Abstract

We present a rigorous differential method describing the conical diffraction of an electromagnetic plane wave by an inclined parallel-plate grating. Above and below the grating, the fields are written with the use of Rayleigh’s expansion. Inside the grating, Maxwell’s equations are used in covariant form, written in a nonorthogonal coordinate system. Therefore the expression of boundary conditions on the perfectly conducting walls as well as on the planes delimiting the grating becomes simplified. In the classical diffraction case the numerical results compared successfully with those obtained with a Wiener–Hopf method [IEEE Trans. Antennas Propag. 36, 1424 (1988)].

© 1997 Optical Society of America

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References

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  1. K. Kobayashi, T. Inoue, “Diffraction of a plane wave by an inclined parallel plate grating,” IEEE Trans. Antennas Propag. 36, 1424–1433 (1988).
    [CrossRef]
  2. J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839–846 (1982).
    [CrossRef]
  3. R. Petit, G. Tayeb, “Numerical study of the symmetrical strip-grating-loaded slab,” J. Opt. Soc. Am. A 7, 373–379 (1990).
    [CrossRef]
  4. Lord Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London, Ser. A 79, 399–416 (1907).
    [CrossRef]
  5. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Heidelberg, 1980), Chap. 3, p. 85.
  6. G. Granet, “Diffraction par des surfaces périodiques: résolution en coordonnées non orthogonales,” Ph.D. dissertation (Université Blaise Pascal, Clermont-Ferrand, France, 1992).
  7. E. Marcellin, “Application des équations de Maxwell covariantes à l’étude de la propagation dans les guides coudés,” Ph.D. dissertation (Université Blaise Pascal, Clermont-Ferrand, France, 1992).
  8. R. N. Brockwell, The Fourier Transform and Its Application (McGraw-Hill, New York, 1986), Chap. 16, p. 211.
  9. R. Dusséaux, E. Marcellin, P. Chambelin, T. Dusseux, “Application of the Maxwell’s equations tensorial form to the propagation in an E-plane taper,” presented at PIERS 1994, European Space Research and Technology Center, Noordwijk, The Netherlands, July 5–7, 1994.
  10. G. Granet, J. P. Plumey, J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
    [CrossRef]
  11. E. J. Post, Formal Structure of Electromagnetics (North-Holland, Amsterdam, 1962).
  12. M. Lichnerowicz, Eléments de Calculs Tensoriel (Armand Collin, Paris, 1950).

1995 (1)

G. Granet, J. P. Plumey, J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

1990 (1)

1988 (1)

K. Kobayashi, T. Inoue, “Diffraction of a plane wave by an inclined parallel plate grating,” IEEE Trans. Antennas Propag. 36, 1424–1433 (1988).
[CrossRef]

1982 (1)

1907 (1)

Lord Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London, Ser. A 79, 399–416 (1907).
[CrossRef]

Brockwell, R. N.

R. N. Brockwell, The Fourier Transform and Its Application (McGraw-Hill, New York, 1986), Chap. 16, p. 211.

Chambelin, P.

R. Dusséaux, E. Marcellin, P. Chambelin, T. Dusseux, “Application of the Maxwell’s equations tensorial form to the propagation in an E-plane taper,” presented at PIERS 1994, European Space Research and Technology Center, Noordwijk, The Netherlands, July 5–7, 1994.

Chandezon, J.

G. Granet, J. P. Plumey, J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839–846 (1982).
[CrossRef]

Cornet, G.

Dupuis, M. T.

Dusséaux, R.

R. Dusséaux, E. Marcellin, P. Chambelin, T. Dusseux, “Application of the Maxwell’s equations tensorial form to the propagation in an E-plane taper,” presented at PIERS 1994, European Space Research and Technology Center, Noordwijk, The Netherlands, July 5–7, 1994.

Dusseux, T.

