Speckle: statistics and interferometric decorrelation effects in complex ABCD optical systems

Implementation of complex-order Fourier transforms in complex ABCD optical systems

Within the framework of complex ABCD-matrix theory, exact theoretical expressions are derived for the space-time-lagged cross-covariance functions of the fields valid for arbitrary (complex) ABCD-optical systems, i.e., systems that include Gaussian-shaped apertures and partially developed speckle. Specifically, we show and discuss the results for the three generic systems, i.e., free-space propagation, Fourier transform configuration, and imaging. To cope with various surface structures of varying rms surface heights, we apply two models in addition to employing the usual model for surfaces giving rise to fully developed speckle. The theoretical results found for free-space propagation are supported by interferometrically obtained data.

A method is proposed for checking the reliability of a quantitative estimate of longitudinal displacements in metallic beam samples when they are investigated by electronic speckle interferometry and phase-shifting speckle interferometry. Theoretical and experimental results are presented for the investigation of longitudinal surface displacements of samples by electronic difference speckle interferometry and phase-shifting speckle interferometry. © 2004 Optical Society of America

20We describe a novel formulation of light beam propagation through any complex optical system that can be described by an ABCD ray-transfer matrix. Within the paraxial approximation, optical propagation can be formulated in terms of a Huygens principle expressed in terms of the ray-transfer ABCD matrix elements of the optical system. We extend and generalize previous treatments to include the effects of finite-sized limiting apertures (i.e., diffractive screens) in the optical train, tilt and random jitter of the optical elements, and distributed random inhomogeneities along the optical path (e.g., clear air turbulence and aerosols). In the presence of limiting apertures the ABCD matrix elements of the optical system are complex. For the case of laser beam propagation and Gaussian-shaped limiting apertures in the optical train, we obtain analytical expressions for both the spot radius and the wave-front radius of curvature at an arbitrary observation plane and give illustrative examples of practical concern. In particular, analytical expressions for the fringe visibility obtained in a coherent laser interferometric system are presented. An analytical expression for the mean spot radius of a laser beam propagating through an optical system in the presence of tilt and random jitter is obtained. We also consider the propagation of partially coherent light through optical systems. In particular, we derive a generalized van Cittert–Zernike theorem that is valid for an arbitrary optical system that can be characterized by an ABCD ray-transfer matrix. Finally, the propagation of laser beams through a general optical system in the presence of distributed random inhomogeneities is considered. An explicit expression for the mean irradiance distribution of a Gaussian-shaped beam is derived that is valid for an arbitrary optical system. In addition, we derive an expression for the mutual-coherence function for wave propagation through an arbitrary optical system. In all cases the results are expressed in terms of the ABCD matrix elements of the complete optical system. The formulation of optical propagation presented here is a rather simple and straightforward way of determining the effects of finite-sized optical elements, tilt and random jitter, and distributed random inhomogeneities along the optical path. It is merely necessary first to multiply the relevant ray matrices together to find the complete system matrix and then to substitute this matrix into the expressions given in this paper.

The interference between a uniform reference wave and a speckle object wave results in variable fringe contrast and background level. Taking these variations into account, we optimize system parameters of phase-shifting speckle interferometry. The results show that the optimal reference intensity should always be equal to one fourth of the detector’s saturation level. The optimal reference to the object-intensity ratio shows an increase from one up to, in most practical cases, six as a chosen interference dynamic range increases from its minimum value. The dependence of a maximal fraction of acceptable measurements on the dynamic range is calculated. Numerical examples indicate that we may hope for a readout accuracy in the range of 1/50th to 1/100th of a fringe period and still cover more than half of the image area with acceptable data. These data are taken without spatial averaging and have maximum resolution.

