MODERN high-average-power solid-state lasers suffer

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1 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 34, NO. 12, DECEMBER Nonlinear Thermal Distortion in YAG Rod Amplifiers David C. Brown, Member, IEEE Abstract Yttrium aluminum garnet (YAG) has thermal, mechanical, and optical properties that vary strongly with temperature. In this paper, we show that when such properties are taken into account the resulting thermally induced distortions in rod amplifiers cannot be described as quadratically varying with rod radius, and thus cannot be completely corrected using a simple spherical lens. In addition, we show that for operation with a coolant temperature at 77 K, thermally induced distortion is minimal and birefringence is substantially reduced when compared to that during operation with a coolant at room temperature. Index Terms Birefringence, nonlinear thermal distortion, thermal distortion, thermal focusing, YAG amplifiers, YAG lasers. I. INTRODUCTION MODERN high-average-power solid-state lasers suffer from thermally induced distortion effects. For the wellknown and often-used material yttrium aluminum garnet (YAG), both wavefront distortion and depolarization occur [1]. The control of wavefront distortion and depolarization in medium or high-average-power (HAP) solid-state lasers is an important issue confronting the laser designer, since it can have a profound effect upon the mode structure and the beam quality of the output beam and in some lasers depolarization losses can seriously compromise the laser efficiency. The magnitude of the thermal distortion and birefringence is determined by both the amount of heat deposited per unit volume into the amplifier as well as by the thermal, mechanical, and optical parameters of the material selected. Measurements of the heat fraction in Nd:YAG and Yb:YAG can be found in a recent paper [2]; a detailed theory of heat generation in Nd:YAG is presented in [3]. A linear classical calculation of the thermal distortion and associated focal lengths, and the depolarization, for YAG, can be found in the summary in [1], where all relevant material parameters are assumed to be constants, independent of temperature. In this paper, we generalize the previous theory by considering the thermal distortion and birefringence obtained when all YAG parameters are allowed to vary with temperature. We have used this nonlinear theory to study the thermal and stress distributions in both rod and slab YAG amplifiers and to predict their average power scaling behavior [4], [5]. We will show, using finite-element results, that the transverse index or wavefront distortion in a rod amplifier does not vary in a quadratic way as predicted using the classical linear theory Manuscript received April 27, This work was supported by Ballistic Missile Defense Organization/Science and Technology Division and managed by the Naval Research Laboratory under Contract N P The author is with LE Systems, Inc., Brackney, PA USA. Publisher Item Identifier S (98) [1], but in addition contains higher order terms characteristic of spherical aberration. We also examine the effects predicted when the coolant temperature is reduced from 300 to 77 K and show that significantly smaller thermal distortion and birefringence are obtained at cryogenic temperatures. This effect was recently observed experimentally in connection with the operation of a Ti:Al O laser [6]. We first review, in Section II, the classical theory of thermally induced aberrations and birefringence in YAG lasers and point out the limiting assumptions made. We then examine a more recent analytical treatment that allows the simple calculation of the index changes in rod amplifiers due to either strains or stresses and discuss modifications to the original classical theory needed to obtain agreement between the two methods. In Section III, we briefly review what is presently known of the temperature dependence of critical YAG material parameters, including the thermal conductivity, thermal expansion coefficient, Young s modulus, Poisson s ratio, and the change in index of refraction with temperature. Section IV presents obtained analytical and finite-element modeling results and a comparison of the two approaches, as well as the results of our investigation of the reduction in thermally induced focusing and birefringence in YAG as the coolant temperature is reduced from 300 to 77 K. II. REVIEW OF THERMALLY INDUCED ABERRATIONS IN YAG ROD AMPLIFIERS In rod amplifiers, there are three distinct contributions to the aberrations that occur when significant heating of the YAG material is present. The first is the change in index of refraction with temperature, which is positive in YAG. Because the temperature of a YAG rod varies with the radius, so then will the index of refraction. The second contribution arises from the stresses or strains that occur in solid-state laser materials and the existence of finite photoelastic or piezooptic coefficients. Stresses and strains in a rod also vary with rod radius, and so will their contribution to the refractive index. The third aberration is due to end effects; in finite rods, the strain displacement of the rod faces or ends distort or bulge, principally because the rod center is hotter than the edge. In this paper, we ignore end effects and concentrate on the changes in the indices of refraction of a rod that occur due to and thermally induced stresses or strains. The radially varying indices of refraction can be written where and refer to the radial and tangential (azimuthal) polarizations respectively, is the linear refractive index, (1) /98$ IEEE

