On anomalous optical beam shifts at near-normal incidence

The reflection and refraction of a plane electromagnetic wave at a dielectric interface are the basic physical processes inherent to all-optical systems and devices. They are rigorously described by the Snell’s law and Fresnel equations. Nevertheless, in practice, we usually deal with optical beams for which the plane wave approximation is oversimplified. This results in the deviation from geometrical optics that manifests itself as spatial shifts of the beam known as the Goos–Hänchen (GH) and Imbert–Fedorov (IF) shifts. These shifts can be explained in terms of weak material-mediated interaction of the beam spectrum (orbit state) and its polarization (spin state).^1–4 

Today, the GH and IF shifts, including the photonic spin Hall effect (PSHE) as a particular example of the IF shift, are well-studied in different systems including atomic optics, optical sensors, graphene, metasurfaces, polarizers, uniaxial crystals, etc.^5–11 In general, the spatial (angular) beam shifts are very small—typically of the order of the light wavelength (the beam angular spectrum variance)—that limits their application.^1–4 The shifts can be enhanced under several specific conditions including the near-Brewster incidence,^12–15 material resonances,^16,17 exceptional points,¹⁸ and output beam polarization post-selection.¹⁹ In all these cases, the enhancement occurs at large angles of incidence (typically, tens of degrees). The standard theory of optical beam shifts⁴ fails to give an accurate explanation of shifts at near-normal incidence, while at the normal incidence they should completely disappear.

In this Letter, we develop a generalized theory of beam shifts covering the case of small incident angles comparable with the angular beam divergence. Usually, the longitudinal (GH) and transverse (IF) shifts are caused by different origins and considered independently.⁴ Here, we show that anomalously large GH and IF shifts have the same nature caused by the geometric Berry phase singularity and appeared simultaneously, for example, in phase plates and uniaxial slabs. While the partial cases of anomalous PSHE enhancement at near-normal incidence have been recently observed due to the spin-to-orbit angular momentum conversion,^20–23 the GH shift is commonly known to exist only under total internal reflection conditions.⁴ Our theory predicts the giant GH shift at near-normal incidence and gives a simple practical guideline for the simultaneous experimental observation of the giant GH and IF shifts in a quarter-wave q-plate (QWQP).^24–26 

We consider first the oblique incidence of a monochromatic optical beam (under angle ϑ to the central plane wave of the beam) on a flat vacuum–medium interface; see Fig. 1.

The values of the shifts could be calculated within the Jones matrix formalism in the paraxial approximation. The Jones matrix $T̂a$ relates the incident and reflected/transmitted plane wave amplitudes in the beam coordinate frame $|Ea〉=T̂a(ϑ,μ,ν)|E〉$⁠, where |E⟩ ∝ |e⟩ · f(μ, ν) is the incident constituent plane-wave Jones vector, index a = r, t denotes reflected and transmitted waves, respectively, $|e〉=(eX,eY)T$ is the Jones vector of the central plane wave in the incident beam, and f(μ, ν) is the incident beam Fourier spectrum expressed in terms of the in-plane (μ) and out-of-plane (ν) deflection angles of the non-central wave vectors.⁴ 

In fact, the polar deflection angle at near-normal incidence (ϑ → 0°) is equivalent to the geometric Berry phase, Φ_B = −ϕ cos ϑ ≃ −ϕ.⁴ Thus, the considered geometric singularity of the polar deflection angle simultaneously means the Berry phase singularity. The large IF shift arises from large transverse momentum derivatives of the Berry phase across the beam spectrum according to its definition.

Another surprising consequence of Eq. (3) is that it leads to the anomalous GH shift. In conventional situation (large angles of incidence), the GH shifts are caused by the spatial dispersion of the scattering coefficients.⁴ In contrast, at near-normal incidence, GH shift arises from the large longitudinal momentum gradient of the Berry phase across the lateral parts of the beam spectrum. Note that such longitudinal gradients caused by strong spin–orbit coupling are negligible at large incident angles. Thus, both GH and IF shifts under near-normal incidence are caused by the spatial gradient of the Berry phase and their giant enhancement at the specified angle of incidence is defined by the Berry phase singularity.

