Abstract

Photonic bandgap (PBG) structures constructed from lossy, dispersive dielectric and metallic materials are characterized in terms of their reflection and transmission properties. Particular emphasis is given to PBG structures with defects. These PBG structures are modeled analytically with an ABCD matrix method for their single-frequency response. They also are modeled numerically with a finite-difference time-domain approach to determine their operating characteristics over a wide set of frequencies in a single simulation. It is shown that material dispersion can significantly alter the characteristics of a PBG structure’s frequency response. Metallic PBG structures at optical frequencies thus exhibit bandgap characteristics significantly different from those of their nondispersive dielectric counterparts. It is shown that microcavities whose mirrors are constructed from dispersive-material PBG structures can be designed to outperform similar nondispersive-mirror microcavities.

© 1999 Optical Society of America

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References

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  1. V. Radisic, Y. Qian, T. Itoh, “Broad-band power amplifier using dielectric photonic bandgap structure,” IEEE Microwave Guided Wave Lett. 8, 13–15 (1998).
    [CrossRef]
  2. V. Radisic, Y. Qian, R. Coccioli, T. Itoh, “Novel 2-D photonic bandgap structure for microstrip lines,” IEEE Microwave Guided Wave Lett. 8, 69–71 (1998).
    [CrossRef]
  3. E. Yablanovich, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
    [CrossRef]
  4. D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
    [CrossRef]
  5. J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonics Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).
  6. R. W. Ziolkowski, “FDTD Modeling of photonic nanometer-sized power splitters and switches,” in Integrated Photonics Research, Vol. 4 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 175–177.
  7. S. Kawakami, “Fabrication processes for 3D periodic nanostructures and photonic crystals,” in Integrated Photonics Research, Vol. 4 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 178–180.
  8. B. D’Urso, O. Painter, A. Yariv, A. Scherer, “Membrane microresonator lasers with 2-D photonic bandgap crystal mirrors for compact in-plane optics,” in Integrated Photonics Research, Vol. 4 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 181–183.
  9. O. Painter, R. Lee, A. Yariv, A. Scherer, “Photonic bandgap membrane microresonator,” in Integrated Photonics Research, Vol. 4 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 221–223.
  10. J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1986), pp. 120–132.
  11. J. B. Judkins, R. W. Ziolkowski, “Finite-difference time-domain modeling of nonperfectly conducting metallic thin-film gratings,” J. Opt. Soc. Am. A 12, 1974–1983 (1995).
    [CrossRef]
  12. M. J. Bloemr, M. Scalora, “Transmissive properties of Ag/MgF2 photonic band gaps,” Appl. Phys. Lett. 72, 1676–1678 (1998).
    [CrossRef]
  13. A. Taflove, Computational Electrodynamics (Artech House, Boston, Mass., 1995), pp. 111–134.

1998 (3)

V. Radisic, Y. Qian, T. Itoh, “Broad-band power amplifier using dielectric photonic bandgap structure,” IEEE Microwave Guided Wave Lett. 8, 13–15 (1998).
[CrossRef]

V. Radisic, Y. Qian, R. Coccioli, T. Itoh, “Novel 2-D photonic bandgap structure for microstrip lines,” IEEE Microwave Guided Wave Lett. 8, 69–71 (1998).
[CrossRef]

M. J. Bloemr, M. Scalora, “Transmissive properties of Ag/MgF2 photonic band gaps,” Appl. Phys. Lett. 72, 1676–1678 (1998).
[CrossRef]

1995 (1)

1994 (1)

D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
[CrossRef]

1987 (1)

E. Yablanovich, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef]

Bloemr, M. J.

M. J. Bloemr, M. Scalora, “Transmissive properties of Ag/MgF2 photonic band gaps,” Appl. Phys. Lett. 72, 1676–1678 (1998).
[CrossRef]

Coccioli, R.

V. Radisic, Y. Qian, R. Coccioli, T. Itoh, “Novel 2-D photonic bandgap structure for microstrip lines,” IEEE Microwave Guided Wave Lett. 8, 69–71 (1998).
[CrossRef]

D’Urso, B.

B. D’Urso, O. Painter, A. Yariv, A. Scherer, “Membrane microresonator lasers with 2-D photonic bandgap crystal mirrors for compact in-plane optics,” in Integrated Photonics Research, Vol. 4 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 181–183.

