Plane Wave

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A plane wave is a wave whose wavefronts (surfaces of constant phase) are infinite parallel planes perpendicular to the direction of propagation.

Mathematical Description

A plane wave solution to the Helmholtz equation:

$$ \hat{u}(\vec{r}) = A e^{j(\vec{k} \cdot \vec{r} + \phi)} $$

where:

  • $A$ is the amplitude
  • $\vec{k} = (k_x, k_y, k_z)$ is the wavevector
  • $\phi$ is the phase at the origin

Direction Cosines

The wavevector can be written as $\vec{k} = k(\gamma_x, \gamma_y, \gamma_z)$ where:

  • $\gamma_x, \gamma_y, \gamma_z$ are direction cosines
  • $\gamma_x^2 + \gamma_y^2 + \gamma_z^2 = 1$
  • $k = 2\pi/\lambda$ is the wavenumber

Spatial Frequency Relationship

A plane wave tilted at angle $\theta$ from the $z$-axis has spatial frequency:

$$ f_x = \frac{\cos\theta}{\lambda} $$

This connects the Fourier transform to physical wave propagation.

Propagating vs. Evanescent

  • Propagating: $\gamma_x^2 + \gamma_y^2 < 1$ → $\gamma_z$ is real
  • Evanescent: $\gamma_x^2 + \gamma_y^2 > 1$ → $\gamma_z$ is imaginary, field decays exponentially

Eigenfunctions of Free Space

Plane waves are eigenfunctions of the free-space propagation operator:

$$ e^{jk(\gamma_x x + \gamma_y y)} \xrightarrow{\text{propagate } \Delta z} e^{jk\gamma_z \Delta z} \cdot e^{jk(\gamma_x x + \gamma_y y)} $$