Abstract

Recently it was shown theoretically [Opt. Commun. 174, 43 (2000)] that a Šolc folded-type anisotropic dielectric structure under certain conditions exhibits omnidirectional reflection at any polarization over a wide spectral range. Here, omnidirectional reflection from Šolc folded-type dielectric periodic structures is further analyzed. Transfer-matrix methodology is applied. Simple expressions are obtained for transfer matrices at interfaces and the unit cell translation matrix. Dispersion relations are determined. Numerical examples are shown comparatively for isotropic and Šolc-type anisotropic periodic structures. The Šolc-type structure has a wider band of omnidirectional reflection. The results demonstrate that anisotropic materials should be useful in photonic bandgap structures.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. M. Bowden, J. P. Dowling, H. O. Everitt, eds., feature on Development and Applications of Materials Exhibiting Photonic Bandgaps, J. Opt. Soc. Am. B 10, 279–413 (1993).
  2. G. Kurizki, J. W. Haus, eds., feature on Photonic Band Structures, J. Mod. Opt. 41, 171–404 (1994).
  3. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2063 (1987).
    [CrossRef] [PubMed]
  4. K. M. Ho, C. T. Chan, C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
    [CrossRef] [PubMed]
  5. 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]
  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. E. Yablonovitch, “Engineered omnidirectional external-reflectivity spectra from one-dimensional layered interference filter,” Opt. Lett. 23, 1648–1649 (1998).
    [CrossRef]
  8. I. Abdulhalim, “Omnidirectional reflection from anisotropic periodic dielectric stack,” Opt. Commun. 174, 43–50 (2000).
    [CrossRef]
  9. I. Šolc, “A new kind of double refracting filter,” Czech. J. Phys. 4, 53–66 (1954).
  10. I. Šolc, “Birefringent chain filters,” J. Opt. Soc. Am. 55, 621–625 (1965).
    [CrossRef]
  11. A. Lakhtakia, “Dielectric sculptured thin films as Šolc filters,” Opt. Eng. 37, 1870–1875 (1998).
    [CrossRef]
  12. H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1957), Chap. 4, pp. 107–109.
  13. G. D. Landry, T. A. Maldonado, “Complete method to determine transmission and reflection characteristics at a planar interface between arbitrarily oriented biaxial media,” J. Opt. Soc. Am. A 12, 2048–2063 (1995).
    [CrossRef]
  14. P. Yeh, “Extended Jones matrix method,” J. Opt. Soc. Am. 72, 507–513 (1982).
    [CrossRef]
  15. C. Gu, P. Yeh, “Extended Jones matrix method. II,” J. Opt. Soc. Am. A 10, 966–973 (1993).
    [CrossRef]
  16. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 14, pp. 665–718.
  17. E. Cojocaru, “Explicit relations for the extraordinary-ray trajectory at the back of a rotating uniaxial birefringent plate,” Appl. Opt. 36, 8886–8888 (1997).
    [CrossRef]
  18. P. Yeh, A. Yariv, C. S. Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67, 423–438 (1977).
    [CrossRef]
  19. P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69, 742–756 (1979).
    [CrossRef]
  20. J. He, M. Cada, “Combined distributed feedback and Fabry–Perot structures with a phase-matching layer for optical bistable devices,” Appl. Phys. Lett. 61, 2150–2152 (1992).
    [CrossRef]
  21. M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, C. M. Bowden, “Thin-film nonlinear optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
    [CrossRef]

2000 (1)

I. Abdulhalim, “Omnidirectional reflection from anisotropic periodic dielectric stack,” Opt. Commun. 174, 43–50 (2000).
[CrossRef]

1998 (4)

A. Lakhtakia, “Dielectric sculptured thin films as Šolc filters,” Opt. Eng. 37, 1870–1875 (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]

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 filter,” Opt. Lett. 23, 1648–1649 (1998).
[CrossRef]

1997 (1)

1995 (2)

1994 (1)

G. Kurizki, J. W. Haus, eds., feature on Photonic Band Structures, J. Mod. Opt. 41, 171–404 (1994).

1993 (2)

C. M. Bowden, J. P. Dowling, H. O. Everitt, eds., feature on Development and Applications of Materials Exhibiting Photonic Bandgaps, J. Opt. Soc. Am. B 10, 279–413 (1993).