To exhibit the near-normal Berry phase singularity as a particular example, we consider transmission through a q-plate with topological charge q = 2. Following the Jones matrix formalism in the k-space (see the supplementary material for the detailed derivation), we calculate analytically the GH and IF shifts of the Gaussian beam transmitted through the q-plate when the beam incidence plane is parallel to the waveplate main optic axis, which lies at the interface (Fig. 1). The Gaussian beam angular spectrum is defined by the Fourier spectrum f(μ, ν) = exp[−κ²(μ² + ν²)/2], where $κ=k0w0/2$ is the inverse beam divergence. The case of q-plate in a real space is considered in the supplementary material by drawing an analogy between the near-normal and off-center beam incidence.

Equations (4)–(8) are the main results of this work applicable for any cases except κϑ ≫ 1. They describe both GH and IF shifts at near-normal incidence for the arbitrary anisotropic slab, q-plate or metasurface. One can see that each shift in Eqs. (5)–(8) is the product of three factors in a good agreement with Eq. (4). The second term in expression (8) for the spatial IF shift is a counterpart of the anomalous PSHE connected with the circular polarization of the incident beam. Moreover, due to the presence of the first term in Eq. (8), the IF shift can be obtained with the arbitrary polarization state except pure TM- and TE-polarization, i.e., along (S₁ = 1) and across (S₁ = −1) the main optic axis.

We also note the remarkable connection of the four shifts to the geometric resonant terms Λ_X,Y. First, there is a proportionality between the spatial and angular shifts, which is only broken by the spin-Hall term in Eq. (8). The simultaneous appearance of the spatial and angular GH and IF shifts at the same angle of incidence is unique. At large incident angles (κϑ ≫ 1), the GH shifts can be simultaneously observed only in special cases such as lossy media²⁸ or vortex beams.²⁹ Second, there is a further similarity of geometric resonant terms along Y and X directions, whereas $ΛX(κϑ)=e−κ2ϑ2ΛY(iκϑ)$⁠. This reflects the fact that all anomalous near-normal incident shifts have the same origin, namely, the geometric Berry phase singularity. As a consequence, all four types of anomalously large optical beam shifts (GH and IF, linear and angular) could be detected simultaneously in the near-normal incidence regime (κϑ ≲ 1), particularly in the vicinity of ϑ = 0.2° using the proposed material platform, beam parameters, and elliptical polarization of incident light. The detection of shifts could be further enhanced with even higher precision by using the weak measurements technique in the darkfield region.³⁰ 

Figure 2 shows the optical beam shift dependencies on the incidence angle for the beam transmitted through the QWQP with q = 2 and the main optic axis lying in the beam plane of incidence. It is important to note that a linear IF shift can change the shift direction to the opposite one at some angle under elliptically-polarized beam illumination [Fig. 2(c)]. This is caused by the interplay between differently polarized terms of the linear IF shift [Eq. (8)]. The developed theory [Eq. (8)] also describes the switching of the IF direction for strongly anisotropic systems due to the polarization mixing, especially when the optical axis is arbitrarily oriented.³¹ 

The optical beam shifts under near-normal incidence depend substantially on the beam waist w₀ via the inverse beam divergence κ, as it is expressed by nonlinear functions Λ_X,Y(κϑ) [Eq. (9)]. The IF shifts are anomalous and have asymptotes ∝1/ϑ at large incidence angles (κϑ ≫ 1), which is hinted by the standard theory.⁴ The GH shifts have higher-order asymptotes at large incident angles ∝1/(ϑ³κ²) (Fig. 3). This explains why the anomaly in GH shifts was not anticipated by the standard theory, in which the spatial GH shift is absent except the case of total internal reflection. Moreover, one can notice that the amplitude of the optical beam shift peak is proportional to the beam waist w₀, while the corresponding critical angle is inversely proportional to it (see the inset in Fig. 3). In general, the critical angle corresponding to the maxima of GH and IF shifts is about a few tenths of a degree for a beam width of several dozen microns at λ = 630 nm. This result completely agrees with the recent experimental observations of the anomalous PSHE under near-normal incidence at hyperbolic metamaterials and anisotropic slabs.^20–23 Finally, we highlight the connection between the Berry phase singularity and optical beam shifts. The resonant angles of G_X(κϑ) and G_Y(κϑ) in Fig. 3 (about 0.15° and 0.25° for κ = 700, respectively) fit well with the giant peaks in angular dependences of all four types of beam shifts shown in Fig. 2. However, for linear IF shift, the peaks may be slightly shifted within the range 0.1° < ϑ < 0.3° due to the non-trivial polarization dependence [Eq. (8)].

