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: