1. Pulse Propagation. Chapter 4.
By Dima Vingurt.
ID:
Master’s thesis: Chemical iodine-oxygen laser: mechanism of iodine dissociation, corona discharge to pre-dissociate iodine.

2. Classical and Quantum Or Upgrade to Chapter 1
One-dimensional case of Maxwell’s equation:
Classical dipole moment: Quantum dipole moment:
ex(t,z) d<average over all dipoles (t,z)>
Where: D-Uniform density of dipoles throughout a small region.
u, v- slowly varying amplitudes.
g- inhomogeneous line shape detuning function.
Electrical field in both cases:

3. Maxwell equation Substitution of Field and Polarization:
(u , v slowly variant, )
The same equation for classical and our case.
First: in phase.
Second: in-quadrature

4. Bloch Equations and Bloch Vector.
The quantum dipole envelope obey the non-linear optical Bloch equaction(Ch.3):
Where: u,v- components of dipole P (in phase and in quadrates)
w- population inversion
T1,T2’- first is lifetime of exited state, phase shift in dipole momentum.
In magnetic theory these times responds to:
T1,T2’- lifetimes which control the decay of magnetic spin (longitudinal and transverse- parallel and perpendicular to the Zeeman field).

5. Bloch Equations and Bloch Vector.
In ch. 2 was shown, that Bloch equation rotating Bloch vector. We got probability conservation law.
u2+v2+w2=const.
If we neglecting T1 and T2’ (experiment is shorter then relation times):
u2+v2+w2=1.
This law is semi classical ( we got it in ch. 2) and shows that Bloch vector ( which is oscillator) can store finite energy. Classical oscillator can store infinite amount of energy.
This is the reason for changing ex0 to d.

6. Solution of reduced Bloch equation.
Assumption:
In case of cooled solids, diluted gases we can neglect T2’( no collision, no radioactive decay).
In case of short pulses we can neglect T1,T2’ .

7. On-resonance solution
Initial condition : all atom in ground states.
Solution ( ch 3):
-pulse area (from ch 1).

8. Now E is time dependent Raby solution from ch . 3 not good.
Assumption: detuning-dependent function
F- dipole spectral response.
w=-1 all atoms in ground state.

9. Pendulum equation. Solution are elliptic function for typical pendulum
Pendulum equation. Solution are elliptic function for typical pendulum. But for our boundary condition (E (and all other parameters) must vanish at plus- minus infinity):

10. We got non-oscillation pendulum: if pendulum was balanced vertically and then falls and balance vertically again in the infinity future.
From our solution it is possible to find field:
It is called hyperbolic secant pulse of McCall and Hahn.
- is a pulse length.
We can also find dipole spectral function F:
It is Lorentzian

11. Solution.
The final solution:

12. Area Theorem
First step: integration of in- quadrature Maxwell equation from minus infinity to the end of the pulse:
This all done exactly as in classical Area theorem. The differences are:
Constants:
Time scale:
Classical: pulse is short (T2’ , T2* is big)
Quantum: no assumption on the pulse time.
Nonlinear on-resonance dipole amplitude (v):
In classical case it was linear function of A.

13. As in classical case , from first step we got:
Where: t0- End of pulse.
- no pulse after this time ( greater then t0).
-absorption coefficient.
In the “weak field” we will get classical result: (Beer’s absorption)

14. Area Theorem
A=n
The derivative is zero for this pulses. The areas that are even multiples are more stable than those that have n odd.

15. Pulse is shorter and higher.
Glibbs and Slusher have used strong pulse in successful attempt to compress and amplify the pulse in absorbing medium.
The will tend toward But if the pulse is short enough (for example 10 nsec) there will be no energy-loss. Because it does not interact with any given group of atoms long enough for either T1 or T2’ to be effective.
Numerical computation gives some justification :
Energy lose for /10 from a.
Assume that we have two pulses with equal energies:
If the pulse is smooth enough (exact for square pulse):
Pulse is shorter and higher.

16. Self-induced Transparency
Pulse corresponds for same angel . This pulse return dipole exactly to its original state. If such pulse returns the dipole exactly to its original stat, then the dipole can take no energy from the pulse. We have stable pulse.
Stable pulse- these pulses behave like matter is transparent. There are area stable and shape-stable.

17. Self-induced Transparency
Stable pulse- in order to show it’s steady state property:
The dipole envelope function have to depend on time and space only through the “local time”.
Where : V- constant velocity.
Derivatives of B :
We also assume small pulse in order to reduce dumping time from Bloch equations. (For integrations times are infinity).

18. Self-induced Transparency
We need to solve Bloch equations without dumping times and Maxwell equations ( we need substitute derivatives).
We will also use factorize assumption:
From Bloch equation for population inversion (w) and from Maxwell in-quadrature equation we could get energy-flux conservation law:
U-energy density.

19. Self-induced Transparency
Let’s define:
From integration of flux-energy equation (dB) and from Maxwell equation we can get:
Where: w0- population inversion with no field.
From first and third Bloch equations we get elimination of v and expression for u:

20. Self-induced Transparency
Let’s find E:
In-quadrature Maxwell equation became:
Then we take derivate with respect to B and using second Bloch equation to eliminate v. We already know w and u in terms of E. So we will get on one side and on other E.
Assumption: initially atoms at ground state:

21. Self-induced Transparency
M is maximum value for E.
Weak field area: exponential growth of E.

22. Self-induced Transparency
For the condition F(0)=1 and Lorentzian. And :
Area of this pulse is 2 and independent of both k and .
This solution is exactly hyperbolic secant pulse of McCall and Hahn.
It is possible to show that u, v and w are same expression as we found in the beginning.
Velocity:
Assumption: K=k.
And from definition of :

23. Self-induced Transparency
Velocity:
Assumption: is big .
g is smooth and broad- we can move it from integral.
Where : a- absorption coefficient.
Delay of pulse traveling though absorber with length L:

24. Self-induced Transparency Summary
Self-induced Transparency is the phenomena of short coherent pulses traveling very long distances, at very low velocities, though resonant absorber.
We have studied only 2pi pulse, but the Area theorem make no great distinction between any even pulses. But it can be shown that there are no steady state single pulse solution other then 2pi. (Additional possibilities is “train” of pulses).
Another possibility was investigated by G.L. Lamb. He has worked with 4pi and 6pi. In such case the electric envelope splits up into discrete pieces, each with area 2pi. But with different length, amplitude and velocities.

25. Self-induced Transparency Summary
Additional possibility is so called “zero pulse” : “bound state” of two pulses +2pi and -2pi. This pulse does not break up on 2pi pulses.

26. Phase Modulation Effects
Let’s take general field:
Where f is phase function, slowly varying in the same sense that E.
And: and for frequency:
In order to see the difference in equations we will work in rotating frame.
Assumption: u and v have same phase shift as field.
Bloch equations in rotating frame:

27. Phase Modulation Effects
Maxwell equations in rotating frame:
First Case.
Assume strong external field: we neglect radiation emitted by atoms (neglecting Maxwell equation) .
We will assume resonance and external field has sech envelope and a tanh frequency sweep (derivative of f) .

28. Phase Modulation Effects First Case
Solution for this case will be well known sech and tanh. Solution is exactly the same as before, instead of detuning we have derivative of f. And we have assumption of short pulses.
These equations are know as solution for adiabatic inversion.
Adiabatic inversion: slowly changing the magnitude of the statistic 3 components of magnetic field so that effective field changes sign. All spins are following the field and also changing sing.
These solutions are analytical , it means than no adiabatic assumption or restriction needed. It mean that inversion is indeed slow process. Process started before pulse peak arrives.

29. Phase Modulation Effects First Case
-called magnitude of the frequency sweep. If then E goes to standard pi pulse:
We expecting to get inversion of atoms.
On other hand :
It has been discover that this pulse does not return atom to their ground state. It happens because in presents of phase modulation the identity between pulse area and dipole turning angel in no longer true. Area theorem is no longer valid in this case.

30. Phase Modulation Effects Second Case
The case with 2pi pulse and phase modulation put some doubts on lossless pulse. But Areas theorem we neglected second derivative term in writing slowly varying envelope in-phase and in-quadrature Maxwell equation. For slowly varying restriction only steady-state is possible, it means that phase is constant.
In order to show steady state we will make substitution We will differentiate in-phase equation:
We will also make factorization assumption for v:

31. Phase Modulation Effects Second Case
We looking only to the first derivative:
Derivative of f is constant,

32. Phase Modulation Effects Third Case
Lamb has find that slowly varying pulse 2pi pulse are not frequency-modulated. However Lamb shows that the collision of two 2pi pulses with different frequencies lead to a 4pi pulse which can be frequency-modulated.
On picture : breakup of such pulse.

33. Circularly Polarize Light.
For circular polarized light: Bloch equations are exact , no RWA needed.
(with no dumping time).
Where : 21/2 is to ensure that wave carries the same energy.
Let’s define operator d (ch 2):
In case of circular polarization:
The factor 21/2 : d+-*d-+=d2

34. Circularly Polarize Light.
This matrix is Hermitian : d-+=(d+-)*.
With this notation with given E Maxwell equations look like:
These are identical with equations in the beginning of lecture.