The angular Goos-Hänchen shift of vortex beam is investigated theoretically when a Laguerre-Gaussian (LG) beam is reflected by an air-metamaterial interface. The upper limit of the angular GH shift is found to be half of the divergence angle of the incident beam, i.e., |Θup| = (|ℓ| + 1)1/2/k0w0, with ℓ, k0, and w0 being the vortex charge, wavenumber in vacuum, and beam waist, respectively. Interestingly, the upper limited angular GH shift is accompanied by the upper-limited spatial IF shift. A parameter F is introduced to compare the total beam shift with the beam size. F varies with the vortex charge ℓ and the propagation distance zr. The values of F at zr = ∞ plane can approach 0.5, which are always larger than those at zr = 0 plane. These findings provide a deeper insight into optical beam shifts, and they may have potential application in precision metrology.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
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 . 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 . The GH effects originate from the dispersion of the Fresnel reflection coefficients , while the IF shift is governed by the conservation law of the total angular momentum . 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 . In 2006, the vortex-induced spatial IF shift was detected experimentally for the reflected beam . 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 . According to their results, the four shifts increase with vortex charge, thus large beam shifts can be obtained . Recently, Jiang and associates gave the upper limits of the spin-dependent shifts (the spatial IF shifts) of Laguerre-Gaussian (LG) beams . 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 . By coating a graphene monolayer on the ENZ metamaterial, Fan et. al. realized an electrically tunable GH shift in the terahertz regime . 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 (xi, yi, zi) and (xr, yr, zr), respectively. The angular spectra of LG beam is , kix and kiy denote the wave vector components in the xi and yi 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 . For a horizontal incident polarization, the angular spectrum of the reflected beam at zr = 0 plane can be written as:22]. The GH and IF shifts can therefore be expressed as δX = ΔX + zrΘx, δY = ΔY + zrΘy, where and , is the energy of the reflected beam [1,22]. After some straightforward calculation, we obtainEqs. (2-5) one finds that, the angular IF Θy vanishes. And the spatial GH ΔX is ℓ-independent, since it originates from Gaussian envelop . 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 Wr is negligible, Θx and ΔY vary linearly with the vortex charge ℓ. Otherwise, they change nonlinearly with ℓ. Now we consider the case when with δp being the argument of rp, i.e, the non-central incident waves contribute much more to the xr components of the reflected beam than to the yr component. ΔY varies with the incident angle θ, and takes peak values among all θ, ± ℓw0 /2(|ℓ| + 1)1/2 when . Therefore, the upper limit of the spatial IF shift is ΔYup = ℓw0/2(|ℓ| + 1)1/2 . Since Θx = (|ℓ| + 1)ΔY/ℓz0, the angular GH shift is limited by Θup = (|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.
The reflected beam will undergo shifts in both xr and yr 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 zr. The upper limit the total beam shift is δ = 0.5w0(|ℓ| + 1)1/2 [|ℓ|2/(|ℓ| + 1)2 + (zr/z0)2]1/2. Due to the diffraction effect, the spot size of the beam increases with zr. For a standard LG beam, the spot size can be described by the second radial moment of the intensity: ,where ϕ is electric field profile of the standard LG beam . It is D(zr) = w0(|ℓ| + 1)1/2[1 + (zr/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(zr) 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 zr.
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 εm = 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 Θy vanishes for any incident angle. The spatial GH shift takes large values near the critical angle for total reflection, θc = 18.435°. The curves of the spatial IF shift ΔY and the angular GH shift Θx changing with θ are identical, although their values are different. ΔY and Θx change signs when the incident angle θ crosses the Brewster angle, θB = 17.548°. They take large values near θB. The maximum spatial IF |ΔY| and angular GH |Θx| 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>θc, both ΔY and Θx vanish owing to the independence of the absolute value of rp on θ. For θ near but below θc, |ΔY| and |Θx| decrease rapidly since θB and θc are close to each other.
For a given beam w0, the normalized spatial shifts ΔY/ΔYup are identical with the normalized angular shifts Θx/Θup. Figure 3 shows ΔY/ΔYup as function of incident angle θ and the permittivity of metamaterial εm, when w0 = 40λ. For each εm, ΔY/ΔYup has a positive and a negative peak near Brewster angle, θB = atan[εm1/2]. The maximum value of |ΔY/ΔYup| varies with εm. When εm = 2.25 (glass), the maximum value of |ΔY/ΔYup| (|Θx/Θup|) 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 εm = 0.1, the maximum value of |ΔY/ΔYup| is up to 0.978. It increases further to 0.994 for εm = 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 , which can be used to control the spatial IF and angular GH shifts.
Most metamaterials are absorptive medium in practical . The lossy metamaterial will reduce the beam shifts significantly . The imaginary part of permittivity of metamaterial εm indicates loss. The normalized spatial IF shifts ΔY/ΔYup as function of incident angle θ and Im[εm] are shown in Fig. 4 for Re[εm] = 0.1. The absolute value ΔY/ΔYup is larger than 0.9 in the slightly loss condition, Im[εm] ≤0.004. For a larger loss with Im[εm] = 0.01, |ΔY/ΔYup| decreases to 0.5. When Im[εm] increases further, |ΔY/ΔYup| decreases gradually, being negligible eventually. To reduce the influence of the loss on the beam shift, low-loss ENZ metamaterial should be chosen . 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  and the measurement of concentrations of glucose and fructose .
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 |Θx| 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/ΔYup and angular Θx/Θup shifts changing with ℓ. For ℓ = −10:1:10, |ΔY| and |Θx| are larger than 0.867 of their upper limits.
After reflected by an interface, a LG beam will propagate forward with its centroid shift changing with the propagation distance zr. 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 (zr<<z0), and the angular shifts are generally small (Θx,y<<Θup) . 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 xr (δX) and yr (δY) axes are determined by the angular GH and spatial IF shifts, respectively. Thus, δY is independent of zr, while δX increase linearly with zr, as shown in Fig. 6(a). At zr = 0 plane, the reflected beam only shifts along yr direction, which is clearly seen by the intensity patterns in Fig. 6(c). With the increase of zr, the intensity pattern of the reflected beam rotates anti-clockwise (sees Fig. 6(c-e)). When zr = 0.75z0, the reflected beam undergoes equal shifts in xr and yr directions. When zr = 5z0, δX is much larger than δY, then reflected beam mainly shifts along xr axis.
From Fig. 6(a) one sees that the spot size of the standard LG beam D/2 increases gradually with zr. In the region of zr>3z0, D/2 is proportional nearly linearly to zr. 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 zr, 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 zr = 0 and ∞ are determined by the spatial and angular shifts, respectively. The values of F at zr = ∞ are larger than those at zr = 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 zr = 0 and ∞. For −10 ≤ ℓ≤10, F is smaller than 0.14. However, in the case of θ = 18.14°, F of zr = ∞ is up to 0.4979 at ℓ = ± 1, while the maximum value of F at zr = 0 is only 0.3671, obtained at ℓ = ± 5. Therefore, the maximum values of F for zr = 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.
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 zr plane are obtained from the angular GH and spatial IF shifts. δ varies with the propagation distance zr and the vortex charge ℓ. A dimensionless parameter F is introduced to compare beam shifts. At zr = ∞ plane, F can approach to 0.5. These results are useful in precision metrology based on beam shifts [6,7,25].
National Natural Science Foundation of China (61705086, 61675092, 61505069); Natural Science Foundation of Guangdong Province (2017A030313375, 2016A030313079, 2016A030310098).
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