Basic Characteristics of Gaussian Beams

In this chapter we will discuss the basic characteristics of Gaussian beams and their transformation by optical elements such as lenses, mirrors, prisms and optical gratings. The following presentation follows that of the recommendable review by Kogelnik and Li [314]. 

A laser beam traveling into the z-direction can be represented by the field amplitude 

While A x,y,z)i constant for a plane wave, it is a slowly varying complex function for a Gaussian beam. Since every wave obeys the general wave equation 

Demtroder, Laser Spectroscopy 1, DOI 10.1007/978-3-642-53859-9 7, Springer-Verlag Berlin Heidelberg 2014 

This illustrates that R(z) represents the radius of curvature of the wavefronts intersecting the axis at z (Fig. 7.1), and w(z) gives the distance r = (x + from the axis where the amplitude has decreased to 1/e and thus the intensity has decreased to 1/e of its value on the axis (Sect. 5.2.3 and Fig. 5.11). Inserting (7.5) into (7.2) and comparing terms of equal power in r yields the relations 

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