Optimal realizable filters and the minimum Euclidean distance principle

Ultrathin van der Waals Metalenses

Ultrathin and flat optical lenses are essential for modern optical imaging, spectroscopy, and energy harvesting. Dielectric metasurfaces comprising nanoscale quasi-periodic resonator arrays are promising for such applications, as they can tailor the phase, amplitude, and polarization of light at subwavelength resolution, enabling multifunctional optical elements. To achieve 2π phase coverage, however, most dielectric metalenses need a thickness comparable to the wavelength, requiring the fabrication of high-aspect-ratio scattering elements. We report ultrathin dielectric metalenses made of van der Waals (vdW) materials, leveraging their high refractive indices and the incomplete phase design approach to achieve device thicknesses down to ∼λ/10, operating at infrared and visible wavelengths. These materials have generated strong interest in recent years due to their advantageous optoelectronic properties. Using vdW metalenses, we demonstrate near-diffraction-limited focusing and imaging and exploit their layered nature to transfer the fabricated metalenses onto flexible substrates to show strain-induced tunable focusing. Our work enables further downscaling of optical elements and opportunities for the integration of metasurface optics in ultraminiature optoelectronic systems.

Eliminating Non-linear Raman Shift Displacement Between Spectrometers via Moving Window Fast Fourier Transform Cross-Correlation

Obtaining consistent spectra by using different spectrometers is of critical importance to the fields that rely heavily on Raman spectroscopy. The quality of both qualitative and quantitative analysis depends on the stability of specific Raman peak shifts across instruments. Non-linear drifts in the Raman shifts can, however, introduce additional complexity in model building, potentially even rendering a model impractical. Fortunately, various types of shift correction methods can be applied in data preprocessing in order to address this problem. In this work, a moving window fast Fourier transform cross-correlation is developed to correct non-linear shifts for synchronization of spectra obtained from different Raman instruments. The performance of this method is demonstrated by using a series of Raman spectra of pharmaceuticals as well as comparing with data obtained by using an existing standard Raman shift scattering procedure. The results show that after the removal of shift displacements, the spectral consistency improves significantly, i.e., the spectral correlation coefficient of the two Raman instruments increased from 0.87 to 0.95. The developed standardization method has, to a certain extent, reduced instrumental systematic errors caused by measurement, while enhancing spectral compatibility and consistency through a simple and flexible moving window procedure.

Image correlation method for DNA sequence alignment

The complexity of searches and the volume of genomic data make sequence alignment one of bioinformatics most active research areas. New alignment approaches have incorporated digital signal processing techniques. Among these, correlation methods are highly sensitive. This paper proposes a novel sequence alignment method based on 2-dimensional images, where each nucleic acid base is represented as a fixed gray intensity pixel. Query and known database sequences are coded to their pixel representation and sequence alignment is handled as object recognition in a scene problem. Query and database become object and scene, respectively. An image correlation process is carried out in order to search for the best match between them. Given that this procedure can be implemented in an optical correlator, the correlation could eventually be accomplished at light speed. This paper shows an initial research stage where results were “digitally” obtained by simulating an optical correlation of DNA sequences represented as images. A total of 303 queries (variable lengths from 50 to 4500 base pairs) and 100 scenes represented by 100 x 100 images each (in total, one million base pair database) were considered for the image correlation analysis. The results showed that correlations reached very high sensitivity (99.01%), specificity (98.99%) and outperformed BLAST when mutation numbers increased. However, digital correlation processes were hundred times slower than BLAST. We are currently starting an initiative to evaluate the correlation speed process of a real experimental optical correlator. By doing this, we expect to fully exploit optical correlation light properties. As the optical correlator works jointly with the computer, digital algorithms should also be optimized. The results presented in this paper are encouraging and support the study of image correlation methods on sequence alignment.

The minimum Euclidean distance principle applied to improve the modulation diffraction efficiency in digitally controlled spatial light modulators

Digital addressing of the electrical signal in spatial light modulators, as it is the case in present liquid crystal on silicon (LCoS) displays, may lead to temporal phase fluctuations in the optical beam. In diffractive optics applications a reduction in the modulation diffraction efficiency may be expected. Experimental work is done characterizing the fluctuations amplitude and phase depth for three different digital addressing sequences. We propose a diffractive model to evaluate the modulation diffraction efficiency of phase diffractive optical elements (DOEs) in the presence of phase fluctuations. Best results are obtained for the most stable electrical sequence even though its phase depth is as small as 280 degrees . The results show good agreement with the numerical calculation given by the model.

