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Lens transformations of high power solid-state laser beams
Optics & Laser Technology 30 (1998) 341±348
Lens transformations of high power solid-state laser beams A. Khizhnyak a, *, M. Lopiitchouk b, 1, I. Peshko b, 2 a
International Centre `Institute of Applied Optics', National Academy of Sciences of Ukraine, 10-G, Kudryavskaya Str, 254053 Kiev, Ukraine b Institute of Physics National Academy of Sciences of Ukraine, 46, Science Ave., 252650 Kiev, Ukraine
Abstract Beam parameters of 300 W-class CW Nd:YAG lasers with double pump cavities have been investigated. The calculations have demonstrated that in the strongly pumped operating laser the amplitude of stresses in the rod can be decreased by a factor of approximately 1.5. A multibeam model of high power beam structure has been proposed. It has been shown theoretically and experimentally that 1.4±1.8 times reduction of the beam parameter product of a complex non-Gaussian beam may be achieved by placing a lens with a strength and location determined by laser parameters. An integrated parameters product has been proposed to describe the complex beams. In the range of possible pumping levels (3±10 kW) the output full divergence angle changed from 2 up to 23±28 mrad but in image space of the lens the variations of this value were no more than 212%. # 1998 Elsevier Science Ltd. All rights reserved. Keywords: Solid-state laser; Beam parameter product
1. Introduction For a high power lamp-pumped solid-state laser, the thermal eects in the cylindrical active rod causes the decrease of output power and the increase of beam parameter product. Because of the temperature dierence between the axis and the surface areas, stresses stimulate the thermal lensing [1, 2]. The thermally induced birefringent and bifocusing phenomena have been suggested as the reasons of the output power reduction [3]. It has been shown that the beam quality factor degrades linearly with the heat power dissipated in the rod because of birefringence [4]. The temperature distribution in Nd:YAG laser rod deviates from a parabolic function due to the temperature dependence of thermal conductivity coecient [5, 6] and due to the inhomogeneity of pumping light distribution [6]. It has been shown that the refractive power of the thermal lens has a parabolic radial dependence. The calculations and the experiments demonstrated that high * Corresponding author. Tel.: +380-44-212-2158; fax: +380-44212-4812; e-mail: [email protected] 1 Tel.: +380-44-265-4069; fax: +380-44-265-1589 2 Tel.: +380-44-265-4069; fax: +380-44-265-1589; e-mail: [email protected] 0030-3992/98/$19.00 # 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 0 - 3 9 9 2 ( 9 8 ) 0 0 0 6 7 - X
output power and good beam quality could not be achieved simultaneously when lens aberrations in a cavity were present. The numerical calculations were supported by analytical investigations based on socalled `intensity moment formalism' [7]. It should be emphasized that this method may be used to describe the beam propagation outside of the cavity only. Wigner function utilized in these calculations does not simulate the real mode structure of a high power multimode beam. To introduce such parameters as coherence, it is necessary to measure at ®rst the parameters of the real beam propagation. The Wigner function may approximate successfully the ®eld of statistically independent light sources or the single mode of any index. However, by applying this method a number of resonance conditions were obtained which established close links between the beam width, far®eld divergence and kurtosis of the laser beam. It was shown that the beam quality can be improved for an aberrated rod. A generalized ABCD-law applied to partially coherent beams (Gaussian±Shell model) has been derived [8]. The Wigner distribution function has been used to study the propagation of partially coherent beams through the optical systems [9]. Namely, it has been demonstrated that the quality factor of such beam does not change during passage through on
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ABCD-system. Naturally, the thermal lens refractive power depends on pump power. This leads to the variations of beam divergence angle and the waist location after passing through an external lens [10]. In the present work we have tried to model in ®rst order approximation the real complex intracavity beam structure to describe its transformation out of the cavity. For the ®rst time, to our knowledge, there is a reduction of the beam parameter product for high power beams with complex transverse intensity distribution. It has been shown that conventional beam parameter product of non-Gaussian beam does not successfully describe the beam properties. We propose to use an integrated beam parameter product which is more precise for the parameters that are under consideration. Stabilization of beam parameter product for a 300 W YAG-laser across the whole range of possible pump powers has been demonstrated. The results of these investigations were used for manufacturing a 1-kW multilaser system, where emissions of four lasers have been collected to the same single ®bre [11]. 2. Anisotropic aberrated thermal lens A double-rod CW YAG laser cavity is usually designed as a thermal lens confocal line at maximum pump power. The cavity, containing two aberrated and changeable lenses, forms the speci®c mode con®guration. Moreover, for the single lamp pump cavity a prism eect in the rod takes place. The Rayleigh range, the direction of beam propagation and divergence angle change with the pump power. Two dominant phenomena in a strongly pumped active rod cause the deformation of beam intensity distribution. These are thermally induced birefringence and radial inhomogeneity of thermal lens refractive power. The ®rst of these processes results in the appearance of dierent thermal lens refractive power for radial and tangential light polarization components. For single ellipse pump cavity (5.5 kW pump power) the calculated focal length of the thermal lens [3] for radial polarization fr is 20 cm and that for tangential polarization ff is 25 cm. The cavity with such a `dual lens' can be imagined as two coinciding cavities with dierent lenses which causes simultaneous appearance of the coaxial beams with dierent divergence angles yr and yf. Using the experimental and theoretical dependencies for thermal lens behavior [1, 2, 10, 12] one may estimate that yr11.3yf at pump powers about 10 kW. As mentioned above, a thermal lens in the active rod has the refractive power with a parabolic radial dependence. The laser cavity con®guration may be simulated as a superposition of some cavities with the
