MIT非线性光学讲义X

Second Harmonic Generation (SHG)
Assume r
gg . Calculate r
ig
(2)
at 2ω.
Two choices for interaction,
Translate A
For steady state, we get
"Resonant term"
Note: If j, g have the same parity, no SHG.
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Population Change
()
2nn ρ,()2gg ρ
Consider only resonant terms.
Translation of Diagrams Directly into Steady State Solutions
For simplicity, consider only terms which start from diagonal states (population, but no initial induced polarization).
ρto account for initial population.
1. Include a factor of ()0gg
2.For each interaction, include an appropriate perturbation matrix
element evaluated at the final time t.
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3. For intermediate state, each pair of vertical lines after the first interaction contributes a resonance factor given by:
(a) the signed sum of all interaction frequencies up to the
intermediate state,
(b) minus one complex frequency associated with the
intermediate state.
4. Include a factor of h for each interaction and a factor of -1 for each interaction with the bra.
Example: Linear polarization
Example: Stimulated Raman Scattering
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Four Wave Mixing
Consider the interaction of three fields,
Nonlinear polarization, P
4(3)
ω
4
= ω
1
−ω
2
+ ω
3
k4= k1- k2+ k3
(phase matching
condition) Assume near resonance, two-level system.
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Additional terms
Degenerate FWM
ω = ω1 = ω2 = ω3
Δk =0
k4 vector for polarization must be the same length as k vector
at incident field.
Several methods to match phase
"Collinear"
"Counter Propagation"
"Phase Conjugation"
"Folded Box"
Consider translation of diagrams,
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Diagram A:
For steady state t o → −∞
Resonance for
ω1= ωng
ω1= ω2
ω1−ω2+ω3= ωng
We can get solution to diagrams C & D from A & B by substituting
ω1→ω3
ω3→ ω1
Steady state diagram C:
For diagram E:
Steady state solution,
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Resonance for
Interpretation of diagrams:
A & B, (C' & D'): population grating at k1 − k2 scatters k3
C & D, (A' & B'): population grating at k3 − k2 scatters k1
E & F: Second harmonic at ω1 +ω3, k3 + k1
mixed with ω2 , k2 .
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§ Transient Four-Wave Mixing
We can use resonance behavior to study damping and lifetime of quantum systems.
Consider transient behavior. Response to pulsed field
where E(t) : envelope.
Diagram A:
For simplicity, assume fields degenerate and on resonance ω = ωng . Recall that T 2 time is short and A ng is fast. Induced polarization will took the applied field envelope.
Thus, the total expression for diagram A, assuming that Γng = 1
2−T is faster than the pulse duration, may replace t 1 and t 3 integral.
Diagram B reduces to the same expression as diagram A.
Diagram C: (ω3 ↔ω1)
Assume E1, E2 are coincident in time and E3 arrived at a later time.
Generated dipole has a momentum in k3 direction and will
change E3's amplitude.
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• Now assume E1, E2 are not coincident in time, but w1 ≅ w2 and on resonance.
Pulse E1(t ) and E2 (t ) are short enough that is comparable to T2 , and T1 is much longer.
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The integration in [ ]
The last two integral become (t2 > t1)
With t2 −t1 ≥ 0
τ ≡ t2 −t1
τ ' ≡ t1 − t2
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