Design and comparison of composite rod crystals for power scaling of diode end-pumped Nd:YAG lasers

Ralf Wilhelm; Denis Freiburg; Maik Frede; Dietmar Kracht; Carsten Fallnich

doi:10.1364/OE.17.008229

1. Introduction

Diode end-pumped solid state rod lasers exhibit a high optical-to-optical efficiency due to a high overlap of the spatial inversion distribution and the resonator’s fundamental eigenmode [1].

For a transversally pumped rod (length L) and under the assumption of a homogeneous pump light distribution in the transversal as well as in the axial direction, the maximum pump power is limited by thermally induced mechanical stress reaching the fracture stress limit for the laser material and is given by [2]

P_{max} = \frac{8 π R L}{η h} .

Here R is the thermal shock parameter and is a function of the thermal expansion coefficient, YOUNG’s module, POISSON’s constant and the fracture stress limit of the laser host material. The heat efficiency η_h accounts for the fraction of pump power being concerted into heat due to the quantum defect and parasitic non-radiative processes. Note that P_max is independent of the crystal diameter.

In conventional end-pumped lasers involving crystals with a single homogeneously doped segment, the maximum absorbed pump power for a rod of given length is limited by the peak values of the temperature and thermally induced mechanical stress distributions resulting from the exponential decay of the pump light intensity. Assuming a monochromatic pump source and an absorption efficiency of 95 %, the maximum applicable pump power for a single-pass end-pumped crystal is reduced by a factor of about 3 when compared to a transversally pumped crystal. By applying a double pass for the pump light or pumping the crystal from both ends, this value can be improved by nearly a factor of 2 because of a more homogenous heat distribution.

For further improvement, a longitudinally varying dopant concentration profile can be employed. A crystal with a continuously (hyperbolically) varying dopant concentration profile has been produced involving the BRIDGMAN-STOCKBARGER-method, and its overall laser performance has been shown to be comparable with conventional end-pumped lasers [3].While these crystals are not yet available on an industrial scale, crystals with non-continuous dopant concentration profiles are available from several manufacturers. Recent experiments with these composite, multi-segmented crystals include the demonstration of laser output powers of 407 W [4] with traditional pumping around 808 nm and 250 W with direct pumping into the upper laser level [5]. A direct comparison of composite crystalline and ceramic Nd:YAG rods with the same geometry has yielded comparable performance for these laser media [6].

In this paper, we introduce a FOURIER-BESSEL approach for solving the stationary heat equation in a cylindrically symmetric geometry. This approach eliminates the need for computationally expensive (e.g. finite element) calculations while taking axial heat transport and temperature dependent material parameters into account and can be easily implemented in a stand-alone program. In the experimental section, the overall multi-mode laser performance with Nd:YAG rod crystals with one, two and three doped segments pumped by 10 fibre coupled high power laser diodes is being compared.

2. Temperature and thermally-induced stress calculations

In order to choose the optimum design for a multi-segmented rod laser crystal, the amount of pump power being absorbed in the crystal as well as the temperature and mechanical stress distributions have to be calculated. Especially at the interfaces between the different segments, axial heat transport cannot be neglected, thus making existing one-dimensional analytic approaches for calculating the temperature distributions inadequate. On the other hand, more appropriate approaches such as relaxation, finite volume and finite element methods are computationally much more expensive. An alternative approach for finding the temperature distributions is described in this section. This approach approximates the heat generation density distribution by an axially symmetric function that can be written as product of two terms, one of which depends solely on the axial coordinate z while the other solely depends on the radial coordinate r. It has been shown earlier [7] that, for long end-pumped rod crystals, the pump light is guided by total internal reflection and tends to concentrate around the crystal axis and that the heat generation density distribution is well described by such an expression.