set of circular thermal lenses with dierent refractive power. Consequently the resulting beam could be presented as a superposition of some beams, but with dierent divergence angles and waist diameters. For the cavity with high Fresnel number such approximation seems quite realistic. Moreover, for such gain medium as garnet the dierent radial areas may work independently. It has been measured experimentally [7] that for 11 kW pump power the radial deviation of thermal lens focal length was about 20%. One can estimate that in this case even the identical mode structure would have about 1.2±1.3 times dierent divergence angle of emission penetrating through the central rod area and near the surface. The procedure of thermal lens calculation has been described in detail [1±3]. The solution of the heat conduction equation for surface cooled cylindrical rod was used as the temperature ®eld in a clamped crystal with free ends. With known temperature distribution the ®elds of stresses skl, of thermal strains ekl and elastic strains eklvE were calculated consecutively. The changes of the components of the relative dielectric impermeability tensor via photoelastic coecients pijkl are connected with elastic strains: DBij pijkl ekl ÿ 2=n3ij @nij =@ T p DT:
1
The changes of refractive index n0 for each light polarization components can be shown to be Dnx,y ÿ n30 =2DBxx,yy :
2
It should be noted that for cubic crystals values nij and @nij/@T are considered isotropic and constant. However, @nij/@T depends on temperature and it is anysotropic in stressed and strained crystal. Unfortunately the value of this anisotropy was not available and we could not calculate the precise numerical value of DBij. The main problem of thermal lens computation is estimation of the real temperature pro®le inside the rod. An approximation has been made by taking into account the temperature dependence of the thermal conductivity of YAG [5, 6] in the form a/T (a = 39 W/ cm for YAG). Moreover, inhomogeneity of pump density was introduce to simulate the radial temperature distribution in the rod [6]. The radial dependence of heat generated per unit volume has been presented as Q r Q0 1 ÿ b=3ÿ1 1 ÿ br2 =b2 ,
3
where b is rod radius, b is heat distribution shape-factor and Q0 is average heat generation per unit volume. For b>0, maximum heat generation is in the centre of the rod and for b < 0 it is on the surface [6]. However, it should be noted that a parabolic distri-
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bution of the heat sources with maximum in the rod centre is valid for pumped but nonoperating laser. It has been shown experimentally [13] that the dierence between dissipated power with and without laser operation is about 20±30%. Real heat sources distribution is proportional to the dierence between the pump and emission power distributions. The resultant function may be ¯at near the rod centre or even with the hole (depending on laser eciency and pump power). The rate of heat degeneration Q1 in the rod centre may be obtained from expression n b r Q1 exp ÿ 2 4 dr ZQ0 b, w 0 where Z is the portion of total heat energy that is `carried out' by the beam, w is radius of the beam. To describe the beams with dierent intensity distributions the shape-coecient n has been used. Generally, the higher pump power is applied, the higher n would be used to model the real beam inside the rod. Combining Eqs. (3) and (4) for n = 2 (i.e. a TEM00 Gaussian beam) it is easy to obtain " r 2 1 ÿ b r=b2 2 2bZ r ÿ exp ÿ 2 : Q r Q0 1 ÿ b=3 p w w 5 Fig. 1 demonstrates dependence of Eq. (5) for dierent beam parameters. The curve (a) represents the radial dependence of heat rate generation for nonoperating laser. The curves (b) and (c) demonstrate Q(r) at medium pump powers (w 1 b/2, Z = 0.1) and at high pump powers (w 1 b, Z = 0.3), respectively. These cal-
Fig. 1. Radial dependence of the rate of heat generation in the gain rod (a) for nonoperating laser, (b) laser at medium pump power and (c) laser at high pump power.
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culations result in surprising conclusion that at high pump power in an operating laser the rate of heat generation is more homogeneous than at medium pump power. The solution of the one-dimensional nonlinear heat equation with heat sources density according to Eq. (5) gives the radial temperature distribution T r T0 exp ( " #) r b2 Q 0 1 b Zw 2 2 4 1 ÿ r ÿ 1 ÿ r F r , a 4k 16k 2b p 6 where k1ÿ
b 3
F r
1 X 2nÿ1 b2n ÿ r2n ÿ1n : nn! w2n n1
To calculate the components of stresses tensor we used the known general expressions [1], but with temperature distribution Eq. (6). Fig. 2 shows the examples of radial (curves a and c) and tangential (curves b and d) stresses computation for nonoperating laser with inhomogeneous heat sources distribution (a and b) and for operating laser (c and d). The calculations demonstrate that the laser emission decreases approximately 1.5 times the stresses amplitude in the rod and consequently it reduces the thermal lens refractive power. It means that, at maximum pump power, switching o the laser operation by introducing a shutter into the cavity may destroy the active rod because of rapid increase of stresses. For double-pump-cavity laser the optimum space between
Fig. 2. Stresses components (a and c are radial and b and d tangential) vs. relative rod radius for nonoperating laser (a and b) and operating laser (c and d).
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the cavities depends on intrinsic losses (rod quality, mirror transparency) and pump power level.
3. Beam parameters product compression There are two main parameters that fully describe laser beam quality: the divergence half angle y and the beam waist radius w. The less the product yw of these values, the higher the power intensity of the focal point. It is well known that if the paraxial approximation is ful®lled for Gaussian beams the parameter product is an invariant in optical transformation and propagation in homogeneous medium. A theory of partially coherent beams transformations demonstrates the law of lens transformations for multimode beams similar to that for the fundamental mode [9]. In the image space of a lens with focal length f the total waist radius w and half divergence angle y may be presented as [10] wi w0 a1=2 , yi y0 aÿ1=2 , Si S0 a, zi z0 a,
7
where a = f 2/(S20+z20), S0 is the distance from the input beam waist to the object focal plane of a lens, z0 is the Rayleigh range of entrance beam de®ned as z0=w0/y0, zi is the Rayleigh range of the output beam and Si is the distance between the output beam waist and the focal plane in image space. For the simplest case the laser irradiation in a strongly pumped solid-state laser may be modelled as a sum of two components with orthogonal polarization, marked in Fig. 3 as beams `a' and `b'. Experimentally measured intensity of the far-®eld distribution (Fig. 7, 7 kW pump power) can be presented as a sum of two bell-shape functions. Let one of the beams have a radius close to the radius of the rod, and the second one have a smaller radius w02y01. Usually, for multimode beam the intensity distribution in near ®eld has super-Gaussian shape. In this case, because of the abrupt beam wings the total width of sum of two incoherent beams with compatible intensity is closer to that one of wider beam (see inserted pictures in Fig. 3). and One can write w02
Fig. 3. Optical model of double-beam transformation. 1 is the laser mirror, 2 the gain medium with thermal lens and 3 the lens-transformer. Intensity distributions of (a) and (b) beam components in the near and far ®elds are shown in the inserted pictures.
