The role of

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The role of Egap in filamentation

• Johanan Odhner
• 3-18-08

johanan_at_temple.edu
http//www.temple.edu/capr/
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Introduction

• When an ultrashort, ultra-intense laser pulse
  travels through a transparent medium many highly
  nonlinear effects occur. Cumulatively these
  effects are described as filamentation when
  they result in dramatic spectral, temporal, and
  spatial changes in the pulse itself.

pclasim47.univ-lyon1.fr/ teramobile.html
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Overview

• Polarizability and the nonlinear refractive index
• Plasma generation in filamentation
• Spectral Broadening
• Conclusion

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The nonlinear refractive index

• The polarizability of a medium can be described
  in general terms by the following equation

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The nonlinear refractive index
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Self-phase modulation and self-focusing

• Two important effects resulting from the
  nonlinear refractive index are
• Self-phase modulation (SPM)
• Self-focusing

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Self-phase modulation

• As an high intensity, ultrashort pulse propagates
  through a medium it accumulates a nonlinear
  phase
• This corresponds to a shift in the instantaneous
  frequency

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SPM and the intensity

• Thus, as a pulse propagates through a material
  where n2 gt 0 it will accumulate new colors
  proportionally to the change in intensity.

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SPM and the spectrum
Nonlinearly induced blue-shifting
Initial Pulse
Nonlinearly induced red-shifting
Initial pulse spectrum
post-SPM pulse spectrum
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Self-focusing

• Index of refraction through a convex lens

• Index of refraction due to the nonlinear
  refractive index

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Catastrophic Pulse Collapse
Self-focusing starts
Catastrophic collapse of pulse!

• As the pulse self-focuses, the contribution of
  the nonlinear refractive index increases. This,
  in turn, increases self-focusing. With no
  limiting mechanism this would lead to
  catastrophic pulse collapse.

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Plasma generation

• As the pulse collapses, the intensity increases
  enormously, which leads to plasma generation.
  This results in defocusing of the pulse.

• Such plasma channels have been observed to
  propagate over distances many times their
  diffraction length.

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A moving-focus model

• The moving-focus model visualizes the pulse as a
  transverse stack of quasi-CW slices, each of
  which focuses at a distance

where
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Intensity Clamping

• Plasma defocusing imposes a limit on the
  intensity which can be contained in a filament.
  For pulses with power in excess of Pcrit, energy
  is redistributed into secondary filaments which
  have the same clamping intensity as the primary
  filament.

Picture from Schroeder and Chin, Opt. Comm. 234,
399406 (2004)
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Mechanisms of intensity clamping

• Different mechanisms which have been proposed for
  intensity clamping are
• Plasma generation by avalanche ionization
• Plasma generation by multi-photon ionization (MPI)

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Multi-photon absorption/ionization

• The transition rate for MPA/MPI can be described
  by the equation
• wheresis the n-photon absorption coefficient and
  I is the intensity. Typically, higher order MPA
  processes require much higher intensities in
  order to occur.

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MPI in different media

• The number of photons required to liberate
  electrons is different for different media.

MPI in water
MPI in carbon disulfide
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Definition of Egap

• Egap, named after the band-gap in semiconductors,
  is generally the HOMO-LUMO gap energy, or the UV
  absorption cutoff. In gases the ionization
  potential is usually used for this quantity.

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MPI in the moving-focus model

• When MPI occurs, slices focusing temporally and
  spatially after plasma formation are effected by
  the plasma contribution to the refractive index.

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The plasma-modified refractive index

• The refractive index at the back of the pulse can
  be written
• where the plasma contribution is negative.
  Because of this there is a drop in the intensity
  at the back of the pulse.

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MPI affects accumulation of ?NL

• Using the moving-focus model we can see that
  phase will be accumulated at the front of the
  pulse, then experience a sudden drop where plasma
  has been formed.

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Self-steepening
Reference Shen, Gaeta

• The nonlinear refractive index can also produce a
  sharp drop in phase through self-steepening. This
  happens when high intensities are reached during
  propagation

high intensity larger nTOT
peak retarded in time
intensity decreases rapidly
lower intensity smaller nTOT
wings travel at normal speed
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MPI-enhanced SPM leads to asymmetric spectral
broadening

• The result is highly asymmetric spectral
  broadening.

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Spectral broadening increases with Egap

• In condensed media it has been shown that higher
  Egap materials induce more spectral broadening.

Brodeur and Chin, J. Opt. Soc. Am. B 16, 637
(1999)
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Conclusion

• The UV absorption cutoff is an important factor
  condensed media filamentation
• It is useful in predicting the spectral extent of
  the continuum generated

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References

• Brodeur and Chin, J. Opt. Soc. Am. B 16, 637
  (1999)
• Chin et al, Can. J. Phys. 83, 863-905 (2005)
• Brabec and Krausz, Phys. Rev. Lett. 78, 3282
  (1997)
• Fuß et al, New J. of Phys. 8, (2006)
• R. W. Boyd. Nonlinear Optics. 2nd ed. Academic
  Press, London. 2003
• DeMartini and Townes, Phys. Rev. 164, 312 (1967)
• Gaeta, Phys. Rev. Lett. 84, 3582 (2000)