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38, NO. 12, DECEMBER 2002. Thermooptical Compensation Methods for. High-Power Lasers. Eduard Wyss, Michelle Roth, Thomas Graf, and Heinz P. Weber.
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IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 38, NO. 12, DECEMBER 2002

Thermooptical Compensation Methods for High-Power Lasers Eduard Wyss, Michelle Roth, Thomas Graf, and Heinz P. Weber

Abstract—Thermally induced optical effects can be exploited to generate adaptive optical devices such as self-adjusting lenses. An adaptive lens in a resonator can be used to compensate for the thermal lens in a high-power solid-state laser rod (LR) and herewith significantly improve the beam quality and increase the output-power range of solid-state lasers. With suitable materials and an appropriate design of the compensating device, resonators with self-balancing thermal lenses can be developed. In this paper, we review the material requirements for a self-adaptive compensating element and discuss a selection of suitable materials (glasses, liquids and curing gels) and schemes to compensate for the thermal lens of a Nd:YAG LR. Finally, we present a very simple and promising design of a thermooptically self-compensated laser amplifier.

scheme exploits the thermal effects themselves by generating a power-dependent (negative) thermal lens in a suitable material that compensates for the pump-power dependent thermal lens in the laser rod (LR). This technique requires a careful optimization of several material parameters in the compensating medium. The selection of the compensating material is, therefore, an important task for a successful implementation of this self-adaptive compensation scheme. The required material properties for the proposed self-adaptive compensating scheme are reviewed in the following section. Subsequently, the suitability of glasses, liquids, and curing gels, as well as different setups, are discussed and compared.

Index Terms—Adaptive optics, fluids, graded index optics, laser beam distortion, laser resonators, laser stability, laser thermal factors, laser tuning, neodymium:glass lasers, neodymium:solid lasers, neodymium:YAG lasers.

II. THEORY In a first-order approximation, the dioptric power of a thermally induced lens in a LR varies linearly with the pump-power

I. INTRODUCTION

T

HE THERMALLY induced lens in the gain material is by far the most critical issue in the development of highpower solid-state lasers and limits the output power of lasers @ 1064 nm with good beam quality (such as for laser cutting of sheet metal) to a narrow power range. The for stable laser oscillation maximum pump-power range is given by [1] (1) is the beam propagation factor, denotes the wavewhere is the specific dioptric power. The quantity length, and depends on the material properties, the cooling method, and the pumping distribution. Equation (1) clearly illustrates that, with a , good beam quality and an arbitrarily large pump given large cannot be achieved simultaneously. Hence, the power range has to be reduced to extend the pump specific dioptric power power range with good beam quality. Recently, we have proposed a self-adaptive compensation scheme that balances the thermally induced lenses in a resonator [2]. Experimentally, the technique was shown to be very effective [3]. The total resulting thermal lens in a Nd:YAG laser was reduced by an order of magnitude, which led to a significantly improved beam quality and extended the power range for stable laser oscillation. The self-adaptive compensating Manuscript received June 7, 2002; revised August 19, 2002. The authors are with the University of Bern, Institute of Applied Physics, CH-3012 Bern, Switzerland (e-mail: [email protected]). Digital Object Identifier 10.1109/JQE.2002.805105

where

(2)

depends on material constants of the LR and on The slope . According to (1), this pump-power the pump spot radius dependent thermal lens limits the power range and the beam quality of a laser system. Deformable mirrors [4], [5], moveable lenses [6], or resonator length adjustments [7] are possible means to compensate for these effects. However, they require sophisticated mechanical arrangements and active external control. Another approach is to exploit the thermal effects themselves to compensate for the thermally induced optical distortions in the gain medium. Preliminary work and ideas on this approach can be found predominantly in early patents [8]–[12]. However, most of these concepts lack an appropriate geometry of the heat flow out of the compensating element (CE). As is pointed out in Section IV, the choice of an optimized geometry was a key to the successful implementation of the technique. In our proposed self-adaptive compensation scheme, a CE—that generates an additional power-dependent thermal lens—is added to the resonator, as shown in Fig. 1. With this setup, the compensating thermal lens is generated in a slightly absorbing rod-shaped material that is heated by the intracavity ). An appropriate design of the compenlaser radiation ( sating element (in particular radial symmetry of the cooling of the thermal scheme) [3] assures that the dioptric power lens is proportional to the intracavity laser power and, therefore, depends on the pump power ( ) of the LR as

0018-9197/02$17.00 © 2002 IEEE

(3)

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TABLE I PROPERTIES OF THE LASER GLASS LG-760 [SCHOTT]

Fig. 1. Self-adaptive compensation scheme for the thermally induced lens in the LR D . A negative thermal lens is induced by the intracavity laser-radiation in the CE. The relay optics (f =2f =f ) is used to optically superimpose the two thermal lenses D and D .

