Abstract

The boundary value techniques in vector–field diffraction theories are generalized to describe electromagnetic scattering of plane waves by a finite number of parallel, rectangular grooves corrugated on a metallic (infinitely conducting) ground plane. Each of the grooves has its own feature size and location for representing a general grating structure in multilevel binary optics. The multiple-scattering matrix is derived for determining the scattering coefficients that lead to a fast convergence in the Bessel-function series representation of the scattered-field angular spectrum. The solution remains stable from the long-wavelength (the Rayleigh limit) to the short-wavelength region (the geometrical optics limit). It is found that any N-groove scattered field can be treated as the sum of N single-groove radiation fields and the cross-groove coupling fields. A coupling index is introduced to measure the coupling effect. Numerical examples of two, six, and twelve grooves are examined in different spectral regions. Plots of the coupling index are generated to show the feasibility of using pattern superposition for an approximate solution.

© 1993 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction gratings,” Proc. IEEE 73, 894–938 (1985).
    [CrossRef]
  2. W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
    [CrossRef]
  3. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
  4. J. A. Kong, “Second-order coupled-mode equations for spatially periodic media,” J. Opt. Soc. Am. 67, 825–829 (1977).
    [CrossRef]
  5. R. Magnusson, T. K. Gaylord, “Analysis of multiwave diffraction by thick gratings,” J. Opt. Soc. Am. 67, 1165–1170 (1977).
    [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. Tamir, H. C. Wang, A. A. Oliner, “Wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microwave Theory Tech. MTT-12, 323–335 (1964).
    [CrossRef]
  8. C. B. Burckhardt, “Diffraction of a plane wave at a sinusoidally stratified dielectric grating,” J. Opt. Soc. Am. 56, 1502–1509 (1966).
    [CrossRef]
  9. R. S. Chu, T. Tamir, “Guided wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 486–504 (1970).
  10. F. Kaspar, “Diffraction by thick periodically stratified gratings with complex dielectric constant,” J. Opt. Soc. Am. 63, 37–45 (1973).
    [CrossRef]
  11. S. T. Peng, T. Tamir, H. L. Betroni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
    [CrossRef]
  12. R. S. Chu, J. A. Kong, “Modal theory of spatially periodic media,” IEEE Trans. Microwave Theory Tech. MTT-25, 18–24 (1977).
  13. R. Kastner, R. Mittra, “Iterative analysis of finite sized planar frequency selective surfaces with rectangular patches or perforations,” IEEE Trans. Antennas Propag. AP-35, 372–377 (1987).
    [CrossRef]
  14. W. L. Ko, R. Mittra, “Scattering by a truncated periodic array,” IEEE Trans. Antennas Propag. 36, 496–503 (1988).
    [CrossRef]
  15. Y.-L. Kok, “Boundary value solution to electromagnetic scattering by a rectangular groove in a ground plane,” J. Opt. Soc. Am A 9, 302–311 (1992).
    [CrossRef]
  16. H. A. Kalhor, “Plane metallic gratings of finite numbers of strips,” IEEE Trans. Antennas Propag. 37, 406–407 (1989).
    [CrossRef]
  17. J. M. Jin, J. L. Volakis, “A finite element-boundary integral formulation for scattering by three-dimensional cavity-backed apertures,” IEEE Trans. Antennas Propag. 39, 97–104 (1991).
    [CrossRef]
  18. T. B. A. Senior, K. Sarabandi, J. R. Natzke, “Scattering by a narrow gap,” IEEE Trans. Antennas Propag. 38, 1102–1110 (1990).
    [CrossRef]
  19. S.-K. Jeng, “Scattering from a cavity-backed slit in a ground plane: TE case,” IEEE Trans. Antennas Propag. 38, 1529–1532 (1990).
  20. S. J. Bever, J. P. Allenbach, “Multiple scattering by a planar array of parallel dielectric cylinders,” Appl. Opt. 31, 3524–3532 (1992).
    [CrossRef] [PubMed]
  21. V. Twersky, “Scattering of wave by two objects,” in Electromagnetic Waves, R. E. Langer, ed. (University of Wisconsin, Madison, Wisc., 1962), pp. 361–389.
  22. J. W. Goodman, F. I. Leonberger, S. Y. Kung, R. A. Athale, “Optical interconnection for VLSI systems,” Proc. IEEE 72, 850–866 (1984).
    [CrossRef]
  23. S. K. Yao, D. E. Thompson, “Chirped-grating lens for guided-wave optics,” Appl. Phys. Lett. 30, 225–226 (1977).
  24. M. A. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1964), Chap. 11, Secs. 11.4.35–11.4.38, pp. 487.
  25. Y.-L. Kok, N. C. Gallagher, R. W. Ziolkowski, “Dual series solution to the scattering of plane waves from a binary conducting grating,” IEEE Trans. Antennas Propagat. 37, 901–917 (1989).
    [CrossRef]
  26. J. D. Kraus, Antennas (McGraw Hill, New York, 1950) Chap. 4, pp. 66–74.
  27. fortran subroutines for mathematical applications, version 1.1 (International Mathematical and Statistical Libraries, Inc.,) 2500 City West Boulevard, Houston, Texas, 1989.