2 2384 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 34, NO. 12, DECEMBER 1998 is the radially varying index due to, and is the index variation due to stress or strain. For constant heat power density, we can write the CW temperature distribution in a rod as are also interested in the birefringence shown to be equal to, which can be (10) (2) where the constant is where is the rod center temperature, the radius, and the thermal conductivity. Equation (2) shows that the temperature is largest in the rod center and varies quadratically with the rod radius. The center temperature can be calculated from where is the rod diameter, the coolant temperature, and the surface or barrel heat transfer coefficient, also known as the film coefficient. The second term in (1) can be written because in (4) is always, the contribution of this term to the change in the index is positive. The largest change in index is on the rod axis where the temperature is largest. This intuitive definition is different from that of [1] where the on-axis change in index is zero and calculated values for are negative, but will not change the final results. Using (3) and (4) gives which shows that ideally, in a rod with constant heat power density, the contribution due to varies quadratically with. The third term in (1) can be written where is the thermal expansion coefficient and are constants involving Poisson s ratio, and are the photoelastic coefficients for YAG. can be shown to be equal to for the radial constant, and for the tangential constant. Calculation shows that the constant is positive, so the change in radial index due to stress or strain will be negative., however, is negative, hence the change in tangential index due to stress or strain is positive. Using (1), (5), and (6), the change in the radial and tangential indices can then be written which shows that when both the effects of and strain or stress are included, the overall index variation is still quadratic. We (3) (4) (5) (6) (7) (8) (9) (11) Equation (9) is close to the form of the equation describing the index variation in a Gaussian duct or gradient index lens, given by (12) The first term of (9) does not contribute to thermal focusing since it contains no dependence and can be ignored. The focal length is given by (13) where is the amplifier length and is a constant which is given by (14) Comparing (9) and (12) and using (13), the inverse focal lengths of a rod can then be shown to be equal to (15) which may be used to calculate the focal lengths for any value of the heat density. If, however, the approximation of (13) is used, we find (16) Equation (16) is identical to that found in [1] and is thus only valid under the condition that, which upon substituting values typical of YAG at room temperature gives W/cm for a rod length of cm. It should be pointed out that even if the approximation leading to (16) is not used, and focal lengths are calculated from (15), that expression is still subject to the approximations inherent in the paraxial ray approach that leads to the concept of a Gaussian duct or lens with an ABCD matrix describing it and having the index variation of (12) [7], [8]. It can be shown that, for the paraxial ray approximation applied to a Gaussian duct, the condition must be satisfied, where is the rod radius. Because most rods have a large aspect ratio where, and because is usually much greater than, the paraxial condition is almost always satisfied. Focal lengths of rod amplifiers are routinely calculated using (16) or an extension of (16) that includes an end-effect approximation [1]. The end-effect approximation is added

3 BROWN: NONLINEAR THERMAL DISTORTION IN YAG ROD AMPLIFIERS 2385 because the model reviewed here is a plane-strain model where everywhere, but we are applying it to rods of finite length in which the plane-strain approximation is not applicable in the end St. Venant regions. From (15) or (16), it can be seen that the and polarization components have different focal lengths, an effect that is referred to as biaxial focusing in the literature. We now examine the classical theory in more detail. The phase accumulated by a light ray of wavelength traveling through a dielectric medium of length and with index is given by (17) using rod strains rather than stresses [11], [12]. We now review in more detail the strain approach to calculating the index changes in rod amplifiers due to thermally induced strain; rederiving the relationships found in [1], [11], [12] was necessary in order to get agreement between that theory and a more recent approach [13] that allows the simple calculation of changes in index of refraction using either a strain or stress approach. The refractive index of a crystal is specified by the optical indicatrix, an ellipsoid whose coefficients are the components of the relative impermeability tensor (23) Taking differentials, we then find that The impermeability tensor components are given by (18) where