Double-multiplexed computer-generated holograms

One of the detour-phase computer-generated-hologram encoding methods was improved for obtaining two objects simultaneously without increasing the plotter resolution. The encoded patterns were reconstructed along two orthogonal diffraction directions. Such a hologram could be useful for two-channel coherent or incoherent optical correlation. Laboratory experiments demonstrate the suggested technique.     

Resolution and intensity distribution of output images reconstructed by sampled computer-generated holograms

We address reconstruction abilities of computer-generated holograms that are fabricated in a sampled form and designed for spherical-wave illumination. Our approach enables one to define the limit of resolution of the object reconstruction and the light-intensity distribution in an output plane. The analysis takes into account a hologram aperture, a sampling aperture, and a curvature radius of spherical illumination. Theoretical results are confirmed by experimental verification.     

Two-dimensional, three-dimensional, and gray-scale images reconstructed from computer-generated holograms designed by use of a direct-search method

A direct-search method for the computer design of holograms is demonstrated. Two-dimensional, three-dimensional, and containing holograms with different levels of intensity (gray scales) were designed, fabricated, and optically reconstructed. The number of effective gray levels and the image-intensity noise and contrast are discussed. A modification of the state-variables cost function used in the direct-search algorithm that permits reliable control of gray scales is presented. Optical reconstructions of two-dimensional, three-dimensional, and gray-scale binary phase computer-generated holograms are presented.     

Modifications of detour phase computer-generated holograms

The detour phase method for the design of computer-generated holograms can be modified to achieve multichannel reconstruction along various diffraction orders. It is shown how a single hologram can be used to display two patterns of different intensities along two diffraction orders. This is achieved by the release of any requirement on the phase distributions of these patterns, thus leaving them as free parameters. Various algorithms are suggested to make possible nonidentical reconstructions along two different off-axis diffraction orders. The two reconstruction orders can be chosen arbitrarily. The case of four-channel reconstructions for generating four different images is discussed as well. Computer simulations and optical experiments were carried out to demonstrate the capabilities of the proposed approaches.     

Interferometric testing for off-axis aspherical mirrors with computer-generated holograms

The interferometers with computer generated holograms (CGHs) have been used to measure off-axis aspherical mirrors. However, the conventional CGH interferometer could not produce a high lateral resolution because of the blur and the distortion of the interferogram of the test mirror caused by the nonzero order diffraction of the CGH. We further develop the application of CGHs concerning interferometers in order to improve the lateral resolution of the interferogram. In particular, we change the application of the CGH in such a way that the returned test beam passes through the CGH with zeroth order diffraction and demonstrates the performance of the CGH interferometer when used for the measurement of the primary mirror of the Okayama Astrophysical Observatory 3.8 m telescope project. As a result, our CGH interferometer produces a good interferogram with high resolution of 2.8 mm for the off-axis aspherical mirror with a size of 1.2 m. With improved usage of the CGH, we successfully demonstrated a high lateral resolution interferometer.     

Design and fabrication of computer-generated beam-shaping holograms

For the design of computer-generated holograms reconstructing certain intensity patterns with phase freedom, we use an object-oriented approach. The given intensity pattern is decomposed into elementary objects for which appropriate phase-only hologram functions can be constructed. The total hologram function is found by the subsequent superposition of its constituents, with a relative amplitude and phase weighting for each of them. Thus, the degrees of freedom are dramatically reduced compared with those of sampling approaches. The design algorithm allows us to compensate on the one hand for the intensity and phase distribution of the impinging laser beam and on the other hand for the shape of the hologram aperture. We report on the computer-aided design of such holograms, as well as their fabrication through the use of laser lithography and reactive ion etching. Optical reconstructions are shown.     

Fast calculation method for spherical computer-generated holograms

The synthesis of spherical computer-generated holograms is investigated. To deal with the staggering calculation times required to synthesize the hologram, a fast calculation method for approximating the hologram distribution is proposed. In this method, the diffraction integral is approximated as a convolution integral, allowing computation using the fast-Fourier-transform algorithm. The principles of the fast calculation method, the error in the approximation, and results from simulations are presented.     

Fast calculation method for computer-generated cylindrical holograms

Since a general flat hologram has a limited viewable area, we usually cannot see the other side of a reconstructed object. There are some holograms that can solve this problem. A cylindrical hologram is well known to be viewable in 360 deg. Most cylindrical holograms are optical holograms, but there are few reports of computer-generated cylindrical holograms. The lack of computer-generated cylindrical holograms is because the spatial resolution of output devices is not great enough; therefore, we have to make a large hologram or use a small object to fulfill the sampling theorem. In addition, in calculating the large fringe, the calculation amount increases in proportion to the hologram size. Therefore, we propose what we believe to be a new calculation method for fast calculation. Then, we print these fringes with our prototype fringe printer. As a result, we obtain a good reconstructed image from a computer-generated cylindrical hologram.     

