Abstract

A design for omnidirectional high reflectors with quarter-wave dielectric stacks in the optical telecommunication band that uses conventional optical thin-film design theory is described. The omnidirectional bandwidth is derived in units of wavelength and investigated as a function of its high- and low-refractive-index values in the near infrared. The results show that the high refractive index should be larger than 2.26 for an omnidirectional high reflector and that the low refractive index for maximum omnidirectional bandwidth should be ∼1.5. It is shown that one can design broad-bandwidth omnidirectional high reflectors for S, C, and L bands for optical telecommunication simply by connecting the band edges of omnidirectional high reflectors.

© 2002 Optical Society of America

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References

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  1. J. N. Winn, Y. Fink, S. Fan, J. D. Joannopoulos, “Omnidirectional reflection from a one-dimensional photonic crystal,” Opt. Lett. 23, 1573–1575 (1998).
    [CrossRef]
  2. E. Yablonovitch, “Engineered omnidirectional external-reflectivity spectra from one-dimensional layered interference filters,” Opt. Lett. 23, 1648–1649 (1998).
    [CrossRef]
  3. Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
    [CrossRef] [PubMed]
  4. K. M. Chen, A. W. Sparks, H.-C. Luan, D. R. Lim, K. Wada, L. C. Kimerling, “SiO2/TiO2 omnidirectional reflector and microcavity resonator via the sol-gel method,” Appl. Phys. Lett. 75, 3805–3807 (1999).
    [CrossRef]
  5. G. V. Morozov, R. Gr. Maev, G. W. F. Drake, “Switching of electromagnetic waves by two-layered periodic dielectric structures,” Phys. Rev. E 60, 4860–4867 (1999).
    [CrossRef]
  6. T. Kawanishi, M. Izutsu, “Coaxial periodic optical waveguide,” Opt. Express 7, 10–22 (2000), http://www.opticsexpress.org .
    [CrossRef] [PubMed]
  7. M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, J. D. Joannopoulos, “An all-dielectric coaxial waveguide,” Science 289, 415–419 (2000).
    [CrossRef] [PubMed]
  8. W. H. Southwell, “Omnidirectional mirror design with quarter-wave dielectric stacks,” Appl. Opt. 38, 5464–5467 (1999).
    [CrossRef]
  9. H. A. Macleod, Thin-Film Optical Filters, 2nd ed. (Macmillan, New York, 1986), pp. 158–187.
  10. C. K. Hwangbo, Thin-Film Optics (Dasung, Seoul, 1999; in Korean), pp. 91–103.
  11. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), pp. 118–128.

2000 (2)

T. Kawanishi, M. Izutsu, “Coaxial periodic optical waveguide,” Opt. Express 7, 10–22 (2000), http://www.opticsexpress.org .
[CrossRef] [PubMed]

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, J. D. Joannopoulos, “An all-dielectric coaxial waveguide,” Science 289, 415–419 (2000).
[CrossRef] [PubMed]

1999 (3)

W. H. Southwell, “Omnidirectional mirror design with quarter-wave dielectric stacks,” Appl. Opt. 38, 5464–5467 (1999).
[CrossRef]

K. M. Chen, A. W. Sparks, H.-C. Luan, D. R. Lim, K. Wada, L. C. Kimerling, “SiO2/TiO2 omnidirectional reflector and microcavity resonator via the sol-gel method,” Appl. Phys. Lett. 75, 3805–3807 (1999).
[CrossRef]

G. V. Morozov, R. Gr. Maev, G. W. F. Drake, “Switching of electromagnetic waves by two-layered periodic dielectric structures,” Phys. Rev. E 60, 4860–4867 (1999).
[CrossRef]

1998 (3)

Chen, C.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef] [PubMed]

Chen, K. M.

K. M. Chen, A. W. Sparks, H.-C. Luan, D. R. Lim, K. Wada, L. C. Kimerling, “SiO2/TiO2 omnidirectional reflector and microcavity resonator via the sol-gel method,” Appl. Phys. Lett. 75, 3805–3807 (1999).
[CrossRef]

Drake, G. W. F.

