Goos-H&#x000E4;nchen and Imbert-Fedorov shifts at gradient metasurfaces

Qian Kong; Han-Yu Shi; Jie-Long Shi; Xi Chen

doi:10.1364/OE.27.011902

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 (l_GH) and transverse IF (l_IF) 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 θ_i, kⁱ, k^r, k^t stand for incidence angle, incident, reflected, and transmitted wave vectors. ∇Φ_x denotes the phase discontinuity.

Download Full Size | PDF

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 θ_i, as shown in Fig. 1. The permittivity, permeability, and refractive index of different regions are denoted by ε_j, μ_j, and $n_{j} = \sqrt{ε_{j} μ_{j}}$ , 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 be

{\vec{E}}^{i} = [c_{s} \hat{z} + c_{p} (sin θ_{i} \hat{y} - cos θ_{i} \hat{x})] e^{i (k_{x}^{i} x + k_{y}^{i} y)},

where the coefficients c_s = |c_s|e^iδ_s and c_p = |c_p|e^iδ_p represent the weight of s and p-polarized components with initial phases, δ_s and δ_p,

k_{x}^{i} = n_{1} k sin θ_{i}

and

k_{y}^{i} = n_{1} k cos θ_{i}

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

{\vec{E}}^{r} = [r_{s} c_{s} \hat{z} + r_{p} c_{p} (sin θ_{r} \hat{y} + cos θ_{r} \hat{x})] e^{i (k_{x}^{r} x - k_{y}^{r} y)},

and

{\vec{E}}^{t} = [t_{s} c_{s} \hat{z} + t_{p} c_{p} (sin θ_{t} \hat{y} - cos θ_{t} \hat{x})] e^{i (k_{x}^{t} x + k_{y}^{t} y)},

where r_s/p and t_s/p are the reflection and transmission coefficients for s and p polarized components,

k_{x}^{r} = n_{1} k sin θ_{r}

,

k_{y}^{r} = n_{1} k cos θ_{r}

,

k_{x}^{t} = n_{2} k sin θ_{t}

,

k_{y}^{t} = n_{2} k cos θ_{t}

, and θ_r/t are the reflection and refraction angles. The magnetic fields in the respective regions can be derived from H⃗ = −i/(ω₀μ_j)∇ × 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],

n_{2} sin θ_{t} - n_{1} sin θ_{i} = \frac{λ_{0}}{2 π} \frac{d Φ}{d x},

n_{1} sin θ_{r} - n_{1} sin θ_{i} = \frac{λ_{0}}{2 π} \frac{d Φ}{d x},

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,

\frac{d ϕ}{d x} \leq - \frac{2 π n_{1}}{λ_{0}} sin θ_{i},

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

θ_{c} = arcsin (\pm \frac{n_{2}}{n_{1}} - \frac{λ_{0}}{2 π n_{1}} \frac{d Φ}{d x}) .

This means that when the incidence angle satisfies

θ_{i} \geq arcsin (\frac{n_{2}}{n_{1}} - \frac{λ_{0}}{2 π n_{1}} \frac{d Φ}{d x}), or θ_{i} \leq arcsin (- \frac{n_{2}}{n_{1}} - \frac{λ_{0}}{2 π n_{1}} \frac{d Φ}{d x}) .

the total internal reflection will happen with the evanescent wave for transmitted wave. The other critical angle is also obtained as

θ_{c}^{'} = arcsin (\pm 1 - \frac{λ_{0}}{2 π n_{1}} \frac{d Φ}{d x}),

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,

θ_{i} \geq arcsin (1 - \frac{λ_{0}}{2 π n_{1}} \frac{d Φ}{d x}), or θ_{i} \leq arcsin (- 1 - \frac{λ_{0}}{2 π n_{1}} \frac{d Φ}{d x}) .

Therefore, we shall combine the two critical angles mentioned above, and obtain the following conditions

arcsin (\frac{n_{2}}{n_{1}} - \frac{λ_{0}}{2 π n_{1}} \frac{d Φ}{d x}) \leq θ_{i} \leq arcsin (1 - \frac{λ_{0}}{2 π n_{1}} \frac{d Φ}{d x}),

or

arcsin (- 1 - \frac{λ_{0}}{2 π n_{1}} \frac{d Φ}{d x}) \leq θ_{i} \leq arcsin (- \frac{n_{2}}{n_{1}} - \frac{λ_{0}}{2 π n_{1}} \frac{d Φ}{d x}),

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,

r_{s, p} = \frac{χ k_{y}^{i} - k_{y}^{t}}{χ k_{y}^{r} + k_{y}^{t}} e^{- i \frac{d Φ}{d x} x},

t_{s, p} = \sqrt{\frac{χ μ_{2}}{μ_{1}}} \frac{k_{y}^{i} + k_{y}^{r}}{χ k_{y}^{r} + k_{y}^{t}} e^{- i \frac{d Φ}{d x} x},

where χ = μ₂/μ₁ for s polarization and χ = ε₂/ε₁ for p polarization. For the convenience, we simply rewrite the coefficients as

r_{s, p} = R_{s, p} exp (i ϕ_{s, p}^{r})

and

t_{s, p} = T_{s, p} exp (i ϕ_{s, p}^{t})

, where R_s,p and T_s,p are the real modulus of reflection and transmission coefficients,

ϕ_{s, p}^{r}

and

ϕ_{s, p}^{t}

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 θ_i = 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

k_{y}^{t} = i κ

and

κ = n_{2} k_{0} \sqrt{{sin}^{2} θ_{t} - 1}

.

