Abstract

The diffracted orders generated by a plane-wave incident upon a multilayered dielectric grating can undergo strong intensity variations known as Wood’s anomalies or scattering resonances. To clarify these effects, we examine the field and power flux inside the grating region by using a rigorous modal solution of the scattering problem. The results show that, at the peak of a scattering resonance, the power flux is almost identical to that of a leaky wave that can be supported by the grating structure. We thus confirm and generalize previous research that has identified the anomalous intensity variations as a forced-resonance effect associated with the free-resonant character of leaky waves. Quantitative data illustrating the behavior of typical gratings are given, and the special case of normal incidence is discussed.

© 1997 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. S. Wang, R. Magnusson, “Multilayer waveguide-grating filters,” Appl. Opt. 34, 2414–2420 (1995).
    [CrossRef] [PubMed]
  2. 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]
  3. S. S. Wang, R. Magnusson, J. S. Bagby, M. G. Moharam, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A 7, 1470–1474 (1990).
    [CrossRef]
  4. L. F. DeSandre, J. M. Elson, “Extinction-theorem analysis of diffraction anomalies in overcoated gratings,” J. Opt. Soc. Am. A 8, 763–777 (1991).
    [CrossRef]
  5. R. Magnusson, S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
    [CrossRef]
  6. S. S. Wang, R. Magnusson, “Design of waveguide-grating filters with symmetrical line shapes and low sidebands,” Opt. Lett. 19, 919–921 (1994).
    [CrossRef] [PubMed]
  7. R. Magnusson, S. S. Wang, “Transmission bandpass guided-mode resonance filters,” Appl. Opt. 34, 8106–8109 (1995).
    [CrossRef] [PubMed]
  8. S. Peng, G. M. Morris, “Efficient implementation of rigorous coupled-wave analysis for surface-relief gratings,” J. Opt. Soc. Am. A 12, 1087–1096 (1995).
    [CrossRef]
  9. S. Peng, G. M. Morris, “Resonant scattering from two-dimensional gratings,” J. Opt. Soc. Am. A 13, 993–1005 (1996).
    [CrossRef]
  10. R. W. Day, S. S. Wang, R. Magnusson, “Filter-response line shapes of resonant waveguide gratings,” J. Lightwave Technol. 14, 1815–1824 (1996).
    [CrossRef]
  11. R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).
    [CrossRef]
  12. A. Hessel, A. A. Oliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. 4, 1275–1297 (1965).
    [CrossRef]
  13. M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
    [CrossRef]
  14. T. Tamir, S. Zhang, “Modal transmission-line theory of multilayered grating structures,” J. Lightwave Technol. 14, 914–927 (1996).
    [CrossRef]
  15. M. Nevière, E. Popov, R. Reinisch, “Electromagnetic resonances in linear and nonlinear optics: phenomenological study of grating behavior through the poles and zeros of the scattering operator,” J. Opt. Soc. Am. A 12, 513–523 (1995).
    [CrossRef]
  16. T. Tamir, “Beam and waveguide couplers,” in Integrated Optics, T. Tamir, eds. (Springer-Verlag, New York, 1985), Chap. 7, pp. 83–137.
  17. M. Nevière, “The homogeneous problem,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 5, pp. 123–158.

1996 (3)

S. Peng, G. M. Morris, “Resonant scattering from two-dimensional gratings,” J. Opt. Soc. Am. A 13, 993–1005 (1996).
[CrossRef]

R. W. Day, S. S. Wang, R. Magnusson, “Filter-response line shapes of resonant waveguide gratings,” J. Lightwave Technol. 14, 1815–1824 (1996).
[CrossRef]

T. Tamir, S. Zhang, “Modal transmission-line theory of multilayered grating structures,” J. Lightwave Technol. 14, 914–927 (1996).
[CrossRef]

1995 (5)

1994 (1)

1992 (1)

R. Magnusson, S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
[CrossRef]

1991 (1)

1990 (1)

1989 (1)

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]

1965 (1)

1902 (1)

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).
[CrossRef]

Bagby, J. S.

