This editorial discusses recognition of papers in Optical Engineering.

Microgrid polarization cameras have historically experienced lower performance than is typical for monolithic polarization systems. Specifically, their polarizer elements have had a lower extinction ratio and a larger orientation error. We show how to use the calibrated parameters of a nonideal polarizer to modify the polarization measurement model, effectively allowing one to generate high-performance measurements from low-performance elements. We demonstrate the effectiveness of this approach on a commercial polarization camera and estimate the signal-to-noise ratio penalty for using nonideal polarizers in this camera as being 1.25  ×   versus a system using ideal polarizers.

This guest editorial introduces the Special Section on Polarization: Systems, Measurement, Analysis, and Remote Sensing.

An approach is proposed to implement diffractive optical elements for the conversion of the polarization state of beams. Calcite crystal etching technology is developed and applied to manufacture a four-sector polarization converter. The fabricated four-sector polarization converter is experimentally investigated. The orthogonal polarization state of beams in opposite sectors is achieved by selecting a wavelength with a tunable laser. The experimental results of focusing the converted beams are consistent with the numerical simulation.

A holographic projection system based on the encoding of computer-generated holograms onto a spatial light modulator is discussed. We show how the size, location, and polarization state of the output can be controlled completely electronically, without physically moving any element in the system. It is finally shown that the system is capable to produce optical logical operations by superimposing two different images encoded onto orthogonal polarization states. We show how these images can be added or subtracted, giving a polarization-based logic system. Experimental results are included in all cases.

We experimentally demonstrate an iterative method for a vectorial optical field generator (VOF-Gen) to achieve accurate amplitude modulation in creating arbitrary complex beams. The method could converge rapidly in several steps and is effective to optimize the patterns applied to the liquid crystal spatial light modulators on a pixel-by-pixel basis to obtain arbitrary desired amplitude distribution. Meanwhile, this method could also mitigate the speckles caused by the defects of the optical components used in the VOF-Gen system and calibrate the wavefront related to the amplitude distribution. Several kinds of optical fields with different intensity distributions are successfully generated to verify the capability and versatility of the presented technique. This effective method may find many important applications in optical tweezers, microscopy, and unidirectional coupling.

Hypernumerical aperture and polarized illumination are the key technologies of resolution enhancement of lithography. When the numerical aperture reaches 0.85 and above, especially in the immersion lithography, polarization effect must be taken into consideration. The performance of the projection lens needs to be characterized by rigorous polarization aberration. The vector polarization imaging system that is suitable for hypernumerical aperture is established, and the distortion effects introduced by polarization aberration are analyzed. Orientation Zernike polynomials-based method and Pauli–Zernike polynomials-based method are adopted to parameterize the polarization aberration represented by Jones pupil. Critical dimension error, placement error, and normal int. log slope index are introduced as the index to value imaging distortion. The proposed method and analysis conclusion would provide meaningful guidance for projection lens design of lithographic tools.

Retarders or waveplates are tools for polarization modification in bulk optical systems. These devices usually have a strong wavelength dependence in their performance, making them suitable for use over a wavelength band on the order of a few percent of the center wavelength for which they are made. Display and tunable laser applications are examples that can require consistent polarization modification over a much broader wavelength range. We discuss new methods and designs for dramatically increasing range of performance and review older methods as well. We show examples of achievable performance using modern polymer and liquid crystal materials.

We report an in-depth experimental characterization and analysis of an infrared active polarimetric imaging system based on the orthogonality breaking polarization-sensing approach. We first recall the principle of this laser scanning polarimetric imaging technique, based on the illumination of a scene by means of a dual-frequency dual-polarization light source. The experimental design is then described, along with measurements on test scenes with known polarimetric properties used to validate/calibrate the imaging system and to characterize its optical properties (sensitivity and resolution). The noise sources that temporally and spatially affect the quality of the orthogonality breaking data are then investigated. Our results show that the raw temporal signals detected at a given location of the scene are perturbed by Gaussian fluctuations, and the spatial information contained in the images acquired through raster scan of the scene are dominated by speckle noise, which is a common characteristic of active polarimetric imaging systems. Finally, the influence of the source temporal coherence on the images is analyzed experimentally, showing that orthogonality breaking acquisitions can still be performed efficiently with a low-coherence source.

We provide a method for calibrating microgrid polarization cameras that is simpler and easier to set up than existing methods. Applying this method to three different commercially available cameras, we compare the mean values and variances in their diattenuation and orientation properties. We derive formulas giving the accuracy with which the pixel polarization properties can be calibrated in both the Gaussian and Poisson noise regimes and demonstrate the statistical instability of the extinction ratio as a parameter. In a series of calibration measurements, we estimate the pixel-to-pixel variation of polarization properties and show how to separate the effects of temporal noise from manufacturing variation.

We describe a numerical method for obtaining a nondepolarizing estimate from an experimental Mueller matrix, a necessary preliminary step in determining the Jones matrix and the polarization properties of the sample under study. The proposed method, being a variant of the general virtual experiment approach, is based on minimizing the least squares distance between the light intensities virtually generated by the effectively measured Mueller matrix of the sample and by its nondepolarizing estimate, while taking into account the exact phenomenological description of the polarimetric instrument used. It can be applied to complete, as well as to partial (12-element) experimental Mueller matrices. The application of the method is illustrated on experimental examples and its performance is compared to that of alternative approaches.

Optimization of polarimeters has historically been achieved using an assortment of performance metrics. Selection of an optimization parameter is, however, frequently made on an ad hoc basis. We rigorously demonstrate that optimization strategies in Stokes polarimetry based on three common metrics, namely the Frobenius condition number of the instrument matrix, the determinant of the associated Gram matrix, or the equally weighted variance, are frequently formally equivalent. In particular, using each metric, we derive the same set of constraints on the measurement states, correcting a previously reported proof, and show that these can be satisfied using spherical 2 designs. Discussion of scenarios in which equivalence between the metrics breaks down is also given. Our conclusions are equally applicable to optimization of the illumination states in Mueller matrix polarimetry.

