[**COUNTERPOSITION AND NEGATIVE REFRACTION DUE TO UNIFORM MOTION** ]{} 0.2cm

[**Tom G. Mackay$^1$ and Akhlesh Lakhtakia$^2$**]{} 0.2cm

[$^1$ School of Mathematics\
University of Edinburgh\
Edinburgh EH9 3JZ, United Kingdom]{} 0.4cm

[$^2$ CATMAS — Computational & Theoretical Materials Sciences Group\
Department of Engineering Science & Mechanics\
212 Earth & Engineering Sciences Building\
Pennsylvania State University, University Park, PA 16802–6812]{} 0.4cm

[**ABSTRACT:**]{} Refraction of obliquely incident plane waves due to the interface of a vacuous half–space and a half–space occupied by a simply moving, nondissipative, isotropic dielectric–magnetic medium is considered, when the medium’s velocity lies parallel to the interface and in the plane of incidence. Counterposition of the refracted wavevector and time–averaged Poynting vector occurs when the medium’s velocity is sufficiently large in magnitude and makes an obtuse angle with the incident wavevector. The counterposition induced by relative motion occurs whether the refraction is negative or positive when the medium is at rest.

0.2cm [**Keywords:**]{} [*Amphoteric refraction, counterposition, negative refraction, positive refraction*]{} 0.4cm

[**1. INTRODUCTION**]{}

When a plane wave is incident upon the planar interface of two homogeneous mediums, the refracted wavevector and time–averaged Poynting vector can emerge on opposite sides of the normal to the interface. This phenomenon is called *counterposition*. Conditions for the occurrence of counterposition in uniaxial dielectric–magnetic mediums have been established O̧ptik. Under the title of amphoteric refraction, counterposition has been confused with negative refraction Zhang. Refraction — whether negative or positive — concerns the orientation of the refracted wavevector relative to the normal to the interface, as per a law often attributed to Willebrord van Snel van Royen Çhen. The orientation of the refracted time–averaged Poynting vector is irrelevant to whether or not the refraction is negative or positive.

In this Letter, we establish the prospects for counterposition due to the interface of a vacuous half–space and a half–space occupied by a simply moving, nondissipative, isotropic dielectric–magnetic medium, when the medium’s velocity lies parallel to the interface and in the plane of incidence. In particular, we show that counterposition and negative refraction are indeed distinct. For background details of planewave propagation in moving mediums, the reader is referred to standard works Çhen,Pappas,ChUn.

[**2. ANALYSIS**]{}

Suppose that a plane wave is launched with wavevector $\mbox{\boldmath$k_i$} = k_i \mbox{\boldmath$\hat{k}_i$}$ from vacuum towards a half–space occupied by an isotropic, nondissipative, dielectric–magnetic medium. Let this medium move at constant velocity $\mbox{\boldmath$v$} = v \mbox{\boldmath$\hat{v}$}
$, parallel to the interface and in the plane of incidence. With respect to an inertial frame of reference that moves with the same velocity $\mbox{\boldmath$v$}$ with respect to the laboratory frame of reference wherein $\mbox{\boldmath$k_i$}$ is specified, the refracting medium is characterized by relative permittivity $\eps_r  $ and relative permeability $\mu_r $. The condition $\eps_r \mu_r \ge 1$ is assumed in order to exclude the possibility of evanescent plane waves.

The angle $\phi_t$ between the refracted wavevector $\mbox{\boldmath$k_t$} = k_t
\mbox{\boldmath$\hat{k}_t$}$, as observed from the laboratory frame of reference, and the unit vector $\mbox{\boldmath$\hat{q}$}$ normal to the interface is related to the angle of incidence $$\phi_i = \cos^{-1} \le \mbox{\boldmath$\hat{k}$}_i
\.\mbox{\boldmath$\hat{q}$}
   \ri$$ by Çhen $$\phi_t = \sin^{-1}  \le \frac{\ko \sin
\phi_i}{k_t} \ri \,,$$ where $$k_t =
\left\{
\begin{array}{lcr}
  \ko \lec 1 + \xi \les 1 -
\beta \le \mbox{\boldmath$\hat{k}_i$} \. \mbox{\boldmath$\hat{v}$}
\ri \ris^2 \ric^{1/2} & \mbox{for} & \eps_r, \mu_r > 0
\\ && \\
  -\ko \lec 1 + \xi \les 1 -
\beta \le \mbox{\boldmath$\hat{k}_i$} \. \mbox{\boldmath$\hat{v}$}
\ri \ris^2 \ric^{1/2} & \mbox{for} & \eps_r, \mu_r < 0
\end{array}\right.$$ is the wavenumber of the refracted wave, $\ko$ is the wavenumber in vacuum, $$\xi =
\frac{\eps_r \mu_r - 1}{1 - \beta^2}\,,$$ and $\beta = v \sqrt{\epso
\muo}$ with $\epso$ and $\muo$ being the permittivity and permeability of vacuum.

