Lecture III Introduction to Fiber Optic Communication …in which we study : Optical Dispersion...

Lecture IIIIntroduction to Fiber Optic Communication

…in which we study: Optical Dispersion

Fiber Kerr Nonlinearity The NonLinear Schroedinger

Equation The Split-Step-Fourier (SSF)

simulation method

Ver 4.2 May 13 2010

CLASS May 9 2010: start from slide15CLASS May 16 : slide 37

Moshe Nazarathy Copyright 2

Review of “Analog Communication”concepts – signals and systemsin the complex representation


Baseband equivalent for filtering bandpass signals

( ) ( ) ( )XHY

( ) ( ) ( ) ( ) ( )2 2Y Xu uH

( ) ( ) ( )a aHY X

0 0 0( ) ( ) ( )a aXHY Left-shift:

0( ) ( ) ( )HY X

( ) ( ) ( )bbY H X

0( ) ( )bbH H

Baseband Equivalent filter in the rms CE domain

( )X

( )Y

( )X ( )Y ( )H ( )h t

0( ) ( )bbj tt t eh h

( ) ( ) ( )bbt ty h x t

( )x t

( )y t

( )x t ( )y t

( )ax t ( )ay t

Note2: Some authors refer to the CE of a signal as its baseband equivalent

( )bbH is simply the filter transfer function shifted to DC

Proof Review from L1


Two equivalent forms for CE propagation through a linear system

There are actually two alternative forms for the propagation of the CEthrough a linear system: ( ) ( ) ( ) ( ) ( )bb lpt t t t thy x xh

0 0( ) ( ) ( )lpH H u 0( ) ( )bbH H

So, we can either throw away the negative frequencies or not,prior to left-shifting the freq. response to the origin – both forms work out for narrowband signals

= the CE of H / sqrt[2]

=the freq. response H left-shifted

Use either or as low-pass equivalent transfer function for the CE

( )bbH ( )lpH

Self-study


Filtering a narrowband signal through an LTI broadband filter

( )H ( )ax t

0

0

0

0

1 1

/22 2

/2

/2 2 2

/2 00 0

( )

( ) ( )

( ) (

( ) ( ) ( )

( ) (

( ))

)

( )

Bj t j t

B

B j t j t

a a

B

a

a a

a a

Ht F F

e d e d

e d e d

H H

H H

y Y X

X X

X X

Narrowbandaround

0

( )ay t0( ) ( )a tH x

( )y t

(0) ( )lpH tx( )x t

Complexenvelope

0( ) ( )a tH x0 0

0/ 2 / 2( ) ( )

B BH H

“Frequency Flat” approximation: NO DISTORTION

0j te 00

( )aX

0 0( ) ( ) ( )lpH H u

locally flat broadbandfilter

0 0 0( ) ( ) ( ) (0) ( )lp lpH H Hu H

Self-study

Slowly-Varying Envelope Optics – Optical Dispersion Theory


Single mode fiber propagation (I)

( ) ( /2)( ) ˆ( , , ) ( )a mj t z zEz t e r rE p

x

y

z

r

At a particular frequency , a monochromatic single mode field pattern (or a particular m-th mode of a multimode fiber)is described (with the pol. unit vector, )

Any narrowband pulse may be described as a superposition of such monochromatic single mode solutions, with freqs. around a center freq. .

( ) ( /2)

0

ˆ( , , ) ( ;0) ( )a mj t z z

az t U e dE

r rE p

Assumed the modal transverse profile is constant over frequency

0z ( ) ( /2)

0

ˆ ( ) ( ;0) j t z zam U eE d

rp

p̂ ( , )x yrby the analytic signal vector

is the complex weight of each monochromatic modal component in the wavepacket

( ;0)aU

0The following integral expression provides a “modal spectrum” expansion of the “wavepacket”:


Single mode fiber propagation (II) x

y

z

r

( /2)

0( , , ) ( ) ( ;0) j z j t

aa mE Ez t U e e d r r

0

( , 0, ) ( ) ( ;0) ( ;0) ( )a mj t

a maz t U e d u tE E E

r r r

FT Pair 0z

( ; )au t zThe modal field patternseparates into a transverse spatial and temporal part

( ;0) ( ;0)a aU Fu t Given the pulse at the input, ( ;0)au t

evaluating its FT,

(*)

Then plug into the integral (*) ( ;0)aU to find the temporal output ( ; )au t z . The mapping from to ( ;0)aU ( ; )au t z

is linear. In fact ( / 2)1( ; ) ( ;0) j za au t z F U e

( / 2)( ; ) ( ;0) j za aFu t z U e ( / 2)( ; ) ( ;0) j z

a aFu t z Fu t e ( )H Fiber transfer function:

The “system” is a piece of fiber with input at z=0 and output at z

obtain the modal complex amplitudes by


Fiber Propagation Transfer Function (I)

x

y

z

r

To generate the output time-waveform, each frequency component of the input modulation is multiplied by

the linear transfer function of the guiding device (fiber or waveguide).

j te

( )j z j te e ( ; ) 0H z

( / 2) ( )z j t ze e Response to harmonic tone:

( / 2)( ; ) ( ;0) j za aFu t z Fu t e

( )H ( ;0)aU ( ; )aU zFiber Transfer Function: ( /2)( ; ) j z j zH z e e

( ;0) j taU e ( / 2)( ; ) ( ; ) j zj t j t

aU z H z e e e

/ 2j

0

( ; 0)aU

Let

Then

The resulting transformed frequency components are then superposed toyield the inverse FT – the time domain output.


Note: The derivation was for (in terms of each monochromatic analytic component) 0 0 For we can extend the transfer function in a hermitian way, i.e.

/ 2

/ 2

; 0( )

; 0

j z j z

j z j z

e eH

e e

Notice that the functions , occuring above are defined only for 0

*( ) ( )H H

such that

Self-study

Fiber Propagation Transfer Function (II)


Low-Pass Equivalent Fiber Transfer Function

( ) ( ) ( )bbt t th xy

0( ) ( )bbH H

We recall that for a system with TF and with input x(t) and output y(t) the complex envelopes propagate according to

( )H

where the low-pass equivalent TF isobtained by downshifting the positive frequencies

1( ) ( )bb bbt Fh H

(such that the carrier is downshifted to the origin), or conversely one can imagine that we bring the origin of freq. to the carriermaking the frequency transformation

0 0 or conversely

i.e. measures the frequency deviation from the carrier

0[ ] ( )S ft Hhi

For fiber propagation ( / 2)( ; ) ( ) ( )j zu eH z u

0 0

0

( /2)( ; ) ( ; )

j

bb

zH z eH z

step function


Dispersion (I) x

y

z

r 0 0( /2)

( ; )bb

j zzH e

For the optical signals of interest (frequency independent loss)

0( /2)( ; )z

b

j

bzz eH e

Apart from the fixed attenuation factor, , which may be ignored, this is an all-pass (phase-shifter only) transfer function:

( ; )( ; ) jb

zb z eH 0

( ; )z z

/2ze

( ); 0( ; ) j zH z e

Expand around

0 0 0

21( ; ) ...

2z z z z

or equivalently around 0 0

0

20 0 0

/2( ; )

j z z j z

bb

jH z e e e

( ; ) ( ; ) ( ;0)bbU z z UH


Dispersion (III) x

y

z

r 0 0( )

( ; )j z

bb z eH ( ; )j ze

20 0 0

/2( ; )

j z z j z

bb

jH z e e e

0 0 0 0

21( ; ) ...

2z z z z z

0g z

00 0

11

gg

z d dv

d d

GROUP VELOCITYVelocity of the envelope:

2nd orderdispersiondistortion

( ;0)u t

( ; )u t z

0 ( / ;0)j z

ge t z vu

g

Retarded time

1

g

zv

01/ gv

gv

[ ]gj

ge Shift


Dispersion (III) x

y

z

r( ;0)u t

( ; )u t z

0 ( ;0)j z

gue t

z

2 /2( ; ) j z j zb

jb

zH z e e e ( ; )u t z

1[sec/ ] gm v

group “slowness”(delay per unit length)

1st order dispersion

00

1

0

1( )g

g

z dv

d

Group velocity of a narrowband optical pulse (wavepacket) centered around freq. 0

Note: a specific spectral component at propagates with the phase velocity0

0

00( )pv

The group velocity is a collective property of a bundle (wavepacket)of spectral components in the vicinity of 0superposing and interfering to generate the appeareance of the envelope(and slowing varying phase) moving at the group velocity (which may differ from the phase velocity

2nd order dispersion

1gzv


( ; ) j zH z e

Dispersion – Interim Review

0( ; ) ( ; )bbH Hz z

z( ;0)u t

( ; )u t z

The Transfer Function (TF)of a piece of fiber of length z(ignoring attenuation)

20 1 2 /2j z j z j ze e e

0j z

e

g

0

2 0( ; ) ( )j z

gbb t z e th

1

1/g gz v

Group slowness(delay per unit length)

21

2

( )( ; ) exp

2bb

t zt z j

zh

15 10 5 5 10 15

1 .0

0 .5

0 .5

1 .0

Re ( ; )bbh t z

t

210 1 22 ...j z

e

0 0

0

0 1

2

;

1z

Time-domain chirp


The peaks and valleys (or zero-crossings) of the fast-varyingquasi-sinusoid move at the phase velocity – the envelope and slow phase moves at the group velocity

0( , ) ( / , )j dz

gt z dz e t dz vu u z

The real-valued signals are

20 0 0

/2( ; )

j z z j z

bb

jH z e e e Ignoring attenuation

Time-domain propagation through a small distance increment for first order dispersion only (neglect quadratic and higher terms):

01 /

gv

00 0 0/ /pv Phase velocity

Group velocity

Phase velocity vs. group velocity – dispersion relation

Dispersionrelationslope

slope 1 / v p

1 / vg

0

00( , ) 2 Re ( , ) 2 ( , ) cos( ( , )j tt z t z e t z t tE u u u z

0 0 0( , ) 2 Re ( , ) 2 Re ( / , )

j dzj t j tgt z dz t z dz e tE u u dz v z e e

002 ( / , ) cos ( / , )g gu t dz v z t dz t dz v zu

0

00

2 ( / , ) cos ( / , )/g g

dzt dz v z t zu t v zu d

02 ( / , ) cos ( / , )g gp

dzt dz v z t t dz z

vu u v


Eval the group index in terms of the effective index:Use

Group index

z( ;0)u t

( ; )u t z

2 /2( ; ) j z j zb

jb

zH z e e e 0 ( ;0)

j z

gue t

z

( ; )u t z

1st order dispersion00

1

0

1( )g

g

z dv

d

( ) /pv ( ) / ( )pn c v Effective index:

In an analogous way we introduce a “group index”relating the group velocity to the vacuum speed of light

/ k /k c / ( )c n vacuum wavenumber:

( ) / ( )g gn c v

0( ) / ( )g gv c n

1( ) /gn c c

( ) ( ) /n k n c

( ) ( ) ( )gn n n

1 alwaysspeed of info 0c

When ( ) const 2 : 0nn n we have (higher-order) “dispersion” (distortion of env.)

Different “freq. components”travel with different speeds

Pure delay doesn’t count as “dispersion” – fixed by timing recovery

( )pv const


Pulse broadening (I)

z( ;0)u t

( ; )u t z2 /2( ; ) j z j z

bj

bzH z e e e

carrierdelay

(complex)envelopedelay

pulse distortion(broadening)

( ) 1/gv if the derivative happens to be constant over freq.,then all narrowband “ wavebundles” at various freqs. propagate at the same group velocity and arrive together – no distortion, just pure delay by

/g gz v z Mathematically 0const

0 When we have pulse distortion, typically broadening (though pulse compression might be possible, e.g. for properly chirped pulses)

In this case ( )gv const The freq. variation of the group velocity leads to pulse broadeningbecause the various spectral components (“wavebundles”) disperse during propagation (move at different group velocities) hence do not arrive simultaneously at the fiber output

Note:We introduced“wavebundles”צרורות-גלים= narrowerin freq. wavepacketscomposinga pulse on afiner freq. scale

Each wavebundle is very narrow in freq., and itsenvelope in time propagates with its own group velocity corresponding to its center freq.. A wavebundle is like a spectral slice of the narrowband spectrum

2 31 12 62 1 1 1 1

19

Pulse broadening (II) z( ;0)u t

( ; )u t z

( ) 1/gv

0 ( )gv const Pulse broadening

Roughly evaluate the extent of pulse broadening for a pulse of spectral content (BW) launched into a fiber of length L:

Consider the spectral wavebundles composing the pulseThe slowest and the fastest wavebundles, move at the two extreme group velocities, hence the increase in the pulse width at the output equals the difference between the group delays of these two wavebundles(make a spacetime diagram to visualize the divergence of the lightlines)

2 1( ) ( ) gg g

d dL L

d d

GVD parameter /g gz v z

L 2 1

2 1[ ] [ ]g g

L LL L L L

v v

Alternatively:

Higher order refinement for the delay per unit length vs. frequency spread:

2 12 1/ 1/ [ ] 1/ [ ]g gL v v 2 31 12 3 42 6

Taylor series expand around : 1

Exercise: Consider 2nd and 3rd order dispersive delay for three wavebundles equi-spaced in freq. how do the wavebundles (starting together) superpose in time and interfere after propagation?


