Heat Generation and Removal in Solid State Lasers

Transcription

1 Chapter 1 Heat Generation and Removal in Solid State Lasers V. Ashoori, M. Shayganmanesh and S. Radmard Additional information is available at the end of the hapter 1. Introdution Based on the type of laser gain medium, lasers are mostly divided into four ategories; gaseous, liquid (suh as dye lasers), semiondutor, and solid state lasers. In reent past years, solid state lasers have been attrated onsiderable attentions in industry and sientifi researhes to ahieve the high power laser devies with good beam quality. In solid state lasers the gain medium might be a rystal or a glass whih is doped with rare earth or transition metal ions. These lasers an be made in the form of bulk [1, ], fiber [3-7], disk [8, 9] and Mirohip lasers [10,11]. Optial pumping is assoiated with the heat generation in solid state laser materials [1]. Moving of heat toward the surrounding medium whih is mostly designed for the ooling management auses thermal gradient inside the medium [13]. This is the main reason of appearane of unwanted thermal effets on laser operation. Thermal lensing [14], thermal stress frature limit [15], thermal birefringene and thus thermal bifousing [16-19] are some examples of thermal effets. Optimizing the laser operation in presene of thermal effets needs to have temperature distribution inside the gain medium. Solving the heat differential equation beside onsidering boundary onditions gives the temperature field. Boundary onditions are diretly related to the ooling methods whih lead to onvetive or ondutive heat transfer from gain to the surrounding medium. In the ase of bulky solid state laser systems (suh as rod shape gain medium), water ooling is the most ommon method whih is almost used in high power regime. Design of optimized ooling avity to ahieve the most effetive heat transfer is the first step to sale up high power laser devies. Then determination of temperature distribution inside the laser gain medium is essential for evaluation of indued thermo-opti effets on laser operation. 01 Ashoori et al., liensee InTeh. This is an open aess hapter distributed under the terms of the Creative Commons Attribution Liense ( whih permits unrestrited use, distribution, and reprodution in any medium, provided the original work is properly ited.

2 34 An Overview of Heat Transfer Phenomena This hapter is organized aording to the requirements of reader with thermal onsiderations in solid state laser whih are mainly dependent on several kinds of laser materials, pumping proedure and ooling system. We hope the subjets inluded in this hapter will be interesting for two guilds of sientifi and engineering researhers. The first ategory relates to the laser sientists, who need enough information about the reent ooling methods, their benefits and disadvantages, thermal management and effets of utilizing a speifi ooling method on laser operation. And the seond one is the mehanial or opto-mehanial engineers who are responsible for designing and manufaturing of the ooling systems. In this hapter our efforts direted suh a way to satisfy both the mentioned ategories of researhers. At First, the priniple of heat generation proess inside the laser gain medium due to the optial pumping are introdued. Then, heat differential equation in laser gain medium and relating boundary onditions are introdued in detail. Formulation of heat problem for a speifi form of gain medium suh as bulk, disk, fiber and Mirohip lasers and details of solution are presented through individual subsetions.. Priniple of heat generation in solid state laser gain medium.1. Laser pumping A laser devie is omposed of three essential omponents whih are the "ative medium", "pumping soure" and the "optial resonator". In the ase of solid state lasers, the ative medium whih is made of a definite glass or rystal, is plaed inside the optial resonator and reeives energy from another external optial soure through the pump beam light. Then it an itself emit an amplified laser beam light delivering a ompletely modified energy and wavelength [0]. The at of energy transfer from the external soure to the ative medium is alled the laser pumping. In reent years, diode lasers [1-] have attrated onsiderable attention between laser sientists as available, high power and beam quality pumping soures. In this hapter we just onentrate on this kind of pumping soures rather than traditional flash lamp pump soures [15]. The pumping proess ommonly performs in two methods, whih are ontinuous wave (CW) and pulse pumping laser systems. Furthermore, diode pumped solid state lasers (DPSS lasers) an be divided into side pumped and end pumped onfigurations. Figure 1, Shows shematially typial solid state lasers, inluding gain medium, optial resonator and diode pumping in the ase of end and side onfigurations. In the end-pumped geometry, the pump light mostly transfers from diode Laser (DL) to the laser material through either optial system or fiber optis whih yields a desirable pump beam shape and size. Then it fouses to the gain medium longitudinally, ollinear to the propagation of laser light. In the side-pumped geometry, the diode arrays loate along the laser material in a definite arrangement around it, suh that the pump light is perpendiular to the propagation of laser light.

3 Heat Generation and Removal in Solid State Lasers 343 The pumping geometry and the resultant pump harateristis (suh as beam shape and size along the gain medium) play an important role in heat generation and therefore thermal gradient inside the gain medium. The issue will review in details in the following subsetions for eah kind of laser gain mediums. Figure 1. Simple drawing for two ommon pumping methods of solid state laser gain medium; a) side pumping b) end pumping. The blue olor used for marking the pump light arrying energy from laser diode to the gain medium, and the red beam onerns to the laser resonator mode... Heat generation In solid state lasers, a fration of the pump energy onverts to heat whih ats as the heat soure inside the laser material [3, 4]. Spatial and time dependene of the heat soure auses important effets on temperature distribution and warming rate of the gain medium, respetively. The spatial form is assumed to be the same shape as pumping light [3, 4] and time dependene relates to the pumping proedure, whih may perform in CW or pulsed regime. Furthermore, depending on the gain medium onfiguration and ooling geometry, deposited heat may mostly flow through a preferable diretion inside the gain medium and therefore auses thermal gradient. For instane, in traditional rod shape laser mediums with water ooling onfiguration and also fiber lasers, the main proportion of heat removal ours through the radial diretion whih leads to the onsiderable radial thermal gradient inside the medium. Figure shows a shemati setup of pumping proedure for several kinds of solid state lasers. The dominant diretions of heat removal whih are assoiated with the gain medium and ooling system geometry are illustrated.

