Abstract

Resonant mirrors introduce large spectral gradients in reflected phase while maintaining high reflectivity, allowing synthesis of optimized reflected phase for many practical applications. In this paper we show theoretically that asymmetry is required for negative group delay in lossless mirrors and explore the limits of reflected phase in resonant mirrors through the use of coupled mode theory and rigorous couple wave analysis. Our coupled mode theory shows that the phase response of resonant mirrors is determined by interacting resonances and gives insight into tradeoffs in design of mirrors with desired phase response.

© 2014 Optical Society of America

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References

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2014 (1)

2013 (2)

2010 (2)

2009 (1)

Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12(12), 60–69 (2009).
[Crossref]

2008 (1)

A. Ahmadi and H. Mosallaei, “Physical configuration and performance modeling of all-dielectric metamaterials,” Phys. Rev. B 77(4), 045104 (2008).
[Crossref]

2004 (2)

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40(10), 1511–1518 (2004).
[Crossref]

O. Kilic, S. Kim, W. Suh, Y. A. Peter, A. S. Sudbø, M. F. Yanik, S. Fan, and O. Solgaard, “Photonic crystal slabs demonstrating strong broadband suppression of transmission in the presence of disorders,” Opt. Lett. 29(23), 2782–2784 (2004).
[Crossref] [PubMed]

2003 (2)

W. Suh, M. F. Yanik, O. Solgaard, and S. Fan, “Displacement-sensitive photonic crystal structures based on guided resonance in photonic crystal slabs,” Appl. Phys. Lett. 82(13), 1999–2001 (2003).
[Crossref]

S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20(3), 569–572 (2003).
[Crossref] [PubMed]

2002 (1)

S. Fan and J. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002).
[Crossref]

2000 (1)

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406(6793), 277–279 (2000).
[Crossref] [PubMed]

1995 (1)

1985 (1)

Ahmadi, A.

A. Ahmadi and H. Mosallaei, “Physical configuration and performance modeling of all-dielectric metamaterials,” Phys. Rev. B 77(4), 045104 (2008).
[Crossref]

Benedick, A. J.

Chang, G.

Chang-Hasnain, C. J.

Chen, L.-J.

Dogariu, A.

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406(6793), 277–279 (2000).
[Crossref] [PubMed]

Fan, S.

O. Kilic, S. Kim, W. Suh, Y. A. Peter, A. S. Sudbø, M. F. Yanik, S. Fan, and O. Solgaard, “Photonic crystal slabs demonstrating strong broadband suppression of transmission in the presence of disorders,” Opt. Lett. 29(23), 2782–2784 (2004).
[Crossref] [PubMed]

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40(10), 1511–1518 (2004).
[Crossref]

S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20(3), 569–572 (2003).
[Crossref] [PubMed]

W. Suh, M. F. Yanik, O. Solgaard, and S. Fan, “Displacement-sensitive photonic crystal structures based on guided resonance in photonic crystal slabs,” Appl. Phys. Lett. 82(13), 1999–2001 (2003).
[Crossref]

S. Fan and J. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002).
[Crossref]

Fedorov, V.

Gellineau, A.

Heritage, J. P.

Joannopoulos, J.

S. Fan and J. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002).
[Crossref]

Joannopoulos, J. D.

Karagodsky, V.

Kärtner, F. X.

Kilic, O.

Kim, S.

Kuzmich, A.

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406(6793), 277–279 (2000).
[Crossref] [PubMed]

Li, C.-H.

Li, J.

Lippens, D.

Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12(12), 60–69 (2009).
[Crossref]

Mosallaei, H.

A. Ahmadi and H. Mosallaei, “Physical configuration and performance modeling of all-dielectric metamaterials,” Phys. Rev. B 77(4), 045104 (2008).
[Crossref]

Pervak, V.

Pervak, Y. A.

Peter, Y. A.

Philips, D. F.

Rakic, A. D.

Sedgwick, F. G.

Solgaard, O.

Sudbø, A. S.

Suh, W.

O. Kilic, S. Kim, W. Suh, Y. A. Peter, A. S. Sudbø, M. F. Yanik, S. Fan, and O. Solgaard, “Photonic crystal slabs demonstrating strong broadband suppression of transmission in the presence of disorders,” Opt. Lett. 29(23), 2782–2784 (2004).
[Crossref] [PubMed]

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40(10), 1511–1518 (2004).
[Crossref]

S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20(3), 569–572 (2003).
[Crossref] [PubMed]

W. Suh, M. F. Yanik, O. Solgaard, and S. Fan, “Displacement-sensitive photonic crystal structures based on guided resonance in photonic crystal slabs,” Appl. Phys. Lett. 82(13), 1999–2001 (2003).
[Crossref]

Thurston, R. N.

Trubetskov, M.

Walsworth, R. L.

Wang, L. J.

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406(6793), 277–279 (2000).
[Crossref] [PubMed]

Wang, Z.

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40(10), 1511–1518 (2004).
[Crossref]

Weiner, A. M.

Wong, Y. P.

Yang, R.

Yanik, M. F.

