thermally induced strain and birefringence calculations for a nd:yag rod encapsulated in a solid...

Thermally induced strain and birefringencecalculations for a Nd:YAG rod encapsulated in asolid pump light collector

Stuart D. Jackson and James A. Piper

Calculations and experimental measurements of the thermally induced strain and birefringence arepresented for a diode-pumped Nd:YAG rod that is encapsulated in a prismatic pump light collector. Anumerical model is developed to determine the spatiotemporal stress-induced strain distribution acrossthe prism, index-matching fixant, and laser rod, and the birefringence that arises from the stress-induced strain within the laser rod. Calculations of the birefringence are compared with polarscopicmeasurements and display good agreement. Support for the rod on all sides is provided by the prismand fixant, and the distribution and degree of the stress-induced strain 1and birefringence2 within thelaser rod are therefore influenced by the geometry and composition of the prism and fixant. Thesestrains are thermomechanical in origin and are primarily a function of the elastic modulus of the fixantand the temperature of the system. Such stress-induced strains are additional to those strains that areproduced from temperature gradients across the laser rod and result from the laser rod beingconstrained from expanding. Collectors utilizing index-matching fluid as the encapsulant display thesmallest measure of birefringence relating to the temperature gradients in the rod. However, forcollectors utilizing solid fixants 1with significant elastic modulus2, an increase in the birefringenceresults. In this case collector designs that have the laser rod located in a symmetrically shaped prismare effective in reducing the nonuniform pressures on the sides of the rod and therefore thebirefringence. r 1996 Optical Society of America

1. Introduction

Diode-pumped solid-state lasers that employ nonim-aging or direct coupling of the light emitted fromquasi-cw diode arrays are receiving continued inter-est in the literature.1,2 Such techniques offer desir-able design and operating characteristics, such assimplicity and tolerance to misalignment of thediode, and they are particularly suitable to side-pumped configurations in order to provide powerscaling. An effective method of pumping the laserrod with this technique is to encapsulate the rodin a solid prismatic collector.3,4 This method pro-vides additional advantages such as rod support and

When this research was performed, the authors were with theCentre for Lasers and Applications, Macquarie University, NorthRyde, Sydney 2109, Australia. S. D. Jackson is now with theLaser Photonics Group, Department of Physics and Astronomy,Schuster Laboratory, University of Manchester, Manchester M139PL, UK.Received 3 January 1995; revised manuscript received 19 May

1995.0003-6935@96@091409-15$06.00@0r 1996 Optical Society of America

improved thermal management5 while producingmoderate output energies in both long-pulse andQ-switched operation.6Figure 1 illustrates the pump geometry under

examination. Physical details of the collector havebeen given elsewhere.4 Nonimaged light from asingle diode laser is incident at the Nd:YAG rod froma number of different directions. Because the laserrod interacts with the surrounding media 1i.e., theprism and fixant2 both mechanically and thermally,the stress-induced strain and hence birefringencedepend in detail on the composition and geometry ofthe prism and fixant.7 When theNd:YAG rod 1whichis fixed with epoxy into a BK-7 glass prism2 ispumped with low to moderate pump energies, thedegree of depolarization suffered by linearly polar-ized light passing through the laser rod is highlydependent on the fixant material, i.e., the measuredvalue and equilibration time of the birefringencevaried in accordance with the type of optical glueused.As a way to investigate the overall effect of the

surrounding material on the birefringence withinthe laser rod, the stress-induced strain distribution

20 March 1996 @ Vol. 35, No. 9 @ APPLIED OPTICS 1409

within the collector and laser rod was determined forvarious compositions of the collector. This was car-ried out by numerical computation of the stress-induced strain within the device and by the use ofthese strains within the laser rod for the determina-tion of the birefringence. The thermally inducedbirefringence within the laser rod can be consideredfor these devices as consisting of two parts. First, aweak contribution results from the temperaturegradients across the rod, and second a strongercontribution results from a thermomechanical effect.The second factor relates to pressures applied on thesides of the rod that arise from the expansion of therod being constrained by the surrounding material.After numerical calculation of the transient tem-

perature distribution 1assuming that the tempera-ture distribution is uncoupled from the determina-tion of the strain2,5 the two-dimensional straindistribution was also calculated by the assumptionof the plane strain condition, i.e., under a long-prismapproximation. The change in the indicatrix for aNd:YAG rod with the z axis of the rod orientedparallel to the 31114 direction of the crystal is thencalculated in a plane that is perpendicular to thisaxis. In general, Nd:YAG laser rods are cut withthe z axis of the rod parallel to the 31114 direction sothat the initial state of the refractive index in a planeperpendicular to the 31114 direction is isotropic. Forother Nd:YAG crystal planes, the polarization 1andelastic2 properties are a function of azimuthal angle.8Calculations of the birefringence are compared

with polarscopic measurements of the birefringencewith the rod for prisms composed of BK-7 glass andsapphire. The fixants that we tested have widelydifferent mechanical properties covering many or-ders of magnitude in elastic modulus. The model isextended to determine the distributions of the stress-induced strain and birefringence for cases in whichthe laser rod is located on the axis of a right-

Fig. 1. Schematic of the single-sided pumping arrangement withthe Cartesian coordinate system used for the numerical modeling.

1410 APPLIED OPTICS @ Vol. 35, No. 9 @ 20 March 1996

rectangular prism with a square cross section. Theresults from these calculations are compared withthe present geometry, and the issues concerning thedesign of lasers that utilize the encapsulation ofNd:YAG rods are discussed.

2. Numerical Calculation of the Stress-Induced Strain

The determination of the stress-induced strain distri-bution within solid-state laser rods has in generalbeen associated with isolated rods, and thus particu-lar symmetry considerations can be applied.8–12For composite problems in which the geometry andtherefore temperature distribution are asymmetric,at least a two-dimensional analysis of the stress-induced strainwithin the devicemust be undertaken.The calculations presented here are restricted to theplane transverse to the rod axis 1i.e., it is assumedthat the rod is uniformly heated and is not inhibitedfrom expanding in the z direction2. Note that thetwo-dimensional temperature distribution for theentire collector is calculated separately and is de-scribed elsewhere.5For the strains within the collector to be deter-

mined, the standard two-dimensional plane equilib-rium equations for the stresses 1in Cartesian coordi-nates and without body forces2 are solved13,14:

≠sxx

≠x1

≠sxy

≠y5 0, 11a2

≠sxy

≠x1

≠syy

≠y5 0, 11b2

where s is the stress with subscripts xx and yyrelating to the normal stress in the x and y direction,respectively, and xy relating to the shearing stress1see orientation of the coordinate system in Fig. 12.Solutions of Eqs. 112 for prismatic-shaped elementsare obtained with the plane strain assumption, i.e.,the strains Exz 5 Eyz 5 Ezz 5 0. The translations ordisplacements, U and V in the x and y directions,respectively, are therefore independent of the z coor-dinate. The components of the stress-induced strainare obtained by removal of the thermal expansionfactor from the total mechanical strain,

Exx 5≠U

≠x2 a1x, y23T 1x, y2 2 T04, 12a2

Eyy 5≠V

≠y2 a1x, y23T 1x, y2 2 T04, 12b2

Exy 51

2 1≠V

≠x1

≠U

≠y 2 , 12c2

where E is defined here as the stress-induced strain1with subscripts xx, yy, and xy relating to the normaland shearing strains, respectively2, a1x, y2 is thethermal expansion coefficient, and 3T1x, y2 2 T04 is thetemperature increment above ambient. Note thatbecause of the composite nature of the problem, all

the mechanical parameters for the system are func-tions of the space variables. The mechanical con-stants of the materials comprising the collectorsused in the modeling and experiments are containedin Table 1.The stress components as a function of the strains

under the plane strain approximation are13

sxx 5E1x, y2

31 1 v1x, y2431 2 2v1x, y24

3 531 2 v1x, y24Exx 1 v1x, y2Eyy6, 13a2

syy 5E1x, y2

31 1 v1x, y2431 2 2v1x, y24

3 5v1x, y2Exx 1 31 2 v1x, y24Eyy6, 13b2

sxy 5E1x, y2

231 1 v1x, y24Exy, 13c2

where E1x, y2 is the elastic modulus and v1x, y2 isPoisson’s ratio. Equations 122 and 132 are coupled,making Eqs. 112 a function of translations U and Vonly. The stress components obtained from thesolution of Eqs. 112 are used to determine the strains1caused by thermoelastic stresses2 by the use of thefollowing equations13:

Exx 51 1 v1x, y2

E1x, y2

3 531 2 v1x, y24sxx 2 v1x, y2syy6, 14a2

Eyy 51 1 v1x, y2

E1x, y2

3 531 2 v1x, y24syy 2 v1x, y2sxx6, 14b2

Exy 5231 1 v1x, y24

E1x, y2sxy. 14c2

Equation 112 is subject to traction-free boundary

Table 1. Mechanical Constants

Material a 1K212 E 1Nm222 v

RodNd:YAGa 7.5 3 1026 1.5 3 1011 0.3

FixantEpoxy 6.2 3 1025 b ,1.3 3 108 c ,0.2Silicone 3.0 3 1024 d ,0.1–1.0 3 108 e ,0.4

PrismBK-7f 8.3 3 1026 8.1 3 1010 0.208Sapphire 1c axis2 5.4 3 1026 g 3.95 3 1011 0.2

aRefs. 10 and 12.bRef. 15.cRef. 16.dRef. 17.eThe elastic modulus of silicone varies significantly, depending

on the exact composition: ,0.1 3 108 N m22 for rubberlike and,1.0 3 108 N m22 for plasticlike silicones.

f Ref. 18.gRef. 19.

conditions at the outer surfaces of the prism in termsof the forces at the surfaces 1i.e., the outer surfaces ofthe prism can expand freely213:

sxxnx 1 sxyny 5 0, 15a2

sxynx 1 syyny 5 0, 15b2

where nx and n y are the normals at the outersurfaces of the prism in the x and y directions,respectively.Because the laser rod is circular and has the

highest temperature, it is appropriate to transformthe numerically determined stress-induced strainsfrom Cartesian coordinates to tangential and radialcomponents. Using the definitions of the polarstrains and the chain rule, one can carry out thetransformation by using the following equations:

Er 5 Exx cos2 u 1 Eyy sin2 u 1 Exy cos u sin u, 16a2

Eu 5cos u U 1 sin uV

Œx2 1 y2

1 Exx sin2 u 1 Eyy cos2 u 2 Exy sin u cos u, 16b2

where u is the angle subtended from the center of therod and the x axis 1see Fig. 12. By using the abovetransformation, we imply that the center of the rod isnot translated in any direction from its initial posi-tion from t . 0. Note that the first term on theright-hand side of Eq. 16b2was set to zero, because weare interested in the stress-induced strain and notthe total mechanical strain.It is convenient to calculate the difference between

the normal components of the stress-induced strainin order to examine more closely the effects of thestress-induced strain throughout the system.Although the value for the stress-induced strain isresponsible for the fracture of materials 1which gen-erally occurs in regions of excessive tension2, thedifference between the components of the stress-induced strain 1or strain gradient, which is defined inour case as the difference between the radial andtangential components of the stress-induced strain,i.e., Er 2 Eu2 is responsible for the photoelastic effect.It can also be used to detect regions where the risk ofstress-related fracture is the greatest. The straingradient defined in this way is essentially a measureof the distortion, i.e., the magnitude of the stress-induced strain in the x direction plus the magnitudeof the stress-induced strain in the y direction of asmall area, as opposed to the sum of the stress-induced strains, which is a measure of the totalchange in area caused by the stress.Equations 112 were numerically evaluated by the

use of the finite-element package PDE2D.20 The300 3 300 matrix of the collector temperature distri-bution supplied by a separate code5 is used as inputto the present strain-calculation program. A ma-trix of this size is required in order to producesufficient resolution in the region of the laser rod.


