1. A Taste of Extreme Nonlinear Optics There are two areas of ultrafast nonlinear optics under intense investigation because the phenomena observed cannot be explained using “classical nonlinear optics”. This has led to the birth of a field called “extreme nonlinear optics. The key processes are: 1.the electromagnetic field-induced ionization of electrons, 2.their subsequent motion under the influence of the field and 3.their recombination with ions. George Stegeman, KFUPM Chair Professor., College of Engineering King Fahd Un. of Petroleum and Minerals, Saudi Arabia Professor Emeritus College of Optics and Photonics/CREOL University of Central Florida, USA Two excellent reviews: T. Brabec and F. Kraus, “Intense Few –Cycle Laser Pulses: Frontiers of Nonlinear Optics”, Rev. Mod. Phys., 72, 545 (2000) F. Kraus and M. Ivanov, “Attosecond Physics”, Rev. Mod. Phys.81, 163 (2009)

2. “First” High Harmonic Generation Paper 1.Only odd harmonics. 2.Plateau regions. 3.Requires terra-watt  petta-watt intensities R. L.Carman, C. K. Rhodes and R. F. Benjamin, “Observation of harmonics in the visible and ultraviolet created in CO 2 -laser-produced plasmas”, Phys. Rev. A, 24, 2649-2663 (1981). 240nm CO 2 laser ( =10.6  m, 10,600nm) Carbon wire Electron plasma Intensity at focus  2.3x10 15 W/cm 2 Authors correctly identified the source of the effect as the interaction of the radiation with free electrons.

3. “Sub-100nm” High Harmonic Generation 1.Only odd harmonics. 2.Plateau regions. 3.Requires terra-watt  petta-watt intensities Ar, Xe, Kr, Ne More experiments followed with other noble gases like Xe, Kr etc., many in capillaries Z. Chang, A. Rundquist, H. Wang, M. M. Murnane and H. C. Kapteyn “Generation of Coherent Soft X Rays at 2.7 nm Using High Harmonics (Ti:SAF laser)” Phys. Rev. Lett., 79, 2967 (1997). Nd:YAG =1.06  m M. Ferray, A. L'Huillier, X. F, Li, L. A. Lompré, G, Mainfray and C. Manus, Let. to Editor. “Multiple-harmonic conversion of 1064 nm radiation in rare gases”, J. Phys. B: Atomic & Molecular Optics, 21, L31-L35, (1988). Ar gas jet

4. Deals with bound electrons in atoms or molecules. The induced polarization which radiates is written as a perturbation expansion in terms of increasing powers of electric field Nonlinear Optics “Classical” Nonlinear Optics Since the noble gases (He, Ar, Xe etc.) gases are centro-symmetric atoms, when excited by a single beam: The range of validity of this expansion is that, i.e. the series converges The usual criterion used is that the optical field should be a small fraction of the atomic field binding the electrons to atoms. The hydrogen atom’s energy levels etc. can be calculated so they are used as an approximate scaling factor.

5. Hydrogen Atom Fields n=1 (ground state) Atomic field E at binding electron to proton (ionization field) in hydrogen atom: Optical Field E optical =0 atomic potential well E optical >E atomic n=1 (ground state) Response (displacements) of free electrons to applied optical fields is much greater than for bound electrons, typically a small fraction of r B !

6. Regimes of Nonlinear Optics Perturbative RegimeStrong Field Regime Extreme Nonlinear Optics Bound Electrons Free Electrons 10 12 W/cm 2 10 14 W/cm 2 10 16 W/cm 2 10 18 W/cm 2 : 1 

7.  E=0 atomic potential well E>0 atomic potential well Optical Field Tunneling: 1  Optical Field   Electron Ionization Mechanisms Multi-Photon Absorption:

8. Electron-Radiation Interaction Electron is “free” since it spends most of its trajectory far from the ion (and its Coulomb field which decreases as 1/x 2 ) +

9. “a” (  =135 0 ) and “c” (  =93 0 ) correspond to the same final energy. “b” (  =107 0 ) is a cut-off trajectory with the highest kinetic energy (3.17I p ). “d” (  =90 0 ) starts at the peak of the electric field where most electrons are produced but returns to the core with zero kinetic energy. “e” (  =45 0 ) never returns to its parent ion. Electrons emitted at ion’s location (0,0). Motion of Free Electrons in Electromagnetic Field  (electric field phase at the instant of ionization) 0  22

10. This recombination process leads to an ultrafast nonlinearity, time scale of an optical period. The electrons which do not recombine with ions inside one cycle recombine on longer time scales. 1.During optical pulse, the electron excursion is still large and recombination occurs on a time scale which depends on the free electron density. Femtoseconds? 2.After the optical pulse, carriers diffuse and time scales of a 100ps – ns have been measured.

12. “Three Step” Process M. Lewenstein, Ph. Balcou, M. Yu. Ivanov, A. L’Huillier and P. B. Corkum, “Theory of High-harmonic Generation by Low Frequency Laser Fields”, Phys. Rev. A, 49, 2117-2132, (1994). 1.Ionized electrons accelerated by strong optical fields. 2.Electrons gain energy from EM field, K – “jitter” energy” 3. Energy released when the trajectory brings electron back to ion and electron recombines with ion. 4. Maximum harmonic generated “m max ” 5.This process repeats every half cycle of the laser field, i.e. T period /2.. Fourier transform of capture event (and emission of radiation)  m=1, 3, 5 etc. 6. Plateaus predicted. Overall good agreement with experiment!