C. Gu, P. Yeh, “Extended Jones matrix method. II,” J. Opt. Soc. Am. A 10, 966–973 (1993).
[CrossRef]

1992 (1)

J. He, M. Cada, “Combined distributed feedback and Fabry–Perot structures with a phase-matching layer for optical bistable devices,” Appl. Phys. Lett. 61, 2150–2152 (1992).
[CrossRef]

1990 (1)

K. M. Ho, C. T. Chan, C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

1987 (1)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2063 (1987).
[CrossRef] [PubMed]

1982 (1)

1979 (1)

1977 (1)

1965 (1)

1954 (1)

I. Šolc, “A new kind of double refracting filter,” Czech. J. Phys. 4, 53–66 (1954).

Abdulhalim, I.

I. Abdulhalim, “Omnidirectional reflection from anisotropic periodic dielectric stack,” Opt. Commun. 174, 43–50 (2000).
[CrossRef]

Bloemer, M. J.

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, C. M. Bowden, “Thin-film nonlinear optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 14, pp. 665–718.

Bowden, C. M.

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, C. M. Bowden, “Thin-film nonlinear optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[CrossRef]

Cada, M.

J. He, M. Cada, “Combined distributed feedback and Fabry–Perot structures with a phase-matching layer for optical bistable devices,” Appl. Phys. Lett. 61, 2150–2152 (1992).
[CrossRef]

Chan, C. T.

K. M. Ho, C. T. Chan, C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

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]

Cojocaru, E.

Dowling, J. P.

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, C. M. Bowden, “Thin-film nonlinear optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[CrossRef]

Fan, S.

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]

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]

Goldstein, H.

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1957), Chap. 4, pp. 107–109.

Gu, C.

He, J.

J. He, M. Cada, “Combined distributed feedback and Fabry–Perot structures with a phase-matching layer for optical bistable devices,” Appl. Phys. Lett. 61, 2150–2152 (1992).
[CrossRef]

Ho, K. M.

K. M. Ho, C. T. Chan, C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Hong, C. S.

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]

Lakhtakia, A.

A. Lakhtakia, “Dielectric sculptured thin films as Šolc filters,” Opt. Eng. 37, 1870–1875 (1998).
[CrossRef]

Landry, G. D.

Maldonado, T. A.

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]

Scalora, M.

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, C. M. Bowden, “Thin-film nonlinear optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[CrossRef]

Šolc, I.

I. Šolc, “Birefringent chain filters,” J. Opt. Soc. Am. 55, 621–625 (1965).
[CrossRef]

I. Šolc, “A new kind of double refracting filter,” Czech. J. Phys. 4, 53–66 (1954).

Soukoulis, C. M.

K. M. Ho, C. T. Chan, C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

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]

Tocci, M. D.

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, C. M. Bowden, “Thin-film nonlinear optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[CrossRef]

Winn, J. N.

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]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 14, pp. 665–718.

Yablonovitch, E.

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

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2063 (1987).
[CrossRef] [PubMed]

Yariv, A.

Yeh, P.

Appl. Opt. (1)

Appl. Phys. Lett. (2)

J. He, M. Cada, “Combined distributed feedback and Fabry–Perot structures with a phase-matching layer for optical bistable devices,” Appl. Phys. Lett. 61, 2150–2152 (1992).
[CrossRef]

M. D. Tocci, M. J. Bloemer, M. Scalora, J. P. Dowling, C. M. Bowden, “Thin-film nonlinear optical diode,” Appl. Phys. Lett. 66, 2324–2326 (1995).
[CrossRef]

Czech. J. Phys. (1)

I. Šolc, “A new kind of double refracting filter,” Czech. J. Phys. 4, 53–66 (1954).

J. Mod. Opt. (1)

G. Kurizki, J. W. Haus, eds., feature on Photonic Band Structures, J. Mod. Opt. 41, 171–404 (1994).

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (2)

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

C. M. Bowden, J. P. Dowling, H. O. Everitt, eds., feature on Development and Applications of Materials Exhibiting Photonic Bandgaps, J. Opt. Soc. Am. B 10, 279–413 (1993).

