Nonlinear ray tracing in focused fields, part 1. Calculation of 3D focused wavefields: tutorial.

In this three-part paper series, we develop a method to trace the lines of flux through a three-dimensional wavefield by following a direction that is governed by the derivative of the phase at each point, a process that is best described as flux tracing but which we interchangeably name "nonlinear ray tracing." In this first part, we provide a tutorial on the high-speed calculation of three-dimensional complex wavefields, which is a necessary precursor to flux tracing. The basis of this calculation is the angular spectrum method, a well-known numerical algorithm that can be used to efficiently and accurately calculate diffracted fields for numerical apertures <0.7. It is known that this approach yields identical predictions to the first Rayleigh-Sommerfeld solution. We employ the angular spectrum method to develop two algorithms that generate the 3D complex wavefield in the region of focus of a lens. The first algorithm is based on the thin lens approximation, and the second is based on the concept of an ideal lens, which can be modeled using an optical Fourier transform. We review how these algorithms can be used to calculate focused laser beams with T E M 00 and T E M 01 laser profiles. The three-dimensional sampling requirements of the focused field are explained in detail, and expressions for the computational and memory efficiency of the two algorithms are developed. These two algorithms generate the 3D scaffold for the flux tracing method developed in the second paper, and in the third paper we highlight the application of the method to understanding monochromatic lens aberration. Disregarding the second and third papers, this first paper will serve as a practical tutorial for anyone seeking to compute focused fields in three dimensions.

Nonlinear ray tracing in focused fields, part 2. Tracing the flux: tutorial.

In this three-part paper series, we develop a method to trace the lines of flux through a three-dimensional wavefield by following a direction that is governed by the derivative of the phase at each point, a process that is best described as flux tracing but which we interchangeably name "nonlinear ray tracing." In the first part we focused on the high-speed calculation of focused three-dimensional complex wavefields in the paraxial approximation for T E M 00 and T E M 01 laser modes. The algorithms developed in the first paper are first used to generate the three-dimensional grid of samples of the complex wavefield in the focal region. In this second part, we focus on tracing a flux through this three-dimensional point cloud. For a given "ray" at an arbitrary position in the 3D volume, interpolation of the three-dimensional samples is applied to determine the derivative of the phase (normal to the direction of propagation) at the ray position, which is then used to direct the ray as it "propagates" forward in a straight line over a short distance to a subsequent plane; the process is repeated between consecutive planes. The initial origin of the ray can be chosen arbitrarily at any point, and the ray can then be traced through the volume with appropriate interpolation. Results are demonstrated for focused wavefields in the absence of aberrations, corresponding to the cases highlighted in the first paper. Some of the most interesting results relate to focused Laguerre-Gaussian beams, for which the rays are found to spiral at different rates of curvature. In the third paper we extend the application of this algorithm to the investigation of lens aberration.

Nonlinear ray tracing in focused fields, part 3. Monochromatic wavefront aberration: tutorial.

Using the flux tracing algorithm developed in the previous two parts, we examine the nonlinear rays that pass through the focus of a lens containing monochromatic aberrations. Lens aberration is modeled differently in the numerical propagation algorithms relating to the thin lens and the ideal lens cases. For the former, an additive phase term is applied to the transmission function of the thin lens, which describes a distortion in the thickness function of the lens, and for the latter an additive phase term is added to the pupil function of the lens (the Fourier transform of the image plane). In both cases, the Zernike polynomials are applied to model various aberrations including spherical, defocus, comatic, astigmatism, trefoil, and quadrafoil. Despite the different methods of modeling aberration for the two types of lenses, remarkably similar results are obtained for both cases. A discussion is also provided on the relationship between classical wavefront aberration theory and nonlinear tracing. This paper demonstrates the extraordinary potential of nonlinear ray tracing to gain insights into complex optical phenomena.

Wavenumber Calibration Protocol for Raman Spectrometers Using Physical Modelling and a Fast Search Algorithm.

