Aperiodic 1-dimensional structures for quasi-phase matching

Andrew H. Norton; C. Martijn de Sterke

doi:10.1364/OPEX.12.000841

Optics Express
Vol. 12,
Issue 5,
pp. 841-846
(2004)
•https://doi.org/10.1364/OPEX.12.000841

Aperiodic 1-dimensional structures for quasi-phase matching

Andrew H. Norton and C. Martijn de Sterke

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Andrew H. Norton and C. Martijn de Sterke, "Aperiodic 1-dimensional structures for quasi-phase matching," Opt. Express 12, 841-846 (2004)

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Abstract

We describe a method for designing 1-dimensional aperiodic poled grating structures of finite length that quasi-phase match multiple χ ⁽²⁾ processes. The poling functions for such gratings are best aligned, in terms of the dot product in Fourier space, with a design target. No restrictions are placed on the quasi-phase matching wave numbers. A grating designed for third harmonic generation is simulated.

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Fig. 1. (a) The target Fourier transform n̂(k) (red) and the Fourier transform of the best aligned poled grating p̂(k) (blue) of length L=1cm. The two QPM peaks were chosen for THG (see Section 5). (b) Close-up of the QPM peak at k=G ₁. (c) The peak at k=G ₂. The target peaks have half-widths ΔG=2π/L.

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Fig. 2. Part of a THG poled grating defined by p(x)=sign(n(x)) where n(x)=w ₁ cos(G ₁ x)+w ₂ cos(G ₂ x). Values for w_j and G_j are given in the text (Section 5). The grating length is L=1cm, of which 0.5mm is shown. Domains below fabrication resolution (≈1 µm for LiNbO₃) can be deleted by inversion before the grating is simulated.

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Fig. 3. THG simulation. Relative energy of the fundamental, 2nd and 3rd harmonic waves (red, green and blue curves respectively) as a function of distance through the poled grating.

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Equations (26)

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p (x) = {\begin{matrix} \pm 1 & for x \in [- L ⁄ 2, L ⁄ 2], \\ 0 & otherwise, \end{matrix}

k_{1} + k_{1} + G_{1} = k_{2}, k_{1} + k_{2} + G_{2} = k_{3} .

u_{k} = \frac{1}{2 π} e^{ikx} .

f (x) = \int_{- \infty}^{\infty} \hat{f} (k) u_{k} d k = \frac{1}{2 π} \int_{- \infty}^{\infty} \hat{f} (k) e^{ikx} d k,

〈 f_{1}, f_{2} 〉 = 2 π \int_{- \infty}^{\infty} f_{1} (x) f_{2} {(x)}^{*} d x .

\hat{f} (k) = 〈 f, u_{k} 〉 = \int_{- \infty}^{\infty} f (x) e^{- ikx} d x .

f = \int_{- \infty}^{\infty} Re {\hat{f} (k)} u_{k} d k + \int_{- \infty}^{\infty} Im {\hat{f} (k)} i u_{k} d k .

f_{1} \cdot f_{2} = \frac{1}{2} (〈 f_{1}, f_{2} 〉 + 〈 f_{2}, f_{1} 〉)

= 2 π \int_{- \infty}^{\infty} (Re {f_{1} (x)} Re {f_{2} (x)} + Im {f_{1} (x)} Im {f_{2} (x)}) d x .

f_{1} \cdot f_{2} = \int_{- \infty}^{\infty} (Re {{\hat{f}}_{1} (k)} Re {{\hat{f}}_{2} (k)} + Im {{\hat{f}}_{1} (k)} Im {{\hat{f}}_{2} (k)}) d k .

p \cdot n = 2 π \int_{- L ⁄ 2}^{L ⁄ 2} p (x) Re {n (x)} d x,

p (x) = {\begin{matrix} sign (Re {n (x)}) & for x \in [- L ⁄ 2, L ⁄ 2], \\ 0 & otherwise . \end{matrix}

\hat{p} (k) = \sum_{j = 1}^{r - 1} p_{j} {\hat{h}}_{j} (k),

h_{j} (x) = {\begin{matrix} 1 & for x \in (x_{j}, x_{j + 1}), \\ 0 & otherwise, \end{matrix}

{\hat{h}}_{j} (k) = \frac{i}{k} (\exp (- i k x_{j + 1}) - \exp (- i k x_{j})) .

H_{j} (k) = {\begin{matrix} 1 & for G_{j} - Δ G_{j} < k < G_{j} + Δ G_{j}, \\ 0 & otherwise, \end{matrix}

\hat{n} (k) = \sum_{j = 1}^{N} (a_{j} (H_{j} (k) + H_{j} (- k)) + i b_{j} (H_{j} (k) - H_{j} (- k))),

n (x) = \frac{1}{2 π} \sum_{j = 1}^{N} \int_{- \infty}^{\infty} (a_{j} (H_{j} (k) + H_{j} (- k)) + i b_{j} (H_{j} (k) - H_{j} (- k))) e^{ikx} d k

= \frac{1}{π} \sum_{j = 1}^{N} \int_{0}^{\infty} (a_{j} \cos (k x) - b_{j} \sin (k x)) H_{j} (k) d k

= \sum_{j = 1}^{N} \frac{2 \sin (Δ G_{j} x)}{π x} (a_{j} \cos (G_{j} x) - b_{j} \sin (G_{j} x))

= \sum_{j = 1}^{N} \frac{2 \sin (Δ G_{j} x)}{π x} w_{j} \cos (G_{j} x + ϕ_{j})

p (x) = {\begin{matrix} sign (Σ_{j = 1}^{N} \frac{\sin (Δ G_{j} x)}{x} w_{j} \cos (G_{j} x + ϕ_{j})) & for x \in [- L ⁄ 2, L ⁄ 2], \\ 0 & otherwise . \end{matrix}

p (x) = {\begin{matrix} sign (Σ_{j = 1}^{N} w_{j} \cos (G_{j} x + ϕ_{j})) & for x \in [- L ⁄ 2, L ⁄ 2], \\ 0 & otherwise . \end{matrix}

\frac{d a_{1}}{dx} = \frac{i ω}{n_{1} c} χ p (x) (a_{2} a_{1}^{*} e^{i G_{1} x} + a_{3} a_{2}^{*} e^{i G_{2} x}),

\frac{d a_{2}}{dx} = \frac{i ω}{n_{2} c} χ p (x) ({2 a}_{3} a_{1}^{*} e^{i G_{2} x} + a_{1}^{2} e^{- i G_{1} x}),

\frac{d a_{3}}{dx} = \frac{3 i ω}{n_{3} c} χ p (x) a_{1} a_{2} e^{- i G_{2} x},

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