Abstract

We consider light scattering from a binary (two-layered) photonic crystal. When the index of refraction of the incident medium exceeds that of either layer, total internal reflection may occur, which is a type of phase transition. At band edges (similar to the case where the index of refraction of the incident medium is less than those of the layers), the two Bloch waves describing propagation become degenerate and a hybrid Floquet mode is required to understand the response. For angles of incidence exceeding the second critical angle (when total internal reflection occurs in both layers), the allowed bands disappear entirely.

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  1. Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
    [CrossRef]
  2. D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, “Observation of total omnidirectional reflection from a one-dimensional dielectric lattice,” Appl. Phys. A 68, 25–28 (1999).
    [CrossRef]
  3. G. V. Morozov and D. W. L. Sprung, “Floquet-Bloch waves in one-dimensional photonic crystals,” Europhys. Lett. 96, 54005 (2011).
    [CrossRef]
  4. G. V. Morozov and D. W. L. Sprung, “Transverse-magnetic-polarized Floquet–Bloch waves in one-dimensional photonic crystals,” J. Opt. Soc. Am. B 29, 3231–3239 (2012).
    [CrossRef]
  5. J. J. Stoker, Nonlinear Vibrations (Waverly, 1950).
  6. V. A. Yakubovich and V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients (Wiley, 1975).
  7. M. S. P. Eastham, The Spectral Theory of Periodic Differential Equations (Scottish Academic, 1975).
  8. W. Magnus and S. Winkler, Hill’s Equation (Dover, 2004).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  12. W. H. Southwell, “Spectral response calculations of rugate filters using coupled-wave theory,” J. Opt. Soc. Am. A 5, 1558–1564 (1988).
    [CrossRef]
  13. G. V. Morozov, D. W. L. Sprung, and J. Martorell, “Semiclassical coupled-wave theory and its application to TE waves in one-dimensional photonic crystals,” Phys. Rev. E 69, 016612 (2004).
    [CrossRef]
  14. G. V. Morozov, D. W. L. Sprung, and J. Martorell, “Semiclassical coupled wave theory for TM waves in one-dimensional photonic crystals,” Phys. Rev. E 70, 016606 (2004).
    [CrossRef]
  15. E. M. Lifshitz and L. P. Pitaevskii, “Part 2: Theory of the condensed state,” in Landau and Lifshitz: Course of Theoretical Physics, 2nd ed., Vol. 9, Statistical Physics (Pergamon, 1980), pp. 228–229.

2012

2011

G. V. Morozov and D. W. L. Sprung, “Floquet-Bloch waves in one-dimensional photonic crystals,” Europhys. Lett. 96, 54005 (2011).
[CrossRef]

2004

G. V. Morozov, D. W. L. Sprung, and J. Martorell, “Semiclassical coupled-wave theory and its application to TE waves in one-dimensional photonic crystals,” Phys. Rev. E 69, 016612 (2004).
[CrossRef]

G. V. Morozov, D. W. L. Sprung, and J. Martorell, “Semiclassical coupled wave theory for TM waves in one-dimensional photonic crystals,” Phys. Rev. E 70, 016606 (2004).
[CrossRef]

1999

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, “Observation of total omnidirectional reflection from a one-dimensional dielectric lattice,” Appl. Phys. A 68, 25–28 (1999).
[CrossRef]

1998

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

1994

1993

D. W. L. Sprung, H. Wu, and J. Martorell, “Scattering by a finite periodic potential,” Am. J. Phys. 61, 1118–1124 (1993).
[CrossRef]

1988

1977

Chen, C.

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

Chigrin, D. N.

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, “Observation of total omnidirectional reflection from a one-dimensional dielectric lattice,” Appl. Phys. A 68, 25–28 (1999).
[CrossRef]

Eastham, M. S. P.

M. S. P. Eastham, The Spectral Theory of Periodic Differential Equations (Scottish Academic, 1975).

Fan, S.

