is the axial propagation constant.

In document Radiation losses in dielectric optical waveguides (Page 92-97)

The total power radiated, P , from the fibre is obtained by

JA.

summing over all the radiation modes, which yields

2 j J

S dK dy , R r r

(5)

where the 2 indicates a summation over the polarisation states.

By converting equation (5) to an integration over the solid angle with the z-asis of the fibre parallel to the z-axis of the spherical co-ordinate system of the solid angle, becomes [24]

P

R (6)

Then, using equations (1) - (4) in equations (5) and (6), the expression for the total power radiated becomes

where

’ y 'H. oo2k 2

32 (tt) 2

^dsvll2 + K,,l2}

AeE exp (ik2R* .r' ) dV' .

(7)

(8)

Thus, equations (7) and (8) provide the expression for the power radiated from dielectric irregularities calculated by Coupled Mode Theory.

From the equations of the Volume Current Method, equations (1.19) - (1.21), the total power radiated, P ', is

P' R

^ * 2

32 (t t) 2

{ dfi I [ RX dV'

(Ae

exp{ik2R * r '})] |2} . (9)

On using equation (4) with the definition, equation (8), this result becomes

P' R

W2k2

32 (TT) 2

asH|Svi

2 {] (10)

which from comparison with equation (7) reveals the equivalence of the two calculations.

We have shown that when the Coupled Mode Theory is used with the radiation modes of the fibre approximated by the free space

radiation modes of the cladding, the calculation is equivalent to that of the Volume Current Method. This proof demonstrates clearly the inherent redundancy in the formalistic treatment of radiative losses by Coupled Mode Theory, when used with such approximations. The Coupled Mode treatment requires explicit incorporation of both independent polarisation states of the radiation field, whereas the Volume Current Method automatically includes both polarisations.

For irregularities whose dielectric permittivity is much greater than the waveguide permittivity, the exact Coupled Mode Theory treatment must be used, since the fields in the vicinity of the

irregularity can no longer be well approximated by the incident electro­ magnetic field. However, the Coupled Mode Theory treatment is not

uniformly valid for all irregularities, and there is much debate in the literature on the applicability of such an analysis when the

imperfections are not slight [25].

3.3 APPLICATIONS OF THE VOLUME CURRENT METHOD

In the preceding sections we discussed the philosophy of the Volume Current Method and considered its relationship with the classical treatment by Coupled Mode Theory for the determination of radiative loss from slight dielectric irregularities. Having established this

practical importance, that demonstrate the facility of the Volume Current Method. This analysis will employ the general modal field

formulation established in Chapter 2 and will be restricted to examining the losses associated with individual modes of the optical fibre.

The two most commonly observed dielectric irregularities are isolated irregularities in the core and variations in the structural geometry of the waveguide. These irregularities are induced during the manufacture of the optical fibre. In the following sections we will determine the modal power losses due to these radiation mechanisms for modes incident with unit power. In the discussion, we shall present quantitative results for the step-index fibre in order to demonstrate the accuracy of the Volume Current Method.

3.3.1(a) Radiation Loss from Isolated Irregularities

When an electromagnetic wave is incident on a dielectric irregularity, a portion of the incident energy is scattered and

redistributed among the bound modes and radiation modes of the waveguide. We will only concern ourselves with that portion redistributed into the radiation field as this is the component of most significance in

determination of the attenuation characteristics of the fibre. The "ultimate" transparent material is limited by the molecular granularity of the material. Such intrinsic fluctuations in the dielectric

permittivity must therefore give rise to radiative loss from any

dielectric structure. In this section we shall determine the effect of such irregularities since they will impose the fundamental limitation on the attenuation characteristics of optical fibres.

The deliberate introduction of dielectric irregularities into the dielectric optical fibre has been postulated as a technique to improve the impulse response of multimode waveguides by inducing

deliberate mode-mixing [26]. This technique, however, proves to be of little significance for step-index fibres as the associated loss penalty on the total power propagating in the pulse is too large for the small gain in impulse response.

In practice, isolated dielectric irregularities appear most frequently at the core-cladding interface due to small foreign particles adhering to the interior of the glass tubing that ultimately forms the inner cladding of the fibre. However, we shall not restrict the

location of the scatterers except that they are in the core of the fibre. Due to the rapidly decaying evanescent field in the cladding, scattering centres beyond the core have little effect on the bound modes of the fibre.

As in fig. 3, we consider a small dielectric irregularity of volume l/, and dielectric permittivity (~ Ej) located at £ = £ 0 in the core of the waveguide. If the dimensions of the irregularity are much less than the wavelength of the incident field, the electric field can be assumed constant over the irregularity and the power radiated from an

incident mode (of the unperturbed fibre) due to this scattering centre is (see Appendix B)

R 2

P = K

r 2tt

dcf> sin 0 d9 S ,

where S is the time averaged Poynting vector

(1)

SR = (U

m

A

e)2

c I R

X E ^ (

r

0 )

I

2 , (2)

where C is defined by equation (1.21) and

Ae =

(£1

-£g).

The modal field, from equations (2.2.42) - (2.2.50) written in cartesian co-ordinates is

E ^ r ) = j ü f ß + i (r) (£ ± i£) exp(-i(£+1)<J>)

- • — ■ f«(r) exp (-i£({)) zl , (3)

where £ is parallel to the axis of the fibre.

In practical weakly guiding fibres, the longitudinal components of the field are of order 0 compared to the transverse

c

Fig. 3: An isolated dielectric irregularity of dielectric permittivity £'3 , and volume l/ located a distance rQ from the fibre axis.

component to this point to determine the effect of a scalar located on the axis of the waveguide. From the power series expansion for f (r) around r = 0, it can be shown that only the functions f Q (r) are non-zero at r = 0.

Thus, irregularities on the axis of the fibre scatter power •f*

from the HE, and TNL modes only. The HE, modes have non-zero

lm 0m lm

transverse electric fields on the fibre axis, whereas from equations (2.2.43) - (2.2.48) the TM ^ modes have non-zero longitudinal electric fields on axis. In weakly guiding fibres, the transverse fields are much larger than the longitudinal fields and we would therefore expect the HE. modes to have a much greater radiation loss than the TM„ modes for scatterers located on the fibre axis. Yip et al. [7] have studied the effect of scatterers located on axis but do not discuss the

scattering from the TM modes although such an effect is easily included in their dyadic Green's function analysis.

In general, the magnitude of the longitudinal fields can be ignored in calculations of the total power radiated. (This is not true, however, of the angular spectrum of the radiation since the longitudinal fields contribute significantly to the power radiated in the transverse direction, however small in comparison to the power flowing in other directions. We will discuss this more fully at a later s t a g e . )

In the practical situation, the interest is centred on the total power radiated and thus, ignoring the longitudinal fields, the

t h R.

total power radiated from the £m mode, P , is (see Appendix B)

R,

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