Optimal quantization by use of an amplitude-weighted probability-density function for diffractive optical elements

We present a method for determining the optimal quantitized phase levels for phase-only diffractive optical elements by taking into account a modified probability-density function (PDF). This modified PDF was obtained from the PDF of the phase distribution weighted by the incident amplitude distribution. We applied the proposed method to construct a multilevel Fresnel zone plate with a focal length of 8 m and a diameter of 6.2 mm upon which was incident a Gaussian beam with a 1/e width of 1.4 mm. The efficiency of a binary design was improved by 26.4% and the signal-to-noise ratio by was improved by 18.2% compared with those obtained by the uniform quantization method.     

Automatic symmetrical iterative Fourier-transform algorithm for the design of diffractive optical elements for highly precise laser beam shaping

A symmetrical iterative Fourier-transform algorithm (IFTA) using a combination of phase and amplitude freedom for the design of diffractive optical elements for highly precise laser beam shaping is presented. We compare this method with the basic IFTA and the symmetrical IFTA exclusively using phase freedom, and the basic IFTA using both phase and amplitude freedom, by employing these methods for super-Gaussian beam shaping. While the latter three methods fail to produce satisfactory solutions, the first method results in a beam non-uniformity error of 0.44% and a theoretical efficiency of 97.2%. Moreover, the new approach avoids the use of the trial-and-error method for finding the appropriate modified Fourier-domain constraints during the iteration, which can be difficult for some beam-shaping problems.     

Rigorous electromagnetic design of finite-aperture diffractive optical elements by use of an iterative optimization algorithm

We propose a rigorous electromagnetic design of two-dimensional and finite-aperture diffractive optical elements (DOEs) that employs an effective iterative optimization algorithm in conjunction with a rigorous electromagnetic computational model: the finite-difference time-domain method. The iterative optimization process, the finite-difference time-domain method, and the angular spectrum propagation method are discussed in detail. Without any approximation based on the scalar theory, the algorithm can produce rigorous design results, both numerical and graphical, with fast convergence, reasonable computational cost, and good design quality. Using our iterative algorithm, we designed a diffractive cylindrical lens and a 1-to-2-beam fanner for normal-incidence TE-mode illumination, thus showing that the optimization algorithm is valid and competent for rigorously designing diffractive optical elements. Concerning the problem of fabrication, we also evaluated the performance of the DOE when the DOE profile is discrete.     

Analytic approach for optimal quantization of diffractive optical elements

One of the most important factors that limit the performance of diffractive optical elements (DOE's) is the depth accuracy of the relief structure. A common procedure for fabricating DOE's is the binary optics procedure, in which binary masks are used for the fabrication of a multilevel relief structure. Here an analytic procedure for calculating the optimal depth levels of DOE's, the phase bias, and the decision levels is presented. This approach is based on the minimization of the mean-squared error caused by the quantization of the continuous profile. As a result of the minimization an optimal value for the etching depth of each photolithographic mask is determined. The obtained depth values are, in general, different from the depth values used by the conventional multilevel approach. Comprehensive mathematical analysis is given, followed by several computer simulations that demonstrate the advantages of the proposed procedure.     

Optimal design method for multi-layer diffractive optical elements with consideration of antireflection coatings

An optimal design method for multi-layer diffractive optical elements (MLDOEs) is put forward with consideration of antireflection coatings in actual applications of hybrid optical systems. Taking this method into account for optimal design, it can ensure 100% diffraction efficiency at the designed wavelengths and maximum polychromatic integral diffraction efficiency over the whole waveband. In addition, it can be obtained not only for mechanical durability of soft polymers improvements but increase antireflection coatings functions for hybrid optical systems. Finally, an example for optimal design algorithm process and simulation of MLDOEs used in visible waveband is presented. The results show that this optimal method perfects the MLDOEs design on the basic theory with great practicability for MLDOEs design.     