R. Dusséaux, E. Marcellin, P. Chambelin, T. Dusseux, “Application of the Maxwell’s equations tensorial form to the propagation in an E-plane taper,” presented at PIERS 1994, European Space Research and Technology Center, Noordwijk, The Netherlands, July 5–7, 1994.

Granet, G.

G. Granet, J. P. Plumey, J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

G. Granet, “Diffraction par des surfaces périodiques: résolution en coordonnées non orthogonales,” Ph.D. dissertation (Université Blaise Pascal, Clermont-Ferrand, France, 1992).

Inoue, T.

K. Kobayashi, T. Inoue, “Diffraction of a plane wave by an inclined parallel plate grating,” IEEE Trans. Antennas Propag. 36, 1424–1433 (1988).
[CrossRef]

Kobayashi, K.

K. Kobayashi, T. Inoue, “Diffraction of a plane wave by an inclined parallel plate grating,” IEEE Trans. Antennas Propag. 36, 1424–1433 (1988).
[CrossRef]

Lichnerowicz, M.

M. Lichnerowicz, Eléments de Calculs Tensoriel (Armand Collin, Paris, 1950).

Marcellin, E.

E. Marcellin, “Application des équations de Maxwell covariantes à l’étude de la propagation dans les guides coudés,” Ph.D. dissertation (Université Blaise Pascal, Clermont-Ferrand, France, 1992).

R. Dusséaux, E. Marcellin, P. Chambelin, T. Dusseux, “Application of the Maxwell’s equations tensorial form to the propagation in an E-plane taper,” presented at PIERS 1994, European Space Research and Technology Center, Noordwijk, The Netherlands, July 5–7, 1994.

Maystre, D.

Petit, R.

Plumey, J. P.

G. Granet, J. P. Plumey, J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

Post, E. J.

E. J. Post, Formal Structure of Electromagnetics (North-Holland, Amsterdam, 1962).

Rayleigh, Lord

Lord Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London, Ser. A 79, 399–416 (1907).
[CrossRef]

Tayeb, G.

IEEE Trans. Antennas Propag. (1)

K. Kobayashi, T. Inoue, “Diffraction of a plane wave by an inclined parallel plate grating,” IEEE Trans. Antennas Propag. 36, 1424–1433 (1988).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Proc. R. Soc. London, Ser. A (1)

Lord Rayleigh, “On the dynamical theory of gratings,” Proc. R. Soc. London, Ser. A 79, 399–416 (1907).
[CrossRef]

Pure Appl. Opt. (1)

G. Granet, J. P. Plumey, J. Chandezon, “Scattering by a periodically corrugated dielectric layer with non-identical faces,” Pure Appl. Opt. 4, 1–5 (1995).
[CrossRef]

Other (7)

E. J. Post, Formal Structure of Electromagnetics (North-Holland, Amsterdam, 1962).

M. Lichnerowicz, Eléments de Calculs Tensoriel (Armand Collin, Paris, 1950).

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Heidelberg, 1980), Chap. 3, p. 85.

G. Granet, “Diffraction par des surfaces périodiques: résolution en coordonnées non orthogonales,” Ph.D. dissertation (Université Blaise Pascal, Clermont-Ferrand, France, 1992).

E. Marcellin, “Application des équations de Maxwell covariantes à l’étude de la propagation dans les guides coudés,” Ph.D. dissertation (Université Blaise Pascal, Clermont-Ferrand, France, 1992).

R. N. Brockwell, The Fourier Transform and Its Application (McGraw-Hill, New York, 1986), Chap. 16, p. 211.

R. Dusséaux, E. Marcellin, P. Chambelin, T. Dusseux, “Application of the Maxwell’s equations tensorial form to the propagation in an E-plane taper,” presented at PIERS 1994, European Space Research and Technology Center, Noordwijk, The Netherlands, July 5–7, 1994.

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Figures (15)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

Incident wave vector characterized by both angles of incidence θ and Φ.

Fig. 3
Fig. 3

Incident wave and the angle of polarization.

