Abstract

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

1. Introduction

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 [5]. Up to now, negative and positive GH shifts have been extensively investigated in different media, for example, “left-handed” metamaterial [6], metal [7,8], and graphene [9,10], with many applications in integrated optics [11–14], optical waveguide switch [15], 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 [24] or chiral metamaterials [25], hyperbolic metamaterials [26–28], metal [29–31] and graphene [32] as well.

Theoretically, the GH shift can be calculated by Artmann’s stationary phase approach [33] and Renard’s energy flux method [34], which also provide the physical explanation on the basis of reshaping effect and energy flux conservation, respectively. As for the IF shift, Imbert [4] once applied Renard’s energy flux argument to calculate and measurement this displacement. However, Beauregard and Imbert [35] commented that there was, strictly speaking, no completely rigorous calculations of the GH or IF shifts. For instance, Yasumoto and Õishi [36] 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 [38] 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 [39] 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 [41]. The analysis of energy flux suggests that the photonic spin Hall effect resembles the transverse IF shift, see the appendix in the literature [41]. 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 [42]. 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 [36], 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 [33] 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.

 

Fig. 1 Schematic diagram of the lateral GH (lGH) and transverse IF (lIF) shifts of reflected light beam at a single gradient metasurface, fabricated by the femtosecond laser self-assembly of nanostructures in silicon substrate, with different refractive indies, n1 and n2, where θi, ki, kr, kt stand for incidence angle, incident, reflected, and transmitted wave vectors. ∇Φx denotes the phase discontinuity.

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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 [39], 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 nj=εjμj, respectively, where j = 1, 2 stand for incident (reflected) and transmitted regions. The electric field of incident beam (the time-dependence exp (−0t) is implied) is assumed to be

Ei=[csz^+cp(sinθiy^cosθix^)]ei(kxix+kyiy),
where the coefficients cs = |cs|es and cp = |cp|ep represent the weight of s and p-polarized components with initial phases, δs and δp, kxi=n1ksinθi and kyi=n1kcosθi are the transverse and longitudinal wave vectors with k = ω0/c, and c is the speed of light in vacuum. The corresponding electric fields for reflected and refracted waves can be further written as
Er=[rscsz^+rpcp(sinθry^+cosθrx^)]ei(kxrxkyry),
and
Et=[tscsz^+tpcp(sinθty^cosθtx^)]ei(kxtx+kyty),
where rs/p and ts/p are the reflection and transmission coefficients for s and p polarized components, kxr=n1ksinθr, kyr=n1kcosθr, kxt=n2ksinθt, kyt=n2kcosθt, and θr/t are the reflection and refraction angles. The magnetic fields in the respective regions can be derived from H⃗ = −i/(ω0μj)∇ × E⃗, where the electric field E⃗ are given in Eqs. (1)(3).

The generalized Snell’s law for anomalous reflection and refraction are given by [39,41,43],

n2sinθtn1sinθi=λ02πdΦdx,
n1sinθrn1sinθi=λ02πdΦdx,
where dΦ/dx is the gradient of phase discontinuity, and λ0 = c/ω is the length in vacuum. Based on these two 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,
dϕdx2πn1λ0sinθi,
derived from 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
θc=arcsin(±n2n1λ02πn1dΦdx).
This means that when the incidence angle satisfies
θiarcsin(n2n1λ02πn1dΦdx),orθiarcsin(n2n1λ02πn1dΦdx).
the total internal reflection will happen with the evanescent wave for transmitted wave. The other critical angle is also obtained as
θc=arcsin(±1λ02πn1dΦdx),
above which the reflected wave becomes evanescent. This is quite different from the usual refraction law. In this case, there will exist the total transmission, when the incidence angle satisfies the above condition,
θiarcsin(1λ02πn1dΦdx),orθiarcsin(1λ02πn1dΦdx).
Therefore, we shall combine the two critical angles mentioned above, and obtain the following conditions
arcsin(n2n1λ02πn1dΦdx)θiarcsin(1λ02πn1dΦdx),
or
arcsin(1λ02πn1dΦdx)θiarcsin(n2n1λ02πn1dΦdx),
from which we can discuss the lateral GH and transverse IF shifts of reflected wave in the case of total internal reflection.

According to the boundary conditions, the reflection and transmission coefficients can be calculated as,

rs,p=χkyikytχkyr+kyteidΦdxx,
ts,p=χμ2μ1kyi+kyrχkyr+kyteidΦdxx,
where χ = μ2/μ1 for s polarization and χ = ε2/ε1 for p polarization. For the convenience, we simply rewrite the coefficients as rs,p=Rs,pexp(iϕs,pr) and ts,p=Ts,pexp(iϕs,pt), where Rs,p and Ts,p are the real modulus of reflection and transmission coefficients, ϕs,pr and ϕs,pt are the phase induced by reflections and refraction, where the phase gradient is also involved. Obviously, the critical angle for total internal reflection, reflection and transmission coefficients are closely related to the phase discontinuity. 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 [39]. In the case of the total reflection, all the results above are valid when kyt=iκand κ=n2k0sin2θt1.

 

Fig. 2 Reflection coefficients, Rs,p, for s-polarization (a) and p-polarization (b). Parameters: θi = 2°, n1 = 3.5 and n2 = 1 represent the refractive indices for silcon and air.

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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 [36], 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, S=12Re[E×H*], from

Syr=kyr2μ1ω0(|cp|2Rp2+|cs|2Rs2),
Sxt=kxte2κy2μ2ω0(|cp|2Tp2+|cs|2Ts2),
Szt=kxtκe2κyn2μ2ω0k0|cp||cs|TpTssin(ϕptϕst+Δ),
where Δ = δpδs is introduced. The self-interference time-averaged Poynting vector is given by
Sxir=kxi+kxr2μ1ω0{Rp|cp|2cos[(kyi+kyr)yϕpr]+Rs|cs|2cos[(kyi+kyr)yϕsr]}.
Therefore, the energy flux can be calculated by integrating the corresponding time-averaged Poynting vectors,
Pxt=kxt4μ2ω0κ(|cp|2Tp2+|cs|2Ts2),
Pzt=|cp||cs|kxt2n2μ2ω0k0TpTssin(ϕptϕst+Δ),
and the self-interference energy flux can be obtained by averaging the integration of the energy flux density in the interference region, yielding as [36,37]
Pxir=(kxi+kxr)(Rp|cp|2sinϕpr+Rs|cs|2sinϕsr)2μ1ω0(kyi+kyr).

