On Special Optical Modes and Thermal Issues in Advanced Gravitational Wave Interferometric Detectors
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Abstract
The sensitivity of present groundbased gravitational wave antennas is too low to detect many events per year. It has, therefore, been planned for years to build advanced detectors allowing actual astrophysical observations and investigations. In such advanced detectors, one major issue is to increase the laser power in order to reduce shot noise. However, this is useless if the thermal noise remains at the current level in the 100 Hz spectral region, where mirrors are the main contributors. Moreover, increasing the laser power gives rise to various spurious thermal effects in the same mirrors. The main goal of the present study is to discuss these issues versus the transverse structure of the readout beam, in order to allow comparison. A number of theoretical studies and experiments have been carried out, regarding thermal noise and thermal effects. We do not discuss experimental problems, but rather focus on some theoretical results in this context about arbitrary order LaguerreGauss beams, and other “exotic” beams.
Keywords
Thermal Noise Thermal Lens Bessel Beam Coupling Loss Rear Face1 Introduction
Gravitational waves (GWs) are a prediction of Einstein’s general theory of relativity, which extends the theory of gravitation by renouncing the instantaneous action at a distance that was shocking to Isaac Newton himself and had already became unacceptable after the special theory of relativity. The gravitational interaction is now carried by a wave messenger at the speed of light, a gravitational wave. However, the efficiency of the conversion of any kind of energy into gravitational radiation is extremely weak, so that emitting and/or detecting such waves has for decades been considered well outside experimental possibilities. The situation changed after the technological expansion in the 1960s. Joseph Weber was the first to propose an experiment aiming to detect GWs of astrophysical origin. However, the initial Weber experiment was still too simple to detect anything of astrophysical interest. This motivated theorists to work out more accurate estimates of the GW signals produced by astrophysical cataclysms such as supernovae, binary coalescences, fast spinning neutron stars etc… Readers interested in this part of the history of the field can consult the review by Thorne [37]. Is was soon noted that optical interferometers of the Michelson type had exactly the right topology with respect to the gravitational wave polarization, had a large potential sensitivity and were able to produce electrical signals analog to the gravitational waveforms, being intrinsically wideband transducers. After several prototypes of various sizes were built (U.S.A., U.K., Germany), and following the pioneering work of Ronald Drever of Caltech, Rainer Weiss of MIT was the first to study the technological issues specific to large size interferometric antennas and to determine the general principles of ground based antennas. This was the seed of the LIGO project in the U.S.A. [27], of the BritishGerman GEO, unfortunately aborted, and of the FrenchItalian Virgo [41]. Despite these efforts and after construction of kilometer size antennas, GWs have yet to be detected because of the still too low sensitivity of present antennas (LIGO, Virgo). It was foreseen from the beginning that technological breakthroughs would allow the sensitivity to be enhanced in the near future. This is the present situation, and the R&D of “advanced detectors” has already begun. One aspect of these advanced detectors is an improved use of light for reading the tiny apparent variations of distances between test masses.
Groundbased interferometers for GW detection are made of silica pieces (the substrates of the mirrors) hanging in a vacuum. Detection of GWs requires the continuous measurement of the flight time of photons between two mirrors facing each other, or, in other words, the reflected phase off a FabryPérot cavity. A passing GW is expected to have a differential effect on the phases of two orthogonal cavities. This is why the Michelson configuration is well adapted to GW antennas. It is classically shown that the sensitivity of a Michelson interferometer ultimately depends on the square root of the light source’s power. This is a strong reason to increase the input laser power. However, there are at least four issues to solve before such an improvement can be made. The first is that, even with high quality materials, a fraction of the power is absorbed by the material (either in the bulk or on the coating); this gives rise to a source of heat at the surface or in the bulk, and there is consequently a temperature field in the material, which results in turn in a refractive index field and a thermal distortion of the substrate. These defects cause mismatching of the interferometer, and therefore, already in the present status of LIGOVirgo, require complex thermal compensation systems. Before increasing the incident power, some new ideas would be welcome. The second issue comes from the fact that in the region of 100 Hz, the sensitivity is not limited by shot noise, but rather by the thermal noise of mirrors. Mirror substrates may be viewed as elastodynamical oscillators, whose modes are excited at room temperature resulting in a fluctuating reflecting face. Increasing the laser power will be of no use in this strategic spectral region unless a means of reducing the effect of thermal noise is found. There is still another source of noise, called thermoelastic, due to temperature fluctuations in the material. The fourth issue is the effect of radiation pressure on the suspended mirrors. Increasing the laser power will cause increasing fluctuations in the radiation pressure, so that there is an optimum in the laser power, dependant on the cavity parameters, giving the standard quantum limit. In the context of the R&D of advanced detectors, several ways of reducing the thermal noise have been proposed: using new highQ materials [34], cooling to cryogenic temperatures [39] or active correction [10]. Changing the geometry of the readout beam, in order to reduce the optical coupling with surface fluctuations, has also been proposed. Regarding this track, there was a proposal [11, 40] to go towards nonspherical mirrors generating a moreorless homogeneous lightintensity profile. There was another proposal [31] in the same spirit but keeping spherical mirrors and using highorder Gaussian modes. Some other proposals are also considered.
Thermal effects, the various thermal noises (Brownian, thermoelastic, thermorefractive) have been extensively studied and reported in the literature. We focus here on their dependence to the transverse structure of the optical readout beam and try particularly to give general formulas for arbitrary order LaguerreGauss modes.
Further material will be added to the present review with coming developments, especially regarding experimental results. But we think it is useful to present already available results during the present R&D phase.
2 Modes of Fabry—Pérot Cavities and Readout Beams
It is well known that a FabryPérot cavity has eigenmodes corresponding to eigenfrequencies determined by its length and geometries defined by the shape of the mirrors. It is also well known that cavities with spherical mirrors are resonant for HermiteGauss or LaguerreGauss modes. Recently more exotic shapes have been proposed in order to reduce the thermal noise.
2.1 LaguerreGauss beams
2.2 Mesa and flat beams
We choose the parameters w_{0} and b in order to have 1 ppm clipping losses. It is possible to reduce clipping losses either by a smaller w_{0} or by a smaller b. However, reducing w_{0} too much leads to distorted wavefronts and unfeasible mirrors. We have found a possible compromise with w_{0} = 3.2 cm and b_{ f } = 10.7 cm, giving exactly 1 ppm clipping losses on both 35 cm diameter mirrors.
Throughout the following discussions, we shall numerically treat three examples. The first, “Ex1”, is the current situation for the Virgo input mirrors, i.e., an LG_{0,0} mode of w = 2 cm. The second, “Ex2”, is the flat mode described above of b = 9.1 cm, or, when needed, the mesa mode with b_{ f } = 10.7 cm (1 ppm clipping loss). The third, “Ex3”, is the LG_{5,5} mode of w = 3.5 cm (1 ppm clipping loss). However, the analytic expressions are general.
2.3 Other exotic modes
Several other types of modes have been or could be proposed in the same spirit of reducing the Brownian thermal noise and/or the thermoelastic noise.
2.3.1 Bessel beams
2.3.2 Conicalmirror or GaussBessel beams
Bondarescu et al. [4] have carried out an optimization of coating thermal noise by combining LG modes. Using the better series of coefficients, they reach a wave analogous to a GaussBessel mode and with a conical wavefront of the same kind. We intend to include these kinds of modes in an update to this review. To be specific, we give, in the section related to coating thermal noise (8.3.2), the figure of merit of the mode described in Figures 5 and 6, which is not optimal, but already exhibits a good value, regarding coating thermal noise in the infinite mirror approximation (see Section 8.3).
3 Heating and Thermal Effects in the Steady State
3.1 Steady temperature field
3.1.1 Coating absorption
3.1.2 Bulk absorption
3.1.3 FourierBessel expansion of the readout beam intensity
3.1.4 Numerical results on temperature fields
Physical constants used in this paper
Symbol  Parameter  Value  units 

