Radiation pressure and photon momentum in negative-index media

Radiation pressure and photon momentum in negative-index media are no different than their counterparts in ordinary (positive-index) materials. This is because the parameters responsible for these properties are the admittance√ e /µ and the group refractive index ng of the material (both positive entities), and not the phase refractive index n =√ µe, which is negative in negative-index media. One approach to investigating the exchange of momentum between electromagnetic waves and material media is via the Doppler shift phenomenon. In this paper we use the Doppler shift to arrive at an expression for the radiation pressure on a mirror submerged in a negative-index medium. In preparation for the analysis, we investigate the phenomenon of Doppler shift in various settings, and show the conditions under which a so-called "inverse" Doppler shift could occur. We also argue that a recent observation of the inverse Doppler shift upon reflection from a negative-index medium cannot be correct, because it violates the conservation laws. 1. Introduction. A negative-index medium (NIM) is a natural or artificial material whose dispersion relation, ω (k), exhibits a negative slope, dω/dk, over a certain frequency interval. Here ω is the angular frequency and k the wave-number of a plane electromagnetic wave propagating within the medium. Since the phase refractive index is given by n (ω )= ck(ω)/ω, whereas the group index is defined as ng(ω) = c/(dω (k)/dk), if we follow tradition and choose a positive sign for the group index, then the phase index will turn out to be negative. The negative-index could come about as a result of the nano-structure of a meta-material (e.g., photonic crystal band structure), or it could be due to the material's permittivity e(ω) and permeability µ(ω) both being real-valued and negative over some frequency range. In the latter case, the phase refractive index n =√ µe will be negative, whereas ng as well as the admittance η =√ e /µ and impedance Z =√ µ /e of the medium will turn out to be positive. It has been suggested that the Doppler shift associated with negative-index media could be the reverse of that observed in ordinary materials (1). Also, anomalous behavior for radiation pressure and photon momentum inside negative-index media has been predicted (2). In this paper we examine the Doppler shift from multiple perspectives, using special relativity as well as the conservation laws of energy and momentum to derive expressions for the Doppler shift under various circumstances. We show that negative-index media, by and large, behave similarly to their positive-index counter- parts, and that a recent claim of experimental observation of the "inverse" Doppler shift upon reflection from a NIM is patently absurd (3). An exception to the above rule occurs in the case of reflection from a mirror submerged in a negative-index dielectric, where, as Veselago originally suggested (1), an inverse Doppler shift is indeed plausible. Photon momentum and radiation pressure are intimately tied to the magnitude and sign of the Doppler shift. We examine the connection between these two phenomena in some detail, and derive expressions for the radiation pressure that are fully consistent with those obtained using the generalized Lorentz force in conjunction with Maxwell's macroscopic equations (4,5). It will be