Efficient optimization of diffractive optical elements based on rigorous diffraction models.

An efficient optimization strategy for the design of diffractive optical elements that is based on rigorous diffraction theory is described. The optimization algorithm combines diffraction models of different degrees of accuracy and computational complexity. A fast design algorithm for diffractive optical elements is used to yield estimates of the optimum surface profile based on paraxial diffraction theory. These estimates are subsequently evaluated with a rigorous diffraction model. This scheme allows one to minimize the need to compute diffraction effects rigorously, while providing accurate design. We discuss potential applications of this scheme as well as details of an implementation based on a modified Gerchberg-Saxton algorithm and the finite-difference time-domain method. Illustrative examples are provided in which we use the algorithm to design Fourier array illuminators.     

Three-dimensional analysis of subwavelength diffractive optical elements with the finite-difference time-domain method

We present a three-dimensional (3D) analysis of subwavelength diffractive optical elements (DOE's), using the finite-difference time-domain (FDTD) method. To this end we develop and apply efficient 3D FDTD methods that exploit DOE properties, such as symmetry. An axisymmetric method is validated experimentally and is used to validate the more general 3D method. Analyses of subwavelength gratings and lenses, both with and without rotational symmetry, are presented in addition to a 2 x 2 subwavelength focusing array generator.     

Design method for small-f-number microlenses based on a finite thickness model in combination with the Yang-Gu phase-retrieval algorithm

We present a fast and general iterative design method for both diffractive and nondiffractive two-dimensional optical elements. The method is based on a finite-thickness model in combination with the Yang-Gu phase-retrieval algorithm. A rigorous electromagnetic analysis (boundary element method) is used to appraise the designed results. We calculate the transverse-intensity distributions, diffraction efficiency, and spot size of the designed microlenses at the focusing plane for microlenses designed using the presented method and the conventional zero-thickness model. The main findings show the superiority of the presented method over the conventional method, especially for nondiffractive optical elements.     

Design and fabrication of continuous-profile diffractive micro-optical elements as a beam splitter

An optimization algorithm that combines a rigorous electromagnetic computation model with an effective iterative method is utilized to design diffractive micro-optical elements that exhibit fast convergence and better design quality. The design example is a two-dimensional 1-to-2 beam splitter that can symmetrically generate two focal lines separated by 80 microm at the observation plane with a small angle separation of +/- 16 degrees. Experimental results are presented for an element with continuous profiles fabricated into a monocrystalline silicon substrate that has a width of 160 microm and a focal length of 140 microm at a free-space wavelength of 10.6 microm.     

Electromagnetic analysis of axially symmetric diffractive optical elements illuminated by oblique incident plane waves.

We present an analysis of axially symmetric diffractive optical elements illuminated by off-axis or oblique incident plane waves. The analysis is performed with a finite-difference time-domain method that has been formulated to exploit axial symmetry yet accommodate off-axis illumination. This approach is compared with a full three-dimensional formulation and is found to be more efficient in both memory requirements and computational time. Validation and applications of this method are presented.     

Diffractive optical element design with memory-matrix-based identification methodology

We present a new technique for the design of diffractive optical elements (DOE's) that is based on previous nonlinear least squares (NLS) and phase-shifting quantization methods [Appl. Opt. 36, 7297-7306 (1997)]. The technique uses a memory-matrix-based identification (MMBI) optimization procedure. We compare results from the MMBI method with those from iterative Fourier transform and NLS methods. In comparison, the MMBI DOE designs produce better-quality reconstructions for DOE's with eight or more fabrication phase levels and generally have a higher signal-to-noise ratio and better uniformity.     

Design of diffractive optical elements with optimization of the signal-to-noise ratio and without a dummy area

We describe a new approach to suppress undesired diffraction orders in the signal area of a Fourier plane diffractive optical element (DOE). We implement this new approach for the DOE design by a two-stage iterative Fourier transform algorithm that incorporates an adaptive optimization of the signal-to-noise ratio and does not require the introduction of a dummy output area outside the field of view. A comparison among this approach and three other approaches are presented on the basis of numerical results from several sample diffraction patterns.     

Diffractive optical elements as raster-image generators

The use of diffractive optical elements (DOEs) to generate complex raster images for a primarily artistic purpose is dealt with. Aspects of human vision that are relevant for the design of such elements are discussed. A design method based on an iterative Fourier transform algorithm and extended with elements from the direct-binary-search and the simulated-annealing algorithms is described. The proposed method provides a large set of parameters that can be adjusted freely to optimize it for any given design task. For demonstration a phase-only DOE was designed that generates an image of a Chinese dragon as a diffraction pattern. It was realized as a surface-relief element on a planar substrate through multilevel binary lithography and reactive-ion etching. Experimental tests confirm the usefulness of the design and the fabrication procedures to achieve excellent image quality.     