All material dependencies in Eqs. (5)–(8) are factored out in four dimensionless coefficients, τ₊, τ₋, τ_×, and t₋. We analyzed the maximum values of these parameters as functions of the retardation and average refractive index of the waveplate (see the supplementary material), and concluded that the low-birefringent QWQP is a perfect candidate for observing all four types of anomalous shifts simultaneously (although PSHE has a maximum for a half-wave q-plate).

To gain deeper insight into the physics of the anomalous beam shifts, we analyze the intensity and helicity (S₃) patterns for the transmitted TM-polarized beam (S₁ = 1) with Jones matrix $T̂t$ (Fig. 4). When the incidence angle ϑ exceeds the beam divergence 1/κ, PSHE naturally occurs owing to the so-called circular birefringence [Fig. 4(a)].⁷ At near-normal incidence ϑ ≲ 1/κ, the spin-Hall doublets from lateral and opposite parts of the beam Fourier spectrum become prominent, and an octupole helicity profile is built up in k-space with a phase singularity at a distance ∼k₀ϑ from the geometric-optics beam axis [Figs. 4(b) and 4(c)]. In real space, at ϑ = 0°, the helicity is carried by a mode resembling a Bessel–Gaussian mode with m = 4³² and null intensity on the beam axis [Fig. 4(d)]. The anomalously large shift and the presence of polarization singularity provide a close connection to the superoscillations.³³ Finally, we demonstrate explicitly the beam shifts of all four kinds in Figs. 4(e)–4(h). The anomalous GH angular shifts are accompanied by noticeable transverse beam spectrum deformations around the polarization singularity [Fig. 4(e)].

To conclude, we have developed a generalized theory of spatial and angular beam shifts applicable to the case of the near-normal incidence, when the angle of incidence becomes comparable with the beam divergence. The principal point of the theory is that anomalously large shifts of all types are shown to have the same fundamental origin—the singularity in the Berry phase appearing in the beam Fourier spectrum. The theory also predicts the anomalous GH shifts at the near-normal incidence. The results have been illustrated for an analytically treated example of a quarter-wave q-plate with q = 2 supporting anomalously large simultaneous near-normal incidence shifts of all kinds in manifold times exceeding shifts observed separately in conventional cases. Thus, the anomalous near-normal shifts are completely feasible for direct experimental verification.

The predicted effects could arise for other types of waves (acoustic, electron beams, etc.), physical platforms (bound states in the continuum,³⁴ q-^24,25 and j-plates^35,36) and material-anisotropy types (biaxial and bianisotropic media). Finally, we conduct the analogy between the beam shifts under near-normal incidence and the near-field spin Hall effect^37–39 manifested as the high-directional polarization-dependent surface wave excitation by means of a dielectric antenna.^40,41 In this case, the role of inverse beam divergence may be played by the ratio of antenna’s characteristic size to the incident wavelength.

The supplementary material contains five sections. Section I provides the detailed discussion on the polar deflection angle and the geometric Berry phase singularity at near-normal incidence. Section II goes through the derivation of the Jones matrix under near-normal incidence for the q-plate. Section III gives the analysis of the impact of the material platform on the optical beam shifts. Section IV covers the case of the isotropic slab and gives the detailed calculations of near-normal optical beam shifts. Sections V and VI go through the detailed derivation of near-normal-incidence shifts for a q-plate in Fourier and real spaces, respectively.

A.B. acknowledges the BASIS foundation, the Ministry of Science and Higher Education of the Russian Federation (No. 075-15-2022-1120), and the academic leadership program Priority 2030.

The authors have no conflicts to disclose.

M. Mazanov: Formal analysis (equal); Investigation (equal); Methodology (equal); Software (lead); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). O. Yermakov: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (lead); Writing – review & editing (equal). A. Bogdanov: Conceptualization (equal); Investigation (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). A. Lavrinenko: Conceptualization (equal); Investigation (equal); Methodology (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal).

The data that support the findings of this study are available within the article and its supplementary material.