Coherent Gaussian Research Articles

A fully digital noncoherent and coherent GMSK receiver architecture with joint symbol timing error and frequency offset estimation

We propose a fully digital noncoherent and coherent Gaussian minimum shift keying (GMSK) receiver architecture with joint frequency offset compensation and symbol timing recovery. Carrier phase offset can be estimated if the coherent demodulation mode is adopted. The converted base-band complex signal is first frequency discriminated and then passed through a digital filter which performs a fast Fourier transform (FFT). The frequency offset can be estimated from the DC component of the FFT, and the symbol timing error can be estimated from the phase angle of the FFT at a specified frequency which is equal to an integral multiple of half the bit rate. These two estimated parameters are then used for frequency offset compensation and symbol timing recovery during a preamble period. Coarse carrier phase can be estimated by averaging sampled in-phase and quadrature-phase signals and finding its phase angle within the preamble period after carrier frequency offset is estimated and compensated. The bit error rate (BER) performance of this GMSK receiver architecture is assessed for an additive white Gaussian noise (AWGN) channel by computer simulation.

Twist phase in Gaussian-beam optics

The recently discovered twist phase is studied in the context of the full ten-parameter family of partially coherent general anisotropic Gaussian Schell-model beams. It is shown that the nonnegativity requirement on the cross-spectral density of the beam demands that the strength of the twist phase be bounded from above by the inverse of the transverse coherence area of the beam. The twist phase as a two-point function is shown to have the structure of the generalized Huygens kernel or Green's function of a first-order system. The ray-transfer matrix of this system is exhibited. Wolf-type coherent-mode decomposition of the twist phase is carried out. Imposition of the twist phase on an otherwise untwisted beam is shown to result in a linear transformation in the ray phase space of the Wigner distribution. Though this transformation preserves the four-dimensional phase-space volume, it is not symplectic and hence it can, when impressed on a Wigner distribution, push it out of the convex set of all bona fide Wigner distributions unless the original Wigner distribution was sufficiently deep into the interior of the set.

The Optimal Growth of Tropical Sea Surface Temperature Anomalies

It is argued from SST observations for the period 1950–90 that the tropical Indo-Pacific ocean-atmosphere system may be described as a stable linear dynamical system driven by spatially coherent Gaussian white noise. Evidence is presented that the predictable component of SST anomaly growth is associated with the constructive interference of several damped normal modes after an optimal initial structure is set up by the white noise forcing. In particular, El Niño–Southern Oscillation (ENSO) growth is associated with an interplay of at least three damped normal modes, with periods longer than two years and decay times of 4 to 8 months, rather than the manifestation of a single unstable mode whose growth is arrested by nonlinearities. Interestingly, the relevant modes are not the three least damped modes of the system. Rather, mode selection, and the establishment of the optimal initial structure from which growth occurs, are controlled by the stochastic forcing. Experiments conducted with an empirical stochastic-dynamical model show that stochastic forcing not only adds energy to the system, but also plays a role in setting up the optimal structure. It is shown that growth from modal interference can occur for as long as 18 months, which within the confines of this model defines a predictability limit for growth events. Growth associated with the stochastic forcing is also possible, but is unpredictable. The timescale on which the predictable and unpredictable components of SST growth become comparable to each other gives a more conservative predictability limit of 15 months. The above scenario is based on empirical evidence obtained from SST anomalies alone. From the results of several tests based on statistical properties of linear and nonlinear dynamical systems, one may conclude that much of the ENSO cycle in nature is dominated by stable, forced dynamics.

Symmetrizing transformation of general astigmatic Gaussian beams

The symmetrizing transformation of coherent general astigmatic Gaussian beams is analysed by the tensor ABCD law. It is shown in this paper that a nonorthogonal astigmatic Gaussian beam (with rotation) can be transformed into a rotationally symmetric one by the combination of four cylindrical lenses. A simple example is given.

Imaging Performance of Confocal Fluorescence Microscopes with Finite-sized Source

Abstract The role of the illumination system of the confocal microscope, and in particular the first pinhole, is discussed. The effects of a finite sized source on the imaging performance, including the three-dimensional optical transfer function, are investigated. Two forms of source are considered, either a coherent Gaussian beam of finite size or an incoherent disc of finite size. For a confocal microscope using optical fibres as source and detector, the optimum geometry is derived for maximizing signal strength.

Decoupling of coherent Gaussian beams with general astigmatism

We show how to decouple a coherent Gaussian beam having general astigmatism (to transform it into a beam having only simple astigmatism, or orthogonal symmetry) by using a single rotated thin cylindrical lens. The resulting coherent orthogonal Gaussian beam may be further transformed into a stigmatic (rotationally symmetric) Gaussian beam by an orthogonal telescopic magnifier. The initial beam parameters must be known; we suggest a procedure for their measurement.

