Probing the Propagation Dynamics in a Femtosecond Laser Filament

Probing the Propagation Dynamics in a Femtosecond Laser Filament Transient Grating XFROG Measurements of Filament Propagation Dynamics On the Mechanism of Negative Birefringence (Nonlinear Polarizability in N 2+ ) AFOSR 2 nd Year MURI Review 18 September 2012 Robert J. Levis Center for Advanced Photonics Research Temple University Philadelphia, PA

Cast DA Romanov M Tarazkar J Odhner E McCole

Recent Submissions and Publications Ionization-Grating-Induced Nonlinear Phase Accumulation in Spectrally-Resolved Transient Birefringence Measurements at 400 nm, J. H. Odhner, D.A. Romanov, E. T. McCole, J.K. Wahlstrand, H.M. Milchberg and R.J. Levis, PRL 2012, 109, 065003 Post-ionization medium evolution in a laser filament: A uniquely non-plasma response, D.A. Romanov* and R.J. Levis, Phys Rev E, Accepted w/ minor revision Direct Phase and Amplitude Characterization of Femtosecond Laser Pulses Undergoing Filamentation in Air, J. H. Odhner, and R. J. Levis, Optics Letters 37, 1775, 2012 Filament-Driven Impulsive Raman Spectroscopy, J.H. Odhner, E.T. McCole, and R.J. Levis, J. Phys. Chem. 2011, 115, 13407. Theoretical Study of the Polarizability and Second-Order Hyperpolarizability for N 2 and N 2+, M. Tarazkar, D.A. Romanov, R.J. Levis, J Chem Phys. For Submission

Intense Laser Pulse-Molecule Interactions Given 1mJ, 60fs focused to 100 micron diameter: photon density ~10 9 photons/cubic wavelength electric field strength ~6 Volts per Angstrom. Butadiene in a Strong Laser Field E Molecule The Laser Molecule E Field of Laser ~V/Angstrom Hijacked System PRL, 1998, 81, 5101 Science, 2001, 292, 709 PRL, 2004, 92, 063001 PRL, 2006, 96, 163002 PRL, 2009, 102, 155004 PRL, 2009, 103, 075005 PRL, 2009, 103, 205001 PRL, 2010, 105, 125001 PNAS, 2011, 108, 12217 PRL, 2012, 109, 065003

Laser Filaments Form Near Single Cycle Pulses Pre-Filament Kerr lensing high intensity High intensity self phase modulation 2 mj 45 fs laser pulse Filament High intensity ionization (10 16 e - cm -3 ) Ionization intensity clamping 10 13 W cm -2 Spatial temporal focusing self shortening Self shortening < 10 fs pulses The Impulsive Source Short intense pulse provides impulsive rotational and vibrational excitation

The Spatio-Temporal Dynamics Forming the Filament electric field envelope diffraction E E E z t k t 2 fil fil i i fil k0 E fil k0 2 2 0 2 group velocity group velocity dispersion 6 i 12 i k c k c t t 2 2 (3) 2 (3) 2 0 0 E 2 fil E fil 2 2 E fil E fil 0 0 20, Kerr lensing Self-steepening Spatial and Temporal Kerr Nonlinear Effects From 100 oscillations of the electro-magnetic field to two!

Self Compression In Pulse As the broad band pulse propagates, the system selfcompresses to ultrashort (~5 fs) features Radial and Temporal Features vs. Position 30 April 2007 / Vol. 15, No. 9 / OPTICS EXPRESS 5396

Fluorescence Intensity Measuring Filament Propagation in Air 2 mj, 40 fs 2 m lens Bree et al. Opt. Ex. 17, 16429, 2009 Probing filamentation dynamics is problematic due to high intensity 10 13 W cm -2 Fluorescence: Talebpour, Optics Communications 171, 285 (1999). Acoustic: Yu, Applied Optics 42, 7117 (2003). Conductivity: Eisenmann, Physical Review Letters 98, (2007). Post Filament SPIDER: Stibenz, Optics Letters 31, 274 (2006).

