Fraunhofer diffraction theory

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... 

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... 

Fraunhofer diffraction theory combines the above results to compute the light scattered at small angles from large particles. Such a particle is pictured in Figure 4.15. 

Early instruments employed low (forward) angle laser light scattering (LALLS) but these have been replaced by multi-angle instruments. MALLS instruments use Lorenz-Mie (often referred to as Mie) theory or Fraunhofer diffraction theory. 

The Crystalsizer is based on the principle of incoherent light diffraction thus turning the traditional optical structure back to front. Thus the same physical effect is measured as with traditional devices, but without using coherent laser light. The size distribution is determined using Fraunhofer diffraction theory. The robust construction of the instrument makes it suitable for use on-site . Typically, samples are removed from the process and fed to the Crystallizer sequentially. 

The relation between equivalent diameter and scattering intensity can be determined from Fraunhofer diffraction theory. The intensity at a point on the receiving plane diffracted by a circular aperture is determined from (Hecht and Zajac 1982) ... 

Laser has been widely used for the measurement of particle size through light scattering [30]. Laser scattering measurements are very accurate and fast. The techniques based on the Fraunhofer diffraction theory can measure the size of particles in the range of 2-100 pm. The Mie theory can extend the measurable size range to 0.1-1000 pm, if special light collection systems are used. 

The powder can be analyzed in the dry state or, more commonly, as a dilute suspension. The measurements can be made accurately and fairly rapidly. For instruments based on the Fraunhofer diffraction theory, a reliable size range is 2-100 xm. However, the use of special light collection systems and the Mie theory can extend the range to —0.1-1000 xm. 

In most modern instruments, the measurement is performed by an array of N detectors. Also, Mie scattering theory is used in many instruments instead of Fraunhofer diffraction theory. In one popular approach the particles are divided into size intervals and each interval is assumed to generate an intensity distribution according to the average size. In this case, the preceding equation becomes... 

We characterized the diffraction properties of the LC grating based on the Jones matrix method and Fraunhofer diffraction theory. By considering that the nematic LC layer consists of M plates with a thickness of dm = d/M, the grating vector is parallel to the x-direction, the substrates are parallel to the ay-plane, and that the substrate normal is parallel to z-direction, the Jones matrix of the LC grating is... 

At the other extreme the particle is large compared with the wavelength (a A) and the concepts employed in geometric optics and Fraunhofer diffraction theory can be used to advantage. Let a beam of parallel radiation be incident on any surface element of the spherical particle, and require the width D of the beam to be much larger than X and much smaller than a i.e., X D a. In geometric optics such a beam is called a ray. 

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