Higher order TEM modes: Why and How?

Andreas Freise

European Gravitational Observatory

17. March 2004

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

5. September 2003 Andreas Freise

Example: Frequency Noise Coupling

Coupling of a frequency calibration peak into the dark fringe output:

Difference between results forTEM00 only and those withhigher-order TEM modes: factor 100 phase 90°

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 opticalcomponent is misaligned and where the beam sizes are notmatched.

5. September 2003 Andreas Freise

Gaussian Beam Parameters

Compute cavity eigenmodes

start node

Trace beam and set beam parameters

5. September 2003 Andreas Freise

Mode Mismatch and Misalignment

Mode mismatch or misalignemt can be described as light scatteringin higher-order spatial modes. Coupling coefficiants for the interferometer matrix are derived by projecting beam 1 on beam 2:

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:

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

5. September 2003 Andreas Freise

Par-axial Approximation

5. September 2003 Andreas Freise

Par-axial Approximation

5. September 2003 Andreas Freise

Par-axial Approximation

5. September 2003 Andreas Freise