Genetic local search algorithm for optimization design of diffractive optical elements

We propose a genetic local search algorithm (GLSA) for the optimization design of diffractive optical elements (DOE's). This hybrid algorithm incorporates advantages of both genetic algorithm (GA) and local search techniques. It appears better able to locate the global minimum compared with a canonical GA. Sample cases investigated here include the optimization design of binary-phase Dammann gratings, continuous surface-relief grating array generators, and a uniform top-hat focal plane intensity profile generator. Two GLSA's whose incorporated local search techniques are the hill-climbing method and the simulated annealing algorithm are investigated. Numerical experimental results demonstrate that the proposed algorithm is highly efficient and robust. DOE's that have high diffraction efficiency and excellent uniformity can be achieved by use of the algorithm we propose.     

Thermal embossing of mid-infrared diffractive optical elements by use of a self-processing photopolymer master

Surface-relief transmissive diffractive elements were fabricated by embossing. The master was made by lithography with a self-developing photopolymer. The highly cross-linked structure exhibited by the polymer has made possible the direct replication by thermal embossing of polyethylene substrates. Fabricated elements are meant to work with mid-infrared radiation. The influence of some process variables, related to the performance of the diffractive elements, is analyzed.     

Design and fabrication of diffractive optical elements by use of gray-scale photolithography

Diffractive optical elements (DOEs) are key components in the miniaturization of optical systems because of their planarity and extreme thinness. We demonstrate the fabrication of DOEs by use of gray-scale photolithography with a high-energy-beam sensitive glass photomask. We obtained DOE lenses with continuous phase profiles as small as 800 microm in diameter and 5.9 microm in the outermost grating pitch by selecting a suitable optical density for each height level and optimizing the process variables. Microlenses patterned with eight levels and replicated by UV embossing with the polymer master mold showed a diffraction efficiency of 81.5%, which was sufficiently high for the devices to be used as optical pickups. The effects of deviations in diffraction efficiency between the DOE height and profile design were analyzed.     

Generating inhomogeneously polarized higher-order laser beams by use of diffractive optical elements

We propose an improved version of the earlier developed optical arrangement for generating inhomogeneously polarized laser light modes with the aid of a diffractive optical element (DOE) with carrier frequency. By eliminating lenses from the optical arrangement, we achieve the miniaturization, reduced light losses, a smaller number of parameters being matched, and a simpler system adjustment procedure. Note that all the capabilities of the previous version, namely, the universality and simple readjustment to different polarization types, are fully retained. The numerical modeling of the polarization mode combiner has made it possible to analyze its performance and capabilities. In the experiments, the quality of the resulting beams is shown to be improved. For generating higher-order cylindrical beams, a lower-order mode at the output of the polarization mode combiner is additionally transformed with a DOE that operates in the zero diffraction order, introducing radial phase changes.     

Multilayer technology for diffractive optical elements

The proposed multilayer technology makes it possible to approximate a continuous phase distribution by discrete phase steps. Compared with binary techniques, a higher diffraction efficiency can be achieved. In most known processes a bulk substrate is used and etched directly; therefore it is difficult to control the height of the phase steps. We propose applying layers of a well-known thickness and structuring them with a selective etching process. In this new multilayer process for reflecting elements a system of metal and dielectric layers is used that can easily be produced by standard methods.     

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.     

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.     

Rapid fabrication of diffractive optical elements by use of image-based excimer laser ablation

We describe a method of fabricating multilevel diffractive optics by excimer laser ablation. A portion of a chrome mask containing many patterns is illuminated by 193-nm laser light and imaged by an objective lens onto a poly(imide) substrate. Ablation of an entire single pattern is achieved in a single laser pulse. Multiple pulses are used to vary the ablation depth, and multiple patterns are used to create a variety of multilevel optics. We have successfully fabricated arrays of eight-level diffractive microlenses with varying focal lengths and decenters. The optics performed with diffraction-limited focusing and near-theoretical diffraction efficiency (92%).     

