the field components at the output of a wave retarder could be written from equation (4.6.11), by normalizing the magnitude of field components to the unity, as follows:
The JR(Δφ) is the Jones matrix of a wave retarder (waveplate) given by equation (4.6.25b). At the input, the incident wave is linearly polarized and at the output of the retarder slab of thickness d, the differential phase Δφ = φz − φy is the relative phase between the field components
Jones Matrix of Half‐waveplate
The Jones matrix of a half‐waveplate and the field components at its output are obtained by taking the relative phase Δφ = π:
(4.6.26)
It is noted that at the output of the half‐wave plate the phase difference between two field components is 180°. Two field components are in‐phase at the input of the half‐waveplate.
Jones Matrix of Quarter‐waveplate
The Jones matrix of a quarter‐wave retarder and also the field components at the output are obtained by taking the relative phase Δφ = − π/2:
(4.6.27)
In the above equation, both field components are equal to E0 = 1. It is noted that at the output of the quarter‐waveplate, the wave is a right‐hand circularly polarized wave. In the case, input wave components are
Equation (4.6.28c) shows that an ellipse is traced by the E‐field in the (y‐z)‐plane. In this case, the quarter‐waveplate produces an elliptically polarized wave. However, for the case m = n, it degenerates into the circularly polarized wave. Further, for the rotated retarders the rotated Jones matrices could be obtained, similar to the case of the rotated polarizer, using equation (4.6.21a).
4.7 EM‐waves Propagation in Unbounded Anisotropic Medium
Two cases of wave propagations in the uniaxial anisotropic media – without off‐diagonal elements and with off‐diagonal elements, are considered in this section. The dispersion relation is also discussed leading to the concept of hypermedia useful for the realization of hyperlens [J.1, J.5–J.7].
4.7.1 Wave Propagation in Uniaxial Medium
The unbounded lossless homogeneous uniaxial medium is considered. The y‐axis is the optical axis, i.e. the extraordinary axis. In the direction of the optic axis, the permittivity is different as compared to the other two directions. The medium is described by a diagonalized matrix with all off‐diagonal elements zero. The permeability of the medium is μ0 and its permittivity tensor is expressed as follows:
(4.7.1)
Figure (4.13a) shows the TEM plane wave propagation in the x‐direction. The TEM waves have Ex = 0, Hx = 0; Ey ≠ 0, Hy ≠ 0; Ez ≠ 0, Hz ≠ 0. Also, the uniform field components in y and z‐directions do not vary, i.e. ∂/∂y(field) = ∂/∂z(field) = 0. Under these conditions, the following Maxwell equations provide the transverse field components of electric and magnetic fields:
On expansion, the above equations provide the following sets of transverse field components:
Figure 4.13 Wave propagation uniaxial media.
On eliminating Hz and Hy from the above equations, wave equations for the electric field transverse components are obtained. Likewise, the wave equations for magnetic field transverse components are obtained on eliminating Ez and Ey:
Equation (4.7.5a) is a 1D wave equation of the y‐polarized electric field Ey ≠ 0 and Ez = 0. The magnetic field component Hz is along the z‐axis. The Ey field component causes polarization in the dielectric medium creating relative permittivity εr‖ along the y‐axis. Further, the electric field Ey is transverse to the (x‐z)‐plane containing the direction of propagation x. Such waves are called the transverse electric (TE) waves. The z‐polarized electric field with Ez ≠ 0 and Ey = 0 follows the wave equation (4.7.5b). The Ez component generates another polarization in the dielectric medium creating relative permittivity εr⊥ along the z‐axis. In this case, Hy‐component is transverse to the (x‐z)‐plane. These waves are called the transverse magnetic (TM) waves. In the present case, also in the case of the oblique incident of the plane waves, the waves are still TEM only. However, the TE and TM terminology is normally used