Multiplexed computer-generated holograms with polygonal-aperture layouts optimized by genetic algorithm

Using a novel genetic algorithm (GA) with a Lamarckian search we optimize the polygonal layout of a new type of multiplexed computer-generated hologram (MCGH) with polygonal apertures. A period ofthe MCGH is divided into cells, and the cell is further divided into polygonal apertures according to a polygonal layout, which is to be optimized. Among an ensemble of 1.21 x 10(24) possible polygonal layouts, we take a population of 102 solutions, which are coded as chromosomes of bits, and find the optimal solution with our GA. We introduce rank-based selection with cumulative normal distribution fitness, double crossover, exponentially decreasing mutation probability and Lamarckian downhill search with a small number of offspring chromosomes into our GA, which shows a rapid convergence to the global minimum of the cost function. In a second step of optimization the phase distributions over the subholograms in the MCGH are determined with our iterative subhologram design algorithm. Our MCGH designs show large-sie reconstructed images with high diffraction efficiency and low reconstruction error.     

Optical Hough-transform processor with a two-dimensional array of computer-generated holograms

An optical implementation of the Hough transform that uses a two-dimensional array of computergenerated holograms based on a direct-binary-search algorithm is investigated. A Hough-transform filter consisting of 16 × 16 Fourier-transform direct-binary-search computer-generated holograms is examined. A novel matrix format, which uses the parameter domain in the Hough transform instead of a conventional orthogonal coordinate system, enables highly flexible fabrication of a Hough-transform filter by reducing constraints for reconstructed sample points of a computer-generated hologram. A completed Hough-transform filter has excellent performance both in the quality of the reconstructed image and in diffraction efficiency.     

Phase-shifting with computer-generated holograms written on a spatial light modulator

We propose a new computer-controlled phase-shifting method based on computer-generated holograms (CGHs) displayed on a spatial light modulator (SLM). In this method the accurate phase shifts required in phase-shifting digital holography or interferometry are induced by a suitable transformation of the encoding patterns of the CGH displayed on a SLM. Both the theoretical analysis and the experimental results demonstrate the feasibility of this approach. We also discuss possible applications of this method in the field of interferometric null testing of aspheres.     

Analysis of phase sensitivity for binary computer-generated holograms

A binary diffraction model is introduced to study the sensitivity of the wavefront phase of binary computer-generated holograms on groove depth and duty-cycle variations. Analytical solutions to diffraction efficiency, diffracted wavefront phase functions, and wavefront sensitivity functions are derived. The derivation of these relationships is obtained by using the Fourier method. Results from experimental data confirm the analysis. Several phase anomalies were discovered, and a simple graphical model of the complex fields is applied to explain these phenomena.     

Polarization-controlled multistage switch based on polarization-selective computer-generated holograms

We describe a polarization-controlled free-space optical multistage interconnection network based on polarization-selective computer-generated holograms: optical elements that are capable of imposing arbitrary, independent phase functions on horizontally and vertically polarized monochromatic light. We investigate the design of a novel nonblocking space-division photonic switch architecture. The multistage-switch architecture uses a fan-out stage, a single stage of 2 x 2 switching elements, and a fan-in stage. The architecture is compatible with several control strategies that use 1 x 2 and 2 x 2 polarization-controlled switches to route the input light beams. One application of the switch is in a passive optical network in which data is optically transmitted through the switch with a time-of-flight delay but without optical-to-electrical conversions at each stage. We have built and characterized a proof-of-principle 4 x 4 free-space switching network using three cascaded stages of arrayed birefringent computer-generated holographic elements. Data modulated at 20 MHz/channel were transmitted through the network to demonstrate transparent operation.     

Dual-wave-front computer-generated holograms for quasi-absolute testing of aspherics

Testing of aspherics by means of computer-generated holograms (CGHs) is well known. To perform a quasi-absolute test of rotationally symmetric aspheric surfaces, two wave fronts must be encoded in the CGH. Both the null lens and a spherical lens have to be stored. This enables successive measurements of the aspheric and of a cat's-eye position without changing the object arm of the interferometer. Two possibilities for encoding both wave fronts have been investigated. A first-order approximation for estimating the influence of disturbing diffraction orders is given.     

