summaryrefslogtreecommitdiff
path: root/source/know/concept/optical-wave-breaking/index.md
blob: 1b6b558c4dc3ac03b61fc2503ea8ee4fc8560104 (plain)
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
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
---
title: "Optical wave breaking"
sort_title: "Optical wave breaking"
date: 2021-02-27
categories:
- Physics
- Optics
- Fiber optics
- Nonlinear optics
layout: "concept"
---

In fiber optics, **optical wave breaking** (OWB) is a nonlinear effect
caused by interaction between
[group velocity dispersion](/know/concept/dispersive-broadening/) (GVD) and
[self-phase modulation](/know/concept/self-phase-modulation/) (SPM).
It only happens in the normal dispersion regime ($$\beta_2 > 0$$)
for pulses meeting a certain criterium, as we will see.

SPM creates low frequencies at the front of the pulse, and high ones at the back,
and if $$\beta_2 > 0$$, GVD lets low frequencies travel faster than high ones.
When those effects interact, the pulse gets temporally stretched
in a surprisingly sophisticated way.

To illustrate this, the instantaneous frequency $$\omega_i(z, t) = -\ipdv{\phi}{t}$$
has been plotted below for a theoretical Gaussian input pulse experiencing OWB,
with settings $$T_0 = 100\:\mathrm{fs}$$, $$P_0 = 5\:\mathrm{kW}$$,
$$\beta_2 = 2\:\mathrm{ps}^2/\mathrm{m}$$ and $$\gamma = 0.1/\mathrm{W}/\mathrm{m}$$.

In the left panel, we see the typical S-shape caused by SPM,
and the arrows indicate the direction that GVD is pushing the curve in.
This leads to steepening at the edges, i.e. the S gradually turns into a Z.
Shortly before the slope would become infinite,
small waves start "falling off" the edge of the pulse,
hence the name *wave breaking*:

{% include image.html file="frequency-full.png" width="100%"
    alt="Instantaneous frequency profile evolution" %}

Several interesting things happen around this moment.
To demonstrate this, spectrograms of the same simulation
have been plotted below, together with pulse profiles
in both the $$t$$-domain and $$\omega$$-domain on an arbitrary linear scale
(click the image to get a better look).

Initially, the spectrum broadens due to SPM in the usual way,
but shortly after OWB, this process is stopped by the appearance
of so-called **sidelobes** in the $$\omega$$-domain on either side of the pulse.
In the meantime, in the time domain,
the pulse steepens at the edges, but flattens at the peak.
After OWB, a train of small waves falls off the edges,
which eventually melt together, leading to a trapezoid shape in the $$t$$-domain.
Dispersive broadening then continues normally:

{% include image.html file="spectrograms-full.png" width="100%"
    alt="Spectrograms of pulse shape evolution" %}

We call the distance at which the wave breaks $$L_\mathrm{WB}$$,
and want to predict it analytically.
We do this using the instantaneous frequency $$\omega_i$$,
by estimating when the SPM fluctuations overtake their own base,
as was illustrated earlier.

To get $$\omega_i$$ of a Gaussian pulse experiencing both GVD and SPM,
it is a reasonable approximation, for small $$z$$, to simply add up
the instantaneous frequencies for these separate effects:

$$\begin{aligned}
    \omega_i(z,t)
    &\approx \omega_\mathrm{GVD}(z,t) + \omega_\mathrm{SPM}(z,t)
    = \frac{tz}{T_0^2} \bigg( \frac{\beta_2 / T_0^2}{1 + \beta_2^2 z^2 / T_0^4}
    + 2\gamma P_0 \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg)
\end{aligned}$$

Assuming that $$z$$ is small enough such that $$z^2 \approx 0$$, this
expression can be reduced to:

$$\begin{aligned}
    \omega_i(z,t)
    \approx \frac{\beta_2 tz}{T_0^4} \bigg( 1 + 2\frac{\gamma P_0 T_0^2}{\beta_2} \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg)
    = \frac{\beta_2 t z}{T_0^4} \bigg( 1 + 2 N_\mathrm{sol}^2 \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg)
\end{aligned}$$

