Abstract

Arbitrary profiled gratings made with anisotropic materials are discussed; the anisotropic character concerns electric and/or magnetic properties. Our aim is to avoid the use of the staircase approximation of the profile, whose convergence is questionable. A coupled first-order differential-equation set is derived by taking into account Li’s remarks about Fourier factorization [J. Opt. Soc. Am. A 13, 1870 (1996)], but the present formulation shows that, in return for a convenient form of the differential system, it is possible to use only the intuitive Laurent rule. Our method, when applied to the simpler case of isotropic gratings, is shown to be consistent with that of previous studies. Moreover, from the numerical point of view, the convergence of our formulation for an anisotropic grating is faster than that of the conventional differential method.

© 2002 Optical Society of America

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References

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  1. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  2. G. Tayeb, R. Petit, M. Cadilhac, “On the theoretical and numerical study of gratings coated with anisotropic layers,” in Optics and the Information Age, H. H. Arsenault, ed., Proc. SPIE813, 407–408 (1987).
    [CrossRef]
  3. R. Petit, “Diffraction d’une onde plane par un réseau métallique,” Rev. Opt. 45, 353–370 (1966).
  4. G. Cerutti-Maori, R. Petit, M. Cadilhac, “Etude numérique du champ diffracté par un réseau,” C. R. Acad. Sci. 268, 1060–1063 (1969).
  5. P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 4, pp. 101–121.
    [CrossRef]
  6. M. Nevière, “Bragg–Fresnel multilayer gratings: electromagnetic theory,” J. Opt. Soc. Am. A 11, 1835–1845 (1994).
    [CrossRef]
  7. M. C. Hutley, J. P. Verrill, R. C. McPhedran, M. Nevière, P. Vincent, “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Opt. 6, 87–95 (1975).
    [CrossRef]
  8. M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
    [CrossRef]
  9. M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
    [CrossRef]
  10. L. Li, “Reformulation of the Fourier modal method for surface-relief grating made with anisotropic materials,” J. Mod. Opt. 45, 1313–1334 (1998).
    [CrossRef]
  11. E. Popov, M. Nevière, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000).
    [CrossRef]
  12. G. Tayeb, “Contribution à l’étude de la diffraction des ondes électromagnétiques par des réseaux. Réflexions sur les méthodes existantes et sur leur extension aux milieux anisotropes,” Ph.D. dissertation, No. 90/Aix 3/0065 (University of Aix-Marseille, France, 1990).
  13. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
    [CrossRef]
  14. F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Cerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1998).
    [CrossRef]
  15. M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).
  16. The programming has been carried out by K. Watanabe.
  17. D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 3, pp. 63–100.
    [CrossRef]
  18. T. O. Korner, J. T. Sheridan, J. Schwider, “Interferometric resolution examined by means of electromagnetic theory,” J. Opt. Soc. Am. A 12, 752–760 (1995).
    [CrossRef]

2000 (1)

1998 (2)

L. Li, “Reformulation of the Fourier modal method for surface-relief grating made with anisotropic materials,” J. Mod. Opt. 45, 1313–1334 (1998).
[CrossRef]

F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Cerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1998).
[CrossRef]

1996 (2)

1995 (1)

1994 (1)

1975 (1)

M. C. Hutley, J. P. Verrill, R. C. McPhedran, M. Nevière, P. Vincent, “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Opt. 6, 87–95 (1975).
[CrossRef]

1974 (1)

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).

1973 (2)

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

1969 (1)

G. Cerutti-Maori, R. Petit, M. Cadilhac, “Etude numérique du champ diffracté par un réseau,” C. R. Acad. Sci. 268, 1060–1063 (1969).

1966 (1)

R. Petit, “Diffraction d’une onde plane par un réseau métallique,” Rev. Opt. 45, 353–370 (1966).

Cadilhac, M.

M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

G. Cerutti-Maori, R. Petit, M. Cadilhac, “Etude numérique du champ diffracté par un réseau,” C. R. Acad. Sci. 268, 1060–1063 (1969).

G. Tayeb, R. Petit, M. Cadilhac, “On the theoretical and numerical study of gratings coated with anisotropic layers,” in Optics and the Information Age, H. H. Arsenault, ed., Proc. SPIE813, 407–408 (1987).
[CrossRef]

Cerutti-Maori, G.

G. Cerutti-Maori, R. Petit, M. Cadilhac, “Etude numérique du champ diffracté par un réseau,” C. R. Acad. Sci. 268, 1060–1063 (1969).

Hutley, M. C.

M. C. Hutley, J. P. Verrill, R. C. McPhedran, M. Nevière, P. Vincent, “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Opt. 6, 87–95 (1975).
[CrossRef]

Korner, T. O.

