Investigation of defect cavities formed in three-dimensional woodpile photonic crystals

Mike P. C. Taverne; Ying-Lung D. Ho; John G. Rarity

doi:10.1364/JOSAB.32.000639

Journal of the Optical Society of America B
Vol. 32,
Issue 4,
pp. 639-648
(2015)
•https://doi.org/10.1364/JOSAB.32.000639

Investigation of defect cavities formed in three-dimensional woodpile photonic crystals

Mike P. C. Taverne, Ying-Lung D. Ho, and John G. Rarity

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Mike P. C. Taverne, Ying-Lung D. Ho, and John G. Rarity, "Investigation of defect cavities formed in three-dimensional woodpile photonic crystals," J. Opt. Soc. Am. B 32, 639-648 (2015)

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Related Topics
Table of Contents Category
- Photonic Crystals and Devices
Optics & Photonics Topics
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The topics in this list come from the Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Microcavities (140.3945)
- Photonic crystals (160.5298)
- Nanophotonics and photonic crystals (350.4238)
- Quantum electrodynamics (270.5580)

About this Article
History
- Original Manuscript: November 19, 2014
- Revised Manuscript: February 6, 2015
- Manuscript Accepted: February 8, 2015
- Published: March 19, 2015

Abstract

We report the optimization of optical properties of single defects in three-dimensional (3D) face-centered-cubic (FCC) woodpile photonic crystal (PC) cavities by using plane-wave expansion (PWE) and finite-difference time-domain (FDTD) methods. By optimizing the dimensions of a 3D woodpile PC, wide photonic band gaps (PBG) are created. Optical cavities with resonances in the bandgap arise when point defects are introduced in the crystal. Three types of single defects are investigated in high refractive index contrast (gallium phosphide–air) woodpile structures, and $Q$ -factors and mode volumes ( $V_{eff}$ ) of the resonant cavity modes are calculated. We show that, by introducing an air buffer around a single defect, smaller mode volumes can be obtained. We demonstrate high $Q$ -factors up to 700,000 and cavity volumes down to $V_{eff} < 0.2 {(λ / n)}^{3}$ . The estimates of $Q$ and $V_{eff}$ are then used to quantify the enhancement of spontaneous emission and the possibility of achieving strong coupling with nitrogen-vacancy (NV) color centers in diamond.