Adaptive phase-input joint transform correlator

An adaptive phase-input joint transform correlator for pattern recognition is presented. The input of the system is two phase-only images: input scene and reference. The reference image is generated with a new iterative algorithm using phase-only synthetic discriminant functions. The algorithm takes into account calibration lookup tables of all optoelectronics elements used in optodigital experiments. The designed adaptive phase-input joint transform correlator is able to reliably detect a target and its distorted versions embedded into a cluttered background. Computer simulations are provided and compared with those of various existing joint transform correlators in terms of discrimination capability, tolerance to input additive noise, and to small geometric image distortions. Experimental optodigital results are also provided and discussed.

Signal-to-noise-ratio analysis for nonlinear N-ary phase filters

The problem of recognizing targets in nonoverlapping clutter using nonlinear N-ary phase filters is addressed. Using mathematical analysis, expressions were derived for an N-ary phase filter and the intensity variance of an optical correlator output. The N-ary phase filter was shown to consist of an infinite sum of harmonic terms whose periodicity was determined by N. For the intensity variance, it was found that under certain conditions the variance was minimized due to a previously undiscovered phase quadrature effect. Comparison showed that optimal real filters produced greater signal-to-noise-ratio values than the continuous phase versions as a consequence of this effect.

Modulation light efficiency of diffractive lenses displayed in a restricted phase-mostly modulation display

We present an analysis of the diffraction efficiency of diffractive lenses displayed on spatial light modulators that depends on the modulation response of the display. An ideal display would produce continuous phase-only modulation, reaching a maximum phase-modulation depth of 2pi. We introduce the concept of modulation diffraction efficiency that accounts for the effect of nonlinearities only in the phase modulation of the display. We review a diffractive model with which to evaluate this modulation efficiency, including modulation defects such as nonlinear phase modulation, coupled amplitude modulation, phase quantization, and a limited modulation depth. We apply this diffractive model to Fresnel lenses and show that these modulation defects produce a lens multiplex effect. Finally we demonstrate that the application of a minimum Euclidean projection principle leads to high modulation diffraction efficiency even if the phase-modulation depth is much less than 2pi. We demonstrate that the modulation efficiency can exceed 90% for a modulation depth of 1.4pi and can exceed 40% (the equivalent for a binary phase element) for a modulation depth of only 0.7pi. Experimental results from use of a twisted nematic liquid-crystal display are presented to confirm these conclusions.

Generalization of the Jared and Ennis method of complex transmittance objects for the generation of synthetic discriminant function filters

We present a simple method of constructing synthetic discriminant function filters optimized to take into account the modulation of liquid-crystal devices. This relaxation algorithm, a generalization of the Jared and Ennis method, is an iterative method that includes arbitrary modulations for both scene and filter, extending the problem to the complex plane. Simulated and experimental results obtained in a VanderLugt correlator are presented for a two-class recognition problem. The optimal number of images needed to describe an object in a filter generated in this way is discussed, and the influence of the spatial light modulation resolution on the correlation is studied.

Generality of matched filtering and minimum Euclidean distance projection for optical pattern recognition

Matched filtering followed by a minimum Euclidean distance projection onto realizable filter values was previously shown to optimize the signal-to-noise ratio for single training images in optical correlation pattern recognition. The algorithm is now shown to solve the combination of (1) standard statistical pattern-recognition metrics with multiple training images, (2) additive input noise of known power spectral density and also additive detection noise that is irreducible by the filter, (3) the building of the filter on arbitrary subsets of the complex unit disk, and (4) the use of observable correlator outputs only. The criteria include the Fisher ratio, the Bayes error and Bayes cost, the Chernoff and Bhattacharyya bounds, the population entropy and expected information, versions of signal-to-noise ratio that use other than second power in their norm, and the area under the receiver operating characteristic curve. Different criteria are optimized by different complex scalar weights.

Optimized quantization for diffractive phase elements by use of uneven phase levels

Many applications of diffractive phase elements involve the calculation of a continuous phase profile, which is subsequently quantized for fabrication. The quantization process maps the continuous range of phase values to a limited number of discrete steps. We present a new scheme with unevenly spaced levels for the design of diffractive elements and apply it to the design of intracavity mode-selecting elements. We show that this modified quantization can produce significantly better results than are possible with a regular or even the bias-phase-optimized quantization scheme that we reported here earlier. In principle this process can be employed to a greater or lesser extent in any quantization process, allowing the fabrication of diffractive elements with much improved performance.