the first term accounts for changes in the physical path length and the second for index changes. Now, since, where is the strain component, we can rewrite (18) as (19) where we have used the general Hooke s law relationship between the strain and the stress components and [9]. It can be seen that the phase change is due to three effects: a change in index of refraction with temperature, a change in the physical path length due to temperature changes, and a change in physical path length due to stresses. If, however, we invoke the plane-strain approximation, in which a long (compared to the diameter) rod is used and the strain is assumed equal to zero everywhere, then one can show that Substitution of (20) into (19) gives (20) (21) Then, in the plane-strain approximation, changes in the physical path length are ignored and only changes in the index of refraction contribute to the phase distortion. This is the reason why changes in physical path lengths are ignored in the classical theory in [1]. In real amplifiers of finite length, of course, both changes in the physical length and index of refraction contribute to the overall phase distortion. It can also be shown that, for the plane-stress approximation, in which it is assumed that, the phase change can be written (22) which is identical to the relationship mentioned in [10]. Note that the plane-stress approximation applies to thin disks, often referred to as face-pumped lasers in the literature. The theory presented in [1] and summarized as (1) (15) is based upon the plane-strain approximation and was derived (24) where the are the indices. Unstressed indices are often designated using, where (25) For a cubic material like YAG which is optically isotropic in the absence of stress, with all other zero, so the indicatrix is a sphere. Thus we set. Changes in the index of refraction are most conveniently specified using the difference between the stressed and unstressed impermeability components using one of two complementary approaches. The first uses the strain components (26) where are the photoelastic coefficients, the the strain components, and thus we can also write (26) as (27) It can also be shown that it is then possible to relate the index change to using the relationship or Similar to (27), we can also write (28) (29) (30) where the are the fourth-rank piezooptic coefficients and the are the stress components. Then (31)

4 2386 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 34, NO. 12, DECEMBER 1998 or (5) to give (32) It is thus possible to describe index changes using either a strain or a stress formulation. We will summarize both approaches but in the remaining sections of this paper we will use the stress formulation, since modern finite-element programs usually return stresses as output and stresses are normally used in the literature. A. Strain Approach We should mention that for computation it is prudent to represent the tensor in the compact form (33) by using the suffix convention of Nye [14]. For YAG, the growth direction is ; the photoelastic components, however, were measured using a crystal orientation [1]. The photoelastic matrix for the orientation is given by [14] (39) Equations (36) (38) are identical to (9) and (10) with the exception of the additional term involving a new constant,. Only the -dependent term in (39) contributes to thermal focusing. This may be seen by using the optical path lengths and then calculating the focal lengths from (40) which, using (39), yields the same expression for the focal lengths as found previously (16). If, however, it is desired to calculate the absolute change in index or the corresponding optical path lengths, all terms must be included. The extra term involving also proved crucial in obtaining agreement between the strain-based model discussed here and the stress approach discussed in the following. The constant in (36) and (37) is given by (41) (34) For we find that, the value we shall use throughout this paper, and. where there are three independent components whose values are, and [1]. The strain matrix in the cylindrical coordinate system along the crystal direction is (35) In order to calculate (33), both and must be in the same coordinate system. In [11], is transformed to the system in which the were measured, (33) is performed, and the results are then transformed back to the system. We give the final results as follows. For the change in the radial index in the system, we find B. Alternative Approach There is another way to approach the calculation of indices in rods or slabs that has been recently employed [13]. Rather than the strain method outlined in the aforementioned, one can simply transform the known photoelastic or piezooptic coefficients from the frame to the frame and then use the results to calculate the index changes in the coordinate system. While there are only three independent or in the frame, each with the same matrix form as (34), after transformation to the laboratory or system one finds that and are given by matrices with eight components, two of which are angular-dependent. For completeness, we will write out the results for the in both strain and stress representations (36) whereas for the tangential index change the result is (37) Similarly, for the birefringence, we find (38) Our final expression for the change in index of refraction with temperature can be found by combining (36) and (37) with (42)