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

Download Full Size | PDF

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, $S = \frac{1}{2} Re [E \times H^{*}]$ , from

S_{y}^{r} = - \frac{k_{y}^{r}}{2 μ_{1} ω_{0}} ({| c_{p} |}^{2} R_{p}^{2} + {| c_{s} |}^{2} R_{s}^{2}),

S_{x}^{t} = \frac{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} = - \frac{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),

where Δ = δ_p − δ_s is introduced. The self-interference time-averaged Poynting vector is given by

S_{x}^{ir} = \frac{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}]} .

Therefore, the energy flux can be calculated by integrating the corresponding time-averaged Poynting vectors,

P_{x}^{t} = \frac{k_{x}^{t}}{4 μ_{2} ω_{0} κ} ({| c_{p} |}^{2} T_{p}^{2} + {| c_{s} |}^{2} T_{s}^{2}),

P_{z}^{t} = - \frac{| 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),

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]

〈 P_{x}^{ir} 〉 = - \frac{(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})} .

As a consequence, the GH shift, $l_{GH} = (〈 P_{x}^{ir} 〉 + P_{x}^{t}) / | S_{y}^{r} |$ , can be obtained by the Yasumoto and Õishi’s energy flux method, which is

l_{GH} = \frac{k_{x}^{t}}{2 κ k_{y}^{r}} \frac{{| c_{s} |}^{2} T_{s}^{2} + {| c_{p} |}^{2} T_{p}^{2}}{{| c_{s} |}^{2} R_{s}^{2} + {| c_{p} |}^{2} R_{p}^{2}} - \frac{k_{x}^{i} + k_{x}^{r}}{k_{y}^{r} (k_{y}^{i} + k_{y}^{r})} \frac{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}} .

Clearly, when s-polarized light, c_p = 0, and c_s = 1, is considered, we can obtain the GH shift of s-polarized light beam totally reflected from such gradient metasurface as

l_{GH}^{s} = \frac{k_{x}^{t}}{2 κ k_{y}^{r}} \frac{T_{s}^{2}}{R_{s}^{2}} - \frac{k_{x}^{i} + k_{x}^{r}}{k_{y}^{r} (k_{y}^{i} + k_{y}^{r})} \frac{sin ϕ_{s}^{r}}{R_{s}},

from Eq. (22). In the same manner, for p-polarized light, c_s = 0, and c_p = 1, we have

l_{GH}^{p} = \frac{k_{x}^{t}}{2 κ k_{y}^{r}} \frac{T_{p}^{2}}{R_{p}^{2}} - \frac{k_{x}^{i} + k_{x}^{r}}{k_{y}^{r} (k_{y}^{i} + k_{y}^{r})} \frac{sin ϕ_{p}^{r}}{R_{p}} .

As a consequence, the GH shift of an arbitrarily polarized light beam can be eventually expressed as

l_{GH} = \frac{{| 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}},

which implies the quantization characteristics of GH shift. The GH shift of arbitrarily polarized light beam is the average value with the weights,

{| c_{s} |}^{2} R_{s}^{2}

and

{| c_{p} |}^{2} R_{p}^{2}

, corresponding to the reflected beams of s and p polarization.

In parallel, the IF shift, $l_{IF} = P_{z}^{t} / | S_{y}^{r} |$ , can be calculated by the energy flux method, see Ref. [41], which is

l_{IF} = - \frac{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})} .

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

ϕ_{p}^{t} - ϕ_{s}^{t} + Δ = 0

, corresponding to circularly polarized reflected beam, instead of s or p linearly polarized beam. In other word, for a given incidence angle, θ_i, the magnitude of IF shift can reach the maximum when the polarization of incident beam is elliptical, due to the phase shifts,

ϕ_{p}^{t}

and

ϕ_{s}^{t}

, resulting from the refraction. The IF shift vanishes when the linearly polarized beam is considered, that is, c_p = 0 or c_s = 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

l_{GH} = - {| c_{s} |}^{2} \frac{\partial ϕ_{s}^{r}}{\partial k_{x}} - {| c_{p} |}^{2} \frac{\partial ϕ_{p}^{r}}{\partial k_{x}},

and

l_{IF} = - \frac{sin (ϕ_{s}^{r} - ϕ_{p}^{r} + Δ)}{n_{1} k_{0} tan θ_{i}} \sim \frac{2 π}{n_{1} λ_{0} tan θ_{i}} .

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 θ_i. 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 θ_c = −2.1° and θ′_c = 42.3°, thus we have total internal reflection, instead of total transmission, satisfying θ_c < θ_i < θ′_c, when θ_i = 2°. Similarly, when dΦ/dx = −3.6 rad/μm, we have θ_c = 2.1° and θ′_c = −42.3°, thus satisfying the condition θ′_c < θ_i < θ_c. 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, θ_i = 2°, $| c_{s} | = | c_{p} | = 1 / \sqrt{2}$ , n₁ = 3.5 and n₂ = 1.

Download Full Size | PDF

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 $ϕ_{p, s}^{t}$ ), 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, θ_i = 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 θ_i = 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 θ_i = 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) θ_i = 2°, and other parameters are the same as those in Fig. 3.

Download Full Size | PDF

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.

Download Full Size | PDF

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.

References

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]

Goos-Hänchen and Imbert-Fedorov shifts at gradient metasurfaces

Abstract

1. Introduction

2. Generalized Snell’s law

3. GH and IF shifts: modified energy flux and stationary phase methods

4. Conclusion

Funding

References

Cited By

Figures (5)

Equations (28)

Optics Express