This document discusses optical solitons, which are pulses of light that propagate without changing shape due to a balance between dispersion and nonlinear effects. It covers self-phase modulation, how pulses propagate in dispersive fiber, and how solitons form under certain conditions. The document also briefly summarizes other soliton types including dark, spatial, and N-solitons. It concludes by noting that while solitons allow for virtually error-free high-speed optical communications, they remain at the prototype stage due to challenges around amplification.

1. Solitons SPM Pulse Propagation Recap Other Soliton Types Conclusion
Soliton Propagation
Shuvan Prashant
Sri Sathya Sai University, Prasanthi Nilayam
June 16, 2014
as part of PHY 1003 Nonlinear Optics Coursework.

2. Solitons SPM Pulse Propagation Recap Other Soliton Types Conclusion
Outline
1 Solitons
Introduction
2 SPM
3 Pulse Propagation
4 Recap
5 Other Soliton Types
6 Conclusion

3. Solitons SPM Pulse Propagation Recap Other Soliton Types Conclusion
Solitary reaper
Soliton
Solitary Solution → Soliton:There exists a single solution to the
propagation equation.
Big deal about Solitons
Soliton suggests particle type behaviour
Solitons travel without any dispersion inside any standard
fiber (even highly dispersive ones)
Result: Single-channel data streams possible of 100 to 200
Gbps
In a WDM system, little (some but small) interaction between
channels using solitons exists.

4. Solitons SPM Pulse Propagation Recap Other Soliton Types Conclusion
Soliton sighted first time 1834
Scott Russell observed a heap of water in a canal that propagated
undistorted over several kilometers.
“a rounded, smooth and well-defined heap of water,
which continued its course along the channel apparently
without change of form or diminution of speed. I
followed it on horseback, and overtook it still rolling on
at a rate of some eight or nine miles an hour, preserving
its original figure some thirty feet long and a foot to a
foot and a half in height. ” [NLFO Agarwal]
Such waves were later called solitary waves.

5. Solitons SPM Pulse Propagation Recap Other Soliton Types Conclusion
Self Phase Modulation is the phase change of optical pulse
due to nonlinearity of medium’s RI
Consider the pulse
˜E(z, t) = ˜Az, tei(k0−ω0t)
+ c.c. (1)
through a medium having nonlinear refractive index
n(t) = n0 + n2I(t) where I(t) = 2n0 0c| ˜A(z, t)|2
(2)
Assumptions: Instantaneous material response and sufficiently
small length of material
Change in phase
φNL(t) = −n2I(t)ω0L/c (3)
Time varying pulse - spectral modification of pulse Instantaneous
frequency of the pulse
ω(t) = ω0 + δω(t) where δω(t) =
d
dt
φNL(t) (4)
[Boyd2003]

6. Solitons SPM Pulse Propagation Recap Other Soliton Types Conclusion
The Case of Curious sech pulse
Pulseshape I(t) = I0sech2
(t/τ0)
Nonlinear phase shift
φNL(t) = −n2I0sech2
(t/τ0)ω0L/c
Change in instantaneous frequency
δω(t) = −n2ω0
dI
dt
L
c
= 2n2
ω0
cτ0
LI0sech2
(t/τ0) tanh(t/τ0)
Leading edge shifted to lower frequencies and trailing edge to
higher frequencies
Max value of freq shift
δωmax n2
ω0
cτ0
LI0
ωmax
∆φmax
NL
τ0
where ∆φmax
NL = n2
ω0
LI0

7. Solitons SPM Pulse Propagation Recap Other Soliton Types Conclusion
The Case of Curious sech pulse
Pulseshape I(t) = I0sech2
(t/τ0)
Nonlinear phase shift
φNL(t) = −n2I(t)ω0L/c
φNL(t) = −n2I0sech2
(t/τ0)ω0L/c
Change in instantaneous frequency
δω(t) = −n2ω0
dI
dt
L
c
= 2n2
ω0
cτ0
LI0sech2
(t/τ0) tanh(t/τ0)
Leading edge shifted to lower frequencies and trailing edge to
higher frequencies
Max value of freq shift
δωmax n2
ω0
cτ0
LI0
ωmax
∆φmax
NL
τ

8. Solitons SPM Pulse Propagation Recap Other Soliton Types Conclusion
The Case of Curious sech pulse
[Boyd2003]

9. Solitons SPM Pulse Propagation Recap Other Soliton Types Conclusion
Chirping of an optical pulse by propagation through a
nonlinear optical Kerr medium
[SalehTeich]

