One of the most fundamental of optical effects is refraction, or the bending of light as it crosses the interface between two materials. The phenomenon of refraction is well-known to most of us: for example, an object under water viewed by an observer in air always appears closer to the surface than it actually is. Refraction is the basic principle behind lenses and other optical elements that focus, steer, guide or otherwise manipulate light. Highly sophisticated and complex optical devices are developed by carefully shaping materials so that light is refracted in desired ways (think of a camera lens or a microscope objective).
The underlying principle of refraction can be easily understood and applies to all electromagnetic waves--not just visible light. Every material, including air, has an index-of-refraction (or refractive index). When an electromagnetic wave traverses the interface from a material with refractive index n1 to another material with refractive index n2, the change in its trajectory can be determined from the ratio of refractive indices n2/n1 by the use of Snell's Law.
To apply Snell's Law, consider an interface between two materials and an imaginary line that runs perpendicular to the interface (the surface normal). The angles in Snell's law are measured away from the surface normal. If the refractive indices of the two materials are not equal, the angle of the transmitted beam will differ from the angle of the incident beam. The beam is then bent at the interface.
A common way to determine the refractive index of a material is to form a prism out of the material, shine a beam of light through it and observe the deflection of the beam on the other side. Light enters the prism through one of the interfaces at direct incidence, striking the opposite interface at an oblique angle. The figure below shows what happens to the beam when the material has the same index as the surrounding medium, or has an index that is greater than the surrounding medium but either positive or negative.
In the figure above, the dashed line represents the surface normal, which is perpendicular to the interface between the prism and the surrounding material. The angle of the prism defines the angle of incidence of the beam to the interface. A measurement of the angle of the exit beam from the surface normal provides a measurement of the refractive index of the prism. Snell's Law shows that a material with a negative refractive index--not a material that exists in nature--would refract a beam to negative angles, as shown in the figure.
Veselago and negative index
All transparent or translucent materials that we know of possess positive refractive index--a refractive index that is greater than zero. However, is there any fundamental reason that there should not be materials with negative refractive index? This question was asked by Victor Veselago, a Russian physicist. In 1968, Veselago published a theoretical analysis of the electromagnetic properties of materials with negative permittivity and negative permeability. The electric permittivity and the magnetic permeability are commonly used material parameters that describe how materials polarize in the presence of electric and magnetic fields. Maxwell's equations relate the permittivity and the permeability to the refractive index as follows:
The sign of the index is usually taken as positive. However, Veselago showed that if a medium has both negative permittivity and negative permeability, this convention must be reversed: we must choose the negative sign of the square root!
This reversal of the refractive index can seem confusing. As an example, it is often said that the velocity of a wave in a material is given by c/n, where c is the speed of light in vacuum. The implication of a negative index, then,is that the wave travels backwards, as indicated in the animation below. An electromagnetic wave can be depicted as a sinusoidally varying function that travels to the right or to the left as a function of time. In the top animation in the figure below, a wave is incident on a positive index material (the reflected wave has been ignored). The greater index of the second medium implies that the wavelength decreases (by a factor of 1/n); however, to maintain the same phase at the interface as a function of time, the speed of the wave must also be reduced, again by a factor of 1/n.
When the refractive index is negative, the speed of the wave--given by c/n--is negative and the wave travels backwards toward the source as in the bottom animation in the figure below. Yet, we would reasonably expect that since energy is incident on the material from the left, the energy in the the material should likewise travel to the right, away from the interface. This seeming paradox is resolved, as Veselago showed, by realizing there are more ways to define the velocity of a wave. The definition c/n is well known as the phase velocity and determines the rate at which the peaks (or zeros) of a wave pass a given point in time. But this is not most relevant definition of a wave's velocity: we can also define the group, energy, signal and front velocities, and these generally differ from the phase velocity.
When the refractive index of a material does not vary with the wavelength of light that travels through it, then all of the velocity definitions above are the same and we can intuitively use the index as a measure of the wave's velocity. However, when a material is dispersive--has an index that varies with wavelength--then the various definitions of velocity no longer agree and we can no longer determine the actual velocity of the wave, or at least the rate at which energy is transported, from the value of the refractive index alone. So, even though the positive and negative index materials in the figure above seem to display drastically different behaviors, a calculation of the group or the energy velocity reveals that energy is actually flowing to the right in both cases. Thus, as Veselago showed, the phase and energy velocities are opposite in a negative index material.
