The Diffraction Efficiency of Double-grating Holographic Fan-out Elements

Abstract

An analytical expression is derived for the diffraction efficiency of a holographic fan-out element containing two superimposed gratings separated by a small angle. In addition to the usual zero and first orders an infinite set of the significant spurious waves is considered. In deriving the expression all these spurious waves are assumed to satisfy the Bragg condition exactly. The results are compared with a direct numerical solution and give very good agreement for angles up to 1°. Furthermore the analysis provides useful information for angles up to 5°. An analogy is drawn with a hybrid hologram consisting of a thick ‘carrier’ grating and a thin ‘modulation’ grating. Using this model the diffraction efficiencies can be calculated using the standard grating formulae. For fan-out applications the spurious waves around the first orders are considered the most important. From this analysis it can be seen that they can be made negligibly small but at the cost of a reduction in total diffraction efficiency.     

Redistribution of the zero order by the use of a phase checkerboard pattern in computer generated holograms

A checkerboard phase plate is proposed to be used together with computer generated holograms to eliminate the zero order by working as a convolution function that shifts the zero order away from the center of a reconstructed pattern. By performing a preshift in the desired hologram pattern, it is possible to obtain a reconstructed pattern that is free of zero order. Simulation results have shown that the technique is tolerant of fabrication errors in the hologram. The technique is also shown to effectively reduce the zero order intensity by two orders in the presence of phase depth errors in the checkerboard. Experimental results using a spatial light modulator support the results shown in the simulation.     

Wavelength-dependent diffraction patterns from a liquid crystal display

Liquid crystal displays (LCDs) are invaluable for a variety of optical applications, including the encoding of programmable diffractive optical elements. The pixel structure in these devices produces a set of diffracted orders of which the central order is the strongest. In most devices that we have examined, the intensity distribution of the diffraction pattern is independent of the wavelength of the illuminating light. Recently we have been examining the performance of LCDs having very small pixel sizes. We compare results for two devices from the same manufacturer. One of them exhibits the normal behavior. For the other, we find surprisingly strong wavelength dependence. The diffraction pattern varies from having most of the energy in the zero order for long wavelengths to having the energy distributed among 50-60 orders as the wavelength decreases. We attribute this behavior to a phase structure over each pixel. We analyze this behavior using a simple two-dimensional model that qualitatively explains the phenomenon. These results can be viewed in two ways--on the positive side this behavior might lead to optical logic or fan-out applications. On the negative side, there is less intensity available in the normally used zero order.     

Frequency and phase swept holograms in spectral hole-burning materials

A new hologram type in spectral hole-burning systems is presented. During exposure, the frequency of narrow-band laser light is swept over a spectral range that corresponds to a few homogeneous linewidths of the spectrally selective recording material. Simultaneously the phase of the hologram is adjusted as a function of frequency-the phase sweep function. Because of the phase-reconstructing properties of holography, this recording technique programs the sample as a spectral amplitude and phase filter. We call this hologram type frequency and phase swept (FPS) holograms. Their properties and applications are summarized, and a straightforward theory is presented that describes all the diffraction phenomena observed to date. Thin FPS holograms show strongly asymmetric diffraction into conjugated diffraction orders, which is an unusual behavior for thin transmission holograms. Investigations demonstrate the advantages of FPS holograms with respect to conventional cw recording techniques in freq ncymultiplexed data storage. By choosing appropriate phase sweep functions, various features of holographic data storage can be optimized. Examples for cross-talk reduction, highest diffraction efficiency, and maximal readout stability are demonstrated. The properties of these FPS hologram types are deduced from theoretical considerations and confirmed by experiments.     

