Power attenuation coefficients

The power attenuation coefficient (2oc) for lower order leaky modes is given by [4] thus. 

Fig. 7 Power attenuation coefficient for a liquid filled capillary for various lower order modes q. Calculations are based on rii = 1.33 and = 1-46...

To describe attenuation, we define P z) to be the power flowing within a narrow ray tube of rays, such as those illustrated in Fig. 4-1 and discussed at the beginning of Chapter 4. We then introduce the power attenuation coefficient y(z) of a ray defined 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... 

The power attenuation coefficient for cladding absorption is found either by averaging T over the ray half-period between successive reflections or... 

The ray half-period is given in Table 2-1, whence we deduce from Eqs. (6-17) and (6-26a) that the power attenuation coefficient can be expressed in the form... 

When both the core and cladding materials of a fiber are absorbing, the power attenuation coefficient y for a ray is given by the sum of the core and cladding absorption coefficients. Hence... 

Thus a fraction of T of ray power is lost at each reflection. The ray power attenuation coefficient y is found either by averaging T over a ray half-period Zp, between successive reflections, or by summing the loss at the N reflections in unit length of the fiber. Either way we have [7]... 

The ray half-period Zp for refracting rays is the same expression, used for bound rays, in Eq. (1-10). Substituting Eq. (7-4) into Eq. (7-2), the power attenuation coefficient is given by... 

Whichever of the above forms is used to derive the transmission coefficient, the power attenuation coefficient follows from Eq. (7-2), where the ray half-period is determined from the general expression of Eq. (2-28) for graded profiles. 

When a leaky ray propagates along a fiber that is absorbing, it is intuitive that the total attenuation is simply the sum of the power attenuation coefficients for... 

At distance z along a fiber, the initial power of each tunneling ray has been attenuated by the factor exp (—yz) of Eq. (7-3), where y is the power attenuation coefficient. The product yz for each ray depends on both the parameters p, 9, V of the fiber and the ray invariants /. If, for a particular ray, yz > 1, then virtually all of its power has been lost to radiation, but if yz < 1, the initial power has been nearly conserved along the fiber. In other words, the attenuation is effectively infinite or zero for the respective cases. We approximate the power of every tunneling ray by assuming it is either zero, if... 

There is a general class of noncircular graded profiles where the fraction of power lost at each reflection does not vary, i.e. T is constant along a given ray path [12]. The power attenuation coefficient is then given by Tj <,Zp >, where <2p > is the average value of the ray half-period along the fiber. 

The imaginary part of the propagation constant, accounts for the attenuation factor exp (—jSjz) of the modal fields in Table 11-2. If Pj z) is the modal power distance z along the waveguide, then, by substituting these fields into Eq. (1 l-21a), we obtain the expression for power attenuation in terms of the power attenuation coefficient jj, and the initial power Pj 0). 

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. 

When the waveguide is only slightly absorbing, as is normally the case in practice, we can obtain approximations to the power attenuation coefficient in explicit form. In one approach, the imaginary part of the propagation constant is determined by setting i) j in the eigenvalue equation for the... 

If the core and cladding materials of the waveguide are slightly absorbing, the ray power attenuation coefficient of Eq. (6-3) is identical to the modal power attenuation coefficient of Table 11-2, page 232, provided the lateral shift is included. Details are presented in Section 36-10. 

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... 

The fractional power loss per unit length, or power attenuation coefficient y, is found by dividing Eq. (23-10) by the length of the loop 2nR. Since y is independent of z, the power at any position along the loop is found by analogy with Section 22-4 to be given by... 

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