Analytic spatial and temporal temperature profile in a finite laser rod with input laser pulses

Abstract: In this communication, we present an analytic expression of the thermal load in a cylindrical laser rod. We consider a pump beam with Gaussian temporal and spatial profile, which permits, using superposition of the single pulse solution, an explicit calculation of the optical path length difference across the radial direction of the rod and of the transient thermal focal length changes for a variable pump repetition rate and pulse width. We have chosen to model Ti:Al2O3 as a specific example, however our solution is completely general and can be applied to any materials with cylindrical geometry employing a stable laser cavity design.

Over the past twenty years, the development of high-repetition, high average power amplifier of solid-state laser materials has revived considerable interest in optically pumped solid-state lasers [1][2][3][4][5][6][7][8][9][10][11][12][13].However, the energy deposited to excite the gain media generates a dynamic heating profile in the material, resulting in a spatial-dependent refractive index.Since the refractive index in a laser material is temperature dependent, the gain material exhibits a spatial index variation, resulting in potentially severe focusing of the seed laser (thermal lensing).Different solutions to this problem have emerged, including the introduction of compensating optics in the amplifier [2] and an amplifier design based on a cryogenically cooled crystal (since thermal conductivity of Ti:Al 2 O 3 increases with decreasing temperature) [1].
Since Koechner pioneered this field (see [3] and references herein), a number of analytic solutions for the thermal lensing properties have been presented [4][5][6][7].For example, the general solution for describing the thermal profile in an "infinite" cylinder is given by [7].However, to our knowledge none of these analyses has solved the exact analytic thermal expression in a finite cylindrical geometry including a transient temperature profile for Gaussian spatial profile and time profile based on the input pulses.This is important because the high heat intensity of such a powerful laser can be used to calculate thermal lensing distortions and beam mode changes in all optical components.
Here, an analytic expression is presented for thermal lensing properties of optical crystals with Gaussian spatial and temporal profile of the source term.The fully analytic expression given in this paper can also be used to establish a stable laser cavity.We focus on the thermal load in Ti:Al 2 O 3 because it is proven material for high-power laser applications [8], but our solution is general and can be applied to any kind of materials with cylindrical geometry.
In this paper, we consider the case in which the pump light absorption generates the heat within the laser rod; this heat is removed by a coolant flowing only along the cylindrical rod surface.The heat loss from the free ends of the crystal is neglected [5,6].With this assumption, the heat flows radially and also longitudinally.For the calculation of the temperature distribution in a laser material, we assume that physical properties of the material (thermal conductivity, absorption coefficient and so forth) are constant over the range of temperatures involved in the heat equation and that the heat transport within the material is isotropic [6].Based on these assumptions, we start our analysis from the classical heat equation, specifying the time evolution of the temperature at all points in a laser load [5,6] 2 ( , ) where C and κ are the laser material's heat capacity and its thermal conductivity.S(r,T) represents the thermal power density of the thermal heat source and can be expressed as a Gaussian spatial time profile of a pulse in case of a single pulse operation, where N, 2ω , 2 h τ and Δt are the normalization factor, a beam diameter, a pulse width, and the temporal center of the Gaussian, respectively.Since S is the power density, i.e. energy per unit time and unit volume, the integration with respect to space (when l is the length of the laser rod) and time is the heat dissipated in the laser rod from a single pulse energy, A: In most end-pumped laser cavity, the time varying source term is a linear sum of multiple Gaussians: Given a geometry in laser materials of cylindrical symmetry with radius a and length l, several boundary conditions could be imposed [6].For the case of an end-pumped crystal in thermal contact with a liquid-cooled holder, constant temperature at the cylindrical surface is an appropriate boundary condition.Neglecting heat loss from the free ends of the crystal, the derivative of the solution vanishes at the end points [5,6].
For the system of cylindrical symmetry, Eq. ( 1) takes the form With the form of a single pulse operation as shown in Eq. ( 8), the general solution can be directly obtained by using the method of separating variables [9].Bessel functions of the first kind 0 ( ) and orthogonal condition [9,10] form an orthogonal complete set of functions, and the general solution of Eq. ( 8) becomes The boundary condition (Eq.( 7)) results only in a z-dependent cosine series ( 0 Inserting Eq. ( 11) into Eq.( 8) and using Eq. ( 9) leads to  Inserting these two expressions (Eqs.( 13) and ( 15)) into Eq.( 12), we find The form of integration is a simple error function defined in [9] where ( ) 0 ( 1) t o r θ ′ = when 0 t′ < (or 0 t′ ≥ ) We thus have the full solution-satisfying Eq. ( 8) If a laser rod is pumped by a single pulse, a transient thermal profile will be established as a result of the fast heating process followed by a slow recovery of the rod to thermal equilibrium as shown in Fig. 1(a) with the boundary condition T(r = a,z) = 0 corresponding a fixed surface temperature, which is taken as zero since the heat transport equation is linear.[12], corresponding to the values when pumping at 532 nm of Ti:sapphire crystal with 0.15 doping.We assumed the input beam radius as 50 m μ .
We now allow the solution T(r,z,t) to evolve to increase the crystal temperature with multiple Gaussian pulses.As seen from Eq. ( 5), multiple Gaussian pulses can be involved to increase the crystal temperature.Since ( ) nm t ϕ is the solution of the differential Eq. ( 24) with one Gaussian pulse input, Therefore the solution of a single-shot operation can be used for multiple Gaussian inputs.
Figure 2 shows a thermal build-up when multiple pulses are introduced to the laser rod with a pulse interval that is smaller than the thermal relaxation time.Under these conditions, a steady-state operating condition will soon be established by the laser (as shown in Fig. 2 in which the radial flow of heat out of the rod equals the total heat being put into the rod.In the extreme case, when the pulse interval time becomes negligibly small as compared to the thermal relaxation time, the transient thermal profile in the laser rod approaches the continuous wave condition.In this region, thermal effects depend only on the average input power.Fig. 2. Thermal buildup when the pulse interval time is less than the thermal relaxation time.For the calculation: (a) three respectively pulsed lasers are introduced to the laser rod with 20μs pulse separation; (b) five thousand pulsed lasers are introduced to the laser with 5-kHz repetition rate.All parameters for the calculation are the same as those given for Fig. 1.
The transient temperature distribution in a laser rod as illustrated in Fig. 2 eventually affects energy output and beam divergence of the laser.The transient thermal lensing properties of a seed pulse sent through the amplifier samples depends on the optical path length of the seed pulse through the crystal and simply takes the form The refractive index near the center of the crystal can be approximated by a first-order expansion around the temperature of the center, T C [5] ( ) ( ) ( ) As shown in [5], the z-dependent thermal profile contains cos( ) m l π and vanishes after integration along z except for an m = 0 constant term Using a zero-order Bessel function around r = 0,   However, with a high enough pulse repetition rate as shown in Fig. 3(c), the focal length become steady and the focal length at the final steady state is less than 5 cm.Due to the simplicity of the focal length expression (Eq.( 31)), it is well suited for the evaluation of the acceptable parameter range for future solid-state laser cavity (by analyzing this thermo-optical property of the system, it is possible to estimate the minimum repetition rate of pump laser to give a stable laser cavity design).All of these types of transient focal length changes in the laser cavity result in serious seed beam distortion through the laser rod, making the establishment of either a high (or low) repetition rate for stable unamplified laser activity impossible.However, thermally induced beam distortion can successfully be removed with the self-adaptive optical elements [13] as described at the beginning of this paper.Placing a suitable material having negative dn/dT after the laser rod, self-balancing thermal lenses can be developed [13].Appropriate design of the compensation devices can be obtained by the above thermal analysis.
In summary, we provide the exact analytic expression, including the transient temperature profile, for a laser-pumped solid-state amplified system.The analysis can directly be applied for the evaluation of thermal-lensing effects for the future development of laser technologies.Since the calculation in this paper assumes that all pulse energy is deposited as heating of the crystal rod, the results presented here are the upper limits of the thermal effects.

