Raymond G. Beausoleil, Eric K. Gustafson, Martin M. Fejer, Erika D’Ambrosio, William Kells, and Jordan Camp, "Model of thermal wave-front distortion in interferometric gravitational- wave detectors. I. Thermal focusing," J. Opt. Soc. Am. B 20, 1247-1268 (2003)

We develop a steady-state analytical and numerical model of the optical response of power-recycled Fabry–Perot Michelson laser gravitational-wave detectors to nonlinear thermal focusing in optical substrates. We assume that the thermal distortions are small enough that we can represent all intracavity fields as linear combinations of basis functions derived from the eigenmodes of a Fabry–Perot arm cavity. We have included the effects of power absorption in optical substrates and coatings, mismatches between laser wave-front and mirror surface curvatures, and aperture diffraction. We demonstrate a detailed numerical example of this model using the matlab program Melody for the initial Laser Interferometer Gravitational Wave Observatory detector.

Kenneth A. Strain, Guido Müller, Tom Delker, David H. Reitze, David B. Tanner, James E. Mason, Phil A. Willems, Daniel A. Shaddock, Malcolm B. Gray, Conor Mow-Lowry, and David E. McClelland Appl. Opt. 42(7) 1244-1256 (2003)

Aidan F. Brooks, Benjamin Abbott, Muzammil A. Arain, Giacomo Ciani, Ayodele Cole, Greg Grabeel, Eric Gustafson, Chris Guido, Matthew Heintze, Alastair Heptonstall, Mindy Jacobson, Won Kim, Eleanor King, Alexander Lynch, Stephen O’Connor, David Ottaway, Ken Mailand, Guido Mueller, Jesper Munch, Virginio Sannibale, Zhenhua Shao, Michael Smith, Peter Veitch, Thomas Vo, Cheryl Vorvick, and Phil Willems Appl. Opt. 55(29) 8256-8265 (2016)

References

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Material Constants for Fused Silica Used in Our Simulations at ${\mathrm{\lambda}}_{0}=1.0642\mu \mathrm{m}$
and ${T}_{0}=300\mathrm{K}$

Constant

Name

Value

Units

η

Refractive index

1.44963

${\mathrm{d}}_{\eta}/\mathrm{d}T$

Thermo-optic coefficient

$8.7\times {10}^{-6}$

K^{-1}

${k}_{T}$

Conductivity

1.38

W/m K

∊

Total spherical emissivity

0.90

${\alpha}_{s}$

Bulk absorption coefficient

$3.5\times {10}^{-4}$

m^{-1}

${\sigma}_{s}$

Bulk scattering coefficient

$0.5\times {10}^{-4}$

m^{-1}

Table 2

Total Absorbed ${\mathrm{TEM}}_{00}$
Powers for Each of the Three Regions Shown in Fig. 4
, Summed Over the Lowest-Order Transverse Modes of all Active Sidebands

The factors of ${\scriptstyle \frac{1}{2}}$
arise from our normalization convention for the laser electric field.

Table 3

Total Absorbed ${\mathrm{TEM}}_{00}$
Powers for Each of the Five Absorption Regions Shown in Fig. 6
, Summed Over the Lowest-Order Transverse Modes of all Active Sidebands^{
a
}

The factors of ${\scriptstyle \frac{1}{2}}$
arise from our normalization convention for the laser electric field.

Table 4

High-Reflection (hr) and Antireflection (ar) Loss Parameters for the Initial LIGO Optical Elements Used in Our Simulations28^{
a
}

Parameter

Loss (ppm)

${a}_{\mathrm{hr}}$

0.5

${a}_{\mathrm{ar}}$

0.5

${s}_{\mathrm{hr}}$

55.0

${s}_{\mathrm{ar}}$

90.0

We assign a scattering loss of 900 ppm to the AR coatings of the ITM mirrors to account for power pick-off monitoring losses. Here ${a}_{c}$
is the optical power absorption that contributes to the substrate thermal loss, and ${s}_{c}$
is the scattering loss.

