Abstract

Wave scattering by a chiral grating is studied in this paper. Numerical results are given, and physical properties are discussed, including the influence of frequency, angle of incidence, and aspect ratio. At high frequencies we find anomalous coupling regions known as Wood’s anomalies, which are explained by the excitation and reradiation of leaky waveguide modes in the periodic layer. The chiral grating can possess both frequency-selection and mode-conversion properties.

© 2004 Optical Society of America

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References

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  1. D. L. Jaggard, A. R. Mickelson, C. H. Papas, “On electromagnetic waves in chiral media,” Appl. Phys. 18, 211–216 (1979).
    [CrossRef]
  2. D. L. Jaggard, N. Engheta, “Chirality in electrodynamics: modeling and applications,” in Directions in Electromagnetic Wave Modeling, H. L. Bertoni, L. B. Felson, eds. (Plenum, New York, 1991), pp. 435–446.
  3. H. N. Kritikos, N. Engheta, D. L. Jaggard, “Symmetry in material media,” in Symmetry in Electromagnetics, C. Baum, H. N. Kritikos, eds. (Taylor & Francis, 1995), Chap. 4.
  4. K. M. Flood, “Distributed feedback in and distributed Bragg reflection from periodic chiral structures,” Ph.D. thesis (University of Pennsylvania, Philadelphia, Pennsylvania, 1995).
  5. K. M. Flood, D. L. Jaggard, “Bandgap structure for periodic chiral media,” J. Opt. Soc. Am. A 13, 1395–1406 (1996).
    [CrossRef]
  6. D. L. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, Y. Kim, “Periodic chiral structures,” IEEE Trans. Antennas Propag. 37, 1447–1452 (1989).
    [CrossRef]
  7. T. X. Wu, D. L. Jaggard, “Scattering of chiral periodic structure,” IEEE Trans. Antennas Propag. 52, 1859–1870 (2004).
    [CrossRef]
  8. H. L. Bertoni, L. S. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
    [CrossRef]
  9. S. T. Peng, “Rigorous formulation of scattering and guidance by dielectric grating waveguides, general case of oblique incidence,” J. Opt. Soc. Am. A 6, 1869–1883 (1989).
    [CrossRef]
  10. J. Jin, The Finite Element Method in Electromagnetics (Wiley, New York, 1993).
  11. D. L. Jaggard, X. Sun, “Theory of chiral multilayers,” J. Opt. Soc. Am. A 9, 804–813 (1992).
    [CrossRef]
  12. J. C. Liu, “Electromagnetic waves scattering from chiral layers,” Ph.D. thesis (University of Pennsylvania, Philadelphia, Pennsylvania, 1996).
  13. A. Hessel, “General characteristics of traveling-wave antennas,” in Antenna Theory, Part II, R. E. Collin, F. J. Zucker, eds. (McGraw-Hill, New York, 1969), p. 184.
  14. A. Hessel, A. A. Oliner, “A new theory of Wood’s anomalies in optical gratings,” Appl. Opt. 4, 1275–1297 (1965).
    [CrossRef]
  15. T. Tamir, “Nonspecular phenomena in beam fields reflected by multilayered media,” J. Opt. Soc. Am. A 3, 558–565 (1986).
    [CrossRef]

2004 (1)

T. X. Wu, D. L. Jaggard, “Scattering of chiral periodic structure,” IEEE Trans. Antennas Propag. 52, 1859–1870 (2004).
[CrossRef]

1996 (1)

1992 (1)

1989 (3)

S. T. Peng, “Rigorous formulation of scattering and guidance by dielectric grating waveguides, general case of oblique incidence,” J. Opt. Soc. Am. A 6, 1869–1883 (1989).
[CrossRef]

H. L. Bertoni, L. S. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

D. L. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, Y. Kim, “Periodic chiral structures,” IEEE Trans. Antennas Propag. 37, 1447–1452 (1989).
[CrossRef]

1986 (1)

1979 (1)

D. L. Jaggard, A. R. Mickelson, C. H. Papas, “On electromagnetic waves in chiral media,” Appl. Phys. 18, 211–216 (1979).
[CrossRef]

1965 (1)

Bertoni, H. L.

H. L. Bertoni, L. S. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

Cheo, L. S.

H. L. Bertoni, L. S. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

Engheta, N.

D. L. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, Y. Kim, “Periodic chiral structures,” IEEE Trans. Antennas Propag. 37, 1447–1452 (1989).
[CrossRef]

H. N. Kritikos, N. Engheta, D. L. Jaggard, “Symmetry in material media,” in Symmetry in Electromagnetics, C. Baum, H. N. Kritikos, eds. (Taylor & Francis, 1995), Chap. 4.

