Abstract

A multiple-interference model to describe double-grating waveguide structures is presented. It is based on the multiple-interference model for single-grating waveguide structures previously established by Friesem and co-workers [J. Opt. Soc. Am. A 14, 2985 (1997)]. We show that double grating waveguide structures, in particular, as well as the usual single-grating waveguide structures can be completely described by our model and that explicit dependences of the resonant conditions on the wavelength, the angle, and the polarization of the incident light as well as on system parameters such as refractive indices, layer thickness, grating depths, and grating period can be given. This multiple-interference model elucidates the resonance behavior of single- and double-grating waveguide structures and predicts reflection and transmission resonance bandwidths in analogy to those of the classic Fabry–Perot interferometer. One can explain and understand the periodicity of resonances by interpreting grating waveguide structures as inverted Fabry–Perot interferometers. Comparison with exact numerical calculations and verification by experimental investigations prove the model to be a powerful tool for design purposes and to provide a deep understanding of the observed resonance phenomena.

© 2004 Optical Society of America

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References

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  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]
  2. C. H. Palmer, “Parallel diffraction grating anomalies,” J. Opt. Soc. Am. 42, 269–276 (1952).
    [CrossRef]
  3. A. Hessel and A. A. Oliner, “A new theory of Wood’s anomalies,” Appl. Opt. 4, 1275–1297 (1965).
    [CrossRef]
  4. M. Nevière, “The homogeneous problem,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 5.
  5. V. A. Sychugov and A. V. Tishchenko, “Propagation and conversion of light waves in corrugated waveguide structures,” Sov. J. Quantum Electron. 12, 923–926 (1982).
    [CrossRef]
  6. G. A. Golubenko, A. S. Svakhin, V. A. Sychugiov, and A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15, 886–887 (1985).
    [CrossRef]
  7. E. Popov, L. Mashev, and D. Maystre, “Theoretical study of anomalies of coated dielectric gratings,” Opt. Acta 32, 607–629 (1986).
    [CrossRef]
  8. S. S. Wang, R. Magnusson, J. S. Bagby, and M. G. Moharam, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A 7, 1470–1474 (1990).
    [CrossRef]
  9. M. Nevière, E. Popov, and R. Reinisch, “Electromagnetic resonances in linear and nonlinear optics: phenomenological study of grating behaviour through the poles and zeros of the scattering operator,” J. Opt. Soc. Am. A 12, 513–523 (1995).
    [CrossRef]
  10. M. G. Moharam, E. B. Grann, D. A. Pommet, and 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]
  11. S. M. Norton, T. Erdogan, and G. M. Morris, “Coupled-mode theory of resonant-grating filters,” J. Opt. Soc. Am. A 14, 629–639 (1996).
    [CrossRef]
  12. T. Tamir and S. Zhang, “Resonant scattering by multilayered dielectric gratings,” J. Opt. Soc. Am. A 14, 1607–1616 (1997).
    [CrossRef]
  13. R. R. Boye, R. W. Ziolkowski, and R. K. Kostuk, “Resonant waveguide-grating switching device with nonlinear optical material,” Appl. Opt. 38, 5181–5185 (1999).
    [CrossRef]
  14. A. Sharon, D. Rosenblatt, A. A. Friesem, H. G. Weber, H. Engel, and R. Steingrueber, “Light modulation with resonant grating-waveguide structures,” Opt. Lett. 21, 1564–1566 (1996).
    [CrossRef] [PubMed]
  15. R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
