Shannon entropy of partially polarized and partially coherent light with Gaussian fluctuations

We propose to analyze Shannon entropy properties of partially coherent and partially polarized light with Gaussian probability distributions. It is shown that the Shannon entropy is a sum of simple functions of the intensity, of the degrees of polarization, and of the intrinsic degrees of coherence that have been recently introduced. This analysis clearly demonstrates the contribution of partial polarization and of partial coherence to the characterization of disorder of the light provided by the Shannon entropy, which is a standard measure of randomness. We illustrate these results on two simple examples.     

Complex degree of coherence for partially coherent general beams in atmospheric turbulence

With the use of the general beam formulation, the modulus of the complex degree of coherence for partially coherent cosh-Gaussian, cos-Gaussian, Gaussian, annular and higher-order Gaussian optical beams is evaluated in atmospheric turbulence. For different propagation lengths in horizontal atmospheric links, the moduli of the complex degree of coherence at the source and receiver planes are examined when reference points are taken on the receiver axis and off-axis. In the on-axis case, it is observed that in propagation, the moduli of the complex degree of coherence are symmetrical and look like the intensity profile of the related coherent beam propagating in a turbulent atmosphere. For all the beams considered, the moduli of the complex degree of coherence profiles turn into Gaussian shapes beyond certain propagation lengths. In the off-axis case, the moduli of complex degree of coherence patterns become drifted at the earlier propagation lengths. Among the beams investigated, the cos-Gaussian beam is found to be almost independent of the changes in the source partial coherence parameter, and the annular beam seems to be affected the most against the variations of the source partial coherence parameter.     

Simulation of partially coherent imaging by outer-product expansion

A method for the numerical simulation of partially coherent imaging systems is introduced. The two-dimensional source and pupil functions are decomposed into outer-product sums by the singular value decomposition algorithm, thus reducing the computation of the corresponding nonlinear transform function. The method is computationally efficient when the source and pupil matrices are of sufficiently low rank. Numerical examples are studied and compared with theoretical results.     

Uniform theory of the Talbot effect with partially coherent light illumination

A uniform formulation for the self-imaging of gratings with any kind of partially coherent illumination is developed in terms of the cross mutual spectral density of the partial coherence theory. The formulation includes the time diffractive intensity distribution and the averaged diffractive intensity distribution at self-imaging distances and can be applied to both continuous and temporal illuminations with any kind of spectra. It is found that the averaged intensity distribution is related only to the intensity spectrum of illumination. The continuous polychromatic illumination and the ultrashort laser pulses with or without frequency chirp are then studied by a numerical stimulation. It is shown that the ultrashort laser pulse and the continuous polychromatic illuminations have similar averaged self-image distributions. Thus the Talbot effect may help in the study of the temporal and spectral characteristics of ultrashort laser pulses. An experiment with an LED is given, as well.     

Beam shaping of focused partially coherent beams by use of the spatial coherence effect

We demonstrate that when a partially coherent beam with a Gaussian intensity distribution is focused by a lens, the desired partially coherent flat-topped intensity distribution or doughnut-shaped intensity distribution at the geometrical focus can be generated by choice of appropriate form of spectral degree of coherence. We provide a novel approach to beam shaping of a partially coherent beam and offer new schemes for their potential applications such as material processing, optical therapy, and optical tweezers.     

Spatial coherence function of partially coherent Gaussian beams in atmospheric turbulence

We introduce a new method of estimating the coherence function of a Gaussian-Schell model beam in the inertial subrange of atmospheric turbulence. It is compared with the previously published methods based on either the quadratic approximation of the parabolic equation or an assumed independence between the source's randomness and the atmosphere using effective beam parameters. This new method, which combines the results of the previous two methods to account for any random source/atmospheric coupling, was shown to more accurately estimate both the coherence radius and coherence functional shape across much of the relevant parameter space. The regions of the parameter space where one method or another is the most accurate in estimating the coherence radius are identified along with the maximum absolute estimation error in each region. By selecting the appropriate estimation method for a given set of conditions, the absolute estimation error can generally be kept to less than 5%, with a maximum error of 7%. We also show that the true coherence function is more Gaussian than expected, with the exponential power tending toward 9/5 rather than the theoretical value of 5/3 in very strong turbulence regardless of the nature of the source coherence.     

Composite coherence vortices in coherent and incoherent superpositions of two off-axis partially coherent vortex beams

The composite coherence vortices by coherently and incoherently superimposing two parallel, off-axis partially coherent vortex beams and their evolution in free space are studied. It is shown that the superposition scheme, off-axis distance, coherence parameter and propagation distance affect the position and number of composite coherence vortices. The motion, creation and annihilation of composite coherence vortices appear by varying the off-axis distance, coherence parameter and propagation distance. The coherent and incoherent superpositions result in the different position and number of composite coherence vortices and their different evolution behavior in the coherent limit.     