As a consequence, the GH shift, lGH=(Pxir+Pxt)/|Syr|, can be obtained by the Yasumoto and Õishi’s energy flux method, which is

lGH=kxt2κkyr|cs|2Ts2+|cp|2Tp2|cs|2Rs2+|cp|2Rp2kxi+kxrkyr(kyi+kyr)Rs|cs|2sinϕsr+Rp|cp|2sinϕpr|cs|2Rs2+|cp|2Rp2.
Clearly, when s-polarized light, cp = 0, and cs = 1, is considered, we can obtain the GH shift of s-polarized light beam totally reflected from such gradient metasurface as
lGHs=kxt2κkyrTs2Rs2kxi+kxrkyr(kyi+kyr)sinϕsrRs,
from Eq. (22). In the same manner, for p-polarized light, cs = 0, and cp = 1, we have
lGHp=kxt2κkyrTp2Rp2kxi+kxrkyr(kyi+kyr)sinϕprRp.
As a consequence, the GH shift of an arbitrarily polarized light beam can be eventually expressed as
lGH=|cs|2Rs2lGHs+|cp|2Rp2lGHp|cs|2Rs2+|cp|2Rp2,
which implies the quantization characteristics of GH shift. The GH shift of arbitrarily polarized light beam is the average value with the weights, |cs|2Rs2and |cp|2Rp2, corresponding to the reflected beams of s and p polarization.

In parallel, the IF shift, lIF=Pzt/|Syr|, can be calculated by the energy flux method, see Ref. [41], which is

lIF=μ1kxt|cp||cs|TpTssin(ϕptϕst+Δ)n2μ2k0kyr(|cs|2Rs2+|cp|2Rp2).
This suggests that the property of IF shifts is quite different from that of GH shifts. For instance, the IF shift depends strongly on the relative initial phase Δ, and the maximum value can be achieved when ϕptϕst+Δ=0, corresponding to circularly polarized reflected beam, instead of s or p linearly polarized beam. In other word, for a given incidence angle, θi, the magnitude of IF shift can reach the maximum when the polarization of incident beam is elliptical, due to the phase shifts, ϕpt and ϕst, resulting from the refraction. The IF shift vanishes when the linearly polarized beam is considered, that is, cp = 0 or cs = 0. This means the left and right circular polarization states are the eigenstates of IF shift, while the s and p linear polarization states are the eigenstates of GH shift.

For comparison, the GH and IF shifts are also calculated in terms of stationary phase approach [38], which are written as

lGH=|cs|2ϕsrkx|cp|2ϕprkx,
and
lIF=sin(ϕsrϕpr+Δ)n1k0tanθi2πn1λ0tanθi.
Interestingly, the GH shift is quantized, which is consistent with the expression obtained from energy flux method. In addition, we should emphasize that the stationary phase approximation is valid when the angular distribution function is sharp [38]. It does not work perfectly when the incidence angle is close to the critical angle for total reflection or zero [44]. 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 [38]. 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 [44], particularly when incidence angle approaches the critical angle for total reflection.

 

Fig. 3 Dependence of GH shifts on the wavelength (a) and the phase gradient (b), where their positive (negative) values correspond the positive (negative) phase gradient. The red solid and blue dashed lines are plotted with energy flux method and stationary phase method respectively. Parameters: (a) dΦ/dx = ±3.6 rad/μm, (b) λ0 = 2 μm, θi = 2°, |cs|=|cp|=1/2, n1 = 3.5 and n2 = 1.

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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 ϕp,st), 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 [38], 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 [41], 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.

 

Fig. 4 Dependence of IF shifts on the wavelength (a,b) and the phase gradient (c). (a) The red solid and blue dashed lines are plotted with energy flux method and stationary phase method respectively, where the positive (negative) values correspond to Δ = π/2 (Δ = −π/2) and incidence angle is θi = 2°. (b) The red solid and blue dashed lines are plotted with energy flux method and stationary phase method respectively, where the positive (negative) values correspond to Δ = π/2 (Δ = −π/2) and incidence angle is θi = 30°. (c) The example given by stationary phase approach, in which the red solid line (Δ = π/2) and the blue dashed line (Δ = −π/2). Parameters: (a) dΦ/dx = −2π/15 rad/μm, (b) θi = 2°, and other parameters are the same as those in Fig. 3.

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As we know, the phase gradient are tailored through metamaterial design [39], which can result in the strong, broadband, and widely tunable GH shift and IF shift, connected with optical spin-orbit coupling [42]. 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 [39]. 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 [45], 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.

 

Fig. 5 Dependence of energy flux on the wavelength with a positive/negative phase gradient in x-(a) and z-(b) directions. Parameters: dΦ/dx = 3.6 rad/μm (solid red), dΦ/dx = 3rad/μm (dotted purple), dΦ/dx = −3.6 rad/μm (dashed blue), dΦ/dx = −3 rad/μm (dashed green), Δ = π/2. Other parameters are the same as those in Fig. 3.

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4. Conclusion

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 [31], 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 [48], or the insertion losses can be considerably reduced thanks to surface-confined wave-matter interactions [49].

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.

Funding

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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37. X. Chen, X. J. Lu, P. L. Zhao, and Q. B. Zhu, “Energy flux and Goos-Hänchen shift in frustrated total internal reflection,” Opt. Lett. 37, 1526–1528 (2012). [CrossRef]   [PubMed]  

38. C.-F. Li, “Unified theory for Goos-Hänchen and Imbert-Fedorov effects,” Phys. Rev. A 76, 013811 (2007). [CrossRef]  

39. N.-F. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with phase discontinuities: Generalized laws of reflection and refraction,” Science 334, 333–337 (2011). [CrossRef]   [PubMed]  

40. A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339, 1289 (2013). [CrossRef]  

41. X.-B. Yin, Z.-L. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic spin Hall effect at metasurfaces,” Science 339, 1405–1407 (2013). [CrossRef]   [PubMed]  

42. Y.-C. Liu, Y.-G. Ke, H.-L. Luo, and S.-C. Wen, “Photonic spin Hall effect in metasurfaces: a brief review,” Nanophotonics 6, 51–70 (2016). [CrossRef]  

43. Z.-N. Wang, Y.-Y. Sun, L. Han, and D.-H. Liu, “General laws of reflection and refraction for subwavelength phase grating,” arXiv 1312.3855 (2013).