Table 5

Physical Parameters for the Initial LIGO Optical Elements Used in Our Simulations28^{
a
}

Mirror

${R}_{M}$
(m)

h
(m)

${r}^{2}$

${t}^{2}$

${\mathcal{M}}_{1},$${\mathcal{M}}_{2}$

14 571.0

0.10

0.971

$1-{r}^{2}-{l}_{\mathrm{hr}}$

${\mathcal{M}}_{3},$${\mathcal{M}}_{4}$

7 400.0

0.10

$1-{t}^{2}-{l}_{\mathrm{hr}}$

$1.5\times {10}^{-5}$

${\mathcal{M}}_{5}$

9 999.8

0.10

0.973

$1-{r}^{2}-{l}_{\mathrm{hr}}$

${\mathcal{M}}_{6}$

∞

0.04

0.495

$1-{r}^{2}-{l}_{\mathrm{hr}}$

Each substrate is fused silica, with parameters given by Table 1
and a radius $a=12.5\mathrm{cm}.$
Here the total loss ${l}_{\mathrm{hr}}={a}_{\mathrm{hr}}+{s}_{\mathrm{hr}},$
where ${a}_{\mathrm{hr}}$
and ${s}_{\mathrm{hr}}$
are the power absorption and scattering losses in the high-reflection coating, respectively.

Tables (5)

Table 1

Material Constants for Fused Silica Used in Our Simulations at ${\mathrm{\lambda}}_{0}=1.0642\mu \mathrm{m}$
and ${T}_{0}=300\mathrm{K}$

Constant

Name

Value

Units

η

Refractive index

1.44963

${\mathrm{d}}_{\eta}/\mathrm{d}T$

Thermo-optic coefficient

$8.7\times {10}^{-6}$

K^{-1}

${k}_{T}$

Conductivity

1.38

W/m K

∊

Total spherical emissivity

0.90

${\alpha}_{s}$

Bulk absorption coefficient

$3.5\times {10}^{-4}$

m^{-1}

${\sigma}_{s}$

Bulk scattering coefficient

$0.5\times {10}^{-4}$

m^{-1}

Table 2

Total Absorbed ${\mathrm{TEM}}_{00}$
Powers for Each of the Three Regions Shown in Fig. 4
, Summed Over the Lowest-Order Transverse Modes of all Active Sidebands

The factors of ${\scriptstyle \frac{1}{2}}$
arise from our normalization convention for the laser electric field.

Table 3

Total Absorbed ${\mathrm{TEM}}_{00}$
Powers for Each of the Five Absorption Regions Shown in Fig. 6
, Summed Over the Lowest-Order Transverse Modes of all Active Sidebands^{
a
}

The factors of ${\scriptstyle \frac{1}{2}}$
arise from our normalization convention for the laser electric field.

Table 4

High-Reflection (hr) and Antireflection (ar) Loss Parameters for the Initial LIGO Optical Elements Used in Our Simulations28^{
a
}

Parameter

Loss (ppm)

${a}_{\mathrm{hr}}$

0.5

${a}_{\mathrm{ar}}$

0.5

${s}_{\mathrm{hr}}$

55.0

${s}_{\mathrm{ar}}$

90.0

We assign a scattering loss of 900 ppm to the AR coatings of the ITM mirrors to account for power pick-off monitoring losses. Here ${a}_{c}$
is the optical power absorption that contributes to the substrate thermal loss, and ${s}_{c}$
is the scattering loss.

Table 5

Physical Parameters for the Initial LIGO Optical Elements Used in Our Simulations28^{
a
}

Mirror

${R}_{M}$
(m)

h
(m)

${r}^{2}$

${t}^{2}$

${\mathcal{M}}_{1},$${\mathcal{M}}_{2}$

14 571.0

0.10

0.971

$1-{r}^{2}-{l}_{\mathrm{hr}}$

${\mathcal{M}}_{3},$${\mathcal{M}}_{4}$

7 400.0

0.10

$1-{t}^{2}-{l}_{\mathrm{hr}}$

$1.5\times {10}^{-5}$

${\mathcal{M}}_{5}$

9 999.8

0.10

0.973

$1-{r}^{2}-{l}_{\mathrm{hr}}$

${\mathcal{M}}_{6}$

∞

0.04

0.495

$1-{r}^{2}-{l}_{\mathrm{hr}}$

Each substrate is fused silica, with parameters given by Table 1
and a radius $a=12.5\mathrm{cm}.$
Here the total loss ${l}_{\mathrm{hr}}={a}_{\mathrm{hr}}+{s}_{\mathrm{hr}},$
where ${a}_{\mathrm{hr}}$
and ${s}_{\mathrm{hr}}$
are the power absorption and scattering losses in the high-reflection coating, respectively.