Zero path difference

The position of ZPD (Zero Path Difference) is critical to the Fourier Transform calculation, since the algorithm assumes that the central burst in the interferogram is in fact the ZPD. However, due to the refractive index properties of the beamsplitter material, the ZPD is not at the same position for every wavelength measured. There are several ways to overcome these phase differences. The most common method is to use a correction factor, which is known as phase correction. This correction factor is calculated for every wavelength, based on a double sided interferogram, since this tends to minimize the effects of phase difference. In practice, most infrared spectrometers collect single sided interferograms, since this halves the mirror movement, and consequently the number of datapoints to be Fourier transformed. 

Figure 1.11 Synchronisation between the collection of FTIR spectra and the potential applied to the working electrode during potential modulation techniques. It is assumed that single-sided interferograms are collected during the forward sweep of the moving mirror (ZPD = zero path difference for the two paths of the interferometer).

Another source of error arises if the sample intervals are not exactly the same on each side of the maxima, corresponding to zero path difference. Phase correction is then required, and this correction procedure ensures that the sample intervals are the same on each side of the first interval and should thus correspond to a path difference of zero. 

The intensity at zero path difference I(0J is equal to the first term on the right hand side of equation A.5... 

The optical path difference (OPD) between the beams that travel to the fixed and movable mirror and back to the beamsplitter is called retardation, 8. When the path length on both arms of the interferometer are equal, the position of the moving mirrors is referred to as the position of zero retardation or zero path difference (ZPD). The two beams are perfectly in phase on recombination at the beamsplitter, where the beams interfere constructively and the intensity of the beam passing to the detector is the sum of the intensities of the beams passing to the fixed and movable mirrors. Therefore, all the light from the source reaches the detector at this point and none returns to the source. To understand why no radiation returns to the source at ZPD one has to consider the phases on the beam splitter. 

Phase effects dispersive phenomena exist in optical elements, and/or due to electronic filters used to reduce the bandwidth of the detector, which induce a wavenumber dependent phase lag. Under these circumstances, the interferogram can be written as... 

With Eq.2.48 one obtains a set of interferograms corresponding to an interfero-gram per baseline, this is, a spectroscopic measurement for each sampled point in the Mv-space. It must be noticed that in this situation, the concept of a spectroscopic zero path difference is not applicable anymore. For example, if the source is a binary consisting of two unresolved point sources, the interferometric phase shift will cause the separation of the two spectroscopic interferograms. This case is similar to the testbed implementation presented in the next chapter. 

When running the spectral arm, data is acquired every 16 p,m of travel, or 32 p,m of optical path difference (OPD), which corresponds to a Nyquist sampling of 4.7 THz. The data presented corresponds to a scanning of the spectral arm from the nominal zero path difference along 3.2cm, at 0.005cm/s. 

The Forman phase correction algorithm, presented in Chap. 2, is shown in Fig. 3.6. Initially, the raw interferogram is cropped around the zero path difference (ZPD) to get a symmetric interferogram called subset. This subset is multiplied by a triangular apodization function and Fourier transformed. With the complex phase obtained from the FFT a convolution Kernel is obtained, which is used to filter the original interferogram and correct the phase. Finally the result of the last operation is Fourier transformed to get the phase corrected spectrum. This process is repeated until the convolution Kernel approximates to a Dirac delta function. 

Figures.19 shows interferograms for the reference sources 1 (blue), 13 (green) and 25 (red) for baselines of 30, 50, 70, 90 110 and 130 mm. First of all, if we look at the level of continuum signal for this 3 sources we can observe that sources 1 and 25 have a similar level of approximately 125 counts while source 13 has a level of approximately 160 counts. This difference is due to a residual background illumination. Looking at the plot for the baseline 30 mm, it can be observed that the distance between the zero path difference (ZPD) of the source 1 and 13 is equal to the distance of ZPD between sources 13 and 25, approximately 10 xm. As the baseline is increased, this distance also increases. It can be observed that at baseline 130 mm...

The next step will be to perform some modifications on the CLEAN algorithm by coding a DFM specific one, in order to include some FTS algorithms such as phase correction and apodization. Apodization requires a unique zero path difference (ZPD), and research would need to be performed to design tailored apodization filters for DFM data. 

The instrument is capable of a 100 cm optical retardation. However, the method by which this path difference is achieved is different from the method of single-mirror motion employed by the previous two instruments. The Kitt Peak interferometer has two cat s-eye reflectors which move in a continuous-scan, reciprocating fashion. One reflector approaches the beam splitter while the other reflector retreats from it. Figure 10 shows the details of this interferometer. Both reflectors move on oil bearings. If the beam splitter is positioned so that the zero path difference occurs when one reflector is near the front of its track while the other... 

It is typical, however, for the system to be used in the symmetric mode, where the zero path difference occurs when both reflectors are in the middle of their tracks. This reduces the... 

Two techniques are useful for locating the zero path difference position of the mirrors. One of these is to use a monochromatic light source and locate that position of the two mirrors for which the central fringe has expanded to fill the field of view as mentioned above. Another is to use a monochromatic light source that is actually a doublet such as the sodium D line. The interference produced by two closely spaced spectral lines will be a sine wave whose amplitude varies at some lower sinusoidal frequency. The output of the interferometer is focused on a photomultiplier and the path length difference of the two mirrors is varied in some uniform manner (i,e. a sawtooth motion). 

If the output of the photomultiplier is displayed on an oscilloscope, the sinusoidal envelope which contains a high frequency sinusoidal modulation is observed and as the path lengths of the two arms approach equality, the modulation depth of the higher frequency sinusoidal modulation in the envelope increases. If the interferometer is adjusted to produce a maximum modulation depth, zero path difference has been located. If a white light is then used to illuminate the interferometer, the photomultiplier should detect the expected eight to ten fringes and present the typical interferogram of a broadband source. Fig, 3. 

Figure 2.2. Phase of the electromagnetic waves from fixed (soUd line) and movable (dashed line) mirrors at different values of the optical retardation (a) zero path difference (b) path difference of one-half wavelength (c) path difference of one wavelength. Note that constructive interference occurs for both (a) and (c) and all other retardations of integral numbers of wavelengths.

F ure 3.10. Dispersion of the zero-path-difference location for two possible chirping configurations. (Reproduced from [6], by permission of the author.)... 

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