Following an analysis of the interference fringes above the plane of a high-efficiency grating, a set of systematic numerical optimizations of slanted trapezoidal gratings aiming at minimizing the maximum electric field inside the grating material was conducted. The results confirm predictions based on the interference fringe analysis. The most important conclusion is that slanting the grating ridges can substantially increase the laser-induced damage threshold of multilayer dielectric gratings.

In a recent Letter [ Opt. Lett. 51 , 1347 ( 2026 ) 10.1364/OL.587829 ], it was theoretically predicted that, besides having appropriate groove depth and duty cycle, slanting the ridges of a multilayer dielectric grating a pre-determined moderate angle away from the incident light beam can substantially increase the grating’s laser-induced damage threshold. We have experimentally verified this prediction. We also have proven that it is feasible to achieve the required slanted grating profile in silica by oblique-incident reactive ion-beam etching through a pure photoresist grating mask.

We present a method to fabricate optical mosaic gratings that aligns the exposure interference field with a previously developed surface-relief grating using the profile symmetry of the latter. Experimental results demonstrate the robustness of this method to variations of the grating groove parameters. Mosaic gratings of size 90 mm x 100 mm and groove depths 180, 300, and 350 nm were successfully fabricated with two exposures (1 x 2 mosaic). The peak-to-valley wavefront errors of the +/- 1st diffraction orders were comparable to those of gratings of the same size fabricated by the monolithic exposure method.

A semi-analytic theory of multilayer dielectric gratings (MLDGs) is presented. Analytic formulas for the -1st-order diffraction efficiency of an MLDG being 100% and greater than a preset value are given in terms of the scattering matrix elements of the top surface-relief grating (TG). The important role of the combined reflection phases is highlighted. The need to secure high reflectances of the multilayer stack at the two angles of incidence below the TG and the need to supply a sufficient number of aperiodic layers in the multilayer substrate for phase matching are emphasized. Three numerical examples of 100% efficient MLDGs, obtained without optimizing the TG, are presented, illustrating the accuracy and generality of the theory.

A general and simple semi-analytic theory of multilayer dielectric gratings is presented. It extends a previous work [J. Opt. Soc. Am. A 41, 252 (2024)] that assumes symmetric grating profile and Littrow mounting to gratings of asymmetric profiles in off-Littrow mounting.

We theoretically present and experimentally validate a method to overlay the interference field (for making straight-line holographic gratings) with a real grating and a derived method to measure the lateral shift between two real gratings. The methods rely only on the real gratings' profile symmetry, and they are robust against variations of the real gratings' profile parameters and functional form.

We present a method to design polarization-independent multilayer dielectric gratings. In this method the reflection phases in transverse electric (TE) and transverse magnetic (TM) polarizations of the multilayer stack thread surface-relief grating at the top and the multilayer stack at the bottom together, allowing the two parts first to be designed separately and efficiently, and then to be combined to achieve simultaneously high diffraction efficiency and large fabrication tolerance. We find numerically that in general a periodic stack is unable to provide the top-grating-demanded phase difference between TM and TE polarizations; adequate aperiodic layers atop of a periodic stack are needed. The analytic diffraction efficiency formula of a recent work [J. Opt. Soc. Am. A 41, 252 (2024). https://doi.org/10.1364/JOSAA.511422] is used at various places of the presented optimization algorithm to save computation time. An example grating with rectangular surface-relief profile and another with trapezoidal profile were successfully designed, validating the effectiveness of this design method.

The work started 20 years ago [Appl. Opt. 42, 6255 (2003)] investigating the physical mechanism of multilayer dielectric reflection gratings to achieve 100% diffraction efficiency is extended to offer much deeper insight than before. How different diffraction amplitudes of the top surface corrugation contribute to the −1st-order efficiency of such a grating is shown analytically using a minimum set of real parameters. The two diffraction amplitudes transmitted through the corrugation play a dominant role in enabling DE = 100%. The necessary and sufficient condition for 100% efficiency is derived, and a very simple sufficient condition is also given. Moreover, the role of the reflection phase of the perfect-reflector, including the contribution due to optical path between the corrugation and the reflector, is emphasized.

An objective function that closely represents the maximum optical pathlength difference on the entire grating surface for designing a flat-field holographic concave grating is proposed. Compared to design methods that minimize certain aberration coefficients of the optical pathlength function, the present objective function has a clearer physical meaning of optical pathlength difference and guarantees a smaller spot size. Compared to design methods that minimize the spot size based on ray tracing, the present method requires less computing time, but is equally effective in minimizing spot size, and it even provides a better image quality in the sagittal direction.

By using the recent theory of elemental scattering matrix, I study the critical-point behaviors of conically mounted and crossed gratings in Littrow mountings. As functions of the projected incident wave vector in the grating plane, the ( m , n )th-order reflected principal diffraction efficiencies (the maximum and minimum diffraction efficiencies of a diffraction order for all possible incident polarizations) of a crossed grating have C 2 symmetry with respect to the ( m , n )th-order Littrow point, and for a conically mounted grating, the symmetry type is increased to C 2 v . In a Littrow mounting, each one of the two principal diffraction efficiencies, independent of the other, can only be at one of the three possible states: a local maximum, a local minimum, or a saddle point of an even (most likely the second) degree. If the grating is mirror-symmetric with respect to a plane parallel to the grating’s mean plane, the results hold for a transmitted diffraction order as well.

