thermally induced birefringence in nd:yag slab lasers

Thermally induced birefringence in Nd:YAG slab lasers

Martin Ostermeyer, Damien Mudge, Peter J. Veitch, and Jesper Munch

We study thermally induced birefringence in crystalline Nd:YAG zigzag slab lasers and the associateddepolarization losses. The optimum crystallographic orientation of the zigzag slab within the Nd:YAGboule and photoelastic effects in crystalline Nd:YAG slabs are briefly discussed. The depolarization isevaluated using the temperature and stress distributions, calculated using a finite element model, forrealistically pumped and cooled slabs of finite dimensions. Jones matrices are then used to calculate thedepolarization of the zigzag laser mode. We compare the predictions with measurements of depolariza-tion, and suggest useful criteria for the design of the gain media for such lasers. © 2006 Optical Societyof America

OCIS codes: 120.6810, 140.3070, 140.3480, 140.3530.

1. Introduction

The use of high brightness diode–laser pump sourcesallow higher gain for a given maximum heat deposi-tion. Together with the reduced cost this has led tonew, high power laser developments that avoid lamppumping, thus avoiding excess heating and notori-ously bad beam quality. The high gain that can beachieved with diode–laser pumping has also allowedthe development of cw Nd:YAG lasers that use un-stable resonators.1–3 To exploit this capability fully,the increased pumping required ultimately leadsback to significant thermal loading and thermallyinduced birefringence, which depolarizes the lasermode and can cause poor efficiency. Slab gain mediaare often used to mitigate these effects, as they min-imize the impact of thermally induced birefringenceand have a higher stress induced fracture limit thanrod configurations.4 They have the additional advan-tage that they can have Brewster-angled entranceand exit faces, removing the need for antireflectioncoatings and providing a polarized laser mode. Fur-thermore, a ��p� plane polarized mode can propagatealong a zigzag path in the yz plane (see Fig. 1), re-ducing the strong “thermal lensing” typically encoun-

tered with high power loadings.5 The zigzag path alsoresults in an efficient overlap of the lasing mode withthe gain distribution and in beam parameters thatare independent of the pump power.

In the ideal slab (see Fig. 1), heat is uniformlygenerated throughout the slab and the large surfaces(or faces) of the slab are uniformly cooled. The idealslab is infinitely high (x direction) and long (z direc-tion). Thus there is heat removal in the y directiononly, resulting in a 1D temperature gradient. Whilethe birefringence would not depolarize a laser modepropagating along the z axis, it could depolarize abeam propagating along a zigzag path in the yzplane.4,6 However, this depolarization can be pre-vented by cutting the slab from the boule with theappropriate crystallographic orientation.6

Real slabs are not infinitely high and long, nor arethey uniformly pumped or uniformly face cooled.Thermally induced shear stresses thus occur, causingadditional birefringence that depolarizes the lasermode.6 This is particularly relevant for the high gaincw laser heads that are necessary for use with unsta-ble resonators.1,3,7

This paper will investigate thermally induced bire-fringence in practical crystalline Nd:YAG slabs. Whileonly Nd:YAG is considered here, the general principlescan be applied to other crystals with a cubic lattice.The stresses are calculated using a finite-elementmodel (FEM) of a Nd:YAG slab that is side pumpedand side cooled over only part of the side face so as tomaximize gain and enable the use of water seals. Themodel includes temperature-dependent thermal con-ductivity and thermal expansion coefficients. We havepreviously used this model to show that the stresses insuch slabs can be a factor of 3 larger than those pre-

M. Ostermeyer ([email protected]) is with the Instituteof Physics, University of Potsdam, 14669 Potsdam, Germany. D.Mudge, P. J. Veitch ([email protected]), and J. Munchare with the Department of Physics, The University of Adelaide,Adelaide, South Australia 5005, Australia.

Received 21 December 2005; accepted 22 January 2006; posted 3February 2006 (Doc. ID 66829).

0003-6935/06/215368-09$15.00/0© 2006 Optical Society of America

5368 APPLIED OPTICS � Vol. 45, No. 21 � 20 July 2006

dicted by the usual simplified models.8 Here we use theresults to calculate the depolarization resulting fromthe thermal stresses within the slab and compare thepredictions with measurements.