44. J.-L. Shi, C.-F. Li, and Q. Wang, “Theory of the Goos-Hänchen displacement in total internal reflection,” Int. J. Mod. Phys. B 21, 2777–2791 (2007). [CrossRef]  

45. H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos-Hänchen effect,” Phys. Rev. E 62, 7330–7739 (2000). [CrossRef]  

46. X. Chen, C.-F. Li, R.-R. Wei, and Y. Zhang, “Goos-Hänchen shifts in frustrated total internal reflection studied with wave packet propagation,” Phys. Rev. A 80, 015803 (2009). [CrossRef]  

47. V. J. Yallapragada, A. P. Ravishankar, G. L. Mulay, G. S. Agarwal, and V. G. Achanta, “Observation of giant Goos-Hänchen and angular shifts at designed metasurfaces,” Sci. Rep. 6, 19319 (2016). [CrossRef]  

48. A. M. Shaltout, A. V. Kildishev, and V. M. Shalaev, “Evolution of photonic metasurfaces: from static to dynamic,” J. Opt. Soc. A. B 33, 501–510 (2016). [CrossRef]  

49. N. M. Estakhri and A. ALú, “Recent progress in gradient metasurfaces,” J. Opt. Soc. A. B 33, A21–A30 (2016). [CrossRef]  

References

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  1. F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur total reflexion,” Ann. der Phys. 436, 333–346 (1947).
    [Crossref]
  2. F. Goos and H. Hänchen, “Neumessung des strahlversetzungseffektes bei total reflexion,” Ann. der Phys. 440, 251–252 (1947).
    [Crossref]
  3. F. I. Fedorov, “K teorii polnovo otrazenija,” Dokl. Akad. Nauk SSSR 105, 465–468 (1955).
  4. C. Imbert, “Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam,” Phys. Rev. D 5, 787–796 (1972).
    [Crossref]
  5. K. Y. Bliokh and A. Aiello, “Goos-Hänchen and Imbert-Fedorov shifts: an overview,” J. Opt. 15, 014001 (2013).
    [Crossref]
  6. P. R. Berman, “Goos-Hänchen shift in negatively refractive media,” Phys. Rev. E 66, 067603 (2002).
    [Crossref]
  7. P. T. Leung, C. W. Chen, and H.-P. Chiang, “Large negative Goos-Hänchen shift at metal surfaces,” Opt. Commun. 276, 206–208 (2007).
    [Crossref]
  8. M. Merano, A. Aiello, G. W. ’t Hooft, M. P. van Exter, E. R. Eliel, and J. P. Woerdman, “Observation of Goos-Hänchen shifts in metallic reflection,” Opt. Express 15, 15928–15934 (2007).
    [Crossref] [PubMed]
  9. J. C. Martinez and M. B. A. Jalil, “Theory of giant Faraday rotation and Goos-Hänchen shift in graphene,” Europhys. Lett. 96, 27008 (2011).
    [Crossref]
  10. Y. Chen, Y. Ban, Q.-B. Zhu, and X. Chen, “Graphene-assisted resonant transmission and enhanced Goos-Hänchen shift in a frustrated total internal reflection configuration, ” Opt. Lett. 41, 4468–4471 (2016).
    [Crossref] [PubMed]
  11. H. K. V. Lotsch, “Beam displacement at total reflection: the Goos-Hänchen effect I,” Optik (Stuttgart) 32, 116–137 (1970).
  12. H. K. V. Lotsch, “Beam displacement at total reflection: the Goos-Hänchen effect II,” Optik (Stuttgart) 32, 189–204 (1970).
  13. H. K. V. Lotsch, “Beam displacement at total reflection: the Goos-Hänchen effect III,” Optik (Stuttgart) 32, 299–319 (1971).
  14. H. K. V. Lotsch, “Beam displacement at total reflection: the Goos-Hänchen effect IV,” Optik (Stuttgart) 32, 553–569 (1971).
  15. T. Sakata, H. Togo, and F. Shimokawa, “Reflection-type 2 × 2 optical waveguide switch using the Goos-Hänchen shift effect,” Appl. Phys. Lett. 76, 2841–2843 (2000).
    [Crossref]
  16. X. B. Yin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, “Goos-Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. 89, 261108 (2006).
    [Crossref]
  17. T.-Y. Yu, H.-G. Li, Z.-Q Cao, Y. Wang, Q.-S. Shen, and Y. He, “Oscillating wave displacement sensor using the enhanced Goos-Hänchen effect in a symmetrical metal-cladding optical waveguide,” Opt. Lett. 33, 1001–1003 (2008).
    [Crossref] [PubMed]
  18. M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93, 083901 (2004).
    [Crossref] [PubMed]
  19. X.-H. Ling, X.-X. Zhou, K. Huang, Y.-C. Liu, C.-W. Qiu, H.-L. Luo, and S.-C. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80, 066401 (2017).
    [Crossref] [PubMed]
  20. K. Yu. Bliokh and Y. P. Bliokh, “Conservation of angular momentum, transverse shift, and spin Hall effect in reflection and refraction of an electromagnetic wave packet,” Phys. Rev. Lett. 96, 073903 (2006).
    [Crossref] [PubMed]
  21. O. Hosten and P. Kwiat, “Observation of the spin-Hall effect of light via weak measurements,” Science 319, 787–790 (2008).
    [Crossref] [PubMed]
  22. A. Aiello and J. P. Woerdman, “Role of beam propagation in Goos-Hänchen and Imbert-Fedorov shifts,” Opt. Lett. 33, 1437–1439 (2008).
    [Crossref] [PubMed]
  23. Y. Qin, Y. Li, H.-Y. He, and Q.-H. Gong, “Measurement of spin Hall effect of reflected light,” Opt. Lett.,  34, 2551–2553 (2009).
    [Crossref] [PubMed]
  24. H.-L. Luo, S.-C. Wen, W.-X. Shu, Z.-X. Tang, Y.-H. Zou, and D.-Y. Fan, “Spin Hall effect of a light beam in left-handed materials,” Phys. Rev. A 80, 043810 (2009).
    [Crossref]
  25. H.-L. Wang and X.-D. Zhang, “Unusual spin Hall effect of a light beam in chiral metamaterials,” Phys. Rev. A 83, 053820 (2011).
    [Crossref]
  26. T. Tang, C.-Y. Li, and L. Luo, “Enhanced spin Hall effect of tunneling light in hyperbolic metamaterial waveguide,” Sci. Rep. 6, 30762 (2016).
    [Crossref] [PubMed]
  27. O. Takayama, J. Sukham, R. Malureanu, A. V. Lavrinenko, and G. Puentes, “Photonic spin Hall effect in hyperbolic metamaterials at visible wavelengths,” Opt. Lett. 43, 4602–4605 (2018).
    [Crossref] [PubMed]
  28. X. Yin, H. Zhu, H.-J. Guo, M. Deng, T. Xu, Z.-J. Gong, X. Li, Z.-H. Hang, C. Wu, H.-Q. Li, S.-Q. Chen, L. Zhou, and L. Chen, “Hyperbolic Metamaterial Devices for Wavefront Manipulation,” Laser Photonics Rev. 13, 1800081 (2019).
    [Crossref]
  29. N. Hermosa, A. M. Nugrowati, A. Aiello, and J. P. Woerdman, “Spin Hall effect of light in metallic reflection,” Opt. Lett. 36, 3200–3202 (2011).
    [Crossref] [PubMed]
  30. X.-X. Zhou, X.-H. Ling, H.-L. Luo, and S.-C. Wen, “Experimental observation of the spin Hall effect of light on a nanometal film via weak measurements,” Phys. Rev. A 85, 043809 (2012).
    [Crossref]
  31. X.-X. Zhou and X.-H. Ling, “Enhanced photonic spin Hall effect due to surface plasmon resonance,” IEEE Photon. J. 8, 4801108 (2016).
    [Crossref]
  32. X.-X. Zhou, X.-H. Ling, H.-L. Luo, and S.-C. Wen, “Identifying graphene layers via spin Hall effect of light,” Appl. Phys. Lett. 101, 251602 (2012).
    [Crossref]
  33. K. V. Artmann, “Berechnung der Seitenversetzung des totalreflektierten strahles,” Ann. Phys. (Leipzig) 2, 87–102 (1948).
    [Crossref]
  34. R. H. Renard, “Total reflection: A new evaluation of the Goos-Hänchen shift,” J. Opt. Soc. Am. 54, 1190–1197 (1964).
    [Crossref]
  35. O. Costa de Beauregard and C. Imbert, “Quantized longitudinal and transverse shifts associated with total internal reflection,” Phys. Rev. D 7, 3555 (1973).
    [Crossref]
  36. K. Yasumoto and Y. Oishi, “A new evaluation of the Goos-Hänchen shift and associated time delay,” J. Appl. Phys. 54, 2170–2176 (1983).
    [Crossref]
  37. X. Chen, X. J. Lu, P. L. Zhao, and Q. B. Zhu, “Energy flux and Goos-Hänchen shift in frustrated total internal reflection,” Opt. Lett. 37, 1526–1528 (2012).
    [Crossref] [PubMed]
  38. C.-F. Li, “Unified theory for Goos-Hänchen and Imbert-Fedorov effects,” Phys. Rev. A 76, 013811 (2007).
    [Crossref]
  39. N.-F. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with phase discontinuities: Generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
    [Crossref] [PubMed]
  40. A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339, 1289 (2013).
    [Crossref]
  41. X.-B. Yin, Z.-L. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic spin Hall effect at metasurfaces,” Science 339, 1405–1407 (2013).
    [Crossref] [PubMed]
  42. Y.-C. Liu, Y.-G. Ke, H.-L. Luo, and S.-C. Wen, “Photonic spin Hall effect in metasurfaces: a brief review,” Nanophotonics 6, 51–70 (2016).
    [Crossref]
  43. Z.-N. Wang, Y.-Y. Sun, L. Han, and D.-H. Liu, “General laws of reflection and refraction for subwavelength phase grating,” arXiv 1312.3855 (2013).
  44. J.-L. Shi, C.-F. Li, and Q. Wang, “Theory of the Goos-Hänchen displacement in total internal reflection,” Int. J. Mod. Phys. B 21, 2777–2791 (2007).
    [Crossref]
  45. H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos-Hänchen effect,” Phys. Rev. E 62, 7330–7739 (2000).
    [Crossref]
  46. X. Chen, C.-F. Li, R.-R. Wei, and Y. Zhang, “Goos-Hänchen shifts in frustrated total internal reflection studied with wave packet propagation,” Phys. Rev. A 80, 015803 (2009).
    [Crossref]
  47. V. J. Yallapragada, A. P. Ravishankar, G. L. Mulay, G. S. Agarwal, and V. G. Achanta, “Observation of giant Goos-Hänchen and angular shifts at designed metasurfaces,” Sci. Rep. 6, 19319 (2016).
    [Crossref]
  48. A. M. Shaltout, A. V. Kildishev, and V. M. Shalaev, “Evolution of photonic metasurfaces: from static to dynamic,” J. Opt. Soc. A. B 33, 501–510 (2016).
    [Crossref]
  49. N. M. Estakhri and A. ALú, “Recent progress in gradient metasurfaces,” J. Opt. Soc. A. B 33, A21–A30 (2016).
    [Crossref]

