Hermite-Gaussian and Laguerre-Gaussian Modes
Intensity of TEMmn and LGpl modes at any distance, with second-moment radius, M², Gouy phase, peak intensity and orbital angular momentum.
Mode
TEM₂₁
Order N = m + n = 3
Rayleigh range z_R
5.067 mm
Same for every mode of this family
Second-moment radius W
157.1 × 121.7 µm
w(z) = 70.24 µm for TEM₀₀; W = w√(2m+1)
Beam quality M²
5 (x), 3 (y)
2m + 1, 2n + 1
Gouy phase (N + 1) arctan(z/z_R)
3.115 rad
2.336 rad more than TEM₀₀
Peak intensity
6.234 W/cm²
At z, for 1.00 mW in the mode
Dark lines
2 vertical, 1 horizontal
Zeros of H_m(√2x/w) and H_n(√2y/w)
umn(x, y) = √(2/π)/w · Hm(√2x/w) Hn(√2y/w) e−(x²+y²)/w² / √(2m+n m! n!)
Normalised to unit power; w = w(z) is the TEM₀₀ radius at the plane shown. The Gouy phase sets the mode spacing in a cavity.
Profiles along x and y
- x: H2
- y: H1
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
Exact unit-power solutions of the paraxial wave equation, indices up to 10. Every mode shares the TEM₀₀ waist, Rayleigh range and w(z); the second-moment radius and M² follow from the indices.
- Kogelnik H., Li T. (1966). Laser beams and resonators. Appl. Opt. 5, 1550–1567. doi:10.1364/ao.5.001550
- Carter W. H. (1980). Spot size and divergence for Hermite Gaussian beams of any order. Appl. Opt. 19, 1027. doi:10.1364/ao.19.001027
- Phillips R. L., Andrews L. C. (1983). Spot size and divergence for Laguerre Gaussian beams of any order. Appl. Opt. 22, 643. doi:10.1364/ao.22.000643
- Allen L., Beijersbergen M. W., Spreeuw R. J. C., Woerdman J. P. (1992). Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes. Phys. Rev. A 45, 8185–8189. doi:10.1103/physreva.45.8185