13. Electron-Ion Recombination

14. High Harmonic Generation From Few Cycle Pulses

15. Optimization Techniques Goals (1) (2) (3)

16. Adaptive Control C. Winterfeldt, Ch. Spielmann and G. Gerber, “Optimal control of high-harmonic generation”, Rev. Mod. Phys., 80, (1980) Control of: 1. Pulse shape 2. Pulse Duration 3. Frequency chirp 4. Interaction geometry

17. Optimization of Specific Orders Suppression of Specific Orders

18. Phase-Matching of Harmonics to Fundamental Dispersion of non-ionized gas Plasma dispersion Waveguide dispersion a – inner capillary radius u pq – p’th root of (q-1)th Bessel function Ion contribution neglected since free electron contribution is much larger and number densities the same Impossible to wave-vector match over large range of harmonics! z I(q)I(q)  k 2 >  k 1

19. Quasi-Phase Matching (QPM) In the flat regions, either zero nonlinearity or phase-mismatch large enough so that no net increase occurs. PPLN: Sign of the nonlinearity is periodically reversed. QPM: Grating or periodically reversed nonlinearity

20. Grating Induced By Counter-Propagating Beam 4541 37 33 29 4541 37 33 29 3 counter pulses 1 counter pulse Harmonic Order Enhancement Factor 1 10 10 2 Intensity (arb. units) 1 10 10 2 10 3 3 counter pulses 1 counter pulse Ar

21. Periodically Modulated Capillary Radius Modulation depth  5-10% 150  m diameter 1.0 mm Straight fiber Helium 2 4 6 8 10 Wavelength (nm) Signal (arb. Units) Modulated Hollow fiber

22. Nonlinear Birefringence in Gases at Ultra-High Intensities z y x Pump Beam Probe Beam 4040 Air molecules N2N2 N2N2 N2N2 N2+N2+ N2N2 N2N2 N2N2 N2N2 N2N2 N2N2 O2O2 O2O2 O2O2 O2O2 - - There have been some recent experiments in which the nonlinear birefringence of air and its major constituents have been measured at very high intensities with 90fs pulses. The interpretation is in dispute. The laser wavelengths was 800nm, non-resonant regime. x y 45 0 I pump >>I probe O 2 and N 2 are linear molecules are randomly oriented and at atmospheric pressure and temperature. The gas can be considered isotropic over a wavelength. Although strong ionization occured at their laser intensities due to multi-photon absorption, they neglect the contribution to the ionized electrons since they are “isotropic”. “Higher Order Kerr” (Optics Express, 17, 13429, 2009)

23. Values for pump-probe Kerr coefficients obtained by fitting to experimental data. Experimental Results

24. Nonlinear Index Change Intensity (Terrawatt/cm 2 ) I is a CW intensity. Is the Interpretation Physical? Linear refractive index of air is 300x10 -6 This represents a 10% change in magnitude and a change in sign! Usual perturbation expansion of NLO cannot work! But, the Keldysh parameter is about 0.1 so that multi-photon absorption dominates the ionization process.

25. Perturbative RegimeStrong Field Regime Extreme Nonlinear Optics Bound Electrons Free Electrons 10 12 W/cm 2 10 14 W/cm 2 10 16 W/cm 2 10 18 W/cm 2 : 1  Higher Order Kerr Experiments

26. Electron-Ion Recombination: Effect on Laser Field Electron motion driven by optical field Appearance of ionized electrons just outside (x 0 ) the atom/molecules Experiments in strong field regime have shown a blue shift in  0 with intensity. 1. Ion recombination process is ultrafast 2. During optical pulse, recombination occurs on a relatively fast time scale. Femtoseconds? 3. After the optical pulse, carriers diffuse and time scales of a 100ps – ns have been measured.

27. Free Electron-Based Alternate Explanations Third harmonic generation has been measured from an electron plasma, and the equivalent quantified by Suntsov et. al., Phys. Rev. A, 81, 033817 (2010). Contributions from electron plasmas was dismissed by the authors of the paper because it is isotropic. However a collective electron plasma should behave like a classical third order nonlinear material and exhibit birefringence, intense optical fields, for example as just discussed previously., via an effective third order nonlinearity n 2eff. At the pulse peak intensity of 50x10 12 W/cm 2,  plasma  0.1  0 with. Can this electron plasma nonlinearity explain the nonlinear birefringence experiment? Pump creates plasma string. Filament produces third harmonic due to air + plasma nonlinearity.

28. Plasma Nonlinearity Assume 1. collective electron effects such as plasma oscillation etc. exist, i.e. acts like a classical isotropic medium, i.e. 2. nonlinear response of electrons is just that of a single electron (in an intense field) X electron density ( N e ). 3. ultrafast response relative to pulse width of femtosecond lasers, i.e. 90 fs k xxxx <0 – nonlinear response Need to estimate response at  0 from measured third order nonlinearity

29. Nonlinear Birefringence Experiment

30. THG Prediction of Electron Plasma n 2 Assume: k xxxx is negative!! (other evidence suggests this.)

31. Experiment VS CW Theory Theory is essentially “plane wave” but does take into account nonlinear beam narrowing. It does not include: 1. plasma saturation and the 2. evolution of the plasma density over the temporal pulse 3. ultrafast nonlinearity which was verified experimentally Because of these factors, we definitely over-estimate the mature plasma contribution. However, it appears that the shape of the curves is due to electron density, i.e. the photo-ionization process.

32. Conclusions 1.There has been a paradigm shift in our understanding of the interaction of matter with intense optical fields when the optical field becomes comparable to the fields inside atoms and molecules. The ionized electrons can dominate the nonlinearity! 2.Coupled with shorter and shorter pulse lasers, dramatic new science and technology will emerge. 3. Characterization (not discussed) and creation of ultrashort pulses depend s on nonlinear optics!