23,204 research outputs found
Spatial coherence of fields from generalized sources in the Fresnel regime
Analytic expressions of the spatial coherence of partially coherent fields
propagating in the Fresnel regime in all but the simplest of scenarios are
largely lacking and calculation of the Fresnel transform typically entails
tedious numerical integration. Here, we provide a closed-form approximation
formula for the case of a generalized source obtained by modulating the field
produced by a Gauss-Shell source model with a piecewise constant transmission
function, which may be used to model the field's interaction with objects and
apertures. The formula characterizes the coherence function in terms of the
coherence of the Gauss-Schell beam propagated in free space and a
multiplicative term capturing the interaction with the transmission function.
This approximation holds in the regime where the intensity width of the beam is
much larger than the coherence width under mild assumptions on the modulating
transmission function. The formula derived for generalized sources lays the
foundation for the study of the inverse problem of scene reconstruction from
coherence measurements.Comment: Accepted for publication in JOSA
Asymptotic expansions and fast computation of oscillatory Hilbert transforms
In this paper, we study the asymptotics and fast computation of the one-sided
oscillatory Hilbert transforms of the form where the bar indicates the Cauchy principal value and is a
real-valued function with analytic continuation in the first quadrant, except
possibly a branch point of algebraic type at the origin. When , the
integral is interpreted as a Hadamard finite-part integral, provided it is
divergent. Asymptotic expansions in inverse powers of are derived for
each fixed , which clarify the large behavior of this
transform. We then present efficient and affordable approaches for numerical
evaluation of such oscillatory transforms. Depending on the position of , we
classify our discussion into three regimes, namely, or
, and . Numerical experiments show that the convergence
of the proposed methods greatly improve when the frequency increases.
Some extensions to oscillatory Hilbert transforms with Bessel oscillators are
briefly discussed as well.Comment: 32 pages, 6 figures, 4 table
SU(2) and SU(1,1) Approaches to Phase Operators and Temporally Stable Phase States: Applications to Mutually Unbiased Bases and Discrete Fourier Transforms
We propose a group-theoretical approach to the generalized oscillator algebra
Ak recently investigated in J. Phys. A: Math. Theor. 43 (2010) 115303. The case
k > or 0 corresponds to the noncompact group SU(1,1) (as for the harmonic
oscillator and the Poeschl-Teller systems) while the case k < 0 is described by
the compact group SU(2) (as for the Morse system). We construct the phase
operators and the corresponding temporally stable phase eigenstates for Ak in
this group-theoretical context. The SU(2) case is exploited for deriving
families of mutually unbiased bases used in quantum information. Along this
vein, we examine some characteristics of a quadratic discrete Fourier transform
in connection with generalized quadratic Gauss sums and generalized Hadamard
matrices
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