Fig. 4
Fig. 4

Reference coordinate system for region 2.

Fig. 5
Fig. 5

(a), (c): F and G values satisfying the boundary conditions on the perfectly conducting walls at u=0 and u=d for TMz polarization. (b), (c): Even or odd extension of F or G from the interval [0, d] onto the interval [-d, 0].

Fig. 6
Fig. 6

Same as Fig. 5, but for TEz polarization.

Fig. 7
Fig. 7

Decimal logarithm of the absolute error between the computed eigenvalues and the analytical eigenmodes versus the truncation order N. 2=2.25, Φ=10°, d/λ=0.7; (a) φ=5°, (b) φ=30°.

Fig. 8
Fig. 8

Definition of scattering matrices. S1 expresses the amplitudes of the fields coming out of the interface y=h cos φ (R and A1) in terms of those coming into it (Δ0m and B1). S2 expresses the amplitudes coming out of the layer (B1 and A2) in terms of those coming into it (A1 and B2). S3 expresses the amplitudes of the fields coming out of the interface y=-h cos φ (B2 and T) in terms of those coming into it (A2 and 0).

Fig. 9
Fig. 9

Error in the conservation of energy versus 1/N (N is the truncation order). The logarithmic (base 10) scale is used for the x axis. 2=2.25, Φ=5°, θ=5°, δ=5°, φ=5°, d/λ=0.7. (a) h/λ=1, (b) h/λ=10.

Fig. 10
Fig. 10

Reflected and transmitted efficiencies in 0 and -1 order versus incidence angle θ: TM polarization. 2=2.25, Φ=0°, δ=0°, φ=0°, d/λ=0.7, h/λ=5.

Fig. 11
Fig. 11

Sum of transmitted efficiencies versus incidence angle θ: (a) polarization E parallel, (b) polarization H parallel.

Fig. 12
Fig. 12

Reflected efficiencies in TEz polarization versus incidence angles θ and Φ. 2=2.25, δ=0°, φ=19.1°, h/λ=0.1, d/λ=0.9.

Fig. 13
Fig. 13

Reflected efficiencies in TEz polarization versus incidence angles θ and Φ. 2=2.25, δ=0°, φ=18°, h/λ=0.1, d/λ=0.9.

Fig. 14
Fig. 14

Reflected efficiencies in TEz polarization versus incidence angles θ and Φ. 2=2.25, δ=0°, φ=20°, h/λ=0.1, d/λ=0.9.

Fig. 15
Fig. 15

Scattering matrices in chain. S1 and S2 are described in terms of quadrupoles. Vectors a represent the incoming waves to the substructure, and vectors b represent the outgoing waves from the substructure.

Tables (1)

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Table 1 Nature of the Elementary Wave versus the Real and Imaginary Parts of rn

Equations (135)