a  half diameter  0.175  m 
h  thickness  0.1  m 
ρ  density  2,202  kg m^{−3} 
K  thermal conductivity  1.38  Wm^{−1} K^{−1} 
C  specific heat cap.  745  Jkg^{−1} K^{−1} 
α  thermal expansion coef.  5.4 × 10^{−7}  K^{−1} 
β  linear absorption  10^{−5}  m^{−1} 
Y  Young’s modulus  7.3 × 10^{10}  Nm^{−2} 
σ  Poisson ratio  0.17  dimensionless 
d n/dT  thermal refractive ind.  1.1 × 10^{−5}  K^{−1} 
3.2 Steady thermal lensing
3.2.1 Thermal lensing from coating absorption
3.2.2 Thermal lens from bulk absorption
3.2.3 Equivalent paraboloid
3.2.3.1 Averaging with LG modes
3.2.3.2 Averaging with flat modes
3.2.4 Coupling losses
3.2.5 Numerical results
Thermal lensing from coating and bulk absorption (abs.)
results (coating abs.)  LG_{0,0} w = 2 cm  Flat b = 9.1 cm  LG_{5,5} w = 3.5 cm 

curvature radius  328 mW  9,682 mW  27,396 mW 
piston  3.23 µm/W  1.43 µm/W  1.08 µm/W 
coupling losses  3.24/W^{2}  0.53/W^{2}  0.51/W^{2} 
results (bulk abs.)  LG_{0,0} w = 2 cm  Flat b = 9.1 cm  LG_{5,5} w = 3.5 cm 
curvature radius  317 mW  9,164 mW  25,926 mW 
piston  3.39 µm/W  1.52 µm/W  1.15 µm/W 
coupling losses  3.47/W^{2}  0.59/W^{2}  0.56/W^{2} 
Thermal lensing curvature radii (R_{ c }) for LG modes having 1 ppm clipping losses, and associated coupling losses (L) in the weak power approximation [Equation (3.83)] (mirror diameter: 35 cm)
order (p, q)  w [cm]  R_{ c } [mW] (coat. abs.)  L [W^{−2}]  R_{ c } [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 
3.3 Thermal distortions in the steady state
3.3.1 Thermal expansion from thermalization on the coating

Ex1 (LG_{0,0}, w = 2 cm): R_{ c } = 5,842 mW

Ex2 (Flat, b = 9.1 cm): R_{ c } = 165,485 mW

Ex3 (LG_{5,5}, w = 3.5 cm): R_{ c } = 477,565 mW.
Curvature radii from thermal expansion due to coating absorption for modes having 1 ppm clipping losses
order (p, q)  w [cm]  R_{ c } of th. aberr. [kmW] 

(0,0)  6.65  77 
(0,1)  5.56  149 
(1,0)  6.06  141 
(0,2)  4.93  210 
(1,1)  5.23  201 
(2,0)  5.65  197 
(0,3)  4.49  258 
(1,2)  4.70  250 
(2,1)  4.97  250 
(3,0)  5.35  250 
(0,4)  4.17  299 
(1,3)  4.32  290 
(2,2)  4.52  292 
(3,1)  4.76  298 
(4,0)  5.11  300 
(0,5)  3.91  332 
(1,4)  4.03  324 
(2,3)  4.18  326 
(3,2)  4.36  332 
(4,1)  4.58  342 
(5,0)  4.91  348 
3.3.2 Thermal expansion from internal absorption
When the linear absorption of light through the bulk material results in an internal heat source, the temperature field is no longer harmonic, and we are bound to solve explicitly the thermoelastic Equations (3.91) and (3.92). As seen earlier, the case of internal absorption leads to a symmetric temperature field. However, we shall derive the general thermoelastic solution, which will also prove useful in Section 4 below.

Ex1 (LG_{0,0}, w = 2 cm): R_{ c } = 22 kmW

Ex2 (flat, b = 9.1 cm): R_{ c } = 325 kmW (mesa: 361 kmW)

Ex3 (LG_{5,5}, w = 3.5 cm): R_{ c } = 937 kmW
Curvature radii from thermal expansion due to bulk absorption for modes having 1 ppm clipping losses
order (p, q)  w [cm]  R_{ c } of th. aberr. [km W] 

(0,0)  6.65  165 
(0,1)  5.56  318 
(1,0)  6.06  290 
(0,2)  4.93  440 
(1,1)  5.23  410 
(2,0)  5.65  404 
(0,3)  4.49  533 
(1,2)  4.70  507 
(2,1)  4.97  504 
(3,0)  5.35  512 
(0,4)  4.17  613 
(1,3)  4.32  586 
(2,2)  4.52  585 
(3,1)  4.76  597 
(4,0)  5.11  615 
(0,5)  3.91  677 
(1,4)  4.03  654 
(2,3)  4.18  650 
(3,2)  4.36  661 
(4,1)  4.58  684 
(5,0)  4.91  713 
Thermal aberrations from coating and bulk absorption
results (coating abs.)  LG_{0,0} w = 2 cm  flat b = 9.1 cm  LG_{5,5} w = 3.5 cm 

