Motivated by recent experimental observation of photonic spin Hall effect at metasurfaces, we study lateral Goos-Hänchen (GH) and transverse Imbert-Fedorov (IF) shifts of an arbitrarily polarized light beam totally reflected from metasurfaces, in terms of stationary phase method and energy flux method. The intriguing phenomenon is that the gradient in phase discontinuity results in anomalous reflection and refraction, and the GH and IF shifts can be thus controlled from negative to positive values by changing the sign of phase discontinuity. The tunable GH and IF shifts have potential applications in nano-optics, with the development of novel functionalities and performances of metasurfaces.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
In classical optics, it is well known that the light beam totally reflected from a single interface between two different dielectric media experiences the lateral Goos-Hänchen (GH) [1, 2] and transverse Imbert-Fedorov (IF) [3,4] shifts from the position predicted by the geometrical optics, see recent review . Up to now, negative and positive GH shifts have been extensively investigated in different media, for example, “left-handed” metamaterial , metal [7,8], and graphene [9,10], with many applications in integrated optics [11–14], optical waveguide switch , and optical sensors [16, 17]. On the other hand, IF shifts, associated with spin Hall effect of light and angular momentum conservation [18–21], have attracted much attention both theoretically and experimentally [22,23] and has been also extended to various media, such as left-handed  or chiral metamaterials , hyperbolic metamaterials [26–28], metal [29–31] and graphene  as well.
Theoretically, the GH shift can be calculated by Artmann’s stationary phase approach  and Renard’s energy flux method , which also provide the physical explanation on the basis of reshaping effect and energy flux conservation, respectively. As for the IF shift, Imbert  once applied Renard’s energy flux argument to calculate and measurement this displacement. However, Beauregard and Imbert  commented that there was, strictly speaking, no completely rigorous calculations of the GH or IF shifts. For instance, Yasumoto and Õishi  have modified Renard’s energy flux method, taking into account the interference between the incidence and reflected light beams, and the modified energy flux method gives the same results as Artmann’s stationary phase approach [36, 37]. In 2009, Li  proposed a unified theory for GH and IF shifts, to calculate the GH and IF shifts and obtain their quantization characteristics.
A slightly different but relevant topic is special so-called metasurfaces, planar, ultrathin metamaterials, which can be served as another important degree of freedom to control the phase and polarization of light with surface-confined, flat components beyond that offered by conventional interface between two natural materials [39–42]. As a pioneering work, Yu and his collaborators  fabricated one metasurface, constructing of V-shaped optical antennas array arranged periodically on a silicon wafer which introduces phase discontinuities varied linearly as a function of position on the surface, demonstrating anomalous reflection and refraction phenomena. Very recently, a photonic spin Hall effect has been experimentally demonstrated at such metasurface with rapidly varying phase discontinuities . The analysis of energy flux suggests that the photonic spin Hall effect resembles the transverse IF shift, see the appendix in the literature . And the Pacharatnam-Berry phases, through space-variant polarization manipulations with metasurfaces, have been further introduced, which enable new approaches for fabricating the spin-Hall devices and for the applications in spin photonics, see recent review . However, the relevant GH shift at such metasurface has not been explored so much, let alone its experimental demonstrations.
In this paper, we will investigate systemically the lateral GH shift and transverse IF shift at the metasurfaces, see Fig. 1, to identify how the gradient of phase discontinuity modulates the GH and IF shifts with potential applications in nano-optics. We first apply modified energy flux method, proposed by Yasumoto and Õishi , to calculate the GH and IF shifts for arbitrarily polarized light beam, reflected from the metasurface. Next, the results obtained from Artman’s stationary phase method  are further compared to clarify the nature of GH and IF shifts. These results are not only interesting to predict anomalous GH shifts, but also useful to understand the experiments on IF shifts for an arbitrarily polarized light beam, with the potential applications of metasurface in nano-scale integrated optics.