2019 (1)

X. Yin, H. Zhu, H.-J. Guo, M. Deng, T. Xu, Z.-J. Gong, X. Li, Z.-H. Hang, C. Wu, H.-Q. Li, S.-Q. Chen, L. Zhou, and L. Chen, “Hyperbolic Metamaterial Devices for Wavefront Manipulation,” Laser Photonics Rev. 13, 1800081 (2019).
[Crossref]

2018 (1)

2017 (1)

X.-H. Ling, X.-X. Zhou, K. Huang, Y.-C. Liu, C.-W. Qiu, H.-L. Luo, and S.-C. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80, 066401 (2017).
[Crossref] [PubMed]

2016 (7)

T. Tang, C.-Y. Li, and L. Luo, “Enhanced spin Hall effect of tunneling light in hyperbolic metamaterial waveguide,” Sci. Rep. 6, 30762 (2016).
[Crossref] [PubMed]

X.-X. Zhou and X.-H. Ling, “Enhanced photonic spin Hall effect due to surface plasmon resonance,” IEEE Photon. J. 8, 4801108 (2016).
[Crossref]

Y. Chen, Y. Ban, Q.-B. Zhu, and X. Chen, “Graphene-assisted resonant transmission and enhanced Goos-Hänchen shift in a frustrated total internal reflection configuration, ” Opt. Lett. 41, 4468–4471 (2016).
[Crossref] [PubMed]

Y.-C. Liu, Y.-G. Ke, H.-L. Luo, and S.-C. Wen, “Photonic spin Hall effect in metasurfaces: a brief review,” Nanophotonics 6, 51–70 (2016).
[Crossref]

V. J. Yallapragada, A. P. Ravishankar, G. L. Mulay, G. S. Agarwal, and V. G. Achanta, “Observation of giant Goos-Hänchen and angular shifts at designed metasurfaces,” Sci. Rep. 6, 19319 (2016).
[Crossref]

A. M. Shaltout, A. V. Kildishev, and V. M. Shalaev, “Evolution of photonic metasurfaces: from static to dynamic,” J. Opt. Soc. A. B 33, 501–510 (2016).
[Crossref]

N. M. Estakhri and A. ALú, “Recent progress in gradient metasurfaces,” J. Opt. Soc. A. B 33, A21–A30 (2016).
[Crossref]

2013 (3)

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339, 1289 (2013).
[Crossref]

X.-B. Yin, Z.-L. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic spin Hall effect at metasurfaces,” Science 339, 1405–1407 (2013).
[Crossref] [PubMed]

K. Y. Bliokh and A. Aiello, “Goos-Hänchen and Imbert-Fedorov shifts: an overview,” J. Opt. 15, 014001 (2013).
[Crossref]

2012 (3)

X.-X. Zhou, X.-H. Ling, H.-L. Luo, and S.-C. Wen, “Identifying graphene layers via spin Hall effect of light,” Appl. Phys. Lett. 101, 251602 (2012).
[Crossref]

