1. §8.4 SHG Inside the Laser Resonator
To improve SHG conversion efficiency, high intensity of fundamental frequency laser is need. However, this situation is changed when SHG inside the laser resonator.
When R approaches 1, the SHG conversion efficiency will be very large. We may expect 100% conversion efficiency.
Optimum power coupling

2. §8.4 SHG Inside the Laser Resonator
Total loss per pass of fundamental beam
Pe is the total power emitted from stimulated emission
The same as the power expression of fundamental frequency as before
and

3. §8.4 SHG Inside the Laser Resonator
Nonlinear coupling constant
Independent of the g0
Numerical Example: Internal SHG

4. §8.5 Photon Model of SHG Output flux at Nonlinear Crystal
Input flux at
Output flux at +

5. §8.6 Parametric Amplification
当频率为 的较强光束（泵浦光）与频率为 的较弱光束（信号光）在二阶非线性介质中相互作用时，泵浦光能量会转移给信号光，并产生频率为 的光波（闲置光）。这一过程称为光学参量放大。
where if
Degenerate parametric amplification
Consider a classical nondriven oscillator:
Solution:
Consider another oscillator which energy storage parameter is modulated at a frequency

6. §8.6 Parametric Amplification
Suppose a solution:

7. §8.6 Parametric Amplification
Steady-state oscillation conditions: ,
In such way, the circuit will break into spontaneous oscillation at frequency
Threshold condition

8. §8.6 Parametric Amplification
Capacitance variation at frequency
Capacitance change by pulling and pushing the parallel plate separation
Charge has an eigen oscillation frequency
Voltage increases twice in each eigen Oscillation cycle
Same phase: energy fed into system Invert phase: system loss energy

9. §8.6 Parametric Amplification
耦合波方程

10. §8.6 Parametric Amplification
At the conditions:
Initial conditions:
Solutions:

11. §8.6 Parametric Amplification
: Signal light
: Idler light
For
Numerical Example

12. §8.7 Phase-Matching in Parametric Amplification
Phase-matching condition:
In the general cases:
Suppose the solutions like:

13. §8.7 Phase-Matching in Parametric Amplification
Exponential coefficient is a function of
The signal and idler are no longer sustained growth, but oscillate as functions of distance z
Phase-Matching realization:

14. §8.8 Parametric Oscillation
Optic axis of crystal
Laser
medium
Nonlinear
crystal
Motivation: convert pump laser power to signal and idler light, with their frequencies can be tuned continuously over larger ranges.
Steady-state

15. §8.8 Parametric Oscillation
Threshold condition for parametric oscillation
Threshold condition as a function of quality factor
Threshold condition as a function of mirror reflectivity

16. §8.9 Frequency Tuning in Parametric Oscillation
Phase-Matching condition:
Frequency
Crystal orientation (E beam)
Indices of refraction:
Electric field (EO crystals)
Temperature
Causes frequency change
Assume at:
we have
After crystal orientation changes from

17. §8.9 Frequency Tuning in Parametric Oscillation
Neglect second-order terms
Extraordinary pump beam
Ordinary signal and idler beams

18. §8.9 Frequency Tuning in Parametric Oscillation

19. §8.10 Power Output and Pump Saturation in Optical Parametric Oscillators
Laser Oscillation: at steady state, the gain could not exceed the threshold value regardless of the intensity of the pump
Parametric Oscillation: very similar to laser oscillation, the gain is “clamped” at its threshold value even if the pump field is increased beyond its threshold.
Pump Saturation
Experimental confirmation

20. §8.11 Frequency Up-Conversion
Parametric interactions can be used to convert a signal from a “low” frequency to a “high” frequency by mixing it with a strong laser beam at , where + +
Nonlinear Crystal
Filter

21. §8.11 Frequency Up-Conversion
Neglect depletion of pump wave A2
Solutions:

22. §8.11 Frequency Up-Conversion
In case of small conversion efficiency: