Abstract

Scalar diffraction theory and electromagnetic vector theory are compared by analyzing plane-wave scattering by a perfectly conducting, rectangular-grooved grating. General field solutions for arbitrary angles of incidence are derived by using scalar and vector theories. Diffraction efficiencies for the scalar and the vector cases as functions of wavelength, grating period, and angles of incidence are determined numerically and plotted. When the wavelength of the incident field is much shorter than the grating period, the diffraction efficiencies match. But when the wavelength is of the order of the grating period, large differences between the scalar and the vector solutions emerge. One general conclusion is that, depending on polarization, scalar theory should not be used when the grating period becomes smaller than ten wavelengths.

© 1993 Optical Society of America

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References

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  1. A. W. Lohmann, D. P. Paris, “Binary Fraunhofer holograms, generated by computer,” Appl. Opt., 6, 1739–1748 (1967).
    [CrossRef] [PubMed]
  2. Y. L. Kok, N. C. Gallagher, “Relative phases of electromagnetic waves diffracted by a perfectly conducting rectangular-grooved grating,” J. Opt. Soc. Am. B 5, 65–73 (1988).
    [CrossRef]
  3. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  4. R. Petit, ed., Electromagnetic Theory of Grating, Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  5. S. Sohail, H. Naqvi, N. C. Gallagher, “Comments on electromagnetic wave scattering by an infinite plane metallic grating in case of oblique incidence and arbitrary polarization,” IEEE Trans. Antennas Propag. (to be published).
  6. 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 Propag. 37, 901–917 (1989).
    [CrossRef]

1989 (1)

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 Propag. 37, 901–917 (1989).
[CrossRef]

1988 (1)

Y. L. Kok, N. C. Gallagher, “Relative phases of electromagnetic waves diffracted by a perfectly conducting rectangular-grooved grating,” J. Opt. Soc. Am. B 5, 65–73 (1988).
[CrossRef]

1967 (1)

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 Propag. 37, 901–917 (1989).
[CrossRef]

Y. L. Kok, N. C. Gallagher, “Relative phases of electromagnetic waves diffracted by a perfectly conducting rectangular-grooved grating,” J. Opt. Soc. Am. B 5, 65–73 (1988).
[CrossRef]

S. Sohail, H. Naqvi, N. C. Gallagher, “Comments on electromagnetic wave scattering by an infinite plane metallic grating in case of oblique incidence and arbitrary polarization,” IEEE Trans. Antennas Propag. (to be published).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Kok, Y. L.

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 Propag. 37, 901–917 (1989).
[CrossRef]

Y. L. Kok, N. C. Gallagher, “Relative phases of electromagnetic waves diffracted by a perfectly conducting rectangular-grooved grating,” J. Opt. Soc. Am. B 5, 65–73 (1988).
[CrossRef]

Lohmann, A. W.

Naqvi, H.

S. Sohail, H. Naqvi, N. C. Gallagher, “Comments on electromagnetic wave scattering by an infinite plane metallic grating in case of oblique incidence and arbitrary polarization,” IEEE Trans. Antennas Propag. (to be published).

Paris, D. P.

Sohail, S.

S. Sohail, H. Naqvi, N. C. Gallagher, “Comments on electromagnetic wave scattering by an infinite plane metallic grating in case of oblique incidence and arbitrary polarization,” IEEE Trans. Antennas Propag. (to be published).

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 Propag. 37, 901–917 (1989).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (1)

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 Propag. 37, 901–917 (1989).
[CrossRef]

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

Y. L. Kok, N. C. Gallagher, “Relative phases of electromagnetic waves diffracted by a perfectly conducting rectangular-grooved grating,” J. Opt. Soc. Am. B 5, 65–73 (1988).
[CrossRef]

Other (3)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

R. Petit, ed., Electromagnetic Theory of Grating, Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980).
[CrossRef]

S. Sohail, H. Naqvi, N. C. Gallagher, “Comments on electromagnetic wave scattering by an infinite plane metallic grating in case of oblique incidence and arbitrary polarization,” IEEE Trans. Antennas Propag. (to be published).

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Figures (16)

Fig. 1
Fig. 1

Lohmann hologram with a rectangular array of apertures.

Fig. 2
Fig. 2

Model of the perfectly conducting rectangular-grooved grating with period d, groove width c, and groove depth h. The grating has infinite extent in the xz plane.

Fig. 3
Fig. 3

Grating parameters: k, normal to the incident plane wave, which indicates the direction of propagation; ϕ, angle between k and its projection in the xy plane; θ, angle between the projection in the xy plane and the y axis.

Fig. 4
Fig. 4

Grating model used in the scalar-theory diffraction analysis, which is equivalent to the grating shown in Fig. 2, with the same period d and groove width c. Ψ and 0 are the relative phase shifts determined by the groove depth h and the wavelength λ.

Fig. 5
Fig. 5

Zeroth-order diffraction efficiency according to scalar theory with normal incidence and with c = d/2 and h = d/4 versus λ/d. Since the equations for diffraction efficiency are independent of θ and ϕ, this plot is valid for arbitrary angles of incidence.

Fig. 6
Fig. 6

Same as Fig. 5 but for first-order diffraction efficiency.

Fig. 7
Fig. 7

Zeroth-order diffraction efficiency for the fast-polarization case and normal incidence according to vector theory with c = d/2 and h = d/4 versus λ/d. The diffraction efficiency equals 1 for λ/d > 1.

Fig. 8
Fig. 8

Same as Fig. 7 but for first-order diffraction efficiency.

