Analysis of temperature and thermo-optical properties in optical materials. 1: Cylindrical geometry

Optical Materials 33 (2010) 48–57

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Analysis of temperature and thermo-optical properties in optical materials. 1: Cylindrical geometry Michael M. Tilleman * Elbit Systems of America, 220 Daniel Webster Highway, Merrimack, NH 03054, USA

a r t i c l e

i n f o

Article history: Received 29 April 2010 Received in revised form 21 June 2010 Accepted 26 July 2010 Available online 24 August 2010 Keywords: Thermo-optic materials Laser Temperature Nonlinear Kirchoff’s transformation

a b s t r a c t Modeling of optical and electro-optical devices requires the implementation of material properties over a broad temperature range. Because optical, thermo-optical, elasto-optical and gain properties of solidstate laser materials depend on temperature, their exact magnitude is needed for designing such optical devices. Derived in this paper is a closed form solution to the problems of nonlinear heat transfer and stress field, resulting in expressions for the local temperature, stress and strain, refractive index, trajectories of propagating rays, optical path difference, thermal lensing, tilt and third order aberrations, induced birefringence and depolarization. In the analysis the temperature dependent coefficients were best fitted to existing experimental data. Some examples are presented for thermally induced optical effects in solid-state laser gain-media in the temperature range of 77–770 K. It is found that: for large heat deposition rates the use of the nonlinear solution is uniquely necessary to assess the thermal and optical characteristics, high pumping loads require cryogenic cooling to maintain low thermal lensing and thermally induced dioptric power depends quadratically on the heat rate. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction The ability to accurately predict the propagation of light beams through an optical material is an imperative tool for laser designers. In this paper an analytical solution, first of its kind to the best of our knowledge, is found to the ray trajectory and fields of refractive index, stress and temperature in solid media spanning a wide temperature range between 77 and 770 K. The formalism is based on solving the nonlinear thermal equation having thermal conductivity dependent on temperature, fitted to measured data. The closed form solution arrived at offers the benefits of accuracy, intuitive insight to the thermo-optical phenomena and fast convergence. Optical and radiative properties of host materials for solid-state lasers, saturable absorbers and other electro-optical activities are strong functions of temperature. Crystals such as yttrium aluminum garnet (Y3Al5O12 or YAG), yttrium lithium fluoride (LiYF4 or YLF), yttrium orthoaluminate (YAlO3 or YALO) and lutetium aluminum garnet (Lu3Al5O12 or LuAG), among others that are one of the main building blocks of high-power lasers, possess a refractive index whose thermal conductivity increases with temperature and emission cross-section that decreases with temperature. The dependence on temperature is further increased at low temperatures, say between cryogenic and room temperature, where many of modern high-power lasers operate [1–4]. That hinges on the * Tel.: +1 603 886 2003. E-mail address: [email protected] 0925-3467/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.optmat.2010.07.015

strong dependence of some laser material properties; such as refractive index (dn/dT), thermal expansion coefficient, specific heat, thermal conductivity and thermal diffusivity on temperature. While dn/dT and thermal expansion increase with temperature, thermal conductivity and diffusivity decrease with temperature, leading to significant improvement in the performance of solidstate lasers operating at low temperatures. Many of the optical, thermo and elasto-optical properties between cryogenic and room temperatures have been measured suggesting dramatically lower thermal lensing and induced birefringence [5–10]. The optical, thermo and elasto-optical properties depend on temperature even at higher temperatures, say between room temperature and 500 °C [11–15]. Therefore, the scalability of such lasers: their output power, efficiency, thermal focusing, induced birefringence and beam quality, depend strongly on the operating temperature. Other electro-optically active materials, for instance those used as saturable absorbers [16] or harmonic generators [17], may also possess thermal conductivity coefficients depending on temperature. In this paper the nonlinear steady-state thermal problem in a cylinder is solved via Kirchoff’s transformation which transforms the nonlinearity of the temperature into a linearly solvable function [18–20]. Further, solved is the elastic compatibility equation permitting the stress and strain analysis in the medium. The solution encompasses the cases of side and end pumping. For each case temperature distribution, refractive index, elastic stresses and elongation, optical path difference, thermally induced lensing and birefringence are formulated.

M.M. Tilleman / Optical Materials 33 (2010) 48–57

The analysis of steady-state radiative and thermal processes begins by defining the governing equation and the boundary conditions. The governing Poisson equation is expressed as:

r  kðrTÞ þ Q ¼ 0

ð1Þ

where k = k(T) is the thermal conductivity and Q is the heat source term defined as deposited power per unit volume. Since most solidstate optical devices are cut either as rectangular parallelepipeds or cylinders either a Cartesian or polar coordinate systems need be considered as necessary. 2.1. Thermal conductivity Thermal conductivity is a product of the material density, thermal diffusivity and specific heat:

k ¼ qcv b

ð2Þ

Since the density of solids varies very little with temperature, it is instructive to consider the dependence of the thermal diffusivity and specific heat on temperature. Thermal diffusivity of crystals is expressed as [5]:

bðTÞ ¼ bR

eT D =bT  1 eT D =bT R  1

ð3Þ

where bR is a reference thermal diffusivity at a reference temperature TR, TD is the Debye temperature (373 K for YAG) and b is a constant (2.05 for YAG). The specific heat is well modeled by the Einstein’s and Debye’s theories for specific heat assuming the form [5]:

cv ¼



T TD

3 Z 0

T D =bT

ex x4 ðex

 1Þ

2

dx

ð4Þ

where x is the variable of integration. It can be seen that for temperatures much lower than TD the coefficient of thermal conductivity k decreases very fast with temperature varying as product of an exponent and cube temperature. Then, k tapers off at temperatures much higher than TD decreasing inversely with temperature. Doping of host crystals by lasent materials such as neodymium, ytterbium or other rare-earth dopants changes the thermo-optic properties including k. This fact has been well established in Yb concentration levels from 0% to 15% by the measurement published in Ref. [5]. As demonstrated in Ref. [23] the dependence on the dopant concentration over the entire range of Yb concentration from 0% to 100% is both strong and non-monotonous. For instance, the thermal conductivity drops by several tens percent reaching a minimum of 30% of the undoped magnitude for 50% doping of YAG by Yb. This has important implications on the optical properties particularly for lasers in which the dopant level is high, such as erbium doping of YAG, YSGG, YLF applicable to laser transition of the Er3+ ion from level 4I11/2 to 4 I13/2 emitting radiation at the wavelength of 2.8–2.94 lm. Note that low dopant concentration, 0.25–0.5% Er:YAG is applicable to laser transition of the Er3+ ion from level 4I11/2 to 4I15/2 emitting radiation at the wavelength of 1.64 lm. The validation of the model is accomplished by comparison with the existing body of measurements in the wide temperature range of 77–770 K [1–15,21–25]. Notwithstanding that in the vicinity of cryogenic temperature thermo-optic properties depend strongly on temperature, the temperature range above room temperature must be also included in the analysis of laser crystals thermal response. This presents an extensive variation of k that well justifies considering its temperature dependent nature in the solution of the Poisson equation. In addition to the dependence

on temperature thermal conductivity of Yb:YAG exhibit inverse dependence on Yb doping at levels up to 40% [23]. Fig. 1 plots the inverse coefficient of thermal conductivity as a function of temperature based on the above cited data and the theory of Eqs. (2)– (4). In inspecting the k data for Yb:YAG at the ambient temperature range of 300–323 K one finds a disagreement by a factor of up to 2. This can be explained by that conductivity decreases with the doping level in YAG, and is further modified by the presence of impurities. In view of the mentioned above it is difficult to fit an analytical curve to the entire body of data. In the temperature range of 50– 300 K both the theoretical calculation and fitting functions used by Brown [8] or Pfistner et al. [26] well agree with the data. However, one observes that the theory of Eqs. (2)–(4) fails to predict measured data below 50 K yet agrees with the data at 770 K while Brown’s approximation agrees excellently with data below 50 K yet strongly disagrees with the data above 300 K. Then, any function reasonably fitting both ranges leaves out some data in the stitching zone between temperatures below and above ambient. Suggested in this paper is an inverse linear approximation:

kðTÞ ¼ k0

T0 T

ð5Þ

where k0 is the coefficient of thermal conductivity at T0 whose values for several materials are summarized Ref. [4]. This simple function, similar to that suggested for Nd:YLF in Refs. [26,27], closely fits the theory of Eqs. (2)–(4) agreeing very well with the data in the range between cryogenic and room temperature as well as with the data at 770 K. It is worth mentioning that the above approximation holds well for doping levels up to 5 at.% Yb, while for higher doping levels power approximations become a better fit. 2.2. Temperature distribution – the nonlinear Poisson equation Let us consider the Poisson equation of Eq. (1) in cylindrical coordinates using the expression for k of Eq. (5):



rr;u;z  k0

 T0 ðrr;u;z TÞ þ Q ðr; u; zÞ ¼ 0 T

ð6Þ

Consistent with Kirchoff’s transformation [12] this equation can be rewritten as:

1 T k0 T 0 rr;u;z  ðrr;u;z TÞ þ Q ðr; u; zÞ ¼ k0 T 0 r2r;u;z ln þ Q ðr; u; zÞ T T0 ð7Þ and finally one arrives at:

K r2r;u;z hðr; u; zÞ þ Q ðr; u; zÞ ¼ 0

ð8Þ

1/k (cm·K/ W)

2. Analysis

49

Fig. 1. Inverse coefficient of thermal conductivity as a function of temperature including data, theoretical curve for undoped YAG, Brown’s model and k / 1/T approximation considered in this study.

