moj seminar P ... - F9

Postgraduate seminar at FMF:PHYSICS OF, AND REQUIREMENTS moj seminarFOR LASER CRYSTALSPut together by:Supervisor:Blaž Kmetec ∗prof. dr. Martin Čopič28th December 2004ABSTRACTThe aim of laser manufacturers is to make lasers with operational characteristics that are fine-tuned forspecific applications. Detailed investigations of various laser crystals have been extremely supportive in makingsolid-state lasers as useful as they are. Lasers employing laser crystals cover a diversity of requirementcombinations these days. An overview of selected requirements is given in this work together with few ideas ofphysical science complexity of the interactions that are relevant for understanding the processes in laser crystalsin order to be able to use them according to these requirements. The discussion on the important problem ofthermal lensing in solid-state lasers is included, the emphasis laid on laser diode-pumping of solid state lasers.Examples of laser crystals are limited to neodymium-doped crystals for compact industrial lasers.KEYWORDS: laser crystals, solid-state lasers, laser material requirements, thermal lensing, Nd:doped lasermaterialsContents0 Foreword 21 Introduction 32 Interactions 43 Material requirements 53.1 Material preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 Representative calculation: nonradiative energy transfer as a result of ion-ion electric dipole interaction 85 Thermal effects in a crystal during laser operation 106 Examples of laser crystals 126.1 Nd,Cr:GSGG opposed to Nd:YAG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126.2 Nd:YAG and Nd:YVO 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Summary 14Literature and index 14∗ blaz.kmetec@lpkf.si1

2 Foreword0 ForewordLaser... inter eximia naturae dona numeratum plurimis compositionibus inseriturThe Laser... is numbered among the most miraculous gifts of natureand lends itself to a variety of applications.Plinius, Naturalis historia, XXII, 49 (first century A.D.)During the Greco-Roman civilization the Laser was well known and much celebrated. Unlike its presentdayhomonym, it was actually a plant (or, more precisely, the plant’s juice), but with no fewer wonderfulproperties. This plant, sometimes also called Laserpitium or Silphion, grew wild over a large area aroundKyrenaika, in present-day Lybia. It was used to cure a variety of diseases and was an effective antidote againstthe poison of snakes, scorpions, and enemy arrows. Its delicate flavour led to its use as an exquisite dressing inthe best cuisine. It was so valuable as to be the main source of Kyrenaikan prosperity and it was exported to bothGreeks and Romans; during the period of Roman domination, it was the only tribute paid by the Kyrenaikans toRomans. Both Greeks and Romans tried hard, but without success, to grow the Laser in various parts of Apuliaand Ionia. The Laser became more and more rare and seems to have disappeared around the second centuryA.D.The equally beneficial properties of crystals as believed in those times were well known too, as was theconnection of crystals with light; Asteria, for example, “having a light enclosed within, ... which has all theappearance of moving within the stone; it transmits according to the angle of inclination at which it is held ...When held facing the sun, it emits white rays like those of a star.” (Plinius, same source as above).Decades of laser research since 1960 have resulted in ability to obtain laser action from countless materialswith a wide variety of operational characteristics, such as power scaling, beam quality, and pulse length. Nowthat we have learnt how to make lasers and seen what they can do for us, the aim is to make lasers with operationalcharacteristics that are fine-tuned for specific applications. Furthermore, manufacturing is becoming moreand more flexible: demands for flexible manufacturing are expressed in tendencies towards shorter product lifecycles or towards small batch production. It is impossible to predict to what extent the future production willdepend on lasers. At least we can speculate though.What we can speculate on is which demands or requirements for laser systems are already expressed andhow justified they are. Some of the demands most frequently alluded to are [6]:1. The demand for lower costs. From the point of view of the machine manufacturer but even more fromthe point of view of the machine user, improved reliability, long-term durability of wear parts and reducedoperating costs are much more important than reduced laser purchase costs. This demand applies to alllaser systems. In addition, bearing in mind that high-power semiconductor laser diodes are mostly used forsolid-state laser pumping, low solid-state laser cost is strongly associated with low costs of laser diodes forsolid-state laser pumping.2. The demand for improved beam quality. The ideal laser beam has the Gaussian transverse intensityprofile. The demand for further improvement of the beam quality is often closely related to the demand forhigher power - such is the case with laser ablation. Then, it is basically a matter of maximizing the quotient:(beam quality) × (power output) / price. In this area there still is plenty room for improvement of solid-statelasers.3. The demand for shorter wavelengths. In the semiconductor chip industry, the demand seems to be clearlyjustified: namely, as the need for UV laser sources.4. The demand for shorter pulses In commercial manufacturing, there is a strong interest in the ideal pulseduration for specific application; this duration can also lie in the range of micro- or millisecond, bearingin mind cutting applications for example. For many applications, however, shorter pulse lengths (tens ofnanoseconds, as an example, for precision laser microprocessing) without implementing costlier pulsingtechniques - such as mode-locking instead of Q-switching 1 - is looked-for.1 Quality-switching, or shortly Q-switching, is a low-demand method for producing pulsed output with temporarily degrading the