Pulse broadening (III)

z( ;0)u t

( ; )u t z

( ) 1/gv

2 .GVD param

/g gz v z

L pulse brodening proportional to

and to the BW-length-product [GHz-Km]

It is customary to formulate the pulse broadening in terms of the wavelength spread

Positive (Negative) Group Velocity Dispersion (GVD):

0 ( 0)Anomalous Normal

( / ) psec/km

n

1

mg

g

d L dD

d d v

psec/nmgdDL

d

In L1 we phenomenologically introduced the DISPERSION PARAMETER

gdDL

d

Pulse brodening:

How are andD related?next page….

1.3 m 1.3 m

2( sec)21.7

p

Km

D = 17 psec/km/nm

Standard G.652 fiber:

(mean)delay of the pulse

2

2 22

c d c

d

Alternatively2 0c const d d

21

Pulse broadening (IV) z( ;0)u t

( ; )u t z

2( sec)

21.7p

Km

D = 17 psec/km/nm

Standard G.652 fiber:

d

d

2 L DL 2D

2

2 c

2

psec/km( / ) 1 2

nmg

g

d L d cD

d d v

12

2g

d d d d d cD v

d d d d d

Alternative Derivation (self-study):

2/ 2 /d d c 2 2 /f c

22 ( )p

g

g

vv

v

q.e.d..

Bitrate limitation due to the dispersion impairment (without dispersion compensation): 1

s sT R A necessary condition for good transmission is

1bD LR or 1bR yielding 1sR L D

Order of magnitude estimate for the baudrate-distance product:

Standard silica fiber near 1.3 m 1 psec/km-nmD 1 Tbaud-kmsR L 1nm

(w/out dispersion compensation)

2 2

2 cD

So at 1.5um D>0Anomalousdispersion

At over standard fiber: 17x worse ! 1.5 m


Dispersion Parameter from refractive index wavelength variation

z( ;0)u t

( ; )u t z

22

2

2ln 2( ) (0) 1

(0)p pp

D zz

c

Broadening of a gaussian pulse (that starts without chirp) [Yariv]:

Relate the Dispersion Parameter D to the wavelength variation of the group index:

1 1 1 ( )( )g g

g

n dnd d dnD n

d v d c c d c d

( ) ( ) ( )gn n n 2 2

2

2 2 ( ) ( )( ) ( ) ( ) ( ) ( )

2g

c d d c dn dn d nn n n n n

d d c d d d

2/ 2 /d d c 2 2 /f c 2/ / 2d d c

2

20

d n

c d

Exercise:Consider the dispersive propagation of chirped pulses

2

2

( ) ( )( )

( ) ( )

gdn d dnn

d d d

dn d dn d n

d d d d

22

_

exp[ / 4 ] 2 ln 2exp[ ] ;

4FWHM PSDt


Dispersion Slope

2

2 2 2

( / ) 1psec/km

n

2

mg

g

d L d c d nD

d d v c d

Dispersion Parameter (characterizes quadratic dispersion - pulse broadening):

D varies with wavelength for any fiber and vanishes at a wavelength called the zero-dispersion wavelength for standard SMF )1.3ZD m

DispersionSlope

2

3 2 3 22 3 2

2 2( 2 )

c cS

Exercise: Show that the dispersion parameterrelates to the 3rd order dispersion coefficient as

Fibers with relatively small values of D in the spectral regions near 1.55 um( ) are called dispersion-shifted fibers(as the minimum of dispersion was shifted from 1.3um towards the 1.5 um)

psec/k nm8 m-D

The second term in vanishes at 1.3ZD m whereat S is proportional to :

32

32

2( )ZD

ZD

cS

different channels have slightly different GDV values.Difficult to compensate GDV for all channels at once.

Near this wavelength (whereat D and the GVD parameter vanish)D varies linearly in wavelength (S>0 for most fibers): ( )ZDD S

2

Fibers with D small and negative (reverse-dispersion fiber) are used for dispersion compensation

S D =0.07 ps nm2 km-1

for standard fiber at 1.5um

22

23 2

S Dc

2

(N)SE Equation for a Dispersive Medium


Derivation of the dispersive (N)SE equation (I)We prove that any solution of Maxwell’s eqs. for a dispersive (homogeneous isotropic) medium,of the form 0 0( )( ; ) ( ; ) j z t

aE t z u t z e satisfies

What we mean by “a solution of Maxwell’s eqs. for a dispersive medium” is a superposition of monochromatic “modes” of various freqs., each satisfying the wave eq.2 2 2( ; ) 0( ; )az a tvE t z E t z

2 2( /( ; ) ( ; )) 0zU z U zv ( ) ( )( ; ) ( ; )a a

j tt z z eE E

We seek monochromatic solutions of the wave eq.:

The monochromatic solution of the the wave eq. is next seen to be a propagating-wave:( / )( ; ) j v zz eU

( ( / ) ) ( )( ) ( ; ) j t j v z j t j z

a eE t z e

0

( ; ) ( ;0) j za

j tE eUt z e d

0

( ;0) ( ;0) { ( ;0)}j ta t e d FE U U

(*)

We express the general narowband solution in terms of its spatiotemporal envelope00

( )( ; ) ( ; )

j z t

aE t z u t z e relative to some suitably selected center freq. 0

(**)

Multiplying (**) by 00( )j z t

e yields

0 0( ) ( )

0

) )( ; ( ;0j z j te e du t z U

2 32 6( ; ) ( ; ) ( ; ) ( ; ) ...t tz tu t z u t z u t z u t zj

with a slowly-varying envelope ( ; )u t z

The analytic signal of a general solution of Maxwell’s eqs. is the superposition of monochromatic modes with suitable weights to be determined by the initial condition at z=0: ( )j t j ze ( ;0)U

0 0 Next take

Self-study


Derivation of the dispersive N)SE equation (II) 0 0( )

0

( ; ) ( ; ) ( ;0)j z j z j ta

tt z e t zu e eU dE

0 0 0 0

( ) ( )( )0

0 0

( ; ) ( ;0) ( ;0)j z j zj t j tt z e e du U U e e d

0 0 0 0( ) ( )1

0

( ; ) ( ;0) ( ;0)j z j zj tu U Ut z e e d F e

Now denote 0( ;0) ( ;0)U U (the justification for why we use the inverted hat will soon follow)

1

0

( ;0) ( ;0) ( ;0)j tt e UFu U d

Set z=0: or ( ;0) ( ;0)FU u t justifying the previously ad-hoc introduced hat notation

(*)

From (*) we get by taking an FT

0 0 0 0( ) ( )

( ; ) ( ;0) { ( ;0)}j z j z

uF t z e F t eU u i.e. 0 0

( )( )

j z

lpH e

is the freq. response (TF) of a systemwith input and output ( ; )u t z( ;0)u t

change of integration variable to 0 where in the last equality we performed a

Self-study


Alternative SVEA derivation of dispersive the (N)SE equation (III)

0 0( )

0 0

( ; ) ( ; ) ( ;0)j zj t j tt z z e d e e du U U

Our generic valid solution of Maxwell’s equations for the dispersive medium is thenexpressible in the form

2 32 6( ; ) ( ; ) ( ; ) ( ; )t tz tu t z u t z j u t z u t z

Let us verify that such solution (for any ) satisfies the equation ( ;0)U

0 0 0 0 0

2 3/ 2 / 6 with the constants being Taylor series coefficients of , ,

Indeed, substituting (*) in the lhs of (**) yields

(*)

(**)

0 0 0 0

0 0

( ) ( )

0 0

( ; ( ;0) ( ;0)( )( ))j z j zj t j t

z zu t z U Ue e d j e e d

0 0

0 0 0

( )2 3

0

( ) / 2 / 6 ( ;0)j z j tUj e e d

0 0 0

2 3

0

/ 2 / 6 ( ; ) j tj j z e dUj

0 0 0

2 31 ( ; ) ( / 2) ( ; ) ( / 6) ( ; )F j z j z j zU U U 2 3

2 6( ; ) ( ; ) ( ; )t ttu t z u t z uj t z q.e.d.

Self-study


zDiscard the attenuation and factor out the accumulated phase introducing the spatio-temporal envelopes, : ( ; ) ( ; ) j zt z t zu eu

( ; )u t z

(Non) Linear Schroedinger Equation (I)

z( ;0)u t

( ; )u t z

2 /2( ; ) j z j zb

z j zb z e e e eH

Next show that this dispersion description (up to 2nd order) in the freq. domain

is equivalent to a time-domain Shroedinger-like linear differential equation…

2( )2( ; ) ( ; ) ( ; ) ( ; )b

j dz

bz dz dz zU Ue zH U

[ , ]( ; ) ( ; )Dj z z dzz d Uz e zU 2( )2[ , ]D dzz z dz

z z+dz

( ; ) ( ; ) j zU Uz z e

2 /2( ; ) ( ; ) j z j zzbb bb

jz e z e eH H ( ; ) ( ; ) j zu ut z t z e

If the previous “classic” derivation was too complex, we do it all over again in a simpler and more insightful way preparing for the SSF


Subtract from both sides and divide by

(Non) Linear Schroedinger Equation (II) x

y

z

r( ;0)u t ( ; )u t z

[ , ]( ; ) ( ; )Dj z z dzz d Uz e zU

2( )2[ , ]D dzz z dz

z z+dz

( ; ) (1 [ , ]) ( ; )Dz dz j z z dzU U z ( ; )U z :dz

[ , ]( ; ) ( ; )( ; ) ( ; )D

zj z z dzz dz zU U

U Uz zdz dz

2( )2( ; ) ( ; )zU zUz j


(Non) Linear Schroedinger Equation (III) x

y

z

r( ;0)u t

( ; )u t z

2( )2[ , ]D dzz z dz

z z+dz2

2( ; ) ( ; ) ( ; )z j jz z zU U U

Inverse FT and use dj dt

22( ; ) ( ; ) ( ; )tz t jt z t z zu u u t 2

2( ) ( ; ) ( ; )tz tj t z t zu u Transform the diff. eq. to retarded time (next pages): / gt z v t z

22( ; ) ( ; )zu zuj z

Linear Schroedinger-like equation,describing dispersion


Dispersion (IX) – retarded time ( ; ) / gt z t z v t z ( ; )t z z

Parked at z, the retarded time at time tis actually the earlier time that the signal should have taken off at, from z=0so that it reaches z at time t.

( ; ) ( ( ; ); ) ( ; )r z t z z z zu u u

Riding on the peak of a wavepacket, i.e. moving along with it, time and distance increase but the Retarded Time (RT) stays constant. RT is a label for the lightlines. It represents the time to launcha “flash” at z=0 so that it reaches at distance z at time t. It is obtained by an oblique projection of thespace-time point (t,z) back to the t axis, drawn with a line of slope

Let’s express the CE of the propagating pulse as a function of retarded time and space(it’s actually a different function, denoted by the r (“retarded”) subscript):

We know that first order dispersion merely delays the CE, with the delay linear in z:

( ; ) ( ;0)u ut z t z Using this expression in (*) yields

(*)

( ; ) ( ; ) ( ;0) ( ;0)r t z t zu u uz t z t z u

i.e. at all z-s the envelope as a function of retarded time is constant(and in fact equal to the envelope as a function of regular time at z=0)

Self-study

@ 0t z

Exercise: Is there a relativistic interpretation?


Dispersion (X) – retarded time

22( ) ( ; ) ( ; )tz tj t z t zu u

( ; ) / gt z t z v t z ( ; )t z z ( ; ) ( ; ) ( ;0) ( ;0)r t z t z

u u uz t z t z u

Let’s write a differential equation for the transformed envelope, ( ; )ru z

22( ; ) ( ; )r rzj z zu u

Evaluate and ( ; )rz zu 2 ( ; )r zu

( ; ) { } ( ; )( ; ) ( ; )r t z

t z t zz z z z

z zt z z t z z

z t zt z

u

z z

uu u

using the chain rule for differentiation:

( ; ) ( ; ) ( ) ( ; )t zt z t zz zt tt z t z t zu u u

Hence ( ; ) ( ) ( ; )r t zz z tj z j t zu u

(*)

=lsh of (*) under the substitution t z

Self-study

Target eq. to prove:


Dispersion (XI) – retarded time Next let us evaluate

( ; ) { } ( ; )( ; ) ( ; )r t z

t z t zz z z z

t z z t z zz

u utu

t zu z

( ; ) { ( ; )}r t t zz t zu u

or

Let us differentiate again

2 { ( ; )} { ( ; )}{ }( ; ) { ( ; )} t tr t t z

t z t zz z z z

t z t zz zu z t z

t z

u uu

22 ( ; ) { ( ; )}r t t ztu z zu

or

Before we got ( ; ) ( ) ( ; )r t zz z tj z j t zu u

22( ; ) ( ; )r rzj z zu u

22( ) ( ; ) ( ; )tz tj t z t zu u

222 2( ; ) { ( ; )}r t t z

u uz t z

Then

And in the t,z domain we have the eq.

