Fringe contrast

We are now ready to derive an expression for the intensity pattern observed with the Young s interferometer. The correlation term is replaced by the complex coherence factor transported to the interferometer from the source, and which contains the baseline B = xi — X2. Exactly this term quantifies the contrast of the interference fringes. Upon closer inspection it becomes apparent that the complex coherence factor contains the two-dimensional Fourier transform of the apparent source distribution I(1 ) taken at a spatial frequency s = B/A (with units line pairs per radian ). The notion that the fringe contrast in an interferometer is determined by the Fourier transform of the source intensity distribution is the essence of the theorem of van Cittert - Zemike. 

The next source we investigate is the surface of an extended, limb-darkened star whose apparent diameter increases from 1 to 25 milli-arcseconds. The visibility function of such a source resembles the Airy function, varying periodically between zero and a maximum value which decreases with increasing frequency. Note how the fringe contrast vanishes repeatedly to rise again without reaching its previous maximum value as the source s apparent diameter... 

The fundamental quantity for interferometry is the source s visibility function. The spatial coherence properties of the source is connected with the two-dimensional Fourier transform of the spatial intensity distribution on the ce-setial sphere by virtue of the van Cittert - Zemike theorem. The measured fringe contrast is given by the source s visibility at a spatial frequency B/X, measured in units line pairs per radian. The temporal coherence properties is determined by the spectral distribution of the detected radiation. The measured fringe contrast therefore also depends on the spectral properties of the source and the instrument. 

The external geometric differential delay (see below) of an off axis source is exactly balanced within a Fizeau interferometer, resulting in fringes with the same phase on top of each source in the field. The position of a source may differ from the position of zero OPD in a Michelson interferometer depending on how dissimilar entrance and exit pupils are. The fringe contrast of off-axis sources also depend on the temporal degree of coherence of the detected light. 

Figure 4. The fringe contrasts measure the resemblance (cross-correlation) of the two mixed fields, (right).

Obviously, the analysis of the correlation between the two fields emerging from the telescope and related devices makes necessary to avoid dissymmetry between the interferometric arms. Otherwise, it may result in confusion between a low correlation due to a low spatial coherence of the source and a degradation of the fringe contrast due to defects of the interferometer. The following paragraphs summarize the parameters to be controlled in order to get calibrated data. 

The fringes contrasts are subject to degradation resulting from dissymmetry in the interferometer. The optical fields to be mixed are characterized by a broadband spectrum so that differential dispersion may induce a variation of the differential phase over the spectrum. Detectors are sensitive to the superposition of the different spectral contributions. If differential dispersion shifts the fringes patterns for the different frequency, the global interferogramme is blurred and the contrast decreases. Fig. 5 shows corresponding experimental results. 

Figure 5. Evolution of the fringes contrast C as a function of the differential dispersion (a.u.) The maximum of this function corresponds to the cancellation of differential dispersion between the fibre arms of the interferometer.

Figure 8. Calibration of the fringes contrasts by means of spatial filtering in monomode...

Particle passes through the center of the beam crossing volume to prevent fringe contrast variations... 

Figure 4.14. Calculated curves showing 7o(a) and /g(a) as a function of thickness z for = 0. These curves correspond to the intensity of thickness fringes in a wedge-shaped crystal in BF and DF, respectively. Note the decrease in the fringe contrast with increasing thickness. tg/tg = 0.05 and (From Hirsch et al.

The minimum value of a for which fringe contrast is observed is about 0.04x = 7.2°, according to Humphreys, Howie, and Booker (1967). 

Stacking faults are a-boundaries for which a = 2xg R. (0gi-0g2) is zero for all g. In some structures, stacking faults and twins are closely related, and different regular sequences of these defects produce various polytypes. Wollastonite is a relatively simple example of such a structure, for which the stacking faults have been studied in some detail by TEM, both by their a-fringe contrast and in two-dimensional high-resolution lattice images. 

Abstract We present a review of recent experiments on molecular coherence and decoherence with fullerene molecules. Nearly perfect quantum interference with high fringe contrast can be observed in far-field diffraction as well as in near-field interferometry, when the molecules are sufficiently well isolated from their environment. This is true for ambient pressures below 10-7 mbar and internal temperatures below 1000 K. The fringe contrast decreases gradually as the interaction with the environment is smoothly turned on by either increasing the ambient pressure or by heating the molecules. 

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