untitled
 1 /115  
On thermal effects in solid state lasers:  
the case of ytterbium-doped materials 
Sébastien Chénais1, Frédéric Druon, Sébastien Forget1, François Balembois and Patrick Georges.  
Laboratoire Charles Fabry de l¹Institut d¹Optique, CNRS, Université Paris-Sud 
Campus Polytechnique 
RD128 
91127 Palaiseau cedex 
Tel : +(33) 1 69 35 88 52; fax : +331 6935 8807 ;  
e-mail : chenais@galilee.univ-paris13.fr, frederic.druon@institutoptique.fr  
PACS codes: 42.55.Xi (Diode-pumped lasers) ; 78.20.Nv (Thermooptical and photothermal 
effects) 
keywords: solid-state lasers, thermal lensing, thermo-optical effects, Ytterbium. 
Abstract: A review of theoretical and experimental studies of thermal effects in solid-state lasers is 
presented, with a special focus on diode-pumped ytterbium-doped materials. A large part of this 
review provides however general information applicable to any kind of solid-state laser. Our aim 
here is not to make a list of the techniques that have been used to minimize thermal effects, but 
instead to give an overview of the theoretical aspects underneath, and give a state-of-the-art of the 
tools at the disposal of the laser scientist to measure thermal effects.  
After a presentation of some general properties of Yb-doped materials, we address the issue of 
evaluating the temperature map in Yb-doped laser crystals, both theoretically and experimentally. 
This is the first step before studying the complex problem of thermal lensing (part III). We will 
focus on some newly discussed aspects, like the definition of the thermo-optic coefficient: we will 
highlight some misleading interpretations of thermal lensing experiments due to the use of the 
dn/dT parameter in a context where it is not relevant. Part IV will be devoted to a state-of-the-art of 
experimental techniques used to measure thermal lensing. Eventually, in part V, we will give some 
concrete examples in Yb-doped materials, where their peculiarities will be pointed out. 
                                                 
1 Current address : Laboratoire de Physique des Lasers, Institut Galilée, Université Paris Nord, 99 avenue Jean-Baptiste 
Clément, 93430 Villetaneuse, France. 
 2 /115  
Contents: 
I. Introduction 
 I.1 The Yb breakthrough 
 I.2. Strategy on the matrix host 
II. Temperature profile of an ytterbium-doped crystal under diode pumping 
 II.1. Theoretical aspects 
II.1.1. The steady-state heat equation 
II.1.2. A review of analytical solutions of the steady state heat equation  
II.1.3. What is special about ytterbium-doped materials? The influence of absorption saturation 
in the temperature distribution. 
II.1.4. Determining the absolute temperature: the boundary conditions 
II.2. Experimental absolute temperature mapping and heat transfer measurements 
using an infrared camera 
II.2.1. Introduction 
II.2.2. Experimental setup for direct temperature mapping 
II.2.3. results and measurements of heat transfer coefficients 
III. Thermal lensing effects : theory  
III.1. Introduction  
III.2. Stress and strain calculations 
III.3. How can we take into account the photoelastic effect ? 
III.4. Simplified account of photoelastic effect in isotropic crystals 
III.5. A consequence of strain-induced birefringence: depolarization losses.  
III.6. Thermally-induced optical phase shift.  
III.6.1. Expression of the optical path 
  III.6.2. The thermal lens focal length 
 III.7. Discussion about the use of the “dn/dT” coefficient 
 3 /115  
III.8. A novel definition for thermo-optic coefficient based on experimentally 
measurable parameters.  
III.9. The aberrations of the thermal lens.  
IV. Thermal lensing techniques  
IV.1. Introduction  
IV.2. Geometrical methods 
IV.3. Methods based on the properties of cavity eigenmodes 
IV.4. Methods based on wavefront measurements 
IV.4.1. Classical interferometric techniques 
IV.4.2. Shearing interferometric techniques 
IV.4.3. Methods based on Shack-Hartmann wavefront sensing 
IV.4.4. Other techniques 
IV.5. Conclusion 
V. thermal lensing measurements in ytterbium-doped materials: the evidence of 
a non radiative path  
V.1. the thermal load in Yb-doped materials  
 V.2. Evidence of nonradiative effects in Yb-doped materials: the example of Yb:YAG  
V.3. Laser wavelength dependence oN the thermal load in Yb-doped broadband 
materials: the example of Yb:Y2SiO5 
IV.4. The influence of the mean fluorescence wavelength on the thermal load: an 
illustration with Yb:KGW 
IV.5. Conclusion 
VI. Conclusion 
Appendix : Calculation of the photoelastic constants Cr, θ  and C’r, θ  using plane strain and plane 
stress approximations.  
 4 /115  
I. Introduction  
I.1. The Yb3+ breakthrough 
Diode-pumped solid-state laser (DPSSL) technology has become a very intense field of 
research in Physics [1,2]. The replacement of flash-lamp pumping by direct laser-diode pumping for 
solid-state materials has brought a very important breakthrough in the laser technology in particular 
for high power lasers [3-4]. In fact, the better matching between absorption wavelength and 
material’s absorption spectra brought by the use of laser diode emission ― compared to the broad 
one of flash-lamps ― has lead to a significant benefit in efficiency and subsequently in simplicity, 
compactness, reliability and cost. This progress has substantial implications on laser applications 
such as fundamental and applied research, laser processing, medical applications …     
 5 /115  
4I9/2
4I11/2
4I13/2
4I15/2
4F3/2
4F5/2
4G5/2
4G7/2
4G9/2
2F5/2
2F7/20
10000
20000
Nd3+ Yb3+
0.8 µm
0.9 µm
0.94 µm
0.98 µm
~1 µm
~ 1 µm
Parasitic effects
4I9/2
4I11/2
4I13/2
4I15/2
4F3/2
4F5/2
4G5/2
4G7/2
4G9/2
2F5/2
2F7/20
10000
20000
Nd3+ Yb3+
0.8 µm
0.9 µm
0.94 µm
0.98 µm
~1 µm
~ 1 µm
Parasitic effects
Figure 1: Energy levels of Yb and Nd ions. Typical laser transition lines are represented for both 
pump absorption and laser emission. Lines of high excited states are also represented including the 
lines involved in the deleterious effects (up-conversion, excited-state absorption or concentration 
quenching).   
In the realm of high average power DPSSL, two rare-earth ions dominate:  neodymium and 
ytterbium [5-6]. Actually they can be efficiently pumped, respectively at 808 nm with 
InGaAsP/GaAs or AlGaAs/GaAs diodes for neodymium, and between 900 and 980 with 
InGaAs/GaAs diodes for ytterbium (fig. 1).  In both case the standard laser emission is around 1 µm, 
corresponding to the transition between the 4F3/2 and 4I11/2 lines for the Nd3+ and between the 2F5/2 
and 2F7/2 for the Yb3+. At the beginning of the high-power-laser development, the Nd-doped 
materials were preferred to the Yb-doped ones mainly because of the four level nature and their 
 6 /115  
many absorption lines, which are more convenient as far as flash-lamp pumping is concerned. 
However it seems obvious, for more than one decade now, that Yb-doped materials are more suited 
for very efficient and very-high-average-power diode-pumped lasers. The main reason for this is the 
very simple electronic level structure of the Yb3+ ion, which consists on two manifolds as shown in 
figure 1. This singular property allows avoiding most of the parasitic effects such as upconversion, 
cross relaxation or excited-state absorption which are present in Nd-doped materials [7] because of 
the existence of higher excited-state levels (4G9/2 for the 1-µm-laser emission). These deleterious 
effects have two main consequences. First, they increase the thermal load and subsequently the 
thermal problems [8] because the main desexcitation paths of the high-excited state levels are non-
radiative (as represented in fig. 1). Secondly, they also alter the gain because they can induce strong 
depopulation of the 4F3/2 level implicated in the laser inversion population. Another advantage of 
Yb-doped materials compared to their neodymium doped counterparts is the very low quantum 
defect (again due to the 2-manifold based electronic structure). In fact, when pumped at 980 nm the 
quantum defect of ytterbium is around 5 % compared to 30 % for neodymium (in YAG). This is a 
real benefit for reducing the thermal problems and thus to attain very high average powers. As an 
example of comparison between Nd and Yb doped materials, we summarized in table 1, the 
different parameters for the same well-known matrix host: YAG (Y3Al5O12) [9-13].  
 7 /115  
Table 1: comparison between Nd:YAG and Yb:YAG 
Crystal Nd:YAG Yb:YAG 
Emission line 
Wavelength 
Cross section 
Broadness (FWHM) 
1064 nm 
28 10-20 cm-2 
0.8 nm 
1031 nm 
2.1 10-20 cm-2 
9 nm 
Lifetime 230 µs 951 µs 
Saturation fluence 0.67 J/cm2 9.2 J/cm2 
Maximum doping rate 2 % 100 % 
Absorption line 
Wavelength 
Cross section 
Broadness (FWHM) 
808 nm 
67 10-20 cm-2 
2 nm 
968 nm 
0.7 10-20 cm-2 
4 nm 
942 nm 
0.75 10-20 cm-2 
18 nm 
Quantum defect  32 % 6.5 % 9.5 % 
I.2. Strategy on the matrix host 
Another advantage of Yb-doped versus Nd-doped materials is the longer lifetime which may 
allow a better storage of the pump energy; and the last but not the least advantage is the generally 
broader bandwidth of the emission lines. This last advantage leads to a potential for femtosecond 
pulse generation which, in the current state-of-the-art has never been demonstrated with 
neodymium. The emergence of ytterbium-based lasers has allowed crucial progress in the 
ultrashort-pulsed laser technology. These materials have been actually the key point for the 
development of the latest generation of “ultrafast” lasers: the all-solid-state femtosecond lasers [14-
36]. Applications for such lasers are abundant and excite a great interest in the scientific community.  
However Yb-doping brings several drawbacks or difficulties. The first one is the very strong 
influence of the matrix on the spectral properties. Actually, as the two levels 2F5/2 and 2F7/2 are split 
in manifolds by the Stark effect due to the electric crystalline field of the host matrix, the ion 
environment strongly models the spectrum. In a simple way, the spectral broadening can be directly 
related to the level of disorder of the matrix [37-48]. On the first hand, if the matrix is relatively 
simple and well-ordered, the spectra would reveal relatively narrow and intense lines (which are a 
 8 /115  
strong disadvantage for short pulse generation). Though, a simple matrix structure generally implies 
a high thermal conductivity which is a key point for developing high power lasers. An example of 
such an Yb-doped material is given in figure 2 with Yb:YAG. On the second hand, if the disorder of 
the host matrix is high, the spectrum will be large and suitable for very-short pulse generation but at 
the expanse of thermal conductivity. An example of such an Yb-doped material is given in figure 3 
with Yb:SYS. The numerous advantages of the Yb3+ ion have led to a strong interest for many host 
matrices but in general favouring either short pulses or high power applications. Table 2 represents 
this diversity of already studied Yb-doped host matrices and their principal properties.  
 9 /115  
Table 2: Comparison between Yb-doped crystals 
Emission line Absorption line Material 
(name and formula) Wavelength 
(nm) 
Cross 
section 
10-20 cm2
Broadness 
(nm) 
Lifetime
(ms) Usual 
wavelength  
(nm) 
Thermal 
conductivity
(undoped) 
(W/m/K) 
Yb:YAG 
Yb:Y3Al5O12 
1031 2.1 9 0.951 942 968 11 
Yb:GGG 
Yb:Gd3Al5O12 
1025 2 10 0.8 971 8 
Yb:Y2O3 
1076 0.4 14.5 0.82 979 13.6 
Yb:Sc2O3 
1041 1.44 11.6 0.8 979 16.5 
Yb:CaF2 
1045 0.25 70 2.4 979 9.7 
Yb:YVO4 
1020 0.9 40 0.25 985 5.1 
Yb:LSO 
Yb:Lu2SiO5 
1040 0.6 35 0.95 978 5.3 
Yb:YSO 
Yb:Y2SiO5 
1042 0.6 40 0.67 978 4.4 
Yb:YLF 
Yb:YLiF4 
1030 0.81 14 2.21 940 4.3 
Yb:KGW 
Yb:KGd(WO4)2 
1023 2.8 20 0.3 981 3.3 
Yb:KYW 
Yb:KY(WO4)2 
1025 3 24 0.3 981 3.3 
Yb:SYS 
Yb:SrY4(SiO4) 30 
1065 0.44 73 0.82 980 2 
Yb:GdCOB 
Yb:Ca4GdO(BO3)3 
1044 0.35 44 2.6 976 2.1 
Yb:BOYS 
Yb:Sr3Y(BO3)3 
1060 0.3 60 1.1 975 1.8 
Yb:glass 
(phosphate glass) 1020 0.05 35 1.3 975 0.8 
Another drawback of the Yb-doped materials is due to the quasi-3-level structure of these lasers. As 
it is apparent on the spectra of figure 2 and 3, there is an overlap between the emission and the 
absorption bands which leads to strong re-absorption effects and to a reduction of the effective 
emission band broadness. Moreover, since the splitting due to Stark effect is relatively small 
(between 200 and 1000 cm-1), the high energy levels within the 2F7/2 manifold (corresponding to the 
 10 /115  
different possible low-energy levels of the laser transition) are somewhat populated at thermal 
equilibrium. This implies two deleterious effects when temperature increases: first, a reduction of 
the laser inversion population, second, an increase of the reabsorption at the laser wavelength. A 
special care concerning the thermal load and thermal management will be then necessary to develop 
efficient lasers based on ytterbium-doped materials, especially in the high power regime. 
 11 /115  
950 970 990 1010 1030 1050 1070 1090
Section efficace
d'absorption
Section efficace
d'émission
Yb:YAG
Wavelength (nm)
Absorption
Emission
Absorption
Emission 
950 970 990 1010 1030 1050 1070 1090
Section efficace
d'absorption
Section efficace
d'émission
Yb:YAG
Wavelength (nm)
Absorption
Emission
Absorption
Emission 
Absorption
Emission 
Figure 2 : Absorption and emission spectra of Yb:YAG.   
850 900 950 1000 1050 1100 1150
Section efficace
d'absorption
Section efficace
d'émission
73 nm
Yb:SYS
Wavelength (nm)
Absorption
Emission
Absorption
Emission 
850 900 950 1000 1050 1100 1150
Section efficace
d'absorption
Section efficace
d'émission
73 nm
Yb:SYS
Wavelength (nm)
Absorption
Emission
850 900 950 1000 1050 1100 1150
Section efficace
d'absorption
Section efficace
d'émission
73 nm
Yb:SYS
Wavelength (nm)
850 900 950 1000 1050 1100 1150
Section efficace
d'absorption
Section efficace
d'émission
73 nm
Yb:SYS
Wavelength (nm)
850 900 950 1000 1050 1100 1150
Section efficace
d'absorption
Section efficace
d'émission
73 nm
Yb:SYS
Wavelength (nm)
Absorption
Emission
Absorption
Emission 
Absorption
Emission 
Figure 3 : Absorption and emission spectra of Yb:SYS.   
 As a first conclusion, Ytterbium-doped crystals are particularly suitable for directly diode-
pumped, solid-state high-power and/or femtosecond lasers. The emergence of new ytterbium-doped 
laser crystals has allowed crucial progress in the DPSSL technology. Nevertheless, a very special 
 12 /115  
care has to be done concerning the thermal properties and thermal effects in the Yd-doped materials 
because of their strong influence on the laser performance. In this paper, we then propose to make a 
review of different thermal-effect studies made on ytterbium-doped laser crystals. 
II. Temperature profile of an ytterbium-doped crystal 
under diode pumping  
 In this part we present a review about how to calculate and measure the temperature 
distribution in an end-pumped laser crystal. We explain in which cases it is possible to obtain an 
analytical expression (otherwise a finite-element analysis would be necessary), and how these well-
known results have to be corrected when we deal with ytterbium-doped crystals, because absorption 
saturation cannot be ignored in this case. Then, we investigate the role of the thermal contact at the 
boundaries, which is an essential parameter for the knowledge of the temperature. This will be 
illustrated, in the last part of this section, by experimental absolute temperature maps, obtained with 
an infrared imaging camera.  
II.1. Theoretical aspects 
II.1.1. The steady-state heat equation 
A study of thermal effects in crystals first requires the calculation of the temperature field at any 
point of the crystal. One has to solve the heat equation: 
)t,z,y,x(Q)t,z,y,x(TK
)t,z,y,x(T
c thcp =∇−
ρ  (II.1.1.) 
with :  - T = T(x,y,z,t) : temperature in K; 
 - ρ : density in kg.m-3; 
 - cp : specific heat in J.kg-1. K-1; 
 13 /115  
 - Kc : thermal conductivity in W.m-1.K-1; 
 - Qth : thermal power (or thermal load) per unit volume in W.m-3.  
The specific heat affects the temperature variation in the pulsed regime or in the transient regime: 
we will ignore it in the following work since we will only consider CW lasers. The thermal 
conductivity governs the temperature gradient inside the crystal, and will have a crucial importance 
for the thermal lens magnitude. The heat transfer coefficient H is arising when writing the boundary 
conditions, and has then an influence only on the absolute value of the temperature inside the 
crystal. We will discuss at the end of this part how to measure it, and some ways to improve its 
value.  
In order to obtain analytical expressions, some assumptions have to be made. We will assume the 
following: (1) The pump profile is axisymmetric. End-pumping by a fiber-coupled diode is a good 
example of such a profile; (2) the thermal conductivity Kc is a scalar quantity, not a tensor; this 
means that we restrict our discussion to glasses and cubic crystals [49] (in practice however the 
anisotropy of the thermal conductivity tensor is often weak); (3) the cooling is isotropic in the z-
plane, which means that the crystal mount does not favour one given direction of cooling. (4) At 
last we assume that the thermal conductivity is not significantly dependant on temperature, so that it 
can be considered as a constant. This approximation is very realistic in YAG around room 
temperature, following the study of Brown et al. [50], and we assume that it is also true in other 
crystals. However this approximation would not be valid anymore at cryogenic temperatures.  
The heat equation, including all these assumptions, becomes: 
)z,r(Q
 (II.1.2.) 
 14 /115  
when r is the radial coordinate of a point inside the crystal, measured with respect to the pump 
distribution axisymmetric axis. To simplify a bit further the equations, the crystal will be considered 
cylindrical, whose axis corresponds to the pump symmetry axis, with a radius r0 and a length L (see 
figure 4) 
Figure 4: geometry of the crystal, taken for all calculations. z=0 is the input face. 
II.1.2. A review of analytical solutions of the steady state heat equation  
An analytical expression of the temperature distribution inside a crystal is calculable only in a 
limited number of simple systems. For more complex geometries, one has to use finite element 
codes. We present in this subsection a brief review of the cases where such an analytical treatment 
is feasible.  
The first study of thermal effects in crystals has been provided by Koechner [51] in the early 
seventies. He considered flash-pumped Nd:YAG rods, within which the thermal load is uniform :  
Q thth 2
=  (II.1.3.)  
where Pth = ηh Pabs is the thermal power (in W) dissipated into the rod. Pabs is the absorbed pump 
power and ηh is the fractional thermal load. The solution writes:  
z r0 
z = 0 
Pump 
 15 /115  
)r(T)r(T
absh 22
+=  (II.1.4.) 
where T(r0) is the temperature at the edge of the crystal, which will be estimated later thanks to the 
boundary conditions. It is useful to write the temperature shift between the centre and the edge of 
the rod: 
( ) ( )
0∆ 0 =−=  (II.1.5.) 
We note that ∆T is independent of the radius r0  of the crystal, but scales inversely with L. 
The previous results can not be applied to end-pumping configurations, because in these latter cases 
the thermal load is localized within a small volume inside the crystal. In a vast majority of practical 
circumstances, the pump beam profile is axisymmetric and can be described by a super-gaussian 
shape. The general solution of the heat equation for a super-gaussian beam of any order has been 
derived by Schmid et al. [52].  
The situation is even simpler in most cases: indeed, the pump is often either a near-diffraction-
limited Gaussian beam (laser or single mode diode), or a “top-hat” beam (that is a super-gaussian 
profile of infinite order). The latter description corresponds quite well to fiber-coupled diode laser 
array pumping.  
The solution of the heat equation in the specific case of Gaussian-beam pumping is treated in [53] 
and [54]. The case of the top-hat shape has been derived by Chen et al. [55].  
Hereafter we give the solution for a top-hat beam profile. Assuming that the thermal load in each 
slice at z coordinate is a disk of radius wp(z), the temperature shift with respect to the edge 
temperature is:  
 16 /115  
)z(wr
)z(wr
)z,r(T)z,r(T
 (II.1.6.) 
where: 
- αNS is the (non-saturated) linear absorption coefficient; saturation absorption is then not 
taken into account in this formula ; 
- The axial heat flux (along z) is ignored, which means in other words that 
∂ is neglected 
in the heat equation. We’ll see in the next section the reasons why we can neglect axial heat 
flux. The formula is then not valid for thin disks.  
- As a consequence of the latter point, the temperature can be computed inside each “slice” of 
thickness dz of the crystal, as if the surrounding slices did not exist.  
The temperature gradient inside the pumped volume is of particular interest because the laser beam 
is usually (and preferably) smaller than the pump volume. One obtains: 
( , ) ( 0, ) ( , )
4 1 ( )
h abs NS
P e r
T r z T r z T r z
K e w z
∆ = = − =
 (II.1.7.) 
The temperature shift turns out to be independent of the crystal radius r0 and of the crystal length L, 
which was not the case in Koechner’s simple model (equation II.1.4). It makes sense since the 
important parameter here is the absorption length NSabsL α1= , and not the  whole length L of the 
crystal.  
In figure 5 we plotted the normalized temperature distribution for a typical ratio wp/r0 = 0.1. 
 17 /115  
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
r/r0 
T( r ) −T( r0 )
Figure 5: Normalized temperature distribution (in a plane perpendicular to the propagation axis)  
for a crystal pumped by a top-hat-profile fiber-coupled laser diode with wp/r0 = 0.1).  
II.1.3. What is special about ytterbium-doped materials? The influence of absorption 
saturation in the temperature distribution. 
An ytterbium-doped material, especially when pumped at the zero-line wavelength (i.e. around 980 
nm), has many common points with a saturable absorber. The absorption rate due to the pump is 
counterbalanced by the spontaneous emission rate, but also (which is far from being negligible) by 
the stimulated emission rate at the pump wavelength. It is essential to take absorption saturation 
effects into account; otherwise the absorbed pump power can be dramatically overestimated. As a 
result, the absorbed pump power is lower under nonlasing than under lasing conditions, since lasing 
provides (hopefully) a very efficient path to carry the excited population back to the fundamental 
level.  
 18 /115  
It is noteworthy that most Finite Element Analysis (FEA) codes (primarily designed for 4-level 
laser systems in which absorption saturation is not a problem) basically assume an exponential 
decay for the pump power inside the crystal. It can lead to large errors, as we illustrate below. 
Let Pp(z) be the pump power through a plane in the crystal at z coordinate. The thermal load density 
generated into a disk of radius wp(z) and thickness dz is: 
dz)z(w
)z(dP
th 2π
=  (II.1.8) 
where -dPp(z) represents the absorbed pump power in the thin slice of thickness dz.  
The temperature field is: 
)z(wr
)z(wr
)z(dP
)z,r(T)z,r(T
 (II.1.9.) 
Inside the pumped volume the temperature shift writes as follows:  
( ) ( )
zrTzT
=−  (II.1.10.)  
Absorption saturation issues are taken into account, under nonlasing conditions, by the following 
equation for the pump irradiance  Ip (which is the pump power divided by the pump spot area):  
 (II.1.11.) 
Where αNS is the absorption coefficient in the non saturated regime.  
The pump saturation irradiance
I is calculated from the spectroscopic properties of the material:  
( ) ( )[ ]τλσλσλ pempabspp
sat +
=  (II.1.12.) 
 19 /115  
where absσ is the absorption cross section, emσ is the emission cross section, λp the pump wavelength, 
and τ  the radiative lifetime. 
The pump power Pp(z) obeys the following equation, for a top hat beam profile (one may find the 
equivalent formulation for a gaussian pump profile in [56]): 
)z(wI)z(P
)z(wI)z(P
)z(dP
pppNSp
=  (II.1.13.) 
A practical way to study absorption saturation issues, and to check the assumptions made so far, is 
to perform fluorescence imaging experiments in a pumped crystal. Using a crystal whose one of the 
edge surfaces (in practice one side not facing the radiator) has been polished, one can make an 
optical image of the fluorescence, under lasing or nonlasing conditions, with a CCD camera and an 
interference filter at a long wavelength (at 1064 nm for instance), required to completely eliminate 
the scattered light at the pump wavelength, as well as to prevent detection of fluorescence photons 
which could have experienced reabsorption. This simple experiment allows visualizing absorption 
saturation (the fluorescence intensity, integrated along the depth of focus of the imaging system, 
does not decay exponentially) and also to measure what is the optimum location for the pump spot 
inside the crystal (figure 6). The experiments we performed with different Yb-doped materials 
taught us that the optimum focus (the one for which the measured laser efficiency was the highest) 
was always located at about one third of the whole crystal length from the input face. This 
parameter is taken into consideration in the following.  
The low brightness of the diode pump beam (compared to the brightness of the laser beam) makes 
the effective Rayleigh distance of the pump beam considerably shorter than the crystal length. For 
this reason, the divergence of the pump beam inside the crystal must also be considered, in order to 
correctly account for saturation issues. Here we describe the pump radius evolution by a relation of 
the type:  
 20 /115  
pp wn
  (II.1.14.) 
where 
w is the pump beam waist radius. The M2 factor is determined experimentally. In our case 
we used a 200µm-diameter core fiber-coupled diode (HLU15F200-980 from LIMO GmbH), whose 
the M2 was measured to be around 80.  
Results shown in figure 6 show experimental data and theoretical predictions in a 15%-at. doped 
Yb:GdCOB crystal [57]. The theoretical profiles are computed assuming that: 1) the pump volume 
has a top-hat profile, and 2) the imaging objective has a very low numerical aperture, so that the rate 
of spontaneous photons detected by one pixel can be calculated by integrating the fluorescence 
yield over one vertical line underneath. The good match between theory and experiments show 
incidentally that the “top hat” hypothesis for the pump beam profile is well justified. 
 21 /115  
Without saturation absorption :
At low power
(Pinc = 1 W
Pabs =200 mW)
With saturation absorption :
At high power
(Pinc = 13.7 W
Pabs =6 W
Ipsat =4.1 kW/cm
experiment
experiment
simulation
simulation
Without saturation absorption :
At low power
(Pinc = 1 W
Pabs =200 mW)
With saturation absorption :
At high power
(Pinc = 13.7 W
Pabs =6 W
Ipsat =4.1 kW/cm
experiment
experiment
simulation
simulation
Theoretical profile 
computed along the
symmetry axis
Experimental
profile measured
along the
symmetry axis  
Figure 6: Fluorescence detected @ 1064 nm on a crystal pumped at 980 nm at low power (top) and at high 
power (bottom), through the optically-polished top face. The influence of absorption saturation is clearly 
visible: at low pump power, the fluorescence yield is higher at the pump waist location, as expected provided 
that both absorption coefficient and absorption saturation are weak; on the contrary, when absorption 
saturation becomes non negligible, the amount of fluorescence photons is minimum at the pump waist. 
Theory and experiments agree very well, except near the exit face of the crystal, a discrepancy which could 
be related to the fact that far from the waist, the pump beam is no longer “top hat”. 
 22 /115  
Fig 7a)  
0 0.5 1 1.5 2 2.5 3
distance z (mm)
Intensité de saturation
décroissante 
position du waist
 de pompe 
Thickness z  (mm) 
Pump waist 
location 
Decreasing pump saturation 
intensity ↓
Fig 7b) 
0 0.5 1 1.5 2 2.5 3
distance z (mm)
position du waist de pompe 
Intensité de saturation
décroissante 
Thickness z (mm)
Pump waist location  
Decreasing pump saturation 
intensity ↓
Figure 7: Evolution of pump power (Fig. 7a) and temperature difference T(0) –T(r0) (Fig. 7b) 
versus crystal thickness z. The pump saturation intensities values are: ∞=
I - 50 - 20 - 10 - 
5 kW/cm2 (for these curves ηh= 0.065 et Kc = 2 W.m-1.K-1, corresponding to the parameters of 
Yb:GdCOB). 
 23 /115  
Fig 8a) 
Thickness z (mm) 
radius r (mm) 
fig. 8b)  
radius r (mm) Thickness z (mm) 
Figure 8: Temperature distribution under nonlasing condition. The pump beam divergence inside 
the crystal is taken into account (M2 = 80). In Fig 8a) the saturation of absorption is ignored; Fig 
 24 /115  
8b) pump absorption saturation is taken into account (
I = 4.1 kW/cm2). The parameters used are 
form Yb:GdCOB. 
Equations (II.1.13) and (II.1.14.) can be solved numerically and injected in (II.1.9.) to obtain the 
temperature distribution. Figure 7 shows the evolution of pump power (fig. 7a) and temperature 
(fig. 7b) at the center of the rod versus crystal thickness, for various values of the pump saturation 
intensity.  
Here we assumed that the pump beam waist was located at z0 = L/3, which is experimentally well 
verified, as far as the laser output is optimized (see figure 6 and above text). In absence of saturation 
I infinite), both the pump power and the temperature experience an exponential decay as 
expected; but for lower values of the pump saturation intensity, the temperature reaches a local 
minimum at the pump beam waist. Figure 8 shows a 3D view of temperature distribution without 
saturation (fig. 8a) and in presence of strong saturation (fig 8b.) corresponding to Ipsat = 4.1 kW/cm2, 
that is the value for Yb:GdCOB. It appears in the latter case that the region where the pump density 
is the strongest (near the pump beam waist) is not the region where the temperature is the highest 
(near the faces of the crystal). Pump beam divergence appears to be an important parameter: it 
makes, for this example, the temperature higher at the exit face than at the entrance face of the 
crystal.  
In presence of laser extraction, the pump intensity evolution through the crystal is given by: 
satsat
  (II.1.15.)  
where  
( ) NlabslNS λσα =   (II.1.16.)  
 25 /115  
( ) ( )[ ]τλσλσλ lemlabsl
sat +
=   (II.1.17.) 
are the non-saturated absorption coefficient at laser wavelength, and the laser saturation intensity, 
respectively.   
When the intracavity laser intensity I largely exceeds 
I , and if reabsorption at laser wavelength 
is small, one can show that (II.1.15) simply becomes: 
α−=   (II.1.18.) 
which means that the ground manifold is repopulated so that absorption is not saturated any more.  
In real cases, as a matter of fact, the absorbed pump power under lasing conditions is intermediate 
between the non saturated regime and the saturated (non lasing) regime: in a first approximation it 
is possible to ignore saturation effects only if the laser extraction is efficient.  
II.1.4. Determining the absolute temperature: the boundary conditions 
In this subsection we deal with the boundary problem. For the moment we have established 
expressions for the temperature gradient, but we have no idea of the absolute temperature inside the 
crystal. Let us assume that the four edge faces of the crystal are in “contact” with a radiator, which 
will be in most cases a piece of cooled copper. The first boundary condition expresses the 
continuity of the thermal flux across these contacts:   
copperinside
copper
crystaltheinside
crystal n
 (II.1.19.) 
where K is the thermal conductivity, n is the surface normal vector, and  ∂/∂n the normal derivative. 