It then follows that

Shroedinger’s equation for dispersion in retarded time

Self-study

In the sequelwe abuse the notationdropping the index r


Alternative SVEA derivation of the dispersionless (N)SE equation 2 3

2 6( ; ) ( ; ) ( ;( ) )t tz tj u t z u t z uj t z

Let us start with deriving first the equation,

0... ( ) ; 0( )z t zj u t

valid when

We start from first E-M principles (the wave equation) and apply the Slowly Varying Envelope Approximation:

2 2 2( ; 0) ( ; )a az p tvE t z E t z ˆ( ; ) ( ; )t zEz tpE

the wave equation

0 0 0 0( ) ( )2 2 2 0( ; ) ( ; )j z t j z tz p tu t z u t ze v e

Equivalently, we consider a non-dispersive situation: phase velocity fixed over freq.

0 0( )( ; ) ;2 ( ) j z taE t z u t z e

Slowly-varying

0 0( ) / .pv const

0 0( ) 2 20 02j z t

z zue j u u 0 0( ) 2 20 02j z t

t tue j u u Cancel, since 0 0 / pv SVEA: Neglect

20

20

z z

t t

u u

u u

20 02 2 0z p tu v j uj

1 1p gv v

2 10 0 0 0 0 0/ p p p pv v v v

10 02 2 0z p tu v j uj

(N)LS eq.for dispersive medium

Self-study

( ) ( ) 2ab a b ab a b a b ab

Fiber non-linearity From the Kerr effect to the NSE



Fiber non-linearity - The Kerr effect (I) x

yz

r

• Kerr effect: Refractive index modulated by optical intensity

• AM PM conversion

2

0 2 0 2( ) ( )n nEn tn In t

210 0 0 0 2 effk n k n k n A u

0

* 20

0

1Re 2 2( )

2( )

n

ZS t E E tH

(3)0 0

2 20 0

3

8

Z Zn

n n

(3)

0

3

4n

from last slide

In an optical fiber the transverse optical intensity varies according to the modal profilehence we must average the NL change in refractive index over the fiber cross-sectionworking with an effective intensity and effective area describing the transverse variation:

0 NLn n n 2 ( )NLn n I t

2

2 2 2( ; ) ( ; ) ( ; ) ( , )/ /NL eff effeff efft z I t z P tn n n z u t zA n A

where we introduced an effective field describing the amplitude of the transverse distribution, in units such that

2( ; ) ( , )effP t z u t z

0 2 2

0

2

eff eff

k n n

A A

(Note: the meaning of the effective transverse area averaging to be clarified)

0 NL NLk n 2

u

and we henceforth discard the eff label

0

2

c

0 NL 2

0 u

1 1W m

repeat from L2


nonlinearity gamma 1.35 W km0.00135 1 1W m

Fiber non-linearity - The Kerr effect (II) x

yz

r

P kNL k0nNL 2 0

n2I 2 0

n2P

Aeff

2 0

n2

Aeff

0 1.55 micro

n2 2.6 1020

Aeff 80 micro2

2 0

n2Aeff

W m 0.00131744=

NL2u Henceforth discard

the prime:

was originally associated with 2| |NL E with field in [V/m]

Now is associated with field normalized to units W

1 1W m such that 1[ ]NL m P

Units and sizes

Another close parameters set

repeat from L2

SPM PL


• Kerr effect:

Refractive index modulated by optical intensity

• AM PM conversion in optically amplified links

• ASE amplifier optical power fluctuations generate “non-linear” phase noise

Preview: The Gordon-Mollenauer effect: Extra phase noise due to fiber nonlinearity

+

OPTICALAMP

2

0 2n n n E

+

OPTICALAMP

1NLje

01

2NL A n

OPTTX

0n 1n

2NLje

0 12

2NL A n n

A


1st derivation of the NSE for a NL dispersionless medium (I)

Apply SVEA:

ˆ( ; ) ( ; )a at zEz tpE

0 0 0 0( ) ( )2 2 2 20 0( ; ( ;) ; )( )j z NLt j z t

z t tu t z ue v e Pt t zz

0 0( )( ; ) ;2 ( ) j z taE t z u t z e

Slowly-varying

0 0 0( ) / .v const

0 0( ) 2 20 02j z t

z zue j u u 0 0( ) 2 20 02j z t

t tue j u u

Cancel, since 0 0 0/ v SVEA: Neglect 2

0

20

z z

t t

u u

u u

0 0( ) 2 20 0 0 0 ( ; )2 2j z t

tL

z tNPuv tj ze j u

1 1 10 p gv v v

0 0 0

2 10 0 0 0 0

0 0 0/

v

v v

v

10 02v j

2 2 2 20 0( ; ) ( ; ) ( ; )a a

Lz t t

Nv P t zE t z E t z Wave Equation for a non-dispersive NL medium

0 0( )20

0

;2

( ) j z tz t t

NLu uj

eP t z

So this is what we got for the lhs(already derived before by other methods)

2 20 tE D

speed in lin. medium (not in vacuum)

( ) 2fg f g f g fg

Self-study


1st derivation of the NSE for NL + dispersionless medium (II)

0 0( )20

0

;2

( ) j z tz t t

NLu uj

eP t z

We may assume the most general input signal (e.g. a multichannel DWDM signal), such that the most general rhs term emerges. The NSE is then capable of modeling SPM+XPM+FWM. For a fiber (centrosymmetric isotropic material) we can take

(1) (3)0 0

3( ; ) ( ;( ; ) )P t z E t z E t z

*( ; ) ( ; ) ( ; / 2)a aE t z E t z E t zExpress Raise this expression to the 3rd power:

0 (3) 3 2 *

8( ; ) (3 ; )a a aE t z E E t z cc

The first 3rd-power term is a 3rd harmonic term which does not generate propagating fields sinceit is severely phase-mismatched, hence it suffices to retain the 2nd term (and its cc)

2( ; ) ( ; )3 a aE t z E t z

Third-order NLpolarization – memoriless3rd order nonlinearity

2(3)3102

(3)

2( ; ) ( ; ) ( ; )a a cEP t z E tt cz z

(3) (3) 30( ; ) ( ( ;; ) )N

aLP t z P t z E t z

* * *0 (3)

8( ; ) ( ; ) ( ; ) ( ; ) ( ; ) ( ; )

a a aa a aE t z E t z E t z E t z E t z E t z

2(3)302

(3) ( ; ;( ; ) ( )) a aa Ez zP t t E t z

(3)P(1)P

Self-study


1st derivation of the NSE for NL + dispersionless medium (III)(1) (3)

0 03( ; ) ( ;( ; ) )P t z E t z E t z

(1)0

(1) ( ; ) ( ; )a aP z Et t z

We also showed the linear term, which might lead to a linear version of the NSE(with a linear refractive index). The overall response is the superposition of the responses of the linear and NL terms, so we can deal with the two terms separately

0 0 0( )( ; ) ( ; ) ( ; )j t j z taE t z E t z u t ze e

*/ 2 / 2( ; ) ( ; ) ( ; )a aE t z E t z E t z

0 02 ((3) )(3)3

02 ( ; ) ( ; )( ; ) j z ta u t z u t zP t z e

0 0

0

0 02 2( ) ( )2 (3)3

0 02(3) { } { }( ; ) ( ; ) ( ; ) ( ;( ; )) j z t j z t

t t t eu t z u t z u t jP t z z eu t z

0 02 2 ( )(3)3

0 02 ( ; ) ( ; )} j z tu t z u t j ez

0

0 0(3 2( )2 2 (3)0 0 30 02

)

0 0

( ;( ; ) ){2

)}2

( ;j z tt

jrhs e j u t z ut z tP z

0

Envelope derivatives neglectedrelative to

0 0( )2 (30

0

) (2

; ) j z tz t t P t z eu

ju

210z tv ju u u u

NSE for NLdispersionless medium:

q.e.d..