π
, integrate from 0 to a and use the orthogonality conditions of the Bessel functions (Eq.(10)the fact that (in most cases) the beam waist ω is smaller than the radius of the laser rod and an oscillation of the Bessel function 0 ( ) ], the coefficient C n can be simplified as 2 , integrate from 0 to l and use the orthogonality conditions of the cosine functions,

Fig. 1 .
Fig. 1.(a) Thermal relaxation of the laser rod of a single input pulse in a cylindrical geometry (0.7-cm long, 0.25-cm radius).For this calculation, we used a time duration, 160ns τ = and a pulse energy of 10mJ.(b) Longitudinal temperature distribution shows the initial exponential (rd curve) and temperature distribution at various times at r = 0.As illustrated in Fig. 1(a), temperature increases without any observable delay with respected to the energy deposited in the rod.The longitudinal temperature evolution is plotted in Fig. 2(b) at varying times.The thermal conductivity, heat capacity, and absorption coefficient applied are 1 1 0.33Wcm K κ

,
the time-dependent solution for many input pulses can be expressed as 0

Fig. 3 .
Fig. 3. (a) Transient focal length change of one single pulse for 1 ms (solid red), 1 s μ (solid green) and 1 ns (solid blue) time duration.Focal length changes of (b) 100 Hz and (c) 5 kHz repetitively pulsed pump laser input for a 1ms, 10mJ pulse.All of the parameters used in this calculation except for the pulse duration and the repetition rate are equal for all three cases and identical to those used in Fig. 1.

Figure 3 (
Figure 3(a) shows the transient focal length change for a single pulse operation.There is a focal length difference between a 1-ms pulse and 1 s μ at earlier time region comparable with the pulse widths shown in Fig.3(a).However, since temperature increases without any observable delay with respected to the energy deposited in the rod for a pulse lasting from 1 ns to 1 s μ (as seen from Fig.3(a)), we found that the transient focal length shows negligible dependency on the pulse width with a pulse lasting for between 1ns to 1 s μ .The transient focal length changes for all these three cases follows the same line after 1 ms.For multiple pulse input, when the time separation between pulses is comparable to the material's thermal relaxation time, we have a result of transient focal length variation as shown in Fig.3(b) as we could expect, and it never becomes stable (it varies between 20 and 1000 cm in Fig.3(b)).However, with a high enough pulse repetition rate as shown in Fig.3(c), the focal length (11)ave obtained the coefficients for spatial components of the general solution, Eq.(11).To obtain the complete solution, including transient heat conduction in the finite rod, we need to 16) #227056 -$15.00USD Received 19 Nov 2014; revised 17 Jan 2015; accepted 19 Jan 2015; published 29 Jan 2015 (C) 2015 OSA 9 Feb 2015 | Vol. 23, No. 3 | DOI:10.1364/OE.23.002591 | OPTICS EXPRESS 2594 κ κ τ