Optical fibers are used to transmit light over long distances, but two effects strongly shape signal quality:
attenuation and chromatic dispersion.
Attenuation reduces the optical power as the signal travels down the fiber, while dispersion spreads out optical pulses in time.
For long-haul communication systems, both effects must be controlled carefully.
Fiber attenuation is usually specified by a coefficient
\(\alpha\)
in
\(\text{dB/km}\).
If a fiber has attenuation
\(\alpha\)
and length
\(L\),
then the total loss in decibels is
\[
\text{Loss}=\alpha L.
\]
Once the loss is known, the output power follows from the decibel relation
\[
P_{\text{out}}=P_{\text{in}}10^{-\alpha L/10}.
\]
This formula is very common in fiber-optic engineering because attenuation is almost always quoted in dB rather than as a simple exponential coefficient.
For example, if
\(\alpha=0.2\,\text{dB/km}\)
and
\(L=100\,\text{km}\),
then the total loss is
\[
\text{Loss}=0.2\times100=20\,\text{dB}.
\]
A 20 dB loss means the output power is only
\(10^{-2}=1\%\)
of the input power.
This is why long-haul links often need optical amplifiers or regeneration stages.
The second major effect is chromatic dispersion.
Different wavelength components of a pulse travel with slightly different group velocities, so the pulse broadens as it propagates.
In fiber optics, this is summarized by the dispersion parameter
\(D\),
usually measured in
\(\text{ps}/(\text{nm}\cdot\text{km})\).
The approximate pulse broadening caused by a source spectral width
\(\Delta\lambda_s\)
over a fiber length
\(L\)
is
\[
\Delta\tau \approx |D|L\Delta\lambda_s.
\]
This means larger dispersion, longer fiber, or a wider optical spectrum all increase pulse spreading.
If the original pulse width is
\(\tau_0\),
then a convenient estimate for the output pulse width is
\[
\tau_{\text{out}}\approx \sqrt{\tau_0^2+\Delta\tau^2}.
\]
This is especially useful when deciding whether neighboring digital pulses in a communication link might begin to overlap and cause intersymbol interference.
Different fiber types have different attenuation and dispersion behavior as functions of wavelength.
Standard single-mode fiber is usually optimized for low loss near
\(1550\,\text{nm}\),
while its dispersion crosses zero near
\(1310\,\text{nm}\).
Dispersion-shifted fibers move the zero-dispersion point closer to
\(1550\,\text{nm}\),
and non-zero dispersion-shifted fibers keep a small but nonzero dispersion there to help reduce nonlinear penalties in advanced systems.
This calculator uses simplified educational preset curves for common silica telecom fibers.
The attenuation is modeled as a wavelength-dependent minimum around a reference wavelength, while the dispersion is modeled as a linear approximation around a zero-dispersion wavelength:
\[
D(\lambda)\approx S(\lambda-\lambda_0),
\]
where
\(S\)
is a dispersion slope and
\(\lambda_0\)
is the approximate zero-dispersion wavelength.
This is not a full manufacturer-grade model, but it is very helpful for understanding the main trends.
For the sample case of standard SMF near
\(1550\,\text{nm}\)
with
\(\alpha\approx0.2\,\text{dB/km}\)
over
\(100\,\text{km}\),
the loss is
\(20\,\text{dB}\),
so only about
\(1\%\)
of the input power remains.
That simple example immediately shows why attenuation is such a central design constraint in long-distance transmission.
At a more advanced university level, one goes beyond this picture and includes nonlinear effects such as self-phase modulation, cross-phase modulation, Raman scattering, and four-wave mixing.
One also distinguishes material dispersion, waveguide dispersion, polarization-mode dispersion, amplifier noise, and dispersion compensation.
But for an introductory fiber-link estimate, the most important ideas remain:
attenuation sets the power budget, and chromatic dispersion sets pulse spreading.
\[
\text{Loss}=\alpha L,
\qquad
P_{\text{out}}=P_{\text{in}}10^{-\alpha L/10},
\qquad
\Delta\tau \approx |D|L\Delta\lambda_s.
\]
These formulas let you estimate how much signal power survives, how strongly a pulse broadens, and why telecom systems are designed around specific transmission windows such as
\(1310\,\text{nm}\)
and
\(1550\,\text{nm}\).