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kx=k sin θ cos Φ,
ky=-k cos θ cos Φ,
kz=k sin Φ.
Exi=(cos δ cos θ-sin δ sin θ sin Φ)Ψi(x, y, z),
Eyi=(cos δ sin θ-sin δ cos θ sin Φ)Ψi(x, y, z),
Ezi=(sin δ cos Φ)Ψi(x, y, z),
ZHxi=(-sin δ cos θ-cos δ sin θ sin Φ)Ψi(x, y, z),
ZHyi=(-sin δ sin θ+cos δ cos θ sin Φ)Ψi(x, y, z),
ZHzi=(cos δ cos Φ)Ψi(x, y, z),
Ψi(x, y, z)=E0 exp[-ik(cos Φsin θ)x-(cos θ)y-ik(sin Φ)z],
Z=μ0/0(waveimpedanceinvacuum).
(k2-k2γ2)ExEyZHxZHy=-ikγx-ikγyiky-ikx-ikyikx-ikγx-ikγyEzZHz,
inregion1:Ez1d=(cos Φ)n=-+ RnTMΨ1d(x, y, z),
inregion3:Ez3d=(cos Φ)n=-+ TnTMΨ3d(x, y, z);
inregion1:ZHz1d=(cos Φ)n=-+ RnTEΨ1d(x, y, z),
inregion3:ZH23d=(cos Φ)n=-+TnTEΨ3d(x, y, z),
Ψ1d=exp(-ik(cos Φ)[βr(y-h cos φ)+αn(x-h sin φ)]-ikz sin Φ),
Ψ3d=exp(ik(cos Φ)[βn(y+h cos φ)-αn(x+h sin φ)]=ikz sin Φ),
αn=sin θ+nλd cos Φ,βn2=1-αn2,
β=1-αn2-iαn2-1ifβ2>0ifβ2<0.
u=x-y tan φ.
Ali=100-tan φ10001,Ail=100tan φ10001,
gij=1tan φ0tan φ1+(tan φ)20001,
gij=gij-1=1+(tan φ)2-tan φ0-tan φ10001.
k2(ν2-γ2)EuEyZ2HuZ2Hy
=-ikγu-ikγyikν(g22y+g12u)-ikν(g11u+g12y)-ikν(g22y+g12u)ikν(g11u+g12y)-ikγu-ikγy
×EzZ2Hz,
1ikFy=-k2(ν2-γ2)k2ν2G+tan φikνF,
1ikGy=-1k2(ν2-γ2)2Fu2-F+tan φikνGu,
F=Ez,G=ZHuinTMzpolarization,
F=ZHz,G=-EuinTEzpolarization.
TMz
EzZHuZHzEu=FG0-ikγ(ν2-γ2)Fu
TEz
+0-ikγ(ν2-γ2)FuF-G.
F(u, y)=p=-+Fp(y)sin(αpu),
Fp(y)=1d0d F(u, y)sin(αpu)du,
G(u, y)=p=-+Gp(y)cos(αpu),
Gp(y)=1d0d G(u, y)cos(αpu)du,
1ikνdFm(y)dy=tan φikνq=1+αmFq(y)(bq+m-bq-m)-k2(ν2-γ2)k2ν2q=1+Gq(y)(bq+m-bq-m)-2G0(y)bm,
1ikνq=1+ dGp(y)dy(bp+m-bp-m)+2 dG0(y)dybm
=αm2k2(ν2-γ2)-1Fm(y)-tan φikναmGm(y),
bp=1d0d sin(αpu)du.
F(u, y)=p=-+ Fp(y)cos(αpu),
Fp(y)=1d0d F(u, y)cos(αpu)du,
G(u, y)=p=-+ Gp(y)cos(αpu),
Gp(y)=1d0d G(u, y)cos(αpu)du.
1ikνEy=1k2(ν2-γ2)Fu-tan φikνG.
1ikνFy=-k2(ν2-γ2)k2ν2G+tan φikνFu,
1ikνGy=-u1k2(ν2-γ2)Fu-tan φikνG-F.