G. V. Morozov, R. Gr. Maev, G. W. F. Drake, “Switching of electromagnetic waves by two-layered periodic dielectric structures,” Phys. Rev. E 60, 4860–4867 (1999).
[CrossRef]

Fan, S.

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, J. D. Joannopoulos, “An all-dielectric coaxial waveguide,” Science 289, 415–419 (2000).
[CrossRef] [PubMed]

J. N. Winn, Y. Fink, S. Fan, J. D. Joannopoulos, “Omnidirectional reflection from a one-dimensional photonic crystal,” Opt. Lett. 23, 1573–1575 (1998).
[CrossRef]

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef] [PubMed]

Fink, Y.

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, J. D. Joannopoulos, “An all-dielectric coaxial waveguide,” Science 289, 415–419 (2000).
[CrossRef] [PubMed]

J. N. Winn, Y. Fink, S. Fan, J. D. Joannopoulos, “Omnidirectional reflection from a one-dimensional photonic crystal,” Opt. Lett. 23, 1573–1575 (1998).
[CrossRef]

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef] [PubMed]

Hwangbo, C. K.

C. K. Hwangbo, Thin-Film Optics (Dasung, Seoul, 1999; in Korean), pp. 91–103.

Ibanescu, M.

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, J. D. Joannopoulos, “An all-dielectric coaxial waveguide,” Science 289, 415–419 (2000).
[CrossRef] [PubMed]

Izutsu, M.

Joannopoulos, J. D.

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, J. D. Joannopoulos, “An all-dielectric coaxial waveguide,” Science 289, 415–419 (2000).
[CrossRef] [PubMed]

J. N. Winn, Y. Fink, S. Fan, J. D. Joannopoulos, “Omnidirectional reflection from a one-dimensional photonic crystal,” Opt. Lett. 23, 1573–1575 (1998).
[CrossRef]

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef] [PubMed]

Kawanishi, T.

Kimerling, L. C.

K. M. Chen, A. W. Sparks, H.-C. Luan, D. R. Lim, K. Wada, L. C. Kimerling, “SiO2/TiO2 omnidirectional reflector and microcavity resonator via the sol-gel method,” Appl. Phys. Lett. 75, 3805–3807 (1999).
[CrossRef]

Lim, D. R.

K. M. Chen, A. W. Sparks, H.-C. Luan, D. R. Lim, K. Wada, L. C. Kimerling, “SiO2/TiO2 omnidirectional reflector and microcavity resonator via the sol-gel method,” Appl. Phys. Lett. 75, 3805–3807 (1999).
[CrossRef]

Luan, H.-C.

K. M. Chen, A. W. Sparks, H.-C. Luan, D. R. Lim, K. Wada, L. C. Kimerling, “SiO2/TiO2 omnidirectional reflector and microcavity resonator via the sol-gel method,” Appl. Phys. Lett. 75, 3805–3807 (1999).
[CrossRef]

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters, 2nd ed. (Macmillan, New York, 1986), pp. 158–187.

Maev, R. Gr.

G. V. Morozov, R. Gr. Maev, G. W. F. Drake, “Switching of electromagnetic waves by two-layered periodic dielectric structures,” Phys. Rev. E 60, 4860–4867 (1999).
[CrossRef]

Michel, J.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef] [PubMed]

Morozov, G. V.

G. V. Morozov, R. Gr. Maev, G. W. F. Drake, “Switching of electromagnetic waves by two-layered periodic dielectric structures,” Phys. Rev. E 60, 4860–4867 (1999).
[CrossRef]

Southwell, W. H.

Sparks, A. W.

K. M. Chen, A. W. Sparks, H.-C. Luan, D. R. Lim, K. Wada, L. C. Kimerling, “SiO2/TiO2 omnidirectional reflector and microcavity resonator via the sol-gel method,” Appl. Phys. Lett. 75, 3805–3807 (1999).
[CrossRef]

Thomas, E. L.

