Abstract

Diffraction gratings are known to exhibit anomalous behavior at certain critical wavelengths or incident angles. These traditional anomalies manifest themselves as abrupt variations in diffracted order efficiency or grating absorption, while their angular position remains unchanged as predicted by the grating equation. Experimental observations have been reported, indicating a diffraction grating anomaly in the angular position of certain diffracted orders that appears to violate the grating equation. Several exotic physical mechanisms have been suggested as possible causes of this intriguing behavior; however, in this paper we show that this angular grating anomaly is the straightforward result of finite beam size on wide-angle diffraction phenomena, as described by simple scalar diffraction theory.

© 1992 Optical Society of America

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References

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  1. R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–410 (1902).
    [CrossRef]
  2. J. W. S. Rayleigh, “Note on the remarkable case of diffraction spectra discovered by Prof. Wood,” Philos. Mag. 14, 60–65 (1907).
    [CrossRef]
  3. M. L. Scott, R. B. Kwong, G. B. Charlton, “Anomalous grating absorption,” Laser Digest AFWL-TR-81-9 (U.S. Air Force Weapons Laboratory, Kirtland Air Force Base, N.M., May1981).
  4. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, New York, 1980).
    [CrossRef]
  5. A. D. Boardman, ed., Electromagnetic Surface Modes (Wiley, New York, 1982).
  6. N. E. Glass, A. A. Maradudin, “Theory of surface-polariton resonances and field enhancements in light scattering from bigratings,” J. Opt. Soc. Am. 73, 1240–1248 (1983).
    [CrossRef]
  7. E. A. Nevis, J. E. Harvey, “Angular grating anomalies: an apparent violation of the grating equation,” in Application, Theory, and Fabrication of Periodic Structures, Diffraction Gratings, and Moire Phenomena II, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.503, 46–52 (1984).
  8. R. A. House, R. D. Petty, J. J. Johnson, B. R. Key, G. J. Denton, “Grating Efficiency Measurement and Automated Scatter Inspection System (GEMASIS),” in Diffraction Phenomena in Optical Engineering Applications, D. M. Byrne, J. E. Harvey, eds., Proc. Soc. Photo-Opt. Instrum. Eng.560, 52–62 (1985).
  9. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3, p. 48.
  10. J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
    [CrossRef]
  11. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965), Chap. 4, p. 51.
  12. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), Chap. 3, p. 40.
  13. R. D. Hudson, Infrared System Engineering (Wiley, New York, 1969), Chap. 1, p. 29.
  14. V. Rehn, “Grazing-incidence optics for synchrotron-radiation insertion-device beams,” in Grazing Incidence Optics, J. F. Osantowski, L. Van Speybroeck, eds., Proc. Soc. Photo-Opt. Instrum. Eng.640, 106–115 (1986).

1983

1979

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

1907

J. W. S. Rayleigh, “Note on the remarkable case of diffraction spectra discovered by Prof. Wood,” Philos. Mag. 14, 60–65 (1907).
[CrossRef]

1902

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–410 (1902).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965), Chap. 4, p. 51.

Charlton, G. B.

M. L. Scott, R. B. Kwong, G. B. Charlton, “Anomalous grating absorption,” Laser Digest AFWL-TR-81-9 (U.S. Air Force Weapons Laboratory, Kirtland Air Force Base, N.M., May1981).

Denton, G. J.

R. A. House, R. D. Petty, J. J. Johnson, B. R. Key, G. J. Denton, “Grating Efficiency Measurement and Automated Scatter Inspection System (GEMASIS),” in Diffraction Phenomena in Optical Engineering Applications, D. M. Byrne, J. E. Harvey, eds., Proc. Soc. Photo-Opt. Instrum. Eng.560, 52–62 (1985).

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), Chap. 3, p. 40.

Glass, N. E.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3, p. 48.

Harvey, J. E.

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

E. A. Nevis, J. E. Harvey, “Angular grating anomalies: an apparent violation of the grating equation,” in Application, Theory, and Fabrication of Periodic Structures, Diffraction Gratings, and Moire Phenomena II, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.503, 46–52 (1984).

House, R. A.

R. A. House, R. D. Petty, J. J. Johnson, B. R. Key, G. J. Denton, “Grating Efficiency Measurement and Automated Scatter Inspection System (GEMASIS),” in Diffraction Phenomena in Optical Engineering Applications, D. M. Byrne, J. E. Harvey, eds., Proc. Soc. Photo-Opt. Instrum. Eng.560, 52–62 (1985).