    [CrossRef]
  16. S. Tibuleac and R. Magnusson, “Reflection and transmission guided-mode resonance filters,” J. Opt. Soc. Am. A 14, 1617–1626 (1997).
    [CrossRef]
  17. S. M. Norton, G. M. Morris, and T. Erdogan, “Experimental investigation of resonant-grating filter line shapes in comparison with theoretical models,” J. Opt. Soc. Am. A 15, 464–472 (1998).
    [CrossRef]
  18. D. L. Brundrett, E. N. Glytsis, and T. K. Gaylord, “Normal-incidence guided-mode resonant grating filters: design and experimental demonstration,” Opt. Lett. 23, 700–702 (1998).
    [CrossRef]
  19. F. Lemarchand, A. Sentenac, and H. Giovannini, “Increasing the angular tolerance of resonant grating filters with doubly periodic structures,” Opt. Lett. 23, 1149–1151 (1998).
    [CrossRef]
  20. A. Mizutani, H. Kikuta, K. Iwata, and H. Toyota, “Guided-mode resonant grating filters with an antireflection structured surface,” J. Opt. Soc. Am. A 19, 1346–1351 (2002).
    [CrossRef]
  21. S. Pereira, J. E. Sipe, M. A. Bader, S. Soria, and G. Marowsky, “Loss-tolerant narrow-band reflector in the UV using a grating-waveguide structure,” Appl. Phys. B 75, 1–6 (2002).
    [CrossRef]
  22. D. Neuschäfer, W. Budach, C. Wanke, and S.-D. Chibout, “Evanscent resonator chips: a universal platform with superior sensitivity for fluorescence-based microarrays,” Biosens. Bioelectron. 18, 489–497 (2003).
    [CrossRef]
  23. S. Peng and G. M. Morris, “Resonant scattering from two-dimensional gratings,” J. Opt. Soc. Am. A 13, 993–1005 (1996).
    [CrossRef]
  24. A. Mizutani, H. Kikuta, K. Nakajima, and K. Iwata, “Nonpolarizing guided-mode resonant grating filter for oblique incidence,” J. Opt. Soc. Am. A 18, 1261–1266 (2001).
    [CrossRef]
  25. L. Li, “Analysis of planar waveguide grating coupler with double surface corrugations of identical period,” Opt. Commun. 114, 406–412 (1995).
    [CrossRef]
  26. O. Parriaux, V. A. Sychugov, and A. Tishchenko, “Coupling gratings as waveguide functional elements,” Pure Appl. Opt. 5, 453–469 (1996).
    [CrossRef]
  27. A. Sharon, S. Glasberg, D. Rosenblatt, and A. A. Friesem, “Metal-based resonant grating waveguide structures,” J. Opt. Soc. Am. A 14, 588–595 (1997).
    [CrossRef]
  28. A. Sharon, D. Rosenblatt, and A. A. Friesem, “Resonant grating-waveguide structures for visible and near-infrared radiation,” J. Opt. Soc. Am. A 14, 2985–2993 (1997).
    [CrossRef]
  29. D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1999).
    [CrossRef]
  30. S. Glasberg, A. Sharon, D. Rosenblatt, and A. A. Friesem, “Spectral shifts and line-shape symmetries in the resonant response of grating waveguide structures,” Opt. Commun. 145, 291–299 (1998).
    [CrossRef]
  31. D. K. Jacob, S. C. Dunn, and M. G. Moharam, “Design considerations for narrow-band dielectric resonant grating reflection filters of finite length,” J. Opt. Soc. Am. A 17, 1241–1249 (2000).
    [CrossRef]
  32. D. K. Jacob, S. C. Dunn, and M. G. Moharam, “Normally incident resonant grating reflection filters for efficient narrow-band spectral filtering of finite beams,” J. Opt. Soc. Am. A 18, 2109–2120 (2001).
    [CrossRef]
  33. D. K. Jacob, S. C. Dunn, and M. G. Moharam, “Flat-top narrow-band spectral response obtained from cascaded resonant grating reflection filters,” Appl. Opt. 41, 1241–1245 (2002).
    [CrossRef] [PubMed]
  34. D. K. Jacob, S. C. Dunn, and M. G. Moharam, “Interference approach applied to dual-grating dielectric resonant grating reflection filters,” Opt. Lett. 26, 1749–1751 (2001).
    [CrossRef]
  35. M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, 2002).

