Appendix C: Equations for Plane Waves, Spherical Waves, and Gaussian Beams

Abstract

In this appendix, we shall derive the equations for plane waves, spherical waves, and Gaussian beams which can be used to represent laser beams. C.1 Equations for Plane Waves Wave fronts are surfaces over which a wave has the same phase. By definition, the wave fronts of plane waves are planes. All of these wave fronts are perpendicular to the direction of wave propagation [1]. Figure C.1 depicts a plane wave which propagates from the origin O in the direction of k (where k is the propagation vector). The mathematical expression for all of these wave fronts is: r ⋅ k = constant (Inner product) (C.1) that is, r cos í µí¼ƒ = constant where í µí¼ƒ = the angle between vector r and vector k (C.2) Figure C.1a depicts wave fronts traveling along the z-axis in the x-z plane, while Figure C.1b depicts the wave fronts traveling in an arbitrary direction in x-y-z space. We can now construct a set of planes over which the value U (r) varies in a sinusoidal fashion in space, namely: U(r) = A sin (r ⋅ k) (C.3) U(r) = A cos (r ⋅ k) (C.4) U(r) = A exp (jk ⋅ r) (C.5) For each of these expressions, U(r) remains constant over every plane defined by the equation k⋅r = constant. Since we are discussing harmonic functions here, we would expect these functions to