50

M.M. Tilleman / Optical Materials 33 (2010) 48–57

This is a linear equation in h where:

hðr; u; zÞ ¼ ln

Tðr; u; zÞ T0

ð9Þ

and K is a constant:

K ¼ k0 T 0

ð10Þ

Obviously though the above notation is cylindrical, this linearization holds true for any orthogonal coordinate system. It should be emphasized that Eq. (8) represents any steady-state heat equation, where h and K may be an arbitrary representation of any compound temperature-heat conduction function. It degenerates to the linear case where the heat conduction coefficient is independent of temperature. Assuming an optically pumped laser device, the heat source is generated by a fraction of the optical radiation absorbed in the material according to Beer’s absorption law. For the purpose of creating a gain zone in a laser medium one plans to maximize the absorbed power in the optical path of the intended laser beam. This can be accomplished by side pumping the active material or by end pumping along the path of the intended laser beam. In many practical instances in order to maximize the generated laser power the pump beam is focused in the optically active material. Ideally a considerable fraction of the absorbed beam at the focus is converted to laser, based on quantum resonant transitions in the optical material of interest. The conversion rules for such process render the generated laser photon having lower photon energy than the pumping photon, a discrepancy referred to as quantum defect. Together with luminescence trapped in the lattice and vibronic relaxation processes, the pump power that has not been converted to laser constitutes the heat source for this problem. This phenomenological heat deposition may be expressed as:

Q ðr; u; zÞ ¼

gPabs f ðr; u; zÞ pr2P Leff 0

ð11Þ

where Pabs is the absorbed pump power in optical material, g is the fraction that has not been converted to the generated laser, rP is the effective radius of the pump zone in the rod, Leff is the effective length of the optical-material rod or slab and f0 is the spatial distribution function of the absorbed power. The boundary conditions for the problem are prescribed by a cooling mechanism, which may consist of a coolant fluid or a solid heat sink, and physical surroundings of the device. Typically at least one side of the optically active crystal is held in good thermal

contact with the cooling substance. Being either held in vacuum or exposed to gas the other sides of the medium may be considered insulated due to lack of any considerable heat transfer. At the areas of the rod through which heat flow occurs, say at r = rW, one may specify the boundary by a known temperature, i.e. a Dirichlet condition:

Tðr W ; u; zÞ ¼ T 0

ð12Þ

Otherwise, in the case the heat flux out of the system is known, the boundary is specified by a Neumann condition:

k

@T ¼q ^ @n

ð13Þ

and on all the insulated sides the heat flux is zero, thus:

@T ¼0 ^ @n

ð14Þ

2.2.1. Solution for cylindrical rods For active optical media such as gain medium having the shape of a cylindrical rod it is most convenient to treat Eq. (8) in polar coordinates, where consistent with Kirchoff’s transformation [18] it is expressed as:

K @ @ K @2 @2 r hðr; u; zÞ þ 2 hðr; u ; zÞ þ K hðr; u; zÞ þ Q ðr; u; zÞ ¼ 0 r @r @r r @ u2 @z2 ð15Þ 2.2.2. Rod of finite length This case is illustrated in Fig. 2(a). Sketched is a rod held by two conducting mounts at the ends, surrounded on its perimeter by laser diodes. In order to approximate cases of: (1) rod with evenly spaced side pumping and varying axial pump power or boundary condition, and (2) end pumping with arbitrarily distributed source or boundary conditions, it is sufficient to assume an axisymmetrical case with a finite rod. Then the governing equation becomes:

K @ @ @2 r hðr; zÞ þ K 2 hðr; zÞ þ Q ðr; zÞ ¼ 0 r @r @r @z

ð16Þ

2.2.2.1. Side pumping. The boundary conditions for a rod may model various cooling configurations including conductive and convective means. A current trend in the solid-state laser industry is to produce lasers in which the rod is cooled by conduction, realized

Fig. 2. Two typical cylindrical devices: (a) rod held by heat sink mounts on both ends with radially symmetrical side pumps, and (b) rod held by a heat sink along its length with end pump.

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M.M. Tilleman / Optical Materials 33 (2010) 48–57

sometimes by a heat sink with a mount holding the rod only over part of its length. These configurations on the radial boundary justify the assumption of either Dirichlet or Neumann boundary conditions or their combination specified around the rod circumference:

 @Tðr; zÞ Tðr W ; zÞ ¼ f1 ðzÞ or kðTÞ @r 

r¼r W

 @hðr; zÞ ¼K @r 

(1) uniform heat source distribution where:

f0 ðrÞ ¼

$

 2 % r f0 ðrÞ ¼ exp 2 rP

r¼r W

ð17Þ The Dirichlet boundary condition is particularly suitable for cases where the boundary is held at a set temperature such as the case of heat sinking to Peltier junction or cryogenic cooling. In terms of the function h it becomes:

Considering a Dirichlet boundary condition for case 1 one may solve the problem for h  hW, whereby the first term in Eq. (21) becomes hW and the solution is expressed as:

hðr; zÞ ¼ hW þ

ð18Þ

¼ hW þ

ð19Þ

where the origin is at the rod center and z = L/2 is half the rod length. Modeling of the source assumes a gain region confined both radially and axially inside the rod:

Qðr; zÞ ¼

8 <

gPabs f ðrÞ; pr2P l=2 0

: 0;