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depends on lens position (S0) and Rayleigh range (z0). Over a range of values of S0, yi1>yi2. Outside this range the inverse is true. Because of its complex internal structure the non-Gaussian beam creates some waists in image space after the lens. The experimentally measured Rayleigh range is the superposition of some displaced Rayleigh ranges and because of this it is longer than that calculated as w0/y0 for the single beam. Among the pair parameters (yi1, yi2 or wi1, wi2) the maximum values yimax and wimax are interpreted as the angular and linear width of the total beam* in the image space. It should be emphasized that experimentally the bigger waist radius in the near ®eld and the higher divergence angle in the far ®eld would be measured. In that way, y02 w01 is the experimentally measured beam parameter product. The lens with focal length f for which the conditions z01
8
For this lens position the `double-beam parameter product' in the image space becomes minimum. Evidently such product y02 w01 is not invariant in the optical transformation. Fig. 4 demonstrates some examples of the relations y02 w01/yimax wimax versus S0/f for dierent pairs of z01, z02. At higher pump power the thermal lens eect is stronger and the bigger dierence between y01 and y02 causes the upper maximum of curve y02 w01/yimax wimax . For lower pump powers this dierence disappears and the above ratio becomes equal to one for the arbitrary S0/f. To characterize non-Gaussian beam distribution the kurtosis parameter has been used [14]. So leptokurtic and platykurtic pro®les imply respectively sharper and ¯atter beams than the Gaussian case. In general, the kurtosis of the beam shape changes along the propagation path. Moreover, for strongly pumped systems the beam pro®le may be nonmonotonic (hole in the centre). Evidently, the proposed de®nition of beam parameter product of such the intensity distribution is not invariant in linear transformation. To simulate the dierent bellshape functions an average transverse intensity distribution of the multimode beam may be described as: I r I0 expÿ2 jrj=wn :
9
For n = 2 it represents the Gaussian beam, for n>2 so-called super-Gaussian distribution. Analogously, for n < 2 it can be named as sub-Gaussian one. In all cases
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Fig. 4. Relative double-beam parameter product vs. relative lens position for set of beam parameters: (a) y01=15 mrad, w01=1 mm, y02=6 mrad, w02=2.5 mm, f = 24 cm. (b) y01=12 mrad, w01=1.25 mm, y02=7.5 mrad, w02=2 mm. (c) y01=10 mrad, w01=1.5 mm, y02=7.5 mrad, w02=2 mm. Circles demonstrate typical experimental data for intensity measurements and the squares for power measurements.
the intensities at the points r = w are the same. However, the power that penetrates inside the circle with r = w be dierent. Sub-Gaussian beams have more heavy wings and for super-Gaussian beams the main part of total power is concentrated inside the core r < w. Generally, in the image space of the lens the beam intensity redistribution takes place [14]. Both calculation and experiment show that platikurtic pro®le before the focal plane transforms to leptokurtic one after the focal plane. Consequently, the proportion of core power also changes. This parameter is interesting in practice, for example for ®bre entrance optics. Fig. 5 demonstrates the relation of core power Pc to total beam power P vs. intensity shape parameter n. As an example, one can estimate that for 1-kW beam with the transverse intensity distribution as like as expvxv the wings have weight about 400 W. Evidently that such power is quite sucient to vaporize the ®bre coating. It seems that in practice because of complex high power beam radial and angular structure, it is more appropriate to estimate the beam width (in near and far ®elds) containing the certain portion of total beam power, for example 0.98 for high power laser. In this case the values of w and y should be measured or calculated, such that y 2p 1 w 2p I r, jdr dj I* W, jdW dj 0:96, 10 P2 0 0 0 0 where I(r,j) and I *(W,j) are intensity distributions in near and far ®eld, respectively.
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Fig. 5. Part of power penetrated inside the beam core with intensity higher then 1/e 2 versus shape factor n in Eq. (9).
Comparison of the experimental beam pro®les before and after the lens (Fig. 8) shows that the beam divergence decreases from 21 to 16 mrad and simultaneously (P ÿ Pc)/P decreases from 4 to 1.8%. Such a transformed beam may be entered to a ®bre easier than that with initial distribution. 4. Experiment The heat generation in a laser rod depends on spectroscopic parameters of the crystal, pump cavity eciency and pump power. These parameters combine to in¯uence the optical beam characteristics. To measure the output laser parameters we used the experimental set-up shown in Fig. 6. As active rods (1) b 5.6 80 mm long Nd:YAG crystals have been used. The arc lamps provided up to 5 kW electrical power each one. The ¯at±¯at cavity is formed by 100 and 75% re¯ecting mirrors (2). Far ®eld distributions were measured on the focal plane of 1-m focal length lens (5) behind a 0.5 mm diameter diaphragm (6), that was moved with a detector (7) by the step-motors in two orthogonal directions. The total power in any divergence angle was measured by a power meter (9) behind an iris diaphragm. Intensity distributions were recorded with an oscilloscope (8).
Fig. 6. Experimental set-up: (1) active rods, (2) laser mirrors, (3) attenuator, (4) lens transformer, (5) divergence measuring lens, (6) diaphragm, (7) detector, (8) oscilloscope and (9) power meter.
The dierent laser units had the maximum output power in the range 220±330 W. According to the documentation provide with the laser, half of output power was emitted in 11 23 mrad angle. Output power was ®rst measured as a function of pump power and is shown in Fig. 7 inside this angle. The curve with rectangles shows the total power propagated in 11 mrad angle. At pump powers over 7 kW saturation and even a decrease of output power takes place. However for 33 mrad angle there is no such eect. In the middle of the pump power range the deformation of far ®eld distribution occurs and beam pro®le becomes non-Schell± Gaussian. In the oscillograms of Fig. 8 the far ®eld distributions for low (3 kW), medium (7 kW) and high (9.5 kW) pump power are shown. For pump power over 5.5 kW the intensity distribution transforms from Gaussian±Schell shape to double-bell shape and to triangular shape at 8±9 kW pump power. As it was mentioned above, the space intensity distribution of the beams had elliptical form with complex internal structure. For some laser specimens three-bell far ®eld distribution was observed in horizontal direction at high pump powers. Fig. 4 demonstrates experimental points for the relation y02 w01/yimax wimax with the lens of f = 24 cm. Among the four lasers investigated the best reduction of wy was about 2±2.2 times in horizontal direction. The typical value was about 1.4±1.5 times. The measurements of the beam parameters by the power meter placed behind the iris diaphragm gave the average value of power emitted in all directions. The maximum product reduction for the power measurements was about 1.4±1.8 times for dierent laser units. The beam width measurement was made for Pc/P = 0.95 20.03.