Rendering of specular surfaces in polygon-based computer-generated holograms

A technique is presented for realistic rendering in polygon-based computer-generated holograms (CGHs). In this technique, the spatial spectrum of the reflected light is modified to imitate specular reflection. The spectral envelopes of the reflected light are fitted to a spectral shape based on the Phong reflection model used in computer graphics. The technique features fast computation of the field of objects, composed of many specular polygons, and is applicable to creating high-definition CGHs with several billions of pixels. An actual high-definition CGH is created using the proposed technique and is demonstrated for verification of the optical reconstruction of specular surfaces.     

Multicriteria optimality for iterative encoding of computer-generated holograms

We propose a new multicriteria method for the determination of computer-generated holograms (CGH's). For this purpose, the direct binary search (DBS) algorithm for computing CGH's has been modified to converge on a new error function that defines the optimal trade-off among different criteria. This approach allows us to control the trade-off between the amplitude error and the diffraction efficiency and to provide a rigorous figure of merit. Comparisons among different encoding methods show that the best results are obtained with a modified version of the DBS method combined with the iterative Fourier transform algorithm.     

Multiplexed computer-generated holograms with irregular-shaped polygonal apertures and discrete phase levels

We propose a novel type of multiplexed computer-generated hologram (MCGH) with irregular-shaped polygonal apertures and discrete phase levels. Each elementary cell forming the new MCGH is divided into a central aperture and several peripheral apertures. The new MCGH allows us to exploit the huge space-bandwidth product provided by standard lithography technologies. With use of the Abbe transform, the Fraunhofer diffraction patterns from the polygonal apertures and, therefore, the layout coefficients can be computed with simple algebraic expressions. Several symmetries related to the polygonal apertures also facilitate the layout-coefficient computation. In the novel iterative subhologram design algorithm (ISDA), we consider all subholograms equally and apply the image-plane constraint to the total reconstructed image, which is the coherent addition of the subimages from the subholograms. We designed MCGHs with several billions of pixels per period, which cannot be achieved with the classical iterative Fourier transform algorithm, because of the prohibitive computational cost and memory limitation. MCGHs with irregular polygonal apertures and discrete phases, which were designed by the ISDA, reconstruct a desired image of large size with high diffraction efficiencies and low reconstruction errors.     

Fabrication error analysis and experimental demonstration for computer-generated holograms

Aspheric optical surfaces are often tested using computer-generated holograms (CGHs). For precise measurement, the wavefront errors caused by the CGH must be known and characterized. A parametric model relating the wavefront errors to the CGH fabrication errors is introduced. Methods are discussed for measuring the fabrication errors in the CGH substrate, duty cycle, etching depth, and effect of surface roughness. An example analysis of the wavefront errors from fabrication nonuniformities for a phase CGH is given. The calibration of these effects for a CGH null corrector is demonstrated to cause measurement error less than 1 nm.     

Fast numerical generation and encryption of computer-generated Fresnel holograms

In the past two decades, generation and encryption of holographic images have been identified as two important areas of investigation in digital holography. The integration of these two technologies has enabled images to be encrypted with more dimensions of freedom on top of simply employing the encryption keys. Despite the moderate success attained to date, and the rapid advancement of computing technology in recent years, the heavy computation load involved in these two processes remains a major bottleneck in the evolution of the digital holography technology. To alleviate this problem, we have proposed a fast and economical solution which is capable of generating, and at the same time encrypting, holograms with numerical means. In our method, the hologram formation mechanism is decomposed into a pair of one-dimensional (1D) processes. In the first stage, a given three-dimensional (3D) scene is partitioned into a stack of uniformed spaced horizontal planes and converted into a set of hologram sublines. Next, the sublines are expanded to a hologram by convolving it with a 1D reference signal. To encrypt the hologram, the reference signal is first convolved with a key function in the form of a maximum length sequence (also known as MLS, or M-sequence). The use of a MLS has two advantages. First, an MLS is spectrally flat so that it will not jeopardize the frequency spectrum of the hologram. Second, the autocorrelation function of an MLS is close to a train of Kronecker delta function. As a result, the encrypted hologram can be decoded by correlating it with the same key that is adopted in the encoding process. Experimental results reveal that the proposed method can be applied to generate and encrypt holograms with a small number of computations. In addition, the encrypted hologram can be decoded and reconstructed to the original 3D scene with good fidelity.     

Space-variant optical interconnection through the use of computer-generated holograms

A space-variant optical interconnection through the use of computer-generated holograms is proposed, and specific configurations to increase the number of parallel channels are analyzed. To this end, the well-known method based on a matrix composed of subholograms is applied. The field diffracted by each channel (assumed to be square apertures) is calculated through the angular-spectrum technique, and the resulting fields are suitably superimposed to obtain a hologram matrix with a reduced bandwidth. Results show that a compact transmissive planar configuration can be handled; in particular, the small interconnect distance between the array planes and the hologram yields a limited system volume.