Fourier space design of high-Q cavities in standard and compressed hexagonal lattice photonic crystals

Kartik Srinivasan; Oskar Painter

doi:10.1364/OE.11.000579

Optics Express
Vol. 11,
Issue 6,
pp. 579-593
(2003)
•https://doi.org/10.1364/OE.11.000579

Fourier space design of high-Q cavities in standard and compressed hexagonal lattice photonic crystals

Kartik Srinivasan and Oskar Painter

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Kartik Srinivasan and Oskar Painter, "Fourier space design of high-Q cavities in standard and compressed hexagonal lattice photonic crystals," Opt. Express 11, 579-593 (2003)

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Abstract

Building upon the results of recent work [1], we use momentum space design rules to investigate high quality factor (Q) optical cavities in standard and compressed hexagonal lattice photonic crystal (PC) slab waveguides. Beginning with the standard hexagonal lattice, the results of a symmetry analysis are used to determine a cavity geometry that produces a mode whose symmetry immediately leads to a reduction in vertical radiation loss from the PC slab. The Q is improved further by a tailoring of the defect geometry in Fourier space so as to limit coupling between the dominant Fourier components of the defect mode and those momentum components that radiate. Numerical investigations using the finite-difference time-domain (FDTD) method show significant improvement using these methods, with total Q values exceeding 10⁵. We also consider defect cavities in a compressed hexagonal lattice, where the lattice compression is used to modify the in-plane bandstructure of the PC lattice, creating new (frequency) degeneracies and modifying the dominant Fourier components found in the defect modes. High Q cavities in this new lattice geometry are designed using the momentum space design techniques outlined above. FDTD simulations of these structures yield Q values in excess of 10⁵ with mode volumes of approximately 0.35 cubic half-wavelengths in vacuum.

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Fig. 1. 2D hexagonal PC slab waveguide structure and cladding light cone.

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Fig. 2. (a) Real and reciprocal space lattices of a standard 2D hexagonal lattice. Refer to Table 5 for identification of key geometrical quantities. (b) Fundamental TE-like (even) guided mode bandstructure for hexagonal lattice calculated using a 2D plane-wave expansion method with an effective index for the vertical guiding; r/a=0.36, n _slab=n _eff=2.65.

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Fig. fig02.1 Table 3. Characteristics of the

B_{A_{2}^{″}}^{a, a 1}

resonant mode in a hexagonal lattice (images are for a PC cavity with r/a=0.35, r′/a=0.45, d/a=0.75, and n _slab=3.4).

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Fig. fig02.2 Table 4. FDTD simulation results for graded hexagonal lattice geometries (images are for the first PC cavity listed below; d/a=0.75 in all designs).

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Fig. 3. (a) Δ͠η(k _⊥) for single enlarged hole design in hexagonal lattice (r/a=0.30, r′/a=0.45). (b) Δ͠η(k _⊥) for graded hexagonal lattice design shown in Table 4.

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Fig. 4. (a) Real and reciprocal space lattices of a compressed 2D hexagonal lattice. Refer to Table 5 for more identification of key geometrical quantities; (b) Fundamental TE-like (even) guided mode bandstructure for a compressed hexagonal lattice, calculated using a 2D plane-wave expansion method with an effective index for the vertical guiding; r/a=0.35, n _slab=n _eff=2.65, γ=0.7.

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Fig. 5. Modal characteristics of a simple defect mode in a compressed hexagonal lattice (d/a=0.75).

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Fig. fig05.1 Table 6. FDTD simulation results for graded compressed hexagonal lattice geometries.

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Tables (3)

Table 1. Symmetry classification and dominant Fourier components for the B-field of conduction band donor modes in a hexagonal lattice.

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Table 2. Symmetry classification and dominant Fourier components for the B-field of valence band acceptor modes in a hexagonal lattice.

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Table 5. Key geometrical quantities associated with the standard and compressed hexagonal lattices.

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Equations (10)

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{\hat{ℒ}}_{H}^{TE} = - \nabla (η_{o} + Δ η) \cdot \nabla - (η_{o} + Δ η) \nabla^{2} .