5 BROWN: NONLINEAR THERMAL DISTORTION IN YAG ROD AMPLIFIERS 2387 TABLE I p ij AND ij VALUES IN THE h111i FRAME and (45) These expressions as well as the corresponding strain expressions from (42) have been carefully compared with the older strain formulation represented by the revised expressions in (36) and (37), and exact agreement was obtained. These expressions do not, however, agree with the early strain index relationships of (9) without the term, but the difference of (44) and (45) agrees well with the birefringence (10) since is not involved. Equations (44) and (45) have been used for calculations of the variation of index of refraction with radius at room temperature and 77 K discussed in Section IV. They are particularly useful since they allow the indices of refraction to be calculated from a knowledge of the temperatures and stresses in the rod which are obtained directly by the finiteelement code we employed. Another issue associated with solid-state lasers is the birefringence, calculated according to (38). The difference in the indices of refraction leads to a phase difference given by for the strain representation, and (46) or using (38) (47) Here is the laser wavelength and the amplifier length. For a birefringent crystal placed between a polarizer and an analyzer, the transmission at any point is (48) (43) for the stress representation. It should be noted that the s of (42) are not the same as those previously defined in connection with the linear classical model. In Table I, we will explicitly list all of the s and s to avoid confusion: The coefficients are dimensionless while the s have the units cm /kg here. It should be noted that two of the and components have an angular dependence; is the azimuthal angle or cut angle, measured with respect to the rod axis [13]. The angularly dependent components have a period of 120. Using (42) and (28) and adding the change in index of refraction with temperature contribution (5), we arrive at the final expressions for the radial and tangential indices in a rod laser using stresses (44) where is the angle between the polarizer and one of the crystal principal axes. Integrating (48) over the aperture of a rod amplifier and using (46), the total transmission is found to be given by (49) Equation (49) can be used to predict the transmission through a system of two crossed polarizers as a function of the heat power density since, from (47), is proportional to. Later in this paper, when and are allowed to vary with temperature and the resulting transverse thermal profiles are not quadratic, to evaluate the transmission (48) must be integrated over the rod aperture numerically. To conclude this section, we have reviewed the classic approach to the calculation of indices of refraction, wavefront distortion, and focal lengths in rod lasers. An additional term, apparently ignored in the original derivations, was found and added to the classic expressions for the changes in the index due to strain. We also reviewed a more recent and straightforward approach to calculating the index changes due to strains or stresses. Complete agreement between the two

6 2388 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 34, NO. 12, DECEMBER 1998 methods was found only when the additional term involving the constant was added to (9) to produce (39). Because of the limitations of defining a rod focal length, and because the original strain approach involves invoking the plane-strain approximation which ignores changes in the rod physical pathlength, a more rational approach to treating the wavefront distortion of a finite rod is to employ a beam propagation code which takes into account all of the important effects. In addition, a three-dimensional (3-D) thermal-stress finite-element code must be used to provide local values of the temperature and stresses at each location and which uses local values for the index of refraction determined by (44) and (45). Refraction at the distorted end faces must also be taken into account. III. VARIATION OF YAG THERMAL AND MECHANICAL PROPERTIES WITH TEMPERATURE Recently, in connection with determining the nonlinear scaling behavior of rod amplifiers [4], we showed that the thermal conductivity and thermal expansion coefficient of YAG are strong functions of temperature, while Poisson s ratio and Young s modulus vary only weakly with temperature. The variation of and with was found to have important consequences for the radial temperature profiles obtained, which are nonquadratic, and in determining the stresses which are underestimated using the classical theory presented in Section II. Here we have employed the same fits to and used in [4]. The fit for is (50) where W/(cm K), (K),, and W/cm. For, weuse (51) where (K) and. It has been pointed out in a recent paper [15] and noted in [4] that (50) underestimates the thermal expansion coefficient of YAG at room temperature but that as, as required by theory [16]. An alternative fit was discussed in [15] and is given by (52) where (K) and (K). This fit produces values near room temperature closer to reported single point measurements, but is nonphysical since becomes negative for. Precise measurements of in YAG are