Abstract

Most authors include a paraxial (small-angle) limitation in their discussion of diffracted wave fields. This paraxial limitation severely limits the conditions under which diffraction behavior is adequately described. A linear systems approach to modeling nonparaxial scalar diffraction theory is developed by normalization of the spatial variables by the wavelength of light and by recognition that the reciprocal variables in Fourier transform space are the direction cosines of the propagation vectors of the resulting angular spectrum of plane waves. It is then shown that wide-angle scalar diffraction phenomena are shift invariant with respect to changes in the incident angle only in direction cosine space. Furthermore, it is the diffracted radiance (not the intensity or the irradiance) that is shift invariant in direction cosine space. This realization greatly extends the range of parameters over which simple Fourier techniques can be used to make accurate calculations concerning wide-angle diffraction phenomena. Diffraction-grating behavior and surface-scattering effects are two diffraction phenomena that are not limited to the paraxial region and benefit greatly from this new development.

© 1999 Optical Society of America

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References

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  1. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).
  2. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), p. 72.
  3. J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
    [CrossRef]
  4. J. A. Ratcliff, “Some aspects of diffraction theory and their application to the ionosphere,” in Reports of Progress in Physics, A. C. Strickland, ed. (The Physical Society, London, 1956), Vol. XIX.
  5. J. E. Harvey, R. V. Shack, “Aberrations of diffracted wave fields,” Appl. Opt. 17, 3003–3009 (1978).
    [CrossRef] [PubMed]
  6. J. J. Muray, F. E. Nicodemus, I. Wunderman, “Proposed supplement to the SI nomenclature for radiometry and photometry,” Appl. Opt. 10, 1465–1468 (1971).
    [CrossRef] [PubMed]
  7. E. L. Dereniak, G. D. Boreman, Infrared Detectors and Systems (Wiley, New York, 1996).
  8. R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, New York, 1983).
  9. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).
  10. R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. 4, 396–410 (1902).
    [CrossRef]
  11. J. E. Harvey, E. A. Nevis, “Angular grating anomalies: effects of finite beam size upon wide-angle diffraction phenomena,” Appl. Opt. 31, 6783–6788 (1992).
    [CrossRef] [PubMed]
  12. D. A. Gremaux, N. C. Gallager, “Limits of scalar diffraction theory for conducting gratings,” Appl. Opt. 32, 1048–1953 (1993).
    [CrossRef]
  13. J. E. Harvey, C. L. Vernold, “Description of diffraction grating behavior in direction cosine space,” Appl. Opt. 37, 8158–8160 (1998).
    [CrossRef]
  14. R. J. Noll, “Effect of mid and high spatial frequencies on optical performance,” Opt. Eng. 18, 137–142 (1979).
    [CrossRef]
  15. J. E. Harvey, E. C. Moran, W. P. Zmek, “Transfer function characterization of grazing incidence optical systems,” Appl. Opt. 27, 1527–1533 (1988).
    [CrossRef] [PubMed]
  16. J. E. Harvey, W. P. Zmek, C. Ftaclas, “Imaging capabilities of normal-incidence x-ray telescopes,” Opt. Eng. 29, 603–608 (1990).
    [CrossRef]
  17. S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
    [CrossRef]
  18. J. C. Stover, Optical Scattering, Measurement and Analysis (McGraw-Hill, New York, 1990).
  19. P. Beckman, A. Spizzichino, The Scattering of Electromagnetic Waves from Rough Surfaces (Pergamon, New York, 1963).
  20. J. E. Harvey, “Light-scattering characteristics of optical surfaces,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1976).
  21. J. E. Harvey, “Surface scatter phenomena: a linear, shift-invariant process,” in Scatter from Optical Components, J. C. Stover, ed., Proc. SPIE1165, 87–99 (1989).
    [CrossRef]
  22. Users Manual for APART/PADE, Version 7 (Breault Research Organization, 4601 East First Street, Tucson, Ariz., 1985) p. 5-2.
  23. ASAP Reference Manual (Breault Research Organization, 4601 East First Street, Tucson, Ariz., 1990).
  24. TracePro User’s Manual, Version 1.3 (Lambda Research Corporation, 80 Taylor Street, Littleton, Mass., 1998), p. 107.
  25. K. A. O’Donnell, E. R. Mendez, “Experimental study of scattering from characterized random surfaces,” J. Opt. Soc. Am. A 4, 1194–1205 (1987).
    [CrossRef]
  26. C. L. Vernold, J. E. Harvey, “A modified Beckmann–Kirchoff scattering theory,” in Scattering and Surface Roughness II, Z. Gu, A. A. Maradudin, eds., Proc. SPIE3426, 51–56 (1998).
    [CrossRef]

