Helmholtz equation: Difference between revisions

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m (Changed the eigenvalue variable to -k^2 to be consistent with what follows. The previous use of \lambda is confusing, as this is the wavelength symbol for light waves, and is the inverse of the wavenumber)
This equation has important applications in the science of [[optics]], where it provides solutions that describe the propagation of [[electromagnetic waves]] (light) in the form of either [[parabola|paraboloidal]] waves or [[Gaussian beam]]s. Most [[laser]]s emit beams that take this form.
The assumption under which the paraxial approximation is valid is that the ''z'' derivative of the amplitude function ''u'' is a slowly- varying function of ''z'':
:<math> \bigg| { \partial^2 u \over \partial z^2 } \bigg| \ll \bigg| { k {\partial u \over \partial z} } \bigg| .</math>