A quantum LIDAR for improving resolution using quantum entanglement in the polarization degree of freedom is described. A thorough mathematical analysis of this device is provided. A mathematical discussion of how to generate other more robust entangled states is developed. Internal loss within the entanglement generator and external loss due to atmosphere, detectors, and targets are modeled. A method using these approaches for imaging is provided giving N times classical resolution, where N is the number of photons entangled with explicit results exhibited for N  =  3. Closed form expressions for the wave function, normalization, density matrix, reduced density matrix, visibility, and probabilities of detection of one through three photons using detectors with general polarization characteristics are provided. An explicit entanglement generator and detector designs are provided in terms of linear and nonlinear photonics devices. The fundamental role of postselection measurement for generating entanglement is included. Discussions of entanglement devices that will produce general M&M states at near-visible frequencies are given. A discussion of a bearing measurement device that exhibits both super sensitivity and resolution is provided. Computational results are provided that compare probabilities of detection for three single photon detectors with −45-deg, 45-deg, and 45-deg linear polarization. Results for detecting one to three photons or the vacuum state are compared. Computational results for detecting three photons with these detectors for various values of internal and atmospheric loss are provided. Resolution improvements born of quantum entanglement are shown not to degrade with loss. Loss degrades probability of detection not resolution.

In polarimetric instruments, it is often necessary to characterize the polarimetric dependence at various polarization states. Frequently, this is done by placing a polarizer between the instrument and light source. Certain polarizer materials (e.g., wire grids as opposed to polymer-based materials) tend to reflect a significant amount of light, which can cause second-order reflections in the region between the two polarizers. We characterize the reflections using Jones calculus and discuss their significance for polarization instruments.

We describe a method of tracing a backward (from camera) ray in a scene that contains birefrigent (uniaxial) media. The physics of scattering of an electromagnetic wave by a boundary between two media is well known and is a base for ray tracing methods; but processing of a backward ray differs from scattering of a “natural” forward ray. Say, when a backward ray refracts by a boundary, besides the energy transfer coefficient like for a forward ray, one must account for the radiance change due to beam divergence. We calculate this factor and prove it must be evaluated only for the first and the last media along the ray path while the contributions from the intermediate media mutually cancel. We present a closed numerical method that allows one to perform transformation of a backward ray on a boundary between two media either of which can be birefrigent. We hope it is more convenient and ready for usage in ray tracing engines than known publications. Calculation utilizes Helmholtz reciprocity to calculate directions of scattered rays and their polarization (i.e., Mueller matrices), which is advantageous over a straightforward “reverse” of forward ray transformation. The algorithm was integrated in the lighting simulation system Lumicept and allowed for an efficient calculation of images of scenes with crystal elements.

Collected sources for the different types of visualizations used in the field of polarization imaging are not extensive. Here, we survey and review the different visualization techniques in passive polarimetric imaging. Analysis of the methods is done by applying various concepts from the field of visualization. We provide recommendations for choosing a visualization based on the data structure, spatial frequency, and analysis goals.

We present the results of the first quantitative multimodal confocal imaging study of methylene blue (MB)-stained cancer and normal human renal cells obtained from fine needle aspiration (FNA) biopsies. Fluorescence emission images provided morphological assessment, and fluorescence polarization (Fpol) images yielded quantitative characterization of each cell in the investigated samples. FNA specimens are obtained from discarded malignant and normal renal specimens following surgery. Prior to imaging, the cells are stained in aqueous MB solution. Our results demonstrate that all the specimens investigated are heterogeneous in terms of size and exhibited Fpol. Cancerous specimens predominantly contain cells of larger size that exhibit higher Fpol as compared to normal specimens. Imaging results correlated well with clinical assessment of the samples. Our results suggest that morphological assessment using fluorescence emission imaging and quantitative information provided by Fpol imaging may be valuable in determining the presence or absence of renal cancer cells in FNA specimens.

Representative examples from 3 years of measurements from JPL’s ground-based multiangle spectropolarimetric imager (GroundMSPI) are compared to a Mueller matrix bidirectional reflectance distribution function (mmBRDF). This mmBRDF is used to model polarized light scattering from solar illuminated surfaces. The camera uses a photoelastic-modulator-based polarimetric imaging technique to measure linear Stokes parameters in three wavebands (470, 660, and 865 nm) with a ±0.005 uncertainty in degree of linear polarization. GroundMSPI measurements are made over a range of scattering angles determined from a fixed viewing geometry and varying sun positions over time. This microfacet mmBRDF model predicts an angle of the linear polarization that is consistently perpendicular to the scattering plane and therefore is only appropriate for rough surface types. The model is comprised of a volumetric reflection term plus a specular reflection term of Fresnel-reflecting microfacets. The following modifications to this mmBRDF model are evaluated: an apodizing shadowing function, a Bréon or Gaussian microfacet scattering density function, and treating the surface orientation as an additional model parameter in the specular reflection term. The root-mean-square error (RMSE) between the GroundMSPI measurements and these various forms of the microfacet mmBRDF model is reported. Four example scenes for which a shadowed-Bréon microfacet mmBRDF model yields realistic estimates of surface orientation, and the lowest RMSE among other model options are shown.

Given its unchallenged capabilities in terms of sensitivity and spatial resolution, the combination of imaging spectropolarimetry and numeric Stokes inversion represents the dominant technique currently used to remotely sense the physical properties of the solar atmosphere and, in particular, its important driving magnetic field. Solar magnetism manifests itself in a wide range of spatial, temporal, and energetic scales. The ubiquitous but relatively small and weak fields of the so-called quiet Sun are believed today to be crucial for answering many open questions in solar physics, some of which have substantial practical relevance due to the strong Sun–Earth connection. However, such fields are very challenging to detect because they require spectropolarimetric measurements with high spatial (sub-arcsec), spectral (<100  mÅ), and temporal (<10  s) resolution along with high polarimetric sensitivity (<0.1  %   of the intensity). We collect and discuss both well-established and upcoming instrumental solutions developed during the last decades to push solar observations toward the above-mentioned parameter regime. This typically involves design trade-offs due to the high dimensionality of the data and signal-to-noise-ratio considerations, among others. We focus on the main three components that form a spectropolarimeter, namely, wavelength discriminators, the devices employed to encode the incoming polarization state into intensity images (polarization modulators), and the sensor technologies used to register them. We consider the instrumental solutions introduced to perform this kind of measurements at different optical wavelengths and from various observing locations, i.e., ground-based, from the stratosphere or near space.

Sunlight becomes partially linearly polarized when scattered from atmospheric gas molecules and can be quantified using the linear Stokes parameters S₀  ,  S₁  ,  S₂ and the derived degree of linear polarization and angle of polarization (AoP). The angle-dependent Stokes parameters S₁ and S₂ and the AoP require a reference plane. Commonly used reference planes for polarimetric applications include the instrument, scattering, and solar principal planes, each of which provides unique insights when analyzing sky polarization data. Methods to transform the parameters between each frame of reference are known; however, previous publications have not shown the results of transforming a time series of all-sky polarization images into these different reference planes clearly showing how this alters the image visualization. We review two methods used to rotate all-sky polarization images from the instrument to the scattering plane and the solar principal plane, and for the first time shows all-sky polarization image sequences recorded from sunrise to sunset of Stokes S₁ and S₂ and AoP for each reference frame.

