1. Theoretical Analysis of the Resonance Four-Wave Mixing Amplitudes: a Fully Nondegenerate Case
Alexander Kouzov
Department of Physics, Saint-Petersburg State University, Ulyanovskaya str. 3, Peterhof, Saint-Petersburg , Russia

2. Outline 1. Key characteristics of 4-wave mixing
2. Resonant 4WM: Pros & cons
3. Particular schemes to excite resonance 4WM:
A/ 3 photons are absorbed, one is emitted (AAAE):
sum frequency generation scheme
B/ 2 photons are absorbed, 2 are emitted
a/ resonance CARS-type schemes (AEAE)
b/ unfolded scheme (AAEE)
C/ 1 photons is absorbed, 3 are emitted (AEEE):
difference frequency generation scheme
4. Decoupling procedure
5. Polarization dependence
6. Looking ahead

3. Basics of 4WM Theory
48 terms
res
TC-RFWM:
CARS:

4. Fully Resonant 4WM: Pros and Cons
High frequency and spatial resolution
High sensitivity (pressures as low as 1 mbar)
Nonintrusive tool
Applies both to isotropic and anisotropic media
TC-RFWM: unique tool to directly trace the state-to-state collisional transfer
Flexible polarization setup allowing to study rotational anisotropy at highest state of art
Radiation coherence
Limitations
Weakly absorbing media (low-pressure gases)

5. Theoretical techniques
Double-sided Feynman diagrams
S.Y. Yee et al, Phys. Rev. A. vol. 18, 1597 (1978).
Line-space (Liouville) techniques
D. Forster “Hydrodynamic Fluctuations, Broken Symmetry and Correlation Functions” (Benjamin, Reading, MA, 1975).
Irreducible spherical tensor (IST) formalism
D. A. Varshalovich, A. N.Moskalev, and V. K. Khersonskii, “Quantum Theory
of Angular Momentum” (World Scientific, Singapore, 1988)

6. Sum frequency generation3 p 2 4 n 1 i 4
time f i 3 p 2 n 1

7. Decoupling procedure (based on IST formalism)
TC-RFWM (Zare’s group, Sandia Labs):
S. Williams et al, J. Chem. Phys

8. 8 diagrams contribute at finite and
CARS-like schemes
8 diagrams contribute at finite and
resulting in:
Conventional CARS (single Raman resonance) 3 4 2 1 n i
Doubly resonant CARS schemes
Triply resonant CARS scheme f f
1-2-3 p p 3 4 3 3 4
2-1-3
2-3-1
3-2-1 1 2 1 2 4 1 2 n n n i i i
4 diagrams: 1( )+3( ) - no cancellation at

9. Unfolded schemes (extended TC-RFWM)4 f f 3 p 3 2 3 2 p n n f 1 4 1 2 p n 4 i i 1 4 4 f 4 3 p 2 f p p n f 2 n 3 1 2 1 n i 3 1

10. Difference frequency generation2 i n 1 p f 4 1 3 2 n 3 n i 4 p 2 i 2 p 3 f 1

11. Difference frequency generation4 4 4 f n p p n 2 2 2 p 2 p n 4 1 3 f n f 3 f 3 1 3 i 1 1 4 4 4 f 4 f p p p f f p 2 1 2 n 1 2 n 2 3 n n 3 1 i 1 3 3

12. Thanks for your attention !
Pros
FR-FWM allows
To study dipole-allowed transitions from/within unpopulated vibronic states
Up and down frequency shifting in a wide spectral range
Cons
Sofisticated techniques (use of 3 lasers, finer frequency tunings)
Looking ahead
Generation of coherent MW; its indicatrix
Extensions to time-domain
Thanks for your attention !