**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^,**

**£1^,**

**and corresponding wavelength**

**X =**

**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**

**0 _{o}**

**(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) "**

**fCT**1**(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 E****1l**** 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)**