The wave propagation likes seismic wave, here’s the energy:

Backscattered energy: \(dE_{BS}(z, z+dz) = (E_p(0)exp\int_0^z-\alpha(u)du)\cdot\alpha_R(z)B(z)dz\cdot exp\int_0^z-\alpha(u)du\)

$\alpha_R(z)$ is Rayleigh Scatter Coefficient. That represents the energy that transferred to Rayleigh wave.

B(z) is the energy that the receiver will receive. Not all the energy will reflect in OTDR.

So the power is:

And \(P_BS(t)\) is OTDR measurement; and \(\eta(z)=\frac{v_g}{2}\alpha_R(z)B(z)\) is returning backscatter coefficient.

We could use \(\frac{v_gt}{2}\) to replace z. Then we have:

This result is beautiful. OTDR curve is exp, while after log it will be a straight line.

The slope of the log curve: \(-2\alpha(z)\)

Stacking

Pulse will inject in a constant velocity.

Limitation

Fiber non-linear effect: if the amplitude is too large.

• Attenuation factor ++ while the Laser amplitude exceeds a certain limitation.

• Normal wave –> Exp
• Single frequency –> Random

  this is caused by interference: random heterogeneity.

Application: check fiber health.

Another method: OFDR

The frequency varies while t is increasing.

Source: \(A_{src}\sin(2\pi\gamma t\cdot t+\phi_1)\) Backscattered energy: E(z)=A_{BS}\sin(2\pi\gamma(t-\frac{2z}{v_g})\cdot t+\phi_2)

Difference between OFDR and OTDR:

• sacrifice range to get higher resolution
• for lab experiment