Upper-limited angular Goos-H&#x00E4;nchen shifts of Laguerre-Gaussian beams

Hai Lin; Wenguo Zhu; Jianhui Yu; Mengjiang Jiang; Linqing Zhuo; Wentao Qiu; Jiangli Dong; Yongchun Zhong; Zhe Chen

doi:10.1364/OE.26.005810

1. Introduction

A light beam with a finite transverse extent may undergo shifts in directions parallel and perpendicular to the plane of incidence when reflected from an interface between two different media [1–3]. These two kinds of shifts are the so-called Goos-Hänchen (GH) and Imbert–Fedorov (IF) shifts, respectively [1]. The GH and IF shifts have both the spatial and angular characters, namely, the reflected beam will be displaced at the interface and deflected upon propagation, respectively [4]. The GH effects originate from the dispersion of the Fresnel reflection coefficients [1], while the IF shift is governed by the conservation law of the total angular momentum [5]. The GH and IF shifts have attracted significant attention owing to their applications in quantum information and precision metrology [6–9].

It has been demonstrated that the GH and IF shifts will be affected by the phase vortex embedded in the incident beam [10,11]. The first discussion of this effect on the transverse beam shifts was given by Fedoseyev in 2001 [12]. In 2006, the vortex-induced spatial IF shift was detected experimentally for the reflected beam [13]. In 2010, all the four shifts (GH and IF, spatial and angular) for vortex beams were observed experimentally by Merano et. al. at the air-glass interface [11]. According to their results, the four shifts increase with vortex charge, thus large beam shifts can be obtained [11]. Recently, Jiang and associates gave the upper limits of the spin-dependent shifts (the spatial IF shifts) of Laguerre-Gaussian (LG) beams [14]. Here, the upper limits of angular shifts are derived, and the influences of both the spatial and angular shifts on the centroid shifts of reflected beams after propagating an arbitrary distance are investigated.

As a kind of artificial medium with permittivity close to zero, the epsilon-near-zero (ENZ) metamaterial has drawn much attention for its intriguing electromagnetic property [15–17]. It was shown that the spatial GH shift is constant for any incident angle when a s polarized Gaussian beam reflected from a ENZ metamaterial [18]. By coating a graphene monolayer on the ENZ metamaterial, Fan et. al. realized an electrically tunable GH shift in the terahertz regime [19]. It was demonstrated that the IF shift can be enhanced by transmitting a Gaussian beam through an ENZ metamaterial [20,21].

In this paper, we focus our attention on the beam shifts of LG beams reflected by an air- metamaterial interface. We find that the upper limit of the angular GH shift is equal to half of the divergent angle of incident LG beam. Near the Brewster angle, the angular GH and spatial IF shifts can approach to their upper limits simultaneously. The total centroid shift of the reflected beam is determined by both the angular and spatial shifts when the propagation distance is not zero. A dimensionless parameter F is introduced to compare the total beam shift with the beam size, as the beam shift depends on the incident beam waist. We find that F is related to the deformation of the reflected beam, and can be used to compare of the centroid shifts of beams with different sizes. F will increase gradually with the propagation distance, but cannot exceed 0.5.

2. Theory and Model

Considering a linearly polarized higher-order LG beam reflected by an interface between air and ENZ metamaterial. As shown by Fig. 1, the incident angle of LG beam is θ, and the beam waist locates at the air-metamaterial interface. the coordinate system attached to the incident and reflected beam are (x_i, y_i, z_i) and (x_r, y_r, z_r), respectively. The angular spectra of LG beam is ${\tilde{ϕ}}_{l} = (w_{0} / \sqrt{2 π | l |}) {(w_{0} / \sqrt{2})}^{| l |} {[- i k_{i x} + s i g n (l) k_{i y}]}^{| l |} \exp [- (k_{i x}^{2} + k_{i y}^{2}) (w_{0}^{2} / 4)]$ , k_ix and k_iy denote the wave vector components in the x_i and y_i directions, respectively. The angular spectrum of the reflected beam can be obtained through the transformation matrix between the incident and reflected angular spectrum given by Ref [1]. For a horizontal incident polarization, the angular spectrum of the reflected beam at z_r = 0 plane can be written as:

{\tilde{E}}_{r} (z_{r} = 0) = {(r_{p} - \frac{k_{r x}}{k_{0}} \frac{\partial r_{p}}{\partial θ}) {\hat{e}}_{r x} - \frac{k_{r y}}{k_{1}} [(r_{p} + r_{s}) \cot θ] {\hat{e}}_{r y}} {\tilde{ϕ}}_{l}

where k₀ is the wavenumber in free space; r_p and r_s are Fresnel reflection coefficients for p and s waves. After propagating a distance of z_r, the reflected beam acquires a propagation phase, thus its angular spectrum become

{\tilde{E}}_{r} (z_{r}) = {\tilde{E}}_{r} (z_{r} = 0) \exp [i k_{0} z_{r} - i z_{r} (k_{r x}^{2} + k_{r y}^{2}) / 2 k_{0}]

. By making inverse Fourier transformation, the electric light field in free space are obtained. Owing to field modulation upon reflection, the centroid of the reflected beam will be shifted, which can be decomposed into z_r-independent and z_r-dependent terms, corresponding to spatial and angular shifts, respectively [22]. The GH and IF shifts can therefore be expressed as δX = ΔX + z_rΘ_x, δY = ΔY + z_rΘ_y, where

Δ X_{r} / Δ Y_{r} = 〈 {\tilde{E}}_{r} | i \partial / \partial k_{r x / r y} | {\tilde{E}}_{r} 〉 / W_{r}^{}

and

Θ_{x / y} = 〈 {\tilde{E}}_{r} | k_{r x / r y} | {\tilde{E}}_{r} 〉 / k_{0} W_{r}^{}

,

W_{r} = 〈 {\tilde{E}}_{r} | {\tilde{E}}_{r} 〉

is the energy of the reflected beam [1,22]. After some straightforward calculation, we obtain

Δ X = \frac{Im [r_{p}^{*} \frac{\partial r_{p}}{\partial θ}]}{k_{0} W_{r}},

Δ Y = - l \frac{Re [r_{p}^{*} \frac{\partial r_{p}}{\partial θ}]}{k_{0} W_{r}},

Θ_{x} = - (| l | + 1) \frac{Re [r_{p}^{*} \frac{\partial r_{p}}{\partial θ}]}{k_{0} Z_{0} W_{r}},

Θ_{y} = 0,

where

W_{r} = {| r_{p} |^{2} + (| l | + 1) [| \partial r_{p} / \partial θ |^{2} + | N |^{2}] / k_{0}^{2} w_{0}^{2}}

and

z_{0} = k_{0} w_{0}^{2} / 2

. From Eqs. (2-5) one finds that, the angular IF Θ_y vanishes. And the spatial GH ΔX is ℓ-independent, since it originates from Gaussian envelop [1]. The spatial IF shift and angular GH shift share similar forms (sees Fig. 2), since the spatial IF shift results from the coupling between the vortex structure and the angular GH shift. If energy correction term (second term) of W_r is negligible, Θ_x and ΔY vary linearly with the vortex charge ℓ. Otherwise, they change nonlinearly with ℓ. Now we consider the case when

{[\partial | r_{p} | / \partial θ]}^{2} > > (| r_{p} |^{2} | \partial δ_{p} / \partial θ |^{2} + | N |^{2})

with δ_p being the argument of r_p, i.e, the non-central incident waves contribute much more to the x_r components of the reflected beam than to the y_r component. ΔY varies with the incident angle θ, and takes peak values among all θ, ± ℓw₀ /2(|ℓ| + 1)^1/2 when

r_{p} = \pm {(| l | + 1)}^{1 / 2} [\partial | r_{p} | / \partial θ] / k_{0}^{} w_{0}^{}

[14]. Therefore, the upper limit of the spatial IF shift is ΔY_up = ℓw₀/2(|ℓ| + 1)^1/2 [14]. Since Θ_x = (|ℓ| + 1)ΔY/ℓz₀, the angular GH shift is limited by Θ_up = (|ℓ| + 1)^1/2/k₀w₀, which is equal to half of divergent angle of the incident LG beam. The upper limit of spatial shift increases with incident beam waist w₀, while that of angular shift decreases with w₀.