4 344 An Overview of Heat Transfer Phenomena Figure. Shemati figure of preferable diretion of heat transfer in three ommon types of solid state lasers; a) disk laser, b) rod, and ) fiber laser Heat differential equation should be solved for evaluation of temperature and thermal gradient indued by optial pumping in solid state lasers. The general form of heat differential equation in ylindrial oordinate system is given by [5] 1 Tr (,, zt,) 1 Tr (,, zt,) Tr (,, zt,) Qr (,, zt,) Tr (,, zt,) r r r r r z k k t (1) Where, Qr (,, zt, ) denotes heat soure density ( W m ), k is thermal ondutivity, and 3 are the density ( Kg m ) and speifi heat ( J Kg. o C) of laser gain medium, respetively. Eq.1 denotes to the transient heat differential equation and an speify the time dependene temperature in the ase of pulsed pumped laser systems. As we mentioned before, Qr (,, zt, ) an be determined aording to the pumping harateristis in several kinds of solid state lasers. The overall thermal load in the laser gain medium due to the optial pumping an be obtained from 3 P Q ( r, z ) dv P () h v In whih, o P is the pump power and is the frational thermal load [1]. In the ase of diode pumping, the frational thermal load, originates from two basi phenomenon, whih show the main role in heat generation; quantum defet heating [15] and energy transfer uponversion (ETU) [6]. In most ases, the first is responsible for the heat generation and therefore has the main ontribution. However it must be noted that, influene of the seond o

5 Heat Generation and Removal in Solid State Lasers 345 phenomena annot be ignored in some ases suh as Er doped laser materials. The frational thermal load in the gain medium is due to the quantum defet and related to the pumping and laser wavelength whih are shown by p and L respetively. p 1 (3) L.3. Bulk solid state lasers.3.1. Temperature distribution.3.. Side pumping Pumping onfiguration performing by one module is illustrated in Figure3.a [7]. The pump beam emitted by diode laser is foused on the rod by the interfaing optis whih onsist of two lenses. End view of side-pumping geometry is depited in Figure 3.b. Transversal diretions of pump and signal beams are easily observable. Figure 3. side pumping geometry; a) Pumping of a laser rod by one module, b) end view of side pumping geometry [7]. In the ase of CW pumping of a laser rod, the steady state heat equation an be written as [7].(,) hrz Qrz (,) (4) Where, h is heat flux and assoiated with the temperature in the rod by hrz (, ) k Trz (, ), (5) And Qrz (, ) is determined aording to spatial variation of pump intensity and is given by r Qrz (, ) I0 exp p (6)

6 346 An Overview of Heat Transfer Phenomena In whih, I0 is the heat irradiane on axis. Integration of Eq.4 over rod ross setion yields r 1 exp pi 0 p hr () (7) 4 r Substituting Eq.7 into Eq. 5 and integrating to the rod radius r o gives the temperature differene inside the rod I 0 p r 0 r 0 r Tr () ln E 1 E 1 8k r p p (8) Where Tr () Tr () Tr ( 0) and E1 is the exponential integral funtion [7] End pumping One of the ommon pumping onfigurations whih are used in diode-pumped solid state lasers is end-pumping or longitudinal pump sheme. The pumping beam is oaxial with the resonator beam in end-pumped lasers; it leads to highly effiient lasers with good beam quality.in this geometry, the pumping beam of diode laser(s) is delivered to the end of the ative medium by optial fousing lenses or optial fibers. In lower power operations (less than a few watts), end-pumping yields more aeptable results [15]. Today's development of diode lasers and new tehniques suh as using miro-lenses in beam shaping of diode laser bars make end-pumped lasers very promising speially in ommerial lasers [8]. Although end-pumping is a ommon onfiguration in solid state lasers whih inlude many types of ative medium geometries suh as slab and mirohip, but it is more ommonly used in rod shape lasers. Many end-pumping investigations and results have reported about rod lasers and most of ommerial end pump systems involve rod lasers [9-33]. Thus, the disussions in this setion are foused on end-pumped rod lasers. Shemati diagram of an end-pumped system is shown in figure 4. Figure 4. End pump systems major elements [15]

7 Heat Generation and Removal in Solid State Lasers 347 In order to establish high mathing effiieny between resonator beam and pumping modes, in end pumped systems the pumping beam is foused in the ative medium with the small beam waist. This issue leads to generation of an intense loal heating and then, reation of refrating index gradient inside the laser rystal [15]. As a onsequene, the laser rod ats as a thermal lens inside the resonator, whih an destroy the beam quality and derease the output effiieny. Additionally, in ontrast to side pumped lasers, heat distribution within the laser material is inhomogeneous in high-power end-pumped lasers, leading to inreased in stress and strain [33]. The restriting fators in end-pumped lasers in high power regime are thermal lensing and thermal frature limit for the laser rystal. The aforementioned restritions make thermal problems very important in end-pumped lasers espeially in highpower systems. From the thermal point of view, the flat top pumping profile is superior to Gaussian profile in high power end-pumped systems, due to the reation of lower thermal gradient inside the laser rystal leading to lower thermal distortions [8]. However, Gaussian pumping profiles is more investigated beause of pratial onsiderations in laser resonator design. One of the earliest thermal analysis in end pumped systems is presented in [15], whih relates to the solution of steady state heat differential equation for Nd:YAG rystal with the assumption of Gaussian pumping profile. (Figure 5). Figure 5. Temperature distribution in an end pumped Nd:YAG laser in the ase of end pumping onfiguration. The pumping and laser beams propagate in z diretion of ylindrial oordinate [15]. The temperature profile and thus assoiated thermal effets in bulky solid state lasers had been the subjet of onsideration in past years by various authors. In the ase of CW pumping, a olletion of exellent literatures disussing numerial and analytial thermal analysis an be found in [13, 3, 34-39]. Similar works onerning the transient heat analysis are available in [40]. A famous work whih presents analytial expressions for temperature and desribes the behavior of temperature inside the rod belongs to Innoenzi [13]. In this work, the laser rod