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

Fink, Y.

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

Gaponenko, S. V.

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, “Observation of total omnidirectional reflection from a one-dimensional dielectric lattice,” Appl. Phys. A 68, 25–28 (1999).
[CrossRef]

Hong, C.-S.

Joannopoulos, J. D.

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

Lavrinenko, A. V.

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, “Observation of total omnidirectional reflection from a one-dimensional dielectric lattice,” Appl. Phys. A 68, 25–28 (1999).
[CrossRef]

Lekner, J.

Lifshitz, E. M.

E. M. Lifshitz and L. P. Pitaevskii, “Part 2: Theory of the condensed state,” in Landau and Lifshitz: Course of Theoretical Physics, 2nd ed., Vol. 9, Statistical Physics (Pergamon, 1980), pp. 228–229.

Magnus, W.

W. Magnus and S. Winkler, Hill’s Equation (Dover, 2004).

Martorell, J.

G. V. Morozov, D. W. L. Sprung, and J. Martorell, “Semiclassical coupled-wave theory and its application to TE waves in one-dimensional photonic crystals,” Phys. Rev. E 69, 016612 (2004).
[CrossRef]

G. V. Morozov, D. W. L. Sprung, and J. Martorell, “Semiclassical coupled wave theory for TM waves in one-dimensional photonic crystals,” Phys. Rev. E 70, 016606 (2004).
[CrossRef]

D. W. L. Sprung, H. Wu, and J. Martorell, “Scattering by a finite periodic potential,” Am. J. Phys. 61, 1118–1124 (1993).
[CrossRef]

Michel, J.

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

Morozov, G. V.

G. V. Morozov and D. W. L. Sprung, “Transverse-magnetic-polarized Floquet–Bloch waves in one-dimensional photonic crystals,” J. Opt. Soc. Am. B 29, 3231–3239 (2012).
[CrossRef]

G. V. Morozov and D. W. L. Sprung, “Floquet-Bloch waves in one-dimensional photonic crystals,” Europhys. Lett. 96, 54005 (2011).
[CrossRef]

G. V. Morozov, D. W. L. Sprung, and J. Martorell, “Semiclassical coupled wave theory for TM waves in one-dimensional photonic crystals,” Phys. Rev. E 70, 016606 (2004).
[CrossRef]

G. V. Morozov, D. W. L. Sprung, and J. Martorell, “Semiclassical coupled-wave theory and its application to TE waves in one-dimensional photonic crystals,” Phys. Rev. E 69, 016612 (2004).
[CrossRef]

Pitaevskii, L. P.

E. M. Lifshitz and L. P. Pitaevskii, “Part 2: Theory of the condensed state,” in Landau and Lifshitz: Course of Theoretical Physics, 2nd ed., Vol. 9, Statistical Physics (Pergamon, 1980), pp. 228–229.

Southwell, W. H.

Sprung, D. W. L.

G. V. Morozov and D. W. L. Sprung, “Transverse-magnetic-polarized Floquet–Bloch waves in one-dimensional photonic crystals,” J. Opt. Soc. Am. B 29, 3231–3239 (2012).
[CrossRef]

G. V. Morozov and D. W. L. Sprung, “Floquet-Bloch waves in one-dimensional photonic crystals,” Europhys. Lett. 96, 54005 (2011).
[CrossRef]

G. V. Morozov, D. W. L. Sprung, and J. Martorell, “Semiclassical coupled-wave theory and its application to TE waves in one-dimensional photonic crystals,” Phys. Rev. E 69, 016612 (2004).
[CrossRef]

G. V. Morozov, D. W. L. Sprung, and J. Martorell, “Semiclassical coupled wave theory for TM waves in one-dimensional photonic crystals,” Phys. Rev. E 70, 016606 (2004).
[CrossRef]

D. W. L. Sprung, H. Wu, and J. Martorell, “Scattering by a finite periodic potential,” Am. J. Phys. 61, 1118–1124 (1993).
[CrossRef]

Starzhinskii, V. M.