Design and fabrication of a high-efficiency extreme-ultraviolet binary phase-only computer-generated hologram

As the development of extreme-ultraviolet (EUV) lithography progresses, interest grows in the extension of traditional optical components to the EUV regime. The strong absorption of EUV by most materials and its extremely short wavelength, however, make it very difficult to implement many components that are commonplace in the longer wavelength regimes. One such component is the diffractive optical element used, for example, in illumination systems to efficiently generate modified pupil fills. The fabrication and characterization of an EUV binary phase-only computer-generated hologram is demonstrated, allowing arbitrary far-field diffraction patterns to be generated. Based on reflective architecture, the fabricated device is extremely efficient. Based on an identically fabricated null hologram, the absolute efficiency into one diffracted order of 22% has been demonstrated. In the case where axially symmetric diffraction patterns are desired (both positive and negative diffraction orders can be used), the efficiency can be twice as high.     

The Diffraction Efficiency of Triple Grating Holographic Fan-out Elements

Operator expressions are derived for the diffraction efficiency of a holographic fan-out element containing an odd number of superimposed gratings each separated by a small fixed angle from its neighbour. An infinite set of spurious waves are considered in the analysis in addition to the usual zero and first orders. Orders which are coincident are combined and the problem is cast in terms of coupled wave equations linking these combined orders. By assuming all the significant orders exactly satisfy the Bragg condition the problem is solved. In the case of three superimposed gratings simple analytical expansions of the operator expressions are obtained. The results when compared with direct numerical calculations give very good agreement for angular separations up to 1°. Useful information is also obtainable for larger angles provided the hologram is only of moderate thickness.     

Suppression of undesired diffraction orders of binary phase holograms

A method to remove undesired diffraction orders of computer-generated binary phase holograms is demonstrated. Normally, the reconstruction of binary Fourier holograms, made from just two phase levels, results in an undesired inverted image from the minus first diffraction order, which is superposed with the desired one. This can be avoided by reconstructing the hologram with a diffuse light field with a pseudorandom, but known, phase distribution, which is taken into account for the hologram computation. As a consequence, only the desired image is reconstructed, whereas all residual light is dispersed, propagating as a diffuse background wave. The method may be advantageous to employ ferroelectric spatial light modulators as holographic display devices, which can display only binary phase holograms, but which have the advantage of fast switching rates.     

Spatial filtering for zero-order and twin-image elimination in digital off-axis holography

Off-axis holograms recorded with a CCD camera are numerically reconstructed with a calculation of scalar diffraction in the Fresnel approximation. We show that the zero order of diffraction and the twin image can be digitally eliminated by means of filtering their associated spatial frequencies in the computed Fourier transform of the hologram. We show that this operation enhances the contrast of the reconstructed images and reduces the noise produced by parasitic reflections reaching the hologram plane with an incidence angle other than that of the object wave.     

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.     

Spatial-phase code-division multiple-access system with multiplexed Fourier holography switching for reconfigurable optical interconnection

A new, to our knowledge, space-variant optical interconnection system based on a spatial-phase code-division multiple-access technique with multiplexed Fourier holography is described. In this technique a signal beam is spread over wide spatial frequencies by an M-sequence pseudorandom phase code. At a receiver side a selected signal beam is properly decoded, and at the same time its spatial pattern is shaped with a Fourier hologram, which is recorded by light that is encoded with the same M-sequence phase mask as the desired signal beam and by light whose spatial beam pattern is shaped to a signal routing pattern. Using the multiplexed holography, we can simultaneously route multisignal flows into individually specified receiver elements. The routing pattern can also be varied by means of switching the encoding phase code or replacing the hologram. We demonstrated a proof-of-principle experiment with a doubly multiplexed hologram that enables simultaneous routing of two signal beams. Using a numerical model, we showed that the proposed scheme can manage more than 250 routing patterns for one signal flow with one multiplexed hologram at a signal-to-noise ratio of ~5.     

