Abstract

Using binary and sinusoidal photonic crystals as examples, it is shown how to construct two independent Floquet–Bloch waves for TM-polarized light in a lossless one-dimensional periodic structure. Particular attention is given to those special situations where either one Bloch wave and a hybrid Floquet mode develop inside the structure (instead of two Bloch waves) or both developed Bloch waves become periodic functions.

© 2012 Optical Society of America

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References

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  1. G. V. Morozov and D. W. L. Sprung, “Floquet–Bloch waves in one-dimensional photonic crystals,” Europhys. Lett. 96, 54005 (2011).
    [CrossRef]
  2. G. V. Morozov, R. G. Maev, and G. W. F. Drake, “Switching electromagnetic waves by two-layered periodic dielectric structures,” Phys. Rev. E 60, 4860–4867 (1999).
    [CrossRef]
  3. I. Nusinsky and A. A. Hardy, “Band-gap analysis of one-dimensional photonic crystals and conditions for gap closing,” Phys. Rev. B 73, 125104 (2006).
    [CrossRef]
  4. J. K. Nurligareev and V. A. Sychugov, “Propagation of light in a one-dimensional photonic crystal: analysis by the floquet-bloch function method,” Quantum Electron. 38, 452–461 (2008).
    [CrossRef]
  5. J. K. Nurligareev, “Floquet–Bloch waves in bound one-dimensional photonic crystals,” J. Surf. Invest. 5, 193–208 (2011).
    [CrossRef]
  6. J. J. Stoker, Nonlinear Vibrations (Waverly, 1950).
  7. V. A. Yakubovich and V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients (Wiley, 1975).
  8. M. S. P. Eastham, The Spectral Theory of Periodic Differential Equations (Scottish Academic, 1975).
  9. W. Magnus and S. Winkler, Hill’s Equation (Dover, 2004).
  10. G. V. Morozov and F. Placido, “High-order bandgaps in one-dimensional photonic crystals,” J. Opt. 12, 045101 (2010).
    [CrossRef]
  11. 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]
  12. B. G. Bovard, “Rugate filter theory: an overview,” Appl. Opt. 32, 5427–5442 (1993).
    [CrossRef]
  13. W. H. Southwell, “Spectral response calculations of rugate filters using coupled-wave theory,” J. Opt. Soc. Am. A 5, 1558–1564 (1988).
    [CrossRef]
  14. N. Perelman and I. Averbukh, “Rugate filter design: an analytical approach using uniform WKB solutions,” J. Appl. Phys. 79, 2839–2845 (1996).
    [CrossRef]
  15. M. Brack and R. K. Bhaduri, Semiclassical Physics (Addison-Wesley, 1997).
  16. L. W. Casperson, “Solvable Hill equation,” Phys. Rev. A 30, 2749–2751 (1984).
    [CrossRef]
  17. L. W. Casperson, “Erratum: solvable Hill equation [Phys. Rev. A 30, 2749 (1984)],” Phys. Rev. A 31, 2743(E) (1985).
    [CrossRef]
  18. S. M. Wu and C. C. Shin, “Construction of solvable Hill equations,” Phys. Rev. A 32, 3736–3738 (1985).
    [CrossRef]
  19. V. Urumov, “Solvable Hill-Harper equations,” Phys. Rev. A 38, 4863–4865 (1988).
    [CrossRef]

2011 (2)

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

J. K. Nurligareev, “Floquet–Bloch waves in bound one-dimensional photonic crystals,” J. Surf. Invest. 5, 193–208 (2011).
[CrossRef]

2010 (1)

G. V. Morozov and F. Placido, “High-order bandgaps in one-dimensional photonic crystals,” J. Opt. 12, 045101 (2010).
[CrossRef]

2008 (1)

J. K. Nurligareev and V. A. Sychugov, “Propagation of light in a one-dimensional photonic crystal: analysis by the floquet-bloch function method,” Quantum Electron. 38, 452–461 (2008).
[CrossRef]

2006 (1)

I. Nusinsky and A. A. Hardy, “Band-gap analysis of one-dimensional photonic crystals and conditions for gap closing,” Phys. Rev. B 73, 125104 (2006).
[CrossRef]

1999 (1)

G. V. Morozov, R. G. Maev, and G. W. F. Drake, “Switching electromagnetic waves by two-layered periodic dielectric structures,” Phys. Rev. E 60, 4860–4867 (1999).
[CrossRef]

1998 (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]

1996 (1)

