Thermo-Optic
Effects
Chapter 7
Solid State Lasers
By W. Keochner
Thermo-Optic Effects
Generation of heat in lasers
(a) the energy difference of the photons between the pump band
and the upper laser level is lost as heat to the host lattice; similarly, the
energy difference between the lower laser level and the ground state
is thermalized. The difference between the pump and laser photon
energies, termed quantum defect heating, is the major source of
heating in solid-state lasers.
(b) In addition, nonradiative relaxation from the upper laser level to the
ground state, and nonradiative relaxation from the pump band to the
ground state will generate heat in the active medium.
(c) In flashlamp-pumped systems, the broad spectral distribution of the
pump source causes a certain amount of background absorption by
the laser host material, particularly in the ultraviolet and infrared
regions of the lamp spectrum. Absorption of lamp radiation by impurity
atoms and color centers can further increase heating.
Thermo-Optic Effects
7.1 Cylindrical Geometry

The combination of volumetric heating of the laser
material by the absorbed pump radiation and
surface cooling required for heat extraction leads
to a nonuniform temperature distribution in the
rod.

This results in a distortion of the laser beam due to
a temperature- and stress-dependent variation of
the index of refraction.

The thermal effects which occur in the laser
Material are thermal lensing and thermal stressinduced birefringence.
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Pump Light
Coolant Flow
r0
Coolant Flow
Pump Light
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7.1.1 Temperature Distribution


We consider the case where the heat generated within the
laser rod by pumplight absorption is removed by a coolant
flowing along the cylindrical-rod surface.
With the assumption of uniform internal heat generation and
cooling along the cylindrical surface of an infinitely long rod,
the heat flow is strictly radial, and end effects and the small
variation of coolant temperature in the axial direction can be
neglected.
The radial temperature distribution in a cylindrical rod is obtained
from the one-dimensional heat conduction equation
Where K is the thermal conductivity, and heat is uniformly
generated at a rate Q per unit volume
Thermo-Optic Effects
With the boundary condition
for
where
is the temperature at the rod surface and
the rod, it follows that
is the radius of
The temperature profile is parabolic, with the highest temperature at the
center of the rod.
The heat generated per unit volume can be expressed as
where
rod.
is the total heat dissipated by the rod and is the length of the
The temperature difference between the rod surface and the center is
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The transfer of heat between the rod and the flowing liquid creates a
temperature difference between the rod surface and the coolant. A
steady state will be reached when the internal dissipation Ph is equal
to the heat removed from the surface by the coolant
where h is the surface heat transfer coefficient and TF is the coolant temperature.
Combining with
For the temperature at the center of the rod
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The boundary conditions for the heat transfer coefficient are a
thermally insulated laser rod (h = 0), or unrestricted heat flow from the
rod surface to a heat sink (h = ∞). For cases of practical interest the
heat transfer coefficient is typically around h = 0.5–2Wcm−2 C−1.
6
T(temperature difference)
5
4
3
the maximum temperature at
the center is 114◦ C
2
The temperature gradient
1
between the center of the
crystal and the surface is 57◦
0
0
C
t=0.00015 s
t=0.03 s
t=0.06 s
t=0.1 s
0.5
1
1.5
Radius(m)
2
2.5
3
-3
x 10
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7.1.2 Thermal Stresses
The temperature gradients generate mechanical stresses in the
laser rod since the hotter inside area is constrained from expansion
by the cooler outer zone.
The stresses in a cylindrical rod, caused by a temperature
distribution T(r), can be calculated from the Stress equations.
Radial Stress
Tangential Stress
Axial Stress
where the factor
contains the material parameters
is Young’s modulus, is Poisson’s ratio, and
coefficient of expansion.
is the thermal
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The stress components represent compression of the material
when they are negative and tension when they are positive.
We notice that the stress distributions also have a parabolic
dependence on r
The center of the rod is under compression.
The radial component of the stress goes
to zero at the rod surface, but the
tangential and axial components are in
tension on the rod surface
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the stresses as a function of radius inside the Nd :YAG rod:
Tensile strength of Nd :YAG
is 1800 to 2100 kg/cm2
As the power dissipation is
increased, the tension on
the rod surface increases
and may exceed the
tensile strength of the rod,
thereby causing fracture.
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It is of interest to determine at what power level this will occur
The total surface stress σmax is the vector sum of σφ and σz :
the tension on the surface of a laser rod depends on the
physical constants of the laser material and on the power
dissipated per unit length of the material, but does not depend
on the cross section of the rod.
Thermo-Optic Effects
Stress Fracture Limit
The mechanical properties of the laser host material determine
the maximum surface stress that can be tolerated prior to
fracture.