Optimization of wavefield intensities in phase-shifting speckle interferometry

Closed-form expressions are derived for various statistical functions that arise in optical propagation through arbitrary optical systems that can be characterized by a complex ABCD matrix in the presence of distributed random inhomogeneities along the optical path. Specifically, within the second-order Rytov approximation, explicit general expressions are presented for the mutual coherence function, the log-amplitude and phase correlation functions, and the mean-square irradiance that are obtained in propagation through an arbitrary paraxial ABCD optical system containing Gaussian-shaped limiting apertures. Additionally, we consider the performance of adaptive-optics systems through arbitrary real paraxial ABCD optical systems and derive an expression for the mean irradiance of an adaptive-optics laser transmitter through such systems.

The reconstruction of surfaces from speckle interferometry data is a demanding data-analysis task that involves edge detection, edge completion, and image reconstruction from noisy data. We present an approach that makes optimal use of the experimental information to minimize the hampering influence of the noise. The experimental data are then analyzed with a combination of wavelet transform and Bayesian probability theory. Nontrivial examples are presented to illustrate the proposed technique.

A theoretical and numerical investigation of the systematic phase errors in phase-shifting speckle interferometry is presented. The theoretical investigation analyzes the behavior of some systematic error induced by intensity variations in two cases of data-computing techniques. The first case deals with the technique in which the phase change is computed, unwrapped, and then linearly filtered; the second case deals with the technique in which the data are linearly filtered before the arctangent calculation and then unwrapped. With the first filtering technique it is shown that it is preferable when the phase change is of relatively low spatial frequency, leading to a particularly accurate method. With the second case it is demonstrated that an important parameter of speckle interferometry is the modulation factor of the interference frame that induces phase errors when the data are filtered before the arctangent calculation. We show that this technique is better than the first when the phase change is composed of high-spatial-frequency variations. The theoretical investigation of the two techniques is compared with numerical simulations, considering the frequency characteristics of the phase change, and this shows a good match between theory and simulations.

It has been shown that complex paraxial optical systems, consisting of various lens and distances of free space propagation, can be described using the Linear Canonical Transform (LCT). Indeed it can be shown that many well know optical transforms such as the Optical Fourier Transform (OFT), Optical Fractional Fourier Transform (OFRT), the effect of a lens or Chirp Modulation Transform (CMT) are all subsets of the more general LCT. Using the <i>ABCD</i> Collins matrix formula it is possible to represent these integral transforms in a simpler form, which facilitates system analysis and design. Speckle Photography (SP) in combination with an optical LCT can be used to measure surface motion of an optically rough body. It has previously been shown that Optical FRT's (OFRT) can be used in speckle based metrology systems to vary the range and sensitivity of a metrology system and also to determine both, the magnitude and direction, of tilting (rotation) and translation motion simultaneously, provided that the motion is captured in two separate OFRT domains. In this paper we extend the OFRT analysis to more general LCT systems and demonstrate how simultaneous tilt and translation measurements can be discerned from the speckle images captured prior to, and after motion. A spherical wavefront can be conveniently described using the Collin's matrix notation. By changing the wavefront of the illuminating light we show that we effectively change the domain of the LCT system without changing the bulk elements in the optical system. Thus the complete motion (in-plane translation and small surface tilting) of a rigid body can be determined using one optical LCT system and illuminating fields of varying curvature.

It has been shown that complex paraxial optical systems, consisting of various lens and distances of free space propagation, can be described using the linear canonical transform (LCT). Indeed it can be shown that many well know optical transforms such as the optical fourier transform (OFT), optical fractional fourier transform (OFRT), the effect of a lens or chirp modulation transform (CMT) are all subsets of the more general LCT. Using the ABCD Collins matrix formula it is possible to represent these integral transforms in a simpler form, which facilitates system analysis and design. Speckle Photography (SP) in combination with an optical LCT can be used to measure surface motion of an optically rough body. It has previously been shown that Optical FRT&apos;s (OFRT) can be used in speckle based metrology systems to vary the range and sensitivity of a metrology system and also to determine both, the magnitude and direction, of tilting (rotation) and translation motion simultaneously, provided that the motion is captured in two separate OFRT domains. In this paper we extend the OFRT analysis to more general LCT systems and demonstrate how simultaneous tilt and translation measurements can be discerned from the speckle images captured prior to, and after motion. A spherical wavefront can be conveniently described using the Collin&apos;s matrix notation. By changing the wavefront of the illuminating light we show that we effectively change the domain of the LCT system without changing the bulk elements in the optical system. Thus the complete motion (in-plane translation and small surface tilting) of a rigid body can be determined using one optical LCT system and illuminating holds of varying curvature.