The temperatures were determined when the diffu-sion equation was numerically solved for particularcollector combinations, with the calculated tempera-tures relating to the heat energies that are producedwithin the rod 1,5 mJ per pulse2 when pumped by asingle 240-W quasi-cw diode operating at a pulse-repetition frequency of 100 Hz. The results for eachstress-induced strain component 1for the same collec-tor configurations whose mechanical parameters arelisted in Table 12 as determined by the present codewere written into a 1003 100matrix and used as theinput for the calculation of the birefringence. Notethat the width of the fixant used in the modeling wasset at 100 µm throughout all stages of the calcula-tion.

3. Calculation of the Birefringence

To analyze the refractive index changes caused bythe thermally induced stresses, one utilizes theequation for the index ellipsoid 1indicatrix221:

Bi jxixj 5 1, 172

whereB is the dielectric impermeability tensor and xrelates to the Cartesian coordinate system coinci-dent with the collector coordinate system. In theunstressed case, the dielectric impermeability tensorcomponents Bi j 5 0 for i fi j. When stresses arepresent within the crystal, the size and shape of theindicatrix changes; the changes give rise to birefrin-gence. Under these conditions the components ofthe dielectric impermeability tensor take on thefollowing form14:

Bii 51

n021 DBii, 18a2

Bi j 5 DBi j, 18b2

where n0 is the refractive index of Nd:YAG 1n0 5 1.822.Dielectric impermeability changes DBi j are found bytransformation of the stress-induced strains 1whichare for the rod axis parallel to the 31114 crystaldirection2 to the system in which the rod axis isparallel to the 30014 direction, then multiplication ofthe photoelastic matrix of the Nd:YAG crystal 1forthe rod axis parallel to the 30014 direction2 with thetransformed stress-induced strain matrix, and thentransformation of the result back to the system forwhich the z axis is parallel with the 31114 direction, asdescribed in Appendix A. Major and minor axes Bxand By of the ellipse 1which are given in terms of theprincipal coordinate system of the ellipse2 are ob-tained when the dielectric impermeability matrix3Eq. 1A924 is made diagonal12:

Bx,y5 B0 1 1⁄21DBxx 1 DByy2

6 1⁄231DBxx 2 DByy22 1 4DBxy

241@2. 192

The change in the optical path length 1DOPL2 forrays parallel to the z axis of the laser rod are then


obtained from12

DOPL 5 1Bx 2 By21Ll2 , 1102

where L is the length of the rod 115 mm2 and l is thewavelength of light passing through the laser rod.Note that the deflection from the z axis of lightpassing through the laser rod is assumed to benegligible. The intensity loss per pass, Ad, for lin-early polarized light passing through the rod 1i.e., therelative intensity of light transmitted when the rodis placed between crossed polarizers2 is obtainedfrom the standard equation22

Ad 5

eS

I1x, y2sin2 23u1x, y2 2 g4sin23d1x, y2@24dS

eS

I1x, y2dS

, 1112

where g is the angle between the polarizer and theanalyzer, d1x, y2 is the phase angle change arisingfrom birefringence, acquired from multiplying Eq.1102 by 2p, and u1x, y2 is angle between the x axis ofthe rod and the x axis of the ellipse. The integralsare taken over the cross section of the beam; thus thevalues obtained from Eq. 1112 can be compared withexperimental measurements.

4. Results of Calculations Relating to theStress-Induced Strain

The numerically calculated radial component of thestress-induced strain across the prism for an epoxy–BK-7 collector combination at thermal equilibrium isshown in Fig. 21a2. Figure 21b2 shows the correspond-ing tangential component of the stress-induced strain.Note that the strains within the fixant and laser rodwere set to zero as a plotting ploy in order to isolatethe stress-induced strains within the prism, and thatthe spatially averaged shearing strains were approxi-mately 1–2% of the value of the normal componentsof the spatially averaged stress-induced strainthroughout all of the calculations.The tension in the tangential direction 1and the

corresponding radial compression2 is strong in aregion immediately surrounding the rod. Theseparticular features can be obtained when Eqs. 112 aresolved analytically for the case of a hole that issubject to uniform internal pressure and that islocated in an infinite plate.23 The stress distribu-tion within the plate for this case is axially symmet-ric, and sr is always negative 1compression2 and su isalways positive 1tension2. The values of the stress-induced strains in the lower region of the prism3underneath the rod as indicated by the arrow in Fig.21a24 are quite significant, and the distribution ischaracterized by a large compression peak in theradial direction and a large tension peak in thetangential direction. In the radial direction 3Fig.21a24, only the regions of compression 1which also peak

1a2

1b2

1c2

Fig. 2. Calculated three-dimensional plots of 1a2 the radial compo-nent of the stress-induced 1S-I2 strain within the prism, excludingthe fixant and rod; 1b2 the tangential component of the stress-induced strain within the prism, excluding the fixant and rod andviewed from the opposite side of 1a2; 1c2 the strain gradient withinthe prism, excluding the fixant and rod.

on the outer two surfaces of the prism2 are comple-mented by regions of tension in the tangentialdirection 3Fig. 21b24. These regions of the prismboundary are also the hottest of the surface and ingeneral represent regions most likely to fracture.The nonuniformity in the stress-induced strain distri-bution 1in both the radial and tangential directions2is solely a consequence of the proximity of the laserrod to the three edges of the prism.In the case of the radial component, see Fig. 21a2,

the two regions of tension that are symmetricallypositioned about the center of the peak in compres-sion are in accordance with analytical solutions toEqs. 112 obtained for the case of a semi-infinite platein which a uniform pressure is applied to a holepositioned close to the straight edge.24 The pointsalong the straight edge at which the sign of thestress changes occur when z 5 6Œ1d2 2 r22, where z isthe distance along the edge from the line perpendicu-lar to the boundary and passes through the center ofthe rod, r is the radius of the rod, and d is thecenter-edge distance. The points where the tensionis a maximum occur at the locations z 5

