Abstract

The diffraction of electromagnetic radiation from one-dimensional planar transmittive screens illuminated by Gaussian-profile beams is examined. The incident Gaussian beam is expressed as a superposition of elementary plane waves that propagate along different directions, with the aid of the plane-wave spectrum technique based on Fourier optics. For each elementary plane wave, the diffracted field is obtained by applying the spectral domain method combined with the method of moments. Results for the cases of normal and oblique incidence are in agreement with theoretically expected properties of such planar screens and are found to be very sensitive to the aspect ratio γ (strip width to period). The diffracted beam is composed of distinct subbeams that propagate along specific directions. These directions coincide with the Floquet harmonics generated by an incident plane wave propagating along the same direction as the initial beam.

© 1994 Optical Society of America

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References

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  1. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [Crossref]
  2. C. C. Chen, “Scattering by a two-dimensional periodic array of conducting plates,”IEEE Trans. Antennas Propag. AP-18, 660–665 (1970).
    [Crossref]
  3. K. Kobayashi, T. Inoue, “Diffraction of a plane wave by an inclined parallel plate grating,”IEEE Trans. Antennas Propag. 36, 1424–1434 (1988).
    [Crossref]
  4. R. Petit, G. Tayeb, “Theoretical and numerical study of gratings consisting of periodic arrays of thin and lossy strips,” J. Opt. Soc. Am. A 7, 1686–1692 (1990).
    [Crossref]
  5. R. S. Chu, T. Tamir, “Bragg diffraction of Gaussian beams by periodically modulated media,”J. Opt. Soc. Am. 66, 220–226 (1976).
    [Crossref]
  6. R. S. Chu, T. Tamir, “Diffraction of Gaussian beams by periodically modulated media for incidence angles close to Bragg angle,”J. Opt. Soc. Am. 66, 1348–1440 (1976).
  7. R. S. Chu, J. A. Kong, T. Tamir, “Diffraction of Gaussian beams by a periodically modulated layer,”J. Opt. Soc. Am. 67, 1555–1561 (1977).
    [Crossref]
  8. M. G. Moharam, T. K. Gaylord, R. Magnusson, “Bragg diffraction of finite beams by thick gratings,”J. Opt. Soc. Am. 70, 300–304 (1980).
    [Crossref]
  9. T. Kojima, “Diffraction of Hermite–Gaussian beams from sinusoidal conducting gratings,” J. Opt. Soc. Am. A 7, 1740–1744 (1990).
    [Crossref]
  10. S. Zhang, T. Tamir, “Spatial modifications of Gaussian beams diffracted by reflection grating,” J. Opt. Soc. Am. A 6, 1368–1381 (1989).
    [Crossref]
  11. A. Yariv, Optical Electronics (Saunders, Philadelphia, Pa., 1991), Chap. 2.
  12. E. M. Khaled, S. C. Hill, P. W. Barber, “Scattered/internal intensity for dielectric object with Gaussian beam illlumination,” in Proceedings of the Eighth Annual Review of Progress in Applied Computational Electromagnetics (U.S. Naval Postgraduate School, Monterey, Calif., 1992), pp. 82–89.
  13. E. E. Kriezis, D. P. Chrissoulidis, A. G. Papagiannakis, Electromagnetics and Optics (World Scientific, Singapore, 1992), Chap. 9.
  14. G. Scott, The Spectral Domain Method in Electromagnetics (Artech, Boston, Mass., 1989), Chaps. 1 and 2.
  15. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 25.
  16. A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering (Prentice-Hall, Englewood Cliffs, N.J., 1991), Chap. 6.

1990 (2)

1989 (1)

1988 (1)

K. Kobayashi, T. Inoue, “Diffraction of a plane wave by an inclined parallel plate grating,”IEEE Trans. Antennas Propag. 36, 1424–1434 (1988).
[Crossref]

1985 (1)

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

1980 (1)

1977 (1)

1976 (2)

R. S. Chu, T. Tamir, “Bragg diffraction of Gaussian beams by periodically modulated media,”J. Opt. Soc. Am. 66, 220–226 (1976).
[Crossref]

R. S. Chu, T. Tamir, “Diffraction of Gaussian beams by periodically modulated media for incidence angles close to Bragg angle,”J. Opt. Soc. Am. 66, 1348–1440 (1976).

1970 (1)

C. C. Chen, “Scattering by a two-dimensional periodic array of conducting plates,”IEEE Trans. Antennas Propag. AP-18, 660–665 (1970).
[Crossref]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 25.

Barber, P. W.

E. M. Khaled, S. C. Hill, P. W. Barber, “Scattered/internal intensity for dielectric object with Gaussian beam illlumination,” in Proceedings of the Eighth Annual Review of Progress in Applied Computational Electromagnetics (U.S. Naval Postgraduate School, Monterey, Calif., 1992), pp. 82–89.