Curvature radius Coupling losses  5.8 kmW 4.3 × 10^{− 2}/W^{2}  167 km W 4.4 × 10^{−3}/W^{2}  478 km W 6.7 × 10^{−3}/W^{2} 
results (bulk abs.)  LG_{0,0} w = 2 cm  flat b = 9.1 cm  LG_{5,5} w = 3.5 cm 
Curvature radius Coupling losses  22 km W 3.0 × 10^{−3}/W^{2}  327 km W 2.2 × 10^{−3}/W^{2}  937 km W 1.8 × 10^{−3}/W^{2} 
3.4 Expansion on Zernike polynomials
Zernike coefficients c_{ n } for LG_{00} w = 2 cm
n  lensing: heating by bulk µm/W  lensing: heating by coating µm/W  aberration: heating by coating nm/W  aberration: heating by bulk nm/W 

0  0.940  0.873  140.76  25.85 
1  −0.633  −0.599  34.79  −17.45 
2  0.470  −0.447  −25.60  12.25 
3  −0.333  −0.319  17.95  −7.80 
4  0.250  0.239  −13.42  5.06 
5  −0.192  −0.185  10.38  −3.35 
6  0.151  0.146  −8.17  2.27 
7  −0.120  −0.116  6.50  −1.57 
8  0.095  0.092  −5.19  1.10 
9  −0.076  −0.074  4.14  −0.79 
10  0.060  0.059  −3.30  0.57 
11  −0.048  −0.047  2.61  −0.41 
12  0.038  0.037  −2.06  −0.30 
13  −0.029  −0.029  1.61  −0.21 
14  0.023  0.022  −1.25  0.16 
Zernike coefficients c_{ n } for LG_{55} w = 3.5 cm
n  lensing: heating by bulk µm/W  lensing: heating by coating µm/W  aberration: heating by coating nm/W  aberration: heating by bulk nm/W 

0  0.817  0.865  16.48  24.21 
1  −0.229  −0.242  13.00  −6.64 
2  0.044  0.046  −2.67  1.38 
3  −0.019  −0.019  1.058  −0.42 
4  0.021  0.022  −1.198  0 
5  −0.002  −0.003  0.135  0 
6  −0.006  −0.006  0.314  0 
7  0.001  0.001  −0.047  0 
8  −0.002  −0.002  0.091  0 
9  0.006  0.007  −0.356  0 
10  −0.006  −0.006  0.315  0 
Zernike coefficients c_{ n } for the mesa mode
n  lensing: heating by bulk µm/W  lensing: heating by coating µm/W  aberration: heating by coating nm/W  aberration: heating by bulk nm/W 

0  0.845  0.895  30.81  25.05 
1  −0.387  −0.408  22.25  −11.19 
2  0.147  0.155  −8.55  4.12 
3  −0.185  −0.020  0  0 
4  −0.011  −0.012  0  0 
5  0.008  0.008  0  0 
4 On Thermal Compensation Systems
It is easily seen that in the case of a TEM_{00} readout beam, even with the largest possible w allowed by clipping losses, the resulting thermal lens is currently a serious spurious effect and will be even greater in advanced interferometers. For a fundamental mode with a width of 6.65 cm on the cavity input mirrors, the curvature radius of the thermal lens is about 4 km for 1 W thermalized, on the same order as the nominal curvature radius of the mirror.
This is why thermal compensation systems (TCS) have been designed. The general principle is to use an auxiliary source of heat for heating the cold parts of the mirror in order to achieve a more homogeneous temperature distribution. The source of heat may be a classical radiator able to radiate infrared energy in a vacuum, for instance, a hot ring near the rear face of the mirror [30, 25, 13, 14, 22, 43]. It may also be an auxiliary laser projector, which can be programmed to scan the mirror surface to produce a given power mask [42].
4.1 Heating the rear face of a mirror
4.2 Simple model of a radiator
Thermal compensation with a heating ring: LG_{0,0} mode of w = 2 cm.
dissipated power  initial losses  compensation power  minimal losses  wavefront curvature 

10 mW  350 ppm  2.7 W  24 ppm  ∞ 
20 mW  1,400 ppm  5.3 W  96 ppm  ∞ 
30 mW  3,100 ppm  8.0 W  220 ppm  ∞ 
100 mW  34,300 ppm  26.5 W  2392 ppm  ∞ 
Thermal compensation with a heating ring: LG_{5,5} mode of w = 3.5 cm.
dissipated power  initial losses  compensation power  minimal losses  wavefront curvature 

