Coherent Gaussian Research Articles

Propagation of a partially coherent hollow vortex Gaussian beam through a paraxial ABCD optical system in turbulent atmosphere

The propagation of a partially coherent hollow vortex Gaussian beam through a paraxial ABCD optical system in turbulent atmosphere has been investigated. The analytical expressions for the average intensity and the degree of the polarization of a partially coherent hollow vortex Gaussian beam through a paraxial ABCD optical system are derived in turbulent atmosphere, respectively. The average intensity distribution and the degree of the polarization of a partially coherent hollow vortex Gaussian beam in turbulent atmosphere are numerically demonstrated. The influences of the beam parameters, the topological charge, the transverse coherent lengths, and the structure constant of the atmospheric turbulence on the propagation of a partially coherent hollow vortex Gaussian beam in turbulent atmosphere are also examined in detail. This research is beneficial to the practical applications in free-space optical communications and the remote sensing of the dark hollow beams.

Coherent Feedback Control of Linear Quantum Optical Systems via Squeezing and Phase Shift

The purpose of this paper is to present a theoretic and numerical study of utiliz- ing squeezing and phase shift in coherent feedback control of linear quantum optical systems. A quadrature representation with built-in phase shifters is proposed for such systems. Fundamen- tal structural characterizations of linear quantum optical systems are derived in terms of the new quadrature representation. These results reveal considerable insights into the issue of the physical realizability of such quantum systems. The problem of coherent quantum linear quadratic Gaussian (LQG) feedback control studied in H. I. Nurdin, M. R. James, and I. R. Petersen, Automatica, IFAC, 45 (2009), pp. 1837-1846; G. Zhang and M. R. James, IEEE Trans. Automat. Control, 56 (2011), pp. 1535-1550 is reinvestigated in depth. First, the optimization methods in these papers are extended to a multistep optimization algorithm which utilizes ideal squeezers. Second, a two-stage optimization approach is proposed on the basis of controller parametrization. Numerical studies show that closed-loop systems designed via the second approach may offer LQG control performance even better than that when the closed-loop systems are in the vacuum state. When ideal squeezers in a closed-loop system are replaced by (more realistic) degenerate parametric amplifiers, a sufficient condition is derived for the asymptotic stability of the resultant new closed-loop system; the issue of performance convergence is also discussed in the LQG control setting.

Propagation factor of partially coherent flat-topped beam array in free space and turbulent atmosphere

Analytical expressions for the propagation factor of partially coherent radial flat-topped (FT) beam array in free-space and turbulent atmosphere are derived. Our formulae can be applied to study the beam propagation factors of partially coherent Gaussian beam, partially coherent FT beam and partially coherent radial Gaussian beam array. The evolution properties of the propagation factors of those beams in turbulent atmosphere are studied numerically and comparatively. From the aspect of the propagation factor, our results show that partially coherent radial FT beam array is less affected by turbulence than partially coherent Gaussian beam, partially coherent FT beam and partially coherent radial Gaussian beam array.

Effects of extremely strong turbulent medium on scintillations of partially coherent annular and flat-topped Gaussian beams

Scintillation index of partially coherent annular and partially coherent flat-topped Gaussian beams propagating in horizontal links is found at the receiver origin when these beams propagate in extremely strong atmospheric turbulence. Scintillation index of coherent versions of such beams attain unity saturation value whereas the decrease in the degree of source coherence results in the decrease of the scintillations. At a fixed degree of partial coherence, thin ring sized annular beams possess smaller scintillations than thick ones. For partially coherent flat-topped Gaussian beams, higher flatness yields smaller intensity fluctuations. In extremely strong turbulence, partially coherent annular and partially coherent flat-topped Gaussian beams have smaller scintillations when compared to partially coherent single Gaussian beam scintillations.

