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-rw-r--r--source/know/concept/dispersive-broadening/index.md19
1 files changed, 11 insertions, 8 deletions
diff --git a/source/know/concept/dispersive-broadening/index.md b/source/know/concept/dispersive-broadening/index.md
index 746eb6d..9642737 100644
--- a/source/know/concept/dispersive-broadening/index.md
+++ b/source/know/concept/dispersive-broadening/index.md
@@ -9,10 +9,10 @@ categories:
layout: "concept"
---
-In optical fibers, **dispersive broadening** is a (linear) effect
+In optical fibers, **dispersive broadening** is a linear effect
where group velocity dispersion (GVD) "smears out" a pulse in the time domain
due to the different group velocities of its frequencies,
-since pulses always have a non-zero width in the $$\omega$$-domain.
+since pulses always have a nonzero width in the $$\omega$$-domain.
No new frequencies are created.
A pulse envelope $$A(z, t)$$ inside a fiber must obey the nonlinear Schrödinger equation,
@@ -29,7 +29,7 @@ and consider a Gaussian initial condition:
$$\begin{aligned}
A(0, t)
- = \sqrt{P_0} \exp\!\Big(\!-\!\frac{t^2}{2 T_0^2}\Big)
+ = \sqrt{P_0} \exp\!\bigg(\!-\!\frac{t^2}{2 T_0^2}\bigg)
\end{aligned}$$
By [Fourier transforming](/know/concept/fourier-transform/) in $$t$$,
@@ -38,7 +38,8 @@ where it can be seen that the amplitude
decreases and the width increases with $$z$$:
$$\begin{aligned}
- A(z,t) = \sqrt{\frac{P_0}{1 - i \beta_2 z / T_0^2}}
+ A(z,t)
+ = \sqrt{\frac{P_0}{1 - i \beta_2 z / T_0^2}}
\exp\!\bigg(\! -\!\frac{t^2 / (2 T_0^2)}{1 + \beta_2^2 z^2 / T_0^4} \big( 1 + i \beta_2 z / T_0^2 \big) \bigg)
\end{aligned}$$
@@ -48,10 +49,12 @@ as the distance over which the half-width at $$1/e$$ of maximum power
(initially $$T_0$$) increases by a factor of $$\sqrt{2}$$:
$$\begin{aligned}
- T_0 \sqrt{1 + \beta_2^2 L_D^2 / T_0^4} = T_0 \sqrt{2}
+ T_0 \sqrt{1 + \beta_2^2 L_D^2 / T_0^4}
+ = T_0 \sqrt{2}
\qquad \implies \qquad
\boxed{
- L_D = \frac{T_0^2}{|\beta_2|}
+ L_D
+ \equiv \frac{T_0^2}{|\beta_2|}
}
\end{aligned}$$
@@ -68,7 +71,7 @@ where $$\phi(z, t)$$ is the phase of $$A(z, t) = \sqrt{P(z, t)} \exp(i \phi(z, t
$$\begin{aligned}
\omega_{\mathrm{GVD}}(z,t)
- = \pdv{}{t}\Big( \frac{\beta_2 z t^2 / (2 T_0^4)}{1 + \beta_2^2 z^2 / T_0^4} \Big)
+ = \pdv{}{t}\bigg( \frac{\beta_2 z t^2 / (2 T_0^4)}{1 + \beta_2^2 z^2 / T_0^4} \bigg)
= \frac{\beta_2 z / T_0^2}{1 + \beta_2^2 z^2 / T_0^4} \frac{t}{T_0^2}
\end{aligned}$$
@@ -76,7 +79,7 @@ This expression is linear in time, and depending on the sign of $$\beta_2$$,
frequencies on one side of the pulse arrive first,
and those on the other side arrive last.
The effect is stronger for smaller $$T_0$$:
-this makes sense, since short pulses are spectrally wider.
+this makes sense, since shorter pulses are spectrally wider.
The interaction between dispersion and [self-phase modulation](/know/concept/self-phase-modulation/)
leads to many interesting effects,