Abstract. Gratings with low diffracted wavefront errors are crucial in many applications. Existing methods to reduce the diffracted wavefront errors of holographically produced gratings mainly address misalignment errors during exposure system adjustment, leaving the inevitable residual errors of collimation lenses uncorrected. The broad-beam scanning exposure method was recently proposed to fabricate gratings with large size and low stray light. We explore its ability to reduce the diffracted wavefront errors caused by both system misalignment and lens errors. During the exposure, the substrate translates in the direction perpendicular to the grating lines to pass through the exposure area, so the aberrations over different parts of the interference field along the scanning path are averaged. We prove the validity of this method by a theoretical analysis and numerical simulations. The averaging effect on different aberrations and with different scanning lengths is discussed. The peak-valley values of diffracted wavefront errors of gratings fabricated with an exposure field of defocus aberration and a more complicated aberration were reduced by 42.5% and 50%, respectively.

We theoretically make plausible and numerically verify that when a high-efficiency, polarization-independent grating, designed to work at or near Littrow mounting, is used in conical Littrow mounting, its high efficiency and polarization independence can be very well maintained at a large off-principal-plane deviation angle. This finding provides a new degree of freedom to the design of high-efficiency, more compact spectral beam combining grating systems.

A systematic and formal study of the global and elemental properties of the propagating-order scattering matrix of conically mounted and crossed gratings is presented. The most general formulation of the scattering matrix is established. Expressions of the global properties (reciprocity and unitarity) of the scattering matrix (S matrix) in the general form previously not available in the literature are presented in the main text, and their full mathematical derivations are given in two appendices. The distinctive contribution of this work is an exposition of the elemental properties of the S matrix. The elemental S tensor and the elemental S matrix, the latter being the linear-space representation of the former, for a pair of an incident plane wave and a diffracted order are defined and studied. The key results of the exposition are two sum rules of diffraction efficiencies and a dot-product-free, vectorial reciprocity theorem.

We recently found that the coordinate transformation method (the C method) equipped with well-established recursive algorithms for solving the system of linear equations is numerically instable when it is applied to thinly coated gratings. The origin of this new kind of numerical instability is not the exponential dependence of the field in the coated layers but the ill condition of the eigenvector matrix of the C method when the truncation number is high. Two simple and effective methods to circumvent the new instability are recommended. We also found that the popular recursive matrix algorithms have different (poor) immunities to the new instability, and they all perform inferiorly to the full matrix (nonrecursive) algorithm.

We designed and fabricated polarization-independent multilayer dielectric reflection gratings of 1300 line/mm for spectral beam combining. Because the designed grating has a duty cycle (or fill factor) much greater than 0.5, we used the lift-off process to invert a less-than-0.5 duty cycle photoresist mask to a greater-than-0.5 duty cycle chromium mask. Because the simultaneous polarization-independence and high-angular-dispersion requirements greatly reduce fabrication tolerance, we employed the end-point detection techniques to control accurately the duty cycle and groove depth of the grating profile to reach the design target. The measured polarization-averaged diffraction efficiency of the fabricated gratings was greater than 97% in the wavelength range of 1050 nm-1080 nm.

Derivation of effective-medium permittivities (in the traditional sense) of one-dimensional gratings is revisited. In the present derivation, carried out in Fourier-space and for conical mounting, systematic treatment and mathematical rigour are emphasized. An error of a previous paper published in this journal concerning the 2nd-order effective permittivity in TM polarization is corrected. The derived formulas of the 4(th)-order effective permittivity and the 2nd-order effective permittivity, when restricted to TE and TM polarizations respectively, agree well with the previous results. The symmetry of the effective permittivities with respect to the sign of the Floquet exponential factor and the sign of grating period is noted and proven.

Spectral beam combining is an effective method to increase the total output power of a laser system while maintaining a high beam quality. The diffraction grating with high diffraction efficiency and high laser damage threshold is the key component in a spectral-beam-combining laser system. To meet the above requirements, we design a grating on top of a high-reflectivity HfO2/SiO2 thin-film stack. The diffraction efficiency of the grating depends mainly on the duty cycle and depth of the grating grooves. The grating was fabricated by using optical interference lithography and reactive ion beam etching. To achieve high efficiency, we controlled the duty cycle by applying an end-point detection technique during development and we controlled the groove depth by adjusting the ion-beam etching time length. In a 50 mm × 50 mm fabricated grating area, the measured diffraction efficiency in TE polarization was 95.8 ± 1.1 % at the center wavelength of 1030 nm. In the wavelength range of 1025 nm to 1035 nm, the measured average diffraction efficiency in TE polarization was 97.7% at center of the grating sample. We describe the details of our work, including design parameters, fabrication processes, and measured diffraction efficiencies.

In view of the fact that inverters using LCL filter have been widely adopted in power systems, resonance caused by LCL filter in parallel-inverter-based system has become more and more common. In order to stabilize the operation of the paralleled inverter system and improve the waveform of the injected currents, a resonance suppression method is proposed. Firstly, based on the stability analysis of the corresponding mathematical model, the resonance frequency that occurs in a parallel-inverter-based system is investigated. Then, a notch filter is selected on the basis of the resonance characteristics, which is adopted to improve the damping of the system and enhance the stability. Simulation tests validate the stability analysis and the performance of the proposed control strategy.

A detailed review of the theory of effective permittivity for one- and two-dimensional periodic structures shows its limited validity for metal-dielectric structures in the visible and near infra-red if the feature dimensions are comparable with the metal skin depth. We propose a phenomenological correction to the static formulae using a realistic assumption for the electric field behavior inside the metal features. This approach allows us to obtain analytical expressions for the effective permittivity in the case when the electric field is not sufficiently homogeneous within the unit cell of the gratings. A comparison with the numerical results of the Fourier modal method demonstrates the validity of the analytical formulae. Additional study is made on the impedance approximation at the outer boundaries of the periodical structure in order to propose analytical formulae for the reflection coefficient that permits better correspondence with the numerical results. The link between the values of effective permittivity and permeability defined as the ratios between the averaged fields, and the metamaterial permittivity and permeability is discussed.