0 6 z 6 2l l 2

Z

hðr; zÞ ¼

þ

1 K

rW

Z

Qðn; fÞGðr; z; n; fÞn dn df

ð21Þ

 z



 f

1 X 1 J 0 ðls;m rÞJ 0 ðls;m nÞ cos 2np L cos 2np L 2 X h   i l2s;m þ 2nLp 2 J2s ðls;m rW Þ

r2W L m¼1 n¼1

ð22Þ where the subscript s assumes the value of either 0 or 1 corresponding to a Neumann or Dirichlet type boundary condition, respectively. The coefficients lm are the roots of the equation:

J0 ðlm rW Þ ¼ 0;

1 X 1 X

p

r 2W r P LK m¼1 n¼1

1;m

qB rW K

Z

1

L

ð24Þ

1;m

L=2

G0 ðr; z; r W ; fÞ df

S=2

0;m



  d J 0 ðlm rÞ ¼ 0; dr rW

G1 ðr; z; n; fÞ dfn dn 0

0;m

0

L

0;m

L=2

where the function f3(f) assumes either f1(f) or f2(f) and the function V(r, z, rW, f) is either oG(r, z, n, f)/@n at n = rW or G(r, z, rW, f), depending on the type of boundary condition around the rod circumference (Dirichlet or Neumann). Green’s function is constructed by solving the homogeneous equation (16), or the Laplace equation, satisfying the specified boundary conditions of Eqs. (17) and (19), and by the function holding throughout the domain [20]. Thus G takes the form as demonstrated in Ref. [20]:

Gs ðr; z; n; fÞ ¼

l=2

Z Z 2gPabs rp l=2 G ðr; z; n; fÞ dfn dn 2 prP lK 0 0 0   1 1 l1;m J0 ðl1;m rÞ q S XX S h i sin c np ¼ B L l2 þ 2np2 J ðl rW Þ K rW L m¼1 n¼1 1 1;m 1;m L 1 X 1

 X z gPabs  cos 2np þ 2 L prW rP LK m¼1 n¼1     J ð l 1 l0;m r p ÞJ 0 ðl0;m rÞ cos 2np Lz h i  sin c np ð25Þ   L l l2 þ 2np 2 J2 ðl rW Þ


0

0

0

Z

þ

ð20Þ

f3 ðfÞVðr; z; r W ; fÞ df Z

rp

Considering a Neumann boundary condition for case 1, and assuming a case where the heat is relieved from the rod by conductive mounts holding the rod perimeter between ±S/2 and ±L/2 in Eq. (21), the solution is:

L=2

0

gP abs

Z

1;m

where l is the length of the gain zone in the rod, i.e. the length of the pump region, such that l 6 L. To solve the set of Eqs. (16)–(19) it is convenient to employ the Green’s function G(r, z) in which case for the mixed boundary conditions the general solution becomes:

hðr; zÞ ¼ rW

2gPabs pr2P lK

    l J 1 ðl1;m rp ÞJ 0 ðl1;m rÞ cos 2np Lz h  sin c np   i L l l2 þ 2np 2 J2 ðl rW Þ

Then, the Neumann condition is suitable for cases where the boundary provides a certain heat flux across it, as specified in Eq. (17). At the rod ends the facets are assumed insulated, specified by Neumann condition, such that:

  @h @h ¼  ¼0  @z z¼0 @z z¼L=2

1; 0 6 r 6 rP 0; r P < r

(2) Gaussian heat source profile:

¼ f2 ðzÞ

f1 ðzÞ hðr W ; zÞ ¼ ln T0



s¼0

ð23Þ

s¼1

To present a complete solution one still needs to define the functions f0(r) used in Eq. (20). Let two cases be considered:

where qB is the heat flux through the cooling mounts. For a very long rod the solutions in Eqs. (24) and (25) approach the asymptotic solution for a two dimensional, axisymmetrical geometry, which for the Dirichlet boundary condition becomes: 

8

 gPabs > > < rW 2pLeff K

exp gPabs rP TðrÞ ¼ T 1 þ gPabs 2pLeff hrW > > : rW 2pLeff K r

;

h

gP abs 4pLeff K



2

1  rr2

i ; 0 6 r 6 rP

P

rP < r 6 rW ð26Þ

For the heat source of case 2 with Gaussian radial distribution the integrand becomes a product of Bessel function of the first kind of order zero and Gauss function. Then, the integral in the second term of Eq. (21) becomes:

Z 0

rW

   n n dn J 0 ðlm nÞ exp 2 rP

ð27Þ

which has a closed form solution strictly for the case of rW ? 1, being [32]:

" # r2P ðlm r P Þ2 exp  4 8

ð28Þ

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M.M. Tilleman / Optical Materials 33 (2010) 48–57

It turns out that for rW/rP = 1, 1.25, 1.5, 1.75 the above result has an error of 15%, 6%, 1.5% and less than 1% relative to a numerical approximation, respectively. Because in real scenarios rW/rP > 1, this result is applicable to the present solution assuming a Dirichlet boundary condition, yielding:

Considering the Neumann boundary condition the solution is expressed as:

hðr; zÞ ¼

1 X 1 gP abs X 2 4pr W LK m¼1 n¼1

hðr; zÞ ¼ hW þ

0 1 X 1 q S X ¼ B K rW L m¼1 n¼1   S h  sin c np L l2

"

