soliton propagation

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.


Outline

1 SolitonsIntroduction

2 SPM

3 Pulse Propagation

4 Recap

5 Other Soliton Types

6 Conclusion


Solitary reaper

Soliton

Solitary Solution → Soliton:There exists a single solution to thepropagation equation.

Big deal about Solitons

F Soliton suggests particle type behaviour

F Solitons travel without any dispersion inside any standardfiber (even highly dispersive ones)

F Result: Single-channel data streams possible of 100 to 200Gbps

F In a WDM system, little (some but small) interaction betweenchannels using solitons exists.


Soliton sighted first time 1834

Scott Russell observed a heap of water in a canal that propagatedundistorted over several kilometers.

“a rounded, smooth and well-defined heap of water,which continued its course along the channel apparentlywithout change of form or diminution of speed. Ifollowed it on horseback, and overtook it still rolling onat a rate of some eight or nine miles an hour, preservingits original figure some thirty feet long and a foot to afoot and a half in height. ” [NLFO Agarwal]

Such waves were later called solitary waves.


Self Phase Modulation is the phase change of optical pulsedue to nonlinearity of medium’s RI

Consider the pulse

E (z , t) = Az , te i(k0−ω0t) + c.c . (1)

through a medium having nonlinear refractive index

n(t) = n0 + n2I (t) where I (t) = 2n0ε0c |A(z , t)|2 (2)

Assumptions: Instantaneous material response and sufficientlysmall length of materialChange in phase

φNL(t) = −n2I (t)ω0L/c (3)

Time varying pulse - spectral modification of pulse Instantaneousfrequency of the pulse

ω(t) = ω0 + δω(t) where δω(t) =d

dtφNL(t) (4)

[Boyd2003]


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) = −n2ω0dI

dt

L

c= 2n2

ω0

cτ0LI0sech

2(t/τ0) tanh(t/τ0)

Leading edge shifted to lower frequencies and trailing edge tohigher frequenciesMax value of freq shift

δωmax w n2ω0

cτ0LI0

ωmax w∆φmax

NL

τ0

where ∆φmaxNL = n2

ω0

cLI0

[Boyd2003]



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) = −n2ω0dI

dt

L

c= 2n2

ω0

cτ0LI0sech

2(t/τ0) tanh(t/τ0)

Leading edge shifted to lower frequencies and trailing edge tohigher frequenciesMax value of freq shift

δωmax w n2ω0

cτ0LI0

ωmax w∆φmax

NL

τ0

where ∆φmaxNL = n2

ω0

cLI0

[Boyd2003]



[Boyd2003]


Chirping of an optical pulse by propagation through anonlinear optical Kerr medium

[SalehTeich]


How do Pulses Propagate in Dispersive Media ???

E (z , t) = A(z , t)e i(k0z−ω0t) + c .c . (5)

where k0 = nlin(ω0)ω0/cHow does pulse envelope function propagate in dispersive media ?Wave Equation

∂2E

∂z2− 1

c2∂2D

∂t2= 0 (6)

Fourier Transforms

E (z , t) =

∫ ∞−∞

E (z , ω)e−iωtdω

2π(7)

D(z , ω) = ε(ω)E (z , ω)Using Fourier transforms in wave equation

∂2E (z , ω)

∂z2+ ε(ω)

ω2

c2E (z , ω) (8)

[Boyd2003]


Pulse Propagation in Dispersive Media

Slowly varying amplitude

A(z , ω′) =

∫ ∞−∞

A(z , t)e iω′tdt

E (z , ω) = A(z , ω − ω0)e ik0z + A∗(z , ω + ω0)e−ik0z

E (z , ω) ' A(z , ω − ω0)e ik0z

On substitution into the wave equation

2ik0∂A

∂z+ (k2 − k20 )A = 0 (9)

where k(ω) =√ε(ω)ω/c ; k2 − k20 ∼ 2k0(k − k0)

∂A(z , ω − ω0)

∂z− i(k − k0)A(z , ω − ω0) = 0

[Boyd2003]



k = k0 + ∆kNL + k1(ω − ω0) +1

2k2(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

v2g

dvgdω

)ω=ω0

[Boyd2003]



