Abstract

Some general properties of one-dimensional photonic crystals are analyzed by using symmetric periodic multilayers. In this way a one-dimensional photonic crystal can be considered a single thin film with an equivalent optical admittance and an equivalent phase thickness. These functions provide important information about the optical properties of the photonic crystal, such as the band structure and the characteristic points, where the bandgaps narrow and close, including Brewster points, which are all determined by analytic expressions. With this formalism it is also possible to study in a simple way different kinds of surface modes that can be present at the interface between photonic crystal and bulk material.

© 2004 Optical Society of America

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References

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  1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, Princeton, N.J., 1995).
  2. S. Johnson and J. D. Joannopoulos, Photonic Crystals: The Road from Theory to Practice (Kluwer Academic Boston, 2002).
  3. K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, Heidelberg, 2001).
  4. F. Ramos-Mendieta and P. Halevi, “Electromagnetic surface modes of a dielectric superlattice: the supercell method,” J. Opt. Soc. Am. B 14, 370–381 (1997).
    [CrossRef]
  5. J. A. Gaspar-Armenta, F. Villa, and T. López-Ríos, “Surface waves in finite one-dimensional photonic crystals: mode coupling,” Opt. Commun. 216, 379–394 (2003).
    [CrossRef]
  6. F. Villa, L. E. Regalado, F. Ramos-Mendieta, J. Gaspar-Armenta, and T. López-Ríos, “Photonic crystal sensor based on surface waves for thin film characterization,” Opt. Lett. 27, 646–648 (2002).
    [CrossRef]
  7. F. Villa, J. A. Gaspar-Armenta, and F. Ramos-Mendieta, “One-dimensional photonic crystals: equivalent systems to single layers with a classical oscillator like dielectric function,” Opt. Commun. 216, 361–367 (2003).
    [CrossRef]
  8. Pochi Yeh, Amnon Yariv, and Chi-Shian Hong, “Electromagnetic propagation in periodic stratified media. I. General theory,” J. Opt. Soc. Am. 67, 423–438 (1977).
    [CrossRef]
  9. Pochi Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).
  10. W. M. Robertson and M. S. May, “Surface electromagnetic wave excitation on one-dimensional photonic band-gap arrays,” Appl. Phys. Lett. 74, 1800–1802 (1999).
    [CrossRef]
  11. W. M. Robertson, “Experimental measurement of the effect of termination on surface electromagnetic waves in one-dimensional photonic bandgap arrays,” J. Lightwave Technol. 17, 2013–2017 (1999).
    [CrossRef]
  12. F. Ramos-Mendieta and P. Halevi, “Surface electromagnetic waves in two-dimensional photonic crystals: effect of the position of the surface plane,” Phys. Rev. B 59, 15112–15120 (1999).
    [CrossRef]
  13. Xianmin Yi, Pochi Yeh, and J. Hong, “Nonexistence of well-confined surface waves on obliquely cut surfaces of one-dimensional photonic crystals,” J. Opt. Soc. Am. B 18, 352–357 (2001).
    [CrossRef]
  14. H. Raether, Excitation of Plasmons and Interband Transitions by Electrons, G. Höhler, ed., Vol. 88 of Springer Tracts in Modern Physics (Springer, Berlin, 1980).
  15. H. A. Macleod, Thin Film Optical Filters (McGraw-Hill, New York, 1989).
  16. C. J. Van der Laan and H. J. Franquena, “Equivalent layers: another way to look at them,” Appl. Opt. 34, 681–687 (1995).
    [CrossRef] [PubMed]

2003 (2)

J. A. Gaspar-Armenta, F. Villa, and T. López-Ríos, “Surface waves in finite one-dimensional photonic crystals: mode coupling,” Opt. Commun. 216, 379–394 (2003).
[CrossRef]

F. Villa, J. A. Gaspar-Armenta, and F. Ramos-Mendieta, “One-dimensional photonic crystals: equivalent systems to single layers with a classical oscillator like dielectric function,” Opt. Commun. 216, 361–367 (2003).
[CrossRef]

2002 (1)

2001 (1)

1999 (3)

W. M. Robertson and M. S. May, “Surface electromagnetic wave excitation on one-dimensional photonic band-gap arrays,” Appl. Phys. Lett. 74, 1800–1802 (1999).
[CrossRef]

W. M. Robertson, “Experimental measurement of the effect of termination on surface electromagnetic waves in one-dimensional photonic bandgap arrays,” J. Lightwave Technol. 17, 2013–2017 (1999).
[CrossRef]

F. Ramos-Mendieta and P. Halevi, “Surface electromagnetic waves in two-dimensional photonic crystals: effect of the position of the surface plane,” Phys. Rev. B 59, 15112–15120 (1999).
[CrossRef]

1997 (1)

1995 (1)

1977 (1)

Franquena, H. J.