Itoh, T.

V. Radisic, Y. Qian, R. Coccioli, T. Itoh, “Novel 2-D photonic bandgap structure for microstrip lines,” IEEE Microwave Guided Wave Lett. 8, 69–71 (1998).
[CrossRef]

V. Radisic, Y. Qian, T. Itoh, “Broad-band power amplifier using dielectric photonic bandgap structure,” IEEE Microwave Guided Wave Lett. 8, 13–15 (1998).
[CrossRef]

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonics Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).

Judkins, J. B.

Kawakami, S.

S. Kawakami, “Fabrication processes for 3D periodic nanostructures and photonic crystals,” in Integrated Photonics Research, Vol. 4 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 178–180.

Kong, J. A.

J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1986), pp. 120–132.

Lee, R.

O. Painter, R. Lee, A. Yariv, A. Scherer, “Photonic bandgap membrane microresonator,” in Integrated Photonics Research, Vol. 4 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 221–223.

Maystre, D.

D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
[CrossRef]

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonics Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).

Painter, O.

B. D’Urso, O. Painter, A. Yariv, A. Scherer, “Membrane microresonator lasers with 2-D photonic bandgap crystal mirrors for compact in-plane optics,” in Integrated Photonics Research, Vol. 4 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 181–183.

O. Painter, R. Lee, A. Yariv, A. Scherer, “Photonic bandgap membrane microresonator,” in Integrated Photonics Research, Vol. 4 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 221–223.

Qian, Y.

V. Radisic, Y. Qian, R. Coccioli, T. Itoh, “Novel 2-D photonic bandgap structure for microstrip lines,” IEEE Microwave Guided Wave Lett. 8, 69–71 (1998).
[CrossRef]

V. Radisic, Y. Qian, T. Itoh, “Broad-band power amplifier using dielectric photonic bandgap structure,” IEEE Microwave Guided Wave Lett. 8, 13–15 (1998).
[CrossRef]

Radisic, V.

V. Radisic, Y. Qian, T. Itoh, “Broad-band power amplifier using dielectric photonic bandgap structure,” IEEE Microwave Guided Wave Lett. 8, 13–15 (1998).
[CrossRef]

V. Radisic, Y. Qian, R. Coccioli, T. Itoh, “Novel 2-D photonic bandgap structure for microstrip lines,” IEEE Microwave Guided Wave Lett. 8, 69–71 (1998).
[CrossRef]

Scalora, M.

M. J. Bloemr, M. Scalora, “Transmissive properties of Ag/MgF2 photonic band gaps,” Appl. Phys. Lett. 72, 1676–1678 (1998).
[CrossRef]

Scherer, A.

O. Painter, R. Lee, A. Yariv, A. Scherer, “Photonic bandgap membrane microresonator,” in Integrated Photonics Research, Vol. 4 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 221–223.

B. D’Urso, O. Painter, A. Yariv, A. Scherer, “Membrane microresonator lasers with 2-D photonic bandgap crystal mirrors for compact in-plane optics,” in Integrated Photonics Research, Vol. 4 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 181–183.

Taflove, A.

A. Taflove, Computational Electrodynamics (Artech House, Boston, Mass., 1995), pp. 111–134.

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonics Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).

Yablanovich, E.

E. Yablanovich, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef]

Yariv, A.

B. D’Urso, O. Painter, A. Yariv, A. Scherer, “Membrane microresonator lasers with 2-D photonic bandgap crystal mirrors for compact in-plane optics,” in Integrated Photonics Research, Vol. 4 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 181–183.

O. Painter, R. Lee, A. Yariv, A. Scherer, “Photonic bandgap membrane microresonator,” in Integrated Photonics Research, Vol. 4 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 221–223.

Ziolkowski, R. W.

J. B. Judkins, R. W. Ziolkowski, “Finite-difference time-domain modeling of nonperfectly conducting metallic thin-film gratings,” J. Opt. Soc. Am. A 12, 1974–1983 (1995).
[CrossRef]

R. W. Ziolkowski, “FDTD Modeling of photonic nanometer-sized power splitters and switches,” in Integrated Photonics Research, Vol. 4 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 175–177.