where is the laser threshold. The slope depends on the material properties, the geometry of the compensating material and also on the laser configuration [2], [3]. The two lenses are superimposed by means of relay optics, and the total dioptric power inside the resonator is given by (4) is constant. For an optimum balance, the slope whereby . With this, of the compensating lens should be neglecting the influences of birefringence1 , the total thermal for all pump lens in the resonator has a constant value of powers above threshold. With the detailed analysis of the slopes and given in [2], it is found that the material parameters of the LR and the CE should fulfill the relation (5) for an optimum balance of the two thermal lenses. The left-hand side of (5) is the total double-pass absorption in the CE needed to generate the compensating lens and is in fact a loss to the is the absorbing length and the laser resonator, where and are the heat absorption coefficient. The quantities conductivities of the compensating element and the LR, respecis the thermal dispersion, and the output coutively, pling of the laser resonator. The slope efficiency of the laser is is the fraction of absorbed pump power denoted by , and that is converted to heat in the LR (both values typically amount to 35% for diode-laser pumped Nd:YAG and can be cancelled and are down). In view of the aberrations, the radii preferably chosen to be the same. The output coupling typically amounts to something between 10% and 20%. It is seen from (5) in that in order to keep the required absorption losses 2 the compensating lens as small as possible, the CE should exhibit a strong thermal dispersion and a low heat conductivity. In the following, we discuss the suitability of different materials and geometries for the proposed compensation scheme. III. CE MADE OF GLASS According to (5), the compensating material should exhibit low heat conductivity (as compared to the LR material Nd:YAG), a strong negative thermal dispersion and a nonvanishing absorption at the laser wavelength. The heat conductivity of glasses is typically an order of magnitude smaller than that of Nd:YAG and glasses with negative thermal dispersion 1The limitations caused by thermally induced birefringence and the solutions to this problem will be discussed elsewhere.

are readily available. Typical values for the total double pass should be about 1% or less. Without absorption 2 further doping, however, the absorption of the available glasses at a wavelength of 1064 nm is much too weak to generate the desired thermally induced lenses within reasonable rod lengths. In collaboration with Schott Glas, Mainz (Germany) we developed new glass samples based on PSK 54 doped with 0.2%, 0.5%, and 1% of Cu. These glasses exhibited an unexpected high absorption. We measured 90% of absorption at 1064 nm within a 5-mm sample of 0.2% Cu:PSK54. To attain the desired absorption of about 1% within a 50-mm-long rod, we estimate that the Cu concentration would have to be reduced to the order of about 10 . Because of the high costs of custom glasses and the availability of other promising materials, we did not pursue the development and the examination of these custom materials any further. However, we note that such custom glasses are potentially suitable materials for an efficient self-adaptive compensation. As an alternative, available active-ion doped laser-glasses such as LHG-8 (Hoya) or LG-760 (Schott) can be used to generate the compensating lens. In the following section we discuss the suitability of LG-760 as one material that is readily available. A. Laser Glass LG-760 LG-760 meets the required properties outlined above. In particular and in contrary to other glasses, LG-760, a Nd-doped glass, absorbs radiation at the laser wavelength of 1064 nm. The material properties of LG-760 are listed in Table I. The absorp) tion at 1064 nm is due to the transition from the lower ( I to the upper ( F ) laser level. Absorption can, therefore, only take place when the lower laser level is (thermally) populated. Since the population of the lower laser level, on the other hand, increases with temperature, the absorption coefficient is also a function of the temperature. The absorption coefficient of 8% Nd doped LG-760 was measured to be C with and

m C

(6)

which holds for temperatures between about 10 C–140 C. The temperature dependence of the absorption has some interesting consequences: On the one hand, the amount of absorpof the dioptric power of the tion and therefore the slope thermally induced lens versus the incident laser power can be set by adjusting the cooling temperature. On the other hand, it can also lead to an exponential increase of the temperature and, consequently, to the thermal destruction of the rod, since an increase in temperature leads to an increase in absorption and

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the increase of absorption on its turn leads to a further increase in temperature. This feedback mechanism can lead to an instability of the temperature and therefore to massive thermal load which eventually results in the fracture of the rod. The behavior of LG-760 was investigated experimentally with a 45–mm-long rod that was irradiated with 76 watts of optical power delivered from a Nd:YAG laser at 1064 nm. The rod was water cooled at a temperature of 55 C. Such a high cooling temperature was chosen to attain a sufficiently high absorption at 1064 nm. The lens was measured by means of a Mach–Zender-interferometer and was found to have a dioptric power of 0.18 m . Most of our LG-760 rods were damaged due to the above-mentioned feedback mechanism when we attempted to generate significantly stronger lenses. This behavior was confirmed with the numerical simulations presented in the following.

