Cristian Manzoni cristian.manzoni@polimi.it

1. Ultrafast Nonlinear optics (I): Basic concepts Cristian Manzoni cristian.manzoni@polimi.it Dipartimento di Fisica, Politecnico di MilanoMilano, Italy Consiglio Nazionale delle Ricerche ,Istituto di Fotonica e Nanotecnologie - Milano, Italy Winter College on Extreme Non-linear Optics, Attosecond Science and High-field Physics

2. Outline Nonlinear processes: wave equations • General equation • Second order processes • Coupled nonlinear equations • Meaning of phase matching (I) Corpuscular view of second order processes • Manley-Rowe equations • Meaning of phase matching (II) Fulfilling phase matching Second order processes with pulses • Temporal overlap • Broadband phase matching

3. w1 + w2 w1 2w1 2w2 w2 Weak fields w1 - w2 Parametric interactions High-intensity fields w1 c(1) c(1) c(2) w2 C. Manzoni and G. Cerullo, “Design criteria for ultrafast optical parametric amplifiers”, J. Opt. 18, 103501 (2016)

4. Weak fields Nonlinear processes: wave equations Starting from Maxwell’s equation for wave propagation: High-intensity fields Polarization:

5. Nonlinear processes: wave equations General wave equation with nonlinear processes: In the following: second order nonlinear processes:

6. w1 + w2 w1 2w1 2w2 w2 w1 - w2 optical rectification - does not oscillate - Second order processes w1 c(1) What is the meaning of E(t)2? + c(1) c(2) • Let’s start from 2 oscillating fields: w2 Second harmonic generation (SHG) + + Sum frequency generation (SFG) Difference frequency generation (DFG)

7. Source of second order processes 3 interacting waves: = with: Monochromatic waves Forcing term for Maxwell’s equation:

8. Coupled nonlinear equations • Slowly varying envelope approximation : = Where: Phase mismatch

9. w1 + w2 w1 w2 Example 1: Sum frequency generation Boundary conditions: • Negligible depletion of A1(z): • No A3field: A3(0)=0 = w1 c(1) c(1) c(2) w2 Phase matching Smallest g Largest efficiency

10. Example 2: Parametric amplification Boundary conditions: • Negligible depletion of A3(z): • No A2field: A2(0)=0 A3 = Phase matching Biggest g Largest efficiency

11. Parametric gain Signal intensity:

12. Meaning of Phase matching (I) Field 3 Propagation velocity Source of field 3 Propagation velocity: PNL efficiently deposits energy into ω3 when they propagate with the same velocity: Phase matching vPNL = v3 = 0

13. Outline Nonlinear processes: wave equations • General equation • Second order processes • Coupled nonlinear equations • Meaning of phase matching (I) Corpuscular view of second order processes • Manley-Rowe equations • Meaning of phase matching (II) Fulfilling phase matching Second order processes with pulses • Temporal overlap • Broadband phase matching

14. Manley Rowe After suitable manipulation: II II I I the sum of the energies of the three waves is conserved (with a lossless medium) If Ni(z)/Δt photons correspond to intensity Ii(z): photon conservation: when one photon at 3 is created, two photons at 1 and 2 are simultaneously annihilated

15. DFG - Difference Frequency Generation SFG - Sum Frequency Generation w1 w1 w2 w1 + w2 w3- w1 w1 w3 OPA - Optical Parametric Amplification Corpuscular view of second order processes w1: signal w2: idler w3: pump

16. w2 w1 + w2 w1 Meaning of Phase matching (II) Nonlinear interaction as a collision of collinear photons: = Energy conservation Momentum conservation Can be also applied to noncollinear interactions:

17. Extension to non-collinear interactions SFG OPA / DFG

18. w1 w1 w3- w1 = w2 w3 w3 Why exponential gain? Role of the idler beam... w2 w2 w3- w2 = w1 ... which gives rise to a positive loop.

19. Outline Nonlinear processes: wave equations • General equation • Second order processes • Coupled nonlinear equations • Meaning of phase matching (I) Corpuscular view of second order processes • Manley-Rowe equations • Meaning of phase matching (II) Fulfilling phase matching Second order processes with pulses • Temporal overlap • Broadband phase matching

20. Phase matching requires k1 + k2 = k3 equivalent to 1n1 + 2n2 = 3n3 How to get phase matching? In a medium with normal dispersion (dn/d > 0): 1< 2 < 3 have refractive indexn1 < n2 < n3 Phase matching can be written as 1n1 + 2n2 = (1+2)n32(n2-n3) = 1(n3-n1) > 0 < 0 no phase matching in isotropic bulk materials

22. Solution: Birefringent crystals o: ordinary axis no , ngo , vgo e: extraordinary axis ne , nge , vge nedepends on θ:

23. Interaction types 1< 2 < 3 can have different polarizations: Finding the phase-matching condition means calculating, for a given Type, the angle θ that satisfies 1n1 + 2n2 = 3n3 When 2 fields are extraordinary: θ to be found numerically Dmitriev V G, Gurzadyan G G, Nikogosyan D N and Lotsch H K V, Optics of nonlinear crystals: Handbook of Nonlinear Optical Crystals, Springer Series in Optical Sciences vol 64 (1999)

24. Example 1: Phase matching curves of a visible OPA

25. Example 2: Phase matching curves of an IR OPA

26. Outline Nonlinear processes: wave equations • General equation • Second order processes • Coupled nonlinear equations • Meaning of phase matching (I) Corpuscular view of second order processes • Manley-Rowe equations • Meaning of phase matching (II) Fulfilling phase matching Second order processes with pulses • Temporal overlap • Broadband phase matching

27. Pump, vp Signal, vs Idler, vi space Nonlinear optics with pulses No need to solve again the nonlinear equations Extend the results of monochromatic waves to pulses, but: • Pulses are limited in time and propagate at different speeds Their interaction vanishes when they are no more overlapped • Pulses are broadband Broadband phase-matching must be fulfilled

28. Pulse duration (I): overlap with the pump Intensity-normalized signal and idler fields

29. Pump duration < Seed duration Pump duration ≈ Seed duration Pump Pump OPA OPA Seed Seed Pulse duration (II): pump vs seed Signal is typically chirped, even if generated by a thin sapphire plate Pump needs to be overlapped with all the seed colors • Long pump pulses for broad bandwidth amplification • Ultrashort laser sources (<50 fs) my not be suitable for OPAs • Short pump pulses allow tunability • A route to get long pulses from OPAs?

30. Broadband gain: general calculation Broadband phase-matching: Δksmall over a large range of frequencies SFG OPA

31. Broadband gain: general calculation λp = 0.4 μm Ip = 60 GW/cm2 θ = 28.9° Collinear

32. Broadband gain: general calculation λp = 0.4 μm Ip = 60 GW/cm2 Collinear

33. Broadband gain: general calculation λp = 0.4 μm Ip = 60 GW/cm2 θ = 31° α = 3.6°

34. Outline Nonlinear processes: wave equations • General equation • Second order processes • Coupled nonlinear equations • Meaning of phase matching (I) Corpuscular view of second order processes • Manley-Rowe equations • Meaning of phase matching (II) Fulfilling phase matching Second order processes with pulses • Temporal overlap • Broadband phase matching

35. Ultrafast Nonlinear Optics (I): Basic concepts Cristian Manzoni cristian.manzoni@polimi.it Dipartimento di Fisica, Politecnico di MilanoMilano, Italy Consiglio Nazionale delle Ricerche ,Istituto di Fotonica e Nanotecnologie - Milano, Italy Winter College on Extreme Non-linear Optics, Attosecond Science and High-field Physics