Enumeration of illumination and scanning modes from real-time spatial light modulators

Using a phase-only spatial light modulator (SLM) in a Fourier transform setup together with fast diffractive optics design algorithms provides a way to automatically generate complex and rapidly changing laser illumination patterns in the far-field. We propose a hierarchical software structure for the adaptive, on-line design of far-field illumination patterns. Using the on-line design system together with camera feedback of the illuminated scene would make it possible to detect and actively laser designate multiple objects in parallel. Possibilities for multispot, arbitrary trajectory scanning and also broad-area speckle-reduced illumination are demonstrated with experimentally measured diffraction pattern sequences from a 120 x 128 pixel phase-only SLM.

Fully complex synthetic discriminant functions written onto phase-only modulators

Synthetic discriminant functions (SDF's) are an effective tool for pattern-recognition applications. However, their experimental implementation is difficult because of the difficulty in writing full complex modulation functions onto spatial light modulators (SLM's) with restricted coding domains. Iterative methods are required for the implementation of SDF filters in real SLM's. A great deal of experimental research has been done with phase-only filters because they can be successfully implemented with liquid-crystal SLM's. We have recently introduced a technique for encoding arbitrary amplitude information onto the phase-only filter, thus allowing us to encode an arbitrary complex function onto a phase-only SLM. We apply this technique to the generation of arbitrary complex SDF filters, thus avoiding the necessity of iterative algorithms. We examine the discrimination capabilities of fully complex SDF filters designed with different parameters and constraints. Experimental results obtained with liquid-crystal SLM's are included.

Improved-fidelity error diffusion through blending with pseudorandom encoding

Error diffusion (ED) and pseudorandom encoding (PRE) methods of designing Fourier transform holograms are compared in terms of their properties and the optical performance of the resulting far-field diffraction patterns. Although both methods produce a diffuse noise pattern due to the error between the desired fully complex pattern and the encoded modulation, the PRE errors reconstruct uniformly over the nonredundant band-width of the discrete-pixel spatial light modulator, while the ED errors reconstruct outside the window of the designed diffraction pattern. Combining the two encoding methods produces higher-fidelity diffraction patterns than either method produces individually. For some designs the fidelity of the ED-PRE algorithm is even higher over the entire nonredundant bandwidth than for the previously reported [J. Opt. Soc. Am. A 16, 2425 (1999)] minimum-distance-PRE algorithm.

Wiener-like correlation filters

We introduce a new, to our knowledge, design for a Wiener-like correlation filter, which consists of cascading a phase-only filter (POF) with a photorefractive Wiener-like filter. Its performance is compared with that of the POF and the Wiener correlation filter (WCF). Correlation results show that for intermediate and higher levels of noise this correlation filter has a peak-to-noise ratio that is larger than that of either the POF or the WCF while still preserving a correlation peak that is almost as high as that of the POF.

Modified minimum-distance criterion for blended random and nonrandom encoding

Two pixel-oriented methods for designing Fourier transform holograms--pseudorandom encoding and minimum-distance encoding-usually produce higher-fidelity reconstructions when combined than those produced by each method individually. In previous studies minimum-distance encoding was defined as the mapping from the desired complex value to the closest value produced by the modulator. This method is compared with a new minimum-distance criterion in which the desired complex value is mapped to the closest value that can be realized by pseudorandom encoding. Simulations and experimental measurements using quantized phase and amplitude modulators show that the modified approach to blended encoding produces more faithful reconstructions than those of the previous method.

Encoding amplitude information onto phase-only filters

We report a new, to our knowledge, technique for encoding amplitude information onto a phase-only filter with a single liquid-crystal spatial light modulator. In our approach we spatially modulate the phase that is encoded onto the filter and, consequently, spatially modify the diffraction efficiency of the filter. Light that is not diffracted into the first order is sent into the zero order, effectively allowing for amplitude modulation of either the first-order or the zero-order diffracted light. This technique has several applications in both optical pattern recognition and image processing, including amplitude modulation and inverse filters. Experimental results are included for the new technique.

Filter implementation technique for multicriteria characterization of coding domains in the joint transform correlator

An improved method for implementing correlation filters in the joint transform correlator architecture is proposed. We derived the method from computer-generated holography techniques. It allows us to use any correlation filters, especially ones that provide an optimal trade-off between noise robustness, peak sharpness, and optical efficiency, with any spatial light modulator (SLM). This method also allows for an objective comparison of the performance of the coding domains of various SLM's.