(a)

B. Numerical Simulations The simulations were performed to determine the limits imposed by the specific properties of LG-760. The time-dependent heat-conduction equation was solved numerically with the method of finite differences. Since the rod is much longer than its radius, the heat flow can be assumed to be purely radial. It is noted that only the temperature gradient in the radial direction contributes to the thermally induced lens. Since only a small percentage of the incident optical power is absorbed, the heating power density can be assumed to be uniform along the rod axis. Taking advantage of the radial symmetry of the rod and the heating power density, the heat-conduction equation in cylindrical coordinates reads (7) the heat capacity at where is the heat diffusion constant, constant volume, the mass density, and the heating-power density. In our model, the boundary condition is [13] (8) is the heat conductivity, is the temperawhere is the temperature of the ture on the surface of the rod, cooling fluid, and is the heat transfer coefficient. Assuming, for simplicity, a uniform distribution of the optical power within the radius of the rod, the (nonuniform) heating-power density can be described by

(9) is the optical power irradiated into the rod, is where the fraction of the absorbed optical power that is converted to is the absorption coefficient as given by (6). A heat, and rod with a length of 45 mm and a diameter of 4 mm was assumed in the simulation. The heat conversion factor is assumed . The heat transfer coefficient at the water-cooled to be

(b) Fig. 2. (a) Temperature evolution in a water-cooled (20 C) LG-760 rod at a pump power of 2000 W. Within a few seconds, the temperature reaches a stable and parabolic profile. (b) Same geometry, but with an optical pump power of 2830 W. The increase of absorption with temperature causes a diverging temperature evolution.

rod surface is W cm K [13]. Fig. 2(a) shows the temperature evolution across the LG-760 rod at an optical power of 2000 W and a cooling temperature of 20 C. The temperature reaches a stationary distribution within a few seconds after the optical power is turned on. The resulting temperature distribution is almost parabolic. For higher pump powers ( 2800 W), the feedback-mechanism between temperature and absorption leads to an instability in temperature, as shown in Fig. 2(b). In the first 10–20 s, the temperature at the center of the rod rises quickly to about 80 C. During the next minute, the temperature increases more slowly up to about 100 C. At this point, the temperature grows exponentially with time due to the feedback mechanism between temperature and absorption. This behavior was observed several times during the experiments. While, for a certain time, the thermally induced lens seemed to be stable, moments later the rod fractured. A summary of the simulation results is shown in Fig. 3. For the three cooling temperatures, 20 C, 40 C, and 55 C, the top graph shows the temperature difference between the center and the cooled surface of the rod and the graph at the bottom shows the calculated dioptric power of the thermally induced lens. For low pump powers, the dioptric power varies almost linearly with temperature. The strongest attainable lens is limited by the temperature instability caused by the feedback mechanism. The most important result of the simulations is that the

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(a)

Fig. 3. Simulation results for a 45-mm-long water-cooled LG-760 rod. The water-cooling temperature is 20 C, 40 C, and 55 C, respectively. Top: Resulting temperature difference between surface and rod center. Bottom: Dioptric power of the resulting thermal lens.

upper limit for the dioptric power of the lens is almost the same for all cooling temperatures. For an LG-760 rod with an uniform pump beam, the maximum that can be attained without the destruction of the rod is 2 diopters/cm rod length. The numerical calculations are in good agreement with the experimental results. For the above-mentioned experiment with 76 W of laser power incident on the 45-mm-long glass rod at 55 C, the simulation results in a dioptric power of 0.2 m (experiment: 0.18 m ). The simulations were carried out with homogeneous pump-beam distributions. With inhomogeneous beams (e.g., Gaussian intensity distribution with a specific beam radius), the stability limit of the temperature will be reached at significantly lower powers. For the compensation of the thermal lens in a high-power Nd:YAG laser, the dioptric power in the CE should be in the order of 10 m . This can only be generated in about 12-cm-long rods to be sufficiently far away from the limits imposed by the temperature instabilities and to stay within the linear behavior of the induced lens. However, long rods are not practical for an intracavity compensation—also because of the problem of beam clipping of the highly divergent beams emerging from the relay optics in the resonator. Hence, LG-760 should mainly be used to compensate for thermally induced distortions in materials with comparatively weak thermal lenses such as Nd:YLF. For gain media such as Nd:YAG, we found a more promising approach based on liquid materials. IV. CE MADE OF LIQUIDS OR CURING GELS Most liquids, such as water, optical-fluids and optical-gels, have a comparatively strong negative thermal dispersion . The absolute values exceed those of Nd:YAG by almost two orders of magnitude. Because of this, even a small amount of absorbed power can lead to a strong thermal lens in a thin layer, sufficient to compensate for the thermal lens in a Nd:YAG laser. A CE can be built by sandwiching a thin layer ( millimeter) of liquid between transparent materials. When using a fluid as the compensating material, the formation of convection has to be prevented either by using materials with high viscosity or