1992

Y.-L. Kok, “Boundary value solution to electromagnetic scattering by a rectangular groove in a ground plane,” J. Opt. Soc. Am A 9, 302–311 (1992).
[CrossRef]

S. J. Bever, J. P. Allenbach, “Multiple scattering by a planar array of parallel dielectric cylinders,” Appl. Opt. 31, 3524–3532 (1992).
[CrossRef] [PubMed]

1991

J. M. Jin, J. L. Volakis, “A finite element-boundary integral formulation for scattering by three-dimensional cavity-backed apertures,” IEEE Trans. Antennas Propag. 39, 97–104 (1991).
[CrossRef]

1990

T. B. A. Senior, K. Sarabandi, J. R. Natzke, “Scattering by a narrow gap,” IEEE Trans. Antennas Propag. 38, 1102–1110 (1990).
[CrossRef]

S.-K. Jeng, “Scattering from a cavity-backed slit in a ground plane: TE case,” IEEE Trans. Antennas Propag. 38, 1529–1532 (1990).

1989

H. A. Kalhor, “Plane metallic gratings of finite numbers of strips,” IEEE Trans. Antennas Propag. 37, 406–407 (1989).
[CrossRef]

Y.-L. Kok, N. C. Gallagher, R. W. Ziolkowski, “Dual series solution to the scattering of plane waves from a binary conducting grating,” IEEE Trans. Antennas Propagat. 37, 901–917 (1989).
[CrossRef]

1988

W. L. Ko, R. Mittra, “Scattering by a truncated periodic array,” IEEE Trans. Antennas Propag. 36, 496–503 (1988).
[CrossRef]

1987

R. Kastner, R. Mittra, “Iterative analysis of finite sized planar frequency selective surfaces with rectangular patches or perforations,” IEEE Trans. Antennas Propag. AP-35, 372–377 (1987).
[CrossRef]

1985

T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction gratings,” Proc. IEEE 73, 894–938 (1985).
[CrossRef]

1984

J. W. Goodman, F. I. Leonberger, S. Y. Kung, R. A. Athale, “Optical interconnection for VLSI systems,” Proc. IEEE 72, 850–866 (1984).
[CrossRef]

1981

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

1977

R. S. Chu, J. A. Kong, “Modal theory of spatially periodic media,” IEEE Trans. Microwave Theory Tech. MTT-25, 18–24 (1977).

S. K. Yao, D. E. Thompson, “Chirped-grating lens for guided-wave optics,” Appl. Phys. Lett. 30, 225–226 (1977).

J. A. Kong, “Second-order coupled-mode equations for spatially periodic media,” J. Opt. Soc. Am. 67, 825–829 (1977).
[CrossRef]

R. Magnusson, T. K. Gaylord, “Analysis of multiwave diffraction by thick gratings,” J. Opt. Soc. Am. 67, 1165–1170 (1977).
[CrossRef]

1975

S. T. Peng, T. Tamir, H. L. Betroni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

1973

1970

R. S. Chu, T. Tamir, “Guided wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 486–504 (1970).

1969

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

1967

W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
[CrossRef]

1966

1964

T. Tamir, H. C. Wang, A. A. Oliner, “Wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microwave Theory Tech. MTT-12, 323–335 (1964).
[CrossRef]

Allenbach, J. P.

Athale, R. A.

J. W. Goodman, F. I. Leonberger, S. Y. Kung, R. A. Athale, “Optical interconnection for VLSI systems,” Proc. IEEE 72, 850–866 (1984).
[CrossRef]

Betroni, H. L.

S. T. Peng, T. Tamir, H. L. Betroni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Bever, S. J.

Burckhardt, C. B.

