Gaussian Beam Propagation
Rayleigh range, divergence, beam radius, wavefront curvature and Gouy phase of a Gaussian beam at any distance from its waist.
Rayleigh range z_R
0.2027 mm
Far-field divergence θ (half angle)
49.34 mrad
Beam parameter product w₀θ
0.4934 mm·mrad
Confocal parameter 2z_R
0.4054 mm
Beam radius w(z)
50.34 µm
Wavefront radius R(z)
1.041 mm
Minimum |R| = 2z_R at z = ±z_R
Gouy phase ψ(z)
78.54°
Total shift through focus: 180° (π)
zR = πw₀²/λ, θ = λ/(πw₀)
w(z) = w₀ √(1 + (z/zR)²), R(z) = z (1 + (zR/z)²)
ψ(z) = arctan(z/zR), the Gouy phase: the extra phase relative to a plane wave in step with the beam at the waist. On axis the phase velocity is slightly above c, so the wavefronts trail that plane wave before the waist and run ahead after it, π in all (some texts call ψ a retardation, but the on-axis phase velocity still exceeds c). A TEMmn mode picks up (m + n + 1)ψ.
Gaussian beam envelope
- Beam radius w(z)
- Waist
- Distance z
Gouy phase through focus
- Gouy phase ψ(z)
- Distance z
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Paraxial TEM₀₀ beam with a 1/e² intensity radius, in air or vacuum: use the wavelength in the medium otherwise. Accurate while the divergence stays below about 0.1 rad.
- Kogelnik H., Li T. (1966). Laser beams and resonators. Appl. Opt. 5, 1550–1567. doi:10.1364/ao.5.001550
- Feng S., Winful H. G. (2001). Physical origin of the Gouy phase shift. Opt. Lett. 26, 485–487. doi:10.1364/ol.26.000485