FOLLOWING THE development of the first kilowatt class

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1 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 33, NO. 5, MAY Ultrahigh-Average-Power Diode-Pumped Nd:YAG and Yb:YAG Lasers David C. Brown, Member, IEEE Abstract Yittrium aluminum garnet (YAG) possesses thermal and mechanical properties that vary significantly with temperature. We show that when temperature variations are accounted for the simple scaling relationships traditionally used for highaverage-power performance predictions fail. We have also found that for room temperature and below, and with uniform heat deposition in a rod, nonquadratic radial temperature profiles result and the magnitude of the thermally induced stresses are seriously underestimated. New nonlinear scaling relationships are presented that properly account for YAG materials property variations with temperature. These results are applied to diode-pumped Nd:YAG and Yb:YAG lasers operating at room temperature and 77 K; we show that significant increases in average power output are possible by operating Nd:YAG and Yb:YAG lasers at 77 K. Index Terms Diode-pumped solid-state lasers, Nd:YAG lasers, ultrahigh-average-power lasers, Yb: YAG lasers. I. INTRODUCTION FOLLOWING THE development of the first kilowatt class solid-state laser, a flashlamp-pumped device [1], the available average power from solid-state laser systems has recently exceeded that power using both rod and slab geometry amplifiers [2] [6]. The excellent average power capability and moderate stimulated emission cross section of Nd:YAG (yittrium aluminum garnet) has made it the material of choice for high-average-power (HAP) lasers, although recent work utilizing Nd:YAP (Nd:YALO) is also promising since the material is naturally birefringent. While it seems probable that further advances in diode pumping and other laser materials (e.g., Yb:YAG [7] [9]) as well as the development of oscillators with multiple amplifiers in a single resonator will significantly increase the available solid-state laser average power, the limited size of available crystalline materials places a fundamental limit on the amount of increase possible. Recently, it was experimentally demonstrated that cooling a Ti:Sapphire laser crystal to cryogenic temperatures resulted in a significant increase in average power when compared to room temperature operation [10]. In addition, thermally induced optical aberrations were significantly reduced. While investigating the variation of the physical and thermal properties of Nd:YAG with temperature, we have discovered that it is also possible to substantially increase the HAP operation of CW and pulsed Nd:YAG and Yb:YAG lasers that are cryo- Manuscript received September 9, 1996; revised January 13, The author is with Renaissance Lasers, Brackney, PA USA. Publisher Item Identifier S (97) genically cooled, as compared to the performance obtainable at room temperature. In this paper, we first review in Section II the HAP scaling law for solid-state lasers, assuming that all mechanical and thermal properties are constant, independent of temperature. In Section III, we present what is known of the variation of Young s Modulus, Poisson s Ratio, thermal conductivity, and thermal expansion coefficient with temperature and show empirical fits to the experimental data that have been used in computer simulations discussed in this paper. We have concentrated primarily on the mechanical and thermal properties of YAG because of their importance in determining limitations to HAP scaling. The low-temperature optical properties of Nd:YAG will not be treated in this article, having been reviewed in the literature [11], [12]. Section IV presents the results of linear and nonlinear modeling of thermal profiles and stress distributions in rod elements. We show that, even at room temperature, the older linear model results in an ideal quadratically varying transverse temperature and distortion variation that does not accurately describe the real, more complicated distributions obtained when variations in properties with temperature are included. In fact, the older model underestimates the magnitude of the rod center temperature and the magnitude of the stresses. The transversely varying temperature and stress distributions become even more complicated as coolant temperature is reduced. In Section V, we describe in detail the results of modeling the HAP performance of rod elements of various diameters; for each diameter, the temperature of the coolant, value of the surface heat transfer coefficient, and heat density deposited in the material are varied. Unlike the previous linear scaling law where HAP performance was independent of coolant temperature and heat transfer coefficient, we show that both are important parameters in determining the performance obtainable at any temperature. Performance curves are presented in Section VI for 4-, 6-, and 8-mm-diameter YAG rods. Using known values for the available sizes of crystals and the ratio of inversion power to heat power for diode-pumped Nd:YAG and Yb:YAG, we show that average power levels of 60 kw are readily achievable from a single crystalline element. Much of our attention during this work has been focused on comparing the HAP performance obtainable at room temperature and at liquid nitrogen LN temperature, around 77 K primarily because water is the coolant fluid most often used, and LN is now commonly available and can easily be /97$ IEEE

2 862 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 33, NO. 5, MAY 1997 circulated as well. In Section VII, we compare the thermal and transport properties of these two fluids and calculate and compare heat transfer coefficients that can be practically obtained. We show that heat transfer coefficients achievable with LN at 77 K are quite comparable to those obtained with water. II. REVIEW OF HIGH-AVERAGE-POWER SCALING OF ROD SOLID-STATE LASERS We begin by reviewing the scaling relationships governing the generation of HAP from solid-state lasers; these relationships were first presented in [13] for lasers utilizing slab and rod amplifiers, and in [14] were extended to include activemirror and disk amplifiers. We include this review to illustrate the important differences between those early results and those presented here. The major limiting factor in scaling solidstate laser average power is thermal heating; such heating is caused by a number of factors and is different for each laser material, however, the common sources of heating are the quantum defect, nonunity quantum efficiency, concentration quenching, upconversion, and other sources. The uniform or nonuniform absorption of pump radiation by a laser crystal and the resulting heating produces thermal profiles that give rise to thermally induced strain, leading to material distortion and stresses. In this article, we concentrate solely upon the true CW case, although the results should apply equally well to any pulsed HAP solid-state laser whose thermal relaxation time is long compared to the time interval between pulses. We discuss rod amplifiers only in this article; the extension of the results to slab, active-mirror, disk, or other amplifier types is straightforward. A. Thermal Effects In Rod Amplifiers We consider rod-type amplifiers, i.e., right circular laser crystal elements that are actively cooled along the barrel. In addition, we assume that the