Analytical characterization of spectral anomalies in polychromatic apertured beams
Introduction
The free-space propagation of partially coherent polychromatic wave fields produces local changes in the spectral density distribution [1]. The observed distortion is attributed to diffraction [2], [3], though the state of spatial coherence is a parameter that should be taken into account [4]. This is not surprising since propagation of fully coherent fields is extremely dependent on the frequency of the radiation [5], [6], [7]. Of particular interest is the spectral shift and spectral switch effects in the focal region of a converging, spatially fully coherent, polychromatic spherical wave diffracted by a circular aperture [8]. Those spectral anomalies are found in the vicinity of intensity zeros, which are located at the transverse focal plane and along the optical axis.
In this short contribution, the spectral shift and spectral switch effects are further investigated within the paraxial regime [9], and no limitations are imposed to the Fresnel number of the focusing arrangement [10]. The power spectrum at points of the optical axis and the transverse focal plane has an analytical expression. Consequently both effects are characterized analytically by means of the power spectrum moments.
We consider a polychromatic focused wave that is diffracted by a circular aperture of radius a and located at a distance f from the geometrical focus. The power spectrum is assumed to have a Gaussian distribution of central frequency ω0 and rms width σ0where S0 is a constant. The power spectrum at a point of the focal volume deviates from the incident spectrum because of diffraction. The resultant power spectrum is evaluated as the product of the incident power spectrum and the spectral modifier [10]whereandare the spatial normalized coordinates expressed in terms of the cylindrical coordinates r and z. Also, N0 = a2/λ0f is the Fresnel number of the focusing geometry, where λ0 = 2πc/ω0. Note that the validity of Eq. (3) is not restricted to high values of the Fresnel number N0.
Section snippets
On-axis spectral anomalies
Eq. (2) establishes formally that diffraction produces a spectral shaping. The on-axis spectral modifier is obtained from Eq. (3) by assuming that v0 = 0,In general, the power spectrum profile resembles that of the incident radiation. However, diffraction induces the power spectrum to vanish for frequencieswhere γn = 4nπ and n is a non-zero integer. Thus, diffraction prohibits the existence of a discrete but infinite number of frequencies.
This effect
Moment analysis
In order to determine the central value of the spectral Gaussian peak and its width around that value, it is appropriate to evaluate the standardized moments of the power spectrum distribution. The kth standardized moment is expressed asfor any non-negative value of the integer k. Obviously, the value of the moments depends on the observation point of the focal region. The zero-, first- and second-order moments of the power spectrum along the optical axis have analytical
Analysis of spectral shifts and spectral broadenings
According to Eq. (14), there are two extrema corresponding to the axial pointsand the values of the relative spectral shift at these points are given byThe fractional bandwidth enhances the maximum and minimum relative spectral shifts following a linear law. The relative spectral switch, which may be defined as the difference between the maximum and minimum values of the relative spectral shift, given as 2Δ, is produced in a region bounded by two axial points which are
Spectral anomalies in the transverse focal plane
The existence of zeros in the spectral modifier justifies some spectral anomalies along the optical axis, but also in the focal plane. Accordingly, now the spectral modifier, given in Eq. (3), is evaluated for points of the transverse focal plane, where u0 = 0. We may find an analytical expression written aswhere J1 is a Bessel function of the first kind of order 1. Since the Bessel function has some zeros at values βn, i.e., J1(βn) = 0, being n a positive integer, the
Discussion
Eqs. (14), (15) provide the axial patterns of the relative spectral shift and the relative spectral broadening in the vicinity of phase singularities for the central frequency ω0. In accordance to Eqs. (22), (23), these analytical spatial dependences remain invariant when we evaluate δω/ω0 and in the transverse focal plane, with the only particularity of precising the spatial coordinate under use and its values for which we locate point phase singularities. Therefore, the presence of a
Acknowledgements
It is a pleasure to thank Pedro Andrés and Juan A. Monsoriu for helpful conversations. I acknowledge the financial support from the Generalitat Valenciana (grants GRUPOS03/227 and GV04B-082), Spain.
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