2. Generalized Snell’s law
Consider an arbitrarily polarized light beam comprised by two orthogonally linear polarization (i.e., s polarization and p polarization) incident upon a gradient metasurface, fabricated from silicon substrate , with the angular frequency ω0 and incident angle θi, as shown in Fig. 1. The permittivity, permeability, and refractive index of different regions are denoted by εj, μj, and , respectively, where j = 1, 2 stand for incident (reflected) and transmitted regions. The electric field of incident beam (the time-dependence exp (−iω0t) is implied) is assumed to beEqs. (1)–(3).Eqs. (4) and (5), the generalized Snell’s law results in different angles of reflection and refraction, provided that a suitable constant, dΦ/dx, gradient of phase discontinuity along the interface, is introduced. Moreover, the condition, Eqs. (4) and (5), induces the negative refraction and reflection. Otherwise there exists positive angles of refraction and reflection, based on the usual Snell’s law. For simplicity, the positive and negative refraction mostly depends on the sign of phase discontinuity, when the incidence angle is small. As a consequence, the two possible critical angles for total reflection, when the light wave hits on the optical sparse material from the optical denser material, n1 > n2. (The case, n2 > n1, can be discussed as well in the similar way.) The critical angle for total internal reflection is
According to the boundary conditions, the reflection and transmission coefficients can be calculated as,Figures 2(a) and 2(b) show the dependence of the reflection coefficients of s-polarized and p-polarized light beam on the phase gradient and wavelength, where θi = 2°, n1 = 3.5 and n2 = 1 are the refractive indices of silicon and air . In the case of the total reflection, all the results above are valid when and .
3. GH and IF shifts: modified energy flux and stationary phase methods
Next, we shall first discuss the GH and IF shifts at metasurface based on Yasumoto and Õishi’s energy flux model , in which the GH shift contributes from two parts, the energy flux within the evanescent wave inside the air layer and the other one carried by the interface between incident and reflected beams. This method allows us to calculate the GH shift for an arbitrarily polarized light beam.
With the straightforward calculations, we can obtain the components of time-averaged Poynting vectors, , from36,37]
As a consequence, the GH shift, , can be obtained by the Yasumoto and Õishi’s energy flux method, which isEq. (22). In the same manner, for p-polarized light, cs = 0, and cp = 1, we have
In parallel, the IF shift, , can be calculated by the energy flux method, see Ref. , which is
For comparison, the GH and IF shifts are also calculated in terms of stationary phase approach , which are written as38]. It does not work perfectly when the incidence angle is close to the critical angle for total reflection or zero . Regarding the IF shift, we can see from Eq. (28) that it depends on the incidence angle, 1/tan θi. The value of IF shift will be divergent when the incidence angle approaches zero. This is relevant to the special polarization distribution, and requires further investigation elsewhere . When the incidence angle is larger than the critical angle for total reflection, its magnitude is of the order of λ0/2π.
In Fig. 3(a), we illustrate the dependence of GH shifts on the wavelength with a positive/negative gradient of phase. With the parameters, dΦ/dx = 3.6 rad/μm, λ0 = 2 μm, n1 = 3.5 and n2 = 1, we can calculate the critical angles θc = −2.1° and θ′c = 42.3°, thus we have total internal reflection, instead of total transmission, satisfying θc < θi < θ′c, when θi = 2°. Similarly, when dΦ/dx = −3.6 rad/μm, we have θc = 2.1° and θ′c = −42.3°, thus satisfying the condition θ′c < θi < θc. The results calculated from energy flux method is in good agreement with those calculated from stationary phase approach. Furthermore, the negative and positive GH shifts are relevant to the sign of phase gradient in metasurface by comparing Figs. 3(a) and 3(b). The GH shifts becomes infinite when the incidence angle is close to the critical angles for total internal reflection and total transmission, see Eqs. (7) and (9). This is consistent with the results in Fig. 2, in which the reflection in these cases is almost zero. As a matter of fact, the stationary phase approximation is not valid , particularly when incidence angle approaches the critical angle for total reflection.
