Spectral Imaging: Methods, Design, and Applications - Biomedical Optical Imaging Technologies: Design and Applications - page 137

To solve this problem, a spectral imaging that works on the Fourier spectroscopy
principle was designed to work with a Sagnac interferometer as shown in Fig. 4.16 b
[ 40 ]. This interferometer belongs to a family of interferometers known as common-
path interferometers. It has the advantage that the splitted beams are traveling
through the same region of space and the same mirrors but in opposite directions.
Therefore, to the first order, mechanical changes that may result from temperature
expansion or other reasons are canceled out, making the interferometer very robust.
The spectral imaging system is constructed of two lenses, two mirrors and a beam
splitter. The lens L1 collimates the light before entering to the interferometer. The
beam splitter splits the beams to a reflected one and a transmitted one. These travel in
opposite directions through two reflections from mirrors M1 and M2, and the light is
combined together and focused by the lens L2 to the array detector. The mirrors and
beam splitter that are shown inside the gray circle (Fig. 4.16 b) are mounted together
on a rotating stage. Note that there are also beams that are reflected (or transmitted)
backward to the light source, but the intensity of this part depends on the position of
the interferometer so that on average only half of the intensity goes to the detector.
The optical geometry of such a system will keep the focusing point always at the
same spot if the stage is rotated, and therefore, the object will always be imaged
exactly on the same pixels without any shift or aberration. On the other hand, it can
be shown by following the optical path of the transmitted and reflected beams that
the OPD depends on the rotation angle. This can be formulated as

OPD. / D C: (4.6)

It means that the interferometer acts as an OPD generator without effecting the
position of the image on the array detector. The effect of the OPD is to modify the
intensity of each pixel while changing the angle .

4.4.4.1 Extracting the Spectrum from the Interferogram

In order to understand the method in which Fourier spectroscopy works, one has to
follow the actual signal that is measured by the detector [ 41 , 42 ].
Recall the light collected from a single pixel on the detector origin from a well-
defined spot in the sample. It is better to describe this light through its electric
field properties. For a single one-dimensional wavelength, this electric field can be
described as E D E 0 cos.2x= !t/where E 0 describes the amplitude of the
field, is the wavelength, ! D 2c= is the angular frequency, c is the speed of
light, x is the distance traveled by the wave from a certain reference point, and t
is the time. In Fourier spectroscopy, it is common to change the wavelength to the
wavenumber units simply described as D 1=. The light would in general contain
many different wavelengths, and therefore the total electric field is

Z

E D A./ cos .2x !t/ d: (4.7)