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.     

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.     

负折射率平板透镜系统中的光束聚焦及相位补偿特性

利用平面角谱理论,建立厄米-高斯光束和拉盖尔-高斯光束在负折射率平板透镜系统中的傍轴传输模型,研究负折射率平板透镜中厄米-高斯光束和拉盖尔-高斯光束强度聚焦及相位补偿特性。结果发现:负折射率平板透镜产生的反向Gouy相移可以补偿空气中Gouy相移导致的相差,因而在满足相位匹配条件下,光束在物平面和像平面之间的相差为零;当强度聚焦和相位匹配条件同时满足的条件下,光束在物平面强度和相位分布可在像平面实现完全重构。     

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.     

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.     

Imaging the Gouy phase shift in photonic jets with a wavefront sensor

A wavefront sensor is used as a direct observation tool to image the Gouy phase shift in photonic nanojets created by micrometer-sized dielectric spheres. The amplitude and phase distributions of light are found in good agreement with a rigorous electromagnetic computation. Interestingly the observed phase shift when travelling through the photonic jet is a combination of the awaited π Gouy shift and a phase shift induced by the bead refraction. Such direct spatial phase shift observation using wavefront sensors would find applications in microscopy, diffractive optics, optical trapping, and point spread function engineering.     

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.     

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.     

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.     

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.     

光束在左手材料中的强度和相位旋转特性

建立了左手材料中拉盖尔高斯光束传输模型,通过研究拉盖尔高斯光束的相位分布来分析光束的强度旋转特性。具体分析了光束参数模式、Gouy相、频率和色散对光束强度和相位旋转特性的影响。研究发现:方位角相差是导致光束强度旋转的主要原因;负折射率引起的逆的Gouy相移将会导致光束强度反转;在正常色散区左手材料中光束的强度和相位旋转方向是相同的,而在反常色散区,左手材料中光束的强度和相位旋转方向是相反的。     

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。     

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.     

Role of the gouy phase in the coherent phase control of the photoionization and photodissociation of vinyl chloride

We demonstrate theoretically and experimentally that the Gouy phase of a focused laser beam may be used to control the photoinduced reactions of a polyatomic molecule. Quantum mechanical interference between one- and three-photon excitation of vinyl chloride produces a small phase lag between the dissociation and ionization channels on the axis of the molecular beam. Away from the axis, the Gouy phase introduces a much larger phase lag that agrees quantitatively with theory without any adjustable parameters.     

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.     

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.     

Gouy wave modes: undistorted pulse focalization in a dispersive medium

Gouy wave modes are linear waves with finite energy that propagate without distortion at any phase and group velocity through a focal region in a dispersive medium. These features make them potentially useful for the onset and control of nonlinear interactions.     

Using Polarization to Control the Phase of Spatial Modes for Application in Quantum Information

Exploiting spatial modes and polarization, we experimentally demonstrate the realization of a conditional π-phase shift. Our approach is based on an optical circuit where the polarization and transverse degrees of freedom control the Gouy phase which is applied on Hermite-Gaussian beams. Our results show good wo-qubit states has beengood agreement with the simulation of the optical circuit. As an application, we propose the implementation of a two-qubit quantum phase gate, where the qubits are encoded on the Hermite-Gaussian modes and linear polarization states.     

Focal transformation and the Gouy phase shift of converging one-cycle surface acoustic waves excited by femtosecond laser pulses

We studied the changes of the pulse shape and the phase of the spectral components in converging-surface acoustic wave pulses. These pulses were excited with a femtosecond laser by a thermoelastic mechanism. To produce converging acoustic pulses, the laser beam was focused with an axicon in a circle on the surface of an aluminum sample. During propagation through the focus, the shape of the pulses of the normal surface velocity changed from two to three polar. The absolute value of the phase of the spectral components experienced a change close to pi/2 rad (Gouy phase shift) after passage of the focal region. These observations were confirmed by analytical and numerical calculations based on the two-dimensional wave equation for surface acoustic waves.     

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.