Steen Gruner Hanson

Optical beam wave propagation through complex optical systems

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.

Second-order statistics for wave propagation through complex optical systems

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.

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

The mean, the variance, and the correlation function of the intensity pattern resulting from an incoherent source that has propagated through a complex ABCD optical system are derived. The number of speckle correlation cells (or modes, N) within an effective measurement area is presented and discussed. Optical field decorrelation effects with respect to secondary fringe formation in phase-shifting speckle interferometry are discussed. In particular, the correlation coefficient resulting from load-induced object tilt, in-plane translations, and displacements parallel to the optic axis as a function of the number of modes propagating through the optical system is derived and discussed. In contrast to previous research, in which the calculation of speckle interferometric decorrelation effects was restricted to direct imaging systems and the special case where N → ∞, the correlation coefficients that we derive are valid for arbitrary complex ABCD systems (i.e., nonimaging configurations) and for all finite values of N ≥ 1.

DYNAMIC LASER SPECKLE IN COMPLEX ABCD OPTICAL SYSTEMS

Within the framework of ABCD matrix theory an exact analytical expression is derived for the space–time-lagged photocurrent covariance that is valid for arbitrary (complex) ABCD optical systems, i.e., systems that include Gaussian-shaped apertures and partially developed speckle. General expressions are derived for the mean spot size and both the mean speckle size and the temporal coherence length. Additionally, a general description of both speckle boiling and speckle translation in an arbitrary observation plane is given. Included in the analysis is the effect of a finite wave-front-curvature radius for the Gaussian-shaped laser beam illumination of the target. The effects of diffraction and wave-front-curvature radius are discussed for both imaging systems and a Fourier transform system. It is shown that, whereas diffraction affects the speckle dynamics in both cases, a finite wave-front curvature affects only the speckle dynamics in the Fourier transform system. Further, the effects of finite detector apertures are considered, in which the effects of speckle averaging are included and discussed. In contrast to previous work, the obtained analytical results are expressed in a relatively compact form yet fully contain all diffraction effects and apply to an arbitrary ABCD optical system.

Laser-speckle angular-displacement sensor: theoretical and experimental study

A novel, to our knowledge, method for the measurement of angular displacement for arbitrarily shaped objects is presented in which the angular displacement is perpendicular to the optical axis. The method is based on Fourier-transforming the scattered field from a single laser beam that illuminates the target. The angular distribution of the light field at the target is linearly mapped on a linear image sensor placed in the Fourier plane. Measuring this displacement facilitates the determination of the angular displacement of the target. It is demonstrated both theoretically and experimentally that the angular-displacement sensor is insensitive to object shape and target distance if the linear image sensor is placed in the Fourier plane. A straightforward procedure for positioning the image sensor in the Fourier plane is presented. Any transverse or longitudinal movement of the target will give rise to partial speckle decorrelation, but it will not affect the angular measurement. Furthermore, any change in the illuminating wavelength will not affect the angular measurements. Theoretically and experimentally it is shown that the method has a resolution of 0.3 mdeg ( approximately 5 murad) for small angular displacements, and methods for further improvement in resolution is discussed. No special surface treatment is required for surfaces giving rise to fully developed speckle. The effect of partially developed speckle is considered both theoretically and experimentally.

Three-dimensional speckle dynamics in paraxial optical systems

Three-dimensional (3D) speckle dynamics are investigated within the paraxial approximation as represented by ABCD-matrix theory. Within the paraxial approximation, exact expressions are derived for the space–time-lagged intensity covariance that results from an in-plane translation, an out-of-plane rotation, or an in-plane rotation of a diffuse scattering object that is illuminated by a Gaussian-shaped laser beam. As illustrative examples we consider the 3D dynamical nature of speckles that are formed in free space and in Fourier transform and imaging systems. The spatiotemporal characteristics of the observed 3D speckle patterns are interpreted in terms of boiling, decorrelation, rotation, translation, and tilting. Experimental results, which support the quantitative theory, are presented and discussed.

Laser Doppler Velocimetry: Analytical Solutions of the Optical System Including the Effects of Partial Coherence of the Source,

Within the framework of ABCD matrix theory, analytical expressions are derived for the time-lagged covariance of a classical laser Doppler velocimetry system as a function of the laser spot size, the limiting aperture, and the measurement aperture size. Both partial and fully developed speckle as well as planar and rotating targets, are considered. Further, error estimates are presented that indicate how well one can determine in practice the velocity of both planar and rotating targets, and a comparison with time-of-flight velocimetry is given.

Effects of receiver optics contamination on the performance of laser velocimeter systems

Comprehensive, general expressions are derived for various statistical functions that characterize the performance of paraxial ABCD optical systems illuminated by partially coherent light in the presence of a collecting aperture that is contaminated by absorbing and/or scattering inhomogeneities. The contaminants are modeled here as consisting of thin random absorbing and/or phase (i.e., scattering) screens. In particular, a novel expression is derived for the autocorrelation function of a random screen consisting of both absorbing and scattering patches that partially cover an aperture. Expressions for the resulting mean irradiance distribution expected in an image plane of laser velocimeter systems are derived and discussed. The effects of contamination on the performance of both laser time-of-flight velocimeters (LTV’s) and laser Doppler velocimeter (LDV) systems are considered. In general, the effects of contamination result in a reduction in magnitude of the time-lagged covariance of the photocurrent compared with the corresponding covariance function in the absence of contamination. It is also shown that contamination effects may lead to bias errors in the estimated mean velocity for LTV systems and an increased variance of the estimated velocity for both LTV and LDV systems.

Laser-time-of-flight velocimetry: analytical solution to the optical system based on ABCD matrices

Within the framework of complex ABCD matrix theory, an analytical expression for the time-lagged covariance of a classical time-of-flight laser velocimetry system is obtained as a function of the laser spot size, the limiting aperture, and the measurement-aperture radii. The decorrelation effects of wave-front tilt and velocity misalignment are derived and discussed. In addition, error estimates are presented that indicate how well one can determine the precise location of the peak of the cross-covariance function.

FIRST-ORDER SPECKLE STATISTICS : AN ANALYTIC ANALYSIS USING ABCD MATRICES

Within the limitations of the generalized ray-matrix method, analytical expressions for the first-order intensity moments are obtained for arbitrary cylindrically symmetric ABCD optical systems, assuming beam illumination of reflective targets with arbitrary values of surface roughness whose heights are a Gaussian process. In contrast to previous work, the results presented here are valid for an arbitrary number of correlation areas of the target that contribute to the observed intensity. Also presented here are analytic closed-form results, as well as a highly accurate approximation based on elementary functions, for the heterodyne signal-to-noise ratio in situations where the scattered light is mixed with a strong local oscillator.