Introduction 3Solid-state lasers have shown the capacity to satisfy several combinations of these requirements such as⋄ high power output at relatively low power consumption, particularly with diode-pumped systems; moreover,high pulse peak power is available together with high beam quality (profile)⋄ high stability, even when frequency doubled or tripled⋄ long life expectancy: twice as high as gas lasers⋄ and other advantages, for instance laser system compactness. Detailed investigations of various laser crystals have been extremely supportive - indeed, they have oftenplayed a primary role - in making solid-state lasers as useful and advanced as they are. The request for lasercrystal research and production is thus to a large extent linked with the demands for improved laser systems.1 IntroductionFollowing the general convention currently in use, the term solid-state lasers will be used to refer to systemsbased on optically active centres (ions) in insulator host materials, distinct from the semiconductor lasers.Insulator host material can be either an insulating crystal or a glass. Within this article, only laser crystals willbe treated. Usually the impurity ion, introduced into an ionic crystal, belongs to one of the series of transitionelements in the periodic table. These are transition metal ions, notably Cr 3+ , or rare earth ions, notably Nd 3+or Er 3+ .The critical components of a solid-state laser includea) the laser material, that is, the host material and active ionsb) the mechanism of optical pumpingc) the cavity configuration. The operational characteristics of a specific laser system are determined by the properties of those individualcomponents and how each of them is integrated into the arrangement. During the past decade, there have beensignificant technological advances for all three components: new materials with wavelength tunability havebeen developed; diode array pump sources have become available; slab and guided wave configurations havebeen demonstrated; and new methods of mode locking have been discovered.A solid-state laser material, or a laser crystal, is a physical system consisting of an ensemble of opticallyactive ions dispersed in a host crystal. While operating as a laser medium, this material absorbs and emitsoptical radiation in a controlled thermal environment, and thus its optical spectroscopic properties are vital toits performance. To fully understand the properties of this system, it is necessary to have extensive knowledgeof the physical properties of the free ions, the host material, and the interaction mechanisms affecting thesystem. This problem is approached by determining the electronic states of the optically active ions and thenconsidering these ions to be immersed in the local electrostatic field of the host, a phonon field, and a photonfield, as well as allowing interactions among active ions. Such a solid-laser system is shown schematically inFig. 1. This total system performs three functions: energy pumping, energy storage, and energy extraction.To perform these functions, the laser material absorbs the excitation energy and reemits it as light at the laserwavelength. This light circulates in the optical cavity and experiences gain each time it passes through thepumped laser material, and part of this light passes through the partially reflecting output coupler (mirror) ofthe cavity to produce the laser emission. Generally, the gain bandwidth of the laser material is broad enough toencompass many different resonating modes of the cavity as depicted schematically in Fig. 2.resonator during pumping so that no oscillation takes place; hence the gain - that is, inversion - can build up to a very high value; whenit reaches its peak, the oscillation is released and the giant pulse is emitted. Mode locking, in contrast, is basically forcing the laserfrequency modes to maintain their relative values; that causes the oscillation frequency to consist of a periodic train.

4 InteractionsFigure 1: Physical system for a solid-state laser [2]. The system consistsof three components: the laser material, which is placed in an optical resonatorand pumped with external source of energy. For a typical novelsolid-state laser, the laser material is a crystal; high reflector is an almostcompletely reflective mirror and output coupler is a partially reflective mirrorpartially reflective); and the pump source is a semiconductor laser diode.Figure 2: Material (crystal) gain curve andcavity resonator modes. Taken from [2].2 InteractionsThe relevant physical processes for this laser system include⋄ static electron-lattice interactions that determine the types and position of the electronic energy levels,⋄ electron-photon interactions that determine the strengths of radiative absorption and emission transitions aswell as as the fluorescence lifetime,⋄ electron-phonon interactions that determine the rates of nonradiative transitions and the temperature-dependentwidths and shifts of spectral lines,⋄ and ion-ion interactions that can cause energy-level splittings and energy transfer between ions.These processes determine the laser operational parameters such as efficiency, threshold, gain, loss, emissionwavelength, maximum power output, and the properties of quality switching and mode locking.There are several contributions to the photon field in the laser cavity. These include⋄ photons injected into the cavity by the pump source⋄ photons generated by the optically active ions through spontaneous emission processes, and⋄ photons generated by the optically active ions through stimulated emission processes.The latter oscillate between the high reflector and the output coupler of the cavity with the partial transmissionof these photons becoming the laser output beam.The optical spectral properties of the laser crystal are determined by the electronic transitions of the activeions in the local field environment of the host. The types of ions that are useful for laser emission in thenear ultraviolet, visible, and near infrared spectral regions are transition-metal ions (Cr 3+ ,Ti 3+ ) and rare-earthions (Nd 3+ ). Both of these types of ions have electron configurations that include unfilled shells and thus haveelectron transitions between energy levels within a specific shell: d-d transitions for transition-metal ions andf-f transitions for rare-earth ions.The key to having efficient absorption of pump radiation is having a strong absorption transition at thewavelength of the pump radiation. If the pump source is a broad-band spectral emitter, then the absorptionband of the ion should be broad in order to absorb the maximum number of pump photons. If narrow-linepump sources are used (such as laser), then the absorption band of the ion can be narrow but must be exactlymatched in frequency with the pump emission.Generally the terminal state of the absorption is not the level from which laser emission occurs. Thusanother important aspect of pump efficiency is that the transition absorbing the pump energy must result in