X.-X. Zhou, X.-H. Ling, H.-L. Luo, and S.-C. Wen, “Experimental observation of the spin Hall effect of light on a nanometal film via weak measurements,” Phys. Rev. A 85, 043809 (2012).
[Crossref]

X. Chen, X. J. Lu, P. L. Zhao, and Q. B. Zhu, “Energy flux and Goos-Hänchen shift in frustrated total internal reflection,” Opt. Lett. 37, 1526–1528 (2012).
[Crossref] [PubMed]

2011 (4)

N.-F. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with phase discontinuities: Generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref] [PubMed]

N. Hermosa, A. M. Nugrowati, A. Aiello, and J. P. Woerdman, “Spin Hall effect of light in metallic reflection,” Opt. Lett. 36, 3200–3202 (2011).
[Crossref] [PubMed]

H.-L. Wang and X.-D. Zhang, “Unusual spin Hall effect of a light beam in chiral metamaterials,” Phys. Rev. A 83, 053820 (2011).
[Crossref]

J. C. Martinez and M. B. A. Jalil, “Theory of giant Faraday rotation and Goos-Hänchen shift in graphene,” Europhys. Lett. 96, 27008 (2011).
[Crossref]

2009 (3)

Y. Qin, Y. Li, H.-Y. He, and Q.-H. Gong, “Measurement of spin Hall effect of reflected light,” Opt. Lett.,  34, 2551–2553 (2009).
[Crossref] [PubMed]

H.-L. Luo, S.-C. Wen, W.-X. Shu, Z.-X. Tang, Y.-H. Zou, and D.-Y. Fan, “Spin Hall effect of a light beam in left-handed materials,” Phys. Rev. A 80, 043810 (2009).
[Crossref]

X. Chen, C.-F. Li, R.-R. Wei, and Y. Zhang, “Goos-Hänchen shifts in frustrated total internal reflection studied with wave packet propagation,” Phys. Rev. A 80, 015803 (2009).
[Crossref]

2008 (3)

2007 (4)

P. T. Leung, C. W. Chen, and H.-P. Chiang, “Large negative Goos-Hänchen shift at metal surfaces,” Opt. Commun. 276, 206–208 (2007).
[Crossref]

M. Merano, A. Aiello, G. W. ’t Hooft, M. P. van Exter, E. R. Eliel, and J. P. Woerdman, “Observation of Goos-Hänchen shifts in metallic reflection,” Opt. Express 15, 15928–15934 (2007).
[Crossref] [PubMed]

C.-F. Li, “Unified theory for Goos-Hänchen and Imbert-Fedorov effects,” Phys. Rev. A 76, 013811 (2007).
[Crossref]

J.-L. Shi, C.-F. Li, and Q. Wang, “Theory of the Goos-Hänchen displacement in total internal reflection,” Int. J. Mod. Phys. B 21, 2777–2791 (2007).
[Crossref]

2006 (2)

K. Yu. Bliokh and Y. P. Bliokh, “Conservation of angular momentum, transverse shift, and spin Hall effect in reflection and refraction of an electromagnetic wave packet,” Phys. Rev. Lett. 96, 073903 (2006).
[Crossref] [PubMed]

X. B. Yin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, “Goos-Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. 89, 261108 (2006).
[Crossref]

2004 (1)

M. Onoda, S. Murakami, and N. Nagaosa, “Hall effect of light,” Phys. Rev. Lett. 93, 083901 (2004).
[Crossref] [PubMed]

2002 (1)

P. R. Berman, “Goos-Hänchen shift in negatively refractive media,” Phys. Rev. E 66, 067603 (2002).
[Crossref]

2000 (2)

T. Sakata, H. Togo, and F. Shimokawa, “Reflection-type 2 × 2 optical waveguide switch using the Goos-Hänchen shift effect,” Appl. Phys. Lett. 76, 2841–2843 (2000).
[Crossref]

H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos-Hänchen effect,” Phys. Rev. E 62, 7330–7739 (2000).
[Crossref]

1983 (1)

K. Yasumoto and Y. Oishi, “A new evaluation of the Goos-Hänchen shift and associated time delay,” J. Appl. Phys. 54, 2170–2176 (1983).
[Crossref]

1973 (1)

O. Costa de Beauregard and C. Imbert, “Quantized longitudinal and transverse shifts associated with total internal reflection,” Phys. Rev. D 7, 3555 (1973).
[Crossref]

1972 (1)

C. Imbert, “Calculation and experimental proof of the transverse shift induced by total internal reflection of a circularly polarized light beam,” Phys. Rev. D 5, 787–796 (1972).
[Crossref]

1971 (2)

H. K. V. Lotsch, “Beam displacement at total reflection: the Goos-Hänchen effect III,” Optik (Stuttgart) 32, 299–319 (1971).

H. K. V. Lotsch, “Beam displacement at total reflection: the Goos-Hänchen effect IV,” Optik (Stuttgart) 32, 553–569 (1971).

1970 (2)

H. K. V. Lotsch, “Beam displacement at total reflection: the Goos-Hänchen effect I,” Optik (Stuttgart) 32, 116–137 (1970).

H. K. V. Lotsch, “Beam displacement at total reflection: the Goos-Hänchen effect II,” Optik (Stuttgart) 32, 189–204 (1970).

1964 (1)

1955 (1)

F. I. Fedorov, “K teorii polnovo otrazenija,” Dokl. Akad. Nauk SSSR 105, 465–468 (1955).

1948 (1)

K. V. Artmann, “Berechnung der Seitenversetzung des totalreflektierten strahles,” Ann. Phys. (Leipzig) 2, 87–102 (1948).
[Crossref]

1947 (2)

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur total reflexion,” Ann. der Phys. 436, 333–346 (1947).
[Crossref]

F. Goos and H. Hänchen, “Neumessung des strahlversetzungseffektes bei total reflexion,” Ann. der Phys. 440, 251–252 (1947).
[Crossref]

’t Hooft, G. W.

Achanta, V. G.

V. J. Yallapragada, A. P. Ravishankar, G. L. Mulay, G. S. Agarwal, and V. G. Achanta, “Observation of giant Goos-Hänchen and angular shifts at designed metasurfaces,” Sci. Rep. 6, 19319 (2016).
[Crossref]

Agarwal, G. S.

V. J. Yallapragada, A. P. Ravishankar, G. L. Mulay, G. S. Agarwal, and V. G. Achanta, “Observation of giant Goos-Hänchen and angular shifts at designed metasurfaces,” Sci. Rep. 6, 19319 (2016).
[Crossref]

Aiello, A.

Aieta, F.

N.-F. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with phase discontinuities: Generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref] [PubMed]

ALú, A.

N. M. Estakhri and A. ALú, “Recent progress in gradient metasurfaces,” J. Opt. Soc. A. B 33, A21–A30 (2016).
[Crossref]

Artmann, K. V.