Creation of line defects in holographic photonic crystals by a double-exposure thresholding method

Recording of periodic variations of amplitude and phase by the interference of coherent laser beams in a hologram offers a natural means for creating one-, two-, and three-dimensional photonic crystals. For device applications such as waveguides in optical communications, one usually needs to create defects in photonic crystals. We present an analysis and an experimental demonstration of a double-exposure method for creating photonic crystals with line defects. The idea is based on the principle of superposition of holographic grating patterns of different spatial periods while the recording medium is held stationary and on the application of a threshold to the recording medium. We use the same symmetrical optical architecture to achieve nondefective and defective holographic photonic crystals. The technique may be extended to the creation of defects based on functional synthesis by means of Fourier series, by use of light sources of other wavelengths with an appropriate high-contrast recording material.     

Holographic Display of Diffraction Patterns Suffering from Third- and Fifth-order Aberrations

The diffraction patterns suffering from third- and fifth-order aberrations are displayed by reconstructing first from the hologram of a point object under proper experimental conditions and secondly from a computer-generated hologram having the desired wavefront aberration. Experiments demonstrating the diffraction patterns with various kinds of aberration of third and fifth orders are presented. The effects of defocusing and stopping down on the aberrated diffraction patterns are observed.     

Plane isoparametric hybrid-stress elements: Invariance and optimal sampling

Formulation and applications of the hybrid-stress finite element model to plane elasticity problems are examined. Conditions for invariance of the element stiffness are established for two-dimensional problems, the results of which are easily extended to three-dimensional cases. Next, the hybrid-stress functional for a 3-D continuum is manipulated into a more convenient form in which the location of optimal stress/strain sampling points can be identified. To illustrate these concepts, 4- and 8-node plane isoparametric hybrid-stress elements which are invariant and of correct rank are developed and compared with existing hybrid-stress elements. For a 4-node element, lack of invariance is shown to lead to spurious zero energy modes under appropriate element rotation. Alternative 8-node elements are considered, and the best invariant element is shown to be one in which the stress compatibility equations are invoked. Results are also presented which demonstrate the validity of the optimal sampling points, the effects of reduced orders of numerical integration, and the behaviour of the elements for nearly incompressible materials.     

Optical pattern recognition using a unidirectional photorefractive oscillator coupled with an angular multiplexing volume hologram

Abstract

We propose and demonstrate a unidirectional photorefractive ring oscillator that couples with an external angular multiplexing volume hologram for pattern recognition. By configuring the hologram externally, a computer-generated hologram (CGH) and a spatial light modulator are utilized to generate reference beams for the hologram. Two-dimensional images are stored in the form of resonating beams in the system, whereby the nonlinear interaction between the beams allows the image that most resembles an input object to be recognized. Five images were used to illustrate the pattern recognition ability of the oscillator. It was found that the input object was successfully recognized within 10s. The fact that CGHs can generate a large number of beamlets enhances significantly the storage capacity in this system.     

Reconfigurable array interconnection by photorefractive correlation

Electronic parallel processors might communicate more effectively by photons sent through glass or air than by electrons sent through wires, but quickly routing thousands of optical signals remains a problem. Previous photorefractive interconnection networks have dedicated one hologram to each input channel. Instead, we compute a control image from the entire network configuration and store it as a single color-keyedvolume hologram. This lets us use hologram superposition for fast switchingbetween multiple prestored patterns. During operation, data signals from the input modulator array, powered by a shared wavelength-tunable laser, are correlated optically with one color-matched connection hologram to produce the output array. This decouples both data rate and interconnect switching speeds from the slow photorefractive response. We can display arbitrary connection weights using simple binary-phase spatial light modulators and gracefully accommodate modulator limitations by trading off control-image bandwidth for output signal-to-noise ratio. Experimental results with color-multiplexed reflection holograms in z-cut LiNbO(3) confirmed our theoretical predictions that this approach works best for densely connected networks with high fan-in to each output. We obtained an average aggregate signal-to-noise ratio of more than 200:1 for 1024 inputs and outputs.     