1998 (1)

1993 (1)

1992 (1)

1990 (1)

J. E. Harvey, W. P. Zmek, C. Ftaclas, “Imaging capabilities of normal-incidence x-ray telescopes,” Opt. Eng. 29, 603–608 (1990).
[CrossRef]

1988 (1)

1987 (1)

1979 (2)

R. J. Noll, “Effect of mid and high spatial frequencies on optical performance,” Opt. Eng. 18, 137–142 (1979).
[CrossRef]

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

1978 (1)

1971 (1)

1951 (1)

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
[CrossRef]

1902 (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]

Beckman, P.

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

Boreman, G. D.

E. L. Dereniak, G. D. Boreman, Infrared Detectors and Systems (Wiley, New York, 1996).

Boyd, R. W.

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, New York, 1983).

Bracewell, R. N.

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

Dereniak, E. L.

E. L. Dereniak, G. D. Boreman, Infrared Detectors and Systems (Wiley, New York, 1996).

Ftaclas, C.

J. E. Harvey, W. P. Zmek, C. Ftaclas, “Imaging capabilities of normal-incidence x-ray telescopes,” Opt. Eng. 29, 603–608 (1990).
[CrossRef]

Gallager, N. C.

Gaskill, J. D.

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

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), p. 72.

Gremaux, D. A.

Harvey, J. E.

J. E. Harvey, C. L. Vernold, “Description of diffraction grating behavior in direction cosine space,” Appl. Opt. 37, 8158–8160 (1998).
[CrossRef]

J. E. Harvey, E. A. Nevis, “Angular grating anomalies: effects of finite beam size upon wide-angle diffraction phenomena,” Appl. Opt. 31, 6783–6788 (1992).
[CrossRef] [PubMed]

J. E. Harvey, W. P. Zmek, C. Ftaclas, “Imaging capabilities of normal-incidence x-ray telescopes,” Opt. Eng. 29, 603–608 (1990).
[CrossRef]

J. E. Harvey, E. C. Moran, W. P. Zmek, “Transfer function characterization of grazing incidence optical systems,” Appl. Opt. 27, 1527–1533 (1988).
[CrossRef] [PubMed]

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

J. E. Harvey, R. V. Shack, “Aberrations of diffracted wave fields,” Appl. Opt. 17, 3003–3009 (1978).
[CrossRef] [PubMed]

J. E. Harvey, “Light-scattering characteristics of optical surfaces,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1976).

C. L. Vernold, J. E. Harvey, “A modified Beckmann–Kirchoff scattering theory,” in Scattering and Surface Roughness II, Z. Gu, A. A. Maradudin, eds., Proc. SPIE3426, 51–56 (1998).
[CrossRef]

J. E. Harvey, “Surface scatter phenomena: a linear, shift-invariant process,” in Scatter from Optical Components, J. C. Stover, ed., Proc. SPIE1165, 87–99 (1989).
[CrossRef]

Mendez, E. R.

Moran, E. C.

Muray, J. J.

Nevis, E. A.

Nicodemus, F. E.

Noll, R. J.

R. J. Noll, “Effect of mid and high spatial frequencies on optical performance,” Opt. Eng. 18, 137–142 (1979).
[CrossRef]

O’Donnell, K. A.

Ratcliff, J. A.

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

Rice, S. O.

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
[CrossRef]

Shack, R. V.

Spizzichino, A.

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

Stover, J. C.