Pseudorandom encoding for real-valued ternary spatial light modulators

Pseudorandom encoding with quantized real modulation values encodes only continuous real-valued functions. However, an arbitrary complex value can be represented if the desired value is first mapped to the closest real value realized by use of pseudorandom encoding. Examples of encoding real- and complex-valued functions illustrate performance improvements over conventional minimum distance mapping methods in reducing peak sidelobes and in improving the uniformity of spot arrays.

Ternary pseudorandom encoding of Fourier transform holograms

Pseudorandom encoding is a statistical method for designing Fourier transform holograms by mapping ideal complex-valued modulations onto spatial light modulators that are not fully complex. These algorithms are notable because their computational overhead is low and because the space-bandwidth product of the encoded signal is identical to the number of modulator pixels. All previous pseudorandom-encoding algorithms were developed for analog modulators. A less restrictive algorithm for quantized modulators is derived that permits fully complex ranges to be encoded with as few as three noncollinear modulation values that are separated by more than 180 degrees on the complex plane.

Operating-curve selection for optical and digital correlation of fingerprints

We present a new method for selecting the optimal spatial light modulator operating curve on which to encode an ideal complex-valued filter function. The method is an extension of the optimization procedure that was used to derive the ideal complex-valued filter. To illustrate the method we consider the selection of a typical operating curve of a commercially available LCD and use this operating curve to encode a filter that was optimized for the task of fingerprint verification. We then extend the method to include the selection of the optimal operating curve from a series of synthetic operating curves.

Full-range, continuous, complex modulation by the use of two coupled-mode liquid-crystal televisions

Switchable, continuous, complex-amplitude modulation is demonstrated with two cascaded, twisted nematic liquid-crystal televisions (LCTV's), both operating in phase- and amplitude-coupled modulation modes. The condition for full-range complex modulation is that one of the LCTV's must provide a 2π-range phase modulation. A look-up table encoding method is proposed that permits the compensation of phase-amplitude coupling and nonlinearity in the two individual LCTV modulations. Experimental techniques for determining the LCTV-device parameters, for maximizing the phase-mostly modulation range and the amplitude-mostly modulation contrast, and for testing the complex-amplitude modulation are developed. Optical complex-amplitude Fresnel holograms are shown.

Random phase encoding of composite fully complex filters

The mapping of complex-valued functions onto phase-only spatial light modulators is examined. Random phase encoding effectively adds amplitude control to the phase-only filter and can be used to trade off systematic errors of the phase-only filter for random errors. This is illustrated for the problem of recognizing a threedimensional object from arbitrary views. The complex-valued composite filters that constitute a filter bank design are encoded by phase-only and pseudorandom methods. The best recognition probabilities are achieved by blending the two methods so that only the smallest amplitudes are randomly encoded.

Analysis of spatial light modulator contrast ratios and optical correlation

We have performed a general analysis of optical correlators with spatal light modulators (SLM's) whose primary defect is a finite contrast ratio (CR). Our mathematical analysis identifies three noise terms that appear in addition to the correlation term. The filter SLM contains either a phase-only filter (POF) or a binary-phase-only filter (BPOF). Insertion of a dc block at the center of the filter SLM decreases the noise background in the correlator plane; this dc block is larger than that required for the same level of performance in a correlator whose SLM's have transmissive (or reflective) dead zones. With a noise-free input and the dc block, our computer simulations that show the peak intensity falling off as the CR decreases are in quantitative agreement with the correlation term of the mathematical model. For a cluttered, disjoint noise input this agreement is only qualitative, and at low CR's the dc block is definitely required for the BPOF correlator if the secondary peaks in the output are to be brought below the correlation peak.

Optical correlation of images with signal-dependent noise using constrained-modulation filter devices

Images with signal-dependent noise present challenges beyond those of images with additive white or colored signal-independent noise in terms of designing the optimal 4-ƒ correlation filter that maximizes correlation-peak signal-to-noise ratio, or combinations of correlation-peak metrics. Determining the proper design becomes more difficult when the filter is to be implemented on a constrained-modulation spatial light modulator device. The design issues involved for updatable optical filters for images with signal-dependent film-grain noise and speckle noise are examined. It is shown that although design of the optimal linear filter in the Fourier domain is impossible for images with signal-dependent noise, proper nonlinear preprocessing of the images allows the application of previously developed design rules for optimal filters to be implemented on constrained-modulation devices. Thus the nonlinear preprocessing becomes necessary for correlation in optical systems with current spatial light modulator technlogy. These results are illustrated with computer simulations of images with signal-dependent noise correlated with binary-phase-only filters and ternary-phase-amplitude filters.