Probing Filament Dynamics via Impulsive Raman Spectroscopy

Criteria for Impulsive Vibrational Excitation Temporal t FT Spectral Irradiating a molecule with a sufficiently short pulse results in: i, a superposition of vibrational states ii, a time-dependent wavepacket iii, a time-dependent polarizability XFROG Characterization of Filament c i 9 fs w/5 fs edge

Measuring the Pulse Duration as a Function of Distance in Filament Using ISRS spectrometer N 2, 14fs, 674nm O 2, 21fs, 712nm ps probe beam filament propagation distance

Filament Profile: Fluorescence and ISRS 2 meter lens focal length Filament Fluorescence Impulsive Raman Scattering Intensity PRL 105, 125001 (2010) oxygen 21 fs (1554.7 cm -1 ) nitrogen 14 fs (2330.7 cm -1 ) Water 9 fs (3657.1 cm -1 )

z (cm) Filament Dynamics Via ISRS Phys. Rev. Lett. 103, 075005 (2009) Phys. Rev. Lett. 105, 125001 (2010) Spectrum 400nm 600nm 900nm Fluorescence Intensity λ (nm) Impulsive Raman Intensity

Filament Phase and Amplitude from Impulsive Raman Scattering θ Gaussian probe pulse 2 ' 2 2 2 2 0 0 sin 2 2 t t t t z 2 2 2 2, ' R R probe probe probe E t E e e E e e The resulting polarization P obs, k, z NE 2 1 probe mn ng ml lg 3 3 sin n, m, l lg ng obs mg probe obs 15 2 R 2 2 R kz k probe cos 2 2sin 2 e d r d ' E ', z, r E ', z, r fil fil obs probe

Characterizing Spectral Phase and Amplitude in a Filament via Transient Grating X-FROG The filament formed using a 2-m lens is diffracted off of an optical grating formed by two reference beams. The spectrallyresolved autocorrelation is then inverted to recover the pulse electric field and spectrum. Measurements as a function of distance from the lens and input power reveal filament formation and propagation dynamics. Measurement Optics Letters 37, 1775, 2012 Reconstruction

Filament XFROG vs. z Distance distance, z, from 2 m lens z = 194.4 cm z = 196.4 cm z = 206.4 cm z = 210.3 cm z = 215.3 cm z = 224.3 cm z = 230.3 cm z = 243.7 cm z = 247.7 cm z = 251.7 cm z = 253.7 cm z = 260.7 cm

Filament XFROG vs. z Distance distance, z, from 2 m lens z = 194.4 cm z = 196.4 cm z = 206.4 cm z = 210.3 cm z = 215.3 cm z = 224.3 cm z = 230.3 cm z = 243.7 cm 5 fs edge z = 247.7 cm z = 251.7 cm z = 253.7 cm z = 260.7 cm

TG-XFROG in Excellent Agreement with Theory

Spectrally-Resolved Transient Birefringence to Determine Mechanism Pump Detect Probe IP gs Odhner, et al. PRL 2012, 109, 065003

The Higher Order Kerr Measurement Loriot, V. et al. Laser Physics, 2011, 21, 1319-1328.

Transient Birefringence in Optical Kerr Effect An optical transient birefringence experiment measures the anisotropic response of a medium due to the presence of a large optical electric field. The electric field distorts the potential experienced by electrons bound to atoms and molecules, which changes the degree to which a weak probe field interacts with the medium depending on its polarization. Filament Polarization Probe Polarization Birefringence e - e -

Ionization-Induced Birefringence Optical Interference Plus Ionization Provides Ionization Grating- Induced Phase Accumulation Wahlstrand and Milchberg proposed that the polarization component of the probe pulse parallel to the pump polarization will selectively participate in ionization, leading to an effective birefringence.