Coherent semiclassical states for loop quantum cosmology

The spatially flat Friedmann-Robertson-Walker cosmological model with a massless scalar field in loop quantum cosmology admits a description in terms of a completely solvable model. This has been used to prove that: (i) the quantum bounce that replaces the big bang singularity is generic; (ii) there is an upper bound on the energy density for all states, and (iii) semiclassical states at late times had to be semiclassical before the bounce. Here we consider a family of exact solutions to the theory, corresponding to generalized coherent Gaussian and squeezed states. We analyze the behavior of basic physical observables and impose restrictions on the states based on physical considerations. These turn out to be enough to select, from all the generalized coherent states, those that behave semiclassical at late times. We study then the properties of such states near the bounce where the most ``quantum behavior'' is expected. As it turns out, the states remain sharply peaked and semiclassical at the bounce and the dynamics is very well approximated by the ``effective theory'' throughout the time evolution. We compare the semiclassicality properties of squeezed states to those of the Gaussian semiclassical states and conclude that the Gaussians are better behaved. In particular, the asymmetry in the relative fluctuations before and after the bounce are negligible, thus ruling out claims of so-called ``cosmic forgetfulness.''

Turbulent Distance of Partially Coherent Hollow Gaussian Beams

For partially coherent hollow Gaussian beam (HGB), the turbulent distance, in which all of the spatial and angular spreading and the beam propagation factor increasing due to turbulence can be neglected, has been investigated in detail. It is shown that the turbulent distance of partial coherent HGBs increases with increasing beam order and wavelength, and decreasing turbulent parameter and coherent parameter. With increasing waist width, the turbulent distance first increases and then decreases. Furthermore, the turbulent distance of a HGB is much larger than that of the corresponding Gaussian Schell-model (GSM) beam when choosing the appropriate value of the waist width, implying that a HGB may be more appropriate to be used in optical communication links than a GSM beam.

Scattering of a partially coherent Gaussian–Schell beam from a diffuse target in slant atmospheric turbulence

On the basis of the extended Huygens-Fresnel principle, the scattering of partially coherent Gaussian-Schell-model (GSM) beams from a diffuse target in slant double-passage atmospheric turbulence is studied and compared with that of fully coherent Gaussian beams. Using the cross-spectral density function of the GSM beams, we derive the expressions of the mutual coherence function, angle-of-arrival fluctuation, and covariance and variance of the intensity of the scattered field, taking into account the fluctuations of both the log-amplitude and phase. The numerical results are presented, and the influences of the wavelength, propagation distance, and waist radius on scattering properties are discussed. The perturbation region of the normalized intensity variance of the partially coherent GSM beam is smaller than that of the fully coherent Gaussian beam at the middle turbulence level. The normalized intensity variance of long-distance beam propagation is smaller than that of beam propagation along a short distance.

Propagation of a General Gaussian Beam by Paraxial Ray Tracing in an Arbitrary Optical System

The propagation of a partially coherent Gaussian beam in a homogeneous medium can be explained with various methods. One of them is Gaussian shell model beams [1–8]. It uses the Gaussian Schell-model source whose irradiance distribution and degree of coherence are both Gaussian. We will describe Gaussian beams as the ray distribution L as functions of linear and angular coordinates. Of course, since the Gaussian Shell model beam can be produced by a superposition of independent coherent Gaussian beams, our concept is similar to the Gaussian Shell model concept. However, the goal of the present paper is to describe the propagation of general Gaussian beams through an arbitrary optical system by using the geometrical paraxial ray tracing method. In other words, many rays propagate through a reference plane, these rays distribute with the Gaussian profile in linear and angular direction cosine coordinates, and every ray propagates according to the laws of geometrical optics. At first, we introduce the ray distribution density of a circular symmetric Gaussian beam and investigate how that Gaussian beam propagates through an arbitrary axially symmetric optical system or in a homogeneous medium under the paraxial approximation, and we determine the physical meanings of the parameters. Also, we introduce the complex parameter q and the Lagrange invariant L for this general Gaussian beam and discover that this Gaussian beam also follows the Kogelnik ABCD law, just as a classical coherent Gaussian beam does.

