formulation. Hence an alternative formulation is proposed, where the grating problem is modeled by taking the electric field as unknown for TM polarization. The application of this idea to both TE and TM polarizations leads to a two-field-component problem. The purpose of the paper is to propose an edge finite-element method to solve this wave problem. A comparison of the results of the proposed formulation and the classical formulation shows improvement and robustness in the new approach.     

Three-port binary reflective grating with high efficiency under second Bragg angle incidence

Abstract

A binary three-port reflective grating under second Bragg angle incidence is designed in this paper. Under second Bragg angle incidence, the grating can separate nearly 33% light wave energy into the 2nd order, the 1st order and the 0th diffractive orders, respectively. Rigorous coupled-wave analysis can give numerical calculation to optimize the three-port grating depths and periods. For the optimized reflective three-port grating, TE polarization and TM polarization can have different values of grating depth and period. Compared with the reported three-port binary grating under Bragg angle incidence, the diffraction efficiencies can be improved. Moreover, the modal method is applied to explain the propagating mechanism. The highly efficient three-port binary reflective grating under second Bragg angle incidence would be manufactured in the emerging industry for its novel performance.     

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.     

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.     

Numerical investigation on the filtering behavior of 2-D PBGs with multiple periodic defects

In this study, the transmission properties of photonic crystals with multiple periodic defects are studied by using a full-wave approach. The high convergence rate of the employed technique has allowed us to accurately and efficiently predict the filtering behavior of the considered structures. Results are presented for both TE and TM polarizations, showing the transmission efficiencies as a function of the involved parameters. In order to give more physical insight, a comparison with a simpler one-dimensional model has been provided. From our numerical investigation, it turns out that, by suitably configuring the photonic bandgap, it is possible to shape the filtering properties in TE polarization in a simple and versatile way.     

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).