J. C. Stover, Optical Scattering, Measurement and Analysis (McGraw-Hill, New York, 1990).

Vernold, C. L.

J. E. Harvey, C. L. Vernold, “Description of diffraction grating behavior in direction cosine space,” Appl. Opt. 37, 8158–8160 (1998).
[CrossRef]

C. L. Vernold, J. E. Harvey, “A modified Beckmann–Kirchoff scattering theory,” in Scattering and Surface Roughness II, Z. Gu, A. A. Maradudin, eds., Proc. SPIE3426, 51–56 (1998).
[CrossRef]

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]

Wunderman, I.

Zmek, W. P.

J. E. Harvey, W. P. Zmek, C. Ftaclas, “Imaging capabilities of normal-incidence x-ray telescopes,” Opt. Eng. 29, 603–608 (1990).
[CrossRef]

J. E. Harvey, E. C. Moran, W. P. Zmek, “Transfer function characterization of grazing incidence optical systems,” Appl. Opt. 27, 1527–1533 (1988).
[CrossRef] [PubMed]

Am. J. Phys. (1)

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

Appl. Opt. (6)

Commun. Pure Appl. Math. (1)

S. O. Rice, “Reflection of electromagnetic waves from slightly rough surfaces,” Commun. Pure Appl. Math. 4, 351–378 (1951).
[CrossRef]

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

Opt. Eng. (2)

R. J. Noll, “Effect of mid and high spatial frequencies on optical performance,” Opt. Eng. 18, 137–142 (1979).
[CrossRef]

J. E. Harvey, W. P. Zmek, C. Ftaclas, “Imaging capabilities of normal-incidence x-ray telescopes,” Opt. Eng. 29, 603–608 (1990).
[CrossRef]

Philos. Mag. (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]

Other (14)

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

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), p. 72.

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

E. L. Dereniak, G. D. Boreman, Infrared Detectors and Systems (Wiley, New York, 1996).

R. W. Boyd, Radiometry and the Detection of Optical Radiation (Wiley, New York, 1983).

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

J. C. Stover, Optical Scattering, Measurement and Analysis (McGraw-Hill, New York, 1990).

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

J. E. Harvey, “Light-scattering characteristics of optical surfaces,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1976).

J. E. Harvey, “Surface scatter phenomena: a linear, shift-invariant process,” in Scatter from Optical Components, J. C. Stover, ed., Proc. SPIE1165, 87–99 (1989).
[CrossRef]

Users Manual for APART/PADE, Version 7 (Breault Research Organization, 4601 East First Street, Tucson, Ariz., 1985) p. 5-2.

ASAP Reference Manual (Breault Research Organization, 4601 East First Street, Tucson, Ariz., 1990).

TracePro User’s Manual, Version 1.3 (Lambda Research Corporation, 80 Taylor Street, Littleton, Mass., 1998), p. 107.

C. L. Vernold, J. E. Harvey, “A modified Beckmann–Kirchoff scattering theory,” in Scattering and Surface Roughness II, Z. Gu, A. A. Maradudin, eds., Proc. SPIE3426, 51–56 (1998).
[CrossRef]

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Figures (17)

Fig. 1
Fig. 1

Geometrical configuration of the incident beam, the diffracting aperture, and the observation hemisphere.

Fig. 2
Fig. 2

Geometrical configuration used to demonstrate the fundamental theory of radiometry from which the quantity radiance is obtained.

Fig. 3
Fig. 3

(a) Profile of an on-axis Gaussian radiance distribution in direction cosine space. (b) The profile of the radiance distribution as a function of the diffraction angle that corresponds to (a). (c) Off-axis (normalized) radiance distribution resulting from a 64° incident angle. (d) The off-axis intensity distribution in direction cosine space that corresponds to (c). (e) Off-axis intensity distribution as a function of the diffraction angle.

Fig. 4
Fig. 4

Diffracted-intensity distribution as predicted by the paraxial model of Eq. (35) for a sinusoidal reflection grating with a period of d = 20λ and operating at normal incidence.

Fig. 5
Fig. 5

Diffracted-intensity distribution as predicted by the paraxial model of Eq. (35) for a sinusoidal reflection grating with a period of d = 1.3λ, which is an invalid operating regime.