rod has a length that is long compared to its diameter. The heat equation describing the temperature in an isotropic medium can be written In writing (3), we have used a cylindrical coordinate system and assumed from the symmetry of the problem that the temperature is a function of the radial coordinate only and that any azimuthal and longitudinal variations in can be ignored. The solution to (3) for constant can be written where is the rod on-axis temperature and varies quadratically with. This situation is rarely achieved in practice since in most practical situations.in obtaining (4), we have assumed Newton s law of cooling at the rod edge-coolant boundary: In (5), is the heat transfer or convective film coefficient coefficient, the coolant temperature, and the first derivative is taken with respect to the normal to the surface. The center temperature can be found by combining (4) with (5) to give The center temperature can thus be seen to be given by three distinct terms, the first representing the ambient or cooling temperature, the second the temperature drop across the cooling boundary (note that is equal to the heat flux incident upon the crystal barrel), and the third due to the finite thermal conductivity of the laser crystal. The average rod temperature can be shown to be equal to Then (4) can be rewritten as (4) (5) (6) (7) (1) (8) where is the heat density, the density, the specific heat at constant pressure, the thermal conductivity, and denotes the gradient operator. In steady state, this equation becomes In the remainder of this paper, we assume that is constant and uniform throughout the laser crystal. As we will show in Section III, however, the thermal conductivity for YAG is not a constant but is strongly dependent upon the temperature. If we assume constant, as has been done in previous work [13] [15], then can simply be factored out of (2) and one obtains (2) (3) The term produces no strain (and thus no stress) in crystals since it represents a uniform temperature rise in the crystal, however, also known as the faired temperature distribution, has a mean temperature of zero and has been shown to account for thermally induced strains and stresses [13]. B. Thermally Induced Stresses in Rod Amplifiers We now invoke the plane strain approximation [16], which assumes that the strain in the direction is zero and effectively eliminates one dimension from the problem of determining the strains and stresses in the laser crystal. This approximation is applicable to long thin rods (large aspect ratio) of the type used in most laser applications. By using Hooke s law relating the

3 BROWN: ULTRAHIGH-AVERAGE-POWER DIODE-PUMPED Nd:YAG AND Yb:YAG LASERS 863 stress and strain components: where is Young s modulus, Poisson s ratio, the thermal expansion coefficient, and the Kronecker delta function, and by invoking the plane strain approximation and converting to cylindrical coordinates, the following expressions can be obtained for the radial tangential and longitudinal stresses in the crystal: (9) (10) (11) (12) Note that and is known as the materials parameter. It is worth noting that the magnitude of the stress produced at the rod barrel is in fact independent of the rod diameter since is inversely proportional to is the total heat power and the rod length). This situation changes when and are allowed to vary with temperature. Because the temperature in the center of the rod is always larger than at the edge, the center region of the rod is in compression while the outer regions and the rod edge are in tension. Typically, the rod barrel is the weakest in terms of material strength due to the presence of scratches, pits, or voids. In addition, since crystalline and glassy optical materials are well known to be weaker in tension than in compression, the rod barrel is typically where catastrophic thermally induced failure of the rod occurs, when the tensile stress exceeds the fracture strength. The relationship between crack size and material strength has been discussed in detail in the literature [17]. In general, however, the presence of large cracks and scratches significantly reduce crystal strength and ultimately the amount of heat and inversion power that can be tolerated before rupture. Rod barrels and slab edges are particularly susceptible to this type of failure; ground barrels are used because of the need to reduce the barrel effective reflectivity to suppress parasitic oscillations. In CW operation, however, because small-signal gains are typically low, polished barrels may be used. Polishing reduces the average scratch or crack size and increases material strength. Another strategy that can be used is to both grind and etch the crystal barrel [18]. C. Average Power Scaling in CW Rod Amplifiers Thermally induced rupture or catastrophic failure of a crystal is then the primary limiting factor to the attainment of higher average powers in solid-state laser materials. If we denote by the CW ratio of the heat power density /inversion power density produced in a crystal, then we can write the CW extractable power from a rod as (13) The parameter must be specified for the laser material and optical pump source of interest. If we want the maximum extractable power available from a crystal, we take the stress equal to the fracture stress and calculate the maximum heat power density using (11), which gives (14) This general equation is applicable to rod amplifiers using any lasing material. Other similar equations can be derived for other types of crystalline elements such as slabs, active mirrors, and disk amplifiers [14]. is known as the rupture modulus for the material and is dependent on the type of crystal used as well as the preparation of the surface. It has been tabulated for a number of crystalline materials and surface preparations [17], [19]. Equation (14) shows that under the assumptions made here, specifically that and other parameters are constant, and that is uniform and constant, that the average power of a rod crystal can be increased by choosing a material with a large, by increasing the fracture strength, and by increasing the rod length and minimizing. Increasing the rod radius does not increase the extractable power. It is important to note that in the formulation summarized in this section, the stresses do not depend upon either the heat transfer coefficient or the coolant temperature. This is a consequence of assuming that the thermal expansion coefficient is independent of temperature. In the analysis presented in Section V, it will be seen that in fact and are important parameters in determining the magnitude of the stresses and the extractable power obtainable from a crystal. III. VARIATION OF YAG THERMAL AND MECHANICAL PROPERTIES WITH TEMPERATURE In this paper, we treat only the crystalline material YAG Y Al O primarily because it is the material most commonly used today and because more is known about its properties than any other laser material. A third reason is because YAG can also be host to two important lasing ions, Nd and Yb. While Nd:YAG is at present in widespread use in a wide variety of applications, Yb:YAG is a relatively new laser material that we believe will become very important because of the significantly lower amount of heat generated (smaller value) due to a small quantum defect. From (14), we can see that a lower value means that more extractable power can be obtained from a rod of a given size.

