4. The evanescent wave [solution]

• The incident wave propagates in the xz plane (i.e this is the plane in which k resides), so the spatial part of the wave in the second medium has the form

where we have made use of . This is a wave that is exponentially decaying in the z direction, but propagating in the x direction. This means that the evanescent wave moves along the interface between the two media rather than penetrating further into the second medium.

• The speed of the wave is

where we recall that this is only valid if the incident angle is larger than the critical angle. Notice that it only depends on the index of the first medium. A graph of v/c for m ₁ = 1.5 is shown here.

• As shown by the expression above, the amplitude of the electric field decreases exponentially as a function of distance into the second medium (i.e. as a function of z). The inverse of the 1/e distance d (i.e. the inverse of the value of z for which the electric field amplitude is 1/e of its value at z = 0) is

A graph of dk₀ as a function of incident angle for the glass-air interface is shown here. As the wavelength of the light decreases (so k₀ increases), the distance will go down (note that the index of refraction of the glass will go up, enhancing the decrease) so that the penetration depth of shorter-wavelength light is less than that at longer wavelengths.
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