The stage and close-up magicians have employed various types of optical effects in the presentation of magic illusions. Although certain magical effects &quot;are done with mirrors&quot;, many other feats of magic employ no mirrors at all, but other optical-type devices in certain instances. The magician has relied on his magical creativity, misdirection, and showmanship, often more than certain optical and engineering-type devices for achievement of the magical effect. However, certain technical devices have always been a part of the magician&apos;s &quot;bag-of-tricks&quot;. A brief overview of the magic, patent, and engineering literature is presented detailing what optical devices and effects magicians have and will continue to use. An important criterion is that the optical effect, however simple or complex, when combined with the intended illusion, give the audience the experience of an entertaining, mysterious, magical effect--not a science class type demonstration. Several examples are presented of optical effects in magic tricks which have been successfully employed. The main purpose of this paper is not to expose the secrets of magic, but to allow the optical engineer to gain an appreciation and understanding of how the magician develops and performs new effects, so that new optical devices and techniques might find their place in a show of illusions.

Figure errors of optical surfaces degrade the performance of optical systems. When predicting the performance and performing system assembly, compensation by clocking of optical components around the optical axis is a conventional but user-dependent method. Commercial optical software cannot optimize this clocking, and existing automatic figure-error balancing methods have limitations. To overcome these limitations, a global and general optimization method based on analyzing the precise relationships between the figure errors and the wavefront error (WFE) is proposed. Using the footprint data of each optical surface, the resulting WFE is calculated. Direct map operation is used for intercepting and rotating the figure-error maps. The simulated annealing algorithm is used to seek the optimal combination of clocking angles for the optical components. This method can be applied to most coaxial optics systems, including dioptric, catoptrics, and catadioptric complex lenses. It is successfully implemented for a catadioptric immersion lithographic optics system with artificial figure errors, and for an experimental lithographic optics system with actual manufacturing figure errors.

Traditional imaging methods often separate the two technical procedures of optical system design and imaging processing, where the lens subsystem is firstly optimized using an optical measure of performance and the image processing subsystem is optimized subsequently, resulting in the failures in efficient cooperation between the optical and digital elements and missing of the optimal solution. Therefore, an innovative approach is presented to combine optical system design and image processing together to obtain high resolution images from the final system. In this paper, the theoretical foundations for this computing optical design method is analyzed by considering the modulation transfer functions (MTF) and aberrations. According to the wave aberration theory and Zernicke formula, the relationship between the point spread function (PSF) and a certain monochromatic aberration can be obtained. Fourier transformation is taken on these PSFs and then modulus value calculation is adopted to get the MTFs. By analyzing the MTFs of idealized optical systems, the influence of each monochromatic aberration on the image can be concluded. Note that some of the MTFs values have zeros and others do not. Considering that imaging processing algorithm cannot recover the information lost at such zeros, thus some aberrations can be easily corrected through imaging processing algorithm, while others not. Aimed at a single monochromatic aberrations respectively, the optical system with single lens is designed in ZEMAX software and the relevant imaging processing algorithm is adopted. From the comparison among the final images, the theoretical foundations for the computing optical design method considering the MTF and aberrations can be verified. The compensation for aberration of imaging processing can be utilized to reduce the pressure of optical system design. The computing optical design method provides a new approach to simplify structure and reduce cost, as well as to gain high resolution images simultaneously, especially for designing certain complex optical system, which shows promising perspective of industrial application.

Design, analysis and preliminary tests of a linear array CCD aerial camera for ground simulation