1ikνq=1+ dFp(y)dy(bp+m-bp-m)+2 dF0(y)dybm
=-tan φikναmFm(y)-k2(ν2-γ2)k2ν2×q=1+Gq(y)(bq+m-bq-m)+2G0(y)bm,
1ikνdGm(y)dy
=αm2k2(ν2-γ2)-1Fm(y)+αm tan φikν×q=1+Gq(y)(bq+m-bq-m)+2G0(y)bm,
1ikνdG0(y)dy=-F0(y).
1ikνddyFn(y)Gn(y)=TFn(y)Gn(y).
Fn(y)Gn(y)=exp(ikνry)FnGn.
rFnGn=TFnGn.
FTMz(u, y)=m=1Nn=1N BnFmn(+) exp[ikνrn(+)y]+n=1N AnFmn(-) exp[ikνrn(-)y]sin(αmu),
GTMz(u, y)=m=0N-1n=1N B0Gmn(+) exp[ikνrn(+)y]+n=1N AnGmn(-) exp[ikνrn(-)y]cos(αmu),
FTEz(u, y)=m=0N-1n=1N BnFmn(+) exp[ikνrn(+)y]+n=1N AnFmn(-) exp[ikνrn(-)y]cos(αmu),
GTEz(u, y)=m=0N-1n=1N BnGmn(+) exp[ikνrn(+)y]+n=1N AnGmn(-) exp[ikνrn(-)y]cos(αmu).
kx2+ky2+kz2-k2ν2=0,
βn=1kν cos φ(kν)2-(nπd cos φ)2-(k sin Φ)2.
Ez(1)(x, y=h cos φ)=Ez(2)(x-h sin φ, y=h cos φ),
Hz(1)(x, y=h cos φ)=Hz(2)(x-h sin φ, y=h cos φ),
Ex(1)(x, y=h cos φ)=Eu(2)(x-h sin φ, y=h cos φ),
Hx(1)(x, y=h cos φ)=Hu(2)(x-h sin φ, y=h cos φ);
Ez(3)(x, y=-h cos φ)=Ez(2)(x+h sin φ, y=-h cos φ),
Hz(3)(x, y=-h cos φ)=Hz(2)(x+h sin φ, y=-h cos φ),
Ex(3)(x, y=-h cos φ)=Eu(2)(x+h sin φ, y=-h cos φ),
Hx(3)(x, y=-h cos φ)=Hu(2)(x+h sin φ, y=-h cos φ).
Δ0mTE=sin δ0ifm=0ifm0,
Δ0mTM=cos δ0ifm=0ifm0.
RA1=S1Δ0mB1,
B1A2=S2A1B2,
B2T=S3A20.
RT=SΔ0m0.
S=L(L(S1, S2), S3),
ξq=-ϕqdϕi,
ξpr=[RpdE(RpdE)*+RpdH(RpdH)*]ReβpIcos θ,
ξpt=[TpdE(TpdE)*+TpdH(TpdH)*]ReβpIcos θ.
ξijkjEk=-Bit,ξijkjHk=Dit-Ji,
iDi=ρ,iBi=0,
i, j, k{1, 2, 3},i=xi,
Ei=AiiEi,Bi=|Aii|-1AiiBi,
Hi=AiiHi,Di=|Aii|-1AiiDi,
Aii=xixi,Aii=xixi,i, i=1, 2, 3,
Di=ijEj,Bi=μijHj.
ij=ggij,μij=μggij,g=|gij|.
gij=AiiAjigij.
(cos Φ)(ΔpTM+RpTM)
=m=1+n=1+Kpm[BnTMFmn+TMΨ(rn+TM)+AnTMFmn-TMΨ(rn-TM)],
(cos Φ)(ΔpTE+RpTE)
=m=1+n=1+K¯pmν[BnTEFmn+TEΨ(rn+TE)+AnTEFmn-TEΨ(rn-TE)],
βp(RpTM-ΔpTM)-(sin Φ)αp(ΔpTE+RpTE)
=m=0+n=1+K¯pm[BnTMGmn+TMΨ(rn+TM)
+AnTMGmn-TMΨ(rn-TM)]+iγk(ν2-γ2)×m=0+n=1+Kpmαm(2)[BnTEFmn+TEΨ(rn+TE)+AnTEFmn-TEΨ(rn-TE)],
-(sin Φ)αp(ΔpTM+RpTM)+βp(ΔpTE-RpTE)
