Phase Velocity - Relation To Group Velocity, Refractive Index and Transmission Speed

Relation To Group Velocity, Refractive Index and Transmission Speed

Since a pure sine wave cannot convey any information, some change in amplitude or frequency, known as modulation, is required. By combining two sines with slightly different frequencies and wavelengths,

the amplitude becomes a sinusoid with phase speed of v_g = Δω/Δk. It is this modulation that represents the signal content. Since each amplitude envelope contains a group of internal waves, this speed is usually called the group velocity. In reality, the v_p = ω/k and v_g = dω/dk ratios are determined by the media. The relation between phase speed, v_p, and speed of light, c, is known as refractive index, n = c/v_p = ck/ω. Taking the derivative of ω = ck/n, we get the group speed,

Noting that c/n = v_p, this shows that group speed is equal to phase speed only when the refractive index is a constant: dn/dk = 0. Otherwise, when the phase velocity varies with frequency, velocities differ and the medium is called dispersive. The phase velocity of electromagnetic radiation may – under certain circumstances (for example anomalous dispersion) – exceed the speed of light in a vacuum, but this does not indicate any superluminal information or energy transfer. It was theoretically described by physicists such as Arnold Sommerfeld and Léon Brillouin. See dispersion for a full discussion of wave velocities.