10 mW  56 ppm  50 mW  6 ppm  ∞ 
20 mW  213 ppm  100 mW  11 ppm  ∞ 
30 mW  474 ppm  130 mW  15 ppm  ∞ 
100 mW  5,218 ppm  450 mW  122 ppm  ∞ 
5 Axicon systems
Thermal compensation with an axicon system
dissipated power  initial losses  compensation power  minimal losses  wavefront curvature 

10 mW  350 ppm  1.2 W  5 ppm  ∞ 
20 mW  1,400 ppm  2.4 W  19 ppm  ∞ 
30 mW  3,100 ppm  3.6 W  43 ppm  ∞ 
100 mW  34,300 ppm  12 W  481 ppm  ∞ 
6 CO_{2} laser compensation by scanning
Thermal compensation with a scanning CO_{2} beam for LG_{00} mode (w = 2 cm)
dissipated power  initial losses  compensation power  minimal losses  wavefront curvature 

10 mW  350 ppm  1.9 W  0.7 ppm  ∞ 
20 mW  1,400 ppm  3.8 W  3 ppm  ∞ 
30 mW  3,100 ppm  5.6 W  6.4 ppm  ∞ 
100 mW  34,300 ppm  18.8 W  71 ppm  ∞ 
Zernike coefficients for three TCS systems compensating LG_{00} mode (w = 2 cm)
c _{ n }  heating ring µm/W  Axicon µm/W  CO_{2} scan µm/W 

0  0.759  0.018  0.774 
1  0.016  −0.008  −0.058 
2  −0.044  0.003  −0.016 
3  −0.012  0.001  0.001 
4  0.002  −0.001  −0.002 
5  0.004  0.001  0.002 
6  0.002  0  −0.001 
7  0  0  0.001 
8  −0.001  0  −0.001 
7 Heating in the Quasistatic Regime: Heating from Cold
It is interesting to study the temperature evolution of a mirror, assuming a constant light power flux switched on at t = 0.
7.1 Transient temperature field
7.1.1 Transient temperature from coating absorption
7.1.2 Transient temperature from bulk absorption
7.2 Transient thermal distortions
7.2.1 Case of coating absorption
7.2.2 Case of bulk absorption
8 Heating and Thermal Effects in the Dynamic Regime: Transfer Functions
We also adopt the new dimensionless parameters \({\xi _s} \equiv {k_s}a \equiv \sqrt {\zeta _s^2 + i\rho C\omega {a^2}/K}\) and η_{ s } ≡ κ_{ s }h/2. With this notation, and by setting the (unchanged since the beginning) boundary conditions, we can specialize the solution to be entirely analogous in form to the static solution.
8.1 Temperature fields and thermal lensing
8.1.1 Coating absorption
8.1.2 Bulk absorption
8.2 Equivalent displacement noise
8.2.1 Asymptotic regime
9 Dynamic Surface Distortion
9.1 Dynamic surface distortion caused by coating absorption
9.1.1 Mean displacement
9.1.2 Under cutoff regime
10 Brownian Thermal Noise
Brownian thermal noise is the phase noise caused at nonzero temperature by random motions of the reflecting faces of mirrors in a GW interferometer. A reflecting face can move either because it is displaced by its suspension system or because it undergoes internal stresses. At finite temperature the two effects are possible. We address here the internal stresses. Consider a massive body at temperature T. If T > 0, the atoms constituting the body are excited and have random motions around their equilibrium position. The fact that they are strongly coupled to neighboring atoms makes possible the propagation of elastic waves of various types, reflecting on the faces, and the onset of stationary waves. One can show that, for a finite body (e.g., a cylinder of silica), there is a discrete infinity of such stationary waves, each corresponding to a particular elastic normal mode. At thermal equilibrium, the state of the body can be represented by a linear superposition of all the modes, with random relative phases, and, due to the energy equipartition theorem, the same energy κ_{ B }T (κ_{ B } is the Boltzmann constant). The motion of atoms near a limiting surface of the body will modify its shape slightly, and, if we consider the reflecting face of a mirror, a surface distortion is a possible cause of phase change in the reflected beam, in other words, of a noise. Estimation of the resulting spectral density of phase noise is the internal thermal noise problem in massive mirrors.
10.1 The FluctuationDissipation theorem and Levin’s generalized coordinate method
10.2 Infinite mirrors noise in the substrate
Some numerical values of g_{0,n,m}
m  0  1  2  3  4  5 

