Q-Switched Laser
Q-switched laser: peak and average power and duty cycle of a pulse train, plus pulse energy, width and peak power from the Q-switch rate equations.
Pulse train from a measured pulse
Peak power
93.94 kW
2√(ln2/π) E/τ
Average power
10.00 W
E × f
Duty cycle
0.01000 %
τ × f; peak/average 9394
If the pulse period 1/f is shorter than the upper-state lifetime (≈ 230 µs in Nd:YAG), the inversion doesn't recover between pulses, so r and E fall at high rates.
Pulse from the cavity (rate equations, defaults: Nd:YAG)
Pulse energy
2.635 mJ
extraction η = 94.05 %; 0.3355 J/cm²
Pulse width (FWHM)
15.49 ns
2.704 τ_c; estimate E/P_p = 17.93 ns
Peak power
146.9 kW
average at this rate 26.35 W
Photon lifetime τ_c
5.729 ns
L_e/(cγ), γ = 0.1985 per pass, L_e = 341.0 mm
Optimum output coupling at this pump (Degnan 1989)
Optimum R_oc
0.7410
z = g₀l/γ_i = 29.48; g₀l = 0.5956, from r·γ
Largest pulse energy
2.655 mJ
η = 1 − 1/z = 96.61 %; 99.23 % of it at this R
Width at the optimum (estimate)
18.12 ns
E/P_p; r = z ln z/(z − 1) = 3.503
Peak power at the optimum
146.5 kW
τ_c = 6.689 ns
Peak power of a measured pulse: shape factor × E/τ, with τ the intensity FWHM.
Rate equations (Wagner–Lengyel, Degnan), four-level medium with an instant switch: rη = −ln(1 − η), E = (hνA/2σ) ln(1/R) rη, P_p = (γ₂/2)(hνA/σ)(r − 1 − ln r)/τ_c with γ₂ = −ln R, γ = γ_i + γ₂/2 and τ_c = L_e/(cγ), L_e = L + (n − 1)l. The plotted shape is integrated numerically; the FWHM comes from it.
With r as the input it stays fixed when you change R_oc. At a fixed pump the initial gain g₀l = rγ is fixed instead, so r = g₀l/γ falls as R_oc drops, and the energy peaks at Degnan's optimum: with z = g₀l/γ_i and γ_i = −ln(1 − internal loss), ln(1/R_opt) = 2γ_i(z − 1 − ln z)/ln z and E_max = (hνA/2σ)·2γ_i(z − 1 − ln z); the final inversion is then 1/z of the initial one. The optimum is computed from g₀l in both modes; the chart holds g₀l fixed.
Pulse energy vs output coupler at fixed g₀l
- Pulse energy at this g₀l
- This R
- Optimum (Degnan)
Q-switched pulse from the rate equations
- Output power
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
Pulse train: shape factor × E/τ for Gaussian, sech² or flat pulses. Cavity model: four-level rate equations with an instant switch and an empty lower level (Wagner–Lengyel, Degnan), top-hat mode, integrated numerically for the shape and FWHM. Pump as r or as g₀l (r = g₀l/γ); Degnan's optimum coupling and E_max in closed form. A slow lower-level decay, a finite switching time and transverse gain variation are not modelled. Nd:YAG σ default from Koechner.
- Wagner W. G., Lengyel B. A. (1963). Evolution of the giant pulse in a laser. J. Appl. Phys. 34, 2040–2046. doi:10.1063/1.1729732
- Degnan J. J. (1989). Theory of the optimally coupled Q-switched laser. IEEE J. Quantum Electron. 25, 214–220. doi:10.1109/3.16265
- Svelto O. (2010). Principles of Lasers, 5th ed., ch. 8. Springer. doi:10.1007/978-1-4419-1302-9
- Koechner W. (2006). Solid-State Laser Engineering, 6th ed., ch. 2 and 8. Springer. doi:10.1007/0-387-29338-8