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Generation and control of high-order harmonics by the Interaction of infrared lasers with a thin Graphite layer. Ashish K Gupta & Nimrod Moiseyev Technion-Israel Institute of Technology, Haifa, Israel. Light – Matter Interaction. Photo-assisted chemical reactions.
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Generation and control of high-order harmonics by the Interaction of infrared lasers with a thin Graphite layer Ashish K Gupta & Nimrod Moiseyev Technion-Israel Institute of Technology, Haifa, Israel
Light – Matter Interaction Photo-assisted chemical reactions Reactant A, product B are chemicals and light is a catalyst. Harmonic Generation Phenomena Reactants and product are photons and chemicals are a catalyst.
Mechanism for generation of high energy photons (high order harmonics) k ħω E Acceleration of electron Probability to get high energy photon ħΩħω: Multi-photon absorption z Radiation ħΩ
Quantum-mechanical solution Time-dependent wave-function of electron (t) Acceleration of electron Hamiltonian with electron-laser interaction Linearly Polarized light: Circularly Polarized light:
Harmonic generation from atoms Experiments Highly nonlinear phenomenon: powerful laser 1015 W/cm2& more Incoming laser frequency multiplied up to 300 times: The intensity of emitted radiation is 6-8 orders of magnitude less than the incident laser intensity.
Molecular systems Graphite Carbon nanotube symmetry C6 symmetry C178 • Our theoretical prediction of Harmonic generation from symmetric molecules: • Strong effect because higher induced dipole • 2) Selective generation caused by structure with high order symmetry Benzene symmetry C6
Why do atoms emit only odd harmonics in linearly polarized electric field ? Non perturbative explanation (exact solution) Selection rules due to the time-space symmetry properties of Floquet operator. CW laser or pulse laser with broad envelope (supports at least 10 oscillations) has 2nd order time-space symmetry:
An Exact Proof for odd Harmonic Generation An exact proof: For atoms: Space symmetry Time symmetry Time-space symmetry:
Floquet Theory An exact proof: - Floquet State Floquet Hamiltonian has time-space symmetry:
Dipole moment: Probability of emitting n-th harmonic: An exact proof: For non-zero probability, the integral should not be zero.
For a non-zero integrand, following equality must hold true: For even n=2m: For odd n=2m+1: An exact proof: Therefore, no even harmonics
Atoms in circularly polarized light Symmetry of the Floquet Hamiltonian: Floquet Hamiltonian has infinite order time-space symmetry, N= Selection rule for emitted harmonics:Ω=(N 1)ω, (2N 1)ω,… Hence no harmonics
Symmetric moleculesCan we get exclusively the very energetic photon??? Circularly polarized light ħω ħΩ, Ω=(N 1)ω, (2N 1)ω,… CNsymmetry YES Low frequency photons are filtered: Systems with N-th order time-space symmetry:
Graphite C6symmetry (6th order time-space symmetry in circularly polarized light) Numerical Method: 1) Choose the convenient unit cell 2) Tight binding basis set 3) Bloch theory for periodic solid structure 4) Floquet operator for description of time periodic system 5) Propagate Floquet states with time-dependent Schrödinger equation.
Graphite Lattice A F B E C D Direct Lattice with the unit vectors
Tight Binding Model A Bloch basis set is used to describe the quasi energy states , F E A D α denotes an atom (A-F) in a unit cell. The summation goes over all the unit cells [n1,n2], generated by translation vectors . B C F E 2py,A A D 2px,B B C σ-basis set: j={2s,2px,2py}, j=1,2,3 π-basis set: j={2pz}, j=1 σ- and π-basis sets do not couple. Only nearest neighbor interactions are included in the calculation.
Formula for calculating HG The probability to obtain n-th harmonicwithin Hartree approximation is given by The triple bra-ket stands for integration over time (t), space (r), and crystal quasi-momentum (k) within first Brillouin zone. The summation is over filled quasi-energy bands. The structure of bands in the field:
Localized (σ) vs. delocalized (π) basis π – electrons are delocalized freely moving electrons, with low potential barriers, hence low harmonics σ – electrons tightly bound in the lattice potential, hence high harmonics
Intensity Comparison Minimal intensity to get plateau: 3.56 1012 W/cm2 Plateau: Intensity remains same for a long range of harmonics (3rd-31st)
Graphite vs. Benzene HG from Benzene-like structure dies faster than HG from Graphite. No enhancement of the intensity using circularly vs. linearly polarized light is obtained, Hence it is a filter, not an amplifier.
Conclusions • High harmonics predicted from graphite. • Interaction ofCNsymmetry molecules/materials with circularly polarized light rather than with linearly polarized light, generates photons with energy ħΩwhere Ω=(N 1)ω, (2N 1)ω,… • 3. Circularly polarized light filters the low energy photons, however no amplification effect is predicted. • 4. Extended structure produces longer plateau as seen in the case of Graphite vs. benzene-like systems . • 5. HG in graphite is stable to distortion of symmetry. For 1% distortion of the polarization the intensity of the emitted 5th (symmetry allowed) harmonic is 100 times larger than the intensity of the 3rd (forbidden) harmonic.
Thanks Prof. Nimrod Moiseyev Prof. Lorenz Cederbaum Dr. Ofir Alon Dr.Vitali Averbukh Dr. Petra Žďánská Dr. Amitay Zohar Aly Kaufman Fellowship
Reciprocal Lattice Potential: V(r)=V(r+d); d=d1a1+d2a2 b1 For the translation symmetry to hold good: n=n1b1+n2b2 b2 Reciprocal lattice: Brillouin zone
Bloch Function d=d1a1+d2a2 Brillouin Zone : k and k+2pi*n correspond to same physical solution hence k could be restricted. For a cubic lattice:
