1. Introduction

Since the left-handed meta-material (LHM) was experimentally verified in 2001 [1], more and more interests in this field were aroused. The lateral displacement of a Gaussian beam transmitted through an isotropic LHM slab has been studied theoretically and experimentally [2,3]. Different from the isotropic LHM or right-handed material (RHM), which are with circular dispersion relation shapes, and the anisotropic RHM, which is with elliptical dispersion relation shape, a one-dimensional (1D) LHM with only one negative principal element in permittivity tensor and one negative principal element in permeability tensor in the orthogonal direction respecting to the negative permittivity element, has a hyperbolic shaped dispersion relation curve [4–6], which can be fabricated using S-shaped or Ω-shaped split ring resonators [7,8]. In this paper, the lateral displacement of a Gaussian beam obliquely incident into an 1D LHM slab is studied. The analytical solutions for the electric and the magnetic fields inside and outside the slab are provided, showing that in the negative pass band of the 1D LHM, the lateral displacement can either shifts to the positive or the negative direction, which is different from isotropic LHM case, whose shift can only be negative [2,3]. Further more, experimental result is also provided to verify the analytical solutions by examining the distribution of the time-averaged power flow at the second interface of the slab where the transmitted Gaussian beam emerges. We see that, the experimental result is in accordance with the analytical result.

2. Electromagnetic fields inside and outside the slab

 

Fig. 1. Illustration of the lateral displacement of an obliquely incident Gaussian beam.

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Consider an 1D LHM slab located in a x-y-z coordinate system depicted in Fig. 1, whose first and second boundaries are located at z=0 and z=z ₀, respectively. Let a Gaussian beam be incident obliquely onto the first boundary at Point “A”, and line “AB” be the normal of the two boundaries. Assume the 1D LHM is with a permittivity tensor diag[ε ₁, ε ₂, ε ₃] and a permeability tensor diag[μ ₁, μ ₂, μ ₃] in an e₁-e₂-e₃ coordinate system, where μ_i,ε_i are along with the principle axes e_i(i=1,2,3), respectively, and the e₁-e₂-e₃, coordinate system can be obtained by anti-clockwise rotating the x-y-z system θ angles around y-axis. The permittivity and permeability tensors described in the x-y-z coordinates are then as follows:

in which only ε ₂ and μ ₁ have negative real parts. For a wave linearly E-polarized along y-axis, the dispersion relation of the 1D LHM is

where α = μ ₁ sin² θ + μ ₃ cos² θ, β = -2sin θ cos θ(μ ₁ - μ ₃)and γ = μ ₁ cos² θ + μ ₃ sin² θ.

Assume a Gaussian beam

is incident from region 1 into the slab at an incident angle φ, where $\psi \left(k_x\right)=\frac{g}{2\sqrt{\pi }}\exp \left\{-\left[\frac{g^2{\left(k_x-k_{\mathrm{ixc}}\right)}^2}{4}\right]\right\}$ is the Gaussian spectrum. The total electric and magnetic fields are expressed as follows:

In region 1,

In region 2,

in which k_z1, k_z2 and k_x satisfy the dispersion relation of Eq. (3); a, b, c and d are the elements of the inversed tensor of μ̿, and a = α/μ ₁ μ ₃, b = c = -β/2μ ₁ μ ₃, d = γ/μ ₁ μ ₃ ; ξ, ζ are the transmission or reflection coefficients of the beams in the 1D LHM slab.

In region 3,

The coefficients R, ξ, ζ, and T can be given by matching the boundary conditions for tangential electric and magnetic fields at z=0 and z=z ₀, respectively:

where p ₁=(-ak _z1 + bk_x + k_tz / μ ₀)exp(ik _z1 z ₀) , p ₂ = (-ak _z2 + bk_x + k_tz / μ ₀)exp(ik _z2 z ₀) , P ₃ = -ak _z1 + bk_x - k_iz / μ ₀ , and P ₄ = -ak _z2 + bk_x - k_iz / μ ₀.

Finally the time-averaged normal and tangential power flow (Poynting vector) can be calculated by

where the subscript n = 1, 2, 3 denotes different regions.

3. Lateral displacement of the Gaussian beam

 

Fig. 2. K-surfaces for the illustration of the lateral shifts for a Gaussian beam incidence.