Chu, R. S.

R. S. Chu, J. A. Kong, “Modal theory of spatially periodic media,” IEEE Trans. Microwave Theory Tech. MTT-25, 18–24 (1977).

R. S. Chu, T. Tamir, “Guided wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 486–504 (1970).

Cook, B. D.

W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
[CrossRef]

Gallagher, N. C.

Y.-L. Kok, N. C. Gallagher, R. W. Ziolkowski, “Dual series solution to the scattering of plane waves from a binary conducting grating,” IEEE Trans. Antennas Propagat. 37, 901–917 (1989).
[CrossRef]

Gaylord, T. K.

T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction gratings,” Proc. IEEE 73, 894–938 (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]

R. Magnusson, T. K. Gaylord, “Analysis of multiwave diffraction by thick gratings,” J. Opt. Soc. Am. 67, 1165–1170 (1977).
[CrossRef]

Goodman, J. W.

J. W. Goodman, F. I. Leonberger, S. Y. Kung, R. A. Athale, “Optical interconnection for VLSI systems,” Proc. IEEE 72, 850–866 (1984).
[CrossRef]

Jeng, S.-K.

S.-K. Jeng, “Scattering from a cavity-backed slit in a ground plane: TE case,” IEEE Trans. Antennas Propag. 38, 1529–1532 (1990).

Jin, J. M.

J. M. Jin, J. L. Volakis, “A finite element-boundary integral formulation for scattering by three-dimensional cavity-backed apertures,” IEEE Trans. Antennas Propag. 39, 97–104 (1991).
[CrossRef]

Kalhor, H. A.

H. A. Kalhor, “Plane metallic gratings of finite numbers of strips,” IEEE Trans. Antennas Propag. 37, 406–407 (1989).
[CrossRef]

Kaspar, F.

Kastner, R.

R. Kastner, R. Mittra, “Iterative analysis of finite sized planar frequency selective surfaces with rectangular patches or perforations,” IEEE Trans. Antennas Propag. AP-35, 372–377 (1987).
[CrossRef]

Klein, W. R.

W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
[CrossRef]

Ko, W. L.

W. L. Ko, R. Mittra, “Scattering by a truncated periodic array,” IEEE Trans. Antennas Propag. 36, 496–503 (1988).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

Kok, Y.-L.

Y.-L. Kok, “Boundary value solution to electromagnetic scattering by a rectangular groove in a ground plane,” J. Opt. Soc. Am A 9, 302–311 (1992).
[CrossRef]

Y.-L. Kok, N. C. Gallagher, R. W. Ziolkowski, “Dual series solution to the scattering of plane waves from a binary conducting grating,” IEEE Trans. Antennas Propagat. 37, 901–917 (1989).
[CrossRef]

Kong, J. A.

R. S. Chu, J. A. Kong, “Modal theory of spatially periodic media,” IEEE Trans. Microwave Theory Tech. MTT-25, 18–24 (1977).

J. A. Kong, “Second-order coupled-mode equations for spatially periodic media,” J. Opt. Soc. Am. 67, 825–829 (1977).
[CrossRef]

Kraus, J. D.

J. D. Kraus, Antennas (McGraw Hill, New York, 1950) Chap. 4, pp. 66–74.

Kung, S. Y.

J. W. Goodman, F. I. Leonberger, S. Y. Kung, R. A. Athale, “Optical interconnection for VLSI systems,” Proc. IEEE 72, 850–866 (1984).
[CrossRef]

Leonberger, F. I.

J. W. Goodman, F. I. Leonberger, S. Y. Kung, R. A. Athale, “Optical interconnection for VLSI systems,” Proc. IEEE 72, 850–866 (1984).
[CrossRef]

Magnusson, R.

Mittra, R.

W. L. Ko, R. Mittra, “Scattering by a truncated periodic array,” IEEE Trans. Antennas Propag. 36, 496–503 (1988).
[CrossRef]

R. Kastner, R. Mittra, “Iterative analysis of finite sized planar frequency selective surfaces with rectangular patches or perforations,” IEEE Trans. Antennas Propag. AP-35, 372–377 (1987).
[CrossRef]

Moharam, M. G.

T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction gratings,” Proc. IEEE 73, 894–938 (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]

Natzke, J. R.

T. B. A. Senior, K. Sarabandi, J. R. Natzke, “Scattering by a narrow gap,” IEEE Trans. Antennas Propag. 38, 1102–1110 (1990).
[CrossRef]

Oliner, A. A.