For determining the spatial temperature profile, the stationary heat conduction equation [8]

(\frac{\partial^{2}}{\partial r^{2}} + \frac{1}{r} \frac{\partial}{\partial r} + \frac{\partial^{2}}{\partial z^{2}}) T (r, z) = - \frac{1}{k (T)} {q (r, z) + \frac{\partial k}{\partial T} [{(\frac{\partial T}{\partial r})}^{2} + {(\frac{\partial T}{\partial z})}^{2}]} \equiv \tilde{Q} (r, z)

has to be solved, where T is the temperature, k is the temperature dependent heat conductivity, and q is heat generation power density, and a cylindrical symmetry of the problem has been assumed.

The heat generation density distribution is assumed to be a truncated GAUSSian function in radial direction (1/e ² value of radius w) and is given by

q (r, z) = \frac{2 P_{pump} ηh}{π w^{2} (1 - \exp (- 2 R^{2} / w^{2}))} α_{eff} (z) \times \exp (- \int_{0}^{z} α_{eff} (z) d z) \exp (- 2 \frac{r^{2}}{w^{2}}),

with R being the crystal radius, α_eff being the effective absorption coefficient and η_h being the heating efficiency, i. e. the fraction of the pump light being converted into heat due to the quantum defect and parasitic non-radiative processes.

For a rod crystal cooled only via its barrel surface, the boundary conditions

\frac{\partial T (r, z)}{\partial z} ∣_{z = 0} = \frac{\partial T (r, z)}{\partial z} ∣_{z = L} = 0 and \frac{\partial T}{\partial r} ∣_{r = R} + \frac{h}{k (T)} (T - T_{k}) = 0 .

apply for the end faces and the barrel surface, respectively. Here, the convective heat transfer coefficient h can be spatially dependent and the temperature of the coolant is given by T_k.

Assuming for the moment that the right side of Eq. (2) is known, Q̃ and T may be expanded into a harmonic series. If the underlying interval of the FOURIER expansion is of length 2L instead of the crystal length L, both a sine and a cosine sum result in the same function on the interval (0,L). Therefore one is free to choose either the sine or the cosine basis set for expanding T and Q̃. Since the cosine functions individually fulfill the boundary conditions on the end surfaces, an expansion of the form

T (r, z) = \frac{T_{0} (r)}{2} + \sum_{n = 1}^{\infty} T_{n} (r) \cos (\frac{nπz}{L}) and \tilde{Q} (r, z) = \frac{{\tilde{Q}}_{0} (r)}{2} + \sum_{n = 1}^{\infty} {\tilde{Q}}_{n} (r) \cos (\frac{nπz}{L})

has been chosen.

Inserting these expressions into Eq. (2), decoupled ordinary differential equations in r result for the different frequency components:

(\frac{\partial^{2}}{\partial r^{2}} + \frac{1}{r} \frac{\partial}{\partial r}) T_{0} (r) = - {\tilde{Q}}_{0} (r) and (\frac{\partial^{2}}{\partial r^{2}} + \frac{1}{r} \frac{\partial}{\partial r} - \frac{π^{2} n^{2}}{L^{2}}) T_{n} (r) = - {\tilde{Q}}_{n} (r) .

The valid solutions have to fulfill the boundary conditions

\frac{d T_{n}}{d r} ∣_{r = 0} = 0 and \frac{d T_{n}}{d r} ∣_{r = R} + \frac{h_{n}}{k (T)} T_{n} = 0 .

The formal solution of these equations involves first and second kind modified BESSEL functions of zeroth order I ₀ and K ₀ [9, 10]. Since I ₀ takes very high values for arguments in the order of a few tens, this approach is efficient but becomes numerically unstable for short, thick rods (length to diameter ratio of less than about 10). The same applies to a numerical solution of the equations by means of a RUNGE-KUTTA approach in combination with a shooting method. The observed numerical instability of these two methods has been traced back to the limited accuracy of the representation of the numerical values in the computer. In the inverse discrete FOURIER transform, differences of large, nearly identical numbers have to be calculated, resulting in large relative numerical errors. In order to enable calculations of short, thick rod crystals, an alternative approach has been chosen. In this approach, the quantities T_n(r) and Q̃_n(r) are expanded into a series of BESSEL functions

T_{n} (r) = \sum_{m = 1}^{\infty} T_{n, m} J_{0} (α_{m, n} r) and {\tilde{Q}}_{n} (r) = \sum_{m = 1}^{\infty} {\tilde{Q}}_{n, m} J_{0} (α_{m, n} r)

with α_{m, n} being a root of α_{m, n} {[\frac{d}{d r} J_{0} (r)]}_{r = R} + \frac{h_{n}}{k} J_{0} (α_{m, n} R) = 0 .