Gaspar-Armenta, J.

Gaspar-Armenta, J. A.

F. Villa, J. A. Gaspar-Armenta, and F. Ramos-Mendieta, “One-dimensional photonic crystals: equivalent systems to single layers with a classical oscillator like dielectric function,” Opt. Commun. 216, 361–367 (2003).
[CrossRef]

J. A. Gaspar-Armenta, F. Villa, and T. López-Ríos, “Surface waves in finite one-dimensional photonic crystals: mode coupling,” Opt. Commun. 216, 379–394 (2003).
[CrossRef]

Halevi, P.

F. Ramos-Mendieta and P. Halevi, “Surface electromagnetic waves in two-dimensional photonic crystals: effect of the position of the surface plane,” Phys. Rev. B 59, 15112–15120 (1999).
[CrossRef]

F. Ramos-Mendieta and P. Halevi, “Electromagnetic surface modes of a dielectric superlattice: the supercell method,” J. Opt. Soc. Am. B 14, 370–381 (1997).
[CrossRef]

Hong, Chi-Shian

Hong, J.

López-Ríos, T.

J. A. Gaspar-Armenta, F. Villa, and T. López-Ríos, “Surface waves in finite one-dimensional photonic crystals: mode coupling,” Opt. Commun. 216, 379–394 (2003).
[CrossRef]

F. Villa, L. E. Regalado, F. Ramos-Mendieta, J. Gaspar-Armenta, and T. López-Ríos, “Photonic crystal sensor based on surface waves for thin film characterization,” Opt. Lett. 27, 646–648 (2002).
[CrossRef]

May, M. S.

W. M. Robertson and M. S. May, “Surface electromagnetic wave excitation on one-dimensional photonic band-gap arrays,” Appl. Phys. Lett. 74, 1800–1802 (1999).
[CrossRef]

Ramos-Mendieta, F.

F. Villa, J. A. Gaspar-Armenta, and F. Ramos-Mendieta, “One-dimensional photonic crystals: equivalent systems to single layers with a classical oscillator like dielectric function,” Opt. Commun. 216, 361–367 (2003).
[CrossRef]

F. Villa, L. E. Regalado, F. Ramos-Mendieta, J. Gaspar-Armenta, and T. López-Ríos, “Photonic crystal sensor based on surface waves for thin film characterization,” Opt. Lett. 27, 646–648 (2002).
[CrossRef]

F. Ramos-Mendieta and P. Halevi, “Surface electromagnetic waves in two-dimensional photonic crystals: effect of the position of the surface plane,” Phys. Rev. B 59, 15112–15120 (1999).
[CrossRef]

F. Ramos-Mendieta and P. Halevi, “Electromagnetic surface modes of a dielectric superlattice: the supercell method,” J. Opt. Soc. Am. B 14, 370–381 (1997).
[CrossRef]

Regalado, L. E.

Robertson, W. M.

W. M. Robertson, “Experimental measurement of the effect of termination on surface electromagnetic waves in one-dimensional photonic bandgap arrays,” J. Lightwave Technol. 17, 2013–2017 (1999).
[CrossRef]

W. M. Robertson and M. S. May, “Surface electromagnetic wave excitation on one-dimensional photonic band-gap arrays,” Appl. Phys. Lett. 74, 1800–1802 (1999).
[CrossRef]

Van der Laan, C. J.

Villa, F.

