Imbert Fedorov Research Articles

Beam shifts for pairs of plane waves

Following Hans Wolter’s treatment of the spatial Goos–Hänchen shift of a totally internally reflected light beam by the superposition of two plane waves, polarized perpendicular to the plane of incidence, we consider the reflection and refraction of several similar pairs of plane waves, with varying geometry and incident polarization. We consider explicitly the partial reflection analogue and the in-plane polarized analogue to Wolter’s example, as well as a pair of plane waves propagating slightly out of their mutual plane of incidence, revealing the transverse, Imbert–Fedorov shift. We find these simple cases have a complicated polarization structure, with a range of polarization singularities and complex orbital and spin current flows, generalizing Wolter’s discovery of an optical vortex and circulating energy flow at the heart of the net scalar interference pattern.

Total internal reflection of orbital angular momentum beams

We investigate how beams with orbital angular momentum (OAM) behave under total internal reflection. This is studied in two complementary experiments: in the first experiment, we study geometric shifts of OAM beams upon total internal reflection (Goos–Hänchen and Imbert–Fedorov shifts, for each the spatial and angular variant), and in the second experiment we determine changes in the OAM mode spectrum of a beam, again upon total internal reflection. As a result we find that, in the first case, the shifts are independent of OAM and beam focusing, while in the second case, modifications in the OAM spectrum occur which depend on the input OAM mode as well as on the beam focusing. This is investigated by experiment and theory. We also show how the two methods, beam shifts on the one hand, and OAM spectrum changes on the other, are related theoretically.

The effect of spectral width on Goos–Hanchen and Imbert–Fedorov shifts

We study the Goos–Hanchen and Imbert–Fedorov shifts for quasi-monochromatic Gaussian beams near the Brewster angle, when the reflection is from a denser to a rarer medium. This is the case of interest in the usual experiments on reflectometry, etc. We have incorporated the effects of the finite linewidth of the quasi-monochromatic light and treated the cases of a Lorentzian and a Gaussian lineshape of the input light spectrum. This study of light with a finite spectral width was carried out for the more frequently studied case, namely reflection from a dense to a rare medium. We found that the shift is increased as compared to the monochromatic Gaussian beam, and is zero at the Brewster angle for a p polarized beam. The shift variation with angle of incidence near the Brewster and critical angles at different values of refractive index ratios is found. We also studied the shift variation for Hermite–Gauss beams around the Brewster angle when the reflection is from a rarer to a denser medium and compare this with our earlier results for the case when the reflection was from a denser to a rarer medium.

Imbert–Fedorov shifts of a Gaussian beam reflected from anisotropic topological insulators

We study the Imbert–Fedorov (IF) shifts of a reflected Gaussian beam from anisotropic topological insulators (TIs). Aiming, respectively, at the two types of TI surface perturbations that can alter the surface optical properties, i.e., a magnetic coating and an applied magnetic field, we discuss the dependences of the IF shifts on the axion field, the TI's extent of anisotropy, the angle of incidence and the applied magnetic field. For a left-elliptically polarized incident beam, a positive (negative) axion field can lead to the decrease (increase) of the IF shifts relative to a conventional insulator in the magnetic coating case, and in the applied magnetic field case, the crossover of the IF shifts between the positive and the negative values can be realized by adjusting some external parameters. Also, the properties of the IF shifts are discussed for linearly and circularly-polarized incident beams. The results show that TI might be a desirable candidate for controlling flexibly the IF shifts.

Generalized shifts and weak values for polarization components of reflected light beams

The simple reflection of a light beam of finite transverse extent from a homogeneous interface gives rise to a surprisingly large number of subtle shifts and deflections which can be seen as diffractive corrections to the laws of geometrical optics (Goos–Hänchen shifts) and manifestations of optical spin–orbit coupling (Imbert–Fedorov shifts), related to the spin Hall effect of light. We develop a unified linear algebra approach to dielectric reflection which allows for a simple calculation of all these effects and lends itself to an interpretation of beam shifts as weak values in a classical analogue to a quantum weak measurement. We present a systematic study of the shifts for the whole beam and its polarization components, finding symmetries between input and output polarizations and predicting the existence of material independent shifts.

Goos–Hanchen and Imbert–Fedorov shifts for Hermite–Gauss beams

We study the lateral Goos-Hanchen and the transverse Imbert-Fedorov shift produced during the reflection of Hermite-Gauss beams H(m0) or H(0m) at a plane interface. The vector angular spectrum method for a light beam in terms of a two-form angular spectrum consisting of the two orthogonal polarized components was used. We have carried out a detailed numerical calculation of these shifts at different angles of incidence, over the whole range of incidence without making the usual approximations. The shift variation as a function of refractive index and order of the Hermite-Gauss beam is studied. We also compare the shift variations with the orientation of the lobes of the Hermite-Gauss beam. We observed that the shifts are nearly equal for the two cases H(m0) (lobe oriented in the plane of incidence) and H(0m) (lobe oriented perpendicular to plane of incidence). These are the first quantitative estimates of the shifts for Hermite-Gauss beams as per our knowledge and are relevant for all cases of slab geometry.

