1. Introduction

In some situations, a reflected beam could experience a lateral shift with respect to the path predicted by the geometric optics. A well-known example is the Goos-Hänchen (GH) effect, which is a kind of lateral shift under the condition of total reflection [1, 2, 3, 4]. There also exist some other lateral shifts, which are quite different from the GH effect, since the magnitude of the reflection coefficient is dependent on the angle of incidence[5]. For instance, Wild et al. [6] found a large yet negative lateral shift near the Brewster angle in a configuration with an absorbing dielectric reflecting medium, which has attracted some attention recently [7, 8]. In particular, Lai et al. [7] analyzed this lateral shift and pointed out its underlying physics to be an abrupt change of the reflection phase. In fact, such a lateral shift occurs around the angle when the real part of the reflection coefficient vanishes. We would like to term this angle as pseudo-Brewster angle, which should be more pertinent than Brewster angle, since an absolutely null reflection cannot exist for a dissipative reflecting medium generally.

In 2001, an experiment accomplished by Shelby et al. [9] realized the hypothesis of double negative medium (DNM) pioneered by Veselago [10]. Since then, the subjects related to DNM have been of great interest [11, 12, 13, 14, 15]. The lateral shift associated with DNM is an important issue, due to its potential applications and rich physics. In some literatures, the GH effect upon reflection from an interface separating DPM (double positive medium) and DNM has also been discussed [16, 17, 18, 19]. In addition, some studies have been devoted to the lateral shifts in a layered structure with a DNM slab [8, 20, 21, 22, 23, 24].

In this work, we explore the lateral shift upon reflection from a semi-infinite DNM with weak absorption. We predict that the lateral shift near the pseudo-Brewster angle could be fairly large, as mentioned in Ref. 7, and both positive and negative lateral shifts are possible. We find that the imaginary parts of the permittivity ε and permeability μ play an important role in determining the lateral shift. A comparison is then given between the cases with DNM and DPM, proving that the lateral shifts are opposite when changing the signs of the real parts of ε and μ for the DNM and DPM. Only TM waves are discussed below, and the results for TE modes can be easily obtained by interchanging ε and μ.

2. Theoretical analysis

Let us consider the case shown in the inset of Fig. 1(a). A wide incident beam propagating in vacuum is reflected from a weakly dissipative DNM, which occupies the region of z > 0. The angle of incidence is denoted as θ. In our discussions, DNM is assumed to be an optically denser medium with respect to vacuum for simplicity. In addition, the incident beam is considered to be well collimated with very narrow angular spectrum so that the angular derivative of reflectivity amplitude could be small enough [6, 7]. Under this assumption, Lai et al. [25] have proved that the Artmann’s formula S = -(λ/2π)(dϕ/dθ) is still in well agreement with the results obtained through the general treatment given by Section 2.4 in Ref. 5. Here S is the lateral shift, λ the wavelength of the incident beam, ϕ the phase of the reflection coefficient r, and θ the angle of incidence, respectively.

Compared with absolutely null reflection at the Brewster angle from a lossless medium, here the reflection coefficient is always nonzero, and only a pseudo-Brewster angle is permitted as mentioned above. The lateral shift at the pseudo-Brewster angle then can be written as [8]

where Re[r] (Im[r]) is the real (imaginary) part of r and θ _pB is the pseudo-Brewster angle. At the interface formed by vacuum and a DNM, the reflection coefficient for TM mode is written as [26]

where ε = ε′ + iε″ and μ = μ′ + iμ″ are the complex permittivity and permeability of the DNM, respectively. For a passive DNM, ε′ and μ′ are simultaneously negative, and ε″,μ″ > 0 is required [27].

In the case of the weak absorption, the pseudo-Brewster angle for TM modes can be approximately determined by

It should be pointed out that expression (3) is tenable under the restriction of ε′/μ′ > 1. So the lateral shift can be presented as

It can be easily found that the direction of the lateral shift depends only on the sign of the denominator of the right side in expression (4).

