Abstract

A numerical integration method based on Kirchhoff’s diffraction integral is used to determine the approximate location and shape of so-called anomalies of a blazed grating in both polarizations. Although a scalar theory is used, the phase shift on reflection accounts for the differences in polarization. Multiple diffraction within the single grating groove accounts for the anomalous behavior. The theory is compared with new experimental data on two microwave gratings.

© 1972 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. W. Wood, Proc. Phys. Soc. (London) 18, 396 (1902); Philos. Mag. 4, 396 (1902).
  2. Rayleigh, Philos. Mag. 14, 60 (1907); Proc. Phys. Soc. (London) A79, 399 (1907).
  3. R. F. Millar, Proc. Cambridge Phil. Soc. 65, 773 (1969); Proc. Cambridge Phil. Soc. 69, 217 (1971).
    [CrossRef]
  4. A. Wirgin, R. Deleuil, J. Opt. Soc. Am. 59, 1348 (1969).
    [CrossRef]
  5. V. Twersky, IRE Trans. Antennas Propagation AP-4, 330 (1956); IRE Trans. Antennas Propagation AP-10, 737 (1962); J. Opt. Soc. Am. 52, 145 (1962).
    [CrossRef]
  6. C. H. Palmer, F. W. Phelps, J. Opt. Soc. Am. 58, 1184 (1968); J. Opt. Soc. Am. 59, 812 (1969).
    [CrossRef]
  7. D. Kerr, C. H. Palmer, J. Opt. Soc. Am. 61,(1971).
    [CrossRef]
  8. H. A. Kalhor, A. R. Neureuther, J. Opt. Soc. Am. 61, 43 (1971).
    [CrossRef]
  9. J. M. Stone, Radiation and Optics (McGraw-Hill, New York, 1963), p. 167.
  10. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 378.
  11. D. S. Jones, Theory of Electromagnetism (Pergamon, New York, 1964), pp. 322–326.
  12. C. H. Palmer, in Proceedings of The Symposium on Quasi-Optics, Jerome Fox, Ed. (Polytechnic Press, Brooklyn, 1964; Wiley, N.Y., Distr.).

1971 (2)

1969 (2)

A. Wirgin, R. Deleuil, J. Opt. Soc. Am. 59, 1348 (1969).
[CrossRef]

R. F. Millar, Proc. Cambridge Phil. Soc. 65, 773 (1969); Proc. Cambridge Phil. Soc. 69, 217 (1971).
[CrossRef]

1968 (1)

1956 (1)

V. Twersky, IRE Trans. Antennas Propagation AP-4, 330 (1956); IRE Trans. Antennas Propagation AP-10, 737 (1962); J. Opt. Soc. Am. 52, 145 (1962).
[CrossRef]

1907 (1)

Rayleigh, Philos. Mag. 14, 60 (1907); Proc. Phys. Soc. (London) A79, 399 (1907).

1902 (1)

R. W. Wood, Proc. Phys. Soc. (London) 18, 396 (1902); Philos. Mag. 4, 396 (1902).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 378.

Deleuil, R.

Jones, D. S.

D. S. Jones, Theory of Electromagnetism (Pergamon, New York, 1964), pp. 322–326.

Kalhor, H. A.

Kerr, D.

D. Kerr, C. H. Palmer, J. Opt. Soc. Am. 61,(1971).
[CrossRef]

Millar, R. F.

R. F. Millar, Proc. Cambridge Phil. Soc. 65, 773 (1969); Proc. Cambridge Phil. Soc. 69, 217 (1971).
[CrossRef]

Neureuther, A. R.

Palmer, C. H.

D. Kerr, C. H. Palmer, J. Opt. Soc. Am. 61,(1971).
[CrossRef]

C. H. Palmer, F. W. Phelps, J. Opt. Soc. Am. 58, 1184 (1968); J. Opt. Soc. Am. 59, 812 (1969).
[CrossRef]

C. H. Palmer, in Proceedings of The Symposium on Quasi-Optics, Jerome Fox, Ed. (Polytechnic Press, Brooklyn, 1964; Wiley, N.Y., Distr.).

Phelps, F. W.

Rayleigh,

Rayleigh, Philos. Mag. 14, 60 (1907); Proc. Phys. Soc. (London) A79, 399 (1907).