N. Perelman and I. Averbukh, “Rugate filter design: an analytical approach using uniform WKB solutions,” J. Appl. Phys. 79, 2839–2845 (1996).
[CrossRef]

1993 (1)

1988 (2)

1985 (2)

L. W. Casperson, “Erratum: solvable Hill equation [Phys. Rev. A 30, 2749 (1984)],” Phys. Rev. A 31, 2743(E) (1985).
[CrossRef]

S. M. Wu and C. C. Shin, “Construction of solvable Hill equations,” Phys. Rev. A 32, 3736–3738 (1985).
[CrossRef]

1984 (1)

L. W. Casperson, “Solvable Hill equation,” Phys. Rev. A 30, 2749–2751 (1984).
[CrossRef]

Averbukh, I.

N. Perelman and I. Averbukh, “Rugate filter design: an analytical approach using uniform WKB solutions,” J. Appl. Phys. 79, 2839–2845 (1996).
[CrossRef]

Bhaduri, R. K.

M. Brack and R. K. Bhaduri, Semiclassical Physics (Addison-Wesley, 1997).

Bovard, B. G.

Brack, M.

M. Brack and R. K. Bhaduri, Semiclassical Physics (Addison-Wesley, 1997).

Casperson, L. W.

L. W. Casperson, “Erratum: solvable Hill equation [Phys. Rev. A 30, 2749 (1984)],” Phys. Rev. A 31, 2743(E) (1985).
[CrossRef]

L. W. Casperson, “Solvable Hill equation,” Phys. Rev. A 30, 2749–2751 (1984).
[CrossRef]

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]

Drake, G. W. F.

G. V. Morozov, R. G. Maev, and G. W. F. Drake, “Switching electromagnetic waves by two-layered periodic dielectric structures,” Phys. Rev. E 60, 4860–4867 (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]

Hardy, A. A.

I. Nusinsky and A. A. Hardy, “Band-gap analysis of one-dimensional photonic crystals and conditions for gap closing,” Phys. Rev. B 73, 125104 (2006).
[CrossRef]

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]

Maev, R. G.

G. V. Morozov, R. G. Maev, and G. W. F. Drake, “Switching electromagnetic waves by two-layered periodic dielectric structures,” Phys. Rev. E 60, 4860–4867 (1999).
[CrossRef]

Magnus, W.

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

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, “Floquet–Bloch waves in one-dimensional photonic crystals,” Europhys. Lett. 96, 54005 (2011).
[CrossRef]

G. V. Morozov and F. Placido, “High-order bandgaps in one-dimensional photonic crystals,” J. Opt. 12, 045101 (2010).
[CrossRef]

G. V. Morozov, R. G. Maev, and G. W. F. Drake, “Switching electromagnetic waves by two-layered periodic dielectric structures,” Phys. Rev. E 60, 4860–4867 (1999).
[CrossRef]

Nurligareev, J. K.

J. K. Nurligareev, “Floquet–Bloch waves in bound one-dimensional photonic crystals,” J. Surf. Invest. 5, 193–208 (2011).
[CrossRef]

J. K. Nurligareev and V. A. Sychugov, “Propagation of light in a one-dimensional photonic crystal: analysis by the floquet-bloch function method,” Quantum Electron. 38, 452–461 (2008).
[CrossRef]

Nusinsky, I.

I. Nusinsky and A. A. Hardy, “Band-gap analysis of one-dimensional photonic crystals and conditions for gap closing,” Phys. Rev. B 73, 125104 (2006).
[CrossRef]

Perelman, N.

N. Perelman and I. Averbukh, “Rugate filter design: an analytical approach using uniform WKB solutions,” J. Appl. Phys. 79, 2839–2845 (1996).
[CrossRef]

Placido, F.

G. V. Morozov and F. Placido, “High-order bandgaps in one-dimensional photonic crystals,” J. Opt. 12, 045101 (2010).
[CrossRef]

Shin, C. C.

S. M. Wu and C. C. Shin, “Construction of solvable Hill equations,” Phys. Rev. A 32, 3736–3738 (1985).
[CrossRef]

Southwell, W. H.

Sprung, D. W. L.

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

Sychugov, V. A.

J. K. Nurligareev and V. A. Sychugov, “Propagation of light in a one-dimensional photonic crystal: analysis by the floquet-bloch function method,” Quantum Electron. 38, 452–461 (2008).
[CrossRef]

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]

Urumov, V.