Fig. 7. Output power emitted into the full angle 11 (rectangles), 22 (circles) and 33 mrad (crosses) vs. pump power.
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Fig. 8. Far-®eld intensity distribution of the laser beam at pump powers (a) 3, (b) 7, (c) 9.5 and (d) after the lens at 9.5 kW pump power.
The exact experimental value of w0 (the waist on the output coupler) has not been measured because of the technical diculties. However w0 may be estimated at the point of lens location S0=0. The beam width on the coupler w01 may be calculated from experimentally measured value wi2 yi1/y02. Such normalization gives 2w0113.7±3.9 mm. Our experimental data are close to the curve with parameters y02/y0111.33. This is in good agreement with above estimates of divergence angles in the thermal lens parameters calculations. In the case where S0=Si=0, the waists of all the laser beam fractions with dierent divergence angles are located at the same distance f from the lens. The coecient a becomes equal to f/z0. The Rayleigh range changes with pump power. It is possible to ®nd the lens with focal length f equal to the average value of z0 in the middle of the pump range. At low pump powers the initial beam divergence is relatively low and the Rayleigh range is bigger than the focal length of the lens. It means that the lens increases the beam divergence angle in the image space. At high pump powers the beam divergence becomes high and the Rayleigh range becomes shorter than the lens focal length and the lens decreases the beam divergence. Fig. 9 demonstrates the experimental dependencies y0=y0(P) and yi=yi(P) after the lens with f = 30 cm. It is evident that the raw beam divergence increases by a factor of approximately 10 over the range of pump power, but after the lens this value changes no more than 12%. This eect gives possibility of successful coupling of the output beam into a ®bre over a wide pump range. For the lens location with S0=0, z0=fyi/y0=w0/y0, yi=w0/f. Consequently if the experimentally measured beam divergence angle in the image space of the exter-
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Fig. 9. Half divergence angle of the laser beam (a) before and (b) after the lens-transformer versus pump power.
nal lens demonstrates the weak deviations from the average value, the waist diameter on the output coupler also would be stable. The above relations give numerical value of the waist of approximately 0.39 cm. This is close to the earlier estimates of waist size using the double-beam model. 5. Conclusions The far and near ®eld beam intensity distributions of a double-pump cavity CW YAG laser becomes obviously non-Gaussian±Schell at pump powers higher 5.5 kW. The calculations demonstrate that the increase in laser emission reduces the amplitude of stresses in the rod and the thermal lens refractive power by a factor of approximately 1.5. The thermal lens can be represented as a set of lenses with the dierent parameters. Analogously, the intracavity beam structure can be imagined as a superposition of some Gaussian±Schell beams with dierent divergence angles and waist diameters. The aberration of the thermal lens and complex beam structure makes it possible to improve the total beam parameter product in the image space of an external lens. The double beam model predicts a factor of 1.5±2.5 decrease of beam parameter product for circular symmetric beams. Experimentally a factor of 1.4±1.8 decrease of beam parameter product has been achieved for power measurement. In the pump range 3±10 kW the half divergence angle of the output beam was changed from 1 to 12±14 mrad. In the image space of 30 cm focal length lens the half divergence angle was 6.5 20.8 mrad across whole pump range. It has been proposed to use an integrated beam parameter product to describe the non-Gausian±Schell beams. The
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maximum laser output power was about 330 W with beam parameter product of 16 mm mrad after the lens with 24 cm focal length. Acknowledgements The authors appreciate the support of the Science and Technology Center in Ukraine, Project No 526. The authors wish to thank G. Galich for technical assistance. References [1] Koechner W. Absorbed pump power, thermal pro®le and stresses in a cw pumped Nd:YAG crystal. Appl. Opt. 1970;9:1429±34. [2] Koechner W. Thermal lensing in a Nd:YAG laser rod. Appl. Opt. 1970;9:2548±53. [3] Forster JD, Osternik LM. Thermal eects in a Nd:YAG laser. J. Appl. Phys. 1970;41:3656±63. [4] Lu Q, Dong S, Weber H. Analysis of TEM00 laser beam quality degradation caused by a birefringent Nd:YAG rod. Opt. Quantum Electron. 1995;27:777±83.
[5] Rozanov AG. A nonlinear model of thermal eects in laser crystals of the YAG:Nd type. Quantum Electron. 1991;18:1185± 8 (in Russian). [6] Hodgson N, Weber H. In¯uence of spherical aberration of the active medium on the performance of Nd:YAG laser. IEEE J. QE 1993;9:2497±507. [7] Martinez-Herrero R, Mejias PM, Hodgson N, Weber H. Beamquality changes generated by thermally-induced spherical aberration in laser cavities. IEEE J. QE 1995;31:2173±6. [8] Simon R, Mukunda N, Sudarshan ECG. Partially coherent beams and a generalized ABCD-law. Opt. Commun. 1988;65:322±8. [9] Lavi S, Prochaska R, Keren E. Generalized beam parameters and transformation laws for partially coherent light. Appl. Opt. 1988;27:3696±703. [10] Jabczynski J, Mindak M. Focus shift of multimode cw YAG laser beam. Opt. Eng. 1992;31:447±52. [11] Peshko I, Galich G, Lopiychuk M, Khizhnyak A, Nakvasuk V. High power CW solid-state multilaser system. Opt. Laser Technol. 1997;29:211±5. [12] Jankiewicz Z, Mindak M, Szydlak J, Wojcik J. Analysis of the thermal focusing eect in a cw Nd:YAG laser. Opt. Appl. 1985;15:125±34. [13] Shokin AA. Dependence of heat dissipation in YAG:Nd3+ on output power. Quantum Electron. 1974;1:423±5 (in Russian). [14] Piquero G, Mejias PM, Martinez-Herrero R. Sharpness changes of gaussian beam induced by spherically aberrated lenses. Opt. Commun. 1994;107:179±83.
