In document Radiation losses in dielectric optical waveguides (Page 182-188)



In this section, we use the formulae of the Surface Current Method, equations (9) and (12) to determine the power radiated from a mode of the straight optical fibre in a length, 2L, of fibre, as in fig. 2, where

Fig. 2: Co-ordinate system for the circular cylindrical dielectric optical fibre.

Although we consider only a finite segment of fibre we will ignore the contributions due to the surfaces at the ends of the fibre, i.e. the end faces of the finite segment. This is ignored since the power radiated from the end faces is both an artificial loss process and


as it is not strongly length dependent, will not contribute to the power attenuation coefficient of the fibre, C L , where


2L • P, (1)

where P , the radiated power, is assumed to be small in comparison to the incident guided power, P Q , i.e.

« P,

In the derivation, we shall be considering both the "proper" bound modes of the structure as well as the analytic continuation of

these modes, together with their modal properties such as the eigenvalue equation, etc., below cutoff, where they become the tunnelling leaky modes of the optical fibre [2-3]. Although in a strict physical and mathematical sense, such an analytic continuation poses problems as to

the interpretation of some of the physical parameters, such as the power normalisation coefficient, A^-^ from equation (2.2.53), we show that the results provide excellent agreement with previous a n a l y s i s .

For the weakly tunnelling leaky modes of the fibre, the interpretations of the analytically continued parameters, mentioned above, have been shown by Snyder and Mitchell to have sound physical meanings. As discussed previously, we will choose the core-cladding interface to be the region over which the surface integrals will be evaluated, as in fig. 2.

k 2L is large enough so that the radiation pattern is sharp but not long enough so that the incident unattenuated modal field is no longer a valid approximation to the field along the length, 2L, of fibre. [This note refers to bottom of p.167.]

The only length dependence of the end face contributions is due to the interference between the radiation from the ends. The absolute magnitude of the radiation is reasonably unaffected by the length of the s e g m e n t .

In the travelling azimuthal wave mode formulation the vector integral, T, defined in equation (2.11), becomes (Appendix C) for the w e a k l y guiding fibre modes defined by equation (2.2.43) - (2.2.48)

r2 TT

rL rr

I - A m p .


c -L dz' <^ ^exp{i (k2p sin 0 sin (j)' + (k2 cos 0 - 3 ) z' - £(j)' ) }

- sin x + B^-^ cos (j)' y + z , (2)

where Aj^_^ is the power normalisation coefficient, which after using the integral representation of Bessel functions and the recurrence relations

[17] ,


£ - Am (41Tp) - kW -

iB«i JJ x +

£ + 1 £ ~ ' q + ' u £ X

where the argument of the Bessel functions is Q, where

Q = k 2 p sin 0 (3) and i„ " l 1 ' <iw > £+1 k 9p „(1) H£il(iW) (5)

where the subscripts (£-1) and (£+1) refer to the HE^ and EH^ modes respectively, and

k 2y = k2 cos 0 - 3 .

Using equation (3) in equation (2.14) and integrating S obtained from p

equation (2.14) to obtain P , the total power radiated, we find (Appendix C) fe if Otherwise t 2 k 2p 2 A £_ 1 cos 0 O = 7 - < 1 • rB£+l Q l Q k 2pj (7) (8) k 2l as k 2L 00 , 0 -5- 0

where from equations (4) and (8)

Q 2 = p2 (k2 - 3 2 ) = - W 2 > 0 .

From equation (8), for the condition for significant length dependent radiation, we can see that only the modes for which

k 2 > 3 (9)

radiate strongly. The bound modes of the fibre satisfy the opposite result to equation (9) and hence do not radiate significantly to contribute to the power attenuation coefficient.

Now, if we seek to find the power radiated from the weakly tunnelling leaky modes which correspond to 3/k2 ^ 1 [1-2], then using this approximation together with the recurrence relations for Bessel functions and their Wronksian relations [17], P becomes (Appendix C)

8L ko (r ^ _ _2_ (A£ + 1 }


- 2 (10)

where we have used

Q k 2p sin 0 Q

> (ID

The modal power attenuation coefficient, a, is for those modes which were incident with unit power

and thus


(A£ + 1 } H m (Q) - 2


Using the expressions for A ^ - . defined by the analytic continuation of equation (3.3.16) to imaginary W via W i Q for the step-index fibre, equation (12) becomes for the power attenuation coefficient, for the step-index fibre, a g ^ 7Tk2p' u 2 ' ( 1 ) (1) , / V H £ + 2 (Q) H £ (Q) - 1 S.I. (13)

where Q = k2p sin0o, with 0Q « 1 . Equation (13) agrees exactly with the results of Snyder and Mitchell, using a wide range of techniques [18]. The upper and lower signs refer, as usual, to the HE^ and EH^ modes respectively.

There is one major difference between the calculations of the Surface Current Method and the Volume Current Method. This difference is due to the former technique retaining the longitudinal components to provide an accurate answer, whereas in the Volume Current Method (due to their small size of the longitudinal field), they did not contribute significantly to the scattered energy. At first sight, this appears to be rather surprising until one realises that the Surface Current Method is equivalent to calculation of the time averaged Poynting vector in the transverse plane evaluated at the core-cladding interface [18]. Once this is realised, it is obvious that the longitudinal fields, though small, cannot be neglected since those components are necessary to contribute to the real transverse component of the time averaged Poynting vector of the mode.

Although we derived the modal power attenuation coefficient for the modes of the fibre by analytically continuing the modal fields below cutoff, we could have used another procedure that removes the need to re-interpret the power normalisation coefficient A^_^. By assuming a distribution over the surface of the guide of the form of the proper modes, but with 3 < k2 , we could have evaluated a temporal attenuation coefficient (i.e. the mode field is everywhere uniform but decaying in time). Using the "equivalence" between the spatial and temporal

attenuation, following Snyder and Mitchell [18], where the only analytic continuation is that of the group velocity of a mode below cutoff, we could then obtain the spatial attenuation coefficient without the necessity of re-interpretation of the power normalisation coefficient for modes below their cutoff.

Due to the abundant literature on these tunnelling leaky modes in step-index fibres, we shall not loiter on this topic. The reader is referred to refs. 1 and 18 for complete information on the physics of the tunnelling leaky modes. Having demonstrated the ability of the Surface Current Method for calculations of curvature loss of both step-

and graded-index waveguides, we shall now discuss in general the applicability of the analysis of the Surface Current Method.

In document Radiation losses in dielectric optical waveguides (Page 182-188)