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## Homework Statement

Given a single mode fibre with non-absorbing core and absorbing cladding where the power propagating in the fundamental mode is split 70%:30% between core and cladding, calculate overall attenuation of the mode along the fibre in dB/km at 1550nm source wavelength where the imaginary part of the cladding refractive index is [itex] 8 \times 10 ^ {-10} [/itex]

## Homework Equations

In a weak guiding approximation the perturbed propagation constant is given as,

[tex] \bar{\beta} = \beta + i \delta \eta k [/tex]

where [itex] \delta [/itex] is the imaginary component of the cladding refractive index and [itex] \eta [/itex] is the fraction of power within the cladding.

Power along a fibre is given as,

[tex] P(z) = P(0) e ^ {-2 \beta^{(i)} z} [/tex]

## The Attempt at a Solution

Substituting the imaginary component of the perturbed propagation constand into the power along the fibre we get,

[tex] P(z) = P(0) e ^ { - 2 \delta \eta k z} = P(0) e^{-2 (8 \times 10 ^{-10}) (0.3) \left(\frac{2 \pi }{1550 \times 10 ^ {-9}}\right) z } [/tex]

To convert this to attenuation I divide both sides by P(0) and then take the log base 10 and multiply by -10 as follows,

[tex] L = -10 \log \left( \frac{P(z)}{P(0)}\right) = 20 (8 \times 10 ^{-10}) (0.3) \left(\frac{2 \pi }{1550 \times 10 ^ {-9}}\right) z \log(e) [/tex]

For the attenuation at 1km I plug [itex] z=1000[/itex] into this equation which should give me the attenuation per km. Doing this I get,

[tex] L = 8.45 \ dB/km[/tex]

My problem is this seems way too high for single mode fibre and so I believe I am missing something here. I was also thinking that with a non absorbing core wouldn't this imply no attenuation in the core and hence after some time you would end up with a power ratio of [itex] \frac{0.7P(0)}{P(0)}[/itex] which is about 1.5dB in loss?

Thanks.