Light transmission in fiber - Part 1

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

0. Loading package

1. Overview

As light is transmitted in fiber, the folllowing phenomena are critical to their use for carrying information:
1- Loss: the light will be weaker, or loss of power or energy, reducing the detectivity of the pulses, resulting in higher error rates and hence limiting the transmission bandwidth.
2- Dispersion: Different frequencies will travel with different speeds and this will cause pulse shape distortion, reducing the temporal distinctivity between pulses and resulting in higher error rates and hence limiting the transmission bandwidth. 
3- Polarization effects: birefringence in fibers can result in polarization mode dispersion (PMD) and affect the transmission bandwidth.
4- Nonlinear effects, including nonlinear scattering, fourwave mixing, that generate spurious signals, increasing the error rates. One special nonlinear phenomenon is soliton that actually can be exploited for transmission, however there are restrictions on its practicality.
In this chapter, we will study these phenomena and their implications.

However, to appreciate these phenomena, we need to understand the basics of light transmission. The next few sections will review these.

2. Concept of a wave packet: carrier and envelope

2.1 Introduction

Hardly do we ever deal with continuous wave (CW) in many applications. Signal is made by modulating a wave: amplitude, frequency, phase modulation. What we deal with is called a wave packet.
The electric field of a mode in fiber, as we have learnt can be described as:
                  (Subscript[E, n])[ρ,ϕ](e^(i ((Subscript[β , n])z -ω t)))=(({{((Subscript[E, ρ])^((n)))[ρ,ϕ], ((Subscript[E, ϕ])^((n)))[ρ,ϕ], ((Subscript[E, z])^((n)))[ρ,ϕ]}}))(({{(OverHat[ρ])}, {(OverHat[ϕ])}, {(OverHat[z])}})) (e^(i ((Subscript[β , n])z -ω t)))
where the superscript or subscript n indicates the mode index. For simplicity, we use just one symbol to include all mode indices, such as (Subscript[HE, m n]) or (Subscript[EH, m n]) modes.
A general light wave is a sum of these modes:
                                             E=(∑_(n=1)^M)(Subscript[a, n]) (Subscript[E, n])(e^(i ((Subscript[β , n])z -ω t)))
But it appears that these waves do not carry any information yet. In fact they are just raw power, they can carry information but must be modulated. They are called carrier waves.

A Review - carrier and modulation

Link to 1100

2.2 Carrier wave

1.2.1 Basic illustration

Carrier is a pure single frequency wave; like this (e^(i (k z - ω t))), we can ignore the transverse profile, which is just a fix function. Such a wave is a pure harmonic, because it has only one spatial frequency k, and one temporal frequency ω.

Manipulate[Plot[Sin[(k z-ω t)+ϕ] /. ω->2 Pi  /. ϕ->0  /. k->1 ,{z,-100,100},
			PlotPoints-> 100, PlotRange->{{-100,100},{-1,1}},
	        ImageSize->{600.,150}
	   , PlotStyle->{RGBColor[0.,0.,1], Thickness[0.005]}
,AspectRatio-> 0.2
	                ], {t,0,1}]

It carries no information, except perhaps its amplitude

1.2.2 Phase velocity

You can see that the wave is moving from left to right. How fast does it move? what speed?
Let's look at the wave expression itself:
(e^(i Φ[z;t])) where Φ[z;t] is the phase function. The movement of this wave is the relation between z (space) and time (t) for which Φ[z;t]=constant.
Thus:
ΔΦ[z,t] =(∂_z) Φ[z,t]Δz+(∂_t) Φ[z,t]Δt =0
or: ((Δz)/(Δt))=-(((∂_t) Φ[z,t])/((∂_z) Φ[z,t]))

-(((∂_t)(k z - ω t))/((∂_z)(k z - ω t)))

Thus, for the carrier wave above, (ω/k) is the wave velocity. It is also called the phase velocity, because it is the velocity that if we move at which, the phase of the carrier would appear to be constant.

1.2.3a Audio instantaneous frequency

Play[Sin[1000 2Pi  t  ],{t,0,2}]

Play[Sin[220 2Pi  t (1+3 t) ],{t,0,2}]

Play[Sin[440 2Pi  t  * 1.4],{t,0,2}]

What is a wave frequency? If we sit at a fixed location, we will see the wave amplitude oscillate in time like this

-(∂_t) (k z - ω t)

(∂_t)(220 2Pi  t (1+3 t))

So we can see that we do get the right answer.

1.2.3b Instantaneous frequency

What is a wave frequency? If we sit at a fixed location, we will see the wave amplitude oscillate in time like this

Manipulate[ListPlot[{{0.,Sin[-ω  t] /.  ω->2 Pi  }},
			                                   PlotRange->{{-1.,1},{-1.,1}}
			                               ,ImageSize->{200,200}
			                          , AspectRatio->1
                                    , PlotStyle->AbsolutePointSize[10]
                                  ,Background->RGBColor[0.7,0.9,0.9]
                                         ] , {t,0,1}]

Manipulate[GraphicsArray[{ListPlot[{{0.,Sin[-ω  t] /.  ω->2 Pi }},
			                                   PlotRange->{{-1.,1},{-1.,1}}
			                               ,ImageSize->{200,200}
			                          , AspectRatio->1
                                    , PlotStyle->AbsolutePointSize[10]
                                  ,Background->RGBColor[0.7,0.9,0.9] ] 
,ListPlot[{{0.,Sin[-ω  t] /.  ω->3.7 Pi }},
			                                   PlotRange->{{-1.,1},{-1.,1}}
			                               ,ImageSize->{200,200}
			                          , AspectRatio->1
                                    , PlotStyle->AbsolutePointSize[10]
                                  ,Background->RGBColor[0.7,0.9,0.9]
                                    ] }], {t,0,1}]

The frequency is the number of cycles per unit time. If the ball moves up down real fast, it's high frequency. If it moves slowly, it's low frequency.

We have a concept: instantaneous frequency. It is the speed of the phase change vs. time:
ω(t)=|((∂ Φ[t])/(∂t))|.

Let's apply the concept to our wave:
(e^(i (k z - ω t))) = (e^(i Φ[z,t]))
Φ[z,t]=(k z - ω t)

-(∂_t) (k z - ω t)

So we can see that we do get the right answer.

2.3 Envelope

2.3.1 Finite temporal extent

We know that in real world, light wave do not extend into infinity, whether time or space. If we take a snap shot, a light wave will have a finite spatial extent. If we sit at one place, the lightwave will be tuned on and off within a finite duration. For example, at t=0, we have a spatial Gaussian extent:
(e^(- (z^2)/(w^(2  ))))

Plot[(E^(- (z^2)/(w^(2  )))) /. w->1,{z,-3,3}]

This is what we call envelope. Within the envelope, we have the carrier wave as above. So, how do we describe such a pulse?