8 348 An Overview of Heat Transfer Phenomena is surrounded by a opper heat sink and exposed to the pump beam with the Gaussian intensity profile as P h exp r I p p exp z (9) Negleting axial heat flux, the steady state heat differential equation an be solved analytially. The derived temperature differene is obtained as Ph exp( z) r 0 r 0 r Trz (,) ln E 1 E 1 4 k r p p (10) As would be expeted, the temperature deays exponentially along the laser rod and have the highest temperature on the pumping surfae (z=0). A ommon ooling method for end-pumped systems onerns to utilize of water jaket or opper tube surrounding the laser rod and keep the ylindrial surfae at the definite temperature. Heat generated inside the gain medium flows to this surfae through the heat ondution proess in radial diretion. Although this method is onsidered as a simple effiient tehni providing onsiderable heat removal from gain medium, but the unooled pumped surfae of the rod whih is in diret ontat with air, performs very week heat transfer. This issue may ause undesirable effets, espeially in high power regimes [34]. The thermal load on this surfae not only inreases thermal lensing effets but also restrits the maximum pumping power beause of the frature limit of the rystal, and damage threshold of optial oatings. One of the effetive strategies to redue thermal effets is based on the ooling of pumping surfae of laser rod. In this respet, three tehnis to ahieve more effiient ooling proess have been presented in [33], whih are shematially shown in figure 6. In the first method (b), the ooling water is diretly in ontat with the pumping surfae. In the seond method (), a ooling plate ooled by water is mounted in lose ontat with the rod pumping surfae. The old plate should have large Yang s module and high heat ondutivity. The other method utilizes an un-doped ap on the pumping surfae (d). The pumping power does not absorb in the undoped ap so there is not heat load in this region, but using this ap inreases the effetive ooling surfae as well as the ratio of ooling surfae to heat generation volume. The influene of thermal effets on laser gain medium in the mentioned methods has been analyzed using FE method [33]. The maximum temperature dereased almost 30%, and 5% using an undoped ap and a sapphire ooling plate in pumping surfae respetively. The maximum stress ourred in the onfiguration with water ooled pumped surfae and redued to half of unooled system value using an undoped ap or ooling plate. Aording to reently developed erami laser materials using omposite rods with undoped ap ould be very promising as the best hoie for high power end pumped systems. The undoped

9 Heat Generation and Removal in Solid State Lasers 349 Figure 6. Different ooling onfigurations in end pumped systems. In these geometries the pumped surfae (a) unooled, (b) water ooled, () sapphire old plate ooled, (d) undoped ap rod [33]. The figures are repainted in olor version just for better realization. end ap onsiderably lowers the thermal stress in the entrane faet of an end-pumped laser. This not only redues the thermal lensing effets and thermal stresses but also lowers the maximum temperature of the laser rod and so removes some onstraints imposed on the oatings. Prominent role of omposite rod in redution of thermal destrutive effets on laser operation have been frequently examined and refleted in literatures [ 41-43]. Figure 7 illustrates the pump model of the dual-end- pumped geometry of omposite Nd:YVO4 Figure 7. Pump modeling of the dual-end-pumped geometry [44].

10 350 An Overview of Heat Transfer Phenomena laser[44]. The Nd:YVO4 as the laser gain medium is onneted to two YVO4 aps at two ends and the pump energy lunhes to it from both ends. As an be seen, every point inside the rod absorbs pump power and experienes heat generation. No absorption inside the aps takes plae and therefore, they an show important role in axial heat transfer from end surfaes. Numerial alulations of temperature distribution in omposite laser rod an be found in [45] In the ase of pulsed pumping laser rod, interesting numerial analysis has been done by Wang published in [46]. In this work the laser rod is surrounded by a ylindrial heat sink whih leads to ondutive heat transfer from rod surfae to the ambient medium. Shemati figure of the rod and ooling system geometry is depited in Figure 8-a. Figure 8. a) simple drawing of laser rod whih is surrounded by a ylindrial heat sink, b) time variation of pump power [46] The laser rod is assumed to be oupled by a fiber-opti to a laser diode; therefore the pump intensity profile with a good approximation has the top-hat shape. Thus the heat soure density an be desribed by Pt () e Qrzt (,, ) p 0, z, r r Where P(t) is a periodi funtion of time desribing pump power in a repetitively pumping regime given by p p (11) Po, 0t Pt () 0, r1 f Pt ( 1 f ) Pt ( ) rep rep (1) In whih Po is the pump peak power, rep f is the repetition frequeny, and τ is the pump duration time. Figure 8.b tries to simply speify the P(t). Detailed information of solving the transient heat differential equation an be found in [46].

11 Heat Generation and Removal in Solid State Lasers 351 Figure 9 shows the temperature distribution inside a Nd:YAG laser rod exposing to the pump beam with pulse duration of 300μs, and repetition frequeny of 50 Hz. Aording to the results, the temperature at first inreases by passing the time and then beomes nearly onstant with small flutuations whih lead the temperature to the steady state ondition. Figure 9. a) Time dependent temperature distribution in laser rod at position z = 0 at 50 Hz repetition rate [46]..4. Thin disk lasers Thin disk lasers are one of the reent frontiers in solid state lasers. The most important features whih make the thin disk laser distinguishable between solid state lasers are power salability, good beam quality and minimal thermal lensing [47,48]. These features are related to the thermal harateristis of the thin disk laser. In disk lasers, ative medium is ooled from the disk fae. The surfae to volume ratio of the disk is large due to the disk geometry, therefore ooling is very effiient and as a result thermal distortion of the ative medium is very low. Considering axial heat flow in a thin disk laser there is no thermal lensing in a first-order approximation. In fat, however weak thermal lensing ours beause of two residual effets: first, pumped diameter is typially smaller than the diameter of the rystal and seond, thermo-mehanial ontribution to the thermal lensing from bending of the disk due to thermal expansion [49]. Thermal lensing is important issue in laser design and operation. This fator an be alulated theoretially using thermo-mehanial modeling softwares. In thin disk lasers, the disk is mounted with a old plate on a heat sink (figure 10). At the same time the other side of disk is radiated by pumping laser, aordingly there is a temperature differene between two faes of the disk. This will ause a temperature distribution through the disk bulk. Generally the refrative index of materials is depended on temperature; aordingly the refrative index of disk will be a funtion of position. The other effet is the expansion of the disk due to the temperature distribution formed in it. Also mounting the disk on heat sink auses deformation and stress in the disk. The stress itself will affet the refrative

