Evanescent Wave

An evanescent wave is a non-propagating electromagnetic field component that decays exponentially with distance from a surface or interface. Evanescent waves arise when the spatial frequency content of a field exceeds the propagating limit.

Origin in Diffraction

In the angular spectrum representation, each spatial frequency $(f_X, f_Y)$ corresponds to a plane wave with axial propagation constant:

$$ k_z = \frac{2\pi}{\lambda}\sqrt{1 - (\lambda f_X)^2 - (\lambda f_Y)^2} $$

When $(\lambda f_X)^2 + (\lambda f_Y)^2 > 1$, the square root becomes imaginary, and $k_z$ is purely imaginary. The corresponding field component decays as $e^{-|k_z|z}$ rather than propagating — this is an evanescent wave.

In Diffraction Gratings

For a grating with spatial frequency $f_0$, the $q$th diffraction order propagates at angle $\sin\theta_q = q\lambda f_0$ (the grating equation). When $|q\lambda f_0| > 1$, the order becomes evanescent. This limits the maximum usable diffraction order to:

$$ q_{\max} = \left\lfloor \frac{1}{\lambda f_0} \right\rfloor $$

Physical Significance

• Evanescent waves carry no energy to the far field — they exist only in the near-field region close to the diffracting object
• They represent spatial detail finer than the wavelength of light, which cannot be resolved by conventional far-field imaging
• Near-field scanning optical microscopy (NSOM) exploits evanescent waves to achieve sub-wavelength resolution
• Total internal reflection generates evanescent waves on the low-index side of the interface