A wavenumber calibration protocol is proposed that replaces polynomial fitting to relate the detector axis and the wavenumber shift. The physical model of the Raman spectrometer is used to derive a mathematical expression relating the detector plane to the wavenumber shift, in terms of the system parameters including the spectrograph focal length, the grating angle, and the laser wavelength; the model is general to both reflection and transmission gratings. A fast search algorithm detects the set of parameters that best explains the position of spectral lines recorded on the detector for a known reference standard. Using three different reference standards, four different systems, and hundreds of spectra recorded with a rotating grating, we demonstrate the superior accuracy of the technique, especially in bands outside of the outermost reference peaks when compared with polynomial fitting. We also provide a thorough review of wavenumber calibration for Raman spectroscopy and we introduce several new evaluation metrics to this field borrowed from chemometrics, including leave-one-out and leave-half-out cross-validation.

Improved Wavelength Calibration by Modeling the Spectrometer.

Wavelength calibration is a necessary first step for a range of applications in spectroscopy. The relationship between wavelength and pixel position on the array detector is approximately governed by a low-order polynomial and traditional wavelength calibration involves first-, second-, and third-order polynomial fitting to the pixel positions of spectral lines from a well known reference lamp such as neon. However, these methods lose accuracy for bands outside of the outermost spectral line in the reference spectrum. We propose a fast and robust wavelength calibration routine based on modeling the optical system that is the spectrometer. For spectral bands within the range of spectral lines of the lamp, we report similar accuracy to second- and third-order fitting. For bands that lie outside of the range of spectral lines, we report an accuracy 12-121 times greater than that of third-order fitting and 2.5-6 times more accurate than second-order fitting. The algorithm is developed for both reflection and transmission spectrometers and tested for both cases. Compared with similar algorithms in the literature that use the physical model of the spectrometer, we search over more physical parameters in shorter time, and obtain superior accuracy. A secondary contribution in this paper is the introduction of new evaluation methods for wavelength accuracy that are superior to traditional evaluation.

Automated Raman Micro-Spectroscopy of Epithelial Cell Nuclei for High-Throughput Classification.

Raman micro-spectroscopy is a powerful technique for the identification and classification of cancer cells and tissues. In recent years, the application of Raman spectroscopy to detect bladder, cervical, and oral cytological samples has been reported to have an accuracy greater than that of standard pathology. However, despite being entirely non-invasive and relatively inexpensive, the slow recording time, and lack of reproducibility have prevented the clinical adoption of the technology. Here, we present an automated Raman cytology system that can facilitate high-throughput screening and improve reproducibility. The proposed system is designed to be integrated directly into the standard pathology clinic, taking into account their methodologies and consumables. The system employs image processing algorithms and integrated hardware/software architectures in order to achieve automation and is tested using the ThinPrep standard, including the use of glass slides, and a number of bladder cancer cell lines. The entire automation process is implemented, using the open source Micro-Manager platform and is made freely available. We believe that this code can be readily integrated into existing commercial Raman micro-spectrometers.

Label-free color staining of quantitative phase images of biological cells by simulated Rheinberg illumination.

Modern microscopes are designed with functionalities that are tailored to enhance image contrast. Dark-field imaging, phase contrast, differential interference contrast, and other optical techniques enable biological cells and other phase-only objects to be visualized. Quantitative phase imaging refers to an emerging set of techniques that allow for the complex transmission function of the sample to be measured. With this quantitative phase image available, any optical technique can then be simulated; it is trivial to generate a phase contrast image or a differential interference contrast image. Rheinberg illumination, proposed almost a century ago, is an optical technique that applies color contrast to images of phase-only objects by introducing a type of optical staining via an amplitude filter placed in the illumination path that consists of two or more colors. In this paper, the complete theory of Rheinberg illumination is derived, from which an algorithm is proposed that can digitally simulate the technique. Results are shown for a number of quantitative phase images of diatom cells obtained via digital holographic microscopy. The results clearly demonstrate the potential of the technique for label-free color staining of subcellular features.

An Algorithm for the Removal of Cosmic Ray Artifacts in Spectral Data Sets.

