Abstract

Scalar diffraction theory is frequently considered inadequate for predicting diffraction efficiencies for grating applications where λd>0.1. It has also been stated that scalar theory imposes energy upon the evanescent diffracted orders. These notions, as well as several other common misconceptions, are driven more by an unnecessary paraxial approximation in the traditional Fourier treatment of scalar diffraction theory than by the scalar limitation. By scaling the spatial variables by the wavelength, we have previously shown that diffracted radiance is shift invariant in direction cosine space. Thus simple Fourier techniques can now be used to predict a variety of wide-angle (nonparaxial) diffraction grating effects. These include (1) the redistribution of energy from the evanescent orders to the propagating ones, (2) the angular broadening (and apparent shifting) of wide-angle diffracted orders, and (3) nonparaxial diffraction efficiencies predicted with an accuracy usually thought to require rigorous electromagnetic theory.

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References

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  1. R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, 1980), p. 98.
  2. P. Beckman and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963).
  3. J. M. Bennett and L. Mattsson, Introduction to Surface Roughness and Scattering, 2nd ed. (Optical Society of America, 1999).
  4. J. C. Stover, Optical Scattering, Measurement and Analysis, 2nd ed. (SPIE, 1995).
    [CrossRef]
  5. D. A. Gremaux and N. C. Gallager, "Limits of scalar diffraction theory for conducting gratings," Appl. Opt. 32, 1048-1953 (1993).
    [CrossRef]
  6. D. A. Pommet, M. G. Moharam, and E. B. Grann, "Limits of scalar diffraction theory for diffractive phase elements," J. Opt. Soc. Am. A 11, 1827-1834 (1994).
    [CrossRef]
  7. E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997).
  8. S. D. Mellin and G. P. Nordin, "Limits of scalar diffraction theory and an iterative angular spectrum algorithm for finite aperture diffractive optical element design," Opt. Express 8, 705-722 (2001).
    [CrossRef] [PubMed]
  9. D. Maystre, "Rigorous vector theories of diffraction gratings," in Progress in Optics XXI, E.Wolf, ed. (Elsevier Science, 1984).
    [CrossRef]
  10. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  11. M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), p. 598.
  12. T. K. Gaylord and M. G. Moharam, "Analysis and applications of optical diffraction by gratings," Proc. IEEE 73, 894-937 (1985).
  13. C. V. Raman and N. S. N. Nath, "The diffraction of light by high frequency sound waves," Proc. Ind. Acad. Sci. A 2, 406-413 (1935).
  14. C. Palmer, Diffraction Grating Handbook, 4th ed. (Richardson Grating Laboratory, 2000), p. 15.
  15. J. E. Harvey, "Fourier treatment of near-field scalar diffraction theory," Am. J. Phys. 47, 974-980 (1979).
    [CrossRef]
  16. J. E. Harvey, C. L. Vernold, A. Krywonos, and P. L. Thompson, "Diffracted radiance: a fundamental quantity in a nonparaxial scalar diffraction theory," Appl. Opt. 38, 6469-6481 (1999).
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  17. J. E. Harvey, C. L. Vernold, A. Krywonos, and P. L. Thompson, "Diffracted radiance: a fundamental quantity in a nonparaxial scalar diffraction theory: errata," Appl. Opt. 39, 6374-6375 (2000).
    [CrossRef]
  18. J. A. Ratcliff, "Some aspects of diffraction theory and their application to the ionosphere," in Reports on Progress in Physics, A.C.Strickland, ed. (Physical Society, 1956), Vol. XIX.
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  22. R. W. Wood, "On a remarkable case of uneven distribution of light in a diffraction grating spectrum," Philos. Mag. 4, 396-410 (1902).
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2002

2001

2000

1999

1998

1994

1993

1992

1979

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

1978

1965

1902

R. W. Wood, "On a remarkable case of uneven distribution of light in a diffraction grating spectrum," Philos. Mag. 4, 396-410 (1902).

