{title2=$\alpha_{\rm dB}$}

<Optical fiber attenuation> is the reduction of transmitted <optical power> along an <optical fiber>. Define a positive loss in <decibels> by comparing <optical power> $P(0)$ already launched into the guide with surviving <optical power> $P(z)$:
$$
A(z)=10\log_{10}\!\left(\frac{P(0)}{P(z)}\right)\,\mathrm{dB}.
$$
The factor ten applies to a <power> ratio; the factor twenty applies to a <wave amplitude> ratio only when <power> is proportional to its square under the same conditions.<Introduction to Optical Fibers, dB, Attenuation and Measurements by Cisco>[Introduction to Optical Fibers, dB, Attenuation and Measurements, What is a Decibel?, Base 10 Logarithm Rules and Power/Voltage Gains: attenuation convention]{ref}

For a uniform illustrative model with constant loss coefficient $\alpha_{\rm dB}$ in $\mathrm{dB/km}$ and distance $z$ in kilometres,
$$
A(z)=\alpha_{\rm dB}z,\qquad
\frac{P(z)}{P(0)}=10^{-A(z)/(10\,\mathrm{dB})}.
$$
For $\alpha_{\rm dB}=0.2\,\mathrm{dB/km}$ over $10\,\mathrm{km}$, the loss is $2\,\mathrm{dB}$ and
$$
\frac{P(10\,\mathrm{km})}{P(0)}=10^{-0.2}\simeq0.630957.
$$
Thus about $63.1\%$ of the launched <optical power> survives and $36.9\%$ is lost. This is an illustrative coefficient, not a property of all <optical fibers>, and is independent of the earlier illustrative <refractive indices>.

\Image[/optical-fiber-distance-and-power-loss.png]
{title=Equal distance increments add decibel loss and multiply surviving power}

The <exponential decay> follows from assuming each equal distance interval transmits the same fraction. For two such intervals, <power> ratios multiply; their <logarithms> add, so the losses in <decibels> add. Two $5\,\mathrm{km}$ sections each transmit $10^{-0.1}\simeq0.794328$; together they transmit $(0.794328)^2\simeq0.630957$. The loss curve is straight in <decibels> and curved on a linear <power> scale.

<Wave absorption> and <scattering of light> can remove guided <optical power> even in a straight fibre, while bends can let it escape. Entrance losses, joints and connectors contribute additional losses; they belong separately from a uniform distance coefficient. This is why a measured end-to-end <power> ratio need not equal the straight-fibre model.<Introduction to Optical Fibers, dB, Attenuation and Measurements by Cisco>[Introduction to Optical Fibers, dB, Attenuation and Measurements, Understand Insertion Loss and Calculate a Power Budget: joint losses, launch dependence and distance contribution]{ref}


 Optical fiber attenuation (source code)