4 864 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 33, NO. 5, MAY 1997 Fig. 1. YAG thermal conductivity data () and fit ( ) as a function of temperature. Fig. 2. YAG thermal expansion data () and fit ( ) as a function of temperature. A. Thermal Conductivity After reviewing the available thermal conductivity data for YAG, the most complete set of data [20], ranging from 30 K 300 K was digitized and used to generate a least-squares analytical fit that was used for the thermal and stress simulations discussed in Section IV. The data as well as the fit are shown in Fig. 1. It can be seen that the thermal conductivity increases by almost an order of magnitude between 300 K and 70 K and continues to increase down to about 30 K where it then decreases again (not shown). We have found no data for YAG above 300 K thus for the simulations described later in this article where rod center temperatures did not exceed 500 K we simply extended the fitting function as shown in Fig. 1 to 500 K decreasing to about W/(cm K) at 500K. The best fit is given by (15) where and are constants with the values W/cm-K 5.33 K 7.14, and 331 W/cm. It should be mentioned that the room temperature value of shown in Fig. 1 or calculated using (17), W/(cm K is significantly less than that given in [21], W/(cm K. While the latter value is often quoted in the literature [11], the data presented there are not nearly as complete as that presented in [20]. A thermal conductivity that increases as temperature is reduced has a number of consequences. First, because the temperature distribution in a crystalline rod is maximum on the rod axis, the thermal conductivity will assume its minimum value there and gradually increase as the rod barrel is approached. As we will see in Section IV, the result is an elevated temperature everywhere in the rod, as well as significantly increased stress, even at room temperature, when compared to the ideal case with constant. Second, the induced thermal distortions produced in the rod are larger than that produced using the constant case. which is directly proportional to reduced. increases as temperature is B. Thermal Expansion Coefficient Another important parameter of interest is the thermal expansion coefficient. As with the thermal conductivity, there are a number of tabulations of data available, mostly quoted at room temperature [11], [15]. Again, we selected the most complete set of data available, in this case spanning the range from about 100 K 1700 K [22]. The linear thermal expansion of YAG from 0 K 500 K is shown in Fig. 2, including the data digitized from [22] as well as the best least-squares fit used in our simulations: (16) The constants in this expression, and are 9.54, K and We can obtain the thermal expansion coefficient by taking the derivative of (16) with respect to is then plotted as a function of in Fig. 3. It can be seen that goes to 0 as approaches 0 in agreement with theory [23]. At 300 K we obtain K which is somewhat lower than a number of values quoted in the literature [11], [15]. Since becomes smaller as is reduced, increases, effectively increasing the rupture modulus for YAG. Because decreases with temperature, we expect that the thermally induced optical distortion in a rod will also decrease as the temperature is lowered. This will be discussed further in Section IV. C. Poisson s Ratio and Young s Modulus Direct measurements of Poisson s ratio and Young s modulus as a function of temperature have not, to our knowledge, been measured. Fortunately, however, the variation in the elastic stiffness matrix constants of YAG with temperature have been reported in the literature between about 160 K and 298 K [24]. In addition, the value of the elastic stiffness

5 BROWN: ULTRAHIGH-AVERAGE-POWER DIODE-PUMPED Nd:YAG AND Yb:YAG LASERS 865 flexibility, for example, in solving for the CW heat and stress distributions when the thermal power density is not uniformly distributed through the rod. Fig. 3. Thermal expansion coefficient (derivative of fit in Fig. 2) as a function of temperature. constants and compliance matrix constants at room temperature are well known [11]. The elastic stiffness matrix and the compliance matrix relate the stress and strain in a crystal according to Hooke s law in the form or. For a cubic crystal like YAG, symmetry arguments show that three independent components are needed to specify or. Because YAG, while cubic, is also nearly isotropic [25], we assume that Young s modulus and Poisson s ratio are given by and. Using the data from [11], we find kg m/cm and These room-temperature values have been used in the simulations discussed in Sections IV VI. While our value for is somewhat lower than that presented in [19], it is consistent with the calculated value and close to that reported in [26]. Using the variation of the with temperature, we have determined that and change by no more than 7% and 2%, respectively, across the entire range calculated, 140 K 298 K. Because of the near constancy of and we have assumed in all that follows that they are constants and have assigned the room-temperature values to them. IV. TEMPERATURE AND STRESS EFFECTS IN ROD AMPLIFIERS: MATERIALS PARAMETERS VARYING To model thermal and stress effects in rod amplifiers, we have chosen to use a finite-element approach, using a program 1 that allows variations in any of the thermal and elastic parameters to be incorporated. In all that follows, we have used the model subject to the