Zigzag Slab Laser

Zigzag slab laser: bounce angle and total internal reflection, threshold and output power, slope efficiency, heat density, temperature profile and the thermal lens the zigzag cancels.

Model: Textbook approximation

Slab and resonator

Slab width W (mm)Slab thickness t (mm, cooled faces)Slab length L (mm)Thickness crossings NCrystal index nOutput coupler reflectance (HR = 1)Scatter loss (1/cm)Absorbed pump power (W)Pump efficiency η_p (absorbed → upper level)

Gain medium (defaults: Nd:YAG)

Laser wavelength λ (nm)Pump wavelength λ_p (nm)Emission cross-section σ (10⁻¹⁹ cm²)Upper-level lifetime τ (µs)Heat fraction of the absorbed pumpThermal conductivity κ (W/(m K))dn/dT (10⁻⁶/K)

Zigzag angle θ

17.74°

face incidence 72.26°; critical 33.33°

Zigzag path L/cos θ

52.50 mm

aperture W t cos θ = 19.05 mm²

Threshold (absorbed pump)

57.43 W

δ = 12.64 % round trip

Output power

72.23 W

slope 50.66 %; P_sat = 552.2 W

Heat density Q

64.00 W/cm³

heat fraction × pump / W t L

Centre temperature rise

2.286 K

Q t²/(8κ); mean 1.524 K

Straight-through thermal lens

0.5993 m

κ/((dn/dT) Q L); the zigzag cancels it to first order

θ = atan(N t/L). Four-level gain: P_th = δ h c A/(2η_p σ τ λ_p), A = W t cos θ, δ = −ln R_oc + 2α_s L/cos θ; output from Rigrod with the mode filling the slab.

ΔT(y) = Q(t²/4 − y²)/(2κ). A straight-through ray sees n(y) = n₀ − (dn/dT)Qy²/(2κ), a lens f = κ/((dn/dT)QL); each zigzag ray samples the whole profile, so this lens and its depolarisation cancel to first order (stress and end effects are not modelled).

Output power versus absorbed pump power

Output power
This pump

Temperature across the slab thickness

Temperature rise above the faces

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 gain with uniform inversion over the slab, the zigzag beam filling the aperture W t cos θ, Rigrod output. Infinite-slab heat conduction with uniform heat density; the straight-through lens keeps only dn/dT (no stress-optic, end-bulging or edge effects). Heat fraction 0.32: Fan's (1993) estimate for Nd:YAG pumped at 808 nm and lasing at 1064 nm (non-lasing, he measured 0.37–0.43).

Eggleston J. M., Kane T. J., Kuhn K., Unternahrer J., Byer R. L. (1984). The slab geometry laser — Part I: Theory. IEEE J. Quantum Electron. 20, 289–301. doi:10.1109/JQE.1984.1072386
Kane T. J., Eggleston J. M., Byer R. L. (1985). The slab geometry laser — Part II: Thermal effects in a finite slab. IEEE J. Quantum Electron. 21, 1195–1210. doi:10.1109/JQE.1985.1072799
Fan T. Y. (1993). Heat generation in Nd:YAG and Yb:YAG. IEEE J. Quantum Electron. 29, 1457–1459. Abstract: 1.04 at.% Nd:YAG pumped at 808 nm, 0.37–0.43 non-lasing, about 0.32 with 1064 nm extraction. doi:10.1109/3.234394
Koechner W. (2006). Solid-State Laser Engineering, 6th ed., ch. 2–3 and 7. Springer. doi:10.1007/0-387-29338-8

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