Modeling of High Power Solid-State Slab Lasers

Transcription

1 Modeling of High Power Solid-State Slab Lasers B. Chen, J. Dong, M. Patel, Y. Chen, A. Kar, and M. Bass 1 * School of Optics/CREOL, University of Central Florida, Orlando, FL USA ABSTRACT We present practical modeling for edged-pumped high power solid-state slab lasers. Both analytic and ray tracing methods were used to analyze the distributions of absorbed pump power using real broad spectrum diode laser pump sources. The resulting thermal and stress distributions were calculated with finite element analysis. Our analyses include the temperature dependence of the gain medium s thermal conductivity and expansion coefficient. Nd:YAG and Yb:YAG were then compared for use as gain media for high power solid state slab laser. Keywords: lasers, solid-state, slab, high-power, YAG 1. INTRODUCTION Solid-state lasers have been used several decades in a wide variety of industrial, military, and medical applications. High-power solid-state lasers have been attracting a lot of interest as the alternate to CO lasers for various industrial applications, such as cutting, drilling, and welding, because of their flexible fiber-delivery capability and compact mechanical design. With the improvements in high-power operational capability of semiconductor laser diode arrays and the development of diode laser divergence and beam control techniques, diode pump arrays have become scalable in absolute power and capable of delivering intensities greater than kw/cm. 1 This makes possible the even higher power solid-state lasers. In this article we present modeling studies for practical solid-state slab lasers with output powers up to 16 kw. Operational limits are discussed. Nd:YAG and Yb:YAG slab lasers were examined in our modeling.. GAIN MATERIALS AND LASER GEOMETRY The choice of gain material for high power solid state lasers is critical. It is generally agreed that the host material, yttrium aluminum garnet, YAG offers the most attractive combination of thermal and mechanical properties for this application. The lasing properties of Nd and Yb ions in YAG are both promising so the question arises as to which to use. The utility of a material is limited by the pulsed laser damage intensity (setting an upper limit to the saturation fluence which will give efficient operation) and the intensity which corresponds to heat deposition leading to thermal stress induced failure (setting an upper limit to the intensity which will give efficient operation). Nd:YAG lies near the intersection of these two limits. It is therefore a likely choice for efficient high average power operation. Yb:YAG has higher saturation fluence and intensity and because of pulsed and CW failure limitations can not be operated as near its most efficient point as Nd:YAG. However, in this material less heat is deposited due to both fluorescent losses and a small quantum defect. In addition, the pump absorption band is wide. This feature makes it easier to match the pump diode spectrum to the excitation spectrum of the gain medium. Thus, we consider modeling both materials in reasonable configurations to determine which offers better performance properties at various operating powers. For 6 kw or higher power lasers, Yb:GGG and Nd:GGG are attractive gain media because larger GGG crystals can be grown than is possible for YAG. Using crystal configurations with long pump absorption paths, direct pumping of Nd:GGG at 88 nm can greatly increase optical-optical efficiency and reduce waste heat deposition. Similar results can be also seen in Yb:GGG lasers pumped by 971 nm light at the zero-phonon line. 3 Our work on high power GGG lasers is in process and will be reported separately. 1 * mbass@creol.ucf.edu; phone ; fax Solid State Lasers XII, Richard Scheps, Editor, Proceedings of SPIE Vol (3) 3 SPIE X/3/$. 1