Fig. 1 The schematic of beam shifts of reflected vortex beam. When reflected by air-metamaterial interface, a horizontally polarized vortex beam will undergo spatial shifts ΔY along y_r direction and angular shift Θ_x along x_r direction.

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Fig. 2 The dependences of spatial GH (ΔX) and IF (ΔY) shifts (a) and angular GH (Θ_x) and IF (Θ_y) shifts (b) on the incident angle θ for w₀ = 40λ, ℓ = 3, and ε_m = 0.1.

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The reflected beam will undergo shifts in both x_r and y_r directions. The total beam shift is defined as δ = [(δX)² + (δY)²]^1/2. Owing to the angular shift, the total beam shift varies with the propagation distance z_r. The upper limit the total beam shift is δ = 0.5w₀(|ℓ| + 1)^1/2 [|ℓ|²/(|ℓ| + 1)² + (z_r_/z₀)²]^1/2. Due to the diffraction effect, the spot size of the beam increases with z_r. For a standard LG beam, the spot size can be described by the second radial moment of the intensity: $D^{2} = \iint | ϕ |^{2} x^{2} d x d y / \iint | ϕ |^{2} d x d y$ ,where ϕ is electric field profile of the standard LG beam [23]. It is D(z_r) = w₀(|ℓ| + 1)^1/2[1 + (z_r_/z₀)²]^1/2. Here, we introduce a dimensionless parameter F = δ/D to compare the centroid shift of the reflected beam and the spot size of the standard LG beam. The spot size of the reflected beam is equal to that of standard LG beam when the beam shift vanishes. Therefore, D(z_r) can be considered as the original spot size of the reflected beam.

In the following, we try to realize the upper-limited spatial and angular shifts by investigating the centroid shifts of vortex beams reflected by an air-metamaterial interface, and study the dependence of parameter F on the propagation distance z_r.

3. Result and discussion

According to Eqs. (2-5), we calculate the four shifts of the reflected vortex beam of ℓ = 3, and show the numerical results in Fig. 2(a) and (b). In the calculations, the permittivity of the metamaterial is ε_m = 0.1, and w₀ = 40λ with λ being the wavelength in free space. As shown in Fig. 2(a) and (b), the angular IF shift Θ_y vanishes for any incident angle. The spatial GH shift takes large values near the critical angle for total reflection, θ_c = 18.435°. The curves of the spatial IF shift ΔY and the angular GH shift Θ_x changing with θ are identical, although their values are different. ΔY and Θ_x change signs when the incident angle θ crosses the Brewster angle, θ_B = 17.548°. They take large values near θ_B. The maximum spatial IF |ΔY| and angular GH |Θ_x| shifts are 29.9λ and 7.9 × 10⁻³ rad, respectively, which reach 97.78% of the theoretical upper limits. In the case of total reflection, θ>θ_c, both ΔY and Θ_x vanish owing to the independence of the absolute value of r_p on θ. For θ near but below θ_c, |ΔY| and |Θ_x| decrease rapidly since θ_B and θ_c are close to each other.

