Abstract

We propose a method for analyzing both theoretically and experimentally the behavior of the phase of the waves diffracted by gratings. The method is applied to the study of resonance phenomena. It is used for determining the optogeometrical parameters of a metallic grating. We show that the experimental setup, which is insensitive to mechanical drifts or thermal fluctuations, can be used for sensing purposes.

© 2001 Optical Society of America

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References

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  1. R. C. McPhedran, D. Maystre, “A detailed theoretical study of the anomalies of a sinusoidal diffraction grating,” Opt. Acta 21, 413–421 (1974).
    [CrossRef]
  2. M. C. Hutley, D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
    [CrossRef]
  3. C. Amra, S. Maure, “Mutual coherence and conical pattern of sources optimally excited within multilayer optics,” J. Opt. Soc. Am. A 14, 3114–3124 (1997).
    [CrossRef]
  4. F. Flory, “Guided wave techniques for the characterization of optical coatings,” in Thin Films for Optical Systems, F. Flory, ed., Vol. 49 of Optical Engineering Series (Marcel Dekker, New York, 1995), pp. 393–454.
  5. P. Vincent, M. Nevière, “Corrugated dielectric waveguides: a numerical study of the second-order stop bands,” Appl. Phys. 20, 345–351 (1979).
    [CrossRef]
  6. G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, A. V. Tischenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quant. Electron. 15, 886–887 (1985).
    [CrossRef]
  7. R. Magnusson, S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
    [CrossRef]
  8. V. Shah, T. Tamir, “Brewster phenomena in lossy structures,” Opt. Commun. 23, 113–117 (1977).
    [CrossRef]
  9. K. Matsubara, S. Kawata, S. Minami, “Optical chemical sensor based on surface plasmon measurement,” Appl. Opt. 27, 1160–1163 (1988).
    [CrossRef] [PubMed]
  10. M. Zhang, D. Uttamchandani, “Optical chemical sensing employing surface plasmon resonance,” Electron. Lett. 24, 1469–1470 (1988).
    [CrossRef]
  11. D. Maystre, “General study of grating anomalies from electromagnetic surface modes,” in Electromagnetic Surface Modes, A. D. Boardman, ed. (Wiley, New York, 1982), pp. 661–724.
  12. R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Phil. Mag. 4, 396–408 (1902).
    [CrossRef]
  13. M. Nevière, “The homogeneous problem,” in Topics in Current Physics, Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 123–157.
    [CrossRef]
  14. M. J. Jory, P. S. Vukusic, J. R. Sambles, “Development of a prototype gas sensor using surface plasmon resonance on gratings,” Sens. Actuators B 17, 203–209 (1994).
    [CrossRef]
  15. H. Giovannini, C. Deumié, H. Akhouayri, C. Amra, “Angle-resolved polarimetric phase measurement for the characterization of gratings,” Opt. Lett. 21, 1619–1621 (1996).
    [CrossRef] [PubMed]
  16. D. Maystre, M. Nevière, “Sur une méthode d’étude théorique quantitative des anomalies de Wood des réseaux de diffraction: application aux anomalies de plasmon,” J. Opt. (Paris) 8, 165–174 (1977).
    [CrossRef]
  17. P. Vincent, “Differential methods,” in Progress in Optics, E. Wolf, ed. (Springer-Verlag, Berlin, 1980), Vol. 22, pp. 101–121.
  18. F. Montiel, M. Nevière, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix algorithm,” J. Opt. Soc. Am. A 11, 3241–3250 (1994).
    [CrossRef]
  19. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996).
    [CrossRef]

1997 (1)

1996 (2)

1994 (2)

F. Montiel, M. Nevière, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix algorithm,” J. Opt. Soc. Am. A 11, 3241–3250 (1994).
[CrossRef]

M. J. Jory, P. S. Vukusic, J. R. Sambles, “Development of a prototype gas sensor using surface plasmon resonance on gratings,” Sens. Actuators B 17, 203–209 (1994).
[CrossRef]

1992 (1)

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

1988 (2)