Observations from the Ground-based Multiangle SpectroPolarimetric Imager (GroundMSPI) are used to relate angle of linear polarization (AoLP) measurements to material properties and illumination conditions in sunlit outdoor environments. GroundMSPI is a push-broom spectropolarimetric camera with an uncertainty in degree of linear polarization (DoLP) of ±0.005. This polarimetric accuracy yields useful AoLP images even when the DoLP is less than 0.02. AoLP images are reported with respect to dependency on surface texture, surface orientation, albedo, and illumination conditions. Agreement with well-known principles of polarized light scattering is illustrated, and several special cases are described. Expected observations of AoLP tangential to surface orientation and AoLP perpendicular to the scattering plane are reported. Significant changes in the AoLP are observed from common variations in outdoor illumination conditions. Also, simple variants in material properties change the dominant polarized light scattering process and thus the AoLP. Measurement examples that isolate a 90 deg AoLP flip are shown for a sunny and cloudy day as well as an object of high and low albedo.

The linear electro-optic effect in lithium niobate is capable of realizing a variety of polarization transformations, including TE-to-TM and left-to-right circular polarization conversion. Several types of devices have been demonstrated in Ti-diffused waveguides. LiNbO₃ devices have historically been based on either Ti-diffused or proton-exchanged waveguides. The proton-exchanged waveguides only guide light polarized along the optic axis, and therefore, are not applicable to polarization transforming devices. Zinc oxide diffusion is an alternative waveguide fabrication technology that guides both e- and o-waves with much higher power-handling capability than Ti  :  LiNbO₃ waveguides. ZnO  :  LiNbO₃ waveguides exhibit a highly circular mode field with lower anisotropy than Ti-diffused waveguides. We report the design, fabrication, and testing of ZnO  :  LiNbO₃ devices for polarization mode conversion.

We advocate a model to effectively detect salient objects in various videos; the proposed framework [spatiotemporal saliency and coherency, (STSC)] consists of two modules, for capturing the spatiotemporal saliency and the temporal coherency information in the superpixel domain, respectively. We first extract the most straightforward gradient contrasts (such as the color gradient and motion gradient) as the low-level features to compute the high-level spatiotemporal gradient features, and the spatiotemporal saliency is obtained by computing the average weighted geodesic distance among these features. The temporal coherency, which is measured by the motion entropy, is then used to eliminate the false foreground superpixels that result from inaccurate optical flow and confusable appearance. Finally, the two discriminative video saliency indicators are combined to identify the salient regions. Extensive quantitative and qualitative experiments on four public datasets (FBMS, DAVIS, SegtrackV2, and ViSal dataset) demonstrate the superiority of the proposed method over the current state-of-the-art methods.

We introduce a navigation filter fused with information of visual optical flow and data collected from an inertial measurement unit during GPS signal degradation. Under the assumption that the tracked feature points are located on a level plane, the feature depth can be explicitly expressed and an exact measurement model was derived. Moreover, an error model analysis for a block-matching-based optical flow algorithm has been investigated. The measurement error follows a Gaussian distribution, which is a prerequisite for leveraging the error-state Kalman filter. Subsequently, a local observability analysis of the proposed filter was performed yielding the expression of three unobservable directions. We emphasize the ability of the proposed filter to estimate the aircraft’s state, especially for accurate altitude estimation, without any help of prior knowledge or extra sensors. Finally, an extensive Monte Carlo analysis was used to verify the findings in the observability results showing that all states can be estimated except the absolute horizontal positions and rotation around the gravity vector. The effectiveness of the proposed filter is demonstrated through experimental hardware used to acquire outdoor flight test data.

An extension of the fusion of interpolated frames superresolution (FIF SR) method to perform SR in the presence of atmospheric optical turbulence is presented. The goal of such processing is to improve the performance of imaging systems impacted by turbulence. We provide an optical transfer function analysis that illustrates regimes where significant degradation from both aliasing and turbulence may be present in imaging systems. This analysis demonstrates the potential need for simultaneous SR and turbulence mitigation (TM). While the FIF SR method was not originally proposed to address this joint restoration problem, we believe it is well suited for this task. We propose a variation of the FIF SR method that has a fusion parameter that allows it to transition from traditional diffraction-limited SR to pure TM with no SR as well as a continuum in between. This fusion parameter balances subpixel resolution, needed for SR, with the amount of temporal averaging, needed for TM and noise reduction. In addition, we develop a model of the interpolation blurring that results from the fusion process, as a function of this tuning parameter. The blurring model is then incorporated into the overall degradation model that is addressed in the restoration step of the FIF SR method. This innovation benefits the FIF SR method in all applications. We present a number of experimental results to demonstrate the efficacy of the FIF SR method in different levels of turbulence. Simulated imagery with known ground truth is used for a detailed quantitative analysis. Three real infrared image sequences are also used. Two of these include bar targets that allow for a quantitative resolution enhancement assessment.

When using a high-resolution full-frame area array charge-coupled device (CCD) FTF4052M as the aerial image sensor of a camera, the frame rate is generally below 1  frame  /  second (fps), which cannot meet the requirements of applications with high frame rates. First, the FTF4052M drive circuit is introduced. Next, an improved driver circuit of the CCD 4052M is proposed. Using the CCD to output four amplifiers simultaneously yielded a maximum frame rate of 3.4 fps. Then, the timing of the CCD driver, the front-end processing circuit, the DC bias circuit, and the interface circuit of the four outputs are designed. The improved driver circuit could meet the application requirements of various aerial cameras. In addition, the FTF4052M nonuniformity is analyzed, and a nonuniformity response detection system is established. Using this system, the FTF4052M array’s nonuniformity among the four quadrants and among the pixels is tested. Based on the CCD’s linear responsivity, two correction algorithms are proposed to correct the nonuniformity. Finally, using the correction, the four quadrants’ standard deviations of the response sensitivity are reduced by a factor of 1  /  13, and the responsive nonuniformity among all pixels is reduced by a factor of 1  /  10. Through the reshooting of the identified rate board, it is found that the nonuniformity of the CCD array is markedly improved.

