| 12 (a) |
In all then, the fields inside the core are given by  and in the cladding are given by  . |
| (b) |
For a guided mode, the propagation constant lies between two limits  and  . |
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If  then a field distribution is generated which will has an oscillatory behavior in the core and a decaying behavior in the cladding. The energy then is propagated along fiber without any loss. |
| (c) |
Field distribution: From the solution of the wave equation we get the longitudinal fields inside the core and the |
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cladding as
Inside Core 
Electric field:
 (11a)
Magnetic field:
 (11b)
In Cladding 
Electric field:
 (11c)
Magnetic field:
 (11d)
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