V. A. Yakubovich and V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients (Wiley, 1975).

Stoker, J. J.

J. J. Stoker, Nonlinear Vibrations (Waverly, 1950).

Thomas, E. L.

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

Winkler, S.

W. Magnus and S. Winkler, Hill’s Equation (Dover, 2004).

Winn, J. N.

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

Wu, H.

D. W. L. Sprung, H. Wu, and J. Martorell, “Scattering by a finite periodic potential,” Am. J. Phys. 61, 1118–1124 (1993).
[CrossRef]

Yakubovich, V. A.

V. A. Yakubovich and V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients (Wiley, 1975).

Yariv, A.

Yarotsky, D. A.

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, “Observation of total omnidirectional reflection from a one-dimensional dielectric lattice,” Appl. Phys. A 68, 25–28 (1999).
[CrossRef]

Yeh, P.

Am. J. Phys.

D. W. L. Sprung, H. Wu, and J. Martorell, “Scattering by a finite periodic potential,” Am. J. Phys. 61, 1118–1124 (1993).
[CrossRef]

Appl. Phys. A

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, “Observation of total omnidirectional reflection from a one-dimensional dielectric lattice,” Appl. Phys. A 68, 25–28 (1999).
[CrossRef]

Europhys. Lett.

G. V. Morozov and D. W. L. Sprung, “Floquet-Bloch waves in one-dimensional photonic crystals,” Europhys. Lett. 96, 54005 (2011).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Phys. Rev. E

G. V. Morozov, D. W. L. Sprung, and J. Martorell, “Semiclassical coupled-wave theory and its application to TE waves in one-dimensional photonic crystals,” Phys. Rev. E 69, 016612 (2004).
[CrossRef]

G. V. Morozov, D. W. L. Sprung, and J. Martorell, “Semiclassical coupled wave theory for TM waves in one-dimensional photonic crystals,” Phys. Rev. E 70, 016606 (2004).
[CrossRef]

Science

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

Other

E. M. Lifshitz and L. P. Pitaevskii, “Part 2: Theory of the condensed state,” in Landau and Lifshitz: Course of Theoretical Physics, 2nd ed., Vol. 9, Statistical Physics (Pergamon, 1980), pp. 228–229.

J. J. Stoker, Nonlinear Vibrations (Waverly, 1950).

V. A. Yakubovich and V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients (Wiley, 1975).

M. S. P. Eastham, The Spectral Theory of Periodic Differential Equations (Scottish Academic, 1975).

W. Magnus and S. Winkler, Hill’s Equation (Dover, 2004).

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Figures (11)

Fig. 1.
Fig. 1.

TE (s) and TM (p) modes incident on a BPC {n1n2}N; n1 and n2 are the refractive indices of homogeneous dielectric layers with thicknesses d1 and d2, respectively, N is the number of periods, nin and nex are the refractive indices of the incident and exit media.

Fig. 2.
Fig. 2.

Transmittance and cos[μ(k)d] for the s mode (gray) and for the p mode (black) incident at angle θincr=arcsin(n1/nin)28.6° on the BPC {n1n2}N, with n1=1.34, n2=2.59, d1=90nm, d2=90nm, and N=6. The incident and exit media have refractive indices nin=nex=2.8. The entire shaded regions delineate s mode bandgaps, while the darker portions are for p modes. Point b is an instance of coincident right band edges, and there is another one at k31.

Fig. 3.
Fig. 3.

Bloch waves F1(z), F2(z), and the absolute value squared of the electric field |E(z)|2 for the wavenumber k15.55μm1 (λ0.404μm), see point a in Fig. 2, for the s mode incident at angle θincr=arcsin(n1/nin)28.6° on a BPC with parameters as in Fig. 2.