Fast and nondestructive quality control tools are important to assess the reliability of photovoltaic plants. On-site inspection is essential to minimize the risk of further damage and electrical yield losses. The most effective way of achieving this is with highly sensitive imaging techniques such as luminescence or infrared thermography imaging. Nowadays, electroluminescence (EL) is used at nighttime to detect defects such as local cell changes, series resistances, and shunts. However, the drawback of this method is low measurement throughput. To increase the throughput, indium gallium arsenide detectors with a resolution of 640  ×  512  pixels are used, for which short integration times are possible to acquire EL images. For such short integration times, even moving image acquisition and movie recording are feasible to detect the mentioned defects. An outdoor EL setup is presented for mobile handheld recording, which can even be used under low-light conditions, below 100  W  /  m², at daytime. Experiments show that 5 ms integration time is a good compromise between low contrasts for lower integration times and motion blur for higher integration times. The camera prototype has an onboard computer to avoid image transmission losses and an external lithium polymer battery power supply for improved mobility.

The design and evaluation of the expected performance of optical systems require sophisticated and reliable information about the surface topography for planned optical elements before they are fabricated. Modern x-ray source facilities are reliant upon the availability of optics with unprecedented quality (surface slope accuracy <0.1  μrad). The problem is especially complex in the case of x-ray optics, particularly for the X-ray Surveyor under development and other missions. The high angular resolution and throughput of future x-ray space observatories requires hundreds of square meters of high-quality optics. The uniqueness of the optics and limited number of proficient vendors makes the fabrication extremely time consuming and expensive, mostly due to the limitations in accuracy and measurement rate of metrology used in fabrication. We discuss improvements in metrology efficacy via comprehensive statistical analysis of a compact volume of metrology data. The data are considered stochastic, and a statistical model called invertible time-invariant linear filter (InTILF) is developed now for two-dimensional (2-D) surface profiles to provide compact description of the 2-D data in addition to one-dimensional data treated so far. The InTILF model captures stochastic patterns in the data and can be used as a quality metric and feedback to polishing processes, avoiding high-resolution metrology measurements over the entire optical surface. The modeling, implemented in our BeatMark™ software, allows simulating metrology data for optics made by the same vendor and technology. The data are vital for reliable specification for optical fabrication, to be exactly adequate for the required system performance.

We present a measurement method for obtaining three-dimensional (3-D) images of the thoracic and abdominal body surface during respiration. This method generates a color pattern composed of R, G, and B cosine stripe patterns to obtain a single image-based 3-D measurement. The 3-D wavelet transform is adopted to extract wrapped phases from a stripe image sequence to improve the accuracy of the wrapped phase extraction. Then, a two-frequency temporal phase unwrapping formula is proposed and extended to a multifrequency formula according to the geometric relationship between the wrapped and the unwrapped phase curves, which enhances its anti-interference performance and calculation speed. Simulations and body measurements reveal that the proposed method can effectively measure the 3-D body surface during respiration. By using multiple characteristic parameters, this method overcomes the limitation of only using feature points to represent respiratory movement on the thoracic and abdominal body surface. This method has important applications for tracking respiratory motion during clinical radiotherapy.

We propose a method for reducing artifactual phase errors inherent to the Fourier transform method (FTM) for fringe analysis. The phase obtained by the FTM is subject to ripple errors at the boundary edges of the fringe pattern where fringes become discontinuous. We note that these artifactual phase errors are found to have certain systematic relations to the form of the phase, amplitude, and background intensity distributions, which can be modeled by low-order polynomials, such as Zernike polynomials, in many cases of practical interest. Based on this observation, we estimate the systematic ripple errors by analyzing a virtual interferogram that is numerically created for a fringe model with known phase, amplitude, and background intensity distributions. Starting from a rough initial guess, the virtual interferogram is sequentially improved by an iterative algorithm, and the estimated errors are finally subtracted from the experimental data. We present the results of simulations and experiments that demonstrate the validity of the proposed method.

We investigated applying the stability of the position on the optical axis direction corresponding to the phase minimum values of different frequencies for signal processing in multiple pulse-train interferometry. We calculated the position on the optical axis direction corresponding to the phase minimum value of different frequencies and obtained accurate peak position measurements of the fringe envelope using our selected frequency components. We chose this method because matching phases of frequencies means that these frequencies are less affected by measurement noise. We used an unbalanced Michelson interferometer with a fixed mirror position measurement to evaluate the performance of our proposed system. This method is expected to be useful for simple and long-range absolute distance measurements.

We propose an automatic calibration method using grid-structured light for the full parameter of a camera-projector system, including principal points, equivalent focal length, image distortion coefficients, the rotation matrix, and translation vectors between the camera and projector. Grid-structured light is projected onto a board, camera image intersection points are extracted, and three-dimensional intersection points are computed according to a homography matrix. Finally, the full parameter of a camera-projector system is solved based on stereo vision. No manual intervention is required during image processing, which simplifies the operations and improves efficiency. The image-processing kernel problem involves both automatic detection and intersection point matching. (1) The proposed intersection point detection method utilizes multiscale fusion. At each level of the image pyramid, intersection points are searched according to gray distribution and geometrical characteristics. With the gray-gravity method, coordinates are achieved with subpixel intersection point precision. Therefore, the location precision exceeds 0.5 pixel. (2) The proposed matching method employs belief propagation. Taking intersection points as nodes, a Bayesian network is established according to the Markov random field hypothesis. The image intersection point matching problem between a camera and the projector is then transformed into a maximum a posteriori estimation problem. Ultimately, 15 images are used to calibrate the full parameter of a camera-projector system. The results indicate that the reprojection error exceeds 0.15 pixel.

The tympanic membrane (TM) is an important hearing component that when ruptured causes severe hearing difficulty. It has been mainly characterized mechanically by measuring its displacements, vibration modes, impedance, and thickness. The objective of this research is to determine the relationship of the TM’s thickness obtained with confocal laser scanning microscopy (with scans along the Z axis) with the magnitude of the TM’s surface displacements measured with sound stimuli of 1.8, 5.2, and 12 kHz using digital holographic interferometry. In order to correlate the data, three regions of four healthy cats’ TMs are studied, with a particular finding that the thickness is not the same in these regions and, among samples, a feature readily noticed in the magnitudes of the displacements. Through the relationship of the data from the TM’s surface displacements with its thickness, it is now possible to confidently detect pathological changes in its structure by simply quantifying the magnitude of the former, a characteristic corroborated by the Pearson correlation coefficient (r).

