Light Propagation
Part 2: Temporal Behavior

ECE 5358/6358                              han q le - copyrighted
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Segment 2

5. Propagation with quadratic or higher order dispersion

In real system, there is no such thing as "zero-higher-order dispersion everywhere". There can be zero-higher-order dispersion at one particular frequency, but not at all frequency. If we include the second order dispersion, we will see that it can cause quite pronounced effects with more significant consequence than for the case discussed above with GV.
Let’s take a look:
-Example 1

5.1 Pulse broadening effect

Now we will substitute to include higher order terms of k[ω] expansion:

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

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

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

exp(I Expand[(z (((d (Ω^2))/2)+(Ω/(Subscript[v, g]))+(Subscript[k, 0]))-t ((Subscript[ω, 0])+Ω))])

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

{exp(I ((Subscript[k, 0]) z-t (Subscript[ω, 0]))),exp(I ((1/2) d (Ω^2) z+FullSimplify[((Ω z)/(Subscript[v, g]))-t Ω]))}

Which can be rewritten: 
                  (e^(i ((Subscript[k, 0]) z-t (Subscript[ω, 0])))) (e^(i   Ω ((z/(Subscript[v, g]))-t))) (e^(i (1/2) d (Ω^2) z))             (5.1.1)
We are familiar with the 1st two terms, which are the carrier and group velocity terms. The 3rd term (e^(i (1/2) d (Ω^2) z)) is the dispersion term and note that it is quadratic in Ω. Because of this quadratic term, does it look like diffusion? Indeed, a way to think about the pulse dispersion effect is to imagine diffusion. Mathematically they are similar. Later on, we’ll see that it’s also quite similar to Fresnel diffraction. We will see the consequence of this quadratic term.

To insert the whole thing of (5.1.1) into Eq. (2.3.12): 
                 E[z;t]=(1/(Sqrt[2 π]))(∫_(-∞)^∞)a[Ω]  (e^(i (k[ω] z - ω t))) dω       
               E[z;t]=(1/(Sqrt[2 π]))(e^(i ((Subscript[k, 0]) z-t (Subscript[ω, 0]))))  (∫_(-∞)^∞)a[Ω]   (e^(i  Ω ((z/(Subscript[v, g]))-t))) (e^(i (1/2) d (Ω^2) z)) dΩ          (5.1.2a)
               E[z;t]=(1/(Sqrt[2 π]))(e^(i ((Subscript[k, 0]) z-t (Subscript[ω, 0]))))  (∫_(-∞)^∞) (a[Ω] (e^(i (1/2) d (Ω^2) z)) )  (e^(i   Ω ((z/(Subscript[v, g]))-t)))  dΩ          (5.1.2b)
              E[z;t]=(e^(i ((Subscript[k, 0]) z-t (Subscript[ω, 0])))) ℱ[a[Ω] (e^(i (1/2) d (Ω^2) z)),Ω,((z/(Subscript[v, g]))-t)]                          (5.1.2c) 
what this means is to take the Fourier transform of function a[Ω] (e^(i (1/2) d (Ω^2) z)) with respect to variable Ω for output variable (z/(Subscript[v, g]))-t. 
Apply the equation to a Gaussian pulse:
                 E[z;t]=(1/(2(Sqrt[π])))τ(e^(i ((Subscript[k, 0]) z-t (Subscript[ω, 0]))))  (∫_(-∞)^∞)(e^(-(1/4) (τ^2) (Ω^2))) (e^(i (1/2) d (Ω^2) z)) (e^(i   Ω ((z/(Subscript[v, g]))-t)))  dΩ      (5.1.3)

k=. ; ω= .;  τ=. ; z=. ; t=  .;
(1/(2(Sqrt[π])))τ(∫_(-∞)^∞)(E^(-(1/4) ((τ^2)-2 I  d  z )  (Ω^2)))  (E^(I   Ω u))  \[DifferentialD]Ω

(E^(-((u^2)/((τ^2)-2 I d z)))) /. u-> ((z/(Subscript[v, g]))-t)

Hence, the solution is:        E[z,t]=((τ )/(Sqrt[(τ^2)-2 i d z])) (e^(i ((Subscript[k, 0]) z-t (Subscript[ω, 0]))))  (e^(-(((((z/(Subscript[v, g]))-t))^2)/((τ^2)-2 i d z))))     (5.1.4)

Let’s compare this with the 1-D spatial Gaussian beam we studied back in Lect Set 1: (please keep in mind that we are comparing temporal Gaussian to spatial Gaussian, which are not the same. We only want to show how similar the mathematical formulism is for both):
                   E[x,z,t]=A (( 1)/(Sqrt[1+((2 i z)/((w^2) (Subscript[k, n])))]))   (e^(i z (Subscript[k, n])-((x^2)/((w^2)+((2 i z)/(Subscript[k, n])))))) (e^(-i ω t))    1(7.2.6)
                 E[x,z,t]=A (( w)/(W[z]))   (e^(i (z (Subscript[k, n])-ω t)))  (e^(-((x^2)/((W[z])^2))))               1(7.2.7a)
Note how similar they are. One is a Gaussian with respect to x, which is spatial, the other is with respect to t, which is temporal. But both have the same form of Gaussian width that increases with travel distance z.

why do we always study Gaussian signal: spatial Gaussian beam, temporal Gaussian beam...?

We can study other pulse shape also, just that the mathematics may not be as neat!

Plot[UnitBox[x],{x,-1,1},Filling->Axis]

Step 1: Fourier transform to get the pulse spectral component function.

FourierTransform[UnitBox[t/τ],t,Ω]

{Plot[sinc(((τ Ω)/2)) /. τ->10, {Ω,-5,5}, PlotRange-> All, Filling->{1-> {Axis,Hue[0,0.6,1,1]}}, Frame-> True] 
,Plot[((sinc(((τ Ω)/2)))^2) /. τ->10, {Ω,-5,5}, PlotRange-> All, Filling->{1-> {Axis,Hue[0.35,1,1,1]}}, Frame-> True] }

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

Not very neat, isn’t it. As it turns out, not many pulse shapes give us neat and convenient form in GVD dispersion effect.

Plot[(1/(Cosh[x])),{x,-5,5},Filling->{1-> {Axis,Hue[0.15,1,0.8,1]}}, Frame-> True]

FourierTransform[(1/(Cosh[t/τ])),t,Ω]

What is (∫_(-∞)^∞)sech(((π τ Ω)/2))  (e^(i   Ω ((z/(Subscript[v, g]))-t))) (e^(i (1/2) d (Ω^2) z)) dΩ?
well... not simple.

Plot[UnitBox[t/τ] (1-Abs[2 t/τ]) /. τ-> 1,{t,-2,2}, PlotRange-> All]

FourierTransform[ UnitBox[t/τ] (1-Abs[2 t/τ]) ,t,Ω]

Since τ >0, θ(-τ)-θ(τ) = -1

-((Sqrt[(2/π)])/(τ (Ω^2)))FullSimplify[ (E^(-(1/2) I τ Ω)) (((-1+(E^((I τ Ω)/2))))^2) ]

Plot[((4 (Sqrt[(2/π)]) (sin^2)(((τ Ω)/4)))/(τ (Ω^2))) /. τ-> 1, {Ω,-50,50}, PlotRange-> All]

Back to Gaussian pulse

The short answer is:         
         E[z;t]=(1/(Sqrt[1-2 i d z/(τ^2)]))(e^(i ((Subscript[k, 0]) z-t (Subscript[ω, 0]))))  (e^(-(((((z/(Subscript[v, g]))-t))^2)/((τ^2) -2 i d z))))          (5.1.3)
Doesn't this look like the diffusion form? Yes, it does!

We see that the expression here:
         E[z;t]=(1/(Sqrt[1-2 i d z/(τ^2)]))(e^(i ((Subscript[k, 0]) z-t (Subscript[ω, 0]))))  (e^(-(((((z/(Subscript[v, g]))-t))^2)/((τ^2) -2 i d z))))         
 is similar to:
          E[z;t]= (e^(i ((Subscript[k, 0]) z-t (Subscript[ω, 0]))))(e^(-(((((z/(Subscript[v, g]))-t ))^2)/(τ^2))))       (3.2.5)
except that the pulse width is not a real constant τ, but a complex number  (Sqrt[(τ^2) -2 i d z]). 
Let’s separate the phase from the amplitude:

-(((((z/(Subscript[v, g]))-t))^2)/((τ^2) -2 I d z))== ((- Expand[(u^2) ((τ^2) +2 I d z)])/(Expand[((τ^2) -2 I d z)((τ^2) +2 I d z)]))

Hence:  (e^(-(((((z/(Subscript[v, g]))-t))^2)/((τ^2) -2 i d z))))=  (e^(-i ((2  d  z)/(4 (d^2) (z^2)+(τ^4)))((((z/(Subscript[v, g]))-t))^2)))   (e^(-((τ^2)/(4 (d^2) (z^2)+(τ^4)))((((z/(Subscript[v, g]))-t))^2)))  (5.1.4)
The first term is the phase chirp term:
              (e^(-i ((2  d  z)/(4 (d^2) (z^2)+(τ^4)))((((z/(Subscript[v, g]))-t))^2)))                     (5.1.5)   
The second shows the pulse broadening:

(e^(-(1/(4 (((d^2) (z^2))/(τ^2))+(τ^2)))((((z/(Subscript[v, g]))-t))^2)))                (5.1.6a)
Note: we can define the “bandwidth” as: BW=(2/τ)
Then (5.1.6) becomes:
                     (e^(-(1/((d^2) (z^2)(BW^2)+(τ^2)))((((z/(Subscript[v, g]))-t))^2)))        (5.1.6b)
It is clear that 3 factors of the term (d^2) (z^2)(BW^2)can effect pulse broadening: d, obviously, the quadratic dispersion coefficient, z: the travel distance, and BW, the pulse bandwidth.

We still have to take care of the factor:
                          (1/(Sqrt[1-2 i d z/(τ^2)]))              (5.1.7)

ComplexExpand[(1/(Sqrt[1-2 I d z/(τ^2)]))]

TrigToExp[%]

Now we can write the whole expression:  
   E[z;t]=(e^(i ((Subscript[k, 0]) z-t (Subscript[ω, 0])))) ( (e^(-(1/2) i arg(1-((2 i d z)/(τ^2))))) (e^(-i ((2  d  z)/(4 (d^2) (z^2)+(τ^4)))((((z/(Subscript[v, g]))-t))^2))) )( (1/(((((4 (d^2) (z^2))/(τ^4))+1))^(1/4)))  (e^(-((τ^2)/(4 (d^2) (z^2)+(τ^4)))((((z/(Subscript[v, g]))-t))^2))))   (5.1.8)
Besides the carrier, The 2 terms in the first bracket are the phase terms and the 2 in the last bracket are the envelope terms.

The dispersion affects both the phase and envelope term, but with different consequences. Dispersion can cause the pulse spreading effect on the envelope, resulting in change of the pulse shape (broadening or narrowing). Dispersion can also cause the phase modulation effect, resulting chirp of the pulse instantaneous frequency. We will see these effects in the follow.

First, we can look at the envelope term alone to see the pulse shape changing effect. We define an envelope function:         
                                        S[z;t]=( (1/(((((4 (d^2) (z^2))/(τ^4))+1))^(1/4)))  (e^(-((τ^2)/(4 (d^2) (z^2)+(τ^4)))((((z/(Subscript[v, g]))-t))^2))))    (5.1.9)

Example 1 tag

Demo: GVD pulse broadening

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.

5.2 Illustration of 2 pulses

5.2 Two pulses with different spectral width simulation

Demo: 2 Pulses of different pulsewidth

What do you observe here? Why so?

5.3 Two pulses with different initial delay

Demo: 2 Pulses of different initial delay

What do you observe here? Why so?

5.3 Chirp

5.3.1 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. 
The pulse spectrum never changes. Dispersion only reshuffle the phase of various pulse spectral components.
We can see what's happening by looking at the oscillatory (wave) component, which, in full  
   E[z;t]=(e^(i ((Subscript[k, 0]) z-t (Subscript[ω, 0])))) ( (e^(-(1/2) i arg(1-((2 i d z)/(τ^2))))) (e^(-i ((2  d  z)/(4 (d^2) (z^2)+(τ^4)))((((z/(Subscript[v, g]))-t))^2))) )( (1/(((((4 (d^2) (z^2))/(τ^4))+1))^(1/4)))  (e^(-((τ^2)/(4 (d^2) (z^2)+(τ^4)))((((z/(Subscript[v, g]))-t))^2))))   (5.3.1)
                  (e^(i ((Subscript[k, 0]) z-t (Subscript[ω, 0])))) ( (e^(-(1/2) i arg(1-((2 i d z)/(τ^2))))) (e^(-i ((2  d  z)/(4 (d^2) (z^2)+(τ^4)))((((z/(Subscript[v, g]))-t))^2))) )                  (3.3.9a)
                  Φ[z;t]=((Subscript[k, 0]) z-t (Subscript[ω, 0]))+(1/2) ArcTan[((2 d z)/(Sqrt[4 (d^2) (z^2)+(τ^4)]))]- ((2  d  z)/(4 (d^2) (z^2)+(τ^4)))((((z/(Subscript[v, g]))-t))^2)   (5.3.2a)   
                  Φ[z;t]=((Subscript[k, 0]) z-t (Subscript[ω, 0]))+(1/2) ArcTan[((2 d z)/(Sqrt[4 (d^2) (z^2)+(τ^4)]))]- ((2  d  z)/(4 (d^2) (z^2)+(τ^4)))(u^2)   (5.3.2b)      
where:                                                  u≡(z/(Subscript[v, g]))-t                                        (5.3.2c)
                                                  
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^(-I ((2  d  z)/(4 (d^2) (z^2)+(τ^4)))((((z/(Subscript[v, g]))-t))^2)))}  /.  z-> s+ (Subscript[v, g]) t

FullSimplify[%]

It doesn't look that pretty, but we can see what's going on in the demo below:

Demo: closed up of chirped pulse

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)

Demo: chirped pulse analyzed: carrier and envelope

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.

5.3.2 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

-(∂_t)(((Subscript[k, 0]) z-t (Subscript[ω, 0]))+(1/2) ArcTan[((2 d z)/(Sqrt[4 (d^2) (z^2)+(τ^4)]))]- ((2  d  z)/(4 (d^2) (z^2)+(τ^4)))((((z/(Subscript[v, g]))-t))^2))

The above expression shows that the "instantaneous" frequency is not always (Subscript[ω, 0]) as we expect, but deviates from it by a term:
                                         Δω=-((4 d z ((z/(Subscript[v, g]))-t))/(4 (d^2) (z^2)+(τ^4)))=-((4 d z )/(4 (d^2) (z^2)+(τ^4)))u .       (5.3.3)
In other words, along the pulse envelope, the frequency deviates by an amount that is proportional to its position relative to the pulse center. The deviation magnitude ((4 d z )/(4 (d^2) (z^2)+(τ^4))) gets larger with the traveled distance.
Note the sign: if d>0, the front part of the pulse (u>0) has lower instantaneous frequency and the back part has higher: this is the case when “red” runs faster than “blue”: shorter wavelength goes slower than longer wavelength.
The opposite happens for d<0. 
If that is the case, we can ask: can we use a medium with d<0 to counter the effect after a pulse travels in a medium with d>0? the answer is yes, certainly as we will see in homework.

Below is illustration:

Demo: Instantaneous frequency of a chirped pulse

Listen to a chirped sound pulse

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".

5.4 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 amplitude term: 
                                                         a[ω] =(e^(-(1/4) (τ^2) (Ω^2)))
and the wave for a given spectral component is:
                                                        (e^(-(1/4) (τ^2) (Ω^2))) (e^(i ((Subscript[k, 0]) z -ω t) ))                              (5.4.1)
which is a product of the amplitude (e^(-(1/4) (τ^2) (Ω^2))) and the carrier (e^(i ((Subscript[k, 0]) z -ω t) )).

Obviously, this can not be the case now. The absolute magnitude 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^(-(1/4) (τ^2) (Ω^2))) (e^(i Ω( (z/(Subscript[v, g]))-t ))) (e^(i z (d/2)  (Ω^2))) ) (e^(i ((Subscript[k, 0]) z -ω t) ))                    (5.4.2)
which indicates that the absolute magnitude term (e^(-(1/4) (τ^2) (Ω^2))) alone is not sufficient, but we need the phase
                                                          (e^(i Ω( (z/(Subscript[v, g]))-t ))) (e^(i z  (d/2)  (Ω^2)))                         (5.4.3)
If we take a spectrum of the pulse, nothing changes because the power spectrum is simply the absolute magnitude squared. The phase is irrelevant.

But all the pulse behavior we see above is due entirely to the phase term in (5.4.3).
Also, we can easily recognize that the group velocity phase term  (e^(i Ω( (z/(Subscript[v, g]))-t ))) is only a linear component, which does not change a pulse shape. This phase term is not important. To cause a change in a pulse shape, it requires higher-order terms (at least 2nd) in the phase:  (e^(i z (d/2)  (Ω^2))).

6. More about dispersion (skip)

5.1 Discussion

5.2 Material dispersion

5.2.1 Lorentz oscillator

5.2.2 Kramers-Kronig model

7. Active modulation (skip)

6.1 Amplitude modulation

6.2 Phase modulation

6.3 Doppler’s shift