Gouy Phase Research Articles

Detection of magnetic impurities using electron vortex beams

Electron vortex beams generated by a transmission electron microscope (TEM) are employed to study magnetic properties of an impurity often embedded in materials. Compared to the optical wave, a higher spatial resolving power of electron waves enables the detection of impurities on the nanoscale. Here, we investigate theoretically the interaction of the twisted electrons and the magnetic impurity in which the magnetic dipole moment is taken as a demonstration element. In addition to the usual optical phase, the inhomogeneous vector potential generated by the magnetic dipole moment makes an additional contribution to the intrinsic orbital angular momentum of the twisted electrons, resulting in a dipole-dependent Gouy phase shift. By interfering the outgoing twisted electron beam with a reference cylindrical wave, one can determine the magnitude and orientation of the magnetic dipole directly via the rotational and deformed interference pattern. Furthermore, the pattern is shown to be sensitive to the width of the beam in the focal plane, which provides an effective way to reveal the influence of impurities on the twisted electrons more intuitively and distinctly. The obtained results demonstrate the usefulness of the twisted electron beams for probing the nanoscale magnetism of impurity by TEM, while the proposed model provides the conceptual basis for future developments of the TEM method.

Tuning of the Gouy phase variation for radially polarized laser beams

We discuss a method to produce a radially polarized laser beam with a tunable on-axis Gouy phase total variation different from the standard value of 2π. It is shown that structuring the derivative of the field near the optical axis of a single-ring-shaped illumination focused by an aplanetic lens or a parabolic mirror, one could obtain any value between 3π/2 and large multiples of π. Our results might prove useful for experiments involving the field pattern of a TM01 beam that requires specific phase matching conditions.

Erratum: Emergent and broken symmetries of atomic self-organization arising from Gouy phase shifts in multimode cavity QED [Phys. Rev. A 99 , 053818 (2019)

Received 17 December 2020DOI:https://doi.org/10.1103/PhysRevA.103.019901©2021 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasAtoms, ions, & molecules in cavitiesBose gasesBose-Einstein condensatesCavity quantum electrodynamicsCold gases in optical latticesPolaritonsQuantum description of light-matter interactionQuantum opticsSuperradiance & subradiancePhysical SystemsQuantum cavitiesPropertiesContinuous symmetriesTechniquesDicke modelCondensed Matter, Materials & Applied PhysicsAtomic, Molecular & Optical

Evolution of spin density vectors in a strongly focused composite field

The evolution of the spin density vectors (SDVs) is studied in a strongly focused composite field. It is found that the SDVs can be spiral along the propagation axis, and they are perpendicular to the ys direction on the ys axis. This behavior is governed by the Gouy phase difference between the field polarization components. The 60° rotation of the spatial distribution of the transverse SDVs is also generated, which is found to be controlled by the Gouy phase difference between the field orbital angular momentum modes. Additionally, the spin density singularities are observed in the evolution of the SDVs.

Second-harmonic generation of single-mode Laguerre-Gaussian beams with an improved quasi-phase-matching method.

In the second-harmonic generation processes involving Laguerre-Gaussian (LG) beams, the generated second-harmonic wave is generally composed of multiple modes with different radial quantum numbers. To generate single-mode second-harmonic LG beams, a type of improved quasi-phase-matching method is proposed. The Gouy phase shift has been considered in the optical superlattice designing and an adjustment phase item is introduced. By changing the structure parameters, each target mode can be phase-matched selectively, whose purity can reach up to 95%. The single LG mode generated from the optical superlattice can be modulated separately and used as the input signals in the mode division multiplexing system.

Electric field measurements in plasmas: how focusing strongly distorts the E-FISH signal

Electric field induced second harmonic generation (E-FISH) has recently demonstrated significant potential as a method for making absolute electric field measurements in non-equilibrium plasmas and gas discharges. Previous studies have relied on the plane-wave approximation in quantifying these measurements, while in reality, focused laser beams are almost always used. In this work, we perform a theoretical and experimental study using focused Gaussian beams, and examine the consequent effects on the E-FISH signal. We show that in addition to important parameters such as the external electric field strength, wave vector mismatch and Rayleigh range, the signal is strongly influenced by the full length and shape of this external field profile. We attribute this to the Gouy phase shift associated with focused beams, and note that analogous effects have been previously observed in second and third harmonic generation microscopy. This dependence of the E-FISH signal on the spatial profile of the external field is worth highlighting since it is often not easily determined a priori in a plasma, and neglecting its influence could lead to an incorrect electric field measurement. To minimize any inaccuracies associated with this issue, we propose several recommendations to consider when using the E-FISH diagnostic with focused beams.

Using the Gouy phase anomaly to localize and track bacteria in digital holographic microscopy 4D images.

