Abstract

Quarter-wave stacks may be designed to reflect both polarizations of a specific wavelength band at all angles of incidences (omnidirectional). Expressions are given for both the omnidirectional band center wavelength and the bandwidth for selected values of the low- and the high-refractive-index layer values. It is shown that selecting the low refractive index near 1.45 maximizes the omnidirectional bandwidth for any value of the high refractive index. It is also shown that the omnidirectional bandwidth may be extended by addition of contiguous quarter-wave stacks.

© 1999 Optical Society of America

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References

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  1. H. A. Macleod, Thin-Film Optical Filters, 2nd ed. (Macmillan, New York, 1986), pp. 164–187.
  2. K. V. Popov, J. A. Dobrowolski, A. V. Tikhonravov, B. T. Sullivan, “Broadband high-reflection multilayer coatings at oblique angles of incidence,” Appl. Opt. 36, 2139–2151 (1997).
    [CrossRef] [PubMed]
  3. W. H. Southwell, Laser Induced Damage in Optical Materials: 1984 (National Bureau of Standards Special Publication 727, U.S. Government Printing Office, Washington, 1986), pp. 322–329.
  4. 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]
  5. E. Yablonovitch, “Engineered omnidirectional external-reflectivity spectra from one-dimensional layered interference filters,” Opt. Lett. 23, 1648–1649 (1998).
    [CrossRef]
  6. 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]
  7. J. P. Dowling, “Mirror on the wall: you’re omnidirectional after all?” Science 282, 1841–1842 (1998).
    [CrossRef]

1998 (4)

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. P. Dowling, “Mirror on the wall: you’re omnidirectional after all?” Science 282, 1841–1842 (1998).
[CrossRef]

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]

E. Yablonovitch, “Engineered omnidirectional external-reflectivity spectra from one-dimensional layered interference filters,” Opt. Lett. 23, 1648–1649 (1998).
[CrossRef]

1997 (1)

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]

Dobrowolski, J. A.

Dowling, J. P.

J. P. Dowling, “Mirror on the wall: you’re omnidirectional after all?” Science 282, 1841–1842 (1998).
[CrossRef]

Fan, S.

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.

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]

Joannopoulos, J. D.

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]

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters, 2nd ed. (Macmillan, New York, 1986), pp. 164–187.

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]

Popov, K. V.

Southwell, W. H.

W. H. Southwell, Laser Induced Damage in Optical Materials: 1984 (National Bureau of Standards Special Publication 727, U.S. Government Printing Office, Washington, 1986), pp. 322–329.

Sullivan, B. T.

Thomas, E. L.

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]

Tikhonravov, A. V.

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.

Appl. Opt. (1)

Opt. Lett. (2)

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]

J. P. Dowling, “Mirror on the wall: you’re omnidirectional after all?” Science 282, 1841–1842 (1998).
[CrossRef]

Other (2)

W. H. Southwell, Laser Induced Damage in Optical Materials: 1984 (National Bureau of Standards Special Publication 727, U.S. Government Printing Office, Washington, 1986), pp. 322–329.

H. A. Macleod, Thin-Film Optical Filters, 2nd ed. (Macmillan, New York, 1986), pp. 164–187.

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Figures (7)

Fig. 1
Fig. 1

Long- and short-wavelength reflection band edges for s (outer darker curves) and p (inner dashed curves) polarization as a function of angle of incidence. The wavelength axis is normalized to that of the normal-incidence band center; n L = 1.7 and n H = 3.4.

Fig. 2
Fig. 2

Reflectance as a function of angle of incidence evaluated at the omnidirectional reflection band center wavelength, 9 µm. The quarter-wave stack is at 10 µm with n L = 1.7 and n H = 3.4. The prescription is AIR (LH)10 SUB, where n SUB = 2.4. The upper solid curve is s-polarized light and the dashed curve is p-polarized light.

Fig. 3
Fig. 3

Equal omnidirectional reflection bandwidth contours plotted as a function of n L and n H . Note that an extremum occurs near n L = 1.45 for any value of n H for maximizing the omnidirectional reflection bandwidth.

Fig. 4
Fig. 4

Effective refractive index as a function of the material refractive index for different angles of incidence in air.

Fig. 5
Fig. 5

Reflectance at normal incidence (solid curve) and for the p polarization at 85° angle of incidence (dashed). The system is a quarter-wave stack at 10 µm, AIR (LH)10 SUB, where n L = 1.7 and n H = 3.4. The overlap of these curves defines the omnidirectional reflection bandwidth.

Fig. 6
Fig. 6

Reflectance at normal incidence (solid curve) and for the p polarization at 85° angle of incidence (dashed curve). The system consists of two contiguous quarter-wave stacks according to AIR (LH)10 1.0992L 1.1984 (HL)9 1.1984H SUB, where n L = 1.7 and n H = 3.4.

Fig. 7
Fig. 7

Reflectance at normal incidence (solid curve) and for the p polarization at 85° angle of incidence (dashed curve). The system consists of three contiguous quarter-wave stacks according to AIR (LH)10 1.0992L 1.1984 (HL)9 1.1984H 1.317L 1.436(HL)9 1.4736H SUB, where n L = 1.7 and n H = 3.4.

Equations (14)

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λC/2=tLηL+tHηH,
ηS=n cos θ,ηP =n/cos θ,
sin θi=nL sin θL =nH sin θH,
λC=λ01-sin2 θi/nL21/2+1-sin2 θi/nH21/2,
tL=λ0/4nL,tH=λ04nH.
λE=λC1±Δg,
Δg=2/πsin-1ηH-ηL/ηH+ηL.
λLONG=λC90°1+Δg90°,
λSHORT=λC0°1-Δg0°.
B=λLONG-λSHORT1/2λLONG+λSHORT.
B=2×12nH2-11/2nH+nL2-11/2nL1+2π sin-1nH2nL2-11/2-nL2nH2-11/2nH2nL2-11/2+nL2nH2-11/2-λ01-2π sin1/2nH-nLnH+nL12nH2-11/2nH+nL2-11/2nL1+2π sin-1nH2nL2-11/2-nL2nH2-11/2nH2nL2-11/2+nL2nH2-11/2+λ01-2π sin1/2nH-nLnH+nL.
ηS=n cos θ=n2-sin2 θi1/2,ηP=n/cos θ=n2/n2-sin2 θi1/2.
nL=5/41/2=1.118nH=51/2=2.236.
λM=λM-11-2π sin-1nH2nL2-11/2-nL2nH2-11/2nH2nL2-11/2+nL2nH2-11/2.

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