Common metals (Copper or indium, the latter being used as an intermediate contacting material) 
have thermal conductivities that are several orders of magnitude higher than the usual conductivities 
of laser crystals: 400 W.m-1.K-1 for copper and 820 W.m-1.K-1 for indium. This means that the 
 26 /115  
temperature gradients inside these metals will always be negligible, so that we consider in the 
following that the temperature inside the radiator is uniform and is noted Tc.   
Let’s see now the second boundary condition. In many papers and FEA codes, the temperature at 
the edge of the crystal is set equal to Tc:  
cTrT =)( 0                                                                                                                         
(II.1.20.) This is actually true only for an ideal contact [58]. But even for flat and polished surfaces pressed 
one against another, this relation is far from reality [59].  
The most realistic condition is surprisingly a Newton-type law of cooling, even if we indeed deal 
with conduction problems here: 
( )cc TrTHn
K −=−= )( 0∂
 (II.1.21.) 
where jq is the thermal density flux. H is the heat transfer coefficient or surface conductance 
(W.cm-2.K-1). H is of course infinite for ideal thermal contact.  
Carslaw et al. [58] have shown that the physical origin of a temperature gap between the edge of the 
rod and the mount was due to the presence of a thin oxide (or air or grease) layer, which acts as a 
very large thermal resistance.  
Measuring the heat transfer coefficient is usually difficult and not found easily in the literature: we 
present in the next section a simple and accurate method to perform this measurement.  
What about the end faces, which are in contact with air most of the time? The heat can flow out of 
the crystal through the two end faces by both convection in free air and thermal radiation. Cousins 
[60] calculated the equivalent H coefficient for the two processes and has shown that both 
coefficients were of the order of 10-3 W.cm-2.K-1.  
Since the measured H coefficients for conduction are typically in the range 1-10 W.cm-2.K-1, this is 
the proof that the assumption of pure radial heat flux made on the previous subsection is correct.  
 27 /115  
Using (II.1.9.) et (II.1.21.) one can calculate the temperature gap between the radiator and the edge 
of the crystal: 
)()(  (II.1.22.) 
The parameter of interest here is the normal derivative of the temperature at the interface. This 
explains why the quality of the thermal contact has a tremendous impact on side or edge-pumped 
slabs or rods; since in these configurations the temperature distribution is described by a formula of 
the type II.1.4, that is a parabolic dependence. In contrast, in end-pumped configurations, where the 
temperature profile is described by equation II.1.6., the thermal gradient at the periphery is smaller 
and the requirement of a good thermal contact can be loosen. 
It is also interesting to know the maximum temperature reached inside the crystal. In order to obtain 
a easy-to-handle scaling formula, we make the strong assumption that absorption saturation is 
absent, and we ignore the divergence of the pump beam inside the crystal. We have:  
cmax w
 (II.1.23.) 
As a conclusion for this section, we will list some conclusions one can make from these two last 
equations, like a list of recipes to reduce the temperature Tmax : 
• Increase wp : obvious and efficient, but at the expense of laser efficiency.  
• Increase  H. As shown in the next experimental section, reducing H does not affect the 
temperature gradient, and will not help to reduce the thermal lens magnitude. All we can get 
is a uniform decrease of temperature. However reducing the absolute temperature is actually 
more interesting in an Yb-doped crystal than in an Nd-doped material for example, in virtue 
of reabsorption losses that are highly temperature-dependant. A better contact can also help 
 28 /115  
reducing fracture risks but this is not directly linked to a decrease of the temperature either: 
it is because a good contact can induce radial components to the stress tensor at the 
periphery, or also because it will decrease the density of high spatial frequency alterations of 
the surface which are the ultimate causes of crack-induced propagating fractures [61-63].   
• Decrease Tc : if the radiator temperature is decreased but still remains around the room 
temperature (that is with a standard thermoelectric or water-flow cooling), the effect is the 
same as increasing H: we only play on the temperature pedestal, not on the gradient. 
However, if the mount is cooled far below room temperature (at cryogenic temperatures for 
instance), the thermal conductivity of the crystals significantly increases, which is highly 
positive for the thermal gradient. This approach has been successively applied to reduce the 
thermal lens in high-energy femtosecond laser chains [64] or in Nd:YAG rods [65] 
• Decrease the crystal size? The crystal size has no influence on the temperature gradient. To 
understand its influence on the maximum temperature, Tmax is plotted versus r0 for different 
values of H in figure 9. We observe that when the radius of the crystal exceeds roughly 10 
times the pumped area radius wp, the temperature becomes independent of r0 . In practice, it 
is possible to reduce the absolute temperature using small crystals, providing they are really 
small (see for example [66]). It is practically very difficult to cut and polish crystals whose 
size is smaller than 2 mm: one can then conclude that the transverse section of a crystal is 
not a parameter on which one can play efficiently. Besides, the effect of a bad thermal 
contact is visible only for crystals whose size would be on the order of the pump spot size. 
As illustrated by figure 9, a small crystal with bad cooling (for example r0 = 0.5 mm and H = 
0.1 W.cm-2.K-1) is far worse than a « reasonably » sized crystal with correct cooling (r0 = 2 
mm ; H = 1 W.cm-2.K-1) since the temperature difference between the two configurations 
reaches  200 °C.  
 29 /115  
• Add an axial component to the heat flux: it does not appear in equation II.1.23 because it has 
been derived with the assumption of a purely radial heat flux. However, one can add a large 
axial heat flux by putting either a « transparent radiator » in front of the input face (this is 
the principle of composite bondings [67] ) or by using thin disks (i.e. L« r0) that are very 
efficiently cooled through the face in contact with the radiator [68]. 
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Crystal radius (mm)
 H = ∞ : perfect contact 
H = 1 W/cm2/K 
H = 0.1 W/cm2/K  
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Crystal radius (mm)
 H = ∞ : perfect contact 
H = 1 W/cm2/K 
H = 0.1 W/cm2/K  
Figure 9: Maximum temperature at the center of the input face of the crystal T(r=0, z=0) versus the 
crystal radius r0. The absorption saturation is neglected, as well as the divergence of the pump 
beam inside the crystal. The parameters are: Tc=15°C, ηh=6.5 %, αNS = 7.4 cm-1, Kc=2.1 W/m/K 
(values for GdCOB), Pinc=15 W, wp=100 µm. 
 30 /115  
II.2. Experimental absolute temperature mapping and heat transfer 
measurements using an infrared camera 
II.2.1. Introduction 
As depicted in the previous paragraph, the temperatures obtained by solving the heat equation are 
only relative temperature distributions, expressed with respect to the rod surface temperature. The 
latter depends on the boundary conditions and is then very difficult to predict. Direct temperature 
mapping could consequently be a helpful measurement to understand pump-induced thermal effects. 
Moreover, we have shown that one of the crucial parameter to uniformly decrease the temperature 
inside the crystal (which can be useful to reduce fracture risks, see above paragraph) is the thermal 
contact between the crystal and its surrounding mount. Consequently, the knowledge of quantitative 
and experimentally measured information as the heat transfer coefficient H is of practical 
importance for high power laser development. 
We herein report on a very simple experimental setup, based on an infrared camera that can perform 
spatially resolved analysis of the absolute temperature on the entrance face of the crystal, where 
temperature reaches generally a very high value (in any case higher than at the beam waist, as 
explained in the subsection II.1).  We can also experimentally measure the heat transfer coefficient 
H between the crystal and its surrounding for different types of commonly used thermal contacts. 
We first describe the experimental setup that allows such measurements, and illustrate it with the 
well-known Yb:YAG crystal [8]. 
II.2.2. Experimental setup for direct temperature mapping 
The experimental setup is presented on figure 10. A fiber-coupled laser diode was focussed inside 
an Yb:YAG laser crystal; the infrared emission of the entrance face of the crystal was observed with 
an infrared camera. A dichroic Zinc selenide (ZnSe) plate was used as a dichroic mirror: it was 
High Reflectivity (HR) coated for 960-1080 nm on one face (at 45° angle of incidence) to direct the 
 31 /115  
pump beam into the crystal, and also coated for High Transmission (HT) in the 8-12 µm spectral 
range on both faces to let the thermal radiation reach the camera.  
Figure 10: Experimental setup for absolute temperature measurements. 
A germanium objective (focal length 50 mm, N.A. 0.7, aberration-corrected for infinite conjugation) 
was appended close to the ZnSe plate to create the intermediate thermal image with high spatial 
resolution. The camera was an AGEMA 570 (Flir Systems Inc.) consisting of 240x320 
microbolometers working at room temperature. The measured noise equivalent temperature 
difference (NETD) of the camera is 0.2 °C. The numerical aperture of the whole imaging system in 
the object plane being around 1, a theoretical spatial resolution of about 10 µm could be achieved; 
however, the resolution is here limited to 60 µm by the size of the pixels of our camera.  The crystal 
used here was a 2-mm long, 4x4 mm2 square cross section, 8-at. % doped Yb:YAG crystal. It was 
 32 /115  
AR-coated on its faces (the lateral ones are polished). Its thermal conductivity, which is lower than 
that of an undoped YAG crystal, was measured to be 7 W.m-1.K-1 (11 W.m-1.K-1 for the undoped 
crystal). The pump source was a high power fiber-coupled diode array (HLU15F200-980 from 
LIMO GmbH) emitting 13.5 W at 968 nm. The fiber had a core diameter of 200 µm and a 
numerical aperture of 0.22. The output face was imaged onto the crystal to a 270-µm-diameter spot 
via two doublets. The crystal absorbed 5.4 watts of pump power in this case. The crystal was 
clamped in a copper block by its four side faces. In addition, on the top surface of the crystal, a 
frictionless copper finger allowed us to apply a well-controlled pressure on the crystal by the use of 
a set of known weights put upon the finger. The heat is finally evacuated from the copper block by a 
flow of circulating water. 
The key issue of infrared absolute temperature measurements is the correct calibration of the system. 
Indeed, neither the crystal nor the copper mount has an infrared luminance which equals that of a 
blackbody at the same temperature. The signal V detected by one pixel for a portion of crystal (or 
copper mount) at temperature T is: 
( ) ( ) ( )∫
ελ dLL
TTrSGTV rt
optr  (II.2.1.) 
where G is the geometric extent; ( )λrS is the spectral sensitivity; Tropt is the whole transmission 
coefficient of the ZnSe plate, Germanium objective and camera optics; 
dLTBB is the spectral 
luminance of a blackbody at temperature T, ε(T) is the emissivity; Lr denotes the infrared luminance 
of the camera itself (and its close surroundings) which is reflected back into it by the Germanium 
objective and by the polished surface of the crystal; Lt is the luminance transmitted through the 
crystal: it is zero in the 8-12 µm range since the crystal is highly opaque in this spectral region. Lr is 
nonzero and makes polished objects look brighter than blackbodies: if Lr is ignored it leads to 
overestimation of the temperature around room temperature. Inversely, the emissivity is less than 
 33 /115  
one and makes objects radiate less than a blackbody. Since the parameters ε and Lr are dramatically 
dependant on the surface quality and flatness, all the visible parts of the heat sink were covered with 
lustreless black painting. Moreover, the evaluation of all those parameters is not straightforward. 
We propose to calibrate the whole system as follows: the crystal and the copper mount were heated 
together to a set of given temperatures using a thermoelectric (Peltier) element, and we then 
compare with the temperature given by the camera to apply the adequate correction. This careful 
calibration allows rigorous and absolute measurement of the temperature with a spatial resolution 
large enough to study with sufficient accuracy the thermal behaviour on the crystal’s input face. 
II.2.3 Results and measurement of heat transfer coefficients 
Figure 11: The temperature map obtained when the crystal is clamped by its four edge faces by 
bare contact with copper without thermal joint (a), with heat sink grease (b), with a thin graphite 
layer (c), and with pressured and non-pressured indium (d). 
 34 /115  
Figure 11 shows the temperature map obtained when the crystal is clamped by its four edge faces 
by bare contact with copper without thermal joint (a), with a thin graphite layer (b), with indium (c) 
and with heat sink grease (d).  
Figure 12 is an enlargement of the two extreme cases, namely the bare contact and heat sink grease 
contact, with a transverse profile (y = 0) that shows the temperature evolution along the crystal 
lateral dimension. 
Figure 12 : Temperature mapping of the crystal (front view) and lateral profile at y=0 for two 
different types of thermal contact (direct copper-crystal contact on the left, with grease on the right).  
In the “bare contact” case, a clear gap is noticeable between the temperatures of the mount and at 
the edge of the crystal. The temperature distribution is parabolic inside the pumped region and then 
 35 /115  
experiences a logarithmic decay until the edge of the crystal, in good agreement with the theory 
described in the previous section in the case of fibre-coupled diode pumping (see equation II.1.9 
and figure 5). As already mentioned, the quality of the heat transfer at the interface between the 
crystal and its mount has an influence on the value of the temperature but not on the thermal 
gradient.  
We consequently studied more in detail the heat contact. The heat transfer coefficient H is defined 
by equation (II.1.22), where the thermal gradient is considered normal to the surface.  
Our system provides a space-resolved temperature mapping of the crystal, with a spatial resolution 
which is far below the crystal size: it then allows the measurement of H. 
By performing a linear fit of the temperature versus position on the points that are closer to the 
crystal edge, the heat flux can be determined: by applying the equation (II.1.22), one can then infer 
the value of H. We found for instance a value of 0.25 W.cm-2.K-1 in the case of bare contact. We 
estimate that the uncertainty on H is about 15%. The order of magnitude obtained is consistent with 
the values evoked by Carslaw [58] and Koechner [69]. The hot spot that can be noticed in figure 12 
betrays the poor contact between the polished face of the crystal and the copper surface. The heat 
transfer is primarily a question of how much two surfaces are in contact with respect to each other; 
we checked experimentally that the temperature inside the crystal did not depend on the applied 
pressure: we did not observe any noticeable variation of the temperature when changing the applied 
pressure in absence of thermal joint between the crystal and the copper mount.  
We summarized in table 3 the results obtained for the different thermal joints used in our set of 
experiments, namely graphite layer, indium foil and heat sink grease (CT40-5 from Circuitworks®). 
Graphite layer (around 0.5 mm thick) does not modify significantly either the maximum 
temperature or the heat transfer coefficient, but it was noticed that the contact was much more 
uniform than in the case of bare contact: in particular no hot spot appeared any more and the contact 
was somewhat independent of the applied pressure. It is not the case with indium foil. For this 
 36 /115  
experiment the crystal was wrapped within a 1-mm thick indium foil. Since Indium is a soft 
material, the quality of the contact is greatly dependant on the applied pressure. The temperature at 
the center of the pumped region experiences a 7°C decrease while the pressure increased from 1.5 
kg/cm2 to 22 kg/cm2 as shown in figure 13 (note that in this case the H coefficient is measured 
across the surface where the pressure is applied and is then an “effective” heat coefficient that takes 
into account the transfer from crystal to indium and then from indium to copper.)  
Figure 13: Evolution of the heat transfer coefficient H (squares) and maximum temperature 
(triangles) versus applied pressure for indium-wrapped crystals. 
The most dramatic change in heat transfer coefficient is obtained with heat sink grease (see table 3). 
The temperature gap drops down to 1°C and H reaches 2 W.cm-2.K-1. The heat contact is here 
independent on the applied pressure. This better heat transfer coefficient is achieved while the 
thermal grease has a much lower thermal conductivity than indium (0.62 W/m.K for 
 37 /115  
CircuitWorks ® CT40-5 thermal grease vs. 82 W/m.K for pure indium). This is an illustration, in 
conjunction with the data about the variation of H with the applied pressure, of the idea that 
achieving a good H is first and foremost a question of decreasing the thermal resistance at the 
interface (eliminating air gaps, maximizing the surface contact…) 
Table 3: Table of measured H coefficients for different contacts. Tmax is the temperature at the 
center of the pumped region; Te is the temperature at the edge of the crystal (averaged on the 4 
sides if not symmetrical), and Tm is the copper mount temperature near the crystal.  
Contact H (W.cm-2.K-1) Tmax (°C) Te(°C) Te-Tm (°C) 
Bare 0.25 49.8 33.5 10.7 
Graphite layer 0.28 46.5 30.5 8.7 
Indium foil 
(applied pressure : 
22 kg/cm2) 
0.9 40.0 25.1 4.9 
Heat sink grease 2.0 37.0 21.6 1.5 
 38 /115  
III. Thermal lensing effects: theory  
III.1. Introduction  
The previous chapter was dedicated to the calculation and measurement of the temperature 
distribution, which is the first essential step for the study of thermal effects. The appearance of 
thermal gradients causes the crystal to be under stress. The presence of inhomogeneous temperature, 
stress, and strain distributions is responsible of many deleterious effects for laser action: the most 
radical effect is fracture, observed when the hoop (tangential) stress at the periphery of the crystal 
exceeds the so-called tensile stress. More subtle effects arise from the stress-induced modification 
of the optical indices of refraction: alteration of the stability domains of the cavity, depolarization, 
losses and degradation in beam quality, all of these four phenomena being largely intermixed. In 
this paper we designate by “thermal lensing” effects all the phenomena resulting in a phase change 
of a beam passing through a pumped crystal; in other words we do not restrict this expression to an 
ideal spherical thin thermal lens, we also include its aberrations and its polarization-dependant 
aberrations. This chapter presents a general and synthetic scope of these effects, and points out how 
they are related to each other. We base our discussion on analytical simple scaling relationships, 
and we point out the validity of these formulas.  
In this review, we come back to well-established theories that have been exposed many 
times in the past [60, 69, 70], but we also bring some new insights, to our knowledge, on some 
points of practical interest.  
In particular, we will point out several inaccuracies generally reported about the values of 
the photoelastic constants in YAG, which are the result of an incorrect use of the Hooke Law; we 
also present what is (still to our knowledge) the first derivation of the photoelastic constants that 
have to be used for end-pumped crystals, that is in other words when the calculation is made using 
the plane stress rather than the plane strain approximation.  
 39 /115  
We will eventually point out that the use of the dn/dT coefficient (temperature derivative of 
the refractive index) is very confusing. In one hand, the classical formula (which reveals the 
existence of three contributions: the “dn/dT” part, the bulging of end faces, and the photoelastic 
effect) which is used since decades is correct provided that the dn/dT appearing in this expression is 
understood as a partial derivative taken at constant strain. In the other hand, the experimentalist 
can measure a quantity which is closer to a partial derivative at constant stress, and the partial 
derivatives are obviously not equal. The dn/dT parameter is then not actually the correct parameter 
to be used in order to estimate the thermal lens focal length: this subtlety means in particular that 
one cannot, in general, make use of a value of dn/dT readily found in handbooks to estimate the 
magnitude of the thermal lens of an operating laser, because the experimental measurement 
conditions are in the two cases mutually inconsistent. We’ll see however that when the dn/dT is 
large and positive, the difference can be ignored. 
We will conclude this review by a synthetic diagram showing all the thermal effects and 
how they are connected together.  
Given that thermo-optical properties pertain more to a crystal host than to a doping ion, this 
section is more general than the others and does not restrict to the case of Ytterbium-doped 
materials.  
III.2. Stress and strain calculations 
Once the temperature field has been computed, the next step is to calculate the stress and strain 
distributions inside the crystal, obtained from the so-called “generalized” Hooke law, because it 
includes the thermal expansion term [49]:  
TS ijTklijklij ∆+= ασε          (III.1) 
 40 /115  
where i, j, k, l = 1,2,3 and the Einstein summation convention is used. ∆T is the temperature shift 
with respect to equilibrium (no strain), (Sijkl) is the compliances tensor, (σkl) is the stress tensor, (εij) 
is the strain tensor, and (
α ) is the thermal expansion coefficients tensor.  
The analytical formulations of thermal stress and strain distributions in end-pumped lasers require a 
large amount of approximations, thoroughly discussed in Cousins’s reference paper published in 
1992 [60].  
In order to obtain an analytical solution to the stress problem, an additional approximation is 
required, which consists in considering the problem in two dimensions. This is either the plane 
strain approximation (valid for long and thin rods) or the plane stress approximation (valid for thin 
disks). Interestingly, Cousins [60] pointed out that the plane stress approximation remained valid 
(within approximately 10%) for aspect ratios up to L/2r0 = 1.5, providing that the stresses were 
considered as mean values integrated along the whole thickness of the rod. In the previous section, 
when we derived the temperature distribution, we had considered the crystal as a stacking of thin 
slices, so that the temperature could be calculated in a single thin slice as if the surrounding material 
did not exist. It is not possible to use this approach for the stress distribution, because a given slice 
is under the mechanical influence of the slices located on both sides, and cannot be considered as 
independent. This is why an attempt to take into account absorption saturation effects and pump 
divergence (as far as thermal stresses are concerned) inevitably requires a finite element analysis.  
As far as diode end-pumping is concerned, the plane stress approximation is then the most 
meaningful approximation that can be done. However, the exact calculation remains possible for a 
given crystal (using FEA codes) provided that all the compliances and thermal expansion 
coefficients are known. To the best of our knowledge, these coefficients have been measured for a 
very restricted number of laser crystals up to now: we may readily find these data for YAG, 
sapphire, YLF and Y2O3 [71]: other data are available in the Handbook of Optics [71] but for 
 41 /115  
crystals which are not commonly used in laser applications. The data for other materials are almost 
inexistent.  
The analytical solution of the generalized Hooke law (eq. III.1) can be found, for example in [60] 
under the plane stress approximation. In this paper, the discussion was restricted to isotropic 
materials (at a mechanical rather than an optical point of view, that is when the compliances can be 
reduced to only two parameters, the Young modulus and the Poisson ratio), absorption saturation 
was not considered, and the divergence of the pump beam was not taken into account either.  
Once this has been calculated, it is possible to study thermal fracture issues. It is generally admitted 
that fracture occurs when the maximum hoop (tangential) stress σmax at the surface periphery of the 
crystal exceeds the tensile stress σTS. The latter depends on both the fracture toughness of the 
material and on its surface flatness. These aspects have been studied in detail by Marion [61-63]. 
Data about fracture toughness of materials can be readily found in the literature for YAG, 
fluoroapatites, sapphire, yttrium orthosilicate YSO and some phosphate glasses [39, 63, 71, 72].  
For a qualitative discussion of fracture issues in Yb-doped materials, the reader is invited to refer to 
a previous publication [73]. 
III.3. How can we take into account the photoelastic effect ? 
We now consider how the temperature, stress and strain fields inside the crystal alter the phase of 
the cavity beam, all these effects being referred as “thermal lensing” phenomena.  
The appearance of stresses in the crystal causes the linear optical indicatrix (related to the linear 
indices of refraction) to change its shape, its size and its orientation. This photoelastic effect is 
accounted by the 4th rank elasto-optical tensor (pijkl): 
klijklij pB ε=∆   (III.2) 
Where (Bij) is the dielectric impermeability 2nd rank tensor. This expression if obviously valid in the 
linear optical regime only, and when piezoelectric effect is neglected [49].  
 42 /115  
The complete computation of thermal effects in a given material requires that we know everything 
about the tensors (Sijkl), (pijkl) and (
α ). The minimum number of independent terms of each tensor 
depends on the crystal symmetry, as discussed by Nye [49]. For instance, let us consider crystals 
like KGW or KYW [74], GdCOB [75], YCOB [76] or YSO [43] which are of particular interest for 
Ytterbium doping. These crystals belong to the monoclinic crystal system: this means that the 
compliances can be “reduced” to 13 independent parameters, and the elasto-optical tensor, once the 
redundant coefficients have been identified, appears to have 20 independent coefficients [49]. 
Adding the 3 thermal expansion coefficients, this means that we need to know no less than 36 
coefficients before to be able to draw the new index ellipsoid at a given point of the crystal. 
Obviously these parameters are not known (for any monoclinic crystal, in fact, to the best of our 
knowledge), which means that a rigorous calculation even with a FEA code is just not possible. 
This simple remark highlights the importance of experimental measurements of thermal effects in 
such crystals, and shows the interest as well as the inherent limitations of a simple analytical model.    
III.4. Simplified account of photoelastic effect in isotropic crystals 
Now we are aware of these difficulties, we focus our discussion on a simpler study case, 
actually the only case where analytical expressions are obtainable, that is:  
- we consider isotropic crystals only, and more particularly the widespread YAG crystal, for which 
all the parameters previously evoked are well known. Generally, cubic crystals belonging to the 
space groups 43 ,432, 3m m m  (like YAG) require 3 independent elastic coefficients ; however the 
remarkable isotropic mechanical properties of YAG enable to think of only two mechanical 
coefficients, that is the Young modulus and the Poisson ratio. In the end, 6 coefficients only are 
needed for YAG. 
- The plane stress approximation is used; 
 43 /115  
- the pump profile is still axisymmetric; 
- we consider what occurs inside the pump volume, that is for r < wp.  
- the pump divergence inside the crystal is neglected. 
- temperature, stress, strain, index are considered integrated along the whole thickness of the rod. 
For a physical quantity A(r,z) , we shall note : 
( ) ( ) dzzrArA
the integrated value of A(r,z) along the rod.  
According to the Neumann-Curie theorem [49], under these assumptions the principal axes of all 
the involved tensors (stress, strain, index ellipsoid) are radial and tangential. The notations used in 
the following are depicted in figure 14. 
nr nθ 
nr nθ 
Figure 14: Orientation of the indices’ ellipsoid in an isotropic crystal under thermal stress.  
 44 /115  
The shift of the principal indices nr and nθ are related to the diagonal coefficients of the 
optical indicatrix by:  
=  (III.3) 
We can also write the indices variation as a function of the strains as follows: 
Bnn ε
=∆−=∆ ,,,,
 (III.4) 
The six coefficients 
(i= r, θ; j=r, θ, z) can be calculated from the pijkl coefficients by a correct 
change of coordinates.  
The complete solutions for the strains εr, εθ and εz can be found, for YAG, in many papers and 
textbooks [60, 69] and can also be found in the appendix. 
Inside the pump volume, it can be readily shown that stresses and strains have a parabolic 
dependence, like the temperature distribution. Since the indices of refraction are linear 
combinations of strains, it turns out that radial and tangential index distributions must also be 
parabolic.   
The (integrated) shift of refractive index may be written as:  
( ) ( ) ( )
rndzzr
rn θθ
=∆ ∑ ∫
 (III.5) 
where Cr and Cθ (or C’r and C’θ)  are constants which will be called, following the pioneering work 
of Koechner, the “photoelastic constants”. Their calculation and expression is given in the appendix. 
We would like to point out two important clarifications about these coefficients (and also justify the 
presence of this appendix in this review):  
♦ W. Koechner published incorrect values of these coefficients in his reference book [69] 
because the temperature term in the Hooke law has been omitted; this omission has first 
 45 /115  
been highlighted by Cousins [60], but the expression of the photoelastic constants remained 
uncorrected in the following editions of this book, and nowadays still remains used under 
this form in many papers. 
♦ Secondly, the derivation of the photoelastic constants requires turning the 3D problem into 
a 2D problem, as discussed in the previous section. Only the plane strain case was 
considered by Koechner. However, we saw that the plane stress case is closer to reality in 
end-pumped rods. Here we denote as Cr and Cθ the photoelastic constants valid for long and 
thin rods (the “Koechner case”, that is when the plane strain approximation is valid), and 
C’r and C’θ the photoelastic constants derived within the framework of the plane stress 
approximation. Since we are only interested in end-pumping, we only consider the ' ,θrC  
constants in the following. 
The above-mentioned relations are derived by making the assumption that the pump beam 
radius is constant through the crystal thickness; however we don’t have to assume a particular 
absorption regime, so that they stay valid in presence of absorption saturation, under lasing as well 
as under nonlasing conditions. 
As we will see in the next section, the most interesting feature for the laser scientist is the 
index shift between the center and the edge of the pumped zone, since it yields the contribution to 
the global thermal lens. From (II.1.10), and using the bracket notation for z-integrated values, we 
can write (III.5) under the form:  
( ) ( ) ( ) ( )rTTCnrnn
−=∆−∆ 020 '30,, ,θαθθ  (III.6) 
III.5. A consequence of strain-induced birefringence: depolarization losses.  
 46 /115  
The birefringence of a crystal submitted to thermal stress has two main consequences for a 
light beam passing through it: both its state of polarization and its phase will be altered. Before 
examining in detail the effect on phase (i.e. thermal lensing effects), let’s first examine the influence 
on polarization. 
We use the restrictions exposed in the previous subsection, given that the following can be 
readily extended to uniaxial crystals provided that the optical axis lies parallel to the propagation 
axis. In these cases, if no polarizing element is added into the cavity, the laser output is not 
polarized, and stress-induced birefringence has no net effect. 
Beam polarized along x 
crystal 
Depolarized beam  
Figure 15: Depolarization of a polarized beam passing through an isotropic crystal under thermal 
stress. 
For many applications however a polarized output is desirable: the situation is depicted 
schematically in figure 15. An incident beam (polarized along the x direction) will have its 
polarization modified differently for every single ray: For a ray crossing the (Ox) or (Oy) axis for 
instance, the polarization is not modified, for all the other rays the polarization becomes elliptical, 
with principal axis that are radial and tangential.  
 47 /115  
At every roundtrip in the laser cavity, the beam meets a polarizing element (such as a plate 
at Brewster angle) and this depolarizing element, yielding to the so-called “depolarization losses”. 
Another effect, consequence of the latter, is a modification of the beam spatial profile: since 
the beam is not altered along the (Ox) and (Oy) directions and suffers losses elsewhere, it tends to 
take the shape of a cross; this aspect is chiefly discussed in Koechner [69].  
In biaxial crystals (or uniaxial crystals with optical axis normal to the direction of 
propagation), the output is naturally polarized along the crystallophysic (not to be confused with 
cristallographic) axis along which the emission cross section is the highest. In this case, stress-
induced birefringence does not generally twist the index ellipsoid enough to significantly modify 
the polarization state.  
Finally, clever solutions have been imagined to compensate for depolarization losses in 
isotropic crystals: for example the use of two rods with a Faraday rotator inbetween [78, 79], or 
even a simple quarter waveplate [70]. This last technique is however limited to a few configurations 
[80]. 
In the following, we present results obtained with Yb-doped crystals which are either 
isotropic (and naturally not polarized), or naturally birefringent, so that this problem was not 
encountered. 
III.6. Thermally-induced optical phase shift.  
III.6.1. Expression of the optical path 
We present now this derivation with some detail, because it makes appear a trivial yet 
fundamental difference between these results and what is currently reported in the literature. The 
derivation is largely reproducing Cousins’s work [60]. 
 48 /115  
n (T, ε j ) 
L+∆L d - ∆L 
z  = 0 
n0 (Tc ) 
z  = 0 
Figure 16: Notations used for the calculation of the optical path 
Consider a crystal whose length is L and index n0 at temperature Tc (temperature of the heat 
sink) and in absence of strain. We consider the optical path of a straight ray (running parallel to the 
crystal axis z) between a plane z = 0 (taken at the entrance face of the crystal) and a plane z = L+d  
(see figure 16).  