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Defect type	D0			D1			D2
Defect size	$(0.25, 0.25, 0.5) \cdot c$			$(0.5, 0.5, 0.25) \cdot c$			$(0.5, 0.5, 0.5) \cdot c$
Air buffer size	$n / a$			$n / a$			$n / a$
$n_{def}$	3.3			3.3			3.3
$n_{wp}$	3.3			3.3			3.3
$n_{bf}$	1			1			1
$c (μm)$	0.3358			0.3358			0.3358
Dipole orientation	$E_{x}$	$E_{y}$	$E_{z}$	$E_{x}$	$E_{y}$	$E_{z}$	$E_{x}$	$E_{y}$	$E_{z}$
$c / λ_{0}$	$n / a$	0.5890	0.5530	0.5255	0.5058	0.5300	0.4938	0.4922	0.5007
$λ_{0} (nm)$	$n / a$	570.08	607.23	638.98	663.88	633.53	679.99	682.22	670.71
$Q_{u c}$	$n / a$	$3.71 \times 10^{2}$	$3.34 \times 10^{4}$	$7.54 \times 10^{5}$	$2.73 \times 10^{5}$	$8.35 \times 10^{4}$	$1.35 \times 10^{5}$	$1.06 \times 10^{5}$	$8.05 \times 10^{4}$
$V_{eff} ({μm}^{3})$	$n / a$	$5.09 \times 10^{- 3}$	$1.19 \times 10^{- 3}$	$1.17 \times 10^{- 3}$	$2.44 \times 10^{- 3}$	$1.37 \times 10^{- 3}$	$2.89 \times 10^{- 3}$	$3.48 \times 10^{- 3}$	$4.56 \times 10^{- 3}$
$V_{n} = \frac{V_{eff}}{{(λ_{0} / n_{def})}^{3}}$	$n / a$	0.987	0.190	0.161	0.299	0.193	0.330	0.394	0.543
$F_{p}$	$n / a$	$2.85 \times 10^{1}$	$1.33 \times 10^{4}$	$3.56 \times 10^{5}$	$6.92 \times 10^{4}$	$3.29 \times 10^{4}$	$3.10 \times 10^{4}$	$2.05 \times 10^{4}$	$1.13 \times 10^{4}$
$κ_{u c} / (2 π) = c_{0} / (λ_{u c} \cdot Q_{u c})$ (GHz)	$n / a$	1418.48	14.79	0.62	1.66	5.66	3.27	4.13	5.56
$τ_{u c} = 2 π / κ_{u c}$ (ns)	$n / a$	$7.05 \times 10^{- 4}$	0.07	1.61	0.60	0.18	0.31	0.24	0.18
Diamond NV centers	Diamond nanocrystals: $n_{os} \sim 2.4$
$λ_{os} (nm)$	637
$V_{eff}^{'} ({μm}^{3}) = V_{n} \times {(\frac{λ_{os}}{n_{def}})}^{3}$	$n / a$	$7.10 \times 10^{- 3}$	$1.37 \times 10^{- 3}$	$1.16 \times 10^{- 3}$	$2.15 \times 10^{- 3}$	$1.39 \times 10^{- 3}$	$2.37 \times 10^{- 3}$	$2.83 \times 10^{- 3}$	$3.91 \times 10^{- 3}$
$κ_{u c}^{'} / (2 π) = c_{0} / (λ_{os} \cdot Q_{u c})$ (GHz)	$n / a$	1269.47	14.09	0.62	1.73	5.63	3.49	4.43	5.85
$τ_{u c}^{'} = 2 π / κ_{u c}$ (ns)	$n / a$	$7.88 \times 10^{- 4}$	0.07	1.60	0.58	0.18	0.29	0.23	0.17
$γ^{ZPL} / (2 π)$ (GHz)	$3.3 \times 10^{- 3}$ ( $2 π / γ^{ZPL} = τ^{ZPL} \sim 300 ns$ )
$d_{EG}^{ZPL} (C \cdot m)$	$3.57 \times 10^{- 30} (= 1.07 D)$
$f_{EG}^{ZPL}$	0.025
$E_{s p}$ (V/m)	$n / a$	$4.77 \times 10^{5}$	$1.09 \times 10^{6}$	$1.18 \times 10^{6}$	$8.67 \times 10^{5}$	$1.08 \times 10^{6}$	$8.25 \times 10^{5}$	$7.55 \times 10^{5}$	$6.43 \times 10^{5}$
$g_{R}^{ZPL} / (2 π)$ (GHz)	$n / a$	2.57	5.85	6.37	4.67	5.81	4.45	4.07	3.46
$τ_{R}^{ZPL} = 2 π / g_{R}^{ZPL}$ (ns)	$n / a$	0.39	0.17	0.16	0.21	0.17	0.22	0.25	0.29

