40m QIL Cryo_Lab CTN SUS_Lab TCS_Lab OMC_Lab CRIME_Lab FEA ENG_Labs OptContFac Mariner WBEEShop
 COMSOL elog Not logged in  Mon Jun 3 21:01:46 2013, Emory Brown, Optics, General, Analytic Calculation of Thermal Noise due to Brownian Motion using Levin and Thorne's Methods    Wed Jun 5 13:31:41 2013, Matt A, Optics, General, Analytic Calculation of Thermal Noise due to Brownian Motion using Levin and Thorne's Methods
Message ID: 5     Entry time: Mon Jun 3 21:01:46 2013     Reply to this: 7
 Author: Emory Brown Type: Optics Category: General Subject: Analytic Calculation of Thermal Noise due to Brownian Motion using Levin and Thorne's Methods

Motivation:

Reduction of Brownian thermal noise in future gravitational wave detectors is of significant interest. It has been suggested that changing the shape of the mirrors used may reduce the Brownian thermal noise. Before we can study how alterations in mirror shape effect Brownian thermal noise, we must be able to calculate the thermal noise analytically in order to compare it to our finite element model.

Methods:
I made Mathematica notebooks to perform calculation of the thermal noise.  The first notebook implemented Levin's method directly.
Sx[f_] := (4*Kb*T/f)*(1 - \[Sigma]^2)*1.87322*\[Phi]/(\[Pi]^3*E0*r0)
To test this against Levin's paper, the same values were used as in the paper, such that
Kb=1.38065*10^-23 (Boltzmann's constant)
T=300 (Temperature)
Sigma=0.16 (Poison's Ratio)
E0=71.8*10^9 (Young's Modulus)
r0=0.0156 (Gaussian Beam Size)
Phi=10^-7 (Loss Angle)
H=0.2 (Mirror Height)
A log-log plot of Sqrt[Sx[f]] with f ranging from 0.1 to 5000 Hz was plotted and is displayed below.  Additionally, the value for 100 Hz was explicitly computed and agreed with Levin's value of 8.7*10^-40 m^2/Hz computed from his equation 15.

The notebook was modified to perform Thorne's method of computing the thermal noise for a finite test mass.  This calculation was performed using equations 28, 35a, 54, 56, and 57.  Equations 56 and 57 require the approximation made in eqn 37 which assumes a relatively low mass mirror.  According to Liu and Thorne, this is a "rather good approximation for realistic parameter values."  Performing the calculations again using this method gave Sx[100 Hz]=7.80081*10^-40 m^2/Hz.  So, for these parameters, the finite test mass provides a correction factor of about 10%.  Another log-log plot of Sqrt[Sx[f]] against f was made using this method.

We can now modify the parameters above to match the values in our finite element model to verify the results of our finite element model.

Levin, Y. (1998). Internal thermal noise in the LIGO test masses: A direct approach. Physical Review D, 57(2), 659.
Liu, Y. T., & Thorne, K. S. (2000). Thermoelastic noise and homogeneous thermal noise in finite sized gravitational-wave test masses. Physical Review D, 62(12), 122002.

 Attachment 1: ComparisonPowerSpectralDensitySurfaceFluctuationsVsFrequency.png  12 kB  Uploaded Mon Jun 17 12:38:06 2013 Attachment 2: DifferencePlot.png  11 kB  Uploaded Mon Jun 17 12:38:17 2013 Attachment 3: ThorneComparisonPowerSpectralDensitySurfaceFluctuationsVsFrequency.nb  135 kB  Uploaded Mon Jun 17 12:39:12 2013
```(* Content-type: application/vnd.wolfram.mathematica *)

(*** Wolfram Notebook File ***)
(* http://www.wolfram.com/nb *)

(* CreatedBy='Mathematica 9.0' *)

(*CacheID: 234*)
(* Internal cache information:
NotebookFileLineBreakTest
```
... 2754 more lines ...
ELOG V3.1.3-