K. V. Artmann, “Berechnung der Seitenversetzung des totalreflektierten strahles,” Ann. Phys. (Leipzig) 2, 87–102 (1948).
[Crossref]

Ban, Y.

Berman, P. R.

P. R. Berman, “Goos-Hänchen shift in negatively refractive media,” Phys. Rev. E 66, 067603 (2002).
[Crossref]

Bliokh, K. Y.

K. Y. Bliokh and A. Aiello, “Goos-Hänchen and Imbert-Fedorov shifts: an overview,” J. Opt. 15, 014001 (2013).
[Crossref]

Bliokh, K. Yu.

K. Yu. Bliokh and Y. P. Bliokh, “Conservation of angular momentum, transverse shift, and spin Hall effect in reflection and refraction of an electromagnetic wave packet,” Phys. Rev. Lett. 96, 073903 (2006).
[Crossref] [PubMed]

Bliokh, Y. P.

K. Yu. Bliokh and Y. P. Bliokh, “Conservation of angular momentum, transverse shift, and spin Hall effect in reflection and refraction of an electromagnetic wave packet,” Phys. Rev. Lett. 96, 073903 (2006).
[Crossref] [PubMed]

Boltasseva, A.

A. V. Kildishev, A. Boltasseva, and V. M. Shalaev, “Planar photonics with metasurfaces,” Science 339, 1289 (2013).
[Crossref]

Cao, Z.-Q

Capasso, F.

N.-F. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with phase discontinuities: Generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref] [PubMed]

Chen, C. W.

P. T. Leung, C. W. Chen, and H.-P. Chiang, “Large negative Goos-Hänchen shift at metal surfaces,” Opt. Commun. 276, 206–208 (2007).
[Crossref]

Chen, L.

X. Yin, H. Zhu, H.-J. Guo, M. Deng, T. Xu, Z.-J. Gong, X. Li, Z.-H. Hang, C. Wu, H.-Q. Li, S.-Q. Chen, L. Zhou, and L. Chen, “Hyperbolic Metamaterial Devices for Wavefront Manipulation,” Laser Photonics Rev. 13, 1800081 (2019).
[Crossref]

Chen, S.-Q.

X. Yin, H. Zhu, H.-J. Guo, M. Deng, T. Xu, Z.-J. Gong, X. Li, Z.-H. Hang, C. Wu, H.-Q. Li, S.-Q. Chen, L. Zhou, and L. Chen, “Hyperbolic Metamaterial Devices for Wavefront Manipulation,” Laser Photonics Rev. 13, 1800081 (2019).
[Crossref]

Chen, X.

Chen, Y.

Chiang, H.-P.

P. T. Leung, C. W. Chen, and H.-P. Chiang, “Large negative Goos-Hänchen shift at metal surfaces,” Opt. Commun. 276, 206–208 (2007).
[Crossref]

Costa de Beauregard, O.

O. Costa de Beauregard and C. Imbert, “Quantized longitudinal and transverse shifts associated with total internal reflection,” Phys. Rev. D 7, 3555 (1973).
[Crossref]

Deng, M.

X. Yin, H. Zhu, H.-J. Guo, M. Deng, T. Xu, Z.-J. Gong, X. Li, Z.-H. Hang, C. Wu, H.-Q. Li, S.-Q. Chen, L. Zhou, and L. Chen, “Hyperbolic Metamaterial Devices for Wavefront Manipulation,” Laser Photonics Rev. 13, 1800081 (2019).
[Crossref]

Eliel, E. R.

Estakhri, N. M.

N. M. Estakhri and A. ALú, “Recent progress in gradient metasurfaces,” J. Opt. Soc. A. B 33, A21–A30 (2016).
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Y.-C. Liu, Y.-G. Ke, H.-L. Luo, and S.-C. Wen, “Photonic spin Hall effect in metasurfaces: a brief review,” Nanophotonics 6, 51–70 (2016).
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Li, H.-Q.

X. Yin, H. Zhu, H.-J. Guo, M. Deng, T. Xu, Z.-J. Gong, X. Li, Z.-H. Hang, C. Wu, H.-Q. Li, S.-Q. Chen, L. Zhou, and L. Chen, “Hyperbolic Metamaterial Devices for Wavefront Manipulation,” Laser Photonics Rev. 13, 1800081 (2019).
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Ling, X.-H.

X.-H. Ling, X.-X. Zhou, K. Huang, Y.-C. Liu, C.-W. Qiu, H.-L. Luo, and S.-C. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80, 066401 (2017).
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X.-H. Ling, X.-X. Zhou, K. Huang, Y.-C. Liu, C.-W. Qiu, H.-L. Luo, and S.-C. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80, 066401 (2017).
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X. B. Yin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, “Goos-Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. 89, 261108 (2006).
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Luo, H.-L.

X.-H. Ling, X.-X. Zhou, K. Huang, Y.-C. Liu, C.-W. Qiu, H.-L. Luo, and S.-C. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80, 066401 (2017).
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X.-X. Zhou, X.-H. Ling, H.-L. Luo, and S.-C. Wen, “Experimental observation of the spin Hall effect of light on a nanometal film via weak measurements,” Phys. Rev. A 85, 043809 (2012).
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X.-X. Zhou, X.-H. Ling, H.-L. Luo, and S.-C. Wen, “Identifying graphene layers via spin Hall effect of light,” Appl. Phys. Lett. 101, 251602 (2012).
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H.-L. Luo, S.-C. Wen, W.-X. Shu, Z.-X. Tang, Y.-H. Zou, and D.-Y. Fan, “Spin Hall effect of a light beam in left-handed materials,” Phys. Rev. A 80, 043810 (2009).
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T. Tang, C.-Y. Li, and L. Luo, “Enhanced spin Hall effect of tunneling light in hyperbolic metamaterial waveguide,” Sci. Rep. 6, 30762 (2016).
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A. M. Shaltout, A. V. Kildishev, and V. M. Shalaev, “Evolution of photonic metasurfaces: from static to dynamic,” J. Opt. Soc. A. B 33, 501–510 (2016).
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J.-L. Shi, C.-F. Li, and Q. Wang, “Theory of the Goos-Hänchen displacement in total internal reflection,” Int. J. Mod. Phys. B 21, 2777–2791 (2007).
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T. Sakata, H. Togo, and F. Shimokawa, “Reflection-type 2 × 2 optical waveguide switch using the Goos-Hänchen shift effect,” Appl. Phys. Lett. 76, 2841–2843 (2000).
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H.-L. Luo, S.-C. Wen, W.-X. Shu, Z.-X. Tang, Y.-H. Zou, and D.-Y. Fan, “Spin Hall effect of a light beam in left-handed materials,” Phys. Rev. A 80, 043810 (2009).
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T. Tang, C.-Y. Li, and L. Luo, “Enhanced spin Hall effect of tunneling light in hyperbolic metamaterial waveguide,” Sci. Rep. 6, 30762 (2016).
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H.-L. Luo, S.-C. Wen, W.-X. Shu, Z.-X. Tang, Y.-H. Zou, and D.-Y. Fan, “Spin Hall effect of a light beam in left-handed materials,” Phys. Rev. A 80, 043810 (2009).
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N.-F. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with phase discontinuities: Generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
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T. Sakata, H. Togo, and F. Shimokawa, “Reflection-type 2 × 2 optical waveguide switch using the Goos-Hänchen shift effect,” Appl. Phys. Lett. 76, 2841–2843 (2000).
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X.-H. Ling, X.-X. Zhou, K. Huang, Y.-C. Liu, C.-W. Qiu, H.-L. Luo, and S.-C. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80, 066401 (2017).
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X.-X. Zhou, X.-H. Ling, H.-L. Luo, and S.-C. Wen, “Identifying graphene layers via spin Hall effect of light,” Appl. Phys. Lett. 101, 251602 (2012).
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X.-X. Zhou, X.-H. Ling, H.-L. Luo, and S.-C. Wen, “Experimental observation of the spin Hall effect of light on a nanometal film via weak measurements,” Phys. Rev. A 85, 043809 (2012).
[Crossref]