V. Urumov, “Solvable Hill-Harper equations,” Phys. Rev. A 38, 4863–4865 (1988).
[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, S. M.

S. M. Wu and C. C. Shin, “Construction of solvable Hill equations,” Phys. Rev. A 32, 3736–3738 (1985).
[CrossRef]

Yakubovich, V. A.

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

Appl. Opt. (1)

Europhys. Lett. (1)

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

J. Appl. Phys. (1)

N. Perelman and I. Averbukh, “Rugate filter design: an analytical approach using uniform WKB solutions,” J. Appl. Phys. 79, 2839–2845 (1996).
[CrossRef]

J. Opt. (1)

G. V. Morozov and F. Placido, “High-order bandgaps in one-dimensional photonic crystals,” J. Opt. 12, 045101 (2010).
[CrossRef]

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

J. Surf. Invest. (1)

J. K. Nurligareev, “Floquet–Bloch waves in bound one-dimensional photonic crystals,” J. Surf. Invest. 5, 193–208 (2011).
[CrossRef]

Phys. Rev. A (4)

L. W. Casperson, “Solvable Hill equation,” Phys. Rev. A 30, 2749–2751 (1984).
[CrossRef]

L. W. Casperson, “Erratum: solvable Hill equation [Phys. Rev. A 30, 2749 (1984)],” Phys. Rev. A 31, 2743(E) (1985).
[CrossRef]

S. M. Wu and C. C. Shin, “Construction of solvable Hill equations,” Phys. Rev. A 32, 3736–3738 (1985).
[CrossRef]

V. Urumov, “Solvable Hill-Harper equations,” Phys. Rev. A 38, 4863–4865 (1988).
[CrossRef]

Phys. Rev. B (1)

I. Nusinsky and A. A. Hardy, “Band-gap analysis of one-dimensional photonic crystals and conditions for gap closing,” Phys. Rev. B 73, 125104 (2006).
[CrossRef]

Phys. Rev. E (1)

G. V. Morozov, R. G. Maev, and G. W. F. Drake, “Switching electromagnetic waves by two-layered periodic dielectric structures,” Phys. Rev. E 60, 4860–4867 (1999).
[CrossRef]

Quantum Electron. (1)

J. K. Nurligareev and V. A. Sychugov, “Propagation of light in a one-dimensional photonic crystal: analysis by the floquet-bloch function method,” Quantum Electron. 38, 452–461 (2008).
[CrossRef]

Science (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]

Other (5)

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).

M. Brack and R. K. Bhaduri, Semiclassical Physics (Addison-Wesley, 1997).

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Figures (14)

Fig. 1.
Fig. 1.

TM mode incident on a binary photonic crystal {n1n2}N. n1 and n2 are the refractive indices of two homogeneous dielectric layers with thicknesses d1 and d2. N is the number of periods, and nin and nex are the refractive indices of the incident and exit media.

Fig. 2.
Fig. 2.

Transmittance and cos[μ(k)d] for a TM mode incident at the angle 65° on the binary photonic crystal {n1n2}N, with n1=4.6, n2=1.6, d1=800nm, d2=1650nm, and N=4. The incident and exit media are air (nin=nex=1). The shaded regions represent bandgaps.

Fig. 3.
Fig. 3.

Bloch waves F1(z), F2(z), and the absolute value squared of the field |Ψ(z)|2 for the wavenumber k1.155μm1 (λ5.440μm) (see point a in Fig. 2) for a TM mode incident at angle 65° on a binary photonic crystal with parameters as in Fig. 2.

Fig. 4.
Fig. 4.

Single Bloch wave F(z), hybrid Floquet mode G(z), and the absolute value squared of the field |Ψ(z)|2 for the wavenumber k1.197μm1 (λ5.251μm) (see point b in Fig. 2) for a TM mode incident at angle 65° on a binary photonic crystal with parameters as in Fig. 2.

Fig. 5.
Fig. 5.

Bloch waves F1(z), F2(z), I[F2(z)]=I[F1(z)], and the absolute value squared of the field |Ψ(z)|2 for the wavenumber k1.327μm1 (λ4.735μm) (see point c in Fig. 2) for a TM mode incident at angle 65° on a binary photonic crystal with parameters as in Fig. 2.

Fig. 6.
Fig. 6.

Transmittance and cos[μ(k)d] for a TM mode incident at the angle 65° on the binary photonic crystal {n1n2}N, with n1=4.6, n2=1.6, d1=648nm, d2=2216nm, and N=4. The incident and exit media are air (nin=nex=1).

