1. FINESSE FINESSE Frequency Domain Interferometer Simulation Andreas Freise European Gravitational Observatory 17. March 2004

2. 5. September 2003 Andreas Freise

3. light power, field amplitudes eigenmodes, beam shape error/control signals (modulation-demodulation) transfer functions, sensitivities, noise couplings alignment error signals, mode matching, etc. Possible Outputs of FINESSE

4. 5. September 2003 Andreas Freise Plane Waves – Frequency Domain Coupling of light fields: Set of linear equations: solved numerically

5. 5. September 2003 Andreas Freise Frequency Domain Simple cavity: two mirrors + one space (4 nodes) Light source (laser) Output signal (detector)

6. 5. September 2003 Andreas Freise Frequency Domain one Fourier frequency one complex output signal

7. 5. September 2003 Andreas Freise Static response phase modulation = sidebands 3 fields, 3 beat signals

8. 5. September 2003 Andreas Freise Frequency Response infenitesimal phase modulation 9 frequencies, 13 beat signals

9. 5. September 2003 Andreas Freise Gaussian Beam Parameters Compute cavity eigenmodes start node Trace beam and set beam parameters

10. 5. September 2003 Andreas Freise Mode Mismatch and Misalignment Mode mismatch or misalignemt can be described as light scattering in higher-order spatial modes. Coupling coefficiants for the interferometer matrix are derived by projecting beam 1 on beam 2:

11. 5. September 2003 Andreas Freise FINESSE: Fast and (fairly) well tested TEM order Omatrix elements(effective) computation time (100 data points) 0~25000340<1 sec 5~1100000083000400 sec Example: Optical layout of GEO 600 (80 nodes) The Hermite-Gauss analysis has been validated by: computing mode-cleaner autoalignment error signals (G. Heinzel) comparing it to OptoCad (program for tracing Gaussian beams by R. Schilling) comparing it to FFT propagation simulations (R. Schilling)

12. 5. September 2003 Andreas Freise Power Recycling Signals

13. 5. September 2003 Andreas Freise FINESSE http://www.rzg.mpg.de/~adf/ /virgo/VCS/1.0/VIRGOSW/Finesse/v0r93/... Windows, Linux Linux, AIX

14. 5. September 2003 Andreas Freise

15. Using Par-Axial Modes Hermite-Gauss modes allow to analyse the optical system with respect to alignment and beam shape. Both misalignment and mismatch of beam shapes (mode mismatch) can be described as scattering of light into higher- order spatial modes. This means that the spatial modes are coupled where an optical component is misaligned and where the beam sizes are not matched.

16. 5. September 2003 Andreas Freise From Plane Waves to Par-Axial Modes The electric field is described as a sum of the frequency components and Hermite-Gauss modes: Example: lowest-order Hermite-Gauss: Gaussian beam parameter q

17. 5. September 2003 Andreas Freise Gaussian Beam Parameters Example: normal incidence transmission through a curved surface: Transforming Gaussian beam parameters by optical elements with ABCD matrices:

18. 5. September 2003 Andreas Freise Frequency Noise Coupling Coupling of a frequency calibration peak into the dark fringe output: Difference between results for TEM 00 only and those with higher-order TEM modes: factor  100 phase  90°

19. 5. September 2003 Andreas Freise Mode Healing 1.00.10.01 T MSR

20. 5. September 2003 Andreas Freise Mode Healing power recycling only: Each recycling cavity minimises the loss due to mode mismatch of the respective other with signal recycling:

21. 5. September 2003 Andreas Freise Error signals, control signals photo detectors, multiple mixers Transfer functions amplitude-, phase- and frequency modulations Shot-noise-limited sensitivities Typical Tasks For FINESSE

22. 5. September 2003 Andreas Freise FINESSE: Versatile simulation software for user-defined interferometer topologies. Fast, easy to use. Higher-order spatial modes: Commissioning of interferometers with high- finesse cavities requires to understand the influences of mode-matching and alignment on control signals and noise couplings.