2 The slab geometry has traditionally been used to scale solid-state lasers to high peak and average powers. Especially after the advent of affordable laser diode arrays as pumping light sources, slab lasers with a variety of pumping geometries have been developed. End-pumped slab lasers exhibit high efficiency and excellent mode match, but the fairly complicate pumping arrangements limit its scalability to higher powers. 4 From the viewpoint of thermal-induced distortion, a face pumped and cooled laser is superior. This results in a one-dimensional temperature distribution and eliminates thermal stress induced depolarization. Further a zig-zag optical path in the slab eliminates first order thermal focusing. The drawbacks of this geometry are low pump efficiency due to the small thickness of slabs and the requirement to cool and pump through the same faces. The separation of the pumping and cooling faces brings the edge-pumped geometry some engineering advantages. Each interface can be optimally arranged for the best performance and because of the longer path for pump absorption the pump efficiency in the edge-pumped geometry can be better than that for face-pumping. 3. PUMPING UNIFORMITY Near the edges of an edge-pumped slab more pumping light is absorbed than near the center (Fig. 1) due to the uniform doping of the crystal. Since the absorption per unit length is the same everywhere more light is absorbed near the edges where more light is present. Optical distortions of laser crystals due to the resulting thermal gradient can not be eliminated by zigzag paths and to avoid beam quality degradation one must strive to achieve uniform pumping in the width of the edge pumped slab. y Heat Removal Pumping x Laser Beam z Pumping Heat Removal Figure 1: Sketch of a slab having parallel Brewster angle end faces. Coordinate axes are defined. For a slab pumped through both edges the pump power absorbed as a function of position in the width of the slab, x, where x = is the center, is given by the hyperbolic cosine function exp( αw αp ) p ρabs (x) = cosh( αx) (1) tl 1 R p exp( αw) where P p : total pump power; R p : average pump reflectivity of the edge faces; α = n d σ α : pump absorption coefficient; n d : dopant concentration; σ α : effective absorption cross section. The pumping efficiency is 1 exp( αw) ηabs = () 1 R p exp( αw) Proc. of SPIE Vol. 4968

3 The pumping efficiency η abs depends on pumping reflectivity R p and αw. Following the procedure of T. S. Rutherford, et al. 6 we define the pumping uniformity U and pump figure of merit F as ρabs() U = = sec h( αw ) (3) ρ (w / ) abs 1 exp( αw) F = Uη sec h( w abs = α ) (4) 1 R exp( w) p α Only αw affects pumping uniformity. Therefore, choices of R p and αw could be made to optimize pumping figure of merit F. Rutherford, et al. 6, allow that some areas on the edges are highly reflective and other small areas are highly transmitting serving as the pump light entrance, and so both high absorption efficiency and pumping uniformity could be achieved. In our cases, antireflection coatings are used to assure pump light coupling into the medium. Fig. is a plot of the pumping efficiency, uniformity and pumping figure of merit as a function of αw for the case of R p =, where the figure of merit is optimized when both absorption efficiency and absorption uniformity are ~78% Absorption Efficiency Pumping Uniformity..4.3 Pumping Figure of Merit (Absorption Efficiency x Pumping Uniformity) Absorption Depth, αw Figure : Pump absorption efficiency (dot line), pump absorption uniformity (dash line) and pumping figure of merit (solid line) as a function of absorption depth for the case R p =. For the edge-pumping of high power solid-state lasers the arrangement of the diode laser pump sources is very important. In our ray tracing simulation of pump distribution the divergence angle of each diode laser is set separately for better pump light coupling into gain medium. Usually the fast-axis divergence angle of the diode laser is about 6 degree. In separate work we have developed a beam control technique that can bring the divergence angle down to 1 degree. 7 This gives us more space to arrange the diode laser array to achieve high power uniform pumping in the YAG slab. In reality, the pump sources, diode lasers, are not ideally monochromatic, so the pump efficiency is dependent of pump spectral overlap with the excitation feature of the gain medium. The absorbed pump power distribution is quite sensitive to any wavelength shift of the pump light if the absorption feature is narrow and less sensitive if it is broad. In our case the broad spectral pump band of the diode laser array is expressed by considering several monochromatic light sources whose associated pump power is P(λ) with total pump power P total = P( λ) λ. For Yb:YAG slab lasers, our λ ASAP simulations show that if the pump band is centered around 941 nm, which accesses a broad absorption band, the absorbed pump power distribution is not much affected by shifts of the peak pumping wavelength (see Fig. 3a). On the. 3. Proc. of SPIE Vol