Changes in the spectrum, in the spectral degree of polarization, and in the spectral degree of coherence of a partially coherent beam propagating through a gradient-index fiber

Expressions are derived for the cross-spectral density matrix of an electromagnetic Gaussian Schell-model beam propagating through a paraxial ABCD system. Using the recently developed unified theory of coherence and polarization of electromagnetic beams and the ABCD matrix for gradient-index fibers, we study the changes of the spectral density, of the spectral degree of polarization, and of the spectral degree of coherence of such a beam as it travels through the fiber. Effects of material dispersion are also considered.     

Cancellation of coherent artifacts in optical coherence tomography imaging

Coherent artifacts in optical coherence tomography (OCT) images can severely degrade image quality by introducing false targets if no targets are present at the artifact locations. Coherent artifacts can also add constructively or destructively to the targets that are present at the artifact locations. This constructive or destructive interference will result in cancellation of the true targets or in display of incorrect echo amplitudes of the targets. We introduce the use of a nonlinear deconvolution algorithm, CLEAN, to cancel coherent artifacts in OCT images of extracted human teeth. The results show that CLEAN can reduce the coherent artifacts to the noise background, sharpen the air-enamel and enamel-dentin interfaces, and improve the image contrast.     

Analyzing the power coupled between partially coherent waveguide fields in different states of coherence

A procedure is described for calculating the power coupled between partially coherent waveguide fields that are in different states of coherence. The method becomes important when it is necessary to calculate the power transferred from a distributed source S to a distributed load L through a length of multimode metallic, or dielectric, waveguide. It is shown that if the correlations between the transverse components of the electric and magnetic fields of S and L are described by coherence matrices M and M', respectively, then the normalized average power coupled between them is (eta) = Tr[MM']/Tr[M]Tr[M'], where Tr denotes the trace. When the modal impedances are equal, this expression for the coupled power reduces to an equation derived in a previous paper [J. Opt. Soc. Am. A 18, 3061 (2001)], by use of thermodynamic arguments, for the power coupled between partially coherent free-space beams.     

Power coupled between partially coherent vector fields in different states of coherence.

A procedure is described for calculating the power coupled between collimated, partially coherent vector fields that are in different states of coherence. This topic is of considerable importance in designing submillimeter-wave optical systems for astronomy. It is shown that if the incoming field S has coherence matrix A, and the outgoing field D has coherence matrix B, then the power coupled is simply Ps = Tr(ATBT), where the elements of T project the basis functions of B onto those of A. A similar technique can be used to calculate the power coupled from the background of S to D. The scheme is illustrated by calculating the power coupled between two scalar, Gaussian Schell-model beams. The procedure can be incorporated into optical design software.     

Estimation of the degree of polarization of coherent light in the presence of uniform and nonuniform illumination

We address estimation of the degree of polarization (DOP) in active images acquired under coherent light when the coherency matrix of the backscattered light is diagonal. We consider cases when the illumination intensity is uniform on the scene and when it is not, and we take into account different parametrizations of the DOP. In all cases, we determine the maximum-likelihood (ML) estimators and compare their variances with the Cramer-Rao lower bound for unbiased estimation. When illumination is not uniform, the ML estimators are solutions of equations that have to be solved numerically. We show that simpler estimators based on the trimmed mean make it possible to approach performance of ML estimators at lower computational cost.     

Cramer-Rao lower bounds on the estimation of the degree of polarization in coherent imaging systems

We analyze the estimation precision of the parameter of the orthogonal state contrast image (OSCI) under coherent illumination. This parameter represents the degree of polarization of the light if the materials that compose the scene are purely depolarizing. Two different estimation modes are considered, depending on the uniformity of the illumination of the scene. We first determine lower bounds on the estimation precision in both cases by computing the Cramer-Rao lower bounds (CRLBs) for unbiased estimation. This allows us to compare the potential precision that can be reached in each mode. We then consider the estimators based on empirical averaging of the data, and we show that there are cases where they are strongly biased. We thus propose and characterize another estimator based on the natural representation of the OSCI, which is asymptotically unbiased and whose variance is close to the unbiased CRLB.     

Numerical model of optical coherence tomographic vibrography imaging to estimate corneal biomechanical properties

Most techniques measuring corneal biomechanics in vivo are biased by side factors. We demonstrate the ability of optical coherence tomographic (OCT) vibrography to determine corneal material parameters, while reducing current prevalent restrictions of other techniques (such as intraocular pressure (IOP) and thickness dependency). Modal analysis was performed in a finite-element (FE) model to study the oscillation response in isolated thin corneal flaps/eye globes and to analyse the dependency of the frequency response function on: corneal elasticity, viscoelasticity, geometry (thickness and curvature), IOP and density. The model was verified experimentally in flaps from three bovine corneas and in two enucleated porcine eyes using sound excitation (100–110 dB) together with a phase-sensitive OCT to measure the frequency response function (range 50–510 Hz). Simulations showed that corneal vibration in flaps is sensitive to both, geometrical and biomechanical parameters, whereas in whole globes it is primarily sensitive to corneal biomechanical parameters only. Calculations based on the natural frequency shift revealed that flaps of the posterior cornea were 0.8 times less stiff than flaps from the anterior cornea and cross-linked corneas were 1.6 times stiffer than virgin corneas. Sensitivity analysis showed that natural vibration frequencies of whole globes were nearly independent from corneal thickness and IOP within the physiological range. OCT vibrography is a promising non-invasive technique to measure corneal elasticity without biases from corneal thickness and IOP.     

