1. Deep Chatterjee IISER Kolkata Mentors: Koji Arai; Matthew Abernathy
Design of a Coating-less Optical Cavity based on Total Internal Reflection
Deep Chatterjee
IISER Kolkata
Mentors: Koji Arai; Matthew Abernathy

2. Why TIR Optical Cavity? Optical cavity
Typically consists of a pair of mirrors, flat or curved, facing each other to create standing wave patterns in light waves. Since only certain modes can exist in the cavity depending on the geometry, they are used for Frequency stabilization.
LIGO Laboratory

3. Why TIR Optical Cavity? Total Internal Reflection (TIR) cavity
It uses the phenomenon of Total Internal Reflection. Injection of light into the cavity is carried out by the means of Evanescent coupling.(see Optics Lett. 17, 5, pp )
Top view
Actual Shape
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4. Why TIR Optical Cavity?
No mirrors used and hence no reflection coatings
Brownian Thermal Noise from coatings is a big issue
Coating removal is better for frequency stabilization
High reflectivity without coatings can be achieved with TIR
Cancelling other Thermal Noise sources is a possibility.
TIR cavity is expected to be more “quiet” and is suitable
for frequency stabilization
LIGO Laboratory

5. Thermal Noise
It is the unwanted, random signal caused by Thermal Fluctuations.
Thermal fluctuations affect the refractive index and the dimensions of the body due to thermal expansion.
These changes adds an extra random phase to the beam which manifests as noise.
Temperature change
causes expansion(contraction)
Beam reflecting off the surface
Temperature change
causes change in refractive index
Optical path φ = nL changes with changing n (TR Noise)
Change in position of reflection causes change in phase (TE Noise)
Beam going through
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6. Noise in TIR cavity TE Noise TR Noise Change in position of faces
Optical path length
The optical path φ = nL is affected which
results in TE and TR Noise
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7. Calculating the Noise – The Fluctuation Dissipation Theorem
General case of Einstein’s theory of Brownian Motion
Relation between them
Fluctuations in position of
Brownian particle
Viscosity of the medium
More general case by Callen and Welton (see Phys. Rev. 83, 34–40 (1951))
Fluctuations in a thermodynamic variable
Dissipation mechanism
associated
Fluctuations can be calculated from the dissipation
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8. Applying the FDT – The Levin’s Approach
Y. Levin applied the FDT to compute the Noise in GW test masses
(see Phys. Rev. D 57, 2 (1998))
Levin’s algorithm to calculate the Noise
Apply an oscillatory “force” to the system that will cause perturbation in the quantity of interest.
Calculate the dissipation caused due to the applied “force”.
Relate the Dissipation to the Fluctuation using the FDT.
We calculate the Dissipation and relate it to the Fluctuation
LIGO Laboratory

9. Calculating the Thermo-Elastic Noise
The procedure of TE noise calculation following Levin’s approach, for cylindrical geometry, goes as
Heat dissipation takes place
Oscillatory pressure
(scaled by beam intensity) is applied to reflecting surface
Stresses develop
Heat is generated
The total power dissipation which is lost as heat is related to the temperature fluctuation by the FDT
LIGO Laboratory

10. Calculating the Thermo-Refractive Noise
The procedure of TR noise calculation following Levin’s approach, for cylindrical geometry, goes as
Oscillatory heat source
(scaled by beam intensity) is applied along beam path
Heat dissipation takes place
Thermal gradients develop
The power dissipation as heat is related to the spectral density of TR noise by the FDT
LIGO Laboratory

11. Finite Element Calculation of TE and TR Noise
Finite Element Analysis (FEA) was performed using COMSOL and MATLAB.
Simpler case of cylindrical cavity was tested out first.
Results were compared to the analytic cases.
Aim to find suitable material properties that minimizes the total TE and TR noise.
Expect to look towards cancellation of TE and TR Noise.
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12. Finite Element Calculation of TE and TR Noise
Cylindrical test mass cavity
Radius = 0.25 [m]
Height = 0.46 [m]
Beam Radius = 0.09 [m]
TR analytic:
Phys. Rev. D 84, (2011)
TE(infinite) analytic:
Phys. Rev. D 62, (2000)
LIGO Laboratory

13. Finite Element Calculation of TE and TR Noise
Cylindrical test mass cavity
Radius = 0.25 [m]
Height = 0.46 [m]
Beam Radius = 0.09 [m]
TR analytic:
Phys. Rev. D 84, (2011)
TE(infinite) analytic:
Phys. Rev. D 63, (2001)
LIGO Laboratory

14. Putting TE and TR together
With the models working fine, the effects are put together to check for any cancellation.
Heat source used
along laser beam path
Pressure applied on the two faces
Heat source and pressure is applied simultaneously
to see the combined effect
LIGO Laboratory

15. Putting TE and TR together
Cylindrical test mass cavity
Radius = 0.25 [m]
Height = 0.46 [m]
Beam Radius = 0.09 [m]
TR analytic:
Phys. Rev. D 84, (2011)
TE(infinite) analytic:
Phys. Rev. D 62, (2000)
LIGO Laboratory

16. Putting TE and TR together
Cylindrical test mass cavity
Radius = 0.25 [m]
Height = 0.1 [m]
Beam Radius = 0.09 [m]
TR analytic:
Phys. Rev. D 84, (2011)
TE(infinite) analytic:
Phys. Rev. D 63, (2001)
LIGO Laboratory

17. Future Work Parametric study to bring TE and TR noise close.
Cancellation occurs when TE and TR are of same order
Building the correct model for TIR cavity
Calculate Noise using FEA for the same
Noise Cancellation – Can it be achieved?
If not cancelled, can the noise be minimized?
LIGO Laboratory

18. Summary and Conclusion
Simpler model has been built successfully.
Special Analytic cases have been tested.
Model is expected to work well for the TIR cavity
Noise cancellation has to be thought about.
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19. Thank you