Modeling photonic crystal interfaces and stacks: impedance-based approaches

Felix J. Lawrence; C. Martijn de Sterke; Lindsay C. Botten; R. C. McPhedran; Kokou B. Dossou

doi:10.1364/AOP.5.000385

Advances in Optics and Photonics
Vol. 5,
Issue 4,
pp. 385-455
(2013)
•https://doi.org/10.1364/AOP.5.000385

Modeling photonic crystal interfaces and stacks: impedance-based approaches

Felix J. Lawrence, C. Martijn de Sterke, Lindsay C. Botten, R. C. McPhedran, and Kokou B. Dossou

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Felix J. Lawrence, C. Martijn de Sterke, Lindsay C. Botten, R. C. McPhedran, and Kokou B. Dossou, "Modeling photonic crystal interfaces and stacks: impedance-based approaches," Adv. Opt. Photon. 5, 385-455 (2013)

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- Optical Devices
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About this Article
History
- Original Manuscript: February 28, 2013
- Revised Manuscript: June 3, 2013
- Manuscript Accepted: June 27, 2013
- Published: September 12, 2013
Virtual Issues
Advances in Optics and Photonics (2013)

Abstract

In many research areas, the reflective properties of a bulk medium are characterized by its impedance or an impedance-like quantity. Such a quantity is essential for the efficient design of stacked structures such as antireflection coatings and thin-film filters. For 2D photonic crystals and metamaterials, the literature contains multiple definitions of impedance, not all of which are consistent. We review these proposed definitions, evaluate their regions of applicability, and numerically test their accuracy in a variety of salient photonic crystal examples.

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	PC 1	PC 2	PC 3	PC 4
Lattice type	square	rectangular	hexagonal	hexagonal
Radius	$0.35 a$	$0.3 a$	$0.3 a$	$0.2 a$
$a_{y}$	$a$	$0.7 a$	$a \sqrt{3} / 2$	$0.7 a \sqrt{3} / 2$

Section	9.1	9.2	9.3	9.4	9.5
Polarization	$E_{z}$	$H_{z}$	$E_{z}$	$H_{z}$	$H_{z}$
Incident angle	0°	0°	30°	0°	20°
PCs studied	1, 2	1	1	3	3, 4

Symbol	Definition	Eq. No.
$Z_{vg}$	$Z_{0} / \frac{d k}{d ω}$	(22)
$Z_{Boscolo}$	$\frac{\int η (x) Re (S_{⊥} (x)) d x)}{\int Re (S_{⊥} (x)) d x}$	(27)
$Z_{Ushida}$	$\pm η (0)$	(28)
$Z_{Biswas}$	$\frac{\int {\| E_{z} \|}^{2} + {\| E_{x} \|}^{2} d x}{\int E_{z}^{} H_{x} - E_{x}^{} H_{z} d x}$	(30)
$Z_{Momeni}$	$\frac{{\| 〈 E_{∥} 〉 \|}^{2}}{2 S_{⊥}}$ , $\frac{2 S_{⊥}}{{\| 〈 H_{∥} 〉 \|}^{2}}$	(31)
$Z_{RE}$	$Z_{vac} \frac{1 + r_{vac / PC}}{1 - r_{vac / PC}}$	(34)
$Z_{B}$	$\frac{\int E_{∥} (x) d x}{\int H_{∥} (x) d x}$	(33)
$Z_{1 PW}$	$\frac{\int e^{- i k_{x} x} E_{∥} (x) d x}{\int e^{- i k_{x} x} H_{∥} (x) d x}$	(32)
$Z$ (square)	$Z_{i, j} = \frac{2}{a_{x}} \int H_{∥ i}^{(0)} (- x) E_{∥ j} (x) d x$	(38)
$Z$ (hex)	See Eq.	(39)

	PC 1	PC 2	PC 3	PC 4
Lattice type	square	rectangular	hexagonal	hexagonal
Radius	$0.35 a$	$0.3 a$	$0.3 a$	$0.2 a$
$a_{y}$	$a$	$0.7 a$	$a \sqrt{3} / 2$	$0.7 a \sqrt{3} / 2$

Section	9.1	9.2	9.3	9.4	9.5
Polarization	$E_{z}$	$H_{z}$	$E_{z}$	$H_{z}$	$H_{z}$
Incident angle	0°	0°	30°	0°	20°
PCs studied	1, 2	1	1	3	3, 4

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