Transmitted field

Consider the waves scattered by isotropic dipole oscillators in the thin slab of matter shown in Fig. 9.3 only part a is of concern at the moment. These waves add vectorially at point P to produce the resultant forward-scattered wave s the important point, which is by no means obvious yet, is that this resultant scattered wave is phase shifted 90° relative to the incident wave (in addition to the phase shift between the oscillators and the incident wave). The background necessary to show this has been presented in Chapter 3 Fig. 3.8 is similar to Fig. 9.3 except that in the former the scatterers were arbitrary particles. The transmitted field t at P is the vector sum of the incident field and the fields scattered by all the oscillators. If we assume that the direction of vibration of the incident wave is not rotated as it propagates through the slab, the transmitted field is given by (3.39) ... 

The transmitted field to the right of the crack can be calculated similarly, with a change of sign in eqn (12.38) and in the x-dependence of wc(x, () and wsc(x). In this way approximate values of 1r and Rr can be found for the two-dimensional crack contrast theory, and possibly for the three-dimensional theory as well. The calculated field is reasonably good near the free surface but not near the crack tip, so the approximations are better the deeper the crack is compared with the wavelength. 

We assume also that all wavefields - the total, the incident, the scattered, and the transmitted fields - satisfy the Sommerfeld radiation conditions, which are... 

Classical and quantum treatments of electromagnetic fields [7], near a plane dielectric interface, show that the probability of photo-ionization or photo-excitation of an atom exhibits a pronounced peak at the critical angle of incidence of a plane wave propagating in the dense medium. The probability of absorption or emission of a photon is proportional to the photometric intensity of the electromagnetic field. In particular, consider a plane wave propagating in the dense medium with refractive index /2i, at angle of incidence Ox. The intensity of the transmitted field at a distance dy from the interface is given by... 

Our notation is the same as that of JACKSON [4.12] except for the vector spherical harmonics for which we use the notation of EDMONDS [4.13]. This will produce transmitted fields inside the particle... 

There will also be a reflected field. The vertical and horizontal components have amplitudes and phases given simply by F and F , respectively. In general this reflected field will be elliptic polarized. What is of concern, however, is the fact that when a single meander-line sheet transmits a perfect circular polarized field, we readily see from the Smith chart that F = t = F = Tft. In other words A single meander-line sheet will produce a reflected field of the same amplitude as the transmitted field that is, the efficiency is only 50%. The remedy for this calamity is to use several meander-line sheets cascaded after each other as will be discussed in the next section. See also Problem C.l, where you are asked to consider a two-sheet meander-line polarizer. 

Let the subscript i denote incident, r reflected and t transmitted field components for beams in isotropic media separated by a flat stratified layer of thickness T. Let the first medium, which contains the incident and reflected rays, have optical dielectric constant 1, and let the last medium have optical dielectric constant 2. Let the angle of incidence within the first medium be Fig. 1). SnelFs... 

Figure 5 shows that at 1 MHz, the reflection coefficient is around 0.7 and at about 10 MHz it drops down to 0.5. The study of its intrinsic attenuation reveals that it will not significantly absorb transmitted fields (Figure 6). This modeling studies do indeed show that the tunable composites are good microwave reflectors.

The transmitted field at the solid-vacuum interface is described by Fresnel s formulas and can be written as... 

Here, E,o is the tangential component of the electric field amplitude at the interface and Ns is the surface number density of adsorbed atoms. It is also assumed that the dielectric response of the adsorbed atoms is isotropic in the surface plane. Accounting for the surface current (7.43) in the boundary condition for the tangential components of the magnetic field, results in a modified expression for the transmitted field amplitude. For example, in the case of s-polarization one obtains, instead of Eq. (3.13) with 62 = 1,... 

Following the above discussion, the fields at the endface are the transmitted fields in Fig. 20-1 (b) associated with the illuminating field. If the beam makes angle with the fiber axis, then it is refracted at z = 0 and makes angle 0 with the fiber axis for z > 0 given by Snell s Law... 

The first term on the right accounts for the change in phase, and the second term for the change in amplitude. To determine the contribution to da -, we examine the reflected and transmitted fields when the k th forward-propagating local mode is incident on the section. 

Once M is found, the transmitted field components can be obtained from the equation. ... 

Let and be the incident field. The transmitted field is then given by... 

Kaiser, T., Schweiger, G., Stable Algorithm for the Computation of Mie Coefficients for Scattered and Transmitted Fields ofa Coated Sphere, Comput Phys., 1993, 7, 682-686. 

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