Described over 100 years ago, the Gouy phase anomaly refers to the additional π phase shift that is accumulated as a wave passes through focus. It is potentially useful in analyzing any type of phase-sensitive imaging; in light microscopy, digital holographic microscopy (DHM) provides phase information in the encoded hologram. One limitation of DHM is the weak contrast generated by many biological cells, especially unpigmented bacteria. We demonstrate here that the Gouy phase anomaly may be detected directly in the phase image using the z-derivative of the phase, allowing for precise localization of unlabeled, micrometer-sized bacteria. The use of dyes that increase phase contrast does not improve detectability. This approach is less computationally intensive than other procedures such as deconvolution and is relatively insensitive to reconstruction parameters. The software is implemented in an open-source FIJI plug-in.

Gouy phase of type-I SPDC-generated biphotons

We consider a double Gaussian approximation to describe the wavefunction of twin photons (also called a biphoton) created in a nonlinear crystal via a type-I spontaneous parametric downconversion (SPDC) process. We find that the wavefunction develops a Gouy phase while it propagates, being dependent of the two-photon correlation through the Rayleigh length. We evaluate the covariance matrix and show that the logarithmic negativity, useful in quantifying entanglement in Gaussian states, although Rayleigh-dependent, does not depend on the propagation distance. In addition, we show that the two-photon entanglement can be connected to the biphoton Gouy phase as these quantities are Rayleigh-length-related. Then, we focus the double Gaussian biphoton wavefunction using a thin lens and calculate a Gouy phase that is in reasonable agreement with the experimental data of D. Kawase et al. published in Ref. [1].

Nonlinear control of polarization rotation of hybrid-order vector vortex beams

Vector vortex beams are widely concerned for their application prospects in various fields. Here we study the polarization rotation of hybrid-order vector vortex beam (HVVB) in hot atomic vapor both theoretically and experimentally. In this work, the HVVB is experimentally generated by combining two q-plates with a polarization beam splitter, and this method is verified by analyzing Jones vector based on the orthogonal circular polarization basis. We have experimentally investigated the polarization rotation of HVVB during the propagation in free space and atomic media. In free space, the polarization rotation characterized by rotation of horizontal components of the beams, has a limitation due to the effect of Gouy phase. In contrast, the polarization rotation angle in atomic media can break through this limitation due to the non-linearity and can be controlled by adjusting the relevant atomic parameters. The experimental results can be simulated by the theoretical model about cross-phase modulation of vector beams in a two-level atomic system.

Origin of Gouy Phase Shift Identified by Laser-Generated Focused Ultrasound

The Gouy phase shift, the phase change of a converging wave passing through a focus, is a general characteristic of waves of any kind, including light and sound. Direct observation of the phase shi...

Reduced volume and reflection for bright optical tweezers with radial Laguerre–Gauss beams

Spatially structured light has opened a wide range of opportunities for enhanced imaging as well as optical manipulation and particle confinement. Here, we show that phase-coherent illumination with superpositions of radial Laguerre-Gauss (LG) beams provides improved localization for bright optical tweezer traps, with narrowed radial and axial intensity distributions. Further, the Gouy phase shifts for sums of tightly focused radial LG fields can be exploited for phase-contrast strategies at the wavelength scale. One example developed here is the suppression of interference fringes from reflection near nanodielectric surfaces, with the promise of improved cold-atom delivery and manipulation.

General laws of the propagation of few-cycle optical pulses with orbital angular momentum in free space

We conduct a theoretical study of the propagation of few-cycle, ultrafast vortices (UFVs) carrying orbital angular momentum (OAM) in free space. Our analysis reveals much more complex temporal dynamics than that of few-cycle fundamental Gaussian-like beams, particularly when approaching the single-cycle regime and the magnitude of the topological charge $l$ is high. The recently described lower bound $\sqrt{|l|}$ to the number of oscillations of UFVs with propagation-invariant temporal shape (isodiffracting UFVs) is found to hold on average also for UFVs of general type, with variations along the propagation direction above and below that bound, even vanishing locally. These variations are determined by the so-called Porras factor or $g_0$-factor characterizing the dependence of the Rayleigh distance of the spectral constituents with frequency. With a given available bandwidth, UFVs must widen temporally with increasing magnitude of the topological charge, and must widen or may shrink temporally during propagation as a result of the axially varying, $g_0$-dependent lower bound. Under very restrictive conditions in their generation, an UFV can be shrunk below the lower bound $\sqrt{|l|}$ at a focus into a kind of locally compressed state of OAM, but it broadens well-above $\sqrt{|l|}$ and distorts in a tiny fraction of the depth of focus because of the dispersions introduced by Gouy's phase and wave front mismatch. These propagation phenomena have implications and should be taken into account in experiments and applications of UFVs, such as the generation of high-harmonics and attosecond pulses with high OAM, or in OAM-based ultrafast communications systems, as well as in other areas of physics such as acoustics or electron waves.

Abrupt λ/2 Demodulation Errors in Spectral Interferometry: Origins and Suppression

An approach for suppressing the abrupt half the wavelength errors in case of spectral interferometric interrogation of optical fiber sensors is proposed. The approach was derived from theoretical analysis of least-squares fitting of interference signal and is based on estimating the additional phase shift in the interference scheme. The approach was experimentally validated through estimation of the Gouy phase in an extrinsic Fabry-Perot interferometer and eliminating abrupt half the wavelength errors which took place when the interferometer was simultaneously heated and bent. The proposed signal processing approach can be applied for increasing sensing resolution and robustness of low-cost fiber-optic interferometric sensors, paving the way for their wide-spread use.