Electromagnetic design of an all-diffractive millimeter-wave imaging system

We present the design and electromagnetic analysis of an all-diffractive millimeter-wave imaging system having a field of view of +/- 15 degrees. This system consists of two 16-level diffractive lenses, with the stop in contact with the first lens. By considering the Seidel aberrations for a diffractive lens and applying the corresponding stop shift formula, we established the expressions of third-order wave aberrations for this system. By setting all primary Seidel aberrations to zero and solving the corresponding system of equations, we obtained two sets of solutions for this two-element all-diffractive system, which totally compensate for all Seidel aberrations. To assess image system performance, we apply the finite-difference time-domain technique and a vector plane-wave spectrum method, in combination, to validate the performance of the system. To reduce the computational cost and thereby enable the complete electromagnetic analysis of the system, a four-step analysis procedure has been developed and applied as an electromagnetic system model.     

遗传算法用于衍射光学元件的优化设计

提出了一种基于遗传算法的衍射光学元件优化设计方法;在衍射光学元件设计中遗传算法运行参数对遗传算法性能有一定的影响:采用较大的群体规模,遗传算法越容易获得最优解;交叉算子越大,遗传算法全局搜索能力越强;选择算子对遗传算法的影响不是太大;如果要进一步提高解的精度,可选取较大的终止代数。数值计算结果表明,用遗传算法优化设计的衍射光学元件,其误差小于 5.2%,衍射效率达到 91.2%。遗传算法很适合衍射光学元件的优化设计。     

Algorithm based on rigorous coupled-wave analysis for diffractive optical element design

Diffractive optical element design is an important problem for many applications and is usually achieved by the Gerchberg-Saxton or the Yang-Gu algorithm. These algorithms are formulated on the basis of monochromatic wave propagation and the far-field assumption, because the Fourier transform is used to model the wave propagation. We propose an iterative algorithm (based on rigorous coupled-wave analysis) for the design of a diffractive optical element. Since rigorous coupled-wave analysis (instead of Fourier transformation) is used to calculate the light-field distribution behind the optical element, the diffractive optical element can thus be better designed. Simulation results are provided to verify the proposed algorithm for designing a converging lens. Compared with the well-known Gerchberg-Saxton and Yang-Gu algorithms, our method provides 7.8% and 10.8%, respectively, improvement in converging the light amplitude when a microlens is desired. In addition, the proposed algorithm provides a solution that is very close to the solution obtained by the simulated annealing method (within 1.89% error).     

Iterative algorithm for the design of free-space diffractive optical elements for fiber coupling

We present a design method based on the Gerchberg-Saxton algorithm for the design of high-performance diffractive optical elements. Results from this algorithm are compared with results from simulated annealing and the iterative Fourier-transform algorithm. The element performance is comparable with those designed by simulated annealing, whereas the design time is similar to the iterative Fourier-transform method. Finally, we present results for a demanding beam-shaping task that was beyond the capabilities of either of the traditional algorithms. The element performances demonstrate greater than 85% efficiency and less than 2% uniformity error.     

GA/FDTD technique for the design and optimisation of periodic metamaterials

An efficient and powerful full-wave electromagnetic technique is presented to characterise and design periodic metamaterial structures. First, the spectral finite-difference time-domain (FDTD) method with periodic boundary conditions and uniaxial perfect matched layer is employed to predict the performance of a mushroom-like artificial magnetic conductor (AMC) surface and further extended to characterise a negative-refractive-index material consisting of lumped and distributed transmission-line elements. Then, a new computational technique is developed to design and optimise periodic metamaterial structures by integrating the spectral FDTD method with a genetic algorithm (GA), namely the micro-genetic algorithm. This computational technique is successfully applied to design and optimise single-band and dual-band AMC structures consisting of a frequency-selective surface and a ground plane. It is demonstrated that the GA/FDTD technique is a very effective approach for the design and optimisation of periodic metamaterial structures consisting of dielectrics and conductors of arbitrary configurations     

Applying a mapped pseudospectral time-domain method in simulating diffractive optical elements

A new technique for the analysis of two-dimensional diffractive optical elements, by use of the pseudospectral time-domain (PSTD) method, is presented. In particular, the method uses a nonuniform (NU) grid and a mapping technique to obtain very accurate spatial derivatives in an efficient manner. To this end, we present the formulation of the PSTD method by using a NU grid and compare its application to the analysis with that of the finite-difference time-domain (FDTD) method. Using only a fraction of the memory and a fraction of the computation time used by FDTD, the mapped PSTD was able to obtain very close results to FDTD.     

