**operator.**

**Now if the transverse fields are all proportional to the one**
**scalar field, we have**

**where c is a constant. ** **For the longitudinal electric and magnetic**
**fields to be of the same functional form, equations (1) and (2) using**
**equation (3) must be consistent. ** **From an investigation of equations (1)**
**and (2), we find such approximations for the fields are consistent only**
**within ignoring all terms in de/dr. ** **The differential equation for the**
**transverse field function, equation (33) of ref. 22 (or equation (2.37)**
**of the previous s e c t i o n ) , is the differential equation for the**

**transverse electromagnetic fields only within ignoring all terms in**
**de/dr [28]. ** **Once it is recognised that the analysis is only correct to**

**zeroth order in d£/dr, the simple modal fields of section 2.2 result.**

**parabolic profile in ref. 22 using the weak guidance approximation is**
**invalid as this approximation is not possible far from the centre of the**
**profile at r = 0. ** **As the behaviour of the electromagnetic fields far**

**from the waveguide axis is employed to determine the eigenvalues of the**
**structure, the consequent eigenvalues derived with the weak guidance**
**approximation are of questionable accuracy.**

**such criticisms as the analysis is applicable only to bounded regions**
**over which d£/dr << 1 everywhere, and is totally inapplicable to the**

**infinite profile.**

**In section 2.1 it was noted that the eigenvalue equation,**
**derived by neglecting terms in d£/dr, indicated that the hybrid modes**

**(3)**

**Together with the above, the analysis of the infinite**

**(i.e. £ ^ 0 ) of cylindrical waveguides with continuous dielectric**
**profiles, could decouple into pure TE^ and TM^ modes. ** **However, upon**

**further investigation (as in Appendix B) into the effect of the (d£/dr)**
**terms, using a perturbation analysis correct to first order in d£/dr,**
**such decoupling cannot occur and the choice of TE^ and TM^ modes would**
**result in a singular perturbation analysis.**

**We see from Appendix B, equation (B.25), that the modes of a**
**fibre with a continuous dielectric profile obey the same relationship**
**between the TE^ and TM^ components as the hybrid modes of fibres with**
**discontinuous dielectric profiles, i.e. equation (2.28). ** **The zeroth**
**order (in d£/dr) eigenvalue equation for continuous dielectric profiles,**
**equation (2.25) with l/(p) = 0, does provide an interesting result. ** **We**
**observe that for such situations, pairs of HE^ and EH^ modes become**
**degenerate, although from the analysis we cannot order these pairs of**
**modes. ** **Such extra degeneracies, above the HE^ and EH^ ** **degeneracy of**
**weakly guiding fibre modes, have been observed in numerical studies on**
**graded-index waveguides [16,27]. ** **However, extension of the preceding**
**analysis is necessary to determine whether these theoretically predicted**
**degeneracies in weakly guiding fibres correspond to the numerically**
**determined modes.**

**Appendix B then justifies the choice of the amplitudes of the**
**longitudinal electric and magnetic fields, equation (2.28). ** **The**

**question as to which dielectric inhomogeneity, i.e. the dielectric**
**discontinuity or the gradient of the dielectric, is the dominant first**
**order perturbation cannot be fully answered without resorting to an**
**evaluation of the integrals defined in equation (B.13). ** **(The results**
**of Appendix B, for the fields correct to first order in **

**A, **

**defined in**

**equation (B.2), can be further simplified, as mentioned in the appendix,**

**thereby simplifying the above comparison.**

**However, as this topic is**

**beyond the scope of this thesis, it will not be considered further.)**

**The dielectric profiles for which closed form solutions of**
**differential equations of the form**

**r dr**

**+ k 2 (r) <j>**

**r** **0** ,

**1 _d_**

**(where**

**is the radial wavenumber), are rare and generally are infinite profiles**
**tending to negative infinity (as in the infinite parabolic profile),**

**f**

**f**

**which are physically unrealistic.**

**Most theoretical studies on graded index fibres use simple**
**W.K.B.J. asymptotic approximations for solutions of equation (4) above,**

**to circumvent the mathematical complexity. ** **However, these approximate**
**solutions are not uniformly valid throughout the core of the fibre, as,**
**in the vicinity of the "turning points" [19], where k ^ = 0 , the solutions**
**diverge. ** **Applying the familiar "connection formulae" across the turning**
**points [19], essentially imposes extra boundaries and therefore the**
**modal relationships of the previous section cannot be readily used,**
**since they are applicable to two region waveguides only. ** **However, for**
**modes whose outer caustic, or turning point, is far from the core**

**cladding interface, at r = p, the mode should behave as if it were in the**
**infinitely extended medium. ** **With these modes the simple W.K.B.J.**

**solutions should provide good estimations of the fields, provided the**
**conditions for the applicability of the asymptotics are satisfied [21].**

**Due to the difficulties involved in differentiation of**

**asymptotic series [21] , any modal field components which are formed from**
**the derivatives of other field components estimated by asymptotic**

**techniques should be viewed with some scepticism. ** **Such a difficulty is**
**incumbent on Kurtz and Streifer's results which leads to longitudinal**
**field components which are expressed in terms of the scalar optical**
**field and its derivative. ** **The analysis of section 2.2 circumvents this**
**diffficulty and allows for direct and simple asymptotic representations**
**of the fields from the asymptotic analysis of equation (4) together with**
**equations (2.43) - (2.48).**

**Landau and Lifshitz confirm the unphysical nature of such static**
**dielectric permittivities by a thermodynamic argument [29] . ** **In**

**dielectric waveguides, the dielectric permittivity is taken to be the**
**static dielectric permittivity unlike the case of plasma waveguides in**
**which the dielectric permittivity can be less than zero.**

**We retain the HE,EH nomenclature for the modal fields in**
**preference to the transverse field representation, denoted by the LP**

**i**

**modes, derived by Gloge for the step-index fibre [30], ** **and more**
**recently extended by Kurtz to graded-index fibres [32] and used**
**exclusively in ref. 31. ** **Although such a modal field solution can be**
**generated for a general class of dielectric profiles of the fibre core,**
**by a linear combination of the HE^ and EH^ ** **modes, this set of "pseudo"**
**modes suffer from major disadvantages. ** **Since the LP mode set is a**

**linear combination of the "proper" modes, i.e. H E ^ E H ^ ^ modes (which**
**are only degenerate within the weakly guiding fibre approximation), they**
**are pseudo-stable modes due to the slight difference between the**

**propagation constants, A3 = 3h e^ “ ^EH^_2• ** **Consequently the two real**
**modes of the fibre lose their intimate phase relationship (necessary for**
**the formation of the LP modes) as they propagate along the waveguide.**
**The distance for which the pseudo modes are stable for step-index wave**
**guides is [33]**

**where 0 is the complement of the critical angle, defined in equation**