Calculation of Mutual Information for Partially Coherent Gaussian Channels With Applications to Fiber Optics

The mutual information between a complex-valued channel input and its complex-valued output is decomposed into four parts based on polar coordinates: an amplitude term, a phase term, and two mixed terms. Numerical results for the additive white Gaussian noise (AWGN) channel with various inputs show that, at high signal-to-noise ratio (SNR), the amplitude and phase terms dominate the mixed terms. For the AWGN channel with a Gaussian input, analytical expressions are derived for high SNR. The decomposition method is applied to partially coherent channels and a property of such channels called “spectral loss” is developed. Spectral loss occurs in nonlinear fiber-optic channels and it may be one effect that needs to be taken into account to explain the behavior of the capacity of nonlinear fiber-optic channels.

Intensity fluctuations of partially coherent cos Gaussian and cosh Gaussian beams inatmospheric turbulence

On-axis and off-axis scintillation indices of partially coherent collimated cos Gaussian andcosh Gaussian beams are evaluated on their propagation in a weakly turbulent atmosphere.In the coherent limits, our results are in quite good agreement with those for the existingRytov scintillation index of coherent cos Gaussian and cosh Gaussian beams,especially for very small and very large sized beams and at weaker turbulence levels.The on-axis scintillation index becomes lower as the degree of source coherencedecreases for both cos Gaussian and cosh Gaussian beams. Comparing to thepartially coherent Gaussian beam on-axis scintillations, partially coherent cosGaussian beam scintillations are generally lower and partially coherent cosh Gaussianbeam scintillations are higher. Scintillations of Gaussian beams are very closeto the scintillations of equal-power cos Gaussian and are quite similar to thescintillations of equal-power cosh Gaussian beams. Off-axis evaluations yield that thescintillations of both cos Gaussian and cosh Gaussian beams increase as the diagonaldistance from the origin increases, being valid for all degrees of partial coherence.

Propagation of partially coherent Lorentz and Lorentz–Gauss beams through a paraxial ABCD optical system in a turbulent atmosphere

Based on the generalized Huygens–Fresnel integral, propagation of partially coherent Lorentz and Lorentz–Gauss beams through a paraxial ABCD optical system in a turbulent atmosphere was investigated. Analytical propagation formulae were derived for the cross-spectral densities of partially coherent Lorentz and Lorentz–Gauss beams. As an application example, the focusing properties of partially coherent Gaussian, Lorentz and Lorentz–Gauss beams in a turbulent atmosphere and in free space were studied numerically and comparatively. It is found that the focusing properties of such beams are closely related to the initial coherence length and the structure constant of turbulence. By choosing a suitable initial coherence length, a partially coherent Lorentz beam can be focused more tightly than a Gaussian or Lorentz–Gauss beam in free space or in a turbulent atmosphere with small structure constant at the geometrical focal plane.

An alternative model for a partially coherent elliptical dark hollow beam

An alternative theoretical model named partially coherent hollow elliptical Gaussian beam (HEGB) is proposed to describe a partially coherent beam with an elliptical dark hollow profile. Explicit expression for the propagation factors of a partially coherent HEGB is derived. Based on the generalized Collins formula, analytical formulae for the cross-spectral density and mean-squared beam width of a partially coherent HEGB, propagating through a paraxial ABCD optical system, are derived. Propagation properties of a partially coherent HEGB in free space are studied as a numerical example.

