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Phase Matching

Phase Matching. Alex Filin. Everything you always wanted to know about it but were afraid to ask. Outline. Introduction: Origin of Optical Nonlinearity Phase Matching in SHG Phase Matching in CARS Conclusion. Origin of optical nonlinearity: mechanical analog. Nonlinear conditions

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Phase Matching

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  1. Phase Matching Alex Filin Everything you always wanted to know about it but were afraid to ask

  2. Outline • Introduction: Origin of Optical Nonlinearity • Phase Matching in SHG • Phase Matching in CARS • Conclusion

  3. Origin of optical nonlinearity:mechanical analog Nonlinear conditions Force: Linear conditions Force: Potential: Potential:

  4. Origin of optical nonlinearity:Polarization

  5. Origin of optical nonlinearity Linear conditions Nonlinear conditions • Where P is polarization • eo is free-space permittivity • is susceptibility E is electric field • Where • (i) is nonlinear susceptibility of ith order

  6. Origin of optical nonlinearity • Third Harmonic Generation • Kerr effect • All types of FWM phenomena, • including CARS • All mixing phenomena, • involving generation of • sum and difference • frequencies (SHG, • parametric amplification) • Pockels’ effect • Optical rectification • c(2) vanishes in media • with inversion symmetry

  7. Second Harmonic Generation Why does phase mismatching happen? w E(z) 2w t1 z w E(z) 2w t2 z w E(z) 2w t3 z

  8. Second Harmonic Generation In an uniaxial crystal • where ne and no are indexes of refraction for extraordinary and • ordinary rays, respectively, • is angle between k and optic axis of the crystal Phase matching conditions: q = q0 and Or n2w=nw, but nw= lkw/2pandn2w= (l/2)k2w/2p So, 2kw= k2w,or Dk = k(2w) - 2k(w) => 0

  9. Second Harmonic Generation One can show, that electric field And Poynting vector Because In ideal case (Dk = 0) =>

  10. Second Harmonic Generation In real case Dk never is equal to 0, So, SHG power oscillates with z Finally, phase matching for SHG requires 2 conditions: a) Correct angle between k and crystal axis to reach n2w=nw Dk = k(2w) - 2k(w) => 0 or b) Correct crystal length to reach maximum SHG power

  11. q1 LaserwP Sample q3 LaserwS q2 2wP-wS Coherent Anti-Stokes Raman Spectroscopy (CARS) wP wCARS wP wS wRaman 2wP = wS + wCARS • q1 and q2 correspond to wP • q3 corresponds to wS • wP–wS= wRamanis the Raman shift (Raman active vibrational mode)

  12. Coherent Anti-Stokes Raman Spectroscopy (CARS) Intensity: Where: is the refractive index at frequency is the intensity of i-th signal Phase matching for BOXCARS is the interaction length kP1 kP2 kS kCARS After Maker and Terhune (1989)

  13. Mask Lens 1 2 |kP1| |kCARS| |kS| = + 2wPump= wStokes + wCARS Principles of BOXCARS Method Geometry of laser beams for BOXCARS Phase matching for BOXCARS Lens 2 kP2 kP1 wCARS jP jS wPump kS kCARS wStokes f For h << f h d or

  14. r0 r0 Phase Matching in fs-BOXCARS h f I f

  15. fs-CARS: Theory normalized CARS frequency and normalized Stokes detuning Where:

  16. Phase Matching in fs-BOXCARS ps-CARS: fs-BOXCARS: So far:

  17. -80 -80 -40 -40 0 0 40 40 80 80 Stokes Detuning, meV Stokes Detuning, meV Phase Matching in fs-BOXCARS Our results Without G-correction With G-correction 1.60 1.60 1.70 1.70 1.80 1.80 CARS Photon Energy, meV CARS Photon Energy, meV

  18. Phase Matching in fs-BOXCARS Comparison theory and experiment 1 - experiment - no correc. - with correc. 0.5 Intensity, arb.units 0 2200 1000 1400 1800 Wavenumber, cm-1

  19. Conclusion • Every nonlinear optical phenomenon requires it’s own unique approach to understand the phase matching conditions • Understanding of phase matching is crucially important to run a nonlinear optical experiment correctly and for interpretation of it’s results.

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