Estimation of thermal fracture limits in quasi-continuous-wave end-pumped lasers through a time-dependent analytical model

E. H. Bernhardi; A. Forbes; C. Bollig; M. J. D. Esser

doi:10.1364/OE.16.011115

Optics Express
Vol. 16,
Issue 15,
pp. 11115-11123
(2008)
•https://doi.org/10.1364/OE.16.011115

Estimation of thermal fracture limits in quasi-continuous-wave end-pumped lasers through a time-dependent analytical model

E. H. Bernhardi, A. Forbes, C. Bollig, and M. J. D. Esser

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E. H. Bernhardi, A. Forbes, C. Bollig, and M. J. D. Esser, "Estimation of thermal fracture limits in quasi-continuous-wave end-pumped lasers through a time-dependent analytical model," Opt. Express 16, 11115-11123 (2008)

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Abstract

A time-dependent analytical thermal model of the temperature and the corresponding induced thermal stresses on the pump face of quasi-continuous wave (qcw) end-pumped laser rods is derived. We apply the model to qcw diode-end-pumped rods and show the maximum peak pump power that can be utilized without fracturing the rod. To illustrate an application of the model, it is applied to a qcw pumped Tm:YLF rod and found to be in very good agreement with published experimental results.The results indicate new criteria to avoid fracture when operating Tm:YLF rods at low qcw pump duty cycles.

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Supplementary Material (2)

Media 1: MOV (754 KB)

Media 2: MOV (329 KB)

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Fig. 1. An example of a measured top-hat transverse intensity profile produced by a fibre coupled diode laser pump (own experimental results).

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Fig. 2. The analytically (red) and numerically (black) predicted temperature in the centre of the Tm:YLF rod as a function of time while the rod is subjected to a qcw pump with a peak power of (a) 200 W at 10 Hz (τ_on =10 ms) and, (b) 90 W at 50 Hz (τ_on =10 ms).

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Fig. 3. The maximum stress on the pump face of the Tm:YLF rod as a function of time while the rod is subjected to a qcw pump with a peak power of (a) 200 W at 10 Hz (τ_on = 10 ms), and (b) 90 W at 50 Hz (τ_on = 10 ms). The analytical (red) and numerical (black) solutions are shown.

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Fig. 4. (0.75 MB and 0.33 MB respectively) Animations of (a) the analytical stress distribution on the pump face [Media 1] and (b) the numerical stress distribution in volume of the Tm:YLF rod while it is subjected to a 90 W peak power qcw pump beam at 50 Hz (τ_on = 10 ms) [Media 2].

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Fig. 5. (a). The average pump power (as a fraction of the cw fracture power P_cw ) at which fracture of the Tm:YLF rod occurs as a function of qcw pump duty cycle (τ_on =10 ms). The green shaded region indicates the average pump power at which the Tm:YLF rod can be pumped without fracturing according to the analytical model. The yellow shaded region indicates the difference between the analytical model and P_cw . (b) The same notation as in (a) but for the peak pump power (in units of P_cw ) at which fracture of the Tm:YLF rod occurs as a function of qcw pump duty cycle.

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Tables (1)

Table 1. Parameter values of the pumped Tm:YLF rod that were implemented in the simulations.

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Equations (13)

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\frac{\partial u (r, t)}{\partial t} - D \nabla^{2} u (r, t) = Q (r, t),

u (r, t) = \int_{0}^{t} \int_{0}^{R} Q (ξ, τ) G (r, ξ, t - τ) dξdτ,

G (r, ξ, t) = \sum_{m = 1}^{\infty} \frac{2 ξ}{R^{2} J_{1}^{2} (μ_{m})} J_{0} (\frac{μ_{m} r}{R}) J_{0} (\frac{μ_{m} r}{R}) \exp (- \frac{D μ_{m}^{2} t}{R^{2}}) .

