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Kim Larsen
November 4, 2008
OPTI 521R
Technical Paper Synopsis
Optical distortion coefficients of high-power laser windows by Claude Klein
Low intensity light incident on a surface will increase its temperature, but usually not
enough to significantly deform the object due to its thermal conductivity dissipating the heat
such that no significant thermal gradient can develop. However, high-powered lasers are so
intense that even for a material with low absorption, the laser beam incident on the material can
cause a thermal gradient. For windows for laser systems, the intensity profile of the laser will
cause a thermal gradient through the window, affecting the optical path length of different parts
of the laser. This effect, called “thermal lensing” causes wavefront aberrations in the laser beam.
Claude Klein describes the well-known analysis under some assumptions in Optical distortion
coefficients of high-power laser windows and derives the following “figure of merit” for the
distortion (FOM)d depending on known material properties for simple geometries is shown in
Equation 1 below,
( FOM ) d 
Cp '
(1)
 V  eff
where Cp’ is the heat capacity per unit volume, βV is the bulk absorption coefficient, and χeff is
the effective optical distortion coefficient, defined in Equation 13a.1 The following summary
will describe the determination of the effective optical distortion coefficient and the figure of
merit’s relation to the Strehl ratio.2
The determination requires a number of assumptions to derive a compact expression, but
that are reasonable for a common laser windows. The assumptions include that the thermal
loading on the window is axially symmetric, the window is isotropic with respect to its elastic
and photoelastic properties in the plane parallel to the surface, and the stresses follow
distributions that are either “thin-plate” type or “long-rod” type. The incident beam is assumed
to be Gaussian causing radially dependent wavefront distortions and that prior to thermal
loading, the window is assumed to be at a uniform temperature and birefringence free. The
“short-time” assumption also allows for ignoring thermal diffusion effects.
From the assumption of radially dependent effects, the stress in the window can be
characterized by radial and azimuthal components, σρ and σθ, which then causes the phase shifts
for light polarized in the radial and azimulthal directions described as δφρ and δφθ. Assuming
weak distortional effects, the phase delay for each polarization direction can be approximated to
the second order and then the Strehl ratio can be approximated as Equation 2 below,
s  1  12   
2
1
2
2
 14   
2
(2)
All other quantities are defined in the appendix.
Klein, Claude A. Optical distortion coefficients of high-power laser windows. Optical Engineering Vol. 29. No. 4.
1990
where the angled brackets denote an average over the aperture of the window.3 This leads to the
definition of new phase variables δφ+ and δφ-, to take advantage of the symmetry in the Strehl
ratio, whose definitions are shown below in Equations 3 and 4.
  12 (   )
(3)
  12 (   )
(4)
An advantage of this formulation is that δφ- is only non-zero if stress-induced birefringence
occurs4, allowing separation of the effects of stess-induced birefringence.
In order to determine the phase delay induced by the laser, the change in the index of
refraction caused by both its dependence on temperature and the stress induced by the thermal
expansion of the window need to be considered. The light travels axially, therefore the axial
length is assumed to be the dominant effect in determining the path length. Also, the amount of
heat deposited is dependent on the axial length due to absorption. In a first-order approximation,
the total index change in axial direction is sum of a stress-free, temperature dependent term and a
stress dependent term shown in Equation 5 below.
 n 
n    T  nstress
 T  0
(5)
Utilizing the linear approximations for axial strain including both thermal expansion and stressstrain5 in all directions, the phase shifts are found to be proportional to two aberration
coefficients χ+ and χ- dependent on stress geometry. The coefficients were calculated for two
common stress models, the plane-stress “thin-disk” geometry and the plane-strain “long-rod”
geometry. The distortion coefficients for the “thin-disk” geometry are shown below.
 
dn
n 3E
 (n  1) (1   ) 
(q||  q )
dT
4
(6)
n 3E
 
(q||  q )
4
(7)
The coefficients for the “long-rod” geometry follows.
 
dn n 3E / 4

(q||  3q )
dT
1 
(8)
1
3
The average of a quantity Y over the aperture is given by
Y
 Y (  ) I (  ) d .

 I (  ) d
0
1
0
4
5
If the phase delay of both polarizations directions δφρ and δφθ are equal, there is no birefringence in the window.
From photo-elastic effects of the temperature increase.
 
n 3E / 4
(q||  q )
1
(9)
The equations above show that the coefficients are dependent on linear combinations of the
stress-optic coefficients6 for the directions normal and parallel to the surface. Equations 6-9
show that the existence of χ- is dependent on the presence of birefringence due to stress, and that
χ+ in the “thin-disk” geometry has a term dependent on the bulging of the disk that is absent from
χ+ in the “long-rod” geometry, as it has been shown experimentally. In both geometries, the
phase variable will vary linearly with the intensity profile as shown in Equations 10 and 11
below.
  k  LT    independent terms  I (  )
(10)
  k  L(T  T )  I (  )
(11)
Using these results to rewrite the Strehl ratio under the condition that the stress induced
birefringence has a minor effect (χ- << χ+), the Strehl ratio for weak distortions can be described
in terms of an effective polarization distortion coefficient shown in Equations 12 and 13 below:
 kF (W )  app L eff I o t 
s  1 