(b) Fig. 4. (a) The compensating material is sandwiched between two glass substrates and heated up by laser radiation. (b) Induced dioptric power in a 1-mm-thick OC-431 A layer. For such a design of the compensating element, the dioptric power of the lens depends more on the shape of the beam rather than on the absorbed optical power.

by using very thin layers. Furthermore, since thermal lenses are caused by temperature gradients in the radial direction, one has to make sure that the heat removal takes place in the radial direction. With thin layers (thin compared to the radius) sandwiched between two glass substrates, the heat will be removed mostly along the beam axis. As a consequence of this longitudinal heat flow, the resulting transversal temperature distribution within the liquid layer is similar to the shape of the incident laser beam. Thus, the shape of the thermally induced OPD—and with it the dioptric power of the thermal lens—depends on the beam shape rather than on the applied optical power. This behavior is illustrated in Fig. 4(a), where the dioptric power does not change proportionally with the absorbed laser power. Hence, in order to generate the desired power-dependant thermal lens in the CE, the temperature profile in the liquid has to correspond to the temperature profile generated by a radial heat removal out of a cylindrical rod. This can be achieved by assembling the CE, as shown in Fig. 5(a). The compensating medium is filled in a thin gap between two edge-cooled glass rods. No matter whether the heat is generated in the compensating medium or in the adjacent rods, the heat is always removed mainly in a radial direction resulting in the desired temperature profile and the corresponding power-dependent lens as shown in Fig. 5(b). In such a composite element, the high thermal dispersion of the liquid is combined with the comparatively high thermal conductivity of the glass. In order to determine the most suitable liquids for the generation of negative thermal lenses, we compared the properties and behavior of several fluids. The relevant material properties are summarized in Table II. The absorption coefficients where

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TABLE II ABSORPTION , REFRACTIVE INDEX n, THERMAL DISPERSION dn=dT , AND DYNAMIC VISCOSITY  OF INVESTIGATED LIQUIDS AND CURING GELS

(a)

*Inaccurate values were previously published in [14] and [15]. Manufacturer: 3M. [Online] Available: http://www.3m.com. Datasheet: Newport. [Online] Available: http://www.newport.com. Manufacturer: Nye. [Online] Available: http://www.nyeoptical.com Manufacturer: Dow Corning http://www.dowcorning.com

Corporation.

[Online]

Available:

(b) Fig. 5. (a) Assembly of the composite compensating element. L is the thickness of the compensating layer. (b) Induced dioptric power in a CE with a 0.5-mm-thick layer of sylgard 184. The dioptric power is proportional to the irradiated optical power.

measured with a spectrophotometer (Perkin Elmer). Absorption within the CE is required in this setup (Fig. 1) but it should not exceed a few tenths of a percent per millimeter. The glass rods were AR coated at the faces in contact with air. The faces in contact with the compensating medium were not coated, as the refractive indices of the liquids almost matched the glass rods. High viscosity is needed to prevent convection in the liquid layer. Fig. 6 shows the thermooptical effects in 1-mm-thick liquid layers irradiated with a Nd:YAG laser beam. Liquids such as fluorinert (FC104) and deionized water show thermally induced convection due to their low viscosity. In the interferometer, the convection results in a strongly distorted interference pattern. No measurable convection was observed with all compensating media that have a dynamic viscosity exceeding 1040 cP (which corresponds to OCF-446). Hence, water, heavy water, fluorinert, and the index-matching fluid F-IMF105 cannot be used due to their low viscosity. The viscosity of glycerin is also high enough, but the absorption is too strong. Formation of bubbles was observed in Glycerin and OC431A after extensive irradiation at high powers. The origin of these bubbles is not yet clear and may be avoided if the air within the liquid is exhausted by means of a vacuum after assembling the CE. Among the studied liquids, the best experimental results were obtained with OCF-446. As an attractive alternative to liquids, curing gels such as OCK-433 or sylgard 184 can also be used. Curing gels com-

Fig. 6. Thermally induced effects in 1-mm-thick liquid layers irradiated with a laser beam at a wavelength of 1064 nm (Nd:YAG).

bine the advantages of a solid material (no convection, simpler handling) with the strong negative thermal dispersion of a fluid. Within the experimental accuracy, these gels led to the same results as obtained with the liquid OCF-446. The details on the experiments with OCF-446 are given in [3]. Currently, sylgard 184 is favored due to its rubber-like consistence after is not known. Howcuring. For sylgard 184, the value of ever, the strong negative thermal dispersion of the materials listed in Table II results essentially from a large expansion coefficient. The expansion coefficients are 2.07 10 K for water [16], 5 10 K for glycerine [16], 8 10 K for OCF-446, approximately 1..2 10 K for OCK-433, and 6 10 K for OC-431 A. With a value of 9.6 10 K sylgard 184 has the largest expansion coefficient of all considered materials, leading to the assumption that the value of the is in the same range as the other mathermal dispersion terials listed in Table II. Since the optimum thickness of the compensating layer is determined experimentally, this uncertainty is not an issue for the present work, but will be of interest for future investigations. Until now, we have no evidence for a possible degradation of compensating elements made of sylgard 184. Long-term tests at high powers are planned for the near future. The effectiveness of OCF-446 to compensate for the thermally induced lenses with the setup shown in Fig. 1 was reported in [3]. In the following, we show, that the use of curing gels leads to the same promising result. The same Nd:YAG laser as described in [3] was used to demonstrate the compensating