The Model for Plasma Grating-Induced Birefringence at 400nm Pump Probe At 400 nm, four photons are required to ionize O 2 At low probe intensity, one photon from probe can add to pump photons to provide an excitation-ionization path. Interference between pump and probe sets up an intensity grating. Interaction or exchange of probe photons in ladder climbing leads to phase accumulation. IP gs

Two-beam phase coupling Pump (and probe) can diffracts into probe (and pump) direction to give energy coupling. If frequencies of pump and probe are not degenerate, the grating moves appreciably on the time scale of pulse duration and diffraction washes out. In low intensity limit, there is also phase coupling. 0 Probe Pump Detect Pump N (, ) t 1 t e r t dt W Ee t e dt Ep t W Ee t e c c N q r 2 Probe i i t ' ' ' ' ' ' '.. Experimental Demonstration PRL 2012, 109, 065003

, 2 Kerr Birefringence Measurements Loriot, et al. showed apparent inversion at high intensity for transient birefringence using 800nm pump and probe. We repeated measurements using both 800nm and 400nm probes to test Milchberg hypothesis. 800nm or 400nm probe 800nm pump

800nm Filament, 400nm Probe Measurement Initial results of experiments performed at ~40 TW/cm 2 show that no inversion of the birefringence is observed at zero delay in nitrogen using a 400 nm probe pulse, though inversion is observed at 800 nm. Time-Resolved Birefringence for Degenerate and Nondegenerate Beams Birefringence Response at t=0 for 800nm, 800nm and 800nm, 400nm 400 nm probe 800 nm probe Difference in nonlinearlties between 800nm and 400nm could account for the observation.

Testing the Ionization Grating Theory using Nearly Degenerate Beams Spectrometer 10μm BBO ~uj 50 fs ~18 THz λ/4 f =20 cm λ/2 ~mj 50 fs λ/2 λ/2 ~10 THz 1mm BBO To test the new theory, the spectrally-resolved transient birefringence of air was measured at 400 nm.

Spectrally-Resolved Measurements at 400 nm Double light in two different length BBOs provides different bandwidths Spectrally-resolving adds new dimension to data, revealing positive sidebands on the leading edge of the pump not otherwise observable. PRL, submitted Where the frequencies of pump and probe are degenerate there is negative contribution Otherwise there is no inversion as would be expected by HOKE

Simulating the Two Beam Coupling Geometry e e E pump k e E probe k e k p k p e k p k p e e 12 n0 ikr e iet E pump ee I Ae t kere c.. c cke 12 ik ri t p p E I e A t, r e A t, r e c. c. probe p e A e Ip I Basic assumption: is not changed by the interaction, and it does not undergo noticeable (self)focusing over the interaction length.

Plasma Nonlinearity I: Conventional Nonlinear Refraction The plasma is produced by the ionization of the medium at the rate depending on the local field intensity Which leads to the corresponding nonlinear variation in the refraction index where t N npl t z n dt R E t z 2 neut, 0 ' ', N dn R E t z N dt c m 4 e 2 e 2 is the critical plasma density e, neut Nc In the two-beam coupling setup, this will instigate an equal negative phase shift in both polarization components of the probe pulse.

Plasma Nonlinearity II: Incorporating Multi Photon Ionization As the ionization rate depends on the total intensity, in the two-beam situation, the generated plasma density will undergo periodic spatial variations due to the small but significant interference term. In the case of multiphoton (m-photon) ionization, m 2m I m p 2m2 * i k p ke zi p ep t R E t, z m Itotal mi Ae 2 m Ae Ae A e c. c I Accordingly, the plasma-related correction to the refraction index will form a grating capable of initiating scattering between the pump and the probe beams: t m N I p, neut i k p ke z f z A ip ep npl t, z n0m I f t, z 2 m e d e c. c 2 N c I. A e where 2 t f t, z d Ae, z m Plasma grating terms

The Combined Transient Birefrigence Signal from Kerr, Transient Plasma Grating and Rotation Keeping only the first term in the sum for the plasma heterodyne signal and adding the Kerr-related terms, we obtain the following expression for the heterodyned signal intensity to be compared with the spectrally resolved birefringence measurements: where:

400/400 Birefringence trend The maximum negative amplitude of the center wavelength component was measured and fit as a function of pump intensity, demonstrating a fourth-order dependence. Time-Resolved Birefringence as a Function of Pump Power Phys. Rev. Lett. 109, 065003 (2012) 400 nm trends are consistent with an ionization-induced birefringence in the multi-photon regime

Simulation Results HOKE Simulation Using a Gaussian input pulse with FWHM =54 fs, no chirp, and I e =71 TWcm -2 we obtain: w/ experimental beam parameters

800/800 Kerr Birefringence trend The trends at 800 nm are consistent with the results of Loriot et al. Measurements are consistent with Loriot et al. Optics Express 17 (16), 13429 (2009)

TG XFROG Analysis of Low Intensity Data at 800nm pump and probe Optical Interference Kerr 10.1 TWcm -2 At low intensity only the Kerr and Rotational alignment are visible. Rotational Alignment

Spectrally resolved data 27.5 TWcm -2 With identical pump and probe pulses it we are unable to distinguish between a pure Kerr and an ionization-driven process at 800 nm. 36.2 TWcm -2

Conclusions for Nonlinear Probing of Filamentation Impulsive Raman measurements reveal evidence for ~8 fs bandwidth TG-XFROG reveals spectral phase and amplitude as a function of propagation length in a filament. Agreement with theory. No evidence for HOKE in filamentation at 400nm

Modification of Polarizability and Hyperpolarizability of Nitrogen Molecules upon Ionization In the process of filamentation, ionization changes optical characteristics of the medium. So far, most attention have been paid to the optical response of the free electrons. However, polarizability and hyperpolarizability, and higher-order hyperpolarizabilities of emerging ions may differ considerably from those properties of neutral molecules, and this change may influence the filamentation process. It is thus desirable to establish a consistent theoretical method for calculating microscopically nonlinear refractive index n= n0 + n2 I + where on the level of individual molecules n2 corresponds to second-order hyperpolarizability g.

Challenge The available suites of quantum-chemical programs (e.g. Gaussian09) are not amenable to calculating dynamic hyperpolarizabilities. N 2 ion is an open-shell system in which correlation effects are expected to be more pronounced and require a special treatment.

Auxiliary Static Field Approach A power-series expression for the molecular quasi energy E in the electric field in terms of polarizability and hyperpolarizability.[1] E( F ) i E 0 i i F i i 1 2! ij ij F F i j 1 3! ijk ijk F F i j F k 1 4! ijkl ijkl F F i j F k F l... Here: E i is the energy of the molecule in the absence of electric field, µ i 0 is dipole moment, α ij is the polarizability, β ijk is the first order hyperpolarizability and ϒ ijkl is the second order hyperpolarizability. These characteristics can be expressed as derivatives of the quasienergy with respect to the electric field. E( F) 4 2 3 E( F) E( F) E( F) i ( ) 0 ij ( ) 0 βijk ( ) ijkl ( ) 0 0 F F i i F j Fi F j F Fi F j Fk Fl k [1]: Hari Singh, Nalwa, Junji Mukai Atsushi Kakata, Journal of Physical Chemistry, 1995, 99, 10766-10774

Auxiliary Static Field Approach We use the results of Gaussian09 calculations for α ij and β ijk as nonlinear functions of auxiliary electric field. In the first approach, these data are numerically differentiated and the result is extrapolated to zero field. In the second approach, the data for polarizability are fitted with a polynomial whose coefficients are related to the hyperpolarizabilities.

