From Equation (3.18) we obtain the thermal lens caused by coating absorption:
From Equation (3.25), one finds
In order to study the consequences of the focusing properties of the thermal lens, we can compute the nearest paraboloid, defined by the apex equation :
where In words, if all averages are performed with the weighting function (i.e., the normalized
beam intensity), then the determination of the best paraboloid amounts to the classical least-squares
formulas:
The differences in thermal lensing between axisymmetric and nonaxisymmetric high-order beams are very
small, because diffusion of heat rapidly produces a quasihomogeneous temperature with respect to the
polar angle. This is why we now restrict the discussions to axisymmetric modes. Existence of a
thermal lens causes a mismatching of the beam, which has passed through the lens,
with the ideal one. The amplitude coupling coefficient is given by the Hermitian scalar product
In Figure 20, we have plotted the evolution of the coupling losses versus the dissipated power on the
coating of a mirror. The solid and dashed curves correspond respectively to the total losses by a numerical
integration of Equation (3.76
) with the overall thermal lens and to harmonic losses. We see that all modes
have almost only harmonic losses for weak dissipated losses (roughly below 100 mW). The anharmonicity
appears soon for the
mode of width 2 cm, and for a fraction of W, nonharmonic losses are
significant. For Ex2 and Ex3, we see that the coupling losses are weaker and practically harmonic. The
curve for
is practically identical to the curve of the flat mode, which is why we did not plot
it.
For weak dissipated power, in the case of LG modes we use the lowest-order approximation of
Equation (3.80), so that we get
The thermal lensing is almost identical for 1 W coating or bulk absorption. In Figure 21, we have plotted
the thermal lens profile and the best fit paraboloid
for the case of heating by internal
absorption.
Some numerical results can be seen in Table 2. The losses are computed from the parabolic
approximation [see Equations (3.83) and (3.85
)] valid for weak dissipated power.
results (coating abs.) | ![]() |
Flat b = 9.1 cm | ![]() |
curvature radius | 328 mW | 9,682 mW | 27,396 mW |
piston | 3.23 ![]() |
1.43 ![]() |
1.08 ![]() |
coupling losses | 3.24 /W2 | 0.53 /W2 | 0.51 /W2 |
results (bulk abs.) | ![]() |
Flat b = 9.1 cm | ![]() |
curvature radius | 317 mW | 9,164 mW | 25,926 mW |
piston | 3.39 ![]() |
1.52 ![]() |
1.15 ![]() |
coupling losses | 3.47 /W2 | 0.59 /W2 | 0.56 /W2 |
Table 3 contains results for some LG modes having parameters tuned for 1 ppm clipping losses on
a 35 cm diameter mirror.
order ![]() |
w [cm] | ![]() |
L [W2] | ![]() |
L [W2] |
(0,0) | 6.65 | 4,400 | 2.20 | 4,184 | 2.43 |
(0,1) | 5.56 | 8,566 | 1.42 | 8,139 | 1.57 |
(1,0) | 6.06 | 8,130 | 0.89 | 7,713 | 0.99 |
(0,2) | 4.93 | 12,113 | 1.14 | 11,497 | 1.27 |
(1,1) | 5.23 | 11,608 | 0.97 | 11,006 | 1.08 |
(2,0) | 5.65 | 11,414 | 0.51 | 10,822 | 0.57 |
(0,3) | 4.49 | 14,870 | 1.00 | 14,106 | 1.11 |
(1,2) | 4.70 | 14,430 | 0.92 | 13,677 | 1.02 |
(2,1) | 4.97 | 14,499 | 0.70 | 13,736 | 0.78 |
(3,0) | 5.35 | 14,484 | 0.34 | 13,729 | 0.38 |
(0,4) | 4.17 | 17,219 | 0.91 | 16,328 | 1.01 |
(1,3) | 4.32 | 16,731 | 0.87 | 15,855 | 0.97 |
(2,2) | 4.52 | 16,889 | 0.73 | 15,997 | 0.82 |
(3,1) | 4.76 | 17,237 | 0.53 | 16,324 | 0.59 |
(4,0) | 5.11 | 17,368 | 0.25 | 16,462 | 0.27 |
(0,5) | 3.91 | 19,117 | 0.85 | 18,123 | 0.95 |
(1,4) | 4.03 | 18,686 | 0.82 | 17,705 | 0.92 |
(2,3) | 4.18 | 18,787 | 0.74 | 17,792 | 0.82 |
(3,2) | 4.36 | 19,204 | 0.60 | 18,182 | 0.67 |
(4,1) | 4.58 | 19,790 | 0.42 | 18,738 | 0.46 |
(5,0) | 4.91 | 20,104 | 0.19 | 19,055 | 0.21 |
The difference between the flat beam and the mesa beam, regarding thermal lensing, is shown
in Figure 22. One sees that using the crude flat beam yields a small overestimation of the
lensing.
http://www.livingreviews.org/lrr-2009-5 | ![]() This work is licensed under a Creative Commons License. Problems/comments to |