Fig. 9
Fig. 9

Same as Fig. 7 but for the slow-polarization case. The small sharp spike at λ/d = 1/3 is due to a singular matrix for that value of λ/d.

Fig. 10
Fig. 10

Same as Fig. 9 but for the first-order diffraction efficiency.

Fig. 11
Fig. 11

Overlay of Figs. 5, 7, and 9 so that easier comparisons can be made between the different cases. The dashed curve is the fast-polarization zeroth-order diffraction efficiency, the solid curve that approaches 1 for λ/d = 1 is the slow-polarization case, and the other solid curve is the scalar solution.

Fig. 12
Fig. 12

Zeroth-order diffraction efficiency according to scalar theory with c = d/2 and h = d/4 versus λ/d. The angles of incidence are θ = 60° and ϕ = 0°. The diffraction efficiency equals 1 for λ/d > 1.50.

Fig. 13
Fig. 13

Overlay of all three cases for the first-order diffraction efficiency with normal incidence. The dashed curve represents fast polarization and the solid curve with the higher peaks represents slow polarization. The other solid curve represents the scalar solution.

Fig. 14
Fig. 14

Zeroth-order diffraction efficiency for the fast-polarization case according to vector theory with c = d/2 and h = d/4 versus λ/d. The angles of incidence are θ = 60° and ϕ = 0°. The diffraction efficiency equals 1 for λ/d > 1.50. θ has a large effect on the overall shape of the plot as compared with that in Fig. 7.

Fig. 15
Fig. 15

Same as Fig. 14 but for the slow-polarization case. θ has a large effect on the overall shape of the plot as compared with that in Fig. 9. Also, the first peak (0.20 < λ/d < 0.33) is only two thirds of the value of the corresponding peak (0.20 < λ/d < 0.33) in Fig. 9.

Fig. 16
Fig. 16

Same as Fig. 15 but for angles of incidence of θ = 60° and ϕ = 70°. This plot is identical to that in Fig. 14 except for the values of λ/d. In fact, if the diffraction efficiency was plotted versus (λ/d)cos(ϕ), the two plots would be identical.

Equations (20)

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Ψ = 4 π h λ cos ϕ sin θ = 2 k h ( cos ϕ sin θ ) .
U inc ( x , y , z ) = exp [ i ( α 0 x - β 0 y + γ 0 z ) ] ,
α 0 2 + β 0 2 + γ 0 2 = k 2 = ( 2 π λ ) 2 .
α 0 = k cos ϕ cos θ , β 0 = k cos ϕ sin θ , γ 0 = k sin ϕ .
U ( x , y , z ) = n = - ρ n exp [ i 2 π ( n d + α 0 2 π ) x + i γ 0 z ] × exp { i 2 π [ 1 λ 2 - ( n d + α 0 2 π ) 2 - ( γ 0 2 π ) 2 ] 1 / 2 y } ,
ρ n = { 1 - c d [ 1 - exp ( i Ψ ) ] , n = 0 1 2 π n [ exp ( i Ψ ) - 1 ] [ exp ( - i 2 π n c d ) - 1 ] , n 0 .
1 λ 2 - ( n d + α 0 2 π ) 2 - ( γ 0 2 π ) 2 0 ,
E fast = ( E x , E y , E z ) H fast = ( H x , H y , 0 ) , E slow = ( E x , E y , 0 ) , H slow = ( H x , H y , H z ) .
E x = i γ 0 a 2 E z x + i ω μ 0 a 2 H z y , H x = i γ 0 a 2 H z x - i ω 0 a 2 E z y , E y = i γ 0 a 2 E z y - i ω μ 0 a 2 H z x , H y = i γ 0 a 2 H z y + i ω 0 a 2 E z x ,
E z ( x , y , z ) = cos 2 ϕ { exp [ i ( α 0 x - β 0 y + γ 0 z ) ] + n = - r n exp [ i ( α n x + β n y + γ 0 z ) ] } ,
H z ( x , y , z ) = cos 2 ϕ { exp [ i ( α 0 x - β 0 y + γ 0 z ) ] + n = - s n exp [ i ( α n x + β n y + γ 0 z ) ] } ,
E z ( x , y , z ) = { n = 1 a n sin ( n π x c ) sin ( A n y ) exp ( i z γ 0 )             0 x c 0             c < x d ,
H z ( x , y , z ) = n = 0 b n cos ( n π x c ) cos ( B n y ) exp ( i z γ 0 ) ,             0 x c ,
cos 2 ϕ [ δ n 0 exp ( - i 2 β 0 h ) + r n ] = m = 1 a m d sin ( A m h ) exp ( - i β n h ) I n m ,
m = - cos 2 ϕ [ - i β 0 δ m 0 exp ( - i β 0 h ) + r m i β m × exp ( i β m h ) ] c n - m = m = 1 a m A m d cos ( A m h ) I n m ,
cos 2 ϕ [ - i β 0 δ n 0 exp ( - i 2 β 0 h ) + i β n s n ] = m = 0 - b m B m d sin ( B m h ) exp ( - i β n h ) J n m ,
m = - cos 2 ϕ [ δ m 0 exp ( - i β 0 h ) + s m exp ( i β m h ) ] c n - m = m = 0 b m d cos ( B m h ) J n m
I n m = 0 c sin ( m π x c ) exp ( - i α n x ) d x , J n m = 0 c cos ( m π x c ) exp ( - i α n x ) d x , c n = { 1 i 2 π n [ 1 - exp ( - i n 2 π c d ) ]             n 0 c d             n = 0 , J n m = { 1 n = m 0 n m .
n = - ρ n 2 = 1
Ψ = 4 π h λ cos ϕ sin θ ,

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