Spatial correlation properties of focused partially coherent light

Fischer, David G.; Visser, Taco D.

doi:10.1364/josaa.21.002097

Cited by 70 publications

(42 citation statements)

References 7 publications

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“…On making use of the Debye approximation [4, Sec. 8.8.1], one can then derive for the crossspectral density the following expression (see [22] for details):…”

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confidence: 99%

Generalized Gouy phase for focused partially coherent light and its implications for interferometry

2012

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The Gouy phase, sometimes called the phase anomaly, is the remarkable effect that in the region of focus a converging wave field undergoes a rapid phase change by an amount of π, compared to the phase of a plane wave of the same frequency. This phenomenon plays a crucial role in any application where fields are focused, such as optical coherence tomography, mode selection in laser resonators, and interference microscopy. However, when the field is spatially partially coherent, as is often the case, its phase is a random quantity. When such a field is focused, the Gouy phase is therefore undefined. The correlation properties of partially coherent fields are described by their so-called spectral degree of coherence. We demonstrate that this coherence function does exhibit a generalized Gouy phase. Its precise behavior in the focal region depends on the transverse coherence length. We show that this effect influences the fringe spacing in interference experiments in a nontrivial manner. © 2012 Optical Society of America OCIS codes: 030. 1640, 050.1960, 180.3170, 120.3940. In 1890 Gouy found that the phase of a monochromatic, diffracted converging wave, compared to that of a plane wave of the same frequency, undergoes a rapid change of 180°near the geometric focus [1][2][3][4] Under many practical circumstances, light is not monochromatic, but rather partially coherent. Examples are light that is produced by a multimode laser or light that has traveled through the atmosphere or biological tissue. In those cases, the phase of the wave field is a random quantity. Therefore, when a partially coherent field is focused (as described in [13][14][15][16][17][18][19][20]), the Gouy phase is undefined; i.e., it has no physical meaning. In the space-frequency domain, a partially coherent optical field is characterized by two-point correlation functions, such as the cross-spectral density or its normalized version, the spectral degree of coherence [21]. These complex-valued functions have a phase that is typically welldefined. As we will demonstrate for a broad class of partially coherent fields, the phase of both correlation functions shows a generalized phase anomaly, which reduces to the classical Gouy phase in the coherent limit. Furthermore, this generalized Gouy phase affects the interference of highly focused fields, in microscopy for example, altering the fringe spacing compared to that of a coherent field.Consider first a converging, monochromatic field of frequency ω that is exiting a circular aperture with radius a in a plane screen (see Fig. 1). The origin O of the coordinate system coincides with the geometrical focus. The amplitude of the field is U 0 r 0 ; ω, r 0 being the position vector of a point Q in the aperture. The field at a point Pr in the focal region is, according to the Huygens-Fresnel principle [4, Chap. 8.2], given by the following expression:where the integration extends over the spherical wave front S that fills the aperture, s jr − r 0 j denotes the distance QP, and k 2π∕λ is the wavenumber a...

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“…On making use of the Debye approximation [4, Sec. 8.8.1], one can then derive for the crossspectral density the following expression (see [22] for details):…”

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confidence: 99%

Generalized Gouy phase for focused partially coherent light and its implications for interferometry

2012

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“…Such correlation singularities, or "coherence vortices," occur at pairs of points at which the field is completely uncorrelated. Coherence vortices have since been found in optical beams [20][21][22][23][24][25][26][27], in focused fields [28], in the far-zone of quasi-homogeneous sources [29], and in fields produced by Mie scattering [30,31]. Some of these studies have been carried out in the space-time domain, others in the space-frequency domain.…”

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confidence: 99%

Topological reactions of correlation functions in partially coherent electromagnetic beams

2013

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It was recently shown that so-called coherence vortices, singularities of the two-point correlation function, generally occur in partially coherent electromagnetic beams. We study the three-dimensional structure of these singularities and show that in successive cross sections of a beam a rich variety of topological reactions takes place. These reactions involve, apart from vortices, the creation or annihilation of dipoles, saddles, maxima and minima of the phase of the correlation function. Since these reactions happen generically, i.e., under quite general conditions, these observations have implications for interference experiments with partially coherent, electromagnetic beams.

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“…Focusing of coherent light under a scalar approximation has been well understood for many years (see, for example, [1]), while even as early as 1919 focusing of coherent, fully polarized electromagnetic waves could be described by what is now known as the Debye-Wolf diffraction integral [2][3][4][5]. In more recent years attention has slowly turned toward focusing of partially coherent light in both scalar [6][7][8][9] and vectorial [10,11] regimes due to its potential use in lithography, laser fusion, and microscopy [12][13][14][15]. Consideration of the full electromagnetic problem has, however, been limited to homogeneous (partial) polarization across the pupil of the focusing lens.…”

Section: Introductionmentioning

confidence: 99%

Focusing of spatially inhomogeneous partially coherent, partially polarized electromagnetic fields

Foreman

Török

2009

J. Opt. Soc. Am. A

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We report a general framework capable of describing the focusing of electromagnetic waves with spatially varying coherence and polarization properties in optical systems of arbitrary numerical aperture and Fresnel number. We also investigate the reduction of the dimensionality of the requisite integrals by use of a coherent mode expansion. We find that coherent mode expansions treating each component of the electric field vector individually are unsuitable for describing focusing systems because of the inter-component mixing that can occur in high numerical aperture systems. In addition, we show that the assumption of harmonic angular dependence allows the azimuthal integration to be performed analytically, providing further simplification of the analysis. We also find that the effective degree of spectral coherence of an electromagnetic beam is unchanged upon focusing. Finally, as an illustration of the developed framework, we calculate the transverse and axial focal distributions for a partially coherent source formed by incoherent superposition of radially and azimuthally polarized Laguerre-Gauss modes.

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Spatial correlation properties of focused partially coherent light

Cited by 70 publications

References 7 publications

Generalized Gouy phase for focused partially coherent light and its implications for interferometry

Generalized Gouy phase for focused partially coherent light and its implications for interferometry

Topological reactions of correlation functions in partially coherent electromagnetic beams

Focusing of spatially inhomogeneous partially coherent, partially polarized electromagnetic fields

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