Abstract

In a recent paper [J. Opt. Soc. Am. A 16, 1108 (1999)] Logofǎtu et al. demonstrated by experimental and numerical evidence that the 0th-order cross-polarization (s to p and p to s) reflection coefficients of isotropic, symmetrical, surface-relief gratings in conical mount are identical. Here an analytical proof is given to show that the observed identity is merely a manifestation of the electromagnetic reciprocity theorem for the 0th-order diffraction of symmetrical gratings. The above result is further generalized to bianisotropic gratings, to the 0th-order cross-polarization transmission coefficients, and to the mth-order reflection and transmission coefficients when the wave vector of the incident plane wave and the negative of the wave vector of the mth reflected order are symmetrical with respect to the plane perpendicular to the grating grooves.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. C. Logofãtu, S. A. Coulombe, B. K. Minhas, J. R. McNeil, “Identity of the cross-reflection coefficients for the symmetric surface-relief gratings,” J. Opt. Soc. Am. A 16, 1108–1114 (1999).
    [CrossRef]
  2. P. Vincent, M. Nevière, “The reciprocity theorem for corrugated surfaces used in conical diffraction mountings,” Opt. Acta 26, 889–898 (1979).
    [CrossRef]
  3. J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1986), Chap. 1, Sec. 1, pp. 1–8, and Chap. 5, Sec. 5, pp. 398–405.
  4. L. Li, “Analysis of planar waveguide grating couplers with double surface corrugations of identical period,” Opt. Commun. 114, 406–412 (1995).
    [CrossRef]

1999 (1)

1995 (1)

L. Li, “Analysis of planar waveguide grating couplers with double surface corrugations of identical period,” Opt. Commun. 114, 406–412 (1995).
[CrossRef]

1979 (1)

P. Vincent, M. Nevière, “The reciprocity theorem for corrugated surfaces used in conical diffraction mountings,” Opt. Acta 26, 889–898 (1979).
[CrossRef]

Coulombe, S. A.

Kong, J. A.

J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1986), Chap. 1, Sec. 1, pp. 1–8, and Chap. 5, Sec. 5, pp. 398–405.

Li, L.

L. Li, “Analysis of planar waveguide grating couplers with double surface corrugations of identical period,” Opt. Commun. 114, 406–412 (1995).
[CrossRef]

Logofãtu, P. C.

McNeil, J. R.

Minhas, B. K.

Nevière, M.

P. Vincent, M. Nevière, “The reciprocity theorem for corrugated surfaces used in conical diffraction mountings,” Opt. Acta 26, 889–898 (1979).
[CrossRef]

Vincent, P.

P. Vincent, M. Nevière, “The reciprocity theorem for corrugated surfaces used in conical diffraction mountings,” Opt. Acta 26, 889–898 (1979).
[CrossRef]

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

Opt. Acta (1)

P. Vincent, M. Nevière, “The reciprocity theorem for corrugated surfaces used in conical diffraction mountings,” Opt. Acta 26, 889–898 (1979).
[CrossRef]

Opt. Commun. (1)

L. Li, “Analysis of planar waveguide grating couplers with double surface corrugations of identical period,” Opt. Commun. 114, 406–412 (1995).
[CrossRef]

Other (1)

J. A. Kong, Electromagnetic Wave Theory (Wiley, New York, 1986), Chap. 1, Sec. 1, pp. 1–8, and Chap. 5, Sec. 5, pp. 398–405.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Multilayer grating and the coordinate system.

Fig. 2
Fig. 2

Topographical symmetries of multilayer gratings: (a) C2y and σx symmetries, (b) C2z and i symmetries, (c) C2x and σy symmetries, (d) C2xT2x and σyT2x symmetries. All symmetry symbols are explained in the text.

Fig. 3
Fig. 3

Two practical grating structures that have high degrees of topographical symmetry: (a) a metallic wire grating, (b) a transmission grating formed by putting face-to-face two isosceles triangular gratings.

Fig. 4
Fig. 4

Schematic aid for proving Theorem 1. As a convention, only the s or p unit vector is shown for an incident plane wave, but both s and p unit vectors are shown for a reflected plane wave because it in general has both components. The unit p vectors are not perpendicular to the plane of the drawing, but for simplicity they are drawn as circles, with a dot at the center indicating that they have a positive y component. (a) Incidence with s polarization. (b) Incidence with p polarization, conjugate to that of (a) and with respect to the 0th reflected order. (c) An incidence configuration obtained by a rotation of both the grating and the total electromagnetic fields in (b) about the y axis by π.

Fig. 5
Fig. 5

Schematic aid for proving Theorem 2. The dashed-line arrows represent plane waves propagating in the transmission side of the grating. (a) Incidence with s polarization. (b) Incidence with p polarization, conjugate to that of (a) and with respect to the 0th transmitted order. (c) An incidence configuration obtained by a spatial inversion of both the grating and the total electromagnetic fields in (b). The unit vectors pˆ in (c) are opposite to what would be given by Eq. (5). The crosses indicate that they have a negative y component.

Fig. 6
Fig. 6

Schematic aid for proving Theorem 3. (a) Incidence with s polarization. (b) Incidence with p polarization, conjugate to that of (a) and with respect to the mth reflected order. (c) An incidence configuration obtained by a mirror reflection of both the grating and the total electromagnetic fields in (b) with respect to the z=0 plane. Note that the unit vector sˆ in (c) is opposite to what would be given by Eq. (5).

Fig. 7
Fig. 7

Schematic aid for proving Theorem 4. (a) Incidence with s polarization. (b) Incidence with p polarization, conjugate to that of (a) and with respect to the mth transmitted order. (c) An incidence configuration obtained by a rotation of both the grating and the total electromagnetic fields in (b) about the x axis by π. The meanings of the unit vectors sˆ and pˆ in Fig. (c) are the same as in Figs. 5(c) and 6(c).

Equations (13)

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

|yˆ·kia|(Eia·Edb)=|yˆ·kd,la|(Ei,lb·Ed,la),l=T, B.
D=ϵ˜·E+ξ˜·H,
B=ζ˜·E+μ˜·H.
ϵ˜=ϵ˜T,μ˜=μ˜T,ξ˜=-ζ˜T,
SdS·(Ea×Hb-Eb×Ha)=0,
sˆ=(k×yˆ)/|k×yˆ|,pˆ=sˆ×k/|k|.
r0sp=-r0ps,
t0sp=t0ps,
sin θ cos ϕ=-mλ2d,
rmsp=rmps,
tmsp=-tmps
tmsp=-(-1)mtmps,
tmss=tmpp=0,mod(m, 2)=1.

Metrics