D. L. Jaggard, N. Engheta, “Chirality in electrodynamics: modeling and applications,” in Directions in Electromagnetic Wave Modeling, H. L. Bertoni, L. B. Felson, eds. (Plenum, New York, 1991), pp. 435–446.

Flood, K. M.

K. M. Flood, D. L. Jaggard, “Bandgap structure for periodic chiral media,” J. Opt. Soc. Am. A 13, 1395–1406 (1996).
[CrossRef]

K. M. Flood, “Distributed feedback in and distributed Bragg reflection from periodic chiral structures,” Ph.D. thesis (University of Pennsylvania, Philadelphia, Pennsylvania, 1995).

Hessel, A.

A. Hessel, A. A. Oliner, “A new theory of Wood’s anomalies in optical gratings,” Appl. Opt. 4, 1275–1297 (1965).
[CrossRef]

A. Hessel, “General characteristics of traveling-wave antennas,” in Antenna Theory, Part II, R. E. Collin, F. J. Zucker, eds. (McGraw-Hill, New York, 1969), p. 184.

Jaggard, D. L.

T. X. Wu, D. L. Jaggard, “Scattering of chiral periodic structure,” IEEE Trans. Antennas Propag. 52, 1859–1870 (2004).
[CrossRef]

K. M. Flood, D. L. Jaggard, “Bandgap structure for periodic chiral media,” J. Opt. Soc. Am. A 13, 1395–1406 (1996).
[CrossRef]

D. L. Jaggard, X. Sun, “Theory of chiral multilayers,” J. Opt. Soc. Am. A 9, 804–813 (1992).
[CrossRef]

D. L. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, Y. Kim, “Periodic chiral structures,” IEEE Trans. Antennas Propag. 37, 1447–1452 (1989).
[CrossRef]

D. L. Jaggard, A. R. Mickelson, C. H. Papas, “On electromagnetic waves in chiral media,” Appl. Phys. 18, 211–216 (1979).
[CrossRef]

H. N. Kritikos, N. Engheta, D. L. Jaggard, “Symmetry in material media,” in Symmetry in Electromagnetics, C. Baum, H. N. Kritikos, eds. (Taylor & Francis, 1995), Chap. 4.

D. L. Jaggard, N. Engheta, “Chirality in electrodynamics: modeling and applications,” in Directions in Electromagnetic Wave Modeling, H. L. Bertoni, L. B. Felson, eds. (Plenum, New York, 1991), pp. 435–446.

Jin, J.

J. Jin, The Finite Element Method in Electromagnetics (Wiley, New York, 1993).

Kim, Y.

D. L. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, Y. Kim, “Periodic chiral structures,” IEEE Trans. Antennas Propag. 37, 1447–1452 (1989).
[CrossRef]

Kowarz, M. W.

D. L. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, Y. Kim, “Periodic chiral structures,” IEEE Trans. Antennas Propag. 37, 1447–1452 (1989).
[CrossRef]

Kritikos, H. N.

H. N. Kritikos, N. Engheta, D. L. Jaggard, “Symmetry in material media,” in Symmetry in Electromagnetics, C. Baum, H. N. Kritikos, eds. (Taylor & Francis, 1995), Chap. 4.

Liu, J. C.

D. L. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, Y. Kim, “Periodic chiral structures,” IEEE Trans. Antennas Propag. 37, 1447–1452 (1989).
[CrossRef]

J. C. Liu, “Electromagnetic waves scattering from chiral layers,” Ph.D. thesis (University of Pennsylvania, Philadelphia, Pennsylvania, 1996).

Mickelson, A. R.

D. L. Jaggard, A. R. Mickelson, C. H. Papas, “On electromagnetic waves in chiral media,” Appl. Phys. 18, 211–216 (1979).
[CrossRef]

Oliner, A. A.

Papas, C. H.

D. L. Jaggard, A. R. Mickelson, C. H. Papas, “On electromagnetic waves in chiral media,” Appl. Phys. 18, 211–216 (1979).
[CrossRef]

Pelet, P.

D. L. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, Y. Kim, “Periodic chiral structures,” IEEE Trans. Antennas Propag. 37, 1447–1452 (1989).
[CrossRef]

Peng, S. T.

Sun, X.

Tamir, T.

H. L. Bertoni, L. S. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

T. Tamir, “Nonspecular phenomena in beam fields reflected by multilayered media,” J. Opt. Soc. Am. A 3, 558–565 (1986).
[CrossRef]

Wu, T. X.

