Analytical thermal model for end-pumped solid-state lasers

Fundamentally power-limited by thermal effects, the design challenge for end-pumped “bulk” solid-state lasers depends upon knowledge of the temperature gradients within the gain medium. We have developed analytical expressions that can be used to model the temperature distribution and thermal-lens power in end-pumped solid-state lasers. Enabled by the inclusion of a temperature-dependent thermal conductivity, applicable from cryogenic to elevated temperatures, typical pumping distributions are explored and the results compared with accepted models. Key insights are gained through these analytical expressions, such as the dependence of the peak temperature rise in function of the boundary thermal conductance to the heat sink. Our generalized expressions provide simple and time-efficient tools for parametric optimization of the heat distribution in the gain medium based upon the material and pumping constraints.


Introduction
The end-pumped solid-state laser is a mature design architecture exploited for many scientific, industrial, and medical laser applications, in the tens-of-watts power regime. Cost effective, compact, and relatively efficient, in recent decades

Temperature dependence of thermal conductivity
We focus our attention on the Nd:YAG crystal. Since YAG is a cubic crystal, the thermal conductivity can be considered as a scalar quantity. The dependence on temperature of the thermal conductivity can be modeled by the following formula: where k 0 , T 0 and m result from a best fit to measured data. Since these parameters strictly depend on the doping level of the rare-earth ion, we choose a particular doping concentration, say ∼1%. We performed two best fits for two different temperature ranges around CT and RT. The first fit is based on measured data on a 1.3 at.% Nd:YAG sample in a temperature range of 40 K≤ T ≤175 K [10], while the second is based on published data of Sato et al. [11] relating to 1.2 at.% Nd:YAG in a temperature range of 300 K≤ T ≤ 475 K. By finding the intersection of the two fitting curves, a common value of T 0 and k 0 = k(T 0 ) for the two temperature ranges can be found. The values of the parameters for ∼1 at.% Nd:YAG are, therefore, as follows: T 0 = 164.17 K, k 0 = 15.09 WK −1 m −1 , m CT = −1.77, and m RT = −0.75. The data used and the corresponding best fits curves are shown in Fig. 1.

Temperature dependence of thermal conductivity for selected materials
In Table 1, the values of k 0 , T 0 , and m are provided for Nd:YAG at different temperatures, along with different doping concentrations and a variety of other interesting host materials, including some that would not normally be considered for an end-pumped rod architecture, rather more appropriate to a thin-disk geometry. These values result from a best fit of Eq. (1) to available data from the published literature [6,11,12]. As previously mentioned, the thermal conductivity can be considered a scalar quantity strictly Measured data Data from Sato et al. Best fit to measured data Best fit to data from Sato et al. Fig. 1 Measured thermal conductivity dependence on temperature, after [10,11], and best fits using Eq. (1) Analytical thermal model for end-pumped solid-state lasers for isotropic laser crystals. However, since, in this thermal model, a pure radial heat flux will be assumed (see Sect. 3), uniaxial laser materials can also be included and, in this case, only the dependence on temperature along the a-axes is considered.

Analytical solution
The steady-state heat conductance equation, with temperature-dependent thermal conductivity and different endpumping profiles, governing the system shown in Fig. 2, is: where k(T) is the temperature-dependent thermal conductivity and S(r, z) is the thermal power per unit volume dissipated in the laser rod.
The edge of the rod is considered to be surrounded by a heat sink, held at a constant temperature, T c , by active cooling. Heat transfer across the boundary between the laser rod and the heat sink is described by the heat transfer coefficient or surface conductance, h. The end faces are instead supposed to be in contact with air, with negligible heat flux through them. An equivalent h ef coefficient can be calculated for the end faces [15], and since typically h >> h ef , the assumption of pure radial heat flux can be made [5]. Furthermore, it is assumed in the following that the diameter of the pumped region is significantly smaller than the inverse of the absorption coefficient, reinforcing the statement of radial heat flux. It follows that the longitudinal derivatives in Eq. (2) with respect to the corresponding radial derivatives [14] can be neglected. Here, it is also assumed that the pump profile is axisymmetric, the behavior of the thermal conductivity k(T) of the crystal is described by Eq. (1), and the cooling is isotropic in the z-plane. These assumptions allow us to write Eq. (2) as follows: Equation (3)  where C is an arbitrary constant of integration. It is important to note that Eq. (7) is not valid if m = −1; however,  the temperature distribution in this case has already been derived by Hodgson and Weber for a side-pumped laser rod [9]. The four different pump distributions studied in this paper, corresponding to four different S(r, z) terms in Eq. (6), are shown in Fig. 3 at z = 0 (see Sect. 3.1 for the value of the parameters considered). They are, respectively, the top-hat (TH), Gaussian (G), super-Gaussian (SG), and donut (D). Two super-Gaussian profiles are reported in the figure, one for the fourth order and the other for the eighth order.

Top-hat pumping
Although not a typically realistic pump distribution for endpumped lasers, the top-hat profile is the simplest starting solution, which can be easily compared with the previous literature studies [5,15]. Consider the following thermal loading distribution: where Q 0 is a normalization constant, is the pump absorption coefficient, a is the radius of the pumping beam, and b is the radius of the laser rod. The normalization constant Q 0 can be calculated using the following relation: where h P is the total heating power ( h is the fractional thermal load) and V is the volume of the pump-photon distribution in the rod, where it is assumed that there are insignificant energy migration effects and the heat source is created at the point of pump-photon absorption, that is where 1 − e − L = abs is the absorption efficiency. Substituting Eqs. (9) and (10) into Eq. (8), the thermal power per unit volume dissipated into the laser rod becomes: where P in is the incident pump power. Equation (6) is solved separately between 0 ≤ r ≤ a and a < r ≤ b, resulting in two functions U 1 and U 2 , respectively, leading to two temperature solutions, T 1 (r, z) and T 2 (r, z) in the respective regions, as shown in Appendix A. These solutions are as follows: for 0 ≤ r ≤ a and for a < r ≤ b, where F 0 is a constant defined as follows: Figure 4 shows the calculated temperature distribution using Eqs. (12) and (13) with the following parameters: b = 1.25, L = 5 mm, a = 300 μm, P in = 25 W, h = 0.25, T c = 300 K, = 350 m −1 and h = 2 WK −1 cm −2 (the choice of this value for h will be discussed in Sect. 4).
The temperature-independent thermal conductivity case can be obtained from Eqs. (12) and (13) setting m = 0. In this way, the temperature distribution inside the rod is given by the following: where ΔT(r, z) = T(r, z) − T(b, z). Equation (15) shows the well-known quadratic radial dependence of the temperature inside the pumped region of the rod and the logarithmic dependence outside [15]. In Fig. 5, a comparison, at the pumped surface of the rod, z = 0, for ΔT(r, z) considering a temperature-dependent thermal conductivity, and Eq. (15), is shown. The parameters for this comparison are the same as those for Fig. 4. The constant thermal conductivity in Eq. (15) is evaluated using Eq. (1) at T = T c .

Temperature profile for different coolant temperatures
Maintaining the same thermal load conditions, whilst lowering the temperature of the heat sink surrounding the Nd:YAG crystal, which is an artificial example as the spectroscopic proprieties of the gain medium also change [13], a decrease in the maximum temperature rise at the center of the crystal is obtained. This is due to the corresponding increase in the thermal conductivity as shown in Fig. 1. Equations (12) and (13) can be used to have a quantitative measure of this effect. In Fig. 6, the temperature rise is calculated with the same parameters above and different coolant temperatures.

Gaussian pumping
A more realistic and often used pumping distribution employing a diffraction-limited pump is a Gaussian, which is defined as follows: where w is the 1/e 2 radius of the intensity profile of the pump beam. The volume of the heated region is given by the following: and the term in the brackets in Eq. (17) can be set to unity in most cases of interest, where the pump beam is significantly smaller than the radius of the laser rod. Using Eqs. (9) and (16), the heat source becomes: Thus, Eq. (6) in the case of Gaussian pumping is where E 1 (u) is the exponential integral defined as follows: Note that if a temperature-independent thermal conductivity (i.e., m = 0) is considered, Eq. (21) becomes which gives a temperature difference between the center and the edge of the rod equal to 24) is the same result as the one published by Innocenzi et al. [14]. Figure 7 shows a comparison between our temperature profile and the one published by Innocenzi et al. with the same parameters used for the tophat case (w = 300 μm is chosen). In addition, in this case, the constant thermal conductivity is evaluated using Eq. (1) at T = T c . In Fig. 8, the analytical temperature difference profiles with the same parameters above, but for different coolant temperatures, are again shown.

