Acceleration of femtosecond pulses to superluminal velocities by Gouy phase shift

The phase and the group velocities are calculated in a three-dimensional neighbourhood of the focus of an aberration-free lens illuminated by a spatially Gaussian beam. The Gouy phase shift caused by the diffraction results in superluminal pulse propagation on the optical axis within the Rayleigh range.     

Observation of Gouy phase anomaly with an interferometer

The intensity variation of bright and dark fringes in Young's double slit experiment shows that a light beam propagating through the focus of a lens experiences a phase shift (called the Gouy phase shift [Gouy CR. Acad Sci Paris 1890;110:1251]) with respect to its plane wave counterpart. The additional phase change of π introduced on focusing the light beam and then further propagating it in one arm of the interferometer changes a bright fringe into a dark fringe and vice-versa. We have, thus, made direct visualization of the Gouy phase shift by a simple experiment.     

Physical origin of the Gouy phase shift

We show explicitly that the well-known Gouy phase shift of any focused beam originates from transverse spatial confinement, which, through the uncertainty principle, introduces a spread in the transverse momenta and hence a shift in the expectation value of the axial propagation constant. A general expression is given for the Gouy phase shift in terms of expectation values of the squares of the transverse momenta. Our result also explains the phase shift in front of the Kirchhoff diffraction integral.     

Propagation of optical vortex solitons due to the Gouy phase in strongly nonlocal nonlinear media

In this paper,we present a study on the propagation of the symmetrical optical vortices formed by two collinear Laguerre-Gauss solitons in strongly nonlocal nonlinear media.The optical vortices,which move along the beam axis as the light propagates,result in a rotation of the beam’s transverse profile.This physical reason of the rotation is the Gouy phase acquired by the component beams.     

Focusing and phase compensation of paraxial beams by a left-handed material slab

On the basis of angular spectrum representation, a formalism describing paraxial beams propagating through an isotropic left-handed material (LHM) slab is presented. The treatment allows us to introduce the ideas of beam focusing and phase compensation by LHM slab. Because of the negative refractive index of LHM slab, the inverse Gouy phase shift and the negative Rayleigh length of paraxial Gaussian beam are proposed. It is shown that the phase difference caused by the Gouy phase shift in right-handed material (RHM) can be compensated by that caused by the inverse Gouy phase shift in LHM. If certain matching conditions are satisfied, the intensity and phase distributions at object plane can be completely reconstructed at the image plane.     

The Gouy phase shift of the highly focused radially polarized beam

The Gouy phase shift in the focal field of high-NA focused radially polarized beam has been investigated in detail. Analytical expression for the Gouy phase shift can be derived using tilted wave interpretation, which provides a reasonable prediction compared to vectorial diffraction numerical simulation. Using this method, irregular wave spacing in the vicinity of the focus can be revealed.     

In situ spatial mapping of Gouy phase slip for high-detail attosecond pump-probe measurements

Attosecond pump-probe experiments routinely utilize extreme ultraviolet (XUV) and IR fields, with relative phase being the variable parameter. However, the Gouy phase slip between the focused IR and XUV pulses inevitably leads to a certain amount of phase averaging and loss of accuracy. By using ion imaging, we establish a one-to-one mapping between the local phase slip and the spatial coordinates of the focal volume, thus performing in situ characterization of the Gouy phase of a complex beam and its role in ionization of He and Xe. We demonstrate that spatially discriminated ion imaging enhances the contrast of a phase-dependent XUV+IR ionization signal. We utilize our technique to unmask a weak ionization asymmetry, thus opening pathways for further high-precision attosecond studies.     

Manifestation of the Gouy phase in vector-vortex beams

Experimental measurements of the twirl and changes in the anisotropy of the constant intensity ellipse, and the rotation of the polarization singular lemon pattern a generalized vector-vortex beam experiences around the two foci due to the converging and diverging conical waves and in between, are presented and interpreted as being due to the universal form of the Gouy phase, φ(G)=mπ/2.     

Interferometry of a single nanoparticle using the Gouy phase of a focused laser beam

We provide a quantitative explanation of the mechanism of the far-field intensity modulation induced by a nanoparticle in a focused Gaussian laser beam, as was demonstrated in several recent direct detection studies. Most approaches take advantage of interference between the incident light and the scattered light from a nanoparticle to facilitate a linear dependence of the signal on the nanoparticle volume. The phase relation between the incoming field and the scattered field by the nanoparticle is elucidated by the concept of Gouy phase. This phase relation is used to analyze the far-field signal-to-noise ratio as a function of exact nanoparticle position with respect to the beam focus. The calculation suggests that a purely dispersive nanoparticle should be displaced from the Gaussian beam focus to generate a far-field intensity change.     

Generalized eikonal treatment of the Gouy phase shift

We use a generalized refractive index that includes diffraction effects to show that the Gouy phase shift can be seen as an intensity averaged optical path difference between the generalized eikonal and the geometrical eikonal. This approach generalizes previous treatments to include the effects of phase distortion and confirms the role of transverse spatial confinement in the Gouy shift.     