〈 H_{l', k'} ∣ {\hat{ℒ}}_{H}^{'} H_{l, k} 〉 = \sum_{G} \sum_{k ″} ({\tilde{Δ η}}_{k ″} K_{l', l} (k', k, G) + {\tilde{Δ η}}_{k ″} (i k ″) \cdot L_{l', l} (k', k, G)) δ_{k' - k ″ + G, k},

B_{{A ″}_{2}}^{a, a 1} = \hat{z} (\cos (k_{J_{1}} \cdot r_{⊥}^{a}) + \cos (k_{J_{3}} \cdot r_{⊥}^{a}) + \cos (k_{J_{5}} \cdot r_{⊥}^{a})),

V B_{a} = \hat{z} (\begin{matrix} \cos (k_{X_{1}} \cdot r_{⊥}^{a}) \\ e^{{- i k}_{J_{1}} \cdot r_{⊥}^{a}} + e^{{- i k}_{J_{3}} \cdot r_{⊥}^{a}} \\ e^{{- i k}_{J_{4}} \cdot r_{⊥}^{a}} + e^{{- i k}_{J_{6}} \cdot r_{⊥}^{a}} \\ e^{{- i k}_{J_{2}} \cdot r_{⊥}^{a}} \\ e^{{- i k}_{J_{5}} \cdot r_{⊥}^{a}} \end{matrix})

P_{A_{2}} = (\begin{matrix} 2 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 & 1 \end{matrix}), P_{B_{2}} = (\begin{matrix} 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & - 1 & 0 & 0 \\ 0 & - 1 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & - 1 \\ 0 & 0 & 0 & - 1 & 1 \end{matrix}) .

{C B}_{a} = \hat{z} (\begin{matrix} \sin (k_{X_{2}} \cdot r_{⊥}^{a}) \\ \sin (k_{X_{3}} \cdot r_{⊥}^{a}) \end{matrix}), {C B}_{b} = \hat{z} (\begin{matrix} \cos (k_{X_{2}} \cdot r_{⊥}^{b}) \\ \cos (k_{X_{3}} \cdot r_{⊥}^{b}) \end{matrix}),

B_{B_{1}}^{a, d 1} = \hat{z} (\sin (k_{X_{2}} \cdot r_{⊥}^{a}) - \sin (k_{X_{3}} \cdot r_{⊥}^{a})),

B_{B_{2}}^{a, d 1} = \hat{z} (\sin (k_{X_{2}} \cdot r_{⊥}^{a}) + \sin (k_{X_{3}} \cdot r_{⊥}^{a})),

B_{A_{1}}^{b, d 1} = \hat{z} (\cos (k_{X_{2}} \cdot r_{⊥}^{b}) - \cos (k_{X_{3}} \cdot r_{⊥}^{b})),

B_{A_{2}}^{b, d 1} = \hat{z} (\cos (k_{X_{2}} \cdot r_{⊥}^{a}) + \cos (k_{X_{3}} \cdot r_{⊥}^{b})),

Defect Center	C _6υ Modes	Fourier Comp.	(σ_d, σ_υ)a	C _2υ Modes	(σ _x , σ _y )a
(0, 0)	$B_{B_{1}^{″}}^{a, d 1}$	$\pm {k_{X_{1}}, k_{X_{2}}, k_{X_{3}}}$	(+,-)	$B_{B_{1}}^{a, d 1, 1}$	(-, +)
(0, 0)	$B_{E_{1}, 1}^{a, d 1}$	$\pm {k_{X_{1}}, k_{X_{2}}, k_{X_{3}}}$	(0, 0)	$B_{B_{1}}^{a, d 1, 2}$	(-, +)
(0, 0)	$B_{E_{1}, 2}^{a, d 1}$	$\pm {k_{X_{2}}, k_{X_{3}}}$	(0, 0)	$B_{B_{2}}^{a, d 1}$	(+,-)
(a/2, 0)	N/Ab	$\pm {k_{X_{2}}, k_{X_{3}}}$	N/A	$B_{A_{1}}^{b, d 1}$	(+, +)
(a/2, 0)	N/A	$\pm {k_{X_{2}}, k_{X_{3}}}$	N/A	$B_{A_{2}}^{b, d 1}$	(-,-)
(a/2, 0)	N/A	$\pm {k_{X_{1}}}$	N/A	$B_{B_{1}^{″}}^{b, d 1}$	(-, +)

Defect Center	C _6υ Modes	Fourier Comp.	(σ _d , σ_υ)	C _2υ Modes	(σ _x , σ _y )
(0, 0)	$B_{A_{2}^{″}}^{a, a 1}$	$\pm {k_{J_{1}}, k_{J_{3}}, k_{J_{5}}}$	(-,-)	$B_{A_{2}}^{a, a 1}$	(-,-)
(0, 0)	$B_{B_{2}^{″}}^{a, a 1}$	$\pm {k_{J_{1}}, k_{J_{3}}, k_{J_{5}}}$	(-, +)	$B_{B_{2}}^{a, a 1}$	(+,-)
(a/2, 0)	N/A	$\pm {k_{J_{1}}, k_{J_{3}}, k_{J_{5}}}$	N/A	$B_{A_{2}}^{b, a 1}$	(-,-)
(a/2, 0)	N/A	$\pm {k_{J_{1}}, k_{J_{3}}, k_{J_{5}}}$	N/A	$B_{B_{2}}^{b, a 1}$	(+,-)

Crystal Parameter(s)	Hexagonal Lattice	Compressed Hexagonal Lattice
$G$ _a a	C _6υ	C _2υ
$G$ _b b	C _2υ	C _2υ
{a₁, a₂}	${(\frac{a}{2}, \frac{a \sqrt{3}}{2}), (a, 0)}$	${(\frac{a}{2}, \frac{a \sqrt{3} γ}{2}), (a, 0)}$
{G ₁,G ₂}	${(0, \frac{4 π}{a \sqrt{3}}), (\frac{2 π}{a}, - \frac{2 π}{a \sqrt{3}})}$	${(0, \frac{4 π}{a \sqrt{3} γ}), (\frac{2 π}{a}, - \frac{2 π}{a \sqrt{3} γ})}$
±X ₁	$(0, \pm \frac{2 π}{a \sqrt{3}})$	$(0, \pm \frac{2 π}{a \sqrt{3} γ})$
±X ₂	$(\pm \frac{π}{a}, \pm \frac{π}{a \sqrt{3}})$	$(\pm \frac{π}{a}, \pm \frac{π}{a \sqrt{3} γ})$
±X ₃	$(\pm \frac{π}{a}, \mp \frac{π}{a \sqrt{3}})$	$(\pm \frac{π}{a}, \mp \frac{π}{a \sqrt{3} γ})$
±J ₁	$(\pm \frac{2 π}{3 a}, \pm \frac{2 π}{a \sqrt{3}})$	$(\pm \frac{π}{a} (1 - \frac{1}{3 γ^{2}}), \pm \frac{2 π}{a \sqrt{3} γ})$
±J ₂	$(\pm \frac{4 π}{3 a}, 0)$	$(\pm \frac{π}{a} (1 + \frac{1}{3 γ^{2}}), 0)$
±J ₃	$(\pm \frac{2 π}{3 a}, \mp \frac{2 π}{a \sqrt{3}})$	$(\pm \frac{π}{a} (1 - \frac{1}{3 γ^{2}}), \mp \frac{2 π}{a \sqrt{3} γ})$
$G_{o, k_{X_{i}}}$ c	C _2υ	C _2υ
$G_{o, k_{J_{1}}}$	C _3υ	C _1υ={e, σ _y }
$G_{o, k_{J_{2}}}$	C _3υ	C ₁={e}
$G_{o, k_{J_{3}}}$	C _3υ	C _1υ={e, σ _y }

Defect Center	C _6υ Modes	Fourier Comp.	(σ_d, σ_υ)a	C _2υ Modes	(σ _x , σ _y )a
(0, 0)	$B_{B_{1}^{″}}^{a, d 1}$	$\pm {k_{X_{1}}, k_{X_{2}}, k_{X_{3}}}$	(+,-)	$B_{B_{1}}^{a, d 1, 1}$	(-, +)
(0, 0)	$B_{E_{1}, 1}^{a, d 1}$	$\pm {k_{X_{1}}, k_{X_{2}}, k_{X_{3}}}$	(0, 0)	$B_{B_{1}}^{a, d 1, 2}$	(-, +)
(0, 0)	$B_{E_{1}, 2}^{a, d 1}$	$\pm {k_{X_{2}}, k_{X_{3}}}$	(0, 0)	$B_{B_{2}}^{a, d 1}$	(+,-)
(a/2, 0)	N/Ab	$\pm {k_{X_{2}}, k_{X_{3}}}$	N/A	$B_{A_{1}}^{b, d 1}$	(+, +)
(a/2, 0)	N/A	$\pm {k_{X_{2}}, k_{X_{3}}}$	N/A	$B_{A_{2}}^{b, d 1}$	(-,-)
(a/2, 0)	N/A	$\pm {k_{X_{1}}}$	N/A	$B_{B_{1}^{″}}^{b, d 1}$	(-, +)

Defect Center	C _6υ Modes	Fourier Comp.	(σ _d , σ_υ)	C _2υ Modes	(σ _x , σ _y )
(0, 0)	$B_{A_{2}^{″}}^{a, a 1}$	$\pm {k_{J_{1}}, k_{J_{3}}, k_{J_{5}}}$	(-,-)	$B_{A_{2}}^{a, a 1}$	(-,-)
(0, 0)	$B_{B_{2}^{″}}^{a, a 1}$	$\pm {k_{J_{1}}, k_{J_{3}}, k_{J_{5}}}$	(-, +)	$B_{B_{2}}^{a, a 1}$	(+,-)
(a/2, 0)	N/A	$\pm {k_{J_{1}}, k_{J_{3}}, k_{J_{5}}}$	N/A	$B_{A_{2}}^{b, a 1}$	(-,-)
(a/2, 0)	N/A	$\pm {k_{J_{1}}, k_{J_{3}}, k_{J_{5}}}$	N/A	$B_{B_{2}}^{b, a 1}$	(+,-)

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