1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
|
---
title: "Dispersive broadening"
date: 2021-02-27
categories:
- Physics
- Optics
- Fiber optics
layout: "concept"
---
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.
No new frequencies are created.
A pulse envelope $A(z, t)$ inside a fiber must obey the nonlinear Schrödinger equation,
where the parameters $\beta_2$ and $\gamma$ respectively
control dispersion and nonlinearity:
$$\begin{aligned}
0
= i \pdv{A}{z} - \frac{\beta_2}{2} \pdvn{2}{A}{t} + \gamma |A|^2 A
\end{aligned}$$
We set $\gamma = 0$ to ignore all nonlinear effects,
and consider a Gaussian initial condition:
$$\begin{aligned}
A(0, t)
= \sqrt{P_0} \exp\!\Big(\!-\!\frac{t^2}{2 T_0^2}\Big)
\end{aligned}$$
By [Fourier transforming](/know/concept/fourier-transform/) in $t$,
the full analytical solution $A(z, t)$ is found to be as follows,
where it can be seen that the amplitude
decreases and the width increases with $z$:
$$\begin{aligned}
A(z,t) = \sqrt{\frac{P_0}{1 - i \beta_2 z / T_0^2}}
\exp\!\bigg(\! -\!\frac{t^2 / (2 T_0^2)}{1 + \beta_2^2 z^2 / T_0^4} \big( 1 + i \beta_2 z / T_0^2 \big) \bigg)
\end{aligned}$$
To quantify the strength of dispersive effects,
we define the dispersion length $L_D$
as the distance over which the half-width at $1/e$ of maximum power
(initially $T_0$) increases by a factor of $\sqrt{2}$:
$$\begin{aligned}
T_0 \sqrt{1 + \beta_2^2 L_D^2 / T_0^4} = T_0 \sqrt{2}
\qquad \implies \qquad
\boxed{
L_D = \frac{T_0^2}{|\beta_2|}
}
\end{aligned}$$
This phenomenon is illustrated below for our example of a Gaussian pulse
with parameter values $T_0 = 1\:\mathrm{ps}$, $P_0 = 1\:\mathrm{kW}$,
$\beta_2 = -10 \:\mathrm{ps}^2/\mathrm{m}$ and $\gamma = 0$:
<a href="pheno-disp.jpg">
<img src="pheno-disp-small.jpg" style="width:100%">
</a>
The **instantaneous frequency** $\omega_\mathrm{GVD}(z, t)$,
which describes the dominant angular frequency at a given point in the time domain,
is found to be as follows for the Gaussian pulse,
where $\phi(z, t)$ is the phase of $A(z, t) = \sqrt{P(z, t)} \exp(i \phi(z, t))$:
$$\begin{aligned}
\omega_{\mathrm{GVD}}(z,t)
= \pdv{}{t}\Big( \frac{\beta_2 z t^2 / (2 T_0^4)}{1 + \beta_2^2 z^2 / T_0^4} \Big)
= \frac{\beta_2 z / T_0^2}{1 + \beta_2^2 z^2 / T_0^4} \frac{t}{T_0^2}
\end{aligned}$$
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.
The interaction between dispersion and [self-phase modulation](/know/concept/self-phase-modulation/)
leads to many interesting effects,
such as [modulational instability](/know/concept/modulational-instability/)
and [optical wave breaking](/know/concept/optical-wave-breaking/).
Of great importance is the sign of $\beta_2$:
in the **anomalous dispersion regime** ($\beta_2 < 0$),
lower frequencies travel more slowly than higher ones,
and vice versa in the **normal dispersion regime** ($\beta_2 > 0$).
## References
1. O. Bang,
*Numerical methods in photonics: lecture notes*, 2019,
unpublished.
|
