Melles Griot Optics Guide - Incorporating M2 Into the Propagation Equations

Real Beam Propagation

Incorporating M ² Into the Propagation Equations

In the previous section we defined the propagation constant M ²

where w_0R and q _R are the beam waist and far-field divergence of the real beam, respectively. For a pure Gaussian beam, M ² = 1, and the beam-waist beam-divergence product is given by

For a real laser beam, we have

where w_0R q _R are the 1/2 intensity waist radius and the far-field half-divergence angle of the real laser beam, respectively. The propagation equations for a real laser beam are now written as

and

where w_R(z ) and R_R( z) are the 1/e ² intensity radius of the beam and the beam wavefront radius at z, respectively. The definition for the Rayleigh range remains the same for a real laser beam and becomes

For M ² = 1, these equations reduce to the Gaussian beam propagation equations

and

In a like manner, the lens equation can be modified to incorporate M ². The standard equation becomes

and the normalized equation transforms to

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