needed to establish the correct functional form; in lieu of such measurements and because (52) gives a nonphysical result for low temperatures, in this paper we have used the same fit to, (51), used in our previous work on rod and slab amplifiers [4], [5]. We have also used the same values for Poisson s ratio and Young s modulus, as used previously [4], [5], and kg/cm, respectively. and were shown to vary by no more than 7% and 2%, respectively, across the range K and are thus taken as constants [4]. Fig. 1. Change in optical path length of YAG with temperature, showing data points from [15] as well as the best linear fit. To compute the indices of refraction variation with temperature, (44) and (45), we must know the variation in index of refraction with temperature and its temperature dependence. However, to our knowledge, there have not been any direct measurements performed of the index of refraction of YAG as a function of temperature. has been measured at a number of discrete wavelengths or inferred from measurements of the optical path length; a good summary of those measurements may be found in [15]. Measurements of the change in the thermooptic coefficient with temperature over a wide temperature range ( K) are reported in [15], where is related to the thermal expansion coefficient and through the relationship (53) where is the linear refractive index. In Fig. 1, we show the measurements of as well as a best linear fit to the data given by (54) where K and K.By combining (52) and (50), we can solve (52) for. The result is shown in Fig. 2, where the derived data points and the best fit are shown; the fit to is given by (55) and where K, K, and K. This fit has been used to calculate the thermal distortion and birefringence results discussed in Section IV. It should be pointed out that the data of [15], shown in Fig. 1, when linearly extrapolated to lower temperatures, indicates that at some critical temperature the change in the optical path length becomes zero. Because for, and it can be shown that as [17], from (53) should also equal for zero temperature. Because was not measured below about 86 K where it can be expected to make a smooth transition to zero for, and due

7 BROWN: NONLINEAR THERMAL DISTORTION IN YAG ROD AMPLIFIERS 2389 Fig. 2. Change in index of refraction of YAG with temperature as a function of temperature. Data points are derived from the data of [15] and (51). The solid curve represents the best fit to the data. to the approximate nature of of (50), shown in Fig. 2 vanishes at about 85 K rather than at 0 K. The correct functional dependence of and as must be determined by more accurate and complete experiments. Nevertheless, the fit of (54) is used in the remainder of this paper to estimate the magnitude of thermally induced distortions in rod amplifiers using coolants at 300 and 77 K. It is clear from the aforementioned and [17] that operating a rod amplifier with cryogenic cooling can result in a large reduction in index variations due to. In Section IV, we will show that this reduction in, when combined with reductions in the center-edge temperature due to the larger thermal conductivity of YAG available at low temperature, and a corresponding reduction in the stresses, can lead to HAP YAG rod amplifiers displaying only small or residual thermal distortion. Fig. 3. Change in total index of refraction as a function of rod radius for a 6-mm-diameter YAG rod, 300 K coolant temperature, a heat power density of 250 W/cm 3, and a heat transfer coefficient of 5 W/cm 2 1K. k;, and values were set equal to their values at 300 K. Discrete points were generated by use of a finite-element code and (44) and (45) for the radial () and tangential ( ) polarizations. Solid curves represent results obtained from the modified classical analytical model [see (39)]. IV. NONLINEAR THERMAL DISTORTION IN ROD AMPLIFIERS: MATERIAL PARAMETERS VARYING WITH TEMPERATURE We first examine a linear case in which and are constant and equal to their values at 300 K, calculated from (50), (51), and (55). Fig. 3 shows the total change in index of refraction as a function of radius for a 6-mm-diameter YAG rod with a heat transfer coefficient of W/cm K, a coolant temperature of K, and a heat power density W/cm. To generate this plot and the remaining plots in this paper, we used a commercial finite-element code 1 that provides temperature and stress as a function of rod radius. The plane-strain approximation was used. In Fig. 3 we show the index change for both the radial and tangential polarizations, calculated using (44) and (45), respectively, and using stresses provided by the finite-element code. Also shown in Fig. 3 as solid lines are the change in the indices of refraction calculated directly from the analytical expressions of (39). Since the finite-element results were calculated using the stress approach reviewed previously in Section IV of this article [13], while the analytical approach is based upon the 1 PDEase, distributed by Macsyma, Arlington, MA USA. Fig. 4. Change in total index of refraction as a function of rod radius for a 6 mm diameter YAG rod, 300 K coolant temperature, a heat power density of 250 W/cm 3, and a heat transfer coefficient of 5 W/cm 2 1K. k;, and values were allowed to vary with temperature. Discrete points represent