Generalized nth order super-Gaussian
Often in the case of higher power solid-state lasers, fibrecoupled diode-laser pumps are employed. For a substantial (23) part of the pump distribution in the laser rod, the pump can be approximated by a super-Gaussian. The heat source considered is as follows: where n is an even integer. The volume of the heated region can be found as follows: for higher values of n, this term is even smaller, it can be safely ignored. Furthermore, by definition, Γ 2 n , 0 = Γ 2 n , so the normalized heat source is  The center temperature is found to be maximum in the case of n = 4. This is quite surprising, since one might have expected that it increases going from the TH (n = ∞) to G pump profile (n = 2). A more in-depth analysis, varying both the transversal dimensions of the crystal and the pump laser waist in Eq. (28), shows that the relationship between the center temperatures in Figs. 10 and 11 remains the same. Indeed, this does not depend on the lab parameters, but rather on how G 0 and depend on n.
However, T(0, 0) is not included in the calculation of the thermal dioptric power of the pumped rod, which does follow the expected trend (see Sect. 5). It will be shown that what matters is the coefficient of the quadratic term in the Taylor expansion of the temperature distribution, whose modulus is maximum in the case of G pumping and decreases in the SG pumping case as n increases.

Gaussian solution as the special case of the super-Gaussian with n = 2
It can be shown for the special case of n = 2 in Eq. (28) that its solution for the temperature profile is the same as that derived from the Gaussian distribution (Eq. 21). In this case, the Generalized Hypergeometric Function in Eq. (28) corresponds to From Eq. (31) and substituting these values into Eq. (32), one obtains By the definition of the exponential integral: Equation (32) becomes: Substituting Eq. (36) into (28) with n = 2, the temperature profile is found to be exactly as Eq. (21).

Donut pumping
Finally, a fourth pump distribution that can be exploited to excite higher-order LG cavity modes is a donut shape [16]. This can be described by the following: Using the same procedure explained in the previous sections, the following temperature distributions can be obtained, respectively, for 0 ≤ r ≤ d 1 , d 1 < r ≤ d 2 , and d 2 < r ≤ b:

A brief digression on the heat transfer coefficient
The heat transfer coefficient, h, and its influence on the temperature distribution in this thermal model deserves a special mention. For the sake of simplicity, we limit our discussion to the pump-input surface, z = 0, where the maximum rise of temperature is expected. In the case of a top-hat pumping distribution, from Eq. (12), the maximum temperature is given by the following: Figures 13 and 14 show, respectively, the dependence of Eq. (41), and its derivative with respect to h, on h. Input parameters chosen are the same as in Sect. 3.1, with T c = 300 K. As one can see from these figures and as one would expect, an increasing h implies that the maximum temperature rise asymptotes to a minimum constant value. Since this constant value is strictly reached at h = ∞ (i.e., in the case of perfect cooling), it is not easy to define an expression for a critical (41) value of h, below which the maximum temperature rises rapidly. However, from the only h-dependent term in Eq. (41), one can define the limiting case, such that if then the dependence of T 1 (0, 0) on h can be ignored. For the parameters used in this paper, the left-hand side of Eq. (42) is approximately equal to 0.093 WK −1 cm −2 . As it can be seen, this value is just off the scale in Figs. 13 and 14, and ultimately a tolerable increase in the peak temperature rise would have to be chosen to find a critical value for h. For example, a 1% increase with respect to T c (e.g. at 300 K) would imply a critical value for h ∼ 9.3 WK −1 cm −2 . This is five times higher than the best value reported by Chénais et al. [5], when employing heat sink grease as the thermal interface to a copper mount.

Dependence of the temperature rise on h
To date, the effect of h has only been related to the temperature difference across the boundary layer, since from the temperature-independent thermal conductivity model, the temperature difference between the center of the rod and the boundary does not depend on h [5]. As can be seen from Eqs. (41) and (13), this is no longer true when considering k(T), as expressed in: (note that setting m = 0, this dependence disappears).
Actually this is quite intuitive when considering k(T), as h, resulting mainly in a different temperature at the boundary of the rod, therefore, changes the thermal conductivity of the rest of the laser crystal.

