Abstract

We examined the properties of photonic crystals that consist of nonoverlapping chiral spheres in a dielectric medium. We considered the effect of the chiral property of the spheres on the frequency band structure of the electromagnetic field in the crystal and on the transmittance properties of a slab of the crystal, and we estimated the optical activity of the crystal.

© 1999 Optical Society of America

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References

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  1. J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).
  2. C. M. Soukoulis, ed., Photonic Band Gap Materials (Kluwer Academic, Dordrecht, The Netherlands, 1996).
  3. A. Lakhtakia, V. V. Varadan, V. K. Varadan, “Field equations, Huygens’s principle, integral equations, and theorems for radiation and scattering of electromagnetic waves in isotropic chiral media,” J. Opt. Soc. Am. A 5, 175–184 (1988).
    [CrossRef]
  4. E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
    [CrossRef]
  5. A. Lakhtakia, ed., Selected Papers on Natural Optical Activity (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1990).
  6. C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974).
    [CrossRef]
  7. A. Modinos, “Scattering of electromagnetic waves by a plane of spheres—formalism,” Physica A 141, 575–588 (1987).
    [CrossRef]
  8. N. Stefanou, V. Karathanos, A. Modinos, “Scattering of electromagnetic waves by periodic structures,” J. Phys. Condens. Matter 4, 7389–7400 (1992).
    [CrossRef]
  9. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 185–194.
  10. N. Stefanou, V. Yannopapas, A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 113, 49–77 (1998).
    [CrossRef]
  11. J. Chongjun, Q. Bai, Y. Miao, Q. Ruhu, “Two-dimensional photonic band structure in the chiral medium-transfer matrix method,” Opt. Commun. 142, 179–183 (1997).
    [CrossRef]
  12. We chose rather large chirality parameters to demonstrate effects that otherwise would be minimal. Although the chirality parameters of most naturally occurring optically active substances are quite small, present technology has raised the prospect of crafting synthetic materials with intrinsically large chiral parameters. See, e.g., I. V. Lindell, M. P. Silverman, “Plane-wave scattering from a nonchiral object in a chiral environment,” J. Opt. Soc. Am. A 14, 79–90 (1997).
    [CrossRef]
  13. K. M. Flood, D. L. Jaggard, “Band-gap structure for periodic chiral media,” J. Opt. Soc. Am. A 13, 1395–1406 (1996).
    [CrossRef]
  14. V. Karathanos, N. Stefanou, A. Modinos, “Optical activity of photonic crystals,” J. Mod. Opt. 42, 619–626 (1995).
    [CrossRef]
  15. D. Stroud, “Generalized effective-medium approach to the conductivity of an inhomogeneous material,” Phys. Rev. B 12, 3368–3373 (1975).
    [CrossRef]
  16. D. Lacoste, B. A. van Tiggelen, G. L. J. A. Rokken, A. Spareberg, “Optics of a Faraday-active Mie sphere,” J. Opt. Soc. Am. A 15, 1636–1642 (1998).
    [CrossRef]

1998 (2)

N. Stefanou, V. Yannopapas, A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 113, 49–77 (1998).
[CrossRef]

D. Lacoste, B. A. van Tiggelen, G. L. J. A. Rokken, A. Spareberg, “Optics of a Faraday-active Mie sphere,” J. Opt. Soc. Am. A 15, 1636–1642 (1998).
[CrossRef]

1997 (2)

1996 (1)

1995 (1)

V. Karathanos, N. Stefanou, A. Modinos, “Optical activity of photonic crystals,” J. Mod. Opt. 42, 619–626 (1995).
[CrossRef]

1992 (1)

N. Stefanou, V. Karathanos, A. Modinos, “Scattering of electromagnetic waves by periodic structures,” J. Phys. Condens. Matter 4, 7389–7400 (1992).
[CrossRef]

1988 (1)

1987 (1)

A. Modinos, “Scattering of electromagnetic waves by a plane of spheres—formalism,” Physica A 141, 575–588 (1987).
[CrossRef]

1975 (1)

D. Stroud, “Generalized effective-medium approach to the conductivity of an inhomogeneous material,” Phys. Rev. B 12, 3368–3373 (1975).
[CrossRef]

1974 (1)

C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974).
[CrossRef]

1937 (1)

E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
[CrossRef]

Bai, Q.

J. Chongjun, Q. Bai, Y. Miao, Q. Ruhu, “Two-dimensional photonic band structure in the chiral medium-transfer matrix method,” Opt. Commun. 142, 179–183 (1997).
[CrossRef]

Bohren, C. F.

C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974).
[CrossRef]

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 185–194.

Chongjun, J.