4 other hand, the distribution may change with the peak pump wavelength if the pumping band is centered around 97 nm, which accesses a narrow absorption band (see Fig. 3b). In general, the broad spectral pumping results in a more uniform absorbed pump power distribution at the expense of pump efficiency. Our ASAP ray tracing simulations for absorbed pump power distributions show that we can achieve uniform pumping profiles on length-thickness planes (see Fig. 4). The absorbed pump power distributions for all those planes are similar except the value of the density. Absorbed pump power density (W/mm 3 ) nm 939 nm 937 nm Width (mm) Absorbed pump power density (W/mm 3 ) nm 97 nm 97. nm 971 nm Width (mm) Figure 3: The pump profile as a function of the width position. a) The pump band is centered around 941 nm. (left); b) The pump band is centered around 97 nm. (right) Figure 4: The absorbed pump power density on length-thickness planes. Here, x is along the length, y along the thickness. 4. THERMAL AND STRESS SIMULATION The main challenge in developing high power solid-state lasers is dealing with thermomechanical distortions caused by waste heat deposited by optical pumping. The waste heat inside the gain medium leads to thermal lensing, mechanical stresses, birefringence and depolarization losses that may result in degraded beam quality, fracture of the gain medium, 4 Proc. of SPIE Vol. 4968

5 and reduced laser power. Heat produced inside gain medium must be removed through the cooled surfaces. If the gain medium is shaped as a slab cooled through its largest face, the area through which the heat can be removed is wl where w is the slab width and l the slab length. The rate of heat removal by the coolant must be at least η h ( η abs Pp ) () wl where η h is the fraction of the absorbed pump power that appears as heat. However, the maximum heat removal rate through a surface is obtained by evaporative spray cooling and is ~ 1 kw/cm. 8 In fact, the heat removal rate by sensible cooling (cooling by the flow of a liquid across a surface) is less than 3 W/cm while for liquid jet impingement cooling it can be about 7 W/cm. 8 In the design of any laser we must first be sure that the heat deposited in the gain medium can be removed by the proposed cooling process. In order to better describe the thermal situation of the slab, we performed thermal and stress analysis by finite element analysis (FEA) with the software program ALGOR. In our analysis, the convection heat transfer coefficient, h, is taken as 3 W/m K. It is found that there is no significant change in temperature when the value of convection heat transfer coefficient is greater than 3 W/m K in the case of one-dimensional heat conduction with uniform density of deposited heat. The slab is pumped from edges and cooled from the top and bottom faces (see Fig. 1). The three dimensional heat conduction equation is, Q(x).( k(t). T) + = (6) k(t) with boundary conditions, T h w w = T w T x = ± at x k(t) T h t t = T t T y = ± at y k(t) T h l L = T l T z = ± at z k(t) w x = ± (7) t y = ± (8) L z = ± (9) where h w = free convection heat transfer coefficient at h t = forced convection heat transfer coefficient at w x = ± t y = ± L h l = free convection heat transfer coefficient at z = ± T l, T w = ambient room temperature along length and width (K) T t = coolant temperature (K) The temperature dependence of the thermal conductivity of YAG is given as 9 a d k(t) = c (ln(bt)) T () where a = W/cm K, b =.33 1/ K, c = 7.14, and d = 331. W/cm. The thermal expansion coefficient of YAG, α, also varies with temperature as, Proc. of SPIE Vol. 4968

6 3 α ( T) = at + bt + ct (11) where a = K -, b = K -3, and c = K -4. The heat deposit is calculated based on the absorbed pump power density (Eq. 1). Due to symmetry of heat flow in the slab, the thermal analysis is done for one-eight of slab. For stress analysis, it is assumed that there is no strain along the length because there is no temperature variation in this direction. The thermal and stress distributions are shown in Figs. and 6 for the slab and pump power indicated. The stress near the center is compressive while near the surfaces it is tensile. The tensile stress on surfaces may initiate failure of the gain medium so it is important to design so that the stress is well below the stress fracture limit. From Fig. 6(a) it is seen that stress does not vary significantly in the width direction. The maximum tensile stress is attained about 1 mm from the outer edge since the absorbed pump power is high in that region. As maximum variation in temperature occurs along the thickness direction the stresses, as shown in Fig. 6(b), vary from compressive at the center of slab to tensile on the top and bottom surfaces. The stress simulation suggests that bonding undoped crystals to the slab edges can move the locations of maximum tensile stress out of the gain medium and therefore improve on the performance of the laser slab. Similar simulations were done for a Nd:YAG slab but with 3% of optical energy resulting in heat energy compared to 11% for Yb:YAG. Thermal and stress simulation results are listed in Table II. Figure : The temperature distribution in an Yb:YAG slab of t 16t 8t pumped with 9.7 kw. (a) along the width. (left); (b) along the thickness (right) Figure 6: The stress distribution in an Yb:YAG slab of t 16t 8t pumped with 9.7 kw. (a) along the width. (left); (b) along the thickness (right) 6 Proc. of SPIE Vol. 4968