Material requirements 5Figure 3: Simplified energy-level diagram and transitions for three-level laser systems [1]. For high laser efficiency oflaser operation the difference between absorption energy and emission energy E 3 − E 2 should be small. The lasing statewith energy E 2 is called a metastable state. It is important that no pump absorption transitions should take place from themetastable state or from any excited level (here the upper one with energy E 3 ) where relaxation to the metastable statenormally occurs. This radiationless relaxation is very fast: τ 32 ≪ τ 21 .populating the metastable state of the laser transition. This requires efficient radiationless relaxation to thedesired level without loss of excitation energy to other emission transitions; see Fig. 3 for a better idea.The key to having efficient emission of radiation for laser applications is having a strong laser transitionat the wavelength of the desired laser output. The absorption of this fluorescence inversion, on the other hand,should be as small as possible. Also, the probability for radiationless decay processes should be low, in otherwords, the quantum efficiency for radiative emission should be high.3 Material requirementsIn order for a material to be useful for solid-state laser application, it must possess a good many (mechanical,thermal, and optical) properties. These are determined by a combination of the inherent properties of the hostmaterial, the properties of the optically active ions, and the mutual interaction between the host and the dopantions. The most fundamental requirement for a laser material is that it can be easily and economically producedwith high quality in large amounts and different sizes. It should also have a high enough hardness to allow fora good optical polishing.In order for the host material to be useful in applications outside the laboratory, it should also be stablewith respect to local environmental changes such as temperature, humidity, and stress. Since the operation ofthe laser requires exposing the material to both light and heat, chemical instabilities (in terms of ion diffusion,the formation of second phases, and so forth) in the material can be induced, either thermally or photo-induced.In addition, internal stress created thermally or optically in the crystal can distort its shape or in extremecircumstances cause it to fracture. Both of these properties depend on parameters such as the thermal expansioncoefficient α T and the thermal conductivity λ T . The details of these thermal effects depend on the pumpinggeometry and will be considered in section 5. In short: the host crystal must have suitable (not too low) thermalconductivity, high hardness and high fracture strength that will permit high-average-power operation.The ions useful for providing the optical dynamics of laser materials must be able to absorb pump radiationefficiently and to emit radiation efficiently at the desired laser wavelength. Some types of ions have excellentabsorption properties but poor emission properties, or vice versa. In this case it is possible to put two types ofions in the same host material, one to absorb the pump energy (called sensitizer ions) and the other to providethe laser emission (called activator ions). The key to making this scheme work is having efficient nonradiativeenergy transfer from the sensitisers to the activators (see section 4). This is achieved through strong overlap ofthe emission spectrum of the sensitizers and the absorption spectrum of the activators. If, however, the couplinginteraction between the two types of ions is too strong, these ions no longer have the properties of independentions, but instead form a coupled ion pair with its own spectral properties.

6 Material requirementsTable 1: Criteria for laser crystals.TOTAL SYSTEMEconomic production and fabrication in large sizeIon-host compatibility:Valence and size of substitutional ion similar to host ionUniform distribution of optical centres in the hostHOST MATERIALStable with respect to operational environmentChemical stability against thermal, photo, and mechanical changesMechanical:High stress-fracture limitHigh threshold for optical damageHardness for good polishingOptical:Minimum scattering centresMinimum parasitic absorption at lasing and pump wavelengthsEfficient absorptionEfficient radiative emissionLow absorptionOPTICALLY ACTIVE CENTRESof pump radiationat the laser wavelength with high quantum efficiencyat the lasing wavelengthThe host crystal must have lattice sites that can accept the dopant ions and that have local fields of symmetryand strength needed to induce the desired spectroscopic properties. One example of compatibility, the Nd:YAGcrystal, will be given in section 6.1.There is no single laser crystal that perfectly meets all of the criteria listed here; in fact, some of thesecriteria are mutually contradictory. Thus in designing solid-laser systems, it is important to have a wide varietyof materials available for use and to understand the optical properties of these materials thoroughly.3.1 Material preparationAlthough laser operation has been demonstrated in a wide variety of materials, only a few types of solid-statelasers have been developed for commercial applications. In many cases the development of a specific type oflaser had been limited through the lack of availability of high-optical-quality material. This can be a result of anumber of reasons including the expense of exotic materials and the difficulty in producing large-size syntheticmaterials with the appropriate properties.The standard techniques for growing laser host crystals are pulling from the melt (Czochralski) and meltgrowth (Bridgman-Stockbarger). The former is generally used for oxide materials that must be grown at hightemperatures, while the latter has been most useful for fluoride crystals. In the Czochralski growth technique,a seed is dipped into the melted material contained in a crucible and slowly withdrawn, resulting in crystalsof approximately cylindrical shape. Both radial and vertical temperature gradients are critically important indetermining crystal size and quality.One of the major problems in crystal growth is not having accurate information about the complete phasediagram of the material [2]. Many laser materials have constituents that are not congruently melting, whichsignificantly complicates the growth procedure. Even if the conditions for ideal crystal growth are known, accuratecontrol of these conditions may be difficult. Any variations in growth conditions can result in pieces withbubbles, multiple phases, and other defects that scatter or distort optical beams passing through the material. Ingeneral, it is easier to obtain laser-quality hosts from materials with the fewest number of chemical constituents.