2(3)302

(3) ( ; ;( ; ) ( )) a aa Ez zP t t E t z Self-study


(3)0

0 20

3

4k n

c n

2 (3) 2 (3) 2 (3)0 0 030 0 0 02 2

0 0 0

2 (3) (3)002

0 0 0

13 3

2 4 4

3 3

4 ( / ) 4

c

n c c n c

found in the previous slide can be further simplified: Self-study

This coincides with the result in L2 and is suitable for describing non-linear propagationof plane waves in a transversely uniform mediumwith the electric field in units of [Volt/meter]

0 2 02 20 2

0 2 0

22

eff eff eff

k n Zn nk n

A A A n n

For fiber propagation we derived a related non-linear coefficient

210z tu uv uj u

1/ 2W

In the sequel we shall often drop the primewriting when we actually mean

as we always assume that the field is measured in units such thatthe total transverse power is

1/ 2W 2u

0 0 0

0 0

/

/n

v

c

0 0 2

1

c


2nd derivation of the NSE for a NL dispersionless medium using the NL index (I)

210 2( ; ) ( ; )effn n n At z u t z

0 0 0( ; ) ( ; ) ta

j z jeE t z u t z e

0 02 2 2 20( ; ) ( ; ) 0( ; )z ta aE t z t z E tc zn

210 00 2

0 0

( ; ) ( ; ) ( ; )effn nt z t z u t zn Ac c

(*) 0 0( )2 202 2 ( ; ) ( ; )( ; ) ( ; ) j z ttat n et z u t zt z E t zn

0 00 0

0 0 0 0

( ) ( )2 2 20

2( ) ( )2 20 0

( ; ) ( ; ) ( ; ) ( ; )

( ; ) ( ; ) (

{ }

; ) (

{

;

}

{ } }){

j z t j z t

t t

j z t j z tt

n e n j e

n j

t z u t z t z u t z

t z u t z te n jz u z et

0 0 0 0( ) ( )2 2 2

0 0( ; ) ( ; ) ( ; ) ( ; )2 { } { }j z t j z tt t z u t z t z u t zj n e n e

0 0 0 0 0 0( ) ( ) ( )2 2 2 2 2 20 0 0( ; ) ( ; 2) ; ; 0( ) ( )j z t j z t j z t

z tu t z t z ue c n e c j nt z z eu t 2 ( ; )t z

0 0 0 0( ) ( )2 2 20 0 0( ; ) (2 { }; ) ( ; )j z t j z t

tu t z t z ue zn n etj

00 0 0 0( ) ( )2 2 2 2 2 2 2 20 0 0( ; ) ( ; ) 2 { }j z t j z t

t tac n c j n et z E t ez c unu

0 0 0 0( ) ( )2 20( ; ) ( ; ) ( ; )} { ; ){ }(j z t j z t

t t n e nt z u t z t z u t z j e

2

0 2 ( ; )un tn z

0

0

0 0n

c

0

2 210 00 2 0 [( ; ) ( ; ) ]eff NLu tn n A t

cz

cz u

Start from a wave equation for SPM affected signal in terms of the effective NL refractive index:

(dropping the prime)

NL

Self-study


2nd derivation…cont. (II) 210 2( ; ) ( ; )effn n n At z u t z

0 0 0( ; ) ( ; ) ta

j z jeE t z u t z e

2 2 2 20 0 0 0( ; ) ( ; ) ( ; ) ( ; 0)2 2z tu t z t z u t z u t zj c j n

2 20 0 0

0 0

2 20

2 2NL

z t

c nu u u

j

j j

2( ; )[ ]NL u t z 02

210z tv ju u u u

0 00 0 0 0( ) ( )( )2 2 2

0 0( ; ) ( ; ) ( ; ) ( ;2 ) 0j z t j z tj z t

z te e j n eu t z t z u t z u t z

0 0( ) 2 20 02j z t

z zue j u u

Set 2 20 0 0 0( ; ) ( ; ) ( ; 2) NLt z t z t z

cancel : 02 jand2

2 0 0

0

0z NL t

nj ucu u

0 0 0/ /c n

0

20 00 2( ; ) ( ; )n n z

c ct z u t

0( ; )NL t z

2 20 0( / )c c n n

10 0c n v

where we used

10

NL

z t ju v u u

NL

2( ; )u t z

q.e.d..

Self-study


Recap of 1st order dispersion(over an infinitesimal distance)

In the absence of NL, and for first-order dispersion only, we have shown that

0( , ) ( / , )j dzgt z dz e t d vu zu z

0 0( )( , ) ( , ) ( , )j dz j dzz dz e z zU U Ue

( , ) ( , )j dza az dz e U zU To quickly recap the proof:

Inverse FT:

0( , ) ( , )j dzt z dz eu t d zu z

0( , ) ( , )j dz j dzU Uz dz e e z

1/ /gv dt dz

(Towards an alternative, more intuitive derivation of the NSE for Kerr NL nondispersive media over the next few slides)

q.e.d..

( , ) ( , )u ut z dz t dz z


3rd rederivation of the NSE (I)x

y

z

r

z z+dz

( , )( , ) ( / , )j z t dzgt z dz e t d vu z zu

Yet another more intuitive rederivation: The NL refractive indexsets a spatial frequency (accumulation of local-instantaneous phase per unit length)

0( ; ) ( ; )t z n t zc

In the absence of NL, and for first order dispersion only, we have shown that

0( , ) ( / , )j dzgt z dz e t d vu zu z

Adding in Kerr NL, we may define a local-instantaneous propagation constant

00 0nc

(proportional to the varying intensity-dependent NL refractive index) extending (*) to

(*)

2

0 ( , )u t z NL

Once we accept this equation, the alternative derivation of the NSE is simple.We first carry out the derivation, then return to justify (**).