T. Tamir, H. C. Wang, A. A. Oliner, “Wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microwave Theory Tech. MTT-12, 323–335 (1964).
[CrossRef]

Peng, S. T.

S. T. Peng, T. Tamir, H. L. Betroni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Sarabandi, K.

T. B. A. Senior, K. Sarabandi, J. R. Natzke, “Scattering by a narrow gap,” IEEE Trans. Antennas Propag. 38, 1102–1110 (1990).
[CrossRef]

Senior, T. B. A.

T. B. A. Senior, K. Sarabandi, J. R. Natzke, “Scattering by a narrow gap,” IEEE Trans. Antennas Propag. 38, 1102–1110 (1990).
[CrossRef]

Tamir, T.

S. T. Peng, T. Tamir, H. L. Betroni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

R. S. Chu, T. Tamir, “Guided wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 486–504 (1970).

T. Tamir, H. C. Wang, A. A. Oliner, “Wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microwave Theory Tech. MTT-12, 323–335 (1964).
[CrossRef]

Thompson, D. E.

S. K. Yao, D. E. Thompson, “Chirped-grating lens for guided-wave optics,” Appl. Phys. Lett. 30, 225–226 (1977).

Twersky, V.

V. Twersky, “Scattering of wave by two objects,” in Electromagnetic Waves, R. E. Langer, ed. (University of Wisconsin, Madison, Wisc., 1962), pp. 361–389.

Volakis, J. L.

J. M. Jin, J. L. Volakis, “A finite element-boundary integral formulation for scattering by three-dimensional cavity-backed apertures,” IEEE Trans. Antennas Propag. 39, 97–104 (1991).
[CrossRef]

Wang, H. C.

T. Tamir, H. C. Wang, A. A. Oliner, “Wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microwave Theory Tech. MTT-12, 323–335 (1964).
[CrossRef]

Yao, S. K.

S. K. Yao, D. E. Thompson, “Chirped-grating lens for guided-wave optics,” Appl. Phys. Lett. 30, 225–226 (1977).

Ziolkowski, R. W.

Y.-L. Kok, N. C. Gallagher, R. W. Ziolkowski, “Dual series solution to the scattering of plane waves from a binary conducting grating,” IEEE Trans. Antennas Propagat. 37, 901–917 (1989).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

S. K. Yao, D. E. Thompson, “Chirped-grating lens for guided-wave optics,” Appl. Phys. Lett. 30, 225–226 (1977).

Bell Syst. Tech. J.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).

IEEE Trans. Antennas Propag.

R. Kastner, R. Mittra, “Iterative analysis of finite sized planar frequency selective surfaces with rectangular patches or perforations,” IEEE Trans. Antennas Propag. AP-35, 372–377 (1987).
[CrossRef]

W. L. Ko, R. Mittra, “Scattering by a truncated periodic array,” IEEE Trans. Antennas Propag. 36, 496–503 (1988).
[CrossRef]

H. A. Kalhor, “Plane metallic gratings of finite numbers of strips,” IEEE Trans. Antennas Propag. 37, 406–407 (1989).
[CrossRef]

J. M. Jin, J. L. Volakis, “A finite element-boundary integral formulation for scattering by three-dimensional cavity-backed apertures,” IEEE Trans. Antennas Propag. 39, 97–104 (1991).
[CrossRef]

T. B. A. Senior, K. Sarabandi, J. R. Natzke, “Scattering by a narrow gap,” IEEE Trans. Antennas Propag. 38, 1102–1110 (1990).
[CrossRef]

S.-K. Jeng, “Scattering from a cavity-backed slit in a ground plane: TE case,” IEEE Trans. Antennas Propag. 38, 1529–1532 (1990).

IEEE Trans. Antennas Propagat.

Y.-L. Kok, N. C. Gallagher, R. W. Ziolkowski, “Dual series solution to the scattering of plane waves from a binary conducting grating,” IEEE Trans. Antennas Propagat. 37, 901–917 (1989).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

T. Tamir, H. C. Wang, A. A. Oliner, “Wave propagation in sinusoidally stratified dielectric media,” IEEE Trans. Microwave Theory Tech. MTT-12, 323–335 (1964).
[CrossRef]

R. S. Chu, T. Tamir, “Guided wave theory of light diffraction by acoustic microwaves,” IEEE Trans. Microwave Theory Tech. MTT-18, 486–504 (1970).