Li, L.

Maystre, D.

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 3, pp. 63–100.
[CrossRef]

McPhedran, R. C.

M. C. Hutley, J. P. Verrill, R. C. McPhedran, M. Nevière, P. Vincent, “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Opt. 6, 87–95 (1975).
[CrossRef]

Montiel, F.

F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Cerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1998).
[CrossRef]

Nevière, M.

E. Popov, M. Nevière, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000).
[CrossRef]

F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Cerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1998).
[CrossRef]

M. Nevière, “Bragg–Fresnel multilayer gratings: electromagnetic theory,” J. Opt. Soc. Am. A 11, 1835–1845 (1994).
[CrossRef]

M. C. Hutley, J. P. Verrill, R. C. McPhedran, M. Nevière, P. Vincent, “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Opt. 6, 87–95 (1975).
[CrossRef]

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).

M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

Petit, R.

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).

M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

G. Cerutti-Maori, R. Petit, M. Cadilhac, “Etude numérique du champ diffracté par un réseau,” C. R. Acad. Sci. 268, 1060–1063 (1969).

R. Petit, “Diffraction d’une onde plane par un réseau métallique,” Rev. Opt. 45, 353–370 (1966).

G. Tayeb, R. Petit, M. Cadilhac, “On the theoretical and numerical study of gratings coated with anisotropic layers,” in Optics and the Information Age, H. H. Arsenault, ed., Proc. SPIE813, 407–408 (1987).
[CrossRef]

Peyrot, P.

F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Cerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1998).
[CrossRef]

Popov, E.

Schwider, J.

Sheridan, J. T.

Tayeb, G.

G. Tayeb, “Contribution à l’étude de la diffraction des ondes électromagnétiques par des réseaux. Réflexions sur les méthodes existantes et sur leur extension aux milieux anisotropes,” Ph.D. dissertation, No. 90/Aix 3/0065 (University of Aix-Marseille, France, 1990).

G. Tayeb, R. Petit, M. Cadilhac, “On the theoretical and numerical study of gratings coated with anisotropic layers,” in Optics and the Information Age, H. H. Arsenault, ed., Proc. SPIE813, 407–408 (1987).
[CrossRef]

Verrill, J. P.

M. C. Hutley, J. P. Verrill, R. C. McPhedran, M. Nevière, P. Vincent, “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Opt. 6, 87–95 (1975).
[CrossRef]

Vincent, P.

M. C. Hutley, J. P. Verrill, R. C. McPhedran, M. Nevière, P. Vincent, “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Opt. 6, 87–95 (1975).
[CrossRef]

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).

M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 4, pp. 101–121.
[CrossRef]

C. R. Acad. Sci. (1)

G. Cerutti-Maori, R. Petit, M. Cadilhac, “Etude numérique du champ diffracté par un réseau,” C. R. Acad. Sci. 268, 1060–1063 (1969).

J. Mod. Opt. (2)

L. Li, “Reformulation of the Fourier modal method for surface-relief grating made with anisotropic materials,” J. Mod. Opt. 45, 1313–1334 (1998).
[CrossRef]

F. Montiel, M. Nevière, P. Peyrot, “Waveguide confinement of Cerenkov second-harmonic generation through a graded-index grating coupler: electromagnetic optimization,” J. Mod. Opt. 45, 2169–2186 (1998).
[CrossRef]

J. Opt. Soc. Am. A (5)

Nouv. Rev. Opt. (2)

M. C. Hutley, J. P. Verrill, R. C. McPhedran, M. Nevière, P. Vincent, “Presentation and verification of a differential formulation for the diffraction by conducting gratings,” Nouv. Rev. Opt. 6, 87–95 (1975).
[CrossRef]

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).

Opt. Commun. (2)

M. Nevière, R. Petit, M. Cadilhac, “About the theory of optical grating coupler-waveguide systems,” Opt. Commun. 8, 113–117 (1973).
[CrossRef]

M. Nevière, P. Vincent, R. Petit, M. Cadilhac, “Systematic study of resonances of holographic thin film couplers,” Opt. Commun. 9, 48–53 (1973).
[CrossRef]

Rev. Opt. (1)

R. Petit, “Diffraction d’une onde plane par un réseau métallique,” Rev. Opt. 45, 353–370 (1966).

Other (5)

G. Tayeb, R. Petit, M. Cadilhac, “On the theoretical and numerical study of gratings coated with anisotropic layers,” in Optics and the Information Age, H. H. Arsenault, ed., Proc. SPIE813, 407–408 (1987).
[CrossRef]

P. Vincent, “Differential methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 4, pp. 101–121.
[CrossRef]

The programming has been carried out by K. Watanabe.

