7.1 The paraxial wave equation

Mathematically, Gaussian modes are solutions to the paraxial wave equation – a specific wave equation for electromagnetic fields. All electromagnetic waves are solutions to the general wave equation, which in vacuum can be given as:
1 ¨ Δ E⃗ − -2⃗E = 0. (109 ) c
But laser light fields are special types of electromagnetic waves. For example, they are characterised by low diffraction. Hence, a laser beam will have a characteristic length w describing the ‘width’ (the dimension of the field transverse to the main propagation axis), and a characteristic length l defining some local length along the propagation over which the beam characteristics do not vary much. By definition, for what we call a beam w is typically small and l large in comparison, so that w ∕l can be considered small. In fact, the paraxial wave equation (and its solutions) can be derived as the first-order terms of a series expansion of Equation (109View Equation) into orders of w ∕l [37].

A simpler approach to the paraxial-wave equation goes as follows: A particular beam shape shall be described by a function u(x,y,z ) so that we can write the electric field as

E(x, y,z) = u(x,y,z) exp (− ikz). (110 )
Substituting this into the standard wave equation yields a differential equation for u:
( 2 2 2) ∂x + ∂y + ∂z u(x,y,z) − 2ik∂zu(x, y,z) = 0. (111 )
Now we put the fact that u(x,y,z ) should be slowly varying with z in mathematical terms. The variation of u(x, y,z) with z should be small compared to its variation with x or y. Also the second partial derivative in z should be small. This can be expressed as
| 2 | | 2 | | 2 | |∂ zu(x,y,z)| ≪ |2k ∂zu(x,y,z)|,|∂xu (x, y,z)|,|∂yu(x,y, z)|. (112 )
With this approximation, Equation (111View Equation) can be simplified to the paraxial wave equation,
( 2 2) ∂x + ∂ y u(x, y,z) − 2ik ∂zu(x,y,z) = 0. (113 )
Any field u that solves this equation represents a paraxial beam shape when used in the form given in Equation (110View Equation).
  Go to previous page Go up Go to next page