Far field diffraction

By deriving or computing the Maxwell equation in the frame of a cylindrical geometry, it is possible to determine the modal structure for any refractive index shape. In this paragraph we are going to give a more intuitive model to determine the number of modes to be propagated. The refractive index profile allows to determine w and the numerical aperture NA = sin (3), as dehned in equation 2. The near held (hber output) and far field (diffracted beam) are related by a Fourier transform relationship Far field = TF(Near field). 

The work by M. Arndt el al. starts with the description of far-field diffraction experiments with fullerene molecules Cm. Since for larger objects the far-field observations become much more challenging, the authors also study the feasibility of near-field Talbot Lau interferometry with 6Vo. This particular technique allows them to work with a spatially incoherent beam and thus with a much increased count rate. Moreover it has a better wavelength scaling in the grating constant and can potentially be applied to much smaller wavelengths at a reasonable grating constant. 

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. 

Based on these historical achievements one may ask how far one might be able to extend such quantum experiments and for what kind of objects one might still be able to show the wave-particle duality. Recently, a new set of experiments exceeding the mass and complexity of the previously used objects by about an order of magnitude has been developed in our laboratory. These far-field diffraction experiments with the fullerene molecule Ce0 will be shown in Sec. 1. 

Figure 1. Setup of the far-field diffraction experiment. Fullerenes are sublimated, collimated by two narrow slits and diffracted at a nanofabricated SiN grating. The ionizing laser is tightly focused and scans over the molecular density distribution.

Figure 2. Far-field diffraction curve recorded with the full thermal distribution of the Cm beam [Nairz 2003]. The velocity is centered around v = 200 m/s and has a FWHM of Av/v = 0.6.

Figure 3. First order far field diffraction at a two slit grating.

In principle, the diffraction patterns can be quantitatively understood within the Fraunhofer approximation of Kirchhoff s diffraction theory as described in any optics textbook (e.g., [Hecht 1994]). However, Fraunhofer s optical diffraction theory misses an important point of our experiments with matter waves and material gratings the attractive interaction between the molecule and the wall results in an additional phase of the molecular wavefunction [Grisenti 1999], Although the details of the calculations are somewhat involved2, the qualitative effect of this attractive force on far-field diffraction can be understood as a narrowing of the real slit width to an effective slit width [Briihl 2002], For our fullerene molecules the reduction can be as big as 20 nm for the unselected molecular beam and almost 30 nm for the slower, velocity selected beam. The stronger effect on slower molecules is due to the longer and therefore more influential interaction between the molecules and the wall. 

Hence the far field diffraction pattern at the point P is related to the aperture function A x, y), by the Fourier transform. The final step is to remove the scaling effect of R in the equation, as it does not affect its structure, just its size. The coordinates [a, jS] are absolute and are scaled by the factor R. For this reason, we normalise the coordinates and define the Fourier transform of the aperture in terms of its spatial frequency components [u, v],... 

It is simple to repeat this calculation in the orthogonal direction to create the far field diffraction pattern in two dimensions. The far field of a square aperture with transmission of A and width a therefore its Fourier transform. 

We have seen that the current distribution in the detector plane at any instant is the far-field diffraction pattern... 

Airy Pattern Far field diffracted by a circular aperture illuminated by a plane wave or field on the focal plane of an axisymmetric imaging system. 

We have recently shown that the presence of phase-separated structures in polymer-blend microparticles can be indicated qualitatively by a distortion in the two-dimensional diffraction pattern. The origin of fringe distortion from a multi-phase composite particle can be understood as a result of refraction at the boundary between domains of different polymers, which typically exhibit large differences in refractive index. Thus, the presence of separate sub-domains introduces optical phase shifts and refraction resulting in a randomization (distortion) in the internal electric field intensity distribution that is manifested as a distortion in the far-field diffraction pattern. 

We find from diffraction theory that, E x,y) is in fact the analytical Fourier transform of A(x,y). The pattern of E x,y) is called the far field diffraction pattern of the original aperture function. Hence we have the relationship between E x,y) and A x,y) linked by the Fourier transform... 

Instrumentation for Particle Size Analysis by Far Field Diffraction Accuracy, Limitations and Future... 

There are several instruments currently available for particle sizing based on far field diffraction. Details of many of them are given in Table 1. This is not intended to be comprehensive, but to illustrate the range of possibilities. The instruments are now easy to use. They provide guidance on correct particle concentration in the circulating dispersant, advice on the compatibility of the sample/instrument size ranges etc. 

For a far field diffraction instrument (Malvern ST1800), amongst the first to publish results of testing were AzzopardP and Hammond. The former used photography and sedimentation to determine the size distribution of samples of glass spheres. Hammond employed commercially available "standard" distributions of polystyrene spheres. The result of the comparison is shown in Figure 3 in terms of the Sauter Mean Diameter. 

A systematic test of particle sizing instruments has been carried out by Allen of Dupont. This covered a wide range of instruments, including those which do not use far field diffraction. Standard BCR silica was used in this exercise. This is supplied by the Community Bureau of Reference in Brussels (part of the CEC) and is available in different median sizes from about 1 to 50 fim. At the 2nd International Congress on Optical Particle Sizing, Allen described a series of tests with BCR 66 (particle size around 1 pm). A summary of the accuracy and reproducibility for far field diffraction instruments is shown in Figure 8. Here equations 13 and 14 are used to define accuracy and reproducibility respectively. It is noted that because of the size of the particles, it is not the diffraction aspect of the instrument that is being tested. This probably explains the apparent poor accuracy of some instruments. 

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