K. Matsubara, S. Kawata, S. Minami, “Optical chemical sensor based on surface plasmon measurement,” Appl. Opt. 27, 1160–1163 (1988).
[CrossRef] [PubMed]

M. Zhang, D. Uttamchandani, “Optical chemical sensing employing surface plasmon resonance,” Electron. Lett. 24, 1469–1470 (1988).
[CrossRef]

1985 (1)

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

1979 (1)

P. Vincent, M. Nevière, “Corrugated dielectric waveguides: a numerical study of the second-order stop bands,” Appl. Phys. 20, 345–351 (1979).
[CrossRef]

1977 (2)

V. Shah, T. Tamir, “Brewster phenomena in lossy structures,” Opt. Commun. 23, 113–117 (1977).
[CrossRef]

D. Maystre, M. Nevière, “Sur une méthode d’étude théorique quantitative des anomalies de Wood des réseaux de diffraction: application aux anomalies de plasmon,” J. Opt. (Paris) 8, 165–174 (1977).
[CrossRef]

1976 (1)

M. C. Hutley, D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
[CrossRef]

1974 (1)

R. C. McPhedran, D. Maystre, “A detailed theoretical study of the anomalies of a sinusoidal diffraction grating,” Opt. Acta 21, 413–421 (1974).
[CrossRef]

1902 (1)

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Phil. Mag. 4, 396–408 (1902).
[CrossRef]

Akhouayri, H.

Amra, C.

Deumié, C.

Flory, F.

F. Flory, “Guided wave techniques for the characterization of optical coatings,” in Thin Films for Optical Systems, F. Flory, ed., Vol. 49 of Optical Engineering Series (Marcel Dekker, New York, 1995), pp. 393–454.

Giovannini, H.

Golubenko, G. A.

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

Hutley, M. C.

M. C. Hutley, D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
[CrossRef]

Jory, M. J.

M. J. Jory, P. S. Vukusic, J. R. Sambles, “Development of a prototype gas sensor using surface plasmon resonance on gratings,” Sens. Actuators B 17, 203–209 (1994).
[CrossRef]

Kawata, S.

Li, L.

Magnusson, R.

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

Matsubara, K.

Maure, S.

Maystre, D.

D. Maystre, M. Nevière, “Sur une méthode d’étude théorique quantitative des anomalies de Wood des réseaux de diffraction: application aux anomalies de plasmon,” J. Opt. (Paris) 8, 165–174 (1977).
[CrossRef]

M. C. Hutley, D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
[CrossRef]

R. C. McPhedran, D. Maystre, “A detailed theoretical study of the anomalies of a sinusoidal diffraction grating,” Opt. Acta 21, 413–421 (1974).
[CrossRef]

D. Maystre, “General study of grating anomalies from electromagnetic surface modes,” in Electromagnetic Surface Modes, A. D. Boardman, ed. (Wiley, New York, 1982), pp. 661–724.

McPhedran, R. C.

R. C. McPhedran, D. Maystre, “A detailed theoretical study of the anomalies of a sinusoidal diffraction grating,” Opt. Acta 21, 413–421 (1974).
[CrossRef]

Minami, S.

Montiel, F.

Nevière, M.

F. Montiel, M. Nevière, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix algorithm,” J. Opt. Soc. Am. A 11, 3241–3250 (1994).
[CrossRef]

P. Vincent, M. Nevière, “Corrugated dielectric waveguides: a numerical study of the second-order stop bands,” Appl. Phys. 20, 345–351 (1979).
[CrossRef]

D. Maystre, M. Nevière, “Sur une méthode d’étude théorique quantitative des anomalies de Wood des réseaux de diffraction: application aux anomalies de plasmon,” J. Opt. (Paris) 8, 165–174 (1977).
[CrossRef]

M. Nevière, “The homogeneous problem,” in Topics in Current Physics, Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 123–157.
[CrossRef]

Sambles, J. R.

M. J. Jory, P. S. Vukusic, J. R. Sambles, “Development of a prototype gas sensor using surface plasmon resonance on gratings,” Sens. Actuators B 17, 203–209 (1994).
[CrossRef]

Shah, V.

