Fraunhofer approximation diffraction

Fraunhofer rules do not include the influence of refraction, reflection, polarization and other optical effects. Early Iziser particle analyzers used Fraunhofer approximations because the computers of that time could not handle the storage cuid memory requirements of the Mie method. For example, it has been found that the Fraunhofer-based instrumentation cannot be used to measure the particle size of a suspension of lactose (R.I. = 1.533) in iso-octane (R.I. = 1.391) because the relative refractive index is 1.10, i.e.- 1.533/1.391. This is due to the fact that diffraction of light passing through the particles is nearly the same as that passing around the particles, creating a combined interference pattern which is not indicative of the true... 

To monitor nanoparticle swelling in salt environment, we employed laser diffraction Mastersizer Micro Particle Analyzer MAF5000 (Malvern) with a dynamic range of 0.3 to 300 pm. This instrument utilizes Mie scattering algorithm with the Fraunhofer approximation. 

The newer la.ser diffraction instrument allows measurement for particle sizes ranging from 0.1 pm to 8 mm (7). Most of the laser diffraction instruments in the pharmaceutical industry use the optical model based on several theories, either Fraunhofer, (near-) forward light scattering, low-angle laser light scattering, Mie, Fraunhofer approximation, or anomalous diffraction. These laser diffraction instruments assume that the particles measured are spherical. Hence, the instrument will convert the scattering pattern into an equivalent volume diameter. A typical laser diffraction instrument consists of a laser, a sample presentation system, and a series of detectors. 

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. 

Figure 7-11 Calculated response curves for the intensity of diffracted light based on Fraunhofer approximation for different minimum collection angles (maximum collection angle 4.76°, laser wave length 514.5 ran)...

For a given C(x ), Equation 3 can be evaluated to produce a diffraction pattern appearing at the screen. The integral may be solved for any set of experimental conditions, but the problem is simplified significantly if the screen is placed at some large distance away from the electrode. This is a standard approximation in solving diffraction problems known as the Fraunhofer approximation,13 and can easily be realized experimentally. When this approximation is made, the expression for f-7 is simplified, and Equation 3 becomes... 

Particle Size Laser Refractometiy is based upon Mie scattering of particles in a liquid medium. Up until about 1985, the power of computers supplied with laser diffraction instruments was not sufficient to utilize the rigorous solution for homogeneous spherical particles formulated by Gustave Mie in 1908. Laser particle instrument manufacturers therefore used approximations conceived by Fraunhofer. 

For light scattering studies, the far held approximation (Fraunhofer diffraction) can be applied to the general vector diffraction equation [2]. For the diffracted electric held it follows that... 

Such an expression can only be solved directly for a few specific aperture functions. To account for an arbitrary aperture, we must approximate, simplify and restrict the regions in which we evaluate the diffracted pattern. If the point P is reasonably coaxial (close to the z axis, relative to the distance R) and the aperture A(x, y) is small compared to the distance R, then the lower section of (1.2) for dE can be assumed to be almost constant and that for all intents and purposes, r = R. The similar expression in the exponential term in the top line of (1.2) is not so simple. It cannot be considered constant as small variations are amplified through the exponential. To simplify this section we must consider only the far field or Fraunhofer region where. 

It should be noted that these two definitions are oversimplified in reality, the boundary between the Fraunhofer and Fresnel approximations is not so clear-cut, since the Fresnel-number criterion is a rather crude delimiter. Regardless, for conceptual reasons and simplicity, we only will discuss Fraunhofer diffraction here and restrict the principal treatment to the two most commonly encountered (useful) diffraction objects, namely an elongated straight slit (e.g. as encountered in spectrometers) and a circular aperture (pin holes, diaphragms, any circular lens). 

The Fraunhofer theory was the basis for the first (approximating) optical model for particle size measurement. For particles with a diameter dp larger than the wavelength A, Fraunhofer diffraction is often assumed [105]. However, only scattering by opaque particles or particles with a large real refractive index ratio m, i.e., the ratio of the refractive index of scattering particles to that of the fluid. 

The Rigorous Solution Mie Theory The Zeroth-order Approximation Rayleigh Scattering The First-order Approximation RDG Scattering The Large-end Approximation Fraunhofer Diffraction Numerical Approaches... 

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