Coherent-mode decomposition of partially polarized,partially coherent sources

It is shown that any partially polarized, partially coherent source can be expressed in terms of a suitable superposition of transverse coherent modes with orthogonal polarization states. Such modes are determined through the solution of a system of two coupled integral equations. An example, for which the modal decomposition is obtained in closed form in terms of fully linearly polarized Hermite Gaussian modes, is given.     

Kullback relative entropy and characterization of partially polarized optical waves

Different properties of partially polarized light are discussed using the Kullback relative entropy, which provides a physically meaningful measure of proximity between probability density functions (PDFs). For optical waves with a Gaussian PDF, the standard degree of polarization is a simple function of the Kullback relative entropy between the considered optical light and a totally depolarized light of the same intensity. It is shown that the Kullback relative entropies between different PDFs allow one to define other properties such as a degree of anisotropy and a degree of non-Gaussianity. It is also demonstrated that, in dimension three, the Kullback relative entropy between a partially polarized light and a totally depolarized light can lead to natural definitions of two degrees of polarization needed to characterize the polarization state. These analyses enlighten the physical meaning of partial polarization of light waves in terms of a measure of disorder provided by the Shannon entropy.     

Numerical experiment of the Shannon entropy in partially coherent imaging by Koehler illumination to show the relationship to degree of coherence

This paper describes the Shannon entropy in a partially coherent imaging system with Koehler illumination. Numerical simulation shows that the entropy has a one-to-one relationship with the normalized mutual intensity given by the van Cittert-Zernike theorem. Analytical evaluation shows that the entropy is consistent with the definition of coherence and incoherence, which is also verified by numerical simulations. Additional numerical experiments confirm that the entropy depends on the source intensity distribution, polarization state of the source, object, and pupil. Therefore, the entropy quantitatively measures the degree of coherence of the partially coherent imaging system.     

Correlated imaging with partially coherent light for remote sensing

Based on the extended Huygens–Fresnel integral and the theory of coherence of light field, we have investigated the correlated imaging by using the transverse normalized second-order intensity fluctuation correlation function with partially coherent light radiation. The imaging for a reflected object with relative long distance is determined by the feature of speckle-to-speckle correlation. By using the correlation function, we study the effects of imaging distance, the sizes of object lens and reference lens, the source’s transverse coherent width and its transverse size on the quality of correlated imaging. Numerical results show that the parameters of imaging system and the properties of partially coherent light source have significant influences on the imaging resolution and visibility. For an object lens with large enough diameter, the resolution is determined by the transverse coherent width of light source. On the contrary, it depends on the aperture of object lens. The magnification of the system depends only on the propagation distance. This speckle-to-speckle correlated imaging with unbalanced arms have potential applications in remote sensing due to its unique features.     

Anisotropic pure-phase plates for quality improvement of partially coherent,partially polarized beams

From a theoretical point of view, the use of anisotropic pure-phase plates (APP) is considered in order to improve the quality parameter of certain partially coherent, partially polarized beams. It is shown that, to optimize the beam-quality parameter, the phases of the two Cartesian components of the field at the output of the APP plate should be identical and should exhibit a quadratic dependence on the radial polar coordinate.     

Coincidence fractional Fourier transform implemented with partially coherent light radiation

We introduce the coincidence fractional Fourier transform (FRT) implemented with incoherent and partially coherent light radiation. Optical systems for implementing the coincidence FRT are designed. The results show that the visibility and quality of the coincidence FRT of an object are closely related to the light source's transverse size, coherence, and spectral width. As an example, we numerically study the coincidence FRT of a single slit.     

Spatial correlation properties of focused partially coherent light

We investigate the spatial coherence properties in the focal region of a converging, spatially partially coherent wave field. In particular, we find that, depending on the effective coherence length of the field in the aperture, the longitudinal and transverse coherence lengths in the focal region can be either larger or smaller than the corresponding width of the intensity distribution. Also, the correlation function is shown to exhibit phase singularities.     

Errors of Mueller matrix measurements with a partially polarized light source

The linear errors of Mueller matrix measurements, using a partially polarized light source, have been formulated for imperfections of misalignment, depolarization, and nonideal ellipsometric parameters of the polarimetric components. The error matrices for a source-polarizer system and a source-polarizer-compensator system are derived. A polarized light source, when used with an imperfect polarizer, generates extra errors in addition to those for an unpolarized source. The compensator redistributes these errors to different elements of the error matrix. The errors of the Mueller matrices for the polarizer-sample-analyzer and the polarizer-compensator-sample-analyzer systems are evaluated for a straight through case. This error analysis is applied to a Stokes method and an experiment was performed to show the errors by a polarized light source. This general analysis can be used to evaluate errors for ellipsometry and polarimetry.     