12 35 An Overview of Heat Transfer Phenomena index of the disk rystal. To omplete analysis of the effets of the disk on the laser and pump beam, one must alulate umulative effets of expansion and deformation of the disk, also thermo-optial and stress dependent variations of refrative index [50]. Total effet of the disk on laser beam phase an be alulated by [51]: () [ h 0 [ n r n0 ( T (,) r z T0 ) ns (,) r z 1].[1 z (,)] r z dzz0 ()] r T (13) In whih n 0 is refrative index of disk at referene temperature T 0, n T is the thermooptial oeffiient, ns is the hanges of refrative index due to stresses, z is the strain in diretion of thikness of the disk, z 0 is the displaement of bak side (High Refletion oated) of the disk and h is the thikness of the disk. As relation (1) shows, optial behavior of the disk is strongly depended to the temperature distribution of disk. The temperature distribution is result of optial pumping and ooling of the disk. Figure 10. Shemati setup of a thin disk laser; end pump onfiguration [5].4.1. Pump and ooling onfigurations There are two onventional methods for pumping disk lasers; first is (quasi) end pumping and the seond is edge pumping. Shemati diagram of end pumping is shown in figure 10. Also figure 11 shows a shemati setup of the edge pumped thin disk laser. In both mentioned methods, the disk is ooled from the fae. The disk an be ooled by jet impingement (figure 1) or ryogeni tehnique. The disk is mounted with a old plate on the heat sink. In jet impingement a jet of ooling liquid is sprayed to the old plate. Different liquids an be onsidered as oolant whih most ommon is water.

13 Heat Generation and Removal in Solid State Lasers 353 Figure 11. Shemati setup of edge pumped thin disk laser [53] Figure 1. Shemati diagram of jet impingement ooling system for thin disk laser [54] Laser ooling has been an important problem from the invention of the first pratial laser in After invention of laser, ryogeni ooling of solid state lasers has been interested and first time proposed by Bowness [55] in 1963 and then by MMahon [56] in In mentioned referenes the ondution ooling is used and laser element was plaed in ontat with a material with very high thermal ondutivity. That material was, in turn, in ontat with a ryogen suh as liquid nitrogen near 77 K, liquid Ne near 7 K, or He near 4 K. At ryogeni temperatures the absorption and emission ross setions inreases and the Yb:YAG absorption band near 941 nm narrows at 77 K, however it is still broad enough for

14 354 An Overview of Heat Transfer Phenomena pumping with pratial diode lasers. At 77 K, Yb:YAG rystal behaves as a four-level ative medium however in room temperature Yb:YAG is a quasi three-level material. Signifiant problems assoiated with quasi-three-level materials like Yb:YAG, suh as the need to provide a signifiant pump density to reah transpareny, a high pump threshold power, and the assoiated loss of effiieny, disappear at 77 K [57]. When the Yb:YAG is ooled from room to ryogeni temperatures, the lasing threshold dereases and slope effiieny inreases. Figure 13 shows drop of lasing threshold from 155 W to near 10 W and inreasing slope effiieny from 54% to a 63% for a typial thin disk laser [58]. The inset spetral peak in Figure 13 is the 80 K laser output and is entered near nm. At room temperatures this peak is near nm. Figure 13. Lasing power versus pump power at 15 C (88 K) and 80 K. [58].4.. Temperature distribution Speifiations of disk laser beam are tightly related to the ative medium geometry. The preise geometry of the ative medium geometry is also strongly depended to the thermomehanial and opto-mehanial properties of the disk and the temperature distribution in the disk. Temperature distribution of the disk an be obtained by solving the heat ondution equation using proper boundary onditions. The flow of heat generated by the pumping diode radiation through a laser gain media in general form is desribed by a non-homogeneous partial differential equation: 1 Tr (,, z) Qr (,, z) Tr (,,) z k ( T, dop) t k ( T, dop) (14) In whih, K is heat ondutivity that is assumed to be isotropi and is the heat soure density in the laser rystal. In CW regime of output laser, the steady-state temperature distribution obeys the heat diffusion equation Qr (,,) z Tr (,,) z k ( T, dop ) (15)

15 Heat Generation and Removal in Solid State Lasers 355 As it is seen the heat ondutivity is depended on the temperature and doping onentration (dop) of the rystal. Temperature dependeny of YAG rystal in room temperature is not signifiant and it an be onsidered as a onstant [37] however this approximation would not be valid anymore at ryogeni temperature. Heat ondutivity of Yb:YAG rystal whih is onventional ative medium of thin disk laser an be given by [59]: dop 04 k( T, dop) ( dop). W / m T 94 1 K 1 (16) Charaterizing the behavior of the thermally indued lensing effet of the thin disk gain medium is not a trivial task. In order to fully analyze the dynamis of the heat flow and thus the indued n T stresses and strains on the gain medium one must solve the 3-D heat equation with appropriate boundary onditions. This an be aomplished in several ways. The most ommon is to employ a finite element analysis (FEA) method. Another method is to solve the 3-D heat equation using the Hankel transform. For more details the reader an refer to [60]. Initial estimation of disk thermal behavior an be arried out by alulation of maximum and average temperature of the disk. In thin disk lasers, the thikness of the disk is very low. When the pump spot size is very larger than the disk thikness, one dimensional heat ondution is a good approximation. If pump power of Ppump radiates to the disk in a pump spot with radius of r p, the heat load per area an be given by [61]: I heat P (17) pump abs rp In whih abs is absorption effiieny and is heat generation oeffiient in the disk. The heat generation oeffiient in the disk is due to the quantum defet and related to the pumping and laser wavelength whih are shown by p and l respetively. p 1 (18) A paraboli temperature profile will be formed along the axis inside the disk due to the loaded heat whih is given by: l z 1 z 0 heat th, disk Tz ( ) T I R ( ) h h (19) in whih R th, disk h / k is the heat resistane of the disk material and T 0 is the temperature of the disk s ooled fae. Also z is the distane along the disk axis in the thikness of the disk and h is the thikness of the disk. In partiular, one an alulate maximum temperature from relation (19) whih is given by