If σmax is the maximum surface stress at which fracture occurs,
then we can rewrite
Where
is a “thermal shock parameter.”
A larger Rs indicates a higher permissible thermal loading before
fracture occurs.
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7.1.3 Photoelastic Effects
A change of refractive index due to strain is given by a small change
in shape, size, and orientation of the indicatrix. The change is specified
by small changes in the coefficients
where Pi jkl is a fourth-rank tensor giving the photoelastic effect. εkl is
a second-rank strain tensor.
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In a cylindrical coordinate system the photoelastic changes in the
refractive index for the r and φ polarizations are given by
the refractive-index changes are given by
where and
Nd :YAG
are functions of the elasto-optical coefficients of
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7.1.4 Thermal lensing
The change of the refractive index can be separated into a
temperature- and a stress-dependent variation.
where n(r) is the radial variation of the refractive index, n0 is the
refractive index at the center of the rod, and n(r )T, n(r )ε are the
temperature- and stress-dependent changes of the refractive
index, respectively.
The temperature-dependent change of the refractive index
the refractive index shows a quadratic variation with radius r
Thermo-Optic Effects
The focal length of a lens-like medium, where the refractive
index is assumed to vary according to
is given by
This expression is an approximation where it was assumed that
the focal length is very long in comparison to the rod length. The
distance f is measured from the end of the rod to the focal
point.
The total variation of the refractive index is obtained by
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Then
The end effects:
The deviation from flatness of the rod ends is obtained from
where
is the length of the end section of the rod over which
expansion occurs.
With
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The focal length of the rod caused by an end-face curvature is
obtained from the thick-lens formula of geometric optics
where the radius of the end-face curvature is
the focal length of the rod caused by a physical distortion of the
flat ends
The combined the temperature- and stress-dependent variation
of the refractive index and the distortion of the end-face
curvature of the rod lead to
where A is the rod cross-sectional area and Ph is the total heat
dissipated in the rod.
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theoretical and measured
thermally induced back focal
lengths of various laser rods
are plotted as a function of
lamp input.
(A) the radially and (B)
tangentially polarized
beam components, and (C,
D) measurements of average
focal length for different
rods and pump cavities
Thermo-Optic Effects
7.1.5 Stress Birefringence
it was shown that the principal axes of the induced birefringence
are radially and tangentially and that the magnitude of the
birefringence increases quadratically with radius r
a linearly polarized beam passing through the laser rod will
experience a substantial depolarization.
is radial refractive index component
is tangential refractive index component
is the polarization vector for incident radiation
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Radiation incident at point P must be resolved into two
components, one parallel to
and the other parallel to
.
Since
, there will be a phase difference between the
two components and the light will emerge elliptically polarized.
Birefringence effects in pumped laser rods can be studied by
collimated light beam from an HeNe laser
Thermal stresses in a
7.5-cm long and
0.63-cm-diameter
Nd : LaSOAP crystal.
Input power (a)
115W, (b) 450W, (c)
590W, and (d) 880W
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When a birefringent crystal is placed between a polarizer and
analyzer that are parallel, the transmitted intensity is given by
where φ is the angle between the polarizer and one of the principal
birefringence axes and ϕ is the polarization phase shift of the light
emerging from the crystal.
The index difference,
, leads to a phase difference
the difference in optical path length normalized to the wavelength
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As can be seen from this figure,
at maximum lamp input power
of 12kW, the path-length
difference is approximately six
wavelengths.
we can calculate the total transmitted intensity by integrating over
the cross-sectional area of the rod:
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with
the intracavity power which is polarized orthogonal to the polarizer will
actually be ejected from the cavity and represents the depolarization
loss of the resonator
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for a TEM00 mode for which it was assumed
that the beam radius
For the same lamp input power the losses for the TEM00 mode are less
than for a highly multimode beam. This is expected since the energy in
the TEM00 mode is concentrated nearer the center of the rod, where
the induced birefringence is smaller.
Thermo-Optic Effects
The interaction of a linearly polarized beam with a birefringent laser
rod and a polarizer not only leads to a substantial loss in power, but
also a severe distortion of the beam shape.
Output beam pattern for a high-power cw Nd :YAG laser (a)
without and (b) with a Brewster plate in the cavity.
Thermo-Optic Effects
7.1.6 Compensation of Thermally
Optical Distortions Induced
Complete compensation of the thermal aberrations produced by a
laser rod is difficult because:
(a) The focal length depends on the operating conditions of the
laser and changes with pump power and repetition rate.
(b) The thermal lens is bifocal due to the stress-dependent variation
of the refractive index.