Goos-Hänchen and Imbert-Fedorov shifts of vortex beams at air–left-handed-material interfaces

In this paper, we present a systematic study of beam shifts and angular momenta of paraxial vortex beams at air--left-handed-material (LHM) interfaces. It is shown that, compared to their counterparts at air--right-handed-material (RHM) interfaces, the spatial Goos-H\"anchen (GH) and Imbert-Fedorov (IF) shifts remain the same, while the angular GH and IF shifts are reversed at air-LHM interfaces. The spatial and angular shifts of paraxial vortex beams have their respective origins in transverse angular momenta and transverse linear momenta. The spatial GH and IF shifts remain unreversed as a result of both reversions of transverse angular momenta and $z$-component linear momentum, while the angular GH and IF shifts are reversed because the $z$-component linear momentum is reversed and the transverse linear momenta are unreversed at air-LHM interfaces. In addition, we perform a quantitative analysis on spin-orbit angular momentum conversion and orbit-orbit angular momentum conversion, which further helps us understand the essence of vortex beam shifts at air-LHM interfaces and their fundamental distinctions from those at air-RHM interfaces.

Goos–Hänchen and Imbert–Fedorov shifts of an electromagnetic wave packet by a moving object

We present a solution to the problem of reflection and refraction of a polarized Gaussian beam at the interface between the transparent medium and the moving object based on Minkowski’s constitutive relations. The Goos–Hanchen (GH) and Imbert–Fedorov (IF) effects are discussed in detail when the object moves parallel to the interface. It is shown that the IF shifts of reflection and transmission beams can reach several wavelengths of the incident beam at some certain incident angles and moving velocities of the object. The transitions from the positive IF shift to the negative IF shift have been observed with the increase of the velocity of the moving object. Furthermore, our calculations show that some unusual GH effects can also be caused by the movement of the medium. The physical origins for these phenomena have been analyzed. It is well known that the IF shift is directly related to the spin-Hall effects of light. Thus, these findings can also provide an alternative pathway for controlling the spin-Hall effects of light.

Role of spatial coherence in Goos–Hänchen and Imbert–Fedorov shifts: reply to comment

Wang and Liu [Opt. Lett. 37, 1056 (2012)] comment on our previous Letter [Opt. Lett. 36, 3151 (2011)] regarding the validity of the theory we presented. We reply to their comment here.

Goos–Hänchen and Imbert–Fedorov shifts: a novel perspective

When a beam of light is reflected by a smooth surface its behavior deviates from geometrical optics predictions. Such deviations are quantified by the so-called spatial and angular Goos–Hänchen (GH) and Imbert–Fedorov (IF) shifts of the reflected beam. These shifts depend upon the shape of the incident beam, its polarization and on the material composition of the reflecting surface. In this paper we suggest a novel approach that allows one to unambiguously isolate the beam-shape dependent aspects of GH and IF shifts. We show that this separation is possible as a result of some universal features of shifted distribution functions which are presented and discussed.

Role of spatial coherence in Goos-Hänchen and Imbert–Fedorov shifts

We present a theory for Goos-Hänchen (GH) and Imbert-Fedorov (IF) shifts for beams of light with arbitrary spatial coherence. By applying the well-known theory of partial spatial coherence, we can calculate explicitly spatial and angular GH and IF shifts for completely polarized beams of any shape and spatial coherence. For the specific case of a Gauss-Schell source, we find that only the angular part of GH and IF shifts is affected by the spatial coherence of the beam. A physical explanation of our results is given.

Geometric Spin Hall Effect of Light at polarizing interfaces

The geometric Spin Hall Effect of Light (geometric SHEL) amounts to a polarization-dependent positional shift when a light beam is observed from a reference frame tilted with respect to its direction of propagation. Motivated by this intriguing phenomenon, the energy density of the light beam is decomposed into its Cartesian components in the tilted reference frame. This illustrates the occurrence of the characteristic shift and the significance of the effective response function of the detector. We introduce the concept of a tilted polarizing interface and provide a scheme for its experimental implementation. A light beam passing through such an interface undergoes a shift resembling the original geometric SHEL in a tilted reference frame. This displacement is generated at the polarizer and its occurrence does not depend on the properties of the detection system. We give explicit results for this novel type of geometric SHEL and show that at grazing incidence this effect amounts to a displacement of multiple wavelengths, a shift larger than the one introduced by Goos–Hänchen and Imbert–Fedorov effects.