7 . LASER OSCILLATION MODELING AND POWER SCALING Based on the work reported by R. J. Beach and T.S. Rutherford, et al., 1, 6 we developed a model for an edge-pumped slab laser. Besides including the effects of depletion of the lower manifold, large gain and output coupling, our model includes more realistic effects, such as the temperature dependence of the stimulated emission cross section and of the level populations, thermal stress, heat removal and cavity loss. For an edge-pumped quasi-three-level slab laser, it is assumed that the slab is completely filled by multiple transverse modes. The excitation rate is Pp R ex = ηabs (1) hν p where ν p is pump light frequency. The overall de-excitation rate is Pout R oc βσ nl βσ nl n u wtl R de = (e eff 1)[1 + (1 δ ) e eff ] + (13) hν L 1 R oc τ where P out = laser output power, ν L = laser frequency, R oc = output coupler reflectivity, β = zigzag overlap factor(1 for no overlap, for complete overlap), σ = emission cross section, n = population inversion density on the laser transition, l eff = geometric length of zigzag length, δ =one-way cavity loss, n u = upper manifold population, and τ = fluorescence life time. It should be noticed that η abs, σ, n, n u are temperature dependent and hence dependent on the absorbed pump power for a specific slab. One-way cavity loss can be taken as the sum of gain medium scattering loss (.3/cm for Yb:YAG and./cm for Nd:YAG) and mirror-related loss. At steady state, the excitation rate must match the deexcitation rate to maintain oscillation. By equating R ex and R de we find an expression for laser output power: ν L 1 R oc Pout (Pp ) = ν p βσ (Pp ) n(pp )leff βσ (Pp ) n(pp )leff R oc[e 1][1 + (1 δ ) e ] (14) hν p [ ηabs(pp ) Pp wtln u (Pp )] τ To maintain the oscillation in the resonator, the overall round-trip gain of the resonator must be one so that βσ (Pp ) n(pp ) leff R oc (1 δ ) e = 1 () Then population inversion density could be expressed as ln[r oc (1 δ ) ] n(pp ) = (16) βσ(pp ) n(pp )leff By definition, n(pp ) = fbnu fanl, where f a and f b represent the fraction of atoms in the ground state manifold which are in the lower laser level and the fraction of atoms in the upper manifold which are in the upper laser level, respectively. Both f a and f b are temperature dependent and therefore pump power dependent. The lower manifold population, n L = n d -n u. The upper manifold population can be calculated as ln[r oc (1 δ ) ] f a (Pp ) n u (Pp ) = + n f (P ) f (P ) d (17) fβσ(pp )l a p + b p eff Taking into account of depopulation of the lower pump level, the pump absorption coefficient used in Eq. (1) could be taken as α=σ α n eff = σ α (f α n L -f b n u ) instead of σ α n d, where f α is the fraction of the lower manifold atoms which are in the lower pump level and f b is the fraction of the upper manifold atoms which are in the upper pump level. From Eq. (17), we find f '(P p ) ln[r oc (1 δ ) ] f a '(Pp )f (P p ) f a (Pp )f '(Pp ) neff (Pp ) = + n f (P ) f (P ) d (18) p βσ(pp )l p eff where f(p p )=f a (P p )+f b (P p ) and f (P p )=f a (P p )+f b (P p ). For the Nd:YAG slab laser, a four-level system, the lifetime of upper pump and lower laser level are relatively short so we can neglect their populations. We may assume f a (P p ) = f b (P p )=, and f b (P p ) = f a (P p ) =1. With these assumptions, the equations above for Yb:YAG slab lasers can also be used for Nd:YAG slab lasers. Proc. of SPIE Vol