O. Kilic, S. Kim, W. Suh, Y. A. Peter, A. S. Sudbø, M. F. Yanik, S. Fan, and O. Solgaard, “Photonic crystal slabs demonstrating strong broadband suppression of transmission in the presence of disorders,” Opt. Lett. 29(23), 2782–2784 (2004).
[Crossref] [PubMed]

W. Suh, M. F. Yanik, O. Solgaard, and S. Fan, “Displacement-sensitive photonic crystal structures based on guided resonance in photonic crystal slabs,” Appl. Phys. Lett. 82(13), 1999–2001 (2003).
[Crossref]

Zhang, F.

Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12(12), 60–69 (2009).
[Crossref]

Zhao, Q.

Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12(12), 60–69 (2009).
[Crossref]

Zhou, J.

Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12(12), 60–69 (2009).
[Crossref]

Zhu, W.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

W. Suh, M. F. Yanik, O. Solgaard, and S. Fan, “Displacement-sensitive photonic crystal structures based on guided resonance in photonic crystal slabs,” Appl. Phys. Lett. 82(13), 1999–2001 (2003).
[Crossref]

IEEE J. Quantum Electron. (1)

W. Suh, Z. Wang, and S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40(10), 1511–1518 (2004).
[Crossref]

J. Opt. Soc. Am. A (1)

Mater. Today (1)

Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12(12), 60–69 (2009).
[Crossref]

Nature (1)

L. J. Wang, A. Kuzmich, and A. Dogariu, “Gain-assisted superluminal light propagation,” Nature 406(6793), 277–279 (2000).
[Crossref] [PubMed]

Opt. Express (4)

Opt. Lett. (3)

Phys. Rev. B (2)

A. Ahmadi and H. Mosallaei, “Physical configuration and performance modeling of all-dielectric metamaterials,” Phys. Rev. B 77(4), 045104 (2008).
[Crossref]

S. Fan and J. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002).
[Crossref]

Other (1)

H. A. Haus, in Waves and Fields in Optoelectronics (Prentice-Hall Englewood Cliffs, 1984), Vol. 464, pp. 59–63.

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Figures (7)

Fig. 1
Fig. 1 Mirror schematic and its effective thickness (teff).
Fig. 2
Fig. 2 Phasor diagram representing the effective thickness possible for the reflected field r 12 in Eq. (20).
Fig. 3
Fig. 3 a) CMT Model b) RCWA Model c) Resulting reflectance and effective thickness of both models. d) Individual Reflectance and phase of guided modes (G.R.) and the direct mode (D.M.). e) Phasors of the resonators present from CMT.
Fig. 4
Fig. 4 a) CMT Model b) RCWA Model c) Resulting reflectance and effective thickness of both models. d) Individual Reflectance and phase of guided resonances and the direct mode.
Fig. 5
Fig. 5 a) Resulting reflectance and effective thickness of both models. b) Individual Reflectance and phase of guided modes (G.R.) and the direct mode (D.M.).
Fig. 6
Fig. 6 a) CMT Model b) RCWA Model c) Resulting reflectance and effective thickness of both models. d) Individual Reflectance and phase of guided resonances and the direct mode.
Fig. 7
Fig. 7 a) Refractive index, n and absorption coefficient, k for Aluminum. b) Reflectance and effective thickness of the bulk material.

Equations (22)

Equations on this page are rendered with MathJax. Learn more.

t e f f = 1 2 d ϕ r ( β ) d β .
( x , y , z , t ) = E 0 U ( x , y ) e j ( ω t β z )
s i = E 0 2 η i e j β z
[ s 1 s 2 ] = S m i r r o r [ s 1 + s 2 + ] = [ r 12 t 21 t 12 r 21 ] [ s 1 + s 2 + ]
t 12 = t 21
| r 12 | = | r 21 |
r 12 * t 21 + t 12 * r 21 = 0
r 21 r 12 * = t 21 t 12 * e j ( ϕ r 21 + ϕ r 12 ) = e j ( π + 2 ϕ t 12 ) ϕ r 21 + ϕ r 12 = π + 2 ϕ t 12      
d ϕ r 21 d β + d ϕ r 12 d β = 2 d ϕ t d β
t e f f 21 + t e f f 12 = d ϕ t d β
ϕ t = 2 π n e f f L λ = n e f f β L
t e f f 21 + t e f f 12 = ( d n e f f d β β + n e f f ) L 2
d a d t = ( j Ω Γ ) a + D T | s +
| s = C | s + + D a
C D * = D ; D D = 2 Γ
A k = ( C 11 e j θ k C 22 e j θ k ) + ( C 11 e j θ k C 22 e j θ k ) 2 + 4 C 12 C 21 2 C 12 e j θ k
j ω a k = ( j ω k γ k ) a k + D T | s +
a k = D T | s + j ( ω ω k ) + γ k
r 12 , k = D 1 k 2 j ( ω ω k ) + γ k
r 12 = C 11 + k D 1 k 2 j ( ω ω k ) + γ k
r 12 = C 11 [ D ( j ( Ω I ω ) Γ ) 1 D T ] 11
t e f f = ( n 2 k 2 1 ) d k d β 2 n k d n d β ( k 2 + n 2 1 ) 2 + 4 k 2

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