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 is the curvature parameter, related to the mean curvature radius of the lens by . is called a piston. We define the nearest paraboloid by requiring the new lens to carry out the best correction to the wavefront distorted by . If is the normalized mode amplitude, the Hermitian scalar product of the perfect incoming field with the distorted-corrected one is where ( being the laser wavelength). For a small difference, this is, at second order, The first integral represents a phase that can always be cancelled out by a suitable choice of . The second integral represents the coupling loss due to imperfect correction of the wavefront. Parameters and are found by requiring a minimum loss. If we define the average of any function byIn 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

where the average is weighted, as usual, by the normalized intensity of the readout beam. Mismatching results in turn in coupling losses due to the distortion of the wavefront of the transmitted beam. These distortions have been summarized by a spurious radius of curvature for which we have given formulas in the preceding paragraph (??). But it is clear from the shape of the lenses that the distortion is not parabolic. In general, there is some departure of the wavefront from a parabola. Thus, we have two contributions to coupling losses: a harmonic part (by reference to the harmonic oscillator potential) and a nonharmonic part. It is possible to compare the two contributions. For the parabolic or harmonic part, we have the following coupling coefficient for a spurious curvature radius : In the case of LG modes, we can be more specific: Assuming negligible clipping losses, we can replace the upper integration bound by and take: (with ) so that we obtain In the case of a flat mode, the integral is trivial and we get: while the power coupling losses are simplyIn 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

We note that, owing to Equation (3.56), this is nothing but a variance of (weighted by the intensity profile) Now for the flat mode, we get which is again NB: The curvature radii are expressed in m.W, the losses in WThe 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.) | w = 2 cm | Flat b = 9.1 cm | w = 3.5 cm |

curvature radius | 328 mW | 9,682 mW | 27,396 mW |

piston | 3.23 m/W | 1.43 m/W | 1.08 m/W |

coupling losses | 3.24 /W^{2} |
0.53 /W^{2} |
0.51 /W^{2} |

results (bulk abs.) | w = 2 cm | Flat b = 9.1 cm | w = 3.5 cm |

curvature radius | 317 mW | 9,164 mW | 25,926 mW |

piston | 3.39 m/W | 1.52 m/W | 1.15 m/W |

coupling losses | 3.47 /W^{2} |
0.59 /W^{2} |
0.56 /W^{2} |

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] | [mW] (coat. abs.) | L [W^{2}] |
[mW] (bulk abs.) | L [W^{2}] |

(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.

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