(**)

0 0( ; ) [ , ]NL NLt z dz dz dz dz z z dz 0 [ , ]( , ) ( / , )NLj dz j z z dz

gu ut z dz e e t dz v z


3rd rederivation of the NSE (II)

x

y

z

r

z z+dz( , )( , ) ( / , )j z t dz

gt z dz e t d vu z zu

0

2( , )NLk n t z du z

0( , )uj z

t z e

Spatio-temporal complexenvelope relative to the linear propagationconstant

0 [ , ] ( / , )NLj dz j z z dzge e t dzu v z

NL

2( , )t z du z

Instant. spatialfreq. can be defined due to the SVEArelated to the instant. variation of the refractive index

The CE picks up some phase related to the instant. refractive index experienced at the space-time location

Note: As the envelope varies very slowly,the difference in intensity (hence in NL phase)between two times that are spaced by a differential is second-order, i.e. is negligible, e.g. ( , ) ( , / )

gz t z t dz v

( , )[ , ]NL NL z t dzz z dz

The NL refractive indexsets a spatial frequency (accumulation of local-instantaneous phase per unit length)

0

0

( ; ) ( ; )t z n t zc


3rd rederivation of the NSE (III)x

y

z

r

z z+dz

0 [ , ]( , ) ( / , )NLj dz j z z dzgt z dz e eu t dz zu v

( ) [ , ]0 0 0( , ) ( / , )

j z dz j dz j zj z z dzNLgt z dz e e e t dz v z eu u

[ , ]

/ ,( , ) ( )NLj z z dz

gt dz v zt z dz eu u 1 / ,( , ) [ , ] ( )NL gt dz v zt z dz j z z dzu u

In the absence of NL, and for first order dispersion only, we have shown that0( , ) ( / , )j dz

gt z dz e t d vu zu z 0 0 0

/ /n c v

In the presence of the SPM (Kerr) effect

2( , )t z du z

( , )[ , ]NL NL z t dzz z dz

0 p gv v v


3rd rederivation of the NSE (IV)x

y

z

r

z z+dz

1 / ,( , ) [ , ] ( )NL gt dz v zt z dz j z z dzu u Subtract from both sides and divide by / ,( )gt dz v zu :dz

/( , ) (( , )

) [ , ]g NLt dz vt

t zz dz j z z dz

dz dz

u uu

2( , )u t z dz

1 /( , ) ( )( , ) ( , )( , ) ( , )

,1

/g

g

gg

t dz v

vz t

tt tt z t z

zu uu uu u

zz dz zlhs v

dz v dz

2

( , ) ( , )j t zu t zurhs

subtract and add identifying partial derivatives:( , )tu zTo simplify the lhs

Using we get the differential equation1gv

2( , ) ( , )( ) ( ; )z t u t z t zj t zu u NSE for Kerr NL:


The non-linear Schroedinger equation in a NL+dispersionless medium

x

y

z

r

z z+dz

Expressing field envelopes in terms of the retarded time / gt z v 2

( , ) ( , ) ( , )uz uz z zj u /

( ; ) ( ; )g

r t z vz t zu u

2( , ) ( , )( ) ( ; )z t u t z t zj t zu u

(similarly to the derivationfor the dispersion case)


4th derivation of the NSE for NL non-dispersive media (I)

NSE2

( , ) ( , ) ( , )zu uuz z zj

( , ) ( , ) ( , )zu u uz dz z z dz

Taylor Series expand around z (strict equality when using diff. increments dz), or alternatively use the derivative definition:

2( , ) ( , ) ( , ) ( , )u u uz dz z z z dzuj Set the derivative above to be equal to the rhs of the NSE:

Revert to regular time:

Work backwards:

(( ), ) ( , ) ( , ) ( , )t z dz z dz t z z j t z z t z z dzu u u u Set ,i oz z z z dz

( , ) ( , ) ( , ) ( , )o i i idt z z t z j t z t z dzu u u u

( ; ) ( ; )z t z zu u

t t z

Linear groupdelay propagationfrom input to outputover the differentialdistance dz

Secondary field at the outputradiated by the NL polarizationin the differential slab dzThe NL (polarization is in turn induced by the input field)

Self-study


4th derivation of the NSE for NL non-dispersive media (II)

NSE2

( , ) ( , ) ( , )zu uuz z zj Equivalenly, must prove:

( , ) ( , ) ( , ) ( , )o i i idt z z t z j t z t z dzu u u u Linear groupdelay propagationfrom input to outputover the differentialdistance dz

Secondary field at the outputradiated by the NL polarizationin the differential slab dzThe NL [polarization is in turn induced by the input field

) 2(0 0(( ; ( ))) ; ; o

II z zt j t t zE zE du

We have derived at the end of L2 the NL pol.-induced secondary field:

0iz z

(*)

t t

We could then apply (**) to justify

(**)

( , ) ( , ) ( , ) ( , )i i iN iL t z t z j t z t z dzu u u u imagining the NL dielectric screen concentrated at iz

( , ) ( , )NLo idt z z t zu u The combination of the last two equation yields (*) – q.e.d.

The next step is the linear non-dispersive propagation moving from to :iz oz

Up to the different notation, taking

the rhs-s are equivalent

Self-study

NSE Equation for a Dispersive, Lossy, NonLinear Medium


The non-linear Schroedinger equation (IV)x

y

z

r

z z+dz

2( , ) ( , ) ( , )zu uz z zuj

Kerr NL

22( ; ) ( ; )zj z zu u

Dispersion LIN

Combine the three components (NL Schroedinger eq.):

2

22

2

( ; )

( ; ) ( , ) ( , ) ( ; )

z

uu u u

j z

z z z j z

u

Loss LIN( ; ) ( / 2) ( ; )zj z j zu u 2( ; ) ( ;0)

zz eu u

The cross-interactions between the three differential effects are 2nd order


The non-linear Schroedinger equation (V)x

y

z

r

z z+dz

Kerr NLLIN Dispersion

2222

( ; )

( ; ) ( , ) ( , ) ( ; )

z

u uu u

j z

z z z j z

u

LIN Loss

Nonlinear Schroedinger Eq. (NSE):

22 3( / 2) ( / 6) / 2z t t tu u u u u uj uj


Rederivation of the NSE with all effects modeled at once (I)

0( ; )j z

bb z eH

0( ; ) ( ;0)j z

z eU U

We recall that the CE of a narrowband optical signal propagatesin the frequency domain according to a TF

( ; )u t z

where we define a complex-valued propagation constant to account for lossesand we also allow for the Kerr non-linearity, as follows:

0 0/ 2Lin NL j

2( ; )tu z

where for narrowband opticalpulses (>1 ps envelope, whilethe optical cycle is about 5 fs)we can assume that the loss and non-linearity are constant over freq. hence equal to their value at the center freq.

