FINESSE 0.99.8 Frequency domain INterferomEter Simulation SoftwarE Andreas Freise Finesse is a fast interferometer simulation program. For a given optical setup, it com- putes the light field amplitudes at every point in the interferometer assuming a steady state. To do so, the interferometer description is translated into a set of linear equa- tions that are solved numerically. For convenience, a number of standard analyses can be performed automatically by the program, namely computing modulation-demodulation error signals, transfer functions, shot-noise-limited sensitivities, and beam shapes. Fi- nesse can perform the analysis using the plane-wave approximation or Hermite-Gauss modes. The latter allows computation of the properties of optical systems like telescopes and the effects of mode matching and mirror angular positions. 21 Jan 2010 Finesse, the accompanying documentation, and the example files have been written by: Andreas Freise School of Physics and Astronomy The University of Birmingham Edgbaston, Birmingham, B15 2TT UK [email protected] Parts of the Finesse source and ‘mkat’ have been written by Gerhard Heinzel, the document ‘sidebands.ps’ by Keita Kawabe, the Octave examples and its description by Gabriele Vajente, part of the Finesse source have been written by Paul Cochrane. The software and documentation is provided as is without any warranty of any kind. Copyright c by Andreas Freise 1999 – 2010. For the moment I only distribute a binary version of the program. You may freely copy and distribute the program for non-commercial purposes only. Especially you should not charge fees or request donations for any part of the Finesse distribution (or in connection with it) without the author’s written permission. No other rights, such as ownership rights, are transferred. ------------------------------------------------------------------------ FINESSE 0.99.8 - Help Screen - A. Freise 21.01.2010 ------------------------------------------------------------------------ ** usage (1) kat [options] infile [outfile [gnufile]] or (2) kat [options] basename in (2) e.g. basename ’test’ means input filename : ’test.kat’, output filename : ’test.out’ and Gnuplot batch filename : ’test.gnu’. ** Available options : -v : prints version number and build date -h : prints this help (-hh prints second help screen) -c : check consistency of interferometer matrix -max : prints max/min --server : starts Finesse in server mode --noheader : suppresses header information in output data files --perl1 : suppresses printing of banner --quiet : suppresses almost all screen outputs -sparse, -klu : switch to SPARSE or KLU library respectively ** Available interferometer components : l name P f [phase] node - laser m name R T phi node1 node2 - mirror (or: m1 name T Loss phi ... m2 name R Loss phi ... ) s name L [n] node1 node2 - space bs name R T phi alpha node1 node2 node3 node4 - beamsplitter (or: bs1 name T Loss phi ... bs2 name R Loss phi ... ) isol name S node1 node2 - isolator mod name f midx order am/pm [phase] node1 node2 - modulator lens f node1 node2 - thin lens ** Detectors : pd[n] name [f1 [phase1 [f2... ]]] node[*] - photodetector [mixer] pdS[n] name [f1 phase1 [f2... ]] node[*] - sensitivity pdN[n] name [f1 phase1 [f2... ]] node[*] - norm. photodetector ad name [n m] f node[*] - amplitude detector shot name node[*] - shot noise bp name x/y parameter node[*] - plots beam parameters cp cavity_name x/y parameter - plots cavity parameters gouy name x/y space-list - plots gouy phase beam name [f] node[*] - plots beam shape qshot name num_demod f [phase] node[*] - quantum shotnoise detector qshotS name num_demod f [phase] node[*] - quantum shotnoise sens. ** Available commands : fsig name component [type] f phase [amp] - signal tem input n m factor phase - input power in TEMs mask detector n m factor - mode mask for outputs pdtype detector type-name - set detector type attr component M value Rcx/y value x/ybeta value - attributes of m/bs (alignment angles beta in [rad]) map component [angle] [mapname] filename - read mirror map file savemap component mapname filename - save coefficients to file maxtem order - TEM order: n+m<=order gauss name component node w0 z [wy0 zy] - set q parameter gauss* name component node q [qy] (q as ’z z_R’) - set q parameter cav name component1 node component2 node - trace beam in cavity startnode node - startnode of trace retrace [off] - re-trace beam on/off deriv_h value - set deriv_h for diff phase 0-7 (default: 3) - change Gouy phases (1: phi(00)=0, 2: gouy(00)=0, 4: switch ad phase) ** Plot and Output related commands : xaxis[*] component param. lin/log min max steps - parameter to tune x2axis[*] component param. lin/log min max steps - second x-axis for 3D plot noxaxis - ignore xaxis commands const name value - constant $name variable name value - variable $name set name component