H.-L. Luo, S.-C. Wen, W.-X. Shu, Z.-X. Tang, Y.-H. Zou, and D.-Y. Fan, “Spin Hall effect of a light beam in left-handed materials,” Phys. Rev. A 80, 043810 (2009).
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Wu, C.

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H. M. Lai, C. W. Kwok, Y. W. Loo, and B. Y. Xu, “Energy-flux pattern in the Goos-Hänchen effect,” Phys. Rev. E 62, 7330–7739 (2000).
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X. Yin, H. Zhu, H.-J. Guo, M. Deng, T. Xu, Z.-J. Gong, X. Li, Z.-H. Hang, C. Wu, H.-Q. Li, S.-Q. Chen, L. Zhou, and L. Chen, “Hyperbolic Metamaterial Devices for Wavefront Manipulation,” Laser Photonics Rev. 13, 1800081 (2019).
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V. J. Yallapragada, A. P. Ravishankar, G. L. Mulay, G. S. Agarwal, and V. G. Achanta, “Observation of giant Goos-Hänchen and angular shifts at designed metasurfaces,” Sci. Rep. 6, 19319 (2016).
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K. Yasumoto and Y. Oishi, “A new evaluation of the Goos-Hänchen shift and associated time delay,” J. Appl. Phys. 54, 2170–2176 (1983).
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X.-B. Yin, Z.-L. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic spin Hall effect at metasurfaces,” Science 339, 1405–1407 (2013).
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X. Yin, H. Zhu, H.-J. Guo, M. Deng, T. Xu, Z.-J. Gong, X. Li, Z.-H. Hang, C. Wu, H.-Q. Li, S.-Q. Chen, L. Zhou, and L. Chen, “Hyperbolic Metamaterial Devices for Wavefront Manipulation,” Laser Photonics Rev. 13, 1800081 (2019).
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X. B. Yin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, “Goos-Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. 89, 261108 (2006).
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X.-B. Yin, Z.-L. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic spin Hall effect at metasurfaces,” Science 339, 1405–1407 (2013).
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N.-F. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light Propagation with phase discontinuities: Generalized laws of reflection and refraction,” Science 334, 333–337 (2011).
[Crossref] [PubMed]

Yu, T.-Y.

Zhang, X.

X.-B. Yin, Z.-L. Ye, J. Rho, Y. Wang, and X. Zhang, “Photonic spin Hall effect at metasurfaces,” Science 339, 1405–1407 (2013).
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X. B. Yin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, “Goos-Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. 89, 261108 (2006).
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Zhang, X.-D.

H.-L. Wang and X.-D. Zhang, “Unusual spin Hall effect of a light beam in chiral metamaterials,” Phys. Rev. A 83, 053820 (2011).
[Crossref]

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X. Chen, C.-F. Li, R.-R. Wei, and Y. Zhang, “Goos-Hänchen shifts in frustrated total internal reflection studied with wave packet propagation,” Phys. Rev. A 80, 015803 (2009).
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Zhao, P. L.

Zhou, L.

X. Yin, H. Zhu, H.-J. Guo, M. Deng, T. Xu, Z.-J. Gong, X. Li, Z.-H. Hang, C. Wu, H.-Q. Li, S.-Q. Chen, L. Zhou, and L. Chen, “Hyperbolic Metamaterial Devices for Wavefront Manipulation,” Laser Photonics Rev. 13, 1800081 (2019).
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Zhou, X.-X.

X.-H. Ling, X.-X. Zhou, K. Huang, Y.-C. Liu, C.-W. Qiu, H.-L. Luo, and S.-C. Wen, “Recent advances in the spin Hall effect of light,” Rep. Prog. Phys. 80, 066401 (2017).
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X.-X. Zhou and X.-H. Ling, “Enhanced photonic spin Hall effect due to surface plasmon resonance,” IEEE Photon. J. 8, 4801108 (2016).
[Crossref]

X.-X. Zhou, X.-H. Ling, H.-L. Luo, and S.-C. Wen, “Experimental observation of the spin Hall effect of light on a nanometal film via weak measurements,” Phys. Rev. A 85, 043809 (2012).
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Zhu, H.

X. Yin, H. Zhu, H.-J. Guo, M. Deng, T. Xu, Z.-J. Gong, X. Li, Z.-H. Hang, C. Wu, H.-Q. Li, S.-Q. Chen, L. Zhou, and L. Chen, “Hyperbolic Metamaterial Devices for Wavefront Manipulation,” Laser Photonics Rev. 13, 1800081 (2019).
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Zhu, Q. B.

Zhu, Q.-B.

Zou, Y.-H.