Lens transformations of high power solid-state laser beams A. Khizhnyak a, *, M. Lopiitchouk b, 1, I. Peshko b, 2 a
International Centre `Institute of Applied Optics', National Academy of Sciences of Ukraine, 10-G, Kudryavskaya Str, 254053 Kiev, Ukraine b Institute of Physics National Academy of Sciences of Ukraine, 46, Science Ave., 252650 Kiev, Ukraine
Abstract Beam parameters of 300 W-class CW Nd:YAG lasers with double pump cavities have been investigated. The calculations have demonstrated that in the strongly pumped operating laser the amplitude of stresses in the rod can be decreased by a factor of approximately 1.5. A multibeam model of high power beam structure has been proposed. It has been shown theoretically and experimentally that 1.4±1.8 times reduction of the beam parameter product of a complex non-Gaussian beam may be achieved by placing a lens with a strength and location determined by laser parameters. An integrated parameters product has been proposed to describe the complex beams. In the range of possible pumping levels (3±10 kW) the output full divergence angle changed from 2 up to 23±28 mrad but in image space of the lens the variations of this value were no more than 212%. # 1998 Elsevier Science Ltd. All rights reserved. Keywords: Solid-state laser; Beam parameter product
1. Introduction For a high power lamp-pumped solid-state laser, the thermal eects in the cylindrical active rod causes the decrease of output power and the increase of beam parameter product. Because of the temperature dierence between the axis and the surface areas, stresses stimulate the thermal lensing [1, 2]. The thermally induced birefringent and bifocusing phenomena have been suggested as the reasons of the output power reduction [3]. It has been shown that the beam quality factor degrades linearly with the heat power dissipated in the rod because of birefringence [4]. The temperature distribution in Nd:YAG laser rod deviates from a parabolic function due to the temperature dependence of thermal conductivity coecient [5, 6] and due to the inhomogeneity of pumping light distribution [6]. It has been shown that the refractive power of the thermal lens has a parabolic radial dependence. The calculations and the experiments demonstrated that high * Corresponding author. Tel.: +380-44-212-2158; fax: +380-44212-4812; e-mail: [email protected] 1 Tel.: +380-44-265-4069; fax: +380-44-265-1589 2 Tel.: +380-44-265-4069; fax: +380-44-265-1589; e-mail: [email protected] 0030-3992/98/$19.00 # 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 0 - 3 9 9 2 ( 9 8 ) 0 0 0 6 7 - X
output power and good beam quality could not be achieved simultaneously when lens aberrations in a cavity were present. The numerical calculations were supported by analytical investigations based on socalled `intensity moment formalism' [7]. It should be emphasized that this method may be used to describe the beam propagation outside of the cavity only. Wigner function utilized in these calculations does not simulate the real mode structure of a high power multimode beam. To introduce such parameters as coherence, it is necessary to measure at ®rst the parameters of the real beam propagation. The Wigner function may approximate successfully the ®eld of statistically independent light sources or the single mode of any index. However, by applying this method a number of resonance conditions were obtained which established close links between the beam width, far®eld divergence and kurtosis of the laser beam. It was shown that the beam quality can be improved for an aberrated rod. A generalized ABCD-law applied to partially coherent beams (Gaussian±Shell model) has been derived [8]. The Wigner distribution function has been used to study the propagation of partially coherent beams through the optical systems [9]. Namely, it has been demonstrated that the quality factor of such beam does not change during passage through on
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ABCD-system. Naturally, the thermal lens refractive power depends on pump power. This leads to the variations of beam divergence angle and the waist location after passing through an external lens [10]. In the present work we have tried to model in ®rst order approximation the real complex intracavity beam structure to describe its transformation out of the cavity. For the ®rst time, to our knowledge, there is a reduction of the beam parameter product for high power beams with complex transverse intensity distribution. It has been shown that conventional beam parameter product of non-Gaussian beam does not successfully describe the beam properties. We propose to use an integrated beam parameter product which is more precise for the parameters that are under consideration. Stabilization of beam parameter product for a 300 W YAG-laser across the whole range of possible pump powers has been demonstrated. The results of these investigations were used for manufacturing a 1-kW multilaser system, where emissions of four lasers have been collected to the same single ®bre [11]. 2. Anisotropic aberrated thermal lens A double-rod CW YAG laser cavity is usually designed as a thermal lens confocal line at maximum pump power. The cavity, containing two aberrated and changeable lenses, forms the speci®c mode con®guration. Moreover, for the single lamp pump cavity a prism eect in the rod takes place. The Rayleigh range, the direction of beam propagation and divergence angle change with the pump power. Two dominant phenomena in a strongly pumped active rod cause the deformation of beam intensity distribution. These are thermally induced birefringence and radial inhomogeneity of thermal lens refractive power. The ®rst of these processes results in the appearance of dierent thermal lens refractive power for radial and tangential light polarization components. For single ellipse pump cavity (5.5 kW pump power) the calculated focal length of the thermal lens [3] for radial polarization fr is 20 cm and that for tangential polarization ff is 25 cm. The cavity with such a `dual lens' can be imagined as two coinciding cavities with dierent lenses which causes simultaneous appearance of the coaxial beams with dierent divergence angles yr and yf. Using the experimental and theoretical dependencies for thermal lens behavior [1, 2, 10, 12] one may estimate that yr11.3yf at pump powers about 10 kW. As mentioned above, a thermal lens in the active rod has the refractive power with a parabolic radial dependence. The laser cavity con®guration may be simulated as a superposition of some cavities with the