Note that these functions form a complete set and fulfill the boundary conditions. Now the temperatures and modified heat generation densities in ’frequency space’ are related to each other by

T_{n, m} = \frac{{\tilde{Q}}_{n, m}}{α_{m, n}^{2} + \frac{π^{2} n^{2}}{L^{2}}} .

The temperature distribution T(r,z) is obtained from T _n,m by performing the appropriate inverse (finite) transforms, i. e. an inverse HANKEL transform in radial and an inverse FOURIER transform in axial directions. For performing the HANKEL transforms, an algorithm derived from the Quasi-Discrete HANKEL Transform (QDHT) algorithm described by Yu et al. [11, 12] has been employed. Although this method has been reported to be a few times slower than the Quasi Fast HANKEL Transform (QFHT) given by Siegman [13], it involves an unitary transformation matrix and does not exhibit the lower end correction (‘hole’) problem of the QFHT that results from the finite integration domain in Siegman’s algorithm and has been addressed by Agrawal [14]. The discrete FOURIER transform is based on the routines given by Press et al. [15]. The roots of Eq. (9) are found by the NEWTON-RAPHSON method. For the most general case of a spatially varying heat transfer coefficient h(z), the roots will be different for all frequency components. Within the scope of this paper, a spatially constant heat transfer coefficient is assumed, i. e. h_n = 0 for n ≠ 0 and the roots will only need to be found for the DC component and one non-zero frequency component.

Due to the temperature dependence of the heat conductivity, the right side of Eq. (2) is not known and contains the temperature field. In order to account for this fact, an iterative approach is used, taking the temperature distribution from the (k - 1)th iteration for computing the temperature profile in the kth iteration. The coolant temperature is used as a starting value for the first iteration. Typically, between 5 and 10 iterations are necessary in order to obtain a relative temperature variation of less than 10^-2 from one iteration to the next.

Table 1. Segment lengths and dopant concentrations of the different laser rod designs.

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In order to evaluate the accuracy of the method, the temperature distribution in a composite rod consisting of two undoped end caps 7 mm in length and a 40 mm long region doped at 0.1 at. % Nd (design 1 in Table 1) has been calculated and compared to the results of a finite element (FEM) analysis obtained with ANSYS 11. A pump power of 300 W has been assumed. For the FEM analysis, the geometry was described as a plane, axisymmetric structure. No significant change has been observed when changing the number of nodes from about 10000 to 20000. While the peak temperature values coincide very well for both models, the location of the peaks is slightly different and the curves obtained from the FOURIER-BESSEL approach are slightly steeper at the segment interfaces (Fig. 1a)).

Fig. 1. a) Difference in temperature distributions obtained from a finite element model and the described FOURIER–BESSEL approach. b) Comparison of the stress distributions obtained from the plain–strain approximation (dashed lines) and a finite element model (solid lines).

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Within the limits of the plain-strain approximation, the non-vanishing components of the stress tensor are given by [16]

σ_{r} (r, z) = \frac{α_{T} E}{1 - ν} (\frac{1}{R^{2}} \int_{0}^{R} Δ T (r, z) rdr - \frac{1}{r^{2}} \int_{0}^{r} Δ T (r, z) rdr),

σ_{t} (r, z) = \frac{α_{T} E}{1 - ν} (\frac{1}{R^{2}} \int_{0}^{R} Δ T (r, z) rdr + \frac{1}{r^{2}} \int_{0}^{r} Δ T (r, z) rdr - Δ T (r, z)),

σ_{z} (r, z) = \frac{α_{T} E}{1 - ν} (\frac{2}{R^{2}} \int_{0}^{R} Δ T (r, z) rdr - Δ T (r, z)) .