8 Besides the introduction of the temperature dependence of the thermal conductivity, thermal expansion coefficient of gain medium and occupational fraction factors, we introduce the temperature dependence of the stimulated emission cross section in our model. For Yb:YAG, we obtained this dependence by fitting the referenced experimental data in literature. 11 The functional form of the fit is TPin ( ) σ( Pin) := e cm - (19) A. Rapaport et al. have measured the temperature dependence for the Nd:YAG cross section 1 as 19 σ (Pp ) = [ T(Pp )] cm () The result of the introduction of the temperature dependence of the stimulated emission cross section is that the output is just slightly reduced with increasing pump power as compared to calculations in which take the stimulated emission cross section to be constant (e.g., the room temperature value). Significant affects appear only when the laser presently considered operates at a level in which the fracture limit has been exceeded. The temperature excursion and the thermal induced stress therefore impose much more stringent limitations on laser operation than do these temperature dependences. Here it is well to remember that we have adjusted the laser parameters such that the maximum temperature excursion for Yb:YAG is below 3 o C. The ratio of the absorbed pump power at the point of stress fracture divided by that required to reach transparency to the pump light is called power scaling figure of merit and is often used to describe the ability of lasers to operate below stress fracture and above transparency. The power scaling figure of merit of an edge-pumped slab laser is found to be proportional to w/t, so to scale to higher power, slab thickness can be reduced, and the width can be independently increased to reach acceptable pump absorption efficiency MODELING EXAMPLES We begin our comparison of Nd and Yb:YAG lasers by developing a practical model for a 3 kw Yb:YAG slab laser and compare its properties to the 3 kw Nd:YAG slab laser reported by Sato et al. 13 In our modeling, the reflectivity of the edges through which the pump light enters the gain medium is taken to be. That is we assume there are antireflection coatings at the pump wavelength. Figs. 7a and 7b show, for Yb:YAG and Nd:YAG, respectively, the stress factor reached and maximum temperature of gain medium at various absorbed pump powers. If we set the operational stress limit at 1% then an Yb:YAG slab of the size considered can handle pump power as high as 13 kw and can produce kw of laser output power. Of course, if one chose to be less conservative, a higher stress factor can be taken and more output power is possible by risking catastrophic failure of the gain medium. Output Power (kw), Stress Factor (%) 4 Operational stress limit Stress factor, b 14 Temperature 16 P out Slab Temperature ( o C) Pump Power (kw) Pump Power (kw) Figure 7: Output laser power, stress factor, and maximum temperature of slab lasers as a function of pump power. a): Yb:YAG (t 16t 8t), (left); b): Nd:YAG (t t t), (right) Output Power (kw), Stress Factor (%) Operational stress limit Stress factor, b Temperature By increasing the aspect ratio and decreasing the dopant concentration, the edge-pumped slab lasers can be scaled to higher powers. 7 6 mm is the largest available YAG crystal slab reported. 13 Sato et al. have obtained 3.3 kw P out Slab Temperature ( o C) 8 Proc. of SPIE Vol. 4968

