Radiation Loss Due to Variations in the Radius of the Core

In document Radiation losses in dielectric optical waveguides (Page 113-118)

In this section we calculate the power radiated from a mode incident on a finite length of fibre with an axial variation in the radius of the core. This case is of general interest for two reasons.

(a) A statistical variation in the radius can be resolved by Fourier analysis into its sinusoidal components and each component can be studied individually before recombination of the resultant fields into the total radiation field.

(b) Machine vibrations during the manufacture and drawing of the fibre can cause small amplitude fluctuations of a constant f r e q u e n c y .

We shall be concerned with variations in the radius that retain the circular symmetry of the dielectric optical fibre, so that the radius of the perturbed fibre, of length 2L, can be written as (see fig. 9)

p = p 0 (1 + a sin ttz) , (29)

where is the spatial frequency of the perturbation and a p Q <<

A,

where

A

is the wavelength of the incident field. The dielectric profile of the unperturbed fibre of radius p Q , as shown in fig. 10, can be described mathematically as /

A

- A

q

r n r c l C O o C l H(P 0 “ r > + e 2 ' £(r) (30)

Fig. 9: Sinusoidal radius variations along the axis of the fibre, with spatial frequency

£1^,

and corresponding wavelength

X =

1 1 p/r -* 1

Fig. 10: The dielectric profile of the circular cylindrical dielectric optical fibre.

where q ( r / P 0) contains the variation of the dielectric with radius and

A

cl

- £2)

is the maximum relative difference between the core and cladding dielectrics, and

co

is the maximum relative difference in the core dielectric variation, and H(x) is the Heaviside step function defined by

H (x) = 0 : x < 0

= 1 : x > 0 .

In the section of the fibre with the perturbed core radius, we assume that in every cross-section, the shape of the dielectric profile, determined by q(x) in equation (30), does not change but is only

confined to a different radius.

Using the formula for the induced volume current density, equation (1.19) due to this perturbation of the radius, we find

J iu) a p 0 sin

l a p J

p

=

p

.

(31)

where we have used a Taylor series expansion around the unperturbed radius pQ , for the dielectric profile of the perturbed waveguide. In a mathematical sense, this expansion does not appear valid around p = p Q , where the derivative of H ( p Q - r) is a delta function.

However, if the fluctuations of the radius are small in

comparison to the wavelength of the incident light, the contribution to the volume currents induced due to the core-cladding discontinuity can be considered to be due to a constant field (in the radial direction) and we take this field to be the field at the unperturbed core-cladding interface. In this way, we can represent the core-cladding contribution by a delta function representation which reduces the volume current

density to a surface current density located at the core-cladding i n t e rface.

When the length of the perturbed fibre section is sufficiently 1*

large so as to produce distinct (sharp) radiation lobes, i.e. when k 2L > > l , the total radiated power , induced by an incident modal field, , is (see Appendix C)

TTL R 2C (ü)apn ) d<j>{|RXl|2 } (32)

where C is defined in equation (1.21). The subscript 0Q on the

integrand indicates that the term is to be evaluated for 0 = 0 O , where 0 Q is the direction of the radiation, to the z-axis

0o (33)

where 3^ is the propagation constant of the bound mode.

The term

I

in the integrand yields the magnitude of the radiation field (see Appendix C) and

l£+1 ±iDoxi (Pn (x 1 iy) "

fCT1(p„) r

r ^ - 1 £+1 where

Dp (P0^ )

r dr 2tt - [V] dcj)' 0 f (r) L p

W

x exp{i (k2r sin0 cos ((f) - <j)' ) - p(f)' ) }

(34)

(35)

From an investigation of equation (33) it is observed that there is negligible length dependent radiation loss from the mode £m [9], unless the spatial frequency of the perturbation, £2, satisfies

ß £ m - k 2 < < ßjlnl+k 2 . (36)

2L cannot be infinitely large as the incident field of the

unperturbed guide would no longer be a valid approximation to the field in the perturbed fibre.

The spatial frequencies, ß, not satisfied by equation (36), do not induce significant radiation from the waveguide. As Snyder points out, such frequencies do induce a small radiation component inversely proportional to L due to the discontinuity at the terminations of this finite element [9]. Such frequencies, however, can still induce

coupling between the bound modes of the waveguide and by this procedure the power of the one mode, unable to couple directly to the radiation field, can eventually couple indirectly.

By inspection of equation (16), the term (3e/8p) ^ can be seen to play a significant role in type of radiation expected. Using equation (11) for £ (r), we observe that,t

3e

13PJ

P=Pr £, (A -A ) 6 (p - r) + e A q' 1 c l c o 0 1 co vP0J H(p_ - r) , (37) P o

where the ' indicates the difference with respect to the argument (r/pn),

The first term on the right hand side of equation (37) gives rise to surface current densities located at the core-cladding interface. This is the only term present for the well-known result of the step-

index fibre [9,5]. If the dielectric profile is continuous across the core boundary, i.e.

A co

Then the surface current contribution to the radiation field does not appear.

The second term on the right hand side of equation (37) gives rise to the volume current density throughout the core of the waveguide. For the continuous profile discussed above, this is the only term that contributes to the radiation field. In general, both surface and volume

The appearance of the delta function appears to invalidate the Taylor series expansion. However, we use this notation for the

particular case of small amplitude variations in comparison to the wave­ length of light in the medium. The effect of the discontinuity across the core-cladding boundary can then be considered as occurring at the boundary itself, so that the boundary contributions in this particular case can be written as in equation (37) as we have discussed previously.

current densities occur. However, in a graded-index fibre, with a discontinuity in the dielectric profile across the core boundary, the modes far from cutoff have negligible intensity at r = p Q , due to the

rapid decay of the evanescent field beyond the outer caustic, and the dominant contribution of the radiation field should be due to the volume current distribution in the core.

Let us now consider the situation of a sinusoidal perturbed radius of a weakly guiding.step-index fibre. Using the complete fields, including the small longitudinal fields, the modal power loss per unit length of the H E1l mode, a i x , is [20]

l l

2 2 2

TTa W V

8 k 2P 2 J 2 (U) Jo (U)

J n (k„ p sin 9 ) +

o 2 ^ o cos0 J (U) J (k p sin 0 )o o o' 2^ o

+ -gjj s in0o J x (U) Jj (k2 p sin 0 O ) (38)

In document Radiation losses in dielectric optical waveguides (Page 113-118)