T. X. Wu, D. L. Jaggard, “Scattering of chiral periodic structure,” IEEE Trans. Antennas Propag. 52, 1859–1870 (2004).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. (1)

D. L. Jaggard, A. R. Mickelson, C. H. Papas, “On electromagnetic waves in chiral media,” Appl. Phys. 18, 211–216 (1979).
[CrossRef]

IEEE Trans. Antennas Propag. (3)

D. L. Jaggard, N. Engheta, M. W. Kowarz, P. Pelet, J. C. Liu, Y. Kim, “Periodic chiral structures,” IEEE Trans. Antennas Propag. 37, 1447–1452 (1989).
[CrossRef]

T. X. Wu, D. L. Jaggard, “Scattering of chiral periodic structure,” IEEE Trans. Antennas Propag. 52, 1859–1870 (2004).
[CrossRef]

H. L. Bertoni, L. S. Cheo, T. Tamir, “Frequency-selective reflection and transmission by a periodic dielectric layer,” IEEE Trans. Antennas Propag. 37, 78–83 (1989).
[CrossRef]

J. Opt. Soc. Am. A (4)

Other (6)

J. Jin, The Finite Element Method in Electromagnetics (Wiley, New York, 1993).

J. C. Liu, “Electromagnetic waves scattering from chiral layers,” Ph.D. thesis (University of Pennsylvania, Philadelphia, Pennsylvania, 1996).

A. Hessel, “General characteristics of traveling-wave antennas,” in Antenna Theory, Part II, R. E. Collin, F. J. Zucker, eds. (McGraw-Hill, New York, 1969), p. 184.

D. L. Jaggard, N. Engheta, “Chirality in electrodynamics: modeling and applications,” in Directions in Electromagnetic Wave Modeling, H. L. Bertoni, L. B. Felson, eds. (Plenum, New York, 1991), pp. 435–446.

H. N. Kritikos, N. Engheta, D. L. Jaggard, “Symmetry in material media,” in Symmetry in Electromagnetics, C. Baum, H. N. Kritikos, eds. (Taylor & Francis, 1995), Chap. 4.

K. M. Flood, “Distributed feedback in and distributed Bragg reflection from periodic chiral structures,” Ph.D. thesis (University of Pennsylvania, Philadelphia, Pennsylvania, 1995).

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Figures (11)

Fig. 1
Fig. 1

(a) Configuration of a chiral grating and (b) the grating’s bisection due to symmetry.

Fig. 2
Fig. 2

Configuration of an unbounded periodic array of chiral slabs.

Fig. 3
Fig. 3

(a) Wave reflection and transmission by a uniform chiral slab. At low frequencies this is a simplified model for wave reflection and transmission by a chiral grating. (b) Process of guided wave excitation and plane wave reradiation when the phase-matching condition k0 sin θ=2π/d-β0 is satisfied. The incident plane wave couples to the n=-1 space harmonic of a waveguide mode traveling in the -x direction.

Fig. 4
Fig. 4

For TM wave incidence, the normalized power versus normalized frequency variable k0h up to 5. Here μ1=μ2=μ0, ε1=2.5ε0, ε2=1.5ε0, κ1=κ2=0.1, θ=45°, d1=d2=d/2, and h/d=1.8. Solid curves, results for the chiral grating in Fig. 1(a). Dashed curves, results for the uniform chiral slab in Fig. 3(a).

Fig. 5
Fig. 5

For TM wave incidence, the normalized power versus normalized frequency variable k0h from 5 to 6.63. Here μ1=μ2=μ0, ε1=2.5ε0, ε2=1.5ε0, κ1=κ2=0.1, θ=45°, d1=d2=d/2, and h/d=1.8. Solid curves, results for the chiral grating in Fig. 3(b). Dashed curves, results for the uniform chiral slab in Fig. 3(a).

Fig. 6
Fig. 6

Dispersion diagrams for eigenmodes of the planar chirowaveguide in Fig. 3(a). These are approximate dispersion diagrams for the n=0 space harmonic of waveguide modes in Fig. 2(b) with μ1=μ2=μ0, ε1=2.5ε0, ε2=1.5ε0, κ1=κ2=0.1, θ=45°, d1=d2=d/2, and h/d=1.8. The dashed curve is (β/k0)2=[(2πh)/(k0hd)-sin θ]2, to check the phase-matching conditions.

Fig. 7
Fig. 7

For TM wave incidence, the normalized power versus incident angle θ, up to 71.5°. Here μ1=μ2=μ0, εc1=2.5ε0, εc2=1.5ε0, κ1=κ2=0.1, d1=d2=d/2, h/d=1.8, and k0h=5.8.

Fig. 8
Fig. 8

Dispersion diagrams for eigenmodes of the planar chirowaveguide in Fig. 3(a). These are approximate dispersion diagrams for the n=0 space harmonic of waveguide modes in Fig. 2(b) with μ1=μ2=μ0, ε1=2.5ε0, ε2=1.5ε0, κ1=κ2=0.1, k0h=5.8, d1=d2=d/2, and h/d=1.8. The dashed curve is (β/k0)2=[(2πh)/(k0hd)-sin θ]2, to check the phase-matching conditions.