Goos–Hänchen and Imbert–Fedorov shifts of a nondiffracting Bessel beam

Goos-Hänchen (GH) and Imbert-Fedorov (IF) shifts are diffractive corrections to geometric optics that have been extensively studied for a Gaussian beam that is reflected or transmitted by a dielectric interface. Propagating in free space before and after reflection or transmission, such a Gaussian beam spreads due to diffraction. We address here the question of how the GH and IF shifts behave for a "nondiffracting" Bessel beam.

How orbital angular momentum affects beam shifts in optical reflection

It is well known that reflection of a Gaussian light beam ($\text{TEM}_{00}$) by a planar dielectric interface leads to four beam shifts when compared to the geometrical-optics prediction. These are the spatial Goos-H\"{a}nchen (GH) shift, the angular GH shift, the spatial Imbert-Fedorov (IF) shift and the angular IF shift. We report here, theoretically and experimentally, that endowing the beam with Orbital Angular Momentum (OAM) leads to coupling of these four shifts; this is described by a $4 \times 4$ mixing matrix.

Reply to “Comment on ‘The Imbert–Fedorov shift of paraxial light beams’”

This is a reply to the Comment by Bliokh on our paper that appeared in Opt. Commun. 281 (2008) 3427. After a brief introduction of the representation theory of vector electromagnetic beams advanced in a recent paper, I point out that the Imbert–Fedorov effect is the evidence of the change of the beam parameter Θ and the polarization ellipticity σ caused by the reflection or transmission process in the linear approximation. Then I explain that it is because the linear approximation of the incident beam that we used in our paper had been assumed in previous works that we reproduced their results.

Comment on “The Imbert–Fedorov shift of paraxial light beams”

Here I argue that Liu and Li [B.-Y. Liu, C.-F. Li, Opt. Commun. 281 (2008) 3427] reproduce calculations of the Imbert–Fedorov transverse shift previously made in a number of other works. However, it has recently been shown that these results are not valid for standard uniformly polarized beams. The corrected values of the Imbert–Fedorov shift were derived in papers [K.Y. Bliokh, Y.P. Bliokh, Phys. Rev. Lett. 96 (2006) 073903; Phys. Rev. E 75 (2007) 066609] and confirmed by recent measurements [O. Hosten, P. Kwiat, Science 319 (2008) 787] with a great accuracy.

Role of beam propagation in Goos–Hänchen and Imbert–Fedorov shifts

We derive the polarization-dependent displacements parallel and perpendicular to the plane of incidence for a Gaussian light beam reflected from a planar interface, taking into account the propagation of the beam. Using a classical-optics formalism we show that beam propagation may greatly affect both Goos-Hänchen and Imbert-Fedorov shifts when the incident beam is focused.

Total internal reflection of vector Bessel beams: Imbert–Fedorov shift and intensity transformation

Total internal reflection of vector Bessel beams is studied. Transmitted beams are described by the evanescent fields, the energy fluxes of which show the shift of the reflected beam. As is well known, the Imbert–Fedorov shift is the lateral displacement of the plane wave leading the reflected beam out from the plane of incidence. Using an analogy with plane waves, the Imbert–Fedorov shift is introduced for the Bessel beams. This shift results in the intensity redistribution of the reflected beam compared with the incident one. The conditions of the Bessel beam intensity transformation from a squared Bessel beam function of the order m−1 (Jm−12-profile) to a squared Bessel beam function of the order m+1 (Jm+12-profile) are derived.

The Imbert–Fedorov shift of paraxial light beams

We present a solution to the problem of reflection and transmission of a polarized paraxial light beam at an interface between two homogeneous media by using a two-form amplitude and an extension matrix to represent the vectorial angular spectrum of a three-dimensional (3D) light beam. We derive general formulas for the Imbert–Fedorov (IF) shift of the reflected and transmitted beams of a polarized paraxial light beam. The IF shift of two different types of polarized beams is calculated, and the influence of the polarization state and the polarization feature of the vectorial angular spectrum on the IF shift is discussed.

Unified theory for Goos-Hänchen and Imbert-Fedorov effects

A unified theory is advanced to describe both the lateral Goos-H\"anchen (GH) effect and the transverse Imbert-Fedorov (IF) effect, through representing the vector angular spectrum of a three-dimensional light beam in terms of a two-form angular spectrum consisting of its two orthogonal polarized components. From this theory, the quantization characteristics of the GH and IF displacements are obtained, and the Artmann formula for the GH displacement is derived. It is found that the eigenstates of the GH displacement are the two orthogonal linear polarizations in this two-form representation, and the eigenstates of the IF displacement are the two orthogonal circular polarizations. The theoretical predictions are found to be in agreement with recent experimental results.