LaserCanvas 5 > Reference >

Gaussian Beams

Fundamental Mode

The beam propagation, i.e., the evolution of the laser mode w(z) as a function of distance z from the waist, is given for the fundamental TEM₀₀ Gaussian beam by

where w₀ is the mode size at the waist, and is the laser wavelength. This equation is sometimes expressed in terms of the Rayleigh length

For large distances z >> 1, this leads to a far-field divergence of

Similarly, the wave front radius of curvature evolves according to

Embedded Gaussian and M²

The propagation of a non-perfect fundamental Gaussian beam can be described using the beam propagation factor M². This parameter is sometimes colloquially referred to as the beam quality. For physical beams, M² > 1, with the limit M² = 1 for perfect TEM₀₀ beams.

The propagation of a beam with beam propagation factor M² is established as follows:

• From the beam waist w₀, we construct a virtual embedded Gaussian beam with waist w₀ = W₀ / M.
• The embedded Gaussian mode w(z) is propagated according to the formula above.
• The full beam mode at any position is given by W(z) = w(z) M.

For laser resonators, deriving the resonant mode using the ABCD self-consistency argument yields the embedded Gaussian mode.

From the definition of the Rayleigh length, that the mode W(z) is (2^1/2) W₀, it can easily be shown that the Rayleigh length is the same for the full beam as for the embedded Gaussian.

Internal Representation

LaserCanvas includes the M² beam propagation factor in its calculations. Since the embedded Gaussian is rarely of interest, the formula

is used directly when calculating mode sizes.

The beam evolution from one optic to the next is performed by calculating the q parameter using the optic and space ABCD matrices.

The spot sizes reported by the  Measurement tool use the formula above directly.

For painting, the mode evolution can be divided into near-field and far-field. In the near field, the mode exhibits a characteristic curving, approaching the far-field asymptote.

To minimise the number of drawing cycles required, LaserCanvas scales a tabulated near-field mode evolution up to a distance of 10z_R. In the far field, a single data point is plotted at the end of each segment.