Beckman, P.

P. Beckman and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963).

Bennett, J. M.

J. M. Bennett and L. Mattsson, Introduction to Surface Roughness and Scattering, 2nd ed. (Optical Society of America, 1999).

Bogunovic, D.

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), p. 598.

Gallager, N. C.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978).

Gaylord, T. K.

T. K. Gaylord and M. G. Moharam, "Analysis and applications of optical diffraction by gratings," Proc. IEEE 73, 894-937 (1985).

Goodman, J. W.

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

Grann, E. B.

Gremaux, D. A.

Harvey, J. E.

Hessel, A.

Krywonos, A.

Loewen, E. G.

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997).

Mattsson, L.

J. M. Bennett and L. Mattsson, Introduction to Surface Roughness and Scattering, 2nd ed. (Optical Society of America, 1999).

Maystre, D.

D. Maystre, "Rigorous vector theories of diffraction gratings," in Progress in Optics XXI, E.Wolf, ed. (Elsevier Science, 1984).
[CrossRef]

Mellin, S. D.

Moharam, M. G.

D. A. Pommet, M. G. Moharam, and E. B. Grann, "Limits of scalar diffraction theory for diffractive phase elements," J. Opt. Soc. Am. A 11, 1827-1834 (1994).
[CrossRef]

T. K. Gaylord and M. G. Moharam, "Analysis and applications of optical diffraction by gratings," Proc. IEEE 73, 894-937 (1985).

Nath, N. S. N.

C. V. Raman and N. S. N. Nath, "The diffraction of light by high frequency sound waves," Proc. Ind. Acad. Sci. A 2, 406-413 (1935).

Nevis, E. A.

Nordin, G. P.

Oliner, A. A.

Palmer, C.

C. Palmer, Diffraction Grating Handbook, 4th ed. (Richardson Grating Laboratory, 2000), p. 15.

Petit, R.

R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, 1980), p. 98.

Pommet, D. A.

Popov, E.

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997).

Raman, C. V.

C. V. Raman and N. S. N. Nath, "The diffraction of light by high frequency sound waves," Proc. Ind. Acad. Sci. A 2, 406-413 (1935).

Ratcliff, J. A.

J. A. Ratcliff, "Some aspects of diffraction theory and their application to the ionosphere," in Reports on Progress in Physics, A.C.Strickland, ed. (Physical Society, 1956), Vol. XIX.

Shack, R. V.

Spizzichino, A.

P. Beckman and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963).

Stover, J. C.

J. C. Stover, Optical Scattering, Measurement and Analysis, 2nd ed. (SPIE, 1995).
[CrossRef]

Thompson, P. L.

Vernold, C. L.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), p. 598.

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).

Am. J. Phys.

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

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Express

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).

Other

D. Maystre, "Rigorous vector theories of diffraction gratings," in Progress in Optics XXI, E.Wolf, ed. (Elsevier Science, 1984).
[CrossRef]

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

M. Born and E. Wolf, Principles of Optics (Pergamon, 1980), p. 598.

T. K. Gaylord and M. G. Moharam, "Analysis and applications of optical diffraction by gratings," Proc. IEEE 73, 894-937 (1985).

C. V. Raman and N. S. N. Nath, "The diffraction of light by high frequency sound waves," Proc. Ind. Acad. Sci. A 2, 406-413 (1935).

C. Palmer, Diffraction Grating Handbook, 4th ed. (Richardson Grating Laboratory, 2000), p. 15.

R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, 1980), p. 98.

P. Beckman and A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, 1963).

J. M. Bennett and L. Mattsson, Introduction to Surface Roughness and Scattering, 2nd ed. (Optical Society of America, 1999).