S. T. Peng, T. Tamir, H. L. Betroni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

R. S. Chu, J. A. Kong, “Modal theory of spatially periodic media,” IEEE Trans. Microwave Theory Tech. MTT-25, 18–24 (1977).

IEEE Trans. Sonics Ultrason.

W. R. Klein, B. D. Cook, “Unified approach to ultrasonic light diffraction,” IEEE Trans. Sonics Ultrason. SU-14, 123–134 (1967).
[CrossRef]

J. Opt. Soc. Am

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

J. Opt. Soc. Am A

Y.-L. Kok, “Boundary value solution to electromagnetic scattering by a rectangular groove in a ground plane,” J. Opt. Soc. Am A 9, 302–311 (1992).
[CrossRef]

J. Opt. Soc. Am.

Proc. IEEE

J. W. Goodman, F. I. Leonberger, S. Y. Kung, R. A. Athale, “Optical interconnection for VLSI systems,” Proc. IEEE 72, 850–866 (1984).
[CrossRef]

T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction gratings,” Proc. IEEE 73, 894–938 (1985).
[CrossRef]

Other

M. A. Abramowitz, I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1964), Chap. 11, Secs. 11.4.35–11.4.38, pp. 487.

V. Twersky, “Scattering of wave by two objects,” in Electromagnetic Waves, R. E. Langer, ed. (University of Wisconsin, Madison, Wisc., 1962), pp. 361–389.

J. D. Kraus, Antennas (McGraw Hill, New York, 1950) Chap. 4, pp. 66–74.

fortran subroutines for mathematical applications, version 1.1 (International Mathematical and Statistical Libraries, Inc.,) 2500 City West Boulevard, Houston, Texas, 1989.

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 (7)

Fig. 1
Fig. 1

Perfectly conducting ground plane corrugated by N parallel rectangular grooves is illuminated by a plane wave with wave vector k with incident angles ϑ and φ. The grooves, uniform in the z axis direction, are positioned by the phase scattering centers dl along the x axis (l = 1, 2,…, N). The groove widths and depths are specified by cl and hl, respectively. Any given point P on the xy plane can be positioned in polar coordinates (r, ϑ′).

Fig. 2
Fig. 2

Far-field amplitude plotted as a function of the polar angle ϑ′ for different numbers of grooves N in multiple scattering and with fast polarization: (a) N = 2, (b) N = 6, (c) n = 12. The other parameters are ϑ = 30°, φ = 0°, Δd = 1.6, hl = 0.4, cl = 0.8, l = 1, 2,…, N, and wavelength λ = 6.5 (kc = 0.7733).

Fig. 3
Fig. 3

Same as Fig. 2 but with wavelength λ = 1.0 (kc = 5.03) and a different scale on the y axis.

Fig. 4
Fig. 4

Same as Fig. 2 but with wavelength λ = 0.1 (kc = 50.27) and a different scale on the y axis.

Fig. 5
Fig. 5

Coupling index ρ calculated versus the incident wave-length ranging from 0.05 to 5.5 for three different numbers of grooves in multiple scattering for the fast polarization case: N = 2, 6, and 12. Other parameters are ϑ = 30°, φ = 0°, Δd = 0.8, hl = 1.0, cl = 0.58, and l = 1, 2,…, N.

Fig. 6
Fig. 6

Far-field amplitude due to first-order scattering by six equally spaced grooves and with fast polarization (single-groove radiation fields) plotted (dashed curve) against the exact multiple-groove scattering (solid curve). With ρ = 0.3148, significant amplitude discrepancies (≈18%) are found at the specular direction ϑ′ = 60° and the +1 diffraction-order direction ϑ′ = 12°. The parameters are ϑ = 30°, φ = 0°, Δd = 0.8, hl = 1.0, cl = 0.58, l = 1, 2,…, 6, and N = 6.

Fig. 7
Fig. 7

(a) Two 12-groove, binary, finite gratings illuminated with incident angles ϑ = 30°, φ = 0°, and wavelength = 1.0, (b) plots of the far-field amplitudes. The amplitude peak at the −1 diffraction-order direction (ϑ′ = 106°), originally generated with the constant groove-to-groove spacing grating, can be suppressed if the groove-to-groove spacing is chirped (or slowly decreased) as depicted in (a) by the solid curve.