Here, E is YOUNG’s modulus, α_T is the thermal expansion coefficient, and ν is POISSON’s ratio. Note that on the rod’s barrel surface, the radial component σ_r = 0 and that α_t = σ_z.

Strictly speaking, the plain–strain approximation assumes a z-independent temperature distribution to be valid. Near the interfaces between two segments, the temperature gradients are relatively large and in this case, the shear stress component α_rz takes a nonzero value. In order to illustrate the effect of the z–dependent temperature distribution, a FEM calculation has been performed for the composite rod with three doped segments (design 3 of Table 1). The maximum value of σ_rz occurs very close to the segment interface at a distance of about 0.9 mm from the crystal axis. Figure 1 b) shows the VON-MISES equivalent stress distributions on the rod’s axis, at a distance of 0.9 mm from the rod’s axis and on the barrel surface of the crystal. Although the stress on the rod’s surface is slightly overestimated when compared to the FEM calculation, the general agreement between the two approaches is satisfactory, given the simplicity of the plain–strain approximation.

A stand-alone program employing the methods described above for calculating the relevant quantities has been developed in Borland Delphi 7.0.

For comparing the laser performance of rod crystals with one, two and three doped segments, three crystals have been optimized with this program. All three crystal use two 7 mm long undoped end caps for eliminating bulging of the end-surfaces [17, 18]. The overall length of the crystals has been set to 54 mm. The lowest available dopant concentration was 0.1 at. % Nd. The segment length and dopant concentration data are given in Table 1. The crystal diameter is 3 mm.

Fig. 2. Comparison of the three simulated rod designs at 300 W of pump power. A rod with single doped segment at 0.2 at. % Nd as been added for comparison. a) on-axis temperature and b) VON-MISES equivalent stress on barrel surface.

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While the crystal with a single doped segment used a pump light double pass, the two other crystals use a pump light single pass.

The on-axis temperatures and the VON-MISES equivalent stress on the rods barrel surface are shown in Fig. 2. The model assumes a truncated GAUSSian radial heat generation distribution with a z-independent shape. The 1/e ² radius of the distribution has been set to 2 mm which coincides well with earlier raytracing calculations [7]. A constant heating efficiency η_h of 0.28 is used in order to reproduce with this model the thermal lensing measurements. The heat transfer coefficient between the rod’s barrel surface and the cooling water has been set to h = 1 W/cm²K and the coolant temperature has been assumed as 20 °C. The temperature dependent heat conductivity and the expansion coefficient have been taken from the data given by Contag [19].

3. Experimental results

Fig. 3. Schematic setup of the short end-pumped multi-mode laser resonator. For details see text.

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In our experimental setup, shown in Fig. 3, the multi-segmented laser rods were longitudinally pumped with a bundle of 10 fibre-coupled laser diodes (JENOPTIK Laserdiode GmbH, type JOLD-30-CPXF-1L) with a nominal output power of 30 W each, resulting in a maximum available pump power of slightly more than 300 W. Each diode was individually temperature stabilized by thermo-electrical cooling, such that the emission of each diode could be spectrally shifted for optimum overlap with the absorption spectrum of Nd:YAG. A bundle of ten fibres was used, each with a core diameter of 400 μm. A system of three lenses (focal lengths 80 mm plano-convex,100 mm bi-convex,60 mm plano-convex)was employed for focusing the radiation emerging from the fibre ends into the laser rods.

The barrel surface of the multi-segmented laser rods was polished to nearly optical quality in order to act as a waveguide for the pump light due to total internal reflection because of the difference in refractive indices for YAG and the ambient cooling water.