Material preparation 7Figure 4: Crystals in lasers are used not only for converting the pumpenergy into stimulated radiation, they are also utilized for nonlineareffects - such as intracavity frequency doubling or tripling, or forquality-switching the resonator to generate pulsed output. Thesesamples, shown above, are Cr 4+ :YAG passive Q-switch elements forproviding high power laser pulses. They are distinguished for theirhigh damage threshold, and they are more robust and stable than dyes.Figure 5: Nonetheless, the term “laser crystal”and the topic of this work will be reservedfor crystals that enable stimulated coherentlaser emission. Among the most popularlaser crystals is high gain, high efficiencycrystal Nd:YAG, shown here. It willbe treated in more detail in section 6.1.Providing for optically active centres - dopant ions that substitutionally replace one of the constituents ofthe host material - is also challenging. Ideally, those should be uniformly distributed throughout the host crystalwithout causing significant distortions of the properties of the host. Then again, dopant ions almost never havethe exact same size, valance, and chemical bonding properties of the host ions they replace. This may resultin nonuniform distributions of dopant ions; these can form local material phases, and dislocations or grainboundaries. The final result are significant spatial variations in lasing properties throughout the crystal. Accuratelyknowing the dopant concentration and spatial distribution is one of the major challenges in characterisingsolid-state materials.After high-quality materials have been obtained, they must be fabricated into rods, slabs, or other configurationsconsistent with laser cavity design requirements. This involves cutting, polishing, and possibly opticalcoating. Obtaining an optical quality surface is important for minimizing losses both in coupling pump lightinto the crystal and in reflection losses of the circulating laser radiation in the cavity.Figure 6: Czochralski method: scheme and temperature distribution of the crystallizing structure, pulled from the melt

8 Representative calculation: nonradiative energy transfer as a result of ion-ion electric dipole interaction4 Representative calculation: nonradiative energy transfer as a result of ionionelectric dipole interactionSince the gain of a laser depends directly on the concentration of lasing centres, it would appear to be beneficialto increase this concentration to as high level as possible. Yet, optimal doping levels of different types of lasercrystals vary between a few hundreds of a percent and several tens of percents. One limitation is posed bythermal lensing - see section 5 - , because the thermal loading in a crystal increases with doping concentration.The other limitation arises from the ion-ion interactions. The interaction strength is a function of the separationof the two ions and the physical mechanisms of interaction [2]. Usually the doping level is not so high toallow the ions to couple strongly, rather the ions maintain their independent optical properties and interact bythe transfer of energy from one to another through nonradiative processes such as multipolar interactions; orthey may be coupled indirectly through the radiative emission from one ion to the other. In this paper only thefirst case, the weak ion-ion radiationless coupling, more precisely dipole-dipole resonant interaction leading toenergy transfer between ions with individual energy levels involved remaining the same as those for isolatedions, will be dealt with.Whatever the type of ion-ion interaction mechanisms, it can have very different effect on the propertiesof the laser performance. It can either enhance or degrade a specific laser performance characteristic. In thissection the general form of weak ion-ion radiationless coupling will be presented, and in section 6.1 one of theeffects - sensitized pumping - will be described.We employ the sensitizer-activator scheme from page 5. The photon energy absorbed by sensitizer moves[7], through the dipole-dipole interaction possibly aided by surrounding lattice relaxation, to the activator inone step without radiation exchange between sensitizer and activator ion. As with any quantum-mechanicalprocess, the rate for energy transfer to occur is described by the golden rule, given byW sa = 2π |M sa| 2 ρ f (E f = E i ) ; (4.1a)M sa = 〈 〈 ∣∣ψ ∣∣H ion-ion〉 ∑ ψf ∣∣H ion-ion〉 〈 ∣∣intψ j ψj ∣∣H ion-ion〉∣intψ if ∣intψ i ++ . . . (4.1b)Eji − E j}{{} }{{}resonant interactionphonon-assisted energy transferIn the second line, the matrix element M sa for the transition was expanded according to time-dependentperturbation theory: the first term describes a resonant (direct) interaction while the higher-order terms describephonon-assisted energy transfer. The former is discussed below, while the theory including the latter term can befound in literature [7, 2]. Hintion-ion is the ion-ion interaction Hamiltonian and ψ i,f,j represents the antisymmetrizedproduct functions of the optically active electrons on the sensitizer and activator ions.Under these circumstances, the main interaction is electromagnetic Coulomb interaction between the electronsof the sensitizer and of the activator ion. This interaction can be expressed as a multipole expansion usinga Taylor’s series about the sensitizer-activator separation ⇀ R sa:H EMint= −e 2∣4π εε ∣∣∣⇀0 r ∣∣ =sae 2= − ∣ ∣∣∣∣⇀ ∣∣∣ =4π εε 0 R sa + ⇀ r a − ⇀ r s= −e 24π εε 0 R 3 sa(⇀r s · ⇀r a − 3R 2 sa(⇀r s · ⇀R sa) (⇀r a · ⇀R sa))+ . . . (4.2)Only the first term in expansion, representing electric dipole-dipole interaction is significant; higher termsinclude quadrupole and higher order interactions. The dielectric constant of the host crystal is given by ε. Thevectors ⇀ r a and ⇀ r s designate the positions of the electrons on the sensitizer and activator as shown in Fig. 7.