2003 (1)

D. Neuschäfer, W. Budach, C. Wanke, and S.-D. Chibout, “Evanscent resonator chips: a universal platform with superior sensitivity for fluorescence-based microarrays,” Biosens. Bioelectron. 18, 489–497 (2003).
[CrossRef]

2002 (3)

2001 (3)

2000 (1)

1999 (2)

D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1999).
[CrossRef]

R. R. Boye, R. W. Ziolkowski, and R. K. Kostuk, “Resonant waveguide-grating switching device with nonlinear optical material,” Appl. Opt. 38, 5181–5185 (1999).
[CrossRef]

1998 (4)

1997 (4)

1996 (4)

1995 (3)

1992 (1)

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

1990 (1)

1986 (1)

E. Popov, L. Mashev, and D. Maystre, “Theoretical study of anomalies of coated dielectric gratings,” Opt. Acta 32, 607–629 (1986).
[CrossRef]

1985 (1)

G. A. Golubenko, A. S. Svakhin, V. A. Sychugiov, and A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15, 886–887 (1985).
[CrossRef]

1982 (1)

V. A. Sychugov and A. V. Tishchenko, “Propagation and conversion of light waves in corrugated waveguide structures,” Sov. J. Quantum Electron. 12, 923–926 (1982).
[CrossRef]

1965 (1)

1952 (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]

Bader, M. A.

S. Pereira, J. E. Sipe, M. A. Bader, S. Soria, and G. Marowsky, “Loss-tolerant narrow-band reflector in the UV using a grating-waveguide structure,” Appl. Phys. B 75, 1–6 (2002).
[CrossRef]

Bagby, J. S.

Boye, R. R.

Brundrett, D. L.

Budach, W.

D. Neuschäfer, W. Budach, C. Wanke, and S.-D. Chibout, “Evanscent resonator chips: a universal platform with superior sensitivity for fluorescence-based microarrays,” Biosens. Bioelectron. 18, 489–497 (2003).
[CrossRef]

Chibout, S.-D.

D. Neuschäfer, W. Budach, C. Wanke, and S.-D. Chibout, “Evanscent resonator chips: a universal platform with superior sensitivity for fluorescence-based microarrays,” Biosens. Bioelectron. 18, 489–497 (2003).
[CrossRef]

Dunn, S. C.

Engel, H.

Erdogan, T.

Friesem, A. A.

Gaylord, T. K.

Giovannini, H.

Glasberg, S.

S. Glasberg, A. Sharon, D. Rosenblatt, and A. A. Friesem, “Spectral shifts and line-shape symmetries in the resonant response of grating waveguide structures,” Opt. Commun. 145, 291–299 (1998).
[CrossRef]

A. Sharon, S. Glasberg, D. Rosenblatt, and A. A. Friesem, “Metal-based resonant grating waveguide structures,” J. Opt. Soc. Am. A 14, 588–595 (1997).
[CrossRef]

Glytsis, E. N.

Golubenko, G. A.

G. A. Golubenko, A. S. Svakhin, V. A. Sychugiov, and A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15, 886–887 (1985).
[CrossRef]

Grann, E. B.

Hessel, A.

Iwata, K.

Jacob, D. K.

Kikuta, H.

Kostuk, R. K.

Lemarchand, F.

Li, L.

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

Magnusson, R.

Marowsky, G.

S. Pereira, J. E. Sipe, M. A. Bader, S. Soria, and G. Marowsky, “Loss-tolerant narrow-band reflector in the UV using a grating-waveguide structure,” Appl. Phys. B 75, 1–6 (2002).
[CrossRef]

Mashev, L.

E. Popov, L. Mashev, and D. Maystre, “Theoretical study of anomalies of coated dielectric gratings,” Opt. Acta 32, 607–629 (1986).
[CrossRef]

Maystre, D.

E. Popov, L. Mashev, and D. Maystre, “Theoretical study of anomalies of coated dielectric gratings,” Opt. Acta 32, 607–629 (1986).
[CrossRef]

Mizutani, A.

Moharam, M. G.

Morris, G. M.

Nakajima, K.

Neuschäfer, D.

D. Neuschäfer, W. Budach, C. Wanke, and S.-D. Chibout, “Evanscent resonator chips: a universal platform with superior sensitivity for fluorescence-based microarrays,” Biosens. Bioelectron. 18, 489–497 (2003).
[CrossRef]

Nevière, M.

Norton, S. M.

Oliner, A. A.

Palmer, C. H.

Parriaux, O.

O. Parriaux, V. A. Sychugov, and A. Tishchenko, “Coupling gratings as waveguide functional elements,” Pure Appl. Opt. 5, 453–469 (1996).
[CrossRef]

Peng, S.

Pereira, S.