J. Chongjun, Q. Bai, Y. Miao, Q. Ruhu, “Two-dimensional photonic band structure in the chiral medium-transfer matrix method,” Opt. Commun. 142, 179–183 (1997).
[CrossRef]

Condon, E. U.

E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
[CrossRef]

Flood, K. M.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 185–194.

Jaggard, D. L.

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).

Karathanos, V.

V. Karathanos, N. Stefanou, A. Modinos, “Optical activity of photonic crystals,” J. Mod. Opt. 42, 619–626 (1995).
[CrossRef]

N. Stefanou, V. Karathanos, A. Modinos, “Scattering of electromagnetic waves by periodic structures,” J. Phys. Condens. Matter 4, 7389–7400 (1992).
[CrossRef]

Lacoste, D.

Lakhtakia, A.

Lindell, I. V.

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).

Miao, Y.

J. Chongjun, Q. Bai, Y. Miao, Q. Ruhu, “Two-dimensional photonic band structure in the chiral medium-transfer matrix method,” Opt. Commun. 142, 179–183 (1997).
[CrossRef]

Modinos, A.

N. Stefanou, V. Yannopapas, A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 113, 49–77 (1998).
[CrossRef]

V. Karathanos, N. Stefanou, A. Modinos, “Optical activity of photonic crystals,” J. Mod. Opt. 42, 619–626 (1995).
[CrossRef]

N. Stefanou, V. Karathanos, A. Modinos, “Scattering of electromagnetic waves by periodic structures,” J. Phys. Condens. Matter 4, 7389–7400 (1992).
[CrossRef]

A. Modinos, “Scattering of electromagnetic waves by a plane of spheres—formalism,” Physica A 141, 575–588 (1987).
[CrossRef]

Rokken, G. L. J. A.

Ruhu, Q.

J. Chongjun, Q. Bai, Y. Miao, Q. Ruhu, “Two-dimensional photonic band structure in the chiral medium-transfer matrix method,” Opt. Commun. 142, 179–183 (1997).
[CrossRef]

Silverman, M. P.

Spareberg, A.

Stefanou, N.

N. Stefanou, V. Yannopapas, A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 113, 49–77 (1998).
[CrossRef]

V. Karathanos, N. Stefanou, A. Modinos, “Optical activity of photonic crystals,” J. Mod. Opt. 42, 619–626 (1995).
[CrossRef]

N. Stefanou, V. Karathanos, A. Modinos, “Scattering of electromagnetic waves by periodic structures,” J. Phys. Condens. Matter 4, 7389–7400 (1992).
[CrossRef]

Stroud, D.

D. Stroud, “Generalized effective-medium approach to the conductivity of an inhomogeneous material,” Phys. Rev. B 12, 3368–3373 (1975).
[CrossRef]

van Tiggelen, B. A.

Varadan, V. K.

Varadan, V. V.

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).

Yannopapas, V.

N. Stefanou, V. Yannopapas, A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 113, 49–77 (1998).
[CrossRef]

Chem. Phys. Lett. (1)

C. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. 29, 458–462 (1974).
[CrossRef]

Comput. Phys. Commun. (1)

N. Stefanou, V. Yannopapas, A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 113, 49–77 (1998).
[CrossRef]

J. Mod. Opt. (1)

V. Karathanos, N. Stefanou, A. Modinos, “Optical activity of photonic crystals,” J. Mod. Opt. 42, 619–626 (1995).
[CrossRef]

J. Opt. Soc. Am. A (4)

J. Phys. Condens. Matter (1)

N. Stefanou, V. Karathanos, A. Modinos, “Scattering of electromagnetic waves by periodic structures,” J. Phys. Condens. Matter 4, 7389–7400 (1992).
[CrossRef]

Opt. Commun. (1)

J. Chongjun, Q. Bai, Y. Miao, Q. Ruhu, “Two-dimensional photonic band structure in the chiral medium-transfer matrix method,” Opt. Commun. 142, 179–183 (1997).
[CrossRef]

Phys. Rev. B (1)

D. Stroud, “Generalized effective-medium approach to the conductivity of an inhomogeneous material,” Phys. Rev. B 12, 3368–3373 (1975).
[CrossRef]

Physica A (1)

A. Modinos, “Scattering of electromagnetic waves by a plane of spheres—formalism,” Physica A 141, 575–588 (1987).
[CrossRef]

Rev. Mod. Phys. (1)

E. U. Condon, “Theories of optical rotatory power,” Rev. Mod. Phys. 9, 432–457 (1937).
[CrossRef]

Other (4)

A. Lakhtakia, ed., Selected Papers on Natural Optical Activity (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1990).

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals (Princeton U. Press, Princeton, N.J., 1995).