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, J. D. Joannopoulos, “An all-dielectric coaxial waveguide,” Science 289, 415–419 (2000).
[CrossRef] [PubMed]

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef] [PubMed]

Wada, K.

K. M. Chen, A. W. Sparks, H.-C. Luan, D. R. Lim, K. Wada, L. C. Kimerling, “SiO2/TiO2 omnidirectional reflector and microcavity resonator via the sol-gel method,” Appl. Phys. Lett. 75, 3805–3807 (1999).
[CrossRef]

Winn, J. N.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef] [PubMed]

J. N. Winn, Y. Fink, S. Fan, J. D. Joannopoulos, “Omnidirectional reflection from a one-dimensional photonic crystal,” Opt. Lett. 23, 1573–1575 (1998).
[CrossRef]

Yablonovitch, E.

Yeh, P.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), pp. 118–128.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

K. M. Chen, A. W. Sparks, H.-C. Luan, D. R. Lim, K. Wada, L. C. Kimerling, “SiO2/TiO2 omnidirectional reflector and microcavity resonator via the sol-gel method,” Appl. Phys. Lett. 75, 3805–3807 (1999).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. E (1)

G. V. Morozov, R. Gr. Maev, G. W. F. Drake, “Switching of electromagnetic waves by two-layered periodic dielectric structures,” Phys. Rev. E 60, 4860–4867 (1999).
[CrossRef]

Science (2)

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef] [PubMed]

M. Ibanescu, Y. Fink, S. Fan, E. L. Thomas, J. D. Joannopoulos, “An all-dielectric coaxial waveguide,” Science 289, 415–419 (2000).
[CrossRef] [PubMed]

Other (3)

H. A. Macleod, Thin-Film Optical Filters, 2nd ed. (Macmillan, New York, 1986), pp. 158–187.

C. K. Hwangbo, Thin-Film Optics (Dasung, Seoul, 1999; in Korean), pp. 91–103.

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988), pp. 118–128.

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Figures (6)

Fig. 1
Fig. 1

Reflectances of a quarter-wave stack of [air|(HL)10|NaCl, where n H = 4.6, n L = 1.6, n S = 1.48, and λ0 = 12.4 µm] for (a) TE- and (b) TM-polarized light as a function of wavelength and incident angle. (c) Reflection-band shift as a function of incident angle. In (c) the dashed curves represent the reflection-band edges obtained from approximate Eqs. (10) and (11) and the solid curves are those calculated from exact Eq. (6). The shaded area in (c) represents an omnidirectional high-reflection band.

Fig. 2
Fig. 2

Normalized omnidirectional bandwidth plotted as a function of low- and high-refractive indices where n 0 = 1.0 (air). Point P represents the zero omnidirectional bandwidth.

Fig. 3
Fig. 3

(a) Normalized omnidirectional bandwidths of 3.77% (solid curve) and 5.23% (dashed curve) as functions of low and high refractive indices. The low- and high-refractive-index values at each bandwidth are represented in parentheses. (b) At n H = 2.45 and Δλomni C = 3.77% there are two sets of materials: [n L , n H ] = [1.362, 2.45] (dashed curves) and [1.722, 2.45] (thinner solid curves). At n H = 2.45 the maximum bandwidth is possible at [1.516, 2.45] (thicker solid curves), as can be determined from (a). The shaded band represents the omnidirectional bandwidth of 5.23% at [1.516, 2.45], and the horizontal lines inside the shaded band represent a bandwidth of 3.77% at both [1.362, 2.45] and [1.722, 2.45]. (c) Normalized omnidirectional bandwidth at n H = 2.45 as a function of available low- and medium-refractive-index coating materials, which are, from left to right, 1.35 (Na3AlF6), 1.38 (MgF2), 1.44 (SiO2), 1.516 (glass), 1.65 (Al2O3), 1.77 (Y2O3), and 1.91 (Gd2O3).