S. Pereira, J. E. Sipe, M. A. Bader, S. Soria, and G. Marowsky, “Loss-tolerant narrow-band reflector in the UV using a grating-waveguide structure,” Appl. Phys. B 75, 1–6 (2002).
[CrossRef]

Pommet, D. A.

Popov, E.

Reinisch, R.

Rosenblatt, D.

Sentenac, A.

Sharon, A.

Sipe, J. E.

S. Pereira, J. E. Sipe, M. A. Bader, S. Soria, and G. Marowsky, “Loss-tolerant narrow-band reflector in the UV using a grating-waveguide structure,” Appl. Phys. B 75, 1–6 (2002).
[CrossRef]

Soria, S.

S. Pereira, J. E. Sipe, M. A. Bader, S. Soria, and G. Marowsky, “Loss-tolerant narrow-band reflector in the UV using a grating-waveguide structure,” Appl. Phys. B 75, 1–6 (2002).
[CrossRef]

Steingrueber, R.

Svakhin, A. S.

G. A. Golubenko, A. S. Svakhin, V. A. Sychugiov, and A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15, 886–887 (1985).
[CrossRef]

Sychugiov, V. A.

G. A. Golubenko, A. S. Svakhin, V. A. Sychugiov, and A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15, 886–887 (1985).
[CrossRef]

Sychugov, V. A.

O. Parriaux, V. A. Sychugov, and A. Tishchenko, “Coupling gratings as waveguide functional elements,” Pure Appl. Opt. 5, 453–469 (1996).
[CrossRef]

V. A. Sychugov and A. V. Tishchenko, “Propagation and conversion of light waves in corrugated waveguide structures,” Sov. J. Quantum Electron. 12, 923–926 (1982).
[CrossRef]

Tamir, T.

Tibuleac, S.

Tishchenko, A.

O. Parriaux, V. A. Sychugov, and A. Tishchenko, “Coupling gratings as waveguide functional elements,” Pure Appl. Opt. 5, 453–469 (1996).
[CrossRef]

Tishchenko, A. V.

G. A. Golubenko, A. S. Svakhin, V. A. Sychugiov, and A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15, 886–887 (1985).
[CrossRef]

V. A. Sychugov and A. V. Tishchenko, “Propagation and conversion of light waves in corrugated waveguide structures,” Sov. J. Quantum Electron. 12, 923–926 (1982).
[CrossRef]

Toyota, H.

Wang, S. S.

Wanke, C.

D. Neuschäfer, W. Budach, C. Wanke, and S.-D. Chibout, “Evanscent resonator chips: a universal platform with superior sensitivity for fluorescence-based microarrays,” Biosens. Bioelectron. 18, 489–497 (2003).
[CrossRef]

Weber, H. G.

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.

Ziolkowski, R. W.

Appl. Opt. (3)

Appl. Phys. B (1)

S. Pereira, J. E. Sipe, M. A. Bader, S. Soria, and G. Marowsky, “Loss-tolerant narrow-band reflector in the UV using a grating-waveguide structure,” Appl. Phys. B 75, 1–6 (2002).
[CrossRef]

Appl. Phys. Lett. (1)

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

Biosens. Bioelectron. (1)

D. Neuschäfer, W. Budach, C. Wanke, and S.-D. Chibout, “Evanscent resonator chips: a universal platform with superior sensitivity for fluorescence-based microarrays,” Biosens. Bioelectron. 18, 489–497 (2003).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33, 2038–2059 (1999).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (14)

A. Mizutani, H. Kikuta, K. Nakajima, and K. Iwata, “Nonpolarizing guided-mode resonant grating filter for oblique incidence,” J. Opt. Soc. Am. A 18, 1261–1266 (2001).
[CrossRef]

D. K. Jacob, S. C. Dunn, and M. G. Moharam, “Normally incident resonant grating reflection filters for efficient narrow-band spectral filtering of finite beams,” J. Opt. Soc. Am. A 18, 2109–2120 (2001).
[CrossRef]

A. Mizutani, H. Kikuta, K. Iwata, and H. Toyota, “Guided-mode resonant grating filters with an antireflection structured surface,” J. Opt. Soc. Am. A 19, 1346–1351 (2002).
[CrossRef]