n  
0  1  .60  .46  .39  .34  .31 
1  .69  .50  .41  .36  .32  .29 
2  .57  .44  .37  .33  .30  .28 
3  .50  .40  .35  .31  .29  .27 
4  .46  .37  .33  .30  .27  .26 
5  .43  .35  .31  .28  .26  .25 
10.3 Infinite mirrors, noise in coating
10.3.1 Coating Brownian thermal noise: LG modes
Some numerical values of g_{1,n,m}
m  0  1  2  3  4  5 

n  
0  1  .50  .34  .27  .22  .19 
1  .50  .31  .23  .19  .16  .14 
2  .38  .25  .19  .16  .14  .12 
3  .31  .21  .17  .14  .12  .11 
4  .27  .19  .15  .13  .11  .10 
5  .25  .17  .14  .12  .11  .10 
10.3.2 Coating Brownian thermal noise: Flat modes
Here are some numerical values for comparison. For the LG_{5,5} mode, we have w_{1,5,5} ∼ 4.14 m^{−2}, For the flat mode (b = 9.1 cm), we have ϖ_{1,flat} ∼ 6.12 m^{−2} (to be discarded, as the sharp edge effect becomes spurious, see the next value), ϖ_{1,mesa} ∼ 4.52 m^{−2} (numerical integration), and for the GaussBessel mode of Figure 5, this is ∼_{1,GB} ∼ 3.56 m^{−2}. A value of 2.34 m^{−2} is reported by [4] after an optimization process involving an expansion on LG modes. It is probably possible to have a not too different result by a fine tuning of the conical mode’s parameters.
10.4 Finite mirrors
10.4.1 Equilibrium equations
10.4.2 Boundary conditions
 No shear on the cylindrical edge, i.e.,This can be satisfied by requiring ζ_{ s } ≡ k_{ s }a to be a strictly positive zero of J_{1}(ζ). The family of all such zeroes defines a family of functions {J_{0}(ζ_{ s }r/a), s ∈ ℕ*} complete and orthogonal on [0, a], on which any function of r may be expanded as a FB series. Note that this family is different from the families encountered in thermal studies. In particular, the orthogonality relation is simpler:$${\Theta _{rz}}(a,z) = 0{.}$$(8.73)$$\int\nolimits_0^a {{J_0}} ({k_s}r)\,{J_0}({k_{s\prime}}r)\,r\,dr = {{{a^2}} \over 2}\,J_0^2({\zeta _s})\,{\delta _{s,s\prime}}.$$(8.74)
 No shear on the two circular faces, i.e.,$${\Theta _{rz}}(a,0) = {\Theta _{rz}}(a,h) = 0{.}$$(8.75)
 The given pressure on the reflecting face:where p(r) is a pressure distribution normalized to an integrated force of 1 N, identical to the normalized optical intensity function I(r).$${\Theta _{zz}}(r,0) =  p(r),$$(8.76)
 No pressure on the rear face:$${\Theta _{zz}}(r,h) = 0{.}$$(8.77)
 No radial stress on the edge:$${\Theta _{rr}}(a,z) = 0{.}$$(8.78)
10.4.3 Strain energy
 LG_{0,0}, w = 2 cm$${U_0} = 1.81 \times {10^{ 10}}\;{\rm{J}}\,{{\rm{N}}^{ 2}}$$(8.117)$$\Delta U = 2.08 \times {10^{ 11}}\;{\rm{J}}\,{{\rm{N}}^{ 2}}$$(8.118)$$U = 2.02 \times {10^{ 10}}\;{\rm{J}}\,{{\rm{N}}^{ 2}}$$(8.119)$$S_x^{1/2}(f) = 1.03 \times {10^{ 18}}\,{\left[ {{{1\;{\rm{Hz}}} \over f}} \right]^{1/2}}\;{\rm{mH}}{{\rm{z}}^{ 1/2}}$$(8.120)
 flat mode, b = 9.1 cm$${U_0} = 2.08 \times {10^{ 11}}\;{\rm{J}}\,{{\rm{N}}^{ 2}}$$(8.121)$$\Delta U = 1.21 \times {10^{ 11}}\;{\rm{J}}\,{{\rm{N}}^{ 2}}$$(8.122)$$U = 3.28 \times {10^{ 11}}\;{\rm{J}}\,{{\rm{N}}^{ 2}}$$(8.123)$$S_x^{1/2}(f) = 4.16 \times {10^{ 19}}\,{\left[ {{{1\;{\rm{Hz}}} \over f}} \right]^{1/2}}\;{\rm{mH}}{{\rm{z}}^{ 1/2}}$$(8.124)
 mesa mode, b_{ f } = 10.7 cm$${U_0} = 1.58 \times {10^{ 11}}\;{\rm{J}}\,{{\rm{N}}^{ 2}}$$(8.125)$$\Delta U = 1.04 \times {10^{ 11}}\;{\rm{J}}\,{{\rm{N}}^{ 2}}$$(8.126)$$U = 2.62 \times {10^{ 11}}\;{\rm{J}}\,{{\rm{N}}^{ 2}}$$(8.127)$$S_x^{1/2}(f) = 3.72 \times {10^{ 19}}\,{\left[ {{{1\;{\rm{Hz}}} \over f}} \right]^{1/2}}\;{\rm{mH}}{{\rm{z}}^{ 1/2}}$$(8.128)
 LG_{5,5}, w = 3.5 cm$${U_0} = 4.24 \times {10^{ 12}}\;{\rm{J}}\,{{\rm{N}}^{ 2}}$$(8.129)$$\Delta U = 4.40 \times {10^{ 12}}\;{\rm{J}}\,{{\rm{N}}^{ 2}}$$(8.130)$$U = 8.64 \times {10^{ 12}}\;{\rm{J}}\,{{\rm{N}}^{ 2}}$$(8.131)It is clear that for modes widely spread on the mirror surface, the SaintVenant correction becomes important. Moreover, if we compare to the values found in the infinite mirror approximation, we see that the first example was underestimated by about 7%, the flat mode by 17%, and the third by a factor of 3. We also see the discrepancy (11%) between the flat estimation and the mesa beam. This leads us to be cautious with the foregoing estimations. Figure 58 summarizes the gain in thermal noise obtained with respect to the current situation on Virgo input mirrors for several beams having 1 ppm clipping losses.$$S_x^{1/2}(f) = 2.13 \times {10^{ 19}}\,{\left[ {{{1\;{\rm{Hz}}} \over f}} \right]^{1/2}}\;{\rm{mH}}{{\rm{z}}^{ 1/2}}$$(8.132)
10.4.4 Explicit displacement and strain tensor
10.5 Coating Brownian thermal noise: finite mirrors
Comparison infinite/finite mirror Strain Energy (SE)
3 examples  coating SE [JN^{−2}]  bulk SE [J N^{−2}] 