Fig. 4
Fig. 4

Reflectances of a quarter-wave stack [air|(HL)15|glass, where n H = 2.45, n L = 1.44, n S = 1.51, and λ0 = 1810.2 nm] at θ0 = 0 and θ0 = 85° as a function of wavelength. The shaded area represents the omnidirectional high-reflection bandwidth of 78.3 nm, from 1550.9 to 1629.1 nm.

Fig. 5
Fig. 5

(a) Coupling of two omnidirectional high reflectors. Dashed curves, omnidirectional high reflector at λ0 = 1810.2 nm; solid curves, second high reflector at λ0 = 1501.9 nm; and shaded area, omnidirectional band of the two-coupled omnidirectional high reflector, where n H = 2.45, n L = 1.44, n S = 1.51. (b) Reflectances of an extended omnidirectional high reflector [air|0.83(HL) S (HL) S |glass, where n H = 2.45, n L = 1.44, n S = 1.51, and λ0 = 1810.2 nm] at θ0 = 0 and θ0 = 85° as a function of wavelength. The shaded area represents the two-coupled omnidirectional high-reflector bandwidth of 342.4 nm from 1286.8 to 1629.1 nm. Note that, as the period increases from S = 15 (dashed curves) to S = 20 (solid curves), the ripple in the omnidirectional band decreases.

Fig. 6
Fig. 6

Reflectances of a wideband omnidirectional high reflector [air|0.69(HL)20 0.83(HL)20(HL)20|glass, where n H = 2.45, n L = 1.44, n S = 1.51, and λ0 = 1810.2 nm] at θ0 = 0 and θ0 = 85° as a function of wavelength. The shaded area represents the bandwidth of a three-coupled omnidirectional high reflector of 561.5 nm, from 1067.7 to 1629.1 nm.

Tables (1)

Tables Icon

Table 1 Pairs of [nH , n L ] for the Maximum Normalized Omnidirectional Bandwidth a

Equations (25)

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

M11M12M21M22=cos δHi sin δH/ηHiηH sin δHcos δH×cos δLi sin δL/ηLiηL sin δLcos δL,
δj=2πλ njdj cos θj,
ηj=nj cos θj
ηj=nj/cos θj
M11+M222=-1.
cos δH cos δL-12ηLηH+ηHηLsin δH sin δL=-1,
cosδH+δL2=±Z-1Z+11/2,
Z=12ηLηH+ηHηL.
δj=π2λ0λcos θj,
λshortθ0λ0=Fθ0cos-1-ηH-ηLηH+ηL-1,
λlongθ0λ0=Fθ0cos-1ηH-ηLηH+ηL-1,
Fθ0=π4×nLnH2-n02 sin2 θ01/2+nHnL2-n02sin2 θ01/2nHnL.
Δλθ0=λlongθ0-λshortθ0.
ΔλomniλC=2λlongTM90°-λshort0°λlongTM90°+λshort0°,
Δλomni=λlongTM90°-λshort0°
λC=λlongTM90°+λshort0°/2.
λshort0°λ0=π2cos-1-nH-nLnH+nL-1,
λlongTM90°λ0=π4nLnH2-n02+nHnL2-n02nHnL ×cos-1nH2nL2-n02-nL2nH2-n02nH2nL2-n02+nL2nH2-n02-1.
nLΔλomniλC=0
λ0=8λCπnLnH2-n02+nHnL2-n02nHnLcos-1nH2nL2-n02-nL2nH2-n02nH2nL2-n02+nL2nH2-n02-1+2cos-1-nH-nLnH+nL-1.
λshort,S1TM90°=λlong,S2TM90°.
Δλomni=λlong,S1TM90°-λshort,S20°.
λ02=λ01cos-1A-BA+Bcos-1-A-BA+B,
Δλomni=λlong,S1TM90°-λshort,Sm0°,
λ0m=λ01cos-1A-BA+Bcos-1-A-BA+Bm-1.

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