1. Chapter 8. Second-Harmonic Generation and Parametric Oscillation
8.0 Introduction
Second-Harmonic generation :
Parametric Oscillation :
Reference :
R.W. Boyd, Nonlinear Optics,
Chapter 1. The nonlinear Optical Susceptibility

2. The Nonlinear Optical Susceptibility
General form of induced polarization :
where,
: Linear susceptibility
: 2nd order nonlinear susceptibility
: 3rd order nonlinear susceptibility
: 2nd order nonlinear polarization
: 3rd order nonlinear polarization
Maxwell’s wave equation :
Source term : drives (new) wave

3. Second order nonlinear effect
Let’s us consider the optical field consisted of two distinct frequency components ;
: Second-harmonic generation
: Sum frequency generation
: Difference frequency generation
: Optical rectification
# Typically, no more than one of these frequency component will be generated
 Phase matching !

4. Nonlinear Susceptibility and Polarization
1) Centrosymmetric media (inversion symmetric) :
Potential energy for the electric dipole can be described as
Restoring force :
Equation of motion :
Damping force
Coulomb force
Restoring force

5. Purtubation expansion method :
Assume,
Each term proportional to ln should satisfy the equation separately
: Damped oscillator
Second order nonlinear effect in centrosymmetric media
can not occur !

6. 2) Noncentrosymmetric media (inversion anti-symmetric) :
Potential energy for the electric dipole can be described as
Restoring force :
Equation of motion :
Damping force
Coulomb force
Restoring force

7. Similarly, : Report Assume,
Each term proportional to ln should satisfy the equation separately
Solution :
: Report

8. Example) Solution for SHG
Put general solution as
Similarly,
: Report

9. Susceptibility Polarization : : linear susceptibility : SHG : SFG
: DFG
: OR

10. <Miller’s rule> - empirical rule, 1964
is nearly constant for all noncentrosymmetric crystals.
# N ~ 1023 cm-3 for all condensed matter
# Linear and nonlinear contribution to the restoring force would be comparable when the displacement
is approximately equal to the size of the atom (~order of lattice constant d) :
mw02d=mDd  D=w02/d : roughly the same for all noncentrosymmetric solids.
(non-resonant case) : used in rough estimation of nonlinear coefficient.
: good agreement with
the measured values

11. Qualitative understanding of Second order nonlinear effect
in a noncentrosymmetric media

12. 2w component

13. General expression of nonlinear polarization and
Nonlinear susceptibility tensor
General expression of 2nd order nonlinear polarization :
where,
2nd order nonlinear susceptibility tensor
# Full matrix form of
: SHG
: SFG
: SFG
: SHG

14. Example 1. SHG
Example 2. SFG

15. Properties of the nonlinear susceptibility tensor
1) Reality of the fields
are real measurable quantities :
2) Intrinsic permutation symmetry

16. 3) Full permutation symmetry (lossless media : c is real)
4) Kleinman symmetry (nonresonant, c is frequency independent)
intrinsic
: Indices can be freely permuted !

17. Define, 2nd order nonlinear tensor,
## If the Kleinman’s symmetry condition is valid, the last two indices can be simplified
to one index as follows ;
and,
can be represented as the 3x6 matrix ;
: 18 elements

18. Again, by Kleinman symmetry (Indices can be freely permuted),
dil has only 10 independent elements :
: Report

19. Example 1. SHG
: Report
Example 2. SFG

20. 8.2 Formalism of Wave Propagation in Nonlinear Media
Maxwell equation
Polarization :
Assume, the nonlinear polarization is parallel to the electric field, then
Total electric field propagating along the z-direction :
where,
and

21. w1 term
(slow varying approximation)
Text

22. Similarly,

23. 8.3 Optical Second-Harmonic Generation
Neglecting the absorption ;
where,
Assume, the depletion of the input wave power due to the conversion is negligible

24. Phase-matching in SHG Output intensity of 2nd harmonic wave :
Conversion efficiency :
Phase-matching in SHG
Maximum
: phase-matching condition If
: decreases with l
Coherence length : measure of the maximum crystal length that is useful in producing the SHG
(separation between the main peak and the first zero of sinc function)

25. Technique for phase-matching in anisotropic crystal
Example) Phase matching in a negative uniaxial crystal

26. Experimental verification of phase-matching
, there exists an angle qm at which ,
so, if the fundamental beam is launched along qm as an ordinary ray,
the SH beam will be generated along the same direction as an extraordinary ray.
Example (p. 289)
Experimental verification of phase-matching
Taylor series expansion
near
: Report

