Two-dimensional poling patterns for 3rd and 4th harmonic generation

Andrew H. Norton; C. Martijn de Sterke

doi:10.1364/OE.11.001008

Optics Express
Vol. 11,
Issue 9,
pp. 1008-1014
(2003)
•https://doi.org/10.1364/OE.11.001008

Two-dimensional poling patterns for 3rd and 4th harmonic generation

Andrew H. Norton and C. Martijn de Sterke

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Andrew H. Norton and C. Martijn de Sterke, "Two-dimensional poling patterns for 3rd and 4th harmonic generation," Opt. Express 11, 1008-1014 (2003)

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Abstract

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

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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.

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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.

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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 U_q (x), q=1, 2, 3 (1st, 2nd and 3rd harmonics) respectively. For the critical value ψ_crit ≈ 41.42°, 100% conversion is attained at x=∞. (a) ψ=41.41°. (b) ψ=41.43°.

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Fig. 4. (1.9MB) THG amplitudes near ψ_crit≈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°.

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Fig. 5. (a) (1.8MB) FHG energy fluxes U_q, 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.

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Equations (17)

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E_{q} (t, x) = A_{q} (x) e^{i (k_{q} \cdot x - q ω t)} + A_{q}^{*} (x) e^{- i (k_{q} \cdot x - q ω t)} .

k_{q} = k_{m} + k_{n} + G,

p (x) = \sum_{a = - \infty}^{\infty} \sum_{b = - \infty}^{\infty} p_{a b} e^{i x \cdot G_{a b}} = \pm 1 .

\frac{d A_{1}}{d x} = \frac{i ω χ}{c n_{1}} (σ_{1}^{*} A_{2} A_{1}^{*} + σ_{2}^{*} A_{3} A_{2}^{*}),

\frac{d A_{2}}{d x} = \frac{i ω χ}{c n_{2} \cos θ_{2}} (σ_{1} A_{1}^{2} + 2 σ_{2}^{*} A_{1}^{*} A_{3}),

\frac{d A_{3}}{d x} = \frac{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}} \mapsto {A_{1} e^{i β}, A_{2} e^{i 2 β}, A_{3} e^{i 3 β}},

{A_{1}, A_{2}, A_{3}, x} \mapsto {μ A_{1}, μ A_{2}, μ A_{3}, \frac{x}{μ}},

{σ_{1}, σ_{2}, A_{2}, A_{3}} \mapsto {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} \mapsto {μ σ_{1}, μ σ_{2}, \frac{x}{μ}} .

σ_{1} \mapsto σ_{1} \exp (- i x_{0} \cdot G_{a_{1} b_{1}}), σ_{2} \mapsto σ_{2} \exp (- i x_{0} \cdot G_{a_{2} b_{2}}) .

p (x) = sign (n \cdot δ_{x}) .

p (x) = sign (\cos ψ \cos (x \cdot G_{a_{1} b_{1}}) + \sin ψ \cos (x \cdot G_{a_{2} b_{2}})) .

\frac{\partial A_{1}}{\partial x} = \frac{i ω χ}{c n_{1}} σ_{1}^{*} A_{2} A_{1}^{*},

\frac{\partial A_{2}}{\partial x} = \frac{i ω χ}{c n_{2} \cos θ_{2}} (σ_{1} A_{1}^{2} + 2 σ_{2}^{*} A_{4} A_{2}^{*}),

\frac{{\partial A}_{4}}{\partial x} = \frac{2 i ω χ}{c n_{4} \cos θ_{4}} σ_{2} A_{2}^{2} .

Abstract

Supplementary Material (5)

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Equations (17)

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