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Fig. 8. Resonator with a thermooptically compensated LR (TOSCA). The sapphire sleeve is only needed for liquid or hygroscopic compensating media. (a)

(b) Fig. 7. Effect of the thermooptical lens compensation with a 0.5-mm-thick layer of sylgard 184. (a) Dioptric power of the thermally induced lenses versus pump power measured over a central region with a diameter of 2 mm. Without compensation, the slope is 17 diopters/kW. With compensation it is 1.6 diopters/kW. (b) Output power versus pump power for the compensated and uncompensated resonator. The pump power is the power delivered by the diode laser bars and does not take into account the losses of the pumping optics used for this experiment.

effect by extending the stability range of the resonator shown and where chosen to be 9 and in Fig. 1. The distances 27.5 cm, respectively. The total dioptric power of the thermal and ) lens inside the resonator (superposition of was measured during laser operation with a Mach–Zehnder interferometer with an accuracy of about 15%. The experimental results are shown Fig. 7(a). Without the CE, the slope of the thermal lens in the Nd:YAG rod versus pump power is 17 diopters/kW. With a 0.5-mm-thick layer of sylgard 184 in the CE, the slope of the total thermal lens was reduced to 1.6 diopters/kW. This reduction of the total specific dioptric power leads to an extended power range, as shown in Fig. 7(b). Without compensation the laser gets unstable at 370-W pump power and stops oscillating at higher powers. With compensation, the resonator remains stable over the whole available pump power range. As can be seen from the two curves at powers below 370 W, the compensating element leads to no measurable deterioration of the laser efficiency. V. THERMOOPTICALLY SELF-COMPENSATED AMPLIFIER (TOSCA) With the set-up presented in Fig. 1, the compensating adaptive lens is generated in a separate element (CE) that absorbs a small fraction of the circulating laser power to generate a thermal lens in a thin liquid layer [3]. This design follows from the original

concept, where the compensating element was proposed to be a single rod of an appropriate material [2]. However, since the compensating adaptive lens is now generated in a single thin layer, the implementation of the self-adaptive compensation can be simplified significantly. In the original set-up, the compensating lens had to be imaged into the LR by means of a relay optics. To avoid the need for relay optics, the compensating layer can be mounted directly within the LR, as shown in Fig. 8. With this setup, the temperature distribution generated in the LR is transferred to the compensating layer by heat contact. Hence, the power-dependent compensating lens is induced at the center of the LR without the need to absorb any laser radiation in the compensating medium. A sapphire sleeve (also shown in Fig. 8) may be used for liquid or hygroscopic compensating materials and to hold the adjacent ends of the LRs. At present we are investigating LRs with a compensating layer made of sylgard 184. The slices of the rubber-like sylgard 184 can be prefabricated and mounted between two appropriately coated LR halves without the need for a sleeve. The experimental tests are being prepared and will be reported at a later date. In the following, we present a theoretical discussion of the proposed compensation scheme. A. Finite-Element Simulations Based on our experimental setup, we have performed finite-element simulations for a typical laser configuration to study the temperature distribution and the resulting optical distortions inside the compensated LR. The optical path difference (OPD)—defined as the change of the optical path length through the LR that is caused by the thermal effects—was calculated along straight parallel lines through the rod at different distances from the rod axis. The results are shown in Fig. 9. The calculations were made with the following assumptions: The Nd:YAG LR with a diameter of 4 mm is pumped over a total length of 40 mm, with an optical power of 1 kW (absorbed). The simulation also took into account the very small, but not negligible, fraction of the pump power that is absorbed in the compensating medium. The pumping distribution across the rod has a parabolic profile and the absorbed power density at the edge of the rod is half the intensity absorbed at the center of the rod. A fraction of 42% of the absorbed power is converted to heat (assuming no laser operation) [17]. The length of the LR exceeds the pumped length by at least 10 mm to avoid bending of the end faces. The dashed line shows the calculated OPD across the Nd:YAG LR without a compensating layer. The shape of this OPD can be approximated with a parabola

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is expressed in the form (11) the resulting temperature distribution is given by [18] (12)

