Power absorption coefficients

Figure 15 Theoretical (a) and experimental (b) power absorption coefficient. (Fig. 13b is reproduced from Vij et al. [32]).

The critical parameters for high-power windows are the dielectric characteristics of the window material the dielectric loss-factor tan 6 and the permittivity 8r (or the refractive index n = r,r -) because they affect power absorption and reflection. The power absorption coefficient a is related to tan 6 by [43]... 

The absorption coefficient at a boundary, also called the absorption factor or sound power absorption coefficient, is defined as the fraction of the incident acoustic power arriving at the boundary that is not reflected, and is therefore regarded as being absorbed by the boundary (Morfey, 2001) ... 

The dielectric loss spectrum at 293 K consists of the main microwave peak at 10 Hz (3.3 cm ) with a barely resolved shoulder at v ca 1.5x10 Hz corresponding to the maximum of the power absorption coefficient at v ca SO cm in the far infrared spectrum. If the measurement temperature is towered, the microwave loss peak, ascribed by the authors to p relaxation process, moves out ccmsidetably to lower frequendes and becomes resolved from the far infrared peak which is shifted with decreasing temperature in the opposite direction. 

The high-frequency part of the loss in the far infrared region is characterized by the authors [60] for the first time as the y-process due to short time torsional oscillations of the solute dipoles. This occurs universally and is the high-frequency adjunct of the a and )S-processes. This part of loss in the far infrared region can be studied in detail because of the relation between the dielectric loss e"(cu) and the optical power absorption coefficient ... 

The power absorption coefficient a (r) does not vary along a cylindrically symmetric fiber. Thus, provided z is large compared to the ray half-period Zp, we can make an accurate approximation to the integrals in Eq. (6-7) by... 

When the power absorption coefficient is uniform, i.e. aco( ) co> power attenuation coefficient of Eq. (6-6) is proportional to the real part of the profile. Ray power attenuates according to Eq. (6-8), which has the form... 

In general, the integration in Eq. (6-8) must be performed numerically when (r) varies across the core. However, an analytical solution is possible for one case of practical interest [4], when the power absorption coefficient is proportional to the real... 

A second example with an analytical solution arises when the power absorption coefficient varies inversely with the profile, i.e. 

We first consider the trivial case of an infinite, uniform medium of constant refractive index n = n + in. The modal fields, which depend implicitly on time through the factor exp (—iojt), are the fields of a plane wave propagating in the z-direction with propagation constant Pj = kn + ikn, where k = InIL Thus the field amplitude attenuates as exp(—Icn z), and the power attenuation coefficient = iP) = 2kn. The quantity 2kn defines the power absorption coefficient a, introduced in Eq. (6-2), and thus jj = a in this case. 

It is clear from Table 13-2, page 292, that the shape of the intensity distribution S on an absorbing fiber does not change as the mode propagates, but its amplitude is reduced to exp (—yz) of its original value everywhere in the cross-section after distance z. If the power absorption coefficient a is uniform over the whole cross-section, then y = a and power lost by attenuation at any position is exactly equal to the power absorbed there. This special case corresponds to a = a = in Eq. (18-16), whence y = a. However, if a is not uniform, then power lost by attenuation and power absorbed will differ at an arbitrary point, and, consequently, there must be a flow of power within the cross-section. This corresponds to the general case in Eq. (18-16), when power flows across the interface to maintain the shape of the intensity distribution. The direction of this flow depends on the relative values of and a, . For example, if a, = 0, power flows into the core. 

If the core material is slightly absorbing with power absorption coefficient a , then Eq. (18-16) shows that the power attenuation coefficient is y z) = rj (z)a o at position z. The modal power P(z) is found by replacing y with y(z) in Eq. (11-62) and integrating to obtain... 

We assume a uniform power absorption coefficient a throughout the array, in which case the power attenuation coefficient y for each fiber satisfies y = a, as we showed in Section 18-8. Thus, by analogy with Eq. (29-21), the power in each fiber is given by the product of Pj (z) or P2 (z) of Eq. (29-39) with exp (—az). We define P+ (z) and P (z) to be the power absorbed over length z of fibers with illumination -f and —, respectively. If there were no cross-talk, then P+(z) - P,(0) and P (z) - P2(0) as z - 00, i.e. each fiber of the array would absorb all of the power illuminating it. However, because of cross-talk, this situation is degraded. If we define a normalized difference in absorption... 

The ratio of to of Eq. (35-34) gives the transmission coefficient. We express in terms of the ray invariant using the relationship in Table 36-1, page 695, and nj, in terms of the power absorption coefficient a., of Eq. (6-19). Hence... 

We relate the modal parameters to the propagation constants by equating real and imaginary parts in the definitions inside the back cover and retaining only the lowest-order terms in each case. In terms of the core and cladding power absorption coefficients, and a, defined in Eqs. (6-10) and (6-19), respectively, and the power attenuation coefficient y of Table 11-2, page 232, we obtain... 

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