SemiDiscrete Solitons in Arrays of Quadratically Nonlinear Waveguides

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Semi-Discrete Solitons in Arrays of Quadratically
Nonlinear Waveguides
Nicolae Panoiu1, Richard Osgood1 and Boris
Malomed2 1Columbia University, Department of
Applied Physics and Applied Mathematics 2Tel Aviv
University, Department of Interdisciplinary
Studies
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• Introduction
• We study soliton formation in arrays of
  quadratically nonlinear waveguides coupled to
  slab waveguides.
• We demonstrate that this structure supports
  semi-discrete solitons, which possess new and
  remarkable physical properties.
• Motivation
• Study nonlinear modes with new physical
  properties.
• Excitation of nonlinear modes in media with both
  discrete and continuous properties.
• Are there new physics? And applications to new
  opto-electronics devices (switches)?

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Device Geometry Mathematical Model
A) Discrete Fundamental Frequency (FF)
Continuous Second Harmonic (SH)
Normalized equations

• Total Power
• Hamiltonian

r coupling constant b wave-vector mismatch
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Device Geometry Mathematical Model
B) Continuous Fundamental Frequency Discrete
Second Harmonic
Normalized equations

• Total Power
• Hamiltonian

r coupling constant b wave-vector mismatch
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A) Discrete FF Continuous SH
Odd solitons

• Stationary solutions

• Both odd and even soliton composites are stable
  if lgt2.

Even solitons

• Odd solitons have smaller power are more
  stable.

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B) Continuous FF Discrete SH
Odd solitons

• Stationary solutions

• Both odd and even soliton composites are stable
  if l 1b/2.

Even solitons

• Odd solitons have smaller power are more
  stable.

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A) Discrete FF Continuous SH

• Field profiles of odd and even soliton composites

• Odd solitons
• b 0
• l 2.75.

• Even solitons
• b 0
• l 3.

• continuous component spikes at waveguides
  position.
• power concentrated in the discrete component.

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B) Continuous FF Discrete SH

• Field profiles of odd and even soliton composites

• Odd solitons
• b 0
• l 1.5.

• Even solitons
• b 0
• l 1.5.

• continuous component spikes at waveguides
  position.
• powers in discrete and continuous components
  about equal.

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A) Discrete FF Continuous SH

• Stable propagation of perturbed odd soliton
  composite

• b 0 l 2.4.
• Initial condition stationary solutions 10
  white noise.

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B) Continuous FF Discrete SH

• Stable propagation of perturbed odd soliton
  composite

• b 0 l 1.1.
• Initial condition stationary solutions 10
  white noise.

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Twisted Soliton Composites

• Stability properties of odd twisted solitons

• In both cases odd twisted solitons are stable in
  a large region of the wave-vector l.
• The domain of stability is larger in case B.

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Field Profiles of Twisted Odd Solitons

• Discrete FF Continuous SH.
• b 0
• l 3.25.

• Continuous FF Discrete SH.
• b 0
• l 2.75.

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Stable propagation of twisted odd soliton
composite

• b 0 l 2.75 (case B).
• Initial condition stationary solutions 10
  white noise.

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Conclusions

• New kinds of nonlinear modes (semi-discrete
  solitons) exist in a WGA coupled to a slab WG.
• Newly found even semi-discrete composite
  solitons are stable in a large parameter domain.
• Twisted (in FF) semi-discrete composite solitons
  are stable in a large parameter domain.
• Comprehensive numerical simulations confirm
  theory.