conditions of plane-strain [16]. It should be noted that since and are continuously varying functions of temperature and the temperature distribution depends upon the coefficient of heat transfer at the material-coolant boundary as well as the coolant temperature, simple universal solutions are not apparent. While undoubtedly one could solve (2) for some analytical functions describing the thermal conductivity, such solutions would have limited utility. The finite-element approach is simple to implement and has a great deal more 1 PDEase TM, distributed by Macsyma, Arlington, MA. A. Comparison of Finite Element and Analytical Approaches We first consider the thermal distribution in a 4-mmdiameter YAG rod with the coolant fluid at room temperature, 500 W/cm and a heat transfer coefficient of 10 W/cm K. In Fig. 4, we show the analytical solution calculated using (4) (6) and compare it with the same solution obtained using the finite element program with and constant and equal to their room-temperature values. It can be seen that the agreement is excellent. We also show the temperature distribution calculated with constant and allowed to vary with temperature. The result is that the rod center temperature is elevated above that obtained using the analytical model. Furthermore, the new temperature distribution is not described by a simple radial dependence; even if it were possible to obtain perfectly uniform heat density in a rod, the resulting optical aberrations cannot be corrected using a simple lens [15]. Finally, we let both and vary; the resulting temperature distribution is identical to the previous case, as would be expected. A larger temperature than that calculated using the analytical model is a consequence of the varying which is minimum in the hotter central region of the rod. This effect becomes more pronounced as the heat power density is increased and as the heat transfer coefficient is decreased. In Figs. 5 and 6, we show the resulting radial and tangential or hoop stress distributions, respectively. Again, excellent agreement is obtained between the analytical solutions given in (10) and (11) and the finiteelement model with constant. When only is allowed to vary and held constant, the magnitude of the radial and hoop stress components increase slightly, reflecting the slight increase in the temperature of the rod. When, however, both and are allowed to vary, there is a dramatic increase in the magnitude of the stresses. This effect is a consequence of varying with temperature. Stresses arise in rods with held constant primarily because the center of the rod expands more due to the higher temperatures in the rod center. This effect is further exacerbated here because not only is the center of the rod hotter than the edge but the thermal expansion coefficient in the center of the rod is also larger, producing additional stress. This result has important consequences for evaluating the stresses in HAP lasers, particularly those that run close to the fracture limit. While we have discussed here only a 4-mm-diameter YAG rod, the results presented are general. The temperature in a rod does not follow that predicted using a simple analytical model, and the resulting rod optical aberrations cannot be corrected using a simple lens. In addition, the thermally induced stresses calculated using the analytical model seriously underestimate the magnitude of the radial and hoop stresses in the rod. It has been noted by one reviewer of this paper that the obtained nonquadradic temperature profiles might help to

6 866 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 33, NO. 5, MAY 1997 Fig. 4. Temperature as a function of radius for D = 4 mm;q o = 500 W=cm 3 ; and h = 10 W=cm 2 1K with k; constant () at 300 K;k variable ( ); and both k and variable (r). Also shown is the analytical curve with k and constant at 300 K ( ). Fig. 6. Tangential stress as a function of radius for D = 4 mm;q o = 500 W=cm 3 ;h = 10 W=cm 2 1K with k; constant ( ) at 300 K;k variable (); and both k and variable (1). Also shown is the analytical curve with k and constant at 300 K ( ). Fig. 5. Radial stress as a function of radius for D = 4 mm;q o = 500 W=cm 3 ;h = 10 W=cm 2 1K with k; constant () at 300 K;k variable ( ); and both k and variable (r). Also shown is the analytical curve with k and constant at 300 K ( ). explain the deviation of experimental results from the oftenused relationship derived using the constant theory with a constant and uniform where it is shown that the inverse focal length of the rod is proportional to the pump power [15]. While it is indeed possible that the effects described here partially contribute to such deviations, other effects such as nonuniform pumping, poor or not well-developed flow, and the change in heatload encountered in flashlamp-pumped solid-state lasers as pump level is changed (thus changing the value) can also be important. Because those parameters were not measured in the data in [15], we cannot assess the importance of the effects described here in the interpretation of such data. The effect that a nonuniform pump or heat profile has on the ideal temperature distribution is also significant, however, such effects are beyond the scope of this article and will be treated in a separate publication. Fig. 7. Temperature as a function of radius for D = 4 mm;q o = 250 W=cm 3 ;T c = 300K; and h = 10 (r); 5 ( ); and 1 () W=cm 2 -K. Solid lines represent the analytical results calculated with k