THE SURFACE CURRENT METHOD

In document Radiation losses in dielectric optical waveguides (Page 177-182)

CURVATURE INDUCED RADIATION: THE SURFACE CURRENT METHOD

5.2 THE SURFACE CURRENT METHOD

In this section we derive the formulae of the Surface Current Method in general and then specialise this result for the practically important case of the weakly guiding dielectric optical fibre of

circular cross section. This technique is founded upon the Equivalence Principle, formulated by Schelkunoff [11], or the Stratton-Chu Integral

[12] for the fields in a volume V , due to the sources in the volume together with the effect of sources outside the volume which can be represented by equivalent surface currents on the surface S' enclosing the volume. The replacement of the sources outside the volume by equivalent currents over the surface is the philosophy of Huygen's

Principle [13] , in which the fields over a surface (generated by sources away from the surface) can be considered as secondary sources for the fields beyond the surface, on the opposite side of S' to the real sources. These pseudo sources generate secondary waves that add to produce distant wavefronts. In the electromagnetic equivalent of Huygen's Principle, these pseudo sources can be considered as electric and magnetic surface current densities distributed over the surface S ' .

From the Stratton-Chu Integral, the electric and magnetic

■f

fields in a volume V at a position R can be written as [12] t

E(R) 1 4tt

V

-ic o y J (j) - j x Vcj) + — p V(j) ~m £o e dV' 1 4tt S'

-ic o y (n x h) ({) + (n x e) x V(j) + (n • E) Vcj) dA' (la)

and H (R) = —1_ 4TT V -iu)£~ J cj) + j x Vcj) + - p Vcj) y m S' —ico2 (n x e) cj) - (n x h) x Vcj) - (n • H) Vcj) dA' f (lb)

where £2 and y are the dielectric permittivity and magnetic permeability of the homogeneous isotropic dielectric of volume V* enclosed by the surface S'. (j) is the Green's function for the homogeneous region and n is the outward normal to the surface. J ,p and J fp are the electric

e m

and magnetic volume current and charge densities respectively.

If all the electric and magnetic sources are enclosed within a sphere of finite radius, then the field is regular at infinity and the contribution of the surface at infinity can be neglected in the surface integrals of equations (1).

From an inspection of equation (1) , we can see that the terms

and

K = - fixH

~e ~ ~ (2a)

K = h x E (2b)

behave as if they were the two-dimensional analogues of the electric and magnetic volume current densities, J and J respectively. Similarly,

~ m

the terms (-n • E) and (-n • H) appear as the two-dimensional counterparts of the volume electric and magnetic charge densities. Thus K and K

~e ~m are referred to as the pseudo surface current densities of electric and magnetic character, respectively. Thus, for a region in which there are no real sources, the field in the volume is generated by the sources beyond the boundary to the volume and these sources can be represented

analysis, in comparison to exp(-icot) variation in ref. 12, there are minor discrepancies in sign between the two formulae.

by pseudo surface electric and magnetic current densities and K^, as defined in equation (2).

In general, the complete description of the field inside the volume V is of little interest. For the special situations for which only the knowledge of the radiation fields that exist far from the surface S' is required, the formulation is greatly simplified and the radiation fields (denoted by the superscript R) become (Appendix A)

:2 G n (k2 ,R) _.k> ( f i x f R x f N y *2 >| K + K 1 J s ' < lb 2J ~ e ~-mJ exp{ik2R*R'} dA' and R

k y.

R x E (3) (4)

Gn (k2 ,R) is obtained from the asymptotic form of the Green's function for the homogeneous isotropic medium of n dimensions [14]

for n = 2 (5)

for n = 3 . (6)

R is the unit vector in the direction of the observation point in the radiation field, from the origin. R' is the position vector of points

lT

on the surface S' and k 2 (= ü)(ye2 ) ) is the wavenumber of light in the c l a dding.

G ( k 2 ,R) 2tt (k2Rj

h

Fro m equations (3) and (4) the time average radial Poynting vector, S , for the power radiating away from the surface S' is [12]

which becomes % E x h* (7) SR h f £ 2) 2 k z-- V G 2 (k , R) 2 {r x • R x r > y ^2 K + K exp{ i k 9 R* R ' } dA' \ l y J 327T2 1 n 2 1 S' lb 2

J

~ e ~m_ ~ ~

J

(8) R R

and the total power radiated P can be determined by integrating S over the surface of a sphere of radius R-*-°° enclosing the surface S' [12],

P R = R 2 S sin 0 d0 d({) : for 3 dimensions

(9)

SR d0 : for 2 dimensions

From equations (8) and (9), it is obvious that the knowledge of the fields over the surface S* is sufficient to determine the total radiated power. Although the form of equation (9) is intimidating for all but the most elementary of surfaces and incident fields, we shall see that for the weakly guiding dielectric optical fibre of circular cross section, the calculation can be considerably simplified.