finite-element results obtained for the radial () and tangential ( ) polarizations. The solid and dashed curves represent the best fits to the discrete points for the tangential and radial polarizations, respectively. modified model of [1], it can be seen that the two distinctly different methods agree very well. It should be noted that the results of Fig. 3 vary quadratically with the radius of the rod as expected for, and constant, and that the index variation is larger for the radial than for the tangential component. As a consequence, the radial distortion is larger than the tangential and the corresponding focal length is thus shorter [1]. In Fig. 4, we show the results of a simulation using the same parameters as Fig. 3, but with, and now allowed to vary with temperature according to (50), (51), and (55) respectively. The magnitude of the index change is significantly larger for both polarizations, due to the larger rod center temperature at-

8 2390 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 34, NO. 12, DECEMBER 1998 tangential polarizations, respectively, where (57) For and fixed, it can be shown by use of (57) and (16) that the following relationship holds: (58) Fig. 5. Ratio of radial center-edge index difference to tangential center to edge index difference as a function of heat power density for YAG rod diameters of 4 (), 6 (), and 8 (r) mm. Coolant temperature was 300 K and the heat transfer coefficient had the value 5 W/cm 2 1K. Discrete points were generated using finite-element analysis with k;, and varying with temperature. Solid lines through the discrete points are for illustrative purposes only. The dashed horizontal line represents the ratio obtained using the classical linear model (a value of 1.13) with k;, and constant and equal to their values at 300 K. tributable to a smaller thermal conductivity there, and because the stresses are also larger [4], [5]. The radial distortion is still larger than the tangential, however, the resulting radially dependent curves are no longer quadratic with. Best fits to the finite-element generated points in Fig. 4 are of the form (56) Terms of even power larger than the quadratic term may be characterized as spherical aberration (fourth-order term) and higher order spherical aberration (sixth-order term). If (56) is converted to phase by use of (21) and a normalized radius defined, then we arrive at a phase aberration power series in which each term in (56) may be related to the circle polynomials of Zernicke [18]. In practice, we have found that terms of up to the eighth-order are required to accurately fit profiles like those shown in Fig. 4, because when and are allowed to vary with temperature, the resulting index profiles are not described by a simple quadratic dependence; it is not possible to completely correct for thermally induced aberrations in rod amplifiers using common spherical lenses. We have found that the same conclusion applies to slab amplifiers as well, thus simple cylindrical lenses cannot completely correct thermal aberrations. This result is a consequence of the significant variations of thermal conductivity, thermal expansion coefficient, and of YAG with temperature. In this paper, we have examined the thermal aberrations under a variety of conditions with, and variable. Because of the difficulties in defining rod focal lengths under high heat loads, and because of the presence of spherical aberration, in Fig. 5 we have simply plotted the ratio of the index variations and between the rod center and edge as a function of the heat power density for the radial and where the focal lengths can be calculated from (16). For small this ratio takes the constant value 1.13, thus the tangential focal length is longer than the radial focal length. This ratio is shown as the dashed horizontal line in Fig. 5 where the ratio of (58) is displayed as a function of heat power density and rod diameter. When, however, we let, and vary with temperature, the results shown in Fig. 5 for rods with diameters of 4, 6, and 8 mm are obtained. This plot displays a number of interesting features. First, the ratio is a function of both rod diameter and the heat density. Using the classical linear theory, the ratio of the rod tangential to radial focal lengths is independent of rod diameter and. Second, each curve converges, for low, to a ratio value of about Thus, for low and as, where (58) holds, for, and variable, the result is that the ratio approaches a significantly larger value than for, and constant. Thus, the ratio also has a larger value. It has been noted [1] that measurements of the ratio are larger than the theoretical ratio value, calculated as 1.2 in [1]. Our lower theoretical ratio value of 1.13 is due to the approximate nature of our fit to the thermal expansion coefficient (51), which it underestimates at room temperature. The generation of better data and fits to the data in the future will increase our ratio value for, and fixed closer to that of [1] and will also increase the ratio value beyond 1.22, shown in Fig. 5 for, and variable. It thus appears that the larger values of the ratio of the radial/tangential focal lengths measured experimentally for YAG may be at least partially attributable to ignoring important physical parameters that vary significantly with temperature. Fig. 