> 1 ,
angles 0^ such that
the leaky rays of a fibre are more strongly the scattering centres at the core-cladding
(2) The total power radiated from an irregularity in the
dielectric waveguide is greater than if the irregularity was in free space with the same incident field.
The former result, (1) above, is obvious from a geometric optics approach (see fig. 4a). One can see that the distance that the outer-cladding interface is from the inner-cladding core interface is immaterial in determining the amount of power radiated from the
structure. The dominant factor in determining the amount of radiated power is the complement, 0^, of the critical angle of the outer
cladding interface, a 0. If a,0 is large then most of the radiation from
the irregularity in the core becomes power propagating in the cladding modes of the multiply clad waveguide.
Although, from the above argument, the total radiated power from the multiply clad optical fibre will not be affected by the
distance between the inner and outer core-cladding interfaces, the rate at which this radiation escapes from the multiply clad fibre will vary. Since the radiation emanating from the inner-cladding boundary at an inclination 0 > tt/2 - a 0 to the axis, where a 0 is, as above, the local
z c c
critical angle at the outer core-cladding boundary, becomes a tunnelling leaky ray relative to the outer-cladding interface, depending on the radius of the outer-cladding surface, this radiation will escape at a rate consistent with the tunnelling coefficient at the outer boundary. As can be seen from fig. 4b, the skewness angle y 1 which critically affects the tunnelling leaky ray loss, decreases as the radius of the outer cladding interface increases relative to the inner-cladding. As the tunnelling leakage increases, with decreasing skewness of the radiation
, the rate of loss of the radiation from the multiply clad fibre should increase as the radius of the outer-cladding is increased, i.e. in the limit of an infinite inner-cladding, the singly clad fibre result should apply.
This geometric optics argument raises doubt as to the correct ness then of the calculation of Yip
et at., showing that the finitely cladded fibre radiates more from an irregularity than the irregularity would, if it were in free space. Snyder states that these studies on multiply clad waveguides confirm the validity of the Volume Current
Fig. 4: Radiation from a dielectric irregularity in the presence of an outer core-cladding interface. In fig. 4b, we have plotted two radii for the outer core-cladding boundary, p 1,p2 .
Method even down to any small outer-cladding radii of the order
p / p . ~ 5, where p. is the radius of core of the waveguide and p is the
o i l o
radius of the inner cladding, as in fig. 4a. However, the author does not agree, since from the above argument, it does not appear that the ratio p /p. plays the dominant role in determining the accuracy of the
Volume Current Method, except for p / p . ~ 1, i.e. there is no inner-
cladding. The significant factor in determining the accuracy of the Volume Current Method is the outer-cladding dielectric permittivity relative to that of the inner-cladding. Assuming that the outer-cladding is air, this factor becomes (£2 - e Q), which is no longer very small since for typical waveguides £2 ~ 2.25 e0 .
In the waveguide with infinite cladding, the core-cladding dielectric difference strongly affects only the power propagating at angles to the axis Thus for diffuse (i.e. wide angle)
scattering processes it can be seen that the Volume Current Method should provide valid estimates of the total power radiated. The calculations are in error by including the power flowing in the 2tt0^
(<< 1) steradians centred about 0 = 0 that constitutes the power scattered into the bound modes (or rays) of the fibre. However, although all the power scattered into the directions with 0 >0
z c eventually escapes from the core of the fibre, the rate at which this power escapes depends on the nature of the ray (i.e. either a leaky tunnelling ray or a refracting ray). This effect introduces additional phase interference between the radiation fields and thus the radiation . pattern of the scattered power will not be accurately determined by the Volume Current Method, as Yip and Martucci have recently shown .
Let us digress, momentarily, and investigate the efficiency of the isolated point dipoles as an excitation mechanism for a monomode weakly guiding step-index waveguide. This simple study provides
valuable insight into the relative accuracy of the above calculations.
3.3.1(b) The Efficiency of Dipole Sources in Excitation of the Fundamental Mode of a Step-Index Fibre
irregularity behaves as a point dipole radiator. To determine the relative accuracy of the Volume Current calculation, we need to
calculate the relative amount of power coupled into the bound modes of the optical fibre in comparison to the total power radiated from the source. This is known as the excitation efficiency of a source. In this short digression, we calculate the efficiency of a point dipole in excitation of the H E ^ mode of a monomode step-index fibre, using the well known excitation formula of waveguide theory , viz.
J dV , (10)
where P is the power coupled into the q mode with electric field E
by a current source, J occupying a volume V ' . The electric field E^ is that of the mode carrying unit power so that
E x h • z dA = 1 ,