Fig. 9
Fig. 9

For TM wave incidence, the normalized power versus normalized frequency variable k0h and angle of incidence θ. Here μ1=μ2=μ0, ε1=2.5ε0, ε2=1.5ε0, κ1=κ2=0.1, d1=d2=d/2, and h/d=1.8. The magenta curves are (β/k0)2=[(2πh)/(k0hd)-sin θ]2, to check the phase-matching conditions.

Fig. 10
Fig. 10

For TM wave incidence, the normalized power versus d1/d ratio, with μ1=μ2=μ0, ε1=2.5ε0, ε2=1.5ε0, κ1=κ2=0.1, h/d=1.8, k0h=5.8, and θ=45°.

Fig. 11
Fig. 11

Dispersion diagrams for eigenmodes of the planar chirowaveguide in Fig. 3(a). These are approximate dispersion diagrams for the n=0 space harmonic of waveguide modes in Fig. 3(b) with μ1=μ2=μ0, ε1=2.5ε0, ε2=1.5ε0, κ1=κ2=0.1, k0h=5.8, θ=45°, and h/d=1.8. The dashed curve is (β/k0)2=[(2πh)/(k0hd)-sin θ]2, to check the phase-matching conditions.

Equations (60)

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kx=k0sin θ,
kz=k0cos θ,
kxn=kx+2nπd,forn=,-2, -1, 0, 1, 2,  ,
F(x+d, y, z)=λF(x, y, z),
Ey(x, z)=nϕn(x)Un(z),
Hy(x, z)=nϕn(x)Un(z),
Ex(x, z)=nϕn(x)Jn(z),
Hx(x, z)=-nϕn(x)Jn(z),
ϕn(x)=exp(ikxnx)/d,
0dϕm(x)ϕn*(x)dx=δmn.
×E=iωμH+ωκμεcE,
×H=-iωεcE+ωκμεcH,
εc=ε+μξc2,
κ=ξcμ/εc,
Ex=ex(x)J(z),
Ey=ey(x)U(z),
Ez=ez(x)U(z),
Hx=hx(x)J(z),
Hy=hy(x)U(z),
Hz=hz(x)U(z),
ddx K˜(x) dΦ(x)dx+ω2K(x)Φ(x)-kz2K˜(x)Φ(x)=0,
Φ(x)=ey(x)hy(x),
K(x)=-iεc(x)κ(x)μ(x)εc(x)κ(x)μ(x)εc(x)iμ(x),
K˜(x)=11-κ(x)2× -iεc(x)-κ(x)μ(x)εc(x)-κ(x)μ(x)εc(x)iμ(x).
F[Φ(x)]=0ddΦT(x)dxK˜(x) dΦ*(x)dx-ω2ΦT(x)K(x)Φ*(x)+kz2ΦT(x)K˜(x)Φ*(x)dx,
Φ(x)=e=1MNeT(x)Ψe,
Ψe=ey1eey2eey3ehy1ehy2ehy3e,
Ne(x)=N1e(x)0N2e(x)0N3e(x)00N1e(x)0N2e(x)0N3e(x).
e=1M(Ae-kz2De)Ψe=0,
Ae=ω2x1x2Ne(x)K(x)NeT(x)dx-x1x2   Ne(x)x   K˜(x) NeT(x)x   dx,
De=x1x2Ne(x)K˜(x)NeT(x)dx.
AΨ=kz2DΨ
Ey(x, z)=neyn(x)Un(z),
Hy(x, z)=nhyn(x)Un(z),
Ex(x, z)=nexn(x)Jn(z),
Hx(x, z)=nhxn(x)Jn(z).
0d(exmhyn*-eyn*hxm)dx=δmn,
U=MU¯,
J=NJ¯.
U=UU,
J=JJ,
M=M1M2,
N=N1N2,
(M1)mn=0dϕm*(x)e¯yn(x)dx,
(M2)mn=0dϕm*(x)h¯yn(x)dx,
(N1)mn=-0dϕm*(x)h¯xn(x)dx,
(N2)mn=0dϕm*(x)e¯xn(x)dx.
NU=U¯,
MJ=J¯.
MN=NM=I,
(Z¯o)n=iZ¯cn cot(k¯znh/2),
(Z¯s)n=-iZ¯cn cot(k¯znh/2),
Zo=MZ¯oM*,
Zs=MZ¯sM*,
Ro=[Zo+Zc]-1[Zo-Zc],
Rs=[Zs+Zc]-1[Zs-Zc],
R=(Ro+Rs)/2,
T=(Ro-Rs)/2.
2π/d-β=k0   sin θ,
βk02=2πhk0hd-sin θ2.

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