Gaussian Schell model sources in one-dimensional first-order systems with loss or gain: errata

A detailed examination of the propagation of Gaussian–Schell model sources in one-dimensional, possibly nonlossless, first-order systems is constructed. The laws of focusing are derived. The conditions for periodicity of the Gaussian–Schell model source are derived. This result generalizes the well-known result −2 ≤ A + D ≤ 2 for confinement of a perfectly coherent Gaussian beam to the partially coherent nonlossless case. When loss or gain is present several conditions must be satisfied simultaneously for periodicity. The self-consistent solutions are derived and the perturbation stability of the solutions is studied. A physical realization of an arbitrary nonlossless one-dimensional ABCD system is derived, which yields a convenient formula for deciding whether the ABCD system has loss or gain. Special attention is devoted to real and ripple systems.

Gaussian–Schell model sources in one-dimensional first-order systems with loss or gain

A detailed examination of the propagation of Gaussian-Schell model sources in one-dimensional, possibly nonlossless, first-order systems is constructed. The laws of focusing are derived. The conditions for periodicity of the Gaussian-Schell model source are derived. This result generalizes the well-known result -2 </= A + D </= 2 for confinement of a perfectly coherent Gaussian beam to the partially coherent nonlossless case. When loss or gain is present several conditions must be satisfied simultaneously for periodicity. The self-consistent solutions are derived and the perturbation stability of the solutions is studied. A physical realization of an arbitrary nonlossless one-dimensional ABCD system is derived, which yields a convenient formula for deciding whether the ABCD system has loss or gain. Special attention is devoted to real and ripple systems.

Gaussian Schell-model beams generated with synthetic acousto-optic holograms

The recently introduced concept of a synthetic acousto-optic hologram [ J. Appl. Phys.67, 49 ( 1990)] is applied to convert a Gaussian laser beam into a partially coherent anisotropic Gaussian Schell-model (AGSM) beam. Real-time reconfigurability of the coherence properties is achieved by this technique, which features scattering of the laser beam by an electronically synthesized, digitally phase-modulated volume grating that propagates in an acousto-optic Bragg cell. The coherence and intensity distributions of the fields obtained by different types of phase modulation are investigated theoretically. We demonstrate some particularly interesting AGSM sources and fields: a secondary elliptical AGSM source with a circularly symmetric far-field intensity distribution and an AGSM field that retains the eccentricity of its intensity profile in the propagation through any centrosymmetric (paraxial) optical system.

Detection of the Object Velocity Using Doubly-scattered Dynamic Speckles Under Gaussian Beam Illumination

Abstract The dynamic properties of doubly-scattered speckles produced at the image plane have been investigated using a scattering model that consists of a coherent Gaussian beam, a cascade of two moving diffusers and a single imaging lens. Dependence of the fluctuation speed of time-varying image speckles on the velocities of the diffusers is found to change with the size of illuminating speckles and the width of the point-spread function of the imaging lens. When the point-spread function and the focal position satisfy certain conditions, three kinds of velocity information—the velocity of each diffuser, the average velocity of the diffusers and the velocity difference between the diffusers—can be obtained by measuring a temporal correlation length of the time-varying speckle intensity. Experimental results confirm the theoretical predictions.

Quantum correlations in high-energy multihadron distributions.

We have used the Jaiswal and Mehta formulation as a model of hadronic multiplicity distributions to evaluate the joint forward-backward multiplicity distribution {ital P}({ital N}{sub {ital F}},{ital N}{sub {ital B}}) in hadronic interactions, where the underlying field includes a mixture of a coherent component and Gaussian noise. This allows us to investigate the effect of the quantum correlation length in rapidity on the strength of the forward-backward correlation, the validity of the binomial distribution for fixed {ital N}{sub {ital F}}+{ital N}{sub {ital B}}, and the Koba-Nielsen-Olesen distribution. The resultant parameters indicated that for the NA22 and CERN ISR regions, the relative noise content rises slowly with energy, while the correlation length also increases with the total width of the rapidity window.

Spectrum-enhanced spreading of partially coherent beams

We investigate the effects of a finite spectral width on the propagation characteristics of partially coherent gaussian beams. Our results predict that quasi-monochromatic and broad band beams exhibit intensity profiles that are appreciably different from those of monochromatic beams. As the beam propagates in free space, the width and the shape of the intensity profile depend not only on the width and the shape of the spectral profile of the light in the source plane, but also on its coherence properties. We illustrate the general phenomenon of spectrum-enhanced spreading of partially coherent beams for several spectral lineshapes.

Gaussian pure states in quantum mechanics and the symplectic group.

