Abstract

Near-field diffraction patterns are merely aberrated Fraunhofer diffraction patterns. These aberrations, inherent to the diffraction process, provide insight and understanding into wide-angle diffraction phenomena. Nonparaxial patterns of diffracted orders produced by a laser beam passing through a grating and projected upon a plane screen exhibit severe distortion (W 311). This distortion is an artifact of the configuration chosen to observe diffraction patterns. Grating behavior expressed in terms of the direction cosines of the propagation vectors of the incident and diffracted orders exhibits no distortion. Use of a simple direction cosine diagram provides an elegant way to deal with nonparaxial diffraction patterns, particularly when large obliquely incident beams produce conical diffraction.

© 2003 Optical Society of America

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References

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  1. J. A. Ratcliff, “Some aspects of diffraction theory and their application to the ionosphere,” in Reports of Progress in Physics, A. C. Strickland, ed. (Physical Society, London, 1956), Vol. 19.
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    [CrossRef]
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    [CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 6th ed. (Pergamon, Oxford, UK, 1980).

Gaskill, J. D.

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

Gaylord, T. K.

Goodman, J. W.

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

Guild, J.

J. Guild, The Interference Systems of Crossed Diffraction Gratings: Theory of Moire Fringes (Clarendon, Oxford, UK, 1956).

Harvey, J. E.

Hopkins, H. H.

H. H. Hopkins, Wave Theory of Aberrations (Clarendon, Oxford, UK, 1950).

Mahajan, V. N.

V. N. Mahajan, “Aberrations of diffracted wave fields. I. Optical imaging,” J. Opt. Soc. Am. A 17, 2216–2222 (2000).
[CrossRef]

V. N. Mahajan, Optical Imaging and Aberrations: Part II. Wave Diffraction Optics, Vol. PM103 of the SPIE Press Monographs (SPIE, Bellingham, Wash., 2001).

V. N. Mahajan, Optical Imaging and Aberrations: Part I. Ray Geometrical Optics, Vol. PM45 of the SPIE Press Monographs (SPIE, Bellingham, Wash., 1998).

Moharam, M. G.

Murty, M. V. R. K.

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. (Physical Society, London, 1956), Vol. 19.

Rowland, H. A.

H. A. Rowland, “Gratings in theory and practice,” Philos. Mag., Suppl. 35(216), 397–419 (1893).
[CrossRef]

Shack, R. V.

Spencer, G. H.

Vernold, C. L.

Wolf, E.

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 6th ed. (Pergamon, Oxford, UK, 1980).

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. (2)

J. Opt. Soc. Am. (2)

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

Philos. Mag., Suppl. (1)

H. A. Rowland, “Gratings in theory and practice,” Philos. Mag., Suppl. 35(216), 397–419 (1893).
[CrossRef]

Other (8)

J. A. Ratcliff, “Some aspects of diffraction theory and their application to the ionosphere,” in Reports of Progress in Physics, A. C. Strickland, ed. (Physical Society, London, 1956), Vol. 19.

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

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

H. H. Hopkins, Wave Theory of Aberrations (Clarendon, Oxford, UK, 1950).

V. N. Mahajan, Optical Imaging and Aberrations: Part I. Ray Geometrical Optics, Vol. PM45 of the SPIE Press Monographs (SPIE, Bellingham, Wash., 1998).

V. N. Mahajan, Optical Imaging and Aberrations: Part II. Wave Diffraction Optics, Vol. PM103 of the SPIE Press Monographs (SPIE, Bellingham, Wash., 2001).

M. Born, E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 6th ed. (Pergamon, Oxford, UK, 1980).

J. Guild, The Interference Systems of Crossed Diffraction Gratings: Theory of Moire Fringes (Clarendon, Oxford, UK, 1956).

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Figures (8)

Fig. 1
Fig. 1

Location of diffracted orders projected upon a plane observation screen: (a) paraxial (coarse grating) diffraction grating behavior, (b) nonparaxial (fine grating) planar diffraction grating behavior, (c) nonparaxial conical (fine grating tilted about the x axis) behavior.

Fig. 2
Fig. 2

(a) Pattern of diffracted orders produced with two coarse crossed gratings. (b) Pattern of diffracted orders produced with two fine crossed gratings. The effects of the aberration known as distortion, W 311, is readily apparent in (b).

Fig. 3
Fig. 3

Percent error in the predicted position of the +1 diffracted order.

Fig. 4
Fig. 4

Percent error in the predicted position of the +1 diffracted order when the effects of various orders of distortion are included in the calculation.

Fig. 5
Fig. 5

Illustration of the position of the diffracted orders in real space and direction cosine space for an arbitrary (skew) obliquely incident beam.

Fig. 6
Fig. 6

Relative position of diffracted orders and incident beam in direction cosine space. Diffracted orders outside the unit circle are evanescent.

Fig. 7
Fig. 7

Diffraction patterns from simple linear gratings are shift invariant only in direction cosine space.

Fig. 8
Fig. 8

Direction cosine diagrams for four orientations of a grating with period d = 3λ illuminated with an obliquely incident beam (α i = -0.3, β i = -0.4).

Tables (2)

Tables Icon

Table 1 Tabulation of Expressions for the Aberration Coefficients for Several Different Geometric Configurations of Incident Wave Front and Observation Space

Tables Icon

Table 2 Percent Error in the Radial Position of Diffracted Orders That Are Due to Paraxial Limitation of Conventional Fourier Theory

Equations (13)

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xˆ=x/λ, yˆ=y/λ, zˆ=z/λ,
α=xˆ/rˆ, β=yˆ/rˆ, γ=zˆ/rˆ
Uxˆ2, yˆ2; zˆ=-i -- U0xˆ1, yˆ1; 0× zˆlˆexpi2πlˆlˆdxˆ1dyˆ1.
Wˆ=Wˆ000+Wˆ200ρ2+Wˆ020aˆ2+Wˆ111ρaˆ cosϕ-ϕ+Wˆ400ρ4+Wˆ040aˆ4+Wˆ131ρaˆ3×cosϕ-ϕ+Wˆ222ρ2aˆ2 cos2ϕ-ϕ+Wˆ220ρ2aˆ2+Wˆ311ρ3aˆ cosϕ-ϕ,
Uα, β; rˆ=γ expi2πrˆirˆ-- U0xˆ, yˆ; α, β×exp-i2παxˆ+βyˆdxˆdyˆ.
U0xˆ, yˆ; α, β=T0xˆ, yˆ; 011+ε2expi2πWˆ
Wˆ131=1.25×10-4, Wˆ222=-2.50×10-1.
rmn=L λdm2+n21/2.
rmn=L λdm2+n21-mλ/d2+nλ/d21/2.
αm+αi=mλ/d, βm+βi=0,
αm=sin θm cos ϕ0, αi=-sin θ0 cos ϕ0, βi=-sin ϕ0.
sin θm+sin θi=mλ/d, m=0, ±1, ±2, ±3.
αm+αi=mλdsin ψ, βm+βi=-mλdcos ψ,

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