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Here we define: if the directions of the tangential power flow in and out of the slab are the same, the lateral shift of the beam is positive, and vise versa. In Fig. 1, a positive shift means the outgoing beam at the second boundary locates in the upper side to the normal line “AB”, and a negative shift means the opposite situation. Notice that for an RHM slab, the shift must be positive, and for an isotropic LHM, the shift must be negative. From Eq. (3), when the real part of α changes from negative to positive with frequency, the solution for k_z changes its sign too, which leads to the change of the direction of the tangential power flow in the 1D LHM, concluded from Eq. (6.c) and Eq. (13). The above discussion could be illustrated by Fig. 2, in which the solid hyperbola represents the k-surface of the 1D LHM when the real part of α is negative, and the dashed one represents the case when the real part of α is positive. Knowing that the Poynting vector is always normal to the k-surface, from Fig. 2, we can see that the solid hyperbola will yield a negative lateral shift, i.e., S̅₁ has a different direction of the tangential component compared to the incident beam’s, while the dashed one will yield a positive shift, or S̅₂ has the same direction of the tangential component as the incident beam’s. When the Lorentz model is applied to μ ₁ and ε ₂, say

where ω_ij = 2πf_ij, i = ε,μ and j = o,p. Assume f_εo = 0GHz , f_εp = 12.25GHz, f_μo = 10.6GHz, f_μp = 12.2GHz , γ_ε = γ_μ = 0.001, ε ₁ = ε ₃ = ε ₀, and μ ₂ = μ ₃ = μ ₀. For θ = 18.4°, we see that the real part of α goes to zero when f = 10.78 GHz, indicating that in the first region of the negative pass band, say 10.6 GHz < f < 10.78 GHz (Re{α} < 0), the direction of the tangential power flow in the 1D LHM slab is toward negative direction; and in the second region, say 10.78 GHz < f < 12.2 GHz (Re{α} > 0), the direction of the tangential power flow is positive. For θ = 45°, the separating frequency changes to 11.43GHz, and the negative and positive shift regions change to 10.6 GHz < f < 11.43 GHz and 11.43 GHz < f < 12.2 GHz, respectively. All above can be illustrated in a 2D form by calculating the analytical result of the normal power flow demonstrated by Eq. (12) at the second interface of the slab when the Gaussian beam is incident from air onto the 1D LHM slab at point “A” with an incident angle φ = 45° and g = 0.2 (the Gaussian spectrum parameter defined in Eq. (4)), as shown in Fig. 1. The results are presented in Fig. 3, where the vertical axes represent the location of the transmitted beam, and the originals 0 correspond to the point “B” in Fig. 1. From Fig. 3, we see that in the lower frequency regions, from 10.6 GHz to 10.78 GHz for θ = 18.4° and from 10.6 GHz to 11.43 GHz for θ = 45°, the beam has negative shifts, while in the higher frequency regions, from 10.78 GHz to 12.2 GHz for θ = 18.4° and from 11.43 GHz to 12.2 GHz for θ = 45°, positive shifts appear.

 

Fig. 3. The normalized power flow calculated at the second interface of the 1D LHM slab. (a)θ = 18.4°, (b)θ = 45°

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4. Experimental verification

To verify the above analytical conclusion, the S-shaped 1D LHM sample with a negative pass band occupied from 10 to 12 GHz is chosen to construct the slab with a rotated angle θ = 18.4° [7]. The schematic of the experimental setup is depicted in Fig. 4, in which the point “A” and “B” correspond to the same points in Fig. 1. For the experimental convenience to detect the transmission power, the detector is moved along axis- X'. Although axis- X' is not parallel to the second interface of the 1D LHM slab, the original point 0 still corresponds to the point “B”, so that we can use the axis- X' to define the positive shift region and negative shift region.

 

Fig. 4. Schematic of the experimental setup for detecting the lateral shift of the incident Gaussian beam.

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The experimental result is shown in Fig. 5. We see that in the negative pass band of the S-shaped 1D LHM, with a separating frequency around 11.1 GHz, there indeed exist two regions, each corresponding to a different lateral shift: one is to the negative direction in the lower frequency region, and the other is to the positive direction in the higher frequency region. Such experimental result clearly shows the phenomenon predicted by previous analysis. However, there also exists difference. The separated beams in fig. 5 seem to have more smearing and therefore not appear exactly similar to fig. 3(a). We conclude this due to the inexactness of the Lorentz model to the real dispersion property of the S-shaped 1D LHM sample. Although artificial LHMs fabricated by dense metallic sub-wavelength resonators are considered to have dispersion relations in accordance with Lorentz model, the actual dispersion curves are with noises caused by some complex issues, such as sub-resonances.

 

Fig. 5. Normalized transmission power detected at the second interface of the S-shaped 1DLHM slab.

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5. Conclusion

Since the 1D LHM studied in this paper has a unique hyperbola dispersion relation, analytically we conclude that in the negative pass band, a rotated 1D LHM slab will bring both negative and positive lateral shifts for an obliquely incident Gaussian beam. This is also verified by the experimental result demonstrated above. The critical frequency, who separates the negative pass band into two parts, is decided by the rotating angle θ and the dispersion property of the negative permeability element μ ₁, if the incident beam is E-polarized. According to the duality, we can also draw a similar conclusion for a H-polarized beam. Such structure can be used as a spatial beam splitter or diplexer, by giving an appropriate dispersion property and rotated angle of the material.