For a given beam w₀, the normalized spatial shifts ΔY/ΔY_up are identical with the normalized angular shifts Θ_x/Θ_up. Figure 3 shows ΔY/ΔY_up as function of incident angle θ and the permittivity of metamaterial ε_m, when w₀ = 40λ. For each ε_m, ΔY/ΔY_up has a positive and a negative peak near Brewster angle, θ_B = atan[ε_m^1/2]. The maximum value of |ΔY/ΔY_up| varies with ε_m. When ε_m = 2.25 (glass), the maximum value of |ΔY/ΔY_up| (|Θ_x/Θ_up|) can reach 0.917. The large angular GH shifts at an air-glass interface have been observed experimentally for both fundamental Gaussian and higher-order LG beams [3,11]. For an air-metamaterial interface with ε_m = 0.1, the maximum value of |ΔY/ΔY_up| is up to 0.978. It increases further to 0.994 for ε_m = 0.01. Therefore, an ENZ metamaterial is better than a glass in the realization of upper-limited beam shifts. Moreover, the permittivity of metamaterial can be flexibly modulated via all-optical method [16], which can be used to control the spatial IF and angular GH shifts.

Fig. 3 The changes of normalized spatial shifts ΔY/ΔY_up as a function of incident angle θ and ε_m.

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Most metamaterials are absorptive medium in practical [15]. The lossy metamaterial will reduce the beam shifts significantly [14]. The imaginary part of permittivity of metamaterial ε_m indicates loss. The normalized spatial IF shifts ΔY/ΔY_up as function of incident angle θ and Im[ε_m] are shown in Fig. 4 for Re[ε_m] = 0.1. The absolute value ΔY/ΔY_up is larger than 0.9 in the slightly loss condition, Im[ε_m] ≤0.004. For a larger loss with Im[ε_m] = 0.01, |ΔY/ΔY_up| decreases to 0.5. When Im[ε_m] increases further, |ΔY/ΔY_up| decreases gradually, being negligible eventually. To reduce the influence of the loss on the beam shift, low-loss ENZ metamaterial should be chosen [24]. One finds from Fig. 3 and 4 that, the spatial IF and angular GH shifts of the reflected beam depend strongly on the permittivity of metamaterial. Thus, the permittivity of a material can be possibly measured based on beam shifts. Optical beam shifts have already been used for the identification of graphene layer [25] and the measurement of concentrations of glucose and fructose [6].

Fig. 4 The changes of normalized spatial shifts ΔY /ΔY_up as a function of the incident angle θ and the image part of permittivity of the metamaterial Im[ε_m] when Re[ε_m] = 0.1.

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The vortex charge of the incident beam will affect the centroid shifts of the reflected beam from the air-metamaterial interface. As shown in Fig. 5, both the spatial IF |ΔY| and angular GH |Θ_x| shifts increase with the vortex charge ℓ. For incident beam with positive and negative vortex charges ± ℓ, the angular GH shifts are identical, while the spatial IF shifts are identical in magnitude but opposite in sign. When ℓ = 0, the spatial IF shift vanishes, however, the angular GH shift does not, since it contains both the vortex-induced shift and shift originated from Gaussian envelop. The inset in Fig. 5 shows the normalized spatial ΔY/ΔY_up and angular Θ_x/Θ_up shifts changing with ℓ. For ℓ = −10:1:10, |ΔY| and |Θ_x| are larger than 0.867 of their upper limits.

Fig. 5 The spatial IF ΔY and angular GH Θ_x shifts changing with the vortex charge ℓ. The inset shows the normalized spatial ΔY/ΔY_up and angular Θ_x/Θ_up shifts.

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After reflected by an interface, a LG beam will propagate forward with its centroid shift changing with the propagation distance z_r. The centroid shift depends on both the spatial and angular shifts. However, the spatial shifts have attached much more attention than the angular shift [26–28], since the centroid shifts contributed from angular shifts can be neglected near interface (z_r<<z₀), and the angular shifts are generally small (Θ_x_,_y<<Θ_up) [1]. At the air-metamaterial interface, the angular GH shifts can approach closely to their upper-limits for both foundational Gaussian and high-order LG incident beams. As will be shown below, these upper-limited angular GH shifts play important role in the centroid shifts of propagated beams.