Representative calculation: nonradiative energy transfer as a result of ion-ion electric dipole interaction 9Figure 7: Spatial geometry for the interaction between sensitizer and activator ions.Using the first term in Eq. (4.1), the square of electric dipole-dipole interaction matrix element becomes|M sa | 2 = 〈 ∣ψ ∣∣H EM: D-D〉 2f ∣intψ i =(e 2 ) 2 ∣ ∣∣∣∣∣ 〈 ⇀〉 〈 ⇀〉 3(〈 ⇀〉 ) ⇀(〈 ⇀〉 ) ∣ 2 ⇀ ∣∣∣∣=4π εε 0 Rsa3 r s · r a −R 2 r s · R sa r a · R sasa}{{}〈 〉spatialaverage= 2 〈 ⇀〉∣ ∣∣∣ 2 ∣ 〈∣ r s ∣∣ ⇀〉∣ ∣∣∣ 2r a3(4.3)(4.4)after denoting the expectation value of the electron position vector by 〈 ⇀r 〉 and, in the second step, assumingthat there is no preferred orientation of the dipoles, spatially averaging over all dipole orientation angles. Theresulting energy transfer rate depends on the ion-to-ion separation asEM: D-DWsa∝ Rsa −6 .An analytic evaluation of all the matrix elements as a function of energy is far from trivial, taking intoaccount that the wave functions are not known in advance. An approach of Dexter [7] was to associate thematrix element directly with experimentally measurable quantities such as decay times, Einstein coefficients,and absorption cross sections 2 . There is a lot of work included in coming to a complete compact expressionfor the energy transfer rate [7, 2, 5](EM: D-DWsa =364 π 5 ) ( 1R 6 sa) ( 1τ sps∫ ∞0( c0) 4gs (ν) σ a (ν) dν)n ν= 1 ( ) 6 R0τ sp . (4.5)s R saHere τ sps and g s (ν) are, respectively, the radiative decay time and the line-shape function of the sensitizer,σ a (ν) is the absorption cross section of the activator, and n is the refractive index of the surrounding medium(of the host crystal). The rate clearly depends upon the frequency overlap between the emission spectrum ofthe sensitizer and the absorption spectrum of the activator (ion-ion compatibility). The quantity R 0 is called theFörster radius; for good overlap, and for electric-dipole allowed transitions, R 0 may typically range between 2and 4 nm [5], thus implying the long-range characteristic of the interaction. An illustration of the interactionwill be given in section 6.1.

10 Thermal effects in a crystal during laser operation5 Thermal effects in a crystal during laser operationDiode-pumping of solid-state lasers has greatly reduced the proportion of wasted pump energy which is depositedas heat in the crystal. With, however, diode laser prices continuously decreasing and consequentlyhigher diode power becoming available, thermal distortion and even fracture, have again become a critical issuein designing diode-pumped lasers [9, 10]. This is especially true for end-pumped crystals where the pumpenergy is deposited into the central region of the crystalline rod. On the one hand, this longitudinal pumpingstrongly favours the excitation of the lowest optical transverse mode and is thus preferable. On the other hand,the heating distribution is highly nonuniform, in particular when the diode pump beam is tightly focused intothe crystal.Figure 8: In contrast to flashlamps, the emission from laser diodes (L.D.) is strictly directional. In the end-pumpedtechnique (A,B), which is unique for to laser diodes, pump radiation is introduced longitudinally into the appropriatelyoptically coated crystal (in (B) scheme the pump light is coupled to the resonator through optical fibre). End pumpingis the more efficient method for generating diffraction limited performance, as the pump radiation is spatially overlappedwith the TEM 00 lasing mode [1]. In side pumping (C), the large interaction surface is available; consequently heat removalis easier which reduces thermally related problems. After [11]We consider the case where the heat generated in the laser rod by pump-light absorption is removed bya coolant flowing along the cylindrical rod-surface. With the assumption of uniform internal heat generationwithin the pumped region and uniform cooling along the cylindrical surface of an infinitely long rod, the heatflow is strictly radial. The radial temperature distribution in a cylindrical rod with the thermal conductivity λand radius R 0 , in which there is a pumped region inside r < R where heat is uniformly generated at a rate q perunit volume, is obtained from the stationary heat conduction equationd 2 Tdr + 1 {dT −q2 r dr = λ; r < R. (5.1)0 ; R ≤ r < R 0With the boundary condition at the rod surface T(R 0 ) = T 0 we can solve the equation (5.1); the easiest way isto separate the equation solving for the two regions; in the r < R region the logarithmic part with ln(r/R) isdiscarded as a result of finite temperature at the rod’s centre, and in the boundary of the two regions the continuityof the temperature and of the energy flow (λT ′ ) must be taken into account. It follows straightforwardlythatT(r) − T 0 =⎧⎪⎨⎪⎩1 q4 λ12( ( )R21 + 2 ln ) R 0R − r2; r < R. (5.2)r; R ≤ r < R 0qλ R2 ln R 0Temperature gradients result in optical distortions in the rod, mostly through the refractive index variationattributable to deformations caused by thermal stress (photoelastic effect)[12, 10]. The temperature-dependent2 We expect the energy transfer rate to be proportional to the normalised overlap between the fluorescence emission line of thesensitizer and the absorption line (or, equivalently, to the absorption cross section frequency distribution) of the activator. Moreover,the probability of a spontaneous radiative transition between two atom states (which is basically the electric dipole transition) is known[8] to be proportional to the squared expected value of the electric dipole moment of the transition. Here the dipole moment of thesensitizer ion e ⇀ r s is applicable through equation (4.4) in view of separate measurement of sensitizer’s decay time, the latter being,by definition, inversely proportional to the probability of a spontaneous radiative transition (the Einstein’s coefficient A). Similarly,the squared expected value of the electric dipole moment of the activator’s induced transition is connected (through the Einstein’scoefficient B) to the absorption cross section of the activator.

