Multiple Parameter Simulation of the Fabry Perot Interferometer

Two simulation programmes have been written as aids to the design of Fabry-Perot interferometers. The use of the Modulation Transfer Function vis-a-vis the Aperture Function is discussed and an example given. Other features of the simulations are described briefly.

Natural broadening and collisional broadening of a spectral line gives rise to a Lorentzian intensity distribution and Doppler broadening gives a Gaussian intensity distribution (Gill 1965). The shape of the airglow spectral line is given by the convolution of a Lorentzian and Gaussian function which is called a Voigt function (Rees M.A. 1989). The Voigt profile is Gaussian from the peak out to about three full widths at half height, from the peak (Thorne, 1974) so for the purpose of this discussion the airglow spectral line may be considered to have a Gaussian shape.

The shape of the recorded spectral line is further broadened by the interferometer instrument function. The instrument function is composed of a function representing the line shape due to the multiple reflections in the etalon and subsequent interference.

An Airy function is normally used to describe the intensity of the interference fringes (Lothian 1975, Hecht 1974) but a combination of Lorenzian functions can be used (Vaughan 1989). The spectrum can be modified by a Gaussian function representing imperfections in the etalon and aberrations in the rest of the optics.

If Lorenzian functions are used to describe the action of the etalon then the recorded spectrum can be modelled with a Voigt function (Vaughan 1989). However, the usual procedure is to model the recorded spectrum using a convolution of Airy and Gaussian functions.

In a none-imaging (scanning) interferometer there is a broadening of the recorded spectral line due to the effect of the sampling aperture or field stop. An aperture function is normally expressed in terms of the fraction of an order of interference d m admitted by the aperture or field stop (Hernandez 1986). Such a function is given by Wilksch (1975) as:

q is the angle between an accepted ray and the normal to the reflecting surfaces of the etalon plates and d q is the range of angles accepted by the aperture.

For a scanning instrument d q is fixed and the fractional change in t is very small so the aperture function is essentially fixed. For an imaging interferometer a d q is defined by each pixel. For a constant d m, d q will change with the distance between each bin or channel of the spectrum. The cone of rays admitted by the field stop is much wider than that accepted by a single pixel and so a better way to describe broadening by the detector is to use the modulation transfer function (MTF) of the imaging detector. The MTF of an Imaging Photon Detector has been modelled numerically and the broadening of spectral peaks is shown below. The bin number is proportional to the square of the radius and the peaks would all have the same width in bins if the MTF of the detector and optics was always unity.

In addition to the MTF the effect of temperature and pressure changes of air (nitrogen) in the etalon chamber have been modelled using the equation (Kaye and Laby 1972).

Other design parameters which can be varied in simulations are focal length, etalon spacing, air pressure, air temperature and instrument finesse. In addition it is possible to simulate change in wavelength and temperature of airglow emissions.

This work has been supported by an Australian Research Council grant at La Trobe University

Gill, T P. The Doppler Effect, Logos Press, London, 1965

Hecht, E, Zajac, A, Optics, Addison-Wesley, 1974

Hernandez, G. Fabry-Perot Interferometers, Cambridge Studies in Modern Optics 1986

Kaye, G W C. Laby, T H. Tables of physical and chemical constants, Longman, London & New York ,1972

Lothian, G F. Optics and its uses, Van Nostrand Reinhold Company, London, 1975

Rees, M.A. Physics and chemistry of the upper atmosphere, Cambridge University Press, Cambridge, 1989

Thorne, A.P. Spectrophysics, Chapman and Hall, London, 1974

Vaughan, J. M. The Fabry-Perot interferometer, Adam Hilger, Bristol,1989

Wilksch, P A Measurement of Thermospheric Temperatures and Winds Using a Fabry-Perot Spectrometer, Thesis for PhD University of Adelaide, 1975