From 1d700ab734aa9b6711eb31796beb25cb7659d8e0 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Tue, 20 Dec 2022 20:11:25 +0100 Subject: More improvements to knowledge base --- source/know/concept/dispersive-broadening/index.md | 19 +++++++++++-------- 1 file changed, 11 insertions(+), 8 deletions(-) (limited to 'source/know/concept/dispersive-broadening') 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, -- cgit v1.2.3