Radiation free-space’ approximation

Within the free-space approximation, the total power radiated by the current dipole of Sections 21-4 and 21-5 is independent of both the position and orientation of the dipole. Thus, for simplicity, we can assume the dipole is parallel to the z-axis at the origin in Fig. 21-4. Substituting Eq. (21-7) into Eq. (21-20), we deduce that the vector potential has only a z-component. Hence... 

Consider a small nonuniformity of volume dV and uniform refractive index n located at radius in the fiber cross-section. An incident,. x-polarized fundamental mode sets up an induced current given by Eq. (22-7)withF = R = r /pandfi = n(r ), where pis the core radius. Within the free-space approximation of Section 21-8, the total power radiated, P ad > independent of polarization of the mode and is obtained by replacing Id with a, bo/PoV k n -n )Fa5Y in Eq. (21-23b). The fraction of incident power P radiated follows from Table 13-2, page 292 as... 

To calculate the power radiated from the bend, it is sufficient to use an approximate model of the fiber consisting of a current-carrying antenna of ir nitesimal thickness which radiates in an infinite medium of index equal to the cladding index n, i.e. the free-space approximation of Section 21-8. The far-field formulae of antenna theory, as expressed by Eq. (21-21), can then be applied to calculate the radiation, once the current on the antenna is specified. 

Another important factor in dielectric heating is the depth of penetration of the radiation because an even field distribution in a material is essential for the imiform heating. The properties that most strongly influence the penetration depth are the dielectric properties of the material. These may vary with the free space wavelength and frequency of the propagating wave. For low loss dielectrics such as plastics (e" l) the penetration depth is given approximately by ... 

In order to demonstrate large SPMs, an approximate free-space fluorescence profile is usually sufficient since the profile of the radiation enhancement factor ) usually dominates. For more subtle effects, like... 

In Section 22-5 we determined the attenuation of the fundamental mode on a weakly guiding, step-profile fiber due to radiation from a sinusoidal perturbation of the interface, using free-space antenna methods and correction factors. Here we consider the situation when the radiation field is well approximated by a single leaky mode, which can be realized by having an on-axis sinusoidal nonuniformity of the form of Eq. (22-14). The azimuthal symmetry ensures that only HEi leaky modes are excited. Further, the direction of radiation should coincide with the direction of the leaky-mode radiation [23]. If we represent the nonuniformity and the incident fundamental-mode fields by the induced current method, as in Section 22-5, the direction condition is satisfied by setting C = in Eq. (24-43), whence... 

Radiation modes with p kn, i e. 0 = n/2, are composed of plane waves at near normal incidence to the core-cladding boundary so that they are only weakly influenced by the core. Thus, radiation modes with P < kn are approximately free-space modes with ej ej and = h. The implicit conditions discussed below in Section 25-11 help facilitate this limiting condition. At the other extreme, when P = kn, i.e. 0, 0, the plane waves that form a radiation mode are at near grazing incidence and are significantly influenced by the fiber core. 

The radiation is characteristic of the cathode material and the incident ions. A dark space arises in front of the cathode glow. In this region the secondary electrons are accelerated away from the cathode. The width of this zone corresponds approximately to the mean free path of the electrons. 

Space away from Earth surface is the closest natural approximation of perfect vacuum. In this condition, force of friction is almost negligible. However, even in the deep vacuum of intergalactic space, there are merely few hydrogen atoms per cubic meter (Borowitz and Beiser 1971). In comparison, the air at Earth surface contains about 10 molecules per cubic meter (Davies 1977). The sparse density of matter in outer space means that electromagnetic radiation can travel great distances without being scattered the mean free path for a photon in intergalactic space is about 10 km (Chapmann 1991). 

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