Hudson, R. D.

R. D. Hudson, Infrared System Engineering (Wiley, New York, 1969), Chap. 1, p. 29.

Johnson, J. J.

R. A. House, R. D. Petty, J. J. Johnson, B. R. Key, G. J. Denton, “Grating Efficiency Measurement and Automated Scatter Inspection System (GEMASIS),” in Diffraction Phenomena in Optical Engineering Applications, D. M. Byrne, J. E. Harvey, eds., Proc. Soc. Photo-Opt. Instrum. Eng.560, 52–62 (1985).

Key, B. R.

R. A. House, R. D. Petty, J. J. Johnson, B. R. Key, G. J. Denton, “Grating Efficiency Measurement and Automated Scatter Inspection System (GEMASIS),” in Diffraction Phenomena in Optical Engineering Applications, D. M. Byrne, J. E. Harvey, eds., Proc. Soc. Photo-Opt. Instrum. Eng.560, 52–62 (1985).

Kwong, R. B.

M. L. Scott, R. B. Kwong, G. B. Charlton, “Anomalous grating absorption,” Laser Digest AFWL-TR-81-9 (U.S. Air Force Weapons Laboratory, Kirtland Air Force Base, N.M., May1981).

Maradudin, A. A.

Nevis, E. A.

E. A. Nevis, J. E. Harvey, “Angular grating anomalies: an apparent violation of the grating equation,” in Application, Theory, and Fabrication of Periodic Structures, Diffraction Gratings, and Moire Phenomena II, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.503, 46–52 (1984).

Petty, R. D.

R. A. House, R. D. Petty, J. J. Johnson, B. R. Key, G. J. Denton, “Grating Efficiency Measurement and Automated Scatter Inspection System (GEMASIS),” in Diffraction Phenomena in Optical Engineering Applications, D. M. Byrne, J. E. Harvey, eds., Proc. Soc. Photo-Opt. Instrum. Eng.560, 52–62 (1985).

Rayleigh, J. W. S.

J. W. S. Rayleigh, “Note on the remarkable case of diffraction spectra discovered by Prof. Wood,” Philos. Mag. 14, 60–65 (1907).
[CrossRef]

Rehn, V.

V. Rehn, “Grazing-incidence optics for synchrotron-radiation insertion-device beams,” in Grazing Incidence Optics, J. F. Osantowski, L. Van Speybroeck, eds., Proc. Soc. Photo-Opt. Instrum. Eng.640, 106–115 (1986).

Scott, M. L.

M. L. Scott, R. B. Kwong, G. B. Charlton, “Anomalous grating absorption,” Laser Digest AFWL-TR-81-9 (U.S. Air Force Weapons Laboratory, Kirtland Air Force Base, N.M., May1981).

Wood, R. W.

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–410 (1902).
[CrossRef]

Am. J. Phys.

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

J. Opt. Soc. Am.

Philos. Mag.

R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–410 (1902).
[CrossRef]

J. W. S. Rayleigh, “Note on the remarkable case of diffraction spectra discovered by Prof. Wood,” Philos. Mag. 14, 60–65 (1907).
[CrossRef]

Other

M. L. Scott, R. B. Kwong, G. B. Charlton, “Anomalous grating absorption,” Laser Digest AFWL-TR-81-9 (U.S. Air Force Weapons Laboratory, Kirtland Air Force Base, N.M., May1981).

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, New York, 1980).
[CrossRef]

A. D. Boardman, ed., Electromagnetic Surface Modes (Wiley, New York, 1982).

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965), Chap. 4, p. 51.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), Chap. 3, p. 40.

R. D. Hudson, Infrared System Engineering (Wiley, New York, 1969), Chap. 1, p. 29.

V. Rehn, “Grazing-incidence optics for synchrotron-radiation insertion-device beams,” in Grazing Incidence Optics, J. F. Osantowski, L. Van Speybroeck, eds., Proc. Soc. Photo-Opt. Instrum. Eng.640, 106–115 (1986).

E. A. Nevis, J. E. Harvey, “Angular grating anomalies: an apparent violation of the grating equation,” in Application, Theory, and Fabrication of Periodic Structures, Diffraction Gratings, and Moire Phenomena II, J. M. Lerner, ed., Proc. Soc. Photo-Opt. Instrum. Eng.503, 46–52 (1984).