Thermal effects in a crystal during laser operation 11Figure 9: Temperature profile in the laser rod, sized R 0 = 1.6 R and R 0 = 5 R, where R is the size of the pumping region.change of of refractive index can be expressed as∆n(r) T = (T(r) − T(0))( ) dndTstress. (5.3)In the pumped region, we obtain∆n(r) T = − 1 4qλ( ) dndTstressr 2(5.4a)And, in a compact form,n(r) = n(0)(1 − 1 2 α2 r 2 ). (5.4b)An optical beam propagating along the rod axis suffers a quadratical spatial phase variation. This perturbationis equivalent to the effect of a spherical lens [4]. The focal length of such medium is well-defined, opposite tothe region r > R with logarithmic temperature profile; the latter cannot be compensated with a negative lensor different resonator configuration. It is therefore vital for the laser resonator to sustain Gaussian transversemodes that the stimulated emission amplification does not take place outside the pumped region; in otherwords, optical pump beam cross-section should be larger than resonator beam cross-section.The focal length of the rod is given by [4, 3]f 1α 2 n 0 l(5.5)n 0 can be taken as the non-perturbed refraction index of the crystal. Clearly, the focal length depends on theoperating conditions of the laser and changes with pump power density; how sensitive f is to the change ofpump power, depends upon crystal material characteristics such as thermal conductivity (compare Eq. (5.4a)).Moreover, there is a contribution to lensing power from the “endeffects”:end-face thermal-stress-induced curvature of the rod causesadditional lensing effect, notably with end-pumping configurations[9, 13, 14]. It is possible to lessen this impact by using compositelaser rods consisting of undoped segments of crystal bonded to dopedsegments [13, 10, 15]. The design separates the rod endfaces from theheat-producing active region, and the undoped segments act as heatsinks. Tight bonding prevents end bulge in heated doped section.Figure 10: Composite YAG rod.To conclude, composite crystals with high conductivity are less likely to encourage substantial focal lengthvariations with pump power change, which is of particular importance under pulsed mode.

12 Examples of laser crystals6 Examples of laser crystals6.1 Nd,Cr:GSGG opposed to Nd:YAGNeodymium-doped yttrium aluminium garnet (Nd:Y 3 Al 5 O 12 , Nd:YAG) laser is the most commonly used typeof solid-state laser [1]. The YAG host is hard, of good optical quality, and has a high thermal conductivity.Pure Y 3 Al 5 O 12 is a colorless, optically isotropic crystal which possesses a cubic structure characteristic ofgarnets. It is grown exclusively by Czochralski method. In Nd:YAG about 1% of Y 3+ is substituted by Nd 3+ .The radii of the two rare earth ions differ by about 3% [1]. Consequently, with the addition of large amountsof neodymium, strains in crystal occur, indicating that the lattice of YAG is seriously distorted by the inclusionof neodymium, which limits the maximum doping concentration to a few atomic weight percent.Figure 11: To illustrate the influence od ion radiuson the doping mechanism, it is convenient to rewritethe formula for YAG in Y 3 Al 2 Al 3 O 12 arrangement.In this manner, dodecahedral Y 3 site, octahedral Al 2site, and tetrahedral Al 3 site, are marked distinctively[2]. The Nd 3+ ions substitute for Y 3+ without theneed for charge compensation. The same goes [16]for Yb 3+ , Tm 3+ , Er 3+ and Ho 3+ , while Cr 3+ and Ti 3+ ,for example, match the octahedral site. The Cr 4+ inCr:YAG from Fig. 3 corresponds to tetrahedral site.Figure 12: The Er:YAG laser emission occurs at 2940 nm,which makes it interesting medical applications because thewavelength coincides with a water absorption line. On theother hand, the energy output of Er:YAG laser is not veryimpressive due to low pump efficiency. Also, because ofvery long lower level lifetime (2 ms) Er:YAG laser cannotbe Q-switched. Besides Er:YAG, other erbium lasers such asEr:YALO 3 , Er:YLF and Er:Cr:YSGG have been investigated[1] because they provide additional laser lines in the infrared.Soon after the invention of the Nd:YAG laser (1964) attempts were made to increase the efficiency oftransferring radiation from the pump source to the laser crystal by utilizing a second, sensitizer dopant. Aparticulary attractive sensitizer was Cr 3+ (incorporated at the six-fold aluminium sites, see Fig. 11) becausethe broad absorption band of chromium can efficiently absorb light throughout the whole visible region of thespectrum 3 . Due to spin-forbidden nature of the transition, however, little improvement was achieved [17, 1];what is more, the Cr 3+ →Nd 3+ transfer time turns out to be much longer than the Nd 3+ decay time [18], andconsecutively laser efficiency for pulsed applications is low.18 years later it was discovered [18] that nearly 100% transfer efficiency could be achieved in the codopedgadolinium gallium scandium garnet crystal GSGG, where Nd and Cr ions are separated by as littleas 1 nm [16]. Experiments performed in several laboratories showed nearly a factor-of-three improvement inoutput-power-to-pump-power efficiency (when pumped by broad-spectrum source such as flashlamp), almostindependent of Cr concentration. On the other hand, Nd,Cr:GSGG 4 exhibits much stronger thermal focusingand stress-induced birefringence as compared to Nd:YAG. Investigations of Nd,Cr:GSGG lasers involve mostlylow power flashlamp pumped Nd,Cr:GSGG lasers.3 Of course, this is of special importance when pumping with flashlamps.4 {Gd 1−x Nd x } 3 [(Sc,Ga) 1−y Cr y ] 2 Ga 3 O 12