and constant at 300 K. B. Comparison of Thermal and Stress Distributions in Rods at Room Temperature and 77 K To further illustrate the effects of and that vary with temperature, we now consider a 4-mm-diameter YAG rod at 300 K with 250 W cm and with different values for. Fig. 7 shows the obtained temperature distributions for values of 1, 5, and 10 W cm K as well as the analytical temperature distributions from (4) (6). It can be seen that as is increased, the center and edge temperatures decrease. In addition, the difference between the finite-element obtained results and the analytical results decrease as well. This difference is plotted in Fig. 8 for the three values. Figs. 9 and 10 show the radial and hoop stress distributions calculated with the finite-element model. As expected, the obtained stress decreases as increases. We have repeated Figs for an assumed cooling fluid LN temperature of 77 K. Fig. 11 shows the temperature

7 BROWN: ULTRAHIGH-AVERAGE-POWER DIODE-PUMPED Nd:YAG AND Yb:YAG LASERS 867 Fig. 8. Temperature difference between finite-element calculated and analytical model as a function of radius for D = 4mm;Q o = 250 W=cm 3 ;T c = 300 K; and h = 10 (r); 5 ( ); and 1 () W=cm 2 1K. Fig. 11. Temperature as a function of radius for D = 4 mm;q o = 250 W=cm 3 ;T c = 77K; and h = 10 (); 5 ( ); and 1 (r) W=cm 2 1K: Also shown ( ) are analytical results with k and values taken at 77 K. Fig. 9. Radial stress as a function of radius for D = 4 mm;q o = 250 W=cm 3 ;T c = 300 K; and h = 10 (r); 5 ( ); and 1 () W=cm 2 1K. Fig. 12. Temperature difference between finite-element calculated and analytical model as a function of radius for D = 4mm;Q o =250 W=cm 3 ;T c = 77K; and h = 10 (r); 5 ( ); and 1 () W=cm 2 1K: Also shown ( ) are analytical results with k and values taken at 77 K. Fig. 10. Tangential stress as a function of radius for D = 4mm;Q o = 250 W=cm 3 ;T c = 300 K; and h = 10 (r); 5 ( ); and 1 () W=cm 2 -K. distributions for the three values and Fig. 12 the difference between the finite-element calculated and analytical model calculated temperatures. For the analytical model, we used values for and calculated at 77 K. In Figs. 13 and 14, we show the finite element calculated radial and hoop stress distributions, respectively. When comparing Figs with Figs , a number of important facts can be noted. First, in both cases, the deviation between the finite-element and analytical results worsens as is decreased. Second, the temperature difference between the rod center and edge is significantly lower at 77 K than at 300 K. For 1Wcm -K for example, at 300 K is about 28 K whereas at 77 K it is only about 3 K almost an order of magnitude in difference. The decrease in the center-edge temperature difference for the same and is explained by the fact that the thermal conductivity near 77 K (0.876 W/cm K) is significantly larger than at 300 K (0.105 W/cm K). Because is much smaller, the resulting stresses are lower as well, thus thermally induced birefringence will be reduced.

8 868 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 33, NO. 5, MAY 1997 where is the laser wavelength, the amplifier length, and the center-edge difference in index of refraction. can then be minimized by reducing the temperature difference between the center and the edge of the rod. The index difference can be written (18) Fig. 13. Radial stress as a function of radius for D = 4 mm;q o = 250 W=cm 3 ;T c = 77K; and h = 10 (r); 5 (); and 1 ( ) W=cm 2 1K: Also shown ( ) are analytical results with k and values taken at 77 K. Fig. 14. Tangential stress as a function of radius for D = 4mm;Q o = 250 W=cm 3 ;T c =77K; and h = 10 (r); 5 ( ); and 1 () W=cm 2 1K: Also shown ( ) are analytical results with k and values taken at 77 K. To summarize, cooling a rod from room temperature to 77 K results in a much smaller temperature differential a similar reduction in thermally induced stresses results. As pointed out in [10], these reductions have important consequences for the magnitude of thermally induced distortions in HAP amplifiers. A very important consequence of these results is that, because the thermally induced stresses are much reduced in rods cooled to low temperature, it is also possible to significantly increase the heat power density and thus the extractable optical power from such a rod, until one again reaches the limit where the hoop stress equals the fracture strength. This effect is discussed in detail in Section V. The phase or optical path difference between the rod center and edge for the radial and tangential components can then be written (17) where is the change in refractive index with temperature, the linear index, and the the photoelastic coefficients for YAG in the direction [15]. Thermal distortions are significantly reduced by decreasing the rod operating temperature; in the aforementioned, the reduction is almost an order of magnitude, assuming does not vary with temperature. Unlike Al O [10], we have not found measurements of the change in index of refraction with temperature for YAG at reduced temperatures, nor have we found any data on changes in the photoelastic coefficients with temperature. Nevertheless, there are theoretical arguments [26] for showing that is expected to decrease for decreasing temperature, further reducing the thermal distortions at low temperature. As we have already seen, decreases by about a factor of 2.5 between 300 K and 77 K. By using the temperature difference between the analytical