The generation and subsequent dynamics of a microbubble in a liquid are investigated, both experimentally and theoretically. When a laser beam is focused into an absorbing liquid comprising colloidal red dye particles in isopropanol alcohol inside a thick quartz cuvette, microbubbles can be generated at around the focus due to nucleation and thermal cavitation. It is experimentally shown that in some cases, the generated microbubble that initially moves away from the focus due to the longitudinal optical gradient force is later attracted to the focus due to the longitudinal thermocapillary force. In our previous work, the thermocapillary force acting on a microbubble is studied two dimensionally. The thermocapillary force acting on the microbubble is determined by directly solving the three-dimensional (3-D) heat equation by 3-D Fourier transform methods. When developing the complete force model in microbubble dynamics, the thermo-capillary force, optical force, buoyancy force, gravity, and the viscous force have been considered.

To minimize dynamic measurement error in an underwater laser scanning system, a shake reduction method is proposed. Feature points are used to construct a dynamic coordinate, estimate the position and attitude of the scanning system synchronously, and correct the scanning results, so as to achieve high-precision measurement under severe shaking conditions. Experimental results show that the dynamic measurement error is <2  mm and the RMS is <2  mm when the target is located 3 m away, which verifies the feasibility of the method.

Flavin mononucleotide (FMN) plays an important role in living systems. Accurate probing and characterizing of FMN is the key to understanding the contribution of FMN coenzyme in biological processes. We demonstrate a way to get a bright surface-enhanced Raman spectroscopy (SERS) signal while simultaneously quenching the fluorescence background through silver nanorod (AgNR) array substrates fabricated by the oblique angle deposition method. The AgNR array substrates show a good SERS activity with an enhancement factor of ∼4.2  ×  10⁴. Meanwhile, the characteristic peaks of FMN are assigned by the density function theory calculation. An exponential relationship is demonstrated between SERS intensity and FMN concentration in the concentration range of 0.1 to 100  mg  /  L. This SERS-based method enabled the rapid and highly sensitive detection of FMN, holding great promise for expanding SERS application in biochemistry and food safety fields.

To improve the control accuracy of optoelectronic imaging equipment, the factors that affect the accuracy of the optoelectronic axis pointing are analyzed. Following Snell’s law, the kinematic coupling equation of the line-of-sight (LOS) angle based on the fast steering mirror (FSM) is established. The calibration method of the FSM system is designed according to the obtained motion characteristics, and a nonlinear correction method is designed to decouple the LOS equation. For a two-axis fast mirror with a stroke of ±20  mrad, the LOS pointing error is <1  μrad when the incident angle is 45 deg, which equates to an improvement of at least 78.9 times compared with linear correction methods. The nonlinear correction method is verified by practical experiments. The method provides a theoretical basis for generating position instructions and thus enables a precise FSM-based pointing system.

We present a system for 360-deg shape measurement based on three-dimensional (3-D) orientation target. The measurement system mainly includes a CCD camera, digital light projector, 3-D orientation target, and the measured object. The global coordinates for all chessboard corners of the target are calculated based on close-range photogrammetry, for which the relative orientation between camera stations is discussed in detail. According to the homography matrix between the checkerboard and its image plane, the position and orientation of the measuring camera relative to the target are determined. A dot-matrix structured light is designed and projected onto the measured object surface. In each image of the structured light deformed by the object shape, the pixel coordinates of the dot-centers are accurately calculated by the ellipse-fitting algorithm, and then they are all sorted in the same order by the beads-stringing matching method. In the experiment, the designated front, rear, left, and right surfaces of a reference object are measured, and their separate point clouds are automatically stitched together by the 3-D orientation target. Compared with coordinate measurement machine data, the accuracy of the system is demonstrated; the maximum deviations are +1.257  /    −  0.977  mm, and the standard deviation is 0.124 mm.

In order to provide a cost-effective indoor positioning and tracking service for autonomous ground vehicles and other important assets in smart factories, we report the theory and experiments for a real-time indoor positioning system using commercially available LED lamps. Inspired by the fundamental theory of the global navigation satellite system, the proposed system uses the phase difference of arrival (PDOA) approach to obtain the time difference of arrival of each carrier transmitted from individual modified LED lamps so as to estimate the receiver position. A prototype of the atto-cellular positioning system covering an area of 2.2  ×  1.8  m² with a height of 2 m was designed and experimentally demonstrated. For the design, we performed a simulation based on the Crámer–Rao bound to achieve optimal LED lamp arrangement, RF power, and other parameters. Furthermore, a virtual local oscillator for the PDOA scheme was applied to reduce the hardware complexity and to ensure the processing speed. In the experiment, the receiver was mounted on a movable material buffer station in a smart workshop, and the positioning performance was validated by tracking the trajectory of the material buffer station moving within the positioning coverage area. The experimental results show that an average positioning accuracy of ∼7  cm was achieved.

Active optics technology eases the requirements for the fabrication and stability of mirrors. Optical replication techniques make it possible to fabricate a mirror of optical quality rapidly and inexpensively. Combining the advantages of these two methods, we designed a lightweight unimorph mirror with a diameter of 160 mm based on an Al-SiC passive layer. A finite element model was established to predict the performance of this mirror and to optimize relevant parameters. A spherical prototype mirror was fabricated and tested to measure its shape-correction capability, thermal stability, and surface roughness; this prototype has been used in an innovation project on lightweight active mirrors from the Chinese Academy of Sciences.

Lens arrays of gradient-index (GRIN) rods that project an erected unit magnification image are used in light sources of LED printers. It is shown how alignment accuracy of the GRIN rod affects imaging performance to maintain the image uniformity. The coordinates and inclinations of the chief and marginal rays of the GRIN rod that has a different object distance or is tilted are analyzed with the ray matrix. We compare the ray coordinate shift with the object distance difference with that with the GRIN rod optical axis tilt. The image shift with the tilted GRIN rod is several times larger than that with the different object distance of the GRIN rod. However, the image by the tilted GRIN rod shifts parallel on the image plane without being almost blurred. Consequently, the GRIN rod tilt should be minimized sufficiently.

We highlight fundamental properties of the dual-contact diffractive optical elements (DOEs) with high-diffraction efficiency throughout the visible spectrum and its fabrication example. To obtain high-diffraction efficiency over the broad wavelength range, the refractive index difference between the two kinds of materials constituting the dual-contact DOE must satisfy the constraint condition (blazing condition) on the wavelength dependence. For this reason, it has been considered difficult to obtain a suitable material for the dual-contact DOE. In recent years, nanocomposite materials have been studied as constituting materials of dual-contact DOE. We find the combination consisting of two kinds of UV curable resins that almost satisfy the blazing condition and have excellent processability. Prototyped dual-contact DOE using those materials showed high-diffraction efficiency and low-noise level throughout the visible spectrum.

