Classical behaviour of CW Optical Parametric Oscillators

1. Classical behaviour of CW Optical Parametric Oscillators T. Coudreau Laboratoire Kastler Brossel, UMR CNRS 8552 et Université Pierre et Marie Curie, PARIS, France also with Pôle Matériaux et Phénomènes Quantiques, Fédération de Recherche CNRS 2437 et Université Denis Diderot , PARIS, France

2. Introduction Definition • An Optical Parametric Oscillator is a device that can • generate two coherent waves (signal and idler) from a pump wave. • It consists in : • an active medium • an optical cavity, Fabry Perotresonator, in which resonates one, two or three frequencies Signal (1) Pump (0) Idler (2)

3. Introduction History • First realised in 1965 : Giordmaine & Miller, • Phys. Rev. Lett 14, 973 (1965) • Important development 1965 - 1975 as a tunable source of coherent radiation • Outdated between 1975-1990 due to the occurrence of dye lasers • Renewal since the 1990s due to • improvements in laser sources and crystals • quantum properties

4. Introduction Outline • Introduction • Definition • History • Basic principles • Optical non linearities • Second order non linearity • Energy conservation and phase matching • Classical Operation • Singly resonant OPO • Doubly resonant OPO • Triply resonant OPO • Conclusion

5. Basic Principles + Optical nonlinearities An electric field applied to an atomic medium displaces the dipole : - - + As the electric field becomes large, one gets :

6. Basic Principles O3 Nb O3 O3 Li Second order non linearity In a non centrosymetric medium, one can get a non zero Lithium Niobate Molecule A D

7. Basic Principles Second order non linearity • With a pump wave at frequency 0, on can get two kinds of behaviour : • Second Harmonic Generation (SHG) where a wave at frequency 20 is generated • Parametric down-conversion where two waves at frequencies 1 and 2 are generated 0 2 20 1+2 0 1 1 0 2

8. Basic Principles Energy and momentum conservation • Two conditions must be fulfilled : • Energy conservation • which must be always fulfilled exactly • Momentum conservation • which has to be fulfilled exactly only in the case of an infinite medium, the useful condition being

9. Basic Principles Phase matching Momentum conservation is often called phase matching : the generated signal and idler remain in phase with the waves generated before in the crystal. If , the phase shift is  after a length called the coherence length. Output power Pump signal, idler Signal, idler  Pump k0 Crystal’s length

10. Basic Principles Realisation of phase matching The natural birefringence of the crystal is generally used to ensure phase matching Extraordinary axis Ordinary axis Inputlight Index of refraction Frequency

11. Basic Principles   T T Tmin Tmin Influence of temperature The phase matching depends on the crystal temperature(and angle) Type I Type II Signal Signal Idler Idler

12. Basic Principles Quasi phase matching • The previous solution is not always chosen : • the most efficient nonlinear coefficient is not always used • some wavelength regions are not reachable • One can revert the sign of the non linearity after a length lc. Single pass output power Crystal’s length

13. Basic Principles Parametric down-conversion : basic eqns where |i|2 is a number of photons and is a field envelope These equations can be solved analytically in terms of elliptic functions.

14. Basic Principles ! Notations For a weak efficiency, we have a linear variation of the amplitudes The variation depends on the relative phase !

15. Basic Principles Pump Laser vs OPO • Laser • The pump creates a population inversion which generates gain through stimulated emission • The system depends on the pumpintensity • OPO • No population inversion, i.e. the medium is transparent • The system depends on the pump amplitude Signal (1) Pump (0) Idler (2)

16. Classical operation Doubly resonant Singly resonant Triply resonant Pump enhanced singly resonant Threshold Different kind of cw OPOs Frequency tuning difficulty

17. Classical operation Singly Resonant OPO Only the signal (or idler) wave resonates inside the cavity. Coupling mirror is the free space round trip length is the crystal length is the amplitude reflection coefficient • Usual assumptions : • Good cavity : with • close to resonance : with • Finally, one gets :

18. Classical operation SROPO - Basic properties • Pump threshold which corresponds to optical powers on the order of 1W 4 • Behaviour above threshold Mean pump intensity constant Signal field at resonance

19. Classical operation SROPO - Output Power The output power is given by the implicit equation 100 % conversion efficiency at times above threshold E. Rosencher, C. Fabre JOSA B 19 1107 (2002)

20. Classical operation SROPO - Frequency tuning • There is a linear variation of the frequency (for small variations of • ).The SROPO is • tunable like a standard laser • has a bandwidth limited by phase-matching, and/or mirror bandwidth

21. Classical operation Doubly Resonant OPO Signal and pump Doubly resonant : Pump enhanced singly resonant Signal and idler Doubly resonant Similar to a SROPO Specific behaviour

22. Classical operation With (normalised detuning) PESROPO - Basic Properties The pump threshold power is diminished with respect to the SROPO case : but the pump-cavity detuning, 0, must be taken into account. The output power is also modified :

23. Classical operation PESROPO - Frequency tuning As in a SROPO, the frequency depends linearly on the cavity length. However, the cavity length region is limited by the pump resonance width.

24. Classical operation DROPO - Basic Properties The system forces the signal and idler detunings :1= 2=  with Output power :

25. Classical operation DROPO - Frequency tuning (1) Since we have 1= 2, the round trip phases are equal (modulo 2) : which gives for the signal frequency As opposed to the previous case, the variation depends on the distance to frequency degeneracy

26. Classical operation DROPO - Frequency tuning (2) m m+1 The resonance width is the signal resonance width which is very narrow : it is almost impossible to tune by length without mode hops

27. Classical operation Triply Resonant OPO The threshold is again lower than for a DROPO : It can be below 1 mW ! The output intensity now obeys a second degree equation : the system can be monostable, bistable or even chaotic...

28. Classical operation TROPO - Stability

29. Classical operation TROPO - Frequency tuning The behaviour is similar to a DROPO with a limitation due to the pump resonance width. m m+1 m+2 ...

30. Conclusion Frequency of emission • OPOs draw their advantage from their very broad tunability since it is not limited by the proximity of a resonance in the active medium. What then limits this tunability ? • The nonlinear coefficient and the reflection coefficients of the mirrors • Phase matching which can be varied using temperature (or orientation) • Recycling of one or more waves inside the cavity • The system oscillates on frequency corresponding to the lowest threshold and only on this frequency (in a cw laser)as an homogeneously broadened laser.

31. Conclusion Summary Triply resonant Singly resonant Doubly resonant Threshold ~ 100s mW Tuning like a laser Threshold ~ 100s µW Tuning by mode hops Threshold ~ 10s mW Tuning by mode hops Pump enhanced singly resonant Threshold ~ 100s mW Tuning like a laser

32. Conclusion Conclusion • The OPO • is a coherent source of radiation • can be tuned over large domains of wavelength • can have a very low threshold • can have a very small linewidth