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]

Day, R. W.

R. W. Day, S. S. Wang, R. Magnusson, “Filter-response line shapes of resonant waveguide gratings,” J. Lightwave Technol. 14, 1815–1824 (1996).
[CrossRef]

DeSandre, L. F.

Elson, J. M.

Gaylord, T. K.

Grann, E. B.

Hessel, A.

Magnusson, R.

Moharam, M. G.

Morris, G. M.

Nevière, M.

Oliner, A. A.

Peng, S.

Pommet, D. A.

Popov, E.

Reinisch, R.

Tamir, T.

T. Tamir, S. Zhang, “Modal transmission-line theory of multilayered grating structures,” J. Lightwave Technol. 14, 914–927 (1996).
[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]

T. Tamir, “Beam and waveguide couplers,” in Integrated Optics, T. Tamir, eds. (Springer-Verlag, New York, 1985), Chap. 7, pp. 83–137.

Wang, S. S.

Wood, R. W.

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).
[CrossRef]

Zhang, S.

T. Tamir, S. Zhang, “Modal transmission-line theory of multilayered grating structures,” J. Lightwave Technol. 14, 914–927 (1996).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

R. Magnusson, S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

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. Lightwave Technol. (2)

R. W. Day, S. S. Wang, R. Magnusson, “Filter-response line shapes of resonant waveguide gratings,” J. Lightwave Technol. 14, 1815–1824 (1996).
[CrossRef]

T. Tamir, S. Zhang, “Modal transmission-line theory of multilayered grating structures,” J. Lightwave Technol. 14, 914–927 (1996).
[CrossRef]

J. Opt. Soc. Am. A (6)

Opt. Lett. (1)

Philos. Mag. (1)

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).
[CrossRef]

Other (2)

T. Tamir, “Beam and waveguide couplers,” in Integrated Optics, T. Tamir, eds. (Springer-Verlag, New York, 1985), Chap. 7, pp. 83–137.

M. Nevière, “The homogeneous problem,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 5, pp. 123–158.

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 (12)

Fig. 1
Fig. 1

Plane-wave incidence in the cover region of a multilayered dielectric grating. Unless otherwise stated, the parameters are εs=2.31, εf=εp1=4.0, εp2=3.61, εc=1.0, tf =0.05 μm, tp=0.15 μm, Λ=0.39 μm, Λ1/Λ=0.6.

Fig. 2
Fig. 2

Dispersion curves for modes guided by the grating configuration in Fig. 1: (a) large-scale diagram in which the thick curve refers to the surface-wave mode of the nonperiodic basic structure, (b) details of the rectangular region around the intersection between the operating lines OP and QR.

Fig. 3
Fig. 3

Possible plane-wave incidence situations: (a) phase matching to q=-1 harmonic of leaky wave progressing along +z, (b) phase matching to q=-2 harmonic of leaky wave progressing along -z.

Fig. 4
Fig. 4

Variation with λ of the fundamental reflected ρ0 =|ρ0|exp(iϕ0) and transmitted τ0=|τ0|exp(iψ0) amplitudes for incidence as in Fig. 1 with fixed θc=38°.

Fig. 5
Fig. 5

Transverse variation with λ of the field magnitude |Ej| = |Ey(x; λ)| for plane-wave incidence as in Figs. 1 and 4.

Fig. 6
Fig. 6

Variation of power flux pz with x in the region around the grating layer for λ=λA and λ=λB. Planar boundaries are indicated by vertical solid lines for the grating and vertical dotted lines for the substrate. The curves show results for plane-wave incidence as in Fig. 1, while the points marked × delineate the variation of the leaky wave supported at λ=λA.

Fig. 7
Fig. 7

Variation of power Pl with λ for plane-wave incidence as in Fig. 1 with fixed θc=38°.

Fig. 8
Fig. 8

Comparison of results by means of Table 1 for the anomaly around λA0.4686 μm in Fig. 4. Exact results are given by solid curves on which points obtained from Eq. (29) are shown by × markers. The dotted curves show the Lorentzian variation obtained from Eq. (29) if the term containing the zero is ignored.