Fig. 9. OPD caused by the inhomogeneous temperature distribution in the Nd:YAG rod (dashed line corresponds to a lens with a focal length of 7 cm). The Nd:YAG rod has a 0.65-mm-thick compensating layer of OCK-433 and a sapphire sleeve (dashed–dotted) and the same compensated rod but with an ideal heat contact between the LR and sapphire sleeve (solid line).

is determined by the boundary conditions. where Neglecting the elastooptical effects caused by the thermally induced stress, the refractive index across the rod is given by

(13) that corresponds to a thin lens with a focal length of 7 cm. The dashed–dotted curve shows the OPD of the LR with a 0.65-mm-thick layer of OCK-433 sandwiched between the two Nd:YAG crystals (as in Fig. 8). The curing gel OCK-433 also serves as the adhesive between the sapphire sleeve and the LRs. The gap between sleeve and rods filled with the gel was assumed to be 5 m. This layer acts as a thermal insulation and makes that the temperature distribution in the compensating slice differs slightly from the temperature distribution in the LRs. This is the reason why the resulting OPD through the compensated rod is not absolutely flat. Despite these small residual aberrations, the thermal effects are reduced very significantly. The solid line in Fig. 9 shows the calculated OPD across the compensated rod assuming an ideal heat contact between the LRs and the sapphire sleeve. Without a sleeve the OPD would be even more flat, showing that this thermooptical compensation technique is, indeed, very effective. The determination of the OPD in the above calculations followed straight rays along the rod axis and the resulting flat OPD profile at all pump powers would suggest that the resonator modes are not affected with varying laser power. However, as the thermal graded-index lens is distributed over the whole length of the pumped LR and the compensation is concentrated in a thin layer in the middle of the rod, the actual electromagnetic rays do not follow straight paths. This fact needs to be taken into account for accurate calculation of the modes in the resonator. B. TOSCA Description With Ray Matrices The temperature distribution in a long LR with cylindrical symmetry is found by solving the one-dimensional heat conduction equation

where, in this case, is the refractive index at the rod axis. For the following, we assume that the LR is pumped homogeneously ) with ( (14) is the total heating power generated in the LR, where is the length of the rod, and is the rod radius. With this, the refractive index is given by (15) The ray propagation along such a lens-like LR can be described with high accuracy by the ray transfer matrix [19] (16) where

is the propagation distance and

is given by (17)

, the 1) Thin-Lens Approximation: For the case matrix elements in (16) can be approximated by power series. Taking into account only terms up to , it can easily be shown that (16) is approximated by a thin lens sandwiched between two unperturbed rods with a length of (18) where the dioptric power of the thin lens is given by

(10) (19) where is the distance from the rod center, the heat conducis tivity (assumed to be independent of temperature), and the local heating power density. When den heating distribution

According to (18), the thermal lens in the LR can indeed be compensated for simply by adding a thin lens in the middle of

WYSS et al.: THERMOOPTICAL COMPENSATION METHODS FOR HIGH-POWER LASERS

Fig. 10. Radii of the fundamental T EM mode throughout the laser resonator at different pump powers with (solid line) and without (dashed) compensating layer. The radius of curvature of the resonator mirrors is 2 m. The elements of the resonator are shown schematically (gray) at their respective position. The compensating layer is located in the middle of the LR.

the rod with a dioptric power given by in Fig. 8

, as shown

(20) which is the same as the propagation through the unperturbed LR. This is the concept behind the proposed self-adaptive compensation with a thin layer (liquid or gel) in the middle of the LR. 2) Resonator Modes: With (17) and (19), the condition means that the focal length of the thermal lens in the LR is significantly longer than the length of the rod. As was pointed out in the above discussion of Fig. 9, the equivalent focal length of the thermal lens in the 4-cm-long Nd:YAG LR was 7 cm, which means that the above approximation fails for high-power laser systems. In the following, we discuss the mode properties in a high-power Nd:YAG laser resonator, taking into account the distributed graded-index nature of the thermal effects in the LRs. The accurate description of the TOSCA is given by the ray transfer matrix. The modeled system consists of a symmetric resonator with a total length of 260 mm. The homogeneously pumped LR has a diameter of 4 mm and a length of 60 mm. The radius of curvature of the two resonator mirrors is 2 m. The fraction of abconverted to heat was assumed to be sorbed pump power )[17]. Without any pump power, the 34.4% ( mode throughout the cavity radius of the fundamental is given by the dashed–dotted line in Fig. 10. Without the compensating layer, the mode properties change very significantly with increasing pump power. First, the mode decreases in size (leading to a deteriorated beam quality as more and more higher order transversal modes start to oscillate) and then becomes very divergent (with increasing radius at the location of the LR) before the resonator turns unstable. This behavior is shown with