We shall begin by briefly reviewing the preferredorientation of the zigzag slab within the Nd:YAGboule in Section 2, and the photoelastic effects incrystalline Nd:YAG slabs6 in Section 3. The ther-mally induced birefringence is calculated using thedielectric impermeability tensor and the temperatureand stress distributions predicted by the FEM. Jonesmatrices are then used to calculate the depolarizationof the laser mode. This procedure is summarized in

Section 4, and the numerical predictions are dis-cussed in Section 5. The predictions are comparedwith measurements in Section 6. The results showthat our model provides important insight into theeffect of the nonideal pumping and�or cooling profileon thermally induced birefringence and laser modedepolarization.

2. Optimum Orientation of Zigzag Slabs Within theNd:YAG Boule

Nd:YAG crystals are usually grown in a boule usingthe Czochralski method with a �111� growth axis, asshown in Fig. 2. When the boule is pulled from themelt, a conelike shape results at the top of the boule,and growth rings or striations are produced due tominor fluctuations in the chemical concentration andtemperature of the melt. The growth striations areparallel to the surface of the cone in the boule. Thereis also usually a radial gradient in the dopant con-centration.

When a zigzag slab is cut from the boule two condi-tions should be satisfied: The growth striations shouldnot be parallel to the beam propagation for any seg-ment of the zigzag path and the impact of thermallyinduced birefringence should be minimized. The slab isusually cut with the long (z) axis, parallel to the �111�boule axis. In the case of zigzag slabs (a) and (c) inFig. 2, the growth striations are almost parallel to thezigzag for some of the path and thus these orienta-tions are unsuitable. The slab location denoted (b)shows a suitable section of the Nd:YAG boule fromwhich zigzag slabs may be harvested. Figure 3 shows

Fig. 1. Ideal slab geometry showing the axes and notation used inthis paper. The axes are the same as those used by Eggleston et al.(Ref. 4) but different from those used by Lü et al. (Ref. 6).

Fig. 2. Suitable and unsuitable orientations for zigzag slabs in a Nd:YAG boule, elevation (left) and top view (right). The slab (b) in theleft-hand diagram is located at the perimeter of the boule. The double-arrowed line in slabs (a) and (b) in the right-hand diagram, representthe zigzag direction. The �110� and �211� crystal axes and the equivalent directions are marked. The section of the boule from wherecylindrical laser rods are typically harvested is also shown for completeness. The cut angle, �, is measured relative to the [101̄] direction.

20 July 2006 � Vol. 45, No. 21 � APPLIED OPTICS 5369

the image of a HeNe laser beam transmitted throughzigzag slabs that have suitable and unsuitable crys-tallographic orientations; the diffraction due to thegrowth striations is clearly evident. The effect of ther-mally induced birefringence also varies strongly withorientation within the boule, as will be discussed inthe Section 3.

3. Photoelastic Effects in Crystalline Nd:YAG

The optical properties of an anisotropic crystal aredetermined by the dielectric tensor �ij, whereDi � �ij Ej and �ij � �o�1 � �ij� are the well-knownrelationships from electromagnetic theory.9,10

Conservation of energy requires that �ij is symmet-ric. For a transparent, nonmagnetic optical crystal, itis possible to write the surfaces of constant energydensity as an ellipsoid in terms of the principal axesof the dielectric tensor,10

x2

�x�

y2

�y�

z2

�z� 1.

With ni2 � �i��o, this is the well-known ellipsoid of

wave normals, or “optical indicatrix.”It is convenient to define the relative dielectric im-

permeability tensor Bij as the inverse of �ij, resultingin an alternative form of the optical indicatrix9:

Bxx2 � Byy2 � Bzz2 � 1,

or more generally,

Bij xi xj � 1.

Changes in the refractive index due to temperatureand stress can be treated as small changes in Bij

6,11:

Bij � Bo,ij � �ijkl �kl, (1)

where

Bo,ij � �ij��no �dndT �T�x, y, z� � To2

,

�ijkl is the piezo-optic tensor, and �kl is the stresstensor. Thus an applied stress will change Bij and

rotate the indicatrix, leading to a change in the bire-fringence of the crystal.

For a Nd:YAG slab in which the z axis is in the [111]direction, the x axis is at an angle of to the�101̄ direction, and the y axis is at an angle of to the�1̄21̄ direction (see Fig. 2). Equation (1) can bewritten6

where the values of �ij for Nd:YAG (measured at632.8 nm) are6,12

�11 � �0.30285 10�12 m2 N�1,�12 � �0.11158 10�12 m2 N�1,�13 � �0.17187 10�12 m2 N�1,�33 � �0.36313 10�12 m2 N�1,�44 � �0.14693 10�12 m2 N�1,�66 � �0.20722 10�12 m2 N�1,�14 � �0.08525 10�12 cos�3� m2 N�1,�15 � �0.08525 10�12 sin�3� m2 N�1,

Thus for a zigzag slab in which the mode is incidentupon the cooled surfaces with an angle � as shown inFig. 4(a), the relative dielectric impermeability tensorcan be determined by rotating B (to B�) by an angle� � ��2 � � around the x axis,6 giving

B11��B11,

B22��B22 cos2 � � 2B23 cos � sin � � B33 sin2 �,

B12� � B21� � B12 cos � � B13 sin �.

Substituting from Eq. (2) gives

B12� � cos ��23�15 � �13�14 � �12�66� sin ��11 � �22��15 � �13�44 � �12�14

� cos ��yz�15 � �xz�14 � �xy�66� sin ��xx � �yy��15 � �xz�44 � �xy�14. (3)

For an ideal zigzag slab, B12� � sin ��11 ��22��15 sin�3�, and thus depolarization can be pre-vented by using a cut angle, , that is a multiple of60°. That is, the y axis of the slab could be in, forexample, the �1̄21̄ direction.

B11

B22

B33

B23

B13

B12

�

B0,11

B0,22

B0,33

000

�

�11 �12 �13 �14 �15 0

�12 �11 �13 ��14 ��15 0�13 �13 �33 0 0 0�14 ��14 0 �44 0 ��15

�15 ��15 0 0 �44 �14

0 0 0 ��15 �14 �66

�11

�22

�33

�23

�13

�12

, (2)


4. Mode Depolarization in a Realistic Zigzag Slab

The temperature and stress distributions in ourreal zigzag slab are determined using a FEM thatincorporates realistic pump light absorption, real-istic cooling, and temperature-dependent thermalconductivity and thermal expansion coefficients.The slab width is chosen to ensure efficient absorp-tion of the pump light, low thermal resistance, andgood mechanical stability. In this paper we considera 1.1 at. % doped Nd:YAG coplanar folded zigzagslab13 (CPFS) that is side pumped and side cooled.Figure 4(a) shows a top view of the pump architec-ture and the beam path through a CPFS. Figure4(b) defines the geometrical parameters used in themodel. The slab is coated with Teflon AF (Ref. 14) toprevent disruption of the total internal reflection(TIR) of the zigzag mode due to water seals at theside faces. The temperature in the slab is greaterthan the cooling water temperature �298 K� becauseof poor thermal conductivity of the Teflon coating andthe limited heat removal efficiency of the flowing wa-ter. This will be discussed further in Section 5. Thedesign of this laser head and comparison of the pre-dictions of the FEM with analytical results have beenreported in detail elsewhere.8,15

We have constructed a 2D and a 3D FEM. The 2Dmodel can be used to calculate the temperature andstress distributions at high resolution for a cross sec-tion perpendicular to the z axis. End effects cannot bemodeled with the 2D model, and we thus use thelower resolution 3D model to investigate the effect ofthe finite length of the slab.

The calculated stress distribution is used to de-termine the components of the relative dielectricimpermeability tensor for the zigzag path. Thechange in the Bij components due to the temperaturedependence is neglected as it is small and will havenegligible effect on the eigenvalues and eigenvectorsof the B submatrix. Jones matrices are used to cal-culate the change in polarization of a mode. In zigzagand CPFS slabs, this mode is usually � polarized. Asdiscussed in Section 3, for an ideal slab there will bea 1D temperature gradient in the y direction. As a

consequence, the y-stress components are zero andthe stresses in the x and z directions are equal.16 Thuswe consider only the depolarization as a function ofheight (x direction). We also assume that the probebeam is collimated in this analysis.