10. Solitons SPM Pulse Propagation Recap Other Soliton Types Conclusion
How do Pulses Propagate in Dispersive Media ???
˜E(z, t) = ˜A(z, t)ei(k0z−ω0t)
+ c.c. (5)
where k0 = nlin(ω0)ω0/c
How does pulse envelope function propagate in dispersive media ?
Wave Equation
∂2 ˜E
∂z2
−
1
c2
∂2 ˜D
∂t2
= 0 (6)
Fourier Transforms
˜E(z, t) =
∞
−∞
E(z, ω)e−iωt dω
2π
(7)
D(z, ω) = (ω)E(z, ω)
Using Fourier transforms in wave equation
∂2E(z, ω)
∂z2
+ (ω)
ω2
c2
E(z, ω) (8)
[Boyd2003]

11. Solitons SPM Pulse Propagation Recap Other Soliton Types Conclusion
Pulse Propagation in Dispersive Media
Slowly varying amplitude
˜A(z, ω ) =
∞
−∞
˜A(z, t)eiω t
dt
E(z, ω) = A(z, ω − ω0)eik0z
+ A∗
(z, ω + ω0)e−ik0z
E(z, ω) A(z, ω − ω0)eik0z
On substitution into the wave equation
2ik0
∂A
∂z
+ (k2
− k2
0 )A = 0 (9)
where k(ω) = (ω)ω/c; k2 − k2
0 ∼ 2k0(k − k0)
∂A(z, ω − ω0)
∂z
− i(k − k0)A(z, ω − ω0) = 0
[Boyd2003]

12. Solitons SPM Pulse Propagation Recap Other Soliton Types Conclusion
Pulse Propagation in Dispersive Media
k = k0 + ∆kNL + k1(ω − ω0) +
1
2
k2(ω − ω0)2
(10)
Nonlinear contribution of propagation constant
∆kNL = ∆nNLω0/c = n2Iω0/c (11)
with I = [nlin(ω0)c/2π]| ˜A(z, t)|2
k1 =
dk
dω ω=ω0
=
1
c
nlin(ω0) + ω
dnlin(ω0)
dω ω=ω0
∼=
1
vg (ω0)
k2 =
d2k
dω2
ω=ω0
=
d
dω
1
vg (ω0) ω=ω0
∼=
−1
v2
g
dvg
dω ω=ω0
[Boyd2003]

13. Solitons SPM Pulse Propagation Recap Other Soliton Types Conclusion
Pulse Propagation in Dispersive Media

14. Solitons SPM Pulse Propagation Recap Other Soliton Types Conclusion
Pulse Propagation in Dispersive Media
∂A
∂z
− i∆kNLA − ik1(ω − ω0)A −
1
2
ik2(ω − ω0)2
A = 0 (12)
Frequency Domain to Time domain transformation
∞
−∞
A(z, ω − ω0)e−i(ω−ω0)t d(ω − ω0)
2π
= ˜A(z, t)
∞
−∞
(ω − ω0)A(z, ω − ω0)e−i(ω−ω0)t d(ω − ω0)
2π
= i
∂
∂t
˜A(z, t)
∞
−∞
(ω − ω0)2
A(z, ω − ω0)e−i(ω−ω0)t d(ω − ω0)
2π
= −
∂2
∂t2
˜A(z, t)
The final equation is
∂ ˜A
∂z
+ k1
∂ ˜A
∂t
+
1
2
ik2
∂2 ˜A
∂t2
− i∆kNL
˜A = 0 (13)
[Boyd2003]

15. Solitons SPM Pulse Propagation Recap Other Soliton Types Conclusion
Pulse Propagation in Dispersive Media
On co-ordinate transformation from t to τ
τ = t −
z
vg
= t − k1z and ˜As(z, τ) = ˜A(z, t) (14)
∂ ˜As
∂z
+
1
2
ik2
∂2 ˜As
∂τ2
− i∆kNL
˜As = 0 (15)
Defining nonlinear propagation as
∆kNL = n2
ω0
c
I =
n0n2ω0
2π
| ˜As|2
= γ| ˜As|2
(16)
Nonlinear Schrodinger Equation
∂ ˜As
∂z
+
1
2
ik2
∂2 ˜As
∂τ2
= iγ| ˜As|2 ˜As (17)
[Boyd2003]

