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

Abstract

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...

Poisson's spot and Gouy phase

“…Further, a complete definition for the Gouy phase for partially coherent light waves was given in Ref. [37]. In this work we use the definition of Ref.…”

“…In this work we use the definition of Ref. [37] to obtain the Gouy phase for partially coherent matter waves as…”

“…The effect of the environment is summarized by "a collision term" in the propagator which takes into account the decoherence, that is to say, the damping of off-diagonal terms of the density matrix in position representation just as in [36]. The Gouy phase for partially coherent light wave was treated in [37] which define a generalized expression for the Gouy phase in terms of the cross-spectral density. For a model of matter waves with loss of coherence we do not have an expression for the Gouy phase.…”

“…For a model of matter waves with loss of coherence we do not have an expression for the Gouy phase. However, since the cross-spectral density and density matrix have analogous meaning, in this contribution we follow the treatment adopted in [37] and define the Gouy phase as the phase of the density matrix.…”

Phase and polarization changes of pulsed Gaussian beams during focusing and propagation

“…So the question is, that was raised already: how focusing influences phase velocity? Of course, as the change of phase velocity on axis is a consequence of the Gouy phase shift of beams, the answer affects linear optical studies as well [165,192,193] (and references therein).…”