Dye Laser Resonator
CW dye jet laser in a two-mirror cavity: TEM₀₀ waist, triplet loss versus flow speed, threshold pump power, small-signal gain, output power and slope efficiency.
g₁g₂
0.9604
g₁ = -0.9800, g₂ = -0.9800; stable
TEM₀₀ waist w₀ (in the jet)
30.57 µm
49.50 mm from mirror 1
Gain left after triplet absorption
71.51 %
Effective triplet lifetime τ_eff = 0.2492 µs (τ_T = 0.2500 µs)
Round-trip loss δ
5.056 %
−ln R_OC + 2 × loss per pass
Threshold pump power
0.8233 W
Overlap area π(w₀² + wₚ²)/2 = 2881 µm²
Small-signal gain g₀l
0.09212
Single pass at 3.000 W; threshold δ/2 = 0.02528
Output power
0.6420 W
I_sat = hc/(λσₑτ) = 728.8 kW/cm², near-threshold Rigrod
Slope efficiency
29.49 %
Mode–pump overlap 75.9 %; quantum defect limit λₚ/λ = 90.17 %
Gain cross-section net of triplets: σₑ − σ_T k_ST τ_eff. Triplets made upstream decay while the jet carries them through the mode, so τ_eff = (c√π/(2v)) erfcx(c/(2vτ_T)), c² = (w₀² + wₚ²)/2: τ_T for slow flow, the transit time √(π/8)√(w₀² + wₚ²)/v for fast flow. The pumped jet (pump fully absorbed) gives g₀l = ηₚ(σₑ − σ_T k_ST τ_eff)τPₚλₚ/(hcA), A = π(w₀² + wₚ²)/2, so the jet thickness cancels.
Threshold g₀l = δ/2. Output from Rigrod with the saturation area πw₀²(w₀² + 2wₚ²)/(2(w₀² + wₚ²)), which gives the near-threshold slope overlap w₀²(w₀² + 2wₚ²)/(w₀² + wₚ²)² of Laporta & Brussard (1991): ¾ for matched spots.
Default dye: rhodamine 6G in ethanol. σₑ(590 nm) = 1.1 × 10⁻¹⁶ cm², read off Fig. 6, and τ = 4.2 ns (Falkenstein, Penzkofer & Kaiser 1978); σ_T(590 nm) = 3.7 × 10⁻¹⁷ cm² from ε_T = 9.7 × 10³ M⁻¹cm⁻¹, measured in a 77 K glass (Pavlopoulos & Golich 1988); k_ST = 3.4 × 10⁶ s⁻¹ and τ_T = 0.25 µs in air-saturated ethanol, where dissolved O₂ quenches the triplets (Webb et al. 1970; degassed solutions keep them much longer). Together K_T = k_ST τ_T σ_T/σₑ = 0.29, against 0.35 ± 0.02 measured in a CW laser (Phillion, Kuizenga & Siegman 1974, water with Triton X-100). With air-saturated dye (τ_T ≈ 0.25 µs) oxygen quenching, not flow, clears the triplets, so the flow speed barely matters; degassed dye (τ_T ≈ 10 µs) needs v ≳ wₚ/τ_T.
Output power versus absorbed pump power
- Output power
- Operating point
Singlet gain left after triplet absorption versus flow speed (below 0 % no CW lasing)
- Gain left after triplet absorption
- This flow speed
TEM₀₀ beam radius between the mirrors
- TEM₀₀ radius w(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
Four-level dye with the pump fully absorbed in a jet at the cavity waist; the Gaussian pump and mode overlap over π(w₀² + wₚ²)/2. Triplets reduce the gain cross-section to σₑ − σ_T k_ST τ_eff; τ_eff = (c√π/(2v)) erfcx(c/(2vτ_T)), c² = (w₀² + wₚ²)/2, is the mode-weighted small-signal triplet lifetime in the flowing jet (derived here; τ_T for slow flow, the transit time for fast flow). Output from Rigrod's homogeneous model with the Laporta–Brussard near-threshold saturation area πw₀²(w₀² + 2wₚ²)/(2(w₀² + wₚ²)), slope overlap ¾ for matched spots. Default dye: rhodamine 6G in ethanol at 590 nm (Falkenstein 1978 σₑ and τ; Pavlopoulos 1988 σ_T, from a 77 K glass; Webb 1970 k_ST and τ_T, air-saturated). Ground-state reabsorption and the gain spectrum are ignored.
- Peterson O. G., Tuccio S. A., Snavely B. B. (1970). CW operation of an organic dye solution laser. Appl. Phys. Lett. 17, 245–247. doi:10.1063/1.1653384
- Webb J. P., McColgin W. C., Peterson O. G. et al. (1970). Intersystem crossing rate and triplet state lifetime for a lasing dye. J. Chem. Phys. 53, 4227–4229. doi:10.1063/1.1673926
- Svelto O. (2010). Principles of Lasers, 5th ed., ch. 6–7. Springer. doi:10.1007/978-1-4419-1302-9
- Rigrod W. W. (1965). Saturation effects in high-gain lasers. J. Appl. Phys. 36, 2487–2490. doi:10.1063/1.1714517
- Falkenstein W., Penzkofer A., Kaiser W. (1978). Amplified spontaneous emission in rhodamine dyes. Opt. Commun. 27, 151–156. Rh6G in ethanol: σₑ spectrum (Fig. 6), τ_F = 4.2 ± 0.3 ns. doi:10.1016/0030-4018(78)90196-7
- Pavlopoulos T. G., Golich D. J. (1988). Triplet extinction coefficients of some laser dyes I. J. Appl. Phys. 64, 521–527. Rh6G ε_T(590 nm) = 9.7 × 10³ M⁻¹cm⁻¹. doi:10.1063/1.341991
- Phillion D. W., Kuizenga D. J., Siegman A. E. (1974). Rotational diffusion and triplet state processes in dye laser solutions. J. Chem. Phys. 61, 3828. K_T = 0.35 ± 0.02 at 596 nm. doi:10.1063/1.1682572
- Laporta P., Brussard M. (1991). Design criteria for mode size optimization in diode-pumped solid-state lasers. IEEE J. Quantum Electron. 27, 2319–2326. Slope overlap (∫s r dV)²/∫s² r dV. doi:10.1109/3.97276
- Shayeganrad G. (2012). Optimum design of end-pumped solid-state lasers. In Solid State Laser, InTech, eqs. 10–12. doi:10.5772/38902
- Hollberg L. (1990). CW dye lasers. In Dye Laser Principles (Duarte F. J., Hillman L. W., eds.), ch. 5, 185–238. Academic Press. Jet flow and pump-spot regime. doi:10.1016/B978-0-12-222700-4.50010-3