Let us consider case where $\eps_r $ and $\mu_r $ are positive–valued. Then, $0 < \phi_t < \pi/2$ for all $\phi_i \in
(0, \pi/2)$. That is, the refraction is always positive $\forall\,\beta \in (-1,1)$, as is illustrated schematically in Fig. \[fig1\]. Plots of $\phi_t$ against $\beta \in (-1,1)$ for three values of $\phi_i$ are provided in Fig. \[fig2\].

The time–averaged Poynting vector of the refracted plane wave is given by Çhen $$\mbox{\boldmath$P$}_t = P_t \, \mbox{\boldmath$\hat{P}$}_t = \le
\frac{1}{\mu_r} \,| C_1 |^2 +  \eps_r  | C_2 |^2 \ri \le
\mbox{\boldmath$k_t$} \times \mbox{\boldmath$\hat{v}$} \ri^2 \les
\mbox{\boldmath$k_t$} + \xi \beta  \le \ko - \beta
\mbox{\boldmath$k_t$} \. \mbox{\boldmath$\hat{v}$} \ri
\mbox{\boldmath$\hat{v}$} \ris,$$ where $C_{1}$ and $C_2$ are constants. The angle $$\phi_P = \tan^{-1} \le \frac{\mbox{\boldmath$\hat{P}$}_t \.\mbox{\boldmath$v$}}{
| v | \, \mbox{\boldmath$\hat{P}$}_t \.\mbox{\boldmath$\hat{q}$}}
   \ri$$ between $\mbox{\boldmath$\hat{q}$}$ and $\mbox{\boldmath$\hat{P}$}_t$ is plotted in Fig. \[fig3\] against $\beta \in (-1,1)$ for three values of $\phi_i$. The orientation of the refracted time–averaged Poynting vector rotates towards the direction of motion as $\beta$ increases from $-1$. Clearly, counterposition arises $\forall\,\beta < \tilde\beta$ where $\tilde\beta$ is some negative number that depends, among other quantities, on $\phi_i$.

Now let us turn to the scenario wherein $\eps_r < 0 $ and $\mu_r <
0$, as schematically represented in Fig. \[fig4\]. As a consequence of both $\eps_r$ and $\mu_r $ being negative–valued, we have $- \pi/2 < \phi_t < 0 $ for all $\phi_i \in (0, \pi/2)$. That is, the refraction is always negative $\forall\,\beta \in (-1,1)$. The plots in Fig. \[fig5\] of $\phi_t$ against $\beta \in (-1,1)$ for three values of $\phi_i$ illustrate that conclusion.

The corresponding orientation angles for the refracted time–averaged Poyting vector are graphed against $\beta \in (-1,1)$ in Fig. \[fig6\]. As is the case for the positively refracting scenario, we see that counterposition arises $\forall\,\beta < \tilde\beta$ where $\tilde\beta$ is negative. However, in contrast to the positively refracting scenario, the refracted time–averaged Poynting vector rotates against the direction of motion as $\beta$ increases.

[**3. CONCLUDING REMARKS**]{}

Thus, counterposition may be induced in an isotropic dielectric–magnetic medium by relative motion at constant velocity — whether the medium is positively or negatively refracting when at rest. Thereby, the distinction between counterposition and negative refraction is further emphasized Zhang,GK. Also, in the scenario considered here, we note that the phase velocity of the refracted plane wave is positive when positive refraction occurs and negative when negative refraction occurs. Hence, negative phase velocity is not induced by relative motion parallel to the medium interface NPV\_STR. In the absence of relative motion (i.e., $\beta = 0$) the refracted wavevector and time–averaged Poynting vector are parallel for $\eps_r, \mu_r > 0$ and anti–parallel for $\eps_r, \mu_r < 0$, and accordingly counterposition cannot occur, as is confirmed by Figs. \[fig2\], \[fig3\], \[fig5\] and \[fig6\].

[**Acknowledgement:**]{} TGM is supported by a *Royal Society of Edinburgh/Scottish Executive Support Research Fellowship*.

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