Ultrafast Nonlinear optics I Basic concepts Cristian Manzoni

Ultrafast Nonlinear optics (I): Basic concepts Cristian Manzoni cristian. manzoni@polimi. it Dipartimento di Fisica, Politecnico di Milano, 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 Cristian Manzoni - ICTP Winter College 1

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 Cristian Manzoni - ICTP Winter College 2

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

Nonlinear processes: wave equations Starting from Maxwell’s equation for wave propagation: High-intensity fields Polarization: Weak fields Cristian Manzoni - ICTP Winter College 4

Nonlinear processes: wave equations General wave equation with nonlinear processes: In the following: second order nonlinear processes: Cristian Manzoni - ICTP Winter College 5

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

Source of second order processes 3 interacting waves: with: Forcing term for Maxwell’s equation: Cristian Manzoni - ICTP Winter College = es v a w c i t a m o r h c o n Mo 7

Coupled nonlinear equations Slowly varying envelope approximation : = Where: Phase mismatch Cristian Manzoni - ICTP Winter College 8

Example 1: Sum frequency generation Boundary conditions: = Negligible depletion of A 1(z): No A 3 field: A 3(0)=0 Largest efficiency Smallest g Cristian Manzoni - ICTP Winter College Phase matching 9

Example 2: Parametric amplification Boundary conditions: Negligible depletion of A 3(z): A 3 = No A 2 field: A 2(0)=0 Largest efficiency Biggest g Cristian Manzoni - ICTP Winter College Phase matching 10

Parametric gain Signal intensity: Cristian Manzoni - ICTP Winter College 11

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: v. PNL = v 3 Cristian Manzoni - ICTP Winter College =0 Phase matching 12

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 Cristian Manzoni - ICTP Winter College 13

Manley Rowe After suitable manipulation: I II 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: II when one photon at 3 is created, two photons at 1 and 2 are simultaneously annihilated Cristian Manzoni - ICTP Winter College 14

Corpuscular view of second order processes SFG - Sum Frequency Generation 2 1 + 2 1 DFG - Difference Frequency Generation 1 3 - 1 1: signal 2: idler 3: pump OPA - Optical Parametric Amplification Cristian Manzoni - ICTP Winter College 15

Meaning of Phase matching (II) 2 1 + 2 1 Nonlinear interaction as a collision of collinear photons: = Energy conservation Momentum conservation Can be also applied to noncollinear interactions: Cristian Manzoni - ICTP Winter College 16

Extension to non-collinear interactions SFG OPA / DFG Cristian Manzoni - ICTP Winter College 17

Why exponential gain? Role of the idler beam. . . 1 3 - 1 = 2 2 3 - 2 = 1 . . . which gives rise to a positive loop. Cristian Manzoni - ICTP Winter College 18

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 Cristian Manzoni - ICTP Winter College 19

How to get phase matching? Phase matching requires k 1 + k 2 = k 3 equivalent to 1 n 1 + 2 n 2 = 3 n 3 In a medium with normal dispersion (dn/d > 0): 1< 2 < 3 have refractive index n 1 < n 2 < n 3 Phase matching can be written as 1 n 1 + 2 n 2 = ( 1+ 2)n 3 2(n 2 -n 3) = 1(n 3 -n 1) < 0 > 0 no phase matching in isotropic bulk materials Cristian Manzoni - ICTP Winter College 20

Cristian Manzoni - ICTP Winter College 21

Solution: Birefringent crystals o: ordinary axis no , ngo , vgo e: extraordinary axis ne , nge , vge ne depends on θ: Cristian Manzoni - ICTP Winter College 22

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 1 n 1 + 2 n 2 = 3 n 3 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) Cristian Manzoni - ICTP Winter College 23

Example 1: Phase matching curves of a visible OPA Cristian Manzoni - ICTP Winter College 24

Example 2: Phase matching curves of an IR OPA Cristian Manzoni - ICTP Winter College 25

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 Cristian Manzoni - ICTP Winter College 26

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 Pump, vp Signal, vs Their interaction vanishes when they are no more overlapped Idler, vi space Pulses are broadband Broadband phase-matching must be fulfilled Cristian Manzoni - ICTP Winter College 27

Pulse duration (I): overlap with the pump Intensity-normalized signal and idler fields Cristian Manzoni - ICTP Winter College 28

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 Pump duration ≈ Seed duration Pump Seed Pump duration < Seed duration Pump OPA Seed OPA Long pump pulses for broad Short pump pulses allow tunability bandwidth amplification A route to get long pulses from OPAs? Ultrashort laser sources (<50 fs) my not be suitable for OPAs Cristian Manzoni - ICTP Winter College 29

Broadband gain: general calculation Broadband phase-matching: Δk small over a large range of frequencies SFG OPA Cristian Manzoni - ICTP Winter College 30

Broadband gain: general calculation λp = 0. 4 μm Ip = 60 GW/cm 2 θ = 28. 9° Collinear Cristian Manzoni - ICTP Winter College 31

Broadband gain: general calculation λp = 0. 4 μm Ip = 60 GW/cm 2 Collinear Cristian Manzoni - ICTP Winter College 32

Broadband gain: general calculation λp = 0. 4 μm Ip = 60 GW/cm 2 θ = 31° α = 3. 6° Cristian Manzoni - ICTP Winter College 33

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 Cristian Manzoni - ICTP Winter College 34

Ultrafast Nonlinear Optics (I): Basic concepts Cristian Manzoni cristian. manzoni@polimi. it Dipartimento di Fisica, Politecnico di Milano, 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 Cristian Manzoni - ICTP Winter College 35