C. M. Soukoulis, ed., Photonic Band Gap Materials (Kluwer Academic, Dordrecht, The Netherlands, 1996).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 185–194.

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Figures (5)

Fig. 1
Fig. 1

Frequency band structure of the EM field along the normal to the (001) surface of a fcc crystal. The crystal consists of chiral spheres (ϵ=1.1, β/a=0.3, S/a=0.21) in a dielectric host medium (ϵh=12.96). The transmittance of light incident normally on a slab of the crystal of 32 planes parallel to the (001) surface is shown on the right-hand side.

Fig. 2
Fig. 2

Frequency band structure of the EM field along the normal to the (001) surface of a fcc crystal. The crystal consists of dielectric spheres (ϵ=1.1, β/a=0, S/a=0.21) in a dielectric host medium (ϵh=12.96). The transmittance of light incident normally on a slab of the crystal of 32 planes parallel to the (001) surface is shown on the right-hand side.

Fig. 3
Fig. 3

Projection of the frequency band structure on the SBZ of a (001) surface of a fcc composite crystal. The gaps in the frequency spectrum, in the considered region of frequency, are shaded. The crystal is constituted of (a) chiral spheres (ϵ=1.1, β/a=0.3, S/a=0.21) or (b) dielectric spheres (ϵ=1.1, β/a=0, S/a=0.21) in a dielectric host medium (ϵh=12.96). The inset shows the SBZ of the (001) surface.

Fig. 4
Fig. 4

Rotatory power of a slab of a fcc photonic crystal that consists of layers of chiral spheres (ϵ=1.1, β/a=0.3, S/a=0.21) in a dielectric host medium (ϵh=12.96). A plane wave is incident normally on a slab of the crystal parallel to the (001) surface. (a) Variation of the angle of rotation of the plane of polarization with the thickness of the slab, when ωa/c=1; (b) variation of the specific rotatory power with the frequency.

Fig. 5
Fig. 5

Frequency band structure at the center of the SBZ of a (001) surface of a fcc crystal of chiral spheres (ϵ=4, nβω/c=0.7, S/a=0.21) in a dielectric host (ϵh=1). On the right-hand side we show the transmittance of normally incident light on a slab of the crystal that consists of 32 planes parallel to the (001) surface.

Equations (25)

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D(r, t)=ϵϵ0[E(r, t)+β×E(r, t)],
B(r, t)=μμ0[H(r, t)+β×H(r, t)].
nL,R=n/(1±nβω/c),
kL,R=nL,Rω/c,
|β|<(c/nω).
E(r)=l,miq[alm0E×jl(qr)Xlm(rˆ)+alm+E×hl+(qr)Xlm(rˆ)]+[alm0Hjl(qr)+alm+Hhl+(qr)]Xlm(rˆ),
H(r)=ϵhϵ0μhμ01/2l,m-iq[alm0H×jl(qr)Xlm(rˆ)+alm+H×hl+(qr)Xlm(rˆ)]+[alm0Ejl(qr)+alm+Ehl+(qr)]Xlm(rˆ),
alm+P=P=E,HTlPPalm0P.
TlEE=UlLClR+UlRClLUlLVlR+UlRVlL,
TlHH=VlLDlR+VlRDlLUlLVlR+UlRVlL,
TlEH=i UlLDlR-UlRDlLUlLVlR+UlRVlL=-TlHE,
UlL,R=ϵϵh1/21kL,RSz[zjl(z)|z=kL,RShl+(qS)-jl(kL,RS)1qSz[zhl+(z)|z=qS,
VlL,R=hl+(qS)1kL,RSz[zjl(z)|z=kL,RS-ϵϵh1/21qSz[zhl+(z)|z=qSjl(kL,RS),
ClL,R=ϵϵh1/21qSz[zjl(z)|z=qSjl(kL,RS)-jl(qS)1kL,RSz[zjl(z)|z=kL,RS,
DlL,R=ϵϵh1/21qSz[zjl(z)|z=qSjl(kL,RS)-jl(qS)1kL,RSz[zjl(z)|z=kL,RS.
En,α(r, t)=Re[En,α(r)exp(-iωt)],
En,α(r)=g{Eg,α+(n)exp[iKg+·(r-An)]
+Eg,α-(n)exp[iKg-·(r-An)]},
Kg±(k+g,±(q2-|k|2)1/2),
Eg,α±(n+1)=exp(ik·a3)Eg,α+(n),
k=(k, kz,α(ω)).
E0,R±(n)=(xˆ+iyˆ)E0,R±(n)
E0,L±(n)=(xˆ-iyˆ)E0,L±(n)
ϕ(ω)=[kz,R(ω)-kz,L(ω)]d/2
ϕ(ω)/dβ¯n¯¯2(ω/c)2,

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