Where we have assumed $$\beta_2 > 0$$,
and $$N_\mathrm{sol}$$ is the **soliton number**,
which is defined as:

$$\begin{aligned}
    N_\mathrm{sol}^2
    \equiv \frac{L_D}{L_N}
    = \frac{\gamma P_0 T_0^2}{|\beta_2|}
\end{aligned}$$

This quantity is very important in anomalous dispersion,
but even in normal dispersion, it is still a useful measure of the relative strengths of GVD and SPM.
As was illustrated earlier, $$\omega_i$$ overtakes itself at the edges,
so OWB occurs when $$\omega_i$$ oscillates there,
which starts when its $$t$$-derivative,
the **instantaneous chirpyness** $$\xi_i$$,
has *two* real roots for $$t^2$$:

$$\begin{aligned}
    0
    = \xi_i(z,t)
    = \pdv{\omega_i}{t}
    &= \frac{\beta_2 z}{T_0^4} \bigg( 1 + 2 N_\mathrm{sol}^2 \Big( 1 - \frac{2 t^2}{T_0^2} \Big) \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big) \bigg)
    \equiv \frac{\beta_2 z}{T_0^4} \: f\Big(\frac{t^2}{T_0^2}\Big)
\end{aligned}$$

Where the function $$f(x)$$ has been defined for convenience. As it turns
out, this equation can be solved analytically using the *Lambert $$W$$ function*,
leading to the following exact minimum value $$N_\mathrm{min}^2$$ for $$N_\mathrm{sol}^2$$,
such that OWB can only occur when $$N_\mathrm{sol}^2 > N_\mathrm{min}^2$$:

$$\begin{aligned}
    \boxed{
        N_\mathrm{min}^2
        = \frac{1}{4} \exp\!\Big(\frac{3}{2}\Big)
        \approx 1.12
    }
\end{aligned}$$

If this condition $$N_\mathrm{sol}^2 > N_\mathrm{min}^2$$ is not satisfied,
$$\xi_i$$ cannot have two roots for $$t^2$$, meaning $$\omega_i$$ cannot overtake itself.
GVD is unable to keep up with SPM, so OWB will not occur.

Next, consider two points at $$t_1$$ and $$t_2$$ in the pulse,
separated by a small initial interval $$(t_2 - t_1)$$.
The frequency difference between these points due to $$\omega_i$$
will cause them to displace relative to each other
after a short distance $$z$$ by some amount $$\Delta t$$,
estimated by:

$$\begin{aligned}
    \Delta t
    &\approx z \Delta\beta_1
    \qquad
    &&\Delta\beta_1
    \equiv \beta_1(\omega_i(z,t_2)) - \beta_1(\omega_i(z,t_1))
    \\
    &\approx z \beta_2 \Delta\omega_i
    \qquad
    &&\Delta\omega_i
    \equiv \omega_i(z,t_2) - \omega_i(z,t_1)
    \\
    &\approx z \beta_2 \Delta\xi_i \,(t_2 - t_1)
    \qquad \quad
    &&\Delta\xi_i
    \equiv \xi_i(z,t_2) - \xi_i(z,t_1)
\end{aligned}$$

Where $$\beta_1(\omega)$$ is the inverse of the group velocity.
For a certain choice of $$t_1$$ and $$t_2$$,
OWB occurs when they catch up to each other,
which is when $$-\Delta t = (t_2 - t_1)$$.
The distance $$L_\mathrm{WB}$$ at which this happens first
must satisfy the following condition for some value of $$t$$:

$$\begin{aligned}
    L_\mathrm{WB} \: \beta_2 \: \xi_i(L_\mathrm{WB}, t)
    = -1
    \qquad \implies \qquad
    L_\mathrm{WB}^2
    = - \frac{T_0^4}{\beta_2^2 \: f(t^2/T_0^2)}
\end{aligned}$$