Now we would like to pay attention to two special cases: (I) DNM without magnetic absorption (i.e., μ″ = 0) and (II) DNM without electric absorption (i.e., ε″ = 0). For Case-I, the sign of the lateral shift depends on ε′[μ′ - 2ε′/(ε′ ² + 1)]. Since ε′ is negative, μ′ > 2ε′/(ε′ ² + 1) and μ′ < 2ε′/(ε′ ² + 1) then give rise to negative and positive shifts, respectively. For Case-II, the lateral shift is always negative, owing to -ε′ ² μ″ < 0.

For more general situations, electric and magnetic absorption should exist simultaneously, thus the lateral shift is certainly from the contribution of combination of the two special cases as stated above. The introduction of nonzero-μ″ in Case-I changes the sign of the lateral shift from positive to negative only. In contrast, for Case-II, the introduction of nonzero-ε″ leads to a negative-to-positive change for the sign of the lateral shift. We should point out that if the denominator in expression (4) is vanishing, it in fact corresponds to absolutely null reflection, i.e., r(θ) = 0, and the lateral shift in this situation cannot be defined by expression (1).

Based on the above discussions, the negative and positive shifts at the pseudo-Brewster angle for a DNM are allowed. It should be noted that ε′,μ′ together with ε″,μ″ determine the lateral shift, including both the magnitudes and signs. Additionally, for a DPM with opposite signs of the real parts of e and μ to those of a DNM, we can easily get r _DPM = $r_{\text{DPM}}^{*}$ and S _DPM = -S _DNM as mentioned in Ref. 17.

As to such a large lateral shift, it is found to be due to an abrupt change of phase across the pseudo-Brewster angle [7]. For a regular reflecting material with μ = 1, because the imaginary part of the reflection coefficient is always positive near the pseudo-Brewster angle and the real part changes from positive to negative across the pseudo-Brewster angle, it can be found from expression (1) that the lateral shift is always negative. For the reflection from an absorbing DNM, however, the imaginary part of the reflection coefficient is possible to be positive or negative that depends on the two parameters ε and μ of DNM, so it leads to the respective positive or negative lateral shift near the pseudo-Brewster angle.

3. Numerical calculations

To illustrate the above analysis and predictions, we make a series of numerical calculations based on the Artmann’s formula. Figure 1 shows the dependence of the lateral shift on the angle of incidence in three different situations for TM waves.

 

Fig. 1. Lateral shift of a TM wave reflected from a weakly absorbing DNM. (a) μ″ = 0 and ε = -1.8 + 0.09i. μ = -1.2, - 1 and -0.7 are shown by the dotted, dashed and dash-dotted curves, respectively. (b) ε = -1.8. μ = -1.2 + 0.09i, -1 + 0.09i and -0.7 + 0.09i are shown by the dotted, dashed and dash-dotted curves, respectively. (c) ε = -1.8 + 0.09i. Dotted, dashed, dash-dotted and solid curves correspond to μ = -1.2 + 0.09i, -0.7 + 0.09i, - 1.2 + 0.01i and -0.7 + 0.01i, respectively. The inset in (a) is the sketch of the structure.

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Figure 1(a), which corresponds to Case-I, shows the lateral shifts near ${\theta }_{\text{pB}}^{\text{TM}}$ with μ″ = 0 (i.e., μ = μ′) and ε = - 1.8 + 0.09i. Three different real values of μ are taken as -1.2, - 1 and -0.7. For the case of μ = - 1, the lateral shift is positive, with a maximum value of 12.44λ at ${\theta }_{\text{pB}}^{\text{TM}}$ = 53.3°. Compared with the result associated with DPM in Ref. 7, a unique difference is that the sign of the lateral shift is changed, just as we have predicted. For the case of μ = - 1.2, the maximum lateral shift occurs at ${\theta }_{\text{pB}}^{\text{TM}}$ = 43.98° and has a positive value of 5.98λ. However, for the case of μ = -0.7, the lateral shift becomes negative, with a value of -9.16λ at ${\theta }_{\text{pB}}^{\text{TM}}$ = 70.08°. These results are in well agreement with the above analysis we have made for Case-I, and the critical value of μ for the sign-changing of the lateral shift is determined by μ ≈ 2ε′/(ε′ ² + 1), which is -0.85 for the case shown in Fig. 1(a). It is noted that since higher order terms of ε″ have been omitted in deducing expression (4), the exact critical value of μ should have a little deviation from -0.85. It is also found that there exists indeed a large lateral shift near the pseudo-Brewster angle, which can be an order of magnitude greater than the wavelength [7].

