Thermal noise and thermal deformations of mirrors Yuri Levin, Monash Fluctuation-dissipation theorem Thermo-mechanical noise; reciprocity relations; general optical readouts; gratings Thermal deformations of mirrors 4. Thermo-refractive noise; standing wave 5. Thermo-chemical noise

1. Fluctuation-dissipation theorem Callen & Welton 1951, Kubo+many others Macroscopic degree of freedom Microscopic degrees of freedom Coupling

1. Fluctuation-dissipation theorem Callen & Welton 1951, Kubo+many others dissipation Macroscopic degree of freedom Microscopic degrees of freedom Coupling

1. Fluctuation-dissipation theorem Callen & Welton 1951, Kubo+many others dissipation Macroscopic degree of freedom Microscopic degrees of freedom Coupling Thermal motion fluctuations

1. Fluctuation-dissipation theorem Callen & Welton 1951, Kubo+many others dissipation Macroscopic degree of freedom Microscopic degrees of freedom Coupling Thermal motion fluctuations

2. Thermo-mechanical noise Callen and Welton 51, Levin 98 Readout variable: Interaction 1. oscillating pressure 2. Compute/measure dissipated power 3.

Two questions: 1. How does one figure out the readout variable in a general opto-mechanical set up? Optical reciprocity Heinert + 13 2. If the dissipation is local, how does the local random stress communicate to macroscopic surface displacement? How do we quantify this communication? dissipation Elastodynamic reciprocity fluctuation

Reciprocity relations Normal computational scheme Compute all consequences of the perturbation Extract the quantity you want to know Using reciprocity relationship Identify the quantity you want to know Solve the reverse problem (often easier). Construct a map between perturbations and readout quantity. Do it once!

Opto-mechanical readout variables Question: how does the mode frequency change when dielectric interface moves? Adiabatic theorem for oscillators so Mode energy Interface displacement Optical pressure on the interface Useful for thermal noise calculations from e.g. gratings (cf. Heinert et al. 2013; correction ~25%)

Opto-mechanical readout variables Linear optical readout, e.g. phase measurements Carrier light + Perturbation Phase Form-factor

Part 6: opto-mechanics with interfaces Linear optical readout, e.g. phase measurements Photo-diode Phase Form-factor

Opto-mechanical readout variable 1. Generate imaginary beam with oscillating dipoles Photo-diode 2. Calculate induced optical pressure on the interface 3. The phase

3. Thermal deformations of mirrors Not an issue for advanced KAGRA. Major issue for LIGO & Virgo High-temperature region cf. Hello & Vinet 1990 New coordinates Basis functions: Zernike polynomials, Hermite-Gauss functions Treat this as a readout variable

Elastodynamic reciprocity relations form-factor Force density Readout variable displacement form-factor

Elastodynamic reciprocity relations form-factor Force density is invariant with respect to interchange of and Readout variable displacement form-factor

How to calculate Apply pressure to the mirror face King, Levin, Ottaway, Veitch to be submitted. Apply pressure to the mirror face Calculate trace of the induced deformation tensor Have to do it only once! Calculate the thermal deformation Young modulus Thermal expansion Temperature perturbation

Computation of scattering coefficients imaginary pressure Impressive Agreement!

Thermo-refractive noise Braginsky, Gorodetsky, & Vyatchanin 2000 medium Index of refraction: beam 3 phaseshift temperature fluctuation intensity readout variable

Thermo-refractive noise Levin 2008 Readout variable: 1. oscillating entropy injection 2. Compute/measure dissipated power, e.g. 3.

Standing waves, GEO600 beamsplitter Image credit: Prof. Nollert effective ineffective Benthem & Levin 2009

Thermochemical and Carrier-Density Noise Intensity profile effective ineffective Thermochemical noise: due to chemical impurities Benthem & Levin 2009 Carrier-density noise: due to semiconductor charge carriers Heinert et al. submitted Both seem to not threaten GW interferometry, but one has to be vigilant, especially for the ET

Conclusions Linear systems (elastic, optomechanical) feature reciprocity relations They give direct computation of readout variable non-trivial geometries, and they allow fast computations of thermal distortions and scattering coefficients Vigilance for new types of thermal noise