6Œ331d2 2 r224. Performing these calculations forthe lower surface of the prism establishes that thestress-induced strains will be zero when z 561.17 mm, and a maximum in tension when z 562.02 mm 3as verified in Fig. 21a24. Importantly, thestresses on the interior 1or circular2 boundary of theprism are not uniform. From the analytical solu-tions the stresses on this boundary are a maximumat points on the boundary that coincide with the linethat is tangent to the circular boundary and passesthrough z 5 0 of the straight boundary. Note thatthe maximum compression in the radial direction is,38% of the maximum tension in the tangentialdirection; further, the calculated maximum tensilestress is ,10% of the maximum tensile strength ofBK-7.Figure 21c2 displays the strain gradient across the

prism for the same collector configuration. Thedistribution indicates that the prism is highly dis-torted in regions immediately surrounding the rod, acondition that arises from the axially symmetricstress, and in a region between the rod and thenearest prism boundary; this is a consequence of thefinite separation between the center of the rod andthe prism surfaces. Note that the tension in thetangential direction is the dominant factor in thedistortion of the prism and that the strong variationin the values of the stress-induced strain for bothcomponents contributes to the strong variation inthe strain gradient.Figures 31a2, 31b2, and 31c2 show the stress-induced

strain within the fixant in the radial direction, thetangential direction, and the strain gradient, respec-tively. Overall, the fixant is under compression inboth directions, with the compression in the radialdirection being approximately 67% of the stress-induced strain in the tangential direction. 1Notethat the stress-induced strain in the fixant is approxi-


1a2

1b2

1c2

Fig. 3. Calculated three-dimensional plots of 1a2 the radial compo-nent of the stress-induced 1S-I2 strain within the fixant, 1b2 thetangential component of the stress-induced strain within thefixant, 1c2 the strain gradient within the fixant.


mately an order of magnitude greater than in theBK-7 prism because of the smaller elastic modulusand dimension of the fixant compared with theprism.2 The stress-induced strain in the tangentialdirection is quite uniform 1maximum variation ,4%2compared with the stress-induced strain in the ra-dial direction 1maximum variation ,36%2, and thusthe radial variation is the primary cause for the,39% variation in the strain gradient; see Fig. 31c2.The asymmetry in the tangential component of thestress-induced strain within the prism 3see Fig. 21b24is not transferred to the tangential direction in thefixant. Note that the magnitude of the stress-induced strain in the radial direction is a maximumin the region of the fixant adjacent to the prism–fixant interface and a maximum in the tangentialdirection in the region of the fixant adjacent to thefixant–rod interface, in accordance with the stressesproduced from axially symmetric stress.25The radial and tangential components of the stress-

induced strain within the rod are displayed in Figs.41a2 and 41b2, respectively. The rod is also undercompression in both directions; however, regions ofmaximum compression in the radial direction coin-cide with regions of minimum compression in thetangential direction, and vice versa. Such a situa-tion gives rise to finite values for the strain gradientwithin the rod 3see Fig. 41c24. Themaximum compres-sion in the tangential direction is approximately 50%of the maximum compression in the radial direction.Qualitatively, both components of the stress-inducedstrain exhibit similar spatial distribution, with thetwo maxima in compression in the tangential direc-tion 1and the two minima in the radial direction2lying on the vertical line through the center of therod. This is a consequence of the pressures appliedto the rod along this line because of the proximity ofthe lower surface of the prism. The correspondingminima 1or maxima in the radial direction2 aredisplaced from the horizontal line through the centerof the rod because of the locality of the other twoouter surfaces of the prism with respect to the rod.The locality of the minima can be verified when linesare drawn from the z 5 0 points on the prismboundaries to the circular boundary of the prism.Note that an approximately 100% variation exists inthe strain gradient within the rod. In general, thespatially averaged magnitude of the stress-inducedstrain 1,10252 represents ,4% of the total mechani-cal strain 1approximately 2.2 3 10242, and theNd:YAGrod is compressed to ,5% of its maximum compres-sion strength.Overall, the finite size of the prism and location of

the laser rod with respect to the surfaces of the prisminduce a significant degree of nonuniform radiallydirected stress on the outer 1peripheral2 surface ofthe laser rod, which translates to a considerablevariation in both polar components of the stress-induced strain within the laser rod. In the case of afluid–BK-7 collector combination, the high degree ofcompliance of the fluid leads to quite small stress-

1a2

1b2

1c2

Fig. 4. Calculated topographical plots of 1a2 the radial componentof the stress-induced strain within the rod, 1b2 the tangentialcomponent of the stress-induced strain within the rod, 1c2 thestrain gradient within the rod. The values on the contoursrepresent the values for stress-induced strain multiplied by 1026.

induced strains within the rod 1,5% and ,4% of thevalues of the radial and tangential components,respectively, of the stress-induced strain within therod for the triangular geometry2. Note that similarvalues for the stress-induced strain within the rodwere obtained from the fluid–sapphire collector com-bination as for the fluid–BK7 combination, althoughthe rod temperatures for the two cases are markedlydifferent.4 In the case of the epoxy–sapphire collec-tor combination, the calculated stress-induced strainswith the rod were determined to be ,16% and ,20%of the values of the radial and tangential compo-nents, respectively, of the stress-induced strainwithinthe rod for the triangular geometry. The abovecalculations highlight the importance of judiciouschoice of collector materials in order to reduce thedegree of stress-induced strain within the laser rod.