Cp '


 eff
2

   
|   | 1  G  

    
2
1/ 2
, G
(12)
(T  T ) 2
T 2  T
2
(13a, b)
where G and F(W) are beam intensity profile dependent quantities, dependent on the beam
truncation parameter W.7 The Strehl ratio is also dependent on the profile of beam, specifically
the variance in the change in temperature over the aperture. For a Gaussian beam with a
reasonable truncation, G is approximately 0.75. Since the temperature increases the more time
the beam is incident on the surface, a useful metric for determining the quality of the window
would be the time until a noticeable drop in the quality of the beam, which is when the Strehl
ratio drops to 0.80, which can be easily found to be Equation 14 below,
t
Cp '
0.2
1


F (W ) kI0 L  app  eff
(14)
6
The stress optic coefficients are composed from piezo-optic tensor elements qij, material properties dependent on
crystal structure and propagation direction. q|| , q   aij qij , where aij are scalar coefficients
The derivation of δT involves separating the time and radial components of the change in temperature to compute
the average effects, while ignoring surface cooling and thermal conduction under the short time approximation.
Specifically, F(W) is related to the radial dependence of the temperature increase.
7
which breaks up the expression into a beam profile dependent term, a beam and path dependent
term, and the figure of merit for the window defined in Equation 1. The only approximation to
completely determine the figure of merit is to assume that the surface absorption effects are
significantly smaller than the bulk absorption through the material due to coatings (2βs << βVL),
so that the apparent absorption is just the bulk absorption (βapp ≈ βV)8.
Analysis of Equation 14 shows how quickly the beam quality will degrade for a given set
of system quantities, and the third term, as described in Equation 1, can be used to determine the
benefits and drawbacks of window material choices, at least in a first order approximation. As
expected, the approximate time to beam degradation increases with lower absorption and lower
distortion sensitivity and higher heat capacity. However, since there were many assumptions
made, these derivations are rough and only a way to determine approximate relationships
between different system variables, specifically the material properties of the window and a way
to compare their relative beam degradation.
Since high powered laser systems will not just heat windows, but also other optical
elements, similar derivations have been done to determine other system design choices. If the
laser is not pulsed, but a CW laser, thermal distortion even with the best materials will be too
high, a system involving cooled windows is required and the choice between side and plane
cooled windows is examined in High Powered CW Laser Windows: Side-Cooled or FaceCooled? also by Klein.9 Similar thermal effects will occur with other optical elements, and
Klein also discusses mirror faceplate materials in high powered systems in High-Power LaserMirror Faceplate Materials: Figures of Merit for Optical Distortion10 although he has another
paper on the subject, specifically for reflectors.11 In particular, Actively cooled silicon mirrors
by Anthony and Hopkins12 detail the advantages of silicon mirrors over molybdenum mirrors in
high power CW laser systems. As for more information on different type of windows for special
applications, Optical Windows by Barnes13 details some uncommon environments and describes
their impact on optical performance.
8
The apparent absorption coefficient is defined as  app  V  2 S .
L
9
Klein, Claude A. High Powered CW Laser Windows: Side-Cooled or Face-Cooled? Proc. of SPIE Vol. 1739, High
Heat Flux Engineering, ed. Feb 1993.
10
Klein, Claude A. High-Power Laser-Mirror Faceplate Materials: Figures of Merit for Optical Distortion. Proc.
of SPIE Vol. 3151, High Heat Flux and Synchrotron Radiation Beamlines, ed. Dec 1997.
11
Klein, Claude A. Mirror figure-of-merit and material index-of-goodness for high power laser beam reflectors.
Proc. of SPIE Vol. 0288, Los Alamos Conference on Optics ’81, ed. Jan 1981.
12
Anthony, F. M, and Hopkins, A. K. Actively cooled silicon mirrors. Proc. of SPIE Vol. 0279, Emerging Optical
Materials, ed. Jan1981.
13
Barnes, W. P. Optical Windows. SPIE Critical Review Vol. CR43, Optomechanical Design, ed. Yoder, Jr. July
1992.
Appendix:
Coefficient Definitions
βV, βS – bulk, surface absorption coefficients
Cp’ – heat capacity per unit volume
χeff – effective optical distortion coefficient
s – Strehl ratio
σρ, σθ, σz – stress components: radial, azimulthal, and axial
δφρ, δφθ – phase delay, radial and azimuthal
δφ+, δφ- – phase delay variables
n – index of refraction before thermal loading
dn/dT – change in index of refraction due to temperature without the presence of stress
α – coefficient of thermal expansion
E – Young’s Modulus
q|| , q  – parallel and perpendicular stress-optic coefficients
qij – piezo-optic tensor elements
ν – Poisson’s ratio
χ+, χ- – optical distortion coefficients
k – laser wavenumber
L – window thickness
I(ρ) – laser beam intensity profile as a function of radius
I0 – peak laser intensity
δT – axial temperature increase
T – mean temperature increase to radial position ρ