Fig. 9
Fig. 9

Variation with λ of the first higher-order amplitudes ρ-1=|ρ-1|exp(iϕ-1) and τ-1=|τ-1|exp(iψ-1) for the configuration shown in Fig. 1 and θc=38°.

Fig. 10
Fig. 10

Variation with θc of the fundamental amplitudes ρ0=|ρ0|exp(iϕ0) and τ0=|τ0|exp(iψ0) for the configuration shown in Fig. 1 and λ=0.47736 μm.

Fig. 11
Fig. 11

Variation with θc of the first-order amplitudes ρ-1=|ρ-1|exp(iϕ-1) and τ-1=|τ-1|exp(iψ-1) for the configuration shown in Fig. 1 and λ=0.47736 μm.

Fig. 12
Fig. 12

Behavior around normal incidence: (a) dispersion curves in the vicinity of the intersection between the q=-1 curves, (b) variation with θc of the fundamental magnitudes |ρ0| and |τ0| for the configuration shown in Fig. 1 and λ = 0.67495 μm.

Tables (2)

Tables Icon

Table 1 Poles and Zeros for Structure Shown in Fig. 1 and for θc=38°

Tables Icon

Table 2 Poles and Zeros for Structure Shown in Fig. 1 and for λ=0.47736

Equations (34)

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

Ej=q=-Fq(x)exp(κqz-ωt),
κq=βq+iα,βq=β0+2qπ/Λ.
βc0=(2π/λ)εc sin θc,
βcr=βc0+2rπ/λ,r=0, ±1, ±2,.
tan γc=(βc0Λ/π)/(Λ/λ)=2εc sin θc.
Ec=fc0 exp[i(kc0x+kz0z)]+ngcn exp[i(-kcnx+kznz)],
Ep=m{fpm exp(ikpmx)+gpm×exp[ikpm(tp-x)]}nanm exp(ikznz),
Ef=n{fn exp[ikfn(x-tp)]+gfn×exp[ikfn(h-x)]}exp(ikznz),
Es=nfsn exp{i[ksn(x-h)+kznz]},
kzn=βn+iα,βn=β0+2nπ/Λ,
kjn=kjn+ikjn=(k02εj-kzn2)forall jp.
gc=R0cu=ρ,
fs=T(u+ρ)=τ,
R0c=(Yc+Y0c)-1(Yc-Y0c),
T=A(I+Rtp)(Ep-1+EpRtp)-1A-1(I+Rtf)×(Ef-1+EfRtf)-1,
Y0c=AYp(I-EpRtpEp)(I+EpRtpEp)-1A-1,
Rtp=(Yp+Ytp)-1(Yp-Ytp),
Ytp=A-1Yf(I-EfRtfEf)(I+EfRtfEf)-1A,
Rtf=(Yf+Ys)-1(Yf-Ys).
Wgc=R0c-1gc=0,
fs=Tgc,
|W| = |R0c-1| =0,
κν=β0ν+iαν,ν=0, 1, 2.
Pl=-lch+lspz(x)dx,
ρr=wr0/|W|,
ρr(kz)=(kz-kρ0)(kz-kρ1)(kz-kρ2)(kz-kP0)(kz-kP1)(kz-kP2) Rr,
kPn=kPq=κ+2qπ/Λ,q=0, ±1, ±2, ;
ρr(kz)=kz-kρrkz-kPq Rkr,
τr=ntrnρ˜n,
τr(kz)=kz-kτrkz-kPq Tkr,
σr(kz)=kz-kσrkz-kPq Skr,
σr(λ)=λ-λσrλ-λPq Sλr,
σr(θ)=θ-θσrθ-θPq Sθr,
σr(θ)=(θ-θσr)(θ+θσr)(θ-θPq)(θ+θPq) Sθr=θ2-θσr2θ2-θPq2 Sθr.

Metrics