Manipulate[ Plot[ Re[(E^(- (z^2)/(w^(2  )))) (E^(I (β z - ω t))) ] /. w-> 10. /. β-> 2 π /. ω->  2 π, {z,-20,20}, PlotRange-> {-1,1}, PlotPoints-> 50], {t,0.,1}]

The above clearly does NOT describe a light pulse (or package) because it doesn't move.

2.3.2 Linear superposition for a light pulse or wave packet

What we need is to describe a light pulse in terms of linear combination of carrier waves:
                     E(z,t)=∫a(ω) (e^(i (k[ω] z - ω t)))dω                                           (1.3)
where a(ω) is a general function such that (1.3) is a finite integral and exists.
Note that the propagation constant k[ω] is explicitly written as a function of frequency ω. We recall that:
                                             k[ω]=((2 π)/λ)(Subscript[n, eff])[ω] =(ω/c)(Subscript[n, eff])[ω]

We can choose    a(ω)=(1/(2 (Sqrt[π])))τ (e^(-(1/4) (τ^2) (Ω^2)))      (1.12)
where: Ω = ω-(Subscript[ω, 0])  and find out what it looks like. 
                                         (1/(2 (Sqrt[π])))τ(e^(-i  (Subscript[ω, 0]) t))(∫_(-∞)^∞)(e^(-(1/4) (τ^2) (Ω^2))) (e^(i ((ω/c)(Subscript[n, eff])[ω] z - Ω t))) dω

We cannot integrate unless we know what (Subscript[n, eff])[ω] is. For simplicity we consider the case of NO dispersion and let (Subscript[n, eff])[ω]  be just a constant

(1/(2 (Sqrt[π])))τ(E^(I  (Subscript[ω, 0]) ((z/c)(Subscript[n, eff]) - t)))(∫_(-∞)^∞)(E^(-(1/4) (τ^2) (Ω^2))) (E^(I  Ω ((z/c)(Subscript[n, eff]) - t))) \[DifferentialD]Ω

So the answer is:
                                               E(z,t)=(e^(i (Subscript[ω, 0]) (((z (Subscript[n, eff]))/c)-t)))(e^(-((((c t-z (Subscript[n, eff])))^2)/((c^2) (τ^2)))))

FourierTransform[(E^(-(1/4) (τ^2) (Ω^2))), Ω, ((z/c)(Subscript[n, eff]) - t)]

2.4 Wave packet

This part: (e^(-((((c t-z (Subscript[n, eff])))^2)/((c^2) (τ^2))))) is the envelope, and this part: (e^(i (Subscript[ω, 0]) (((z (Subscript[n, eff]))/c)-t))) is the carrier wave. They both move with the same speed (c). Let's define Efield

Efield[z_,t_,w0_,τ_,n_]:=
		                                       		                                Re[(E^(-((((c t-n z))^2)/((c^2) (τ^2))))) (E^(-((I   w0(c t-n z) )/(c ))))]

c=2.99792458*10^10 ;
n=1.48 ;
τ=3*10^-14 ;
ω0 =2 π  2*10^14 ; 
Manipulate[Plot[Re[(E^(-((((c t-n z))^2)/((c^2) (τ^2))))) (E^(-((I   ω0  (c t-n z) )/(c ))))],{z,-0.001,0.005}, PlotRange-> {-1,1}, PlotPoints-> 100,
AspectRatio-> 0.2, Ticks-> None],{t,0,1.8*10^-13} ]

We can plot the intensity only of the light pulse:

c=2.99792458*10^10 ;
n=1.48 ;
τ=3*10^-14 ;
ω0 =2 π  2*10^14 ; 
Manipulate[Plot[Abs[(E^(-((((c t-n z))^2)/((c^2) (τ^2))))) (E^(-((I   ω0  (c t-n z) )/(c ))))]^2,{z,-0.001,0.005}, PlotRange-> {0,1}, PlotPoints-> 100,
AspectRatio-> 0.2, Ticks-> None],{t,0,1.8*10^-13} ]

Summary:
- For transmission in fiber, we don't just deal with continuous carrier wave; we deal with modulated light wave
- The most common form of modulation in OFC is intensity modulation (amplitude modulation). A signal is made up of light packets (light pulse)
- Fiber transmission depends both on intrinsic properties of the fiber as well as the properties of light packet

3. Transmission loss and noises

3.1 Loss models and mechanisms

As light propagates, the amplitude is not necessarily the same:

c=2.99792458*10^10 ;
τ=3*10^-14 ;
ω0 =2 π  2*10^14 ; 
Manipulate[n=1.48+I n1 ;
Plot[Abs[(E^(-((((c t-n z))^2)/((c^2) (τ^2))))) (E^(-((I   ω0  (c t-n z) )/(c ))))]^2,{z,-0.001,0.01}, PlotRange-> {0,1}, PlotPoints-> 100,
AspectRatio-> 0.2, Ticks-> None],{t,0,4*10^-13} ,{n1,0.,0.0015}]

This is known as linear distributed loss, because the loss is uniformly distributed along the length of the fiber. We also use the term linear to indicate that this loss has NOTHING to do with power intensity. In other words, it is linearly proportional with power. We'll see there is non-linear loss that can be ∝ (P^2) or higher.
We describe this linear distributed loss as an attenuation:
                                        (e^(i (β z -ω t)))   with β->β +i(α/2) 
                                                    (e^(i (β z -ω t))) (e^(- α z/2))
The unit of loss coefficient is usually expressed in dB/km. Defined as:
                                     -(1/L)10×(Subscript[Log, 10])[((P[L])/(P[0]))]=-(1/L)10×(Subscript[Log, 10])[(e^(- α L))]
                                                                      =α × 4.34

There are the kinds of loss that can happen at a local points or section of a fiber: fiber end face, spliced, fused joint & connector,  local defects, or local bending. With these, we described as just a net loss factor, also in dB (but not per km):
                                                            Loss=10×(Subscript[Log, 10])[((Subscript[P, in])/(Subscript[P, out]))]
The bending loss can be also described as distributed loss, as a function of length and curvature. However, in practice, the bending section of a fiber is usually shorter than the straight section, and the loss can be lumped into a local net loss.

More important are the distributed losses, which can be separated into 2 categories: intrinsic and extrinsic. Intrinsic losses are those that are fundamental to the nature of the fiber: part of the physical process. Nothing can be done. Extrinsic losses are those caused by mechanisms that are NOT fundamental to fibers, but from external sources such as impurities that cause aborption loss.