The Gouy phase of Airy beams

The phase behavior of Airy beams is studied, and their Gouy phase is defined. Analytic expressions for the idealized, infinite-energy type beam are derived. They are shown to be excellent approximations for finite-energy beams generated under typical experimental conditions.     

The origin of the Gouy phase anomaly and its generalization to astigmatic wavefields

One of the most poorly understood subjects in physical optics is the origin of the Gouy phase (sometimes called “the phase anomaly near focus”). This is evident from the large number of publications on the subject, many of which attribute it to quite different causes. In this paper we show that the Gouy phase anomaly can be clearly understood from elementary properties of normal congruences of light rays and from the relationship between geometrical optics and physical optics. We also show that the Gouy phase anomaly may be regarded as a degenerate case of a rapid π/2 phase change that is found to occur at each focal line of an astigmatic pencil of rays. The intensity distribution in the region of the phase changes is also presented. Furthermore, symmetry relations for both the phase anomaly and the intensity distribution are derived.     

Fractional Gouy phase

In general, the total Gouy phase shift has the form n pi, where n need not be an integer. As a result of the Fourier transforming property of a lens, the Gouy phase is found to be related to the types of discontinuities at the upper or lower range of the pupil function Q(c) resulting from the asymptotic order of the Fourier transform. The sign of the Gouy phase is also related to the slope of the pupil function. The oscillations of the Gouy phase shift arise from the strength of the nondominant discontinuity.     

Dynamic and geometric phase accumulation by Gaussian-type modes in first-order optical systems

Based on the ray transformation matrix formalism, we propose a simple method for the identification of the dynamic and geometric parts of the Gouy phase, acquired by an appropriate Gaussian-type beam while propagating through a first-order optical system.     

Observing quantum correlation of photons in Laguerre-Gauss modes using the Gouy phase

The effect of the Gouy phase, which is one of the geometrical phases of photons, is observed through quantum correlation in Laguerre-Gaussian (LG) modes. In an experiment, the relative phase of two different LG modes of measurement basis states is manipulated via the Gouy phase, and the observed coincidence count rates agree well with theoretical predictions. This result suggests that the Gouy phase can be used as a new tool to manipulate multidimensional photonic quantum states.     

Revisiting Gouy phase

The aim of this paper is to highlight the added value of the generalized Gouy phase shift introduced by Siegman. Although suited for optical systems study, including those more complex than free space, we note that it did not meet the use that it deserves so far. The analysis of the whole of the ideas and analytical approaches associated to the important concept of the Gouy phase proves its effectiveness.

Usually, the resonance condition is systematically built on the basis of the equivalent empty cavity. Unfortunately, this approach does not cover some of the useful parameters of the real resonator. By means of the generalized Gouy phase and the self-consistent complex parameter q, we derive here a new approach for the calculation of the resonance condition for the real cavity. Moreover, the use of the generalized Gouy phase clearly simplifies the study of resonators, while making it possible to avoid the use of the Huygens’ Fresnel integral.     

Gouy phase and visibility in the double-slit experiment

We study a matter wave double-slit experiment with unequal aperture widths in order to evaluate the effect of the Gouy phase on the visibility/predictability. While the predictability changes as one increases the width of one of the slits, the visibility receives a contribution from the Gouy phase at a specific point in the detection screen. Consequently such apparatus constitutes a simple device for measuring the Gouy phase of matter waves. We illustrate it numerically for neutrons.     

Creation of a three-dimensional optical chain for controllable particle delivery

We propose a design for producing a conveyable quasi-periodic optical chain that can stably trap and deliver multiple individual particles in three dimensions at different planes near the focus. A diffractive optical element (DOE) is designed to spatially modulate the phase of an incoming radially polarized beam. For a tighly focused beam, a three-dimensional (3D) optical chain can be formed because of the difference in the Gouy phase shift from two concentric regions of the DOE. A desired number of particles can be stably tweezed one by one with individual 3D volumes in this trapping structure. By controlling the phase modulation of the incident beam, one can manipulate the interference pattern to accelerate and transport trapped particles along the optical axis in a prescribed way.     

超连续谱干涉方法测量古依相移

利用两束超连续光干涉得到的信号,测量了透镜焦点附近飞秒激光脉冲的古依(Gouy)相移。根据得到的光谱干涉信号,利用傅里叶变换得到相对相位值。激光光束在聚焦透镜后的束腰半径可以由成像方法测得。根据测量得到的激光光束束腰半径,用非线性拟合的方法得到了古依相移曲线,拟合曲线与实验结果符合得非常好。给出了古依相位在焦点前后1 mm区域内的移动量。     

Gouy phase for full-aperture spherical and cylindrical waves

We investigate the Gouy phase shift for full-aperture waves converging to a focal point from all directions in two and three dimensions. We find a simple interpretation for the Gouy phase in this situation and show that it has a dramatic effect on reshaping sharply localized pulses.