16 356 An Overview of Heat Transfer Phenomena 1 Tmax T0 IheatRth, disk (0) also the average temperature in the disk thikness an be given by 1 Tav T0 IheatRth, disk (1) 3 In this way using relations (19) to (1) one an evaluate, temperature distribution and maximum and average temperature in the disk in one dimensional heat ondution approximation..5. Fiber laser and fiber amplifiers.5.1. Introdution to fiber geometry and ooling methods In reent years, design and manufaturing of high effiient fiber lasers whih deliver exellent beam quality, has made them as the main adversary of other types of high power solid state lasers, suh as bulk and thin-disk lasers. Ahieving to multi-kilowatt output powers [7, 6] with diffration limited laser beam ould be onsidered as the unique reord in laser tehnology. This progress an be attributed diretly to the apability of more effiient ooling proedure in fiber lasers, whih originates from inherent large surfae to volume ratio. In fat, thermal load spreads over meters or tens of meters of fibers, whih auses onvenient and effiient ooling management and therefore avoids thermo-opti problems. Fiber lasers are onsists of fiber ore whih is mostly surrounded by two oaxial fiber ladding (double lad fiber lasers) and is pumped by diode bars or diode laser at one or both ends. The laser light an only propagate through the fiber ore and doesn t have any role in heat generation inside the fiber. There are two general methods for lunhing pump light into the fiber laser whih are alled as "ore pumping" and "ladding pumping". The onventional ore pumping was initially used to ahieve single mode output laser, in whih the pump light was oupled into the small ore. On the other hand, small ore auses a serious restrition on pump power level [63]. Furthermore, the ore size leads to highly loalized pump intensity whih usually indues thermal damage at the fiber ends. Therefore, ladding pumping has been developed as the proper solution whih ensures lunhing high pump power into the double lad fiber lasers. In this method the pump light ouples into the inner ladding and propagates through it and gradually absorbs in doped ore. In both ases, the pump light only absorbs within the ore, where heat generation takes plae. Figure 14 shows a simple diagram of ladding pumping of fiber laser geometry [63]. In most ases, ooling proedure in fiber lasers does not need any speial ooling system and are alled passive ooling, whih easily an be performed by the air through the onvetional proess [6-64,65]. However, in modern fiber lasers an ative ooling system is

17 Heat Generation and Removal in Solid State Lasers 357 Figure 14. Cladding pumped fiber amplifier [63] onsidered to sale up high power lasers, whih ensures a fored heat dissipation proess. Liquid ooled fiber lasers [66] is an example of new ooling methods in whih, the whole or some part of the fiber is plaed inside a liquid with a definite temperature. Therefore, heat removal ours through the onvetion from the fiber periphery to the ooling liquid. This tehnique is usually applied to the long fiber lasers. Another tehnique whih is often utilized for ooling of short fiber lasers, onerns to the thermoeletri ooling system (TEC). In this ase, the fiber medium surrounds by a opper heat sink and therefore heat removal performs through the heat ondution proess. The three ommon methods whih imply to the passive and ative ooling tehniques are introdued in more details in follow. Example of ondutive boundary ondition in whih a short length fiber is surrounded by a temperature ontrolled opper heat sink an be found in [67]. Ignoring of axial heat flux as an approximation, heat differential equation an be solved numerially by means of Finite Element (FE) method. Figure 15, shows a drawing of a TEC-ooled fiber assembly. Figure 15. side view of a TEC-ooled short-length fiber laser [67]. Cooling of long fiber laser based on the ondutive heat transfer is reported in [67]. The fiber is plaed between aluminum plates with onstant temperature aused by water ooling. Pratial models of high power fiber lasers with the unfored onvetive heat transfer from fiber to air are reported in [6-64, 65]. Figure16, illustrates the experimental set up for an Er:

18 358 An Overview of Heat Transfer Phenomena ZBLAN double-lad fiber laser. Fiber is plaed inside the resonator and is pumped from one end by a diode laser after passing the pump beam from the designed optis. Figure 16. experimental set up for high power Er: ZBLAN fiber laser [64]. The fiber is pumped by a diode laser at one end. Figure 17 shows another example relating to high power Yb doped Fiber laser (YDFL), whih is pumped from both ends [6]. Convetional ooling proess from fiber to air is freely established. Pump power is delivered from two diode staks propagate from the both ends toward the fiber enter and ause two individual heat soures inside the fiber. Figure 17. experimental arrangement of double lad YDFL pumping with two diode staks [6]. Freely onvetive heat transfer to the surrounding air is onsidered. Effiient fiber ooling leads to sale up high power lasers without thermal damage and avoiding destrutive thermal effets on laser operation. A new tehnique for thermal management of fiber was examined in [66], whih alled diret liquid ooling. In this method the fiber was in diret ontat with the fluoroarbon liquid. Furthermore, the both ends of fiber faet are in physial ontat with the CaF windows. This leads to ondutive heat transfer from fiber faet to the window and onsiderable axial heat removal, whih allows inreasing pump power without thermal damage. This tehnique was already used in omposite bulky solid state laser mediums [4, 41-43] and had found highly operative [68, 69]. Figure 18, shows a drawing of system assembly. The fiber is pumped by two fiber oupled diode lasers from both ends.

19 Heat Generation and Removal in Solid State Lasers 359 Figure 18. Liquid ooling of long fiber laser. Ld1 and Ld fiber oupled diode lasers, L1 and L aspheri lenses, W, CaF windows, DM dihroi mirrors [66]..5.. Continuous Wave (CW) pumping onditions Using CW pump soures lead to generation of time independent heat soure density in fiber ore. Therefore, Eq.1 turns to steady state heat differential equation as 1 Trz (,) Trz (,) Qrz (,) r r r r z k () In whih, the azimuthal part of the temperature is omitted due to the ylindrial symmetry of pumping spatial distribution. As we mentioned before, spatial form of heat soure density obeys from the spatial form of pump intensity profile lunhed to the fiber. "Top hat" and "Gaussian" are two ommon shapes for the pump beam profile, whih are usually onsidered as the spatial form of heat soure density in thermal analysis. Different onsiderations lead to different differential equations and therefore, needs different solutions. Analytial and Numerial solutions of Eq.1, to speify the temperature behavior inside fiber medium, are reported in various literatures with different approximations and methods. As we mentioned before, different ooling arrangements lead to different boundary onditions whih are ondutive or onvetive heat transfer from fiber periphery to the surrounding medium. In the ase of fiber oupled fiber laser, pump intensity distribution with a good approximation has a top hat profile aross the beam. Entering the pump beam inside the fiber ore and propagating along the fiber length auses exponential deay in axial diretion. Therefore, the heat soure density Q(r,z), inside the fiber an be expressed by [70] 1 P - z 0e ; r a Qrz (,) a Leff 0 ; ar < b (3) Where, P o is the pump power, a is radius of the fiber ore, b is the radius of outer ladding, (1 e L ) Leff is the effetive fiber length, L is the geometrial fiber length and is the

