Abstract

We find globally optimal poling patterns for 2-dimensional χ(2) photonic crystals for 3rd and 4th harmonic generation.

© 2003 Optical Society of America

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References

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  1. V. Berger,�??Nonlinear photonic crystals,�?? Phys. Rev. Lett. 81, 4136-4139 (1998).
    [CrossRef]
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    [CrossRef] [PubMed]
  3. A. Chowdhury, C.Staus, B.F. Boland, T.F. Kuech and L. McCaughan, �??Experimental demonstration of 1535 �??1555-nm simultaneous optical wavelength interchange with a nonlinear photonic crystal,�?? Opt. Lett. 26, 1353 �??1355 (2001).
    [CrossRef]
  4. A. Chowdhury, C.Staus, B.F. Boland, T.F. Kuech and L. McCaughan, �??Experimental demonstration of 1535 �??1555-nm simultaneous optical wavelength interchange with a nonlinear photonic crystal,�?? Opt. Lett. 26, 1353 �??1355 (2001).
    [CrossRef]
  5. M.M. Fejer, G.A. Magel, D.H. Jundt, and R.L. Byer, �??Quasi-phase-matched second harmonic generation:tuning and tolerances,�?? IEEE J. Quantum Electron. 28, 2631 �??2654 (1992).
    [CrossRef]
  6. A.H. Norton and C.M. de Sterke, �??Optimal poling of nonlinear photonic crystals for frequency conversion,�?? Opt. Lett. 28, 188 (2003).
    [CrossRef] [PubMed]
  7. J. Kevorkian and J.D. Cole, Perturbation methods in applied mathematics (Springer-Verlag, New York, 1981).
  8. R.W. Boyd, Nonlinear Optics (Academic Press, San Diego,1992).
  9. D.H. Jundt, �??Temperature-dependent Sellmeier equation for the index of refraction,ne incongruent lithium niobate,�?? Opt. Lett. 22, 1553 (1997).
    [CrossRef]
  10. C. Zhang,Y. Zhu, S. Yang, Y. Qin, S. Zhu, Y. Chen, H. Liu, and N. Ming, �??Crucial effects of coupling coeficients on quasi-phase-matched harmonic generation in an optical superlattice,�?? Opt. Lett. 25, 436 (2000).
    [CrossRef]

IEEE J. Quantum Electron. (1)

M.M. Fejer, G.A. Magel, D.H. Jundt, and R.L. Byer, �??Quasi-phase-matched second harmonic generation:tuning and tolerances,�?? IEEE J. Quantum Electron. 28, 2631 �??2654 (1992).
[CrossRef]

Opt. Lett. (5)

Phys. Rev. Lett. (2)

V. Berger,�??Nonlinear photonic crystals,�?? Phys. Rev. Lett. 81, 4136-4139 (1998).
[CrossRef]

N.G.R. Broderick, G.W. Ross, H.L. Offerhaus, D.J. Richardson, and D.C. Hanna,�??Hexagonally poled Lithium Niobate:a two-dimensional nonlinear photonic crystal,�?? Phys. Rev. Lett. 84, 4345 �?? 4348 (2000).
[CrossRef] [PubMed]

Other (2)

J. Kevorkian and J.D. Cole, Perturbation methods in applied mathematics (Springer-Verlag, New York, 1981).

R.W. Boyd, Nonlinear Optics (Academic Press, San Diego,1992).

Supplementary Material (5)

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Figures (5)

Fig. 1.
Fig. 1.

Generic QPM schemes for (a) THG, (b) FHG. Reciprocal lattice vectors G 1 and G 2 phase match quadratic interactions between jth harmonic wave vectors k j .

Fig. 2.
Fig. 2.

(a) Each point on the curve Σ corresponds to a Σ-pattern. (b) (2.4MB) Video of the Σ-patterns defined by Eq. (14) for varying angle ψ.

Fig. 3.
Fig. 3.

(1.9MB) Energy fluxes for standard THG solutions plotted against distance into the photonic crystal. The video shows how the energy fluxes change as the poling parameter ψ is varied. The red, green and blue curves are Uq (x), q=1, 2, 3 (1st, 2nd and 3rd harmonics) respectively. For the critical value ψcrit ≈ 41.42°, 100% conversion is attained at x=∞. (a) ψ=41.41°. (b) ψ=41.43°.

Fig. 4.
Fig. 4.

(1.9MB) THG amplitudes near ψcrit≈41.42°. The red, green and blue curves are Re{A 1}, Im{A 2} and Re{A 3} respectively (the other components are zero for standard THG solutions). (a) ψ=41.41°. (b) ψ=41.43°.

Fig. 5.
Fig. 5.

(a) (1.8MB) FHG energy fluxes Uq, q=1, 2, 4. (b) (1.8MB) FHG amplitudes Re{A 1}, Im{A 2}, and Im{A 4}. Red, green and blue curves are for 1st, 2nd and 4th harmonics respectively.

Equations (17)

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E q ( t , x ) = A q ( x ) e i ( k q · x q ω t ) + A q * ( x ) e i ( k q · x q ω t ) .
k q = k m + k n + G ,
p ( x ) = a = b = p a b e i x · G a b = ± 1 .
d A 1 d x = i ω χ c n 1 ( σ 1 * A 2 A 1 * + σ 2 * A 3 A 2 * ) ,
d A 2 d x = i ω χ c n 2 cos θ 2 ( σ 1 A 1 2 + 2 σ 2 * A 1 * A 3 ) ,
d A 3 d x = 3 i ω χ c n 3 cos θ 3 σ 2 A 1 A 2 .
( σ 1 , σ 2 ) = σ ( ϕ ) ( cos ϕ , sin ϕ ) ,
{ A 1 , A 2 , A 3 } { A 1 e i β , A 2 e i 2 β , A 3 e i 3 β } ,
{ A 1 , A 2 , A 3 , x } { μ A 1 , μ A 2 , μ A 3 , x μ } ,
{ σ 1 , σ 2 , A 2 , A 3 } { σ 1 e 1 α 1 , σ 2 e i α 2 , A 2 e i α 1 , A 3 e i ( α 1 + α 2 ) } ,
{ σ 1 , σ 2 , x } { μ σ 1 , μ σ 2 , x μ } .
σ 1 σ 1 exp ( i x 0 · G a 1 b 1 ) , σ 2 σ 2 exp ( i x 0 · G a 2 b 2 ) .
p ( x ) = sign ( n · δ x ) .
p ( x ) = sign ( cos ψ cos ( x · G a 1 b 1 ) + sin ψ cos ( x · G a 2 b 2 ) ) .
A 1 x = i ω χ c n 1 σ 1 * A 2 A 1 * ,
A 2 x = i ω χ c n 2 cos θ 2 ( σ 1 A 1 2 + 2 σ 2 * A 4 A 2 * ) ,
A 4 x = 2 i ω χ c n 4 cos θ 4 σ 2 A 2 2 .

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