In absence of temperature and stress fields (pump off), the optical path is: 
dLTn c
off +×= )(0δ  (III.7) 
With the pump on, the optical path is dependant on the lateral shift r of the ray with respect 
to the crystal axis, but also on the direction of polarization. As a consequence there will be two 
distinct thermal lens focal lengths, one for a (virtual) radially polarized beam and another for a 
tangentially polarized beam, what is commonly named “bifocussing”.  
We may write the optical path with the pump beam “on” as (see figure 16):  
  ( ) ( ) ( )rLddzTnr
∆−+= ∫
, ,, εδ θθ  (III.8) 
where ∆L(r) is the crystal length shift due to the inner compression, responsible for the 
bulging of the end faces.  
 49 /115  
Assuming small variations in the refractive index, one may expand ( )jr Tn εθ ,,  in Taylor 
series, and discard the second and higher-order terms:  
( ) ( ) ( )( ) ( ) ( )rLddzzr
+= ∫ ∑
=0 ,,
0 ,,,
 (III.9) 
Note that the temperature derivative of the refractive index appearing in this equation is a 
partial derivative calculated at constant strain. As we will discuss in the next subsection, this is not 
the usual dn/dT parameter. 
The rod length change ∆L(r) can be written as a function of the axial strain εz and equals  
( ) ( ) ( )rdzz,rrL z
z εε∆
== ∫  (III.10) 
From the strain-stress relationships featured in the appendix, it can be easily shown that the 
axial strain, under the plane stress approximation, is equal to: 
 ( ) ( ) ( ) ( )rTrTr Tz −=+−= 01 ναε  (III.11) 
Given that the first-order terms in the integral appearing in (III.9) are much smaller than 
n0(Tc) and given that ∆L « L, we can write that for the first order terms  ( ) ( )∫∫ ≈
∆+ LLL
......
The relative optical path is then:  
( ) ( )
( ) ( ) ( )( ) ( ) ( )0110
TrTnr
offon
−+−+⎟
 (III.12) 
III.6.2. The thermal lens focal length 
The thermal lens is related to the optical path difference (OPD or ∆ in the following) 
between an on-axis central ray (r = 0) and an outer parallel ray passing inside the pumped region, 
defined by a radius r < wp.  
 50 /115  
We note : 
( ) ( ) ( )rr relrrelr θθ δδ ,, 0 −=∆  (III.13) 
The expression of the optical path difference is, from (III.6) and (III.12):  
 ( ) ( ) ( ) ( ) ( )rTTnCn
r TrT
+−++⎟⎟
=∆ 0112 0
, ναα θ
θ    (III.14) 
We assume that the thermal derivative of the refractive index is equal for radial and tangential index, 
so we can write:  
( ) ( )( ) ( ) ( )
( ) ( )rTT
rTTCnn
++−+⎟
 (III.15) 
where θχ ,r is usually called the “thermo-optic” coefficient. We remind the reader that this 
expression is valid only under some restrictive conditions that have been presented in detail in the 
section 4 of this chapter.   
Given that the integrated temperature shift is given by (from eq. II.1.11):  
( ) ( ) ( ) ( )( )
dzzrTzTrTT
∫ =−=−  (III.16) 
it appears that the optical path difference also follows a quadratic dependence in r. This 
means that in the paraxial approximation, the pumped crystal acts as a thin lens whose focal length 
is given by :  
),( 2∆
=  (III.17) 
In the following the difference between fr and fθ (responsible for bifocussing) will be 
omitted (realistic if the photoelastic effect is negligible, or if the laser beam is not polarized.) 
 51 /115  
The thermal lens dioptric power is thus defined by: 
( )( )
th Kw
002 2
++−+⎟
==  (III.18) 
where χ is a polarization-averaged thermo-optic coefficient. 
Let’s notice that this formula still holds in Yb-doped materials, with strong absorption 
saturation, since no assumption was made concerning absorption.  
If the pump profile is gaussian (e.g. end-pumping by another laser), one can show [70] that 
the thermal lens dioptric power is twice as large as (III.18), meaning that the thermal load is more 
spiky around the center of the beam than in the case of an uniform “top hat” pump beam energy 
deposition.  
To conclude, let’s address some orders of magnitude. To be correct, the previous derivation 
has to be performed within the paraxial approximation, which means Lf th >> . 
Taking some typical values (ηh ∼ 0.1, χ ∼ 10-5 , Kc ∼ 5 W/m/K, wp = 100 µm, et L = 3 mm), 
it appears that the paraxial conditions are met provided that  Pabs << 100 W. This corresponds to 
most of practical cases. 
        
III.7. Discussion about the use of the “dn/dT” coefficient. 
Let’s start this discussion by the expression of the thermo-optic coefficient: 
( )( ) 3 ', 0 0 ,1 1 2r T T r
n n C
Tθ θ ε
χ ν α α
∂⎛ ⎞= − + + + ⎜ ⎟∂⎝ ⎠
 (III.19)  
 52 /115  
In this subsection, we would like to point out how this expression can be misleading if one 
carelessly uses, to evaluate the magnitude of a thermal lens, the so-called dn/dT instead of
The error is especially important when all terms except the dn/dT are discarded for the sake of 
simplification. This represents, to our knowledge, a discussion that has never been published so far.  
The three contributions appearing in (III.19) may be understood as follow: 
♦ The term ( )( ) Tn αν110 +−  is clearly related to the bulging of end faces, and is the direct 
consequence of the inner compression of the crystal, which causes the optical path to 
increase (if αT>0). It is strictly true for an infinitely thin crystal, since plane stress 
approximation is used to derive it. It is reported by Cousins [60] that for a rod whose ratio 
length/diameter is 1.5, this term overestimates the actual bulging by around 35%. In general 
this can be taken as an upper limit for end faces bulging in DPSSLs. 
♦ The term ' ,
02 θα rT Cn accounts for the photoelastic effect only, as already discussed in 
subsections III.3 and 4. It explains bifocussing, depolarization and polarization-dependant 
astigmatism.  
♦ As for the first term (III.19), it represents the partial derivative of refractive index at 
constant strain, which is the thermo-optic coefficient of a virtual perfectly rigid crystal.  
It is noteworthy that it is actually not the usual dn/dT parameter that one can measure and 
find easily in the literature. The “usual techniques” for measuring dn/dT are based either on 
geometrical optics (e.g. measurement of the minimum-of-deviation angle of a prism cut in the 
material under study [72]) or on interferometric techniques (e.g. measuring fringe patterns 
displacements [81]). In all cases, the sample is put into an oven and exposed to different well-
known temperatures. The crystal is free to expand, and the temperature rise into the material is 
uniform, which is incidentally an essential condition for valuable measurements. In this case, 
 53 /115  
obviously, the crystal experiences thermal expansion, a phenomenon which causes the index to 
change (to decrease, in general) in virtue of a pure photoelastic effect. In all these practical 
circumstances, the strain tensor relates directly to the temperature shift by the thermal expansion 
tensor, in other words the stress terms in equation III.1 are zero. The coefficient measured 
experimentally can then be regarded as a partial derivative at constant stress: 
measured
 (III.20) 
On the other hand, the reality experienced by the laser crystal while optically pumped is 
radically different. We know that in all cases (transverse as well as end pumping, thin disks as well 
as long and thin rods) the pumped area inside the crystal is under compression (negative stresses 
and strains), which is true as soon as the thermal expansion coefficient is positive. It can be 
explained qualitatively by saying that the central region of the crystal, yet hotter than the edges, is 
prevented from expanding by the expanding (cooler) outer parts of the rod, which eventually causes 
the central region to be under compression.  
As a result, we see that if we consider the measured dn/dT instead of the partial derivative at 
constant strain in eq. (III.19), the photoelastic contribution to the thermal lens, already fully taken 
into account with the term 2 ' ,
0 θα rT Cn , which accounts for thermal expansion if any, is partially 
cancelled by the photoelastic (thermal expansion) term hidden in the measured dn/dT.  
In a first attempt to correct for this, we thus have to evaluate precisely the thermal expansion 
contribution to the measured dn/dT. It can be done in a simple way by considering the Clausius-
Mossotti model for refractive index, which writes: 
 54 /115  
( ) ( )( )
    (III.21) 
where ρ is the specific mass (density), M the molecular weight, Na the Avogadro number, and αe the 
polarizability. This expression is valid strictly speaking for isotropic ionic crystals (for covalent 
crystals the local field correction is smaller and atomic polarizabilities loose their meaning due to 
the very nature of covalent bonding) 
The modification of polarizability with temperature results from the change in thermal 
occupancies and spectra of the energy levels. Tsay et al. [82] have developed a two-oscillator model 
where they consider the contribution of both electronic and lattice vibration terms. The discussion 
about the different origins of dn/dT is beyond the scope of this review and will not be exposed in 
detail here.  
We can differentiate (III.20), assuming that changes in density only originate from isotropic 
thermal expansion. This is consistent with the experimental procedure used to measure this 
coefficient. 
( ) ( )( ){ }, ,e e e
measured
dn T T T n n n n
dT T T Tσ
ρ α ρ α αρ
ρ α ρ α
⎛ ⎞ ⎛ ⎞∂ ∂∂ ∂ ∂ ∂ ∂⎛ ⎞⎜ ⎟ = = + +⎜ ⎟⎜ ⎟⎜ ⎟ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠
    (III.22) 
From (III.20) and isotropic expansion assumption we obtain: 
( )( )
12 22
    (III.23) 
This expression puts into the light three contributions to the measured dn/dT:  
- a pure effect of thermal expansion : 
( )( )
12 22 −+
 (III.24) 
 55 /115  
- the influence of thermal expansion on polarizability,  
              
( )( )2 22 1
− + − ⎡ ⎤∂
⎢ ⎥∂⎣ ⎦
 (III.25) 
- a thermal expansion-independent contribution, which can be assimilated to the partial 
derivative at constant strain: 
( )( )
12 22
 (III.26) 
Since the second and third terms do not breakdown easily into a set of available material physical 
parameters, no general formula based on the measured dn/dT can be derived for the last partial 
derivative. We see also that thermal expansion appears in both (III.24) and (III.25), so that we 
cannot in a straightforward way dissociate “pure” thermal expansion from strain-related 
polarizability effects. 
This formulation brings some questions (rather than answers) about the interpretation of 
some experimental results, in particular those obtained with materials whose measured dn/dT is 
negative.  
It is often reported that in such materials (e.g. LiCAF [72, 83], FAP [39], YLF [84]), the 
thermal lens is weak or even divergent (case observed in YLF crystals) because the negative dn/dT 
counterbalances (or even surpasses) the other positive terms in the expression of the thermo-optic 
coefficient. 
Although, because of the lack of data about these materials, the photoelastic term is just 
supposed to be positive, or evaluated from other materials whose properties are believed to be 
similar (Woods et al. [72] use data from CaF2 to evaluate photoelastic constants of LiCAF, Payne et 
al. [39] used data from LG-750 phosphate glass to approximate that of FAP). In some cases (e.g. 
thermal lensing measurements in Nd:YLF crystals [84]), discrepancies are reported  between theory 
and experiment, which do not occur with YAG which has a positive measured dn/dT. 
 56 /115  
A large and negative dn/dT coefficient means that the thermal expansion is dominating 
polarizability contribution for an unstressed crystal; it is observed that such behaviour is generally 
associated with a large thermal expansion coefficient. This also means that the photoelastic term, 
proportional to αT, can be expected to be greater. In contrast, we can say nothing about the sign of 
this term, whose knowledge requires that we know all of the pijkl coefficients of the crystal. 
This means that there is no obvious relationship between the sign and magnitude of the 
measured dn/dT and the sign and magnitude of the thermal lens. 
To go further, the only term which is truly always positive is the end faces bulging term. The 
polarizability dependence on temperature (eq. III.26) is mostly positive too [82]. We can then assess 
that negative thermal lensing is more likely to be explained by negative photoelastic terms, and/or 
possibly, by a negative e
 term. 
In conclusion, the crystals with negative measured dn/dT have to be considered very 
carefully as far as simulations are concerned: photoelastic terms must not be neglected. However, it 
remains that photoelastic contributions, whatever calculated or measured, tend to be small in many 
crystals. This means that the rude approximation made by replacing χ by the measured dn/dT (+ the 
end bulging term) will be all the more close to reality that the dn/dT is large and positive.   
III.8. A novel definition for thermo-optic coefficient based on experimentally 
measurable parameters.  
In the precedent subsection, we saw that there are many problems related to the definition of 
the thermo-optic coefficient, which are above all related to the abusive use of the parameter (dn/dT) 
in a context where it is not relevant.  
 57 /115  
Actually, a better way to describe the phenomena is to start from measurable data that are 
relevant as far as solid-state lasers are concerned. The partial derivative at constant strain is a formal 
parameter which has not a real physical meaning since it is impossible to prevent the crystal from 
any strain, compression or thermal expansion. Furthermore, photoelastic and polarizability effects 
are so strongly intermingled, that one cannot imagine easily an experiment that could separate 
clearly the effect of one from the other. That’s why we propose to base solid state laser thermal 
characterization for high power applications on measurable data and separate the thermo-optic 
coefficient χ in three truly independent contributions, as follows: 
lgn bu ing birefringenceχ χ χ χ= + +  (III.27) 
where  
( )3 ' '0n T r
n C C
= + +⎜ ⎟∂⎝ ⎠
 (III.28) 
( )( )lg 0 1 1bu ing Tnχ ν α= − +  (III.29) 
( )3 ' '0birefringence T rn C Cθχ α= ± −  (III.30) 
χbulging accounts for curvature of end faces, and is measurable by performing, for example, 
interferometric or wavefront measurements on a probe beam reflected on each side of the crystal, as 
done by Baer et al. [85] or Kleine et al. [86]. The expression given in (III.29) applies to the thin 
disk ideal model, but its real value can be computed depending on every special geometry, quite 
easily with a finite element code, since this is a pure thermomechanical problem, where data are 
more readily accessible or measurable. 
χbirefringence accounts for strain-induced birefringence (“+” for radial polarization, “-“ for 
tangential). It can be measured separately by performing measurements of polarization-dependant 
 58 /115  
astigmatism, for instance, thanks to a wavefront measurement method sensitive to aberrations (see 
next section). 
Eventually, the term χn which accounts for all the refractive index variations with 
temperature, is not rigorously calculable for all the reasons exposed above, but its exact value can 
be deduced for each material and for each pumping configuration, from the separate measurement 
of χ (global thermo-optic coefficient), χbirefringence and χbulging.  
To conclude, let us give some orders of magnitude of different terms in widespread YAG. 
The measured value of the global thermo-optic coefficient (see next section of this article for details) 
for this material under diode pumping is 10 10-6 K-1. 
The different contributions calculated from tabulated data are: 
6 19.10
measured
− −⎛ ⎞ =⎜ ⎟
( )( )2 2 6 12 1 31.5 10
measured
n ndn
+ −⎛ ⎞ + =⎜ ⎟
( )( ) 6 1lg 0 1 1 7.210bu ing Tn Kχ ν α − −= − + =  
3 6 1
3 6 1
2 0.27 10
2 0.9310
n C K
n C Kθ
(see appendix for the detail of the calculation of photoelastic constants within the plane stress 
approximation). 
The thermo-optic coefficient are small, but the bulging term is far from being negligible. 
Weber et al. [87] have shown that in Nd:YAG, bulging represented 30% of the global thermal lens.  
We calculated also for information a “corrected” dn/dT, which is obtained after subtraction of the 
inappropriate thermal expansion term (III.24): this coefficient turns out to be very large here.   
 59 /115  
In diode-end-pumped Nd:YLF and Nd:YVO4, Baer et al. [85] and Kleine et al. [86] have 
shown experimentally that the bulging term represented half of the total thermo-optic coefficient. A 
finite-element analysis, performed by Peng et al. [88], leads to a similar conclusion for a Nd:YVO4 
crystal.  
III.9. The aberrations of the thermal lens.  
In conclusion to this chapter, we will introduce the thermal lens higher-order distortions id 
est the thermal lens aberrations. 
For a perfectly parabolic distortion of the wave front or equivalently a pure thermal lens the 
thermal distortion can be easily compensated by addition in the laser cavity of the opposite 
divergent lens or by adjusting the distance of the different cavity elements. But, if aberrations are 
present the compensation is very difficult and requires complex systems [89]. While uncorrected, 
these aberrations lead to degradation in beam quality (brightness), and also to losses due to 
diffraction of the beam high spatial frequencies [90]. 
The aberrations are present when the wavefront distortions induced by the absorbed pump 
beam are not perfectly parabolic. This occurs when the longitudinal pump beam has not a true top-
hat profile, for example in the case of a Gaussian pump beam profile [91]. Moreover, if the laser 
beam size is larger than the pumped area, the aberrations also become important as shown in figure 
17. The rays, far from the pumped area are almost not deviated. This is the signature of spherical 
aberration (which is sometimes referred as a “thermal lens varying with radius r”.)   
 60 /115  
Pumped areaPumped area
Figure 17: When the laser beam is larger than the pumped area, spherical aberration is observed. 
In general the aberrations affect the laser modes of the cavity in a way that to degrade the 
beam quality. This degradation can be evaluated by the M2 factor or the Strelh ratio for exemple 
[92]. We are just giving here an exemple of the influence of aberrations on the beam quality, some 
results given by Clarkson [70].  
Let us consider that only 3rd order spherical aberration is present. In this case, the optical 
path difference ∆(r) can be written as:  
( ) 44
−=  (III.31) 
if the laser initial beam M2 factor is 2iM , the M
2 with the added aberations is :  
( ) ( )22222 qif MMM +=  (III.32) 
 61 /115  
with 
π8 442 L
M =  (III.33) 
where λ is the wavelength and wL the laser beam waist. 
One can show that obviously C4=0 for a top-hat pump profile provided that wL< wp. But for 
a gaussian pump-beam profile (beam waist wp) we obtain [70]:  
M  (III.34) 
In that case, the wL/wp is present to the power of 4, which means a strong increase of M2 
even for small mode mismatch. 
Another classical aberration to be considered is the thermal astigmatism.  In this case, the 
analytical solution is not as simple as with the 3rd order spherical aberration and we will only focus 
on the qualitative approach, answering this simple question: in which conditions does the thermal 
lens exhibit astigmatism? In practice it will occur whenever: 
- the thermal conductivity is anisotropical ;  
- the thermal expansion tensor 
ijTα does not reduce to a scalar quantity; 
- the cooling is inhomogeneous; 
- the laser beam is  polarized, even for an isotropic material. This is the so-called polarization-
dependant astigmatism, which can be used a sa probe to evaluate the strain-induced 
birefringence (see figure 18.) 
 62 /115  
crystal 
(nr-1)L 
(nθ-1)L 
Figure 18: Illustration of polarization-dependant astigmatism. Here a vertically polarized strikes 
the crystal from the left. The ray  #    sees the radial index of refraction nr while ray #  sees the 
tangential index of refraction nθ . If for the sake of simplicity the indices are considered constant 
over the whole crystal length L, the astigmatism is (nr- nθ)L. 
As a conclusion for this part, we present a schematic diagram (fig. 19) summing up the different 
thermal effects arising in a solid state laser medium, and how they are related to each other.  
 63 /115  
Figure 19: Summary of the thermal effects in solid-state lasers. The observable consequences 
observable are presented in full rectangles. The aberrations can be split in two classes: the ones 
Pumping ∆T 
    temperature 
deformations 
Modification of crystal lattice (thermal 
expansion)  
(stress) 
Elastic 
tensor 
Modification of refraction indices 
Thermal lens 
Induced birefringence 
Photoelastic 
effect 
depolarization 
Modification of 
polarizability, 
related to 
⎜ ⎟∂⎝ ⎠
FRACTURE 
if σ > σTS
Curvature (bulging) of both faces of 
the crystal 
focussing  
For an ideal 
lens 
Changes in 
stability range of 
the laser cavity, 
and in the size 
and the 
divergence of the 
laser beam. 
Losses   
potentially 
Spherical 
aberration or/and 
astigmatism  
Beam quality degradation  
ABERRATIONS of the thermal lens 
Polarization-
dependant 
Astigmatism 
Leading to 
 64 /115  
that do not depend on the polarization (lined rectangles)  and the ones that  depend on polarization 
(dotted rectangles)  which come from the strain-induced birefringence. 
(*): 2 types of losses are induced by aberrations: the diffraction losses (associated with degradation 
in beam quality) and the losses induced by the eventual presence of a diaphragm in the cavity to 
prevent from oscillation of higher order laser modes. 
IV. Thermal lensing techniques  
IV.1. Introduction  
The first evidence of thermal effects in lasers were demonstrated in 1965 by Gordon, Leite 
and Whinnery [93] working at Bell Labs on He-Ne lasers for Raman spectroscopy applications. The 
use of liquids to Q-switch the laser lead to the observation of unexpected effects such as relaxation 
oscillation of jump of modes. The exceptionally long time constant of this phenomenon (several 
seconds) lead to the conclusion that a thermal lens was at the origin of the observed effects.  This 
lens was created by the small absorption occurring in the liquids. The first application proposed by 
the authors was then to use it to measure very small absorption coefficients down to 10-4 cm-1. As a 
matter of fact, the so-called “thermal lens method” allows nowadays to measure absorption 
coefficient lower than 10-7 cm-1 [94]. Since 1965, the photothermal-methods panel available to 
physicians has grown in diversity [95]. Actually, due to the complexity of simulating thermal 
effects in lasers (see previous subsections), the experimental determination is often the only 
accurate method.  
Since the 70’s, numerous attempts have been done to measure the thermal lens in solid-state 
laser media. They can be classified in three categories: the geometrical methods based on the 
deflexion of a beam, the methods based on the properties of cavity modes, and the methods based 
on the wavefront measurements. 
 65 /115  
IV.2. Geometrical methods 
These methods are probably the simplest methods to measure the thermal lenses. They can 
be separated in two sub-groups: one can exploit either the defocussing or the deflection of a beam 
passing through the pumped medium  
crystal 
Powermeter 
Figure 20 : Example of thermal lens measurement based on the displacement of the focal point. 
The principle of the first category of methods is very simple. Considering a probe beam 
going through the crystal, the measurement of the axial shift of the focal point position allows to 
retrieve directly the thermal lens using simple geometrical optics. When the rod is relatively large, 
one can use a collimated beam of comparable size to directly measure the position of this focal 
point by Z-scan measurement for example as shown in figure 20 (a small aperture is longitudinally 
translated in order to find the maximum of probe-beam transmission [96].) This method is based on 
the assumption that the lens is perfect (without aberrations) and then is especially suited to 
transverse pumping with large-size materials . In fact, the method is not easily applicable to end-
pumped lasers because of the very large depth-of-focus associated with a very small and low solid-
angle–probe beam, and obviously does not yield the thermal lens aberrations. 
Moreover, for small beam diameters (10-100 µm), this method needs to be generalized using 
Gaussian beam optics. Hu and Whinnery [97] described a simple method to evaluate the thermal 
lens with a probe beam whose size is comparable to the cavity beam in end-pumping schemes. This 
 66 /115  
last method is described in figure 21: it can be based either on the measurement of the divergence of 
the probe beam [97], or in the measurement of the beam diameter in an appropriate plane.  
figure 21 : Example of measurement based on focal point displacement for tightly focused probe 
beam according to Hu and Whinnery’s method. 
To conclude on the methods based on the focal point displacement, we can say that their 
main drawback is their low accuracy. The relative precision is only 20-30% on the focal lens 
measurement. 
We can use geometrical optics in another way by measuring the deflection of a probe beam. 
Instead of using a beam covering the whole pumped area, we can measure the deflection of a small 
beam slightly off-axis as shown in figure 22.  
figure 22 : Example of measurement based on deflection measurement [98]. 
detector 
Laser 
crystal 
With thermal lens 
Gaussian 
beam 
Without thermal lens
 67 /115  
The whole surface can be scanned in order to measure not only the focal lens but also the 
aberrations due to the thermal effects [98]. However this method is complex and it can only be used 
for large transversally-pumped crystals (even if the resolution can be lower than the probe beam 
size [99]). Moreover this method can be considered as a point-to-point measurement of the simpler 
Shack-Hartmann technique described later.  
IV.3. Methods based on the properties of cavity eigenmodes 
After the development of end-pumped lasers (particularly thanks to the increasing 
performance of laser-diodes), alternative methods appeared. These less straightforward methods are 
based on the properties on the laser cavity eigenmodes, in particular on the fact that the thermal lens 
affects the stability zones of a laser cavity. All these methods are based on the theory of paraxial 
beam propagation theory in cavities presented by Kogelnik and Li [100] and can be formalized 
using the ABCD matrices. An example of this influence of the thermal lens on the stability of a 
cavity is given in figure 23. As a direct consequence, these methods consider ideal lens (aberration-
free) and do not give any information on the thermally-induced aberrations. Nevertheless these 
methods remain easy to implement since there is no probe beam (the laser itself is used to measure 
the thermal lens). In counterpart, it is impossible to measure the thermal lens in absence of laser 
extraction (with pumping but no laser emission). Here are three different examples of these methods 
based respectively on the work of Frauchiger and al., Neuenschwander and al. and Ozygus et al.   
 68 /115  
No thermal lens
thermal lens
Unstable cavity
Stable cavity
No thermal lens
thermal lens
Unstable cavity
Stable cavity
Figure 23: example of this influence of the thermal lens on the stability using a plano-plano cavity 
[101]. 
- Frauchiger and al. [102] measured the divergence of the output laser beam in a diode-pumped 
Nd :YAG and retrieved the thermal lens by a paraxial calculation. This example allows us to 
perceive the limitation of these techniques using the properties of cavity modes. Indeed, they are 
very sensitive to the beam quality. The measured divergence is actually directly proportional to the 
M2 factor of the beam. If the laser mode is not perfectly TEM00 the reliability of the method is 
strongly affected.  
- Neuenschwander and al. [101] used a plano-plano laser cavity stabilised by the thermal lens 
(figure 23). Adding two extra-cavity lenses and measuring the beam diameter in different 
longitudinal positions they found the waist in the cavity and therefore the thermal lens. This method 
is interesting because it allows the simultaneous measurement of the M2 factor which reduces the 
 69 /115  
limitations due to the beam quality. In that case, the limitation is due to the precision on the 
distances between the different optical components which is not always perfectly known.  
- Ozygus et al. [103] proposed an alternative method that consisted in using the frequencies of 
different transverse modes. In fact, the frequencies of the different transverse modes (in a plano-
concave laser cavity) depend not only on the length of the cavity but also on the radius of curvature 
of the mirrors (considering the flat mirror and the crystal as another concave mirror). If one 
achieves to have two modes lasing simultaneously in the cavity, the measurement of the beating 
frequency with an optical spectrum analyser allows retrieving the thermal lens. Another upgraded 
technique based on the same effect was also presented by Ozygus et al.later in 1997 [104]. In the 
last one, one translates one of the mirrors to find the positions of the spectral degeneracy (when two 
eigenmodes have the same frequency). The advantage of this method is its capacity to be used for 
measuring long-focal-length thermal lenses. It allows the measurement of focal lengths as long as 5 
In conclusion, the methods based on the properties of the cavity modes allows to have 
relative precisions on the thermal focal lens of 15 % for a TEM00 beam but down to 60 % for an 
non-diffraction limited beam [101].  
IV.4. Methods based on  wavefront measurements 
In this part, we will distinguish three wavefront measurement techniques: “classical” 
interferometry (based on fringe measurements on Michelson, Fizeau, … interferometers),  shearing 
interferometry, and Shack-Hartmann sensing. 
IV.4.1. Classical interferometric techniques 
 70 /115  
As far as “classical” interferometry is concerned, one can consider equal-thickness fringes 
between the parallel end faces of the rod (this is the so-called Fizeau interferometer); one can also 
insert the rod under study in an arm of a Michelson-type [105] or Mach-Zehnder -type 
interferometer [106] for example. The first type of method is simpler since it does not require a 
second interferometrically adjusted arm.   
These methods are particularly well suited for large amplifier rods but in counterpart, they 
are not convenient for end-pumping. Indeed, even if there is no fundamental contraindication to use 
this method for small spots, in practice in this case the number of fringes is too small to obtain an 
exploitable interferogram. As an example, if we consider a 200-µm diameter probe beam, a focal 
length of 5 cm only induces a phase shift between the centre and the edge of the beam of ∆ = h2/2f 
= λ/6 (with h the radius of the beam, f the focal length, and λ the wavelength of the probe beam, 
here λ =670 nm). In this case, of course, no fringes are visible.  
To overcome this problem, one can choose to take a probe beam larger than the pumped 
area, given that in these conditions, as shown in figure 17, some spherical aberration will be present, 
which requires an additional numerical model to fit the data and retrieve the thermal lens, as done 
by Pfistner et al. [107]. The weakness of this method relies precisely in the retrieving algorithm 
since the whole interferogram consists on only a few fringes. The precision is then a hard point and 
in the best case this precision is evaluated to λ/4. This work [107] has also been done on YAG, 
GSGG and YLF crystals.   
IV.4.2. Shearing interferometric techniques 
A classical solution to obtain information on the phase “between two fringes” is phase shift 
interferometry. This technique has been used by Khizhnyak et al [108] with longitudinally-pumped 
Nd :YAG lasers. This method is easy to implement since it’s based on commercial products, but it 
 71 /115  
remains quite unused due to its important cost. One generally prefers lateral-shearing-interferometer 
methods.  
Methods based on lateral shearing interferometers are particularly well suited to end 
pumping. The principle is the following: the beam is duplicated in several replicas, typically 2 [109-
110],3 [111] or 4 [112-113] (as presented in figures 24 and 25 with a tri-wave lateral shearing 
interferometer setup).  
Figure 24 : Example of tri-wave lateral shearing interferometer (courtesy of J.C. Chanteloup). 
 72 /115  
Figure 25: Example of tri-wave lateral shearing interferometer setup (courtesy of J.C. Chanteloup). 
The replicas are slightly shifted from each other in the lateral direction which provides an 
interferogram whose fringes (for 2 waves) or dots (for 3, 4 waves) separation give information on 
the derivative of the wavefront.  For example in the absence of wave-front distortion the lines are 
rectilinear for the 2-wave shearing interferometer, they form an homogenous honey-comb for the 3-
wave shearing interferometer and perfect squares for the 4-wave shearing interferometer. This 
technique is more sensitive than classical interferometry since the sensitivity is tunable by adjusting 
the shearing distance. The larger the shift, the more precise the technique . The precision of this 
method is excellent since it is in the order of λ/50 [109] and can even reach λ/200 [111].  
The method using 3 (or 4)-wave « trilateral shearing interferometry » [111] has the 
advantage over the 2-wave shearing interferometry to allow the cartography in the 2 dimensions in 
one acquisition.  This method is simple to implement (figure 25) since the splitting of the beam can 
be realized readily with a 3D grating.  Moreover the use of a grating makes this method totally 
achromatic which allows its use for broad-band lasers such as femtosecond lasers. In 1998, 
 73 /115  
Chanteloup et al.  [112] reported on the wavefront distortions of a terawatt-class femtosecond laser 
system with an accuracy of λ/50. 