Appl. Phys. Lett. (1)

M. J. Bloemr, M. Scalora, “Transmissive properties of Ag/MgF2 photonic band gaps,” Appl. Phys. Lett. 72, 1676–1678 (1998).
[CrossRef]

IEEE Microwave Guided Wave Lett. (2)

V. Radisic, Y. Qian, T. Itoh, “Broad-band power amplifier using dielectric photonic bandgap structure,” IEEE Microwave Guided Wave Lett. 8, 13–15 (1998).
[CrossRef]

V. Radisic, Y. Qian, R. Coccioli, T. Itoh, “Novel 2-D photonic bandgap structure for microstrip lines,” IEEE Microwave Guided Wave Lett. 8, 69–71 (1998).
[CrossRef]

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

Phys. Rev. Lett. (1)

E. Yablanovich, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef]

Pure Appl. Opt. (1)

D. Maystre, “Electromagnetic study of photonic band gaps,” Pure Appl. Opt. 3, 975–993 (1994).
[CrossRef]

Other (7)

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonics Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).

R. W. Ziolkowski, “FDTD Modeling of photonic nanometer-sized power splitters and switches,” in Integrated Photonics Research, Vol. 4 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 175–177.

S. Kawakami, “Fabrication processes for 3D periodic nanostructures and photonic crystals,” in Integrated Photonics Research, Vol. 4 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 178–180.

B. D’Urso, O. Painter, A. Yariv, A. Scherer, “Membrane microresonator lasers with 2-D photonic bandgap crystal mirrors for compact in-plane optics,” in Integrated Photonics Research, Vol. 4 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 181–183.

O. Painter, R. Lee, A. Yariv, A. Scherer, “Photonic bandgap membrane microresonator,” in Integrated Photonics Research, Vol. 4 of OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), pp. 221–223.

J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1986), pp. 120–132.

A. Taflove, Computational Electrodynamics (Artech House, Boston, Mass., 1995), pp. 111–134.

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Figures (16)

Fig. 1
Fig. 1

Refractive index versus normalized driving frequency in the case in which ω0=ωp=ωC, Γ=0.01ωC and χL=1.

Fig. 2
Fig. 2

PBG structures (a) without defect and (b) with defect.

Fig. 3
Fig. 3

Reflection and transmission coefficients versus normalized driving frequency for an 11-layer structure (a) without defect and (b) for a microcavity consisting of a defect layer sandwiched between a pair of 11-layer structures. The parameters of medium 1 are ω01=100ωC, Γ1=0.01ω01 and χL1=1. The parameters of medium 2 are ω02=100ωC, Γ2=0.01ω02, and χL2=3.

Fig. 4
Fig. 4

(a) Reflection and (b) transmission coefficient versus normalized driving frequency for a three-layer structure. The parameters of media 1 and 2 are ω01=100ωC and ω02=100ωC, 3ωC, 2ωC, ωC, and 0.5ωC.

Fig. 5
Fig. 5

(a) Reflection and (b) transmission coefficient versus normalized driving frequency for a three-layer structure. The parameters of media 1 and 2 are ω01=100ωC, 3ωC, 2ωC, ωC, and 0.5ωC, and ω02=100ωC.

Fig. 6
Fig. 6

(a) Reflection and (b) transmission coefficient versus normalized driving frequency for 3-, 5-, 7-, 9-, and 11-layer structures. The parameters of media 1 and 2 are ω01=100ωC and ω02=ωC.

Fig. 7
Fig. 7

(a) Reflection and (b) transmission coefficient versus normalized driving frequency for 3-, 5-, 7-, 9-, and 11-layer structures. The parameters of media 1 and 2 are ω01=ωC and ω02=100ωC.

Fig. 8
Fig. 8

Transmission coefficient versus normalized driving frequency for a three-layer-mirror PBG microcavity with an air defect. The parameters of media 1 and 2 are ω01=100ωC and ω02=100ωC, 3ωC, 2ωC, ωC, and 0.5ωC.

Fig. 9
Fig. 9

Transmission coefficient versus normalized driving frequency for a three-layer-mirror PBG microcavity with an air defect. The parameters of media 1 and 2 are ω01=100ωC, 3ωC, 2ωC, ωC, and 0.5ωC and ω02=100ωC.

