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Quasi-phase-matching of high-order-harmonic generation using multimode polarization beating

Lewis Z. Liu, Kevin O'Keeffe, and Simon M. Hooker
Phys. Rev. A 87, 023810 – Published 13 February 2013
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  • INTRODUCTION
  • DERIVATION OF ENVELOPE EQUATION
  • ANALYSIS OF GROWTH EQUATION
  • SIMULATION RESULTS
  • CONCLUSION
  • ACKNOWLEDGEMENTS
    • References

    Abstract

    The generalization of quasi-phase-matching using polarization beating and of multimode quasi-phase-matching (MMQPM) for the generation of high-order harmonics is explored, and a method for achieving polarization beating is proposed. If two (and in principle more) modes of a waveguide are excited, modulation of the intensity, phase, and/or polarization of the guided radiation will be achieved. By appropriately matching the period of this modulation to the coherence length, quasi-phase-matching of high-order-harmonic radiation generated by the guided wave can occur. We show that it is possible to achieve efficiencies with multimode quasi-phase-matching greater than the ideal square wave modulation. We present a Fourier treatment of QPM and use this to show that phase modulation, rather than amplitude modulation, plays the dominant role in the case of MMQPM. The experimental parameters and optimal conditions for this scheme are explored.

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    • Received 6 October 2012

    DOI:https://doi.org/10.1103/PhysRevA.87.023810

    ©2013 American Physical Society

    Authors & Affiliations

    Lewis Z. Liu*, Kevin O'Keeffe, and Simon M. Hooker

    • Clarendon Laboratory, University of Oxford Physics Department, Parks Road, Oxford OX1 3PU, United Kingdom

    • *l.liu1@physics.ox.ac.uk

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    Vol. 87, Iss. 2 — February 2013

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    Images

    • Figure 1
      Figure 1
      Simulation results for columns (1) “Pure MMQPM” (c2M2=0.002, Ω=0∘), (2) “Hybrid MMPBQPM” (c2M2=0.01, Ω=60∘), and (3) “Pure PBQPM” (c2M2=0.05, Ω=90∘); all for Lb=2Lc and q=51. Row (a) shows the relative HHG intensity. Dot-dashed cyan line shows ideal square-wave QPM, solid red line shows the indicated form of MMPBQPM, solid black line shows the intensity for no phase matching, and the dashed blue line shows the harmonic intensity calculated from Eq. (35). Row (b) shows the modulus of the intensity-dependent source term G[I(z)] (solid red line), ellipticity-dependent source term H[ɛ(z)] (solid green line), and the combined source terms |F|=GH (dashed black line). Row (c) shows Ψx(z), the total phase of dξx/dz (solid black line), and Ψx for large z (dotted light-red line). Row (d) shows the UσVκ distribution as a function of σ and κ for n=σ+κ for n=2.Reuse & Permissions
    • Figure 2
      Figure 2
      Simulation results for columns (1) the first resonance (c2M2=0.0009, Ω=0∘) and (2) the second resonance (c2M2=0.0091, Ω=0∘); all for Lb=Lc and q=51. Row (a) shows the relative HHG intensity. Dot-dashed cyan line shows ideal square-wave QPM, solid red line shows the indicated form of MMPBQPM, solid black line shows the intensity for no phase matching, and dashed blue line shows the harmonic intensity calculated from Eq. (35). Row (b) shows the modulus of the intensity-dependent source term G[I(z)] (solid red line), ellipticity-dependent source term H[ɛ(z)] (solid green line), and the combined source terms |F|=GH (dashed black line). Row (c) shows Ψx(z), the total phase of dξx/dz (solid black line), and Ψx for large z (dotted light-red line). Row (d) shows the UσVκ distribution as a function of σ and κ for n=σ+κ for n=1.Reuse & Permissions
    • Figure 3
      Figure 3
      Relative HHG amplitude |ξx(z)| for q=51 at large z (z=10Lc,1) as a function of Lb/Lc,1 and m=2 mode-coupling strength c22M22 where c12M12=1−c22M22, normalized to ideal square-wave QPM for (a) Ω=0∘ and (b) Ω=90∘.Reuse & Permissions
    • Figure 4
      Figure 4
      Relative HHG intensity at z=Lb for Lb=2Lc,1 as a function of Θmax=tan−1(c2M2c1M1) and q, normalized to ideal QPM.Reuse & Permissions
    • Figure 5
      Figure 5
      Relative HHG amplitude for q=51 after 2Lb as a function of coupling angle Ω and m=2 mode-coupling strength c22M22 where c12M12=1−c22M22, normalized to ideal QPM for (a) Lb=Lc,1 and (b) Lb=2Lc,1.Reuse & Permissions
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