5 also shows that the focal length ratio is not constant but shows a strong variation with and the rod diameter. The focal length ratio when, and are variable also varies with the heat transfer coefficient. We have also simulated the variation in the rod indices when the coolant temperature is varied. Of particular interest is the decrease in thermal distortion when a rod is cooled to the vicinity of 77 K with a cryogenic coolant such as liquid nitrogen [4], [5]. Fig. 6 shows the total change in index of refraction as a function of rod radius for the tangential and radial polarizations for a 6-mm-diameter YAG rod with W/cm and W/cm K., and were held fixed at 77 K. The small distortion obtained is a consequence of the reduced at low temperature, the smaller radial temperature gradient, and the decreased stresses. The slightly negative values of the change in index are attributable to the negative value of at 77 K. The fact that the tangential polarization has an increasing value with radius is not physically significant since the value of obtained at any temperature should be positive [17], and when better data are available the tangential index variation will likely remain focusing rather than defocusing in nature. A comparison with

9 BROWN: NONLINEAR THERMAL DISTORTION IN YAG ROD AMPLIFIERS 2391 Fig. 6. Change in total index of refraction as a function of the YAG rod radius for a rod diameter of 6 mm, heat power density of 250 W/cm 3, heat transfer coefficient of 5 W/cm 2 1K, and a coolant temperature of 77 K. k;, and are constant and equal to their values at 77 K. Discrete points for the radial () and tangential ( ) polarizations were generated using finite-element analysis. Solid curves are drawn through discrete points for illustrative purposes only. Fig. 8. Transmission through a pair of crossed polarizers as a function of heat power density for a 6-mm-diameter YAG rod with length cm, cooling at 300 K, and a heat transfer coefficient of 5 W/cm 2 1K. The solid curve is for k;, and constant and equal to their values at 300 K, calculated according to (49). Also shown are discrete points calculated by numerically integrating (48) for k;, and variable at 300 ( ) and 77 (r) K for a heat power density of 250 W/cm 3. Fig. 7. Change in total index of refraction as a function of the YAG rod radius for a rod diameter of 6 mm, heat power density of 250 W/cm 3, heat transfer coefficient of 5 W/cm 2 1K, and a coolant temperature of 77 K. k;, and are allowed to vary with temperature. Discrete points for the radial () and tangential ( ) polarizations were generated using finite-element analysis. Solid curves are drawn through discrete points for illustrative purposes only. Fig. 3, where the same rod simulation is shown for a coolant temperature of 300 K, indicates that in the rod center the reduction in the change in index is by a factor of over 300. The reduction in the center-edge index variation for the radial polarization is over 200. Clearly, a rod run with a large heat power density at room temperature will display only small thermal effects with cooling at 77 K. In Fig. 7, we show results for the same rod but with, and allowed to vary with temperature. The variation in both indices is increased because of the larger temperature gradient and increased stresses. Comparing the more realistic results of Fig. 7 to Fig. 3 shows that the on-axis variation in the index is reduced by a factor of 70 while the radial center-edge index variation is reduced by a factor of over 50. Since most YAG amplifiers have heat power density values smaller than the 250 W/cm assumed here, it seems likely that a 100-fold decrease in thermal distortion effects can be obtained in common YAG rod amplifiers cooled at 77 K. The near vanishing of thermally induced aberrations in YAG rod amplifiers cooled at 77 K can have important consequences for solid-state laser systems. HAP lasers can be demonstrated of which resonator characteristics will only be slightly perturbed by thermal focusing; output beam quality and mode content will then remain near-constant as average power is changed, a desirable characteristic for many important laser applications. The near vanishing of thermal distortion for 77 K coolant temperature is also predicted in slab amplifiers. Slab amplifiers are often employed in HAP lasers to eliminate thermal effects in the total-internal-reflection plane, but suffer due to the lack of thermal compensation in the transverse plane. Because practical HAP rod amplifiers can be constructed with cooling at 77 K with no significant thermal aberrations, it seems likely that in the future the use of slab amplifiers could be eliminated entirely. We have also investigated the birefringence losses in rod amplifiers operating at 300 and 77 K. Fig. 8 shows the transmission through crossed polarizers as a function of heat power density calculated by use of (49), for a 6-mm-diameter cm-long YAG rod, assuming an operating wavelength of 1064 nm. For W/cm, it can be seen that the transmission is calculated as about and were taken equal to their values at 300 K. If, however,, and are allowed to vary with temperature and the coolant temperature remains at 300 K, numerical integration of (48) shows that the transmission drops to about 0.62, or less than the theoretical minimum allowed by the classic treatment. This is a consequence of the larger stresses obtained. If the same rod is

