( 2 ) where is the transverse component of V, the vector differential

In document Radiation losses in dielectric optical waveguides (Page 60-63)

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

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

In document Radiation losses in dielectric optical waveguides (Page 60-63)