In detail, the calculation includes the followingsteps:

(i) The beam propagation between adjacent totalinternal reflections is divided into 20 propagationsteps of length �. At each step, the relative dielectricimpermeability tensor, B�, is calculated from thestress distribution at 101 locations along the vertical(x) direction.

(ii) At each location, the eigenvalues, ��, and(unnormalized) eigenvectors, uh�, of the submatrix ofB� that is transverse to the beam propagation direc-tion,

Fig. 3. Imaged transmission at the exit face of a zigzag slab withsuitable (left) and unsuitable (right) orientations. The growth stri-ations in the right image are visible as some parts of the zigzagpath is almost parallel to the growth striations.

Fig. 4. (a) Pump architecture and beam path through a side-pumped, side-cooled CPFS laser. The waveguides smooth theintensity profile of the pump distribution in the plane of the pagebut preserve the numerical aperture of the pump light in theperpendicular (x) direction. (b) Geometrical parameters used inthe FEM of the side-pumped, side cooled CPFS. The slab ispumped and cooled on both sides. wpump is the full height of thepump profile in the slab. Lp is the length of the pumped region.The cooling water flows along a channel that is wcool high and Lp

long. All other slab surfaces are assumed to be insulated.


�B11� B12�

B12� B22��,

are calculated using

�� 12 ��B11� � B22��

� �B11� � B22��2 � 4B11�2, and

u� �� 1

�� B11�

B12��.

The refractive indices corresponding to these eigen-values are given by n� � 1� ��.

(iii) The Jones matrices in the coordinate systemdefined by the eigenvectors are calculated at eachlocation within the step using

J � �exp��ikn�� 00 exp��ikn��,

where k is the wavenumber.(iv) These Jones matrices are transformed into

the zigzag (primed) coordinate system using

J� � R��JR��,

where

R�� cos � sin �

�sin � cos ��and �, the angle between the x axis and the u� di-rection, is calculated using

cos � �

u� · �10��u��

� 1��1 � �� B11�

B12��2�1�2

�� 0 corresponds to the �u�, u�� coordinate systembeing rotated anticlockwise relative to the �x�, y�� co-ordinate system].

(v) We assume that the Jones matrix for the TIR,JTIR, is given by

where nr is the isotropic refractive index ofNd:YAG.

(vi) The total Jones matrix, Jtotal, for the propaga-tion though the gain medium is calculated using

Jtotal � � �m�p

1�J20

m�J19m�, . . . , J1

m�JTIR�� J20

m�, . . . , J11m�JTIRJ10

m�, . . . , J1m��

� �m�1

p�JTIRJ20

m�J19m�, . . . , J1

m��,where p is the number of total internal reflections oneach side face [eg, p � 9 in Fig. 4(a).] The total Jonesmatrix consists of three parts: From the Brewster-angled input face to the TIR just before the blunt endof the slab, the TIR at the blunt end, and from the TIRjust after the blunt end to the output Brewster face.

(vii) The transmitted electric field is calculated usingEout � JtotalEin and the depolarization loss evaluated.

5. Numerical Results

As discussed in Section 3, there will be no depolarizationof a �-polarized mode in an ideal zigzag slab that hasa suitable crystallographic orientation. However, thefinite height and nonuniform pumping of a practicalzigzag slab result in shear stresses that cause depo-larization. In this section, we discuss the impact ofthe finite slab height (or aspect ratio), cut angle, andthe nonuniform pumping and cooling of a zigzag slab.This is done by incorporating the stresses predictedby a 2D FEM17 into the birefringence calculation de-scribed in Section 4. We define the transmission asthe fraction of a �-polarized input beam transmittedthrough a �-oriented polarizer (analyzer) placed afterthe slab. The �-polarized probe laser beam is as-sumed to have a top-hat profile and a diameter equalto that of the pump beam.