∂A

∂z− i∆kNLA− ik1(ω − ω0)A− 1

2ik2(ω − ω0)2A = 0 (12)

Frequency Domain to Time domain transformation∫ ∞−∞

A(z , ω − ω0)e−i(ω−ω0)td(ω − ω0)

2π= A(z , t)∫ ∞

−∞(ω − ω0)A(z , ω − ω0)e−i(ω−ω0)t

d(ω − ω0)

2π= i

∂

∂tA(z , t)∫ ∞

−∞(ω − ω0)2A(z , ω − ω0)e−i(ω−ω0)t

d(ω − ω0)

2π= − ∂2

∂t2A(z , t)

The final equation is

∂A

∂z+ k1

∂A

∂t+

1

2ik2∂2A

∂t2− i∆kNLA = 0 (13)

[Boyd2003]



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

2ik2∂2As

∂τ2− i∆kNLAs = 0 (15)

Defining nonlinear propagation as

∆kNL = n2ω0

cI =

n0n2ω0

2π|As |2 = γ|As |2 (16)

Nonlinear Schrodinger Equation

∂As

∂z+

1

2ik2∂2As

∂τ2= iγ|As |2As (17)

[Boyd2003]


Solitary Solution

Nonlinear Schrodinger Equation

∂As

∂z+

1

2ik2∂2As

∂τ2= iγ|As |2As (18)

SolitonAs(z , τ) = A0

s sech(τ/τ0)e iκz (19)

where pulse amplitude an pulse width

I0 = |A0s |2 =

−k2γτ20

=−2πk2

n0n2ω0τ20(20)

and phase shift experienced by the pulse upon propagation

κ = −k2/2τ20 =1

2γ|A0

s |2 (21)

This is the Fundamental Soliton solution [Boyd2003]


How it happens? A Recap in words

Anomalous dispersion regime λ > 1310nm

Chromatic dispersion ( remember GVD ) causes shorterwavelengths to travel faster.

Thus a spectrally wide pulse disperses → the shorterwavelengths got to the leading edge of the pulse


What makes it work?

High intensity pulses → Change in RI → Phase change andfrequency change

Non-linear Kerr effect → self-phase modulation (SPM)

SPM causes a chirp effect where longer wavelengths tend tomove to the beginning of a pulse

Opposite direction to the direction of GVD in anomalousdispersion regime

If the pulse length and the intensity are right, negative GVDand SPM strike a balance and the pulse will stay together.


What makes it work?

The faster (high-frequency components) at the beginning ofthe pulse are slowed down a bit and the slower (low-frequencycomponents) in the back are speeded up.


Soliton

launch a pulse of right energy and right duration into a fibremedium → short travel → evolves into the characteristicsech(hyperbolic secant) shape of a soliton.

[IBM]


N-soliton


Dark Soliton

If you have a small gap within an unbroken high power opticalbeam or a very long pulse, the gap in the beam can behave exactlylike a regular soliton! Such gaps are called dark solitons.


Spatial Soliton

Temporal Optical Solitons → Spatial Solitons

intense beam of light → a beam which holds together in thetransverse direction without spatial (lateral) dispersion

It travels in the material as though it was in a waveguidealthough it is not!

Beam constructs its own waveguide

Diffraction effects and SPM

Application

Potentially to be used as a guide for light at other wavelengthsFast optical switches and logic devices by carrying beams ofdifferent wavelength .

[IBM]


Spatial Soliton


Pros and Cons

Amplifiers at regular intervals and working with an intensesignal → Practical problem

Low maximum power limits on the signal imposed by effectslike SBS and SRS

To Retain the soliton shape and characteristics amplificationneeded at intervals of 10 to 50 km!

Virtually error-free transmission over very long distances atspeeds of over 100 Gbps.

Optical TDM needed as the electronic systems to which thelink must be interfaced cannot operate at these very highspeeds.

Laboratory prototype stage right now

Attractive for long-distance links in the future. [IBM]


To sum up

Solitons are solitary solution to the NLS equation in theanomalous dispersion regime for a material having positiveSPM

Solitons evolve into sech pulses

Different types of solitons have been explored

Laboratory Stage only


References

Nonlinear Optics, R Boyd, II Edition(2003), ElsevierPublications

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


Thank You for your patient listening

soliton propagation

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