Fig. 10
Fig. 10

Transmission coefficient versus normalized driving frequency for an n-layer-mirror PBG microcavity with an air defect. The parameters of media 1 and 2 are ω01=100ωC and ω02=ωC.

Fig. 11
Fig. 11

Transmission coefficient versus normalized driving frequency for an n-layer-mirror PBG microcavity with an air defect. The parameters of media 1 and 2 are ω01=ωC and ω02=100ωC.

Fig. 12
Fig. 12

Transmission coefficient versus normalized driving frequency for a nondispersive three-layer-mirror PBG microcavity with a dispersive defect. The parameters of media 1 and 2 are ω01=100ωC and ω02=100ωC; for the defect region they are ω0d=100ωC, 3ωC, 2ωC, ωC, and 0.5ωC.

Fig. 13
Fig. 13

Transmission coefficient versus normalized driving frequency in the case of an n-layer, nondispersive-mirror-microcavity PBG structure with a dispersive defect. The mirrors are composed of 3, 7, 11, and 21 layers. The parameters of media 1 and 2 are ω01=100ωC and ω02=100ωC; for the defect region they are ω0d=ωC.

Fig. 14
Fig. 14

Reflection and transmission coefficients versus normalized driving frequency for 3-, 7-, and 11-layer-mirror PBG structures. The PBG is composed of (a) aluminum and dielectric layers and (b) gold and dielectric layers.

Fig. 15
Fig. 15

Transmission coefficient versus normalized driving frequency for an n-layer-mirror PBG microcavity with an air defect. The PBG mirrors are composed of (a) aluminum and dielectric layers and (b) gold and dielectric layers.

Fig. 16
Fig. 16

Transmission coefficient versus normalized driving frequency for a PBG microcavity formed by an aluminum or a gold defect sandwiched between a pair of 11-layer nondispersive mirrors.

Tables (2)

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Table 1 FDTD Dispersion-Model Parameters

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Table 2 Lorentz Material-Model Properties of Aluminum and Gold at 780 nm

Equations (18)

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2Px t2+ΓPx t+ω02Px=0ωp2χLEx,
Px,ω=0ωp2χLω02-ω2+jωΓEx,ω
Hyn+1/2k+12=Hyn-1/2k+12-Δtμ0Exn(k+1)-Exn(k)Δ z,
Exn+1(k)=Exn(k)-Δt0×Hyn+1/2(k+1/2)-Hyn+1/2(k-1/2)Δ z+Jxn+1/2(k).
Jx=Px t,
 Jx t+ΓJx=0ωp2χLEx-ω02Px.
Jxn+1/2(k)=1-ΓΔt21+ΓΔt2Jxn+1/2(k)+Δt1+ΓΔt2×[0ωp2χLExn(k)-ω02Pxn(k)],
Pxn+1(k)=Pxn(k)+Δ tJxn+1/2(k).
R=(2/1)N(0/1)-1(2/1)N(0/1)+1exp(j2k0d0),
T=2j(-1)N(2/1)N/2(0/1)1/2(2/1)N(0/1)+1×exp[jk0(d2N+1-d0)].
R=(2/1)M+N[(0def)/12]-1(2/1)M+N[(0def)/12]+1exp(j2k0d0),
T=2j(-1)M+N+1(2/1)(M+N)/2[(0def)/12]1/2(2/1)M+N[(0def)/12]+1×exp[jk0(d2M+2+2N+1-d0)],
R=exp(-j2k0d0)f(ω)-g(ω)exp-jπn2(ω)ωn2ωC1+R01 R12 exp-jπωωC2-R01 exp-jπωωC+R122 exp-jπn2(ω)ωn2ωC,
f(ω)=R01+R12(1+R012)exp[-jπ(ω/ωC)]+R01R122 exp[-j2π(ω/ωC)],
g(ω)=R01R122+R12(1+R012)exp[-jπ(ω/ωC)]+R01 exp[-j2π(ω/ωC)],
R01=n0-n1n0+n1,
R12=n1-n2(ωC)n1+n2(ωC),
Rexp(-j2k0d0)f(ω){1+R01R12 exp[-jπ(ω/ωC)]}2

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