Real-time three-dimensional object reconstruction by use of a phase-encoded digital hologram

A three-dimensional (3D) object reconstruction technique that uses only phase information of a phase-shifting digital hologram and a phase-only spatial-light modulator is proposed. It is well known that a digital hologram can store both amplitude and phase information of an optical electric field and can reconstruct the original 3D object in a computer. We demonstrate that it is possible to reconstruct optically 3D objects using only phase information of the optical field calculated from phase-shifting digital holograms. The use of phase-only information enables us to reduce the amount of data in the digital hologram and reconstruct optically the 3D objects using a liquid-crystal spatial light modulator without optical power loss. Numerical evaluation of the reconstructed 3D object is presented.     

An Inventory System with Periodic Regular Review and Flexible Emergency Review

We analyze a discrete time inventory model with two supply modes. Regular orders are placed periodically and arrive after a deterministic lead time. Emergency orders, characterized by a shorter deterministic lead time but higher purchase costs, can be placed in case of imminent stock-outs. We propose and analyze flexible order-up-to policies with emergency orders that arrive one, two,⃛, up to some fixed number of time units before a regular order arrives. Based on an approximate cost model, we develop two sets of approximate explicit optimality conditions. The first set of conditions are rather complicated, but simulation shows that they do lead to near-optimal policies in most cases. The second set of conditions are very simple and seem more practical, but they only lead to a near-optimal policy if there are enough emergency supply opportunities in a review period to prevent most backorders.     

Complex Fresnel hologram display using a single SLM

We propose a novel optical method to display a complex Fresnel hologram using a single spatial light modulator (SLM). The method consists of a standard coherent image processing system with a sinusoidal grating at the Fourier plane. Two or three position-shifted amplitude holograms displayed at the input plane of the processing system can be coupled via the grating and will be precisely overlapped at the system's output plane. As a result, we can synthesize a complex hologram that is free of the twin image and the zero-order light using a single SLM. Because the twin image is not removed via filtering, the full bandwidth of the SLM can be utilized for displaying on-axis holograms. In addition, the degree of freedom of the synthesized complex hologram display can be extended by involving more than three amplitude holograms.     

Polarization-selective computer-generated holograms: design, fabrication, and applications

We constructed polarization-selective computer-generated holograms that apply an independent phase profile during readout by horizontal and vertical light polarizations. These elements are composed of two surface-relief-etched birefringent substrates joined face to face. We describe the design methodology for arbitrary birefringent substrate and gap materials. We show how these holograms are fabricated with standard microelectronics technology and discuss the effects of etching and alignment errors on performance. We demonstrated a diffraction efficiency of 60% with a polarization contrast ratio of >100:1 using a multilevel phase hologram made from two birefringent lithium niobate substrates. We also showed that a single-layer SiO(2) thin-film antireflection coating on all surfaces can reduce reflections from the high-index substrates without significant effect on hologram performance. We also consider some possible applications of this technology and demonstrate experimentally a dual focal-length lens and a self-interconnecting binary 2 × 2 polarization switch.     

Multichannel holographic recording method for three-dimensional displays

A multichannel holographic recording method is presented for three-dimensional (3D) displays, utilizing pixel-based recording instead of image-based recording in order to realize parallel processing. The proposed approach is composed of two main stages. In the first stage, each two-dimensional (2D) image acquired from multiple viewpoints is partitioned by holographic recording channels (HRC) into nonoverlapping subimages. In the second stage, the corresponding pixels of the subimages are rearranged to constitute an encoding image. The encoding images are recorded simultaneously by each HRC, respectively, so the recording speed is improved significantly. The experimental results have demonstrated that the three-channel system is feasible and the full-parallax hologram reconstructed with white light is acceptable in quality. The three-channel system saves approximately 60% of the recording time in comparison with the single-channel system. More importantly, the proposed method can accomplish a large-scale final hologram composed of multichannel holograms without sacrificing the hologram quality. Several 3D imaging applications such as medical diagnosis and advertisements could benefit from this research.