Defect type	A0			A1			A2
Defect size	$(0.5, 0.5, 0.25) \cdot c$			$(0.5, 0.5, 0.25) \cdot c$			$(0.5, 0.5, 0.25) \cdot c$
Air buffer size	$(a - 0.5 \cdot w, a - 0.5 \cdot w, 0.25 \cdot c)$			$(a, a, 0.25 \cdot c)$			$(a + 0.5 \cdot w, a + 0.5 \cdot w, 0.25 \cdot c)$
$n_{def}$	3.3			3.3			3.3
$n_{w p}$	3.3			3.3			3.3
$n_{bf}$	1			1			1
$c (μm)$	0.3358			0.3358			0.3358
Dipole orientation	$E_{x}$	$E_{y}$	$E_{z}$	$E_{x}$	$E_{y}$	$E_{z}$	$E_{x}$	$E_{y}$	$E_{z}$
$c / λ_{0}$	0.5409	0.5303	0.5377	0.5513	0.5391	0.5449	0.5715	0.5440	0.5593
$λ_{0}$ (nm)	620.86	633.28	624.46	609.13	622.88	616.23	587.59	617.28	600.35
$Q_{u c}$	$3.67 \times 10^{5}$	$2.50 \times 10^{5}$	$6.14 \times 10^{4}$	$1.48 \times 10^{5}$	$1.69 \times 10^{5}$	$4.25 \times 10^{4}$	$3.80 \times 10^{3}$	$1.17 \times 10^{5}$	$1.35 \times 10^{4}$
$V_{eff} ({μm}^{3})$	$6.66 \times 10^{- 4}$	$2.09 \times 10^{- 3}$	$1.13 \times 10^{- 3}$	$7.76 \times 10^{- 4}$	$1.39 \times 10^{- 3}$	$1.26 \times 10^{- 3}$	$1.73 \times 10^{- 3}$	$9.88 \times 10^{- 4}$	$7.22 \times 10^{- 4}$
$V_{n} = \frac{V_{eff}}{{(λ_{0} / n_{def})}^{3}}$	0.100	0.296	0.167	0.123	0.207	0.193	0.306	0.151	0.120
$F_{p}$	$2.78 \times 10^{5}$	$6.41 \times 10^{4}$	$2.79 \times 10^{4}$	$9.14 \times 10^{4}$	$6.21 \times 10^{4}$	$1.67 \times 10^{4}$	$9.45 \times 10^{2}$	$5.92 \times 10^{4}$	$8.58 \times 10^{3}$
$κ_{u c} / (2 π) = c_{0} / (λ_{u c} \cdot Q_{u c})$ (GHz)	1.32	1.90	7.82	3.32	2.85	11.45	134.14	4.13	36.91
$τ_{u c} = 2 π / κ_{u c}$ (ns)	0.76	0.53	0.13	0.30	0.35	0.09	0.01	0.24	0.03
Diamond NV centers	Diamond nanocrystals: $n_{os} \sim 2.4$
$λ_{os} (nm)$	637
$V_{eff}^{'} ({μm}^{3}) = V_{n} \times {(\frac{λ_{os}}{n_{def}})}^{3}$	$7.20 \times 10^{- 4}$	$2.13 \times 10^{- 3}$	$1.20 \times 10^{- 3}$	$8.87 \times 10^{- 4}$	$1.49 \times 10^{- 3}$	$1.39 \times 10^{- 3}$	$2.20 \times 10^{- 3}$	$1.09 \times 10^{- 3}$	$8.62 \times 10^{- 4}$
$κ_{u c}^{'} / (2 π) = c_{0} / (λ_{os} \cdot Q_{u c})$ (GHz)	1.28	1.88	7.66	3.17	2.78	11.08	123.73	4.01	34.78
$τ_{u c}^{'} = 2 π / κ_{u c}$ (ns)	0.78	0.53	0.13	0.32	0.36	0.09	0.01	0.25	0.03
$γ^{ZPL} / (2 π)$ (GHz)	$3.3 \times 10^{- 3}$ ( $2 π / γ^{ZPL} = τ^{ZPL} \sim 300 ns$ )
$d_{EG}^{ZPL} (C \cdot m)$	$3.57 \times 10^{- 30} (= 1.07 D)$
$f_{EG}^{ZPL}$	0.025
$E_{s p}$ (V/m)	$1.50 \times 10^{6}$	$8.71 \times 10^{5}$	$1.16 \times 10^{6}$	$1.35 \times 10^{6}$	$1.04 \times 10^{6}$	$1.08 \times 10^{6}$	$8.57 \times 10^{5}$	$1.22 \times 10^{6}$	$1.37 \times 10^{6}$
$g_{R}^{ZPL} / (2 π)$ (GHz)	8.07	4.69	6.24	7.27	5.61	5.82	4.62	6.57	7.38
$τ_{R}^{ZPL} = 2 π / g_{R}^{ZPL}$ (ns)	0.12	0.21	0.16	0.14	0.18	0.17	0.22	0.15	0.14