(c) Nonuniform pumping leads to nonspherical aberrations.
Thermo-Optic Effects
In many pump configurations pump radiation is more intense at
the center than at the periphery of the rod.
The focal length of a given area in the rod is inversely proportional
to the intensity of the absorbed pumped radiation.
the focal length at the center of the rod is shorter than at the edges.
the thermally induced refractive index profile contains terms that
are higher than quadratic.
A negative lens will remove the quadratic term; however higherorder effects cannot be compensated.
Thermo-Optic Effects
The most common approaches are the insertion of a negative lens
in the resonator
Bifocusing (if thermal lensing compensation is combined with
birefringence compensation) is eliminated and depolarization losses
are minimized in resonators containing polarized beams.
The objective of birefringence compensation is to achieve equal
phase retardation at each point of the rod’s cross section for radially
and tangentially polarized radiation.
This can be accomplished by rotating the polarizations, either
between two identical laser rods or in the same rod on successive
passes, such that the radial and tangential components of the
polarizations are exchanged.
Thermo-Optic Effects
For example, birefringence compensation in an oscillator containing
two identical laser heads can be achieved by inserting a 90◦ quartz
rotator between the laser rods. The rotator produces a 90◦ rotation of
every component of the electric field of the laser beam. The part of a
mode that is radially polarized in the first rod, is tangentially polarized
in the second rod. Since each part of the beam passes through nearly
identical regions of the two rods, the retardation induced by one rod
is reversed by the other.
Thermo-Optic Effects
7.2 Slab and Disk Geometries
The upper and lower surfaces are maintained at a constant
temperature by water-cooling, and the sides are uncooled.
thermal gradients are negligible in the x- and z-directions and the
thermal analysis is reduced to a one-dimensional case, y axis.
Thermo-Optic Effects
The maximum temperature that
occurs between the surface and
the center of the slab (y = d/2) is
given by
where Q is the heat deposition, d
is the thickness, and K is the
thermal conductivity of the slab.
The temperature rise causes stress in
the slab according to
The surfaces are in tension and the
center is under compression
Thermo-Optic Effects
the maximum temperature difference allowed between the surface
and the center before thermal fracture occurs
With
obtains
(thermal-shock parameter) for Nd : glass, one
For slabs of finite width w, the power per unit length at the
stress fracture limit is given by
where
is the aspect ratio of a finite slab.
It is interesting to compare the surface stress of a rod and slab for the
same thermal power absorbed per unit length:
Thermo-Optic Effects
The temperature and stress profile leads to a birefringent cylindrical
lens. The focal lengths of the birefringent lens are
for x and y polarized light, respectively. The parameter
contribution from thermal focusing, that is
the parameters
and
is the
are related to stress-induced focusing
where and
are the stress optic coefficients for stress applied
parallel and perpendicular to the polarization axis.
Thermo-Optic Effects
7.3 End-pumped Configurations
In contrast to transversally pumped systems, heat deposition in
end-pumped lasers is very inhomogeneous.
The very localized heat deposition leads to highly nonuniform and
complex temperature and stress profiles.
Besides the temperature and stress-dependent variations of the
refractive index, the contribution of end bulging to the formation
of a thermal lens can be substantial in end-pumped lasers.
Thermo-Optic Effects
Inhomogeneous local heating and nonuniform temperature
distribution in the laser crystal lead to a degradation of the beam
quality due to the highly aberrated nature of the thermal lens.
An end-pumped laser rod has a temperature profile across the
pumped region which is a function of the distribution of pump
radiation.
Thermo-Optic Effects
An Nd :YAG crystal with, a 15W pump beam from a diode array was
assumed to be focused onto an Nd :YAG rod of 4.75mm radius. The
pump beam, which enters the laser crystal from the left along the zaxis, has a Gaussian intensity distribution and a spot-size radius of
0.5mm in the x-direction. It was assumed that 32% of the incident
pump radiation is converted to heat.
Thermo-Optic Effects
A Gaussian pump beam incident on the crystal has been
assumed
where α0 is the absorption coefficient and wp is the (1/e2) Gaussian
radius of the pump beam. With Ph the fraction of the pump power
that results in heating, the effective focal length for the entire rod
can be expressed by
where K is the thermal conductivity of the laser material and dn/dT
is the change of refractive index with temperature.
Thermo-Optic Effects
An end-pumped Nd :YAG rod with a length of 20 mm and a radius
of 4.8mm was pumped with a fiber-coupled laser-diode array. The
output from the fiber bundle was imaged onto the crystal surface
into a pump spot with radius wp = 340μm.
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