Gaussian pure states of systems with n degrees of freedom and their evolution under quadratic Hamiltonians are studied. The Wigner-Moyal technique together with the symplectic group Sp(2n,openR) is shown to give a convenient framework for handling these problems. By mapping these states to the set of n\ifmmode\times\else\texttimes\fi{}n complex symmetric matrices with a positive-definite real part, it is shown that their evolution under quadratic Hamiltonians is compactly described by a matrix generalization of the M\obius transformation; the connection between this result and the ``abcd law'' of Kogelnik in the context of laser beams is brought out. An equivalent Poisson-bracket description over a special orbit in the Lie algebra of Sp(2n,openR) is derived. Transformation properties of a special class of partially coherent anisotropic Gaussian Schell-model optical fields under the action of Sp(4, openR) first-order systems are worked out as an example, and a generalization of the ``abcd law'' to the partially coherent case is derived. The relevance of these results to the problem of squeezing in multimode systems is noted.

Partially coherent beams and a generalized ABCD-law

The beam width and the angular spread of a partially coherent beam, and their transformation as the beam passes through Sp(2, R) first order optical systems are studied using the method of generalized rays. A generalized abcd-law which governs this transformation is derived. Kogelnik's abcd-law for coherent gaussian beams and its later generalization to gaussian Schell-model beams are shown to be special cases of this law.

Density-matrix formalism for partially coherent optical fields propagating in slightly inhomogeneous media

The integral of motion method and the density matrix formalism are used to describe the propagation of partially coherent radiation in slightly inhomogeneous media. Expressions for the parameters of partially coherent gaussian beams in media with general square-law refractive-index distributions are obtained using operator algebra. New equations for the correlation function are formulated and beam invariants are obtained.

Propagation parameters of gaussian Schell-model beams

The propagation of partially coherent light beams generated by gaussian Schell-model sources is found to be conveniently characterized by two parameters that are analogous to the beam radius and the radius of wavefront curvature of fully coherent gaussian laser beams. Gaussian quasi-homogeneous beams are examined as a limiting case.

Heterodyne detection of partially coherent optical signal

AbstractThe SN ratio is studied for heterodyne detection of partially coherent plane wave optical signals when they are focused by a lens after passing through a circular aperture and are added to a coherent Gaussian beam local oscillator signal. We treat the case in which the intensity distribution of the signal over the aperture is uniform and the natural logarithm of its spatial coherence is expressed as the vth power of the distance between two arbitrary points on the aperture. Using a two‐dimensional Fourier transform, we express the mutual coherence function of the signal on the detector surface in a series form. The results are used to present an expression for the SN ratio suitable for numerical calculations. The SN ratio is investigated numerically with the correlation length on the aperture and the detector radius as parameters. It is found that, regardless of the correlation length, the detector radius of the order of the Airy disk is sufficient for obtaining an SN ratio almost equal to the maximum value determined for each correlation length. We also find that the maximum SN ratio is approached in the case of partially coherent signals if we use the optimum beam parameter of local oscillator signals, which has already been determined.

Coherent reflection of a Gaussian beam at optical frequencies from a large convex mirror

In some modern laser-system application and performance evaluations, an accurate formula is needed for the spatial distribution of intensity in the reflected or backscattered field that is produced by a coherent Gaussian beam incident upon an optical-grade convex mirror. This paper presents the development of such an analytical formula. The formula is rigorous within the Fresnel approximation, is easy to evaluate, and consists of only a few terms. However, it requires that the mirror size and distance to satisfy the Fresnel condition and the radius of curvature of the mirror be large compared with the wavelength of radiation. With the Gaussian beam transformed to a plane uniform beam and the mirror approximated to a flat circular disk, the formula reduces to the well-known Airy diffraction-pattern formula when subjected to the Fraunhofer condition. Although the formula presented here is strictly applicable only to spherical mirrors with a small aperture-radius-to-radius-of-curvature ratio, when pushed to the full sphere limit, it yields results that have good order-of-magnitude agreement with the machine-calculated results that use the rigorous Mie theory of scattering from a full sphere.

Spot Size of Focused Truncated Gaussian Beams

The separation of half-maximum intensity points of a focused truncated coherent Gaussian beam is evaluated. The results differ appreciably from the conventional values used in the past. This result is important in areas where the resolution of two optical beams is required, as, for instance, in determining the optimal separation of elements in a detector array used to identify these beams.

Partially coherent diffraction by a defocused slit aperture in the presence of image motion and transverse vibrations

This paper presents an analysis of the far-field diffraction pattern of partially spatially coherent light by an optical system with a defocused slit aperture in the presence of linear image motion and sinusoidal transverse vibrations. For partially coherent light sinc and gaussian forms of correlations in the two points over the diffracting aperture are employed. The results for the irradiance distribution have been presented graphically for typical values of the coherence, defocusing and vibrational parameters.