Defect type	D0			D1			D2
Defect size	$(0.25, 0.25, 0.5) \cdot c$			$(0.5, 0.5, 0.25) \cdot c$			$(0.5, 0.5, 0.5) \cdot c$
Air buffer size	$n / a$			$n / a$			$n / a$
$n_{def}$	3.3			3.3			3.3
$n_{wp}$	3.3			3.3			3.3
$n_{bf}$	1			1			1
$c (μm)$	0.3358			0.3358			0.3358
Dipole orientation	$E_{x}$	$E_{y}$	$E_{z}$	$E_{x}$	$E_{y}$	$E_{z}$	$E_{x}$	$E_{y}$	$E_{z}$
$c / λ_{0}$	$n / a$	0.5890	0.5530	0.5255	0.5058	0.5300	0.4938	0.4922	0.5007
$λ_{0} (nm)$	$n / a$	570.08	607.23	638.98	663.88	633.53	679.99	682.22	670.71
$Q_{u c}$	$n / a$	$3.71 \times 10^{2}$	$3.34 \times 10^{4}$	$7.54 \times 10^{5}$	$2.73 \times 10^{5}$	$8.35 \times 10^{4}$	$1.35 \times 10^{5}$	$1.06 \times 10^{5}$	$8.05 \times 10^{4}$
$V_{eff} ({μm}^{3})$	$n / a$	$5.09 \times 10^{- 3}$	$1.19 \times 10^{- 3}$	$1.17 \times 10^{- 3}$	$2.44 \times 10^{- 3}$	$1.37 \times 10^{- 3}$	$2.89 \times 10^{- 3}$	$3.48 \times 10^{- 3}$	$4.56 \times 10^{- 3}$
$V_{n} = \frac{V_{eff}}{{(λ_{0} / n_{def})}^{3}}$	$n / a$	0.987	0.190	0.161	0.299	0.193	0.330	0.394	0.543
$F_{p}$	$n / a$	$2.85 \times 10^{1}$	$1.33 \times 10^{4}$	$3.56 \times 10^{5}$	$6.92 \times 10^{4}$	$3.29 \times 10^{4}$	$3.10 \times 10^{4}$	$2.05 \times 10^{4}$	$1.13 \times 10^{4}$
$κ_{u c} / (2 π) = c_{0} / (λ_{u c} \cdot Q_{u c})$ (GHz)	$n / a$	1418.48	14.79	0.62	1.66	5.66	3.27	4.13	5.56
$τ_{u c} = 2 π / κ_{u c}$ (ns)	$n / a$	$7.05 \times 10^{- 4}$	0.07	1.61	0.60	0.18	0.31	0.24	0.18
Diamond NV centers	Diamond nanocrystals: $n_{os} \sim 2.4$
$λ_{os} (nm)$	637
$V_{eff}^{'} ({μm}^{3}) = V_{n} \times {(\frac{λ_{os}}{n_{def}})}^{3}$	$n / a$	$7.10 \times 10^{- 3}$	$1.37 \times 10^{- 3}$	$1.16 \times 10^{- 3}$	$2.15 \times 10^{- 3}$	$1.39 \times 10^{- 3}$	$2.37 \times 10^{- 3}$	$2.83 \times 10^{- 3}$	$3.91 \times 10^{- 3}$
$κ_{u c}^{'} / (2 π) = c_{0} / (λ_{os} \cdot Q_{u c})$ (GHz)	$n / a$	1269.47	14.09	0.62	1.73	5.63	3.49	4.43	5.85
$τ_{u c}^{'} = 2 π / κ_{u c}$ (ns)	$n / a$	$7.88 \times 10^{- 4}$	0.07	1.60	0.58	0.18	0.29	0.23	0.17
$γ^{ZPL} / (2 π)$ (GHz)	$3.3 \times 10^{- 3}$ ( $2 π / γ^{ZPL} = τ^{ZPL} \sim 300 ns$ )
$d_{EG}^{ZPL} (C \cdot m)$	$3.57 \times 10^{- 30} (= 1.07 D)$
$f_{EG}^{ZPL}$	0.025
$E_{s p}$ (V/m)	$n / a$	$4.77 \times 10^{5}$	$1.09 \times 10^{6}$	$1.18 \times 10^{6}$	$8.67 \times 10^{5}$	$1.08 \times 10^{6}$	$8.25 \times 10^{5}$	$7.55 \times 10^{5}$	$6.43 \times 10^{5}$
$g_{R}^{ZPL} / (2 π)$ (GHz)	$n / a$	2.57	5.85	6.37	4.67	5.81	4.45	4.07	3.46
$τ_{R}^{ZPL} = 2 π / g_{R}^{ZPL}$ (ns)	$n / a$	0.39	0.17	0.16	0.21	0.17	0.22	0.25	0.29

