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Counterposition and negative refraction due to uniform motion

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.


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 [1]. Under the title of amphoteric refraction, counterposition has been confused with negative refraction [2]. 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 [3]. 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 dielectricmagnetic 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 [3,4,5].

ANALYSIS
Suppose that a plane wave is launched with wavevector k i = k iki from vacuum towards a half-space occupied by an isotropic, nondissipative, dielectric-magnetic medium. Let this medium move at constant velocity v = vv, parallel to the interface and in the plane of incidence.
With respect to an inertial frame of reference that moves with the same velocity v with respect to the laboratory frame of reference wherein k i is specified, the refracting medium is characterized by relative permittivity ǫ r and relative permeability µ r . The condition ǫ r µ r ≥ 1 is assumed in order to exclude the possibility of evanescent plane waves.
The angle φ t between the refracted wavevector k t = k tkt , as observed from the laboratory frame of reference, and the unit vectorq normal to the interface is related to the angle of where and β = v √ ǫ 0 µ 0 with ǫ 0 and µ 0 being the permittivity and permeability of vacuum.
The time-averaged Poynting vector of the refracted plane wave is given by [3] where C 1 and C 2 are constants. The angle betweenq andP t is plotted in Fig. 3 against β ∈ (−1, 1) for three values of φ i . The orientation of the refracted time-averaged Poynting vector rotates towards the direction of motion as β increases from −1. Clearly, counterposition arises ∀ β <β whereβ is some negative number that depends, among other quantities, on φ i . Now let us turn to the scenario wherein ǫ r < 0 and µ r < 0, as schematically represented in Fig. 4. As a consequence of both ǫ r and µ r being negative-valued, we have −π/2 < φ t < 0 for all φ i ∈ (0, π/2). That is, the refraction is always negative ∀ β ∈ (−1, 1). The plots in Fig. 5 of φ t against β ∈ (−1, 1) for three values of φ i illustrate that conclusion.
The corresponding orientation angles for the refracted time-averaged Poyting vector are graphed against β ∈ (−1, 1) in Fig. 6. As is the case for the positively refracting scenario, we see that counterposition arises ∀ β <β whereβ is negative. However, in contrast to the positively refracting scenario, the refracted time-averaged Poynting vector rotates against the direction of motion as β increases.

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 [2,6]. 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 [7]. In the absence of relative motion (i.e., β = 0) the refracted wavevector and time-averaged Poynting vector are parallel for ǫ r , µ r > 0 and anti-parallel for ǫ r , µ r < 0, and accordingly counterposition cannot occur, as is confirmed by Figs. 2, 3, 5 and 6.
moving medium Figure 1: A plane wave with wavevector k i is incident from vacuum onto a half-space occupied by a simply moving medium at an angle φ i with respect to the unit vectorq normal to the planar interface. The moving medium is characterized by relative permittivity ǫ r > 0 and relative permeability µ r > 0 in a comoving frame of reference. As observed in the non-comoving (laboratory) frame of reference wherein the incident plane wave is specified, the refracted wavevector k t makes an angle φ t withq.