The transformation of an edge dislocation in atmospheric turbulence

The explicit expression for the cross-spectral density of partially coherent elliptical Gaussian beams carrying an edge dislocation propagating through atmospheric turbulence along a slant path is derived, and used to study the transformation of the edge dislocation in atmospheric turbulence. We find that the edge dislocation disappears and transforms to a noncanonical coherence vortex, when partially coherent elliptical Gaussian beams with edge dislocation propagate through atmospheric turbulence. The inversion of the topological charge of the coherence vortex may take place. The ellipticity of the beam and slope of the edge dislocation play a dominant role in the evolution of the coherence vortex. In the coherent limit the coherence vortex in turbulence becomes an intensity vortex, however, differing from the case in free-space propagation, the position of the intensity vortex depends on the choice of the reference point. The results are illustrated analytically and numerically.

Diffraction properties of partially coherent electromagnetic four-petal Gaussian beams

The concept of existing scalar coherent four-petal Gaussian (FPG) beam is extended to the more general partially coherent electromagnetic four-petal Gaussian (PCFPG) beam, and the cross-spectral density matrix of the latter is derived in analytical expressions. Intensity distributions and degree of polarization in free space are investigated by numerical examples. Results show that propagation properties of the PCFPG beam are different from their scalar coherent and scalar partially coherent counterparts. Both intensity distributions and polarization strongly depend on transversal coherence lengths. When the propagation distance increases to a large value, intensity distributions of PCFPG beams convert into the Gauss profile. Polarization of PCFPG beams shows an oscillating characteristic upon propagation, and the oscillation would enhance with the increase of coherence lengths. These results may find potential applications in beam shaping and free space optical communications.

Partially coherent off-axis Gaussian beam scintillations

The scintillation index at the receiver origin is formulated for a partially coherent off-axis Gaussian beam in atmospheric turbulence by employing the extended Huygens–Fresnel principle. Our formula correctly reduces to the existing coherent and partially coherent Gaussian beam scintillation indices in the limiting cases. For off-axis Gaussian beams with imaginary, real and complex displacement parameters, the scintillation index reduces when the incidence becomes more incoherent. When the source size of the off-axis Gaussian beam increases, the scintillations increase for partially coherent sources and decrease for incoherent sources, the tendency being observed for imaginary, real and complex displacement parameters. For the fully coherent off-axis Gaussian beams, increase in the source size first causes an increase in the scintillations, eventually reaching saturation at large source sizes, the increase is not monotonic and may exhibit a peak around the Fresnel zone sized off-axis Gaussian sources. For all degrees of partial coherence, off-axis beams possessing imaginary displacement parameters exhibit larger scintillations when the displacement parameter increases for large sized incidences, however, for small sized incidences, scintillations stay at the same level when the imaginary displacement parameter increases, the fixed scintillation value being lower for more incoherent sources. For off-axis Gaussian sources possessing real displacement parameters, this behavior is reciprocal with respect to the source size, i.e. for all degrees of partial coherence, off-axis beams possessing real displacement parameters exhibit larger scintillations when the displacement parameter increases for small sized incidences, and for large sized incidences, scintillations stay at the same level when the real displacement parameter increases, the fixed scintillation value again being lower for more incoherent sources. For all degrees of partial coherence, off-axis Gaussian beams possessing imaginary displacement parameters exhibit larger scintillations than the off-axis Gaussian beams possessing real displacement parameters when the absolute value of displacement parameter increases for large sized incidences.

AVERAGE INTENSITY AND SPREADING OF PARTIALLY COHERENT FOUR-PETAL GAUSSIAN BEAMS IN TURBULENT ATMOSPHERE

The concept of partially coherent four-petal Gaussian (PCFPG) beam is introduced and described in analytical forms. Based on the Huygens-Fresnel integral formula, average intensity and beam spreading in turbulent atmosphere are derived in analytical expressions. Efiects of beam parameters and atmospheric structure constant on intensity distributions and efiective beam sizes are investigated in detail, respectively. Results show that PCFPG beams carrying larger coherence lengths or higher beam orders would be less afiected by turbulence. It is also indicated that, when the propagation distance increases, the PCFPG beam would convert into the Gauss- like proflle sooner or later, but this degradation can be reduced by modulating beam parameters. Results in this paper may provide potential applications in free-space optical communications.

