Abstract

The guided-mode resonance behavior of the evanescent and propagating fields associated with an unslanted, planar diffraction grating is studied by means of the rigorous coupled-wave theory. For weakly modulated gratings, the condition on the guided-mode wave number of the corresponding unmodulated dielectric-layer waveguide may be used to predict the range of the incident angle or wavelength within which the resonances can be excited. Furthermore, the locations of the resonances are predicted approximately by the eigenvalue equation of the waveguide. As the modulation amplitude increases, the location and shape of the resonances are described in detail by the rigorous coupled-wave theory. The results presented demonstrate that the resonances can cause rapid variations in the intensity of the external propagating diffracted waves.

© 1990 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. W. Wood, “Remarkable spectrum from a diffraction grating,” Philos. Mag. 4, 396–402 (1902).
  2. A. Hessel, A. A. Oliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. 10, 1275–1297 (1965).
    [Crossref]
  3. T. Tamir, “Scattering of electromagnetic waves by a sinusoidally stratified half-space, II. Diffraction aspects at the Rayleigh and Bragg wavelengths,” Can. J. Phys. 44, 2461–2494 (1966).
    [Crossref]
  4. M. Nevière, “The homogeneous problem,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, New York, 1980), pp. 123–157.
    [Crossref]
  5. S. Zhang, T. Tamir, “Spatial modifications of Gaussian beams diffracted by reflection gratings,” J. Opt. Soc. Am. A 6, 1368–1381 (1989).
    [Crossref]
  6. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [Crossref]
  7. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [Crossref]
  8. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).
  9. T. Tamir, “Inhomogeneous wave types at planar interfaces: II—surface waves,” Optik 37, 204–228 (1973).
  10. T. Tamir, “Inhomogeneous wave types at planar interfaces: III—leaky waves,” Optik 38, 269–297 (1973).

1989 (1)

1985 (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[Crossref]

1981 (1)

1973 (2)

T. Tamir, “Inhomogeneous wave types at planar interfaces: II—surface waves,” Optik 37, 204–228 (1973).

T. Tamir, “Inhomogeneous wave types at planar interfaces: III—leaky waves,” Optik 38, 269–297 (1973).

1966 (1)

T. Tamir, “Scattering of electromagnetic waves by a sinusoidally stratified half-space, II. Diffraction aspects at the Rayleigh and Bragg wavelengths,” Can. J. Phys. 44, 2461–2494 (1966).
[Crossref]

1965 (1)

A. Hessel, A. A. Oliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. 10, 1275–1297 (1965).
[Crossref]

1902 (1)

R. W. Wood, “Remarkable spectrum from a diffraction grating,” Philos. Mag. 4, 396–402 (1902).

Gaylord, T. K.

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[Crossref]

M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
[Crossref]

Hessel, A.

A. Hessel, A. A. Oliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. 10, 1275–1297 (1965).
[Crossref]

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

Moharam, M. G.

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[Crossref]

M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
[Crossref]

Nevière, M.

M. Nevière, “The homogeneous problem,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, New York, 1980), pp. 123–157.
[Crossref]

Oliner, A. A.

A. Hessel, A. A. Oliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. 10, 1275–1297 (1965).
[Crossref]

Tamir, T.

S. Zhang, T. Tamir, “Spatial modifications of Gaussian beams diffracted by reflection gratings,” J. Opt. Soc. Am. A 6, 1368–1381 (1989).
[Crossref]

T. Tamir, “Inhomogeneous wave types at planar interfaces: II—surface waves,” Optik 37, 204–228 (1973).

T. Tamir, “Inhomogeneous wave types at planar interfaces: III—leaky waves,” Optik 38, 269–297 (1973).

T. Tamir, “Scattering of electromagnetic waves by a sinusoidally stratified half-space, II. Diffraction aspects at the Rayleigh and Bragg wavelengths,” Can. J. Phys. 44, 2461–2494 (1966).
[Crossref]

Wood, R. W.

R. W. Wood, “Remarkable spectrum from a diffraction grating,” Philos. Mag. 4, 396–402 (1902).

Zhang, S.

Appl. Opt. (1)

A. Hessel, A. A. Oliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. 10, 1275–1297 (1965).
[Crossref]

Can. J. Phys. (1)

T. Tamir, “Scattering of electromagnetic waves by a sinusoidally stratified half-space, II. Diffraction aspects at the Rayleigh and Bragg wavelengths,” Can. J. Phys. 44, 2461–2494 (1966).
[Crossref]

J. Opt. Soc. Am. (1)

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

Optik (2)

T. Tamir, “Inhomogeneous wave types at planar interfaces: II—surface waves,” Optik 37, 204–228 (1973).

T. Tamir, “Inhomogeneous wave types at planar interfaces: III—leaky waves,” Optik 38, 269–297 (1973).

Philos. Mag. (1)

R. W. Wood, “Remarkable spectrum from a diffraction grating,” Philos. Mag. 4, 396–402 (1902).

Proc. IEEE (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[Crossref]

Other (2)

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974).

M. Nevière, “The homogeneous problem,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, New York, 1980), pp. 123–157.
[Crossref]

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 (6)

Fig. 1
Fig. 1

Planar grating diffraction model. The angles θi represent the angles of the wave vector of the ith backward-diffracted wave with respect to the z axis; θi are the corresponding angles for the forward-diffracted waves. The angle of incidence θ is arbitrary.

Fig. 2
Fig. 2

Graph showing which diffracted orders would be guided (in an unmodulated waveguide) with respect to the specific incident angle. The parameters are 1 = 3 = 1, 0 = 2, and m = 1.

Fig. 3
Fig. 3

Amplitudes of the Si(0) and Si(d/Λ) evanescent waves calculated by using step size Δ(d/Λ) = 6.25 × 10−3: (a) i = −1; (b) i = 2.

Fig. 4
Fig. 4

Amplitudes of the Si(0) and Si(d/Λ) evanescent waves calculated by using step size Δ(d/Λ) = 2.5 × 10−7: (a) i = −1; (b) i = 2.

Fig. 5
Fig. 5

Diffraction efficiency of the propagating waves calculated by using step size Δ(d/Λ) = 2.5 × 10−7: (a) i = 0; (b) i = 1.

Fig. 6
Fig. 6

Effect of increasing the modulation Δ on the amplitudes of the Si(0) and Si(d/Λ) evanescent waves calculated by using step size Δ(d/Λ) = 1.0 × 10−2: (a) i = 0; (b) i = 1.

Equations (14)

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

2 E y ( x , z ) + k 2 2 ( x ) E y ( x , z ) = 0 ,
E y ( x , z ) = i = S i ( z ) exp ( j σ i · r ) = i = S i ( z ) exp { j [ ( k 2 x i K ) x + ( k 2 z ) z ] } ,
1 2 π 2 d 2 S i ( z ) d z 2 j 2 π ( 0 cos θ λ ) d S i ( z ) d z + 2 i ( m i ) Λ 2 S i ( z ) + Δ λ 2 [ S i 1 ( z ) + S i + 1 ( z ) ] = 0 ,
E 1 = exp ( j K 1 · r ) + i = R i exp ( j K 1 i · r ) = exp { j [ k 1 ( sin θ x + cos θ z ) ] } + i = R i exp { j ( k 2 sin θ i K ) x + j [ k 1 2 ( k 2 sin θ i K ) 2 ] 1 / 2 z }
E 3 = i = T i exp [ j k 3 i · ( r d z ˆ ) ] = i = T i exp { j ( k 2 sin θ i K ) x j [ k 3 2 ( k 2 sin θ i K ) 2 ] 1 / 2 ( z d ) } ,
D E 1 i = Re [ ( K 1 i · z ˆ ) / ( K 10 · z ˆ ) ] R i R i * ] = Re ( { 1 [ sin θ i λ / ( 1 Λ ) ] 2 } 1 / 2 / cos θ ) R i R i * ,
D E 3 i = Re [ ( K 3 i · z ˆ ) / ( K 10 · z ˆ ) ] T i T i * = Re ( { ( 3 / 1 ) [ sin θ i λ / ( 1 Λ ) 2 ] 1 / 2 / cos θ ) T i T i * .
E y ( x , z ) = i = S ˆ i ( z ) exp [ j ( k 2 x i K ) x ] .
1 2 π 2 d 2 S ˆ i ( z ) d z 2 2 [ ( 0 sin θ λ i Λ ) 2 0 λ 2 ] S ˆ i ( z ) + Δ λ 2 [ S ˆ i 1 ( z ) + S ˆ i + 1 ( z ) ] = 0.
1 2 π 2 d 2 S ˆ i ( z ) d z 2 2 [ ( 0 sin θ λ i Λ ) 2 0 λ 2 ] S ˆ i ( z ) = 0 ,
κ 2 = k 2 [ 0 ( 0 sin θ i λ / Λ ) 2 ]
max { 1 , 3 } | 0 sin θ i λ / Λ | < 0 .
tan ( κ d ) = κ ( γ + δ ) / ( κ 2 γ δ ) ,
i = ( D E 1 i + D E 3 i ) = 1.

Metrics