This method was also used to characterize thermal lensing in Ytterbium-doped materials, 
namely in Yb:YAB by J.L. Blows et al. [110], who used the 2-wave shearing interferometer 
technique to measure thermal lenses, and then thermal conductivities and fractional thermal 
loadings. The experimental setup used is reproduced on figure 26.  
Figure 26: setup for measuring thermal lensing in diode-pumped Yb:YAB crystal with lateral 
shearing interferometry from [110] (courtesy of J. Dawes, Centre for lasers and Applications, 
Sydney) 
The use of a probe beam at 530 nm allowed the thermal lens to be evaluated under lasing 
and nonlasing conditions (that is with a shutter inside the cavity), which is an essential requirement 
to perform radiative quantum efficiencies measurements. 
IV.4.3. Methods based on Shack-Hartmann wavefront sensing 
Although it can be seen also as a multiple beam interferometric method, we separate this method 
from the previously mentioned ones since it can be understood easily with geometrical optics. 
 74 /115  
In 1900, Hartman proposed to use a drilled plate [115] to measure wavefronts. The principle is 
simple: since light rays run perpendicular to the wavefront, one can retrieve the local wavefront 
slope as soon as the direction of the ray is measured. The Hartmann plate is made of small apertures 
which scatter the beam into regularly-spaced diffraction patterns, and behind which is located a 
detector (typically a CCD camera nowadays). In 1971, Roland Shack and Ben Platt improved the 
Hartmann setup by replacing the array of holes by an array of microlenses (figures 27, 26).  
Single unit 
Figure 27: example of intensity pattern observed in the focal plane of the Shack-Hartmann 
microlenses. 
 75 /115  
Rays 
propagationIncident 
Focal length
Microlens
array
Rays 
propagationIncident 
Focal length
Microlens
array
Figure 28: Shack-Hartmann wavefront sensor setup. 
Reconstructed 
wave front
Measured 
local slopes
Input wave 
front
Reconstructed 
wave front
Measured 
local slopes
Input wave 
front
Measured 
local slopes
Input wave 
front
Figure 29: Shack-Hartmann sampling. 
 76 /115  
The small axial shift of the microlens diffraction pattern centroid (see figure 28) is directly 
proportional to the average local slope of the wavefront on the aperture of the micro-lenses. The 
displacement vectors allow having 2D information. The standard sensitivity of such systems is 
typically λ/100 RMS for commercial products, which settles this technique as a competitor of  
lateral shearing interferometry. One of the advantages of this method, compared to interferometric 
ones, is its insensitivity to mechanical vibrations or thermal fluctuations. In counterpart, the 
principal limitation of this kind of sensor is the discrete sampling that limits the transverse 
resolution (figure 29) and thus prevents from obtaining information about high-spatial-frequency 
phase distortions. Nevertheless, it’s noteworthy that this point is not a problem for thermal lensing 
and aberration measurements because in virtue of the general heat equation (II.1.1), temperature 
variations in a crystal are smooth even if the thermal load exhibits sharp variations.  
In 1998, Armstrong [116] reported the measurement of thermal lens in transversally-pumped 
Nd:YAG and Nd:YAP rods. More recently Ito et al. [117] and Pittman et al. [118] reported the use 
of a Hartmann-Shack sensor to measure the thermal lens in Ti:sapphire rods of terawatt-class 
femtosecond lasers.   
In 2001, reports of temporal changes of thermal lens effects on high power pulsed Yb:glass 
lasers have been done by Nishimura et al. using a Shack-Hartmann wavefront sensor [120]. Here 
the sensor was used for its ability to yield real-time (100 Hz sampling rate) estimation of Zernike 
coefficients of aberrations, and allowed to measure characteristic thermal relaxation times. 
Our group reported recently [73] a derivation of Armstrong’s work to measure thermal lensing 
in various diode-pumped Yb:doped crystals, under lasing or nonlasing conditions. The setup 
appears in figure 30. The probe beam was a laser diode at 670 nm coupled in a single mode fiber, 
chosen for its high spatial coherence (an essential feature to correctly define the “reference 
wavefront”) and its low temporal coherence (necessary to avoid coherent cross talk, that is 
interference between two neighbouring microlens diffraction patterns.)  After collimation by a 
 77 /115  
microscope objective, the probe beam was focused onto the crystal and superimposed with the 
pump beam. The crystal was then imaged upon the microlens array using a magnifying relay 
imaging system. An uncoated glass plate was inserted in the pump beam path to reflect the probe 
beam towards the sensor. A selective interference filter at 670 nm was added in front of the sensor 
to eliminate any unwanted signal at the pump or laser wavelengths. A « reference wavefront » is 
recorded when the pump diode is turned off, which includes all static aberrations of the optical 
elements and of the cold crystal itself. It is then subtracted to the measured wavefront when the 
pump is on. Thus, only phase distortions originating in thermal effects are recorded. The phase front 
was then reconstructed by projection over the set of the orthogonal Zernike polynomials [70].   
Figure 30: Experimental setup used to measure thermal lensing in diode-pumped Yb-doped crystals 
with a Shack-Hartmann wavefront sensor, from [73]. 
IV.4.4  Other techniques 
There exists some more marginal and less used techniques to measure thermal lensing. For 
example, it is possible to achieve phase reconstruction basing on Fourier optics. One can show that 
it is possible to retrieve the phase by knowing the intensity profile in different planes linked by 
Fiber-coupled pump 
laser diode 
Fiber-coupled 
Shack-Hartmann sensor 
Interference 
Filter @ 670 nm 
Laser cavity Glass plate 
crystal 
 78 /115  
Fourier transformation. For instance it is quite obvious that a uniform intensity over a circular 
aperture and an Airy pattern appearing in two Fourier-related planes implies that the wavefront 
propagating from one plane to the other is purely spherical. Nevertheless the general inverse 
problem is far from being obvious. Grossard and al. [114] proposed a technique for measuring 
thermal lensing aberrations in a diode-pumped Nd:YVO4 crystal using intensity profiles in 3 planes 
and a complex phase retrieval algorithm derived from Gerchberg and Saxton’s work.      
IV.5. Conclusion 
In conclusion, we made a review of the main methods used to experimentally measure the 
thermal effects in lasers, putting in emphasis the advantages and the limitations of each method. A 
summary of this review is presented in table 4. 
 79 /115  
Methods 
Based on  focal point 
displacement no 
Very 
difficult yes yes yes yes yes 
Very easy / 
very cheap 
Based on cavity 
properties 
no yes Only with 
Almost 
impossible
probe yes yes 
Low  
≈ 15 à 60 %
Difficult in 
general / 
cheap 
« classical » 
Interferometry 
yes Very difficult 
yes in 
theory no no no yes 
σ∆ ∼λ/4 for the 
interferometry 
without phase 
shift. 
Difficult / 
Cheap 
(except phase 
shift interf.) 
lateral shearing 
interferometry yes yes yes yes yes yes 
yes  
for the 3-4-wave 
shearing 
high 
σ∆∼λ/50 
to λ/200 
medium / 
expensive 
reconstruction de la 
phase from different 
position intensity 
profiles 
yes yes yes yes yes yes no unknown Difficult /no commercial 
Hartmann- Shack yes yes yes yes yes yes yes 
high 
σ ∆ ∼λ/100 
easy/ 
expensive 
Table 4: summary of the different techniques used to measure thermal lensing. σ∆ denotes the 
reported RMS deviation on phase shift. 
V. Thermal lensing measurements in ytterbium-doped 
materials: the evidence of a non radiative path  
We will conclude this review by giving some examples of experimental thermal lensing 
measurements in diode-pumped Yb-doped crystals. Most of the examples are taken from our 
previous publications [73,121]: the reader interested in the technical details of the experiments, as 
well as by the theoretical considerations underneath, is invited to refer to these works. 
 80 /115  
V.1. the thermal load in Yb-doped materials  
What are the heat sources in a laser medium ? Following T. Y. Fan and using the same notations 
[8], the fractional thermal load ηh (that is the fraction of the absorbed pump power converted into 
heat) can be written:  
rlph λ
ηηηη 11   (V.1) 
Where: 
- λp, λl, λf are the pump wavelength, the observed laser wavelength, and the mean fluorescence 
spectrum wavelength, respectively; 
- ηp is the pump quantum efficiency, which is the fraction of absorbed pump photons contributing 
to inversion. Non-unity pump quantum efficiency accounts for residual absorption of the undoped 
crystal, or can be related to the presence of nonradiative sites. 
- ηr  is the radiative quantum efficiency for the upper manifold: it represents the fraction of excited 
atoms that decay by a radiative path (in absence of stimulated emission). Non-unity radiative 
quantum efficiency can be related to multiphonon relaxation (although it is very unlikely since a 
large number of phonons are necessary to bridge the 10 000 cm-1 gap separating the excited and 
ground manifolds) or more probably to concentration quenching. The latter phenomenon 
corresponds to the trapping of the energy by a color center, an impurity, or a lattice defect (Yb2+, 
rare-earth impurities or hydroxyl groups have been evoked as possible “dark sites” [122-124]) after 
several transfers of excitation between neighbouring ions.  
- ηl is the laser extraction efficiency, defined as the fraction of excited ions that are extracted by 
stimulated emission. An expression of the laser extraction efficiency can be derived by writing the 
stimulated, spontaneous and nonradiative relaxation rates [73]. In most cases an approximate 
 81 /115  
relation can be used by neglecting reabsorption at laser wavelength. In the latter case we obtain the 
simplified expression: 
≈   (V.2) 
where I is the intracavity laser intensity, and ( )lem λσ  the emission cross section at laser 
wavelength. As shown by eq. (V.2) the laser extraction efficiency tends towards 1 for intracavity 
laser intensities that surpass the laser saturation intensity. Generally, cw oscillators based on Yb-
doped materials work with high reflectivity output couplers: as a consequence the intracavity laser 
intensity is very high, at least one order of magnitude higher than the laser saturation intensity, so 
that ηl is typically close to unity in an operating Yb laser. In this case, the thermal load becomes 
nearly independent on the radiative quantum efficiency, and is only given by the quantum defect. 
Nevertheless, the quantum efficiency directly affects the excited state population and has 
subsequently crucial importance for Q-switched lasers or low repetition rate amplifiers. 
Incidentally, the relation (V.2) also illustrates that the performance of an Yb-based cw oscillator 
becomes nearly independent of the emission cross section at laser wavelength, provided that the 
pump intensity is far higher than the pump saturation intensity.  
V.2. Evidence of nonradiative effects in Yb-doped materials: the example of 
Yb:YAG  
During the past decade, it has been assumed and claimed many times that nonradiative effects did 
not exist in Yb-doped materials, in virtue of the very simple electronic structure that prevented 
deleterious effects that are well known with other ions, such as cross relaxation, excited state 
absorption or upconversion. Nevertheless, all the measurements performed in the past few years 
 82 /115  
have all brought a contradiction to this statement. Blows et al. have demonstrated clear evidence of 
a nonradiative path in Yb:YAB [110],  Ramirez et al. [125] in Yb:MgO:LiNbO3 , and as for YAG, 
non-unity quantum efficiencies have been reported by Barnes et al. [126], Patel et al. [127], 
Ramirez et al. [125] and Chenais et al. [121]. This recent work has also shown the existence of 
nonradiative effects in Yb:GGG, Yb:YSO, Yb:KGW, Yb:YCOB, Yb:GdCOB.  
An example of quantum efficiency measurement thanks to the thermal lens method is shown in 
figure 31 with an Yb:YAG crystal. A simple qualitative explanation is given in figure 32. The clear 
difference between the thermal lens dioptric power under lasing and nonlasing conditions can be 
modelled using eq. (V.1) and eq. (V.2), provided that laser power is measured simultaneously. 
Given some additional approximations, detailed in [121], one can retrieve both radiative quantum 
efficiency and thermo-optic coefficient χ. The results recently reported for different crystals have 
been summarized in table 5. 
 83 /115  
figure 31: Thermal lens dioptric power (here, aberrations of the thermal lens were negligible) 
under lasing and nonlasing conditions. On the right (same graph) the measured laser power, useful 
to compute the laser extraction efficiency. (from [121]) 
laserfluorescence
NON lasing lasing
2 F 5/2 
2 F 7/2 
Non Radiative 
(NR) decay 
Energy 
fluorescence
NR decay  
NON LASING 
LASING
543210
Absorbed pump power (W)
8- at.% Yb:YAG
 TL, non lasing : experiment
 TL, non lasing : model
 TL, lasing : experiment
 TL, lasing : model
 Laser power
 84 /115  
figure 32 : Simple qualitative explanation of the observed difference between thermal lens dioptric 
power under lasing and nonlasing conditions. When the laser is on, stimulated emission short-
circuits the nonradiative path, causing the thermal load to be lower. 
It is seen that for a crystal like YAG, many different techniques have been used (for the details of 
each method, refer to the cited publications), and that a large dispersion of reported quantum 
efficiencies appears. This dispersion tends to assess the conjecture of concentration quenching as 
the nonradiative source in Yb-doped materials. This conjecture has been very alleviated in highly-
doped Yb:YAG samples and was attributed to cooperative processes between two Yb3+ ions 
towards Yb2+ impurities [122]. Owing to the intrinsic nature of concentration quenching and the 
major role played by impurities, it is clear that the radiative quantum efficiency is a parameter that 
pertains to a single given sample, characterized by its doping concentration, the growth technique 
and its associated environment (in particular the nature of the crucible), and of course the degree of 
purity of the compounds.  
 85 /115  
Table 5: Reported values of measured thermo-optic coefficients and radiative quantum efficiencies 
in the literature for different Yb-doped materials. 
crystal Mean 
fluorescence 
wavelength 
λf (nm) 
Thermo-optic 
coefficient (×10-6 K-1)
(measured, under 
diode-pumping 
conditions) 
Radiative 
quantum 
efficiency ηr 
(measured) 
Method used 
 (for quantum 
efficiency 
measurement) 
Reference 
0.70 Thermal lensing 
(Shack-Hartmann) 
[121] 
0.874 photometric [126] 
0.835 calorimetric [126] 
0.97 lifetime [127] 
Yb:YAG 1007 10.0 
(from [121]) 
0.85 Direct temperature 
measurement 
[125] 
Yb:GGG 1013 31 0.90 Thermal lensing 
(Shack-Hartmann) 
[121] 
Yb:GdCOB 1011 6.5 0.71 idem [121] 
Yb:YCOB 1035 17 0.90 idem [121] 
Yb:KGW 993 7.5 0.96 idem [121] 
Yb:YSO 1001 15 0.89 idem [121] 
Yb:YAB 996 Non reported 0.88 Thermal lensing 
(lateral shearing) 
[110] 
V.3. Laser wavelength dependence on the thermal load in Yb-doped broadband 
materials: the example of Yb:Y2SiO5 
 86 /115  
We saw that the fractional thermal load (eq. V.1) is dependant on the operating wavelength of the 
laser. In most practical circumstances however, this dependence is hidden by the fact that the laser 
extraction efficiency also greatly depends on the laser wavelength, since it is linked to the emission 
cross section. The Yb:Y2SiO5 (Yb:YSO [128]) crystal exhibits two maxima of comparable 
amplitude in its emission spectrum, at 1042 and 1058 nm respectively. In addition the output is 
naturally linearly polarized along the crystallophysic axis Y for both wavelengths. It is thus a good 
candidate to put clearly into evidence the influence of laser wavelength on thermal lensing. To 
perform the experiment, we added a SF6 dispersive prism cut at Brewster angle in the collimated 
arm of the three-mirror cavity appearing in figure 30, so that identical laser efficiencies were 
achieved at both wavelengths (2.1 Watts were obtained for 8.5 Watts of absorbed pump power). 
The results are shown in figure 33. It appears clearly that the thermal lens is weaker when the laser 
oscillates at 1042 nm, as expected since quantum defect is lower at this wavelength. The theoretical 
curve derived for the 1058-nm laser oscillation was obtained from the 1042-nm curve by just 
modifying the wavelength and the emission cross section in eq. (V.1) and (V.2), without any 
adjustable parameter (see [121]). This simple experiment shows the interest of multiwavelength 
thermal lensing measurements in broadband materials. Indeed, we have considered here a simple 
formulation of the fractional thermal loading (given by eq. V.1) which gives here satisfactory 
results; but it is possible, from the work of Patel et al. [127] for instance, to derive a more accurate 
expression of the fractional thermal load, which takes into account the probability of excitation 
transfer to a neighbouring ion [129]. If measurements are performed versus absorbed power at 
different (more than 2) laser wavelengths, for which the laser extraction efficiency is also known, 
this means that we have the possibility to infer other spectroscopic parameters (such as the transfer 
probability to a neighbouring ion) involved in the expression of the thermal load. 
 87 /115  
86420
Absorbed pump power (W)
1058 nm
1042 nm
1058 nm
1042 nm
 TL, lasing at 1042 nm : experiment
 TL, lasing at 1058nm : experiment
 TL, lasing at 1042 nm : model
 TL, lasing at 1058 nm : model
5-at. % Yb:YSO 
Figure 33: Thermal lens dioptric power at 1042 and 1058 nm (left) and laser power (right in 
Yb:YSO. The effect of wavelength on quantum defect is clearly visible for this material. (from [121]) 
IV.4. The influence of the mean fluorescence wavelength on the thermal load: an 
illustration with Yb:KGW 
The fact that the mean fluorescence wavelength affects the fractional thermal load is a very 
interesting feature of broadband Yb-doped materials. There are even some materials whose mean 
fluorescence wavelength is below the tail of the absorption spectrum. More precisely, according to 
eq (V.1) it is clear that if we may find a pump wavelength verifying:  
 88 /115  
f p r pλ η η λ<  (V.3) 
then the thermal load will be negative (in absence of laser extraction) and “radiative cooling” will 
be achieved. Such radiative cooling has been reported in Yb-doped ZBLANP glass in 1995 [130] 
and in a KGW crystal by Bowman et al. in 2002 [131]. The latter author has also proposed to use 
this phenomenon to realize a thermal load-free (radiation-balanced) laser [132]. The key idea is to 
correctly adjust the laser intensity so that the spontaneous emission rate, source of cooling 
providing the pump wavelength is long enough, exactly balances the stimulated emission rate, 
source of heating. 
We show here how the mean fluorescence wavelength plays a key role in the interpretation of the 
results obtained when measuring the thermal lens in a Yb:KGW crystal. KGW is now a well-
known crystal, suited for ultrafast laser applications [32-33, 17, 46, 74]. When used with 
wavevector k//c, the polarization-averaged mean fluorescence wavelength is 993 nm while the 
observed laser wavelength is 1030 nm and the pump wavelength tuned to the zero-line absorption 
peak, i.e. 979 nm. Results are shown in figure 34, where it can be seen that unlike previously 
reported results, thermal lensing is actually stronger under lasing conditions. A simple explanation 
is given with the schematic picture appearing in figure 35. The simple model suggested above fits 
well with experimental data and yields a high quantum efficiency for this sample (0,96), consistent 
with the fact that tungstate crystals are grown with the flux method, which is known to carry less 
impurities during growth than the Czochralski technique.  
 89 /115  
figure 34: Thermal lensing measurements in Yb:KGW (from [121]) 
NON LASING 
LASING 
6543210
Absorbed pump power (W)
5- at.% Yb:KGW
 TL, non lasing : experiment
 TL, non lasing : model
 TL, lasing : experiment
 TL, lasing : model
 Laser power
2 F 5/2 
2 F 7/2 
993 nm 1030 nm 
laser fluo fluorescence 
Non lasing lasing 
Energy 
 90 /115  
Figure 35: Qualitative explanation of the paradoxical behaviour of the Yb:KGW crystal. Under 
nonlasing conditions (left), the thermal defect is low due to a low mean fluorescence wavelength 
(993 nm). When laser extraction at 1030 nm becomes the dominant way down for excitation, then 
the quantum defect is higher. This simple picture assumes that ηr = 1. 
IV.5. Conclusion 
As a conclusion for this final part of this paper, we have shown some examples of thermal lensing 
measurements, which allowed us to highlight several points of interest pertaining to Yb-doped laser 
media : 
- all measurements show a difference between thermal lensing dioptric power with and without 
laser action: this provides the proof that in all the crystals under investigation (here YAG, GGG, 
YSO, GdCOB, YCOB, KGW, YAB)  there is a nonradiative return path for excited state 
population. This is all the more detrimental for laser performance in pulsed regime, since in cw 
oscillators the laser extraction efficiency can be large enough to dissimulate this effect. 
- Since Yb-doped materials have broader spectra than their Nd-doped counterparts, thermal lensing 
has two specific properties, which have been illustrated by two experiments: 
1) the thermal load depends on the laser oscillation wavelength: the lower the wavelength and the 
lower the quantum defect; 
2) the mean fluorescence wavelength λf plays a key role. Materials exhibiting a low λf will be less 
sensitive to heating under nonlasing conditions, and in KGW we saw that the thermal lens was 
even higher under lasing action.  
 91 /115  
The peculiarities of Yb-doped materials are not limited to these two latter facts. Chenais et al. [121] 
have reported the appearance of a roll-off in the thermal lens dioptric power at high pumping 
densities, under nonlasing conditions, with several different materials [121]. Furthermore, it has 
been observed by several groups ([110], [124]) a green luminescence that can be related to 
cooperative luminescence.  
The detailed interpretation of these phenomena and their connection to the thermal load are still 
works in progress. 
VI. Conclusion 
Ytterbium-doped materials have brought new prospects and deep changes in the area of high power 
solid state lasers. Associated to new and clever pumping concepts (fiber lasers, thin disk lasers, 
spinning disk lasers...) they are now well-established competitors of Nd-doped materials for high 
power applications. This paper has been an effort to make a review of the recently published works 
dealing with thermal effects in solid state lasers, with a particular scope on the special case of Yb-
doped crystals.  
In the first section we have presented the general properties of Ytterbium-doped media, and 
pointed out the crucial role of the matrix host on the properties of the laser material. The part II was 
devoted to a detailed presentation of the temperature distribution in a diode-pumped Yb-doped 
crystal: how to calculate it and how to measure it. We pointed out the importance of boundary 
conditions, and gave some practical information about the role of the thermal contacts in the 
temperature profile. We have shown that it was possible to easily include pump absorption 
saturation effects and pump beam divergence inside the crystal, exploiting the fact that the heat 
transfer coefficient towards end faces was far smaller than towards edge faces.  
In the third part of this review we focussed on the thermo-optical properties and made a quite 
detailed study of the so-called thermal lensing phenomenon. A comprehensive understanding of 
 92 /115  
this aspect requires a good knowledge about both thermo-mechanical and thermo-optical properties 
of the materials under consideration. This lead us to point out several inaccuracies reported in 
previous works, concerning the calculation of photoelastic constants for isotropic materials, and 
more generally the abusive employment of the dn/dT parameter when it is used to estimate the 
magnitude of the thermal lens: we have shown that the partial derivative of index with temperature 
taken at constant strain is the most appropriate figure, this is because the dn/dT is classically 
measured under experimental conditions that are not consistent with the usual situation of a diode-
pumped crystal under mechanical stress.  We proposed an alternative way to split the thermo-optic 
coefficient into three truly independent terms, and addressed in conclusion a schematic diagram of 
thermal effects showing how all the different apparent consequences (lensing, depolarization, 
strain-induced birefringence, astigmatism, fracture...) are connected to each other.  
Given the high complexity of these thermo-optical phenomena, and the unfeasibility of precise 
calculations as far as all the properties of a crystal are not known (that is the case for the majority 
of laser crystals), we then focussed our attention on thermal lensing measurement techniques, 
which was the topic of part IV.  
We presented in that section a review of what are, to our knowledge, the main different techniques 
that can be employed to measure thermal lensing in end-pumped laser media, and discussed their 
relative advantages and drawbacks.  
Finally, we presented some examples of thermal lensing measurements that have been reported 
recently in Yb-doped crystals. All these measurements agree to find non-unity radiative quantum 
efficiencies for the Yb-doped materials under investigation. This non-ideal behaviour, presumably 
related to concentration quenching, provides contradiction to the general consideration that Yb-
doped materials are totally free of deleterious nonradiative effects. We concluded this review by 
giving some examples of the influence of the laser operating wavelength on the thermal load, as 
well as the influence of the mean fluorescence wavelength. 
 93 /115  
Acknowledgements: We wish to acknowledge all the people involved in the realization of this 
review: at first Romain Gaumé (Stanford University) for fruitful discussions about issues related to 
dn/dT and characterization of materials; Bruno Viana, Gerard Aka and Daniel Vivien from the 
Laboratoire de Chimie Appliquée de l’Etat Solide (Ecole Nationale Supérieure de Chimie Paris, 
France), Alain Brenier and Georges Boulon from the Laboratoire de Physico-Chimie des 
Materiaux Luminescents (Lyon, France) and at last Bernard Ferrand from CEA-LETI (Grenoble, 
France), for growing the crystals used in most of the experimental results reported here. We 
gratefully acknowledge the Ecole Supérieure d’Optique (Orsay, France) for loaning us the Shack-
Hartmann wavefront sensor as well as the infrared camera. The experiments related to our previous 
works have been done under the auspices of the CNRS and the Délégation Générale de 
l’Armement (DGA). We also thank Jean-Christophe Chanteloup and Judith Dawes for allowing us 
to reproduce some figures taken from their works.  
VII. Appendix: Calculation of the photoelastic constants Cr, θ  and 
C’r, θ  using plane strain and plane stress approximations.  
The Cr, θ constants are useful parameters to account for photoelastic effects in the most 
simple cases (optically isotropic crystals or glasses, and parabolic dependence of strain inside the 
crystal).  When Koechner published for the first time the derivation of these constants in 1970, the 
term accounting for thermal dilatation in the generalized Hooke law was omitted [51]. In 1992, 
Cousins pointed out the mistake but did not correct, at that time, the expressions of the constants Cr, 
θ, whose derivation however requires the use of the generalized Hooke law. In the last version of 
 94 /115  
Koechner’s reference book (Solid State Laser engineering, Springer Verlag ed.), the faulty 
equations were actually removed or corrected, but the formulation of the photoelastic constants Cr, θ 
remained unchanged since the first edition. 
Because the generalized Hooke law is used to infer an expression of the constants, it means 
that their expression depends on which assumption has been made about stresses and strains. In 
Koechner’s calculation, the plane strain approximation was made. In the case of end-pumping, it is 
well known however that the plane stress approximation is closer to reality. In this appendix we 
derive the expression of the photoelastic constants, within the framework of the two approximations.  
VII.1. Basic equations  
We restrict our discussion to cubic crystals. In this case, only the elasto-optical coefficients 
p11, p12, p44 are nonzero. These coefficients are given in the [100] system linked to the crystal.  
In optically isotropic materials, the principal axes of strain are given by the cylindrical coordinate 
system axes (radial, tangential, axial). After a change of coordinates one obtains [77] : 
[ ]441211
441211
441211
 (A.1.) 
for the radial index, and : 
 95 /115  
[ ]441211
441211
441211
 (A.2.) 
for the tangential index.  
As stated by Shoji et al.[119], these expressions are valid only for propagation along the [111] axis. 
For YAG we have :  
p11 = - 0.029 
p22 = + 0.0091 
p44 = - 0.0615. 
The strains are related to stresses by the generalized Hooke law, which in our case (whatever 
the approximation we make) writes:  
( )[ ] ( )
( )[ ] ( )
( )[ ] ( )cTrzz
cTzrr
−++−=
−++−=
−++−=
ασσνσ
ασσνσ
ασσνσ
 (A.3.) 
VII.2. Derivation of Cr,θ  using the plane strain approximation (Koechner’s 
case)  
 96 /115  
We consider a homogeneously pumped rod. Only the radial dependence of stresses and 
temperature is considered, since the additional constants represent a constant phase shift which does 
not affect the phase profile.  
We define:  
( )ν116
and  
Q absh
=  . 
The stresses and temperature fields, within the plane strain approximation, are given by 
Koechner [69] :  
( ) cteQSrr
cteQSrr
.cteQSrr
θ   (A.4.) 
( ) .cter
+−= 2
 (A.5.) 
Omitting the constant terms, the generalized Hooke Law yields :  
( )[ ] ( ) 0
ν14ασσνσ
ε 22θ =⎥
−−=++−= r
Trzz  (A.6.) 
which checks the plane strain condition. We have also: 
( )[ ] ( ) 22θ ν134
ν71ασσνσ
ε QSr
−−=++−=   (A.7.)  
 97 /115  
( )[ ] ( ) 22θθ ν14
ν53ασσνσ
ε QSr
−−=++−=  (A.8.) 
The radial index variation is related to the strains by:  
=  (A.9.) 
and the  
are given by (A.1) and (A.2). 
Following Koechner, we can write the index variations under the form :  
nn ,r
,r −=  (A.10.) 
With :  
( )( )
( )1ν48
875ν1 441211
C r  (A.11.) 
( )( )
( )1ν16
3ν1 1211
C  (A.12.) 
For YAG, we find Cr = + 0.020 et Cθ = + 1.77 10-4. The valued computed by Koechner are: Cr = 
0.017 et Cθ = -0.0025 (with the same values taken for pmn and the same Poisson ratio: ν = 0.25). 
The error made by Koechner is about 20%.  
 98 /115  
These coefficients have been extensively used for the purpose of evaluating depolarization losses in 
Nd:YAG. In this latter case, the relevant parameter is [60]: 
( )rB CCC −= θ2
 (A.13.) 
With (A.11) and (A.12), we find:  
( )441211 4148
pppCB +−−
 (A.14.) 
which is coincidentally the same formula obtained by Koechner from incorrect expressions of Cr et 
Cθ.  
VII.3. Derivation of C’r,θ  using the plane stress approximation (end pumping 
case) 
We consider a thin disk, pumped by a top-hat pump beam profile of radius wp  equal to the rod 
radius. This is also true in the conditions defined in Section III.4, provided that we are only 
concerned in the area r < wp and that integrated values of parameters along the crystal length are 
considered. 
Let’s define :   
=     
The temperature distribution is given by:  
 99 /115  
( ) 2' .
T r r cte
= − +  (A.15.) 
where we use Cousins’ notation (see Part III.4) , meaning that the bracketed quantity is integrated 
along the rod length L, and is then homogeneous to the said quantity times a length.  
We have: 
3 ' '
0 ( )
r Q S r Cte
r Q S r Cte
r plane stress
 (A.16.) 
From (A.3) we obtain :  
3 ' '
4 ' '
Q S r
Q S r
Q S r
 (A.17.) 
By analogy with (A.10.), the index shift can be written under the form:  
3 ' 2
n n C r
∆ = −  
which yields, using (A.1), (A.2), and (A.9) :  
( )( )11 12' 1 9 15
ν− + +
=  (A.18.) 
 100 /115  
( )( )11 12 44' 1 7 17 8
p p p
ν− + + −
=  (A.19.) 
For YAG : C’r = + 0.0032 et C’θ = - 0.011.  
The optical indicatrix does not deform the same way compared to the case of the long thin rod: here 
the tangential index becomes greater than n0. An interesting feature is the stress-induced 
birefringence term, defined previously (eq. A.13):  
( )' 11 12 44
C p p p
= − +  (A.20.) 