5. Results of Calculations of the Birefringence

The elastic properties of the fixant surrounding therod essentially determine the interaction betweenthe rod and the prism. As a way to show this, Adwas plotted for a number of collector combinations asa function of time from diode switch on, as shown inFig. 5. The temporal characteristics of Ad for collec-tors utilizing fixants of moderate elastic modulus1see insert in Fig. 5, which relates to collector configu-rations involving silicone fixant2 follow the change inthe thermal expansion of the rod and thus theaverage temperature within the rod. Because thetransmitted intensity ratio 1Ad2 is proportional to thesquare of the strain gradient for small strains 1which

Fig. 5. Calculations of Ad for a number of collector combinationsas a function of time from diode switch on. Calculations 1solidcurves2 and experimental measurements 1circles and triangles arefor BK-7 and sapphire prisms, respectively2 of Ad as a function oftime from diode switch on are for the fluid–BK-7 and for thefluid–sapphire collector combinations. Included in the inset arethe calculations of Ad for silicone–BK-7 and silicone–sapphirecollector combinations.


are known to be approximately a factor of 7 less forthe silicone–sapphire collector combination2, thesteady-state birefringence for the silicone–sapphiresapphire collector combination is approximately 2orders of magnitude lower than for the silicone–BK-7 collector combination. The change in Ad withtime observed for collector combinations involvingfluid fixant at the diode switch-on 1and turn-off 2stages is quite sudden, with the value for Ad equili-brating in less than 40 s for the fluid–sapphirecollector combination. Note that the value for Adfor the fluid was set at 1.0 3 101 N m22, an estimatebased on its low viscosity. The changes in Ad withtime at the turn-on and turn-off points for thefluid–BK-7 collector combination are similar to thosefor the fluid–sapphire case, but Ad equilibrates in alonger period of time 1,700 s later2. Note that thesteady-state value for Ad in both cases is approxi-mately the same 1to within 10%2 because the thermalgradients for these systems are similar in magnitude1167 Km21 and 147 Km21 for the fluid–sapphire andfluid–BK-7 collector combinations, respectively2.The transmitted intensity pattern after linearly

polarized probe light has passed through the laserrod and a crossed analyzer is shown topographicallyin Fig. 6. For low Ef 3,106 N m22; Fig. 61a24, thepattern is similar to the Maltese Cross patternassociated with the transmitted intensity for iso-lated rods placed between crossed polarizers,25 and itis related to the parabolic temperature distributionalone.12 The optical path-length change 1DOPL2 dis-tribution arising from birefringence shown in Fig.61a2 is mapped in Fig. 61b2. As expected, the DOPLdistribution is approximately concentric with thecenter of the rod and increases to the perimeter ofthe rod with the familiar r2 dependence. Such adistribution is consistent with rods that are cooledevenly around the curved edge, demonstrating againthat when Ef is low, the birefringence is dependenton the temperature distribution across the rod only.The corresponding transmitted intensity profile

for higher Ef 11.3 3 108 N m222 is shown in Fig. 61c2.The associated DOPL distribution mirrors the straingradient distribution 3Fig. 41c24, because DOPL isproportional to the strain gradient within the rod.

6. Polarscopic Experiment

For the measurement of the birefringence within therod for different collector combinations, a polarscopicexperiment was set up as shown schematically inFig. 7. Light from an unpolarized green 1543.5-nm2He–Ne laser was polarized with a calcite polarizer1Karl Lambrecht double escape window, MGLA-DW2and then apertured with a 2.5-mm-diameter holebefore passing through the encapsulated laser rod.The probe light emerging from the laser rod wasanalyzed with an identical polarizer orientated per-pendicular to the polarizing plane of the first polar-izer. Note that the probe beamdiameterwas smaller1to ,2.8 mm2 than the rod diameter so that edgediffraction effects12 would be avoided. The transmit-


1a2

1b2

1c2

Fig. 6. Calculated contour plots of 1a2 the transmitted intensityfor the fluid–BK-7 collector combination, 1b2 the steady-stateDOPL distribution across the laser rod for the fluid–BK-7 collectorcombination, 1c2 the transmitted intensity for the epoxy–BK-7collector combination. Note that the plane of the polarizer isoriented with the x axis.

ted light was detected with a silicon photodiodepower meter 1Newport 8152, and the results wereregistered on a chart recorder. This effectively re-cords the spatially averaged depolarization 1Ad2 suf-fered by the polarized light incident at the laser rod.So that noise from scattered laser pump diode lightwould be avoided, the signal from the power meterpassed through a low-pass filter with a cutoff at,10 Hz.The rod was optically pumped with a quasi-cw,

240-W, four-bar stacked array 1SDL-3230-ZC2 operat-ing with 200-µs pulses at a pulse-repetition fre-quency of 100 Hz throughout. The diode laser stackwas driven at a 70-Apeak current, providing,210 Wof peak optical power 1or 42 mJ2 per pulse. Thediode was placed approximately 1 mm from theprism hypotenuse in the uncollimated diode light-collection geometry.4The thermally induced cavity length instabilities

associated with the warm up of He–Ne laser pro-duced a rapid modulation of the light that wastransmitted through the first polarizer; conse-quently, the He–Ne laser was turned on some 12 hprior to the experiments.

7. Experimental Results

The spatially averaged transmitted intensity, Ad, asa function of time from diode switch on for a silicone–BK-7 collector combination is presented in Fig. 81a2.The residual stress-induced strain within the laserrod produced approximately 0.1% transmission.The residual strain is related to the curing of thefixant and is discussed in detail later in this paper.Ad is observed to increase until ,240 s, where it

equilibrates to a value of ,0.56%. When the diodelaser was turned off, the transmission decreasedrelatively quickly 1in less than 180 s2 to the residualvalue. Included in Fig. 81a2 is the calculation of Adthat uses the numerical model 1with Ef set at1.0 3 108 N m22; see Table 12 and with a 0.1% offsetin accordance with the measured birefringencecaused by the residual stress-induced strain. Notethat the calculated value for Ad increases with timebeyond the experimental value for Ad. Because the

Fig. 7. Schematic of the polarscopic experiment: TM, A, PM,LPF, and CR represent the turning mirror, aperture, power meter,low-pass filter, and chart recorder, respectively.