set of circular thermal lenses with dierent refractive power. Consequently the resulting beam could be presented as a superposition of some beams, but with dierent divergence angles and waist diameters. For the cavity with high Fresnel number such approximation seems quite realistic. Moreover, for such gain medium as garnet the dierent radial areas may work independently. It has been measured experimentally [7] that for 11 kW pump power the radial deviation of thermal lens focal length was about 20%. One can estimate that in this case even the identical mode structure would have about 1.2±1.3 times dierent divergence angle of emission penetrating through the central rod area and near the surface. The procedure of thermal lens calculation has been described in detail [1±3]. The solution of the heat conduction equation for surface cooled cylindrical rod was used as the temperature ®eld in a clamped crystal with free ends. With known temperature distribution the ®elds of stresses skl, of thermal strains ekl and elastic strains eklvE were calculated consecutively. The changes of the components of the relative dielectric impermeability tensor via photoelastic coecients pijkl are connected with elastic strains: DBij pijkl ekl ÿ 2=n3ij @nij =@ T p DT:
1
The changes of refractive index n0 for each light polarization components can be shown to be Dnx,y ÿ n30 =2DBxx,yy :
2
It should be noted that for cubic crystals values nij and @nij/@T are considered isotropic and constant. However, @nij/@T depends on temperature and it is anysotropic in stressed and strained crystal. Unfortunately the value of this anisotropy was not available and we could not calculate the precise numerical value of DBij. The main problem of thermal lens computation is estimation of the real temperature pro®le inside the rod. An approximation has been made by taking into account the temperature dependence of the thermal conductivity of YAG [5, 6] in the form a/T (a = 39 W/ cm for YAG). Moreover, inhomogeneity of pump density was introduce to simulate the radial temperature distribution in the rod [6]. The radial dependence of heat generated per unit volume has been presented as Q r Q0 1 ÿ b=3ÿ1 1 ÿ br2 =b2 ,
3
where b is rod radius, b is heat distribution shape-factor and Q0 is average heat generation per unit volume. For b>0, maximum heat generation is in the centre of the rod and for b < 0 it is on the surface [6]. However, it should be noted that a parabolic distri-
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bution of the heat sources with maximum in the rod centre is valid for pumped but nonoperating laser. It has been shown experimentally [13] that the dierence between dissipated power with and without laser operation is about 20±30%. Real heat sources distribution is proportional to the dierence between the pump and emission power distributions. The resultant function may be ¯at near the rod centre or even with the hole (depending on laser eciency and pump power). The rate of heat degeneration Q1 in the rod centre may be obtained from expression n b r Q1 exp ÿ 2 4 dr ZQ0 b, w 0 where Z is the portion of total heat energy that is `carried out' by the beam, w is radius of the beam. To describe the beams with dierent intensity distributions the shape-coecient n has been used. Generally, the higher pump power is applied, the higher n would be used to model the real beam inside the rod. Combining Eqs. (3) and (4) for n = 2 (i.e. a TEM00 Gaussian beam) it is easy to obtain " r 2 1 ÿ b r=b2 2 2bZ r ÿ exp ÿ 2 : Q r Q0 1 ÿ b=3 p w w 5 Fig. 1 demonstrates dependence of Eq. (5) for dierent beam parameters. The curve (a) represents the radial dependence of heat rate generation for nonoperating laser. The curves (b) and (c) demonstrate Q(r) at medium pump powers (w 1 b/2, Z = 0.1) and at high pump powers (w 1 b, Z = 0.3), respectively. These cal-
Fig. 1. Radial dependence of the rate of heat generation in the gain rod (a) for nonoperating laser, (b) laser at medium pump power and (c) laser at high pump power.
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culations result in surprising conclusion that at high pump power in an operating laser the rate of heat generation is more homogeneous than at medium pump power. The solution of the one-dimensional nonlinear heat equation with heat sources density according to Eq. (5) gives the radial temperature distribution T r T0 exp ( " #) r b2 Q 0 1 b Zw 2 2 4 1 ÿ r ÿ 1 ÿ r F r , a 4k 16k 2b p 6 where k1ÿ
b 3
F r
1 X 2nÿ1 b2n ÿ r2n ÿ1n : nn! w2n n1
To calculate the components of stresses tensor we used the known general expressions [1], but with temperature distribution Eq. (6). Fig. 2 shows the examples of radial (curves a and c) and tangential (curves b and d) stresses computation for nonoperating laser with inhomogeneous heat sources distribution (a and b) and for operating laser (c and d). The calculations demonstrate that the laser emission decreases approximately 1.5 times the stresses amplitude in the rod and consequently it reduces the thermal lens refractive power. It means that, at maximum pump power, switching o the laser operation by introducing a shutter into the cavity may destroy the active rod because of rapid increase of stresses. For double-pump-cavity laser the optimum space between
Fig. 2. Stresses components (a and c are radial and b and d tangential) vs. relative rod radius for nonoperating laser (a and b) and operating laser (c and d).
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the cavities depends on intrinsic losses (rod quality, mirror transparency) and pump power level.
3. Beam parameters product compression There are two main parameters that fully describe laser beam quality: the divergence half angle y and the beam waist radius w. The less the product yw of these values, the higher the power intensity of the focal point. It is well known that if the paraxial approximation is ful®lled for Gaussian beams the parameter product is an invariant in optical transformation and propagation in homogeneous medium. A theory of partially coherent beams transformations demonstrates the law of lens transformations for multimode beams similar to that for the fundamental mode [9]. In the image space of a lens with focal length f the total waist radius w and half divergence angle y may be presented as [10] wi w0 a1=2 , yi y0 aÿ1=2 , Si S0 a, zi z0 a,
7
where a = f 2/(S20+z20), S0 is the distance from the input beam waist to the object focal plane of a lens, z0 is the Rayleigh range of entrance beam de®ned as z0=w0/y0, zi is the Rayleigh range of the output beam and Si is the distance between the output beam waist and the focal plane in image space. For the simplest case the laser irradiation in a strongly pumped solid-state laser may be modelled as a sum of two components with orthogonal polarization, marked in Fig. 3 as beams `a' and `b'. Experimentally measured intensity of the far-®eld distribution (Fig. 7, 7 kW pump power) can be presented as a sum of two bell-shape functions. Let one of the beams have a radius close to the radius of the rod, and the second one have a smaller radius w02
Fig. 3. Optical model of double-beam transformation. 1 is the laser mirror, 2 the gain medium with thermal lens and 3 the lens-transformer. Intensity distributions of (a) and (b) beam components in the near and far ®elds are shown in the inserted pictures.