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.     

Fractional Fourier transform for partially coherent off-axis Gaussian Schell-model beam

The fractional Fourier transform (FRT) is applied to a partially coherent off-axis Gaussian Schell-model (GSM) beam, and an analytical formula is derived for the FRT of a partially coherent off-axis GSM beam. The corresponding tensor ABCD law for performing the FRT of a partially coherent off-axis GSM beam is also obtained. As an application example, the FRT of a partially coherent linear laser array that is expanded as a sum of off-axis GSM beams is studied. The derived formulas are used to provide numerical examples. The formulas provide a convenient way to analyze and calculate the FRT of a partially coherent off-axis GSM beam.     

Dynamics of the linearly polarized fundamental Gaussian light wave

A continuous planar array of dipoles that are oriented in a particular direction and have an amplitude distribution that is Gaussian in the paraxial limit is introduced as a source for the fundamental Gaussian light wave. The radiation intensity of the Gaussian light wave is determined and its characteristics are analyzed. The universal Gaussian beam factor is deduced and identified as the radiation intensity of the scalar Gaussian wave. The total radiated power, the mean center of the localized wave, and the beam widths of the intensity distribution are obtained. The ratio of the power in the Gaussian wave to that in the corresponding paraxial Gaussian beam is used as a measure of the quality of the paraxial beam approximation. A limiting factor for the power ratio is introduced as an indicator for the acceptability of the paraxial beam approximation. The cross section and the beam widths of the localized light wave are investigated in the large and small kw0 limits, where k is the wavenumber and w0 is the beam waist at the input plane. The beam width of the full Gaussian wave is found to be less than that of the corresponding paraxial Gaussian beam both for the scalar Gaussian wave and for the Gaussian light wave.     

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.     

Polarization recording of holograms in partially polarized recording light

A theoretical approach is considered for media whose scalar and anisotropic responses are of opposite sign. The nondiffracted beam, and the imaginary and real images formed by a polarization hologram are analyzed under these conditions. It is shown that the imaginary image has its polarization transformed compared with the object field while a pseudoscopic object field reconstructed in terms of polarization state and degree is formed in the real image. Pis’ma Zh. Tekh. Fiz. 23, 38–42 (February 12, 1997)     

Evolution properties of partially coherent four-petal Gaussian beams in oceanic turbulence

Based on the extended Huygens–Fresnel integral formula, the analytical expressions for partially coherent four-petal Gaussian beam propagating in oceanic turbulence are derived, and the influences of coherence length, beam order N and the parameters of oceanic turbulence (the rate of dissipation of turbulent kinetic energy per unit mass of fluid, the rate of dissipation of mean square temperature and the relative strength of temperature and salinity fluctuations) on average intensity properties are investigated using numerical examples in detail. The results show that the beam with the higher beam order N or coherence length will lose its initial four-petal profiles slower. It is also indicated that the beam will evolve into a Gauss-like beam more rapidly with increasing oceanic turbulence strength. The results have the potential application in underwater laser communication using a partially coherent four-petal Gaussian beam.     

Propagation properties of partially coherent radially polarized beam in a turbulent atmosphere

Based on the Huygens–Fresnel principle and the unified theory of coherence and polarization of partially coherent beams, we investigate the propagation characteristics of a partially coherent radially polarized doughnut (PCRPD) beam in a turbulent atmosphere. It is found that, after propagating through a turbulent atmosphere, the doughnut beam spot is changed into a circular Gaussian beam. Moreover, the degree of coherence, the degree of polarization and the degree of cross-polarization of the beam will change on propagation, and this change is dependent upon the degree of coherence of the source and atmospheric turbulence.     

Propagation properties of a partially coherent radially polarized beam in atmospheric turbulence

Based on the extended Huygens–Fresnel integral, second-order moments of the Wigner distribution function of a partially coherent radially polarized beam propagating through atmospheric turbulence are derived. Besides, propagation properties such as the mean-squared beam width, angular width, effective radius of curvature, beam propagation factor and Rayleigh range can also be obtained and calculated numerically. It is shown that the propagation properties are dependent on the spatial correlation length, refraction index structure constant and propagation distance.