Fresnel length

The Fraunhofer approximation is useful when sF 5 1, i.e. well beyond the Fresnel distance F = a jX. Closer to the transducer the field must be calculated numerically (Zemanek 1971). If aj/X > 1, the amplitude and phase fluctuate considerably in the region between the face of the transducer and the plane sF = 1 in particular there are nulls along the axis in the region 0 < s <0.5. The final maximum in the amplitude on the axis occurs at sp = 1, i.e. F = a /X, which is known as the Fresnel length. In the plane perpendicular to the axis at sp = 1 the field is reasonably well behaved in both amplitude and phase. 

In the simplest case of the interface between two bulk media, the reflectivity is related to the refractive index difference across the interface, and is described by Fresnel s Law [18]. The refractive index, n, is related to the neutron scattering length density, such that... 

A nice qualitative derivation of the Gladstone-Dale equation was given by Schoorl (1920). The Huygens-Fresnel wave optics leads to the conclusion that the refractive index is equal to the ratio of the light velocities in the two media of transmission and also to the ratio of the respective wave-lengths so Snellius law can be extended to... 

In this case, properly accounting for diffraction effects reduces the required focal length by a factor of 2. The distance Zg is called the confocal distance, which we introduced in Section III. It separated the near-field (z Zg) and far-field (z Zg) regions, or equivalently, the Fresnel and Fraunhofer diffraction regions. 

In this expression Rp is the Fresnel reflectivity of the water-air interface, i.e., the reflectivity of an infinitely sharp interface between homogeneous media, p is the scattering-length density of the subphase and p(z) is that of the monolayer. In the measurements, the ratio R/Rp is determined as a function of... 

Figure 7. Top to bottom Fresnel reflectivity, penetration length, evanescent wave intensity and reflected beam plane calculations for a model system. The parameter bu = 2/uk i Q], where is the scattering wavevector, Qc is the critical angle wavevector, and p is the linear absorption coefficient. [From Elements of Modem X-ray Physics by Als-Neilson and DesMorrow, with permission from the editors at John Wiley and Sons.]...

In some configurations of dryers the length of zones and thus the sound intensity distribution may become important. For example, the Fresnel zone for the plane source 10 cm in diameter is negligible (couple of millimeters) for sound at 100 Hz (cf. frequency of pulse combustion) but extends for 15.6 cm in the range of ultrasound at 20 kHz and 31.2 cm at 40 kHz. 

When considering external attenuation affecting the transmission of light through optical fibres, one finds that, for short lengths of fibres, external attenuation normally is substantially larger than internal attenuation. At the entrance and exit face, both Fresnel reflection Lr and Fraunhofer diffraction Ld losses occur. 

From Fresnel s approximation, the focal length of an LC lens is related to the lens radius r and Sn = ncenter-nborder which is the refractive index difference between the lens center and borders, as described in Equation (12.5). 

FIGURE 13.30 First Fresnel zone Reflected path length equals direct path length plus 180". 

Fresnel zone A locus of points along a path where a reflection results in a change of overall path length... 

The meaning of this is as follows (Fig. 5.6a). If cones around the resonator axis are constructed with the side length = ( + m)Xfl and the apex point A on a resonator mirror they intersect the other resonator mirror at a distance d = qX11 in circles with radii = q- -m)-X. The annular zone on mirror Mi between two circles is called Fresnel zone. The quantity Ap gives the number of Fresnel zones [306, 307] across a resonator mirror with diameter 2a, as seen from the center A of the opposite mirror. For the mirror separation d these zones have radii pm = JmXd and the distances = 2 (m - -q)X (m = 0,1,2,... < ) from A (Fig. 5.6). 

In a binary phase Fresnel lens, the focal length/is related to the innermost zone radius rj as / = r lE, where X is the wavelength of the incident beam. The primary focal length of the lens was -50 cm (for X = 633 nm). Due to the higher order Fourier components, a Fresnel zone lens has multiple foci a.t f, f/3, f/5, etc. Flowever, most of the incident light diffracts into the primary focus. The theoretical diffraction efficiency of the primary focus for the binary-phase Fresnel lens is 41%. 

Figure16.12 Fresnel lens (a) Principleofcreatinga refractive Fresnel lens by segmenting a normal plano-convex lens, (b) Large-area AFM image of the center of an embossed micro Fresnel lens (side length is 150 pm).

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