Diffraction and the Wave Theory of Light — Question 10

Question 10

Subwavelength spatial structure can be evanescent rather than propagating

For a monochromatic angular-spectrum component in vacuum, $k_x^2+k_z^2=k^2$ with $k=2\pi/\lambda$. Its transverse period is $\Lambda=\lambda/2$, so $k_x=2\pi/\Lambda$. Choose the z dependence that decays away from the aperture.

1. Find $k_z$ and the amplitude decay length.

2. For $\lambda=600\,\mathrm{nm}$, find the distance over which intensity falls by a factor $e^{-2}$.

Question 10 - Solution

Subwavelength spatial structure can be evanescent rather than propagating

(a) Method and calculation

$$k_x=2k\Longrightarrow k_z^2=-3k^2\Longrightarrow\boxed{k_z=i\sqrt3k}.$$$$e^{ik_zz}=e^{-\sqrt3kz},\quad\boxed{\ell_E=1/(\sqrt3k)=\lambda/(2\pi\sqrt3)}.$$

(b) Interpretation and further calculation

$$I(z)/I(0)=e^{-2z/\ell_E};\quad I(z)/I(0)=e^{-2}\Longrightarrow z=\ell_E.$$$$\boxed{z=55.1329\,\mathrm{nm}}.$$

Physical check

This component decays exponentially and cannot appear as an ordinary far-field ray; the real longitudinal-wave-vector assumption has failed.