parameter - variable $name func name = function-string - function $name lock[*] name $var gain accuracy - lock: make $var to 0 put component parameter $var/$x1/$x2 - updates parameter noplot output - no plot for ’output’ trace verbosity - verbose tracing yaxis [lin/log] abs:deg/db:deg/re:im/abs/db/deg - y-axis definition scale factor [output] - y-axis rescaling diff component parameter - differentiation deriv_h value - step size for diff ** Auxiliary plot commands : gnuterm terminal [filename] - Gnuplot terminal pause - pauses after plotting multi - plots all surfaces GNUPLOT \ ... \ END - set of extra commands for plotting. ------------------------------------------------------------------------ FINESSE 0.99.8 - Help Screen (2) - A. Freise 21.01.2010 ------------------------------------------------------------------------ ** Some conventions: names (for components and nodes) must be less than 15 characters long angles of incidence, phases and tunings are given in [deg] (a tuning of 360 deg corresponds to a position change of lambda) misalignment angles are given in [rad] ** Geometrical conventions: tangential plane: x, z (index n), saggital plane: y, z (index m) xbeta refers to a rotation in the x, z plane, i.e. around the y-axis R<0 when the center of the respective sphere is down beam (the beam direction is defined locally through the node order: i.e. mirror: node1 -> node2, beam splitter: node1 -> node3) beam parameter z<0 when waist position is down beam ** trace n: ‘n’ bit coded word, the bits give the following output: trace 1: list of TEM modes used trace 2: cavity eigenvalues and cavity parameters like FWHM, FSR optical length and Finesse trace 4: mode mismatch parameters for the initial setup trace 8: beam parameters for every node, nodes are listed in the order found by the tracing algorithm trace 16: Gouy phases for all spaces trace 32: coupling coefficients for all components trace 64: mode matching parameters during calculation, if they change due to a parameter change, for example by changing a radius of curvature. trace 128: nodes found during the cavity tracing ** phase 0-7: also bit coded, i.e. 3 means ‘1 and 2’ phase 1: phase of coupling coefficients k_00 set 0 phase 2: Gouy phase of TEM_00 set to 0 phase 4: ‘ad name n m f’ yields amplitude without Gouy phase (default: phase 3) ** kmn n: ‘n’ bit coded word, set computation of coupling coeffs.: kmn 0 : use approximation from Bayer-Helms kmn 1 : verbose, i.e. print coupling coefficients. kmn 2 : use numerical intergration if x and y misalignment is set. kmn 4 : use numerical intergration if x or y misalignment is set. kmn 8 : use always numerical intergration. (default: kmn 0) ** bp, possible parameters for this detector: w : beam radius w0 : waist radius z : distance to waist zr : Rayleigh range g : Gouy phase q : Gaussian beam parameter ** isol S, suppression given in dB: amplitude coefficient computed as 10^-(S/20) ** maxtem O/off : O=n+m (TEM_nm) the order of TEM modes, ‘off’ switches the TEM modus off explicitly vi Contents 1 Introduction 1 1.1 Motivation . 2 1.2 How does it work? . 3 1.3 Quickstart ................................... 3 1.3.1 Installation . 4 1.3.2 How to perform a simulation . 4 2 The program files 15 2.1 kat—the main program . 15 2.2 kat.ini—the init file for kat . 15 2.3 *.kat—the input files (how to do a calculation) . 18 2.4 *.out—the output files . 19 2.5 *.gnu—the Gnuplot batch files . 19 2.6 *.m—the Matlab script files . 20 3 Mathematical description of light beams and optical components 21 3.1 Introduction . 21 3.1.1 Static response and frequency response . 21 3.1.2 Transfer functions and error signals . 22 3.1.3 The interferometer matrix . 25 3.2 Conventions and concepts . 26 3.2.1 Nodes and components . 26 3.2.2 Mirrors and beam splitters . 27 3.3 Frequencies and wavelengths . 28 3.3.1 Phase change on reflection and transmission . 29 3.3.2 Lengths and tunings . 29 3.4 The plane-wave approximation . 30 3.4.1 Description of light fields . 31 3.4.2 Photodetectors and mixers . 32 3.4.3 Modulation of light fields . 32 3.4.4 Coupling of light field amplitudes . 38 3.4.5 Input fields or the ‘right hand side’ vector . 49 3.4.6 Photodetectors and demodulation . 54 3.5 Shot-noise-limited sensitivity . 57 3.5.1 Simple Michelson interferometer on a half fringe . 58 3.5.2 Simple Michelson interferometer on a dark fringe . 61 4 Higher-order spatial modes, the paraxial approximation 65 4.1 Finesse with Hermite-Gaussian beams . 65 4.2 Gaussian beams . 66 4.3 Higher order Hermite-Gauss modes . 68 4.3.1 Gaussian beam parameter . 69 vii Contents 4.3.2 Tangential and sagittal plane . 70 4.3.3 Gouy phase shift . 71 4.3.4 ABCD matrices . 71 4.4 Tracing the beam . 74 4.5 Interferometer matrix with Hermite-Gauss modes . 76 4.6 Coupling of Hermite-Gauss modes . 77 4.6.1 Coupling coefficients for TEM modes . 78 4.6.2 Mirror surface maps .