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

M. Nevière, E. Popov, and R. Reinisch, “Electromagnetic resonances in linear and nonlinear optics: phenomenological study of grating behaviour through the poles and zeros of the scattering operator,” J. Opt. Soc. Am. A 12, 513–523 (1995).
[CrossRef]

M. G. Moharam, E. B. Grann, D. A. Pommet, and 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]

D. K. Jacob, S. C. Dunn, and M. G. Moharam, “Design considerations for narrow-band dielectric resonant grating reflection filters of finite length,” J. Opt. Soc. Am. A 17, 1241–1249 (2000).
[CrossRef]

S. M. Norton, G. M. Morris, and T. Erdogan, “Experimental investigation of resonant-grating filter line shapes in comparison with theoretical models,” J. Opt. Soc. Am. A 15, 464–472 (1998).
[CrossRef]

A. Sharon, S. Glasberg, D. Rosenblatt, and A. A. Friesem, “Metal-based resonant grating waveguide structures,” J. Opt. Soc. Am. A 14, 588–595 (1997).
[CrossRef]

S. M. Norton, T. Erdogan, and G. M. Morris, “Coupled-mode theory of resonant-grating filters,” J. Opt. Soc. Am. A 14, 629–639 (1996).
[CrossRef]

T. Tamir and S. Zhang, “Resonant scattering by multilayered dielectric gratings,” J. Opt. Soc. Am. A 14, 1607–1616 (1997).
[CrossRef]

S. Tibuleac and R. Magnusson, “Reflection and transmission guided-mode resonance filters,” J. Opt. Soc. Am. A 14, 1617–1626 (1997).
[CrossRef]

A. Sharon, D. Rosenblatt, and A. A. Friesem, “Resonant grating-waveguide structures for visible and near-infrared radiation,” J. Opt. Soc. Am. A 14, 2985–2993 (1997).
[CrossRef]

S. S. Wang, R. Magnusson, J. S. Bagby, and M. G. Moharam, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A 7, 1470–1474 (1990).
[CrossRef]

Opt. Acta (1)

E. Popov, L. Mashev, and D. Maystre, “Theoretical study of anomalies of coated dielectric gratings,” Opt. Acta 32, 607–629 (1986).
[CrossRef]

Opt. Commun. (2)

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

S. Glasberg, A. Sharon, D. Rosenblatt, and A. A. Friesem, “Spectral shifts and line-shape symmetries in the resonant response of grating waveguide structures,” Opt. Commun. 145, 291–299 (1998).
[CrossRef]

Opt. Lett. (4)

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]

Pure Appl. Opt. (1)

O. Parriaux, V. A. Sychugov, and A. Tishchenko, “Coupling gratings as waveguide functional elements,” Pure Appl. Opt. 5, 453–469 (1996).
[CrossRef]

Sov. J. Quantum Electron. (2)

V. A. Sychugov and A. V. Tishchenko, “Propagation and conversion of light waves in corrugated waveguide structures,” Sov. J. Quantum Electron. 12, 923–926 (1982).
[CrossRef]

G. A. Golubenko, A. S. Svakhin, V. A. Sychugiov, and A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15, 886–887 (1985).
[CrossRef]

Other (2)

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, 2002).

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

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Figures (8)

Fig. 1
Fig. 1

Propagating waves and multiple interference in a single GWS: (a) grating on top of the waveguide layer, (b) grating under the waveguide layer.

Fig. 2
Fig. 2

Propagating waves and multiple-interference in a DGWS (see text).

Fig. 3
Fig. 3

TE transmission for 800 nm and 34° incident angle as a function of phase and waveguide layer thickness of the investigated DGWS. Resonances occur at waveguide layers of thickness h=134, 453, and 772 nm.

Fig. 4
Fig. 4

Model calculation of spectral and angular TE0 resonance of the investigated DGWS at 800 nm and 34° incident angle.

Fig. 5
Fig. 5

Spectral resonances of the investigated DGWS (Exp.) and the RCWA calculation at 34° incident angle for TE and at 44° incident angle for TM transmission.

Fig. 6
Fig. 6

Angular resonances of the investigated DGWS (Exp.) and the RCWA calculation at 800 nm for TE and TM transmission.