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depends on lens position (S0) and Rayleigh range (z0). Over a range of values of S0, yi1>yi2. Outside this range the inverse is true. Because of its complex internal structure the non-Gaussian beam creates some waists in image space after the lens. The experimentally measured Rayleigh range is the superposition of some displaced Rayleigh ranges and because of this it is longer than that calculated as w0/y0 for the single beam. Among the pair parameters (yi1, yi2 or wi1, wi2) the maximum values yimax and wimax are interpreted as the angular and linear width of the total beam* in the image space. It should be emphasized that experimentally the bigger waist radius in the near ®eld and the higher divergence angle in the far ®eld would be measured. In that way, y02 w01 is the experimentally measured beam parameter product. The lens with focal length f for which the conditions z01
8
For this lens position the `double-beam parameter product' in the image space becomes minimum. Evidently such product y02 w01 is not invariant in the optical transformation. Fig. 4 demonstrates some examples of the relations y02 w01/yimax wimax versus S0/f for dierent pairs of z01, z02. At higher pump power the thermal lens eect is stronger and the bigger dierence between y01 and y02 causes the upper maximum of curve y02 w01/yimax wimax . For lower pump powers this dierence disappears and the above ratio becomes equal to one for the arbitrary S0/f. To characterize non-Gaussian beam distribution the kurtosis parameter has been used [14]. So leptokurtic and platykurtic pro®les imply respectively sharper and ¯atter beams than the Gaussian case. In general, the kurtosis of the beam shape changes along the propagation path. Moreover, for strongly pumped systems the beam pro®le may be nonmonotonic (hole in the centre). Evidently, the proposed de®nition of beam parameter product of such the intensity distribution is not invariant in linear transformation. To simulate the dierent bellshape functions an average transverse intensity distribution of the multimode beam may be described as: I r I0 expÿ2 jrj=wn :
9
For n = 2 it represents the Gaussian beam, for n>2 so-called super-Gaussian distribution. Analogously, for n < 2 it can be named as sub-Gaussian one. In all cases
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Fig. 4. Relative double-beam parameter product vs. relative lens position for set of beam parameters: (a) y01=15 mrad, w01=1 mm, y02=6 mrad, w02=2.5 mm, f = 24 cm. (b) y01=12 mrad, w01=1.25 mm, y02=7.5 mrad, w02=2 mm. (c) y01=10 mrad, w01=1.5 mm, y02=7.5 mrad, w02=2 mm. Circles demonstrate typical experimental data for intensity measurements and the squares for power measurements.
the intensities at the points r = w are the same. However, the power that penetrates inside the circle with r = w be dierent. Sub-Gaussian beams have more heavy wings and for super-Gaussian beams the main part of total power is concentrated inside the core r < w. Generally, in the image space of the lens the beam intensity redistribution takes place [14]. Both calculation and experiment show that platikurtic pro®le before the focal plane transforms to leptokurtic one after the focal plane. Consequently, the proportion of core power also changes. This parameter is interesting in practice, for example for ®bre entrance optics. Fig. 5 demonstrates the relation of core power Pc to total beam power P vs. intensity shape parameter n. As an example, one can estimate that for 1-kW beam with the transverse intensity distribution as like as expvxv the wings have weight about 400 W. Evidently that such power is quite sucient to vaporize the ®bre coating. It seems that in practice because of complex high power beam radial and angular structure, it is more appropriate to estimate the beam width (in near and far ®elds) containing the certain portion of total beam power, for example 0.98 for high power laser. In this case the values of w and y should be measured or calculated, such that y 2p 1 w 2p I r, jdr dj I* W, jdW dj 0:96, 10 P2 0 0 0 0 where I(r,j) and I *(W,j) are intensity distributions in near and far ®eld, respectively.
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Fig. 5. Part of power penetrated inside the beam core with intensity higher then 1/e 2 versus shape factor n in Eq. (9).
Comparison of the experimental beam pro®les before and after the lens (Fig. 8) shows that the beam divergence decreases from 21 to 16 mrad and simultaneously (P ÿ Pc)/P decreases from 4 to 1.8%. Such a transformed beam may be entered to a ®bre easier than that with initial distribution. 4. Experiment The heat generation in a laser rod depends on spectroscopic parameters of the crystal, pump cavity eciency and pump power. These parameters combine to in¯uence the optical beam characteristics. To measure the output laser parameters we used the experimental set-up shown in Fig. 6. As active rods (1) b 5.6 80 mm long Nd:YAG crystals have been used. The arc lamps provided up to 5 kW electrical power each one. The ¯at±¯at cavity is formed by 100 and 75% re¯ecting mirrors (2). Far ®eld distributions were measured on the focal plane of 1-m focal length lens (5) behind a 0.5 mm diameter diaphragm (6), that was moved with a detector (7) by the step-motors in two orthogonal directions. The total power in any divergence angle was measured by a power meter (9) behind an iris diaphragm. Intensity distributions were recorded with an oscilloscope (8).
Fig. 6. Experimental set-up: (1) active rods, (2) laser mirrors, (3) attenuator, (4) lens transformer, (5) divergence measuring lens, (6) diaphragm, (7) detector, (8) oscilloscope and (9) power meter.
The dierent laser units had the maximum output power in the range 220±330 W. According to the documentation provide with the laser, half of output power was emitted in 11 23 mrad angle. Output power was ®rst measured as a function of pump power and is shown in Fig. 7 inside this angle. The curve with rectangles shows the total power propagated in 11 mrad angle. At pump powers over 7 kW saturation and even a decrease of output power takes place. However for 33 mrad angle there is no such eect. In the middle of the pump power range the deformation of far ®eld distribution occurs and beam pro®le becomes non-Schell± Gaussian. In the oscillograms of Fig. 8 the far ®eld distributions for low (3 kW), medium (7 kW) and high (9.5 kW) pump power are shown. For pump power over 5.5 kW the intensity distribution transforms from Gaussian±Schell shape to double-bell shape and to triangular shape at 8±9 kW pump power. As it was mentioned above, the space intensity distribution of the beams had elliptical form with complex internal structure. For some laser specimens three-bell far ®eld distribution was observed in horizontal direction at high pump powers. Fig. 4 demonstrates experimental points for the relation y02 w01/yimax wimax with the lens of f = 24 cm. Among the four lasers investigated the best reduction of wy was about 2±2.2 times in horizontal direction. The typical value was about 1.4±1.5 times. The measurements of the beam parameters by the power meter placed behind the iris diaphragm gave the average value of power emitted in all directions. The maximum product reduction for the power measurements was about 1.4±1.8 times for dierent laser units. The beam width measurement was made for Pc/P = 0.95 20.03.