calculated stress-induced strain within the rod isproportional to the temperature rise, the calculatedvalues for Ad do not reach a steady-state value untilt , 700 s. The discrepancy between the experimen-tal results and the model calculations for Ad isbelieved to result from the effects of heating thesilicone fixant, whereby it is apparent that for anincrease in the temperature of the fixant, the fixantbecomesmore flexible, i.e.,Ef decreases with increas-ing temperature. The temperature dependence ofEf on silicone is also evident when the system coolsdown, because the measured time 1,420 s2 for thesystem to reach the transmission at the residualstress-induced strain level is shorter than the modelprediction.The temperature dependence of Ef on silicone was

1a2

1b2

Fig. 8. Ad 1a2 calculations 1solid curve2 and experimental measure-ments 1circles2 from diode switch on for the silicone–BK-7 collectorcombination and 1b2 experimental measurements for an epoxy–BK-7 collector combination from diode switch on.


also apparent when the rod was pumped with differ-ent average powers, because the equilibration timeof the transmitted intensity decreased with an in-crease in the average pump power.7 It was ob-served that as the repetition rate of the diode laserwas increased from 10 to 50 Hz, the equilibrationtime of the birefringence decreased 1going from.12 min at 10 Hz to ,5 min at 50 Hz2.Collector configurations involving fluid fixants in

conjunction with both BK-7 and sapphire prismswere also tested. Included in Fig. 5 are the experi-mental values for Ad for the fluid–BK-7 collectorcombination. The experimental curve displays fairlygood agreement with the calculated curve: themea-sured steady-state value for Ad is ,0.014%, and thecalculated value is ,0.013%. Good agreement isalso achieved between the measured and calculatedvalues of Ad for the fluid–sapphire collector combina-tion; the measured steady-state value for Ad of,0.011% is close to the calculated result of 0.012%.The results of the polarscopic measurements of Ad

as a function of time for an epoxy–BK-7 collectorcombination are illustrated in Fig. 81b2. At t 5 01before diode switch on2, a very high residual birefrin-gence is present 1Ad < 1.36%2; with the onset of diodepumping, a sudden sharp increase in birefringence isobserved. On the basis of the above results, thissudden increase in the birefringence can be attrib-uted to the relatively quick equilibration of thetemperature gradient. However, unlike the charac-teristics of Ad for the fluid–BK-7 case, Ad for thissystem decreases with time, reaching zero in approxi-mately 50 s. When the pump diode is turned off thedepolarization of the linearly polarized probe lightincreases slowly, reaching the residual value approxi-mately 250 s later. The birefringence characteris-tics measured with this collector combination areattributed to the strong residual stresses within therod that result from the curing of the epoxy. Duringthe curing process it is evident that the epoxycontracts slightly, creating a residual pressure onthe perimeter of the rod that is directed away fromthe center of the rod, resulting in overall tensionwithin the rod 1i.e., the pressures at each of theinterfaces are initially directed away from eachinterface2. Consequently, as the rod heats up theresidual and thermomechanical strain contributionsbegin to cancel, and the resultant force on the laserrod at thermal equilibrium is approximately zero.Note that the birefringence across the rod equili-

brates in a shorter time than the overall tempera-ture of the rod. This again may be credited to thetemperature dependence of the elastic modulus ofthe epoxy, because it is known26 that the modulus ofelasticity of most thermoplastics 1which includessilicones2 decreases significantly with increasing tem-perature. After the diode laser has been switchedoff, the temporal characteristics of the birefringencefollowmore closely with the temporal characteristicsof the decreasing temperature. Note that in prac-


tice there are many factors that contribute to theresidual stresses on curing, such as thickness 1andthickness variation2 of the fixant layer and small airpockets that may exist. Therefore, exact compensa-tion in the steady state for the thermomechanicallyinduced birefringence with these particular collec-tors may not be possible in some specific cases. Ofcourse for a given collector arrangement, exact com-pensation only occurs for certain pumping 1i.e., ther-mal2 conditions.

8. Discussion

The experimental measurements and theoreticalcalculations of the stress-induced strain and birefrin-gence indicate that the additional thermomechani-cal birefringence displays considerable dependenceon the fixant material. This dependence not onlyrelates to the rigidity of the fixant material but alsoin practice to the apparent temperature dependenceof the elastic modulus 1for some of the materials2, andto residual stress-induced strain that is initiallypresent for some collector configurations. However,overall agreement between the modeling results andthe experimental measurements is good, with mea-surements taken from collectors that use fluid fixantdisplaying the best agreement 1where there is noresidual strain or birefringence2.The model can be used to investigate a number of

prism and fixant combinations as an aid to theestablishment of optimum configurations. Figure91a2 displays the numerically calculated stress-induced strain within the rod as a function of theelastic modulus of the fixant for prisms made up ofBK-7 and sapphire 1the numerically determinedstress-induced strains presented here represent thespatially averaged stress-induced strain over thecross section of the laser rod2.The results of Fig. 91a2 that relate to a BK-7 prism

indicate that at low Ef 1for the fixant2, the spatiallyaveraged radial component of the stress-inducedstrain is negative and the spatially averaged tangen-tial component is positive; this is consistent with thestress-induced strains associated with a parabolictemperature distribution.12 As Ef increases, bothcomponents become increasingly negative until at Ef,5 3 107 N m22, a significant increase in thecompression within the rod for both polar compo-nents is observed. At this point the fixant materialbecomes sufficiently rigid to restrict effectively thethermal expansion of the rod. The increase in thecompression within the rod is proportional to theincrease in Ef, because the pressures on the rod areproportional to the force of the rod 1and thus thespring constant of the fixant2 arising from compres-sion of the fixant.Included in Fig. 91a2 is the spatially averaged

stress-induced strain within the rod as a function ofEf for a collector employing a sapphire prism. Thedifference between the numerically determinedstress-induced strains within the rod for each prism

material is primarily related to the difference inoverall temperature of the rod in each case. Thepoint at which compression within the rod becomessignificant occurs at Ef , 108 Nm22, a value approxi-mately twice that for the BK-7 case. The differencebetween these values is somewhat reduced becauseof the higher elastic modulus of the sapphire prism.The trends in the spatially averaged stress-inducedstrain as a function of Ef in this case are similar tothose for BK-7 prisms, although the average tangen-tial component is observed to undergo greater varia-