28. Second-Harmonic Generation with Focused Gaussian Beams
If z0>>l, the intensity of the incident beam is nearly independent of z within the crystal
Total power of fundamental beam with Gaussian beam profile :

29. So, Conversion efficiency :
: identical to (8.3-5) for the plane wave case
(*) P(2w) can be increased by decreasing w0
until z0 becomes comparable to l
# It is reasonable to focus the beam until l=2z0 (confocal focusing)
(**)
Example (p. 292)

30. Second-Harmonic Generation with a Depleted Input
Considering depletion of pump field,
Define,
(8.2-13) 
where,
SHG :
Let’s consider
a transparent medium :
, and perfect phase-matching case :

31. : Total energy conservation
Define,
: Total energy conservation
Initial condition : #
: 100% conversion
[2N(w photons)  N(2w photons)]

32. Conversion efficiency :

33. 8.4 Second-Harmonic generation Inside the Laser Resonator
# Second-harmonic power
Pump beam power
# Laser intracavity power :
 Efficient SHG
SH output power :

34. 8.5 Photon Model of SHG
Annihilation of two Photons at w and a simultanous creation of a photon at 2w
- Energy : w+ w=2w
- Momentum :

35. 8.6 Parametric Amplification:
# Special case : w1=w2 (degenerate parametric amplification)
Analogous Systems :
- Classical oscillators
- Parasitic resonances in pipe organs(1883, L. Rayleigh) :
- RLC circuits
Example) RLC circuit

36. Steady-state solution :
Assuming
Put,
where,
Steady-state solution :
(degenerate parametric oscillation)
Phase matching
Threshold condition

37. Optical parametric Amplification
Polarization of 2nd order nonlinear crystal :

38. (phase-matching), and also depletion of field due to
(8.2-13), 
where,
When
(lossless),
(phase-matching), and also depletion of field due to
the conversion is negligible,
where,

39. Qualitative understanding of parametric oscillation :
Solution :
Qualitative understanding of parametric oscillation :
# Initially w1(or w2) is generated by two photon spontaneous fluorescence
or by cavity resonance
# w2(or w1) wave increases by difference frequency generation
between w3 and w1(or w2)
# w1(or w2) wave also increases by difference frequency generation
between w3 and w1(or w2)
# w2(or w1) wave : Signal [A(0)=0]
# w2(or w1) wave : Idler [A(0)>0]

40. Initial condition :
Photon flux :

41. 8.7 Phase-Matching in Parametric Amplification
Put,

42. General solution :

43. Phase-Matching : Report
Example) Phase-matching by using a negative uniaxial crystal
: Report

44. 8.8 Parametric Oscillation
where,
(8.8-1)

45. Think of propagation inside a cavity as a folded optical path.
Even though Eq. (8.8-1) describe traveling-wave parametric interaction, it is still valid if we
Think of propagation inside a cavity as a folded optical path.
If the parametric gain is equal to the cavity loss (threshold gain),
So,
absorption in crystal, reflections on the interfaces,
cavity loss(mirrors, diffraction, scattering), …
Condition for nontrivial solution :
: Threshold condition for parametric oscillation

46. Threshold pump intensity :
If we choose to express the mode losses at w1 amd w2 by the quality factors, respectively,
Decay time (photon lifetime) of a cavity mode :
(4.7-5)
Temporal decay rate :
and
Threshold pump intensity :
Pump intensity :
Threshold pump intensity :

47. Example) Absorption loss = 0
(4.7-5), (4.7-3) 
: given by only the cavity mirror’s reflectivity
Example (p. 311)

48. 8.9 Frequency Tuning in Parametric Oscillation
Phase-Matching condition :
If the phase matching condition is satisfied at the angle, q=q0
And, we have

49. Neglecting the second order terms,
(w3 is a fixed frequency, and if we use an extraordinary ray for the pump)
(If we use ordinary rays for the signal and idler)
Parametric oscillation frequency with the angle :

50. Example) Frequency tuning by using a negative uniaxial crystal

51. 8.11 Frequency Up-Conversion
: Sum Frequency Generation
Phase-matching condition :
Solution :
where,

52. # Oscillating function with z (cf : parametric oscillation)
therefore
Power :
# Oscillating function with z (cf : parametric oscillation)

53. Conversion efficiency :
Typically, conversion efficiency is small
Example (p. 318)