Multilevel phase- and amplitude-encoded modified-filter synthetic-discriminant-function filters

The performance of the modified-filter synthetic-discriminant-function (MfSDF) filter with multilevel phase and amplitude (MLAP) constraints is investigated with various in-plane rotated images from an in-class Bradley armored personnel carrier vehicle and an out-of-class Abram MI tank; this is of interest because of the commercial availability of liquid-crystal televisions, which are able to encode the gray-level amplitude and/or the discrete multilevel phase information. The evaluation is performed to explain better the image-distortion range that can be covered effectively by MLAP/MfSDF filters. The results show that the MLAP/MfSDF filter offers much-improved correlator system performance with a greater allowable image-distortion range while maintaining 100% discrimination between in-class and out-ofclass images; furthermore, it shows an improved ability to accommodate the input image noise when compared with the MfSDF filter with a binary phase-only constraint. Thus the MLAP/MfSDF can be implemented effectively by a hybrid optical/digital correlator system to track a vehicle or a tank dynamically as it moves along a random trajectory across the input field.

Theory and optical implementation of the geometrical approach of multiple circular harmonic filters

The circular harmonic filter contains only one component of the image. Its discrimination capability has been questionable. The geometrical approach of multiple circular harmonic filters uses relative locations of the correlation peaks as the rotation-, shift-, and intensity-invariant features for pattern recognition. Each feature depends on the entire image. This approach has a good discrimination capability. Optical real-time implementation of the on-axis continuous phase-only circular harmonic filters by the use of a commercial liquid-crystal television is shown. A harmonic analysis shows that the phase-mostly filter can tolerate coupled amplitude modulation at the acceptable expense of the output signal-to-noise ratio. An optical experiment of the geometrical approach of multiple circular harmonic filters for a multiple-image input is described. The cross-correlation peaks between the individual filters and the clutter are eliminated, because they are not in good locations.

Computation of arbitrarily constrained synthetic discriminant functions

An algorithm for computing correlation filters based on synthetic discriminant functions that can be displayed on current spatial light modulators is presented. The procedure is nondivergent, computationally feasible, and capable of producing multiple solutions, thus overcoming some of the pitfalls of previous methods.

Phase-coding technique for one-way image transmission through aberrating media

We propose and demonstrate one-way image transmission through an aberrating medium. In contrast to previous techniques, which usually use exact phase compensation for correcting phase distortion, our technique uses the phase- or amplitude-coded form of the exact phase. This technique allows us to utilize a wide variety of spatial light modulators for image correction in one-way image transmission through aberrating media.

Optical correlation: influence of the coding of the input image

We analyze the influence of different optical coding methods of the input image in optical correlators. The noise robustness and the optical efficiency of the correlator are investigated. We show in particular that the signal-to-noise ratio is greatly dependent on the coding method. It decreases drastically for large phase modulation.

Optimal trade-off synthetic discriminant function filters for arbitrary devices

A new correlation-filter design methodology is presented for achieving two objectives: synthetic discriminant function filters that can be implemented on arbitrary various criteria of interest. devices and that can provide optimal trade-off among various criteria of interest.

Effects of systematic phase errors on phase-only correlation

The performance of phase-only optical correlators is usually reduced if the filter-plane phase differs from that prescribed for the classical matched filter. Current spatial light modulators, which frequently produce less than 2π phase modulation, and interface circuits, which quantize or incorrectly amplify signals placed on the spatial light modulator, both can produce systematic phase errors. We examine these effects using a model of correlation-peak amplitude as a function of phase error. The correlation peak is reasonably approximated as the product of an average of unity-amplitude error phasors multiplied by the average amplitude across the filter plane. The trends predicted by this new model compare favorably with computer simulations that use gray-scale images.

Multicriteria characterization of coding domains with optimal Fourier spatial light modulator filters

A multicriteria optimization method is introduced in order to find optimal filters for implementation on arbitrary spatial light modulators in the Fourier plane of an optical correlator. This method is applied to the trade-offs between noise robustness, sharpness of the correlation peak, and optical efficiency. A fast and simple algorithm is given in this case, which is independent of the particular form of the spatial light modulator coding constraint. It is used to characterize and to compare typical coding domains through the performances of their associated optimal filters.