This expression differs from (A.14) only by a factor of 1-ν (=0.75 for YAG, but also for most 
materials).  
It is of special interest to calculate the average value of the photoelastic constants:  
0.0039
rC Cθ+ = −  
which gives an order of magnitude of the role of photoelastic effect in the thermal lens.  
Here, in the end pumping case, the thermal stresses yield a divergent contribution to the 
thermal lens, whereas it was convergent in the case of side-pumping.  
See subsections 6 to 8 of Section III for a detailed discussion about the use of these 
parameters.  
 101 /115  
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  A review of theoretical and experimental studies of thermal effects in
solid-state lasers is presented, with a special focus on diode-pumped
ytterbium-doped materials. A large part of this review provides however general
information applicable to any kind of solid-state laser. Our aim here is not to
make a list of the techniques that have been used to minimize thermal effects,
but instead to give an overview of the theoretical aspects underneath, and give
a state-of-the-art of the tools at the disposal of the laser scientist to
measure thermal effects. After a presentation of some general properties of
Yb-doped materials, we address the issue of evaluating the temperature map in
Yb-doped laser crystals, both theoretically and experimentally. This is the
first step before studying the complex problem of thermal lensing (part III).
We will focus on some newly discussed aspects, like the definition of the
thermo-optic coefficient: we will highlight some misleading interpretations of
thermal lensing experiments due to the use of the dn/dT parameter in a context
where it is not relevant. Part IV will be devoted to a state-of-the-art of
experimental techniques used to measure thermal lensing. Eventually, in part V,
we will give some concrete examples in Yb-doped materials, where their
peculiarities will be pointed out.

Introduction 
 I.1 The Yb breakthrough 
 I.2. Strategy on the matrix host 
II. Temperature profile of an ytterbium-doped crystal under diode pumping 
 II.1. Theoretical aspects 
II.1.1. The steady-state heat equation 
II.1.2. A review of analytical solutions of the steady state heat equation  
II.1.3. What is special about ytterbium-doped materials? The influence of absorption saturation 
in the temperature distribution. 
II.1.4. Determining the absolute temperature: the boundary conditions 
II.2. Experimental absolute temperature mapping and heat transfer measurements 
using an infrared camera 
II.2.1. Introduction 
II.2.2. Experimental setup for direct temperature mapping 
II.2.3. results and measurements of heat transfer coefficients 
III. Thermal lensing effects : theory  
III.1. Introduction  
III.2. Stress and strain calculations 
III.3. How can we take into account the photoelastic effect ? 
III.4. Simplified account of photoelastic effect in isotropic crystals 
III.5. A consequence of strain-induced birefringence: depolarization losses.  
III.6. Thermally-induced optical phase shift.  
III.6.1. Expression of the optical path 
  III.6.2. The thermal lens focal length 
 III.7. Discussion about the use of the “dn/dT” coefficient 
 3 /115  
III.8. A novel definition for thermo-optic coefficient based on experimentally 
measurable parameters.  
III.9. The aberrations of the thermal lens.  
IV. Thermal lensing techniques  
IV.1. Introduction  
IV.2. Geometrical methods 
IV.3. Methods based on the properties of cavity eigenmodes 
IV.4. Methods based on wavefront measurements 
IV.4.1. Classical interferometric techniques 
IV.4.2. Shearing interferometric techniques 
IV.4.3. Methods based on Shack-Hartmann wavefront sensing 
IV.4.4. Other techniques 
IV.5. Conclusion 
V. thermal lensing measurements in ytterbium-doped materials: the evidence of 
a non radiative path  
V.1. the thermal load in Yb-doped materials  
 V.2. Evidence of nonradiative effects in Yb-doped materials: the example of Yb:YAG  
V.3. Laser wavelength dependence oN the thermal load in Yb-doped broadband 
materials: the example of Yb:Y2SiO5 
IV.4. The influence of the mean fluorescence wavelength on the thermal load: an 
illustration with Yb:KGW 
IV.5. Conclusion 
VI. Conclusion 
Appendix : Calculation of the photoelastic constants Cr, θ  and C’r, θ  using plane strain and plane 
stress approximations.  
 4 /115  
I. Introduction  
I.1. The Yb3+ breakthrough 
Diode-pumped solid-state laser (DPSSL) technology has become a very intense field of 
research in Physics [1,2]. The replacement of flash-lamp pumping by direct laser-diode pumping for 
solid-state materials has brought a very important breakthrough in the laser technology in particular 
for high power lasers [3-4]. In fact, the better matching between absorption wavelength and 
material’s absorption spectra brought by the use of laser diode emission ― compared to the broad 
one of flash-lamps ― has lead to a significant benefit in efficiency and subsequently in simplicity, 
compactness, reliability and cost. This progress has substantial implications on laser applications 
such as fundamental and applied research, laser processing, medical applications …     
 5 /115  
4I9/2
4I11/2
4I13/2
4I15/2
4F3/2
4F5/2
4G5/2
4G7/2
4G9/2
2F5/2
2F7/20
10000
20000
Nd3+ Yb3+
0.8 µm
0.9 µm
0.94 µm
0.98 µm
~1 µm
~ 1 µm
Parasitic effects
4I9/2
4I11/2
4I13/2
4I15/2
4F3/2
4F5/2
4G5/2
4G7/2
4G9/2
2F5/2
2F7/20
10000
20000
Nd3+ Yb3+
0.8 µm
0.9 µm
0.94 µm
0.98 µm
~1 µm
~ 1 µm
Parasitic effects
Figure 1: Energy levels of Yb and Nd ions. Typical laser transition lines are represented for both 
pump absorption and laser emission. Lines of high excited states are also represented including the 
lines involved in the deleterious effects (up-conversion, excited-state absorption or concentration 
quenching).   
In the realm of high average power DPSSL, two rare-earth ions dominate:  neodymium and 
ytterbium [5-6]. Actually they can be efficiently pumped, respectively at 808 nm with 
InGaAsP/GaAs or AlGaAs/GaAs diodes for neodymium, and between 900 and 980 with 
InGaAs/GaAs diodes for ytterbium (fig. 1).  In both case the standard laser emission is around 1 µm, 
corresponding to the transition between the 4F3/2 and 4I11/2 lines for the Nd3+ and between the 2F5/2 
and 2F7/2 for the Yb3+. At the beginning of the high-power-laser development, the Nd-doped 
materials were preferred to the Yb-doped ones mainly because of the four level nature and their 
 6 /115  
many absorption lines, which are more convenient as far as flash-lamp pumping is concerned. 
However it seems obvious, for more than one decade now, that Yb-doped materials are more suited 
for very efficient and very-high-average-power diode-pumped lasers. The main reason for this is the 
very simple electronic level structure of the Yb3+ ion, which consists on two manifolds as shown in 
figure 1. This singular property allows avoiding most of the parasitic effects such as upconversion, 
cross relaxation or excited-state absorption which are present in Nd-doped materials [7] because of 
the existence of higher excited-state levels (4G9/2 for the 1-µm-laser emission). These deleterious 
effects have two main consequences. First, they increase the thermal load and subsequently the 
thermal problems [8] because the main desexcitation paths of the high-excited state levels are non-
radiative (as represented in fig. 1). Secondly, they also alter the gain because they can induce strong 
depopulation of the 4F3/2 level implicated in the laser inversion population. Another advantage of 
Yb-doped materials compared to their neodymium doped counterparts is the very low quantum 
defect (again due to the 2-manifold based electronic structure). In fact, when pumped at 980 nm the 
quantum defect of ytterbium is around 5 % compared to 30 % for neodymium (in YAG). This is a 
real benefit for reducing the thermal problems and thus to attain very high average powers. As an 
example of comparison between Nd and Yb doped materials, we summarized in table 1, the 
different parameters for the same well-known matrix host: YAG (Y3Al5O12) [9-13].  
 7 /115  
Table 1: comparison between Nd:YAG and Yb:YAG 
Crystal Nd:YAG Yb:YAG 
Emission line 
Wavelength 
Cross section 
Broadness (FWHM) 
1064 nm 
28 10-20 cm-2 
0.8 nm 
1031 nm 
2.1 10-20 cm-2 
9 nm 
Lifetime 230 µs 951 µs 
Saturation fluence 0.67 J/cm2 9.2 J/cm2 
Maximum doping rate 2 % 100 % 
Absorption line 
Wavelength 
Cross section 
Broadness (FWHM) 
808 nm 
67 10-20 cm-2 
2 nm 
968 nm 
0.7 10-20 cm-2 
4 nm 
942 nm 
0.75 10-20 cm-2 
18 nm 
Quantum defect  32 % 6.5 % 9.5 % 
I.2. Strategy on the matrix host 
Another advantage of Yb-doped versus Nd-doped materials is the longer lifetime which may 
allow a better storage of the pump energy; and the last but not the least advantage is the generally 
broader bandwidth of the emission lines. This last advantage leads to a potential for femtosecond 
pulse generation which, in the current state-of-the-art has never been demonstrated with 
neodymium. The emergence of ytterbium-based lasers has allowed crucial progress in the 
ultrashort-pulsed laser technology. These materials have been actually the key point for the 
development of the latest generation of “ultrafast” lasers: the all-solid-state femtosecond lasers [14-
36]. Applications for such lasers are abundant and excite a great interest in the scientific community.  
However Yb-doping brings several drawbacks or difficulties. The first one is the very strong 
influence of the matrix on the spectral properties. Actually, as the two levels 2F5/2 and 2F7/2 are split 
in manifolds by the Stark effect due to the electric crystalline field of the host matrix, the ion 
environment strongly models the spectrum. In a simple way, the spectral broadening can be directly 
related to the level of disorder of the matrix [37-48]. On the first hand, if the matrix is relatively 
simple and well-ordered, the spectra would reveal relatively narrow and intense lines (which are a 
 8 /115  
strong disadvantage for short pulse generation). Though, a simple matrix structure generally implies 
a high thermal conductivity which is a key point for developing high power lasers. An example of 
such an Yb-doped material is given in figure 2 with Yb:YAG. On the second hand, if the disorder of 
the host matrix is high, the spectrum will be large and suitable for very-short pulse generation but at 
the expanse of thermal conductivity. An example of such an Yb-doped material is given in figure 3 
with Yb:SYS. The numerous advantages of the Yb3+ ion have led to a strong interest for many host 
matrices but in general favouring either short pulses or high power applications. Table 2 represents 
this diversity of already studied Yb-doped host matrices and their principal properties.  
 9 /115  
Table 2: Comparison between Yb-doped crystals 
Emission line Absorption line Material 
(name and formula) Wavelength 
(nm) 
Cross 
section 
10-20 cm2
Broadness 
(nm) 
Lifetime
(ms) Usual 
wavelength  
(nm) 
Thermal 
conductivity
(undoped) 
(W/m/K) 
Yb:YAG 
Yb:Y3Al5O12 
1031 2.1 9 0.951 942 968 11 
Yb:GGG 
Yb:Gd3Al5O12 
1025 2 10 0.8 971 8 
Yb:Y2O3 
1076 0.4 14.5 0.82 979 13.6 
Yb:Sc2O3 
1041 1.44 11.6 0.8 979 16.5 
Yb:CaF2 
1045 0.25 70 2.4 979 9.7 
Yb:YVO4 
1020 0.9 40 0.25 985 5.1 
Yb:LSO 
Yb:Lu2SiO5 
1040 0.6 35 0.95 978 5.3 
Yb:YSO 
Yb:Y2SiO5 
1042 0.6 40 0.67 978 4.4 
Yb:YLF 
Yb:YLiF4 
1030 0.81 14 2.21 940 4.3 
Yb:KGW 
Yb:KGd(WO4)2 
1023 2.8 20 0.3 981 3.3 
Yb:KYW 
Yb:KY(WO4)2 
1025 3 24 0.3 981 3.3 
Yb:SYS 
Yb:SrY4(SiO4) 30 
1065 0.44 73 0.82 980 2 
Yb:GdCOB 
Yb:Ca4GdO(BO3)3 
1044 0.35 44 2.6 976 2.1 
Yb:BOYS 
Yb:Sr3Y(BO3)3 
1060 0.3 60 1.1 975 1.8 
Yb:glass 
(phosphate glass) 1020 0.05 35 1.3 975 0.8 
Another drawback of the Yb-doped materials is due to the quasi-3-level structure of these lasers. As 
it is apparent on the spectra of figure 2 and 3, there is an overlap between the emission and the 
absorption bands which leads to strong re-absorption effects and to a reduction of the effective 
emission band broadness. Moreover, since the splitting due to Stark effect is relatively small 
(between 200 and 1000 cm-1), the high energy levels within the 2F7/2 manifold (corresponding to the 
 10 /115  
different possible low-energy levels of the laser transition) are somewhat populated at thermal 
equilibrium. This implies two deleterious effects when temperature increases: first, a reduction of 
the laser inversion population, second, an increase of the reabsorption at the laser wavelength. A 
special care concerning the thermal load and thermal management will be then necessary to develop 
efficient lasers based on ytterbium-doped materials, especially in the high power regime. 
 11 /115  
950 970 990 1010 1030 1050 1070 1090
Section efficace
d'absorption
Section efficace
d'émission
Yb:YAG
Wavelength (nm)
Absorption
Emission
Absorption
Emission 
950 970 990 1010 1030 1050 1070 1090
Section efficace
d'absorption
Section efficace
d'émission
Yb:YAG
Wavelength (nm)
Absorption
Emission
Absorption
Emission 
Absorption
Emission 
Figure 2 : Absorption and emission spectra of Yb:YAG.   
850 900 950 1000 1050 1100 1150
Section efficace
d'absorption
Section efficace
d'émission
73 nm
Yb:SYS
Wavelength (nm)
Absorption
Emission
Absorption
Emission 
850 900 950 1000 1050 1100 1150
Section efficace
d'absorption
Section efficace
d'émission
73 nm
Yb:SYS
Wavelength (nm)
Absorption
Emission
850 900 950 1000 1050 1100 1150
Section efficace
d'absorption
Section efficace
d'émission
73 nm
Yb:SYS
Wavelength (nm)
850 900 950 1000 1050 1100 1150
Section efficace
d'absorption
Section efficace
d'émission
73 nm
Yb:SYS
Wavelength (nm)
850 900 950 1000 1050 1100 1150
Section efficace
d'absorption
Section efficace
d'émission
73 nm
Yb:SYS
Wavelength (nm)
Absorption
Emission
Absorption
Emission 
Absorption
Emission 
Figure 3 : Absorption and emission spectra of Yb:SYS.   
 As a first conclusion, Ytterbium-doped crystals are particularly suitable for directly diode-
pumped, solid-state high-power and/or femtosecond lasers. The emergence of new ytterbium-doped 
laser crystals has allowed crucial progress in the DPSSL technology. Nevertheless, a very special 
 12 /115  
care has to be done concerning the thermal properties and thermal effects in the Yd-doped materials 
because of their strong influence on the laser performance. In this paper, we then propose to make a 
review of different thermal-effect studies made on ytterbium-doped laser crystals. 
II. Temperature profile of an ytterbium-doped crystal 
under diode pumping  
 In this part we present a review about how to calculate and measure the temperature 
distribution in an end-pumped laser crystal. We explain in which cases it is possible to obtain an 
analytical expression (otherwise a finite-element analysis would be necessary), and how these well-
known results have to be corrected when we deal with ytterbium-doped crystals, because absorption 
saturation cannot be ignored in this case. Then, we investigate the role of the thermal contact at the 
boundaries, which is an essential parameter for the knowledge of the temperature. This will be 
illustrated, in the last part of this section, by experimental absolute temperature maps, obtained with 
an infrared imaging camera.  
II.1. Theoretical aspects 
II.1.1. The steady-state heat equation 
A study of thermal effects in crystals first requires the calculation of the temperature field at any 
point of the crystal. One has to solve the heat equation: 
)t,z,y,x(Q)t,z,y,x(TK
)t,z,y,x(T
c thcp =∇−
ρ  (II.1.1.) 
with :  - T = T(x,y,z,t) : temperature in K; 
 - ρ : density in kg.m-3; 
 - cp : specific heat in J.kg-1. K-1; 
 13 /115  
 - Kc : thermal conductivity in W.m-1.K-1; 
 - Qth : thermal power (or thermal load) per unit volume in W.m-3.  
The specific heat affects the temperature variation in the pulsed regime or in the transient regime: 
we will ignore it in the following work since we will only consider CW lasers. The thermal 
conductivity governs the temperature gradient inside the crystal, and will have a crucial importance 
for the thermal lens magnitude. The heat transfer coefficient H is arising when writing the boundary 
conditions, and has then an influence only on the absolute value of the temperature inside the 
crystal. We will discuss at the end of this part how to measure it, and some ways to improve its 
value.  
In order to obtain analytical expressions, some assumptions have to be made. We will assume the 
following: (1) The pump profile is axisymmetric. End-pumping by a fiber-coupled diode is a good 
example of such a profile; (2) the thermal conductivity Kc is a scalar quantity, not a tensor; this 
means that we restrict our discussion to glasses and cubic crystals [49] (in practice however the 
anisotropy of the thermal conductivity tensor is often weak); (3) the cooling is isotropic in the z-
plane, which means that the crystal mount does not favour one given direction of cooling. (4) At 
last we assume that the thermal conductivity is not significantly dependant on temperature, so that it 
can be considered as a constant. This approximation is very realistic in YAG around room 
temperature, following the study of Brown et al. [50], and we assume that it is also true in other 
crystals. However this approximation would not be valid anymore at cryogenic temperatures.  
The heat equation, including all these assumptions, becomes: 
)z,r(Q
 (II.1.2.) 
 14 /115  
when r is the radial coordinate of a point inside the crystal, measured with respect to the pump 
distribution axisymmetric axis. To simplify a bit further the equations, the crystal will be considered 
cylindrical, whose axis corresponds to the pump symmetry axis, with a radius r0 and a length L (see 
figure 4) 
Figure 4: geometry of the crystal, taken for all calculations. z=0 is the input face. 
II.1.2. A review of analytical solutions of the steady state heat equation  
An analytical expression of the temperature distribution inside a crystal is calculable only in a 
limited number of simple systems. For more complex geometries, one has to use finite element 
codes. We present in this subsection a brief review of the cases where such an analytical treatment 
is feasible.  
The first study of thermal effects in crystals has been provided by Koechner [51] in the early 
seventies. He considered flash-pumped Nd:YAG rods, within which the thermal load is uniform :  
Q thth 2
=  (II.1.3.)  
where Pth = ηh Pabs is the thermal power (in W) dissipated into the rod. Pabs is the absorbed pump 
power and ηh is the fractional thermal load. The solution writes:  
z r0 
z = 0 
Pump 
 15 /115  
)r(T)r(T
absh 22
+=  (II.1.4.) 
where T(r0) is the temperature at the edge of the crystal, which will be estimated later thanks to the 
boundary conditions. It is useful to write the temperature shift between the centre and the edge of 
the rod: 
( ) ( )
0∆ 0 =−=  (II.1.5.) 
We note that ∆T is independent of the radius r0  of the crystal, but scales inversely with L. 
The previous results can not be applied to end-pumping configurations, because in these latter cases 
the thermal load is localized within a small volume inside the crystal. In a vast majority of practical 
circumstances, the pump beam profile is axisymmetric and can be described by a super-gaussian 
shape. The general solution of the heat equation for a super-gaussian beam of any order has been 
derived by Schmid et al. [52].  
The situation is even simpler in most cases: indeed, the pump is often either a near-diffraction-
limited Gaussian beam (laser or single mode diode), or a “top-hat” beam (that is a super-gaussian 
profile of infinite order). The latter description corresponds quite well to fiber-coupled diode laser 
array pumping.  
The solution of the heat equation in the specific case of Gaussian-beam pumping is treated in [53] 
and [54]. The case of the top-hat shape has been derived by Chen et al. [55].  
Hereafter we give the solution for a top-hat beam profile. Assuming that the thermal load in each 
slice at z coordinate is a disk of radius wp(z), the temperature shift with respect to the edge 
temperature is:  
 16 /115  
)z(wr
)z(wr
)z,r(T)z,r(T
 (II.1.6.) 
where: 
- αNS is the (non-saturated) linear absorption coefficient; saturation absorption is then not 
taken into account in this formula ; 
- The axial heat flux (along z) is ignored, which means in other words that 
∂ is neglected 
in the heat equation. We’ll see in the next section the reasons why we can neglect axial heat 
flux. The formula is then not valid for thin disks.  
- As a consequence of the latter point, the temperature can be computed inside each “slice” of 
thickness dz of the crystal, as if the surrounding slices did not exist.  
The temperature gradient inside the pumped volume is of particular interest because the laser beam 
is usually (and preferably) smaller than the pump volume. One obtains: 
( , ) ( 0, ) ( , )
4 1 ( )
h abs NS
P e r
T r z T r z T r z
K e w z
∆ = = − =
 (II.1.7.) 
The temperature shift turns out to be independent of the crystal radius r0 and of the crystal length L, 
which was not the case in Koechner’s simple model (equation II.1.4). It makes sense since the 
important parameter here is the absorption length NSabsL α1= , and not the  whole length L of the 
crystal.  
In figure 5 we plotted the normalized temperature distribution for a typical ratio wp/r0 = 0.1. 
 17 /115  
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
r/r0 
T( r ) −T( r0 )
Figure 5: Normalized temperature distribution (in a plane perpendicular to the propagation axis)  
for a crystal pumped by a top-hat-profile fiber-coupled laser diode with wp/r0 = 0.1).  
II.1.3. What is special about ytterbium-doped materials? The influence of absorption 
saturation in the temperature distribution. 
An ytterbium-doped material, especially when pumped at the zero-line wavelength (i.e. around 980 
nm), has many common points with a saturable absorber. The absorption rate due to the pump is 
counterbalanced by the spontaneous emission rate, but also (which is far from being negligible) by 
the stimulated emission rate at the pump wavelength. It is essential to take absorption saturation 
effects into account; otherwise the absorbed pump power can be dramatically overestimated. As a 
result, the absorbed pump power is lower under nonlasing than under lasing conditions, since lasing 
provides (hopefully) a very efficient path to carry the excited population back to the fundamental 
level.  
 18 /115  
It is noteworthy that most Finite Element Analysis (FEA) codes (primarily designed for 4-level 
laser systems in which absorption saturation is not a problem) basically assume an exponential 
decay for the pump power inside the crystal. It can lead to large errors, as we illustrate below. 
Let Pp(z) be the pump power through a plane in the crystal at z coordinate. The thermal load density 
generated into a disk of radius wp(z) and thickness dz is: 
dz)z(w
)z(dP
th 2π
=  (II.1.8) 
where -dPp(z) represents the absorbed pump power in the thin slice of thickness dz.  
The temperature field is: 
)z(wr
)z(wr
)z(dP
)z,r(T)z,r(T
 (II.1.9.) 
Inside the pumped volume the temperature shift writes as follows:  
( ) ( )
zrTzT
=−  (II.1.10.)  
Absorption saturation issues are taken into account, under nonlasing conditions, by the following 
equation for the pump irradiance  Ip (which is the pump power divided by the pump spot area):  
 (II.1.11.) 
Where αNS is the absorption coefficient in the non saturated regime.  
The pump saturation irradiance
I is calculated from the spectroscopic properties of the material:  
( ) ( )[ ]τλσλσλ pempabspp
sat +
=  (II.1.12.) 
 19 /115  
where absσ is the absorption cross section, emσ is the emission cross section, λp the pump wavelength, 
and τ  the radiative lifetime. 
The pump power Pp(z) obeys the following equation, for a top hat beam profile (one may find the 
equivalent formulation for a gaussian pump profile in [56]): 
)z(wI)z(P
)z(wI)z(P
)z(dP
pppNSp
=  (II.1.13.) 
A practical way to study absorption saturation issues, and to check the assumptions made so far, is 
to perform fluorescence imaging experiments in a pumped crystal. Using a crystal whose one of the 
edge surfaces (in practice one side not facing the radiator) has been polished, one can make an 
optical image of the fluorescence, under lasing or nonlasing conditions, with a CCD camera and an 
interference filter at a long wavelength (at 1064 nm for instance), required to completely eliminate 
the scattered light at the pump wavelength, as well as to prevent detection of fluorescence photons 
which could have experienced reabsorption. This simple experiment allows visualizing absorption 
saturation (the fluorescence intensity, integrated along the depth of focus of the imaging system, 
does not decay exponentially) and also to measure what is the optimum location for the pump spot 
inside the crystal (figure 6). The experiments we performed with different Yb-doped materials 
taught us that the optimum focus (the one for which the measured laser efficiency was the highest) 
was always located at about one third of the whole crystal length from the input face. This 
parameter is taken into consideration in the following.  
The low brightness of the diode pump beam (compared to the brightness of the laser beam) makes 
the effective Rayleigh distance of the pump beam considerably shorter than the crystal length. For 
this reason, the divergence of the pump beam inside the crystal must also be considered, in order to 
correctly account for saturation issues. Here we describe the pump radius evolution by a relation of 
the type:  
 20 /115  
pp wn
  (II.1.14.) 
where 
w is the pump beam waist radius. The M2 factor is determined experimentally. In our case 
we used a 200µm-diameter core fiber-coupled diode (HLU15F200-980 from LIMO GmbH), whose 
the M2 was measured to be around 80.  
Results shown in figure 6 show experimental data and theoretical predictions in a 15%-at. doped 
Yb:GdCOB crystal [57]. The theoretical profiles are computed assuming that: 1) the pump volume 
has a top-hat profile, and 2) the imaging objective has a very low numerical aperture, so that the rate 
of spontaneous photons detected by one pixel can be calculated by integrating the fluorescence 
yield over one vertical line underneath. The good match between theory and experiments show 
incidentally that the “top hat” hypothesis for the pump beam profile is well justified. 
 21 /115  
Without saturation absorption :
At low power
(Pinc = 1 W
Pabs =200 mW)
With saturation absorption :
At high power
(Pinc = 13.7 W
Pabs =6 W
Ipsat =4.1 kW/cm
experiment
experiment
simulation
simulation
Without saturation absorption :
At low power
(Pinc = 1 W
Pabs =200 mW)
With saturation absorption :
At high power
(Pinc = 13.7 W
Pabs =6 W
Ipsat =4.1 kW/cm
experiment
experiment
simulation
simulation
Theoretical profile 
computed along the
symmetry axis
Experimental
profile measured
along the
symmetry axis  
Figure 6: Fluorescence detected @ 1064 nm on a crystal pumped at 980 nm at low power (top) and at high 
power (bottom), through the optically-polished top face. The influence of absorption saturation is clearly 
visible: at low pump power, the fluorescence yield is higher at the pump waist location, as expected provided 
that both absorption coefficient and absorption saturation are weak; on the contrary, when absorption 
saturation becomes non negligible, the amount of fluorescence photons is minimum at the pump waist. 
Theory and experiments agree very well, except near the exit face of the crystal, a discrepancy which could 
be related to the fact that far from the waist, the pump beam is no longer “top hat”. 
 22 /115  
Fig 7a)  
0 0.5 1 1.5 2 2.5 3
distance z (mm)
Intensité de saturation
décroissante 
position du waist
 de pompe 
Thickness z  (mm) 
Pump waist 
location 
Decreasing pump saturation 
intensity ↓
Fig 7b) 
0 0.5 1 1.5 2 2.5 3
distance z (mm)
position du waist de pompe 
Intensité de saturation
décroissante 
Thickness z (mm)
Pump waist location  
Decreasing pump saturation 
intensity ↓
Figure 7: Evolution of pump power (Fig. 7a) and temperature difference T(0) –T(r0) (Fig. 7b) 
versus crystal thickness z. The pump saturation intensities values are: ∞=
I - 50 - 20 - 10 - 
5 kW/cm2 (for these curves ηh= 0.065 et Kc = 2 W.m-1.K-1, corresponding to the parameters of 
Yb:GdCOB). 
 23 /115  
Fig 8a) 
Thickness z (mm) 
radius r (mm) 
fig. 8b)  
radius r (mm) Thickness z (mm) 
Figure 8: Temperature distribution under nonlasing condition. The pump beam divergence inside 
the crystal is taken into account (M2 = 80). In Fig 8a) the saturation of absorption is ignored; Fig 
 24 /115  
8b) pump absorption saturation is taken into account (
I = 4.1 kW/cm2). The parameters used are 
form Yb:GdCOB. 
Equations (II.1.13) and (II.1.14.) can be solved numerically and injected in (II.1.9.) to obtain the 
temperature distribution. Figure 7 shows the evolution of pump power (fig. 7a) and temperature 
(fig. 7b) at the center of the rod versus crystal thickness, for various values of the pump saturation 
intensity.  
Here we assumed that the pump beam waist was located at z0 = L/3, which is experimentally well 
verified, as far as the laser output is optimized (see figure 6 and above text). In absence of saturation 
I infinite), both the pump power and the temperature experience an exponential decay as 
expected; but for lower values of the pump saturation intensity, the temperature reaches a local 
minimum at the pump beam waist. Figure 8 shows a 3D view of temperature distribution without 
saturation (fig. 8a) and in presence of strong saturation (fig 8b.) corresponding to Ipsat = 4.1 kW/cm2, 
that is the value for Yb:GdCOB. It appears in the latter case that the region where the pump density 
is the strongest (near the pump beam waist) is not the region where the temperature is the highest 
(near the faces of the crystal). Pump beam divergence appears to be an important parameter: it 
makes, for this example, the temperature higher at the exit face than at the entrance face of the 
crystal.  
In presence of laser extraction, the pump intensity evolution through the crystal is given by: 
satsat
  (II.1.15.)  
where  
( ) NlabslNS λσα =   (II.1.16.)  
 25 /115  
( ) ( )[ ]τλσλσλ lemlabsl
sat +
=   (II.1.17.) 
are the non-saturated absorption coefficient at laser wavelength, and the laser saturation intensity, 
respectively.   
When the intracavity laser intensity I largely exceeds 
I , and if reabsorption at laser wavelength 
is small, one can show that (II.1.15) simply becomes: 
α−=   (II.1.18.) 
which means that the ground manifold is repopulated so that absorption is not saturated any more.  
In real cases, as a matter of fact, the absorbed pump power under lasing conditions is intermediate 
between the non saturated regime and the saturated (non lasing) regime: in a first approximation it 
is possible to ignore saturation effects only if the laser extraction is efficient.  
II.1.4. Determining the absolute temperature: the boundary conditions 
In this subsection we deal with the boundary problem. For the moment we have established 
expressions for the temperature gradient, but we have no idea of the absolute temperature inside the 
crystal. Let us assume that the four edge faces of the crystal are in “contact” with a radiator, which 
will be in most cases a piece of cooled copper. The first boundary condition expresses the 
continuity of the thermal flux across these contacts:   
copperinside
copper
crystaltheinside
crystal n
 (II.1.19.) 
where K is the thermal conductivity, n is the surface normal vector, and  ∂/∂n the normal derivative. 
Common metals (Copper or indium, the latter being used as an intermediate contacting material) 
have thermal conductivities that are several orders of magnitude higher than the usual conductivities 
of laser crystals: 400 W.m-1.K-1 for copper and 820 W.m-1.K-1 for indium. This means that the 
 26 /115  
temperature gradients inside these metals will always be negligible, so that we consider in the 
following that the temperature inside the radiator is uniform and is noted Tc.   
