Abstract
Our point of departure is a hypothetical substance for which, in a certain frequency range, both the dielectric constant (DC) and the magnetic permeability (MP) become negative. In this range the group velocity and the phase velocity of electromagnetic waves have opposite direction, and the refractive index becomes negative. Such negative refraction (NR) would be of both fundamental and practical interest, especially if it could be obtained at optical (infrared and higher) frequencies. New optical imaging elements would be possible. For example, a simple plate with a refractive index n = –1 would provide a perfect 1:1 imaging. That would be of interest for microscopy, photolithography, integrated optics, and other applications.
A negative DC at optical frequencies is in principle achievable in a weakly damped electron plasma. But because of the absence of magnetic monopoles, a negative MP would require a suitable magnetic dipole interactions. That is much harder to achieve, and the necessary parameters are inaccessible at optical frequencies, where NR would be of greatest interest.
At microwave frequencies, usable magnetic dipole interactions can be achieved via metallic resonators. A periodic lattice of suitably-designed metallic resonators, with a sufficiently small lattice constant, can simulate a uniform medium with negative n, and thereby make possible NR. But if such structures are scaled down in size for optical wavelengths, they have hopelessly large losses.
An alternative route to NR at optical frequencies, which does not involve magnetic interactions at all, draws on the properties of purely dielectric periodic lattice structures (so-called Photonic Crystals), inside which only the DC changes periodically. Wave propagation in periodic media invariably exhibits band structures, with allowed and forbidden bands of propagation, regardless of the nature of the waves This is a familiar phenomenon for the propagation of electron waves through the periodic potential inside a crystal, where it forms the basis of the physics of the electrical transport properties of crystals. It is equally true for the propagation electromagnetic waves through a periodically varying DC.
To discuss the refraction properties of photonic crystals, it becomes necessary to introduce a new distinction between two kinds of refractive index: a longitudinal and a transverse index. A negative transverse index leads to NR, even when the longitudinal index remains positive. The allowed photonic bands may contain regions inside which the transverse index becomes negative, permitting negative refraction. This approach does not suffer from the problems with achieving a negative MP at optical frequency, and hence is the appropriate way towards achieving NR in the optical range.
A negative DC at optical frequencies is in principle achievable in a weakly damped electron plasma. But because of the absence of magnetic monopoles, a negative MP would require a suitable magnetic dipole interactions. That is much harder to achieve, and the necessary parameters are inaccessible at optical frequencies, where NR would be of greatest interest.
At microwave frequencies, usable magnetic dipole interactions can be achieved via metallic resonators. A periodic lattice of suitably-designed metallic resonators, with a sufficiently small lattice constant, can simulate a uniform medium with negative n, and thereby make possible NR. But if such structures are scaled down in size for optical wavelengths, they have hopelessly large losses.
An alternative route to NR at optical frequencies, which does not involve magnetic interactions at all, draws on the properties of purely dielectric periodic lattice structures (so-called Photonic Crystals), inside which only the DC changes periodically. Wave propagation in periodic media invariably exhibits band structures, with allowed and forbidden bands of propagation, regardless of the nature of the waves This is a familiar phenomenon for the propagation of electron waves through the periodic potential inside a crystal, where it forms the basis of the physics of the electrical transport properties of crystals. It is equally true for the propagation electromagnetic waves through a periodically varying DC.
To discuss the refraction properties of photonic crystals, it becomes necessary to introduce a new distinction between two kinds of refractive index: a longitudinal and a transverse index. A negative transverse index leads to NR, even when the longitudinal index remains positive. The allowed photonic bands may contain regions inside which the transverse index becomes negative, permitting negative refraction. This approach does not suffer from the problems with achieving a negative MP at optical frequency, and hence is the appropriate way towards achieving NR in the optical range.