The mechanisms of intrinsic loss include:
- intrinsic absorption or fiber materials, which for GeO2 and SiO2 fiber, consists of a UV band and an IR band
- Rayleigh scattering ((λ^(-4)))
- other scattering loss including non-linear optics

The mechanisms of extrinsic loss include impurity absorption, mostly ions (including metallic) absorption. The largest absorption loss in the past has been caused by the (SuperMinus[OH]) ion. Improved manufacturing process has largely eliminated this problem. Others are fiber imperfection: there are random variation that causes the light to be scattered

3.2  Illustrations of distributed loss

3.3 Scattering loss: Rayleigh scattering and Mie scattering

The Rayleigh scattering is due to microscopic (subwavelength) random variation inside the silica material itself, although improved manifacturing can reduce to a fundamental level. (Note that silica fiber is an amorphous medium and not a crystal, its density, and hence index is NOT perfectly uniform). The glass is described as a frozen state of amorphous SiO2 molecules and an analogy is to look at frosted glass or glass with random distortion. Likewise, we can look through the atmostphere on a hot day and see the thermal fluctuation of the air index. There is a similar fluctuation in fiber but it is "frozen".
The model of Rayleigh scattering loss is the same as above
                                                         P[z]=P[0] (e^(-(Subscript[γ, R]) z ))
with the loss coefficient is given:
                                                         (Subscript[γ, R])=((8 (π^3))/3)(((n^8)(p^2))/(λ^4)) (Subscript[β, c]) (Subscript[k, B]) (Subscript[T, F])  
where λ is the wavelength, n is the index, p is the mean photoelastic coefficient (describing photoelastic scattering), (Subscript[β, c]) is the isothermal compressibility at some fictitious temperature (Subscript[T, F]) (depending on the fiber manufacturing property, to give rise to the random density fluctuation), and (Subscript[k, B]) is the Boltzmann's constant.

B=0.2 ;
LogPlot[ 10*Log[10.,E] B/λ^4,{λ,0.7,2.}, FrameLabel-> {"Wavelength (μm)", "Loss (dB)" }, Frame-> True, GridLines-> Automatic]

When the defects are large, near or larger λ, the scattering process is called Mie scattering. This happens for typical fiber manufacturing defects such as rough interface between core-cladding, or macroscopic defects and impurities within the layers.

3.4  Nonlinear scattering effects

Two very important nonlinear scattering effects are Stimulated Brillouin Scattering (SBS) and Stimulated Raman Scattering (SRS). 
Both Brillouin Scattering (BS) and Raman Scattering (RS) (without the "stimulated" ) are nonlinear optical processes. Nonlinear optical processes are those that are not linearly proportional with the input power.
Both processes generate new light of different frequencies than the input. In other words, you put in frequency ω and light of a different frequencies ω' can be generated.

Brillouin scattering involves the fact that sound waves cause periodic index-of-refraction variation, acting like a grating (later we'll see that it is called Bragg grating). Light can be scattered from these gratings.
Raman scattering involves electronic vibration (which belong to the electronic states in the SiO2 molecule in glass for fiber).

Inside a fiber, these processes generate photons that are not part of the intended signals, hence can cause errors. However, just the BS and RS alone are NOT bad, as these processes produce very weak effects. The problem arises when the main signal power is sufficiently strong that "stimulation" happens: this is when these weakly generated photons are amplified and become strong, hence generating significant background noises. The amplification can happen to signals, and hence these noises are unintendedly become strong.
The main feature of both processes is that there is a threshold "pump" power, above which there is strong a SRS or SBS light coming out of the fiber. Hence, these processes limit how much signal power one can put into a fiber.

4. Dispersion effects

4.0 Introductory discussion: signal spectrum and PSD

Review concept: power spectral density and light wave spectrum.

If we have a cw laser signal that can be approximated as a pure single-frequency light wave:
                                                 E=(Subscript[E, 0]) (e^(i (β z-ω t)))
and we put the signal through a spectrum analyzer, what is its spectrum? (power spectral density).

If we chop this light with a very fast chopper, what does its spectrum look like?

4.1 Spectral decomposition of a pulse

Consider a signal: S[t], what is its power or energy spectrum (or spectral density)?
                                               P[ω]=(Underscript[Lim, T-> ∞])(1/T)(((|(∫_(-T/2)^(T/2))(e^(i ω t))S[t]dt|))^2) (if power signal)
                                               E[ω]=(((|(∫_(-∞)^∞)(e^(i ω t))S[t]dt|))^2)      (if energy signal)
The spectral component of the signal is:
                                                    a[ω]=(1/(Sqrt[2 π]))(∫_(-∞)^∞)(e^(i ω t))S[t]dt
And by FT theorem:
                                                    S[t]=(1/(Sqrt[2 π]))(∫_(-∞)^∞)a[ω](e^(-i ω t))dt  
which is a linear combination of harmonic (e^(-i ω t)) with spectrum a[ω]

Consider a pure wave  (e^(i (k z -(Subscript[ω, 0]) t))). Its spectrum is δ[ω-(Subscript[ω, 0])]
Suppose we have a shutter that can open and close really fast and let a light pulse of (e^(i (k z -(Subscript[ω, 0]) t))) of any arbitrary duration to pass through. What is the light wave spectral component? Let S[t] be the envelop signal, the pulse at z=0 is
                                                        S[t] (e^(-i (Subscript[ω, 0]) t))
The spectral component is:       A[ω]=(1/(Sqrt[2 π]))(∫_(-∞)^∞)(e^(i (ω-(Subscript[ω, 0])) t))S[t]dt                     
                                                  A[ω]=a[ω-(Subscript[ω, 0])]                                                  
This is NOT δ[ω-(Subscript[ω, 0])] any more.
Earlier, we look at a Gaussian pulse:
                                           E(z,t)=(e^(i (Subscript[ω, 0]) (((z (Subscript[n, eff]))/c)-t)))(e^(-((((c t-z (Subscript[n, eff])))^2)/((c^2) (τ^2)))))
We see that it is constructed from multi-spectral components of cw signal:
                                        E(z,t)=∫a(ω) (e^(i (k[ω] z - ω t)))dω
                                                   a(ω)=(1/(2 (Sqrt[π])))τ (e^(-(1/4) (τ^2) (Ω^2)))

Consider the following square pulse

a=1  ; n=1.45  ; ω=2 π*14 ;
Plot[ Re[(E^(-I  ω  t)) (UnitStep[t+a/2] -UnitStep[t-a/2])/a],{t,-1,1},PlotStyle-> {{RGBColor[1,0,0], Thickness[0.005]}},ImageSize-> {600,250},
AspectRatio->0.3 ]

What is its energy spectrum? (or energy spectral density)?