#     ðl1;m rP Þ2 J 0 ðl1;m rÞ cos 2np Lz l h i sin c np  exp  L l2 þ 2np2 J 2 ðl rW Þ 8 1;m

L

1

l1;m J0 ðl1;m rÞ  2np2 i

J 1 l1;m rW 1;m þ L

1;m

ð29Þ

1 X 1

z gP abs X  cos 2np þ 2 L prW rP LK m¼1 n¼1   J ðl r p ÞJ ðl rÞ l h 1 0;m 0 i 0;m  sin c np 2np 2 2 2 L l J 0 ðl0;m r W Þ 0;m l0;m þ L

z  cos 2np L

The temperature in the cylinder for each case is derived via Eq. (9) using the values of h in Eqs. (24), (25), and (29). 2.2.2.2. End pumping. This case is illustrated in Fig. 2(b). Treating the problem of end pumping is not much different than the above model for side pumping. The main difference is in defining the heat source as an exponentially decaying function along the rod axis. Thus in end pumping the temperature becomes maximized at the rod facet. The temperature surge is milder if an undoped

h

2

ðl r Þ 1 X 1 exp  1m p 8 ga2 LPabs X hðr; zÞ ¼ hW þ 2pr 2W K m¼1 n¼1

8 > 0; > <

L  gPabs f0 ðrÞ; 2 a exp a 2  z  zC > prP > :

0;

L 2

 zC < z

0 6 z 6 2L  zC \ 0 6 r 6 r P rP < r ð30Þ

where zC is the thickness of the end-cap, becoming zero for the case with no cap. As in the previous section also here considered are two cases of radial pump laser beam distribution: flat-top pump beam and Gaussian. Also cooling of the rod is modeled by assuming either setting the radial rod wall at a constant temperature or by a conductive mount holding the rod between the lengths of S/2 and L/2 on each side. The solution for the flat top beam pumping with Dirichlet conditions on the cylinder circumference becomes: 1 X 1 2ga2 LP abs X hðr; zÞ ¼ hW þ 2 prP rW K m¼1 n¼1        ð1Þn cos 2np zLC þ 2naLp sin 2np zLC  exp½a 2L  zC  h i   2 ½ð2npÞ2 þ ðaLÞ2  l21;m þ 2nLp



J1 ðl1;m rP ÞJ0 ðl1;m rÞ 2 1;m J 1 ð

l

l1;m rW Þ

z cos 2np L

ð32Þ

Finally, the solution for a Gaussian beam pumping with Dirichlet conditions on the cylinder circumference becomes:

i        zC p

 exp a 2L  zC ð1Þn cos 2np zLC þ 2n J 0 ðl1;m rÞ aL sin 2np L z h i cos 2np   2 2 2 2 L J 1 ðl1;m r W Þ ½ð2npÞ þ ðaLÞ  l21;m þ 2nLp

end-cap, made up of undoped host material, is bonded to the rod at the entrance. The undoped end-cap does not absorb the pump radiation yet conducts heat, contributing to removing heat from the optically active rod facet. Assumed in the present model is an identical coefficient of thermal conductivity in the doped and undoped materials. For the sake of smooth transition from the previous section a notation is selected such where L/2 expresses the entire rod length. In sum the heat source in the rod becomes:

Q ðr; zÞ ¼

Z Z L=2 qB 2gPabs rp G0 ðr; z; r W ; fÞ df þ 2 rW K prP lK 0 S=2 Z l=2  G0 ðr; z; n; fÞ dfn dn

ð33Þ

2.3. Optical propagation 2.3.1. Index of refraction Optical propagation through a medium is characterized by the refractive index of the medium. Both in isotropic and crystalline media the index of refraction is modified by temperature change, induced stress and deformation. The propagation of an optical beam in this case is approximated by a two-dimensional thermal stress problem since no closed form solution for a three-dimensional thermal stress is available [30]. Over 20 years ago Eggleston et al. [28] showed that refractiveindex modification depends linearly on the change of the index of refraction with temperature, the stress optic tensor and the thermal expansion coefficient. They demonstrated that for an infinitely long cylinder the change is:

dn TðrÞ þ B? ½rz ðrÞ  rh ðrÞ þ Bk rr ðrÞ dT dn TðrÞ þ B? ½rz ðrÞ  rr ðrÞ þ Bk rh ðrÞ Dnh ¼ dT

Dnr ¼

ð34Þ

where dn/dT is the change of the index of refraction with temperature which has been measured and fitted to a polynomial by Aggarwal et al. [5]: 3 X dn ðTÞ ¼ Mj T j dT j¼0

ð35Þ

B? and Bk are the perpendicular and parallel components of the stress optic tensor determined by:

1 ðp  2mp12 Þ E 11 1 B? ¼ ½p12  mðp11 þ p12 Þ E

Bk ¼ ð31Þ

ð36Þ

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M.M. Tilleman / Optical Materials 33 (2010) 48–57

where E is Young’s modulus, m is Poisson’s ratio and p are components of the elasto-optic tensor. It is well known that the first term on the right hand side of Eq. (34) dominates the change of refractive index to the extent of about 80% [29]. Still, even the smaller contribution needs to be considered for completion and to predict induced birefringence. Next, in Eq. (34) rr, rh and rz are the stress components in the radial, tangential and axial direction, respectively. In polar coordinates they are defined by the Airy stress function as:

rr ¼

2

1 d/ ; r dr

rh ¼

d / dr

2

;

rz ¼

1 d d/ r r dr dr

ð37Þ

and: 3 B? þ Bk X Nj E ðjþ1ÞgP abs 1  m 2  j¼0 j¼0 2pLeff K    1 1 jþ1 2  T jþ1 ðrÞ þ T jþ1  2 W T P rP 2 r rW ( ) 3 3 X X B? þ Bk B? þ Bk Dnh ðrÞ ¼ Mj TðrÞjþ1  E Nj T jþ1 ðrÞ þ E 1  m 1m j¼0 j¼0    3 X Nj 1 jþ1 2 1  T jþ1 ðrÞ  T jþ1 þ 2 W þT P rP ðjþ1ÞgPabs 2 r rW j¼0 2  2pL K

Dnr ðrÞ ¼

3 X

Mj TðrÞjþ1 þ

eff

ð43Þ where / is the Airy stress function governed by the compatibility equation which in the infinitely long axisymmetrical case is expressed as:

1 d d 1 d d E 1 d d r r /ðrÞ ¼  r aT ½TðrÞTðrÞ r dr dr r dr dr 1  m r dr dr

ð38Þ

where aN is the thermal expansion coefficient dependent on temperature which has been measured and fitted to a polynomial by Aggarwal et al. [5]:

aT ðTÞ ¼

3 X

Nj T j

ð39Þ

Note that the first term in the curled parenthesis of Eq. (42), thus the entire expression, converge at the origin. What is instructive to determine at this point is the thermally induced OPD and dioptric power. For this case in which neither the temperature nor the refractive index depends on the axial position the OPD simply becomes the product of the refractive-index change and the effective rod length. The phase contributed to a beam with a radius rL over a single pass through the rod is evaluated by the difference between the OPD at that radius and the rod center:

du ¼

j¼0

Assumed in Eq. (38) is that Young’s modulus and the Poisson ratio are independent of temperature. This assumption is corroborated by Brown [9], who showed that these parameters are constant over the temperature range of interest within 7% and 2%, respectively. The boundary conditions completing the problem are:

rr ðrW Þ ¼

 1 d/ ¼ 0; r dr rW

rh ðrW Þ ¼ rz ðrW Þ

ð40Þ

ð41Þ

Since this equation is applicable to the case of a very long rod, one considers the temperature expression for a like geometrical condition. On inspecting the temperature expression in Eq. (26) one observes that the equation is integrable over r. Two respective expressions are derived for the ranges of 0 6 r < rP and rP 6 r < rW, as follows:

Dnr ðrÞ ¼

3 X

M j TðrÞjþ1 

j¼0



Dnh ðrÞ ¼

3 B? þ Bk 2pLeff K X Nj E 1m gPabs j¼0 j þ 1

( )  2

r 2 rP P ½T jþ1 ðrÞ  T jþ1 ð0Þ  ½T jþ1 ðrP Þ  T jþ1 ð0Þ r rW

3 X j¼0

M j TðrÞjþ1 

ð44Þ

2.3.2. Induced focusing Radial modulation of the refractive index modulates the phase of a propagating beam thus inducing optical focusing. Ray propagation in such medium can be treated as that propagating in an inhomogeneous medium, described by [31]:

d dr n ¼ rn ds ds

ð45Þ

where r is the position vector and s is the distance along the ray. For paraxial rays this equation can be rewritten as:

With these the Eq. (38) is simplified to: 3 1 d d 1 d d E 1 d d X r r /ðrÞ ¼  r Nj T jþ1 ðrÞ r dr dr r dr dr 1  m r dr dr j¼0

2pLeff ½Dnðr L Þ  Dnð0Þ k

3 B? þ Bk X E Nj TðrÞjþ1 1m j¼0

3 B? þ Bk 2pLeff K X Nj E 1m gPabs j¼0 j þ 1 ( )  2

r 2 rP P  ½T jþ1 ðrÞ  T jþ1 ð0Þ þ ½T jþ1 ðrP Þ  T jþ1 ð0Þ r rW



ð42Þ

d dr dnðrÞ n½rðzÞ ¼ dz dz dr

ð46Þ

with initial conditions of a specified ray radius and slope at the origin. Considering that r = r(z), the solution is approached via a new variable defined as: y(r) = dr/dz and some algebraic operations thus solving Eq. (46) in the form below, 2

dnðrÞ dr d r dnðrÞ  þ nðrÞ 2 ¼ dz dz dr dz dnðrÞ dy dnðrÞ y2 þ nðrÞ ¼ dr dz ffi dr pffiffiffiffiffiffiffiffiffiffiffiffiffi nðrÞ ¼ nðH0 Þ 1  y2