Analytical expression for the thermal-lens power
Consider first the case of top-hat pumping and assume h = ∞, i.e., use the approximation of perfect cooling. A Taylor expansion can be performed at r ≈ 0, giving and the approximation of T 2 (r, z) at r ≈ 0 is instead Figure 15 shows the agreement of the approximate function (44) in the pumped region (r ≤ a) of the laser rod at z = 0 for the RT. (43) In a similar manner as detailed by Hodgson et al. [19], we consider the thermo-optic coefficient increasing linearly with temperature, that is Then, utilizing the definition for the optical path difference (OPD) [17] and substituting Eq. (46) into Eq. (47), one obtains Furthermore, using the thermal-lens dioptric power is found to be which gives (see Appendix B for details) Note that, thanks to the particular dependence on r and z on the temperature distribution (Eq. 44), a higher order polynomial fit equation for the thermo-optic coefficient could also be used, that is in which case the thermal-lens dioptric power is found to be Finally, in the case of temperature-independent thermal conductivity (m = 0), and, thermo-optic coefficient ( 1 = 0), it is remarkable that Eq. (51) reduces to the well-known result for the thermal-lens dioptric power generated in a TH endpumped laser rod [5]: A similar analysis can also be followed in the case of SG pumping (and then of G if n = 2 is considered), whereby the thermal-lens dioptric power is given by the following: where, for the sake of simplicity, the constant (51) Again, this result contracts to the well-known one for the thermal-lens dioptric power generated by a Gaussianbeam end-pumped laser rod in the case of constant thermal conductivity and thermo-optic coefficient (n = 2, m = 0 and 1 = 0) [14]: Figure 16 shows the thermal-lens powers as a function of the coolant temperature in both ranges (CT and RT), defined in Sect. 2, for four different pump distributions with the same parameters of Fig. 4. Two different linear expressions for the thermo-optic coefficient are used, for the CT and RT ranges, corresponding to the valid temperature ranges for the thermal conductivity fits. The values utilized for 0 and 1 for each range have been found as follows.
Consider the thermo-optic coefficient defined as follows [17]: where dn∕dT(T) is the thermal dispersion, C is a parameter between 0 and 1 that includes the limitation of a heated element of the rod in the free expansion along the longitudinal direction, due to the colder surrounding, if a transversely localized temperature increase occurs, is the Poisson's ratio, n 0 is the refractive index, and T (T) the thermal expansion coefficient. Supposing both the thermal dispersion and the thermal expansion coefficient linearly depend upon temperature, that is and Equation (57) becomes that is, equivalent to Eq. 46, by defining and For the sake of simplicity, C is set equal to 1 [17], and, since for YAG = 0.25 and n 0 ∼ 1.8, Eqs. (61) and (62), therefore, become  58) and (59) to measured data, as reported by Furuse et al. [20] for a single-crystal YAG (except the dn∕dT coefficient at RT, the data of which were for ceramic YAG). Their values are listed in Table 2.
Note that the dn∕dT data were obtained using a 633 nm He-Ne laser [20]; however, for a real understanding of the lens power, it would need to be evaluated with data measured at the laser wavelength. As reported by Sato and Taira [22] at RT, for example, at a wavelength of 1064 nm, the dn∕dT value is 12.1 10 −6 ∕K, ∼50% higher than the value at 633 nm.
The lens power in Fig. 16 at a heatsink temperature T c > 125 K in the CT range should be taken with caution, since, in this case, the maximum temperature of the rod no longer belongs to the CT range and the fit parameters T 0 , k 0 and m CT are not strictly valid anymore (see Figs. 6 and 8).

Conclusion
In this paper, we have drawn on the fact that the thermal conductivity of laser gain media is dependent upon their temperature. Consequently, solving the heat conductance equation, using the Kirchhoff integral transform, analytical expressions were derived for a number of realistic pumping sources, typically used in end-pumped rod-laser configurations. We compare the solutions obtained with exemplars from the literature and show that in the limit of a constant thermal conductivity, there is a natural convergence.
Through the power of having an analytical solution, we show that the temperature rise at the center of the laser rod has a dependence on the thermal impedance of the boundary layer, as is intuitively expected. In contrast with accepted expressions, these solutions predict an additional temperature rise associated with the thermal conductance parameter h. With this expression, guidance on the minimum acceptable value for h for a set of design parameters can be obtained.
Finally, analytical expressions for the thermal-lens dioptric power are derived in the limit of perfect boundary conditions. Intimately connected with the temperature dependence of the thermal conductivity, in the limit of a constant k 0 , the expressions once again converge to their simplified analogues for Gaussian or top-hat pumping distributions.
It is evident that the solutions derived herein, for the typical parameters chosen, demonstrate a higher temperature rise than those obtained with temperature-independent thermal conductivity reported previously. Having analytical expressions that can be used for a range of temperatures provide a powerful set of tools for designing future end-pumped laser systems. This will be especially relevant to laser systems operating at cryogenic temperatures. mus+ programme of the European Union and Professor Mauro Tonelli, whose encouragement is greatly appreciated. The authors are indebted to Dr N. Hodgson for his invaluable critic and input, which have dramatically enhanced the final version of this manuscript. J.I. Mackenzie would like to acknowledge support by the Engineering and Physical Sciences Research Council support via the Grant (EP/ M020932/1). The data derived from calculations and measurements and the MATLAB codes can be found at doi:10.5258/SOTON/D0117.
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