Aiming to address the low efficiency of the traditional calibration method for star sensors, we propose a calibration method using two-dimensional (2-D) Dammann grating, which can generate a dot array with uniform intensity and a certain interval. This method is based on theoretical analysis of the diffraction angles of 2-D Dammann grating relative to the tilt angles of incident light. We calculate the angles using a fitting method and further calculate the geometric parameters, such as principal point and principal distance of the star sensor. Finally, we establish a calibration system using 2-D Dammann grating and a nanostar sensor with a 25-mm focal length and 15-deg full field of view has been calibrated. The results show that the calibration accuracy for single-star measurement is better than 6 arc sec, which satisfies the requirement for high-efficiency star sensor calibration and validates the proposed method.

We present the general formulae to compute an aspheric mirror such as it corrects the spherical aberration generated by an arbitrary number of preceding refractive surfaces.

We present a general analytic and close-form formula to determine the shape of a surface that corrects the spherical aberration generated by an arbitrary number of preceding surfaces. The equation is tested for a variety of optical systems, including doublets, triplets, and more sophisticated systems with at least six interfaces.

The optical design of a compact 2  ×   zoom lens in a lens barrel for miniature cameras is reported. The magnification in the optical system is achieved by means of changing positions of lens elements inside the lens barrel. Six lens elements are employed in the optical system, where up to four elements are actuated. Maximum travel distances for the moving parts are determined as 1.67 mm to reach a magnification of 2.06  ×  , and the total track length for the telephoto zoom lens state is 9.16 mm at its most extended configuration. Manufacturability of the proposed system is also discussed using a tolerance analysis. The designed optical system is suitable for use with image sensors up to 13 MP, and it can be employed in an endoscopic camera, where the optical zoom and compact design are beneficial. It is also suitable for smartphone cameras with further enhancements and integration of voice-coil-motors and/or MEMS actuators.

We report a mid-infrared (mid-IR or 3.5 to 5.0  μm) laser source for testing jamming code effectiveness against heat-seeking missiles in open-field conditions. We developed an optical parametric oscillator (OPO) based on a ZnGeP₂ (ZGP) crystal for converting a 2.1-μm pump laser into mid-IR. The pump laser is an acousto-optically Q-switched Ho:YAG laser pumped by a continuous-wave (cw) Tm:fiber laser. The maximum possible average mid-IR power obtained is 6.5 W at a pump power of 20 W with a 33% power conversion efficiency. The pump pulses resulting from the Q-switched operation of the Ho:YAG laser have a pulse width of 30 ns (FWHM) at a pulse repetition frequency (PRF) of 50 kHz. We characterized the output beam in terms of its output power and wavelength spectrum. The output power of the pump laser can be modulated in a pattern of on/off pulses with a smaller frequency than the PRF. The frequency and the duty cycle of these on/off pulses can be adjusted. Identical on/off pulse modulation appears on the mid-IR output beam after the OPO. It is possible to create jamming patterns and codes in mid-IR band using this feature. We have presented examples of on/off modulation of the OPO output at arbitrary pattern frequencies and duty cycles.

A stability improved optical frequency comb (OFC) generation system based on a single integrated polarization multiplexing dual-drive Mach–Zehnder modulator is proposed. Due to the influence of the modulator bias drift on the spectra of the OFC, the amplitude of the OFC spectra would fluctuate with time, which seriously deteriorates the stability of the OFC. The effect of the modulator bias drift on the stability of the OFC is analyzed by theoretical analyses and numerical simulations. A feedback system is proposed to improve the OFC stability. The ratio between the total power of the OFC and the optical carrier power is used as the feedback parameter. In the proof-of-concept experiment, by using the feedback structure, the seven-line OFC stability is experimentally improved from being large than 3 dB within 1.5 h to being below 2 dB more than 4 h.

External modulation, such as with a Mach–Zehnder modulator (MZM), is commonly used in optical communications to reduce nonlinear signal distortions. However, in addition to the signal distortion due to chromatic dispersion, the MZM itself is another source of nonlinear distortion. Despite the fact that dispersion is a linear distortion that develops in the field domain, in low-cost direct-detection links, the detector destroys the system’s linearity. Therefore, in these channels, dispersion is another source of nonlinearity. We suggest the implementation of a dispersion module to substantially reduce the MZM-induced nonlinearity. This technique is possible only because linear dispersion mitigation filters can be implemented post detection. We show experimentally that chromatic dispersion can indeed substantially reduce the channel’s nonlinear distortion, in good agreement with theoretical predictions.

Fiber optical parametric amplifiers (FOPAs) are regarded as one of the candidate optical amplifiers for future ultralong-distance, large-capacity, and high-speed optical fiber communication systems. FOPAs provide high signal gain and low noise figures compared with other optical amplifiers. However, they suffer from gain fluctuation, which restricts commercial application. Optimization of the dispersion parameters of highly nonlinear fibers (HNLFs) employed as gain media is essential because they have an immediate impact on the flatness of the amplifier. An optimization method for multiple HNLF segments using the differential evolution algorithm is proposed to minimize spectral gain variation. We theoretically yield an FOPA with an average signal gain of 20 dB and gain fluctuation <0.5  dB within 400 nm.

A hybrid shaping scheme, carrier-less amplitude and phase modulation geometric-probabilistic hybrid shaping 16QAM (CAP-GPS-16QAM), is proposed for data center networks. The probabilistic shaping is achieved by applying the proposed probabilistic subset shaping to the constellation after geometric shaping (GS). An intensity modulation/direct detection experimental system of 5 GBaud is built to verify the proposal. The results show that the gains of the proposed scheme are 2.4, 2.6, and 3 dB compared with CAP-GS-16QAM, CAP-PAS-16QAM, and CAP-square-16QAM, respectively, at BER  =  10^−  3. Meanwhile, in terms of generalized mutual information, the proposal improves the performance by 0.078, 0.125, and 0.153 bits/symbol compared with CAP-GS-16QAM, CAP-PAS-16QAM, and CAP-square-16QAM, respectively.

A compact single-frequency master oscillator power amplifier laser system composed of three-stage thulium-doped fiber amplifiers was developed. At a repetition rate of 10 Hz, >100-μJ pulse energy at 2050.5-nm wavelength, with ∼431-ns pulse width, was achieved successfully. The pulse profile could be actively controlled by adjusting the drive signal of an acoustic-optical modulator. This all-fiber laser system could be utilized as a seeder laser for a solid-state power amplifier system.