A. Depolarization Due to Finite Slab Height

The predicted maximum temperature, Tmax, stresses�xx,max, �yy,max, and shear stress �xy,max for infinitelylong, optimum cut angle, uniformly pumped, andface-cooled slabs with heights of h � 4.3 mm andh � 43 mm are listed in Table 1. Contour plots of thetemperature, �xx and �xy for the h � 4.3 mm slab areshown in Fig. 5.

JTIR � �sin�� i cos��2 � nr

2

sin�� i cos��2 � nr2

0

0nr

2 sin�� i cos��2 � nr2

nr2 sin�� i cos��2 � nr

2�,


The edge temperature of 354 K is higher than thecoolant due to the poor thermal conductivity of thecooling interface as discussed earlier. While it is in-teresting to note the large reduction in �yy,max owing tothe increase in slab aspect ratio, we are more con-cerned with the maximum value of the shear stress

�xy,max, which decreases only by a factor of approxi-mately 2. However, the nonzero shear stresses in thelarge aspect-ratio example are concentrated near thetop and bottom edges of the slab, as discussed furtherin the following sections. The shear stresses occupy asmaller fraction of the slab cross section and are fur-ther from the center, compared to those for the smallaspect-ratio slab. That is, the depolarization lossesdue to shear stresses can be reduced by increasingthe slab aspect ratio.

B. Depolarization Due to Nonoptimum Cut Angle

The effect of the cut angle on the transmitted in-tensity for infinitely long, uniformly pumped, andface-cooled slabs with heights of h � 43 mm andh � 4.3 mm is plotted in Figs. 6 and 7, respectively.The same pump density was used for both slabs. Notein Fig. 6 the concentration of the depolarization nearthe top and bottom edges of the slab, which is inagreement with the previous discussion, and the lowdepolarization near the center for the 0° cut angle.

Fig. 5. (Color online) (a) temperature, (b) �xx, and (c) �xy for theh � 4.3 mm high slab.

Fig. 6. Predicted transmission through parallel polarizers for aninfinitely long, uniformly pumped, and cooled slab, h � 43 mm andw � 3.0 mm. Cut angles � � 0° and 30°.

Fig. 7. Predicted transmission through parallel polarizers for aninfinitely long, uniformly pumped, and cooled slab, h � 4.3 mm andw � 3 mm. Cut angles � � 0°, 10°, and 20°.

Table 1. Maximum Temperature and Stresses for Two Infinitely Long,Optimum Cut Angle, Uniformly Pumped and Face-Cooled Slabsa

Slab height(h)

Tmax

[K]�xx,max[Mpa]

�yy,max[Mpa]

�xy,max[Mpa]

43 mm 431 130 16.4 13.34.3 mm 431 146 108 28

aThe slabs have heights of 4.3 and 43 mm. The width is w �3 mm and the pump density is 672 MW�m3. �xz � 0 and �yz � 0 asexpected.


There is no extended low depolarization regionwithin the small aspect-ratio slab, even for � 0°,and the depolarization increases as approaches30°. For � 60° (not shown in Figs. 6 and 7),the result of � 0° is reproduced, as expected.

The dependence of the integrated depolarizationloss on cut angle for the small �h � 4.3 mm� and large�h � 43 mm� aspect-ratio slabs is plotted in Fig. 8. Inboth instances a cut angle of 0° or 60° is best. For theh � 4.3 mm slab, the transmission is limited to �90%due to the depolarization resulting from the high xyshear stress. For the h � 43 mm slab, the 3 depen-dence can be observed, and almost 100% transmis-sion is achieved at multiples of 60°.

C. Depolarization Due to Nonuniform Pumpingand Cooling

Until now, the examples used have assumed uniformpumping and face cooling. However, in practice, theside pumping is not uniform, and in our case moreappropriately represented by a Gaussian profile. Fur-thermore, if water cooled, the slab requires the use of

a gasket or O-ring, and thus the entire slab face is notuniformly cooled. This section considers the impacton shear stress and birefringence resulting from thistype of pumping and cooling, assuming that � 0°.