20 360 An Overview of Heat Transfer Phenomena effetive pump absorption oeffiient. Substitution of Qrz (, ) from Eq. 3 into Eq., the heat differential equation for two regions an be written as r Pe ; 0 r a 1 Trz (, ) Trz (, ) 1 - z 0 r r r z a k L eff (4-a) 1 Trz (, ) Trz (, ) r r r r z 0 ; a r< b (4-b) Equation 4-a orresponds to the fiber ore, where exposes to the pump beam and therefore experienes the heat generation. Equation 4-b relates to fiber ladding that is just responsible for heat transmission to the surrounding medium. Solution of these differential equations give the steady state three dimensional temperature at any point in fiber ore T1(r,z) and ladding T(r,z).Figure 19, illustrates shematially double lad irular fiber geometry under pumping proess [70]. a a b b Figure 19. End (a) and side (b) view of irular fiber laser. Absorption of pump power inside the fiber ore auses heat generation whih redues exponentially along the fiber axis [70]. Assuming the outer surfae of fiber is in diret ontat with a liquid or gas suh as air, the onvetive boundary ondition ould be defined aording to Newton s law as T (,) r z k h T T b z r rb ( (, )) (5) Where, h is heat transfer oeffiient [71],T is the oolant temperature andt (b,z) represents the temperature along the fiber on its ylindrial surfae and the first derivative is taken with respet to the surfae normal. This equation expresses that the radial thermal flux whih arrives at the fiber periphery via the ondution method, removes by the oolant through the heat onvetion proess. Further information about the analytial solution of Equation and some tehnial points on numerial approah an be found in [70]. The resultant temperature expressions are

21 Heat Generation and Removal in Solid State Lasers 361 Pexp( z) 1 T(,) r z ( CJ ( r) ) T ; ra Ka(1exp( L)) (6-a) Pexp( z) T (,) r z ( A J ( r) AY( r)) T ; ra Ka(1exp( L)) (6-a) Where C, A1 and A are onstant oeffiients whih are derived from the boundary onditions as follows Y( a) 1 C A A (7-a) J ( a) J ( a) a A ( hy. ( b) K.. Y( b)) (7-b) h a A ( hj ( b) K J ( b)) (7-) 0 1 4h The temperature is redued exponentially along the fiber axis due to the exponential deay of heat deposition. Radial dependene in the ore inludes zero order Bessel funtion and inside the ladding, linear ombination of the first and seond kind of zero order Bessel funtions are established. This is illustrated in Figure 0 whih shows the alulated temperature distribution in the r-z plane of ZBLAN (ZrF4-BaF-LaF3-AlF3-NaF) double lad fiber. More information about the fiber geometry and pumping harateristis is given in Table 1. Figure 0. Temperature distribution in r-z plane for the double lad ZBLAN fiber laser [70].

22 36 An Overview of Heat Transfer Phenomena Quantity Magnitude Core radius, a 15 μm Clad Radius, b 370 μm Fiber length, L 4 m thermal ondutivity, K 0.68 W/(m.K) absorption oeffiient, α.3db/m (0.54m -1 ) Pump Power, P0 4.8 W heat transfer oeffiient, h 50 W m - K -1 ambient air temperature, T 300 K Table 1. amount of physial quantities are used in temperature alulations [70]. In the ase of short length fiber laser, analytial approah of temperature expressions for an Er 3+/ Yb 3+ o-doped fiber laser an be found in [6]. Shemati drawing of fiber pumping is illustrated in Figure 1. R 1 P ( z ) p P ( z ) p R Z 0 P ( z ) s P ( z ) s Z L Figure 1. Shemati figure of short length fiber laser with propagation of traveling pump and signal light into positive and negative axial diretion[6]. In short fiber lasers, the pumping energy doesn't absorb ompletely by one passing of light along the fiber length and thus, the refleted part from the oupler ats as a pump beam delivers inside the fiber at the end fae. Heat soure funtion is omposed of the proportion of the both the left and right traveling pump beams. The heat soure density funtion with the assumption of Gaussian pump beam shape is given by [6] ( )[ P ps p ( z) P p ( z)] [ P s s ( z) P s ( z)] Qrz (, ) exp r p p (8) where Pp ( z) and Ps ( z) are the pump power and signal power in the positive and negative diretions, and are sattering loss oeffiients at the pump wavelength and signal ps s wavelength respetively and is the Gaussian radius of the pump light. Figure p illustrates an end view drawing of the modeled fiber. As it an be seen, the ative fiber is surrounded by a erami ferrule whih is plaed inside a opper tube. Heat ondution from fiber ladding to erami ferrule ours through the fiber ylindrial surfae. The steady state heat differential equations are r r r z k 1 Trz (, ) Trz (, ) Qrz (, ) [ r ],0ra (9)

23 Heat Generation and Removal in Solid State Lasers 363 Copper tube Core Cerami ferrule Clad Figure. Geometry of modeled fiber laser 1 Trz (, ) Trz (, ) [ r ] 0, arb r r r z (30) and orresponding boundary onditions are Trz (,) k h[ T T( r, z)], r b r 1 1, T T T T, r a r r Trz (,) 0, z0, z L z (31) (3) (33) Where, T 1 and T are the temperatures in fiber ore and ladding regions. Eq. 30 relates to the heat removal into the ambient air whih is hold at the onstant temperature of T, Eq. 31 ensures the same temperature value at the fiber ladding and ore joint boundary (r = a) and Eq.35 expresses the negligible heat transfer from fiber ends into the air beause of the small amount of air heat transfer oeffiient.. Solution of equations (9)-(33) gives the temperature expressions in fiber ore and ladding are given by 0 m n T1( r, z) T C0lnaD0 Anmos zj0 r n1m0 L a m m CmI0 aos z m1 L L m m m T(,) r z T C0lnrD0 CmI0 rdmk0 ros z m1 L L L (34) (35) 0 n Where, is nth zero point of the zero order Bessel funtion of the first kind, J0 and J 1 are the zero and first order Bessel funtion of the first kind, I 0, I1 and K 0, K1 are the zero and

