THE BOUND MODES OF THE IDEAL STRAIGHT DIELECTRIC OPTICAL FIBRE

In document Radiation losses in dielectric optical waveguides (Page 37-41)

In the proceeding chapters, the bound modes of the ideal

straight dielectric optical fibre of circular cross-section will be used as the basis for analysis of radiative loss mechanisms in the dielectric waveguide. The present state of the art of dielectric waveguides

requires techniques that are useful for both homogeneous (step-index) and inhomogeneous dielectric cored waveguides. To this end, this

chapter presents a general set of of bound modes for the simplistic (two region) form of the dielectric optical fibre, with an arbitrary radial variation of the dielectric profile £(r) in the core. With this set of modes the analysis of radiative loss mechanisms will be studied and only

for quantitative discussions will a definite form for the profile be considered. For these quantitative results the modes of the step-index fibre will be used.

Studies of the propagation behaviour of a cylindrical dielectric embedded in an infinite medium of different dielectric

permittivity began with a theoretical treatment by Hondros and Debye [1] at the turn of the century. These were followed much later by

theoretical and experimental investigations relevant to the propagation of microwave signals along dielectric rod waveguides and to the

preliminary analysis of circular waveguides containing ferrite rods [2-6]. The possible use of the structure as an optical waveguide is attributed to Snitzer [7] whose investigations stimulated interest in optical fibre waveguides for telecommunication applications.

Due to the complexity of the exact solutions of Maxwell's equations for cylindrical structures, analysis using these modes proves

cumbersome. The computational analysis of the modes far from cutoff, in Snitzer's theoretical study, first suggested the existence of a

simplified set of modes for these dielectric structures but it was Snyder [8] in 1969 who first derived such a simplified modal electro­ magnetic field for step-index waveguides based on the assumption of a weakly guiding dielectric optical fibre in which the dielectric

difference between the core and cladding dielectrics is small. This approximate set of modes facilitate calculations of all modal

characteristics. These modal forms together with the simplified eigen­ value equation have been shown to be excellent approximations for the fibre with infinite cladding for all frequencies [9]. This "weak

guidance" approximation can be extended to include a more general class of dielectric profiles and in section 2.2 such a set of approximate modes are presented. In the final section of this chapter a discussion of the results obtained in the previous sections is given along with the modal fields for the step-index fibre, which will be used to discuss quantitatively the radiative loss mechanisms to be studied in the following chapters.

We begin with a brief resume of some classical electromagnetic theory with specific reference to cylindrical structures of circular cross-section.

2.1 PRELIMINARY RESULTS AND NOTATION

The fundamental quantities involved in electromagnetic

phenomena are the electric field intensity

E,

the electric displacement

V,

the magnetic field intensity

H

and the magnetic flux density B. The equations governing their behaviour in a linear, source-free medium of dielectric permittivity £ and magnetic permeability y are Maxwell's equations [10]

9B

= -

a?-

VxH

a t '

= 0 , (2)

V • V

(3)

V • B =

o

, (4)

and the constitutive relations

V =

(5)

B

= y H , (6)

where £ and y are tensors in an anisotropic medium.

At a dielectric interface (a discontinuity in the dielectric permittivity) the electromagnetic fields must also satisfy the boundary

conditions

(i) components of iE and H tangential to the interface are continuous across the interface, and

(ii) components of

V

and B normal to the interface are continuous.

In the absence of static electric charges or magnetic fields, equations (3) - (4) and boundary conditions (ii) are a direct consequence of equations (1) - (2) so that, in the time-varying situations with which we are concerned, only the latter equations and boundary conditions (i) need be considered [11] .

Any well-behaved complex field £(.r,t) (electric or magnetic) which satisfies these equations and conditions can be Fourier-

transformed in time to give

F(r,w) 1 / 2 7 • f°° F_(.r,t) e dt — 00 r°° (7) F ( r,t) 1 / 2tt _ , . lU)t , F(r,oo) e da) — OO (8)

so that it is only necessary to consider time-harmonic fields of the form F(r,üo) e ^ ^ , where w is the angular frequency of the field.

Throughout this thesis, the fields to be studied will be complex, time- harmonic fields, it being understood that the physical field is given by the real part of this quantity.

For these harmonic fields, equations (1) - (2) can be rewritten using equations (5) - (6) to give

V x e = - iwyH (9)

V x h = iU)£E , (10)

where E and H are the Fourier transforms (in time) of

E

and

H.

Taking the curl of equation (9), substituting equation (10) and using the identity (in a cartesian co-ordinate system),

V x (V x E) = V ( V * E ) - V 2E, we find

V 2E + V[E • V(Zn £)] +V(Zn y) x (V x e) + Q)2yeE = 0

and similarly the equation for H is obtained by replacing E by H and interchanging e and y. The ( ) indicates that the V operator acts on the term in the brackets only.

In graded-index fibres, where £,y are not homogeneous, we can consider the inhomogeneity of the dielectric permittivity and magnetic permeability as perturbations in homogeneous waveguides. From an

investigation of equation (12) above, the inhomogeneity manifests itself in two ways, viz. (a) in the term (0)2y£E) and (b) V (E • Ve) and

V(Z-n y) x Vx e. When the dielectric permittivity and magnetic

permeability vary much more slowly than the wavelength of light the latter terms may be ignored in comparison with the former [12,13], and equation (12) then reduces to the Helmholtz equation

[V2 + k 2 ] E = 0 , (13)

, s h 2TT

= u)(ye) = — (14)

is the magnitude of the propagation vector, X being the wavelength of light in the medium in question. We can determine the higher order corrections to the electromagnetic by an iteration procedure,

demonstrated in Appendix B.

(ID

If we choose a co-ordinate system in which the z-axis lies in

In document Radiation losses in dielectric optical waveguides (Page 37-41)