16. Solitons SPM Pulse Propagation Recap Other Soliton Types Conclusion
Solitary Solution
Nonlinear Schrodinger Equation
∂ ˜As
∂z
+
1
2
ik2
∂2 ˜As
∂τ2
= iγ| ˜As|2 ˜As (18)
Soliton
˜As(z, τ) = A0
s sech(τ/τ0)eiκz
(19)
where pulse amplitude an pulse width
I0 = |A0
s |2
=
−k2
γτ2
0
=
−2πk2
n0n2ω0τ2
0
(20)
and phase shift experienced by the pulse upon propagation
κ = −k2/2τ2
0 =
1
2
γ|A0
s |2
(21)
This is the Fundamental Soliton solution [Boyd2003]

17. Solitons SPM Pulse Propagation Recap Other Soliton Types Conclusion
How it happens? A Recap in words
Anomalous dispersion regime λ > 1310nm
Chromatic dispersion ( remember GVD ) causes shorter
wavelengths to travel faster.
Thus a spectrally wide pulse disperses → the shorter
wavelengths got to the leading edge of the pulse

18. Solitons SPM Pulse Propagation Recap Other Soliton Types Conclusion
What makes it work?
High intensity pulses → Change in RI → Phase change and
frequency change
Non-linear Kerr effect → self-phase modulation (SPM)
SPM causes a chirp effect where longer wavelengths tend to
move to the beginning of a pulse
Opposite direction to the direction of GVD in anomalous
dispersion regime
If the pulse length and the intensity are right, negative GVD
and SPM strike a balance and the pulse will stay together.

19. Solitons SPM Pulse Propagation Recap Other Soliton Types Conclusion
What makes it work?
The faster (high-frequency components) at the beginning of
the pulse are slowed down a bit and the slower (low-frequency
components) in the back are speeded up.

20. Solitons SPM Pulse Propagation Recap Other Soliton Types Conclusion
Soliton
launch a pulse of right energy and right duration into a fibre
medium → short travel → evolves into the characteristic
sech(hyperbolic secant) shape of a soliton.
[IBM]

21. Solitons SPM Pulse Propagation Recap Other Soliton Types Conclusion
N-soliton

22. Solitons SPM Pulse Propagation Recap Other Soliton Types Conclusion
Dark Soliton
If you have a small gap within an unbroken high power optical
beam or a very long pulse, the gap in the beam can behave exactly
like a regular soliton! Such gaps are called dark solitons.

23. Solitons SPM Pulse Propagation Recap Other Soliton Types Conclusion
Spatial Soliton
Temporal Optical Solitons → Spatial Solitons
intense beam of light → a beam which holds together in the
transverse direction without spatial (lateral) dispersion
It travels in the material as though it was in a waveguide
although it is not!
Beam constructs its own waveguide
Diffraction effects and SPM
Application
Potentially to be used as a guide for light at other wavelengths
Fast optical switches and logic devices by carrying beams of
different wavelength .
[IBM]

24. Solitons SPM Pulse Propagation Recap Other Soliton Types Conclusion
Spatial Soliton

25. Solitons SPM Pulse Propagation Recap Other Soliton Types Conclusion
Pros and Cons
Amplifiers at regular intervals and working with an intense
signal → Practical problem
Low maximum power limits on the signal imposed by effects
like SBS and SRS
To Retain the soliton shape and characteristics amplification
needed at intervals of 10 to 50 km!
Virtually error-free transmission over very long distances at
speeds of over 100 Gbps.
Optical TDM needed as the electronic systems to which the
link must be interfaced cannot operate at these very high
speeds.
Laboratory prototype stage right now
Attractive for long-distance links in the future. [IBM]

26. Solitons SPM Pulse Propagation Recap Other Soliton Types Conclusion
To sum up
Solitons are solitary solution to the NLS equation in the
anomalous dispersion regime for a material having positive
SPM
Solitons evolve into sech pulses
Different types of solitons have been explored
Laboratory Stage only

27. Solitons SPM Pulse Propagation Recap Other Soliton Types Conclusion
References
Nonlinear Optics, R Boyd, II Edition(2003), Elsevier
Publications
Photonics, Saleh & Teich , II Edition(2007), Wiley Interscience
Understanding Optical Communications, I Edition(1998),
International Technical Support Organization
Nonlinear Fiber Optics, III Edition(2004), AP

28. Solitons SPM Pulse Propagation Recap Other Soliton Types Conclusion
Thank You for your patient listening