0 0 0 0

2 30 0 0( ) ( ) / 2 ( ) / 6

0 0 0 0 0

2 3/ 2 / 6

( ; ) ( ; ) ( ; ) ( ;0)bbFu U Ut z z H z

Differentiate both sides by z:

0

0 0( ; ) ( ;0) ( ; )

j z

zU U Uz j e j z

( ; ) j zH z e stemming from the TF:

Note: The NLbeta does notstem from the TFThis is the weaknessof this compact proof


Rederivation of the NSE with all effects modeled at once (II)

0( ; ) ( ; )z z zU j U

The CE always satisfies the following 1st-order differential equation

0( ; ) ( ; )j z

t z tu zu e

Next we investigate the spatiotemporal envelope, ( ; )u t z0( ; ) ( ; )

j zU Uz z e FT

0 0

0( ; ) ( ; )

j z j z

z U Uz e j z e

Substituting into (*) yields:

which simplifies toa differential equationfor the spatiotemporal CE:

0 0

( ; ) ( ; )z zU j U z Note: A more direct rederivation of this differential equation is available:

0( ; ) ( ;0)j z

z eU U

Start from:

(*)

(**)

Substituting (**) 0 0( ; ) ( ;0)j z j z

z e eU U Or, moving the phase-factor to the rhs:

0 0( ; ) ( ;0)j z j z

z eU Ue


Rederivation of the NSE with all effects modeled at once (III)

0 0 0 0

Lin NL j Now

0 0 0 0

2 3/ 2 / 6

0 0( )

( ; ) ( ;0)j z

U Uz e TF for the spatiotemporal CE

Differentiate both sides with respect to z:

0 0

0 0 0 0

( )( ; ) ( ) ( ;0) ( ) ( ; )

j z

z zU j e j zU U

0 0

( ; ) ( ; )z zU j U z q.e.d..

0 0 0 0 0 0 0

2 3/ 2 / 6 / 2NL j

0 0 0 0 0

2 3( ; ) / 2 / 6 ( ; )NLzU z j zUj

0 0 0 0 0

2 3( ; ) ( ) / 2 ( ) / 6 ( ; ) / 2 ( ; )NLzU Uz j j j j z zUj


Substituting into (*) and, rearranging yields (for SPM)

2( ; )NL u t z

In the case of SPM, we have, the refractive index (hence ) modulatedby the pulse itself, i.e.

NL

Rederivation of the NSE with all effects modeled at once (IV)

0 0( )

( ; ) ( ;0)j z

U Uz e e

0 0 0 0 0

2 3( ; ) ( ) / 2 ( ) / 6 ( ; ) / 2 ( ; )NLzU Uz j j j j z zUj

2 3( ; ) ( / 2) ( / 6) ( ; ) / 2 ( ; )NLz t t tt z j t zu u j t zu

22 3( / 2) ( / 6)2z t t tu u u u uj j u u

NSE:

tj Inverse FT both sides, using and discard the subscripts 0

This is our 1st form of the NSE, applicable to XPM as well, with proper ID of NL


Solving the NSE for NL non-dispersive media + loss (I)NSE with just loss term: 2

2( , ) ( , ) ( , ) ( , )zu u uz z zu zj

2

2

( , ) ( , )( , )zuu

uz zz

j

2

2ln ( , ) ( , )z u uz zj ( , )( , ) ( , ) j zu z u z e

( , ) ( , )ln ( , ) ln ln( , ) ( , )j z zz zu u ju ze 2

ln ( , ) ln ( , ) ( , )2

( , )( , )

( , ) z z zz z z j zu

uu z

uz

z ju

Separate into real and imaginary parts:

ln ln ln2 2

( , ) ( , ) ( ,0)z zz zu u u

2( , ) ( , )z z u z

( /2)0( , ) ( , ) zeu uz 0

2( , ) ( , )

z

z zzu d Self-phasemodulation

0

2( , ) ( , ) ( , )

z

t z t z z zu t z z d

Retarded intensityat all space-time locations along the lightline determines a phase-shift per unit length integrated along the way

( , )zu

(*)

The intensity profile along the way is fixed (*)…

( /2) 02( , )( , ) ( ,0) z

zj t z z dze

uu z eu


Solving the NSE for NL non-dispersive media + loss (II)2

2( , ) ( , ) ( , ) ( , )zu u uz z zu zj

( /2)0( , ) ( , ) zeu uz

0 0

2 2( , ) 0( , ) ( , )

L LzL z e zu z d du

SPM phase at the output

( , )( , ) ( , ) j zu z u z e

The magnitude equation gets decoupledfrom the phase equation (which in turndoes depend on the magnitude): 2 2

0( , ) ( , ) zeu uz

0

1( , ) ( ,0) ( ,0)

L LzP P

eL e zd

( , ) ( ,0)effL L P 1

eff

LeL

Effective NL length(e.g. for a 80 Km fiber at 0.21 dB/KmLeff is about 20 Km)

Split-Step Fourier Numerical Simulation of Fiber Propagation


Split-Step Fourier (SSF) numerical simulation (I)x

y

z

r

z z+dz

The NSE derivation contained two key steps:I. Linear Dispersive Propagation: Mult. by a quadratic phase factor in the freq. domain

II. Intensity-dependent Phase-shift: Mult. by a phase factor in the time domain

[ , ]( ; ) ( ; )Dj z z dzz d Uz e zU (1 [ , ]) ( ; )Dj z z dz zU

[ , ]( , ) ( / , )j z z dzNLgt z dz e t dz v zu u

1 [ , ] ( / , )NL gj z z dz t dz v zu Actually, since the delaying (and distorting) effect of dispersion has already been accounted for in II, it suffices to apply the NL phase without any delay:

1( , ) [ , ] ( , )NLt z dz j z z dz t zu u


Split-Step Fourier (SSF) numerical simulation (II)x

y

z

r

1kz kz 1 1( , )( ; )k kz t zFU u

1 1( ; ) (1 [ , ]) ( ; )k D k k kz z z zU Uj 2

1( / 2)( )k kz z 1( , ) ( ; )D k kt z zFu U

11( , ) [ , ] ( , )NL k kk D kt z j z z t zu u

1

2( )( , ) k kzkD zt zu 1k k

1( , )kt zu ( , )kt zu

Notice that without the quadratic term:

1 1( , ) [ ( , ])k k k kD Dt z t z z zu u

corresponding to

( / , )gt dz v zu


Split-Step Fourier (SSF) numerical simulation (II)x

y

z

r

1kz kz 1 1( , )( ; )k kzuz FU

1 1( ; ) (1 [ , ]) ( ; )retk D k k kz z z zU Uj

21( / 2)( )k kz z

1( , ) ( ; )D k kz zFu U

11( , ) [ , ] ( , )NL k k Dk kz j z z zu u

1

2( )( , ) k kD z zku z 1k k

1( , )kzu

( , )ku z RETARDED-TIME VERSION

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Lecture III Introduction to Fiber Optic Communication …in which we study : Optical Dispersion...

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