9 laser power using a Nd:YAG slab of 6 6 mm. In their work, face pumping was used so the crystal was thicker than it needs to be for edge pumping. Therefore, we consider a thinner slab to determine the maximum power output capability of a single-slab laser. In our case of edge pumping, the maximal aspect ratio of the slab used is 6, and the output laser power is scaled up to 7 kw for Nd:YAG slab lasers. Figs. 8a and 8b show the pump power dependence of fraction of fracture strength and maximum temperature of Yb:YAG and Nd:YAG slab lasers, respectively. Our modeling is summarized in the following paragraphs and in Tables I and II. Output Power (kw), Stress Factor (%) 3 Operational stress limit 3 4 Pump Power (kw) 6 P out Stress factor, b Temperature Slab Temperature ( o C) Output Power (kw), Stress Factor (%) 3 Operational stress limit Stress factor, b 3 Pump Power (kw) 4 P out Temperature Figure 8: Output laser power, stress factor, and maximum temperature of slab lasers as a function of pump power. a): Yb:YAG (t 6t t), (left); b): Nd:YAG (t 6t t), (right) Slab Temperature ( o C) Table I: Input parameters of the modeling of edge-pumped slab lasers Input Parameters Nd:YAG Yb:YAG Desired output power (kw) Pump light wavelength (nm) Lasing wavelength (nm) Coolant temperature ( o C) Heat transfer coefficient (W/cm K) Scatter loss (cm -1 ) Output coupling reflectivity Table II: Output parameters of the modeling of edge-pumped slab lasers Output Parameters Nd:YAG Yb:YAG Output power (kw) Dimensional ratios (w/t, l/t) Dopant concentration (% at) Slope efficiency (%) Pump threshold (kw) Overall optical efficiency (%) Heat extraction requirement (W/cm ) Maximum temperature ( o C) Fraction of fracture strength (%) Pump intensity (kw/cm ), , , , It is worth noting that the smaller crystals required for the Yb:YAG design result from the smaller heat deposition in this material (11% of pump power as compared to 3 % for Nd:YAG) and the practical requirement that the heat deposited can be removed by whichever cooling procedure is selected. Mode analysis and resonator design are in process and will be reported separately. Proc. of SPIE Vol

10 In addition, using ASAP, we simulated the fluorescence allocation from Yb:YAG. 14 We also analyzed the amplified spontaneous emission (ASE) and parasitic oscillations in the slab laser. It is found that to limit ASE effects the gain coefficient and length product should be kept less than. Our ASAP simulations shows that canting both edge faces by 1 o is sufficient to suppress parasitic oscillation inside the slab. So far, our analysis is for multimode lasers. We are working to obtain near diffraction-limited laser beams. We are also calculating thermal lensing and stress-induced depolarization losses to include them in our resonator design. We will refine our models to include these effects 7. CONCLUSION We present analyses of high power solid-state slab lasers. Both analytic and ray tracing simulations were used to explore means to achieve pump uniformity. By including realistic absorbed pump power distributions based on realistic pump source spectra, the temperature dependence of the fractional level populations, stimulated emission cross section, thermal conductivity and thermal expansion coefficient, as well as heat removal capability, we developed realistic models for both Nd:YAG and Yb:YAG edge-pumped slab lasers and discussed practical limits. It was found that the gain medium required for Yb:YAG is smaller than required for Nd:YAG for the same output power. This together with the more relaxed demand on the pump laser wavelength for Yb:YAG makes it attractive for such high power lasers. This work was funded by Raytheon Corporation ACKNOWLEDGEMENTS REFERENCES R. J. Beach, Opt. Commun. 13, 38 (1996). G. T. Liu et al., CLEO Proc. (1). S. Chenais et al., CLEO Proc. (1). N. Hodgson et al., CLEO, 64 (). T. Rutherford, W. M. Tulloch, S. Sinha, et al., Opt. Lett. 6, 986 (1). T. Rutherford, W. M. Tulloch, E. K. Gustafson, et al., IEEE J. Quantum Electron. 36, (). T. Y. Chung, M. Bass, L. Chow, et al., in SSDLTR, Albuquerque, NM, (). L. Chow, Orlando, Private Communications. (). P. H. Klein and W. J. Croft, J. Appl. Phys. 38, 163 (1967). R. Wynne, J. L. Danue, and T. Y. Fan, Appl. Opt. 38, (1999). 11 D. S. Sumida and T. Y. Fan, OSA Proceedings on Advanced Solid-State Lasers, (1994). 1 A. Rapaport et al., Orlando, Private Communications. (). 13 M. Sato et al., Proc. of SPIE 3889, 18 (). 14 Y. Chen, A. Rapaport, T. Y. Chung, et al., Manuscript (3). B. Chen, Y. Chen, and M. Bass, Manuscript. (3). Proc. of SPIE Vol. 4968