Diffraction Fraunhofer

In practical appHcations, diffraction instmments may exhibit certain problems. Eor example, there may be poor resolution for the larger droplets. Also, it is not possible to obtain an absolute measure of droplet number density or concentration. Furthermore, the Fraunhofer diffraction theory cannot be appHed when the droplet number density or optical path length is too large. Errors may also be introduced by vignetting, presence of nonspherical... 

Instrument Based on Fraunhofer Diffraction of Laser Light... 

In order to observe fringes, the screen should be placed in the regime of Fraunhofer diffraction where F/B B/X. In practice, such an interferometer can be realized by placing the stop immediately in front of a collecting optics, e. g., a lens or a telescope, and by observing the fringes in its focal plane (F = fes). 

For instance, with the introduction of SR sources, particles with a radius of a few nanometers can be studied with conventional methods. This has also stimulated a new kind of microscopy, named diffraction microscopy, where the Fraunhofer diffraction intensity patterns are measured at fine intervals in reciprocal space. By means of this oversampling a computer assisted solution of the... 

Figure 13. Fraunhofer diffraction pattern of a single slit illuminate with coherent monochromatic light the intensity distribution is shown for two...

Weiner, B. B. Particle and Droplet Sizing Using Fraunhofer Diffraction. In... 

Capillary hydrodynamic chromatography Fraunhofer diffraction Light-scattering photometry Phase Doppler anemometry Ultrasonic spectroscopy... 

If the linear dimensions of the aperture are small compared with r, we can expand (4.69) in powers of /r and i /r Fraunhofer diffraction results if we terminate this expansion at linear terms ... 

This is called a point-spread function, because it describes how what should be a point focus by geometrical optics is spread out by diffraction. The expression in the curly brackets is the one that is of interest. The other terms are phase and overall amplitude terms, as are usual with Fraunhofer diffraction expressions. The function Ji is a Bessel function of the first kind of order one, whose values can be looked up in mathematical tables. 2Ji(x)/x, the function in the curly brackets, is known as jinc(x). It is the axially symmetric equivalent of the more familiar sinc(x) = sin(x)/x (Hecht 2002), the diffraction pattern of a single slit, usually plotted in its squared form to represent intensity. Just as sinc(x) has a large central maximum, and then a series of zeros, so does jinc(x). Ji(x) = 0, but by L Hospital s rule the value of Ji(x)/x is then the ratio of the gradients, and jinc(0) = 1. The next zero in Ji(x) occurs when x = 3.832, and so that gives the first zero in jinc(x). This occurs at r = (3.832/n) x (q/2a)Xo in (3.2), which is the origin of the numerical factor in (3.1). 

Fraunhofer diffraction phenomena are observed when both the source and the point of observation are effectively at infinite distance from the diffracting ohject, obstacle, or aperture. This condition is sometimes brought about by passing the light from the source through a collimator before it is diffracted, and then focusing the parallel diffracted rays at the point of observation. 

Figure 10.24 Angular distribution for 12C +160 elastic scattering, showing Fraunhofer diffraction and the elastic scattering of 160 with 208Pb, which shows Fresnel diffraction. [From Valentin (1981).]...

The sizing methods involve both classical and modem instrumentations, based on a broad spectrum of physical principles. The typical measuring systems may be classified according to their operation mechanisms, which include mechanical (sieving), optical and electronic (microscopy, laser Doppler phase shift, Fraunhofer diffraction, transmission electron miscroscopy [TEM], and scanning electron microscopy [SEM]), dynamic (sedimentation), and physical and chemical (gas adsorption) principles. The methods to be introduced later are briefly summarized in Table 1.2. A more complete list of particle sizing methods is given by Svarovsky (1990). 

In this book, particles larger than 1 pm are of primary interest, and thus, only the Fraunhofer diffraction method, which can account for particles larger than 2-3 pm, is discussed here. The Fraunhofer diffraction theory is derived from fundamental optical principles that are not concerned with scattering. To obtain the Fraunhofer diffraction, two basic requirements must be satisfied. First, the area of the particle or aperture must be much smaller than the product of the wavelength of light and the distance from the light source to the particle or aperture. Second, this area must also be smaller than the product... 

Figure 1.6. Fraunhofer diffraction system for particle size analysis (a) Diffraction by a circular aperture (b) Diffraction by a particle cloud.

The transmittance of Fraunhofer diffraction for a circular aperture or spherical particles of diameter d can be expressed by... 

Figure 1.7. Fraunhofer diffraction pattern for circular aperture or opaque disk (from Weiner, 1984).

The number density function is usually obtained by using microscopy or other optical means such as Fraunhofer diffraction. The mass density function can be acquired by use of sieving or other methods which can easily weigh the sample of particles within a given size range. 

Weiner, B. B. (1984). Particle and Droplet Sizing Using Fraunhofer Diffraction. In Modem Methods of Particle Size Analysis. Ed. H. G. Barth. New York John Wiley Sons. 

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