Benchmark: Nonlinear Polarizability versus Electric Field for N 2 Molecule 2 2 1 0 * F F 2 2 2 2 1 ) ( F F F F E

Second-order Hyperpolarizability for N 2 Ion Obtained in the two Approaches xxxx zzzz xxzz HF 1318.82 1309.1 ± 11.0 2049.1 2032.7 ± 40.8 1795.4 1777.9 ± 42.1 MP2 15323.98 14535.6 ± 217.5 15400.9 15187.8 ± 409.9 15439.0 15349.3 ± 493.6 MP4(SDQ) 10976.52 10336.7 ± 148.4 12048.1 11864.3 ± 301.7 11298.6 11246.4 ± 356.5 CCSD 929.66 705.3 ± 29.5 6770.0 6628.5± 183.6 2194.0 2137.3 ± 94.2 All values are reported in atomic units. As it was the case with N 2 molecule, there is a good agreement between the results obtain from numerical differentiation and the results obtain from polynomial fitting.

N 2 Molecule vs. N 2 Ion xx zz zzzz xxxx xxzz Molecule Ion 10.09 9.84 14.67 18.67 4.59 8.83 11.61 12.78 1146.7 6770.07 604.5 929.66 296.4 2194.00 788.9 3605.03

Change in Hyperpolarizability: Root Causes The change in polarizability upon ionization depends on two competing factors: 1) when one of the electrons contributing to the polarizability is removed, the overall response of electron system to the external field is expected to become proportionally smaller; 2) the electronic excitation energies in the ion are typically smaller than those in the neutral molecule resulting in enhanced linear and nonlinear response. Indeed, the calculations of electronic excited states for N 2 molecule and ion indicate that the excitation energies undergo a considerable decrease upon ionization, compensating for the decrease in the number of electrons leading to an increase in hyperpolarizability.

Conclusions The developed methodology allows for calculation of static second-order hyperpolarizabilities for ionized molecules. The calculated hyperpolarizability values are expected to be relevant in long-wavelength limit. At realistic strong electric field magnitude of 2.7 V/Å, the calculated nonlinear contribution to the induced dipole moment of nitrogen ion is several times greater than that of the neutral molecule. The results obtained indicate the enhanced role of excitations in the polarization response of ion as compared with its relative neutral molecule. The developed polynomial-fit approach may be extended to higher-order hyperpolarizabilities.

Impulsive Raman Time-Domain Detection laser vib CO 2 spectrometer laser, 1 = laser, 2 < vib Fourier Transform

Intensity (arb. units) Impulsive Raman Frequency-Domain Detection laser, probe laser, pump vib probe probe - vib N 2 spectrometer laser, pump < vib << laser, probe O2 N2 1554.7 cm -1 2330.7 cm -1 probe probe ± vib pump prepares coherence Stokes and anti-stokes Raman Shift (cm -1 )

(arb. units) Impulsive Stimulated Raman Spectrum in Air An 800 nm drives both the impulsive filament pulse (2 mj) and the weak (1.65 uj) 400 nm probe pulse Impulsive Raman Spectrum ~1.5 ps delay O2 N2 1554.7 cm -1 2330.7 cm -1 BBO delay stage 400 nm 10 μj 1.5 ps 800 nm 2 mj 55 fs spectrometer spectral filter Raman Shift (cm -1 ) Spatial filter removes 400 nm probe Impulsive Raman provides high sensitivity, rapid acquisition time (10 ms), few laser shots (10).

Time Scales for Molecular Motion Determine Pulse Duration for Impulsive Excitation Protein Backbone 500 femtosecond to ps CO 2 vibration 30 femtosecond O 2 vibration 21 femtosecond N 2 vibration 14 femtosecond C-H, O-H Stretch 10 fs H 2 vibration 8 femtosecond

fluorescence intensity A Primer on Femtosecond Laser Technology phase Oscillator 20fs, 84 MHz, 5 nj Stretch ~ 5000 X Emission Bandwidth 100 ps Amplify 10 6 X 1 khz 5 mj 100 ps 1 KHz 2.5 mj 30 fs 700 900 wavelength, nm Compress ~ 5000 X

Intensity dependence of the negative birefringence The maximum negative amplitude of the center wavelength component was measured as a function of energy, demonstrating a third-order dependence. Time-Resolved Birefringence as a Function of Pump Power Order of Negative Birefringence from Signal vs Laser Intensity dotted line =3rd-order