Opt. Commun. (1)

I. Abdulhalim, “Omnidirectional reflection from anisotropic periodic dielectric stack,” Opt. Commun. 174, 43–50 (2000).
[CrossRef]

Opt. Eng. (1)

A. Lakhtakia, “Dielectric sculptured thin films as Šolc filters,” Opt. Eng. 37, 1870–1875 (1998).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. Lett. (2)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2063 (1987).
[CrossRef] [PubMed]

K. M. Ho, C. T. Chan, C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Science (1)

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]

Other (2)

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1957), Chap. 4, pp. 107–109.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), Chap. 14, pp. 665–718.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Šolc folded-type anisotropic periodic dielectric structure consisting of alternating birefringent uniaxial layers that have the same principal refractive indices and different orientations of the principal axes with respect to the laboratory axes. The azimuth angle takes values +ϕ and -ϕ in the two layers of one unit cell. Layers have the same thickness d. The period is Λ = 2d. The phase velocity indices of ordinary and extraordinary waves are n o and n eo , respectively. Interfaces lie in the xy plane. k 0 is the wave vector of the incident wave in the isotropic medium of incidence of refractive index n 0 at angle φ0 with respect to the normal positive z axis.

Fig. 2
Fig. 2

Isotropic periodic structure with alternating isotropic layers of equal thickness d and constant refractive indices n 1 = n and n 2 = n . The period is Λ = 2d.

Fig. 3
Fig. 3

Numerical examples for the isotropic periodic structure shown in Fig. 2 that comprises N = 20 periods of alternating layers with refractive indices n = 1.7 and n = 2.3 in air (n 0 = 1): (a) The smallest values of Ω that satisfy condition (31) at various incidence angles φ0 for the p(*) and s(×) modes. The solid curve represents Eq. (34) with N = 1. (b) The widths ΔΩ p and ΔΩ s of the lowest forbidden gap for the p(*) and s(×) modes versus φ0. (c) Variations of R pt = R pp and (d) R st = R ss versus φ0 and Ω. Ω is the normalized, dimensionless frequency defined by Eq. (32); φ0 is varied in increments of 2.5° and Ω in increments of 0.008, except in (b), where Ω is varied in increments of 0.0002.

Fig. 4
Fig. 4

Numerical examples for the Šolc folded-type anisotropic periodic structure shown in Fig. 1 that comprises N = 20 periods of alternating birefringent layers with azimuth angles +π/4 and -π/4 and the same principal refractive indices, n = 1.7 and n = 2.3, in air (n 0 = 1): (a) The smallest values of Ω that satisfy condition (31) at various incidence angles φ0 for the p(*) and s(×) modes. The lowest forbidden gap is the same for the p and the s modes. Solid curve, Eq. (33) with N = 1; (b) variation of R pt , (c) R st , (d) R ps = R sp , (e) R ss , and (f) R pp - R ss versus φ0 and Ω. φ0 is varied in increments of 2.5°; Ω, in increments of 0.008.

Fig. 5
Fig. 5

Width ΔΩ of the lowest forbidden gap versus φ0 and ϕ for a Šolc folded-type periodic structure with n = 1.7, n = 2.3, and n 0 = 1; φ0 is varied in increments of 10°, ϕ in increments of 5°, and Ω in increments of 0.002.

Fig. 6
Fig. 6

Upper and lower limits of the Ω interval where the lowest forbidden gaps at φ0 = 0 and φ0 = π/2 overlap versus azimuth ϕ for a Šolc folded-type periodic structure with n = 1.7, n = 2.3, and n 0 = 1. Shaded area, regimes of omnidirectional reflection irrespective of the light polarization; Ω is varied in increments of 0.0002 and ϕ in increments of 1°.

Equations (44)

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

P±=cos ϕ±sin ϕ0001±sin ϕ-cos ϕ0.
kvkxx+kvzz=ω/cξx+ζvz,  v=0, o, eo,
ζo=n2-ξ21/2,
ζeo=n2-ξ21+sin2 ϕn2/n2-11/2.
E0=E0ss+E0ppexpjk0r-ωt,
cos ηeo=n2+n2-n2cos2 βeo/n4+n4-n4cos2 βeo1/2,
tan ηeo=n2-n2sinβeo cosβeo/n2+n2-n2cos2 βeo,
cos χo=-cos ϕ/sin βo,  sin χo=-sin ϕ cos φo/sin βo,
cos χeo=-sin ϕ cos φeo/sin βeo,  sin χeo=cos ϕ/sin βeo.
E0sf, E0sb, E0pf, E0pbT=C˜˜±Eof, Eob, Eeof, EeobT.
¯0=C˜˜±¯,
C˜˜±=1/2±A˜±˜C˜D˜,
A˜=sin χo1+Zo-1-Zo1-Zo-1+Zo,
˜=sin χeo cos ηeo1+Zeo1-Zeo1-Zeo1+Zeo,
C˜=no/n0 cos χo1+Yo1-Yo1-Yo1+Yo,
D˜=neo/n0 cos χeo cos ηeo1+Yeo-1-Yeo1-Yeo-1+Yeo.
Zv=ζv/ζ0,  Yv=ρvn02ζv/nv2ζ0,  v=o, eo,
ZoYo=ZeoYeo.
¯0i+1=C˜˜±X˜˜¯i.
X˜˜=diagXo, Xo*, Xeo, Xeo*,
¯0i+1=C˜˜±X˜˜C˜˜±-1¯0i.
M±=C±XC±-1.
¯02N+1=M¯01
r˜rssrpsrsprpp=-M22M24M42M44-1M21M23M41M43,
RptRpp+Rsp=|rpp|2+|rsp|2,
RstRss+Rps=|rss|2+|rps|2.
=C+-1C-XC--1C+X.
S2=C+-1C-X2.
α=no2/ζoζeo tan2 ϕ,  τv=kvzd v=o, eo,  μ=cos ηeo sin χeo/sin χo.
S=1/Zo+αZeoS110S13S140S11*-S14*-S13*S31S32S330-S32*-S31*0S33*,
S11=-expjτoZo-αZeo,  S13=-μ expjτeoZo+Zeo,  S14=μ exp-jτeoZeo-Zo,  S31=-α/μ expjτoZo+Zeo,  S32=α/μ exp-jτoZeo-Zo,  S33=expjτeoZo-αZeo.
Mp=C+S2C+-1,  M=C+S2NC+-1.
C+-1=GU1C+TU1/sin2 χoZo+αZeo,  C-=U2C+, M-=U2M+U2, Mp=U2M+2.
EKx, z, t=EKzexpjKzexpjω/cξx-ωt.
detS-σI=0,
σ4+c1σ3+c2σ2+c1σ+1=0,
c1=-2Zo-αZeo/Zo+αZeocos τeo-cos τo,
c2=2/Zo+αZeo2Zo-αZeo2+2αZo2+Zeo2sin τo sin τeo-2Zo2+α2Zeo2cos τo cos τeo.
|realexp-jKΛ|>1.
ΩωΛ/2πc=Λ/λ0,
Ωc=N/ζo+ζeo,  N=1, 2, ,
Ωc=N/ζ1+ζ2,  N=1, 2, ,
KΛ=lπ+jKi,  l=0, 1,
KΛ=qπ+jKi,

Metrics