Chen, C. C.

C. C. Chen, “Scattering by a two-dimensional periodic array of conducting plates,”IEEE Trans. Antennas Propag. AP-18, 660–665 (1970).
[Crossref]

Chrissoulidis, D. P.

E. E. Kriezis, D. P. Chrissoulidis, A. G. Papagiannakis, Electromagnetics and Optics (World Scientific, Singapore, 1992), Chap. 9.

Chu, R. S.

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, R. Magnusson, “Bragg diffraction of finite beams by thick gratings,”J. Opt. Soc. Am. 70, 300–304 (1980).
[Crossref]

Hill, S. C.

E. M. Khaled, S. C. Hill, P. W. Barber, “Scattered/internal intensity for dielectric object with Gaussian beam illlumination,” in Proceedings of the Eighth Annual Review of Progress in Applied Computational Electromagnetics (U.S. Naval Postgraduate School, Monterey, Calif., 1992), pp. 82–89.

Inoue, T.

K. Kobayashi, T. Inoue, “Diffraction of a plane wave by an inclined parallel plate grating,”IEEE Trans. Antennas Propag. 36, 1424–1434 (1988).
[Crossref]

Ishimaru, A.

A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering (Prentice-Hall, Englewood Cliffs, N.J., 1991), Chap. 6.

Khaled, E. M.

E. M. Khaled, S. C. Hill, P. W. Barber, “Scattered/internal intensity for dielectric object with Gaussian beam illlumination,” in Proceedings of the Eighth Annual Review of Progress in Applied Computational Electromagnetics (U.S. Naval Postgraduate School, Monterey, Calif., 1992), pp. 82–89.

Kobayashi, K.

K. Kobayashi, T. Inoue, “Diffraction of a plane wave by an inclined parallel plate grating,”IEEE Trans. Antennas Propag. 36, 1424–1434 (1988).
[Crossref]

Kojima, T.

Kong, J. A.

Kriezis, E. E.

E. E. Kriezis, D. P. Chrissoulidis, A. G. Papagiannakis, Electromagnetics and Optics (World Scientific, Singapore, 1992), Chap. 9.

Magnusson, R.

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, R. Magnusson, “Bragg diffraction of finite beams by thick gratings,”J. Opt. Soc. Am. 70, 300–304 (1980).
[Crossref]

Papagiannakis, A. G.

E. E. Kriezis, D. P. Chrissoulidis, A. G. Papagiannakis, Electromagnetics and Optics (World Scientific, Singapore, 1992), Chap. 9.

Petit, R.

Scott, G.

G. Scott, The Spectral Domain Method in Electromagnetics (Artech, Boston, Mass., 1989), Chaps. 1 and 2.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 25.

Tamir, T.

Tayeb, G.

Yariv, A.

A. Yariv, Optical Electronics (Saunders, Philadelphia, Pa., 1991), Chap. 2.

Zhang, S.

IEEE Trans. Antennas Propag. (2)

C. C. Chen, “Scattering by a two-dimensional periodic array of conducting plates,”IEEE Trans. Antennas Propag. AP-18, 660–665 (1970).
[Crossref]

K. Kobayashi, T. Inoue, “Diffraction of a plane wave by an inclined parallel plate grating,”IEEE Trans. Antennas Propag. 36, 1424–1434 (1988).
[Crossref]

J. Opt. Soc. Am. (4)

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

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 (6)

A. Yariv, Optical Electronics (Saunders, Philadelphia, Pa., 1991), Chap. 2.

E. M. Khaled, S. C. Hill, P. W. Barber, “Scattered/internal intensity for dielectric object with Gaussian beam illlumination,” in Proceedings of the Eighth Annual Review of Progress in Applied Computational Electromagnetics (U.S. Naval Postgraduate School, Monterey, Calif., 1992), pp. 82–89.

E. E. Kriezis, D. P. Chrissoulidis, A. G. Papagiannakis, Electromagnetics and Optics (World Scientific, Singapore, 1992), Chap. 9.

G. Scott, The Spectral Domain Method in Electromagnetics (Artech, Boston, Mass., 1989), Chaps. 1 and 2.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 25.

A. Ishimaru, Electromagnetic Wave Propagation, Radiation and Scattering (Prentice-Hall, Englewood Cliffs, N.J., 1991), Chap. 6.

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Figures (8)

Fig. 1
Fig. 1

Diffraction of a 2D Gaussian-profile beam from an array of conducting strips.

Fig. 2
Fig. 2

Relative error versus number (Nr) of plane waves. The grid consists of Np = 1250 points.

Fig. 3
Fig. 3

Equiamplitude contour map of the exact Gaussian beam.