Nd:YAG and Nd:YVO 4 13Figure 13: Energy level diagrams of the sensitizer (Cr 3+ )-activator (Nd 3+ ) system in YAG (left, [17]; the archaic unit cm −1is equal to 0.124 meV) and in GSGG (right, [18]) hosts. The Cr 3+ photoluminescence (the relevant process is actually theinverted process - absorption, see (4.1)), the radiationless ion-ion dipole energy transfer and the Nd 3+ laser emission areindicated by arrows. Spectrum overlap is evidently better in Cr,Nd:GSGG.6.2 Nd:YAG and Nd:YVO 4Neodymium-doped yttrium vanadate has several spectroscopic properties that are particularly relevant to laserdiode pumping, most notably large stimulated cross section which is five times larger than in Nd:YAG [1](which is due to shorter fluorescence lifetime of Nd metastable level in a vanadate host), and a strong broadbandabsorption as indicated in Fig. 14. In the past, the major obstacle proved to be the unavailability of high-qualitycrystals. In the last decade, Nd:YVO 4 is becoming a leading competitor to Nd:YAG crystals. Actually, thehighest efficiency TEM 00 performance has been demonstrated in this Nd:YVO 4 [1].Figure 14: Output from a Nd:YVO 4 and Nd:YAG laseras a function of diode pump temperature and wavelength[1]; apparently, Nd:YVO 4 laser output power is less affectedby wavelength drifts of the pump diode.Figure 15: left: just made Nd:YVO 4 crystal before polishing(see section 3.1). Right: finished Nd:YVO 4 crystals, readyfor use. Crystals are from Beijing Opto-Electronic Technologycorporation.

14 SummaryThe vanadate crystal is naturally birefringent, as opposed to Nd:YAG. One of the most prominent advantagesof Nd:YVO 4 is that the impact of thermally induced birefringence is negligible in this crystal. Forrepetitively Q-switched operation, the relatively short upper state lifetime of Nd:YVO 4 requires high pulse ratesof 100 Hz in order to achieve average power close to continuous-wave performance. Pulsewidth in repetitivelyQ-switched systems is typically around 10 ns, which is relatively short compared to Nd:YAG. In applicationswhere longer pulses are required (such as laser drilling or cutting), Nd:YAG still plays the key role.7 SummaryThe treatment of laser crystals is a distinctively industrious application. The requirements for laser crystalscome directly from the laser industry, and the requirements for lasers come principally from other industriessuch as microelectronic industry. On the other hand, the physics of laser crystals is quite complex, merginginteractions between electrons, light and lattice; implementing optical, solid-state and other sciences such asphysics of deformations. The research on laser crystals is thus as beneficial for the manufacturing domain asappealing as a scientific research challenge.Figure 16: One of the crystal groups we haven’t touched is the group of laser crystals for tunable lasers. The absorbanceand fluorescence spectra of representative Ti:Al 2 O 3 , the titan:sapphire crystal, are shown on the right, as opposed toNd:YAG fluorescence spectra on the left. The term tunability of the laser emission denotes the possibility to modify theoutput wavelength of the laser over the broad range such as 400 nm in Ti:sapphire (670 nm -1070 nm), which is muchbroader than the Nd:YAG width of the single fluorescence line (2 nm).We have reviewed some of the important properties of laser crystals that are covered by repeatedly expressedrequirements. The requirements are dictated by the application. To satisfy a vast range of applications,it is valuable to have a multitude of laser crystal species on hand. The short fluorescence time of Nd:YVO 4crystal, as an example, is an advantage for high-repetition-rate Q-switched laser operation. On the other hand,the short lifetime means low energy storage, with other words, low energy per pulse. The Nd:YVO 4 lasermedium is therefore most usable in continuous-wave or high-repetition-rate Q-switched operation. If, however,strong thermal lensing is undesired or when unpolarized output is preferred, other type of crystal such asNd:YAG might be the right choice - provided, of course, that the induced birefringence phenomena of Nd:YAGis not an obstacle . . . .