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the dashed lines in Fig. 10 for the modes at W and W of pump power. The resonator is unstable above 1930 W. With the compensating layer in the middle of the LR, the behavior of the fundamental mode is shown by the solid lines. At 0 W of pump power, the mode is the same as in the resonator without compensation (dashed–dotted line). With the thermooptical compensation, the changes of the mode properties are negligible over a large pump power range. In contrast to the simplified theory discussed above, the ideal compensation is not . When the reached for the case where graded-index nature of the thermal lens spread along the LR is taken into account, it is found that with increasing pump power the variation of the lens in the compensating layer should be slightly stronger than the change of the graded-index lens in the LR. The exact conditions depend on the actual configuration of the resonator. The results shown in Fig. 10 were calculated with a dioptric power of the compensating lens that amounted to , where is given by (19). This is why the mode first slightly increases in size and then decreases in size at pump powers above about 1 kW. According to theory, in this specific resonator, the beam propagation factor would , have a minimum of vary from initially around 900 W of pump power, and then increase to at 2.5 kW [1]. Thanks to the self-adaptive compensation, the resonator can further be optimized (resonator length, curvature of mirrors, telescope, etc.) to generate an even better beam quality without significant restrictions of the power range. The efficacy of the proposed compensation technique is evident (Fig. 10). Over a pump power range of 2.5 kW—which is well beyond the fracture limit imposed by the thermally induced stress [20]—the mode properties undergo almost negligible changes. The main effect is a small constriction of the mode at the location of the compensating layer in the middle of the LR. This constriction also means that the small deformation of the OPD at the edge of the LR (see Fig. 9) caused by a compensating layer within a sleeve can be neglected. Without the compensation, the mode properties undergo significant changes with varying power, and the resonator is unstable at pump powers exceeding 1930 W. VI. CONCLUSIONS In this paper, we have discussed a selection of glasses, fluids, and gels for the thermooptical compensation of the thermally induced distortions in high-power LRs as well as different designs of the compensation element. We found that glasses are, in principle, also potential materials for the proposed compensation scheme. The investigations show that the laser glass LG-760 can compensate for up to a maximum of 2 diopters/cm rod length and, therefore, is suitable to compensate for the thermally induced distortions in materials with comparatively weak thermal lenses. Fluids and (curing) gels with low absorption and high viscosity such as OCF-446 and sylgard 184 are very suitable materials to compensate for strong thermally induced lenses. The significant advantage of liquids and (curing) gels is that a comparatively small layer ( 1 mm) is sufficient to generate a strong negative thermal lens. This circumstance allows a very simple

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design for the TOSCA by sandwiching the compensating layer between two adjacent LRs. With the investigated thermooptical compensation schemes, the resonator modes are almost unaffected by the varying pump power, and the attainable power is only limited by the fracture of the LR caused by the thermally induced stress. Present research is focused on the optimization of the setup and the improvement of the technology to manufacture the compensating disks made of sylgard 184 or similar materials. Furthermore, the limits of the compensation scheme—imposed by the thermally induced birefringence and the distribution of the GRIN lens in the LR—are being studied.

REFERENCES [1] T. Graf, E. Wyss, and H. P. Weber, “Self-adaptive compensation for the thermal lens in high-power lasers,” in Proc. ASSL 2001, vol. 50, OSA Tops, 2001, pp. 688–692. [2] R. Weber, T. Graf, and H. P. Weber, “Self-adjusting compensating thermal lens to balance the thermally induced lens in solid-state lasers,” IEEE J. Quantum Electron., vol. 36, pp. 757–764, 2000. [3] T. Graf, E. Wyss, M. Roth, and H. P. Weber, “Laser resonator with balance thermal lenses,” Opt. Commun., vol. 190, pp. 327–331, 2001. [4] U. J. Greiner and H. H. Klingenberg, “Thermal lens correction of a diode-pumped Nd:YAG laser of high T EM power by an adjustablecurvature mirror,” Opt. Lett., vol. 19, pp. 1207–1209, 1994. [5] A. V. Kudryashov, “Intracavity laser beam control,” in SPIE Proc., vol. 3611, 1999, pp. 32–41. [6] S. Jackel, I. Moshe, and R. Lavi, “High performance oscillators employing adaptive optics comprised of discrete elements,” in SPIE Proc., vol. 3611, 1999, pp. 42–49. [7] D. C. Hanna, C. G. Sawyers, and M. A. Yuratich, “Telescopic resonators for large-volume T EM -mode operation,” Opt. Quantum Electron., vol. 13, pp. 493–507, 1981. [8] R. Koch, “Self-adaptive optical elements for compensation of thermal lensing effects in diode end-pumped solid state lasers – proposal and preliminary experiments,” Opt. Commun., vol. 140, pp. 158–164, 1997. [9] Kahan et al., “Variable lens and birefringece compensator,” U.S. Patent 4 848 881, 1989. [10] J. Zayhowski, “Optically-stabilized plano-plano optical resonators,” U.S. Patent 5 048 051, 1991. [11] , “Thermally controlled lenses for lasers,” U.S. Patent 5 386 427, 1995. [12] Friede et al., “Laser system and method with thermally compensated optics,” U.S. Patent 5 751 750, 1998. [13] R. Weber, B. Neuenschwander, and H. P. Weber, “Thermal effects in solid-state laser materials,” Opt. Mater., vol. 11, pp. 245–254, 1999. [14] E. Wyss, M. Roth, T. Graf, and H. P. Weber, “Self-adaptive compensation for thermal lenses in high-power laser resonators,” in Proc. 3rd Int. Conf. Adaptive Optics , Albuquerque, NM, July 23–27, 2001. [15] T. Graf, E. Wyss, M. Roth, and H. P. Weber, “Adaptive thermal optics in high-power laser resonators,” in SPIE-Proc. Laser Resonators and Beam Control V, vol. 4629, Photonics West ’02, San Jose, CA, 1 21–25, 2002. [16] H. Kuchling, “Taschenbuch der physik,” Fachbuchverlag Leipzig, p. 611, 617, 644, 1996. [17] T. Y. Fan, “Heat generation in Nd:YAG and Yb:YAG,” IEEE J. Quantum Electron., vol. 29, pp. 1457–1459, June 1993. [18] M. Schmid, T. Graf, and H. P. Weber, “Analytical model of the temperature distribution and the thermally induced birefringence in laser rods with cylindrically symmetric heating,” J. Opt. Soc. Amer. B, vol. 17, no. 8, pp. 1398–1404, Aug. 2000. [19] A. E. Siegman, Lasers. Mill Valley, CA: Univ. Science Books, 1986. [20] M. Gerber and T. Graf, “Optimum paramters to etch Nd:YAG crystals with orthophosphoric acid H PO ,” Opt. Laser Technol., vol. 33, pp. 449–453, Oct. 2001.

