Maximum path difference

Whatever the method, the cutoff is finite and absolute. Similar considerations prevail in the Fourier interferometer, where the maximum path difference determines Q. It would seem foolish to suggest that the information at these frequencies could be restored when it is not even present in the data. 

Fig. 24 Interferogram of the two monochromatic sources that would be obtained for a finite maximum path difference of the interferometer. (a) Finite interferogram. (b) Recorded spectrum. The two lines are completely merged into one.

We can conclude from this that the resolution of a Michelson interferometer is proportional to the maximum path difference up to which the interferogram has been measured. When we now consider the case of three narrow lines, we must remember that we have to calculate I (v) from I (s) by means of a Fourier transform [see Eqs. (2.10) and (2.12)]. However, the Fourier integral cannot be executed over s from — oo to - -oo, since the interferogram I s) can be determined experimentally only over a finite range ( —Smax s -j-Smax)- Therefore, the integration too can be performed only over a finite range. 

From the required value of Av, the maximum path difference Smax up to which the interferogram must be recorded is obtained (cf. Table 1). 

Spectral range min— inax (cm t) Sampling interval As (P-jm) Resolution Av (cm-l) Maximum path difference Smax ) (mm) Number of interferogram points N = S]oax/ds Number of spectrum points JVf = 2 f max Vmin ) Av... 

It should be noted further that an increase in resolution is easily achieved in this case by increasing the maximum path difference and the scanning time. The power flux is not influenced by an increase of Smax- However, there will be an increase in noise, as we shall see later. An increase in resolution means for a grating instrument a reduction of slit width and hence, a reduction of the power flux, which is proportional to the square of the slit width [see Eq. (5.12)]. It also seems worth mentioning that the Jacquinot or throughput advantage exists not only in the Michelson interferometer but also in other instruments, e.g. a Fabry-Perot interferometer. 

A real instrument delivers yr(T) for the interval [0, L/c], where L is the maximum path difference which depends on the maximum displacement of the moving mirror of the FTS. The L-dependent instrumental function, that is the response of an instrument to a monochromatic incident light E(t)-coscobt, can be written as... 

The resolution is limited by the maximum path difference between the two beams. The limiting resolution in wavelength (cm ) is the reciprocal of the pathlength difference (cm). For example, a... 

The Fourier spectrometer measures the interferogram, whose variable part is proportional to the autocorrelation function of the studied radiation. The spectrum is obtained by the Fourier transformation of that part. If L is the maximum path-difference between interfering beams it means that the autocorrelation function is known in the interval from 0 to T max- = L/c, which determines the resolution (the minimal resolved spectral interval 6f),... 

For a given solid angle through the interferometer, each optical frequency, then, has a specific value of the optical retaradation where the sine function is zero. If the interferometer is driven beyond this path difference, the phase of the modulation for that frequency is reversed, and energy at that frequency is removed from the spectrun rather than added to it as the optical retardation continues to increase. Therefore, in order to insure that this does not occur for any frequencies within the spectral bandwidth even at maximum path difference, the absolute maximum solid angle which can be used is... 

Using the smaller value of the solid angle will improve the fringe contrast of the highest spectral frequencies at maximum path difference and will produce higher resolution spectra for all frequencies in the bandwidth. 

The resolution for an FTIR instrument is limited by the maximum path difference between the two beams. The limiting resolution in wavenumbers (cm ) is the reciprocal of the pathlength difference (cm). For example, a pathlength difference of 10 cm is required to achieve a limiting resolution of 0.1 cm . This simple calculation appears to show that it is easy to achieve high resolution. Unfortunately, this is not the case since the precision of the optics and mirror movement mechanism become more difBcnlt to achieve at longer displacements of pathlengths. 

The basic principle of all interferometers may be summarized as follows (Fig. 4.24). The indicent lightwave with intensity Iq is divided into two or more partial beams with amplitudes Ak, which pass different optical path lengths Sic = nxk (where n is the refractive index) before they are again superimposed at the exit of the interferometer. Since all partial beams come from the same source, they are coherent as long as the maximum path difference does not exceed the coherence length (Sect. 2.8). The total amplitude of the transmitted wave, which is the superposition of all partial waves, depends on the amplitudes Ak and on the phases 0 = 0o + 27r A / of the partial waves. It is therefore sensitively dependent on the wavelength X. 

The maximum path difference A5 that still gives interference fringes in the plane B is limited by the coherence length of the incident radiation (Sect. 2.8). Using spectral lamps, the coherence length is limited by the... 

Doppler width of the spectral lines and is typically a few centimeters. With stabilized single-mode lasers, however, coherence lengths of several kilometers can be achieved. In this case, the maximum path difference in the MI is, in general, not restricted by the source but by technical limits imposed by laboratory facilities. 

The spectral resolving power X/AX of the Michelson interferometer equals the maximum path difference As/X measured in units of the wavelength X. 

The finesse is a measure for the effective number of interfering partial waves in the interferometer. This means that the maximum path difference between interfering waves is A5max = F 2nd. 

A comparison with the resolving power v/Av = mN = NAs/X of a grating spectrometer with N grooves shows that the finesse F can indeed be regarded as the effective number of interfering partial waves and F As can be regarded as the maximum path difference between these waves. 

The spectral resolving power discussed for the different instruments in the previous sections can be expressed in a more general way, which applies to all devices with spectral dispersion based on interference effects. Let As be the maximum path difference between interfering waves in the instrument, e.g., between the rays from the first and the last groove of a grating (Fig. 4.62a) or between the direct beam and a beam reflected m times in a Fabry-Perot interferometer (Fig. 4.62b). Two wavelengths X and X2 = + AX can still be... 

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