The transmission profiles for the three differentpump and cooling heights are shown in Fig. 9. Theseexamples use Gaussian pump profiles of heights 1.4,2, and 3 mm, and are labeled gw11, gw22, and gw33,respectively. The width of the central, low depolar-ization zone in Fig. 9 broadens as the pump andcooling channel becomes larger. However, the broad-ening is small compared to the increase in the pumpand cooling dimension. Predicted values of shearstress within the slab and the transmission of a top-hat probe beam are shown in Table 2. The width ofthe probe beam used is the same as the full width ofthe Gaussian pump beam, wpump. Although the xyshear stress decreases as wpump increases, the trans-mission also decreases. This is due to the probe beamtraveling through regions of depolarization near theedges of the pump region where the shear stressesbuild up. Thus to minimize the depolarization loss,pump and probe beams that are less than 50% of theheight of the slab should be used, providing that theslab does not fracture.

Fig. 8. Integrated transmission through parallel polarizers as afunction of the cut angle, for the h � 43 mm and h � 4.3 mm slabs.

Fig. 9. Predicted transmission through parallel polarizers for dif-ferent pump and cooling heights and a Gaussian pump profile. Theparameters for gw11, gw22, and gw33 are given in Table 2. Slab h� 4.3 mm and w � 3 mm.

Fig. 10. Predicted transmission through parallel polarizers re-sulting from a top-hat pump profile (uw22) and a Gaussian pumpprofile (gw22), for slab h � 4.3 mm. The parameters for uw22 andgw22 are given in Table 3.

Table 2. Maximum Stress Levels for Slabs with Different Pump andCooling Heights, for slab h � 4.3 mm and w � 3.0 mma

SlabPumpprofile

wcool

[mm]wpump

[mm]�xx,max[MPa]

�xy,max[MPa]

T[%]

gw11 Gaussian 1.4 1.4 144 22.6 98.5gw22 Gaussian 2 2 128 20.7 98.5gw33 Gaussian 3 3 109 17.7 91.5

aThe model assumes an incident pump power of 500 W, with354 W being absorbed in the single pass resulting in 85 W of heating(due to the quantum defect of Nd:YAG). wcool denotes the height ofthe water channel, and wpump is the diameter of the Gaussian pumpprofile in the vertical (x) direction [see Fig. 4(b)]. T denotes theintegrated transmission of a top-hat beam with diameter wpumppassing through the zigzag slab located between parallel polarizers.


Changing the pump profile to a top hat moves theshear stresses further from the center of the slab, asshown in Fig. 10, but does not significantly changethe depolarization loss for the small aspect-ratio slab.The slab with the top-hat pump profile is labeleduw22. The improvement is more significant in higheraspect-ratio slabs with higher pump regions, asshown in Table 3 and Fig. 11. This large reduction indepolarization loss for the top-hat pump will enablethe lasing mode to extract energy more efficientlyfrom the edges of the pumped volume, thus improv-ing lasing efficiency.

6. Comparison with Measurements

A zigzag CPFS slab, shown in Fig. 4, with parametersgiven in Table 4, was used for the measurements. Itwas pumped using fiber-coupled diode lasers, andplanar waveguides were used to smooth the intensityprofile of the pump distribution in the z direction. Thedesign of this laser has been reported in detail else-where.8,15 The pump light emerging from the wave-guide was measured and is well described by a

Gaussian profile with a divergence equal to the 0.22numerical aperture of the pump fibers.

The experiment configuration used to measure thedepolarization is shown in Fig. 12. To distinguishbetween the gain and depolarization loss we measurethe depolarization using a 632.8 nm HeNe probebeam that filled the slab. While the HeNe wavelengthwill not experience identical depolarization to the las-ing wavelength (owing to a small shift in refractiveindex and possible weak wavelength dependence ofthe �ij terms9), it is a useful and convenient method todetect the presence and spatial distribution of depo-larization and thus provide qualitative validation ofthe FEM. The slab is located between crossed polar-izers, P1 and P2, and the transmitted probe beamwas imaged onto the CCD. The depolarization wasmeasured while the gain medium was lasing to en-sure that the heat deposition was dominated by pumpabsorption and to relate the measurements to ob-served lasing efficiency.

