Higher-order Laguerre-Gauss interferometry for gravitational-wave detectors with in situ mirror defects compensation

The use of higher-order Laguerre-Gauss modes has been proposed to decrease the influence of thermal noise in future generation gravitational-wave interferometric detectors. The main obstacle for their implementation is the degeneracy of modes with same order, which highly increases the requirements on the mirror defects, beyond the state-of-the-art polishing and coating techniques. In order to increase the mirror surface quality, it is also possible to act in situ, using a thermal source, sent on the mirrors after a proper shaping. In this paper we present the results obtained on a tabletop Fabry-Pérot Michelson interferometer illuminated with a ${\mathrm{LG}}_{3,3}$ mode. We show how an incoherent light source can reduce the astigmatism of one of the mirrors, increasing the quality of the beam in one of the Fabry-Pérot cavities and then the contrast of the interferometer. The system has the potential to reduce more complex defects and also to be used in future gravitational-wave detectors using conventional Gaussian beams.

Figure 1

Scheme of the experimental setup.

Figure 2

Normalized intensity distribution recorded at the mode cleaner output (a) and theoretical ${\mathrm{LG}}_{3,3}$ distribution at the same distance from waist (b).

Figure 3

${\mathrm{LG}}_{3,3}$ measured transverse intensity distributions at (a) cavity transmission, (b) reflection and (c) at the interferometer output when the cavity is at resonance and the Michelson is set on the dark fringe. The window side is 6 mm (same color scale for the three figures).

Figure 4

${\mathrm{LG}}_{3,3}$ simulated transverse intensity distributions at (a) cavity transmission, (b) reflection and (c) at the interferometer output when the cavity is at resonance and the Michelson is set on the dark fringe. The window side is 6 mm (same color scale for the three figures).

Figure 5

Transmitted beam intensity decomposition in terms of the ten polynomials of 9th order.

Figure 6

Profile of mirror deformation induced by the projection of a round spot and 8 mW of absorbed power. Comparison between simulation (blue line) and experimental results (red stars).

Figure 7

(a) Pattern imaged on the mirror and used as an absorption map in the COMSOL simulation. (b) Simulated mirror deformation obtained with heat pattern (a) for an absorbed intensity of $550\mathrm{W}/{\mathrm{m}}^2$. The color bar is in meters.

Figure 8

${\mathrm{LG}}_{3,3}$ transverse intensity distributions measured (a) and simulated (b) at cavity transmission. Left column: reference condition (presented in Sec. 3). Central column: astigmatism enhancement. Right column: astigmatism reduction. The color scale is the same for the three configurations.

Figure 9

${\mathrm{LG}}_{3,3}$ transverse intensity distributions measured (a) and simulated (b) at cavity reflection. Left column: reference condition (presented in Sec. 3). Central column: astigmatism enhancement. Right column: astigmatism reduction. The color scale is the same for the three configurations.

Figure 10

${\mathrm{LG}}_{3,3}$ transverse intensity distributions measured (a) and simulated (b) at the interferometer output when the cavity is at resonance and the Michelson is set on the dark fringe. Left column: reference condition (presented in Sec. 3). Central column: astigmatism enhancement. Right column: astigmatism reduction. The color scale is the same for the three configurations.

Figure 11

Transmitted beam decomposition in modes of order 9. The astigmatism thermal correction reduces the scattering of ${\mathrm{LG}}_{3,3}$ into modes ${\mathrm{LG}}_{2,5}$ and ${\mathrm{LG}}_{4,1}$ while more power is coupled in the ${\mathrm{LG}}_{3,3}$ mode. On the other hand, couplings into modes ${\mathrm{LG}}_{2,5}$ and ${\mathrm{LG}}_{4,1}$ are enhanced when mirror astigmatism is increased, and less power is found in the ${\mathrm{LG}}_{3,3}$ mode.

Figure 12

Amplitude of the coefficients relative to the mode ${\mathrm{LG}}_{4,1}$ while varying the heating power. A linear fit shows that having about twice the power from the compensation system allows to completely correct astigmatism.