F. Villa, J. A. Gaspar-Armenta, and F. Ramos-Mendieta, “One-dimensional photonic crystals: equivalent systems to single layers with a classical oscillator like dielectric function,” Opt. Commun. 216, 361–367 (2003).
[CrossRef]

J. A. Gaspar-Armenta, F. Villa, and T. López-Ríos, “Surface waves in finite one-dimensional photonic crystals: mode coupling,” Opt. Commun. 216, 379–394 (2003).
[CrossRef]

F. Villa, L. E. Regalado, F. Ramos-Mendieta, J. Gaspar-Armenta, and T. López-Ríos, “Photonic crystal sensor based on surface waves for thin film characterization,” Opt. Lett. 27, 646–648 (2002).
[CrossRef]

Yariv, Amnon

Yeh, Pochi

Yi, Xianmin

Appl. Opt. (1)

Appl. Phys. Lett. (1)

W. M. Robertson and M. S. May, “Surface electromagnetic wave excitation on one-dimensional photonic band-gap arrays,” Appl. Phys. Lett. 74, 1800–1802 (1999).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (1)

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

Opt. Commun. (2)

F. Villa, J. A. Gaspar-Armenta, and F. Ramos-Mendieta, “One-dimensional photonic crystals: equivalent systems to single layers with a classical oscillator like dielectric function,” Opt. Commun. 216, 361–367 (2003).
[CrossRef]

J. A. Gaspar-Armenta, F. Villa, and T. López-Ríos, “Surface waves in finite one-dimensional photonic crystals: mode coupling,” Opt. Commun. 216, 379–394 (2003).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. B (1)

F. Ramos-Mendieta and P. Halevi, “Surface electromagnetic waves in two-dimensional photonic crystals: effect of the position of the surface plane,” Phys. Rev. B 59, 15112–15120 (1999).
[CrossRef]

Other (6)

Pochi Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

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

S. Johnson and J. D. Joannopoulos, Photonic Crystals: The Road from Theory to Practice (Kluwer Academic Boston, 2002).

K. Sakoda, Optical Properties of Photonic Crystals (Springer-Verlag, Heidelberg, 2001).

H. Raether, Excitation of Plasmons and Interband Transitions by Electrons, G. Höhler, ed., Vol. 88 of Springer Tracts in Modern Physics (Springer, Berlin, 1980).

H. A. Macleod, Thin Film Optical Filters (McGraw-Hill, New York, 1989).

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Figures (9)

Fig. 1
Fig. 1

Periodic multilayer of symmetric periods. η0 and ηs are the optical admittance of the incidence and transmission media, respectively.

Fig. 2
Fig. 2

Band structure for the TE polarization of an infinite periodic system. The points corresponding to optical holes are indicated by spheres for np=2.22, dp=90 nm, nq=1.46, and dq=150 nm.

Fig. 3
Fig. 3

Band structure of the 1DPC for the TM polarization. The Brewster points are indicated by diamonds, while optical holes are indicated by spheres as in Fig. 2. The material parameters are those given in Fig. 2.

Fig. 4
Fig. 4

Real (thick solid curve) and imaginary (thin solid curve) parts of the equivalent phase thickness δe as a function of the reduced frequency under normal incidence (β=0). The total phase thickness of the period Δ is given as a dashed curve. The real-part phase thickness is given in radians and normalized to π.

Fig. 5
Fig. 5

Squared amplitude of the electric field for the point (β¯=0, ω¯=0.31) located inside the first bandgap in the TE case (Fig. 3).

Fig. 6
Fig. 6

Curves in the dispersion diagram, where each layer of the period is an absentee independently. Dashed curves correspond to the condition δq=mqπ, and dotted–dotted–dashed curves correspond to the condition δp=mpπ.

Fig. 7
Fig. 7

Optical admittances of each layer of the period when ω¯=1.019. The intersection denotes the Brewster point in the first bandgap of Fig. 3.

Fig. 8
Fig. 8

Band structure for the TM case with the thin-film parameters np=2.22, dp=127.75 nm, nq=1.46, and dq=150 nm. The thicknesses were chosen to produce a coincidence of a Brewster point and an optical hole at the third bandgap (indicated by a sphere). Surface modes for the air–1DPC interface are given as dashed lines appearing in the first three bandgaps. Brewster points are indicated by diamonds.

Fig. 9
Fig. 9

Band structure for the TE case of the system given in Fig. 8. As above, the regions where the equivalent admittance is positive are shaded with a mesh within the bandgaps, and those regions where it is imaginary negative are in white. Surface modes are indicated with dashed lines, appearing in this case in the first, third, and fourth bandgaps.