According to Eqs. (2-5), the centroid shifts along x_r (δX) and y_r (δY) axes are determined by the angular GH and spatial IF shifts, respectively. Thus, δY is independent of z_r, while δX increase linearly with z_r, as shown in Fig. 6(a). At z_r = 0 plane, the reflected beam only shifts along y_r direction, which is clearly seen by the intensity patterns in Fig. 6(c). With the increase of z_r, the intensity pattern of the reflected beam rotates anti-clockwise (sees Fig. 6(c-e)). When z_r = 0.75z₀, the reflected beam undergoes equal shifts in x_r and y_r directions. When z_r = 5z₀, δX is much larger than δY, then reflected beam mainly shifts along x_r axis.

Fig. 6 (a) The comparisons of the centroid shifts along x_r (δX) and y_r (δY) axes of the reflected LG beam with the spot size D/2 of standard LG beam for ℓ = 3. (b) The parameter F as a function of z_r/z₀ for ℓ = 0, 1, 3, 5, respectively. Intensity patterns at z_r = 0 (c,f), 0.75z₀ (d,g), 5z₀ (e,h) planes for ℓ = 3 (c-e) and 1 (f-h), where green dotted lines represent the x_r axis, and the white dotted lines denote the rotation the intensity patterns.

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From Fig. 6(a) one sees that the spot size of the standard LG beam D/2 increases gradually with z_r. In the region of z_r>3z₀, D/2 is proportional nearly linearly to z_r. The parameter F compares the total beam shifts δ with the spot size of the standard LG beam D/2. For each ℓ, F increase gradually with z_r, and will tend to an asymptotic value eventually. Parameter F cannot exceed 0.5 owing to the upper limit of the beam shift.

Parameter F is related to the deformation of the intensity pattern of the reflected beam. When F = 0, the intensity pattern keeps its initial shape. With the increase of F, the intensity pattern becomes more and more asymmetric, as shown in Fig. 6(f-h). It is in meniscus shape when F≈0.5. Parameter F is significant in the measurement of beam shifts, since the spot size and the beam shifts can be amplified by an optical system with their ratio F unchanging.

It is worth to note that the values of F at z_r = 0 and ∞ are determined by the spatial and angular shifts, respectively. The values of F at z_r = ∞ are larger than those at z_r = 0, as shown in Fig. 7(a) and (b). This indicates that the angular GH shifts can lead to larger deformation in the intensity pattern. In the case of θ = 15°, F increases gradually with |ℓ| for z_r = 0 and ∞. For −10 ≤ ℓ≤10, F is smaller than 0.14. However, in the case of θ = 18.14°, F of z_r = ∞ is up to 0.4979 at ℓ = ± 1, while the maximum value of F at z_r = 0 is only 0.3671, obtained at ℓ = ± 5. Therefore, the maximum values of F for z_r = 0 and ∞ are obtained at different ℓ, since the upper limits of the spatial IF shift and angular GH shift have different dependence on ℓ.

Fig. 7 Dependence of parameter F on the vortex charge ℓ for z_r = 0 (red dots) and ∞ (blue dots), when the incident angles θ = 15°(a) and 18.14° (b), respectively.

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4. Conclusions

We have demonstrated the upper limits of the angular GH and spatial IF shifts of vortex beams reflected by an air-metamaterial interface. These upper limits can be approached closely to near Brewster incidence. For a horizontal incident polarization, the beam shifts δ at z_r plane are obtained from the angular GH and spatial IF shifts. δ varies with the propagation distance z_r and the vortex charge ℓ. A dimensionless parameter F is introduced to compare beam shifts. At z_r = ∞ plane, F can approach to 0.5. These results are useful in precision metrology based on beam shifts [6,7,25].

Funding

National Natural Science Foundation of China (61705086, 61675092, 61505069); Natural Science Foundation of Guangdong Province (2017A030313375, 2016A030313079, 2016A030310098).

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Upper-limited angular Goos-Hänchen shifts of Laguerre-Gaussian beams

Abstract

1. Introduction

2. Theory and Model

3. Result and discussion

4. Conclusions

Funding

References and links

Cited By

Figures (7)

Equations (5)

Optics Express