Abstract

The role of the excitation of guided waves propagating in a corrugated dielectric waveguide is discussed in view of the resonance anomalies in reflectivity. Narrow-wavelength filtering properties that are due to these sharp anomalies have been a topic of interest for some time, but a proper understanding of device performances requires an analysis of tolerances with respect to the incident-beam collimation and to waveguide losses. Such an analysis is proposed in this paper, and the conclusion is that the incident-beam divergence plays a critical role in reducing the maximum reflectivity for narrow-band filters.

© 2001 Optical Society of America

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References

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  1. R. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–402 (1902).
    [CrossRef]
  2. U. Fano, “The theory of anomalous diffraction gratings and of quasi-stationary waves on metallic surfaces (Sommerfeld’s waves),” J. Opt. Soc. Am. A 31, 213–222 (1941).
    [CrossRef]
  3. A. Hessel, A. A. Oliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. 4, 1275–1297 (1965).
    [CrossRef]
  4. M. Neviere, “The homogeneous problem,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 5.
  5. E. Popov, “Light diffraction by relief gratings: a microscopic and macroscopic view,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1993), Vol. XXXI, pp. 139–187.
  6. M. Neviere, E. Popov, R. Reinisch, “Electromagnetic resonances in linear and nonlinear optics: phenomenological study of grating behavior through the poles and the zeros of the scattering operator,” J. Opt. Soc. Am. A 12, 513–523 (1995).
    [CrossRef]
  7. T. Tamir, S. Zhang, “Resonant scattering by multilayered dielectric gratings,” J. Opt. Soc. Am. A 14, 1607–1616 (1997).
    [CrossRef]
  8. S. M. Norton, G. M. Morris, 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]
  9. L. Mashev, E. Popov, “Zero order anomaly of a dielectric coated grating,” Opt. Commun. 55, 377–380 (1985).
    [CrossRef]
  10. R. Magnusson, D. Shin, Z. S. Liu, “Guided-mode resonance Brewster filter,” Opt. Lett. 23, 612–614 (1998).
    [CrossRef]
  11. E. Popov, L. Mashev, D. Maystre, “Theoretical study of anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
    [CrossRef]
  12. L. Mashev, E. Popov, “Diffraction efficiency anomalies of multicoated dielectric gratings,” Opt. Commun. 51, 131–136 (1984).
    [CrossRef]

1998 (2)

1997 (1)

1995 (1)

1986 (1)

E. Popov, L. Mashev, D. Maystre, “Theoretical study of anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

1985 (1)

L. Mashev, E. Popov, “Zero order anomaly of a dielectric coated grating,” Opt. Commun. 55, 377–380 (1985).
[CrossRef]

1984 (1)

L. Mashev, E. Popov, “Diffraction efficiency anomalies of multicoated dielectric gratings,” Opt. Commun. 51, 131–136 (1984).
[CrossRef]

1965 (1)

1941 (1)

U. Fano, “The theory of anomalous diffraction gratings and of quasi-stationary waves on metallic surfaces (Sommerfeld’s waves),” J. Opt. Soc. Am. A 31, 213–222 (1941).
[CrossRef]

1902 (1)

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

Erdogan, T.

Fano, U.

U. Fano, “The theory of anomalous diffraction gratings and of quasi-stationary waves on metallic surfaces (Sommerfeld’s waves),” J. Opt. Soc. Am. A 31, 213–222 (1941).
[CrossRef]

Hessel, A.

Liu, Z. S.

Magnusson, R.

Mashev, L.

E. Popov, L. Mashev, D. Maystre, “Theoretical study of anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

L. Mashev, E. Popov, “Zero order anomaly of a dielectric coated grating,” Opt. Commun. 55, 377–380 (1985).
[CrossRef]

L. Mashev, E. Popov, “Diffraction efficiency anomalies of multicoated dielectric gratings,” Opt. Commun. 51, 131–136 (1984).
[CrossRef]

Maystre, D.

E. Popov, L. Mashev, D. Maystre, “Theoretical study of anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

Morris, G. M.

Neviere, M.

Norton, S. M.

Oliner, A. A.

Popov, E.

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

E. Popov, L. Mashev, D. Maystre, “Theoretical study of anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

L. Mashev, E. Popov, “Zero order anomaly of a dielectric coated grating,” Opt. Commun. 55, 377–380 (1985).
[CrossRef]

L. Mashev, E. Popov, “Diffraction efficiency anomalies of multicoated dielectric gratings,” Opt. Commun. 51, 131–136 (1984).
[CrossRef]

E. Popov, “Light diffraction by relief gratings: a microscopic and macroscopic view,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1993), Vol. XXXI, pp. 139–187.

Reinisch, R.

Shin, D.

Tamir, T.

Wood, R.

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

Zhang, S.

Appl. Opt. (1)

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

Opt. Acta (1)

E. Popov, L. Mashev, D. Maystre, “Theoretical study of anomalies of coated dielectric gratings,” Opt. Acta 33, 607–619 (1986).
[CrossRef]

Opt. Commun. (2)

L. Mashev, E. Popov, “Diffraction efficiency anomalies of multicoated dielectric gratings,” Opt. Commun. 51, 131–136 (1984).
[CrossRef]

L. Mashev, E. Popov, “Zero order anomaly of a dielectric coated grating,” Opt. Commun. 55, 377–380 (1985).
[CrossRef]

Opt. Lett. (1)

Philos. Mag. (1)

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

Other (2)

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

E. Popov, “Light diffraction by relief gratings: a microscopic and macroscopic view,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1993), Vol. XXXI, pp. 139–187.