Organic materials are attracting attention for their potential applications due to their large third-order optical nonlinearities. A passive fiber Q-switching laser has been demonstrated based on an erbium/ytterbium codoped amplifier and an acetone solution saturable absorber (SA). When pump power was increased from 120 to 155 mW, the repetition rate increased from 5.68 to 16.67 kHz, while the pulse width decreased from 24.1 to 15  μs. These results indicate that acetone solution can be used as an effective SA for pulsed lasers.

Orbital angular momentum (OAM) is generated using circular-photonic crystal fiber (C-PCF), which is used as a key solution for higher data communication. This C-PCF steadily supports 80 OAM modes and 2 linear polarization modes from 1- to 2.5-μm wavelengths with low confinement loss (CL), very high nonlinearity, and flat dispersion. The average CL is <1  ×  10^−  6  dB  /  m and the dispersion of each mode is controlled in the range of 30 to 500  ps  ·  nm^−  1  ·  km^−  1. The average nonlinearity value is >1.26  ×  10⁵  W^−  1 km^−  1 at 1-μm wavelength that is essential for the generation of the supercontinuum, biomedical imaging, and several nonlinear optical applications. With all these good features, this proposed optical fiber is promising for application in fiber-based OAM high-capacity communication systems.

A chloride ion sensor based on a multitaper modulated fiber fabricated via arc discharge is proposed and studied experimentally and theoretically. The sensing unit consists of four periodic tapers with a diameter of 28.24  μm, a period of 676  μm, and a total length of 2.7 mm. The results show that the concentration sensitivity values are 170 pm/(g/L) at a wavelength of 1303.60 nm and 220 pm/(g/L) at a wavelength of 1389.84 nm when the chloride ion concentration is increased from 0 to 40 g/L. The proposed sensor is characterized by a simple manufacturing process, a compact structure, and a low cost, and this sensing unit has great potential for application in marine chloride detection and environmental safety monitoring, especially for monitoring building corrosion and water pollution.

We propose and investigate cooperative visible light communications (VLC) enabled vehicular network architecture with an unmanned aerial vehicle (UAV) relay. In the proposed system, a UAV equipped with an onboard photodetector and light-emitting diodes is used as a relay in a vehicular VLC link. The advantage of having a flying UAV relay is its great mobility and ease of deployment. The performance of UAV-assisted cooperative VLC is investigated for amplify and forward and decode and forward protocols with half duplex and full duplex modes. Moreover, location of the UAV relay is optimized in order to maximize the signal-to-noise ratio at the destination vehicle. The active light beam tracking technique is further employed to increase the reliability of the proposed VLC system. Our results reveal that a UAV relay can significantly improve the transmission performance of conventional VLC-based vehicular networks.

Laser ablation of aluminum, silicon, titanium, germanium, and indium antimonide at 1064 nm in ambient laboratory air with pulse durations ranging from 100 ps to 100  μs has been characterized with optical microscopy. Highly focused spots of 10  μm yields fluences of 0.004 to 25  kJ  /  cm² and irradiances spanning 4  ×  10⁶-10¹⁴  W  /  cm². Single pulse hole depths range from 84 nm to 147  μm. A one-dimensional thermal model establishes a set of nondimensional variables for hole depth, fluence, and pulse duration. For pulse durations shorter than the radial diffusion time, the hole depth exceeds the thermal diffusion length by a factor of 1 to 30 for more than 90% of the data. For pulses longer than this critical time, transverse heat conduction losses dominate and holes as small as 10^−  3 times the thermal diffusion depth are produced. For all cases, the ablation efficiency, defined as atoms removed per incident photon, is 10^−  2 or less, and is inversely proportional to volume removed for pulse durations less than 100 ns. At high fluences, more than 10 to 100 times ablation threshold, explosive boiling is identified as the likely mass removal mechanism, and hole depth scales approximately as fluence to 0.3 to 0.4 power. The power-law exponent is inversely proportional to the shielding of the laser pulse by ejected material, and shielding is maximum at the 1-ns pulse duration and minimum near the 1-μs pulse duration for each material. Using the thermal scaling variables, the high-fluence behavior for each material becomes strikingly similar.

The demand for wireless bandwidth is increasing rapidly; however, the throughput of short-range radio frequency (RF) solutions, such as WiFi, are quickly reaching limits due to limited bandwidth in the RF spectrum. This problem can be overcome by utilizing unregulated infrared wavelengths in the optical spectrum. On the other hand, free-space optical (FSO) communication presents a challenge for mobility due to the line-of-sight nature of optics. We present a dual-channel optical link for short-range FSO communication, which consists of a 100  Mb  /  s, large diameter (60 cm at 3 m distance) optical femtocell with a 1.5  Gb  /  s, small diameter (1 cm at 3 m distance) optical attocell embedded within the femtocell. This hybrid femtocell–attocell optical link addresses the needs of various users by providing a high mobility femtocell link combined with an enhanced bandwidth attocell hotspot. A prototype hybrid system was assembled and evaluated in terms of bandwidth, range, and bit error rate (BER). In addition, the relationship between the incident angle of the transmitted beam relative to the receiver and the BER of the received data was evaluated. At a communication distance of 3 m, the BER is <10^−  4 when the incident angle is smaller than 30 deg for the optical femtocell, while the incident angle must be smaller than 10 deg for the optical attocell to achieve the same level of BER. In addition, we experimentally demonstrated simultaneous, error-free recovery of the data from both the femtocell and the attocell when these two cells are overlapped, which proves the feasibility of the dual-channel optical link for FSO communication.

Laser-based displays have gathered much attention because only the displays can express full color gamut of Ultra-HDTV, ITU-R BT.2020. There are many types of laser-based displays. A projector with a single spatial light modulator is mainstream in the displays so far. This projector uses the laser light sources under pulse with duty of 20% to 50%. We improve a triple-emitter red color 638-nm high-power broad area laser diode (LD) for this application. Countermeasures are adopted to improve the output power characteristics and strengthen the facet to the fatal degradation. The improved triple emitter shows the output of 5.0 W at the injection current of 5 A under pulse with duty of 30%, case temperature of 25°C. The output is increased by 4.1% compared to our former one. The peak wall plug efficiency reaches to 41.6% under 25°C. The long-term aging test is also performed. Mean time to failure of the improved LD is estimated 72,000 h under the output of 3.5 W, pulse condition with duty of 30%. The value is ∼3.3 times longer than that of the former.