Fig. 7. Output power emitted into the full angle 11 (rectangles), 22 (circles) and 33 mrad (crosses) vs. pump power.
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Fig. 8. Far-®eld intensity distribution of the laser beam at pump powers (a) 3, (b) 7, (c) 9.5 and (d) after the lens at 9.5 kW pump power.
The exact experimental value of w0 (the waist on the output coupler) has not been measured because of the technical diculties. However w0 may be estimated at the point of lens location S0=0. The beam width on the coupler w01 may be calculated from experimentally measured value wi2 yi1/y02. Such normalization gives 2w0113.7±3.9 mm. Our experimental data are close to the curve with parameters y02/y0111.33. This is in good agreement with above estimates of divergence angles in the thermal lens parameters calculations. In the case where S0=Si=0, the waists of all the laser beam fractions with dierent divergence angles are located at the same distance f from the lens. The coecient a becomes equal to f/z0. The Rayleigh range changes with pump power. It is possible to ®nd the lens with focal length f equal to the average value of z0 in the middle of the pump range. At low pump powers the initial beam divergence is relatively low and the Rayleigh range is bigger than the focal length of the lens. It means that the lens increases the beam divergence angle in the image space. At high pump powers the beam divergence becomes high and the Rayleigh range becomes shorter than the lens focal length and the lens decreases the beam divergence. Fig. 9 demonstrates the experimental dependencies y0=y0(P) and yi=yi(P) after the lens with f = 30 cm. It is evident that the raw beam divergence increases by a factor of approximately 10 over the range of pump power, but after the lens this value changes no more than 12%. This eect gives possibility of successful coupling of the output beam into a ®bre over a wide pump range. For the lens location with S0=0, z0=fyi/y0=w0/y0, yi=w0/f. Consequently if the experimentally measured beam divergence angle in the image space of the exter-
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Fig. 9. Half divergence angle of the laser beam (a) before and (b) after the lens-transformer versus pump power.
nal lens demonstrates the weak deviations from the average value, the waist diameter on the output coupler also would be stable. The above relations give numerical value of the waist of approximately 0.39 cm. This is close to the earlier estimates of waist size using the double-beam model. 5. Conclusions The far and near ®eld beam intensity distributions of a double-pump cavity CW YAG laser becomes obviously non-Gaussian±Schell at pump powers higher 5.5 kW. The calculations demonstrate that the increase in laser emission reduces the amplitude of stresses in the rod and the thermal lens refractive power by a factor of approximately 1.5. The thermal lens can be represented as a set of lenses with the dierent parameters. Analogously, the intracavity beam structure can be imagined as a superposition of some Gaussian±Schell beams with dierent divergence angles and waist diameters. The aberration of the thermal lens and complex beam structure makes it possible to improve the total beam parameter product in the image space of an external lens. The double beam model predicts a factor of 1.5±2.5 decrease of beam parameter product for circular symmetric beams. Experimentally a factor of 1.4±1.8 decrease of beam parameter product has been achieved for power measurement. In the pump range 3±10 kW the half divergence angle of the output beam was changed from 1 to 12±14 mrad. In the image space of 30 cm focal length lens the half divergence angle was 6.5 20.8 mrad across whole pump range. It has been proposed to use an integrated beam parameter product to describe the non-Gausian±Schell beams. The
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maximum laser output power was about 330 W with beam parameter product of 16 mm mrad after the lens with 24 cm focal length. Acknowledgements The authors appreciate the support of the Science and Technology Center in Ukraine, Project No 526. The authors wish to thank G. Galich for technical assistance. References [1] Koechner W. Absorbed pump power, thermal pro®le and stresses in a cw pumped Nd:YAG crystal. Appl. Opt. 1970;9:1429±34. [2] Koechner W. Thermal lensing in a Nd:YAG laser rod. Appl. Opt. 1970;9:2548±53. [3] Forster JD, Osternik LM. Thermal eects in a Nd:YAG laser. J. Appl. Phys. 1970;41:3656±63. [4] Lu Q, Dong S, Weber H. Analysis of TEM00 laser beam quality degradation caused by a birefringent Nd:YAG rod. Opt. Quantum Electron. 1995;27:777±83.
[5] Rozanov AG. A nonlinear model of thermal eects in laser crystals of the YAG:Nd type. Quantum Electron. 1991;18:1185± 8 (in Russian). [6] Hodgson N, Weber H. In¯uence of spherical aberration of the active medium on the performance of Nd:YAG laser. IEEE J. QE 1993;9:2497±507. [7] Martinez-Herrero R, Mejias PM, Hodgson N, Weber H. Beamquality changes generated by thermally-induced spherical aberration in laser cavities. IEEE J. QE 1995;31:2173±6. [8] Simon R, Mukunda N, Sudarshan ECG. Partially coherent beams and a generalized ABCD-law. Opt. Commun. 1988;65:322±8. [9] Lavi S, Prochaska R, Keren E. Generalized beam parameters and transformation laws for partially coherent light. Appl. Opt. 1988;27:3696±703. [10] Jabczynski J, Mindak M. Focus shift of multimode cw YAG laser beam. Opt. Eng. 1992;31:447±52. [11] Peshko I, Galich G, Lopiychuk M, Khizhnyak A, Nakvasuk V. High power CW solid-state multilaser system. Opt. Laser Technol. 1997;29:211±5. [12] Jankiewicz Z, Mindak M, Szydlak J, Wojcik J. Analysis of the thermal focusing eect in a cw Nd:YAG laser. Opt. Appl. 1985;15:125±34. [13] Shokin AA. Dependence of heat dissipation in YAG:Nd3+ on output power. Quantum Electron. 1974;1:423±5 (in Russian). [14] Piquero G, Mejias PM, Martinez-Herrero R. Sharpness changes of gaussian beam induced by spherically aberrated lenses. Opt. Commun. 1994;107:179±83.