Paraxial approximation

The paraxial approximation simplifies wave propagation when waves travel mainly along one direction and deviate from it by only small angles. It reduces the full wave equation to a form suited to describing slowly varying transverse fields.

Helmholtz equation: A partial differential equation describing time-harmonic wave fields in a medium. The paraxial wave equation is derived by simplifying this frequency-domain wave equation.
Wave equation: A partial differential equation governing the propagation of waves through space and time. The paraxial approximation begins by simplifying a wave equation for a chosen field.
ABCD matrix: A two-by-two matrix that represents the first-order transformation of ray height and angle by an optical system. Paraxial ray optics uses these matrices to calculate propagation through lenses and other components.
Nonparaxial optics: Optical analysis that retains effects from large propagation angles or strong focusing beyond paraxial assumptions. It accounts for angular effects discarded by the paraxial approximation.
Slowly varying envelope approximation: An approximation that treats a wave’s amplitude envelope as changing slowly compared with its carrier oscillation. It formalizes the gradual longitudinal variation assumed when deriving paraxial propagation.
Wavenumber: The spatial frequency of a wave, equal to its phase change per unit distance. It sets the longitudinal carrier scale against which envelope variation is judged.
Ray transfer matrix analysis: A method for tracing paraxial rays through optical systems using linear transformations. It applies the same first-order, small-angle regime to geometrical optics.
Debye-Wolf integral: A vectorial diffraction integral used to describe strongly focused optical fields. It handles high-numerical-aperture focusing where scalar paraxial models become inadequate.
Paraxial wave equation: A first-order propagation equation for the slowly varying transverse envelope of a paraxial wave. This is the governing equation that results from the approximation.
Transverse wave vector: The component of a wave vector perpendicular to a chosen propagation axis. Small transverse components encode the small angular deviations required by the approximation.

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