Fig. 4.
Fig. 4.

Bloch waves F1(z), F2(z), and the absolute value squared of the magnetic field |B(z)|2 for the wavenumber k15.55μm1 (λ0.404μm), see point a in Fig. 2, for the p mode incident at angle θincr=arcsin(n1/nin)28.6° on a BPC with parameters as in Fig. 2.

Fig. 5.
Fig. 5.

Single Bloch wave F(z), a hybrid Floquet mode G(z), and the absolute value squared of the electric field |E(z)|2 for the wavenumber k15.75μm1 (λ0.399μm), see point b in Fig. 2, for the s mode incident at angle θincr=arcsin(n1/nin)28.6° on a BPC with parameters as in Fig. 2.

Fig. 6.
Fig. 6.

Single Bloch wave F(z), a hybrid Floquet mode G(z), and the absolute value squared of the electric field |B(z)|2 for the wavenumber k15.75μm1 (λ0.399μm), see point b in Fig. 2, for the p mode incident at angle θincr=arcsin(n1/nin)28.6° on a BPC with parameters as in Fig. 2.

Fig. 7.
Fig. 7.

Bloch waves F1(z), F2(z), I[F2(z)]=I[F1(z)], and the absolute value squared of the field |E(z)|2 for the wavenumber k17.01μm1 (λ0.369μm), (see point c in Fig. 2), for the s mode incident at angle θincr=arcsin(n1/nin)28.6° on a BPC with same parameters as in Fig. 2.

Fig. 8.
Fig. 8.

Bloch waves F1(z), F2(z), I[F2(z)]=I[F1(z)], and the absolute value squared of the field |B(z)|2 for the wavenumber k17.01μm1 (λ0.369μm), (see point c in Fig. 2), for the p mode incident at angle θincr=arcsin(n1/nin)28.6° on a BPC with same parameters as in Fig. 2.

Fig. 9.
Fig. 9.

Transmittance and cos[μ(k)d] for the s mode (gray) and for the p mode (black) incident at angle θincr=arcsin(n2/nin)67.67° on a BPC {n1n2}N, with n1=1.34, n2=2.59, d1=90nm, d2=90nm, and N=6. The incident and exit media have refractive indices nin=nex=2.8.

Fig. 10.
Fig. 10.

Bloch waves F1(z), F2(z), and the absolute value squared of the electric field |E(z)|2 for the wavenumber k=0.8μm1 (λ7.85μm), see point d in Fig. 9, for the s mode incident at angle θincr=arcsin(n2/nin)67.67° on a BPC with parameters as in Fig. 9.

Fig. 11.
Fig. 11.

Bloch waves F1(z), F2(z), and the absolute value squared of the magnetic field |B(z)|2 for the wavenumber k=0.8μm1 (λ7.85μm), see point d in Fig. 9, for the p mode incident at angle θincr=arcsin(n2/nin)67.67° on a BPC with parameters as in Fig. 9.

Equations (43)