In addition, Figs. 4(a) and 4(b) demonstrate the transverse IF shift and its dependence on wavelength, phase discontinuity, and polarization. Obliviously, the negative and positive IF shifts can be modulated by changing the sign of phase gradient. In this case, the polarization of incident light beam determines the sign of IF shift. For instance, when Δ = ±π/2 correspond to left (right) circularly polarized incident light beam (The reflected light is elliptically polarized due to the phase shift ), the IF shifts are negative and positive, see Figs. 4(a) and 4(b). Obviously, the difference between two methods are significant when the incidence angle is small, since the stationary phase approximation could be problematic. Particularly, the result given by stationary phase method in Eq. (28) goes infinite when the incidence angle approaches zero. In fact, the representation of light beam, relevant to stationary phase method, will give different beams with peculiar polarization distributions , when the incidence angle is zero. Moreover, when incidence angle becomes larger, θi = 30°, these two results from stationary phase method and energy flux method are consistent, as shown Fig. 4(b). We shall emphasize that the difference between the stationary phase method and energy flux method is somehow relevant to the physical mechanism of beam shifts. In detail, GH shifts result from the beam reshaping, that is, the multiple interference of different plane components undergoing various phase shift. But IF shifts resemble the spin Hall of light , resulting from the conversion of angular momentum, and thus have nothing to do with total reflection. Thus, the energy flux method is more precise, especially when the stationary phase approximation is not valid.
As we know, the phase gradient are tailored through metamaterial design , which can result in the strong, broadband, and widely tunable GH shift and IF shift, connected with optical spin-orbit coupling . In Fig. 3(b), the GH shift can be modulated by phase gradient. In addition, we can achieve the maximum IF shift by adjusting gradient shift, as shown in Fig. 4(c). All the values of GH and IF shifts are the dependence of sign of gradient phase shift, which can be experimentally adjusted by various length and angle between the rods of V-antennas, fabricated on a silicon wafer . These results might lead to new applications for optical sensing, optical switch, and optical beam splitting.
Finally, to understand the anomalous shifts from the energy flux, we finally check the dependence of energy flux on the wavelength with a positive/negative phase gradient in x- and z-directions. Clearly, from the definition of GH and IF shifts in terms of energy flux method , the energy flux components are relevant to GH and IF shifts respectively. For example, when the phase gradient is positive, the energy flux along x-direction is positive, and thus the GH shift is definitely positive, vice versa, see Fig. 5(a). Also when the energy flux along the z-direction changes from negative to positive values, the IF shifts will changes again, as shown in Fig. 5(b). Evidently, different value of phase gradients provides the different energy flux, which results in the modulation of GH and IF shifts.
In conclusion, we apply the stationary phase and modified energy flux methods to study the GH and IF shifts of an arbitrarily polarized light beam reflected from a metasurfaces systematically. What we obtained here is that both GH and IF shifts for different polarized light beam can be calculated and understood by energy flux, which is consistent with the results obtained from stationary phase method. Particularly, we show the GH shift is quantized for different polarization. In addition, metasuface, as the novel artificial material for controlling the light propagation, leads to the modulation of GH and IF shifts from negative to positive values by changing the sign of phase discontinuity.
However, there are various pending issues not addressed here which we believe deserve further exploration elsewhere. For instance, the magnitude of such GH and IF shifts at single metasurface is the order of wavelength of incidence light. One can try to amplify them by using surface plasmon resonance , for further measurement and applications. To extend these results presented here to resonant tunneling and multi-layer configurations, and other non-specular effects, including angular deflection (angular GH shift), focal shift, and waist-width modification [46,47] by gradient phase is also interesting. In addition, large absorptive losses is in fact challenging in metamaterial, which is inevitable. With the development of the state-of-the-art technique, the dielectric metasurfaces with high index can avoid lossy metals , or the insertion losses can be considerably reduced thanks to surface-confined wave-matter interactions .
In a word, all results presented here provide not only the deeper understanding of GH and IF shifts, particularly for an arbitrarily polarized light beam, but also the potential applications of GH and IF shifts in nano-optics and integrated optics, with the development of novel functionalities and performances of metasurfaces.
National Natural Science Foundation of China (NSFC) (11504226, 11474193); Science and Technology Commission of Shanghai Municipality (STCSM) (18010500400, 18ZR1415500); Ramón y Cajal grant (RYC-2017-22482); Shanghai Program for Eastern Scholar.
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