Equations (30)

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

E z ( x , y ) = exp [ i ( α 0 x β 0 y ) ] + r ( u ) × exp ( i { [ k 2 ( 2 π u ) 2 ] 1 / 2 y + 2 π u x } ) d u for y 0 ,
E z ( x , y ) = n = 1 a l , n sin ( A l , n ( y + h l ) ) sin [ n π c l ( x d l + c l 2 ) ] for | x d l | c l 2 , h l y 0 .
r ( u ) = r ( u ) + δ ( u α 0 2 π ) = l = 1 N r l ( u ) exp ( i 2 π u d l ) ,
r l ( u ) = n = 1 c l , n J n ( π c l u ) u for l = 1 , 2 , , N .
n = 1 c l , n φ l , n ( x d l ) = m = 1 a l , m sin ( A l , m h l ) sin [ m π c l ( x d l + c l 2 ) ] , 0 | x d l | c l 2 ,
j = 1 N n = 1 c j , n Φ j , n ( x ) = β 0 exp ( i α 0 X ) + m = 1 a l , m A l , m cos ( A l , m h l ) 2 i × sin [ m π c l ( x d l + c l 2 ) ] , 0 | x d l | c l 2 ,
φ l , n ( x d l ) = { 2 n cos [ n sin 1 ( x d l c l / 2 ) ] n = 1 , 3 , 5 2 i n sin [ n sin 1 ( x d l c l / 2 ) ] n = 2 , 4 , 6 ,
Φ l , n ( x ) = { u = 0 [ k 2 ( 2 π u ) 2 ] 1 / 2 J n ( π c l u ) u cos [ 2 π u ( x d l ) ] d u n = 1 , 3 , 5 i u = 0 [ k 2 ( 2 π u ) 2 ] 1 / 2 J n ( π c l u ) u sin [ 2 π u ( x d l ) ] d u n = 2 , 4 , 6 .
n = 1 p m n l c l , n = n = 1 s m n l a l , n ,
j = 1 N n = 1 q m n l j c j , n = b m l + n = 1 d m n l a l , n ,
P l · c l = S l · a l ,
j = 1 N Q i , j · c j = b ι + D l · a l .
[ Q l , l D l ( S l ) 1 P l ] · c l + j l , j = 1 N Q i , j · c j = b l .
M · c = b ,
M = [ Q 1 , 1 D l ( S l ) l P 1 Q 1 , 2 Q 1 , N Q 2 , 1 Q 2 , 2 D 2 ( S 2 ) 1 P 2 Q 2 , N Q N , 1 Q N , 2 Q N , N D N ( S N ) 1 P N ] ,
c t = [ c 1 t , c 2 t , , c N t ] ,
b t = [ b 1 t , b 2 t , , b N t ] .
M d = [ Q 1 , 1 D 1 ( S 1 ) 1 P 1 0 0 0 Q 2 , 2 D 2 ( S 2 ) 1 P 2 0 0 0 Q N , N D N ( S N ) 1 P N ] ,
F = [ 0 Q 1 , 2 Q 1 , N Q 2 , 1 0 Q 2 , N Q N , 1 Q N , 2 0 ] ,
c = [ I + ( M d 1 F ) ] 1 · c 0 = c 0 + ( M d 1 F ) c 0 + ( M d 1 F ) 2 c 0 + .
E z d ( r , ϑ ) = 2 π exp [ i ( k r π / 4 ) ] k r P F ( ϑ ) ,
P F ( ϑ ) = l = 1 N n = 1 c l , n exp ( i k d l cos ϑ ) J n ( k c l cos ϑ 2 ) tan ϑ
ρ = | Δ c | | c 0 | ,
p m m l = { 4 c l 0 c l / 2 φ l , n ( x ) sin [ m π c l ( x + c l 2 ) ] d x for ( m + n ) even 0 for ( m + n ) odd ,
q m n l , j = 2 c l c l / 2 c l / 2 Φ j , n ( x + d l ) sin [ m π c l ( x + c l 2 ) ] d x ,
s m n l = δ m n sin ( A l , n h l ) ,
d m n l = δ m m A l , n cos ( A l h l ) 2 i ,
b m l = i β 0 exp ( i α 0 d l ) [ exp ( i m π 2 ) sinc ( α 0 c m π 2 ) exp ( i m π 2 ) sinc ( α 0 c + m π 2 ) ] ,
Δ c = M d 1 F · c .
c = c 0 + ( M d 1 F · c ) .

Metrics