J. C. Stover, Optical Scattering, Measurement and Analysis, 2nd ed. (SPIE, 1995).
[CrossRef]

J. A. Ratcliff, "Some aspects of diffraction theory and their application to the ionosphere," in Reports on Progress in Physics, A.C.Strickland, ed. (Physical Society, 1956), Vol. XIX.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, 1978).

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, 1997).

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Figures (7)

Fig. 1
Fig. 1

Diffraction grating efficiency of the first order of a sinusoidal phase grating ( h d = 0.20 ) in the Littrow condition as predicted by the nonrigorous Beckmann–Kirchhoff theory, the paraxial scalar theory, and the rigorous integral vector theory. EM, electromagnetic.

Fig. 2
Fig. 2

Geometric configuration when the incident beam strikes the diffracting aperture at an arbitrary angle.

Fig. 3
Fig. 3

Illustration of the peak-to-peak phase variation introduced into a given diffracted order by reflection from a sinusoidal surface.

Fig. 4
Fig. 4

Diffraction grating efficiency of the first order of a perfectly conducting sinusoidal phase grating ( h d = 0.20 ) in the Littrow condition as predicted by the Beckmann–Kerchhoff theory, the paraxial scalar theory, the nonparaxial (NP) scalar diffraction theory presented in this paper, and a rigorous integral vector theory. EM, electromagnetic.

Fig. 5
Fig. 5

Diffraction grating efficiency of the first order of a sinusoidal phase grating ( h d = 0.05 ) in the Littrow condition as predicted by the Beckmann–Kerchhoff theory, the paraxial scalar theory, our nonparaxial (NP) scalar diffraction theory, and a rigorous integral vector theory. EM, electromagnetic.

Fig. 6
Fig. 6

Diffraction grating efficiency of the first order of a perfectly conducting sinusoidal phase grating ( h d = 0.15 ) in the Littrow condition as predicted by the Beckmann–Kerchhoff theory, the paraxial scalar theory, our nonparaxial (NP) scalar diffraction theory, and a rigorous integral vector theory. EM, electromagnetic.

Fig. 7
Fig. 7

Diffraction grating efficiency of the first order of a perfectly conducting sinusoidal phase grating ( h d = 0.30 ) in the Littrow condition as predicted by the Beckmann–Kerchhoff theory, the paraxial scalar theory, our nonparaxial (NP) scalar diffraction theory, and a rigorous integral vector theory. EM, electromagnetic.

Equations (35)