10 2392 IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 34, NO. 12, DECEMBER 1998 then cooled to 77 K with, and still allowed to vary with temperature, then the transmission recovers to about Thus, cooling a YAG rod to cryogenic temperatures will result in a significant reduction of thermally induced birefringence and an increase in transmission. V. CONCLUSION In this paper, we began by first reviewing in Section II the classical linear theory of thermally induced aberrations in rod amplifiers. Two distinct approaches were summarized: the strain approach of [1], [11], and [12] which involves tensor transformations from the laboratory frame to the frame in which the photoelastic constants were measured and back again, and a more recent [13] stress or strain method in which the photoelastic constants were transformed from the frame to the frame. It was found that, to achieve agreement between the two approaches, it was necessary to add an additional derived term involving a new constant to the equations for the change in the indices of refraction in the original strain formulation. The new term does not affect the derived radial or tangential focal lengths of a rod but correctly predicts the absolute change in the indices of refraction. In Section III, we discussed what is presently known of the variation of the thermal conductivity, thermal expansion coefficient, and change in index of refraction with temperature as temperature is varied and showed fits to all three parameters used in our finite-element modeling. In Section IV, we first showed in Fig. 3 the excellent agreement obtained for a YAG rod amplifier between the modified strain model of [1] and a stress model of [13] when, and are held constant at 300 K. In Fig. 4, we showed results obtained for the same amplifier at 300 K but with, and varying with temperature. The resulting change in index of refraction profiles were shown to be nonquadratic for both radial and tangential polarizations; spherical aberration and higher order spherical aberration was found for both profiles. The resultant phase aberrations can then be expressed as a series of Zernicke polynomials. This is one of the central results of this paper; except for rod and slab amplifiers in which the heat power density is very low, the transverse phase aberrations in a rod amplifier are not completely correctable using spherical or cylindrical lenses as predicted using the linear classical theory [1]. We also examined, and summarized in Fig. 5, the center-edge index differences for radial and tangential polarization. The ratio of the center-edge index difference for radial to tangential polarizations is equal to 1.13 for the linear model and is independent of the heat power density, rod diameter, and the heat transfer coefficient. Fig. 5 shows that when, and are allowed to vary with temperature, the index ratio is a function of rod diameter and heat power density. Additional simulations show that the index ratio is also a function of the heat transfer coefficient. For, the index ratio approaches about 1.22 for all rod diameters examined, thus the ratio of the tangential to the radial focal lengths also approaches that value. These results may partially explain why values of the focal length ratio larger than those predicted theoretically are measured experimentally. In Figs. 6 and 7, we showed the expected index variations for a YAG rod amplifier with a coolant temperature of 77 K, for both and constant and variable. respectively. Thus, another of the central conclusions of this paper is that thermally induced distortions can be reduced to insignificantly small levels in YAG rod amplifiers, even in those operated with HAP. Fig. 8 shows that substantial decreases in thermally induced birefringence can be obtained in rod amplifiers operated with cryogenic cooling. ACKNOWLEDGMENT The author is grateful to Dr. J. McMahon of the Naval Research Laboratory for his guidance during this project. REFERENCES [1] W. Koechner, Solid-State Laser Engineering, Springer Series in Optical Sciences, D. L. MacAdam, Ed., 4th ed. Berlin, Germany: Springer- Verlag, [2] T. Y. Fan, Heat generation in Nd:YAG and Yb:YAG, IEEE J. Quantum Electron., vol. 29, pp , [3] D. C. Brown, Heat, fluorescence, and stimulated-emission power densities and fractions in Nd:YAG, IEEE J. Quantum Electron., vol. 34, pp , Mar [4], Ultrahigh-average-power diode-pumped Nd:YAG and Yb:YAG lasers, IEEE J. Quantum Electron., vol. 33, pp , May [5], Nonlinear thermal and stress effects and scaling behavior of YAG slab amplifiers, see this issue, pp [6] P. A. Shulz and S. R. Henion, Liquid-nitrogen-cooled Ti:Al 2 O 3 laser, IEEE J. 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Daneu, Thermal coefficients of the optical path length and refractive index in YAG, Appl. Opt., vol. 37, pp , [16] R. S. Krishnan, R. Srinivasan, and S. Devanarayanan, Thermal Expansion of Crystals. New York: Pergamon, [17] D. C. Brown, Thermo-optic, thermal expansion, and dn/dt: Variation with temperature in YAG, unpublished. [18] M. Born and E. Wolf, Principles of Optics, 6th ed. New York: Pergamon, David C. Brown (M 87), for photograph and biography, see p. 572 of the March 1998 issue of this JOURNAL.