The time $$t$$ of OWB must be where $$\omega_i(t)$$ has its steepest slope,
which is at the minimum value of $$\xi_i(t)$$, and by extension $$f(x)$$.
This turns out to be $$f(3/2)$$:

$$\begin{aligned}
    f_\mathrm{min}
    = f(3/2)
    = 1 - 4 N_\mathrm{sol}^2 \exp(-3/2)
    = 1 - N_\mathrm{sol}^2 / N_\mathrm{min}^2
\end{aligned}$$

Clearly, $$f_\mathrm{min} \ge 0$$ when $$N_\mathrm{sol}^2 \le N_\mathrm{min}^2$$,
which, when inserted above, leads to an imaginary $$L_\mathrm{WB}$$,
confirming that OWB cannot occur in that case.
Otherwise, if $$N_\mathrm{sol}^2 > N_\mathrm{min}^2$$, then:

$$\begin{aligned}
    \boxed{
        L_\mathrm{WB}
        = \frac{T_0^2}{\beta_2 \, \sqrt{- f_\mathrm{min}}}
        = \frac{L_D}{\sqrt{N_\mathrm{sol}^2 / N_\mathrm{min}^2 - 1}}
    }
\end{aligned}$$

This prediction for $$L_\mathrm{WB}$$ appears to agree well
with the OWB observed in the simulation:

{% include image.html file="simulation-full.png" width="100%"
    alt="Optical wave breaking simulation results" %}

Because all spectral broadening up to $$L_\mathrm{WB}$$ is caused by SPM,
whose $$\omega$$-domain behaviour is known,
it is in fact possible to draw some analytical conclusions
about the achieved bandwidth when OWB sets in.
Filling $$L_\mathrm{WB}$$ in into $$\omega_\mathrm{SPM}$$ gives:

$$\begin{aligned}
    \omega_{\mathrm{SPM}}(L_\mathrm{WB},t)
    = \frac{2 \gamma P_0 t}{\beta_2 \sqrt{4 N_\mathrm{sol}^2 \exp(-3/2) - 1}} \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big)
\end{aligned}$$

Assuming that $$N_\mathrm{sol}^2$$ is large in the denominator, this can
be approximately reduced to:

$$\begin{aligned}
    \omega_\mathrm{SPM}(L_\mathrm{WB}, t)
    \approx \frac{2 \gamma P_0 t}{\beta_2 N_\mathrm{sol}} \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big)
    = 2 \sqrt{\frac{\gamma P_0}{\beta_2}} \frac{t}{T_0} \exp\!\Big(\!-\!\frac{t^2}{T_0^2}\Big)
\end{aligned}$$

The expression $$x \exp(-x^2)$$ has its global extrema
$$\pm 1 / \sqrt{2 e}$$ at $$x^2 = 1/2$$. The maximum SPM frequency shift
achieved at $$L_\mathrm{WB}$$ is therefore given by:

$$\begin{aligned}
    \omega_\mathrm{max}
    = \sqrt{\frac{2 \gamma P_0}{e \beta_2}}
\end{aligned}$$

Interestingly, this expression does not contain $$T_0$$ at all,
so the achieved spectrum when SPM is halted by OWB
is independent of the pulse width,
for sufficiently large $$N_\mathrm{sol}$$.



## References
1.  D. Anderson, M. Desaix, M. Lisak, M.L. Quiroga-Teixeiro,
    [Wave breaking in nonlinear-optical fibers](https://doi.org/10.1364/JOSAB.9.001358),
    1992, Optical Society of America.
2.  A.M.  Heidt,  A.  Hartung,  H.  Bartelt,
    [Generation of ultrashort and coherent supercontinuum light pulses in all-normal dispersion fibers](https://doi.org/10.1007/978-1-4939-3326-6_6),
    2016, Springer Media.