model and finite-element generated results shown in Fig. 8- for a 4-mm-diameter rod using a 300 K coolant, 250 W cm 1 W cm K a cm length, and the room-temperature value [11], we can calculate the number of waves of noncorrectable distortion between the center and the edge of the beam. Using (17) (18), and taking the value of corresponding to the temperature at the center of the rod, we find that over four waves of distortion are produced, and a consequence of both the temperature difference as well as the increased value of as temperature increases. Detailed calculations of thermally induced aberrations and birefringence will appear in a separate publication. C. Heat Power Density Effects at 77 K Another important determinant of the temperature distribution in rod amplifiers is the magnitude of. In Fig. 15, we show the temperature distributions for a 4-mm-diameter YAG rod with h constant at 10 W cm K and taking the values 1, 2, and 3 kw cm. In Fig. 15, we have also plotted the analytical temperature solution to the heat equation with assuming a constant value of W/cm Kat77K.As increases, the deviation between the finite-element results and the analytical model becomes more pronounced. For the 3- kw cm curve, it should be noted that pronounced changes in the temperature distribution have taken place. Whereas previously shown temperature distributions have a shape that is convex downward, reminiscent of a quadratic type of profile, the 3-kW cm curve takes on a pronounced bell-shape or Gaussian-like form. In our simulations, this type of curve occurs at 77 K when the center temperature becomes elevated

9 BROWN: ULTRAHIGH-AVERAGE-POWER DIODE-PUMPED Nd:YAG AND Yb:YAG LASERS 869 Fig. 15. Temperature as a function of radius for D = 4mm;T c =77 K;h= 10 W=cm 2 1K; and Q o = 1 (r); 2 ( ); and 3 () kw=cm 3. Also shown ( ) are the analytical curves with k constant and equal to it s value at 77 K. Fig. 16. Maximum tangential stress (r = D=2) as a function of heat density for D = 4mm;T c = 300 K; and h = 1 (r); 5 ( ); and 10 () W=cm 2 1K. Solid lines ( ) drawn through discrete points for illustrative purposes only. with respect to the cooling boundary and is a consequence of a thermal conductivity that becomes small in the rod center and remains significantly larger at the edge. The effect can be caused by a large by increasing the rod diameter, or by decreasing. Simulations show that the magnitude of the stresses also increase with and that for the 3-kW cm case, take on the Gaussian-like distribution as well. V. SCALING EFFECTS IN HAP ROD LASERS WITH VARYING COOLANT TEMPERATURE An important goal of this work was to determine the average power obtainable from both room-temperature and cryogenically cooled laser elements operating at 77 K. Here, we show the results of our finite-element modeling in which both and are varying functions of temperature. We begin by showing in Fig. 16 the maximum hoop stress for a 4-mmdiameter YAG rod at 300 K at the location of the rod barrel, as a function of heat density, for three heat transfer coefficients. was increased until a maximum stress near 5 kg m cm was achieved. It should be noted that the obtained curves are nonlinear; in the analytical model with constant, recall that stress is a linear function of. The magnitude of the stress is reduced as increases. In Fig. 17, we show stress curves for the same rod, but with a coolant temperature of 77 K. Again, we have limited the allowable stress to 5 kg m cm. For any however, it can be seen that significantly larger can be dissipated before a stress level identical to that at 300 K is achieved. To further illustrate the difference between operating at 300 K and 77 K we show in Fig. 18 the ratio K K of power density achievable at 77 K to that achievable at 300 K where the stress values are identical to that at 300 K for a 4-mm-diameter YAG rod and the same three values. The ratio is maximized for low and large and decreases as is increased. In general, the interpretation of these results is complicated, however, we can make the observation that as is increased or decreased, the central region of the rod becomes hotter because the thermal conductivity is decreasing. A rise in temperature Fig. 17. Maximum tangential stress (r = D=2) as a function of heat density for D = 4mm;T c = 300 K; and h = 1 ( ); 5 (r); and 10 () W=cm 2 1K. Solid lines ( ) drawn through discrete points for illustrative purposes only. in the rod center is also accompanied by an increased thermal expansion coefficient and the values of all stress components increase. For a 4-mm rod operating at 250 W cm an increase in heat density and extractable power in the range of 6 8 can be obtained for values in the range of 5 10 W cm K at 77 K. Smaller heat densities lead to even greater increases in the ratio. The results shown in Fig. 18 can be used to predict HAP performance. Using (15) and calculating the normalized extractable power per unit length, we obtain the results shown in Fig. 19, where K is plotted as a function of K for three values of. Also shown for convenience, on the upper abscissa, are the corresponding values of at room temperature. For convenience, in Fig. 19 we have also plotted the normalized power/length obtainable at 300 K. It is clear from Figs that substantial increases in extractable laser power can be obtained by cooling YAG to 77 K. The amount of the increase is dependent upon the value of and and as we will see in what follows, the rod diameter.