The Equation for the Probe Envelope In the moving and stretched frame, e e z' A t z A t z z' zn z z, t t c and in the slow varying envelope approximation, the equation for the wave propagating in k p direction reads: ', ' e ', ' ee ', ' e ', ' ' i A t z A t z f t 2 ' p i p e t ip e t e e ' e e t ' f A t, z ' t A t me A t e dt e 0 where: A t ', z ' A t, z ; A t ', z ' A t, z ; 2 k p mnneut m I ; f t d Ae n N 2 0 c t 2 m

Adding Regular Kerr Nonlinearities electronic rotational (delayed) When we keep the interference term in the expression for the total intensity The nonlinear correction to the refraction index duly exhibits terms of two kinds Smooth Grating J. F. Ripoche, et al. Opt. Commun. 135, 310 (1997)

The Solution for the Propagation Equations in the Degenerate Case For the sake of simplicity, we assume izf t A t, z e A t,0 m l1 A,,,0 The propagation equation for iz f t A t z A t e t z p e is solved immediately: The exponential factor signifies the accumulation of the negative phase delay caused by the isotropic modification of the refraction index. t m! l A,0 A t, z What is more surprising, the integro-differential equation for can also be solved analytically by means of the Laplace transform to produce: isotropic term l 1 f Ae t iz d f t f l! l 1! m l! A e Grating terms

The Spectrum of the Heterodyne Signal For heterodyning, the quarter-wave-rotated initial probe signal is used as the local oscillator signal. Then, since for air and 400 nm light m=4, the plasma-related component of the heterodyne signal is explicitly obtained as it * Re d itizf t IH e A,0 dte Ae t t * 4 iz d A,0 A A t 2 * e e t 3 * 6 E e e probe e 6 z d f t f A,0 A A 2 e e t 4 * E LO 2 i z d f t f A,0 A A 1 6 e e 3 6,0 e e z d f t f A A A 6 6 k p

Spatio-Spectral Evolution glass flat transmitted beam 0.01% spectrometer filament 204cm 221cm 262cm deflected filament Integrated intensity from 470nm to 330nm as a function of propagation length reproduces gain in H2O stretch

Filament Profile: Fluorescence and ISRS 2 meter lens focal length Filament Fluorescence Impulsive Raman Scattering Intensity PRL 105, 125001 (2010) oxygen 21 fs (1554.7 cm -1 ) nitrogen 14 fs (2330.7 cm -1 ) Water 9 fs (3657.1 cm -1 )

Temporal Dynamics of N2 Raman Reveals Temp Optical Response as a function of timedelay between pump and probe 400 nm probe delay stage BBO spectrometer spectra filter Line out reveals temporal dynamics Nitrogen Raman Line

Temporal dynamics of N2 e = 2330.7 cm -1 x e = 14.45 cm -1 N2 e - 4 e x e e - 2 e x e rotational rotational revivals Autocorrelation from revivals collinear from quantum wakes collinear quantum wakes Time-dependent polarization:

Dispersion of the Rotational-Vibrational Wavepacket Excite multiple ro-vibrational states that are initially in phase. With increasing time the coherent wave packet disperses via: rotational constant rotation-vibration coupling vibrational frequency anharmonicity

Temporal Dynamics at High Temperature Temporal Dispersion of Diatomic Vibrational Wavepackets N2 decay ~300 K 400nm probe N2 decay ~2950 K Bunsen Burner Phys. Rev. Lett. 103, 075005 (2009) 1/e 2 dispersion times are on the order of ~10 ps.

Near Single Cycle Features in Filaments Enables Detection of All Raman Active Molecules Filament With H 2 Source Single Shot Measurement from 300cm -1 to 4155cm -1 H 2 Source H 2 vibration at 4155cm -1 requires A <8 femtoseconds feature for impulsive excitation CENTER FOR ADVANCED PHOTONICS RESEARCH CAPR