D. Maystre, “Integral methods,” in Electromagnetic Theory of Gratings, Vol. 22 of Topics in Current Physics, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 3, pp. 63–100.
[CrossRef]

G. Tayeb, “Contribution à l’étude de la diffraction des ondes électromagnétiques par des réseaux. Réflexions sur les méthodes existantes et sur leur extension aux milieux anisotropes,” Ph.D. dissertation, No. 90/Aix 3/0065 (University of Aix-Marseille, France, 1990).

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Figures (2)

Fig. 1
Fig. 1

Geometry of the anisotropic grating under consideration.

Fig. 2
Fig. 2

Comparison of convergence of diffraction efficiencies computed by the conventional and present formulations of the differential theory for a sinusoidal deep grating. The grating depth is h=0.5 μm, and the values of other parameters are the same as those in Table 1. The incident polarization is TM.

Tables (1)

Tables Icon

Table 1 Comparison of the Zeroth- and -1st-Order Diffraction Efficiencies Computed by the Conventional Formulation (Ref. 12 ) and the Present Formulation with Different Truncation Orders N for an Anisotropic Shallow Gratinga

Equations (139)

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

y=p(x),
ε=(x, y)=xx(x, y)xy(x, y)xz(x, y)yx(x, y)yy(x, y)yz(x, y)zx(x, y)zy(x, y)zz(x, y), 
μ=(x, y)=μxx(x, y)μxy(x, y)μxz(x, y)μyx(x, y)μyy(x, y)μyz(x, y)μzx(x, y)μzy(x, y)μzz(x, y).
Ex(x, y)=ν=-NNEx,ν(y)exp(iανx),
αν=ω1μ1sin θ+νK,
K=2π/d.
fN(x)=n=-NNfnexp(inKx),
gN(x)=n=-NNgnexp(inKx).
hN(x)=n=-NNhn(N)exp(inKx),
hn(N)=m=-NNfn-mgm,
[h]=f [g],
g=1f h;
[g]=1f[h],
[h]=1f-1[g].
EtDnEn=cos ϕsin ϕ0-xxsin ϕ+yxcos ϕ-xysin ϕ+yycos ϕ-xzsin ϕ+yzcos ϕ001ExEyEz,
HtBnHz=cos ϕsin ϕ0-μxxsin ϕ+μyxcos ϕ-μxysin ϕ+μyycos ϕ-μxzsin ϕ+μyzcos ϕ001HxHyHz,
Ex(x, y)Ey(x, y)Ez(x, y)=axt(e)(x, y)axn(e)(x, y)axz(e)(x, y)ayt(e)(x, y)ayn(e)(x, y)ayz(e)(x, y)001×Et(x, y)Dn(x, y)Ez(x, y),
Hx(x, y)Hy(x, y)Hz(x, y)=axt(h)(x, y)axn(h)(x, y)axz(h)(x, y)ayt(h)(x, y)ayn(h)(x, y)ayz(h)(x, y)001×Ht(x, y)Bn(x, y)Hz(x, y),
 axt(e)=-xysin ϕ-yycos ϕξ,
 axn(e)=-sin ϕξ,