Tables (1)

Tables Icon

Table 1 Multiple Optical Holes Where the Bandgaps Narrow for the System with np=2.22, dp=90 nm, nq=1.46, and dq=150 nm

Equations (62)

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m(ηj, δj)=cos(δj)iηj sin(δj)iη sin(δj)cos(δj).
δj=2πΛk¯zjdj
ηj=yk¯zj/ω¯TEpolarizationynj2ω¯/k¯zjTMpolarization
k¯zj=(nj2ω¯2-β¯2)1/2
M(ηe, δe)=cos(δe)iηe sin(δe)iηe sin(δe)cos(δe),
M11=cos(δp)cos(δq)-ρ+ sin(δp)sin(δq),
M22=M11,
M21=iηp[sin(δp)cos(δq)+ρ+×cos(δp)sin(δq)-ρ- sin(δq)],
M12=iηp[sin(δp)cos(δq)+ρ+×cos(δp)sin(δq)+ρ- sin(dq)],
ρ+=12ηpηq+ηqηp,
ρ-=12ηpηq-ηqηp.
cos(δe)=cos(δp)cos(δq)-ρ+ sin(δp)sin(δq),
ηe=ηpsin(δe)[sin(δp)cos(δq)+ρ+ cos(δp)sin(δq)-ρ- sin(δq)].
Re(δe)=mπ,
Re(δp+δq)=mπ.
ω¯m=mΛ21Re(np2-ζ2dp+nq2-ζ2dq),
β¯m=ω¯mζ,
M11=(-1)m{cos(δp)cos[Re(δp)-i Im(δq)]+ρ+ sin(δp)sin[Re(δp)-i Im(δq)]}.
M11=[1+(ρ+-1)sin2(δp)](-1)m.
M11=(-1)m cosh|δp|.
M11=(-1)m cosh|δq|,
β¯Bm=mΛ2npnqnp2dp+nq2dq,
ω¯Bm=mΛ2np2+nq2np2dp+nq2dq,
ηe=ηp,
δe=δp+δq=mπ,
Re(δp)=mpπ,
Re(δq)=mqπ,
m=mp+mq,
ω¯mp=1npdpmpΛ22+dp2β¯21/2,
ω¯mq=1nqdqmqΛ22+dq2β¯21/2.
β¯mp,mq=Λ21dpdqmq2np2dp2-mp2nq2dq2nq2-np21/2,
ω¯mp,mq=Λ21dpdqmq2dp2-mp2dq2nq2-np21/2.
mqnqdqnpdpmp.
mqnqdqnpdpmp.
dp=λ2mpnp2-ζ2,
dq=λ2mqnq2-ζ2;
ω¯mp,mq=12mpnp2-ζ2dp+mqnq2-ζ2dq,
β¯mp,mq=ω¯mp,mqζ.
det M=M11M22-M21M12=1.
|M11|=1.
M21=iηp[(1-ρ+)cos(δp)-ρ-](-1)m sin(δp),
1-ρ+0,
ρ-εηq.
M21iηpεηq(-1)m sin(δp),
ηe=M21i sin(δe),
δp(β¯mp,mq+Δβ¯)=mpπ+dδpdβ¯Δβ¯,
dω¯=β¯mp,mqω¯mp,mqmqdp2+mpdq2mqnp2dp2+mpnq2dq2 dβ¯.
δpβ¯=-2Λ2πβ¯mp,mqmpdp2,
δpω¯=2Λ2πnp2ω¯mp,mqmpdp2.
dδp=δpβ¯ dβ¯+δpω¯ dω¯,
dδpdβ¯=2Λ2πβ¯mp,mqdp2mpdp2mq+dq2mpnp2dp2mq+nq2dq2mpnp-1.
[sin(δp)]β¯m+Δβ¯sinmpπ+dδpdβ¯Δβ¯;
[sin(δp)]β¯m+Δβ¯(-1)mpdδpdβ¯Δβ¯.
η0+ηs+iη0ηsηe+ηetan(σδe)=0;
ηe(ω¯, β¯)=-η0(ω¯, β¯),
ηe(ω¯, β¯)=-ηs(ω¯, β¯).
ηe=i(ηp-ηq)22ηq sinh(Im|δe|)cos(δp)-ηp+ηqηp-ηqsin(δp).
δp=2πΛnpdpω¯m.
δp=mπnpdpnpdp+nqdq.
sign(x)=1ifx0-1ifx<0.
sign[Im(ηe)]=sign[(np-nq)sin(mπfp)].
dp=mpnq2mqnp2dq,

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