Figure 1(b) shows the lateral shifts for ε = -1.8 and μ″ = 0.09, while μ′ also takes the three different values of -1.2, -1 and -0.7, respectively. The lateral shifts are always negative, which is also in agreement with our prediction for Case-II.

So far, we deal with the lateral shift effect for the two situations only with electric or magnetic absorption. We now would like to explore the more general cases, in which electric and magnetic absorption is taken into account simultaneously. We display some results in Fig. 1(c), in which ε = -1.8 + 0.09i, while μ takes four different values of -1.2 + 0.09i, -1.2 + 0.01i, -0.7 + 0.09i and -0.7 + 0.01i. The results indicate that both negative and positive lateral shifts are possible. For instance, for μ′ = - 1.2, the lateral shift is negative if μ″ = 0.09, but becomes positive when μ″ = 0.01, which agrees with the above analytical conclusion.

To give a comprehensive understanding and confirm the above analysis, we simulate the lateral shift near the pseudo-Brewster angle on reflection from an absorbing medium using the finite-difference time-domain (FDTD) method. In the simulations, we consider an incident beam with a spatial Gaussian profile as Ref. 22 and the half-width of the beam is about 7.5λ. The results are shown in Fig. 2, in which Fig. 2(a) is for a DPM, while Figs. 2(b) and 2(c) are for two different DNMs. We find that the lateral shifts for the three cases have the same order as the width of the incident beam, thus the term “large shift” can be utilized. From Figs. 2(a) and 2(b), we can see the lateral shifts for DPM and DNM exhibit of opposite directions, while from Figs. 2(b) and 2(c), it is evident that both negative and positive shifts are observable, and the absorption plays an important role in determining the sign and magnitude of the lateral shift. It is indubitable that the FDTD simulations manifest our theoretical predictions.

 

Fig. 2. FDTD simulations of the lateral shifts near the pseudo-Brewster angle for a TM wave reflected from an absorbing medium. (a) ε = 1.8 + 0.09i and μ = 1.2. (b) ε = -1.8 + 0.09i and μ = -1.2. (c) ε = -1.8 + 0.09i and μ = -1.2 + 0.09i.

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Finally, we would like to compare our numerical results with the results given by the momentum method [5, 22], which is commonly used to calculate the lateral shifts for cases of the reflection coefficient varying with the angle of incidence. We only take the case of ε = -1.8 + 0.09i and μ = -1.2, as example. The results are shown in Fig. 3(a), and the FDTD results are added for seven different angles of incidence (40°, 41.5°, 43°, 44°, 45°, 46.5° and 48°), for the convenience of comparison. We also take a spatial-Gaussian-shaped beam with a width of 15λ in consideration. Figure 3(b) displays the spatial profiles of the incident and reflected beams at an angle of incidence of 44°. Figure 3 distinctly indicates that our numerical results are indeed reliable.

4. Conclusion

In conclusion, we study the lateral shift upon reflection from a semi-infinite DNM with weak absorption. We observe a large shift near the pseudo-Brewster angle due to an abrupt change of reflection phase as mentioned in Ref. 7. We show that both negative and positive lateral shifts are possible, since DNMs possess simultaneously electric and magnetic responses. In particular, it should be noted that the absorption, even though very weak, has great impact on the lateral shift. An analytical expression regarding the lateral shift at the pseudo-Brewster angle is presented, which offers a distinct explanation of the obtained numerical results and allows us to determine the critical transition values of ε and μ when the lateral shift changes the sign. As evidences, the results of the FDTD simulations are provided. We also compare our numerical results with the calculations by means of the momentum method, implying that our analysis is credible.

 

Fig. 3. (a) A comparison for the results given by the Artmann’s formula, the momentum method and the FDTD method when ε = -1.8 + 0.09i and μ = -1.2. The solid and dotted curves correspond to the results of the Artmann’s formula and the momentum method, respectively. Seven solid circles represent the FDTD results for seven different angles of incidence (40°, 41.5°, 43°, 44°, 45°, 46.5° and 48°). (b) Spatial profiles of the incident (dotted) and reflected (solid) beams at a 44° angle of incidence, and the amplitude of the reflected beam is magnified by a factor of 5000.

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