=-m=0+n=1+K¯pmν[BnTEGmn+TEΨ(rn+TE)
+AnTEGmn-TEΨ(rn-TE)]-iγνk(ν2-γ2)×m=1+n=1+K¯pmαm(2)[BnTMFmn+TMΨ(rn+TM)+AnTMFmn-TMΨ(rn-TM)],
(cos Φ)TpTM=m=1+n=1+Kpm[BnTMFmn+TMΨ(-rn+TM)+AnTMFmn-TMΨ(-rn-TM)],
(cos Φ)TpTE=m=1+n=1+K¯pmν[BnTEFmn+TEΨ(-rn+TE)+AnTEFmn-TEΨ(-rn-TE)],
βpTpTM-(sin Φ)αpTpTE
=m=0+n=1+K¯pm[BnTMGmn+TMΨ(-rn+TM)
+AnTMGmn-TMΨ(-rn-TM)]+i sin Φkν(cos Φ)2×m=0+n=1+Kpmαm(2)[BnTEFmn+TE×Ψ(-rn+TE)+AnTEFmn-TEΨ(-rn-TE)],
βpTpTE-(sin Φ)αpTpTM
=-m=0+n=1+K¯pmν[BnTEGmn+TEΨ(-rn+TE)
+AnTEGmn-TEΨ(-rn-TE)]-i sin Φk(cos Φ)2×m=1+n=1+K¯pmαm(2)[BnTMFmn+TM×Ψ(-rn+TM)+AnTMFmn-TMΨ(-rn-TM)],
αp=sin θ+nλd cos Φ,αp(2)=n πd,
Ψ(r)=exp(ikνrh cos φ),
Kpm=1d0d exp[ik(cos Φ)αpx]sin[αm(2)x]dx,
K¯pm=1d0d exp[ik(cos Φ)αpx]cos[αm(2)x]dx.
RTMRTEA1TMA1TE=S1ΔTMΔTEB1TMB1TE,
S1=I1-1I2,
B1TM,TE=BTM,TE exp(ikνr+h cos φ),
A1TM,TE=ATM,TE exp(ikνr-h cos φ),
I1=cos Φ0β-(sin Φ)α0cos Φ-(sin Φ)α-β-KF-TM0-K¯G-TMiγνk(ν2-γ2)K¯α(2)F-TM0-νK¯F-TE-iγk(ν2-γ2)Kα(2)F-TEK¯G-TE,
I2=-cos Φ0β(sin Φ)α0-cos Φ(sin Φ)α-βKF+TM0-K¯G+TM-iγνk(ν2-γ2)K¯α(2)F+TM0νK¯F+TE-iγk(ν2-γ2)Kα(2)F+TE-K¯G+TE.
B1TMB1TEA2TMA2TE=S2A1TMA1TEB2TMB2TE,
B2TM,TE=BTM,TE exp(-ikνr+h cos φ),
A2TM,TE=ATM,TE exp(-ikνr-h cos φ),
S2=00exp(-ikν2hr-TM)0000exp(-ikν2hr-TE)exp(ikν2hr+TM)0000exp(ikν2hr+TE)00.
B2TMB2TETTMTTE=S3A2TMA2TE00,
[S3]=J1-1J2,
J1=-KF+TM0-K¯G+TMiγνk(ν2-γ2)K¯α(2)F+TM0-νK¯F+TE-iγk(ν2-γ2)Kα(2)F+TEK¯G+TEcos Φ0-B-(sin Φ)α0cos Φ-(sin Φ)αβ,
J2=KF-TM0000νK¯F-TE00K¯G-TMiγk(ν2-γ2)Kα(2)F-TE00-iγνk(ν2-γ2)K¯α(2)F-TM-K¯G+TE00.
b1b2=S111S112S121S122 a1a2forS1,
b1b2=S211S221S212S222 a1a2forS2,
a2=b1,b2=a1.
b1b2=Sg11Sg12Sg21Sg22 a1a2.
[Sg11]=[S111]+[S112]([I]-[S211][S122])[S211]-1[S121],
[Sg12]=[S112]([I]-[S211][S122])[S212]-1,
[Sg21]=[S221]([I]-[S122][S211])[S12]-1,
[Sg22]=[S222]+[S221]([I]-[S122][S211])[S122]-1[S212],

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