10 870 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 33, NO. 5, MAY 1997 Fig. 18. Ratio of heat density at 77 K to that at 300 K producing equal maximum tangential stress (r = D=2); as a function of heat density at 300 K; for D = 4 mm and h = 1 (r); 5 ( ); and 10 () W=cm 2 1K. Solid lines ( ) drawn through discrete points for illustrative purposes only. Fig. 20. Normalized extractable average power at 77 K as a function of normalized extractable average power at 300 K (lower abscissa) or heat density (upper abscissa) for D = 6 mm, and h = 1 (r); 5 ( ); and 10 () W=cm 2 1K: For comparison, the normalized extractable power at 300 K is also plotted ( ). Solid lines drawn through discrete points for illustrative purposes only. Fig. 19. Normalized extractable average power at 77 K as a function of normalized extractable power at 300 K (lower abscissa) or heat density (upper abscissa) for D = 4 mm, and h = 1 (r); 5 ( ); and 10 () W=cm 2 1K. For comparison the normalized extractable power at 300 K is also plotted ( ). Solid lines drawn through discrete points for illustrative purposes only. In Fig. 20, we show the normalized extractable power at 77 K for a 6-mm-diameter YAG rod, obtained in the same way as with the 4-mm rod. When compared to Fig. 19, it shows that for any identical stresses are reached in the 6- mm rod for significantly lower than the 4-mm rod. It is also apparent that for the identical K greater extractable power is obtained from the 6 mm rod, for any value of. For completeness, in Fig. 21 we show the normalized extractable power at 77 K for an 8-mm-diameter rod, obtained in exactly the same way. It is clear from the results presented in Figs that the scaling laws for YAG rod devices are nonlinear and a complicated function of and. It is also apparent that as one increases the rod diameter, the increase in performance at 77 K decreases for any as the center of the rod heats up. Also, as rod diameter is increased, the heat Fig. 21. Normalized extractable average power at 77 K as a function of normalized extractable average power at 300 K (lower abscissa) or heat density (upper abscissa) for D = 8 mm, and h = 1 (r); 5 ( ); and 10 () W=cm 2 1K: For comparison, the normalized extractable power at 300 K is also plotted ( ). Solid lines drawn through discrete points for illustrative purposes only. density to reach a specified stress level decreases. In scaling CW YAG lasers operated with a cooling fluid temperature of 77 K to higher average powers, rather than increasing rod diameter, one strategy may be to use the smallest rod diameter compatible with the efficient absorption of diode light and to use the longest possible element or multiple elements. VI. HAP PERFORMANCE PREDICTIONS FOR DIODE-PUMPED Nd:YAG AND Yb:YAG It is apparent that Figs of the normalized extractable power for various rod diameters can be used to predict the

11 BROWN: ULTRAHIGH-AVERAGE-POWER DIODE-PUMPED Nd:YAG AND Yb:YAG LASERS 871 power output of YAG lasers operating at 300 K or 77 K. We will briefly illustrate those results by presenting a specific example. We assume that a rod length of cm and 8-mm diameter is used and examine what extractable power can be obtained for both Nd:YAG and Yb:YAG. For Nd:YAG, we use acw value of 0.43, while for Yb:YAG we use 0.12 [27]; both correspond to the case where no laser extraction occurs. For strong laser extraction, the value for Nd:YAG would decrease about 25% while Yb:YAG would remain nearly constant [27]. We assume that the rod barrel is polished and use a mean fracture strength of 2.5 kg m cm corresponding to a 64- m-diameter surface defect [17]. Then, by using Fig. 22, one can calculate that Nd:YAG can theoretically approach the kW power level for 8-mm-diameter cm-long rods, and for Yb:YAG can approach kw, both for reasonable heat transfer coefficients in the range 5 10 W cm K and operation at 77 K. At 300 K it appears that kw can be obtained from a single Nd:YAG rod and kw from an Yb:YAG rod, both using the same size rod and heat transfer coefficients. These extractable power values could be increased further by lengthening the rod to the practical limit of about 22 cm, and by a factor of 10 by acid etching of the rod barrel [17]. Clearly, in the absence of other limitations such as optically induced damage and parasitic oscillations, it is possible to predict the development of 100-kW class CW lasers using the techniques discussed in this paper. In practice, a safety factor would be used, and the rod barrel may have a different finish or defect structure. For a ground barrel, for example, the surface defect size increases and a typical fracture strength value would be 1.9 kg m cm [17]. On the other hand, etching a polished surface to remove subsurface cracks can significantly increase the barrel strength, and such surfaces can reach a strength of 33.5 kg m cm [17]. Such scenarios can easily be evaluated by using the approach taken in this article. It should be mentioned that in this article we have not considered rod end-effects or thermally induced birefringence. End effects are ignored using the finite-element model we have chosen, however, we do expect that such effects will play an important role in any experimental work contemplated and must be investigated in the future. Birefringence, which has been modeled using the constant model [15] and a quadratically varying temperature like thermal focusing, will be modified by the straightforward application of the methods discussed in this article, and will be treated in a separate publication. VII. COMPARISON OF HEAT TRANSFER COEFFICIENTS OBTAINABLE WITH WATER AND LIQUID NITROGEN In spite of the considerable technical challenges involved, we have chosen in this article to examine the situation in which LN is flowed directly over the barrel of the rod (or slab). The reason for this is to be able to achieve large heat-transfer coefficients, thus reducing the cooling boundary temperature drop and the rod average temperature, leading to the situation in which stresses are minimized at 77 K. This is unlike previous work where the laser crystal was first placed in intimate contact with a metal (Cu) whose