E. C. G. Sudarshan and symmetry in classical dynamics, optics and quantum mechanics

We review work initiated and inspired by Sudarshan in relativistic dynamics, beam optics, partial coherence theory, Wigner distribution methods, multimode quantum optical squeezing, and geometric phases. The 1963 No Interaction Theorem using Dirac's instant form and particle World Line Conditions is recalled. Later attempts to overcome this result exploiting constrained Hamiltonian theory, reformulation of the World Line Conditions and extending Dirac's formalism, are reviewed.Dirac's front form leads to a formulation of Fourier Optics for the Maxwell field, determining the actions of First Order Systems (corresponding to matrices of Sp(2,R) and Sp(4,R)) on polarization in a consistent manner. These groups also help characterize properties and propagation of partially coherent Gaussian Schell Model beams, leading to invariant quality parameters and the new Twist phase.The higher dimensional groups Sp(2n,R) appear in the theory of Wigner distributions and in quantum optics. Elegant criteria for a Gaussian phase space function to be a Wigner distribution, expressions for multimode uncertainty principles and squeezing are described.In geometric phase theory we highlight the use of invariance properties that lead to a kinematical formulation and the important role of Bargmann invariants. Special features of these phases arising from unitary Lie group representations, and a new formulation based on the idea of Null Phase Curves, are presented.

Scintillations of partially coherent multiple Gaussian beams in turbulence

For an incidence composed of partially coherent multiple Gaussian beams, Huygens-Fresnel principle-based on-axis scintillation index is formulated in a weakly turbulent homogeneous horizontal atmospheric path. Our general formulation is applied to two examples of partially coherent annular and partially coherent flat-topped Gaussian beams. Compared to partially coherent single Gaussian beam scintillations, annular beam scintillations seem to possess higher values for all partial coherence levels, whereas flat-topped Gaussian beam intensity fluctuations are slightly larger, especially at lower coherence levels and at larger source sizes. At the same source partial coherence, annular beams exhibit smaller scintillations for larger ring sizes. For flat-topped Gaussian beams, except for very small and very large source sizes, as the number of Gaussian beams forming the flatness increases, intensity fluctuations also increase, a trend applicable for different degrees of coherence. A trend valid for both single and multiple Gaussian incidence, except for certain annular beams of large primary beam sizes, is that the scintillations decrease as the source becomes less coherent. Being applicable for all degrees of source coherences, for both beams examined, scintillations increase steadily as the Rytov plane wave scintillation index increases.

Comparison of wave structure functions for intensity profiles

We give a list of available wave structure functions (WSFs) of a spherical wave, simultaneously providing some derivation details. The aim is to assess the impact of these WSFs on intensity profiles of various beams propagating in turbulence. For comparisons, coherent and partially coherent fundamental Gaussian, hyperbolic, sinusoidal and annular Gaussian beams are chosen. Comparisons are made by calculating the difference between the intensity profiles of the analytic solution that uses the normalized WSF with quadratic approximation and the intensity profiles obtained by numerically solving the quadruple extended Huygens–Fresnel integral containing other WSFs. The graphical results show that in general the differences arising from the use of different WSFs are not substantial. Such differences become much greater however toward the complete incoherence limit and at relatively higher structure constant values. Even then, at these extremes, the receiver intensity levels are much reduced, making such big differences immaterial.

Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere

Analytical formulas are derived for the propagation of a partially coherent twisted anisotropic Gaussian Schell-model (TAGSM) beam in a turbulent atmosphere. Propagation properties of a TAGSM beam in a turbulent atmosphere are investigated in detail. It is found that a TAGSM beam will become a circular beam in a turbulent atmosphere, and low coherence and larger twist have an effect of anticircularization of the beam spot. The beam spot spreads more rapidly for lower coherence, larger twist, or stronger turbulence.