Cavity Mode Spacing and Stability
Two-mirror resonator: g-parameters and stability diagram, free spectral range, transverse mode spacing, TEM₀₀ waist size and position, and spot sizes on the mirrors.
g₁ / g₂
-0.5000 / -0.5000
g = 1 − L/R
g₁g₂
0.2500
Stable
Free spectral range c/(2nL)
0.9993 GHz
8.008 pm at λ
Transverse mode spacing
0.6662 GHz
One-way Gouy phase θ = 120.0°
Waist radius w₀ (1/e²)
146.2 µm
Rayleigh range 43.30 mm in the medium
Waist distance from mirror 1
75.00 mm
Towards mirror 2; negative is behind mirror 1
Spot radius on mirror 1
292.3 µm
Spot radius on mirror 2
292.3 µm
g₁ = 1 − L/R₁, g₂ = 1 − L/R₂; stable for 0 ≤ g₁g₂ ≤ 1. νqmn= FSR [q + (m + n + 1) θ/π], θ = arccos(±√(g₁g₂)), + for g > 0 and − for g < 0.
zR² = L² g₁g₂(1 − g₁g₂)/(g₁ + g₂ − 2g₁g₂)², w₀² = λzR/(nπ), z₁ = L g₂(1 − g₁)/(g₁ + g₂ − 2g₁g₂), w₁² = (λL/nπ)√(g₂/(g₁(1 − g₁g₂))).
For a dispersive filling the mode spacing uses the group index: FSR = c/(2ngL); the index input here is the phase index.
Mode frequencies relative to TEM₀₀ of order q (stick height marks the order, not the power)
- m + n = 0
- m + n = 1
- m + n = 2
- m + n = 3
- m + n = 4
Stability diagram: stable between the axes and the hyperbola
- g₁g₂ = 1 (boundary)
- g₁g₂ = 1, negative branch
- g₁ = 0 and g₂ = 0 (boundary)
- Symmetric: planar, confocal, concentric
- This cavity
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 Gaussian modes of two spherical mirrors (Kogelnik–Li), with a medium of index n filling the cavity (phase index; a dispersive filling spaces the modes by c/(2n_gL)). The transverse spacing takes the Gouy phase on the branch set by the sign of g, so cavities with g < 0 space their modes by more than FSR/2. Aperture losses and mirror tilt are ignored.
- Kogelnik H., Li T. (1966). Laser beams and resonators. Appl. Opt. 5, 1550–1567. doi:10.1364/ao.5.001550
- Siegman A. E. (1986). Lasers, ch. 19. University Science Books.