Through a transparent plate with no angle of incidence, a plane wave continues to propagate through the plate, but with an altered wavenumber n*k0, where n is the refractive index and k0 is the wavenumber of the plane wave before transmission through the plate.
Where d is the thickness of the plate and U(x,y,z) is the complex amplitude of the wave, the transmittance of a plane wave in a homogenous, transparent plate is described as:
t(x,y) = U(x,y,d)/U(x,y,0) = exp(-j*n*k0*d).
Given the scenario of a plane wave with wavevector k and angle of incidence θ, the formula is altered and may be modeled using Snell’s Law:
sinθ = n*sinθ,
exp(-j*k1 • r) = exp[-jnk0(zcosθ1 + xsinθ1),
t(x,y) = exp(-j*n*k0*d*cosθ1).
A prism may be used in the following manner to direct the propagation of a plane wave:
Further, a thin lens may be used to focus a plane wave, converting it into a paraboloidal wave.
A graded-index (GRIN) lens may be used to produce the same effect:
Diffraction Grating is a method of modulating either the phase or amplitude of an incident wave. An incident plane wave is split into multiple plane waves. They may also be used as filters or spectrum analyzers.
The grating equation is as follows:
θq = θi + q*λ/Λ,
where θq is the angle of resultant wave(s), θi is the angle of incidence, q is the diffraction order (0,1,2…) and Λ is the period of thickness variation in the diffraction grating. Since the device is dependent on the wavelength, it may be used to produce a polychromatic wave, separating it’s spectral components in the following manner:
(1) B. E. A. Saleh and M. C. Teich, Fundamentals of photonics. Hoboken: Wiley, 2019.