Equations (8) and (9) form the basic formulae of the Surface Current Method by which the effect of the core of the optical waveguide and all sources within the core are replaced by pseudo-surface currents

distributed over the core-cladding interface, expressed in terms of the modal fields on the fibre which are excited by the sources in the core.

By a judicious choice of the electromagnetic modes of the weakly guiding optical fibre of circular cross section, the analysis,

using the integrals defined in equation (8), can be considerably reduced. In fact, these calculations via the Surface Current Method were the determining factors in the choice of the azimuthal travelling wave modal form, developed in Chapter 2. These azimuthal travelling waves correspond to rays travelling in a skew path in the ±(f) directions around the axis, as the rays propagate along the fibre, depending on whether it is ±£ modes considered.

The enormous reduction in the volume of calculations necessary in the Surface Current Method when the azimuthal travelling wave modes are used arises from the simple relation between the electric and magnetic fields of the mode, i.e. equation (2.2.46), viz.

H ± i

|M|

( e_

j

2_

[

y J

E , (10)

where the azimuthal variation is exp{-i£(j)}. The + and - signs refer to

To be exact, these currents must be considered over a surface just outside the core-cladding interface, in the cladding so that the

the HEß and E H^ modes respectively.

Using equation (10) in equation (8) for the time averaged Poynting vector for the radiation from the azimuthal travelling wave, we

find that we need only calculate one vector integral, T, in comparison to the two previously necessary, and (Appendix B)

S'

K exp{ik. R » R ' } dA'

~ m 2 ~ ~ (11)

and the time averaged Poynting vector S is

sR =

2

16 (TTR)

x ± i — ß-'- (y cos 0 - z sin 0) (12)

where we have used the azimuthal symmetry of the modes and the optical fibre itself, so that we need only calculate the radiation at one position in the azimuthal plane, viz. = TT/2, see fig. 2. The + and -

signs in equation (12) refer to the HE^ and EH^ modes respectively.

F rom a comparison of equation (12) and equation (8) one can readily see why the azimuthal travelling wave mode is chosen in

preference to the standing wave mode form, i.e. cos £(j), sin Z(p modes. However, the conclusions drawn from equation (12) cannot be extrapolated to the more physically realistic mode set, i.e. the azimuthal standing wave modes, without justification. The recent experience with the non-uniformity of the attenuation coefficients due to bends in optical fibres for the different polarisations of the LP modes [15,16] should be sufficient motivation for the need for justification of this extrapolation.

In Appendix B, the time averaged Poynting vector of an incident azimuthal standing wave mode is shown to be comprised of two t e r m s , namely the individual power radiated from each of the two azimuthal travelling wave modes (equation B.12). Thus the azimuthal travelling wave components of the standing wave mode radiate

independently. As equation (B.12) illustrates, the radiative loss of the sin Z < p and cos £(j) modes for either the HE^ or EH^ modes are identical.

In circumstances in which the symmetric optical waveguide and its distortion are such that there is no discrimination between the ±£

travelling wave modes, these two modes should radiate equally and hence the power radiated from a standing wave mode should be twice the

radiation from each azimuthal travelling wave mode, and thus the attenuation coefficients should be identical.

We have now derived the formulae for the Surface Current Method, equations (8), (9) and (12). Theoretically, the technique is exact with the proviso that the fields of the surface S' are known

exactly. In the derivation of the formulae the only restriction is that the volume in which the fields are required must be a homogeneous

isotropic region [12] . Thus for multiply clad dielectric waveguides one must be careful to apply the integrals over the outermost surface.

Similarly, the analysis is such as to be able to study graded-index fibres with an homogeneous cladding, and equation (8) is not confined to round optical waveguides although equation (12) is specialised to wave­ guides of circular cross section.

Before we begin to apply the Surface Current Integrals to radiative loss processes, we shall recall one crucial result. From equation (la), (lb), the electromagnetic fields in the volume V are linearly related to fields at the surface S' so that the accuracy of these surface fields, which are used in the pseudo surface currents, is the fundamental limitation to the accuracy of the fields in the volume

V .

Let us now consider one of the few problems in which the field at the waveguide surface is known to a high degree of accuracy, i.e. radiation from the modes of the straight ideal dielectric optical fibre.

5.3 RADIATION FROM THE IDEAL

In document Radiation losses in dielectric optical waveguides (Page 177-182)