Let’s see now the second boundary condition. In many papers and FEA codes, the temperature at 
the edge of the crystal is set equal to Tc:  
cTrT =)( 0                                                                                                                         
(II.1.20.) This is actually true only for an ideal contact [58]. But even for flat and polished surfaces pressed 
one against another, this relation is far from reality [59].  
The most realistic condition is surprisingly a Newton-type law of cooling, even if we indeed deal 
with conduction problems here: 
( )cc TrTHn
K −=−= )( 0∂
 (II.1.21.) 
where jq is the thermal density flux. H is the heat transfer coefficient or surface conductance 
(W.cm-2.K-1). H is of course infinite for ideal thermal contact.  
Carslaw et al. [58] have shown that the physical origin of a temperature gap between the edge of the 
rod and the mount was due to the presence of a thin oxide (or air or grease) layer, which acts as a 
very large thermal resistance.  
Measuring the heat transfer coefficient is usually difficult and not found easily in the literature: we 
present in the next section a simple and accurate method to perform this measurement.  
What about the end faces, which are in contact with air most of the time? The heat can flow out of 
the crystal through the two end faces by both convection in free air and thermal radiation. Cousins 
[60] calculated the equivalent H coefficient for the two processes and has shown that both 
coefficients were of the order of 10-3 W.cm-2.K-1.  
Since the measured H coefficients for conduction are typically in the range 1-10 W.cm-2.K-1, this is 
the proof that the assumption of pure radial heat flux made on the previous subsection is correct.  
 27 /115  
Using (II.1.9.) et (II.1.21.) one can calculate the temperature gap between the radiator and the edge 
of the crystal: 
)()(  (II.1.22.) 
The parameter of interest here is the normal derivative of the temperature at the interface. This 
explains why the quality of the thermal contact has a tremendous impact on side or edge-pumped 
slabs or rods; since in these configurations the temperature distribution is described by a formula of 
the type II.1.4, that is a parabolic dependence. In contrast, in end-pumped configurations, where the 
temperature profile is described by equation II.1.6., the thermal gradient at the periphery is smaller 
and the requirement of a good thermal contact can be loosen. 
It is also interesting to know the maximum temperature reached inside the crystal. In order to obtain 
a easy-to-handle scaling formula, we make the strong assumption that absorption saturation is 
absent, and we ignore the divergence of the pump beam inside the crystal. We have:  
cmax w
 (II.1.23.) 
As a conclusion for this section, we will list some conclusions one can make from these two last 
equations, like a list of recipes to reduce the temperature Tmax : 
• Increase wp : obvious and efficient, but at the expense of laser efficiency.  
• Increase  H. As shown in the next experimental section, reducing H does not affect the 
temperature gradient, and will not help to reduce the thermal lens magnitude. All we can get 
is a uniform decrease of temperature. However reducing the absolute temperature is actually 
more interesting in an Yb-doped crystal than in an Nd-doped material for example, in virtue 
of reabsorption losses that are highly temperature-dependant. A better contact can also help 
 28 /115  
reducing fracture risks but this is not directly linked to a decrease of the temperature either: 
it is because a good contact can induce radial components to the stress tensor at the 
periphery, or also because it will decrease the density of high spatial frequency alterations of 
the surface which are the ultimate causes of crack-induced propagating fractures [61-63].   
• Decrease Tc : if the radiator temperature is decreased but still remains around the room 
temperature (that is with a standard thermoelectric or water-flow cooling), the effect is the 
same as increasing H: we only play on the temperature pedestal, not on the gradient. 
However, if the mount is cooled far below room temperature (at cryogenic temperatures for 
instance), the thermal conductivity of the crystals significantly increases, which is highly 
positive for the thermal gradient. This approach has been successively applied to reduce the 
thermal lens in high-energy femtosecond laser chains [64] or in Nd:YAG rods [65] 
• Decrease the crystal size? The crystal size has no influence on the temperature gradient. To 
understand its influence on the maximum temperature, Tmax is plotted versus r0 for different 
values of H in figure 9. We observe that when the radius of the crystal exceeds roughly 10 
times the pumped area radius wp, the temperature becomes independent of r0 . In practice, it 
is possible to reduce the absolute temperature using small crystals, providing they are really 
small (see for example [66]). It is practically very difficult to cut and polish crystals whose 
size is smaller than 2 mm: one can then conclude that the transverse section of a crystal is 
not a parameter on which one can play efficiently. Besides, the effect of a bad thermal 
contact is visible only for crystals whose size would be on the order of the pump spot size. 
As illustrated by figure 9, a small crystal with bad cooling (for example r0 = 0.5 mm and H = 
0.1 W.cm-2.K-1) is far worse than a « reasonably » sized crystal with correct cooling (r0 = 2 
mm ; H = 1 W.cm-2.K-1) since the temperature difference between the two configurations 
reaches  200 °C.  
 29 /115  
• Add an axial component to the heat flux: it does not appear in equation II.1.23 because it has 
been derived with the assumption of a purely radial heat flux. However, one can add a large 
axial heat flux by putting either a « transparent radiator » in front of the input face (this is 
the principle of composite bondings [67] ) or by using thin disks (i.e. L« r0) that are very 
efficiently cooled through the face in contact with the radiator [68]. 
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Crystal radius (mm)
 H = ∞ : perfect contact 
H = 1 W/cm2/K 
H = 0.1 W/cm2/K  
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Crystal radius (mm)
 H = ∞ : perfect contact 
H = 1 W/cm2/K 
H = 0.1 W/cm2/K  
Figure 9: Maximum temperature at the center of the input face of the crystal T(r=0, z=0) versus the 
crystal radius r0. The absorption saturation is neglected, as well as the divergence of the pump 
beam inside the crystal. The parameters are: Tc=15°C, ηh=6.5 %, αNS = 7.4 cm-1, Kc=2.1 W/m/K 
(values for GdCOB), Pinc=15 W, wp=100 µm. 
 30 /115  
II.2. Experimental absolute temperature mapping and heat transfer 
measurements using an infrared camera 
II.2.1. Introduction 
As depicted in the previous paragraph, the temperatures obtained by solving the heat equation are 
only relative temperature distributions, expressed with respect to the rod surface temperature. The 
latter depends on the boundary conditions and is then very difficult to predict. Direct temperature 
mapping could consequently be a helpful measurement to understand pump-induced thermal effects. 
Moreover, we have shown that one of the crucial parameter to uniformly decrease the temperature 
inside the crystal (which can be useful to reduce fracture risks, see above paragraph) is the thermal 
contact between the crystal and its surrounding mount. Consequently, the knowledge of quantitative 
and experimentally measured information as the heat transfer coefficient H is of practical 
importance for high power laser development. 
We herein report on a very simple experimental setup, based on an infrared camera that can perform 
spatially resolved analysis of the absolute temperature on the entrance face of the crystal, where 
temperature reaches generally a very high value (in any case higher than at the beam waist, as 
explained in the subsection II.1).  We can also experimentally measure the heat transfer coefficient 
H between the crystal and its surrounding for different types of commonly used thermal contacts. 
We first describe the experimental setup that allows such measurements, and illustrate it with the 
well-known Yb:YAG crystal [8]. 
II.2.2. Experimental setup for direct temperature mapping 
The experimental setup is presented on figure 10. A fiber-coupled laser diode was focussed inside 
an Yb:YAG laser crystal; the infrared emission of the entrance face of the crystal was observed with 
an infrared camera. A dichroic Zinc selenide (ZnSe) plate was used as a dichroic mirror: it was 
High Reflectivity (HR) coated for 960-1080 nm on one face (at 45° angle of incidence) to direct the 
 31 /115  
pump beam into the crystal, and also coated for High Transmission (HT) in the 8-12 µm spectral 
range on both faces to let the thermal radiation reach the camera.  
Figure 10: Experimental setup for absolute temperature measurements. 
A germanium objective (focal length 50 mm, N.A. 0.7, aberration-corrected for infinite conjugation) 
was appended close to the ZnSe plate to create the intermediate thermal image with high spatial 
resolution. The camera was an AGEMA 570 (Flir Systems Inc.) consisting of 240x320 
microbolometers working at room temperature. The measured noise equivalent temperature 
difference (NETD) of the camera is 0.2 °C. The numerical aperture of the whole imaging system in 
the object plane being around 1, a theoretical spatial resolution of about 10 µm could be achieved; 
however, the resolution is here limited to 60 µm by the size of the pixels of our camera.  The crystal 
used here was a 2-mm long, 4x4 mm2 square cross section, 8-at. % doped Yb:YAG crystal. It was 
 32 /115  
AR-coated on its faces (the lateral ones are polished). Its thermal conductivity, which is lower than 
that of an undoped YAG crystal, was measured to be 7 W.m-1.K-1 (11 W.m-1.K-1 for the undoped 
crystal). The pump source was a high power fiber-coupled diode array (HLU15F200-980 from 
LIMO GmbH) emitting 13.5 W at 968 nm. The fiber had a core diameter of 200 µm and a 
numerical aperture of 0.22. The output face was imaged onto the crystal to a 270-µm-diameter spot 
via two doublets. The crystal absorbed 5.4 watts of pump power in this case. The crystal was 
clamped in a copper block by its four side faces. In addition, on the top surface of the crystal, a 
frictionless copper finger allowed us to apply a well-controlled pressure on the crystal by the use of 
a set of known weights put upon the finger. The heat is finally evacuated from the copper block by a 
flow of circulating water. 
The key issue of infrared absolute temperature measurements is the correct calibration of the system. 
Indeed, neither the crystal nor the copper mount has an infrared luminance which equals that of a 
blackbody at the same temperature. The signal V detected by one pixel for a portion of crystal (or 
copper mount) at temperature T is: 
( ) ( ) ( )∫
ελ dLL
TTrSGTV rt
optr  (II.2.1.) 
where G is the geometric extent; ( )λrS is the spectral sensitivity; Tropt is the whole transmission 
coefficient of the ZnSe plate, Germanium objective and camera optics; 
dLTBB is the spectral 
luminance of a blackbody at temperature T, ε(T) is the emissivity; Lr denotes the infrared luminance 
of the camera itself (and its close surroundings) which is reflected back into it by the Germanium 
objective and by the polished surface of the crystal; Lt is the luminance transmitted through the 
crystal: it is zero in the 8-12 µm range since the crystal is highly opaque in this spectral region. Lr is 
nonzero and makes polished objects look brighter than blackbodies: if Lr is ignored it leads to 
overestimation of the temperature around room temperature. Inversely, the emissivity is less than 
 33 /115  
one and makes objects radiate less than a blackbody. Since the parameters ε and Lr are dramatically 
dependant on the surface quality and flatness, all the visible parts of the heat sink were covered with 
lustreless black painting. Moreover, the evaluation of all those parameters is not straightforward. 
We propose to calibrate the whole system as follows: the crystal and the copper mount were heated 
together to a set of given temperatures using a thermoelectric (Peltier) element, and we then 
compare with the temperature given by the camera to apply the adequate correction. This careful 
calibration allows rigorous and absolute measurement of the temperature with a spatial resolution 
large enough to study with sufficient accuracy the thermal behaviour on the crystal’s input face. 
II.2.3 Results and measurement of heat transfer coefficients 
Figure 11: The temperature map obtained when the crystal is clamped by its four edge faces by 
bare contact with copper without thermal joint (a), with heat sink grease (b), with a thin graphite 
layer (c), and with pressured and non-pressured indium (d). 
 34 /115  
Figure 11 shows the temperature map obtained when the crystal is clamped by its four edge faces 
by bare contact with copper without thermal joint (a), with a thin graphite layer (b), with indium (c) 
and with heat sink grease (d).  
Figure 12 is an enlargement of the two extreme cases, namely the bare contact and heat sink grease 
contact, with a transverse profile (y = 0) that shows the temperature evolution along the crystal 
lateral dimension. 
Figure 12 : Temperature mapping of the crystal (front view) and lateral profile at y=0 for two 
different types of thermal contact (direct copper-crystal contact on the left, with grease on the right).  
In the “bare contact” case, a clear gap is noticeable between the temperatures of the mount and at 
the edge of the crystal. The temperature distribution is parabolic inside the pumped region and then 
 35 /115  
experiences a logarithmic decay until the edge of the crystal, in good agreement with the theory 
described in the previous section in the case of fibre-coupled diode pumping (see equation II.1.9 
and figure 5). As already mentioned, the quality of the heat transfer at the interface between the 
crystal and its mount has an influence on the value of the temperature but not on the thermal 
gradient.  
We consequently studied more in detail the heat contact. The heat transfer coefficient H is defined 
by equation (II.1.22), where the thermal gradient is considered normal to the surface.  
Our system provides a space-resolved temperature mapping of the crystal, with a spatial resolution 
which is far below the crystal size: it then allows the measurement of H. 
By performing a linear fit of the temperature versus position on the points that are closer to the 
crystal edge, the heat flux can be determined: by applying the equation (II.1.22), one can then infer 
the value of H. We found for instance a value of 0.25 W.cm-2.K-1 in the case of bare contact. We 
estimate that the uncertainty on H is about 15%. The order of magnitude obtained is consistent with 
the values evoked by Carslaw [58] and Koechner [69]. The hot spot that can be noticed in figure 12 
betrays the poor contact between the polished face of the crystal and the copper surface. The heat 
transfer is primarily a question of how much two surfaces are in contact with respect to each other; 
we checked experimentally that the temperature inside the crystal did not depend on the applied 
pressure: we did not observe any noticeable variation of the temperature when changing the applied 
pressure in absence of thermal joint between the crystal and the copper mount.  
We summarized in table 3 the results obtained for the different thermal joints used in our set of 
experiments, namely graphite layer, indium foil and heat sink grease (CT40-5 from Circuitworks®). 
Graphite layer (around 0.5 mm thick) does not modify significantly either the maximum 
temperature or the heat transfer coefficient, but it was noticed that the contact was much more 
uniform than in the case of bare contact: in particular no hot spot appeared any more and the contact 
was somewhat independent of the applied pressure. It is not the case with indium foil. For this 
 36 /115  
experiment the crystal was wrapped within a 1-mm thick indium foil. Since Indium is a soft 
material, the quality of the contact is greatly dependant on the applied pressure. The temperature at 
the center of the pumped region experiences a 7°C decrease while the pressure increased from 1.5 
kg/cm2 to 22 kg/cm2 as shown in figure 13 (note that in this case the H coefficient is measured 
across the surface where the pressure is applied and is then an “effective” heat coefficient that takes 
into account the transfer from crystal to indium and then from indium to copper.)  
Figure 13: Evolution of the heat transfer coefficient H (squares) and maximum temperature 
(triangles) versus applied pressure for indium-wrapped crystals. 
The most dramatic change in heat transfer coefficient is obtained with heat sink grease (see table 3). 
The temperature gap drops down to 1°C and H reaches 2 W.cm-2.K-1. The heat contact is here 
independent on the applied pressure. This better heat transfer coefficient is achieved while the 
thermal grease has a much lower thermal conductivity than indium (0.62 W/m.K for 
 37 /115  
CircuitWorks ® CT40-5 thermal grease vs. 82 W/m.K for pure indium). This is an illustration, in 
conjunction with the data about the variation of H with the applied pressure, of the idea that 
achieving a good H is first and foremost a question of decreasing the thermal resistance at the 
interface (eliminating air gaps, maximizing the surface contact…) 
Table 3: Table of measured H coefficients for different contacts. Tmax is the temperature at the 
center of the pumped region; Te is the temperature at the edge of the crystal (averaged on the 4 
sides if not symmetrical), and Tm is the copper mount temperature near the crystal.  
Contact H (W.cm-2.K-1) Tmax (°C) Te(°C) Te-Tm (°C) 
Bare 0.25 49.8 33.5 10.7 
Graphite layer 0.28 46.5 30.5 8.7 
Indium foil 
(applied pressure : 
22 kg/cm2) 
0.9 40.0 25.1 4.9 
Heat sink grease 2.0 37.0 21.6 1.5 
 38 /115  
III. Thermal lensing effects: theory  
III.1. Introduction  
The previous chapter was dedicated to the calculation and measurement of the temperature 
distribution, which is the first essential step for the study of thermal effects. The appearance of 
thermal gradients causes the crystal to be under stress. The presence of inhomogeneous temperature, 
stress, and strain distributions is responsible of many deleterious effects for laser action: the most 
radical effect is fracture, observed when the hoop (tangential) stress at the periphery of the crystal 
exceeds the so-called tensile stress. More subtle effects arise from the stress-induced modification 
of the optical indices of refraction: alteration of the stability domains of the cavity, depolarization, 
losses and degradation in beam quality, all of these four phenomena being largely intermixed. In 
this paper we designate by “thermal lensing” effects all the phenomena resulting in a phase change 
of a beam passing through a pumped crystal; in other words we do not restrict this expression to an 
ideal spherical thin thermal lens, we also include its aberrations and its polarization-dependant 
aberrations. This chapter presents a general and synthetic scope of these effects, and points out how 
they are related to each other. We base our discussion on analytical simple scaling relationships, 
and we point out the validity of these formulas.  
In this review, we come back to well-established theories that have been exposed many 
times in the past [60, 69, 70], but we also bring some new insights, to our knowledge, on some 
points of practical interest.  
In particular, we will point out several inaccuracies generally reported about the values of 
the photoelastic constants in YAG, which are the result of an incorrect use of the Hooke Law; we 
also present what is (still to our knowledge) the first derivation of the photoelastic constants that 
have to be used for end-pumped crystals, that is in other words when the calculation is made using 
the plane stress rather than the plane strain approximation.  
 39 /115  
We will eventually point out that the use of the dn/dT coefficient (temperature derivative of 
the refractive index) is very confusing. In one hand, the classical formula (which reveals the 
existence of three contributions: the “dn/dT” part, the bulging of end faces, and the photoelastic 
effect) which is used since decades is correct provided that the dn/dT appearing in this expression is 
understood as a partial derivative taken at constant strain. In the other hand, the experimentalist 
can measure a quantity which is closer to a partial derivative at constant stress, and the partial 
derivatives are obviously not equal. The dn/dT parameter is then not actually the correct parameter 
to be used in order to estimate the thermal lens focal length: this subtlety means in particular that 
one cannot, in general, make use of a value of dn/dT readily found in handbooks to estimate the 
magnitude of the thermal lens of an operating laser, because the experimental measurement 
conditions are in the two cases mutually inconsistent. We’ll see however that when the dn/dT is 
large and positive, the difference can be ignored. 
We will conclude this review by a synthetic diagram showing all the thermal effects and 
how they are connected together.  
Given that thermo-optical properties pertain more to a crystal host than to a doping ion, this 
section is more general than the others and does not restrict to the case of Ytterbium-doped 
materials.  
III.2. Stress and strain calculations 
Once the temperature field has been computed, the next step is to calculate the stress and strain 
distributions inside the crystal, obtained from the so-called “generalized” Hooke law, because it 
includes the thermal expansion term [49]:  
TS ijTklijklij ∆+= ασε          (III.1) 
 40 /115  
where i, j, k, l = 1,2,3 and the Einstein summation convention is used. ∆T is the temperature shift 
with respect to equilibrium (no strain), (Sijkl) is the compliances tensor, (σkl) is the stress tensor, (εij) 
is the strain tensor, and (
α ) is the thermal expansion coefficients tensor.  
The analytical formulations of thermal stress and strain distributions in end-pumped lasers require a 
large amount of approximations, thoroughly discussed in Cousins’s reference paper published in 
1992 [60].  
In order to obtain an analytical solution to the stress problem, an additional approximation is 
required, which consists in considering the problem in two dimensions. This is either the plane 
strain approximation (valid for long and thin rods) or the plane stress approximation (valid for thin 
disks). Interestingly, Cousins [60] pointed out that the plane stress approximation remained valid 
(within approximately 10%) for aspect ratios up to L/2r0 = 1.5, providing that the stresses were 
considered as mean values integrated along the whole thickness of the rod. In the previous section, 
when we derived the temperature distribution, we had considered the crystal as a stacking of thin 
slices, so that the temperature could be calculated in a single thin slice as if the surrounding material 
did not exist. It is not possible to use this approach for the stress distribution, because a given slice 
is under the mechanical influence of the slices located on both sides, and cannot be considered as 
independent. This is why an attempt to take into account absorption saturation effects and pump 
divergence (as far as thermal stresses are concerned) inevitably requires a finite element analysis.  
As far as diode end-pumping is concerned, the plane stress approximation is then the most 
meaningful approximation that can be done. However, the exact calculation remains possible for a 
given crystal (using FEA codes) provided that all the compliances and thermal expansion 
coefficients are known. To the best of our knowledge, these coefficients have been measured for a 
very restricted number of laser crystals up to now: we may readily find these data for YAG, 
sapphire, YLF and Y2O3 [71]: other data are available in the Handbook of Optics [71] but for 
 41 /115  
crystals which are not commonly used in laser applications. The data for other materials are almost 
inexistent.  
The analytical solution of the generalized Hooke law (eq. III.1) can be found, for example in [60] 
under the plane stress approximation. In this paper, the discussion was restricted to isotropic 
materials (at a mechanical rather than an optical point of view, that is when the compliances can be 
reduced to only two parameters, the Young modulus and the Poisson ratio), absorption saturation 
was not considered, and the divergence of the pump beam was not taken into account either.  
Once this has been calculated, it is possible to study thermal fracture issues. It is generally admitted 
that fracture occurs when the maximum hoop (tangential) stress σmax at the surface periphery of the 
crystal exceeds the tensile stress σTS. The latter depends on both the fracture toughness of the 
material and on its surface flatness. These aspects have been studied in detail by Marion [61-63]. 
Data about fracture toughness of materials can be readily found in the literature for YAG, 
fluoroapatites, sapphire, yttrium orthosilicate YSO and some phosphate glasses [39, 63, 71, 72].  
For a qualitative discussion of fracture issues in Yb-doped materials, the reader is invited to refer to 
a previous publication [73]. 
III.3. How can we take into account the photoelastic effect ? 
We now consider how the temperature, stress and strain fields inside the crystal alter the phase of 
the cavity beam, all these effects being referred as “thermal lensing” phenomena.  
The appearance of stresses in the crystal causes the linear optical indicatrix (related to the linear 
indices of refraction) to change its shape, its size and its orientation. This photoelastic effect is 
accounted by the 4th rank elasto-optical tensor (pijkl): 
klijklij pB ε=∆   (III.2) 
Where (Bij) is the dielectric impermeability 2nd rank tensor. This expression if obviously valid in the 
linear optical regime only, and when piezoelectric effect is neglected [49].  
 42 /115  
The complete computation of thermal effects in a given material requires that we know everything 
about the tensors (Sijkl), (pijkl) and (
α ). The minimum number of independent terms of each tensor 
depends on the crystal symmetry, as discussed by Nye [49]. For instance, let us consider crystals 
like KGW or KYW [74], GdCOB [75], YCOB [76] or YSO [43] which are of particular interest for 
Ytterbium doping. These crystals belong to the monoclinic crystal system: this means that the 
compliances can be “reduced” to 13 independent parameters, and the elasto-optical tensor, once the 
redundant coefficients have been identified, appears to have 20 independent coefficients [49]. 
Adding the 3 thermal expansion coefficients, this means that we need to know no less than 36 
coefficients before to be able to draw the new index ellipsoid at a given point of the crystal. 
Obviously these parameters are not known (for any monoclinic crystal, in fact, to the best of our 
knowledge), which means that a rigorous calculation even with a FEA code is just not possible. 
This simple remark highlights the importance of experimental measurements of thermal effects in 
such crystals, and shows the interest as well as the inherent limitations of a simple analytical model.    
III.4. Simplified account of photoelastic effect in isotropic crystals 
Now we are aware of these difficulties, we focus our discussion on a simpler study case, 
actually the only case where analytical expressions are obtainable, that is:  
- we consider isotropic crystals only, and more particularly the widespread YAG crystal, for which 
all the parameters previously evoked are well known. Generally, cubic crystals belonging to the 
space groups 43 ,432, 3m m m  (like YAG) require 3 independent elastic coefficients ; however the 
remarkable isotropic mechanical properties of YAG enable to think of only two mechanical 
coefficients, that is the Young modulus and the Poisson ratio. In the end, 6 coefficients only are 
needed for YAG. 
- The plane stress approximation is used; 
 43 /115  
- the pump profile is still axisymmetric; 
- we consider what occurs inside the pump volume, that is for r < wp.  
- the pump divergence inside the crystal is neglected. 
- temperature, stress, strain, index are considered integrated along the whole thickness of the rod. 
For a physical quantity A(r,z) , we shall note : 
( ) ( ) dzzrArA
the integrated value of A(r,z) along the rod.  
According to the Neumann-Curie theorem [49], under these assumptions the principal axes of all 
the involved tensors (stress, strain, index ellipsoid) are radial and tangential. The notations used in 
the following are depicted in figure 14. 
nr nθ 
nr nθ 
Figure 14: Orientation of the indices’ ellipsoid in an isotropic crystal under thermal stress.  
 44 /115  
The shift of the principal indices nr and nθ are related to the diagonal coefficients of the 
optical indicatrix by:  
=  (III.3) 
We can also write the indices variation as a function of the strains as follows: 
Bnn ε
=∆−=∆ ,,,,
 (III.4) 
The six coefficients 
(i= r, θ; j=r, θ, z) can be calculated from the pijkl coefficients by a correct 
change of coordinates.  
The complete solutions for the strains εr, εθ and εz can be found, for YAG, in many papers and 
textbooks [60, 69] and can also be found in the appendix. 
Inside the pump volume, it can be readily shown that stresses and strains have a parabolic 
dependence, like the temperature distribution. Since the indices of refraction are linear 
combinations of strains, it turns out that radial and tangential index distributions must also be 
parabolic.   
The (integrated) shift of refractive index may be written as:  
( ) ( ) ( )
rndzzr
rn θθ
=∆ ∑ ∫
 (III.5) 
where Cr and Cθ (or C’r and C’θ)  are constants which will be called, following the pioneering work 
of Koechner, the “photoelastic constants”. Their calculation and expression is given in the appendix. 
We would like to point out two important clarifications about these coefficients (and also justify the 
presence of this appendix in this review):  
♦ W. Koechner published incorrect values of these coefficients in his reference book [69] 
because the temperature term in the Hooke law has been omitted; this omission has first 
 45 /115  
been highlighted by Cousins [60], but the expression of the photoelastic constants remained 
uncorrected in the following editions of this book, and nowadays still remains used under 
this form in many papers. 
♦ Secondly, the derivation of the photoelastic constants requires turning the 3D problem into 
a 2D problem, as discussed in the previous section. Only the plane strain case was 
considered by Koechner. However, we saw that the plane stress case is closer to reality in 
end-pumped rods. Here we denote as Cr and Cθ the photoelastic constants valid for long and 
thin rods (the “Koechner case”, that is when the plane strain approximation is valid), and 
C’r and C’θ the photoelastic constants derived within the framework of the plane stress 
approximation. Since we are only interested in end-pumping, we only consider the ' ,θrC  
constants in the following. 
The above-mentioned relations are derived by making the assumption that the pump beam 
radius is constant through the crystal thickness; however we don’t have to assume a particular 
absorption regime, so that they stay valid in presence of absorption saturation, under lasing as well 
as under nonlasing conditions. 
As we will see in the next section, the most interesting feature for the laser scientist is the 
index shift between the center and the edge of the pumped zone, since it yields the contribution to 
the global thermal lens. From (II.1.10), and using the bracket notation for z-integrated values, we 
can write (III.5) under the form:  
( ) ( ) ( ) ( )rTTCnrnn
−=∆−∆ 020 '30,, ,θαθθ  (III.6) 
III.5. A consequence of strain-induced birefringence: depolarization losses.  
 46 /115  
The birefringence of a crystal submitted to thermal stress has two main consequences for a 
light beam passing through it: both its state of polarization and its phase will be altered. Before 
examining in detail the effect on phase (i.e. thermal lensing effects), let’s first examine the influence 
on polarization. 
We use the restrictions exposed in the previous subsection, given that the following can be 
readily extended to uniaxial crystals provided that the optical axis lies parallel to the propagation 
axis. In these cases, if no polarizing element is added into the cavity, the laser output is not 
polarized, and stress-induced birefringence has no net effect. 
Beam polarized along x 
crystal 
Depolarized beam  
Figure 15: Depolarization of a polarized beam passing through an isotropic crystal under thermal 
stress. 
For many applications however a polarized output is desirable: the situation is depicted 
schematically in figure 15. An incident beam (polarized along the x direction) will have its 
polarization modified differently for every single ray: For a ray crossing the (Ox) or (Oy) axis for 
instance, the polarization is not modified, for all the other rays the polarization becomes elliptical, 
with principal axis that are radial and tangential.  
 47 /115  
At every roundtrip in the laser cavity, the beam meets a polarizing element (such as a plate 
at Brewster angle) and this depolarizing element, yielding to the so-called “depolarization losses”. 
Another effect, consequence of the latter, is a modification of the beam spatial profile: since 
the beam is not altered along the (Ox) and (Oy) directions and suffers losses elsewhere, it tends to 
take the shape of a cross; this aspect is chiefly discussed in Koechner [69].  
In biaxial crystals (or uniaxial crystals with optical axis normal to the direction of 
propagation), the output is naturally polarized along the crystallophysic (not to be confused with 
cristallographic) axis along which the emission cross section is the highest. In this case, stress-
induced birefringence does not generally twist the index ellipsoid enough to significantly modify 
the polarization state.  
Finally, clever solutions have been imagined to compensate for depolarization losses in 
isotropic crystals: for example the use of two rods with a Faraday rotator inbetween [78, 79], or 
even a simple quarter waveplate [70]. This last technique is however limited to a few configurations 
[80]. 
In the following, we present results obtained with Yb-doped crystals which are either 
isotropic (and naturally not polarized), or naturally birefringent, so that this problem was not 
encountered. 
III.6. Thermally-induced optical phase shift.  
III.6.1. Expression of the optical path 
We present now this derivation with some detail, because it makes appear a trivial yet 
fundamental difference between these results and what is currently reported in the literature. The 
derivation is largely reproducing Cousins’s work [60]. 
 48 /115  
n (T, ε j ) 
L+∆L d - ∆L 
z  = 0 
n0 (Tc ) 
z  = 0 
Figure 16: Notations used for the calculation of the optical path 
Consider a crystal whose length is L and index n0 at temperature Tc (temperature of the heat 
sink) and in absence of strain. We consider the optical path of a straight ray (running parallel to the 
crystal axis z) between a plane z = 0 (taken at the entrance face of the crystal) and a plane z = L+d  
(see figure 16).  
In absence of temperature and stress fields (pump off), the optical path is: 
dLTn c
off +×= )(0δ  (III.7) 
With the pump on, the optical path is dependant on the lateral shift r of the ray with respect 
to the crystal axis, but also on the direction of polarization. As a consequence there will be two 
distinct thermal lens focal lengths, one for a (virtual) radially polarized beam and another for a 
tangentially polarized beam, what is commonly named “bifocussing”.  
