Fibre attenuation with absoring cladding

[Particle Head]

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.

 

[mark726]

Your calculations seem to be correct. It is important to note that the imaginary part of the cladding refractive index is very small (8x10^-10), which results in a very small attenuation per km (8.45 dB/km). This may seem high, but it is actually a very small amount of attenuation for a single mode fiber at 1550 nm.

Regarding your concern about the non-absorbing core, the power ratio you mentioned is only valid for a perfect single mode fiber with no imperfections or losses. In reality, there will always be some level of attenuation due to various factors, such as scattering, bending losses, and imperfections in the fiber itself. Therefore, the overall attenuation of the mode along the fiber will still be affected by the absorbing cladding, even though the core is non-absorbing.

I hope this helps clarify your doubts. Keep up the good work!