# Fibre attenuation with absoring cladding

## 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 $8 \times 10 ^ {-10}$

## Homework Equations

In a weak guiding approximation the perturbed propagation constant is given as,
$$\bar{\beta} = \beta + i \delta \eta k$$
where $\delta$ is the imaginary component of the cladding refractive index and $\eta$ is the fraction of power within the cladding.

Power along a fibre is given as,

$$P(z) = P(0) e ^ {-2 \beta^{(i)} z}$$

## The Attempt at a Solution

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

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

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,

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

For the attenuation at 1km I plug $z=1000$ into this equation which should give me the attenuation per km. Doing this I get,
$$L = 8.45 \ dB/km$$

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 $\frac{0.7P(0)}{P(0)}$ which is about 1.5dB in loss?

Thanks.