7 Beam Shapes: Beyond the Plane Wave Approximation

In previous sections we have introduced a notation for describing the on-axis properties of electric fields. Specifically, we have described the electric fields along an optical axis as functions of frequency (or time) and the location z. Models of optical systems may often use this approach for a basic analysis even though the respective experiments will always include fields with distinct off-axis beam shapes. A more detailed description of such optical systems needs to take the geometrical shape of the light field into account. One method of treating the transverse beam geometry is to describe the spatial properties as a sum of ‘spatial components’ or ‘spatial modes’ so that the electric field can be written as a sum of the different frequency components and of the different spatial modes. Of course, the concept of modes is directly related to the use of a sort of oscillator, in this case the optical cavity. Most of the work presented here is based on the research on laser resonators reviewed originally by Kogelnik and Li [35Jump To The Next Citation Point]. Siegman has written a very interesting historic review of the development of Gaussian optics [5251] and we use whenever possible the same notation as used in his textbook ‘Lasers’ [50Jump To The Next Citation Point].

This section introduces the use of Gaussian modes for describing the spatial properties along the transverse orthogonal x and y directions of an optical beam. We can write

∑ ∑ E (t,x, y,z) = ajnm unm (x,y,z) exp (i(ωjt − kjz )), (107 ) j n,m
with unm as special functions describing the spatial properties of the beam and ajnm as complex amplitude factors (ωj is again the angular frequency and kj = ωj∕c). For simplicity we restrict the following description to a single frequency component at one moment in time (t = 0), so
∑ E (x,y,z) = exp (− ikz ) anm unm (x,y,z ). (108 ) n,m
In general, different types of spatial modes unm can be used in this context. Of particular interest are the Gaussian modes, which will be used throughout this document. Many lasers emit light that closely resembles a Gaussian beam: the light mainly propagates along one axis, is well collimated around that axis and the cross section of the intensity perpendicular to the optical axis shows a Gaussian distribution. The following sections provide the basic mathematical framework for using Gaussian modes for analysing optical systems.

 7.1 The paraxial wave equation
 7.2 Transverse electromagnetic modes
 7.3 Properties of Gaussian beams
 7.4 Astigmatic beams: the tangential and sagittal plane
 7.5 Higher-order Hermite–Gauss modes
 7.6 The Gaussian beam parameter
 7.7 Properties of higher-order Hermite–Gauss modes
 7.8 Gouy phase
 7.9 Laguerre–Gauss modes
 7.10 Tracing a Gaussian beam through an optical system
 7.11 ABCD matrices
  Transmission through a mirror:
  Reflection at a mirror:
  Transmission through a beam splitter:
  Reflection at a beam splitter:
  Transmission through a thin lens:
  Transmission through a free space:

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