**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{ik**2

**R * 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,**