We may write the optical path with the pump beam “on” as (see figure 16):  
  ( ) ( ) ( )rLddzTnr
∆−+= ∫
, ,, εδ θθ  (III.8) 
where ∆L(r) is the crystal length shift due to the inner compression, responsible for the 
bulging of the end faces.  
 49 /115  
Assuming small variations in the refractive index, one may expand ( )jr Tn εθ ,,  in Taylor 
series, and discard the second and higher-order terms:  
( ) ( ) ( )( ) ( ) ( )rLddzzr
+= ∫ ∑
=0 ,,
0 ,,,
 (III.9) 
Note that the temperature derivative of the refractive index appearing in this equation is a 
partial derivative calculated at constant strain. As we will discuss in the next subsection, this is not 
the usual dn/dT parameter. 
The rod length change ∆L(r) can be written as a function of the axial strain εz and equals  
( ) ( ) ( )rdzz,rrL z
z εε∆
== ∫  (III.10) 
From the strain-stress relationships featured in the appendix, it can be easily shown that the 
axial strain, under the plane stress approximation, is equal to: 
 ( ) ( ) ( ) ( )rTrTr Tz −=+−= 01 ναε  (III.11) 
Given that the first-order terms in the integral appearing in (III.9) are much smaller than 
n0(Tc) and given that ∆L « L, we can write that for the first order terms  ( ) ( )∫∫ ≈
∆+ LLL
......
The relative optical path is then:  
( ) ( )
( ) ( ) ( )( ) ( ) ( )0110
TrTnr
offon
−+−+⎟
 (III.12) 
III.6.2. The thermal lens focal length 
The thermal lens is related to the optical path difference (OPD or ∆ in the following) 
between an on-axis central ray (r = 0) and an outer parallel ray passing inside the pumped region, 
defined by a radius r < wp.  
 50 /115  
We note : 
( ) ( ) ( )rr relrrelr θθ δδ ,, 0 −=∆  (III.13) 
The expression of the optical path difference is, from (III.6) and (III.12):  
 ( ) ( ) ( ) ( ) ( )rTTnCn
r TrT
+−++⎟⎟
=∆ 0112 0
, ναα θ
θ    (III.14) 
We assume that the thermal derivative of the refractive index is equal for radial and tangential index, 
so we can write:  
( ) ( )( ) ( ) ( )
( ) ( )rTT
rTTCnn
++−+⎟
 (III.15) 
where θχ ,r is usually called the “thermo-optic” coefficient. We remind the reader that this 
expression is valid only under some restrictive conditions that have been presented in detail in the 
section 4 of this chapter.   
Given that the integrated temperature shift is given by (from eq. II.1.11):  
( ) ( ) ( ) ( )( )
dzzrTzTrTT
∫ =−=−  (III.16) 
it appears that the optical path difference also follows a quadratic dependence in r. This 
means that in the paraxial approximation, the pumped crystal acts as a thin lens whose focal length 
is given by :  
),( 2∆
=  (III.17) 
In the following the difference between fr and fθ (responsible for bifocussing) will be 
omitted (realistic if the photoelastic effect is negligible, or if the laser beam is not polarized.) 
 51 /115  
The thermal lens dioptric power is thus defined by: 
( )( )
th Kw
002 2
++−+⎟
==  (III.18) 
where χ is a polarization-averaged thermo-optic coefficient. 
Let’s notice that this formula still holds in Yb-doped materials, with strong absorption 
saturation, since no assumption was made concerning absorption.  
If the pump profile is gaussian (e.g. end-pumping by another laser), one can show [70] that 
the thermal lens dioptric power is twice as large as (III.18), meaning that the thermal load is more 
spiky around the center of the beam than in the case of an uniform “top hat” pump beam energy 
deposition.  
To conclude, let’s address some orders of magnitude. To be correct, the previous derivation 
has to be performed within the paraxial approximation, which means Lf th >> . 
Taking some typical values (ηh ∼ 0.1, χ ∼ 10-5 , Kc ∼ 5 W/m/K, wp = 100 µm, et L = 3 mm), 
it appears that the paraxial conditions are met provided that  Pabs << 100 W. This corresponds to 
most of practical cases. 
        
III.7. Discussion about the use of the “dn/dT” coefficient. 
Let’s start this discussion by the expression of the thermo-optic coefficient: 
( )( ) 3 ', 0 0 ,1 1 2r T T r
n n C
Tθ θ ε
χ ν α α
∂⎛ ⎞= − + + + ⎜ ⎟∂⎝ ⎠
 (III.19)  
 52 /115  
In this subsection, we would like to point out how this expression can be misleading if one 
carelessly uses, to evaluate the magnitude of a thermal lens, the so-called dn/dT instead of
The error is especially important when all terms except the dn/dT are discarded for the sake of 
simplification. This represents, to our knowledge, a discussion that has never been published so far.  
The three contributions appearing in (III.19) may be understood as follow: 
♦ The term ( )( ) Tn αν110 +−  is clearly related to the bulging of end faces, and is the direct 
consequence of the inner compression of the crystal, which causes the optical path to 
increase (if αT>0). It is strictly true for an infinitely thin crystal, since plane stress 
approximation is used to derive it. It is reported by Cousins [60] that for a rod whose ratio 
length/diameter is 1.5, this term overestimates the actual bulging by around 35%. In general 
this can be taken as an upper limit for end faces bulging in DPSSLs. 
♦ The term ' ,
02 θα rT Cn accounts for the photoelastic effect only, as already discussed in 
subsections III.3 and 4. It explains bifocussing, depolarization and polarization-dependant 
astigmatism.  
♦ As for the first term (III.19), it represents the partial derivative of refractive index at 
constant strain, which is the thermo-optic coefficient of a virtual perfectly rigid crystal.  
It is noteworthy that it is actually not the usual dn/dT parameter that one can measure and 
find easily in the literature. The “usual techniques” for measuring dn/dT are based either on 
geometrical optics (e.g. measurement of the minimum-of-deviation angle of a prism cut in the 
material under study [72]) or on interferometric techniques (e.g. measuring fringe patterns 
displacements [81]). In all cases, the sample is put into an oven and exposed to different well-
known temperatures. The crystal is free to expand, and the temperature rise into the material is 
uniform, which is incidentally an essential condition for valuable measurements. In this case, 
 53 /115  
obviously, the crystal experiences thermal expansion, a phenomenon which causes the index to 
change (to decrease, in general) in virtue of a pure photoelastic effect. In all these practical 
circumstances, the strain tensor relates directly to the temperature shift by the thermal expansion 
tensor, in other words the stress terms in equation III.1 are zero. The coefficient measured 
experimentally can then be regarded as a partial derivative at constant stress: 
measured
 (III.20) 
On the other hand, the reality experienced by the laser crystal while optically pumped is 
radically different. We know that in all cases (transverse as well as end pumping, thin disks as well 
as long and thin rods) the pumped area inside the crystal is under compression (negative stresses 
and strains), which is true as soon as the thermal expansion coefficient is positive. It can be 
explained qualitatively by saying that the central region of the crystal, yet hotter than the edges, is 
prevented from expanding by the expanding (cooler) outer parts of the rod, which eventually causes 
the central region to be under compression.  
As a result, we see that if we consider the measured dn/dT instead of the partial derivative at 
constant strain in eq. (III.19), the photoelastic contribution to the thermal lens, already fully taken 
into account with the term 2 ' ,
0 θα rT Cn , which accounts for thermal expansion if any, is partially 
cancelled by the photoelastic (thermal expansion) term hidden in the measured dn/dT.  
In a first attempt to correct for this, we thus have to evaluate precisely the thermal expansion 
contribution to the measured dn/dT. It can be done in a simple way by considering the Clausius-
Mossotti model for refractive index, which writes: 
 54 /115  
( ) ( )( )
    (III.21) 
where ρ is the specific mass (density), M the molecular weight, Na the Avogadro number, and αe the 
polarizability. This expression is valid strictly speaking for isotropic ionic crystals (for covalent 
crystals the local field correction is smaller and atomic polarizabilities loose their meaning due to 
the very nature of covalent bonding) 
The modification of polarizability with temperature results from the change in thermal 
occupancies and spectra of the energy levels. Tsay et al. [82] have developed a two-oscillator model 
where they consider the contribution of both electronic and lattice vibration terms. The discussion 
about the different origins of dn/dT is beyond the scope of this review and will not be exposed in 
detail here.  
We can differentiate (III.20), assuming that changes in density only originate from isotropic 
thermal expansion. This is consistent with the experimental procedure used to measure this 
coefficient. 
( ) ( )( ){ }, ,e e e
measured
dn T T T n n n n
dT T T Tσ
ρ α ρ α αρ
ρ α ρ α
⎛ ⎞ ⎛ ⎞∂ ∂∂ ∂ ∂ ∂ ∂⎛ ⎞⎜ ⎟ = = + +⎜ ⎟⎜ ⎟⎜ ⎟ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎝ ⎠
    (III.22) 
From (III.20) and isotropic expansion assumption we obtain: 
( )( )
12 22
    (III.23) 
This expression puts into the light three contributions to the measured dn/dT:  
- a pure effect of thermal expansion : 
( )( )
12 22 −+
 (III.24) 
 55 /115  
- the influence of thermal expansion on polarizability,  
              
( )( )2 22 1
− + − ⎡ ⎤∂
⎢ ⎥∂⎣ ⎦
 (III.25) 
- a thermal expansion-independent contribution, which can be assimilated to the partial 
derivative at constant strain: 
( )( )
12 22
 (III.26) 
Since the second and third terms do not breakdown easily into a set of available material physical 
parameters, no general formula based on the measured dn/dT can be derived for the last partial 
derivative. We see also that thermal expansion appears in both (III.24) and (III.25), so that we 
cannot in a straightforward way dissociate “pure” thermal expansion from strain-related 
polarizability effects. 
This formulation brings some questions (rather than answers) about the interpretation of 
some experimental results, in particular those obtained with materials whose measured dn/dT is 
negative.  
It is often reported that in such materials (e.g. LiCAF [72, 83], FAP [39], YLF [84]), the 
thermal lens is weak or even divergent (case observed in YLF crystals) because the negative dn/dT 
counterbalances (or even surpasses) the other positive terms in the expression of the thermo-optic 
coefficient. 
Although, because of the lack of data about these materials, the photoelastic term is just 
supposed to be positive, or evaluated from other materials whose properties are believed to be 
similar (Woods et al. [72] use data from CaF2 to evaluate photoelastic constants of LiCAF, Payne et 
al. [39] used data from LG-750 phosphate glass to approximate that of FAP). In some cases (e.g. 
thermal lensing measurements in Nd:YLF crystals [84]), discrepancies are reported  between theory 
and experiment, which do not occur with YAG which has a positive measured dn/dT. 
 56 /115  
A large and negative dn/dT coefficient means that the thermal expansion is dominating 
polarizability contribution for an unstressed crystal; it is observed that such behaviour is generally 
associated with a large thermal expansion coefficient. This also means that the photoelastic term, 
proportional to αT, can be expected to be greater. In contrast, we can say nothing about the sign of 
this term, whose knowledge requires that we know all of the pijkl coefficients of the crystal. 
This means that there is no obvious relationship between the sign and magnitude of the 
measured dn/dT and the sign and magnitude of the thermal lens. 
To go further, the only term which is truly always positive is the end faces bulging term. The 
polarizability dependence on temperature (eq. III.26) is mostly positive too [82]. We can then assess 
that negative thermal lensing is more likely to be explained by negative photoelastic terms, and/or 
possibly, by a negative e
 term. 
In conclusion, the crystals with negative measured dn/dT have to be considered very 
carefully as far as simulations are concerned: photoelastic terms must not be neglected. However, it 
remains that photoelastic contributions, whatever calculated or measured, tend to be small in many 
crystals. This means that the rude approximation made by replacing χ by the measured dn/dT (+ the 
end bulging term) will be all the more close to reality that the dn/dT is large and positive.   
III.8. A novel definition for thermo-optic coefficient based on experimentally 
measurable parameters.  
In the precedent subsection, we saw that there are many problems related to the definition of 
the thermo-optic coefficient, which are above all related to the abusive use of the parameter (dn/dT) 
in a context where it is not relevant.  
 57 /115  
Actually, a better way to describe the phenomena is to start from measurable data that are 
relevant as far as solid-state lasers are concerned. The partial derivative at constant strain is a formal 
parameter which has not a real physical meaning since it is impossible to prevent the crystal from 
any strain, compression or thermal expansion. Furthermore, photoelastic and polarizability effects 
are so strongly intermingled, that one cannot imagine easily an experiment that could separate 
clearly the effect of one from the other. That’s why we propose to base solid state laser thermal 
characterization for high power applications on measurable data and separate the thermo-optic 
coefficient χ in three truly independent contributions, as follows: 
lgn bu ing birefringenceχ χ χ χ= + +  (III.27) 
where  
( )3 ' '0n T r
n C C
= + +⎜ ⎟∂⎝ ⎠
 (III.28) 
( )( )lg 0 1 1bu ing Tnχ ν α= − +  (III.29) 
( )3 ' '0birefringence T rn C Cθχ α= ± −  (III.30) 
χbulging accounts for curvature of end faces, and is measurable by performing, for example, 
interferometric or wavefront measurements on a probe beam reflected on each side of the crystal, as 
done by Baer et al. [85] or Kleine et al. [86]. The expression given in (III.29) applies to the thin 
disk ideal model, but its real value can be computed depending on every special geometry, quite 
easily with a finite element code, since this is a pure thermomechanical problem, where data are 
more readily accessible or measurable. 
χbirefringence accounts for strain-induced birefringence (“+” for radial polarization, “-“ for 
tangential). It can be measured separately by performing measurements of polarization-dependant 
 58 /115  
astigmatism, for instance, thanks to a wavefront measurement method sensitive to aberrations (see 
next section). 
Eventually, the term χn which accounts for all the refractive index variations with 
temperature, is not rigorously calculable for all the reasons exposed above, but its exact value can 
be deduced for each material and for each pumping configuration, from the separate measurement 
of χ (global thermo-optic coefficient), χbirefringence and χbulging.  
To conclude, let us give some orders of magnitude of different terms in widespread YAG. 
The measured value of the global thermo-optic coefficient (see next section of this article for details) 
for this material under diode pumping is 10 10-6 K-1. 
The different contributions calculated from tabulated data are: 
6 19.10
measured
− −⎛ ⎞ =⎜ ⎟
( )( )2 2 6 12 1 31.5 10
measured
n ndn
+ −⎛ ⎞ + =⎜ ⎟
( )( ) 6 1lg 0 1 1 7.210bu ing Tn Kχ ν α − −= − + =  
3 6 1
3 6 1
2 0.27 10
2 0.9310
n C K
n C Kθ
(see appendix for the detail of the calculation of photoelastic constants within the plane stress 
approximation). 
The thermo-optic coefficient are small, but the bulging term is far from being negligible. 
Weber et al. [87] have shown that in Nd:YAG, bulging represented 30% of the global thermal lens.  
We calculated also for information a “corrected” dn/dT, which is obtained after subtraction of the 
inappropriate thermal expansion term (III.24): this coefficient turns out to be very large here.   
 59 /115  
In diode-end-pumped Nd:YLF and Nd:YVO4, Baer et al. [85] and Kleine et al. [86] have 
shown experimentally that the bulging term represented half of the total thermo-optic coefficient. A 
finite-element analysis, performed by Peng et al. [88], leads to a similar conclusion for a Nd:YVO4 
crystal.  
III.9. The aberrations of the thermal lens.  
In conclusion to this chapter, we will introduce the thermal lens higher-order distortions id 
est the thermal lens aberrations. 
For a perfectly parabolic distortion of the wave front or equivalently a pure thermal lens the 
thermal distortion can be easily compensated by addition in the laser cavity of the opposite 
divergent lens or by adjusting the distance of the different cavity elements. But, if aberrations are 
present the compensation is very difficult and requires complex systems [89]. While uncorrected, 
these aberrations lead to degradation in beam quality (brightness), and also to losses due to 
diffraction of the beam high spatial frequencies [90]. 
The aberrations are present when the wavefront distortions induced by the absorbed pump 
beam are not perfectly parabolic. This occurs when the longitudinal pump beam has not a true top-
hat profile, for example in the case of a Gaussian pump beam profile [91]. Moreover, if the laser 
beam size is larger than the pumped area, the aberrations also become important as shown in figure 
17. The rays, far from the pumped area are almost not deviated. This is the signature of spherical 
aberration (which is sometimes referred as a “thermal lens varying with radius r”.)   
 60 /115  
Pumped areaPumped area
Figure 17: When the laser beam is larger than the pumped area, spherical aberration is observed. 
In general the aberrations affect the laser modes of the cavity in a way that to degrade the 
beam quality. This degradation can be evaluated by the M2 factor or the Strelh ratio for exemple 
[92]. We are just giving here an exemple of the influence of aberrations on the beam quality, some 
results given by Clarkson [70].  
Let us consider that only 3rd order spherical aberration is present. In this case, the optical 
path difference ∆(r) can be written as:  
( ) 44
−=  (III.31) 
if the laser initial beam M2 factor is 2iM , the M
2 with the added aberations is :  
( ) ( )22222 qif MMM +=  (III.32) 
 61 /115  
with 
π8 442 L
M =  (III.33) 
where λ is the wavelength and wL the laser beam waist. 
One can show that obviously C4=0 for a top-hat pump profile provided that wL< wp. But for 
a gaussian pump-beam profile (beam waist wp) we obtain [70]:  
M  (III.34) 
In that case, the wL/wp is present to the power of 4, which means a strong increase of M2 
even for small mode mismatch. 
Another classical aberration to be considered is the thermal astigmatism.  In this case, the 
analytical solution is not as simple as with the 3rd order spherical aberration and we will only focus 
on the qualitative approach, answering this simple question: in which conditions does the thermal 
lens exhibit astigmatism? In practice it will occur whenever: 
- the thermal conductivity is anisotropical ;  
- the thermal expansion tensor 
ijTα does not reduce to a scalar quantity; 
- the cooling is inhomogeneous; 
- the laser beam is  polarized, even for an isotropic material. This is the so-called polarization-
dependant astigmatism, which can be used a sa probe to evaluate the strain-induced 
birefringence (see figure 18.) 
 62 /115  
crystal 
(nr-1)L 
(nθ-1)L 
Figure 18: Illustration of polarization-dependant astigmatism. Here a vertically polarized strikes 
the crystal from the left. The ray  #    sees the radial index of refraction nr while ray #  sees the 
tangential index of refraction nθ . If for the sake of simplicity the indices are considered constant 
over the whole crystal length L, the astigmatism is (nr- nθ)L. 
As a conclusion for this part, we present a schematic diagram (fig. 19) summing up the different 
thermal effects arising in a solid state laser medium, and how they are related to each other.  
 63 /115  
Figure 19: Summary of the thermal effects in solid-state lasers. The observable consequences 
observable are presented in full rectangles. The aberrations can be split in two classes: the ones 
Pumping ∆T 
    temperature 
deformations 
Modification of crystal lattice (thermal 
expansion)  
(stress) 
Elastic 
tensor 
Modification of refraction indices 
Thermal lens 
Induced birefringence 
Photoelastic 
effect 
depolarization 
Modification of 
polarizability, 
related to 
⎜ ⎟∂⎝ ⎠
FRACTURE 
if σ > σTS
Curvature (bulging) of both faces of 
the crystal 
focussing  
For an ideal 
lens 
Changes in 
stability range of 
the laser cavity, 
and in the size 
and the 
divergence of the 
laser beam. 
Losses   
potentially 
Spherical 
aberration or/and 
astigmatism  
Beam quality degradation  
ABERRATIONS of the thermal lens 
Polarization-
dependant 
Astigmatism 
Leading to 
 64 /115  
that do not depend on the polarization (lined rectangles)  and the ones that  depend on polarization 
(dotted rectangles)  which come from the strain-induced birefringence. 
(*): 2 types of losses are induced by aberrations: the diffraction losses (associated with degradation 
in beam quality) and the losses induced by the eventual presence of a diaphragm in the cavity to 
prevent from oscillation of higher order laser modes. 
IV. Thermal lensing techniques  
IV.1. Introduction  
The first evidence of thermal effects in lasers were demonstrated in 1965 by Gordon, Leite 
and Whinnery [93] working at Bell Labs on He-Ne lasers for Raman spectroscopy applications. The 
use of liquids to Q-switch the laser lead to the observation of unexpected effects such as relaxation 
oscillation of jump of modes. The exceptionally long time constant of this phenomenon (several 
seconds) lead to the conclusion that a thermal lens was at the origin of the observed effects.  This 
lens was created by the small absorption occurring in the liquids. The first application proposed by 
the authors was then to use it to measure very small absorption coefficients down to 10-4 cm-1. As a 
matter of fact, the so-called “thermal lens method” allows nowadays to measure absorption 
coefficient lower than 10-7 cm-1 [94]. Since 1965, the photothermal-methods panel available to 
physicians has grown in diversity [95]. Actually, due to the complexity of simulating thermal 
effects in lasers (see previous subsections), the experimental determination is often the only 
accurate method.  
Since the 70’s, numerous attempts have been done to measure the thermal lens in solid-state 
laser media. They can be classified in three categories: the geometrical methods based on the 
deflexion of a beam, the methods based on the properties of cavity modes, and the methods based 
on the wavefront measurements. 
 65 /115  
IV.2. Geometrical methods 
These methods are probably the simplest methods to measure the thermal lenses. They can 
be separated in two sub-groups: one can exploit either the defocussing or the deflection of a beam 
passing through the pumped medium  
crystal 
Powermeter 
Figure 20 : Example of thermal lens measurement based on the displacement of the focal point. 
The principle of the first category of methods is very simple. Considering a probe beam 
going through the crystal, the measurement of the axial shift of the focal point position allows to 
retrieve directly the thermal lens using simple geometrical optics. When the rod is relatively large, 
one can use a collimated beam of comparable size to directly measure the position of this focal 
point by Z-scan measurement for example as shown in figure 20 (a small aperture is longitudinally 
translated in order to find the maximum of probe-beam transmission [96].) This method is based on 
the assumption that the lens is perfect (without aberrations) and then is especially suited to 
transverse pumping with large-size materials . In fact, the method is not easily applicable to end-
pumped lasers because of the very large depth-of-focus associated with a very small and low solid-
angle–probe beam, and obviously does not yield the thermal lens aberrations. 
Moreover, for small beam diameters (10-100 µm), this method needs to be generalized using 
Gaussian beam optics. Hu and Whinnery [97] described a simple method to evaluate the thermal 
lens with a probe beam whose size is comparable to the cavity beam in end-pumping schemes. This 
 66 /115  
last method is described in figure 21: it can be based either on the measurement of the divergence of 
the probe beam [97], or in the measurement of the beam diameter in an appropriate plane.  
figure 21 : Example of measurement based on focal point displacement for tightly focused probe 
beam according to Hu and Whinnery’s method. 
To conclude on the methods based on the focal point displacement, we can say that their 
main drawback is their low accuracy. The relative precision is only 20-30% on the focal lens 
measurement. 
We can use geometrical optics in another way by measuring the deflection of a probe beam. 
Instead of using a beam covering the whole pumped area, we can measure the deflection of a small 
beam slightly off-axis as shown in figure 22.  
figure 22 : Example of measurement based on deflection measurement [98]. 
detector 
Laser 
crystal 
With thermal lens 
Gaussian 
beam 
Without thermal lens
 67 /115  
The whole surface can be scanned in order to measure not only the focal lens but also the 
aberrations due to the thermal effects [98]. However this method is complex and it can only be used 
for large transversally-pumped crystals (even if the resolution can be lower than the probe beam 
size [99]). Moreover this method can be considered as a point-to-point measurement of the simpler 
Shack-Hartmann technique described later.  
IV.3. Methods based on the properties of cavity eigenmodes 
After the development of end-pumped lasers (particularly thanks to the increasing 
performance of laser-diodes), alternative methods appeared. These less straightforward methods are 
based on the properties on the laser cavity eigenmodes, in particular on the fact that the thermal lens 
affects the stability zones of a laser cavity. All these methods are based on the theory of paraxial 
beam propagation theory in cavities presented by Kogelnik and Li [100] and can be formalized 
using the ABCD matrices. An example of this influence of the thermal lens on the stability of a 
cavity is given in figure 23. As a direct consequence, these methods consider ideal lens (aberration-
free) and do not give any information on the thermally-induced aberrations. Nevertheless these 
methods remain easy to implement since there is no probe beam (the laser itself is used to measure 
the thermal lens). In counterpart, it is impossible to measure the thermal lens in absence of laser 
extraction (with pumping but no laser emission). Here are three different examples of these methods 
based respectively on the work of Frauchiger and al., Neuenschwander and al. and Ozygus et al.   
 68 /115  
No thermal lens
thermal lens
Unstable cavity
Stable cavity
No thermal lens
thermal lens
Unstable cavity
Stable cavity
Figure 23: example of this influence of the thermal lens on the stability using a plano-plano cavity 
[101]. 
- Frauchiger and al. [102] measured the divergence of the output laser beam in a diode-pumped 
Nd :YAG and retrieved the thermal lens by a paraxial calculation. This example allows us to 
perceive the limitation of these techniques using the properties of cavity modes. Indeed, they are 
very sensitive to the beam quality. The measured divergence is actually directly proportional to the 
M2 factor of the beam. If the laser mode is not perfectly TEM00 the reliability of the method is 
strongly affected.  
- Neuenschwander and al. [101] used a plano-plano laser cavity stabilised by the thermal lens 
(figure 23). Adding two extra-cavity lenses and measuring the beam diameter in different 
longitudinal positions they found the waist in the cavity and therefore the thermal lens. This method 
is interesting because it allows the simultaneous measurement of the M2 factor which reduces the 
 69 /115  
limitations due to the beam quality. In that case, the limitation is due to the precision on the 
distances between the different optical components which is not always perfectly known.  
- Ozygus et al. [103] proposed an alternative method that consisted in using the frequencies of 
different transverse modes. In fact, the frequencies of the different transverse modes (in a plano-
concave laser cavity) depend not only on the length of the cavity but also on the radius of curvature 
of the mirrors (considering the flat mirror and the crystal as another concave mirror). If one 
achieves to have two modes lasing simultaneously in the cavity, the measurement of the beating 
frequency with an optical spectrum analyser allows retrieving the thermal lens. Another upgraded 
technique based on the same effect was also presented by Ozygus et al.later in 1997 [104]. In the 
last one, one translates one of the mirrors to find the positions of the spectral degeneracy (when two 
eigenmodes have the same frequency). The advantage of this method is its capacity to be used for 
measuring long-focal-length thermal lenses. It allows the measurement of focal lengths as long as 5 
In conclusion, the methods based on the properties of the cavity modes allows to have 
relative precisions on the thermal focal lens of 15 % for a TEM00 beam but down to 60 % for an 
non-diffraction limited beam [101].  
IV.4. Methods based on  wavefront measurements 
In this part, we will distinguish three wavefront measurement techniques: “classical” 
interferometry (based on fringe measurements on Michelson, Fizeau, … interferometers),  shearing 
interferometry, and Shack-Hartmann sensing. 
IV.4.1. Classical interferometric techniques 
 70 /115  
As far as “classical” interferometry is concerned, one can consider equal-thickness fringes 
between the parallel end faces of the rod (this is the so-called Fizeau interferometer); one can also 
insert the rod under study in an arm of a Michelson-type [105] or Mach-Zehnder -type 
interferometer [106] for example. The first type of method is simpler since it does not require a 
second interferometrically adjusted arm.   
These methods are particularly well suited for large amplifier rods but in counterpart, they 
are not convenient for end-pumping. Indeed, even if there is no fundamental contraindication to use 
this method for small spots, in practice in this case the number of fringes is too small to obtain an 
exploitable interferogram. As an example, if we consider a 200-µm diameter probe beam, a focal 
length of 5 cm only induces a phase shift between the centre and the edge of the beam of ∆ = h2/2f 
= λ/6 (with h the radius of the beam, f the focal length, and λ the wavelength of the probe beam, 
here λ =670 nm). In this case, of course, no fringes are visible.  
To overcome this problem, one can choose to take a probe beam larger than the pumped 
area, given that in these conditions, as shown in figure 17, some spherical aberration will be present, 
which requires an additional numerical model to fit the data and retrieve the thermal lens, as done 
by Pfistner et al. [107]. The weakness of this method relies precisely in the retrieving algorithm 
since the whole interferogram consists on only a few fringes. The precision is then a hard point and 
in the best case this precision is evaluated to λ/4. This work [107] has also been done on YAG, 
GSGG and YLF crystals.   
IV.4.2. Shearing interferometric techniques 
A classical solution to obtain information on the phase “between two fringes” is phase shift 
interferometry. This technique has been used by Khizhnyak et al [108] with longitudinally-pumped 
Nd :YAG lasers. This method is easy to implement since it’s based on commercial products, but it 
 71 /115  
remains quite unused due to its important cost. One generally prefers lateral-shearing-interferometer 
methods.  
Methods based on lateral shearing interferometers are particularly well suited to end 
pumping. The principle is the following: the beam is duplicated in several replicas, typically 2 [109-
110],3 [111] or 4 [112-113] (as presented in figures 24 and 25 with a tri-wave lateral shearing 
interferometer setup).  
Figure 24 : Example of tri-wave lateral shearing interferometer (courtesy of J.C. Chanteloup). 
 72 /115  
Figure 25: Example of tri-wave lateral shearing interferometer setup (courtesy of J.C. Chanteloup). 
The replicas are slightly shifted from each other in the lateral direction which provides an 
interferogram whose fringes (for 2 waves) or dots (for 3, 4 waves) separation give information on 
the derivative of the wavefront.  For example in the absence of wave-front distortion the lines are 
rectilinear for the 2-wave shearing interferometer, they form an homogenous honey-comb for the 3-
wave shearing interferometer and perfect squares for the 4-wave shearing interferometer. This 
technique is more sensitive than classical interferometry since the sensitivity is tunable by adjusting 
the shearing distance. The larger the shift, the more precise the technique . The precision of this 
method is excellent since it is in the order of λ/50 [109] and can even reach λ/200 [111].  
The method using 3 (or 4)-wave « trilateral shearing interferometry » [111] has the 
advantage over the 2-wave shearing interferometry to allow the cartography in the 2 dimensions in 
one acquisition.  This method is simple to implement (figure 25) since the splitting of the beam can 
be realized readily with a 3D grating.  Moreover the use of a grating makes this method totally 
achromatic which allows its use for broad-band lasers such as femtosecond lasers. In 1998, 
 73 /115  
Chanteloup et al.  [112] reported on the wavefront distortions of a terawatt-class femtosecond laser 
system with an accuracy of λ/50. 
This method was also used to characterize thermal lensing in Ytterbium-doped materials, 
namely in Yb:YAB by J.L. Blows et al. [110], who used the 2-wave shearing interferometer 
technique to measure thermal lenses, and then thermal conductivities and fractional thermal 
loadings. The experimental setup used is reproduced on figure 26.  
Figure 26: setup for measuring thermal lensing in diode-pumped Yb:YAB crystal with lateral 
shearing interferometry from [110] (courtesy of J. Dawes, Centre for lasers and Applications, 
Sydney) 
The use of a probe beam at 530 nm allowed the thermal lens to be evaluated under lasing 
and nonlasing conditions (that is with a shutter inside the cavity), which is an essential requirement 
to perform radiative quantum efficiencies measurements. 