The data rate in a single-mode fiber is approaching the capacity limit given by the Shannon theory. Mode division multiplexing, such as few modes, orbital angular momentum, and cylindrical vector beam (CVB) multiplexing, has shown great potential to further increase data capacity in both free-space and fiber communication. We propose and demonstrate high-order CVB multiplexing communication in an air-core photonics crystal fiber (PCF). The simulation results show that the 19-cell air-core PCF supports transmission of CVB modes from ±1 to ±4 orders. In the experiment, ±1- to ±4-orders CVBs are transmitted in 8.25-m-long air-core PCF with the mode purities higher than 76.5%. We demonstrate four coaxial CVB channel communication by multiplexing the ±2- and ±3-orders CVB modes. Each CVB channel carries 10-Gbit/s on–off keying signals and the measured bit error rates satisfy the forward error correction threshold. CVB communications based on air-core PCF can be used in short-distance optical communication with high capacity and low optical latency.

One design of the state-of-the-art laser scanner systems in automotive applications is based on oscillating mirror modules. The requirement of a large mirror surface for eye-safe transmission beams and long measurement distances is a major drawback for fast and reproducible scanning. Tolerances of angular positioning, position sensing, and vibrational perturbations limit the position accuracy of such a mirror and, thus, the accuracy of the transmission spot position in the field of view (FoV). Our approach for a scanner module with maximum transmission beam diameter combines a microlens array with an objective lens for generating one optical telescope assembly for each angular scan position exclusively. Aperture stops define the beam positions in the FoV and avoid positioning errors caused by angle deviations of the scanner mirror. This increases the reliability of the angular position accuracy of the scanner module significantly. To minimize the shadings between adjacent scan spots in the target distance, created by beam cutoffs at the aperture stop of the objective lens, an array of optimized microwedge prisms is provided in combination with the microlens array. Therefore, we can increase the throughput of transmission power into the FoV and improve the measurement distance, especially at large scan angles.

An optical fiber refractive index (RI) sensor based on a single-mode–multimode–single mode (SMS) fiber structure fabricated with a coreless low-index multimode fiber (CLIMF) tapered to several diameters is presented. Both the response to changes in the external RI and its spectral dependence were strongly influenced by the length and diameter of the CLIMF. Different SMS sensors with diameters of 120, 100, 80, 60, 40  μm and the corresponding reimaging lengths at 1550 nm are immersed in RI of 1.32, 1.34, 1.36, 1.39, and 1.42 to study the response to changes in the external RI. The diameter of 80  μm yields the best sensitivity (540.7  nm  /  RIU) with a resolution of 6  ×  10^−  5 in the RI range 1.32 to 1.39. By duplicating the reimaging length at 1550 nm for this diameter, the RI range increased to 1.42 and a maximum sensitivity of 1022  nm  /  RIU is achieved in the RI range from 1.39 to 1.42. Also the simulation results corroborate the wavelength positions of the peaks of the experimental spectrums. We show that the response range to changes in the external RI is controlled with the SMS geometry. This advantage makes the SMS sensor very suitable for biological applications.

A distributed fiber sensing system based on weak fiber Bragg gratings (wFBGs) enhanced phase-sensitive optical time-domain reflectometry (Φ-OTDR) has improved signal-to-noise ratio (SNR) and sensitivity. The effects of wFBGs on a Φ-OTDR system are investigated theoretically and experimentally. The SNR improvement induced by wFBGs, which have higher reflectivity compared with the Rayleigh scattering, has been analyzed in systems with different laser source linewidths. The experimental results show that Φ-OTDR assisted by wFBGs with −35-dB reflectivity has a lower noise level and its SNR is improved by 27 dB. By comparing laser sources with different linewidths, it is proved that the enhanced system no longer requires an ultranarrow linewidth laser. The SNR in different locations of a 2.4-km-wFBG-array remains at a high level.

Fresnel zone plates (FZPs) are significant optical objects for extreme ultraviolet and x-ray microscopy, both of which are powerful tools due to their high spatial resolution and large penetration depth. Fabrication of FZPs is challenging as it requires high accuracy to achieve the narrow outmost zone and large aspect ratio. The focused ion beam (FIB) has been practically demonstrated as a potential alternative to electron beam lithography for its precision, rapidness, and mask-free essence. However, the common redeposition effect in the FIB process is a major obstacle to distort actual patterns from designed ones, especially for FZPs with outmost zone width of 100 nm and simultaneously aspect ratio beyond 2. We proposed an efficient method noted as single-pixel line-assisted sputtering (SLAS) to eliminate redeposition in situ during FZP fabrication. Through systematically optimizing parameters predominating ion dosages and, particularly, the positions of SLAS, we successfully obtained high-quality FZPs with outmost zone width down to 100 nm, perpendicular side walls, aspect ratio of up to 3 and diameter of 38  μm in 1.5 h. The application of SLAS also provides instructive guidance for other FIB processes to finely modulate the nanostructures.

With the wide-spread availability of rigorous electromagnetic (vector) analysis codes for describing the diffraction of electromagnetic waves by specific periodic grating structures, the insight and understanding of nonparaxial parametric diffraction grating behavior afforded by approximate methods (i.e., scalar diffraction theory) is being ignored in the education of most optical engineers today. Elementary diffraction grating behavior is reviewed, the importance of maintaining consistency in the sign convention for the planar diffraction grating equation is emphasized, and the advantages of discussing “conical” diffraction grating behavior in terms of the direction cosines of the incident and diffracted angles are demonstrated. Paraxial grating behavior for coarse gratings (d  ≫  λ) is then derived and displayed graphically for five elementary grating types: sinusoidal amplitude gratings, square-wave amplitude gratings, sinusoidal phase gratings, square-wave phase gratings, and classical blazed gratings. Paraxial diffraction efficiencies are calculated, tabulated, and compared for these five elementary grating types. Since much of the grating community erroneously believes that scalar diffraction theory is only valid in the paraxial regime, the recently developed linear systems formulation of nonparaxial scalar diffraction theory is briefly reviewed, then used to predict the nonparaxial behavior (for transverse electric polarization) of both the sinusoidal and the square-wave amplitude gratings when the +1 diffracted order is maintained in the Littrow condition. This nonparaxial behavior includes the well-known Rayleigh (Wood’s) anomaly effects that are usually thought to only be predicted by rigorous (vector) electromagnetic theory.

In our original manuscript, the equation for estimating the polarizer orientation angle was incorrect, causing incorrect angle estimates. We correct the error and show revised figures.