Q (r, t) = {\begin{matrix} \frac{αηE}{π w^{2} ρ C_{p} τ_{on}} \\ 0 \end{matrix} \begin{matrix} ; nT \leq t \leq nT + τ_{on} \\ ; nT + τ_{on} \leq t \leq (n + 1) T \end{matrix},

u (r, pT + t) = \frac{2 αηER}{kπw τ_{on}} \sum_{m = 1}^{\infty} \frac{J_{0} (\frac{μ_{m} r}{R}) J_{1} (\frac{μ_{m} w}{R}) f (p, t, μ_{m})}{μ_{m}^{3} J_{1}^{2} (μ_{m})},

f (p, t, μ_{m}) = \exp (- μ_{m}^{2} \frac{t}{τ_{D}}) {\frac{[\exp (μ_{m}^{2} \frac{τ_{on}}{τ_{D}}) - 1] [\exp (- μ_{m}^{2} \frac{pT}{τ_{D}}) - 1]}{1 - \exp (μ_{m}^{2} \frac{T}{τ_{D}})} - [1 - \exp (μ_{m}^{2} \frac{τ}{τ_{D}})]},

σ_{r} (r, t) = c [\frac{1}{R^{2}} \int_{0}^{R} u (r, t) rdr - \frac{1}{r^{2}} \int_{0}^{r} u (r, t) rdr];

σ_{θ} (r, t) = C [\frac{1}{R^{2}} \int_{0}^{R} u (r, t) rdr + \frac{1}{r^{2}} \int_{0}^{r} u (r, t) rdr - u (r, t)],

σ_{r} (r, pT + t) = \frac{2 CαηER}{kπw τ_{on}} \sum_{m = 1}^{\infty} \frac{J_{1} (\frac{μ_{m} w}{R})}{μ_{m}^{3} J_{1}^{2} (μ_{m})} [\frac{J_{1} (μ_{m})}{μ_{m}} - \frac{R J_{1} (\frac{μ_{m} r}{R})}{{rμ}_{m}}] f (p, t, μ_{m});

σ_{θ} (r, pT + t) = \frac{2 CαηER}{kπw τ_{on}} \sum_{m = 1}^{\infty} \frac{J_{1} (\frac{μ_{m} w}{R})}{μ_{m}^{3} J_{1}^{2} (μ_{m})} [\frac{J_{1} (μ_{m})}{μ_{m}} + \frac{R J_{1} (\frac{μ_{m} r}{R})}{{rμ}_{m}} - J_{0} (\frac{μ_{m} r}{R})] f (p, t, μ_{m}) .

σ_{T} (r, pT + t) = ∣ σ_{θ} - σ_{r} ∣

= ∣ \frac{2 CαηER}{kπw τ_{on}} \sum_{m = 1}^{\infty} \frac{J_{1} (\frac{μ_{m} w}{R})}{μ_{m}^{3} J_{1}^{2} (μ_{m})} J_{2} (\frac{μ_{m} r}{R}) f (p, t, μ_{m}) ∣,

P_{av \leq P_{cw}} .

Parameter	Thermal Model	Reference
Pump beam radius (w) [mm]	0.47	[15]
Rod radius (R) [mm]	1.5	[15]
Absorption coefficient (α) [cm^-1]	1.43	[15]
Thermal conductivity (k) [W.m^-1.K^-1]	7.2 (a-axis), 5.8 (c-axis)	[1,16]
Linear expansion coefficient (γ) [10^-6 K^-1]	13 (a-axis), 8.0 (c-axis)	[1,17]
Fractional heat load (η)	0.33	estimated
Poisson’s ratio ( V )	0.33	[1,16]
Young’s modulus (Y) [GPa]	75	[1,2]
Density (ρ) [g.cm^-3]	3.9	[1,17]
Specific heat capacity (C_p ) [J.g^-1.K^-1]	0.79	[1,17]

Abstract

Supplementary Material (2)

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Tables (1)

Equations (13)

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