IV.4.3. Methods based on Shack-Hartmann wavefront sensing 
Although it can be seen also as a multiple beam interferometric method, we separate this method 
from the previously mentioned ones since it can be understood easily with geometrical optics. 
 74 /115  
In 1900, Hartman proposed to use a drilled plate [115] to measure wavefronts. The principle is 
simple: since light rays run perpendicular to the wavefront, one can retrieve the local wavefront 
slope as soon as the direction of the ray is measured. The Hartmann plate is made of small apertures 
which scatter the beam into regularly-spaced diffraction patterns, and behind which is located a 
detector (typically a CCD camera nowadays). In 1971, Roland Shack and Ben Platt improved the 
Hartmann setup by replacing the array of holes by an array of microlenses (figures 27, 26).  
Single unit 
Figure 27: example of intensity pattern observed in the focal plane of the Shack-Hartmann 
microlenses. 
 75 /115  
Rays 
propagationIncident 
Focal length
Microlens
array
Rays 
propagationIncident 
Focal length
Microlens
array
Figure 28: Shack-Hartmann wavefront sensor setup. 
Reconstructed 
wave front
Measured 
local slopes
Input wave 
front
Reconstructed 
wave front
Measured 
local slopes
Input wave 
front
Measured 
local slopes
Input wave 
front
Figure 29: Shack-Hartmann sampling. 
 76 /115  
The small axial shift of the microlens diffraction pattern centroid (see figure 28) is directly 
proportional to the average local slope of the wavefront on the aperture of the micro-lenses. The 
displacement vectors allow having 2D information. The standard sensitivity of such systems is 
typically λ/100 RMS for commercial products, which settles this technique as a competitor of  
lateral shearing interferometry. One of the advantages of this method, compared to interferometric 
ones, is its insensitivity to mechanical vibrations or thermal fluctuations. In counterpart, the 
principal limitation of this kind of sensor is the discrete sampling that limits the transverse 
resolution (figure 29) and thus prevents from obtaining information about high-spatial-frequency 
phase distortions. Nevertheless, it’s noteworthy that this point is not a problem for thermal lensing 
and aberration measurements because in virtue of the general heat equation (II.1.1), temperature 
variations in a crystal are smooth even if the thermal load exhibits sharp variations.  
In 1998, Armstrong [116] reported the measurement of thermal lens in transversally-pumped 
Nd:YAG and Nd:YAP rods. More recently Ito et al. [117] and Pittman et al. [118] reported the use 
of a Hartmann-Shack sensor to measure the thermal lens in Ti:sapphire rods of terawatt-class 
femtosecond lasers.   
In 2001, reports of temporal changes of thermal lens effects on high power pulsed Yb:glass 
lasers have been done by Nishimura et al. using a Shack-Hartmann wavefront sensor [120]. Here 
the sensor was used for its ability to yield real-time (100 Hz sampling rate) estimation of Zernike 
coefficients of aberrations, and allowed to measure characteristic thermal relaxation times. 
Our group reported recently [73] a derivation of Armstrong’s work to measure thermal lensing 
in various diode-pumped Yb:doped crystals, under lasing or nonlasing conditions. The setup 
appears in figure 30. The probe beam was a laser diode at 670 nm coupled in a single mode fiber, 
chosen for its high spatial coherence (an essential feature to correctly define the “reference 
wavefront”) and its low temporal coherence (necessary to avoid coherent cross talk, that is 
interference between two neighbouring microlens diffraction patterns.)  After collimation by a 
 77 /115  
microscope objective, the probe beam was focused onto the crystal and superimposed with the 
pump beam. The crystal was then imaged upon the microlens array using a magnifying relay 
imaging system. An uncoated glass plate was inserted in the pump beam path to reflect the probe 
beam towards the sensor. A selective interference filter at 670 nm was added in front of the sensor 
to eliminate any unwanted signal at the pump or laser wavelengths. A « reference wavefront » is 
recorded when the pump diode is turned off, which includes all static aberrations of the optical 
elements and of the cold crystal itself. It is then subtracted to the measured wavefront when the 
pump is on. Thus, only phase distortions originating in thermal effects are recorded. The phase front 
was then reconstructed by projection over the set of the orthogonal Zernike polynomials [70].   
Figure 30: Experimental setup used to measure thermal lensing in diode-pumped Yb-doped crystals 
with a Shack-Hartmann wavefront sensor, from [73]. 
IV.4.4  Other techniques 
There exists some more marginal and less used techniques to measure thermal lensing. For 
example, it is possible to achieve phase reconstruction basing on Fourier optics. One can show that 
it is possible to retrieve the phase by knowing the intensity profile in different planes linked by 
Fiber-coupled pump 
laser diode 
Fiber-coupled 
Shack-Hartmann sensor 
Interference 
Filter @ 670 nm 
Laser cavity Glass plate 
crystal 
 78 /115  
Fourier transformation. For instance it is quite obvious that a uniform intensity over a circular 
aperture and an Airy pattern appearing in two Fourier-related planes implies that the wavefront 
propagating from one plane to the other is purely spherical. Nevertheless the general inverse 
problem is far from being obvious. Grossard and al. [114] proposed a technique for measuring 
thermal lensing aberrations in a diode-pumped Nd:YVO4 crystal using intensity profiles in 3 planes 
and a complex phase retrieval algorithm derived from Gerchberg and Saxton’s work.      
IV.5. Conclusion 
In conclusion, we made a review of the main methods used to experimentally measure the 
thermal effects in lasers, putting in emphasis the advantages and the limitations of each method. A 
summary of this review is presented in table 4. 
 79 /115  
Methods 
Based on  focal point 
displacement no 
Very 
difficult yes yes yes yes yes 
Very easy / 
very cheap 
Based on cavity 
properties 
no yes Only with 
Almost 
impossible
probe yes yes 
Low  
≈ 15 à 60 %
Difficult in 
general / 
cheap 
« classical » 
Interferometry 
yes Very difficult 
yes in 
theory no no no yes 
σ∆ ∼λ/4 for the 
interferometry 
without phase 
shift. 
Difficult / 
Cheap 
(except phase 
shift interf.) 
lateral shearing 
interferometry yes yes yes yes yes yes 
yes  
for the 3-4-wave 
shearing 
high 
σ∆∼λ/50 
to λ/200 
medium / 
expensive 
reconstruction de la 
phase from different 
position intensity 
profiles 
yes yes yes yes yes yes no unknown Difficult /no commercial 
Hartmann- Shack yes yes yes yes yes yes yes 
high 
σ ∆ ∼λ/100 
easy/ 
expensive 
Table 4: summary of the different techniques used to measure thermal lensing. σ∆ denotes the 
reported RMS deviation on phase shift. 
V. Thermal lensing measurements in ytterbium-doped 
materials: the evidence of a non radiative path  
We will conclude this review by giving some examples of experimental thermal lensing 
measurements in diode-pumped Yb-doped crystals. Most of the examples are taken from our 
previous publications [73,121]: the reader interested in the technical details of the experiments, as 
well as by the theoretical considerations underneath, is invited to refer to these works. 
 80 /115  
V.1. the thermal load in Yb-doped materials  
What are the heat sources in a laser medium ? Following T. Y. Fan and using the same notations 
[8], the fractional thermal load ηh (that is the fraction of the absorbed pump power converted into 
heat) can be written:  
rlph λ
ηηηη 11   (V.1) 
Where: 
- λp, λl, λf are the pump wavelength, the observed laser wavelength, and the mean fluorescence 
spectrum wavelength, respectively; 
- ηp is the pump quantum efficiency, which is the fraction of absorbed pump photons contributing 
to inversion. Non-unity pump quantum efficiency accounts for residual absorption of the undoped 
crystal, or can be related to the presence of nonradiative sites. 
- ηr  is the radiative quantum efficiency for the upper manifold: it represents the fraction of excited 
atoms that decay by a radiative path (in absence of stimulated emission). Non-unity radiative 
quantum efficiency can be related to multiphonon relaxation (although it is very unlikely since a 
large number of phonons are necessary to bridge the 10 000 cm-1 gap separating the excited and 
ground manifolds) or more probably to concentration quenching. The latter phenomenon 
corresponds to the trapping of the energy by a color center, an impurity, or a lattice defect (Yb2+, 
rare-earth impurities or hydroxyl groups have been evoked as possible “dark sites” [122-124]) after 
several transfers of excitation between neighbouring ions.  
- ηl is the laser extraction efficiency, defined as the fraction of excited ions that are extracted by 
stimulated emission. An expression of the laser extraction efficiency can be derived by writing the 
stimulated, spontaneous and nonradiative relaxation rates [73]. In most cases an approximate 
 81 /115  
relation can be used by neglecting reabsorption at laser wavelength. In the latter case we obtain the 
simplified expression: 
≈   (V.2) 
where I is the intracavity laser intensity, and ( )lem λσ  the emission cross section at laser 
wavelength. As shown by eq. (V.2) the laser extraction efficiency tends towards 1 for intracavity 
laser intensities that surpass the laser saturation intensity. Generally, cw oscillators based on Yb-
doped materials work with high reflectivity output couplers: as a consequence the intracavity laser 
intensity is very high, at least one order of magnitude higher than the laser saturation intensity, so 
that ηl is typically close to unity in an operating Yb laser. In this case, the thermal load becomes 
nearly independent on the radiative quantum efficiency, and is only given by the quantum defect. 
Nevertheless, the quantum efficiency directly affects the excited state population and has 
subsequently crucial importance for Q-switched lasers or low repetition rate amplifiers. 
Incidentally, the relation (V.2) also illustrates that the performance of an Yb-based cw oscillator 
becomes nearly independent of the emission cross section at laser wavelength, provided that the 
pump intensity is far higher than the pump saturation intensity.  
V.2. Evidence of nonradiative effects in Yb-doped materials: the example of 
Yb:YAG  
During the past decade, it has been assumed and claimed many times that nonradiative effects did 
not exist in Yb-doped materials, in virtue of the very simple electronic structure that prevented 
deleterious effects that are well known with other ions, such as cross relaxation, excited state 
absorption or upconversion. Nevertheless, all the measurements performed in the past few years 
 82 /115  
have all brought a contradiction to this statement. Blows et al. have demonstrated clear evidence of 
a nonradiative path in Yb:YAB [110],  Ramirez et al. [125] in Yb:MgO:LiNbO3 , and as for YAG, 
non-unity quantum efficiencies have been reported by Barnes et al. [126], Patel et al. [127], 
Ramirez et al. [125] and Chenais et al. [121]. This recent work has also shown the existence of 
nonradiative effects in Yb:GGG, Yb:YSO, Yb:KGW, Yb:YCOB, Yb:GdCOB.  
An example of quantum efficiency measurement thanks to the thermal lens method is shown in 
figure 31 with an Yb:YAG crystal. A simple qualitative explanation is given in figure 32. The clear 
difference between the thermal lens dioptric power under lasing and nonlasing conditions can be 
modelled using eq. (V.1) and eq. (V.2), provided that laser power is measured simultaneously. 
Given some additional approximations, detailed in [121], one can retrieve both radiative quantum 
efficiency and thermo-optic coefficient χ. The results recently reported for different crystals have 
been summarized in table 5. 
 83 /115  
figure 31: Thermal lens dioptric power (here, aberrations of the thermal lens were negligible) 
under lasing and nonlasing conditions. On the right (same graph) the measured laser power, useful 
to compute the laser extraction efficiency. (from [121]) 
laserfluorescence
NON lasing lasing
2 F 5/2 
2 F 7/2 
Non Radiative 
(NR) decay 
Energy 
fluorescence
NR decay  
NON LASING 
LASING
543210
Absorbed pump power (W)
8- at.% Yb:YAG
 TL, non lasing : experiment
 TL, non lasing : model
 TL, lasing : experiment
 TL, lasing : model
 Laser power
 84 /115  
figure 32 : Simple qualitative explanation of the observed difference between thermal lens dioptric 
power under lasing and nonlasing conditions. When the laser is on, stimulated emission short-
circuits the nonradiative path, causing the thermal load to be lower. 
It is seen that for a crystal like YAG, many different techniques have been used (for the details of 
each method, refer to the cited publications), and that a large dispersion of reported quantum 
efficiencies appears. This dispersion tends to assess the conjecture of concentration quenching as 
the nonradiative source in Yb-doped materials. This conjecture has been very alleviated in highly-
doped Yb:YAG samples and was attributed to cooperative processes between two Yb3+ ions 
towards Yb2+ impurities [122]. Owing to the intrinsic nature of concentration quenching and the 
major role played by impurities, it is clear that the radiative quantum efficiency is a parameter that 
pertains to a single given sample, characterized by its doping concentration, the growth technique 
and its associated environment (in particular the nature of the crucible), and of course the degree of 
purity of the compounds.  
 85 /115  
Table 5: Reported values of measured thermo-optic coefficients and radiative quantum efficiencies 
in the literature for different Yb-doped materials. 
crystal Mean 
fluorescence 
wavelength 
λf (nm) 
Thermo-optic 
coefficient (×10-6 K-1)
(measured, under 
diode-pumping 
conditions) 
Radiative 
quantum 
efficiency ηr 
(measured) 
Method used 
 (for quantum 
efficiency 
measurement) 
Reference 
0.70 Thermal lensing 
(Shack-Hartmann) 
[121] 
0.874 photometric [126] 
0.835 calorimetric [126] 
0.97 lifetime [127] 
Yb:YAG 1007 10.0 
(from [121]) 
0.85 Direct temperature 
measurement 
[125] 
Yb:GGG 1013 31 0.90 Thermal lensing 
(Shack-Hartmann) 
[121] 
Yb:GdCOB 1011 6.5 0.71 idem [121] 
Yb:YCOB 1035 17 0.90 idem [121] 
Yb:KGW 993 7.5 0.96 idem [121] 
Yb:YSO 1001 15 0.89 idem [121] 
Yb:YAB 996 Non reported 0.88 Thermal lensing 
(lateral shearing) 
[110] 
V.3. Laser wavelength dependence on the thermal load in Yb-doped broadband 
materials: the example of Yb:Y2SiO5 
 86 /115  
We saw that the fractional thermal load (eq. V.1) is dependant on the operating wavelength of the 
laser. In most practical circumstances however, this dependence is hidden by the fact that the laser 
extraction efficiency also greatly depends on the laser wavelength, since it is linked to the emission 
cross section. The Yb:Y2SiO5 (Yb:YSO [128]) crystal exhibits two maxima of comparable 
amplitude in its emission spectrum, at 1042 and 1058 nm respectively. In addition the output is 
naturally linearly polarized along the crystallophysic axis Y for both wavelengths. It is thus a good 
candidate to put clearly into evidence the influence of laser wavelength on thermal lensing. To 
perform the experiment, we added a SF6 dispersive prism cut at Brewster angle in the collimated 
arm of the three-mirror cavity appearing in figure 30, so that identical laser efficiencies were 
achieved at both wavelengths (2.1 Watts were obtained for 8.5 Watts of absorbed pump power). 
The results are shown in figure 33. It appears clearly that the thermal lens is weaker when the laser 
oscillates at 1042 nm, as expected since quantum defect is lower at this wavelength. The theoretical 
curve derived for the 1058-nm laser oscillation was obtained from the 1042-nm curve by just 
modifying the wavelength and the emission cross section in eq. (V.1) and (V.2), without any 
adjustable parameter (see [121]). This simple experiment shows the interest of multiwavelength 
thermal lensing measurements in broadband materials. Indeed, we have considered here a simple 
formulation of the fractional thermal loading (given by eq. V.1) which gives here satisfactory 
results; but it is possible, from the work of Patel et al. [127] for instance, to derive a more accurate 
expression of the fractional thermal load, which takes into account the probability of excitation 
transfer to a neighbouring ion [129]. If measurements are performed versus absorbed power at 
different (more than 2) laser wavelengths, for which the laser extraction efficiency is also known, 
this means that we have the possibility to infer other spectroscopic parameters (such as the transfer 
probability to a neighbouring ion) involved in the expression of the thermal load. 
 87 /115  
86420
Absorbed pump power (W)
1058 nm
1042 nm
1058 nm
1042 nm
 TL, lasing at 1042 nm : experiment
 TL, lasing at 1058nm : experiment
 TL, lasing at 1042 nm : model
 TL, lasing at 1058 nm : model
5-at. % Yb:YSO 
Figure 33: Thermal lens dioptric power at 1042 and 1058 nm (left) and laser power (right in 
Yb:YSO. The effect of wavelength on quantum defect is clearly visible for this material. (from [121]) 
IV.4. The influence of the mean fluorescence wavelength on the thermal load: an 
illustration with Yb:KGW 
The fact that the mean fluorescence wavelength affects the fractional thermal load is a very 
interesting feature of broadband Yb-doped materials. There are even some materials whose mean 
fluorescence wavelength is below the tail of the absorption spectrum. More precisely, according to 
eq (V.1) it is clear that if we may find a pump wavelength verifying:  
 88 /115  
f p r pλ η η λ<  (V.3) 
then the thermal load will be negative (in absence of laser extraction) and “radiative cooling” will 
be achieved. Such radiative cooling has been reported in Yb-doped ZBLANP glass in 1995 [130] 
and in a KGW crystal by Bowman et al. in 2002 [131]. The latter author has also proposed to use 
this phenomenon to realize a thermal load-free (radiation-balanced) laser [132]. The key idea is to 
correctly adjust the laser intensity so that the spontaneous emission rate, source of cooling 
providing the pump wavelength is long enough, exactly balances the stimulated emission rate, 
source of heating. 
We show here how the mean fluorescence wavelength plays a key role in the interpretation of the 
results obtained when measuring the thermal lens in a Yb:KGW crystal. KGW is now a well-
known crystal, suited for ultrafast laser applications [32-33, 17, 46, 74]. When used with 
wavevector k//c, the polarization-averaged mean fluorescence wavelength is 993 nm while the 
observed laser wavelength is 1030 nm and the pump wavelength tuned to the zero-line absorption 
peak, i.e. 979 nm. Results are shown in figure 34, where it can be seen that unlike previously 
reported results, thermal lensing is actually stronger under lasing conditions. A simple explanation 
is given with the schematic picture appearing in figure 35. The simple model suggested above fits 
well with experimental data and yields a high quantum efficiency for this sample (0,96), consistent 
with the fact that tungstate crystals are grown with the flux method, which is known to carry less 
impurities during growth than the Czochralski technique.  
 89 /115  
figure 34: Thermal lensing measurements in Yb:KGW (from [121]) 
NON LASING 
LASING 
6543210
Absorbed pump power (W)
5- at.% Yb:KGW
 TL, non lasing : experiment
 TL, non lasing : model
 TL, lasing : experiment
 TL, lasing : model
 Laser power
2 F 5/2 
2 F 7/2 
993 nm 1030 nm 
laser fluo fluorescence 
Non lasing lasing 
Energy 
 90 /115  
Figure 35: Qualitative explanation of the paradoxical behaviour of the Yb:KGW crystal. Under 
nonlasing conditions (left), the thermal defect is low due to a low mean fluorescence wavelength 
(993 nm). When laser extraction at 1030 nm becomes the dominant way down for excitation, then 
the quantum defect is higher. This simple picture assumes that ηr = 1. 
IV.5. Conclusion 
As a conclusion for this final part of this paper, we have shown some examples of thermal lensing 
measurements, which allowed us to highlight several points of interest pertaining to Yb-doped laser 
media : 
- all measurements show a difference between thermal lensing dioptric power with and without 
laser action: this provides the proof that in all the crystals under investigation (here YAG, GGG, 
YSO, GdCOB, YCOB, KGW, YAB)  there is a nonradiative return path for excited state 
population. This is all the more detrimental for laser performance in pulsed regime, since in cw 
oscillators the laser extraction efficiency can be large enough to dissimulate this effect. 
- Since Yb-doped materials have broader spectra than their Nd-doped counterparts, thermal lensing 
has two specific properties, which have been illustrated by two experiments: 
1) the thermal load depends on the laser oscillation wavelength: the lower the wavelength and the 
lower the quantum defect; 
2) the mean fluorescence wavelength λf plays a key role. Materials exhibiting a low λf will be less 
sensitive to heating under nonlasing conditions, and in KGW we saw that the thermal lens was 
even higher under lasing action.  
 91 /115  
The peculiarities of Yb-doped materials are not limited to these two latter facts. Chenais et al. [121] 
have reported the appearance of a roll-off in the thermal lens dioptric power at high pumping 
densities, under nonlasing conditions, with several different materials [121]. Furthermore, it has 
been observed by several groups ([110], [124]) a green luminescence that can be related to 
cooperative luminescence.  
The detailed interpretation of these phenomena and their connection to the thermal load are still 
works in progress. 
VI. Conclusion 
Ytterbium-doped materials have brought new prospects and deep changes in the area of high power 
solid state lasers. Associated to new and clever pumping concepts (fiber lasers, thin disk lasers, 
spinning disk lasers...) they are now well-established competitors of Nd-doped materials for high 
power applications. This paper has been an effort to make a review of the recently published works 
dealing with thermal effects in solid state lasers, with a particular scope on the special case of Yb-
doped crystals.  
In the first section we have presented the general properties of Ytterbium-doped media, and 
pointed out the crucial role of the matrix host on the properties of the laser material. The part II was 
devoted to a detailed presentation of the temperature distribution in a diode-pumped Yb-doped 
crystal: how to calculate it and how to measure it. We pointed out the importance of boundary 
conditions, and gave some practical information about the role of the thermal contacts in the 
temperature profile. We have shown that it was possible to easily include pump absorption 
saturation effects and pump beam divergence inside the crystal, exploiting the fact that the heat 
transfer coefficient towards end faces was far smaller than towards edge faces.  
In the third part of this review we focussed on the thermo-optical properties and made a quite 
detailed study of the so-called thermal lensing phenomenon. A comprehensive understanding of 
 92 /115  
this aspect requires a good knowledge about both thermo-mechanical and thermo-optical properties 
of the materials under consideration. This lead us to point out several inaccuracies reported in 
previous works, concerning the calculation of photoelastic constants for isotropic materials, and 
more generally the abusive employment of the dn/dT parameter when it is used to estimate the 
magnitude of the thermal lens: we have shown that the partial derivative of index with temperature 
taken at constant strain is the most appropriate figure, this is because the dn/dT is classically 
measured under experimental conditions that are not consistent with the usual situation of a diode-
pumped crystal under mechanical stress.  We proposed an alternative way to split the thermo-optic 
coefficient into three truly independent terms, and addressed in conclusion a schematic diagram of 
thermal effects showing how all the different apparent consequences (lensing, depolarization, 
strain-induced birefringence, astigmatism, fracture...) are connected to each other.  
Given the high complexity of these thermo-optical phenomena, and the unfeasibility of precise 
calculations as far as all the properties of a crystal are not known (that is the case for the majority 
of laser crystals), we then focussed our attention on thermal lensing measurement techniques, 
which was the topic of part IV.  
We presented in that section a review of what are, to our knowledge, the main different techniques 
that can be employed to measure thermal lensing in end-pumped laser media, and discussed their 
relative advantages and drawbacks.  
Finally, we presented some examples of thermal lensing measurements that have been reported 
recently in Yb-doped crystals. All these measurements agree to find non-unity radiative quantum 
efficiencies for the Yb-doped materials under investigation. This non-ideal behaviour, presumably 
related to concentration quenching, provides contradiction to the general consideration that Yb-
doped materials are totally free of deleterious nonradiative effects. We concluded this review by 
giving some examples of the influence of the laser operating wavelength on the thermal load, as 
well as the influence of the mean fluorescence wavelength. 
 93 /115  
Acknowledgements: We wish to acknowledge all the people involved in the realization of this 
review: at first Romain Gaumé (Stanford University) for fruitful discussions about issues related to 
dn/dT and characterization of materials; Bruno Viana, Gerard Aka and Daniel Vivien from the 
Laboratoire de Chimie Appliquée de l’Etat Solide (Ecole Nationale Supérieure de Chimie Paris, 
France), Alain Brenier and Georges Boulon from the Laboratoire de Physico-Chimie des 
Materiaux Luminescents (Lyon, France) and at last Bernard Ferrand from CEA-LETI (Grenoble, 
France), for growing the crystals used in most of the experimental results reported here. We 
gratefully acknowledge the Ecole Supérieure d’Optique (Orsay, France) for loaning us the Shack-
Hartmann wavefront sensor as well as the infrared camera. The experiments related to our previous 
works have been done under the auspices of the CNRS and the Délégation Générale de 
l’Armement (DGA). We also thank Jean-Christophe Chanteloup and Judith Dawes for allowing us 
to reproduce some figures taken from their works.  
VII. Appendix: Calculation of the photoelastic constants Cr, θ  and 
C’r, θ  using plane strain and plane stress approximations.  
The Cr, θ constants are useful parameters to account for photoelastic effects in the most 
simple cases (optically isotropic crystals or glasses, and parabolic dependence of strain inside the 
crystal).  When Koechner published for the first time the derivation of these constants in 1970, the 
term accounting for thermal dilatation in the generalized Hooke law was omitted [51]. In 1992, 
Cousins pointed out the mistake but did not correct, at that time, the expressions of the constants Cr, 
θ, whose derivation however requires the use of the generalized Hooke law. In the last version of 
 94 /115  
Koechner’s reference book (Solid State Laser engineering, Springer Verlag ed.), the faulty 
equations were actually removed or corrected, but the formulation of the photoelastic constants Cr, θ 
remained unchanged since the first edition. 
Because the generalized Hooke law is used to infer an expression of the constants, it means 
that their expression depends on which assumption has been made about stresses and strains. In 
Koechner’s calculation, the plane strain approximation was made. In the case of end-pumping, it is 
well known however that the plane stress approximation is closer to reality. In this appendix we 
derive the expression of the photoelastic constants, within the framework of the two approximations.  
VII.1. Basic equations  
We restrict our discussion to cubic crystals. In this case, only the elasto-optical coefficients 
p11, p12, p44 are nonzero. These coefficients are given in the [100] system linked to the crystal.  
In optically isotropic materials, the principal axes of strain are given by the cylindrical coordinate 
system axes (radial, tangential, axial). After a change of coordinates one obtains [77] : 
[ ]441211
441211
441211
 (A.1.) 
for the radial index, and : 
 95 /115  
[ ]441211
441211
441211
 (A.2.) 
for the tangential index.  
As stated by Shoji et al.[119], these expressions are valid only for propagation along the [111] axis. 
For YAG we have :  
p11 = - 0.029 
p22 = + 0.0091 
p44 = - 0.0615. 
The strains are related to stresses by the generalized Hooke law, which in our case (whatever 
the approximation we make) writes:  
( )[ ] ( )
( )[ ] ( )
( )[ ] ( )cTrzz
cTzrr
−++−=
−++−=
−++−=
ασσνσ
ασσνσ
ασσνσ
 (A.3.) 
VII.2. Derivation of Cr,θ  using the plane strain approximation (Koechner’s 
case)  
 96 /115  
We consider a homogeneously pumped rod. Only the radial dependence of stresses and 
temperature is considered, since the additional constants represent a constant phase shift which does 
not affect the phase profile.  
We define:  
( )ν116
and  
Q absh
=  . 
The stresses and temperature fields, within the plane strain approximation, are given by 
Koechner [69] :  
( ) cteQSrr
cteQSrr
.cteQSrr
θ   (A.4.) 
( ) .cter
+−= 2
 (A.5.) 
Omitting the constant terms, the generalized Hooke Law yields :  
( )[ ] ( ) 0
ν14ασσνσ
ε 22θ =⎥
−−=++−= r
Trzz  (A.6.) 
which checks the plane strain condition. We have also: 
( )[ ] ( ) 22θ ν134
ν71ασσνσ
ε QSr
−−=++−=   (A.7.)  
 97 /115  
( )[ ] ( ) 22θθ ν14
ν53ασσνσ
ε QSr
−−=++−=  (A.8.) 
The radial index variation is related to the strains by:  
=  (A.9.) 
and the  
are given by (A.1) and (A.2). 
Following Koechner, we can write the index variations under the form :  
nn ,r
,r −=  (A.10.) 
With :  
( )( )
( )1ν48
875ν1 441211
C r  (A.11.) 
( )( )
( )1ν16
3ν1 1211
C  (A.12.) 
For YAG, we find Cr = + 0.020 et Cθ = + 1.77 10-4. The valued computed by Koechner are: Cr = 
0.017 et Cθ = -0.0025 (with the same values taken for pmn and the same Poisson ratio: ν = 0.25). 
The error made by Koechner is about 20%.  
 98 /115  
These coefficients have been extensively used for the purpose of evaluating depolarization losses in 
Nd:YAG. In this latter case, the relevant parameter is [60]: 
( )rB CCC −= θ2
 (A.13.) 
With (A.11) and (A.12), we find:  
( )441211 4148
pppCB +−−
 (A.14.) 
which is coincidentally the same formula obtained by Koechner from incorrect expressions of Cr et 
Cθ.  
VII.3. Derivation of C’r,θ  using the plane stress approximation (end pumping 
case) 
We consider a thin disk, pumped by a top-hat pump beam profile of radius wp  equal to the rod 
radius. This is also true in the conditions defined in Section III.4, provided that we are only 
concerned in the area r < wp and that integrated values of parameters along the crystal length are 
considered. 
Let’s define :   
=     
The temperature distribution is given by:  
 99 /115  
( ) 2' .
T r r cte
= − +  (A.15.) 
where we use Cousins’ notation (see Part III.4) , meaning that the bracketed quantity is integrated 
along the rod length L, and is then homogeneous to the said quantity times a length.  
We have: 
3 ' '
0 ( )
r Q S r Cte
r Q S r Cte
r plane stress
 (A.16.) 
From (A.3) we obtain :  
3 ' '
4 ' '
Q S r
Q S r
Q S r
 (A.17.) 
By analogy with (A.10.), the index shift can be written under the form:  
3 ' 2
n n C r
∆ = −  
which yields, using (A.1), (A.2), and (A.9) :  
( )( )11 12' 1 9 15
ν− + +
=  (A.18.) 
 100 /115  
( )( )11 12 44' 1 7 17 8
p p p
ν− + + −
=  (A.19.) 
For YAG : C’r = + 0.0032 et C’θ = - 0.011.  
The optical indicatrix does not deform the same way compared to the case of the long thin rod: here 
the tangential index becomes greater than n0. An interesting feature is the stress-induced 
birefringence term, defined previously (eq. A.13):  
( )' 11 12 44
C p p p
= − +  (A.20.) 
This expression differs from (A.14) only by a factor of 1-ν (=0.75 for YAG, but also for most 
materials).  
It is of special interest to calculate the average value of the photoelastic constants:  
0.0039
rC Cθ+ = −  
which gives an order of magnitude of the role of photoelastic effect in the thermal lens.  
Here, in the end pumping case, the thermal stresses yield a divergent contribution to the 
thermal lens, whereas it was convergent in the case of side-pumping.  
See subsections 6 to 8 of Section III for a detailed discussion about the use of these 
parameters.  
 101 /115  
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