Ex1: LG_{00}, w = 2 cm  
U_{∞}  2.10 × 10^{−13}  1.88 × 10^{−10} 
U  2.38 × 10^{−13}  2.02 × 10^{−10} 
Ex2: flat  
U_{∞}  1.02 × 10^{−14}  3.95 × 10^{−11} 
U  2.07 × 10^{−14}  3.28 × 10^{−11} 
Ex3: LG_{55}, w = 3.5 cm  
U_{∞}  6.87 × 10^{−15}  2.68 × 10^{−11} 
U  7.47 × 10^{−15}  8.64 × 10^{−12} 
11 Thermoelastic Noise
11.1 Introduction
11.2 Case of infinite mirrors
11.2.1 Gaussian beams
Some numerical values of g_{2,n,m}
m  0  1  2  3  4  5 

n  
0  1  .75  .64  .57  .53  .49 
1  .44  .39  .36  .33  .31  .30 
2  .33  .31  .29  .27  .26  .25 
3  .28  .26  .25  .24  .23  .22 
4  .24  .23  .22  .21  .21  .20 
5  .22  .21  .20  .20  .19  .19 
11.2.2 Flat beams
11.2.3 Thermoelastic noise in the coating
Some numerical values of g_{3,n,m}
m  0  1  2  3  4  5 

n  
0  1  1.5  1.72  1.86  1.96  2.05 
1  .50  .81  .98  1.10  1.20  1.27 
2  .37  .62  .77  .88  .96  1.03 
3  .31  .53  .66  .76  .83  .90 
4  .27  .46  .58  .67  .75  .81 
5  .25  .42  .53  .61  .68  .74 
It seems clear that, as already mentioned, the modes LG_{0,m}, having a sharp peak on the axis, become worse and worse as the order m increases. On the other hand, the reduction factor for the noise in the best cases is much less than for the Brownian thermal noise.
11.2.4 Scaling laws