Stone, J. M.

J. M. Stone, Radiation and Optics (McGraw-Hill, New York, 1963), p. 167.

Twersky, V.

V. Twersky, IRE Trans. Antennas Propagation AP-4, 330 (1956); IRE Trans. Antennas Propagation AP-10, 737 (1962); J. Opt. Soc. Am. 52, 145 (1962).
[CrossRef]

Wirgin, A.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 378.

Wood, R. W.

R. W. Wood, Proc. Phys. Soc. (London) 18, 396 (1902); Philos. Mag. 4, 396 (1902).

IRE Trans. Antennas Propagation (1)

V. Twersky, IRE Trans. Antennas Propagation AP-4, 330 (1956); IRE Trans. Antennas Propagation AP-10, 737 (1962); J. Opt. Soc. Am. 52, 145 (1962).
[CrossRef]

J. Opt. Soc. Am. (4)

Philos. Mag. (1)

Rayleigh, Philos. Mag. 14, 60 (1907); Proc. Phys. Soc. (London) A79, 399 (1907).

Proc. Cambridge Phil. Soc. (1)

R. F. Millar, Proc. Cambridge Phil. Soc. 65, 773 (1969); Proc. Cambridge Phil. Soc. 69, 217 (1971).
[CrossRef]

Proc. Phys. Soc. (London) (1)

R. W. Wood, Proc. Phys. Soc. (London) 18, 396 (1902); Philos. Mag. 4, 396 (1902).

Other (4)

J. M. Stone, Radiation and Optics (McGraw-Hill, New York, 1963), p. 167.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1959), p. 378.

D. S. Jones, Theory of Electromagnetism (Pergamon, New York, 1964), pp. 322–326.

C. H. Palmer, in Proceedings of The Symposium on Quasi-Optics, Jerome Fox, Ed. (Polytechnic Press, Brooklyn, 1964; Wiley, N.Y., Distr.).

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

(A) Geometry for Kirchhoff’s diffraction formula in two dimensions. P0(r) is the source and P(s) the point of observation. Normals are drawn outward from the area between Λa and Λb (i.e., n ˆ b inward toward P). (B) Geometry as applied to a plane surface indicating the direction of the normal, the real source P0, image source P0′, and point of observation P.

Fig. 2
Fig. 2

Grating groove geometry with ray incident from P0 diffracted to P. Incident and diffracted rays are assumed plane.

Fig. 3
Fig. 3

Various multiple diffraction terms from the grating grooves. Incident field U0 on Λ1 at right gives rise to U1, U3, and U5 (single, double, and triple diffractions). Incident U0 on Λ2 at left gives rise to U2, U4, and U6.

Fig. 4
Fig. 4

Theoretical (- - -) and experimental (——) curves of radiance for blazed grating; zero order, S polarization as a function of angle of incidence θi.

Fig. 5
Fig. 5

Theoretical (- - -) and experimental (——) curves of radiance for blazed grating; +1 order, S polarization.

Fig. 6
Fig. 6

Theoretical (- - -) and experimental (——) curves of radiance for blazed grating; zero order, P polarization.

Fig. 7
Fig. 7

Theoretical (- - -) and experimental (——) curves of radiance for blazed grating; +1 order, P polarization.

Equations (28)

