Level-crossing densities in random wave fields

Kessler, David A.; Freund, Isaac

doi:10.1364/josaa.15.001608

Cited by 9 publications

(6 citation statements)

References 31 publications

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“…Previously, we calculated for normal speckle the quantities N x (ϕ x , L) (N x (ϕ y , L)), which are the density of intersections of crossings at level L of phase derivatives ϕ x (ϕ y ) with the x-axis [17]. Extending by inspection these results to the zero crossings Z x and Z y of speckled speckle, we have…”

Section: General Resultsmentioning

confidence: 99%

“…General results. Level crossing densities for R, and its first, R x , R y , and second, R xx , R xy , R yy , derivatives have long been known for normal speckle [17,21], and we obtain by inspection for level zero for speckled speckle,…”

Section: B Zero Level Crossingsmentioning

confidence: 86%

“…We discuss here the dimensionless parameter η that plays a central role in the level crossing statistics of derivatives of the phase, of the real and imaginary parts of the wavefunction, and of the intensity. Extending [17], we write for speckled speckle…”

Section: Appendix B: Derivativesmentioning

confidence: 99%

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Singularities in speckled speckle: Statistics

Freund

Kessler

2008

Optics Communications

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Random optical fields with two widely different correlation lengths generate far field speckle spots that are themselves highly speckled. We call such patterns speckled speckle, and study their critical points (singularities and stationary points) using analytical theory and computer simulations. We find anomalous spatial arrangements of the critical points and orders of magnitude anomalies in their relative number densities, and in the densities of the associated zero crossings. arXiv:0806.3656v1 [physics.optics]

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Section: General Resultsmentioning

confidence: 99%

Section: B Zero Level Crossingsmentioning

confidence: 86%

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Singularities in speckled speckle: Statistics

Freund

Kessler

2008

Optics Communications

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“…The regression term (6) in the Slepian model is almost equal to the classical Taylor expansion around the singularity as seen in (7); the only difference is the covariance-dependent factor c(|r|) = − C (|r|) λ 2 |r| , which tends to 1 as |r| → 0. The second term, 0 res (r) = ( ξ (r), η (r)) T , is a Gaussian residual term.…”

Section: The Slepian Model Around Phase Singularitiesmentioning

confidence: 99%

“…The complex isotropic Gaussian random field is a fundamental mathematical ingredient in the theory of singular optics, [1], introduced already by Berry, [2], and further studied, theoretically and experimentally in, for example, [3][4][5][6][7]. The singularities are defined as lines in space, or points in the plane, where both the real and the imaginary parts of the complex field are zero, and hence the intensity of the light is zero and the phase is undefined.…”

Section: Introductionmentioning

confidence: 99%

A detailed statistical representation of the local structure of optical vortices in random wavefields

Lindgren

2012

J. Opt.

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The statistical properties near phase singularities in a complex wavefield are here studied by means of the conditional distributions of the real and imaginary Gaussian components, given a common zero crossing point. The exact distribution is expressed as a Slepian model, where a regression term provides the main structure, with parameters given by the gradients of the Gaussian components at the singularity, and Gaussian non-stationary residuals that provide local variability. This technique differs from the linearization (Taylor expansion) technique commonly used. The empirically and theoretically verified elliptic eccentricity of the intensity contours in the vortex core is a property of the regression term, but with different normalization compared to the classical theory. The residual term models the statistical variability around these ellipses. The radii of the circular contours of the current magnitude are similarly modified by the new regression expansion and also here the random deviations are modeled by the residual field.

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Short- and long-range screening of optical phase singularities and C points

Kessler

Freund

2008

Optics Communications

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Screening of topological charges (singularities) is discussed for paraxial optical fields with short and with long range correlations. For short range screening the charge variance˙Q 2¸i n a circular region A with radius R grows linearly with R, instead of with R 2 as expected in the absence of screening; for long range screening˙Q 2¸g rows faster than R: for a field whose autocorrelation function is the zero order Bessel function J0,˙Q 2¸∼ R ln R. A J0 correlation function is not attainable in practice, but we show how to generate an optical field whose correlation function closely approximates this form; screening in such a field is well described by our theoretical results for J0.˙Q 2¸c an be measured by counting positive and negative singularities inside A, or more easily by counting signed zero crossings on the perimeter P of A. For the first method˙Q 2¸i s calculated by integration over the charge correlation function C (r), for the second by integration over the zero crossing correlation function Γ (r). Using the explicit forms of C (r) and of Γ (r) we show that both methods of calculation yield the same result. We show that for short range screening the zero crossings can be counted along a straight line whose length equals P , but that for long range screening this simplification no longer holds. We also show that for realizable optical fields, for sufficiently small R,˙Q 2¸∼ R 2 , whereas for sufficiently large R,˙Q 2¸∼ R. These universal laws are applicable to both short and pseudo-long range correlation functions.

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Level-crossing densities in random wave fields

Cited by 9 publications

References 31 publications

Singularities in speckled speckle: Statistics

Singularities in speckled speckle: Statistics

A detailed statistical representation of the local structure of optical vortices in random wavefields

Short- and long-range screening of optical phase singularities and C points

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