24 364 An Overview of Heat Transfer Phenomena first order modified Bessel funtion of the first and seond kind, respetively [6 ]. C m, D m and Anm are the onstants whih derived from boundary onditions, via solution of oupled equations and are given in [6]. The above equations are plotted in figure 3, illustrating the temperature distribution in fiber ore and ladding in a phosphate fiber. As would be expeted, the maximum temperature ahieved at the enter of the fiber faet and is K. Amount of some parameters used in alulations are summarized in Table. Figure 3. Temperature distribution in r-z plane of the short-length o-doped phosphate fiber laser [6]. Quantity Magnitude Core radius, a.7μm Clad Radius, b 6.5 μm Fiber length, L 1m thermal ondutivity, K 0.55 W/(m.K) Pump Power, P0 100 W heat transfer oeffiient, h 10 W m - K -1 ambient air temperature, T 300 K Table. Amount of physial quantities are used in temperature alulations [6] Pulsed pumping onditions An example of pulsed pumped fiber laser whih auses transient temperature field is presented in [7]. This work relates to short length fiber laser and introdues analytial temperature expressions for solution of transient heat differential equations whih are Tzrt (,, ) 1 Tzrt (,, ) Tzrt (,, ) Qzrt (,, ) Tzrt (,, ) ],0ra r r r z k k t Tzrt (,, ) 1 Tzrt (,, ) Tzrt (,, ) Tzrt (,, ) ], arb r r r z k t (36) (37)

25 Heat Generation and Removal in Solid State Lasers 365 In fat, Eq. 36 and Eq. 37 are is written for the fiber ore and ladding regions. In this ase, the heat soure density funtion is defined as Qzrt (,, ) exp Pin r p p zgt (38) Where, Pin is the pulse energy and g(t) is temporal shape of pump pulse. The other parameters were previously introdued. The equations (38)-(40) are appliable here as the boundary onditions. Furthermore, the laser system is assumed to be in thermal equilibrium with the ambient air before pumping proess whih expresses by Tzrt (,, ) T, t 0 (39) Solution of equations (36) and (37) with the aid of boundary onditions through the integral transform method introdued by Özisik [73] gives the temperature expressions as TT p1 0 k k k J ( r)exp( t) g ( )exp( ) d ln t 0 p p 0 0p p p k k k J ( r)os( z)exp[ ( ) t] g ( )exp[ ( ) ] d t 0 p m p m 0 mp p m m1p1lnp (40) Where Qzrt (,, ) g ( ) J ( r) rdrdz (41) l r1 0p k p l r Qzrt (,, ) m 1 gmp( ) 0 0 os( z) J0( pr) rdrdz (4) k L t k 0 g0p( )exp( p) d Pin[1exp( l)] r1 t k 0 exp( r / p) J0( pr) rdr0g( )exp( p) d k p (43) k ( )exp[ ( ) ] d t 0 gmp p m lpin[1exp(1 l)os( m )] r1 0 exp( r / p) J0( pr) rdr k ( l m ) p t k 0 g( )exp[ ( p m) ] d (44)

26 366 An Overview of Heat Transfer Phenomena m m / l (45) p ( p ) 0( p )/ p N r k h J r k (46) hj ( r ) k J ( r ) 0 (47) 0 p p 1 p Where J 0 and J 1 are the zero and first rank Bessel funtion of the first kind, respetively. Assuming square shape for the temporal dependene of pump pulses as bellow, temperature distribution was alulated. 1 ( n 1) T t( n1) Tt gt () 0 ( n1) T0 t0 tnt0 0 0 (48) In whih, n denotes number of pulses, t 0 is the pulse duration and T 0 is pulse period. Figure 4, shows the temperature distribution ahieved from the analytial temperature expressions for a typial Er 3+ /Yb 3+ o-doped fiber laser. Charateristis of the fiber laser whih was introdued in Table were onsidered in alulations. The fiber is in enounter of pulses with the pulse pump of P 1W. in Figure 4. Temperature distribution in r-z plane of a pulsed pumped fiber laser at the time of 0.1s [7]. Entering of eah pulse into the fiber ore auses definite heat load and thus leads to inrease of temperature. Figure 5-a, shows the temperature evolution at fiber end faet during the spae time of pulse generation. The urves orrespond to the time inrement of 1ms. If the pulse period is longer than the spae time of heat removal, the temperature will return to the initial value before loading the next pulse. But if the pulse period is shorter than the time whih is neessary for ompletely remove of generated heat, the heat will gradually deposit inside the fiber medium and leads to inrease of the fiber temperature. Figure5-b illustrates redution of fiber faet temperature after the first pulse. At the time of 15ms

27 Heat Generation and Removal in Solid State Lasers 367 temperature profile has nearly flat shape but the temperature in all positions is C over the initial temperature. Figure 5. a) radial temperature distribution inside fiber laser at the pump faet during the first pulse pump time, b) redution of temperature generated by the first pulse during 5ms [73]. Figure 6, illustrates the temperature evolution of the fiber faet at the enter versus time. Aording to that, temperature rises gradually with inreasing the time until reahing to almost onstant temperature with the small and regular flutuations. Eah flutuation orresponds to heat generation and then heat removal to outer spae during the spae time of pulse formation and pulse period time. Figure 6. a) time evaluation of temperature at the end surfae of fiber, b) enlarged drawings of the part of graph a whih shows regular flutuations in temperature profile [73]..6. Mirohip lasers In mirohip lasers the oated surfae of ative medium is used as laser's mirror. Popular types of mirohip lasers have less than 1 Watt output power. However, reently some reports are demonstrated mirohip lasers with several hundred Watts [74-78]. Due to the importane of thermal problems in high power operation, we fous our investigation on this