Fig. 6
Fig. 6

Plot of the diffracted-radiance distribution (in direction cosine space) as a function of the groove depth. The distribution was produced by a sinusoidal reflection grating with a period of d = 1.95λ.

Fig. 7
Fig. 7

Plot of the diffracted-intensity distribution as a function of the groove depth and the diffraction angle. The distribution was produced by a sinusoidal reflection grating with a period of d = 1.95λ.

Fig. 8
Fig. 8

Experimental data for modest incident angles compared with the predictions from the B-K theory with λ = 0.6328, 10.6 µm and θ i = 20°. The experimental data always exhibited a slightly narrower distribution. Reprinted from Ref. 25 with the permission of the authors.

Fig. 9
Fig. 9

Illustration of the drastic departure exhibited between the experimental data and the theoretical predictions for large incident angles with λ = 0.6328 µm and θ i = 70°. Reprinted from Ref. 25 with the permission of the authors.

Fig. 10
Fig. 10

Peak of the diffusely scattered component, which lies inside of the specular direction for both the experimental data and the B-K theory with λ = 10.6 µm and θ i = 70°. Reprinted from Ref. 25 with the permission of the authors.

Fig. 11
Fig. 11

Off-axis angle-spread function associated with the surface ACV function of Eq. (40) with λ = 10.6 µm, θ0 = 70°, and B = 0.5714.

Fig. 12
Fig. 12

Three-dimensional isometric plot of the truncated and the renormalized angle-spread function for an incident angle of 70° with λ = 10.6 µm, B = 0.5714, and K = 1.4112.

Fig. 13
Fig. 13

Comparison of the scattered-radiance profiles in the plane of incidence before and after renormalization. K = 1.4177.

Fig. 14
Fig. 14

Scattered intensity associated with the surface ACV function of Eq. (40) for an incident angle of 70° with λ = 10.6 µm.

Fig. 15
Fig. 15

Predicted curves of the scattered radiance and the scattered intensity for normal incidence, λ = 0.6328 µm, θ0 = 0.0°, and B = 0.00.

Fig. 16
Fig. 16

Comparison of the predicted scattered radiance and the predicted scattered intensity for a large incident angle with λ = 10.6 µm and θ0 = 70°.

Fig. 17
Fig. 17

Comparison of the Harvey–Shack theoretical predictions and the experimental data for large incident angles with λ = 10.6 µm and θ0 = 70°.

Tables (1)

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Table 1 Diffraction Efficiencies for a Coarse (Paraxial Regime) Sinusoidal Phase Grating Optimized for Maximum Efficiency in the ±1 Orders

Equations (49)