Fig. 7
Fig. 7

Spectral and angular resonance shifts of RCWA calculations as functions of grating depths with respect to model calculation.

Fig. 8
Fig. 8

Comparison of RCWA and the multiple-interference model for a DGWS with very small gratings: The difference between the exact numerical calculation and the model prediction is only 0.05 nm.

Equations (40)

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

ϕ=2kwh-2ϕsp-2ϕsb=M2π,
ϕsTE=arctan[nw2 cos2(ψ)-ns2]1/2nw sin(ψ),s=sp, sb
ϕsTM=arctannw2ns2 [nw2 cos2(ψ)-ns2]1/2nw sin(ψ),
s=sp, sb
nspk sin(θ)+mK=nwk cos(ψ),
ϕd=2kwh-2ϕsp-2ϕsb-π=ϕ-π.
S00=E00t/E0,S01=E01w/E0,
S10=E10t/E01w,S11=E11w/E01w,
|S11|=1-S,S|S01||S10|,
ET=E00t-S exp(iϕ)q=0[|S11|exp(iϕ)]qE0.
ETE0=1-S exp(iϕ)1-(1-S)exp(iϕ),
ER=E00r+S exp(iϕ)q=0[|S11|exp(iϕ)]qE0.
ERE0=S exp(iϕ)1-(1-S)exp(iϕ).
ET=E00t#-S# exp(iϕ)q=0[|S11#|exp(iϕ)]qE0,
ER=E00r#+S# exp(iϕ)q=0[|S11#|exp(iϕ)]qE0.
ETE0=1-S# exp(iϕ)1-(1-S#)exp(iϕ),
ERE0=S# exp(iϕ)1-(1-S#)exp(iϕ).
ET=S00S00#E0-S|S11#|exp(iϕ)q=0[|S11||S11#|exp(iϕ)]qE0-S#|S11|exp(iϕ)q=0[|S11#||S11|exp(iϕ)]qE0,
ETE0=S00S00#-(S|S11#|+S#|S11|)exp(iϕ)1-|S11||S11#|exp(iϕ),
ETE0=1-[S(1-S#)+S#(1-S)]exp(iϕ)1-(1-S)(1-S#)exp(iϕ).
ETE0=1-(S+S#)exp(iϕ)1-[1-(S+S#)]exp(iϕ).
ER=E00r+E00r#+S|S11#|exp(iϕ)q=0[|S11||S11#|exp(iϕ)]qE0+S#|S11|exp(iϕ)q=0[|S11#||S11|exp(iϕ)]qE0.
ERE0=(S+S#)exp(iϕ)1-[1-(S+S#)]exp(iϕ).
ITI0={1/[1-(S+S#)]}sin2(ϕ/2)(S+S#)2/4[1-(S+S#)]+sin2(ϕ/2).
F=4R/(1-R)2,
R=|S11||S11#|=(1-S)(1-S#)[1-(S+S#)],
ITI0=F sin2(ϕ/2)1+F sin2(ϕ/2).
IRI0=11+{4[1-(S+S#)]/(S+S#)2}sin2(ϕ/2).
IRI0=11+F sin2(ϕ/2)
ϕFP=k2L cos(θ),
n(λ/2)=L cos(θ).
h=ϕsp+ϕsbnwk sin(ψ)+λ2nw sin(ψ)M.
ΔϕFWHM=4 arcsin1-R2R2(1-R)R=4F.
ΔϕFWHM=dϕdλΔλFWHM=dϕdθΔθFWHM=,
ΔλFWHM=ΔϕFWHMdϕGWSdλ-12(1-R)R dϕGWSdλ-1,
ΔθFWHM=ΔϕFWHMdϕGWSdθ-12(1-R)R dϕGWSdθ-1.
|S01||S10|=S(Δk2ξ±1d)2,
|S01#||S10#|=S#(Δ#k2ξ±1d#)2,
ΔλFWHMDGWS2(S+S#)[1-(S+S#)]1/2 dϕGWSdλ-1ΔλFWHM+ΔλFWHM#,
ΔθFWHMDGWS2(S+S#)[1-(S+S#)]1/2 dϕGWSdθ-1ΔθFWHM+ΔθFWHM#.

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