axz(e)=-(xzsin ϕ-yzcos ϕ)sin ϕξ,
ayt(e)=xxsin ϕ-yxcos ϕξ,
ayn(e)=cos ϕξ,
ayz(e)=(xzsin ϕ-yzcos ϕ)cos ϕξ,
axt(h)=-μxysin ϕ-μyycos ϕξμ,
axn(h)=-sin ϕξμ,
axz(h)=-(uxzsin ϕ-μyzcos ϕ)sin ϕξμ,
ayt(h)=μxxsin ϕ-μyxcos ϕξμ,
ayn(h)=cos ϕξμ,
ayz(h)=(μxxsin ϕ-μyzcos ϕ)cos ϕξμ,
ξ=(xxsin ϕ-yxcos ϕ)sin ϕ-(xysin ϕ-yycos ϕ)cos ϕ, 
ξμ=(μxxsin ϕ-μyxcos ϕ)sin ϕ-(μxysin ϕ-μyycos ϕ)cos ϕ.
×E=iωμ=H,
×H=-iωε=E
Ezy=iω(bxt(h)Ht+bxn(h)Bn+bxz(h)Hz),
Ezx=-iω(byt(h)Ht+byn(h)Bn+byz(h)Hz), 
Exy-Eyx=-iω(bzt(h)Ht+bzn(h)Bn+bzz(h)Hz),
Hzy=-iω(bxt(e)Et+bxn(e)Dn+bxz(e)Ez), 
Hzx=iω(byt(e)Et+byn(e)Dn+byz(e)Ez),
Hxy-Hyx=iω(bzt(e)Et+bzn(e)Dn+bzz(e)Ez),
bpq(e)=pxaxq(e)+pyayq(e),
bpz(e)=pxaxz(e)+pyayz(e)+pz,
bpq(h)=μpxaxq(h)+μpyayq(h),
bpz(h)=μpxaxz(h)+μpyayz(h)+μpz
ddy [Ez]=iω{bxt(h)[Ht]+bxn(h)[Bn]+bxz(h)[Hz]},
iα[Ez]=-iω{byt(h)[Ht]+byn(h)[Bn]+byz(h)[Hz]},
ddy [Ex]-iα[Ey]
=-iω{bzt(h)[Ht]+bzn(h)[Bn]+bzz(h)[Hz]},
ddy [Hz]=-iω{bxt(e)[Et]+bxn(e)[Dn]+bxz(e)[Ez]},
iα[Hz]=iω{byt(e)[Et]+byn(e)[Dn]+byz(e)[Ez]},
ddy [Hx]-iα[Hy]
=iω{bzt(e)[Et]+bzn(e)[Dn]+bzz(e)[Ez]},
[Ex][Ey][Ez]=axt(e)axn(e)axz(e)ayt(e)ayn(e)ayz(e)00I[Et][Dn][Ez],
[Hx][Hy][Hz]=axt(h)axn(h)axz(h)ayt(h)ayn(h)ayz(h)00I[Ht][Bn][Hz],
[Et][Dn][Ez]=Ctx(e)Cty(e)Ctz(e)Cnx(e)Cny(e)Cnz(e)00I[Ex][Ey][Ez],
 [Ht][Bn][Hz]=Ctx(h)Cty(h)Ctz(h)Cnx(h)Cny(h)Cnz(h)00I[Hx][Hy][Hz],
ddy [Ez]=iωDxx(h)[Hx]+iωDxy(h)[Hy]+iωDxz(h)[Hz],
iα[Ez]=-iωDyx(h)[Hx]-iωDyy(h)[Hy]-iωDyz(h)[Hz],
ddy [Ex]=iα[Ey]-iωDzx(h)[Hx]-iωDzy(h)[Hy]-iωDzz(h)[Hz],
ddy [Hz]=-iωDxx(e)[Ex]-iωDxy(e)[Ey]-iωDxz(e)[Ez],
iα[Hz]=iωDyx(e)[Ex]+iωDyy(e)[Ey]+iωDyz(e)[Ez],
ddy [Hx]=iωDzx(e)[Ex]+iωDzy(e)[Ey]+iωDzz(e)[Ez]+iα[Hy],
Dpq(f )=bpt(f )Ctq(f )+bpn(f )Cnq(f ),
Dpz(f )=bpt(f )Ctz(f )+bpn(f )Cnz(f )+bpz(f )
[Hy]=-1ω {Dyy(h)}-1α[Ez]-{Dyy(h)}-1Dyx(h)[Hx]-{Dyy(h)}-1Dyz(h)[Hz],
[Ey]=-{Dyy(e)}-1Dyx(e)[Ex]-{Dyy(e)}-1Dyz(e)[Ez]+1ω {Dyy(e)}-1α[Hz].
ddy[Ex][Ez][Hx][Hz]=iM11M12M13M140M22M23M24M31M32M33M34M41M420M44[Ex][Ez][Hx][Hz],
M11=-α{Dyy(e)}-1Dyx(e),
M12=-α{Dyy(e)}-1Dyz(e)+Dzy(h){Dyy(h)}-1α,
M13=-ωDzx(h)+ωDzy(h){Dyy(h)}-1Dyx(h),
M14=1ωα{Dyy(e)}-1α+ωDzy(h){Dyy(h)}-1Dyz(h)-ωDzz(h),
M22=-Dxy(h){Dyy(h)}-1α,
M23=ωDxx(h)-ωDxy(h){Dyy(h)}-1Dyx(h),
M24=-ωDxy(h){Dyy(h)}-1Dyz(h)+ωDxz(h),
M31=ωDzx(e)-ωDzy(e){Dyy(e)}-1Dyx(e),
M32=-1ωα{Dyy(h)}-1α-ωDzy(e){Dyy(e)}-1Dyz(e)+ωDzz(e),
M33=-α{Dyy(h)}-1Dyx(h),
M34=-α{Dyy(h)}-1Dyz(h)+Dzy(e){Dyy(e)}-1α,
M41=-ωDxx(e)+ωDxy(e){Dyy(e)}-1Dyx(e),
M42=ωDxy(e){Dyy(e)}-1Dyz(e)-ωDxz(e),
M44=-Dxy(e){Dyy(e)}-1α.
EtDnEz=cos ϕsin ϕ0- sin ϕ cos ϕ0001ExEyEz,
ExEyEz=cos ϕ-sin ϕ0sin ϕcos ϕ0001EtDnEz.
Ezy=iωμ0Hx,
Ezx=-iωμ0Hy,
Exy-Eyx=-iωμ0Hz,
Hzy=-iω(cos ϕ)Et+iω(sin ϕ)Dn,
Hzx=iω(sin ϕ)Et+iω(cos ϕ)Dn,
Hxy-Hyx=iωEz.
ddy [Ez]=iωμ0[Hx],
iα[Ez]=-iωμ0[Hy],
ddy [Ex]-iα[Ey]=-iωμ0[Hz],
ddy[Hz]=-iω cos ϕ[Et]+iωsin ϕ[Dn],
iα[Hz]=iω sin ϕ[Et]+iωcos ϕ[Dn],
ddy [Hx]-iα[Hy]=iω[Ez].
[Ex][Ey][Ez]=cos ϕ-sin ϕ0sin ϕcos ϕ000I[Et][Dn][Ez].
[Et][Dn][Ez]=cos ϕsin ϕ0-1-1sin ϕ1-1cos ϕ000I×[Ex][Ey][Ez],
ddy [Hz]=-iωcos2 ϕ+sin ϕ1-1sin ϕ[Ex]-iωcos ϕ sin ϕ-sin ϕ×1-1cos ϕ[Ey],
iα[Hz]=iωcos ϕ sin ϕ-cos ϕ1-1sin ϕ×[Ex]+iωsin2 ϕ+cos ϕ×1-1cos ϕ[Ey].
ddy [Hz]=-iωA+1-1[Ex]-iωB[Ey],
iα[Hz]=iωB[Ex]+iω(-A)[Ey],
A=-1-1cos2 ϕ,
B=-1-1cos ϕ sin ϕ.
ddy[Ex][Ez][Hx][Hz]=iM1100M1400M2300M3200M4100M44[Ex][Ez][Hx][Hz],
M11=-α(-A)-1B,
M14=1ωα(-A)-1α-ωμ0I,
M23=ωμ0I,
M32=ω-1ωμ0α2,
M41=-ωA+1-1-B(-A)-1B,
M44=-B(-A)-1α.
ddy [Ez]=iM23[Hx],
ddy [Hx]=iM32[Ez],
ddy [Ex]=iM11[Ex]+iM14[Hz],
ddy [Hz]=iM41[Ex]+iM44[Hz]
 pq(x, y)=1(1-w(x,y))δp,q+2,pqw(x, y),
μpq(x,y)=μ1(1-w(x, y))δp,q+μ2,pqw(x,y),
w(x, y)=0fory>p(x)1fory<p(x).
1(x)=I
s(x)=f(x)+g(x)
f(x)g(x)=f(x)g(x)=g(x)f(x)
[u(x)]=[f(x)g(x)v(x)]=f(x)g(x)[v(x)].
[u(x)]=[f(x)g(x)v(x)]=f(x)[g(x)v(x)]
=f(x)g(x)[v(x)].
f(x)g(x)=g(x)f(x)=g(x)f(x).
f(x)g(x)-1=g(x)-1f(x)
fg-1=(g-1fg)g-1=g-1f.
[Ex]=cos ϕ[Et]-sin ϕ[Dn],
[Ey]=sin ϕ[Et]+cos ϕ[Dn].
cos ϕ[Ex]=cos ϕcos ϕ[Et]-cos ϕsin ϕ[Dn],
sin ϕ[Ey]=sin ϕsin ϕ[Et]+sin ϕcos ϕ[Dn].
(cos ϕcos ϕ+sin ϕsin ϕ)[Et]
=cos ϕ[Ex]+sin ϕ[Ey].
[Et]=cos ϕ[Ex]+sin ϕ[Ey].
sin ϕ[Ex]=sin ϕcos ϕ[Et]-sin ϕsin ϕ[Dn],
cos ϕ[Ey]=cos ϕsin ϕ[Et]+cos ϕcos ϕ[Dn].
cos ϕcos ϕ+sin ϕsin ϕ[Dn]
=-sin ϕ[Ex]+cos ϕ[Ey].
1[Dn]=-sin ϕ[Ex]+cos ϕ[Ey].
[Dn]=-1-1sin ϕ[Ex]+1-1cos ϕ[Ey].

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