ð47Þ

where H0 is the height of an arbitrary ray at z = 0. One identifies in Eq. (47) the equation of motion with the restoring force term @n=@r, which is indeed positive since the radial gradient of the refractive index is negative. In media that have a quadratic index gradient such as GRIN lens or heat dissipation in a cylinder with linear thermal conduction coefficient equation (47) becomes linear, having a harmonic solution. However, in a medium as presented herein this equation becomes nonlinear therefore one anticipates a nonlinear oscillatory behavior of the ray. From Eq. (47) it follows for an initially collimated beam that:

dr ¼ dz

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 nðH0 Þ 1 n

ð48Þ

54

M.M. Tilleman / Optical Materials 33 (2010) 48–57

Substituting n with n0 + Dn, where Dn is expressed in Eq. (43), into the right hand side of Eq. (48) presents the phase plane y–r that gives insight to the ray oscillation. Namely, the ray focuses within the unbounded medium time and again at a frequency determined by the temperature distribution. The fully closed loop indicates oscillatory behavior of which the period is calculated by integrating along the path over a complete circumference. Thus the period between focal points becomes:

Lf f ¼ 2

Z 0

H0

dr rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i2ffi 0Þ 1  nn00þþDDnðH nðrÞ

z¼

Z r

H0

" 1

tan sin

HL 1

n0 sin tan

!#) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h iffi 1

n0 þDnðH0 Þ n0 þDnðHL Þ

2

ð51Þ

Note that HL = HL(H0) is obtained transcendentally from Eq. (50) by setting r = HL and z = L. For laser resonators that are much longer than the laser rod it is a common assumption that the rod acts as a thin lens, in which case its effective focal length may be considered:

ð49Þ EFLðHL Þ 

In a similar fashion one obtains r at various distances along the medium from the following transcendental equation:

(

BFLðHL Þ ¼

L þ BFLðHL Þ 2

ð52Þ

Because the focal length depends on the ray height, the system introduces spherical aberration unless a certain condition is met, which for paraxial approximation becomes:

0

dr rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h iffi 1

n0 þDnðH0 Þ n0 þDnðr 0 Þ

2

ð50Þ

Note that Dn may assume the subscript of either r or u, thus manifesting the possible induced astigmatism in the rod. Consider for instance a collimated beam incident on the medium entrance facet. The thermally induced dioptric power is arrived at by considering the back-focal-length (BFL) computed assuming a radius HL equal to the ray height obtained at the exit from the medium having a length L and finding the slope of that ray at the exit from the medium. Recalling the ray slope in the rod expressed in Eq. (48) and considering refraction on exiting the rod into air the BFL is:

HL nðHL Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ const: n2 ðHL Þ  n2 ðH0 Þ

ð53Þ

In addition to astigmatism, if the beam passing through the medium propagates at an angle to the axis of symmetry then also coma is induced. Finally, if the pump does not form a symmetric stress field, the medium will introduce tilt to the passing optical beam. Neglecting the effect of thermal end-facet deformation the OPD through a stressed medium is expressed as:

OPDðrÞ ¼

Z

L

Dnðr; zÞ dz

ð54Þ

0

Fig. 3. Temperature in a rod: (a) axial distribution for heat density of 410 W/cm3, (b) radial distribution for heat density of 410 W/cm3, comparing rod with finite and infinite length and with the linear solution, and (c) radial distribution for heat density of 245, 410 and 815 W/cm3, comparing the nonlinear with linear solutions.

M.M. Tilleman / Optical Materials 33 (2010) 48–57

Next the phase contributed to a beam with a radius rL over a single pass through the rod is evaluated based on the difference between the OPD at that radius and the rod center which takes the form of Eq. (44). Thermally induced dioptric power and birefringence are given by Eqs. (49)–(56). 2.3.3. Induced birefringence Radial modulation of the refractive index can couple power from a given state into its orthogonal state of polarization. Conveniently the polarization phase shift is proportional to the index difference Dnh  Dnr found above, such that:

Leff ðDnh  Dnr Þ k 3 Leff B? þ Bk 2pLeff K rP 2 X Nj ¼ 2p E ½ðj  1ÞT jþ1 ðrÞ  2T jþ1 0  k 1m gPabs r j¼0 j þ 1

dðrÞ ¼ 2p

ð55Þ For a beam propagating through the medium the polarization extinction is evaluated by integrating over space the expression given in Ref. [29] as follows:

Ik  I? 1 ¼ Ik þ I? pr 20

Z 2p Z 0

3 1 ¼ þ 2 4 2r0

Z 0

r0

0

  2 2 dðrÞ dr dw r 1  sin 2w sin 2

r0

r cos½dðrÞ dr

ð56Þ

55

where Ik and I? are the respectively p and s polarized components of the input intensity. For the phase shift defined in Eq. (55) the integral in Eq. (56) has a close form solution only in the case of a long rod and small polarization phase shift, say up to d(r)  1 rad. 3. Example Several cases are calculated showing the optical effects of heating nonlinear materials. Predicted are profiles of temperature, refractive indices and thermally induced effects on passing optical beams in the medium. These include focusing, tilt, spherical aberration, astigmatism, depolarization and optical path difference. To allow a reasonable comparison between the various cases the following parameters are set: material Yb:YAG, rod diameter 5 mm, pumped zone diameter 2.5 mm and cryogenic cooling at 77 K. Assumed is a circumferential, radially directed pump forming a uniform gain and heat zone. To estimate the effect of the rod aspect ratio on the axial temperature distribution let a heat density of 410 W/cm3 be set while varying the rod length. Plotted in Fig. 3(a) is the axial temperature for half a rod from center to facet with a varying length. Observe that for short rods the temperature is relatively small, growing with length to an asymptotic value. Further length increase causes the temperature profile to flatten out reaching an asymptotic value set by an infinitely long rod. A length of 50 mm may be considered as the value at which the rod is well approximated by an infinitely long rod. Then, the temperature profile is calculated assuming

Fig. 4. Distribution in a rod of the refractive index (a), trajectory of optical rays in the rod (b) and in free space on exiting the rod (c).

56

M.M. Tilleman / Optical Materials 33 (2010) 48–57

Fig. 5. Distribution of the refractive index in a water-cooled rod (a), trajectory of optical rays in the rod (b) and in free space on exiting the rod (c).

power magnitudes of Pabs = 600, 1000 and 2000 W with g = 0.4 and a rod length of 50 mm, corresponding to heat density rates of 245, 410 and 815 W/cm3, respectively. Shown in Fig. 3(b) is the radial distribution through a rod center for using finite and infinite rod models yielding essentially identical results. Also in Fig. 3(b) a comparison is made with a calculation based on the linear solution where k is assumed constant equal to that for the median temperature in the rod. In Fig. 3(c) plotted are the temperature radial profiles for all three heat density levels, compared again to the linear approach, exhibiting a gradually growing discrepancy between the two for large power levels. It follows that the linear approach is unjustifiable for large heat loads, say above 200 W/cm3. Fig. 4(a) plots the relative refractive index Dn radial distribution, both radial and tangential components, for the above three heat magnitudes. It turns out that for the low and medium heat levels the profiles closely match a quadratic polynomial, while for the high heating level this proximity no longer holds. This indicates a gradually increasing spherical aberration of the thermally induced lens. Fig. 4(b) and (c) plot the ray trajectories of three optical beams traversing the rod and propagating in free space, respectively. Observe that the focal length depends strongly on the heatsource magnitude and that some spherical aberration is induced. For the case of the rod wall held at 77 K the average BFLs are 770, 335 and 65 mm and depolarization contrasts are 0.85, 0.79 and 0.71 for Pabs = 600, 1000 and 2000 W, respectively. The prediction for cryogenically cooled rod is compared with a water-cooled rod represented by a wall kept at 300 K. Fig. 5(a) plots the relative refractive index Dn radial distribution for the three heating magnitudes. Observe that the magnitude of Dn for

the water-cooled rod is 10–40 times that of the cryogenically cooled rod. Next, Fig. 5(b) and (c) plot the ray trajectories of three optical beams traversing the rod and then propagating in free space, respectively. The average BFLs for this case is much smaller than for the cryogenically cooled rod. For the cases with Pabs = 600 and 1000 W the BFLs are 55 and 25 mm, respectively, whereas for the high heat case the spherical aberration becomes so large that no meaningful focus can be found.

4. Conclusions A closed form solution has been derived for the nonlinear steady-state heat equation in a cylindrical rod. Hinging on using Kirchoff’s transformation, this solution is applicable to any material in which the coefficient of thermal conductivity is integrable in temperature. In turn, the solution enables further solving analytically the nonlinear equation of elasticity and that of a propagating beam in an inhomogeneous medium where the refractive index is radially modulated. The resulting set of solutions predicts the temperature and refractive-index distributions as well as the thermally induced effects on a beam propagating through such optically active media. Exemplary calculations are made for a Yb:YAG crystal with a typical doping level of 4 at.% Yb, where the temperature spans the range of 77–770 K. It is shown that under the condition of high pumping levels, i.e. large heating loads, say above 200 W/cm3, predicting the temperature according to a linear approximation underestimates the temperature. Thus, in order to obtain accurate

M.M. Tilleman / Optical Materials 33 (2010) 48–57

results one must use the nonlinear solution arrived at in this study. The predicted temperatures escalate rapidly with thermal loading reaching 120 K and almost 200 K on the optical path for a respective pumping by 1 and 2 kW, or respectively heating by 400 and 800 W, in a cryogenically cooled rod. If the cooling level is raised to room temperature, then by heating the rod with 400 W the temperature on the optical path will exceed 700 K. This renders cryogenic cooling uniquely imperative for high pumping levels. Following the temperature field the refractive index distribution in the medium is found exhibiting a profile straying from a quadratic curve, thus warping a passing optical beam to the extent of contributing appreciable aberrations. The thermally induced dioptric power depends on the heating load quadratically. Then, the effect of spherical aberration increases with the heat. A prediction assuming heating the rod by 400 W deduced is a thermal lensing of 3 diopter and a spherical aberration below 20 mm. Finally, thermal birefringence is appreciable in the rod corresponding to a polarization contrast of 0.8.

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