Beam shaping for multicolour light-emitting diodes with diffractive optical elements

An improved particle swarm optimization method is proposed for the design of ultra-thin diffractive optical elements (DOEs) enabling multicolour beam shaping functionality. We employ pre-optimized initial structures and adaptive weight strategy in the algorithm to achieve better and identical shaping performance for multiple colours. Accordingly, a DOE for shaping light from green and blue LEDs has been designed and fabricated. Both experiment and numerical simulations have been conducted and the results agree well with each other. 15.66% average root mean square error (RMSE) and 0.22% RMSE difference are achieved. In addition, the parameters closely related to the performance of the optimization are analysed, which can provide insights for future application designs.     

Iterative optimization of diffractive phase elements simultaneously implementing several optical functions

The design of diffractive optical elements that incorporate several optical functions in a single element is discussed. The technique used involves iterative optimization. Aprevious paper is continued, in which initial results with few sampling points were reported. Here new results that involve a large number of sampling points are reported. Because the algorithm is computationally intensive with a large number of data points, the parallel implementation of the algorithm on a MASPAR machine is described. MASPAR is a single-instruction multiple-data machine with 16,384 processors. The computer simulations discussed involve simultaneous wavelength demultiplexing, focusing, and the filtering out of a particular wavelength component. It is shown that satisfactory designs of diffractive optical elements can be achieved by the assignment of only a small number of sampling points on the output plane that adequately specify what is required at each wavelength.     

Diffraction of electromagnetic waves by periodic arrays of rectangular cylinders

Reflection, transmission, and absorption of electromagnetic waves by periodic arrays of conducting or dielectric rectangular cylinders are studied by a finite-difference time-domain technique. Truncated gratings made of lossless and lossy conducting and dielectric elements are considered. Results for surface current density, transmission, and reflection coefficients are calculated and compared with corresponding results in the literature, which are obtained by approximate or rigorous methods applicable only to idealized infinite models. An excellent agreement is observed in all cases, which demonstrates the accuracy and efficacy of our proposed analysis technique. Additionally, this numerical method easily analyzes practical gratings that contain a finite number of elements made of lossless, lossy, or even inhomogeneous materials. The results rapidly approach those for the idealized infinite arrays as the number of elements is increased. The method can also solve nested gratings, stacked gratings, and holographic gratings with little analytical or computational effort.     

Optimal quantization method for uneven-phase diffractive optical elements by use of a modified iterative Fourier-transform algorithm

A method based on an iterative Fourier-transform algorithm (IFTA) with phase optimization for phase-only multilevel diffractive optical elements (DOEs) is presented. Phase optimization by minimizing the mean-squared error composed of an amplitude-weighted probability-density function is performed in the IFTA iterations to ensure that the wavefront difference is minimized and thus to improve the DOEs' performance. Using the proposed method, we obtained a small standard deviation of diffraction efficiency of 20 uneven-phase DOEs, showing that DOEs with high diffraction efficiency did not vary significantly with the initial conditions. The simulation results of even- and uneven-phase four-level DOEs designed by use of direct and stepwise quantization IFTA methods are compared.     

Iterative Fourier transform algorithm with regularization for the optimal design of diffractive optical elements

There is a trade-off between uniformity and diffraction efficiency in the design of diffractive optical elements. It is caused by the inherent ill-posedness of the design problem itself. For the optimal design, the optimum trade-off needs to be obtained. The trade-off between uniformity and diffraction efficiency in the design of diffractive optical elements is theoretically investigated based on the Tikhonov regularization theory. A novel scheme of an iterative Fourier transform algorithm with regularization to obtain the optimum trade-off is proposed.     

Implementation of phase-shift patterns using a holographic projection system with phase-only diffractive optical elements

We proposed a method to implement spatial phase-shift patterns with subdiffraction limited features through a holographic projection system. The input device of the system displayed phase-only diffractive optical elements that were calculated using the iterative Fourier-transform algorithm with the dummy-area method. By carefully designing the target patterns to the algorithm, the diffractive optical elements generated the Fourier-transformed images containing the phase-shift patterns in which the widths of dark lines were smaller than the diffraction limit. With these demonstrations, we have successfully shown that the near-field phase-shift lithographic technique can be realized through an inexpensive maskless lithographic system and can still achieve subdiffraction limited images.