Light transmission in fiber - Part 2

ECE 6323                                                            han q le - copyrighted
Use solvely for students registered for UH ECE 6323 during courses - DO NOT distributed (copyrighted materials)

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6. Polarization effects

6.1 Polarization- review

6.1.1 Concept

Photon is a spin-1 particle J=1 but with zero rest mass, therefore it can have only 2 spin states (Subscript[m, J])=+- 1. For a monochromatic plane wave moving in z-direction, these spin states for J projected along the k-vector correspond to the right-handed circular polarized wave and left-handed circularly polarized waves.
Consider the plane wave electric field:
Ef = (OverVector[A])  (e^(i (k z - ω t) )) where  (OverVector[A]) is a vector that can be projected in a Cartesian coordinate system:
(OverVector[A]) = (OverHat[x]) (Subscript[A, x]) + (OverHat[y]) (Subscript[A, y])
Putting:
(Subscript[A, x])= |(Subscript[A, x])| (e^(i (Subscript[φ, x]))) ;  (Subscript[A, y])= |(Subscript[A, y])| (e^(i (Subscript[φ, y]))) 
then Ef = ( (OverHat[x])  |(Subscript[A, x])| (e^(i (Subscript[φ, x]))) +  (OverHat[y]) |(Subscript[A, y])| (e^(i (Subscript[φ, y]))) )  (e^(i (k z - ω t) )) 
Another way to express Ef is using a vector:
Ef  = (({{(OverHat[x]), (OverHat[y])}})) Re[ (({{|(Subscript[A, x])| (e^(i (Subscript[φ, x])))}, {|(Subscript[A, y])| (e^(i (Subscript[φ, y])))}})) (e^(i (k z - ω t) ))]
The light intensity is proportional to:
<|Ef((|)^2)>  = (({{|(Subscript[A, x])| (e^(-i (Subscript[φ, x]))), |(Subscript[A, y])| (e^(-i (Subscript[φ, y])))}})) . (({{|(Subscript[A, x])| (e^(i (Subscript[φ, x])))}, {|(Subscript[A, y])| (e^(i (Subscript[φ, y])))}}))=(Subscript[|A, x])((|)^2)+(Subscript[|A, y])((|)^2)

We describe the polarization state of an electric field with(1/(Sqrt[(Subscript[|A, x])((|)^2)+(Subscript[|A, y])((|)^2)  ])) (({{(Subscript[A, x])}, {(Subscript[A, y])}})), or (1/(Sqrt[(Subscript[|A, x])((|)^2)+(Subscript[|A, y])((|)^2)  ]))(({{(Subscript[|A, x])|(e^(i (Subscript[φ, x])))}, {(Subscript[|A, y])|(e^(i (Subscript[φ, y])))}})): we call it a Jones vector.

6.1.2 Example

Ef[Axy_,t_,Dphase_]:={Cos[2*Pi*t],Axy*Cos[2*Pi*t+Dphase]} ;

vec={} ;head={};
Axy= 1.; phase =1.2;
For[i=0,i<50, {
	      AppendTo[head, Ef[Axy,i*0.02,phase]];
	          AppendTo[vec,
      Graphics[ 			                {Hue[0,1,1], Arrow[{{0,0},Ef[Axy,i*0.02,phase]} ]
				                    ,Hue[0.6] , Line[head] }
			, PlotRange->{{-2,2},{-2,2}} 
		, AspectRatio-> 1
		, Axes-> Automatic
		,GridLines-> Automatic] ]
		                              },i++];

Manipulate[ Show[vec[[i]] ],{i,1,Length[vec],1}]

The ellipse is swept by the parametric plot of the E field for a fixed z position, e. g. z=0, as time varies:
Ef  = (({{(OverHat[x]), (OverHat[y])}})) Re[ (({{|(Subscript[A, x])| (e^(i (Subscript[φ, x])))}, {|(Subscript[A, y])| (e^(i (Subscript[φ, y])))}})) (e^(-i ω  t ))]
One way to describe this ellipse is as follow. Let
Ef  = (({{(OverHat[x]), (OverHat[y])}})) Re[ (({{|(Subscript[A, x])| }, {|(Subscript[A, y])| (e^(i ((Subscript[φ, y]) - (Subscript[φ, x]) )))}})) (e^(-i ω  t + i  (Subscript[φ, x ])))]
= (({{(OverHat[x]), (OverHat[y])}})) Re[ (({{1, 0}, {0,  (e^(i  ϕ))}})) (({{|(Subscript[A, x])|}, {|(Subscript[A, y])|}}))  (e^(-i ω  t  ))]
= (({{(OverHat[x]), (OverHat[y])}})) (({{Cos[ω  t], 0}, {0,  Cos[ω  t + ϕ]}})) (({{|(Subscript[A, x])|}, {|(Subscript[A, y])|}}))

Other states of polarization include:
    Linear: e. g. (Subscript[φ, x]) = (Subscript[φ, y]) =φ
    Ef = ( (OverHat[x])  |(Subscript[A, x])|  +  (OverHat[y]) |(Subscript[A, y])|  )  (e^(i (k z - ω t + φ) )) 
    This vector makes an angle ArcTan[|(Subscript[A, y])|/|(Subscript[A, x])| ] with the x axis. Linear polarization is also called P-polarization or π-polarization
    Circular, right handed and left handed:
    |(Subscript[A, x])| = |(Subscript[A, y])| = |A| ; (Subscript[φ, x]) - (Subscript[φ, y]) =+ or - π/2
    which are also called (SuperPlus[σ]) or (SuperMinus[σ]) polarization.

vec={} ;head={};
Axy= 1.6; phase =0.;
For[i=0,i<50, {
		        AppendTo[head, Ef[Axy,i*0.02,phase]];
	          AppendTo[vec,
      Graphics[  {Hue[0], Arrow[{{0,0},Ef[Axy,i*0.02,phase]}]
				                    ,Hue[0.6] , Line[head] }
			, PlotRange->{{-2,2},{-2,2}} 
		, AspectRatio->1
		, Axes->Automatic
		,GridLines->Automatic]  ] },i++];

Manipulate[ Show[vec[[i]] ],{i,1,Length[vec],1}]

Linear polarization

vec={} ;head={};
Axy= 1.; phase =Pi/2.;
For[i=0,i<50, {
		        AppendTo[head, Ef[Axy,i*0.02,phase]];
	          AppendTo[vec,
      Graphics[ {Hue[0], Arrow[{{0,0},Ef[Axy,i*0.02,phase]}],
					               Hue[0.8,1.,1.], Arrow[{{0,0},Ef[Axy,i*0.02,-phase]}]
				                    ,Hue[0.6] , Line[head] }
			, PlotRange->{{-2,2},{-2,2}} 
		, AspectRatio->1
		, Axes->Automatic
		,GridLines->Automatic]  ]
		                              },i++];

Manipulate[ Show[vec[[i]] ],{i,1,Length[vec],1}]

Circular polarization

6.2 Birefringence in fiber

6.2.1 Birefringence

As light travels in a medium, different polarizations may have different phase velocities. This is known as birefringence. The polarizations can be linear or circular or any pairs...

Example:                        (Subscript[E, x])  (OverHat[x]) (e^(i ((Subscript[β, 1]) z -ω t))) + (Subscript[E, y])  (OverHat[y]) (e^(i ((Subscript[β, 2]) z -ω t))) 
The phase difference between the 2 components is:
                                                  Δϕ=((Subscript[β, 1])-(Subscript[β, 2])) z 
 This means that as light travels, different state of polarization happens at different location of the fiber.

EB[Axy_,Bire_,t_]:={Cos[2*Pi*t],Axy*Cos[2*Pi*t+2*Pi*Bire*t]} ;

vec={} ;head={};
Axy= 1.; bire=0.1;
For[i=0,i<500, {step=0.01 ;
		        AppendTo[head, EB[Axy,bire,i*step]];
	          AppendTo[vec,
      Graphics[   {Hue[0], Arrow[{{0,0},EB[Axy,bire,i*step]}]
		,Hue[0.6] , Line[head] }
			, PlotRange->{{-2,2},{-2,2}} 
		, AspectRatio->1
		, Axes->Automatic
		,GridLines->Automatic]
		                                                    ]
		                              },i++];

Manipulate[Show[vec[[i]] ], {i,1, Length[vec],1}, AnimationRunning-> False, DisplayAllSteps-> True]

The beat length is defined as the shortest length overwhich the polarization variation repeats itself. Thus:
                                              (Subscript[L, B])=((2 π)/((Subscript[β, 1])-(Subscript[β, 2]))) 
How many cycles of light in a beat length?
The time number of wavelengths in a beat length is approximately:
                                            N= (1/(2 π))(Subscript[L, B]) ((((Subscript[β, 1])+(Subscript[β, 2]))/2))     
                                            N= (1/(2 π))((2 π)/((Subscript[β, 1])-(Subscript[β, 2])))((((Subscript[β, 1])+(Subscript[β, 2]))/2))
                                             N= ((OverBar[β])/((Subscript[β, 1])-(Subscript[β, 2])))  
Example, if (Subscript[β, 1])-(Subscript[β, 2]) ~ 5×(10^(-5)) (OverBar[β]) , then it takes 20,000 cycles of light to repeat itself. (this is a typical number)
In practice beat length can be observed based on Rayleigh scattering, since it is a function of polarization.

The origin of birefringence can be:
 1- Materials: some crystals are not isotropic. The electronic structure appears differently for different crystal planes, and light polarized along these plane have different phase velocities.

2- Structure: we have seen that TE and TM modes in a planar waveguide have different (Subscript[n, eff]), this is waveguide modal birefringence.

6.2.2 Birefringence in fiber

In principle, as discussed in previous chapter, a perfectly cylindrical fiber would have two degenerate components of every mode, corresponding to any pair of polarization states. However, no fiber is perfect. There is usually a slight asymmetry in manufacturing process, stress, or simply bend that make the fiber becomes slightly elliptic .. Once this happens, there is no more cylindrical symmetry, and the degenerate modes HE becomes separated. Each polarization will have its own mode index. Generally, any asymmetric effect on fiber can cause birefringence.
How large is the birefringence? A practical measure is the beat length. Typical fiber beat length can be from cm to 10's of cm (for those unintentional birefringence).
But one can also intentionally design a fiber that is birefringent. The purpose of such a fiber is to preserve the input polarization instead of letting it becoming random. These are called polarization preserving or maintaining fibers.

6.3 Polarization mode dispersion

If a pulse of light is a combination of 2 polarization modes:
                    (Subscript[(OverHat[ϵ]), 1]) c∫(e^(i ((Subscript[β, 1])[ω] z -ω t)))dω+(Subscript[(OverHat[ϵ]), 2]) (Sqrt[1-(c^2)])∫a[ω](e^(i ((Subscript[β, 2])[ω] z -ω t)))dω
in which we assume the same spectral component a[ω] for the original pulse, then from previous result we can write:
         (Subscript[(OverHat[ϵ]), 1]) c (e^(i (z (Subscript[k, 1;0])-t (Subscript[ω, 0]))))∫a[Ω](e^(i Ω( (z/(Subscript[v, g1]))-t )))(e^(i z ((Subscript[d, 1])/2) (Ω^2)))  dΩ
                +   (Subscript[(OverHat[ϵ]), 2]) (Sqrt[1-(c^2)]) (e^(i (z (Subscript[k, 2;0])-t (Subscript[ω, 0]))))∫a[Ω](e^(i Ω( (z/(Subscript[v, g2]))-t )))(e^(i z ((Subscript[d, 2])/2) (Ω^2)))  dΩ

We notice that besides (Subscript[d, 1]) and (Subscript[d, 2]) which cause broadening within each pulse, the difference in group velocity (Subscript[v, g1]) and (Subscript[v, g2]) also causes the stretch between the two pulses. This effect is known as polarization mode dispersion (PMD). Note that this exists as along as (Subscript[v, g1])≠(Subscript[v, g2]) , even if (Subscript[d, 1]) and (Subscript[d, 2]) are insignificant.
In reality, usually (Subscript[d, 1]) and (Subscript[d, 2]) are comparable, hence, both pulses are broadened equally. It is only the PMD that stretches further the two polarization component. This is similar to inter-modal dispersion in multi-mode fibers.
Hence, we have the concept of differential group delay (DGD) to describe the time separation of the 2 pulses over a given length of fiber:
                                         Δτ =|(L/(Subscript[v, g1]))-(L/(Subscript[v, g2]))| 
We define the PMD coefficient of DGD per unit length:
                           (Subscript[δτ, g]) =((Δτ)/L) =|(1/(Subscript[v, g1]))-(1/(Subscript[v, g2]))|=((d ((Subscript[β, 1])-(Subscript[β, 2])))/(dω))

It may appear that the PMD is also linear vs. length, similarly to GVD. However, this happens only if this birefringence is consistent through out the fiber. In reality, it is not because the axis of birefringence is random. In some section, one polarization is faster than the other, but it might be coupled into the slower mode in the next section and the PMD effect is reverse. Thus, only a statistical approach can be used to estimate realistic PMD in actual fibers. 
One key result is that PMD is not as bad as linear vs. L but only as (Sqrt[L]). The PMD coefficient  (Subscript[δτ, g])  is not a deterministic number, but a random variable whose mean square is approximately:
                                         (Sqrt[<((Subscript[δτ, g])^2)>]) =((Subscript[δτ, B])/(Subscript[L, B]))(Sqrt[2 L (Subscript[L, c])]) 
 where (Subscript[δτ, B]) is the DGD over one beat length, and (Subscript[L, c]) some length, known as the correlation length over which the two polarization modes are still correlated (before being randomized independently and no longer related with each other).
Typical values of PMD coefficient are from less than a tenth to few tenths of ps/km, e. g. ~0.1 ps/km. This is still rather small compared with GVD, and PMD is generally considered as a smaller problem of the two.

6.4 Polarization-maintaining fibers

Some fibers can be designed with intentional birefringence. The two P modes are distinctive from each other. Thus, a signal can be launched into either modes and will remain so, hence its polarization is maintained and not randomized.

These fibers can be made by introducing an asymmetry on the fiber cylindrical geometry, making it elliptical. The effect is much larger than unintentional birefringence, of course.
Assigment: a brief survey of PM fibers, and their design.

7. Solitons and other nonlinear effects

7.1 General nonlinear effects

We have learned about nonlinear scattering effects that are related to loss. Here, we will study nonlinear effects more generally, which cause more than just loss, but affecting pulse shapes, dispersion, chirp, signal interference, generating new frequencies, and even amplification (not loss but amplification).
We have to be aware of them in designing a transmission system in order to avoid problems.
Not all nonlinear effects are bad; some have applications.
They are:
- Stimulated Brillouin and Raman scattering (we discussed this already - see section above)
- Self-phase and cross-phase modulation
- 4-wave mixing
- And we will have a separate section on soliton

7.1.1 Self-phase modulation

In many materials, if the light intensity is strong enough, it can change the index of refraction. In fact, the index can be expressed as:
                             n[I]=(Subscript[n, 0]) +(Subscript[n, 2])  I
where I is the light intensity, and (Subscript[n, 2]) is called the nonlinear index coefficient.
In a fiber,                       I=(P/A)
where P is the signal power, and A is the mode effective cross section (area)
For SiO2 fiber, (Subscript[n, 2])=2.6 × (10^(-16)) (cm^2)/W, which seems very small. Yet, it does have an effect on light propagation. The reason is the length of the fiber.
One of the effect is self-phase modulation.

Plot[Exp[-z^2], {z,-5,5}]

The nonlienar index varies along the pulse profile, as it is proportional to intensity (power). Thus, the phase at each location within the pulse is modulated by the intensity at that location.
The consequence? new frequencies are generated (spectral broadening) and chirp. This affects the pulse dispersion and its shape in fiber.
When we have this effect, the wave equation from the Maxwell's equations is modified to include the presence of the nonlinear polarization field:
                                             (∇^2)E-((ω^2)/(c^2))ϵ E=((4 π (ω^2))/(c^2)) (Subscript[P, NL ])
Although it is generally impossible to solve exactly for all cases, approximation can be made and we can obtain reasonable solutions. We express the electric field:
                                               Ef= (e^(i ((Subscript[β, 0]) z - (Subscript[ω, 0]) t))) A[z,t]  
where (e^(i ((Subscript[β, 0]) z - (Subscript[ω, 0]) t))) is the carrier wave as before, and  A[z,t]  is the envelop to be solved. One approach is the adiabatic approximation (slowly varying) of the envelop. By substitute this expression in the nonlinear wave equation, and drop the higher order term (((∂^2)A[z,t])/(∂(z^2))) (which is the slowly-varying envelop approximation) 
           ((∂A[z,t])/(∂z))+(1/(Subscript[v, g]))((∂A[z,t])/(∂t))-(i/2) d(((∂^2)A[z,t])/(∂(t^2)))+(α/2) A[z,t] == i (Subscript[k, 2]) (Subscript[P, c]) ((Abs[A[z,t]])^2) A[z,t]  
 where (Subscript[P, c]) is just a coefficient such that the power is related to the field:
                                                   P=(Subscript[P, c]) (((|Ef|))^2)
and (Subscript[k, 2]) is a coefficient proportional to (Subscript[n, 2]).
 We can substitute a variable transformation:  t->τ-(z/(Subscript[v, g]))
 We obtain
                      ((∂A[z,τ])/(∂z))==-(1/2) α A(z,τ)+(1/2) i d (((∂^2)A[z,τ])/(∂(t^2)))+i (Subscript[k, 2]) (Subscript[P, c])  A(z,τ) ((|A(z,τ)|)^2)
We'll see later on that this can give rise to soliton. Here, we study only the SPM.
First, let's neglect the dispersion term, letting: (1/2) i d (((∂^2)A[z,τ])/(∂(t^2))) -> 0
                                ((∂A[z,τ])/(∂z))==-(1/2) α A(z,τ)+ i (Subscript[k, 2]) (Subscript[P, c])  A(z,τ) ((|A(z,τ)|)^2)
The solution is:
                                         (Subscript[A, 0])(t-(z/(Subscript[v, g]))) (e^(-(1/2)  α z +i (Subscript[k, 2]) (Subscript[P, c]) ϕ))
 where:
                                         ϕ=((|(Subscript[A, 0])(t-(z/(Subscript[v, g])))|)^2) (( (1-(e^(-α z))) )/α)
and (Subscript[A, 0])(t-(z/(Subscript[v, g]))) is the input pulse envelop (initial condition).
The full solution:
                            Ef=(Subscript[A, 0])(t-(z/(Subscript[v, g]))) (e^(i ((Subscript[β, 0]) z - (Subscript[ω, 0]) t)))(e^(-((z α)/2))) (e^(i (((1-(e^(-z α))))/α) (Subscript[k, 2]) (Subscript[P, c]) ((|(Subscript[A, o])(t-(z/(Subscript[v, g])))|)^2)))

For simplicity of illustration, we can let the loss  α be zero
                            Ef=(Subscript[A, 0])(t-(z/(Subscript[v, g]))) (e^(i ((Subscript[β, 0]) z - (Subscript[ω, 0]) t))) (e^(i z  (Subscript[k, 2]) (Subscript[P, c]) ((|(Subscript[A, 0])(t-(z/(Subscript[v, g])))|)^2)))    
Assuming real input envelope (transform limited pulse), the phase term is:
                                   Φ=(z (Subscript[k, 2]) (Subscript[P, c]) ((|(Subscript[A, o])(t-(z/(Subscript[v, g])))|)^2)+z (Subscript[β, 0])-t (Subscript[ω, o]))
We see that  Instantaneous_frequency=-(∂_t) Φ
                                  (Subscript[ω, o])-2 z (Subscript[k, 2]) (Subscript[P, c]) (Subscript[A, 0])(t-(z/(Subscript[v, g]))) (Subscript[A, 0]^′)(t-(z/(Subscript[v, g])))
 The frequencty shift is:
                                       Δω=-2 z (Subscript[k, 2]) (Subscript[P, c]) (Subscript[A, 0])(t-(z/(Subscript[v, g]))) (Subscript[A, 0]^′)(t-(z/(Subscript[v, g])))
Consider a Gaussian example:

ESPM[z_,t_,β_,ω_,VG_,T_,χ_]:=(E^(I (z β+(E^(-2 ((((t-(z/VG)))^2)/(T^2)))) z χ-t ω)-((((t-(z/VG)))^2)/(T^2))));
Instantdelom[ z_,t_,VG_,T_,χ_ ]:=((4 (E^(-((2 (((t-(z/VG)))^2))/(T^2)))) z χ (t-(z/VG)))/(T^2)) ;

Allplot={};vg=1.;chi=0.0002 ;
Do[{ itime=i  ;
		         AppendTo[Allplot,
			                 Plot[{
				                            0.005*Re[ESPM[z,itime,15.,10,vg,3.,chi]]
,Instantdelom[z,itime,vg,3.,chi]
				                               }
				
					                           ,{z,-6+itime*vg ,6+itime*vg}
			                     , PlotRange->{{0,80},{-0.005,0.005}}
				, PlotStyle->{{RGBColor[0,0.5,0],Thickness[0.001]},
						                              {RGBColor[1,0.,0],Thickness[0.004]}}
			                      , AspectRatio-> 0.25]]}
		          ,{i,0,50}]

Manipulate[Allplot[[i]],{i,1,Length[Allplot],1}]

Allplot={};vg=1.;chi=0.4 ;
Do[{ itime=1*i  ;
		         AppendTo[Allplot,
			                 Plot[{
				                            2*Re[ESPM[z,itime,15.,10,vg,3.,chi]]
,0.25*Instantdelom[z,itime,vg,3.,chi]
				                               }
				
					                           ,{z,-4+itime*vg ,4+itime*vg}
			                     , PlotRange->{{-6+itime*vg,6+itime*vg},{-2,2}}
				, AxesOrigin->{-6+itime*vg,0}
				, PlotStyle->{{RGBColor[0,0.5,0],Thickness[0.002]}
,{RGBColor[1,0.,0],Thickness[0.004]}}
			                      , AspectRatio-> 0.25
, Axes-> {False, True}]

]}
		          ,{i,0,30}]

Manipulate[Allplot[[i]],{i,1,Length[Allplot],1}, AnimationRunning-> False, DisplayAllSteps-> True]

Self-phase modulation alters the frequency chirp profile of the pulse.
Observe the chirp. The spectrum actually changes.

Plot spectrum broadening

Allplot={} ; vg=1.;chi=0.1 ;
Do[  {z=10 + j*10 ;spec={};
Do[spec=AppendTo[spec,{10+i*0.05, Abs[NIntegrate[ Exp[I*(10+i*0.05)*t]* ESPM[z,t,15.,10,1,17.,chi],{t,z-40,z+40}]]^2}
		                                   ],{i,-35,35}];
		AppendTo[Allplot,
			            ListPlot[spec,Joined->True
				   ,  PlotRange->{{8.,12},{0.,1000}}
				,PlotStyle->{Hue[0.2+j*0.03,1,0.6],Thickness[0.005]}
			                             ]
		                       ]},{j,0,10}];
Show[Allplot]

Manipulate[Allplot[[i]],{i,1,Length[Allplot],1}, AnimationRunning-> False, DisplayAllSteps-> True]

Key points:
1- A third-order nonlinearity
2- Causing: 
	-chirp of the pulse
	-spectral broadening (new frequencies)
	-change of pulse shape (depending on the fiber dispersion)
3- The effect is accumulative over length

7.1.2 Cross-phase modulation

Cross phase modulation (XPM) is the same as SPM except that the PM is done by one pulse on another, usually a powerful pulse on a weak pulse. (The reverse is too small to matter).
Strictly speaking, it is complicated if all pulses are involved (here we assume same group velocity):
                  ((∂(Subscript[A, 1])[z,τ])/(∂z))==i (Subscript[k, 2]) (Subscript[P, c])  (Subscript[A, 1])(z,τ) ((|(Subscript[A, 1])(z,τ)|)^2)+i (Subscript[k, 2]) (Subscript[P, c])  (Subscript[A, 1])(z,τ) ((|(Subscript[A, 2])(z,τ)|)^2) 
                  ((∂(Subscript[A, 2])[z,τ])/(∂z))==i (Subscript[k, 2]) (Subscript[P, c])  (Subscript[A, 2])(z,τ) ((|(Subscript[A, 1])(z,τ)|)^2)+i (Subscript[k, 2]) (Subscript[P, c])  (Subscript[A, 2])(z,τ) ((|(Subscript[A, 2])(z,τ)|)^2)  
These are called coupled-mode equations. But if we consider one pulses weaker than the other, for example (Subscript[A, 1]) is much weaker than (Subscript[A, 2]), then we can just treat (Subscript[A, 2]) on its own:
                                           ((∂(Subscript[A, 2])[z,τ])/(∂z))≈i (Subscript[k, 2]) (Subscript[P, c])  (Subscript[A, 2])(z,τ) ((|(Subscript[A, 2])(z,τ)|)^2)
which is the SPM of (Subscript[A, 2])    
                                  (Subscript[A, 2])(t-(z/(Subscript[v, g])))=(Subscript[A, 0;2])(t-(z/(Subscript[v, g])))  (e^(i z  (Subscript[k, 2]) (Subscript[P, c]) ((|(Subscript[A, 0;2])(t-(z/(Subscript[v, g])))|)^2)))    
                               (Subscript[Ef, 2])=(Subscript[A, 0;2])(t-(z/(Subscript[v, g]))) (e^(i ((Subscript[β, 0;2]) z - (Subscript[ω, 2]) t))) (e^(i z  (Subscript[k, 2]) (Subscript[P, c]) ((|(Subscript[A, 0;2])(t-(z/(Subscript[v, g])))|)^2)))    
and thus (dropping the SPM term of (Subscript[A, 1]), as it is << than the XPM)
                                              ((∂(Subscript[A, 1])[z,τ])/(∂z))≈i (Subscript[k, 2]) (Subscript[P, c])  (Subscript[A, 1])(z,τ) ((|(Subscript[A, 2])(z,τ)|)^2)
This is the XPM.
                                            (Subscript[A, 1])[z,τ]≈(Subscript[A, 0;1])(t-(z/(Subscript[v, g1]))) (e^(i (Subscript[k, 2]) (Subscript[P, c]) (∫_0^z)((|(Subscript[A, 2])(ζ,τ)|)^2)dζ)) 
In essence, similar effects as SPM happens, except that the modulation depends on the other pulse rather by itself. Spectral broadening, chirp also happen, but depend on the other pulse rather than itself.

7.1.3 Four-wave mixing

Some times, new frequencies can be generated from existing frequencies as shown below.

These are called 4WM. The lower energy signal is called Stokes signal, and the higher is called anti-Stokes.
  We can also have case:             (Subscript[ω, 2])=(Subscript[ω, 1])-(Subscript[ω, 1])+(Subscript[ω, 2])
  which is cross-phase modulation: the signal (Subscript[ω, 1]) modulates phase of (Subscript[ω, 2]).
  We also have:                           (Subscript[ω, 1])=(Subscript[ω, 1])-(Subscript[ω, 1])+(Subscript[ω, 1])      
  which is SPM.
  Hence 4WM is the most general description and SPM, XPM are just special cases.  
  In essence: 4WM that involves Stokes and AS signals cause noises in a system.

7.2 Solitons

http://www.youtube.com/watch?v=Ud7STKWNmQw 
http://www.youtube.com/watch?v=MADng1fqECY

http://www.youtube.com/watch?v=SAbQ4MvDqEE

http://www.youtube.com/watch?v=wEbYELtGZwI

7.2.1 Basic discussion

Soliton is a special wave that is a direct result of BOTH self-phase modulation and dispersion.
                        (1/2) α A(z,τ)-(1/2) i d (((∂^2)A[z,τ])/(∂(t^2)))+((∂A[z,τ])/(∂z))==i (Subscript[k, 2]) (Subscript[P, c])  A(z,τ) ((|A(z,τ)|)^2)
We'll see that this equation has certain special solution for some condition on the dispersion.
It is illustrative however, to see this equation from a different approach.
Earlier, we write:
                                         β[ω]=n[ω]  ((ω )/c)
and we expand around a center frequency  (Subscript[ω, 0]) :
                       β[ω]=β[(Subscript[ω, 0])]+ ((d β)/(d ω)) (ω-(Subscript[ω, 0])) +(1/2) (((d^2) β)/(d (ω^2))) (((ω-(Subscript[ω, 0])))^2) +...  
            β[ω]=n[(Subscript[ω, 0])] ((Subscript[ω, 0])/c)+ (((n[(Subscript[ω, 0])])/c)+((Subscript[ω, 0])/c) ((d n)/(d ω)))  (ω-(Subscript[ω, 0])) 
                                          +(1/2) ((2/c) ((d n)/(d ω))+((Subscript[ω, 0])/c) (((d^2)n)/(d (ω^2)))) (((ω-(Subscript[ω, 0])))^2) +...  
 We recall that:
                                          (((n[(Subscript[ω, 0])])/c)+((Subscript[ω, 0])/c) ((d n)/(d ω)))=(1/(Subscript[v, g]))       
                                          ((2/c) ((d n)/(d ω))+((Subscript[ω, 0])/c) (((d^2)n)/(d (ω^2))))= d (GVD)
What if now that the index depends on power?
                                           n[(Subscript[ω, 0])]=(Subscript[n, 0])[(Subscript[ω, 0])]+(Subscript[n, 2]) ((P )/A)
Neglecting the dependence of (Subscript[n, 2])  on ω, which is small, we see that the equation is now:
            β[ω]=n[(Subscript[ω, 0])] ((Subscript[ω, 0])/c)+ (1/(Subscript[v, g]))(ω-(Subscript[ω, 0]))  +(1/2) d (((ω-(Subscript[ω, 0])))^2) +(Subscript[n, 2]) ((Subscript[ω, 0])/c)((P )/A) +...
Or:       β[ω]=n[(Subscript[ω, 0])] ((Subscript[ω, 0])/c)+ (1/(Subscript[v, g]))(ω-(Subscript[ω, 0]))  +(1/2) d (((ω-(Subscript[ω, 0])))^2) +(Subscript[β, 2]) P +...
where we define:                        (Subscript[β, 2]) =(Subscript[n, 2]) ((Subscript[ω, 0])/(c A))      
Now, we see that theses two terms, (1/2) d (((ω-(Subscript[ω, 0])))^2) ; (Subscript[β, 2]) P  can together produce certain interesting results.
Similarly to the SPM discussion above, we can write:
                                             Ef= (e^(i ((Subscript[β, 0]) z - (Subscript[ω, 0]) t))) ψ[z,t]  
Where the function  ψ[z,t]  is whatever that must satisfy the nonlinear wave equation. Since this is a nonlinear equation, the power or amplitude becomes a part of the equation, we will introduce a proportionality term to relate real power to the amplitude:
                                                P=(Subscript[P, c]) (((|Ef|))^2)
Then, if we take the general from (e^(i (β z - ω t))) and substitute, we will find out that in order for the expression β[ω] above valid, we have the following requirement of  ψ[z,t] :
                               -i (((∂ψ)/(∂z))+(1/(Subscript[v, g]))((∂ψ)/(∂t))) =-(1/2)d(((∂^2)ψ)/(∂(t^2)))+(Subscript[k, 2]) (Subscript[P, c]) (((|ψ|))^2)ψ

or:
                            -i (1/((Subscript[k, 2]) (Subscript[P, c])))(((∂ψ)/(∂z))+(1/(Subscript[v, g]))((∂ψ)/(∂t)))=- (1/2)(d/((Subscript[k, 2]) (Subscript[P, c])))(((∂^2)ψ)/(∂(t^2)))+ (((|ψ|))^2)ψ 
Again, some algebraic manipulation to simplify various expression 
 we define some variable:
                                     ζ=z(Subscript[k, 2]) (Subscript[P, c])    or =z/(Subscript[z, c]) ; (Subscript[z, c]) =(1/((Subscript[k, 2]) (Subscript[P, c])))
                                    τ=(1/(Subscript[t, c]))(t-(1/(Subscript[v, g]))z ) ;    (d/((Subscript[k, 2]) (Subscript[P, c])))=-((Subscript[t, c])^2) 
We'll see that (Subscript[t, c]) is roughly a parameter describing the pulse width, and (Subscript[z, c]) is just a characteristic length relating the dispersion effect and the nonlinear coefficient.
The critical requirement here is that:              
                                                    ((Subscript[t, c])^2) = -(d/((Subscript[k, 2]) (Subscript[P, c]))) > 0 
In other words, the dispersion d and the nonlinearities MUST HAVE OPPOSITE EFFECT. 
The Eq. becomes:
                             -i (Subscript[z, c]) (((∂ψ)/(∂z))+(1/(Subscript[v, g]))((∂ψ)/(∂t)))= (1/2)((Subscript[t, c])^2)(((∂^2)ψ)/(∂(t^2)))+ (((|ψ|))^2)ψ 
Or, with the new variables:
                                          -i ((∂ψ)/(∂ζ))= (1/2)(((∂^2)ψ)/(∂(τ^2)))+ (((|ψ|))^2)ψ    
A bit more generally, the equation that also includes loss is:
                                        -i ((∂ψ)/(∂ζ))= (1/2)(((∂^2)ψ)/(∂(τ^2)))+ (((|ψ|))^2)ψ -i  (α/2) ψ 

 The significant of this equation is the follow:
      -i ((∂ψ)/(∂ζ)): This is how the pulse shape varies as a function of its travel distance
       (1/2)(((∂^2)ψ)/(∂(τ^2))) : this is the effect of dispersion
        (((|ψ|))^2)ψ : the non-linear effect.
The interesting thing occurs when the dispersion and nonlinear effect work against each other and cancel each other's effect out: dispersion retard the red, hasten teh blue, but the nonlinear does the opposite: result: the pulse maintain its shape!

We cannot use linear technique to solve a nonlinear equation. We must "guess" a solution and substitute.
It turns out that a solution is:
                                                     ψ[ζ,τ]=((e^(i ζ/2))/(Cosh[τ]))

ψ[ζ_,τ_]:=((E^(I ζ/2))/(Cosh[τ]))  ;

I  (∂_ζ)ψ[ζ,τ]
(1/2)(∂_(τ,τ))ψ[ζ,τ]
 ((Abs[ψ[ζ,τ]])^2) ψ[ζ,τ]

FullSimplify[-(1/2) (E^((I ζ)/2)) sech(τ) +(1/2) ((E^((I ζ)/2)) (tanh^2)(τ) sech(τ)-(E^((I ζ)/2)) (sech^3)(τ))+(E^((I ζ)/2)) (sech^3)(τ) ]

This is known as the canonical solution of solitons. (lowest order)
The full expression:
                           Ef= (e^(i ((Subscript[β, 0]) z - (Subscript[ω, 0]) t))) ψ[z,t]  
                                 = (e^(i ((Subscript[β, 0]) z - (Subscript[ω, 0]) t))) (e^((i z)/(2 (Subscript[z, c])))) sech(((t-(z/(Subscript[v, g])))/(Subscript[t, c])))

Plot[Re[ψ[0.,τ] ], {τ,-10,10}]

ESol[λ_, SolParm_List,z_, t_]:=Module[{neff,ng, zc, tc, c=299.792458} ,
{neff,ng, zc, tc}=SolParm  ;
(E^(I 2 π  (neff  z /λ - t  c/ λ))) (E^((I z)/(2 zc))) sech(((t-((z  ng)/( c)))/tc))   ] ;

SolParm={1.45, 1.8,20. ,0.02}  ;
Manipulate[Plot[Re[ESol[1.55, SolParm, z,t]],{z,-10,100}, PlotRange->{-1,1}, PlotPoints-> 200],{t,0,0.5,0.005} ]

7.2.2 Plot comparison of typical dispersion vs. Soliton

TypicalDisp[z_,t_,w0_,τ_,n_, VG_,d_]:= (τ/(((4 (d^2) (z^2)+(τ^4)))^(1/4))) (E^(-((((τ^2)/4)((((( z)/VG) -t))^2))/((d^2) (z^2)+((τ^4)/4))))) ;
Soliton[z_,t_,w0_,τ_,n_, VG_,d_, χ_]:=(((E^((I z d)/(2 (T^2)))) sech((((( z)/VG) -t)/T)) (Sqrt[d]))/(T (Sqrt[χ]))) /. T->τ/2  ;

allEplot={};
w0=2 π 2*10^14; tau=3*10^(-14) ; VG=(3*10^10)/2. ;
steptime=0.5*10^(-13) ;
nplot=60 ;
da=2*tau^2/(nplot*steptime*VG) ;
xmin[i_,tau_,VG_,steptime_,d_]:=i*steptime*VG-
		                                                            3*Sqrt[tau^2+4*d*i*steptime*VG]*VG ;
xmax[i_,tau_,VG_,steptime_,d_]:=i*steptime*VG+
		                                                            3*Sqrt[tau^2+4*d*i*steptime*VG]*VG ;
Do[{ AppendTo[allEplot,
			    Plot[{(TypicalDisp[x,i*steptime,w0,tau,1.47,VG,da])^2,
					                0.5*Abs[0.01*Soliton[x,i*steptime,w0,tau,1.47,VG,da,0.01]]^2}
				                         ,{x,xmin[i,tau,VG,steptime,da],xmax[i,tau,VG,steptime,da] }
			                     , PlotRange->{{0.,xmax[nplot,tau,VG,steptime,da]},{0,1}} 
					,PlotStyle->{{RGBColor[0.,0.,1.],Thickness[0.003]},
							                                          {RGBColor[1.,0.,0.],Thickness[0.003]}}
			                      , AspectRatio-> 0.2]]} ,{i,0,nplot}]

Manipulate[Show[allEplot[[i]] ],{i,1,Length[allEplot],1}]

Notice that the blue pulse is broadened via regular dispersion. The nonlinear SPM works against the dispersion tendency and recompress the pulse, resulting in the red pulse that does not change shape.
So what is the implication of soliton?
1- Pulse spreading because of dispersion can be compensated with the nonlinearities
2- Transmission of high bit rate without dispersion penalty
However, there are issues:
3- Signal power must be well controlled for soliton system to be effective (this is a nonlinear phenomenon, we can't arbitrarily scale the signal)
4- Dispersion varied for existing fibers, must manage for DWDM (multiple wavelengths, dispersion is different for each wavelength..)