Biomedical Engineering Reference
In-Depth Information
Having this function, the spectrum I./ can be extracted by an inverse Fourier
transformation. Since the intensity at the detector is sampled for discrete OPDs, the
inverse FT is also a discrete operation. It is performed in the complex space
X
N
I./ D
I.x i / exp. i2x i /
(4.11)
i D1
Here N is the number of OPD's that were measured.
Equation 4.10 shows that the interferogram is a symmetric function with respect
to the OPD. It is most common to measure a symmetric interferogram by measuring
both positive and negative OPDs [ 41 , 43 ]. This means moving the mirror in both
directions around the zero OPD in the Michelson interferometer (Fig. 4.16 a) or
rotating the Sagnac interferometer in both positive and negative values around zero
OPD (Fig. 4.16 b).
Three additional operations are performed before or after the Fourier trans-
formation:
1. Phase correction . If the interferogram given by Eq. 4.10 is symmetric, then the
Fourier transformation (Eq. 4.11 ) will be a real function. Nevertheless, the inter-
ferogram will never be perfectly symmetric. Due to noise, as an example, there
will always be a deviation from symmetry even for perfect optics. Moreover, the
use of imperfect lenses can result in a phase shift of the two splitted beams so
that the intensity is an antisymmetric function. The phase correction solves that
in different possible ways. The most common way is to multiply the complex
measured spectrum I./ by the cosine of the phase at every wavenumber ,or
simply to take into account only the normal of the complex spectrum.
2. Apodization . Fourier transformation (as described in Eq. 4.11 ) is valid given an
infinite interferogram function, which is certainly impossible. The finite nature of
I.x/ can be observed as a multiplication of the infinite function with a square
window of width 2D,whereD is the maximal measured OPD. One of the known
traits of Fourier transformation [ 41 ] is that a multiplication gives a convolution
of the FT of the two functions. The FT of a square window is a sinc function
sin./= that has significant oscillations that will alter the spectrum, usually
in an unacceptable manner. Apodization is used to eliminate this. It comes on
cost of a small reduction of the spectral resolution, and different functions can
give different trade-offs [ 44 ]. The apodization can be performed either as a pre-
process by multiplying the interferogram with the apodization function or by
deconvolution of the FT as a post-process.
3. Zero filling . This operation has the effect of interpolation of the calculated
spectrum. It is a valuable function and it emphasizes the position of maxima
as measured by the system. It is a simple operation that adds zeros to the
interferogram tails. It is also common to use it in order to ensure that the
interferogram has a number of points that is equal to a power of 2 (2 n ). This
is required in order to use fast Fourier transformation (FFT) algorithms.
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