ω= . ;
(1/(2 π))(∫_(-τ/2)^(τ/2))(E^(I ω t))(E^(-I  ω0  t))\[DifferentialD]t

ω0=100. ;  τ=1  ;
LogPlot[(((((sin((1/2) τ (ω-ω0)))/(π (ω-ω0)))))^2),{ω,50,150}, PlotRange-> All, GridLines-> Automatic, PlotRange-> {10^-6,1}]

Per Fourier transform theorem, we can construct a square pulse
                                         (∫_(-∞)^∞)((sin((1/2) τ (ω-ω0)))/(π (ω-ω0)))(e^(i (k[ω] z -ω t)))dω

FourierTransform[((sin((1/2) τ Ω))/(π Ω)), Ω, t]

4.2 Propagation of an intensity modulated pulse

In the above example, we perform the superposition as if wave vector k is a constant of frequency ω. This is what we did to integrate: 
                            (∫_(-∞)^∞)(e^(-(1/4) (τ^2) (Ω^2))) (e^(i (k z - Ω t))) dω=(∫_(-∞)^∞)(e^(-(1/4) (τ^2) (Ω^2))) (e^(i ((ω/c)(Subscript[n, eff])[ω] z - Ω t))) dω
and we let (Subscript[n, eff])[ω]  be just a constant: 
                                               (∫_(-∞)^∞)(e^(-(1/4) (τ^2) (Ω^2))) (e^(i  Ω ((z/c)(Subscript[n, eff]) - t))) dΩ
In all media, k(ω) is never as simple as n ω/c. We can expand k(ω):
k(ω)= k((Subscript[ω, 0]))+(dk/(dω))(ω-(Subscript[ω, 0])) +(1/(2!))(((d^2)k)/((dω)^2))(((ω-(Subscript[ω, 0])))^2)+ ...     (3.1)
as we know, the inverse of (dk/(dω)) is called the group velocity (GV):
(Subscript[v, g])=((((dk/(dω))))^(-1))                                                                (3.2)
The second order term is called the dispersion term, and the term 2π(((d^2)k)/((dω)^2))is called the dispersion coefficient. There are other definitions and names of this term, such as group velocity dispersion (GVD), but all is proportional to the second order derivatives as we can see later.

4.3 Group velocity

If  k(ω)=n (ω/c) and n is constant, then the GV is c/n which is the same as phase velocity. The phase and the envelope move together as the first example above. If there is no dispersion term, it's still easy to see that the pulse maintains its shape, from Eq. (1.3)
E(x,t)=∫a(ω) (e^(i (k z - ω t)))dω           (3.3)

a(ω) (E^(I (k z - ω t))) /. k->(Subscript[k, 0])+(ω-(Subscript[ω, 0]))/(Subscript[v, g])

Which can be rewritten:
(e^(i z ((Subscript[k, 0])-((Subscript[ω, 0])/(Subscript[v, g]))))) (e^(i ω (  (z/(Subscript[v, g])) -t) )) a(ω).  The first factor has no ω dependence so, to insert the whole thing into Eq. (3.3)  
E(x,t)=(e^(i z ((Subscript[k, 0])-((Subscript[ω, 0])/(Subscript[v, g])))))∫(e^(i ω ((z/(Subscript[v, g])) -t) )) a(ω)dω           (3.4)
Let's plot this out. Again, use a Gaussian pulse. From Eq. (1.12)
Ef[ω]=(((e^(-(1/4) (τ^2) (Ω^2))) τ)/(2 (Sqrt[π])))

(E^(I ω (  (z/(Subscript[v, g])) -t) )) (((E^(-(1/4) (τ^2) (Ω^2))) τ)/(2 (Sqrt[π]))) /. ω->(Subscript[ω, 0])+Ω

Take out this term: (e^(i (-t+(z/(Subscript[v, g]))) (Subscript[ω, 0]))), we have:

(((E^(-(1/4) (τ^2) (Ω^2)+I (-t+(z/(Subscript[v, g]))) (Ω+(Subscript[ω, 0])))) τ)/(2 (Sqrt[π]))) /. (Subscript[ω, 0])->0

Now we can integrate as before

(τ/(2 (Sqrt[π])))(∫_(-∞)^∞)(E^(-(1/4) (τ^2) (Ω^2)+I Ω (-t+(z/(Subscript[v, g]))))) \[DifferentialD]Ω

Let's put back all phase terms to get the full expression:  
(e^(i z ((Subscript[k, 0])-((Subscript[ω, 0])/(Subscript[v, g])))))(e^(i (-t+(z/(Subscript[v, g]))) (Subscript[ω, 0])))(e^(-((((z-t (Subscript[v, g])))^2)/((τ^2) (Subscript[v, g]^2)))))
and get: (e^(i ((Subscript[k, 0]) z  - (Subscript[ω, 0]) t))) (e^(-((((z-t (Subscript[v, g])))^2)/((τ^2) (Subscript[v, g]^2)))   ))                                                     (3.5)
Note that the carrier (e^(i ((Subscript[k, 0]) z  - (Subscript[ω, 0]) t))) travels with the phase velocity:
                                         (Subscript[v, ph])=((Subscript[ω, 0])/(Subscript[k, 0]))=(c/(Subscript[n, eff])) 
while the envelop (e^(-((((z-t (Subscript[v, g])))^2)/((τ^2) (Subscript[v, g]^2)))   ))travels with the group velocity:
                                          (Subscript[v, g])=((dω)/(dk))

Efield[z_,t_,w0_,τ_,n_, VG_]:= Module[{k0 },
k0=n w0 /(2.99792458*10^10) ; (* unit is cm/s  *)
		                                Re[ (E^(-((((z-t VG))^2)/((τ^2) (VG^2))))) (E^(I (k0 z - w0 t)))]          ]

n=1.4 ;  w0=2 π  2*10^14 ; 
c=2.99792458*10^10  ;
tau1=3*10^(-14) ; tau2=3*10^(-14)1.4/2.2  ;
VG1=c/2.2 ;VG2=c /1.4;
allEplot={} ;
For[i=-30,i≤ 30,i++,
 t=i*1.*10^(-15) ;
		      x0= t*VG1;
		     delx=tau1*c/n ;
		       AppendTo[allEplot,
GraphicsArray[{{
			    Plot[Efield[x,t,w0,tau1,n, VG1],{x,x0-2*delx,x0+2*delx}
	 , PlotRange->{{-0.002,0.002},{-1,1}}, PlotPoints-> 50
	 , PlotStyle->{RGBColor[0.,0.,1],Thickness[0.001]}
,AspectRatio-> 0.1, Axes-> False
			               ]},
{
			    Plot[Efield[x,t,w0,tau2,n, VG2],{x,x0-2*delx,x0+2*delx}
	 , PlotRange->{{-0.002,0.002},{-1,1}}, PlotPoints-> 50
	 , PlotStyle->{RGBColor[1.,0.,0],Thickness[0.001]}
,AspectRatio-> 0.1, Axes-> False
			               ]}}
		                            ]
]
];

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

Observe and compare the pulses.