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Figures (10)

Fig. 1
Fig. 1

Schematic representations of different types of corrugated waveguides together with some notation used in the text: d is the groove period, h is the total groove depth, and t is the dielectric-layer thickness. (a) The flat lower interface and the symmetrically corrugated upper interface. (b) Identical corrugations of the two interfaces of (a). The corrugations have symmetry with respect to a vertical plane. When n1=n3 the waveguide has an axis of symmetry. (c) The corrugations of (b) but with a horizontal shift of d/2; the waveguide has a vertical plane of symmetry. When n1=n3 the waveguide has a horizontal plane of symmetry. (d) A horizontal shift between identical corrugations of d/4; neither type of symmetry exists. (e) An echelette grating as the upper interface; neither type of symmetry exists.

Fig. 2
Fig. 2

Spectral dependence of the reflectivity of three corrugated waveguides with identical parameters but with different corrugations. Parameters: TE polarization, d=0.303 μm, t=0.7 μm, h=0.12 μm, n1=1.5115, n2=1.542, n3=1, and θi=34.785°. The solid curve with markers represents the waveguide shown in Fig. 1(a); the dashed curve corresponds to the geometry presented in Fig. 1(c); the plain solid curve represents the grating shown in Fig. 1(d).

Fig. 3
Fig. 3

Spectral dependence of the reflectivity of a corrugated waveguide with a plane as the lower boundary and an echelette as the upper boundary [Fig. 1(e)]. Parameters: TE polarization, d=0.303 μm, t=0.65 μm, h=0.12 μm, φB=25°, n1=1.5115, n2=1.542, n3=1, and θi=34.785°.

Fig. 4
Fig. 4

Spectral dependence of the reflectivity of a multilayer dielectric mirror that consists of 17 layers of dielectrics with alternating higher (n=1.7) and lower (n=1.5) refractive indices. Each layer has a λ/4 thickness at λ=0.63 μm. There is normal incidence and TE polarization. The thick solid curve represents a plane as the upper interface; the thin solid curve represents an upper interface that has a sinusoidal modulation with a period of d=0.4117 μm and a modulation depth of h/d=0.2; the dashed curve represents a case with a 2× modulation rate of h/d=0.4. Shown are (a) the general view, (b) the zoom around the region λ=0.645 μm, and (c) the zoom around λ=0.655 μm.

Fig. 5
Fig. 5

Spectral dependence of the reflectivity of the corrugated waveguide shown in Fig. 1(b) with TE polarization, d=0.303 μm, t=0.7 μm, h=0.12 μm, n1=1.5115, n2=1.542+iγ, n3=1, and θi=34.785°. The three curves are calculated for different losses in the middle guiding layer that correspond to different values of the extinction coefficient γ. The thick solid curve represents γ=5×10-6; the thin solid curve represents γ=10-5; the dashed curve represents γ=5×10-5.

Fig. 6
Fig. 6

Spectral dependence of the reflectivity of a lossless waveguide with the parameters of Fig. 5 for three different angles of incidence: θi=34.806° (triangles), θi=34.785° (plain curve), and θi=34.771° (squares).

Fig. 7
Fig. 7

Spectral dependence of the reflectivity of a lossless waveguide with the parameters of Fig. 5 for three different incident- beam-divergence values. The principal incidence direction is θi=34.785°.

Fig. 8
Fig. 8

Values of the reflectivity maxima plotted as a function of the angular beam width. The waveguide parameters are the same as those for Fig. 5.

Fig. 9
Fig. 9

Comparison of the theoretical and the experimental reflectivities. The theoretical parameters are those given for Fig. 5, and the beam divergence is 1.2 mrad. The experimental values are given in the text.

Fig. 10
Fig. 10

Angular dependence of the reflectivity of the corrugated waveguide. The thin solid curve represents the theoretical results under the assumption of plane-wave incidence. The thin dashed curve represents the reflectivity of the plane waveguide. The thick dashed curve represents the measurements made with a He–Ne laser with a 0.3-mrad beam divergence. The thick solid curve represents measurements made with the dye laser with a fixed wavelength of 632.8 nm. The logarithmic vertical scale better reveals the form of the anomaly.

Equations (16)

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ncsin θi=βgk0+m λd,
rm=r0, m(α-αmr, z)(α-α p),
tm=t0, m(α-αmt, z)(α-α p),
1β0rmU(βmr|rm|2+βmt|tm|2)=1,
(βmr, t)2=nc,s2-αm2,
sin θi=1nc α0t,z.
min(t0)Im(αmt,z)Im(αp).
Ezi(x, y)=-+p(δ)exp[i(αx-βry)]dδ,
p(δ)=exp-δ-δ0Δ2.
Ezr(x, y)=-+r(δ)p(δ)exp[i(α x+βry)]dδ,
η0=-+|r(δ)p(δ)|2βr(δ)dδ-+|p(δ)|2βr(δ)dδ.
Ii, r-+Pyi,r(x, y)|y=dx.
Py(x, y)EzH¯xEzE¯zdy,
Pyi(x, y)-+dδdδp(δ)p¯(δ )β×exp[i(α-α)x-i(β-β)x],
Pyr(x, y)-+dδdδ r(δ)r¯(δ )p(δ)p¯(δ )β×exp[i(α-α)x-i(β-β)x].
-+exp[i(α-α)x]dx=2πδ(α-α),

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