V. Shah, T. Tamir, “Brewster phenomena in lossy structures,” Opt. Commun. 23, 113–117 (1977).
[CrossRef]

Svakhin, A. S.

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

Sychugov, V. A.

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

Tamir, T.

V. Shah, T. Tamir, “Brewster phenomena in lossy structures,” Opt. Commun. 23, 113–117 (1977).
[CrossRef]

Tischenko, A. V.

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

Uttamchandani, D.

M. Zhang, D. Uttamchandani, “Optical chemical sensing employing surface plasmon resonance,” Electron. Lett. 24, 1469–1470 (1988).
[CrossRef]

Vincent, P.

P. Vincent, M. Nevière, “Corrugated dielectric waveguides: a numerical study of the second-order stop bands,” Appl. Phys. 20, 345–351 (1979).
[CrossRef]

P. Vincent, “Differential methods,” in Progress in Optics, E. Wolf, ed. (Springer-Verlag, Berlin, 1980), Vol. 22, pp. 101–121.

Vukusic, P. S.

M. J. Jory, P. S. Vukusic, J. R. Sambles, “Development of a prototype gas sensor using surface plasmon resonance on gratings,” Sens. Actuators B 17, 203–209 (1994).
[CrossRef]

Wang, S. S.

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

Wood, R. W.

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Phil. Mag. 4, 396–408 (1902).
[CrossRef]

Zhang, M.

M. Zhang, D. Uttamchandani, “Optical chemical sensing employing surface plasmon resonance,” Electron. Lett. 24, 1469–1470 (1988).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. (1)

P. Vincent, M. Nevière, “Corrugated dielectric waveguides: a numerical study of the second-order stop bands,” Appl. Phys. 20, 345–351 (1979).
[CrossRef]

Appl. Phys. Lett. (1)

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

Electron. Lett. (1)

M. Zhang, D. Uttamchandani, “Optical chemical sensing employing surface plasmon resonance,” Electron. Lett. 24, 1469–1470 (1988).
[CrossRef]

J. Opt. (Paris) (1)

D. Maystre, M. Nevière, “Sur une méthode d’étude théorique quantitative des anomalies de Wood des réseaux de diffraction: application aux anomalies de plasmon,” J. Opt. (Paris) 8, 165–174 (1977).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Acta (1)

R. C. McPhedran, D. Maystre, “A detailed theoretical study of the anomalies of a sinusoidal diffraction grating,” Opt. Acta 21, 413–421 (1974).
[CrossRef]

Opt. Commun. (2)

M. C. Hutley, D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. 19, 431–436 (1976).
[CrossRef]

V. Shah, T. Tamir, “Brewster phenomena in lossy structures,” Opt. Commun. 23, 113–117 (1977).
[CrossRef]

Opt. Lett. (1)

Phil. Mag. (1)

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Phil. Mag. 4, 396–408 (1902).
[CrossRef]

Sens. Actuators B (1)

M. J. Jory, P. S. Vukusic, J. R. Sambles, “Development of a prototype gas sensor using surface plasmon resonance on gratings,” Sens. Actuators B 17, 203–209 (1994).
[CrossRef]

Sov. J. Quant. Electron. (1)

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

Other (4)

F. Flory, “Guided wave techniques for the characterization of optical coatings,” in Thin Films for Optical Systems, F. Flory, ed., Vol. 49 of Optical Engineering Series (Marcel Dekker, New York, 1995), pp. 393–454.

P. Vincent, “Differential methods,” in Progress in Optics, E. Wolf, ed. (Springer-Verlag, Berlin, 1980), Vol. 22, pp. 101–121.

M. Nevière, “The homogeneous problem,” in Topics in Current Physics, Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 123–157.
[CrossRef]

D. Maystre, “General study of grating anomalies from electromagnetic surface modes,” in Electromagnetic Surface Modes, A. D. Boardman, ed. (Wiley, New York, 1982), pp. 661–724.