Equivalence of Gouy and Courant-Snyder phase

The generation of electron vortex beams and the conversion of these beams into beams without angular momentum by means of astigmatic optical systems, or vice versa, has been pursued in the optical and the electron microscopy community, but also in the accelerator community in the past decades. Despite different conceptual approaches similar results have been achieved. By adapting the Courant-Snyder theory, which was originally developed for the description of optical properties of accelerators, to the description of laser modes it is shown, that identical mode converters have been developed for charged particle and for light beams, and that the Courant-Snyder phase and the Gouy phase are equivalent.

Clarifications on the Gouy phase of radially polarized laser beams.

The Gouy phase, which describes how a field accumulates an additional phase shift around the focus compared to a spherical wave, is investigated for the strongly focused TM01 beam in free space. It is shown, using both analytical and numerical solutions, that when the effects of the edges of focusing optics are small, the complete variation of the on-axis Gouy phase of the longitudinal electric field is 2π. It is true even in the strongly focused limit. Moreover, we show, using an appropriate effective wavevector, how to retrieve the Gouy phase when the diffraction caused by the edges is dominant at the focus. We further investigate the off-axis Gouy phase of the TM01 beam and show that any slight variation from the optical axis causes a variation of the Gouy phase of the longitudinal electric field to reduce to π.

Propagation effects in multipass high harmonic generation from plasma surfaces

Multipass high harmonic generation from plasma surfaces is a promising technique to enhance the efficiency of the generation process. In this paper it is shown that there is an optimal distance between two targets where the efficiency is maximized, depending on the laser and plasma parameters. This can be explained by the Gouy phase shift, which leads to the relative phase between the colours being changed with propagation in free space. A simple model is used to mimic the propagation of light from one target to another and to observe this effect in 1D particle-in-cell (PIC) simulations. The results are also verified using 2D PIC simulations.

Off-Axis Vortex Beam Propagation through Classical Optical System in Terms of Kummer Confluent Hypergeometric Function

The analytical solution for the propagation of the laser beam with optical vortex through the system of lenses is presented. The optical vortex is introduced into the laser beam (described as Gaussian beam) by spiral phase plate. The solution is general as it holds for the optical vortex of any integer topological charge, the off-axis position of the spiral phase plate and any number of lenses. Some intriguing conclusions are discussed. The higher order vortices are unstable and split under small phase or amplitude disturbance. Nevertheless, we have shown that off-axis higher order vortices are stable during the propagation through the set of lenses described in paraxial approximation, which is untypical behavior. The vortex trajectory registered at image plane due to spiral phase plate shift behaves like a rigid body. We have introduced a new factor which in our beam plays the same role as Gouy phase in pure Gaussian beam.

Atoms in axially shifted tightly focused counter-propagating beams: the role of the Gouy and curvature phases

We consider the interaction of atoms with two tightly focused and axially shifted counter-propagating optical beams. At sub-wavelength focusing, we find that the scattering force potential in the three-dimensional space between the shifted focal planes changes from a feature with a saddle-point to a three-dimensional trapping potential. Further analysis shows that due to the tight focusing, the trapping depends on significant contributions arising from the Gouy and curvature phase gradients of the interfering beams. The physics and its effects are illustrated with reference to the sub-wavelength trapping of sodium atoms.

Complex shaping of the depth of focus

In this manuscript an exact solution to the inverse problem of axial beam shaping along the focus of a convergent lens is found. This allows to extend, within the framework of the scalar theory of diffraction, the mathematical formalism of complex pupils to include axial phase modulation. Numerical simulations based on Fourier transform as well as convolution operations indicate that amplitude and phase modulation can be performed simultaneously. It is also shown that include or not phase modulation in the beam shaping process can increase its efficiency more than three times. In addition, an analytical expression for the Gouy phase that depends on the introduced phase modulation was also derived. It is expected that obtained results benefit many photonic applications involving the control and manipulation of light along the focal region.

The photon vortex beam in rotating medium

In this paper, we demonstrate that there are vortex beam solutions for the photon in the rotating medium. By constructing the photon wave function with Riemann–Silberstein vector, we derive the dynamic equation of the photon in moving medium from the Maxwell equations and the non-relativistic Minkowski relations. In case of the stationary state, the dynamic equation of the photon can be written as a Dirac-like equation, where the velocity of the medium plays the role of a vector potential. By giving the medium different forms of rotating velocity fields, we obtain different vortex beam solutions of the photon, such as the diffracting and non-diffracting Laguerre–Gaussian (LG) beam solutions via proper approximations. For the diffracting LG beam solution, we acquire a new term arising from the medium rotation that can change the Gouy phase, and then accordingly predict the rotation behavior of the photon interference pattern. In addition, the rotation of the medium can lead to the change of the relative intensity distribution of the interference pattern. Furthermore, our theory predicts the existence of the Landau levels of transverse photon energy in the nondiffracting LG beam solution.