The measured vertical (x) transmission throughcrossed polarizers is compared to the predictions inFig. 13, showing good qualitative agreement. Regionsof high transmission indicate strong depolarization ofthe probe beam. Both show a double lobe structure withsignificant birefringence at the edge of the pumped re-gion producing �50% transmission through crossedpolarizers. The asymmetry between the measuredtransmission at x � �1.0 mm and x � 1.0 mm is dueto asymmetric thermal lensing within the slab, re-sulting from nonideal boundary conditions. The mea-surements also indicate the emergence of two smallerlobes of birefringence near the top and bottom edgesof the slab, as is predicted by the FEM. The two smalllobes measured near x � 0 emerge at lower pumppowers than predicted by the FEM. The distortion inthe horizontal scale of the measurements is probably

Fig. 11. Predicted transmission through parallel polarizers re-sulting from a top-hat pump profile (uw66) and a Gaussian pumpprofile (gw66), for slab h � 8.6 mm. The parameters for uw66 andgw66 are given in Table 3.

Fig. 12. Experiment used to measure the depolarization of aHeNe probe beam. P1 and P2, input and output crossed polarizers;IF, interference filter; PM, power meter; IL, imaging lens; HR, highreflectivity laser mirror at 1064 nm; CCD, charge coupled-devicecamera; PR, partially reflective laser mirror at 1064 nm; HRHT,dichroic beam splitter having high reflectivity at 1064 nm and hightransmission at 632.8 nm.

Table 3. Maximum Stresses for Slabs (h � 4.3 mm and h � 8.6 mm)with Different Pump and Cooling Geometries Using Top-Hat (uw) and

Gaussian (gw) Pump Profilesa

SlabPumpprofile

h[mm]

wcool

[mm]wpump

[mm]�xx,max[MPa]

�xy,max[MPa]

T[%]

uw22 Top hat 4.3 2 2 122 20.2 98gw22 Gaussian 4.3 2 2 128 20.7 98.5uw66 Top hat 8.6 6.0 6.0 144 22.6 99gw66 Gaussian 8.6 6.0 6.0 109 17.7 90

aThe model assumes an incident pump power of 500 W, with354 W being absorbed in the single pass resulting in 85 W ofheating (due to the quantum defect of Nd:YAG). h the height of thecrystal, wcool denotes the height of the water channel, and wpump isthe diameter of the pump profile in the vertical (x) direction [seeFig. 4(b)]. T denotes the integrated transmission of a polarizedtop-hat beam with diameter wpump passing through the zigzag slablocated between parallel polarizers.

Table 4. Dimensions of the Side-Pumped, Side-Cooled Zigzag CPFSSlab Used in the Experiment

Pumpprofile

L[mm]

Lp

[mm]h

[mm]w

[mm]wcool

[mm]wpump

[mm]Cut

angle �

Gaussian 32.9 22 4.3 3.0 2.0 2.0 0°


due to focusing of the probe beam by the thermal lens,which was significant at pump powers greater thanseveral tens of watts. This lensing was not includedin the model.

7. Conclusion

Thermally induced birefringence can significantly re-duce the efficiency of high power zigzag slab lasers.The FEM described in this paper provides useful cri-teria for the design of the gain medium for such la-sers. The loss due to depolarization is minimized byusing slabs that have the correct crystallographic ori-entation and have large aspect ratios. Further im-provement is gained using top-hat pump profiles andlaser modes that do not extend to the top and bottomedges of the slab. Using a top-hat pump profile ratherthan a Gaussian pump profile significantly reducesthe depolarization encountered by the laser mode andincreases the volume of the slab in which the depo-larization is low. If a Gaussian pump profile is usedthen it is essential that the pump and laser mode areconcentrated near the center of the gain medium,

away from the regions where the shear stresses areconcentrated.

This research was supported by the AustralianResearch Council. M. Ostermeyer was supportedby the Alexander von Humboldt Foundation. Wegratefully acknowledge useful discussions concern-ing the propagation algorithms with AlexanderHemming of The University of Adelaide.

References and Notes1. D. Mudge, P. J. Veitch, J. Munch, D. Ottaway, and M. W.

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Fig. 13. Predicted (upper) and measured (lower) transmissionprofiles through crossed polarizers for the Nd:YAG slab shown inFig. 4. The measured depolarization used the setup shown inFig. 12. High transmission indicates strong depolarization.


thermally induced birefringence in nd:yag slab lasers

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