Defect type	A0			A1			A2
Defect size	$(0.5, 0.5, 0.25) \cdot c$			$(0.5, 0.5, 0.25) \cdot c$			$(0.5, 0.5, 0.25) \cdot c$
Air buffer size	$(a - 0.5 \cdot w, a - 0.5 \cdot w, 0.25 \cdot c)$			$(a, a, 0.25 \cdot c)$			$(a + 0.5 \cdot w, a + 0.5 \cdot w, 0.25 \cdot c)$
$n_{def}$	3.3			3.3			3.3
$n_{w p}$	3.3			3.3			3.3
$n_{bf}$	1			1			1
$c (μm)$	0.3358			0.3358			0.3358
Dipole orientation	$E_{x}$	$E_{y}$	$E_{z}$	$E_{x}$	$E_{y}$	$E_{z}$	$E_{x}$	$E_{y}$	$E_{z}$
$c / λ_{0}$	0.5409	0.5303	0.5377	0.5513	0.5391	0.5449	0.5715	0.5440	0.5593
$λ_{0}$ (nm)	620.86	633.28	624.46	609.13	622.88	616.23	587.59	617.28	600.35
$Q_{u c}$	$3.67 \times 10^{5}$	$2.50 \times 10^{5}$	$6.14 \times 10^{4}$	$1.48 \times 10^{5}$	$1.69 \times 10^{5}$	$4.25 \times 10^{4}$	$3.80 \times 10^{3}$	$1.17 \times 10^{5}$	$1.35 \times 10^{4}$
$V_{eff} ({μm}^{3})$	$6.66 \times 10^{- 4}$	$2.09 \times 10^{- 3}$	$1.13 \times 10^{- 3}$	$7.76 \times 10^{- 4}$	$1.39 \times 10^{- 3}$	$1.26 \times 10^{- 3}$	$1.73 \times 10^{- 3}$	$9.88 \times 10^{- 4}$	$7.22 \times 10^{- 4}$
$V_{n} = \frac{V_{eff}}{{(λ_{0} / n_{def})}^{3}}$	0.100	0.296	0.167	0.123	0.207	0.193	0.306	0.151	0.120
$F_{p}$	$2.78 \times 10^{5}$	$6.41 \times 10^{4}$	$2.79 \times 10^{4}$	$9.14 \times 10^{4}$	$6.21 \times 10^{4}$	$1.67 \times 10^{4}$	$9.45 \times 10^{2}$	$5.92 \times 10^{4}$	$8.58 \times 10^{3}$
$κ_{u c} / (2 π) = c_{0} / (λ_{u c} \cdot Q_{u c})$ (GHz)	1.32	1.90	7.82	3.32	2.85	11.45	134.14	4.13	36.91
$τ_{u c} = 2 π / κ_{u c}$ (ns)	0.76	0.53	0.13	0.30	0.35	0.09	0.01	0.24	0.03
Diamond NV centers	Diamond nanocrystals: $n_{os} \sim 2.4$
$λ_{os} (nm)$	637
$V_{eff}^{'} ({μm}^{3}) = V_{n} \times {(\frac{λ_{os}}{n_{def}})}^{3}$	$7.20 \times 10^{- 4}$	$2.13 \times 10^{- 3}$	$1.20 \times 10^{- 3}$	$8.87 \times 10^{- 4}$	$1.49 \times 10^{- 3}$	$1.39 \times 10^{- 3}$	$2.20 \times 10^{- 3}$	$1.09 \times 10^{- 3}$	$8.62 \times 10^{- 4}$
$κ_{u c}^{'} / (2 π) = c_{0} / (λ_{os} \cdot Q_{u c})$ (GHz)	1.28	1.88	7.66	3.17	2.78	11.08	123.73	4.01	34.78
$τ_{u c}^{'} = 2 π / κ_{u c}$ (ns)	0.78	0.53	0.13	0.32	0.36	0.09	0.01	0.25	0.03
$γ^{ZPL} / (2 π)$ (GHz)	$3.3 \times 10^{- 3}$ ( $2 π / γ^{ZPL} = τ^{ZPL} \sim 300 ns$ )
$d_{EG}^{ZPL} (C \cdot m)$	$3.57 \times 10^{- 30} (= 1.07 D)$
$f_{EG}^{ZPL}$	0.025
$E_{s p}$ (V/m)	$1.50 \times 10^{6}$	$8.71 \times 10^{5}$	$1.16 \times 10^{6}$	$1.35 \times 10^{6}$	$1.04 \times 10^{6}$	$1.08 \times 10^{6}$	$8.57 \times 10^{5}$	$1.22 \times 10^{6}$	$1.37 \times 10^{6}$
$g_{R}^{ZPL} / (2 π)$ (GHz)	8.07	4.69	6.24	7.27	5.61	5.82	4.62	6.57	7.38
$τ_{R}^{ZPL} = 2 π / g_{R}^{ZPL}$ (ns)	0.12	0.21	0.16	0.14	0.18	0.17	0.22	0.15	0.14

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Journal of the Optical Society of America B

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