The facet facing the pump fibres was provided with an antireflection coating for the pump and the laser wavelength. For the crystal with a single doped segment (design 1 in Table 1), a coating being highly transmissive for 1064 nm and highly reflective around 808 nm was used on the opposite surface in order to realize a pump light double-pass. For the other two rods, an antireflection coating for both wavelengths was used.

Using this setup, the laser properties of the three different composite rod crystals were examined in a short plane-plane resonator (length 70 mm) and the cw multi-mode laser output power was measured for optimized output power transmission of 15 %.

The results are shown in Fig. 4. A maximum laser output power of 167.5 W was realized with the three-segment rod. At an absorbed pump power of 312.4 W, this corresponds to an optical-to-optical efficiency of 53.6 %. Based on a linear fit for absorbed pump powers lower than 250 W, a slope efficiency in excess of 60 % was realized. A summary of the data of the three crystals is given in Table 2.

Table 2. Laser performance for the studied three rod crystals.

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Based on the determination of the resonator stability range by varying the distance between the laser crystal and the output coupler for a given pump power, a thermal lens coefficient of 40, 45 and 38 diopters per kilowatt of absorbed pump power has been determined for the crystal with one, two and three doped segments, respectively.

Fig. 4. Multi-mode laser output power versus absorbed pump power for the crystals with 1, 2 and 3 doped segments.

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4. Summary

In summary, we have employed a method based on a FOURIER-BESSEL series expansion for solving the stationary heat conduction equation in cylindrical coordinates for designing multi-segmented rod laser crystals for cw high-power end-pumping applications.

A comparison of composite Nd: YAG laser rods with one, two and three doped segments for high power end-pumping has been presented. Maximum output powers of 150.6, 152.3 and 167.5 W at optical-to-optical efficiencies of 49.6, 50.3 and 53.6 % have been achieved for the rod crystals with one, two and three doped segments, respectively.

In conclusion, we demonstrated that the properties of the three examined crystal are nearly identical and that no sacrifice is to be expected from employing multi-segmented rods in high-power end-pumped lasers.

Acknowledgments

This work was partly funded by the German Ministry of Education and Research under contract 13N8299.

References and links

1. S. C. Tidwell, J. F. Seamans, M. S. Bowers, and A. K. Cousins, “Scaling CW Diode-End-Pumped Nd:YAG Lasers to High Average Powers,” IEEE J. Quantum Electron. 28, 997–1009 (1992). [CrossRef]

2. W. Koechner, Solid-State Laser Engineering (Springer, New York, 1996).

3. R. Wilhelm, D. Freiburg, M. Frede, and D. Kracht, “End-pumped Nd:YAG laser with a longitudinal hyperbolic dopant concentration profile,” Opt. Express 16, 20106–20116 (2008) http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-24-20106 [CrossRef] [PubMed]

4. D. Kracht, R. Wilhelm, M. Frede, K. Dupré, and L. Ackermann, “407 W End-Pumped Multi-Segmented Nd:YAG Laser,” Opt. Express 13, 10140–10144 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-25-10140. [CrossRef] [PubMed]

5. M. Frede, R. Wilhelm, and D. Kracht, “250 W end-pumped Nd:YAG laser with direct pumping into the upper laser level,” Opt. Lett. 31, 3618–3619 (2006) http://www.opticsinfobase.org/abstract.cfm?URI=ol-31-24-3618. [CrossRef] [PubMed]

6. D. Kracht, M. Frede, R. Wilhelm, and C. Fallnich, “Comparison of crystalline and ceramic composite Nd:YAG for high power diode end-pumping,” Opt. Express 13, 6212–6216 (2005) http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-16-6212. [CrossRef] [PubMed]

7. R. Wilhelm, M. Frede, and D. Kracht, “Power Scaling of End-Pumped Solid-State Rod Lasers by Longitudinal Dopant Concentration Gradients,” IEEE J. Quantum Electron. 44, 232–244 (2008). [CrossRef]

8. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Oxford University Press, New York, 1959).

9. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, New York, 1964).

10. R. Wilhelm, M. Frede, D. Freiburg, D. Kracht, and C. Fallnich, “Thermal Design of Segmented Rod Laser Crystals,” in Advanced Solid-State Photonics 2005 Technical Digest on CD-ROM (The Optical Society of America, Washington, DC, 2005), MB46.

11. L. Yu, M. Huang, M. Chen, W. Chen, W. Huang, and Z. Zhu, “Quasi-Discrete Hankel Transform,” Opt. Lett. 23, 409–411 (1998). [CrossRef]

12. M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Computation of Quasi-Discrete Hankel Transforms of Integer Order for Propagating Optical Wave Fields,” J. Opt. Soc. Am. A 21, 53–58 (2004). [CrossRef]

13. A. E. Siegman, “Quasi Fast Hankel Transform,” Opt. Lett. 1, 13–15 (1977). [CrossRef] [PubMed]

14. G. P. Agrawal and M. Lax, “End Correction in the Quasi-Fast Hankel Transform for Optical Propagation Problems,” Opt. Lett. 6, 171–173 (1981). [CrossRef] [PubMed]

15. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in PASCAL (Cambridge, New York, 1989).

16. S. P. Timoshenko and J. N. Goodier, Theory of Elasticity (McGraw-Hill, New York, 1970).

17. R. Weber, B. Neuenschwander, M. Mac Donald, M. B. Roos, and P. Weber, “Cooling Schemes for Longitudinally Diode Laser-Pumped Nd:YAG Rods,” IEEE J. Quantum Electron. 34, 1046–1053 (1998). [CrossRef]

18. M. Tsunekane, N. Taguchi, T. Kasamatsu, and H. Inaba, “Analytical and Experimental Studies on the Characteristics of Composite Solid-State Laser Rods in Diode-End-Pumped Geometry,” IEEE J. Sel. Top. Quantum Electron. 3, 9–18 (1998).

19. K. Contag, Modellierung und numerische Auslegung des Yb:YAG-Scheibenlasers (Munich, Herbert Utz Verlag, 2002).

Design	Segment 1	Segment 2	Segment 3	Segment 4	Segment 5
1	7 mm	40 mm	7 mm	—	—
1	0.0 at. %	0.0 at. %	0.1 at. %	—	—
2	7 mm	25 mm	15 mm	7 mm	—
2	0.0 at. %	0.1 at. %	0.3 at. %	0.0 at. %	—
3	7 mm	22 mm	11 mm	7 mm	7 mm
3	0.0 at. %	0.1 at. %	0.23 at. %	0.6 at. %	0.0 at. %

Design	Absorbed pump power (W)	Output power (W)	Optical-to-optical efficiency (%)	slope efficiency (%)
1	303.7	150.6	49.6	54.1
2	302.8	152.3	50.3	55.4
3	312.4	167.5	53.6	60.5

Design	Segment 1	Segment 2	Segment 3	Segment 4	Segment 5
1	7 mm	40 mm	7 mm	—	—
1	0.0 at. %	0.0 at. %	0.1 at. %	—	—
2	7 mm	25 mm	15 mm	7 mm	—
2	0.0 at. %	0.1 at. %	0.3 at. %	0.0 at. %	—
3	7 mm	22 mm	11 mm	7 mm	7 mm
3	0.0 at. %	0.1 at. %	0.23 at. %	0.6 at. %	0.0 at. %

Design	Absorbed pump power (W)	Output power (W)	Optical-to-optical efficiency (%)	slope efficiency (%)
1	303.7	150.6	49.6	54.1
2	302.8	152.3	50.3	55.4
3	312.4	167.5	53.6	60.5

Design and comparison of composite rod crystals for power scaling of diode end-pumped Nd:YAG lasers

Abstract

1. Introduction

2. Temperature and thermally-induced stress calculations

3. Experimental results

4. Summary

Acknowledgments

References and links

Cited By

Figures (4)

Tables (2)

Equations (13)

Optics Express