Carrierpart[x_,t_,w0_,τ_,n_, VG_]:= Module[{k0 },
				                                                                         k0=n w0 /(3*10^(10)) ; (* unit is cm/s  *)
		                                                               
		                                Re[  (E^(I (k0 x - w0 t)))]          ] ;
Envelopepart[x_,t_,w0_,τ_,n_, VG_]:= Module[{k0 },
				                                                                         k0=n w0 /(3*10^(10)) ; (* unit is cm/s  *)
					                                                               
		                              (E^(-((((x-t VG))^2)/((τ^2) (VG^2)))))        ]

allEplot={};
Do[{ AppendTo[allEplot,
			    Plot[{ Carrierpart[x,1*i*10^(-15),2 π 2*10^14,3*10^(-14),1.47,(3*10^10)/2.],
					Envelopepart[x,1*i*10^(-15),2 π 2*10^14,3*10^(-14),1.47,(3*10^10)/2.2]},
				   {x,-0.002,0.002}
			                     , PlotRange->{-1,1}
			,PlotStyle->{{RGBColor[0,0,1],Thickness[0.002]},
						                              {RGBColor[1,0,0],Thickness[0.004]}}
				, ImageSize->{700.,200}, AspectRatio-> 0.2
			]]} ,{i,-30,30}]

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

4.4 Dispersion

In real system, there is no such thing as "zero-dispersion everywhere". There can be zero-dispersion at one particular frequency, but not at any frequency.
Now we have:
E(x,t)=∫Ef(ω) (e^(i (k [ω] z - ω t)))dω    from Eq. (3.3), where
k[ω] is now has higher order nonlinear dependence on ω.
Dispersion can cause many different effects. Let's examine them

4.4.1 Pulse broadening effect

Envelop expression

Ef(ω) (E^(I (k z - ω t))) /. k-> (Subscript[k, 0])+(ω-(Subscript[ω, 0]))/(Subscript[v, g]) + (d/2) (ω-(Subscript[ω, 0]))^2

PowerExpand[%]

(E^(I (z ((1/2) d (((ω-(Subscript[ω, 0])))^2)+((ω-(Subscript[ω, 0]))/(Subscript[v, g]))+(Subscript[k, 0]))-t ω))) Ef(ω) /. (Subscript[ω, 0])-> ω-Ω

(E^(I (z (((d (Ω^2))/2)+(Ω/(Subscript[v, g]))+(Subscript[k, 0]))-t ω))) Ef(ω) /.  ω -> (Subscript[ω, 0])+Ω

Which can be rewritten:
 (e^(i (z (Subscript[k, 0])-t (Subscript[ω, 0])))) (e^(i Ω( (z/(Subscript[v, g]))-t ))) (e^(i z (d/2) (Ω^2)))Ef(Ω+(Subscript[ω, 0]))            (3.6)

The first factor which is the cw wave of the central wavelength has no ω dependence.  The second  is simply the envelope term as usual. Only the third term is interesting which is the dispersion term. Does it look like diffusion to you? Indeed, a way to think about the pulse dispersion effect is to imagine diffusion,  but the broadening term has an imaginary component. Mathematically they are similar. Later on, we'll see that it 's also quite similar to Fresnel diffraction.

To insert the whole thing into Eq. (3.6)  
	E(z,t)= (e^(i (z (Subscript[k, 0])-t (Subscript[ω, 0]))))∫(e^(i Ω( (z/(Subscript[v, g]))-t )))(e^(i z (d/2) (Ω^2))) Ef[Ω+(Subscript[ω, 0])] dΩ           (3.7)
Let's plot this out, again, use a Gaussian pulse. From Eq. (1.12)
Ef[Ω+(Subscript[ω, 0])]=(((e^(-(1/4) (τ^2) (Ω^2))) τ)/(2 (Sqrt[π])))

Now we can integrate as before

(1/(2 (Sqrt[π])))(∫_(-∞)^∞)(E^(-(1/4) (τ^2) (Ω^2)+I Ω (-t+(z/(Subscript[v, g])))))(E^(I z (d/2) (Ω^2))) τ\[DifferentialD]Ω

The answer is: (1/(Sqrt[(τ^2)-2 i d z])) (e^(-((i (((z-t (Subscript[v, g])))^2))/((i (τ^2)+2 d z) (Subscript[v, g]^2))))) 
Say, doesn't that look like the diffusion form? Yes, it does!

denominat=Expand[4 (-(1/2) I d z+((τ^2)/4))((1/2) I d z+((τ^2)/4))]

The exponent part is
((-((1/2) I d z+((τ^2)/4))((((( z)/(Subscript[v, g])) -t))^2))/((d^2) (z^2)+((τ^4)/4)))= ((-(1/2) I d z((((z/(Subscript[v, g])) -t))^2))/((d^2) (z^2)+((τ^4)/4))) -((((τ^2)/4)((((( z)/(Subscript[v, g])) -t))^2))/((d^2) (z^2)+((τ^4)/4)))
and the whole exponent term is:
(e^((-(1/2) i d ((z((( z)/(Subscript[v, g])) -t))^2))/((d^2) (z^2)+((τ^4)/4))))  (e^(-((((τ^2)/4)((((z/(Subscript[v, g])) -t))^2))/((d^2) (z^2)+((τ^4)/4)))))
The first term is the phase chirp term, the second shows the pulse broadening.
we still have to take care of the denominator:
(τ/(Sqrt[-2 i d z+(τ^2)]))

ComplexExpand[(τ/(Sqrt[-2 I d z+(τ^2)]))]

TrigToExp[%]

(( τ)/(Sqrt[|(τ^2)-2 i d z|]))=((τ )/(((4 (d^2) (z^2) + (τ^2)))^(1/4)))

Now we can write the whole expression:  
 (e^(-(1/2) i Arg[-2 i d z+(τ^2)])) (e^((-(1/2)i d ((z((( z)/(Subscript[v, g])) -t))^2))/((d^2) (z^2)+((τ^4)/4)))) ((τ )/(((4  (d^2)  (z^2) + (τ^2)))^(1/4))) (e^(-((((τ^2)/4)((((( z)/(Subscript[v, g])) -t))^2))/((d^2) (z^2)+((τ^4)/4)))))  
The first 2 terms are the phase term and the last 2 are the envelope term.
Phase term:

(E^(-(1/2) I Arg[-2 I d z+(τ^2)]))(E^((-(1/2) I d ((z((( z)/(Subscript[v, g])) -t))^2))/((d^2) (z^2)+((τ^4)/4))))

We can first look at the envelope term alone to see the pulse spreading effect. We define an envelope function

Envelope[x_,t_,w0_,τ_,n_, VG_,d_]:= (τ/(((4 (d^2) (x^2)+(τ^4)))^(1/4))) (E^(-((((τ^2)/4)((((( x)/VG) -t))^2))/((d^2) (x^2)+((τ^4)/4)))))

allEplot={};
w0=2 π 2*10^14; tau=3*10^(-14) ; VG=(3*10^10)/2. ;
steptime=0.25*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[(Envelope[x,i*steptime,w0,tau,1.47,VG,da])^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]}
			                      , ImageSize->{700.,200}, AspectRatio-> 0.3]]} ,{i,0,nplot}]

Fig. 3.3.1

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

Notice that in the above plot, we plot envelope square to reflect the pulse intensity, NOT the electric field envelope.
The value of dispersion coefficient is exaggerated quite a bit in this example in order to illustrate the pulse broadening effects. Typical material dispersion is much smaller than this example. But we have to do this to see the pulse width and its broadening on the same scale as the travel distance. Otherwise, realistic dispersion in many media will require a large distance relative to the pulse width for this type of pulse to broaden significantly.

4.4.2 Two pulses with different spectral width simulation

allEplot={};
w0=2 π 2*10^14; VG=(3*10^10)/2. ;
 tau1=2*10^(-14) ; tau2=5*10^(-14)  ;
steptime=0.5*10^(-13) ;
nplot=50 ;
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[{(Envelope[x,i*steptime,w0,tau1,1.47,VG,da])^2,
					                  (Envelope[x,i*steptime,w0,tau2,1.47,VG,da])^2 *( tau1/tau2)},
				                         {x,xmin[i,tau2,VG,steptime,da],xmax[i,tau2,VG,steptime,da] }

			,PlotRange->{{0.,xmax[nplot,tau2,VG,steptime,da]},{0,1}} 
						                      
				,PlotStyle->{{RGBColor[0,0,1],Thickness[0.004]},
						                              {RGBColor[1,0,0],Thickness[0.004]}}
			                      , ImageSize->{700.,200}, AspectRatio-> 0.25]]} ,{i,0,nplot}]

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

What do you observe here? Why so?

4.4.3 Two pulses with different initial delay

allEplot={};
w0=2 π 2*10^14; VG=(3*10^10)/2. ;
 tau1=2*10^(-14) ; tau2=tau1  ;delay=10^(-13);
steptime=0.5*10^(-13) ;
nplot=50 ;
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+delay) *VG+
		  3*Sqrt[tau^2+4*d*i*steptime*VG]*VG ;
Do[{ AppendTo[allEplot,
			    Plot[{(Envelope[x,i*steptime,w0,tau1,1.47,VG,da])^2,
					                  (Envelope[x,i*steptime+delay,w0,tau2,1.47,VG,da])^2
					,2*(Envelope[x,i*steptime,w0,tau1,1.47,VG,da])^2
				+ 2*(Envelope[x,i*steptime+delay,w0,tau2,1.47,VG,da])^2
					 },
				                         {x,xmin[i,tau2,VG,steptime,da],xmax[i,tau2,VG,steptime,da]}
				

			,PlotRange->{{0.,xmax[nplot,tau2,VG,steptime,da]},{0,2}} 
						                      
				,PlotStyle->{{RGBColor[0,0,1],Thickness[0.004]},
						                              {RGBColor[1,0,0],Thickness[0.004]}
			,{RGBColor[0,0,0],Thickness[0.004]}}
			                      , AspectRatio-> 0.25]]

     } ,{i,0,nplot}]

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

What do you observe here? Why so? What is the implication if the two pulses represent bits of information? After certain distance, what happens to your signals?

4.5 Chirp

4.5.1 Illustrations of chirp

We learnt in example 2 that a transform-limited broader pulse has a narrower spectrum. The plot above does not mean that the pulse spectrum becomes narrower and narrower as the pulse broadens as it travels. 
We can see what's happening by looking at the oscillatory (wave) component, which, in full  
 (e^(i((Subscript[k, 0]) z  - (Subscript[ω, 0]) t))) (e^(-(1/2) i Arg[-2 i d z+(τ^2)]))(e^((-(1/2) i d ((z((( z)/(Subscript[v, g])) -t))^2))/((d^2) (z^2)+((τ^4)/4))))
It's hard to see the oscillatory component of the electric field on the same scale as the pulse broadening and the travel distance. So, we will look at the pulse by traveling along with it! In other words, we move our frame of reference along the pulse at the same speed as the VG. To do this, we need to introduce a new variable:

(E^(I ((Subscript[k, 0]) z  - (Subscript[ω, 0]) t))) (E^(-(1/2) I Arg[-2 I d z+(τ^2)]))(E^((-(1/2) I d z ((((( z)/(Subscript[v, g])) -t))^2))/((d^2) (z^2)+((τ^4)/4)))) /. z->s+ (Subscript[v, g]) t

FullSimplify[%]

It doesn't look that pretty, but we can see what's going on. Now,let's define a carrier term:

Carrierterm[s_,t_,w0_,τ_,n_, VG_,d_]:= Module[{k0},
                                                           k0=n w0 /(3*10^(10)) ; (* unit is cm/s  *)
Re[(E^(-(1/2) I (Arg[(τ^2)-2 I d (s+t VG)]-2 k0 (s+t VG)+((d (s^2) (s+t VG))/((VG^2) (((τ^4)/4)+(d^2) (((s+t VG))^2))))+2 t w0)))]
	                                                                                   	 ] ;

We also need to define a traveling envelope:

d= .;VG= .;
(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)))))  /. z->s + VG *t

FullSimplify[%]

EnvelopeTravel[s_,t_,w0_,τ_,n_, VG_,d_]:=
                                          (((E^(-(((s^2) (τ^2))/((VG^2) (4 (d^2) (((s+t VG))^2)+(τ^4)))))) τ)/(((4 (d^2) (((s+t VG))^2)+(τ^4)))^(1/4))) ;

allEplot={};
w0=2 π 2*10^14; tau=2*10^(-14) ; VG=(3*10^10)/1.6 ;
steptime=  10^(-14) ;
nplot=50 ;
da=1.5*tau^2/(nplot*steptime*VG) ;
smin[i_,tau_,VG_,steptime_,d_]:=-3*Sqrt[tau^2+4*d*i*steptime*VG]*VG ;
smax[i_,tau_,VG_,steptime_,d_]:=
		                                                            3*Sqrt[tau^2+4*d*i*steptime*VG]*VG ;

Do[{ AppendTo[allEplot,
			    Plot[EnvelopeTravel[s,i*steptime,w0,tau,1.47,VG,da]
				              *Carrierterm[s,i*steptime,w0,tau,1.47,VG,da],
				                         {s,smin[i,tau,VG,steptime,da],smax[i,tau,VG,steptime,da] }
			                     , PlotRange->{{smin[nplot,tau,VG,steptime,da],smax[nplot,tau,VG,steptime,da]},{-1,1}} 
                                           , PlotStyle->{RGBColor[0.,0.8,0]}
, PlotPoints->30
			                       ,AspectRatio-> 0.2]]} ,{i,0,nplot}]

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

What we observe here is that the "red" portion of the pulse seems to be in front of the "blue" portion of the pulse. Another clear look is to plot the phase alone (without the envelope)

allEplot={};
Off[General::"spell1"]  ;
w0=2 π 2*10^14; tau=2*10^(-14) ; VG=(3*10^10)/1.6 ;
steptime=  10^(-14) ;
nplot=50 ;
da=1.5*tau^2/(nplot*steptime*VG) ;
smin[i_,tau_,VG_,steptime_,d_]:=-3*Sqrt[tau^2+4*d*i*steptime*VG]*VG ;
smax[i_,tau_,VG_,steptime_,d_]:=
		                                                            3*Sqrt[tau^2+4*d*i*steptime*VG]*VG ;
Do[{ AppendTo[allEplot,
			    Plot[Carrierterm[s,i*steptime,w0,tau,1.47,VG,da],
				                         {s,smin[nplot,tau,VG,steptime,da],smax[nplot,tau,VG,steptime,da] }
			                     , PlotRange->{{smin[nplot,tau,VG,steptime,da],smax[nplot,tau,VG,steptime,da]},{-1,1}} 
, PlotStyle->{RGBColor[0.,0.7,0]}
				, PlotPoints->100
			          ,AspectRatio-> 0.2]]} ,{i,0,nplot}]

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

Clearly, the carrier here is not a pure single frequency. It actually contains many components of the packet. But by plotting them all out here, we can see how the frequency chirp clearly happens as they travel.

4.5.2 Instantaneous frequency

Instantaneous frequency:
Here is where the concept of "instantaneous" frequency above is useful. The pulse now has different frequencies at different parts of the pulse. If we can find the "instantaneous" frequency anywhere in the pulse, we should find that the pulse is indeed "chirped".
Let's apply this concept to our phase function in the dispersion example above

(E^(I ((Subscript[k, 0]) z  - (Subscript[ω, 0]) t))) (E^(-(1/2) I Arg[-2 I d z+(τ^2)]))  (E^((-(1/2) I d z ((((( z)/(Subscript[v, g])) -t))^2))/((d^2) (z^2)+((τ^4)/4))))

Carrier[x_,t_,w0_,τ_,n_, VG_,d_]:= Module[{k0},
                                                           k0=n w0 /(3*10^(10)) ; (* unit is cm/s  *)
		Re[(E^(-(1/2) I Arg[-2 I d x+(τ^2)]-((I d x (((-t+(x/VG)))^2))/(2 ((d^2) (x^2)+((τ^4)/4))))+I (x k0-t w0)))]            ] ;
		
Phase=(-(1/2) I Arg[-2 I d x+(τ^2)]-((I d x (((-t+(x/(Subscript[v, g]))))^2))/(2 ((d^2) (x^2)+((τ^4)/4))))+I (x (Subscript[k, 0])-t (Subscript[ω, 0])))/I

Phase=FullSimplify[%] /. x->z

-(∂_t) Phase

FullSimplify[%]

The above expression shows that the "instantaneous" frequency is not always (Subscript[ω, 0]) as we expect, but deviates from it by a term:
((4 d z (z-t (Subscript[v, g])))/(((τ^4)+4 (d^2) (z^2)) (Subscript[v, g]))).
The parameter z-t (Subscript[v, g]) is the position within the pulse envelop. We can define:
 z-t (Subscript[v, g]) = ξ
then: -((4 d z )/((4 (d^2) (z^2)+(τ^4)) (Subscript[v, g])))  ξ 
In other words, along the pulse envelope, the frequency deviates by an amount that is proportional to that the distance of that position to the point of the pulse center that moves with the group velocity. The deviation gets worse with distance.
To plot thing out more accurately, we should write the full expression:
(e^(i (z (Subscript[k, 0])-t (Subscript[ω, 0]))-(1/2) i Arg[-2 i d z+(τ^2)])) (e^((-(1/2) i d ((z((z/(Subscript[v, g])) -t))^2))/((d^2) (z^2)+((τ^4)/4)))) ((τ )/(((4  (d^2)  (z^2) + (τ^2)))^(1/4))) (e^(-((((τ^2)/4)((((( z)/(Subscript[v, g])) -t))^2))/((d^2) (z^2)+((τ^4)/4)))))
and take only the exponent term:
i (z (Subscript[k, 0])-t (Subscript[ω, 0]))-(1/2) i Arg[-2 i d z+(τ^2)] + ((-(1/2) i d ((z((( z)/(Subscript[v, g])) -t))^2))/((d^2) (z^2)+((τ^4)/4))) -((((τ^2)/4)((((( z)/(Subscript[v, g])) -t))^2))/((d^2) (z^2)+((τ^4)/4)))+ Log[((τ )/(((4  (d^2)  (z^2) + (τ^2)))^(1/4)))] 
We can drop the time-independent term: Log[((τ )/(((4  (d^2)  (z^2) + (τ^2)))^(1/4)))] and take out the -i term:
-(z (Subscript[k, 0])-t (Subscript[ω, 0])) + (1/2)  Arg[-2 i d z+(τ^2)]+(((1/2)  d ((z((( z)/(Subscript[v, g])) -t))^2))/((d^2) (z^2)+((τ^4)/4))) - i ((((τ^2)/4)((((( z)/(Subscript[v, g])) -t))^2))/((d^2) (z^2)+((τ^4)/4)))