Cosmic ray artifacts may be present in all photo-electric readout systems. In spectroscopy, they present as random unidirectional sharp spikes that distort spectra and may have an affect on post-processing, possibly affecting the results of multivariate statistical classification. A number of methods have previously been proposed to remove cosmic ray artifacts from spectra but the goal of removing the artifacts while making no other change to the underlying spectrum is challenging. One of the most successful and commonly applied methods for the removal of comic ray artifacts involves the capture of two sequential spectra that are compared in order to identify spikes. The disadvantage of this approach is that at least two recordings are necessary, which may be problematic for dynamically changing spectra, and which can reduce the signal-to-noise (S/N) ratio when compared with a single recording of equivalent duration due to the inclusion of two instances of read noise. In this paper, a cosmic ray artefact removal algorithm is proposed that works in a similar way to the double acquisition method but requires only a single capture, so long as a data set of similar spectra is available. The method employs normalized covariance in order to identify a similar spectrum in the data set, from which a direct comparison reveals the presence of cosmic ray artifacts, which are then replaced with the corresponding values from the matching spectrum. The advantage of the proposed method over the double acquisition method is investigated in the context of the S/N ratio and is applied to various data sets of Raman spectra recorded from biological cells.

Multicomponent analysis using a confocal Raman microscope.

Measuring the concentration of multiple chemical components in a low-volume aqueous mixture by Raman spectroscopy has received significant interest in the literature. All of the contributions to date focus on the design of optical systems that facilitate the recording of spectra with high signal-to-noise ratio by collecting as many Raman scattered photons as possible. In this study, the confocal Raman microscope setup is investigated for multicomponent analysis. Partial least-squares regression is used to quantify physiologically relevant aqueous mixtures of glucose, lactic acid, and urea. The predicted error is 17.81 mg/dL for glucose, 10.6 mg/dL for lactic acid, and 7.6 mg/dL for urea, although this can be improved with increased acquisition times. A theoretical analysis of the method is proposed, which relates the numerical aperture and the magnification of the microscope objective, as well as the confocal pinhole size, to the performance of the technique.

Multiwavefront digital holographic television.

This paper presents the full technology chain supporting wide angle digital holographic television from holographic capture of real world objects/scenes to holographic display with an extended viewing angle. The data are captured with multiple CCD cameras located around an object. The display system is based on multiple tilted spatial light modulators (SLMs) arranged in a circular configuration. The capture-display system is linked by a holographic data processing module, which allows for significant decoupling of the capture and display systems. The presented experimental results, based on the reconstruction of real world, variable in time scenes, illustrates imaging dynamics, viewing angle and quality.

Digital computation of the complex linear canonical transform.

An efficient algorithm for the accurate computation of the linear canonical transform with complex transform parameters and with complex output variable is presented. Sampling issues are discussed and the requirements for different cases given. Simulations are provided to validate the results.

Twin removal in digital holography using diffuse illumination.

A method to numerically remove the twin image for inline digital holography, using multiple digital holograms, is discussed. Each individual hologram is recorded by using a statistically independent speckle field to illuminate the object. If the holograms are recorded in this manner and then numerically reconstructed, the twin image appears as a different speckle pattern in each of the reconstructions. By performing speckle-reduction techniques the presence of the twin image can be greatly reduced. A theoretical model is developed, and experimental results are presented that validate this approach. We show experimentally that the dc object intensity term can also be removed by using this technique.

Additional sampling criterion for the linear canonical transform.

The linear canonical transform describes the effect of first-order quadratic phase optical systems on a wave field. Several recent papers have developed sampling rules for the numerical approximation of the transform. However, sampling an analog function according to existing rules will not generally permit the reconstruction of the analog linear canonical transform of that function from its samples. To achieve this, an additional sampling criterion has been developed for sampling both the input and the output wave fields.

Introducing secure modes of operation for optical encryption.

We analyze optical encryption systems using the techniques of conventional cryptography. All conventional block encryption algorithms are vulnerable to attack, and often they employ secure modes of operation as one way to increase security. We introduce the concept of conventional secure modes to optical encryption and analyze the results in the context of known conventional and optical attacks. We consider only the optical system "double random phase encoding," which forms the basis for a large number of optical encryption, watermarking, and multiplexing systems. We consider all attacks proposed to date in one particular scenario. We analyze only the mathematical algorithms themselves and do not consider the additional security that arises from employing these algorithms in physical optical systems.

Numerical sampling rules for paraxial regime pulse diffraction calculations.