Fig. 7.
Fig. 7.

Functions F1(z) and F2(z) and the absolute value squared of the field |Ψ(z)|2 for the wavenumber k1.075μm1 (λ5.844μm) (see point d in Fig. 6) for a TM mode incident at the angle 65° on a binary photonic crystal with parameters as in Fig. 6.

Fig. 8.
Fig. 8.

Transmittance and cos[μ(k)d] for a TM mode incident at the angle θBr49° on the binary photonic crystal {n1n2}N, with n1=4.6, n2=1.6, d1=800nm, d2=1650nm, and N=4. The incident and exit media have the indices nin=nex=2.0.

Fig. 9.
Fig. 9.

Functions F1(z), F2(z), and the absolute value squared of the field |Ψ(z)|2 for the wavenumber k0.723μm1 (λ8.686μm) (see point e in Fig. 8) for a TM mode incident at the angle θBr49° on a binary photonic crystal with parameters as in Fig. 8.

Fig. 10.
Fig. 10.

Functions F1(z), F2(z), and the absolute value squared of the field |Ψ(z)|2 for the wavenumber k1.447μm1 (λ4.343μm) (see point f in Fig. 8) for a TM mode incident at angle θBr49° on a binary photonic crystal with parameters as in Fig. 8.

Fig. 11.
Fig. 11.

Transmittance and cos[μ(k)d] for a TM mode incident at the angle 65° on the rugate of N=4 periods, with nav=2.58, na=1.5, and d=2.45μm. The incident and exit media are air (nin=nex=1). The shaded regions represent bandgaps.

Fig. 12.
Fig. 12.

Bloch waves F1(z) and F2(z), and the absolute value squared of the field |Ψ(z)|2 for the wavenumber k1.155μm1 (λ5.440μm) (see point g in Fig. 11) for a TM mode incident at angle 65° on a rugate with parameters as in Fig. 11.

Fig. 13.
Fig. 13.

Bloch waves F1(z), F2(z), I[F2(z)]=I[F1(z)], and the absolute value squared of the field |Ψ(z)|2 for the wavenumber k1.327μm1 (λ4.735μm) (see point l in Fig. 11) for a TM mode incident at angle 65° on a rugate with parameters as in Fig. 11.

Fig. 14.
Fig. 14.

Bloch waves F1(z), F2(z). Upper panel: for the wavenumber k1.16165μm1 (λ5.4088μm) approaching point h in Fig. 11 from the bandgap; lower panel: for the wavenumber k1.16200μm1 (λ5.4072μm) approaching point h in Fig. 11 from the allowed band.

Equations (65)