A coherent trough in the envelope of a signal formed by the interferometer with a partially coherent radiation source

When a mixed radiation from physically different sources with substantially different widths of spectral lines is used in an interferometer, a single trough can appear under certain conditions in the envelope of the time-dependent interference signal. This trough can be used as a measurement signal.     

Theory of partially coherent electromagnetic fields in the space-frequency domain

We construct the coherent-mode representation for fluctuating, statistically stationary electromagnetic fields. The modes are shown to be spatially fully coherent in the sense of a recently introduced spectral degree of electromagnetic coherence. We also prove that the electric cross-spectral density tensor can be rigorously expressed as a correlation tensor averaged over an appropriate ensemble of strictly monochromatic vectorial wave functions. The formalism is demonstrated for partially polarized, partially coherent Gaussian Schell-model beams, but the theory applies to arbitrary random electromagnetic fields and can find applications in radiation and propagation and in inverse problems.     

Efficient numerical representation of the optical field for the propagation of partially coherent radiation with a specified spatial and temporal coherence function

We propose a method to narrow the gap between the rigorous methods for the propagation of partially coherent light, which require excessive computational capacity, and the numerical methods used in practical engineering applications, where it is not clear how to handle spatial and temporal coherence in a statistically correct manner. As is the case for the latter methods, the numerical method described can deal with fields with a large spatial and temporal extent, which is necessary in practical applications such as laser fusion or optical lithography. However, the method also takes a few steps toward a more rigorous, yet efficient, representation of the optical field, which depends on detailed specified coherence properties of the radiation. The described method uses a set of independent monochromatic fields at different oscillation frequencies. The frequencies are chosen such that the statistical properties of the integrated intensity closely resemble those from a full-time trace treatment. Finally, we demonstrate the capabilities and limitations of the method with a few numerical examples of the propagation of a large field with a specified spatial and temporal coherence.     

Representing the behavior of partially coherent optical systems by using overcomplete basis sets

A technique is described for representing the behavior of partially coherent optical systems by using overcomplete basis sets. The scheme is closely related to Gabor function theory. Through singular-value decomposition it is shown that if E is a matrix containing the sampled basis functions, then all of the information needed for optical calculations is contained in S = EE(dagger) and R = E(dagger)E. For overcomplete sets, S can be inverted to give a dual basis set, E = S(-1)E, which can be used to find the correlation matrix elements A of a sampled bimodal expansion of the spatial coherence function. Overcomplete correlation matrices can be scattered easily at optical components. They can be used to determine (i) the natural modes of a field; (ii) the total power in a field, Pt = Tr[RA]; (iii) the power coupled between two fields, A and B, that are in different states of coherence, Pc = Tr[RARB]; and (iv) the entropy of a field, Q = Tr[Zsigmar(I-Z)r/r], where Z = RA/Tr[RA].     

Two-beam interference experiments in the frequencydomain to measure the complex degree of spectral coherence

Abstract

Correlation-induced spectral changes in two-beam interference experiments in the space-frequency domain are used to determine the amplitude and phase of the complex degree of spectral coherence. The spectral modifications are observed either as a shift in the peak wavelength or as sinusoidal modulations within the bandwidth of the white light spectrum due to the complex degree of spectral coherence of the secondary source and the path difference between the interfering beams. These correlation-induced spectral changes were analysed using a theoretical model to establish the behaviour of the real and imaginary parts of the complex degree of spectral coherence over the entire visible region of the spectrum.     

Estimation of the degree of polarization in active coherent imagery by using the natural representation

We address the problem of degree of polarization estimation in polarization diversity images. We consider active imaging techniques with laser illumination, which have the appealing feature of revealing contrasts that do not appear in conventional intensity images. These techniques provide two images of the same scene that are perturbed with speckle noise. Because of the presence of nonhomogeneity in the reflected intensity, it can be preferable to perform image analysis of the orthogonal-state contrast image, which is a measure of the degree of polarization of the reflected light when the coherency matrix is diagonal. It has been shown that a simple nonlinear transformation of this orthogonal-state contrast image leads to an image perturbed with additive symmetrical noise on which simple and efficient estimation and detection techniques can be applied. We propose to precisely analyze estimation properties of the degree of polarization using this natural representation. In particular, we determine the Cramer-Rao bound of the polarization degree estimation and the variance of the proposed estimator, and we study the estimator's efficiency as a function of the speckle order for different measurement strategies.