thermal conductivity degrades at low temperature, and used to conduct heat to an LN bath [10]. We have also chosen 77 K the boiling point of LN at atmospheric pressure, as a convenient temperature at which we have calculated the results presented in this paper; actual operating coolant temperatures will be somewhat lower. It is worth briefly summarizing the cooling properties of rods cooled by either water or LN. To accomplish this, we have calculated the transport properties of LN by using a computer code. 2 In Table I, we show a comparison of the properties of water at 300 K and LN at 77 K. The densities and specific heats are comparable while the thermal conductivity is nearly five times smaller for LN and the dynamic viscosity six times smaller. The Prandtl numbers, however, are very comparable, being about three times smaller for LN. For lower temperatures, for example, 65 K the Prandtl number increases to 3.5. The Reynold s number is given by (19) where is the mass flow rate, the hydraulic diameter, the flow cross-sectional area, and the dynamic viscosity [28]. For equal mass flow rates through the same channel, it can be seen that the Reynold s number for LN will be six times larger than for water. Equal mass flow rates means that about a 25% greater volume flow rate is needed for LN. The heat transfer coefficient can be calculated from where is the Nusselt number. Using the Petukov correlation [28], valid for large Reynold s numbers (turbulent regime), the Nusselt number is given by where is the friction coefficient: (20) (21) Assuming that the Petukov correlation applies equally well to LN as to water, using (19) (21) we have calculated coefficients for both fluids under a wide variety of flow conditions, changing the mass flow rate and the hydraulic diameter for a cooling annulus surrounding the rod barrel. We ignored the change in pressure associated with changing the flow rate through such an annulus, and assumed temperatures of 300 K and 77 K for water and LN, respectively. It is well known [15] that coefficients for water cooling a rod typically fall in the range of 1 10 W cm K for water cooling at room temperature and typical flow rates of 1 10 GPM ( gr/s), the exact value depending upon the radii of the inner and outer annulus surfaces. We have found that for LN flowing 2 Gaspak TM, Version 3.20, distributed by Cryodata, Louisville, CO.

12 872 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 33, NO. 5, MAY 1997 TABLE I COMPARISON OF WATER AND LIQUID NITROGEN TRANSPORT PROPERTIES Parameter Parameter H 2 O LN 2 Pressure Atm 1 1 Temperature K Density gr/cm Specific Heat J/(gr- K) Dynamic Viscosity gr/(cm-s) Thermal Conductivity W/(cm- K) Prandtl Number Unitless at the same mass flow rate, Reynolds and Nusselt numbers are substantially larger for LN than for water, and that the ratio of the water coefficient to that of LN2 is approximately The somewhat lower value for LN is a consequence of a lower thermal conductivity. Nevertheless, we have also found that when the friction factors are compared, the ratio of the LN friction factor to that of water is approximately 1.43, hence the pressure drop associated with pumping LN through a channel or annulus will be 43% lower than water, and as a consequence we can flow more LN through the same channel to obtain equal pressure drops. The larger flow rates associated with LN can then for the most part offset the lower coefficient obtained for equal flow rates. As a result, the coefficients obtainable from LN and water can be nearly equal. VIII. CONCLUSION In this paper, we have examined the consequences of thermal and mechanical properties that vary with temperature for the well-known laser material YAG. The results presented here have been for YAG-based rod lasers only and can be easily extended to slab, active-mirror, and disk amplifiers as well. Preliminary calculations have shown that when extended to slab lasers, even larger increases in average power can be expected. We have shown that when a thermal conductivity varying with temperature is taken into account and for uniform heat deposition in the rod, the resulting thermal profile is nonquadratic. Rod distortions cannot be corrected with a simple lens; a more advanced approach is needed that takes into account higher order optical aberrations. Clearly, only an adaptive optics or phaseconjugation approach will completely correct such aberrations over a wide range of operating conditions, although, as shown in Section IV, thermally induced aberrations can be significantly reduced at lower coolant temperatures. In addition, we have shown that the stresses associated with a thermal expansion coefficient that varies with temperature are larger than those predicted using the older steadystate model in which and are held constant at 300 K. Thus, the hoop and axial stresses are seriously underestimated when thermally varying properties are not taken into account. We have also shown that at 77 K stresses and edge-center temperature differences are much reduced when compared to operation at room temperature and that substantial increases in laser average power can be obtained at LN temperature. Another important result of this work is that when properties varying with temperature are taken into account, the temperature distribution and resulting stresses are no longer independent of and the coolant temperature. We have presented normalized average power/length nonlinear curves that describe the obtainable power at 77 K relative to that obtainable at room temperature, for a number of rod diameters. We have shown that for two laser materials, Nd:YAG and Yb:YAG, ultra-hap performance can be obtained even from commonly available sized single elements, at both room temperature and at 77 K. The newer material, Yb:YAG, seems particularly promising in that regard since it has about onethird the quantum defect produced in Nd:YAG and utilizes a material, YAG, that is well developed and about which much is known. Finally, it is obvious that the technical challenges associated with demonstrating ultra-high-average-power lasers are formidable. 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