Helmholtz Equation

The Helmholtz equation is the time-independent form of the wave equation for monochromatic (single-frequency) fields:

$$ \nabla^2 U(\mathbf{r}) + k^2 U(\mathbf{r}) = 0 $$

where $k = 2\pi/\lambda$ is the wavenumber and $U(\mathbf{r})$ is the complex amplitude of the field.

Derivation

Starting from the scalar wave equation:

$$ \nabla^2 u(\mathbf{r}, t) = \frac{1}{c^2} \frac{\partial^2 u}{\partial t^2} $$

For a monochromatic field $u(\mathbf{r}, t) = \operatorname{Re}\{U(\mathbf{r}) e^{-j\omega t}\}$, the time derivatives yield $-\omega^2$, giving:

$$ \nabla^2 U + \frac{\omega^2}{c^2} U = 0 $$

Since $k = \omega/c$, this is the Helmholtz equation.

Significance in Optics

• Solutions of the Helmholtz equation include plane waves $e^{j\mathbf{k} \cdot \mathbf{r}}$ and spherical waves
• The angular spectrum representation decomposes any field into plane-wave solutions
• The Fresnel and Fraunhofer approximations are derived from the Helmholtz equation under paraxial conditions
• Each spatial frequency $(f_X, f_Y)$ in the angular spectrum corresponds to a plane-wave solution with propagation constant $k_z = k\sqrt{1 - (\lambda f_X)^2 - (\lambda f_Y)^2}$