Fig. 4
Fig. 4

Equiamplitude contour map of the diffracted electric-field intensity. φinc = 0°, dz = 0, dx = a/2, kLx = 6.0, w0/Lx = 1.6667. (a) γ = 0.9, (b) γ = 0.5.

Fig. 5
Fig. 5

Equiamplitude contour map of the diffracted electric-field intensity: φinc = 0°, dz = 0, dx = a/2, γ = 0.75, w0/Lx = 1.6667. (a) kLx = 6.0, (b) kLx = 12.0, (c) kLx = 18.0.

Fig. 6
Fig. 6

Equiamplitude contour map of the diffracted electric-field intensity: φinc = 45°, dz = 0, dx = a/2, kLx = 6.0, w0/Lx = 1.6667. (a) γ = 0.9, (b) γ = 0.5, (c) γ = 0.3.

Fig. 7
Fig. 7

Far-field radiation patterns. Parameters are set as in Figs. 5.

Fig. 8
Fig. 8

Far-field radiation patterns. Parameters are set as in Figs. 6.

Equations (26)

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E y inc ( x , z ) = E 0 inc w 0 w ( z ) exp [ - x 2 w 2 ( z ) ] exp [ - j k x 2 2 R ( z ) ] × exp { - j [ k z - tan - 1 ( z / z 0 ) ] } ,
w ( z ) = w 0 [ 1 + ( z z 0 ) 2 ] 1 / 2 ,             R ( z ) = z [ 1 + ( z 0 z ) 2 ] .
E y inc ( x , 0 ) = E 0 inc exp ( - x 2 w 0 2 ) .
F [ E y inc ( x , 0 ) ] = E 0 inc π w 0 exp ( - π 2 w 0 2 f x 2 ) .
E y inc ( x , y ) = E 0 inc π w 0 - exp ( - π 2 w 0 2 f x 2 ) × exp [ j 2 π ( f x x + f z z ) ] d f x .
E y inc ( x , y ) = E 0 inc w 0 2 π - k k exp ( - w 0 2 α 2 4 ) × exp [ - j ( α x + β z ) ] d α = E 0 inc k w 0 2 π - π / 2 π / 2 exp ( - w 0 2 k 2 sin 2 ϑ 4 ) × exp [ - j k ( sin ϑ x + cos ϑ z ) ] cos ϑ d ϑ .
E y inc ( x , y ) E 0 inc w 0 2 π n exp ( - w 0 2 α n 2 4 ) × exp [ - j ( α n x + β n z ) ] Δ α n = E 0 inc k w 0 2 π n exp ( - w 0 2 k 2 sin 2 ϑ n 4 ) × exp [ - j k ( sin ϑ n x + cos ϑ n z ) ] cos ϑ n Δ ϑ n .
x = ( x - d x ) cos φ inc + ( z - d z ) sin φ inc ,
z = ( x - d x ) sin φ inc - ( z - d z ) cos φ inc .
E y , n inc ( x , z ) = A n ( α n ) exp [ - j ( α n x - β n z ) ] , β n = ( k 2 - α n 2 ) 1 / 2 .
E y , n d , I ( x , z ) = m = - B n m exp [ - j ( α n m x + β n m z ) ] ,             z > 0 ,
E y , n d , II ( x , z ) = m = - C n m exp [ - j ( α n m x - β n m z ) ] ,             z < 0 ,
α n m = α n + 2 m π / L x , β n m = ( k 2 - α n m 2 ) 1 / 2 .
B n m + A n δ ( m ) = C n m ,
[ E y , n d , I ( x , z ) + E y , n inc ( x , z ) ] on strips = 0.
H = - 1 j ω μ × E ,
J = z ^ × [ H ( z = 0 + ) - H ( z = 0 - ) ] .
J n ( x ) = y ^ m = - J y , n m exp ( - j α n m x ) ,
B n m = - k 2 2 ω ɛ 1 ( k 2 - α n m 2 ) 1 / 2 J y , n m .
- k 2 2 ω ɛ m = - 1 ( k 2 - α n m ) 1 / 2 J y , n m exp ( - j α n m x ) = - A n exp ( - j α n x ) .
J n ( x ) = y ^ p F n p Ψ p ( x ) .
J y , n m = p F n p Ψ p , m ,
Ψ p . m = 1 L x strip Ψ p ( x ) exp ( j α n m x ) d x .
p F n p [ m = - Ψ r , m * - k 2 2 ω ɛ ( k 2 - α n m 2 ) 1 / 2 Ψ p , m ] = - A n Ψ r , 0 * ,
error = 1 N p 1 N p E ˜ y inc - E y inc 2 E y inc 2 ,
E y s ( r ) = screen exp ( - j k R ) 2 π R ( j k + 1 R ) z R E y ( r ) d s ,             r + R = r .

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