Instantomega[x_,t_,w0_,τ_,VG_,d_]:=Abs[w0-((4 d x (x-t VG))/((4 (d^2) (x^2)+(τ^4)) VG))] ;
Delomega[x_,t_,w0_,τ_,VG_,d_]:=-((4 d x (x-t VG))/((4 (d^2) (x^2)+(τ^4)) VG)) ;
DelomegaTravel[s_,t_,w0_,τ_,VG_,d_]:=-((4 d s (s+t VG))/((4 (d^2) (((s+t VG))^2)+(τ^4)) VG)) ;

allChirp={};
w0=2 π 2*10^14; tau=2*10^(-14) ; VG=(3*10^10)/1.6 ;
steptime=  10^(-14) ;
nplot=50 ;
da=1.5*tau^2/(nplot*steptime*VG) ;
smin[i_,tau_,VG_,steptime_,d_]:=-3*Sqrt[tau^2+4*d*i*steptime*VG]*VG ;
smax[i_,tau_,VG_,steptime_,d_]:=
		                                                            3*Sqrt[tau^2+4*d*i*steptime*VG]*VG ;

Do[{ AppendTo[allChirp,
			    Plot[ DelomegaTravel[s,i*steptime,w0,tau,VG,da],
				                    {s,smin[i,tau,VG,steptime,da],smax[i,tau,VG,steptime,da] }
			                     , PlotRange->{{smin[nplot,tau,VG,steptime,da],smax[nplot,tau,VG,steptime,da]},{-1*10^(15),1*10^(15)}} 
                                           , PlotStyle->{RGBColor[1.,0.0,0]}
, PlotPoints->30
			                  , AspectRatio-> 0.25]]} ,{i,0,nplot}]

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

Clearly, without higher-order (nonlinear) k-ω term, this chirp would not happen. The net spectral content of the pulse remains the same, only the phase has shifted relatively with each other. As a result, the spectral width is the same, but the pulse width has broadened dramatically. The pulse is no longer "transform limited".

4.6 Phase shift term

This observation brings us back to the question of pulse spectrum. As we discussed earlier, Gaussian pulse that we studied has a simple real spectral term: Ef[ω'] =(((E^(-(1/4) (τ^2) (Ω^2))) τ)/(2 (Sqrt[π])))
Obviously, this can not be the case now. The amplitude of each spectral component is the same as above, but the phase is not. From the discussion above, we obtain the term for individual spectral component: 
 (e^(i  (x  (Subscript[k, 0])-t  (Subscript[ω, 0])))) (e^(i Ω( (x/(Subscript[v, g]))-t ))) (e^(i x  (d/2)  (Ω^2)))Ef[Ω+(Subscript[ω, 0])]  
from which we can drop the constant carrier term  (e^(i  (x  (Subscript[k, 0])-t  (Subscript[ω, 0])))) to have only the phase term:  
(e^(i Ω( (x/(Subscript[v, g]))-t ))) (e^(i x  (d/2)  (Ω^2))) 
Now, we can easily recognize that the group velocity phase term  (e^(i Ω( (x/(Subscript[v, g]))-t ))) is only a linear component, which DOES NOT change a pulse shape. To change a pulse shape, it requires HIGHER-ORDER terms (at least 2nd) in the phase:  (e^(i  x  (d/2)  (Ω^2))).

5. Dispersion in fiber (exclude polarization)

5.1 Summary and Overview

Section 4 above discussion shows how dispersion affects pulse shape.
                          E(z,t)= (e^(i (z (Subscript[k, 0])-t (Subscript[ω, 0]))))∫(e^(i Ω( (z/(Subscript[v, g]))-t )))(e^(i z (d/2) (Ω^2))) A[Ω+(Subscript[ω, 0])] dΩ 
We apply to a Gaussian pulse and obtain:
                       (e^(-(1/2) i Arg[-2 i d z+(τ^2)])) (e^((-(1/2)i d ((z((( z)/(Subscript[v, g])) -t))^2))/((d^2) (z^2)+((τ^4)/4)))) ((τ )/(((4  (d^2)  (z^2) + (τ^2)))^(1/4))) (e^(-((((τ^2)/4)((((( z)/(Subscript[v, g])) -t))^2))/((d^2) (z^2)+((τ^4)/4)))))  
We see that the pulse power envelop is: 
                                               (((τ^2) )/(((4  (d^2)  (z^2) + (τ^2)))^(1/2))) (e^(-((2((((( z)/(Subscript[v, g])) -t))^2))/((4 (((d^2) (z^2))/(τ^2))+(τ^2))))))  
Although the result was obtained for Gaussian, the general property applies to others:
The pulse width increases as  (Sqrt[4 (((d^2) (z^2))/(τ^2))+(τ^2)]) or (Sqrt[4 (d^2) (z^2)(Ω^2)+(τ^2)]) ≈d z Ω
This means that the pulse broadens ~ linearly vs distance vs the spectral width
The spectral width is, inturn related to the bit rate. Generally, we can use these simple rules for estimation: Let the pulse duration be τ, if we require that the pulse must be separated at least by τ also, then the maximum bit rate is:
                                                       (Subscript[B, T])=(1/(2τ )) 
However, this is a bit too strict, we can use a different measure, if we simply assume that the transmitted pulse has an approximate Gaussian duration σ, then the maximum bit rate can be shown to be:
                                                      (Subscript[B, T])=(0.2/σ)      
Hence, the transmission capability, defined as distance x bandwidth (bitrate) is limited by dispersion. (Conversely, given a fiber dispersion, we can calculate σ for a given length, and hence the max bitrate).
There are different definitions and units to express the dispersion coefficients:  ps/nm . km or (cm^(-1))/(THz^2)  depending on whether we take derivative vs. wavelength or frequency or mix. Consider the expression:  d z Ω, if we express Ω (pulse bandwidth) in nm an z in km, then the unit for d is: ps/nm . km,  which is convenient to estimate how many ps pulse spread per unit of wavelength bandwidth in nm per km.

5.1.1. Material Dispersion

Any materials have intrinsic dispersion.

5.1.2 Modal dispersion or waveguide dispersion

Already discussed

5.1.3 Intermodal dispersion (multimode)

Already discussed

5.2 Fiber dispersion considerations and design

The total dispersion is simple a sum of all other dispersions.
                                                (Subscript[D, T])=(Subscript[D, M])+(Subscript[D, WG])
where (Subscript[D, WG]) implicitly contains both modal and inter-modal dispersion (for multimode only)
The dispersion of a fiber can be designed using optical materials and then engineering of index profile. The major categories are:
1- Dispersion shifted
2- Dispersion-flattened
3- Non-zero Dispersion shifted and Negative Dispersion shifted