R. A. House, R. D. Petty, J. J. Johnson, B. R. Key, G. J. Denton, “Grating Efficiency Measurement and Automated Scatter Inspection System (GEMASIS),” in Diffraction Phenomena in Optical Engineering Applications, D. M. Byrne, J. E. Harvey, eds., Proc. Soc. Photo-Opt. Instrum. Eng.560, 52–62 (1985).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3, p. 48.

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Figures (9)

Fig. 1
Fig. 1

(a) No angular anomaly observed for any order; (b) angular anomaly observed for the +1 diffracted order; (c) angular anomaly observed for the −2 diffracted order.

Fig. 2
Fig. 2

Magnitude of the measured angular anomaly versus the predicted diffraction angle.

Fig. 3
Fig. 3

Schematic representation of the plane-wave spectrum produced by (a) an incident beam of infinite lateral extent; (b) an incident beam of finite lateral extent.

Fig. 4
Fig. 4

Geometrical relationship between the incident beam, diffracting aperture, and the observation hemisphere.

Fig. 5
Fig. 5

Illustration of a narrow Gaussian beam incident on a diffraction grating at an arbitrary angle, and the resulting broadened diffracted order at large diffracted angles on the observation hemisphere.

Fig. 6
Fig. 6

Experimentally measured diffraction profiles (d = 0.8333 μm, λ = 0.6328 μm, θ0 = 13.00°) illustrating increased width at large diffracted angles.

Fig. 7
Fig. 7

Asymmetrical irradiance distribution of wide-angle diffracted orders exhibits a substantial angular shift of its centroid from the diffracted angle predicted by the grating equation.

Fig. 8
Fig. 8

Angular position of the detector (measured position of diffracted order) yielding a maximum signal for the situation in which θm = 89°.

Fig. 9
Fig. 9

Comparison of scalar diffraction predictions of angular grating anomaly with measured observations.

Equations (17)

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sin θ m + sin θ i = m λ / d
U ( α , β ; r ^ ) = γ [ exp ( i 2 π r ^ ) / i r ^ ] - U 0 ( x ^ , y ^ ; 0 ) × exp [ i 2 π ( α x ^ + β y ^ ) ] d x ^ d y ^ ,
U ( α , β - β 0 ; r ^ ) = γ [ exp ( i 2 π r ^ ) / i r ^ ] × F { U 0 ( x ^ , y ^ ; 0 ) exp ( i 2 π β 0 y ^ ) } ,
β m + β i = m / d ^ ,
α m + α i = 0.
t 0 ( x ^ , y ^ ; 0 ) = rect ( y ^ a ^ ) * 1 d ^ comb ( y ^ d ^ ) .
U 0 - ( x ^ , y ^ ; 0 ) = exp [ - ( x ^ / w ^ ) 2 ] exp [ - ( y ^ cos θ 0 / w ^ ) 2 ] × exp ( i 2 π β 0 y ^ ) .
U 0 + ( x ^ , y ^ ; 0 ) = U 0 - ( x ^ , y ^ ; 0 ) t 0 ( x ^ , y ^ ; 0 ) .
U ( α , β ; r ^ ) = γ [ exp ( i 2 π r ^ ) / i r ^ ] m ( B m exp [ - ( π w ^ α ) 2 ] × exp { [ π w ^ cos θ 0 ( β - β m ) ] 2 } ) ,
I ( α , β ) = d p d ω = ( r ^ 2 / γ ) U ( α , β ; r ^ ) 2 .
I ( α , β ) = γ m B m 2 exp [ - 2 ( π w ^ α ) 2 ] × exp { - 2 [ π w ^ ( β - β m ) / cos θ 0 ] 2 } .
I ( 0 , β ) = γ m B m 2 exp { - 2 [ ( β - β m ) / Δ β ] 2 } ,
Δ β = ( cos θ 0 ) / ( π w ^ ) .
I ( θ ) = γ m B m 2 exp { - 2 [ ( sin θ - sin θ m ) / Δ β ] 2 } .
sin θ - sin θ m = 2 sin ½ ( θ - θ m ) cos ½ ( θ + θ m ) ( θ - θ m ) cos θ m ,
I ( θ ) = γ m B m 2 exp { - 2 [ ( θ - θ m ) / Δ θ m ] 2 } ,
Δ θ m = Δ β / cos θ m = 1 π w ^ cos θ 0 cos θ m .

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