REFERENCES 15References[1] W. Koechner, Solid-State Laser Engineering, vol. 1 of Springer series in optical sciences ; 1, Springer, Berlin,5th ed., 1999. 3, 8, 6.1, 12, 3, 6.2, 14[2] R. C. Powell, Physics of Solid-State Laser Materials, Atomic, Molecular, and Optical Physics, Springer-Verlag, NewYork, USA, 1998. 1, 2, 3.1, 4, 4, 2, 11[3] M. Vittorio, “Resonators for solid-state lasers with large-volume fundamental mode and high alignment stability,”Applied Optics 25, pp. 107–117, January 1986. 5[4] H. Kogelnik, “Imaging of optical modes - resonators with internal lenses,” Bell System Technologies 44, pp. 455–494, March 1965. 5[5] O. Svelto, Principles of lasers, Plenum Press, New York, 3rd ed., 1989. 2, 2[6] G. Eßer, A. Otto, and M. Geiger, “Requirements towards future laser systems for flexible manufacturing,” in LaserAssisted Net Shape Engineering 4, M. Geiger and A. Otto, eds., LANE 2004 1, pp. 37–45, Bavarian Laser Center,Meisenbach-Verlag Bamberg, (Bamberg, Germany), 2004. 0[7] D. L. Dexter, “A theory of sensitized luminescence in solids,” The Journal of Chemical Physics 21, pp. 836–850,May 1953. 4, 4, 4, 2[8] W. Demtröder, Laser Spectroscopy - Basic Concepts and Instrumentation, "Advanced Texts in Physics", Springer,Berlin, 3 ed., 2003. 2[9] R. Weber, B. Neuenschwander, and H. P. Weber, “Thermal effects in solid-state materials,” Optical materials 11,pp. 245–254, January 1999. 5, 5[10] M. P. MacDonald, J. E. Graf, J. E. Balmer, and H. P. Weber, “Reducing thermal lensing in diode-pumped laser rods,”Optics Communications 178, pp. 383–393, May 2000. 5, 5, 5[11] D. W. Hughes and J. R. M. Barr, “Laser diode pumped solid state lasers,” Journal of Physics D: Applied Physics 25,pp. 563–586, April 1992. 8[12] A. K. Cousins, “Temperature and thermal stress scaling in finite-length end-pumped laser rods,” Journal of QuantumElectronics 28, pp. 1057–1069, April 1992. 5[13] R. Weber, B. Neuenschwander, M. MacDonald, M. B. Roos, and H. P. Weber, “Cooling schemes for longitudinallydiode laser-pumped Nd:YAG rods,” IEEE Journal of Quantum Electronics 34, pp. 1046–1053, June 1998. 5[14] W. Koechner, “Absorbed Pump Power, Thermal Profile and Stresses in a cw Pumped Nd:YAG crystal,” AppliedOptics 9, pp. 1429–1434, June 1970. 5[15] M. Tsunekane, N. Taguchi, and H. Inaba, “Efficient 946-nm laser operation of a composite Nd:YAG rod with undopedends,” Applied Optics 37, pp. 5713–5719, August 1998. 5[16] D. Nikogosyan, Properties of optical and laser-related materials; A HANDBOOK, John Wiley & Sons, West Sussex,England; Cork, Ireland, 1998. 11, 3[17] Z. J. Kiss and R. C. Duncan, “Cross-Pumped Cr 3+ -Nd 3+ :YAG laser system,” Applied Physics Letters 5, pp. 200–202,November 1964. 3, 13[18] D. Pruss, G. Huber, and A. Beimowski, “Efficient Cr 3+ Sensitized Nd 3+ :GdScGa-Garnet Laser at 1.06µm,” AppliedPhysics B 28, pp. 355–358, April 1982. 3, 13

Indexbeam quality, 2TEM 00 , 10, 13compatibilityion-host, 6, 12ion-ion, 9, 12Cr 3+ , 3, 4, 12, 13crystal growth, 6decay time, 9, 12diode lasers, 3, 4, 10as pumping sources, 2, 10, 13diode temperature, 13doping of host material, 7, 12codoping with sensitizer ion, 12maximum concentration, 12optimal, 8Er:YAG, 12gain, 3, 8high in Nd:YAG, 7GSGG, 12laser crystalcharacterization, 3, 7growth, 6index of refraction, 11natural birefringence in Nd:YVO 4 , 14optical isotropy, 12spectra, 4metastable state, 5relaxation to, 5Nd:YAG, 12–14Nd:YVO 4 , 13–14pulse length, 2, 14pulsewidth, see pulse lengthrelaxationradiationless, 5requirementsfor future laser systems, 2material properties, 5semiconductor laser, 3solid-state laserdefinition, 3spectrumoverlap, emission and absorption, 9, 13visible, 12thermal conductivity, 5, 11, 12thermal lensing, 8, 10–1116