1. Coherent Nonlinear Optics1.
Nonlinear Optics 2.
Romeo and Juliet 3.
Cascade multiphoton excitation
Application to the inversion of CH3F 4.
From the cascade to the Bloch vector model
Adiabatic approximation

2. From “Coherent Interactions” to “Nonlinear Optics”
Bloch’s equations: near a resonance, time dependent population transfers,
Modulation of the absorption and dispersion, resulting in a modulation of the pulse
(effects: pulse reshaping, changes in spectrum around the pulse carrier frequency)
Traditional nonlinear optics: creation of harmonics of the field because of a
power expansion of the polarization: P = eo(ce + c(2)e )
The nonlinearity is enhanced by a resonance.
Procedure: calculate the expectation value of the dipole moment P = <y|m|y>
For a two-level system at w0 perturbed by a field at w.
Result: modified Bloch’s equations.
What is the traditional nonlinear optics limit?
Harmonic generation? How to have maximum efficiency?

3. Nonlinear optics and short pulses
Limitations of the traditional approach
P = P(E)
1) It is a Taylor series. A Taylor series has to converge
A highly controversed example:
V. Loriot, E. Hertz, O. Fauchet and B. Lavorel, "Measurement of high order Kerr
refractive index of major air components", Optics Express, 17:13439—13434 (2009)
2) Estimate of
Thus, to have
of the order of 0.1
: pm/V
requires a field of the order of TV/m
(intensity required?)
2 solutions: either short pulse high intensity,
or very high Q resonance (narrow bandwidth, long times)
(long crystals, long propagation, dispersion problems, etc)

4. Nonlinear optics and short pulses
Short pulse implies broad spectrum
For linear polarization:

5. Nonlinear optics and short pulses
Short pulse implies broad spectrum
For linear polarization:
It gets more complex for the next order:
The Fourier transform of this correlation is also a correlation:
If we introduce the Fourier transform of the nonlinear susceptibility:

6. Nonlinear optics and short pulses
In the case of an instantaneous response:
Inverse Fourier transform:

7. Nonlinear optics and short pulses
Nonlinear optics involves generally
interaction of a photon
with a higher level of energy
It is generally multiphoton
with eventual intermediate
near or exact resonances.

8. Nonlinear optics and short pulses
In most cases involving a resonance of order n, with detuning
, there will be a “driving vector”:
and a “driven” vector
representative of the atomic system, with components
in quadrature with
in phase with
= energy stored in the atomic system
The “equations of motion” will be that of a gyroscope
- (relaxation)

9. TWO representations for nonlinear optics and short pulses
Instead of
A complex vector
with components along real and imaginary axis, and
and a complex

10. Interaction of light with a cascade of levels near resonance
We will NOT start with a system having ONE near-resonance of order n,
but with a cascade of n levels all near resonance.
We will restrict ourselves to n = 3. Generalization to n > 3 is straightforward.

11. Interaction of light with a cascade of levels near resonance
Rotating frame approximation:

12. Interaction of light with a cascade of levels near resonance
This systems takes a simpler form is we define the Rabi frequencies as:
Solve, and construct
This is the density matrix
the matrix

13. Interaction of light with a cascade of levels near resonance
The approach presented here applies to different level systems.
The different configurations of 3 level systems are:
Two-photon resonant
Interaction with resonant
Intermediate level
L-level systems
V-level systems

14. Example of application: the ladder to heaven!
Climbing an anharmonic ladder of levels
CH3F

15. Example of application: the ladder to heaven!
Climbing an anharmonic ladder of levels
Solve, and construct
5 levels instead of 3
This is the density matrix
the matrix
Density matrix 5 x 5

16. Example of application: the ladder to heaven!
Climbing an anharmonic ladder of levels
-0.2
0.23
0.41
time
0.45 x x x
x x x x x x
0.89 x x x 0.44

17. Example of application: the ladder to heaven!
Climbing an anharmonic ladder of levels
Diels, Besnainou, J. Chem. Phys. 85: 6347 (1986)
-0.2
0.23
0.41
time
0.45 x x x
x x x x x x
0.89 x x x 0.44

18. Example of application: the ladder to heaven!
There can be more steps to the ladder to heaven!
Diels, Besnainou, J. Chem. Phys. 85: 6347 (1986)
Instead of:
Rotational levels:
R branch: m = 1, 2, 3…
P branch: m = 1, 2, 3…
The levels labeled with quantum number
are detuned by
Initial Boltzman distribution:

19. Example of application: the ladder to heaven!
There can be more steps to the ladder to heaven,
but you get there anyway.
Diels, Besnainou, J. Chem. Phys. 85: 6347 (1986) J 5 10
Rotational quantum number
Level population
0.1
0.2

20. Coherent Nonlinear Optics1.
Nonlinear Optics 2.
Romeo and Juliet 3.
Cascade multiphoton excitation
Application to the inversion of CH3F 4.
From the cascade to the Bloch vector model
Adiabatic approximation

21. Adiabatic approximation
Detuning of the intermediate level 1 larger than the transition rates:
the second equation can be considered to be in steady state, and one
can solve for the coefficient c1:

22. Adiabatic approximation
Defining:
Leads to:

23. Adiabatic approximation
Bloch vector model

24. Adiabatic approximation
Bloch vector model
Contained in these expressions: Third order susceptibility, two-photon absorption,
third harmonic generation, stimulated Raman, phase conjugated FWM, etc...