Figure 1: Gaussian Beam Parameters for defining the Electric Field Amplitude.
Diagram copied from Laser Fundamentals, Silfvast pg. 339.
Gaussian Beam Waist & Divergence Angle
A Gaussian beam is referred to as a diffraction-limited beam since it can be focused to
the smallest theoretical spot size or beam waist. The beam waist (or beam radius) is
defined two ways:
1.
The beam waist (or beam radius), wo, of the Electric Field Amplitude of a
Gaussian beam is defined at the 1/e or 63.21% of the amplitude as shown in
Figure (1).
2.
The beam waist (or beam radius), wo, of the Intensity of a Gaussian beam is
defined at the 1/e2 or 86.50% of the energy.
Note: Figure (1) places the waist, wo, at 1/e since it is describing the Electric Field
Amplitude. All Gaussian beam parameters are defined the same for the Intensity except
the waist, wo, is located at 1/e2.
The general equation to calculate the angular spread of a Gaussian beam is as follows:

 
Equation (1)
tan   
 2   wo
Where, θ = full angle of divergence
(Note: θ / 2 = θhalf which defines the half angle of divergence)
λ = wavelength
wo = beam waist (or beam radius) defined at 1/e for Field Amplitude and
1/e2 for Intensity.
Note: There are discrepancies in various textbooks defining the beam waist, wo, as either
the beam diameter or beam radius. Also some textbooks define the angular spread of a
Gaussian beam, equation (1), as θ/2 = λ/(Л* wo) which is only the case for small
divergence angles.
Rayleigh Range
The Rayleigh range, ZR, of the Gaussian beam is defined as the distance between the
beam waist location, wo, and location where the beam radius is equal to
approximately 2wo . The distance, b, which equals two times the Rayleigh range is
called the confocal parameter as shown on Figure (2).
Figure 2: Rayleigh range for Gaussian beam. Diagram copied from Laser Fundamentals, Silfvast pg.
338.
The equation to calculate the Rayleigh range for a beam waist (or beam radius) is:
 wo2
ZR 

Equation (2)
Where, λ = wavelength
wo = beam waist (or beam radius) defined at 1/e for Field Amplitude and
1/e2 for Intensity.
Real Laser Beams
A real laser beam has both a waist and divergence angle which is larger than the ideal
Gaussian parameters described in the above equations. Real beams can be defined by a
factor M2. This factor, M2, describes the relationship between the real beam and the ideal
Gaussian beam.
W0 real
4
 M
M2 
Equation (3)
 real
W0  M 0
Where, M = factor relating real beam and ideal Gaussian beam
W0 = real beam waist (or beam radius) defined at 1/e2 for Intensity
ω0= Gaussian beam waist (or beam radius) defined at 1/ e2 for Intensity
θreal = real half angle of divergence
θ = Gaussian half angle of divergence
λ = wavelength
Here is a nice picture from Thorlabs Volume 18 catalog, pg. 600. I am not sure the
parameters are defined as above, I haven’t checked. Added 1.8.2007, STS.