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d2Ψ(z)dz22n(z)dn(z)dzdΨ(z)dz+q2(z)Ψ(z)=0,q2(z)=k2n2(z)β2,n(z+d)=n(z),
β=kninsinθin,andq2(z)=k2[n2(z)nin2sin2θin].
B(r,t)=R[By(r)eiωt]y^,By(r)=Ψ(z)eiβx,
E(r,t)=R[Ex(r)eiωt]x^+R[Ez(r)eiωt]z^,Ex(r)=ikn2(z)dΨ(z)dzeiβx,Ez(r)=βkn2(z)Ψ(z)eiβx.
d2f(z)dz2+a(z)df(z)dz+b(z)f(z)=0,a(z)=a(z+d),b(z)=b(z+d).
F1(z+d)=ρ1F1(z),
F2(z+d)=ρ2F2(z).
G(z+d)=ρG(z)+ρdF(z).
ρ1ρ2=exp[0da(z)dz].
ρ1ρ2=exp[0d2n(z)dn(z)dzdz]=n2(d)n2(0)=1,
u(0)=1,u(0)n2(0)=0,v(0)=0,v(0)n2(0)=1.
u(z+d)=u(d)u(z)+u(d)n2(d)v(z),v(z+d)=v(d)u(z)+v(d)n2(d)v(z).
Ψ(z)=Au(z)+Bv(z).
Ψ(d)=ρΨ(0),Ψ(d)n2(d)=ρΨ(0)n2(0).
[u(d)ρ]A+v(d)B=0,u(d)n2(d)A+[v(d)n2(d)ρ]B=0
ρ2[u(d)+v(d)n2(d)]ρ+1=0,
u(d)v(d)u(d)v(d)n2(d)w(d)n2(d)=1
w(d)=w(0)exp(0da(z)dz),
w(d)=w(0)n2(d)n2(0)=w(0)=n2(0).
ρ1+ρ2=u(d)+v(d)n2(d),ρ1ρ2=1.
exp(±iμd)=ρ1,2,2cos(μd)=ρ1+ρ2,
h(k)2cos[μ(k)d]=u(d)+v(d)n2(d).
|h(k)|>2,or|cos[μ(k)d]|>1,
|h(k)|<2,or|cos[μ(k)d]|<1.
qav=mπd,qav=1d0dkm2n2(z)β2dz
km=mπ0dn2(z)nin2sinθin2dz.
ρ1=ρ2=±1,orh(k)=±2,orcosμ(k)d=±1.
h(km)=±2,orcosμ(km)d=±1,
h(k)=2cos[μ(k)d]=u(d)+v(d)n2(d)±2,
ρ1,2=u(d)+v(d)n2(d)±[u(d)+v(d)n2(d)]242,
F1,2(z)u(z)+ρ1,2u(d)v(d)v(z)ρ1,2v(d)/n2(d)u(d)/n2(d)u(z)+v(z).
R[F1(z)]=R[F2(z)],I[F1(z)]=I[F2(z)].
u(d)v(d)n2(d)v(d)u(d)n2(d)=1.
ρ1=u(d),ρ2=v(d)n2(d).
F1(z)=u(z),F2(z)=u(z)+v(d)/n2(d)u(d)v(d)v(z).
F1(z)=u(d)v(d)/n2(d)u(d)/n2(d)u(z)+v(z),F2(z)=v(z).
F1(z)=u(z),F2(z)=v(z).
h(k)=2cos[μ(k)d]=u(d)+v(d)n2(d)=±2,
ρ2=ρ1ρ=u(d)+v(d)n2(d)2=±1.
F2(z)=F1(z)F(z)u(z)+ρu(d)v(d)v(z),ρv(d)/n2(d)u(d)/n2(d)u(z)+v(z).
F(z)=u(z)+ρu(d)v(d)v(z),
v(z+d)=ρv(z)+v(d)F(z),
G(z)=ρdv(d)v(z),
F(z)=ρv(d)/n2(d)u(d)/n2(d)u(z)+v(z),
u(z+d)=ρu(z)+u(d)n2(d)F(z),
G(z)=ρdu(d)/n2(d)u(z),
ρ=u(d)=v(d)n2(d)=±1,F(z)=u(z),
ρ=u(d)=v(d)n2(d)=±1,F(z)=v(z),
G(z)=[P2(z)+zP1(z)]exp(iμz),
Ψ(z+d)=Au(z+d)+Bv(z+d)=ρΨ(z)+Bv(d)[u(z)+ρu(d)v(d)v(z)]+Au(d)n2(d)[ρv(d)/n2(d)u(d)/n2(d)u(z)+v(z)],
Ψ(z+d)=ρΨ(z)+θF(z),
h(k)=2cos[μ(k)d]=u(d)+v(d)n2(d)=±2,v(d)=u(d)=0,
Ψ(z)=Au(z)+Bv(z)=C1F1(z)+C2F2(z),
1+r=Au(0)+Bv(0),iqin(1r)nin2=Au(0)+Bv(0)n2(0),Au(Nd)+Bv(Nd)=t,Au(Nd)+Bv(Nd)n2(Nd)=iqexnex2t,
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,
u(z)={cosq1z,0<zd1,cosq1d1cosq2(zd1)q1q2n22n12sinq1d1sinq2(zd1),d1<zd,
v(z)={n12q1sinq1z,0<zd1,n22q2cosq1d1sinq2(zd1)+n12q1sinq1d1cosq2(zd1),d1<zd,
cos[μ(k)d]=cosq1d1cosq2d212(n22q1n12q2+n12q2n22q1)sinq1d1sinq2d2.
km=mπη1d1+η2d2,η1,2=n1,22nin2sinθin2,
r12cos[mπ2(η1d1η2d2η1d1+η2d2)]=0,m=1,3,r12sin[mπ2(η1d1η2d2η1d1+η2d2)]=0,m=2,4.
r12=n22η1n12η2n22η1+n12η2.
η1d1=η2d2,orn1cosθ1d1=n2cosθ2d2.
sinθBr=1ninn1n2n12+n22,kBr=mπη1d1+η2d2.
n(z)=nav+nasin(2πzd),
a(z)=22πdnacos(2πdz)nav+nasin(2πdz),b(z)=k2[(nav+nasin(2πzd))2nin2sin2θin].

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