Applications and Metrology at Nanometer Scale 1

AppendixPropagation of a Light Ray

In light diffraction problems, the amplitude of a wave E_z(x,y) on a surface located in a plane z is determined by the amplitude E₀(x,y) of a wave in the plane z = 0.

In Chapter 3 of [DAH 16], equation 3.6 gives the expression of an electromagnetic wave traveling along Oz, in the form u(z-vt)=acos(k(z-ct)), where v is the speed of propagation of the wave in a medium of index n, a is its amplitude and k=2πn/λ. In a vacuum, n=1 and in a three-dimensional space the monochromatic plane wave of the angular frequency ω=kc and the wave vector k = (k_x, k_y, k_z) can be expressed in the form E = E₀ exp i(2π/λ (αx+βy+γz) - ωt), where the components of the wave vector have the form: k_x=2πα/λ, k_y=2πβ/λ and k_x=2πγ/λ, with α, β, γ being the direction cosines of the wave vector k and λ being the wavelength. This expression can be obtained by solving the Helmholtz equation (equation 3.7, [DAH 16]):

[A.1]

where , using Green’s function of the Helmholtz equation that verifies that:

[A-2]

given that , where the unit vector is . The solution is readily obtained, since ...