The _[Helmholtz equation](Helmholtz%20equation.md)_ in optics models the propagation of a monochromatic [light wave](Electromagnetic%20wave.md) where the [light-field](Light-field.md) is in the form of a [[plane wave]], $\mathbf{E}(\mathbf{r},t)=\mathbf{E}(\mathbf{r})e^{-i\omega t},$ where $\omega$ is the light frequency. The equation itself is written as $\nabla^2\mathbf{E}(\mathbf{r})+k^2\mathbf{E}(\mathbf{r})=0$ where $k=\frac{n\omega}{c}$ and we notice that the time-dependence disappears. This is also a form of the [eikonal equation](Eikonal%20equation.md) for a constant [index of refraction](Index%20of%20refraction.md). # Scalar diffraction theory The Helmholz equation in optics commonly appears in [[scalar diffraction theory]] and thus it may be given in terms of a scalar [light-field](Light-field.md)s and be written as $\nabla^2E(\mathbf{r})+k^2E(\mathbf{r})=0$ ^27a30d # Derivation of the optical Helmholtz equation Take the [vacuum wave equation](Electromagnetic%20wave%20equations.md#In%20a%20vacuum) for an electric field associated with the [light-field](Light-field.md), which is given as  and consider the solution to this equation given by the [electric field](Electromagnetic%20wave.md) written as a wave $\mathbf{E}(\mathbf{r})e^{-i\omega t}$. If one plugs in that solution and evaluates the double time derivative we are left with a form of the [[Helmholtz equation]] written as $\nabla^2\mathbf{E}(\mathbf{r})+k^2\mathbf{E}(\mathbf{r})=0$ where we wrote the equation in terms of the wave vector $k=\frac{n\omega}{c}.$ # Solutions to the optical Helmholtz equation An important class of solutions to the optical Helmholtz equation are [Gaussian beam](Gaussian%20beam.md)s. #Electromagnetism/Optics/waveOptics