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)
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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>