Eduard W. Wyss was born in Bern, Switzerland, in 1971. After graduating from business school, he received the M.Sc. degree in physics from the University of Bern, Bern, Switzerland, in 2001. He is currently working toward the Ph.D. degree at the Institute of Applied Physics. His research interests include the exploitation of thermooptical effects for high-power laser resonators.

Michelle S. Roth was born in Bern, Switzerland, in 1974. She received the Diploma in physics from the University of Bern, Bern, Switzerland, in 2002, where she is currently working toward the Ph.D. degree in the Institute of Applied Physics. Her thesis was on thermal effects in solid-state lasers. Her current research involves fluids that are convenient for the compensation of thermal lenses in highpower solid-state lasers.

Thomas Graf was born in Lugano, Switzerland, in 1966. He received the M.Sc. degree in physics in 1993 and the Ph.D. degree in 1996, both from the University of Bern, Bern, Switzerland. As a post-doctoral Research Associate, he was engaged in research on high-power solid-state lasers with high beam quality, beam-shaping for diode-laser bars, and thermodynamics of optical systems at the University of Bern. In 1997, he joined the University of Strathclyde, Glasgow, Scotland, U.K., where he was engaged in research on nonlinear optics, passively mode-locked multi-watt all-solid-state lasers, and collaborated in an experiment on dipole traps for laser-cooled atoms. In April 1999, he was appointed Head of the High-Power Lasers and Material Science Group, Laser Department, Institute of Applied Physics, University of Bern, where he has been an Assistant Professor since April 2002. His research currently includes in high-power all-solid-state laser systems with adaptive resonators, novel fiber lasers with phase-locked core-arrays, and in the generation of custom modes with intra-cavity graded-phase optics. Dr. Graf serves as a board member of the Swiss Society for Optics and Microscopy (SSOM) and is a regular member of the Optical Society of America (OSA) and the Swiss Physical Society (SPS). He was awarded the Venia Docendi in 2001 by the University of Bern.

Heinz P. Weber received the Diploma in physics from ETH Zürich, Zürich, Switzerland, in 1964, and the Ph.D. degree from the University of Bern, Bern, Switzerland, in 1968. His thesis was on generation and measurement of picosecond lightpulses. His main contribution was the discovery of the intensity correlation method for time-duration measurement. In 1969, he joined Bell Laboratories, Holmdel, NJ, where he was involved in research on integrated optics with organic materials and also in investigations of new rare-earth laser crystals. In 1975, he became Professor and Head of the Laser Department, Institute of Applied Physics, University of Bern, Bern. His research interests include modelocking and nonlinear effects in propagation of ultrashort lightpulses through optical fibers, in material processing, X-ray lasers, in diode-pumped solid-state lasers with high output power and solid-state and fiber lasers emitting in the 1-, 2-, and 3- wavelength regions, as well as laser surgery employing such lasers. Dr. Weber is member of the Swiss Academy of Technical Sciences and a Fellow of the Optical Society of America.