28 368 An Overview of Heat Transfer Phenomena regime. Generally, omposite ative media are used in high power mirohip lasers. The laser material inludes a entral doped region as the ative medium surrounded by an undoped rim as pumping waveguide. The mirohip laser has a thin ative medium with geometry very similar to that of thin disk lasers. Considering the uniform absorbed power distribution, the thermal onsiderations assumed in thin disk lasers are still valid in mirohip lasers. Most ommon mirohip lasers in high power operations are omposite mirohips with an un-doped rim surrounding ative region ating as pump wave guide. The main drawbak in this geometry is the non-uniformity of absorbed power distribution whih leads to non-uniform temperature distribution. Regarding boundary onditions and non-uniform heat deposition, the temperature distribution in the ative medium should be investigated. An example of thermal analysis in high power mirohip laser is presented in [79]. Utilizing optimum geometry and dimensions for ative medium, one an onsider uniform absorbed power distribution in mirohip lasers whih leads to relatively uniform temperature distribution and an overome the ommon disadvantage of mirohip lasers. A ommon geometry of the Yb:YAG/YAG omposite mirohip lasers is illustrated in figure 7 [79]. Figure 7. Shemati Diagram of a omposite Yb:YAG/YAG side pumped mirohip laser elements [79] In this arrangement, a thin disk Yb:YAG ore enlosed by an irregular symmetri eight sided un-doped YAG rim is bonded on a water-ooled Cu-W old-plate. Fration of absorbed pump power in the gain medium produes heat in the ore and raises the ore temperature. The temperature distribution in the laser material is alulated using the, whih is.( k T) Q 0 (49) Thermal ondutivity is strongly dependent on temperature and atomi doping onentration of Yb ions CYb as follows [80]

29 Heat Generation and Removal in Solid State Lasers K ktc (, Yb) CYb T 96 K C Yb (50) The absorbed power distribution in ative medium is alulated using Mont-arlo ray traing of pumping photons through the optial elements of the system. Heat generation Q is estimated based on the energy differene between the pump and laser photons, given by [81 ] p Q(1 ) Pp (51) In whih Pp is loal absorbed power density. The heat transfer between the gain medium and air is week, whih is desribed by l dt 0 (5) dz z t In whih z axis oinides on mirohip optial axis and t denotes the mirohip thikness. Moreover, the temperature of the bak fae of the old-plate and ontat ooling water were onsidered as the same. The numerial values used in alulations are given in Table 3. Quantity Pumping wavelength Laser wavelength ative medium thikness old plate thikness thermal ondutivity [9-8] old plate old plate heat transfer oeffiient magnitude 94nm 1030nm 0.3mm 1mm 1.9 W/m.K 1 W/m.K Table 3. The mirohip laser parameters used in numerial alulations. Heat differential equation (49), was alulated in numerial Finite- Differene method for the desribed gain medium [79]. This ode is used to determine the temperature distribution in the ative medium. Figure 8 shows the maximum temperature variation versus a/w ratio for various ore diameters with 9 at.% Yb3+ doping onentration. Moreover, the temperature distribution profiles of these ores are also shown in Figure 9. Aording to figure 9, the laser gain mediums with the smaller ore size experiene higher temperatures at the same pump power, whih is attributed to the higher thermal load density.

30 370 An Overview of Heat Transfer Phenomena Figure 8. Maximum temperature of different mirohip ore sizes [79]. Figure 9. Temperature distributions of diffrent ores[79].

31 Heat Generation and Removal in Solid State Lasers 371 Summary In this work, at first the basi priniples of heat generation and removal in all kind of solid state lasers whih are Bulk, Disk, Fiber and Mirohip onfigurations are introdued. Then we tried to present a omplete olletion of solution for heat transfer equation in these types of gain medium regarding to the modern ooling systems, whih are available in the literatures. Therefore, this work gives the reader a straight forward way to reah the latest progress in thermal problem in solid state lasers appliable for high power laser design. Author details V. Ashoori and M. Shayganmanesh Department of physis, Iran University of siene & Tehnology, Narmak, Tehran, Iran S. Radmard Iranian National Center for Laser Siene and Tehnology (INLC), Tehran, Iran 3. Referenes [1] Guangyuan H, Jing G, Biao W, and Zhongxing J (011) Generation of radially polarized beams based on thermal analysis of a working avity. Opt. Express. 19: [] Wan-Jun H, Bao-Quan Y, You-Lun J, and Yue-Zhu W (006) Diode-pumped effiient Tm,Ho:GdVO4 laser with near-diffration limited beam quality. Opt. Express. 14: [3] Zhu X, Peyghambarian N (010) High-Power ZBLAN Glass Fiber Lasers. Review and Prospet Advanes in OptoEletronis.1-3. [4] Rihardson D, Nilsson J, and Clarkson W (010) High power fiber lasers: urrent status and future perspetives. J. Opt. So. Am. B.7: B63-B9. [5] Chaitanya Kumar S,Samanta G and Ebrahim-Zadeh M. (009) High-power, singlefrequeny, ontinuous-wave seond-harmoni-generation of ytterbium fiber laser in PPKTP and MgO:sPPLT. Opt. Express. 17: [6] Brilliant N. and Lagonik K (001) Thermal effets in a dual-lad ytterbium fiber laser. Optis Letters. 6: [7] Jeong Y, Boyland A, Sahu J, Chung S, Nilsson J, and Payne D (009) Multi-kilowatt Single-mode Ytterbium-doped Large-ore Fiber Laser. J. Opt.So. Korea. 13: [8] Ahmed M, Haefner M, Vogel M, Pruss M, Voss A, Osten W and Graf T (011) Highpower radially polarized Yb:YAG thin-disk laser with high effiieny. Opt. Express. 19: [9] Beil K, Fredrih-Thornton S, Tellkamp F, Peters R, Kränkel C, Petermann K and Huber G (010)Thermal and laser properties of Yb:LuAG for kw thin disk lasers. Opt. Express. 18:

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