Analysis of non-quarter-wave grating by a modified Fourier-transform method

When unity reflectance is approached, the Fourier-transform method of calculating the reflectance spectrum of an optical grating modulated by a slowly varying envelope becomes unacceptably inaccurate. The modified Fourier transform method of Bovard [Appl. Opt. 29, 24 (1990)] can achieve complete accuracy for quarter-wave gratings. We report herein the extension of Bovard's method to non-quarter-wave gratings. We demonstrate the accurate deployment of our simplified modified Fourier-transform method to apodized linear gratings and optically apodized nonlinear gratings.     

Extended algorithm for the design of diffractive optical elements around the focal plane

We present a multiplane algorithm for three-dimensional uniform illumination. The large-diameter diffractive optical element simulated by this algorithm homogeneously concentrates more than 86.5% of the incident energy into a 200 microm length of columnar space around the focal plane. The intensity profile in the whole space is nearly flattop, and the beam's quality measured by the root mean square is less than 20.6%. The algorithm is very useful if a great deal of tolerance is required for the installation error of the optical system or if it is used for some particular application, such as uniform illumination on an incline plane.     

Robust design method for highly efficient beam-shaping diffractive optical elements using an iterative-Fourier-transform algorithm with soft operations

Abstract

Beam-shaping diffractive optical elements give a desired intensity distribution in the diffraction plane over areas much larger than the diffraction limited spot size. Such elements can be designed using geometrical optics methods or iterative-Fourier-transform algorithms (IFTAs). The usefulness of geometrical optics methods is considerably limited for two reasons: first the number of cases for which a solution exists is small and second the design solution, if it exists, often does not work in practice. Then IFTAs can be used. They are applicable for any desired intensity distribution in the diffraction plane with any intensity cross-section of the incident beam. The IFTA presented in this paper uses a novel set of operations that introduce a minimum disturbance of the fields while still leading to an improved performance. This makes the method robust, insensitive to stagnation and capable of iteratively distributing an increasing portion of the light in the diffraction plane into the desired areas thus leading to a high efficiency (～95%). Three design examples are given and one is also tested experimentally.     

Polarization-multiplexed diffractive optical elements fabricated by subwavelength structures

Polarization-multiplexed phase-only diffractive optical elements with subwavelength structures are proposed and fabricated. The differences among the phase modulations result from the differences among the effective indices exhibited in the subwavelength structures with various filling factors and surface profiles, and the phase retardations are obtained by the relief depth of the structures. The polarization-selective property is achieved by the polarization dependence of the effective indices exhibited in the one-dimensional subwavelength structures and the polarization independence exhibited in the two-dimensional structures. Additionally, the polarization contrast of our polarization-multiplexed elements, defined as the cross talk between the two polarization incidences, is independent of the relief depth. The principle of the polarization multiplexing by use of the subwavelength structures is described, and the fabrication results for the polarization-multiplexed computer-generated holograms are demonstrated.     

Efficient encoding algorithms for computer-aided design of diffractive optical elements by the use of electron-beam fabrication

One of the general requirements of a computer-aided design system is the existence of efficient (in data size and running time) algorithms that are generally reliable for the broadest range of design instances. The restricted data formats of the electron-beam machines impose difficulties in developing algorithms for the design of diffractive optical elements (DOE's) and computer-generated holograms (CGH's). Issues that are related to the development of CGH algorithms for e-beam fabrication of DOE's and CGH's are discussed. We define the problems the CGH algorithms need to solve, then introduce general curve drawing algorithms for the e-beam data generation of diffractive optical components. An efficient algorithm for general aspherical DOE's is proposed. Actual design and fabrication examples are also presented.     

Design of diffractive optical elements for multiple wavelengths

A method for producing diffractive optical elements (DOE's) for multiple wavelengths without chromatic aberration is described. These DOE's can be designed for any distinct wavelength. The DOE's are produced from two different optical materials, taking advantage of their different refractive indices and dispersions.     

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.