1.2 The Basic Mathematics of Interferometry

Consider a monochromatic light source with wavenumber ( = 1/, where is the wavelength of the source) and intensity , directed into the classical Michelson interferometer shown in Figure 1.1. In the ideal case, half of the incident signal is transmitted by the beam splitter towards the movable mirror along path A while half is reflected towards the stationary mirror along path B. If the difference in lengths, , of the two light paths to the detector is zero or an integer multiple of , then the signals from both arms of the interferometer will combine constructively after the beam splitter, and the output measured by the detector will be

If the path difference is nonzero and not an integer multiple of then interference will occur, and the detected signal will depend on the path difference such that

Clearly, to conserve energy a signal of strength

is directed out of the interferometer in the direction of the input. The phase difference of between the two output beams arises from the different number of reflections and transmissions that the light experiences as it traverses the two paths.

The above expressions can be generalized to sources with continuous spectral distributions, which can be thought of as the sum of an infinite number of monochromatic sources. Therefore, for a source with a spectral intensity distribution (), the interference signal detected at a given path difference will be

Note that at zero path difference (ZPD) and

so that

while for

In other words, the strength of the DC component of the signal is about one half that of the burst signal at ZPD.

By employing a different design than that shown in Figure 1.1, the CFHT FTS makes use of both outputs of the interferometer. The signals from the two detectors, () and () (1.2 and 1.3), are subtracted, after the appropriate gains have been applied to each to balance the DC component, and as a result the DC component is not recorded. This has the advantage of partially compensating for source fluctuations. The remaining AC component is the interferogram and is given by

so that the recorded signal is essentially the Fourier transform of the source's spectrum. Consequently, the source's spectral distribution can be recovered by applying an inverse Fourier transform to the recorded interferogram:

In practice, the exact spectral distribution of the source can not be recovered since an interferometer is restricted to finite path differences. Because of this restriction, a truncated version of the interferogram is actually recorded:

where () is the so-called `boxcar' function:

and is the maximum path difference. Therefore, the spectral distribution which is recovered from a measured interferogram is:

Now, according to the convolution theorem:

where indicates a Fourier transform, and are arbitrary functions, and the symbol `' indicates convolution. Applying the convolution theorem, the spectral distribution which is recovered is then

The Fourier transform of a boxcar function is a sinc function:

and thus

The effect of convolving the actual spectral distribution with the sinc function is to lower the spectral resolution. As the maximum path difference, , increases the sinc function becomes narrower and the maximum resolution which can be recovered increases.

The manner in which the path difference is varied is another practical consideration which can affect the recovery of the spectral distribution. The CFHT FTS varies the path difference in discrete steps rather than in a continuous fashion; such a scheme makes it possible to monitor the position of the moving mirror with a high degree of accuracy. However, the interferogram is then digitized, and an appropriate sampling interval must be selected to avoid aliasing. According to the sampling theorem, aliasing in the wave number interval between and can be prevented by using a step size for the path difference which is no more than:

Please send comments and suggestions to: veillet@cfht.hawaii.edu