1a2

1b2

Fig. 9. Calculations of the radial component of the stress-induced 1S-I2 strain 1solid curves2, the tangential component of thestress-induced strain 1long-dashed curves2, and strain gradient1short-dashed curves2 for variation in the elastic modulus of 1a2 thefixant material, 1b2 the prism material.

tion than the average radial component when com-pared with those of the BK-7 prism. Note that thestrain gradient within the rod is significant for bothcollector prisms when Ef . 108 N m22.As mentioned above, the prism material does

influence the amount of stress-induced strain withinthe rod, as shown in Fig. 91b2, where the spatiallyaveraged stress-induced strains within the rod areplotted against the elastic modulus of the prism.Note that Ef was set in these calculations at thevalue corresponding to the epoxy fixant. The elas-tic properties of the fixant are significantly moreimportant than those of the prism, because the pointat which the compression within the rod signifi-cantly increases occurs at a greater value of theelastic modulus of the prism than of the fixant.This is also exemplified by the fact that at high Ep1,1011 N m222, the values for the average stress-induced strains are less than 50% of the values forthe average stress-induced strains for the identicalvalue Ef. Note that the variation in the straingradient with Ep is relatively minor.Parameter Ad as a function of the elastic modulus

of the fixant 1Ef2 is displayed in Fig. 10 for twocollectors made up of BK-7 and sapphire prisms.In the case of BK-7 prisms, an increase in thebirefringence is noted for Ef . 108 N m22, though thebirefringence roughly stabilizes for Ef . 1010 N m22.Within this region, birefringence increases by ap-proximately 3 orders of magnitude. A similar effectoccurs for sapphire prisms, demonstrating the pri-mary dependence of the birefringence on the me-chanical properties of the fixant.Note that even though the rod temperature incre-

ment above ambient is approximately 7 times smaller

Fig. 10. Calculations of the steady-state value for Ad as afunction of the elastic modulus of the fixant material for collectorsmade up of both BK-7 and sapphire prisms. Note that the othermaterial parameters for the fixant are consistent with the epoxymaterial.


for sapphire prisms than for BK-7 prisms, the bire-fringence is approximately the same for both cases inthe limit of high- and low-fixant elastic modulus.Further, for collectors made from BK-7 prisms, smallchanges in Ef of a few percent or more beyond 108 Nm22 produce significant changes in the transmittedintensity 1of the order of a 100–200% increase2. It istherefore apparent that only a relatively minormodification in Ef of a few percent need occur inorder to produce the experimentally observed charac-teristics of Ad for collector combinations that utilizeepoxy or silicone fixant. Note that the dependenceof the birefringence on vf and vp is only minor.The predominant cause for the thermomechanical

birefringence within the laser rod for fixants 1andprisms2 of significant rigidity results from the lack ofsymmetry in the present geometry of the prism.The displacements on the interior surface of theprism that result from the expansion of the rod arenonuniform because of the proximity and position ofthe laser rod with respect to the outer edges of theprism. Because the displacements on this surfaceare transmitted to the rod when the fixant has amoderate elastic modulus, the strain gradient withinthe laser rod is significant. For the strain gradientto be reduced, the rod must be placed further fromthe edges of the collector. According to theory,24ratio d@r 1where r and d are the radius and center-edge distance, respectively2must be greater than 4 inorder to establish a more uniform stress distributionon the surface of the hole and therefore on theperimeter of the rod. Under these conditions, theanalytical solutions that are associated with a holein an infinite plate23 can be applied. In the case of arod of radius 1.5 mm, the center to edge distance ofthe prism must therefore be greater than 6 mm.As a way to gain better insight into the effects of

the geometry on the stress-induced strain, the stress-induced strains across the prism and rod for asymmetric prism geometry were undertaken. Theradial component of the stress-induced strain isplotted in Fig. 111a2 for a laser rod placed on the axisof a right-rectangular prism with a square crosssection in which the side dimension of the square is12 mm. It can be observed that in a region of theprism close to the rod, the radial component of thestress-induced strain is quite axially symmetric com-pared with that for the triangular prism geometry.In the case of the tangential component of thestress-induced strain 3see Fig. 111b24, the stress-induced strain is less symmetric in the region of theprism close to the laser rod and is the primary causefor the nonuniform distribution in the strain gradi-ent 3see Fig. 111c24. However, the values of bothcomponents of the stress-induced strain on the facesof the prism are a maximum in the central region ofthe face, corresponding to the point of the face closestto the laser rod. The maxima in the radial compo-nent of the stress-induced strain are identical inmagnitude to the tangential component, which is aconsequence of the symmetrical geometry of this


1a2

1b2

1c2

Fig. 11. Calculated topographical plots 1for the right-rectangularprism with a square cross section2 of 1a2 the radial component and1b2 the tangential component of the stress-induced strain withinthe prism, 1c2 the strain gradient within the prism. Note that thevalues on the contours represent the values for stress-inducedstrain multiplied by 1026.

1a2

1b2

1c2

Fig. 12. Calculated topographical plots 1for the right-rectangularprism with a square cross section2 of 1a2 the radial component and1b2 the tangential component of the stress-induced strain withinthe rod, 1c2 the strain gradient within the rod. Note that thevalues on the contours represent the values for stress-inducedstrain multiplied by 1026.