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

τ ( U 2 V V 2 U ) d τ = s ( U V n V U n ) d s .
Λ ( U V n V U n ) d l = 0 ,
V ( s ) = i π H 0 ( 2 ) ( k s ) .
U ( P ) = 1 4 π Λ [ U ( r ) V ( s ) n V ( s ) U ( r ) n ] d l .
U ( P ) = 1 4 π Λ [ U ( r ) V ( s ) n + V ( s ) U ( r ) n ] d l .
r = r 0 ξ sin ( β + θ i ) , s = s 0 ξ sin ( β + θ r ) , r = r 0 η sin ( α θ i ) , s = s 0 η sin ( α θ r ) ,
cos ( n 1 , r ) = cos ( β + θ i ) , cos ( n 1 , s ) = cos ( β + θ r ) , cos ( n 2 , r ) = cos ( α θ i ) , cos ( n 2 , s ) = cos ( α θ r ) .
U 1 = ( k K / 4 ) 0 ξ 0 exp [ i k ( r + s ) ] [ cos ( n 1 , s ) + cos ( n 1 , r ) ] d ξ .
U 1 = ( k K / 4 ) 0 ξ 0 exp ( i k ξ A ) [ cos ( β + θ i ) + cos ( β + θ r ) ] d ξ ,
A sin ( β + θ i ) + sin ( β + θ r ) .
U 1 = ( i K / 4 A ) [ cos ( β + θ i ) + cos ( β + θ r ) ] × [ cos ( k ξ 0 A ) 1 + i sin ( k ξ 0 A ) ] .
U 2 = ( i K / 4 B ) [ cos ( α θ i ) + cos ( α θ r ) ] × [ cos ( k η 0 B ) 1 + i sin ( k η 0 B ) ] ,
B sin ( α θ i ) + sin ( α θ r ) .
t or υ = [ ξ 2 + η 2 + 2 ξ η cos ( α + β ) ] 1 2 .
U ( η ) = ( i k K / 4 ) I A ,
I A 0 ξ 0 [ H 1 ( k t ) cos ( n 1 , t ) + i H 0 ( k t ) cos ( β + θ i ) ] × exp [ i k ξ sin ( β + θ i ) ] d ξ .
U ( η ) / n 2 = + ( i k 2 K / 4 ) I B ,
I B 0 ξ 0 [ F ( k t ) cos ( n 1 , t ) cos ( n 2 , t ) + i H 1 ( k t ) cos ( n 2 , t ) cos ( β + θ i ) ] × exp [ i k ξ sin ( β + θ i ) ] d ξ ,
F ( k t ) ( 1 2 ) ( H 2 ( k t ) H 0 ( k t ) ] .
U 3 = ( i k 2 K / 16 ) 0 η 0 [ I A cos ( α θ r ) + i I B ] × exp [ i k η sin ( α θ r ) ] d η .
U ( ζ ) = ( k 2 K / 16 ) I C
U ( ζ ) / n 1 = ( k 3 K / 16 ) I D ,
I C = 0 η 0 [ I A H 1 ( k υ ) cos ( n 2 , υ ) + I B H 0 ( k υ ) ] d η
I D = 0 η 0 [ I A F ( k υ ) cos ( n 2 , υ ) cos ( n 1 , υ ) + I B H 1 ( k υ ) cos ( n 1 , υ ) ] d η .
U 5 = ( k 3 K / 64 ) 0 ξ 0 [ I C cos ( β + θ r ) + i I D ] × exp [ i k ξ sin ( β + θ r ) ] d ζ .
Re H 0 ( k x ) = J 0 ( k x ) , Im H 0 ( k x ) = N 0 ( k x ) , Re H 1 ( k x ) cos ( n 1 , x ) , Im H 1 ( k x ) cos ( n 1 , x ) , Re H 1 ( k x ) cos ( n 2 , x ) , Im H 1 ( k x ) cos ( n 2 , x ) , Re [ F ( k x ) ] cos ( n 1 , x ) cos ( n 2 , x ) , Im [ F ( k x ) ] cos ( n 1 , x ) cos ( n 2 , x ) .
U 3 = ( i k 2 K / 16 ) 0 ξ 0 0 η 0 [ H 1 ( k t ) cos ( n 1 , t ) cos ( α θ r ) + i H 0 ( k t ) cos ( β + θ i ) cos ( α θ r ) i F ( k t ) cos ( n 1 , t ) cos ( n 2 , t ) + H 1 ( k t ) cos ( n 2 , t ) cos ( β + θ i ) ] × exp { i k [ ξ sin ( β + θ i ) + η sin ( α θ r ) ] } d η d ξ .
U 3 * = ( i k 2 K / 16 ) 0 ξ 0 0 η 0 [ H 1 ( k t ) cos ( n 1 , t ) cos ( α θ i ) + i H 0 ( k t ) cos ( β + θ r ) cos ( α θ i ) i F ( k t ) cos ( n 1 , t ) cos ( n 2 , t ) + H 1 ( k t ) cos ( n 2 , t ) cos ( β + θ r ) ] × exp { i k [ ξ sin ( β + θ r ) + η sin ( α θ i ) ] } d η d ξ .

Metrics