1. Higher order TEM modes: Why and How? Andreas Freise European Gravitational Observatory 17. March 2004

2. 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

3. 5. September 2003 Andreas Freise Example: 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°

4. 5. September 2003 Andreas Freise 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.

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

6. 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:

7. 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:

8. 5. September 2003 Andreas Freise Par-axial Approximation Par-axial approximation can be considered as a first order approximation In the parameter: To compute quantitative limits we can request: Diffraction angle of any beam should not be larger than 30 degrees Any misalignment between two beams should not differ by more than a factor of three The waist sizes of the beams should not differ by more than a factor of three The distances between the waist positions should not be larger than three times the Rayleigh range

9. 5. September 2003 Andreas Freise Par-axial Approximation

10. 5. September 2003 Andreas Freise Par-axial Approximation

11. 5. September 2003 Andreas Freise Par-axial Approximation

12. 5. September 2003 Andreas Freise