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θincr=arcsin(n1,2/nin),
d2Ψ(z)dz2+q2(z)Ψ(z)=0,q(z+d)=q(z)
d2Ψ(z)dz22n(z)dn(z)dzdΨ(z)dz+q2(z)Ψ(z)=0
q2(z)=k2n2(z)β2,
[EB]=[Ψs(z)Ψp(z)]ei(βxωt)y^
F1(z+d)=ρ1F1(z),
F2(z+d)=ρ2F2(z).
G(z+d)=ρG(z)+ρdF(z).
ρ1ρ2=1.
q1,2=kn1,2cosθ1,2=kn1,22nin2sinθin2
F1,2(z)=P1,2(z)exp(±iμz),P1,2(z+d)=P1,2(z),
exp(±iμd)=ρ1,2,2cos(μd)=ρ1+ρ2.
us(0)=1,us(0)=0,vs(0)=0,vs(0)=1,up(0)=1,up(0)n2(0)=0,vp(0)=0,vp(0)n2(0)=1
ρ2[us(d)+vs(d)]ρ+1=0,ρ2[up(d)+vp(d)n2(d)]ρ+1=0.
hs(k)2cos[μs(k)d]=us(d)+vs(d),hp(k)2cos[μp(k)d]=up(d)+vp(d)n2(d).
|hs,p(k)|>2,or|cos[μs,p(k)d]|>1,
|hs,p(k)|<2,or|cos[μs,p(k)d]|<1.
F1,2(z)=A1,2u(z)+B1,2v(z).
[us(d)ρ1,2]A1,2+vs(d)B1,2=0,us(d)A1,2+[vs(d)ρ1,2]B1,2=0
[up(d)ρ1,2]A1,2+vp(d)B1,2=0,up(d)n2(d)A1,2+[vp(d)n2(d)ρ1,2]B1,2=0
h(k)=2cos[μ(k)d]±2,
ρ1,2=us(d)+vs(d)±[us(d)+vs(d)]242
ρ1,2=up(d)+vp(d)n2(d)±[up(d)+vp(d)n2(d)]242
h(k)=2cos[μ(k)d]=±2,
ρ2=ρ1ρ=±1
G(z)=[P2(z)+zP1(z)]exp(iμz),
h(k)=2cos[μ(k)d]=±2,withv(d)=u(d)=0,
Ψ(z)=Au(z)+Bv(z)=C1F1(z)+C2F2(z),
1+r=C1F1(0)+C2F2(0),iqin(1r)=C1F1(0)+C2F2(0),C1F1(Nd)+C2F2(Nd)=t,C1F1(Nd)+C2F2(Nd)=iqext
1+r=C1F1(0)+C2F2(0),iqin(1r)nin2=C1F1(0)+C2F2(0)n2(0),C1F1(Nd)+C2F2(Nd)=t,C1F1(Nd)+C2F2(Nd)n2(Nd)=iqexnex2t
us,p(z)={cosq1z,0<zd1,cosq1d1cosq2(zd1)q1q2[1n22/n12]sinq1d1sinq2(zd1),d1<zd,
vs,p(z)={1q1[1n12]sinq1z,0<zd1,1q2[1n22]cosq1d1sinq2(zd1)+1q1[1n12]sinq1d1cosq2(zd1),d1<zd,
cos(μs,pd)=cosq1d1cosq2d212(q1q2[1n22/n12]+q2q1[1n12/n22])sinq1d1sinq2d2
us,p(z)={1,0<zd1,cosq2(zd1),d1<zd,
vs,p(z)={[1n12]z,0<zd1,1q2[1n22]sinq2(zd1)+[1n12]d1cosq2(zd1),d1<zd,
cos(μs,pd)=cosq2d2[1n12/n22]q2d12sinq2d2.
ks,pr=mπn22n12d2,m=1,2,3,4,,
[1n12/n22](q2d12)tan(q2d22)1=0,m=1,3,,[1n12/n22](q2d12)cot(q2d22)+1=0,m=2,4,.
cos(μs,pd)=coshQ1d1cosq2d2+12(Q1q2[1n22/n12]q2Q1[1n12/n22])sinhQ1d1sinq2d2
us,p(z)={coshQ1z,0<zd1,coshQ1d1+Q1[1n22/n12](zd1)sinhQ1d1,d1<zd,
vs,p(z)={1Q1[1n12]sinhQ1z,0<zd1,[1n22](zd1)coshQ1d1+1Q1[1n12]sinhQ1d1,d1<zd,
cos(μs,pd)=coshQ1d1+Q1d22[1n22/n12]sinhQ1d1
cos(μs,pd)=coshQ1d1coshQ2d2+12(Q1Q2[1n22/n12]+Q2Q1[1n12/n22])sinhQ1d1sinhQ2d2

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