Modulation and Bessel functions

(An optics simulation, part of the Finesse examples)

The following shows a simple example for a Finesse simulation, including the text input file, a brief explanation and the resulting plot. This example has been used in the review article Interferometer Techniques for Gravitational-Wave Detection (see section 2) and is meant to introduce some of the syntax an usage of Finesse.

This example demonstrates the basic use of modulators and amplitude detectors for detecting specific frequency components of the light field.

The input file

%------------------------------------------------------------------------
% Finesse input file to plot the amplitude of modulation
% sidebands, effectively the Bessel function of the modulation index. 
% Andreas Freise 15.08.2009
%------------------------------------------------------------------------
				
laser i1 1 0 n0               % laser P=1W f_offset=0Hz   
mod   eom1 40k .05 5 pm n0 n1 % phase modulator f_mod=40kHz, modulation index=0.05
ad    bessel1 40k n1          % amplitude detector f=40kHz        
ad    bessel2 80k n1          % amplitude detector f=80kHz        
ad    bessel3 120k n1         % amplitude detector f=120kHz       
xaxis eom1 midx lin 0 10 1000 % x-axis: modulation index of eom1 

yaxis abs                     % y-axis: plot `absolute' amplitude 
 

A laser beam is passed through an electro-optical modulator (EOM) which applies a phase modulation with a frequency of 40 kHz and a modulation index of 0.05. The modulator is set such that up to five orders or sidebands will be created (at plus/minus 40, 80, 120, 160 and 200 kHz respectively). The modulated beam is then detected with three amplitude detectors, each of which is set to a particular frequency. This means for example, the detector `bessel2' can only `see' the sideband at +80 kHz and nothing else of the beam.

The optical layout

The optical layout simply has the laser, the EOM and the three amplitude detectors. Note that all three detectors are connected to the same note and thus probe the same field amplitude.

Output graphs

Phase modulation (with up to five higher harmonics is applied to a laser beam and amplitude detectors are used to measure the field at the first three harmonics. The three traces show the amplitude of the single sidebands at 40kHz, 80 kHz and 120 kHz as a function of the modulation index at the modulator. The sideband amplitude are given as Bessel functions of the first kind J_n(m) with n being the sideband order and m the modulation index.

Finesse simulation result: Each trace shows the field amplitude of a single phase modulation sideband.