Sampling rules for numerically calculating ultrashort pulse fields are discussed. Such pulses are not monochromatic but rather have a finite spectral distribution about some central (temporal) frequency. Accordingly, the diffraction pattern for many spectral components must be considered. From a numerical implementation viewpoint, one may ask how many of these spectral components are needed to accurately calculate the pulse field. Using an analytical expression for the Fresnel diffraction from a 1-D slit, we examine this question by varying the number of contributing spectral components. We show how undersampling the spectral profile produces erroneous numerical artifacts (aliasing) in the spatial-temporal domain. A guideline, based on graphical considerations, is proposed that determines appropriate sampling conditions. We show that there is a relationship between this sampling rule and a diffraction wave that emerges from the aperture edge; comparisons are drawn with boundary diffraction waves. Numerical results for 2-D square and circular apertures are presented and discussed, and a potentially time-saving calculation technique that relates pulse distributions in different z planes is described.

Extended focused imaging for digital holograms of macroscopic three-dimensional objects.

When a digital hologram is reconstructed, only points located at the reconstruction distance are in focus. We have developed a novel technique for creating an in-focus image of the macroscopic objects encoded in a digital hologram. This extended focused image is created by combining numerical reconstructions with depth information extracted by using our depth-from-focus algorithm. To our knowledge, this is the first technique that creates extended focused images of digital holograms encoding macroscopic objects. We present results for digital holograms containing low- and high-contrast macroscopic objects.

Reduction of speckle in digital holography by discrete Fourier filtering.

We present a digital signal processing technique that reduces the speckle content in reconstructed digital holograms. The method is based on sequential sampling of the discrete Fourier transform of the reconstructed image field. Speckle reduction is achieved at the expense of a reduced intensity and resolution, but this trade-off is shown to be greatly superior to that imposed by the traditional mean and median filtering techniques. In particular, we show that the speckle can be reduced by half with no loss of resolution (according to standard definitions of both metrics).

Generalized Yamaguchi correlation factor for coherent quadratic phase speckle metrology systems with an aperture.

In speckle-based metrology systems, a finite range of possible motion or deformation can be measured. When coherent imaging systems with a single limiting aperture are used in speckle metrology, the observed decorrelation effects that ultimately define this range are described by the well-known Yamaguchi correlation factor. We extend this result to all coherent quadratic phase paraxial optical systems with a single aperture and provide experimental results to support our theoretical conclusions.

Paraxial speckle-based metrology systems with an aperture.

Digital speckle photography can be used in the analysis of surface motion in combination with an optical linear canonical transform (LCT). Previously [D. P. Kelly et al. Appl. Opt.44, 2720 (2005)] it has been shown that optical fractional Fourier transforms (OFRTs) can be used to vary the range and sensitivity of speckle-based metrology systems, allowing the measurement of both the magnitude and direction of tilting (rotation) and translation motion simultaneously, provided that the motion is captured in two separate OFRT domains. This requires two bulk optical systems. We extend the OFRT analysis to more general LCT systems with a single limiting aperture. The effect of a limiting aperture in LCT systems is examined in more detail by deriving a generalized Yamaguchi correlation factor. We demonstrate the benefits of using an LCT approach to metrology design. Using this technique, we show that by varying the curvature of the illuminating field, we can effectively change the output domain. From a practical perspective this means that estimation of the motion of a target can be achieved by using one bulk optical system and different illuminating conditions. Experimental results are provided to support our theoretical analysis.

Reconstruction of partially occluded objects encoded in three-dimensional scenes by using digital holograms.

We propose a task-specific digital holographic capture system for three-dimensional scenes, which can reduce the amount of data sent from the camera system to the receiver and can effectively reconstruct partially occluded objects. The system requires knowledge of the object of interest, but it does not require a priori knowledge of either the occlusion or the distance the object is from the camera. Subwindows of the camera-plane Fresnel field are digitally propagated to reveal different perspectives of the scene, and these are combined to overcome the unknown foreground occlusions. The nature of the occlusions and the effect of subwindows are analyzed thoroughly by using the Wigner distribution function. We demonstrate that a careful combination of reconstructions from subwindows can reveal features that are not apparent in a reconstruction from the whole hologram. We provide results by using optically captured digital holograms of real-world objects and simulated occlusions.