23,204 research outputs found

    Spatial coherence of fields from generalized sources in the Fresnel regime

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    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

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    In this paper, we study the asymptotics and fast computation of the one-sided oscillatory Hilbert transforms of the form H+(f(t)eiωt)(x)=−int0∞eiωtf(t)t−xdt,ω>0,x≥0,H^{+}(f(t)e^{i\omega t})(x)=-int_{0}^{\infty}e^{i\omega t}\frac{f(t)}{t-x}dt,\qquad \omega>0,\qquad x\geq 0, where the bar indicates the Cauchy principal value and ff is a real-valued function with analytic continuation in the first quadrant, except possibly a branch point of algebraic type at the origin. When x=0x=0, the integral is interpreted as a Hadamard finite-part integral, provided it is divergent. Asymptotic expansions in inverse powers of ω\omega are derived for each fixed x≥0x\geq 0, which clarify the large ω\omega behavior of this transform. We then present efficient and affordable approaches for numerical evaluation of such oscillatory transforms. Depending on the position of xx, we classify our discussion into three regimes, namely, x=O(1)x=\mathcal{O}(1) or x≫1x\gg1, 0<x≪10<x\ll 1 and x=0x=0. Numerical experiments show that the convergence of the proposed methods greatly improve when the frequency ω\omega 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

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    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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