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Figures (9)

Fig. 1
Fig. 1

Schematic representation of incident beams 1 and 2 and of the diffracted orders. θ1 and θ2 are the angles of incidence of beams 1 and 2, respectively.

Fig. 2
Fig. 2

Schematic of the experimental setup. The two detectors, the grating, and the optical fiber can be rotated independently around a vertical axis. The length of the optical path delay may be adjusted. The choice of an appropriate optical density allows one to maximize the amplitude modulation to continuous signal ratio. BS, beam splitter; EOPM, electro-optic lithium niobate phase operator.

Fig. 3
Fig. 3

Diffracted orders for different values of the angle of incidence. Grating period d = 1.082 µm and wavelength λ = 0.633 µm. Shaded areas correspond to the variation range of i 1 and i 2 around a resonance. Measurements are made on the highlighted orders.

Fig. 4
Fig. 4

Measured efficiencies of the orders involved in the determination of differential phase ψ. The measurements are made in the vicinity of two resonances. Incidence angles θ1 and θ2 are linked by Eq. (9). (a) Efficiencies of diffracted orders 1 and 2 as a function of θ1 (incident beam 1). (b) Efficiencies of diffracted orders 0 and 1 as a function of θ2 (incident beam 2).

Fig. 5
Fig. 5

Measured phase difference ψ as a function of θ1.

Fig. 6
Fig. 6

Comparison between experimental results (plain solid curve) and numerical results (marked curve). (a) Efficiencies of orders 1 and 2 as a function of θ1 (incident beam 1). (b) Efficiencies of orders 0 and 1 as a function of θ2 (incident beam 2). (c) Phase difference ψ as a function of θ1.

Fig. 7
Fig. 7

Comparison between experimental results (plain solid curve) and numerical results (marked curve) in the vicinity of the resonance. The efficiencies are calculated from Eq. (3). For the calculation the values of the parameters of Table 1 have been taken. (a) Efficiency of order 1 produced by beam 1 as a function of θ1 (incident beam 1). (b) Efficiency of order 2 as a function of θ1 (incident beam 1). (c) Efficiency of order 0 produced by beam 2 as a function of θ2 (incident beam 2). (d) Efficiency of order 1 produced by beam 2 as a function of θ2 (incident beam 2).

Fig. 8
Fig. 8

Comparison between measured phase ψ (plain solid curve) and phase ψ calculated from Eqs. (2) and (7) (marked curve). For the calculation the values of the parameters of Table 1 have been taken.

Fig. 9
Fig. 9

Sensitivity to the thickness of the region located near the grating’s surface. Numerical results obtained for h = 0.11 µm, ν = 1.23 + i9.6, ν ox = 1.63, 924 grooves/mm, and h ox = 13 nm (plain solid curve) or h ox = 15 nm (marked curve). (a) Efficiencies of orders 1 and 2 as a function of θ1 (incident beam 1). (b) Efficiencies of orders 0 and 1 as a function of θ2 (incident beam 2). (c) Differential phase ψ as a function of θ1.

Tables (1)

Tables Icon

Table 1 Values of the Complex Pole and Zero and of the Square Modulus of Constant C for Each Order Involved in the Calculation of Differential Phase ψa

Equations (12)

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

B0α0rα0α0-αzα0-αp,
argB0α0=arc tan-αZα0-αZ-arc tan-αPα0-αP+argrα0.
B=SA,
Sn,mα0=Cn,mα0-αn,mZα0-αP,
Ia=|B2,1α0+B1,2α0|2,
Ia=I2,1+I1,2+2I2,1I1,21/2cos ψa,
ψa=2πΔλ+argB2,1-argB1,2.
ψθ1=±ψaθ1, θ2-ψbθ1, θ2=±(argB2,1θ1-argB1,2θ2-argB1,1θ1-argB0,2θ2)
θ1L=arc sin±1-n λd.
sin θ2=sin θ1+k λd,
arc sinsin θ1min+λd; arc sinsin θ1max+λd.
k=-n2-1+2n 1+R1-R1/2.

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