Gas Laser Resonator
Stability g₁g₂, TEM₀₀ waist and spot sizes against the bore, Fresnel number, threshold gain, output power and the optimum output coupler of a HeNe, Ar⁺ or CO₂ laser.
g₁g₂
0.4500
g₁ = 0.5000, g₂ = 0.9000; stable
TEM₀₀ waist w₀
316.6 µm
450.0 mm from the HR
Fresnel number a²/(λL)
8.091
Largest mirror spot 426.8 µm; bore radius 1600 µm
Power outside the bore
6.25e-11 %
e^(−2a²/w²) at the larger mirror spot (not in the loss)
Threshold gain g_th
0.03115 1/m
δ/(2L), δ = 3.115 % round trip; g₀ = 0.1500 1/m
Output power
0.9399 mW
P_sat = I_sat π⟨w²⟩/2 = 49.02 mW
Optimum coupler reflectance
0.96545
Rigrod optimum for this gain and loss
Output at the optimum
1.436 mW
Rigrod: P = P_sat (1 − R₂)√R₁′(g₀L + ln√(R₁′R₂))/((√R₁′ + √R₂)(1 − √(R₁′R₂))), R₁′ = R_HR(1 − loss)²; threshold g₀L = δ/2. For small losses the optimum transmission is T ≈ √(2g₀L δᵢ) − δᵢ, with δᵢ = 2L_i + (1 − R_HR) the round-trip internal loss.
Mode: Kogelnik–Li, w₀² = (λ/π)L√(g₁g₂(1 − g₁g₂))/|g₁ + g₂ − 2g₁g₂|. Fresnel number N = a²/(λL).
Output power versus output-coupler reflectance
- Output power
- Optimum coupler
- This coupler
TEM₀₀ beam radius along the tube
- TEM₀₀ radius w(z)
- Bore radius
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
Kogelnik–Li TEM₀₀ mode; gain filling the whole mirror spacing; output and optimum coupler from Rigrod's homogeneous plane-wave model with the saturation power I_sat π⟨w²⟩/2 over the mode's mean-square radius. HeNe and Ar⁺ lines are Doppler (inhomogeneously) broadened, so the output is an estimate. The default g₀ = 0.15 /m and I_sat ≈ 25 W/cm² are values measured on one 3.2 mm-bore HeNe tube (Ng et al.); enter your tube's values for other lines. Bore clipping is shown but not added to the loss.
- Rigrod W. W. (1965). Saturation effects in high-gain lasers. J. Appl. Phys. 36, 2487–2490. doi:10.1063/1.1714517
- 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. 12 and 19. University Science Books.
- Ng E., Onodera T., Stansbury C., Zhang L. (2015). Light amplification and oscillation in the HeNe resonator. Stanford University course report. web.stanford.edu