Perspectives on beam-shaping optimization for thermal-noise reduction in advanced gravitational-wave interferometric detectors: Bounds, profiles, and critical parameters

Suitable shaping (in particular, flattening and broadening) of the laser beam has recently been proposed as an effective device to reduce internal (mirror) thermal noise in advanced gravitational-wave interferometric detectors. Based on some recently published analytic approximations (valid in the infinite-test-mass limit) for the Brownian and thermoelastic mirror noises in the presence of arbitrary-shaped beams, this paper addresses certain preliminary issues related to the optimal beam-shaping problem. In particular, with specific reference to the Laser Interferometer Gravitational-wave Observatory (LIGO) experiment, absolute and realistic lower bounds for the various thermal-noise constituents are obtained and compared with the current status (Gaussian beams) and trends (mesa beams), indicating fairly ample margins for further reduction. In this framework, the effective dimension of the related optimization problem, and its relationship to the critical design parameters are identified, physical-feasibility and model-consistency issues are considered, and possible additional requirements and/or prior information exploitable to drive the subsequent optimization process are highlighted.

Figure 1

Problem schematic: A perfectly symmetric Fabry-Perot optical cavity composed of two nearly flat mirrors [with profile $h(r)$] attached on cylindrical test masses of radius a separated by a distance $L$ along the $z$ axis. (a): Side view. (b): Front view.

Figure 2

Optimal (minimum-noise) compact-support beam profiles in (21). Continuous curve: substrate Brownian ($q=-1$); dashed curve: coating ($q=0$); dotted curve: substrate thermoelastic ($q=1$).

Figure 3

PSWF eigenvalues (magnitude) as a function of order $m$, for various values of $N_D$. The semilog scale highlights the step behavior with exponential tail (see the discussion in Sec. 3b2). Squares: $N_D=1$; circles: $N_D=5$; up-triangles: $N_D=10$; down-triangles: $N_D=20$.

Figure 4

PSWF profiles for $N_D=12$ and various orders (solid curves). (a): $m=0$; (b): $m=2$; (c): $m=5$; (d): $m=8$; (e): $m=10$; (f): $m=13$. Also shown, as reference (dashed curves), is the behavior of the infinite-mirror (LG-type) solutions.

Figure 5

Truncation index $M_T$ in the PSWF expansion [cf. (32)] as a function of number of electromagnetic degrees of freedom $N_D$.

Figure 6

Realistic noise bounds obtained from band-limited beam profiles [cf. (28)] and truncation index $M_T$ as in (32) [cf. Fig. 5], as a function of $N_D$. (a): Substrate Brownian ($q=-1$); (b): Coating ($q=0$); (c): Substrate thermoelastic ($q=1$). Also shown, as references, are the corresponding mirror-radius scale (top axis, assuming $L=4\mathrm{km}$ and $\lambda =1064\mathrm{nm}$), the absolute bounds [dotted lines, cf. (22)], and the noise values for $a=16\mathrm{cm}$ (i.e., $N_D=12.03$).

Figure 7

As in Fig. 6, but representative band-limited beam profiles for various values of $N_D$. Continuous curve: $N_D=6.77$ ($a=12\mathrm{cm}$); dashed line: $N_D=13.58$ ($a=17\mathrm{cm}$); dotted line: $N_D=22.74$ ($a=22\mathrm{cm}$).

Figure 8

Relationship between the MB parameters ${\overline{R}}_0$ and ${\overline{w}}_0$ arising from the clipping-approximated diffraction-loss constraint (${\mathcal{L}}_T=1\mathrm{ppm}$), for various values of $N_D$ (assuming $L=4\mathrm{km}$ and $\lambda =1064\mathrm{nm}$). Squares: $N_D=6.77$ ($a=12\mathrm{cm}$); circles: $N_D=9.21$ ($a=14\mathrm{cm}$); up-triangles: $N_D=12.03$ ($a=16\mathrm{cm}$); down-triangles: $N_D=15.23$ ($a=18\mathrm{cm}$); diamonds: $N_D=18.80$ ($a=20\mathrm{cm}$); stars: $N_D=22.74$ ($a=22\mathrm{cm}$). White bullets mark the minimum-spreading configurations [cf. (40)].

Figure 9

As in Fig. 8, but coating ($q=0$) noises as a function of ${\overline{w}}_0$ (scaled to its minimum-spreading value).