H.-L. Luo, S.-C. Wen, W.-X. Shu, Z.-X. Tang, Y.-H. Zou, and D.-Y. Fan, “Spin Hall effect of a light beam in left-handed materials,” Phys. Rev. A 80, 043810 (2009).
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Appl. Phys. Lett. (3)

T. Sakata, H. Togo, and F. Shimokawa, “Reflection-type 2 × 2 optical waveguide switch using the Goos-Hänchen shift effect,” Appl. Phys. Lett. 76, 2841–2843 (2000).
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X. B. Yin, L. Hesselink, Z. Liu, N. Fang, and X. Zhang, “Goos-Hänchen shift surface plasmon resonance sensor,” Appl. Phys. Lett. 89, 261108 (2006).
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X.-X. Zhou, X.-H. Ling, H.-L. Luo, and S.-C. Wen, “Identifying graphene layers via spin Hall effect of light,” Appl. Phys. Lett. 101, 251602 (2012).
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Figures (5)

Fig. 1
Fig. 1 Schematic diagram of the lateral GH (lGH) and transverse IF (lIF) shifts of reflected light beam at a single gradient metasurface, fabricated by the femtosecond laser self-assembly of nanostructures in silicon substrate, with different refractive indies, n1 and n2, where θi, ki, kr, kt stand for incidence angle, incident, reflected, and transmitted wave vectors. ∇Φx denotes the phase discontinuity.
Fig. 2
Fig. 2 Reflection coefficients, Rs,p, for s-polarization (a) and p-polarization (b). Parameters: θi = 2°, n1 = 3.5 and n2 = 1 represent the refractive indices for silcon and air.
Fig. 3
Fig. 3 Dependence of GH shifts on the wavelength (a) and the phase gradient (b), where their positive (negative) values correspond the positive (negative) phase gradient. The red solid and blue dashed lines are plotted with energy flux method and stationary phase method respectively. Parameters: (a) dΦ/dx = ±3.6 rad/μm, (b) λ0 = 2 μm, θi = 2°, | c s | = | c p | = 1 / 2, n1 = 3.5 and n2 = 1.
Fig. 4
Fig. 4 Dependence of IF shifts on the wavelength (a,b) and the phase gradient (c). (a) The red solid and blue dashed lines are plotted with energy flux method and stationary phase method respectively, where the positive (negative) values correspond to Δ = π/2 (Δ = −π/2) and incidence angle is θi = 2°. (b) The red solid and blue dashed lines are plotted with energy flux method and stationary phase method respectively, where the positive (negative) values correspond to Δ = π/2 (Δ = −π/2) and incidence angle is θi = 30°. (c) The example given by stationary phase approach, in which the red solid line (Δ = π/2) and the blue dashed line (Δ = −π/2). Parameters: (a) dΦ/dx = −2π/15 rad/μm, (b) θi = 2°, and other parameters are the same as those in Fig. 3.
Fig. 5
Fig. 5 Dependence of energy flux on the wavelength with a positive/negative phase gradient in x-(a) and z-(b) directions. Parameters: dΦ/dx = 3.6 rad/μm (solid red), dΦ/dx = 3rad/μm (dotted purple), dΦ/dx = −3.6 rad/μm (dashed blue), dΦ/dx = −3 rad/μm (dashed green), Δ = π/2. Other parameters are the same as those in Fig. 3.

Equations (28)

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E i = [ c s z ^ + c p ( sin θ i y ^ cos θ i x ^ ) ] e i ( k x i x + k y i y ) ,
E r = [ r s c s z ^ + r p c p ( sin θ r y ^ + cos θ r x ^ ) ] e i ( k x r x k y r y ) ,
E t = [ t s c s z ^ + t p c p ( sin θ t y ^ cos θ t x ^ ) ] e i ( k x t x + k y t y ) ,
n 2 sin θ t n 1 sin θ i = λ 0 2 π d Φ d x ,
n 1 sin θ r n 1 sin θ i = λ 0 2 π d Φ d x ,
d ϕ d x 2 π n 1 λ 0 sin θ i ,
θ c = arcsin ( ± n 2 n 1 λ 0 2 π n 1 d Φ d x ) .
θ i arcsin ( n 2 n 1 λ 0 2 π n 1 d Φ d x ) , or θ i arcsin ( n 2 n 1 λ 0 2 π n 1 d Φ d x ) .
θ c = arcsin ( ± 1 λ 0 2 π n 1 d Φ d x ) ,
θ i arcsin ( 1 λ 0 2 π n 1 d Φ d x ) , or θ i arcsin ( 1 λ 0 2 π n 1 d Φ d x ) .
arcsin ( n 2 n 1 λ 0 2 π n 1 d Φ d x ) θ i arcsin ( 1 λ 0 2 π n 1 d Φ d x ) ,
arcsin ( 1 λ 0 2 π n 1 d Φ d x ) θ i arcsin ( n 2 n 1 λ 0 2 π n 1 d Φ d x ) ,
r s , p = χ k y i k y t χ k y r + k y t e i d Φ d x x ,
t s , p = χ μ 2 μ 1 k y i + k y r χ k y r + k y t e i d Φ d x x ,
S y r = k y r 2 μ 1 ω 0 ( | c p | 2 R p 2 + | c s | 2 R s 2 ) ,
S x t = k x t e 2 κ y 2 μ 2 ω 0 ( | c p | 2 T p 2 + | c s | 2 T s 2 ) ,
S z t = k x t κ e 2 κ y n 2 μ 2 ω 0 k 0 | c p | | c s | T p T s sin ( ϕ p t ϕ s t + Δ ) ,
S x ir = k x i + k x r 2 μ 1 ω 0 { R p | c p | 2 cos [ ( k y i + k y r ) y ϕ p r ] + R s | c s | 2 cos [ ( k y i + k y r ) y ϕ s r ] } .
P x t = k x t 4 μ 2 ω 0 κ ( | c p | 2 T p 2 + | c s | 2 T s 2 ) ,
P z t = | c p | | c s | k x t 2 n 2 μ 2 ω 0 k 0 T p T s sin ( ϕ p t ϕ s t + Δ ) ,
P x ir = ( k x i + k x r ) ( R p | c p | 2 sin ϕ p r + R s | c s | 2 sin ϕ s r ) 2 μ 1 ω 0 ( k y i + k y r ) .
l GH = k x t 2 κ k y r | c s | 2 T s 2 + | c p | 2 T p 2 | c s | 2 R s 2 + | c p | 2 R p 2 k x i + k x r k y r ( k y i + k y r ) R s | c s | 2 sin ϕ s r + R p | c p | 2 sin ϕ p r | c s | 2 R s 2 + | c p | 2 R p 2 .
l GH s = k x t 2 κ k y r T s 2 R s 2 k x i + k x r k y r ( k y i + k y r ) sin ϕ s r R s ,
l GH p = k x t 2 κ k y r T p 2 R p 2 k x i + k x r k y r ( k y i + k y r ) sin ϕ p r R p .
l GH = | c s | 2 R s 2 l GH s + | c p | 2 R p 2 l GH p | c s | 2 R s 2 + | c p | 2 R p 2 ,
l IF = μ 1 k x t | c p | | c s | T p T s sin ( ϕ p t ϕ s t + Δ ) n 2 μ 2 k 0 k y r ( | c s | 2 R s 2 + | c p | 2 R p 2 ) .
l GH = | c s | 2 ϕ s r k x | c p | 2 ϕ p r k x ,
l IF = sin ( ϕ s r ϕ p r + Δ ) n 1 k 0 tan θ i 2 π n 1 λ 0 tan θ i .

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