Brownian noise, substrate: ϖ_{0}

Brownian noise, coating: ϖ_{1}

Thermoelastic noise, substrate: ϖ_{2}

Thermoelastic noise, coating: ϖ_{3}. However, in this case there is a more refined analysis [29], taking into account the heat flow. Attention must be paid to this theory (see also [17, 8]). However, the approximate character of the semiinfinitemirror approach reduces its practical interest.
Some values of ϖ_{ n }
n  LG_{00} w = 2 cm  LG_{55} w = 3.5 cm  flat b = 9.1 cm  mesa b_{ f } = 10.7 cm  units 

0  2.245  0.321  0.473  0.426  m^{−1} 
1  126.65  4.13  6.12  4.52  m^{−2} 
2  1.122 × 10^{4}  398  *  76.7  m^{−3} 
3  1.27 × 10^{6}  10^{5}  *  1800  m^{−2} 
11.2.5 Numerical results
 Spectral density of thermoelastic noise in the substrate:$${S_x}{(f)^{1/2}} = 8.53 \times {10^{ 20}}\;{\rm{mH}}{{\rm{z}}^{ 1/2}}\left[ {{{1\;{\rm{Hz}}} \over f}} \right].$$(9.46)Spectral density of noise in the coating:$${S_x}{(f)^{1/2}} = 1.32 \times {10^{ 19}}\;{\rm{mH}}{{\rm{z}}^{ 1/2}}\left[ {{{1\;{\rm{Hz}}} \over f}} \right].$$(9.47)
 Spectral density of thermoelastic noise in the substrate:$${S_x}{(f)^{1/2}} = 1.61 \times {10^{ 20}}\;{\rm{mH}}{{\rm{z}}^{ 1/2}}\left[ {{{1\;{\rm{Hz}}} \over f}} \right].$$(9.48)Spectral density of noise in the coating:$${S_x}{(f)^{1/2}} = 3.71 \times {10^{ 20}}\;{\rm{mH}}{{\rm{z}}^{ 1/2}}\left[ {{{1\;{\rm{Hz}}} \over f}} \right].$$(9.49)
 Spectral density of thermoelastic noise in the substrate:$${S_x}{(f)^{1/2}} = 7.05 \times {10^{ 21}}\;{\rm{mH}}{{\rm{z}}^{ 1/2}}\left[ {{{1\;{\rm{Hz}}} \over f}} \right].$$(9.50)
 Spectral density of noise in the coating:$${S_x}{(f)^{1/2}} = 4.99 \times {10^{ 21}}\;{\rm{mH}}{{\rm{z}}^{ 1/2}}\left[ {{{1\;{\rm{Hz}}} \over f}} \right].$$(9.51)
11.3 Case of finite mirrors
11.3.1 Case of the bulk material
In the case of the flat beam, one should note the peaks at the location of the sharp edges of the intensity distribution. This was the cause of the divergence of the infinite mirror approach. However, the estimation for the flat and mesa beams are not very different.
11.3.2 Case of the coatings
11.3.3 Numerical results
12 Generation of High Order Modes
13 Conclusion
We have discussed some of the main issues regarding mirrors to be used in a highopticalpower interferometer. These issues are thermal lensing, thermal aberration, thermal noise and thermoelastic noise. These spurious effects do not act at the same level. Thermal issues arise directly from the laser power and make necessary compensation systems more and more difficult as the power increases. Thermal noises (standard and thermoelastic) are not related to the laser power, but dominate the shot noise in the central spectral region, spoiling any gain of sensitivity expected from a higher laser power. An interesting approach to reduce these effects is to change the readout beam from the fundamental TEM_{0,0} currently used, to the more widely spread (“exotic”) light power distributions, (either mesa, conical or highorder LaguerreGauss). We have given the formulas for estimating the gains with respect to the above cited issues for these different modes. Thus, we hope to contribute to the design of advanced instruments. It is already possible to point out that exotic beams provide a high gain (up to a factor of five) in thermal noise and thermoelastic noise and a huge gain in spurious thermal effects (up to two orders of magnitude).
Notes
Acknowledgement
I am indebted to Kip S. Thorne for his extensive review work, and for a huge number of suggestions among which I tried to address as much as I could.
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