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Ex2, y2=1λ2z2U0+x1, y1|ξ=x2λz, η=y2λz2,
U0+x1, y1=-- U0+x1, y1exp-i2πx1ξ+y1ηdx1 dy1.
xˆ=x/λ,  yˆ=y/λ,  zˆ=z/λ,  etc.,
α=xˆ/rˆ,  β=yˆ/rˆ,  γ=zˆ/rˆ,
rˆ=xˆ2+yˆ21/2
Uxˆ2, yˆ2; zˆ=-i -- U0xˆ1, yˆ1; 0×zˆlˆexpi2πlˆlˆdxˆ1 dyˆ1.
Uα, β; rˆ=γexpi2πrˆ/irˆU0xˆ, yˆ; 0.
Uα, β-β0; rˆ=γexpi2πrˆ/irˆU0xˆ, yˆ; 0×expi2πβ0yˆ,
U0xˆ, yˆ; 0=γ0U0xˆ, yˆ; 0.
IrradianceE=PAc,  watts/area.
Radiant IntensityI=Pωc,  watts/steradian.
RadianceL=2PωcAs cos θs, watts/steradian projected area.
2P=LAs cos θsωc=LAs cos θsAc cos θcr2=LωsAc cos θc,
I=Pωc=As L cos θsAs.
Ec=PAc=ωs Lθ, ϕ, x, yωs=1r2As Lθ, ϕ, x, ycos θs=1r2 Iθ, ϕ.
α=sin θ cos ϕ, β=sin θ sin ϕ, γ=cos θ,
dωc=sin θ dθ dϕ=dα dβ/γ.
PT=ω=02π Iθ, ϕdωc=ϕ=02πθ=0π/2 Iθ, ϕsin θ dθ dϕ=α=-11β=-1-α21/21-α21/2 Iα, βdα dβγ.
PT=λ2-- |U0xˆ, yˆ|2dxˆ dyˆ=λ2a=-11β=-1-α21/21-α21/2 |U0xˆ, yˆ|2dα dβ.
Iα, β=γλ2|U0xˆ, yˆ|2.
As Lα, β, x, ycos θsAs=γλ2|U0xˆ, yˆ; 0|2.
Iα, β=As Lα, βcos θsAs=Lα, βAs cos θsAsLα, βAs cos θs.
Lα, β=λ2As |U0xˆ, yˆ; 0|2.
Lα, β-β0=γ0λ2As |U0xˆ, yˆ; 0expi2πβ0yˆ|2.
Lα, β=exp-ρ/Δρ2, ρ=α2+β21/2, Δρ=0.2.
Iθ=γAsLα, β,  θ=sin-1ρ=1-ρ21/2.
-- |U0xˆ, yˆexpi2πβ0yˆ|2dxˆ dyˆ=α=-11β=-1-α21/21-α21/2 |U0xˆ, yˆexpi2πβ0yˆ|2dα dβ.
|U0xˆ, yˆexpi2πβ0yˆ|2=|U0xˆ, yˆ|2;
PTλ2=-- |U0xˆ, yˆ|2dxˆ dyˆ=Asλ2α=-11β=-1-α21/21-α21/2 Lα, βdα dβ.
PTλ2=-111-α21/21-α21/2 |U0xˆ, yˆ; 0expi2πβ0yˆ|2dα dβ=1γ0Asλ2-11-1-α21/21-α21/2 Lα, β-β0dαdβ.
PTAs=α=-11β=-1-α21/21-α21/2 Lα, βdα dβ=1γ0α-11β=-1-α21/21-α21/2 Lα, β-β0dα dβ,
Lα, β-β0=KLα, β-β0
K=α=-β=- Lα, β-β0dα dβα=-11β=-1-α21/21-α21/2 Lα, β-β0dα dβ=1.36.
Lα, β-β0=Kγ0λ2As |U0xˆ, yˆ; 0expi2πβ0yˆ|2,α2+β210α2+β2>1.
Iα, β=As Lα, βcos θsAs.
Iα, β-β0=Kγ0γλ2|U0xˆ, yˆ; 0expi2πβ0yˆ|2,α2+β210α2+β2>1.
Iθx, θy=I0m=- Jm2a/2Gaus2bλθx-mλ/d, θy.
Gausbθλ=exp-πbθλ2.
Lα, β=K2E0bˆ2m=- Jm2a/2Gaus2bˆα-m/dˆ, β,α2+β210α2+β2>1.
Iα, β=As Lα, βGaus2r/bcos θ dAs=Lα, βcos θ AsGaus2r/bdAs.
Iα, β=KγE0bˆ4λ2m=- Jm2a/2Gaus2bˆα-m/dˆ, β,α2+β21.0α2+β2>1
Cxl, yl=σs2 exp-rl2.
Hsxˆ, yˆ=exp-4π cos θ0σˆs2×1-Csxˆlˆ, yˆlˆ cos θ0σs2.
Hsxˆ, yˆ=A+BQxˆ, yˆ,
A=exp-4π cos θ0σˆs2,
B=1-exp-4π cos θ0σˆs2,
Qxˆ, yˆ=exp4π cos θ0σˆs2Csxˆlˆ, yˆlˆ cos θ0σs2-1exp4π cos θ0σˆs2-1.
Sα, β; rˆ=Hsxˆ, yˆ=Aδα, β; rˆ+Sα, β; rˆ,
Sα, β; rˆ=BQxˆ, yˆ.

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