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F = sec θ m 1 + cos ( θ i + θ m ) cos θ i + cos θ m = 1 cos 2 θ m ,
η 1 = J 1 2 ( k h 2 cos θ m ) cos 4 θ m ,
η m = J m 2 ( a 2 ) ,
x ̂ = x λ , y ̂ = y λ , z ̂ = z λ , etc . ,
α = x ̂ r ̂ , β = y ̂ r ̂ , γ = z ̂ r ̂
U ( x ̂ 2 , y ̂ 2 ; z ̂ ) = i U 0 ( x ̂ 1 , y ̂ 1 ; 0 ) z ̂ ̂ exp ( i 2 π ̂ ) ̂ d x ̂ 1 d y ̂ 1 ,
U ( α , β ; r ̂ ) = γ [ exp ( i 2 π r ̂ ) ( i r ̂ ) ] F { U 0 ( x ̂ , y ̂ ; 0 ) } ,
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 ̂ ) }
U 0 ( x ̂ , y ̂ ; 0 ) = γ i U 0 ( x ̂ , y ̂ ; 0 ) .
L ( α , β β 0 ) = λ 2 A s F { U 0 ( x ̂ , y ̂ ; 0 ) exp ( i 2 π β 0 y ̂ ) } 2 .
U 0 ( x ̂ , y ̂ ; 0 ) exp ( i 2 π β 0 y ̂ ) 2 d x ̂ d y ̂
= F { U 0 ( x ̂ , y ̂ ; 0 ) exp ( i 2 π β 0 y ̂ ) } 2 d α d β .
U 0 ( x ̂ , y ̂ ; 0 ) exp ( i 2 π β 0 y ̂ ) 2 d x ̂ d y ̂
= A s λ 2 L ( α , β β 0 ) d α d β .
U 0 ( x ̂ , y ̂ ; 0 ) exp ( i 2 π β 0 y ̂ ) 2 d x ̂ d y ̂ = A s λ 2 1 1 1 α 2 1 α 2 L ( α , β β 0 ) d α d β ,
U 0 ( x ̂ , y ̂ ; 0 ) exp ( i 2 π β 0 y ̂ ) 2 d x ̂ d y ̂ = P T ( θ i ) λ 2 = E 0 A s γ i λ 2 .
P T = A s L ( α , β β 0 ) d α d β = A s 1 1 1 α 2 1 α 2 L ( α , β β 0 ) d α d β ,
L ( α , β β 0 ) = K L ( α , β β 0 ) .
K = α = β = L ( α , β β 0 ) d α d β α = 1 1 β = 1 α 2 1 α 2 L ( α , β β 0 ) d α d β .
L ( α , β β 0 ) = K λ 2 A s F { U 0 ( x ̂ , y ̂ ; 0 ) exp ( i 2 π β 0 y ̂ ) } 2 for α 2 + β 2 1
L ( α , β β 0 ) = 0 for α 2 + β 2 > 1 .
t ( x ̂ , y ̂ ) = rect ( x ̂ b ̂ , y ̂ b ̂ ) exp [ i a 2 sin ( 2 π y ̂ d ̂ ) ] .
exp [ i a 2 sin ( 2 π y ̂ d ̂ ) ] = m = J m ( a 2 ) exp ( i 2 π m y ̂ d ̂ ) ,
L ( α , β β 0 ) = γ i E 0 λ 2 A s m = F { rect ( x ̂ b ̂ , y ̂ b ̂ ) exp ( i 2 π β i y ̂ ) J m ( a 2 ) exp ( i 2 π m y ̂ d ̂ ) } 2 .
β m + β i = m d ̂ ,
L ( α , β β 0 ) = γ i E 0 λ 2 A s m = F { rect ( x ̂ b ̂ , y ̂ b ̂ ) exp ( i 2 π β i y ̂ ) } F { J m ( a 2 ) exp ( i 2 π m y ̂ d ̂ ) } 2 ,
L ( α , β β 0 ) = γ i E 0 λ 2 A s m = 1 1 b ̂ 2 sinc ( α 1 b ̂ , β + β i 1 b ̂ ) J m ( a 2 ) δ ( α , β m d ̂ ) 2 .
L ( α , β β 0 ) = γ i E 0 λ 2 A s m = J m ( a 2 ) 1 1 b ̂ 2 sinc ( α 1 b ̂ , β + β i m d ̂ 1 b ̂ ) 2 .
L ( α , β β 0 ) = γ i E 0 λ 2 A s m = J m ( a 2 ) 1 1 b ̂ 2 sinc ( α 1 b ̂ , β β m 1 b ̂ ) 2 .
L ( α , β β 0 ) = γ i E 0 m = J m 2 ( a 2 ) [ 1 1 b ̂ 2 sinc 2 ( α 1 b ̂ , β β m 1 b ̂ ) ] .
P T = γ i E 0 K A s m = min max J m 2 ( a 2 ) 1 1 1 α 2 1 α 2 [ 1 1 b ̂ 2 sinc 2 ( α 1 b ̂ , β β m 1 b ̂ ) ] d α d β ,
P T = γ i E 0 K A s m = min max J m 2 ( a 2 ) .
P m = γ i E 0 K A s J m 2 ( a 2 ) ,
η m = P m P T = J m 2 ( a 2 ) m = min max J m 2 ( a 2 ) .
a = ( 2 π λ ) ( h 1 + h 2 ) = 2 π h ̂ ( cos θ i + cos θ m ) .

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