particular prism. Further, the value of the radialcomponent of the stress-induced strain is ,50% ofthe corresponding stress-induced strain for the trian-gular prism; however, in our case it is the distribu-tion of the stress-induced strain that is important atthese levels of pump power.The significance of the near axially symmetric

stress-induced strain distribution in the region ofthe rod can be examined when the stress-inducedstrain across the rod is plotted for this prism geom-etry, as displayed in Fig. 121a2 for the radial compo-nent and in Fig. 121b2 for the tangential component.Note that the temperature of the laser rod for thisparticular prism geometry was similar to the rodtemperature for the triangular geometry. The ra-dial component is seen to be quite uniform across therod 1,6% variation2, and the tangential component isseen to be quite axially symmetric 1,14% variation2.These more uniform distributions in the stress-induced strain are a direct consequence of the uni-form displacements transmitted from the prism tothe rod. Such uniformity leads to uniformity in thestrain gradient distribution 3see Fig. 121c24, with thevariation in the strain gradient in this case being,20% of the variation in the rod for the triangulargeometry 1even though the overall values for thestress-induced strains within the rod are similar,,10252. In the central region of the laser rod thestrain gradient is small and increases incrementallyto the edge of the rod, and the sign of the gradient1which is constant across the laser rod2 indicates thatcompression in the radial direction is dominant.Note that the maximum compression in the radialdirection is ,95% of the maximum compression inthe radial direction. A crosslike pattern exists inthe strain gradient distributionwithin the rod, whicharises from the shape of the prism in the transverseplane 1because the outer surfaces of the prism arestill reasonably close to the rod2. In the limit ofinfinite center to edge distance, the strain gradientbecomes axially symmetric 1i.e., commensurate withthe temperature gradient contribution alone2. Thebirefringence for this particular prism geometry wascalculated to be ,0.02% and primarily was due tothe birefringence caused from temperature gradi-ents across the rod.

9. Conclusion

We developed a numerical model for the calculationof the stress-induced strain and the birefringence fora Nd:YAG rod encapsulated in a solid prismaticpump light collector. In conjunction with polar-scopic measurements, the model was able to distin-guish the dominant factors leading to birefringencefor a variety of fixants and prism materials. Over-all, the experimental measurements and model cal-culations of the stress-induced strain and birefrin-gence indicate that with the present collectorconfiguration, the birefringence within the Nd:YAGrod is quite low 1,1% per pass through the rod for1.064-µm radiation2. We established that there are


two major processes leading to the birefringence.First, there exists a relatively small contribution tothe overall birefringence, which arises from thethermal gradients that are present across the laserrod. Second, the major contribution to the overallbirefringence in our case relates to the mechanicalconstriction of the laser rod to thermal expansion,and it is primarily a function of the degree of rigidityof the surrounding fixant. Systems utilizing fluidfixants display weak birefringence related to thefirst effect, and for systems incorporating mechani-cally less flexible fixants, such as silicone and epoxy,the strong birefringence effects observed are primar-ily produced by the second effect. Silicone fixant inconjunction with a sapphire prism provides theoptimum system combination because of the low rodtemperatures and moderate fixant flexibility. Itwas determined that the rod and fixant 1in bothradial and tangential directions2 are under compres-sion, whereas the prism is under compression in theradial direction and under tension in the tangentialdirection. Further, it was also established thatcollector geometries that have a higher symmetrythan the present design are effective in reducingsignificantly the second, larger contribution to thebirefringence.

Appendix A.

The stress-induced strains obtained from the solu-tion of Eq. 112 are given in matrix form:

E1x, y2 5 3Exx Exy 0

Exy Eyy 0

0 0 04 , 1A12

where the z axis is parallel to the 31114 direction.To transform this strain matrix so that the z axis isparallel to the 30014 direction, one takes the followingtransformation:

E81x8, y82 5 U21E1x, y2U, 1A22

where x and y relate to a Cartesian coordinatesystem oriented with the z axis parallel to the 31114direction, and x8 and y8 relate to the system with thez axis oriented in the 30014 direction. Transforma-tion matrixU is given by8

U 5 3cos a cos b sin a cos b 2sin b

2sin a cos a 0

sin b cos a sin a sin b cos b4 . 1A32

Angles a 5 45° and b 5 54.8° for a rod axis areparallel with the body diagonal 31114 of the cubicNd:YAG crystal. The 3 3 3 matrix produced fromthematrixmultiplication of Eq. 1A22 can be converted


to a 6 3 1 matrix by the use of Nye’s notation,21

En81x8, y82 5 3E11E22E332E232E132E12

4 , 1A42

where Ei j are the components of the 3 3 3 matrixobtained from Eq. 1A22.The small changes in the dielectric impermeability

that result from the stress-induced strain are givenby

DBm1x8, y82 5 pmn1x8, y82En81x8, y82, 1A52

where photoelastic matrix pmn is for a Nd:YAGcrystal with a Cartesian coordinate system in whichthe z axis is parallel to the 30014 direction and is of theform12

pmn1x8, y82 5 3p11 p12 p12 0 0 0

p12 p11 p12 0 0 0

p12 p12 p11 0 0 0

0 0 0 p44 0 0

0 0 0 0 p44 0

0 0 0 0 0 p44

4 , 1A62

where the elasto-optic coefficients of the photoelastictensor are given by12

p11 5 20.0290, p12 5 10.0091,

p44 5 20.0615. 1A72

The result from matrix multiplication 1A52 is con-verted to a 3 3 3 matrix and is transformed back tothe Cartesian coordinate system in which the z axisis parallel to the 31114 direction of the crystal bymeans of

DB1x, y2 5 UDB1x8, y82U21, 1A82

where the components of the tensor representing thechanges in the dielectric impermeability tensor arenow given in terms of the cartesian coordinatesystem coincident with the collector frame, and aregiven by

DB1x, y, z2 5 3DBxx DBxy DBxz

DBxy DByy DByz

DBxz DByz DBzz4 . 1A92

The authors appreciate the generous support fromG. Sewell in the development of the computer pro-grams, and they gratefully acknowledge the finan-

cial assistance of the Industrial Research and Devel-opment Board, Department of Industry Technologyand Commerce 1Commonwealth of Australia2, andThe BHPCompany, as well as contributions from theDefence Science and Technology Organisation 1Com-monwealth ofAustralia2. Independent financial sup-port was also given by the Australian ResearchCouncil. S. Jackson held an Australian Postgradu-ate Research Award 1Industry2 supported by theAustralian Research Council and The BHP Com-pany.

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thermally induced strain and birefringence calculations for a nd:yag rod encapsulated in a solid...

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