Abstract

It is well known that the angular separation of non-paraxial diffracted orders from a linear grating varies drastically with incident angle. Furthermore, for oblique incident angles (conical diffraction), it is rather cumbersome both analytically and graphically to describe the number and angular position of the various propagating orders. One can readily demonstrate that wide-angle diffraction phenomena (including conical diffraction from gratings) are shift-invariant with respect to incident angle in direction cosine space. Only when the grating equation is expressed in terms of the direction cosines of the propagation vectors of the incident beam and the diffracted orders can we apply the Fourier techniques resulting from linear systems theory. This formulation has proven extremely useful for small-angle diffraction phenomena and in modern, image formation theory. New insight and an intuitive understanding of diffraction grating behavior results from a simple direction cosine diagram.

© 1998 Optical Society of America

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References

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  1. J.D. Gaskill, Linear Systems, Fourier Transforms, and Optics (John Wiley Sons Inc., New York, NY, 1978).
  2. J.W. Goodman, Introduction to Fourier Optics, second ed., (McGraw-Hill, New York, NY, 1996).
  3. F.L. Pedrotti, L. S. Petrotti, Introduction to Optics (Prentice-Hall Inc., Englewood Cliffs, NJ. 1987).
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  6. M.C. Hutley, Diffraction Gratings (Academic Press, London, UK., 1982).
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  8. B.E.A. Saleh, M.C. Teich, Fundamentals of Photonics (John Wiley Sons Inc., New York, NY, 1994).
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    [CrossRef]
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    [CrossRef]
  11. J.E. Harvey, E.A. Nevis, “Angular grating anomalies: Effects of finite beam size on wide-angle diffraction phenomena,” Appl. Opt. 31,6783–6788 (1993).
    [CrossRef]

1993 (1)

1983 (1)

1979 (1)

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

Born, M.

M. Born, E. Wolf, Principles of Optics, sixth ed. (Pergamon Press, Oxford, U.K., 1980).

Gaskill, J.D.

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

Gaylord, T.K.

Goodman, J.W.

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

Harvey, J.E.

Hecht, E.

E. Hecht, Optics, second ed., (Addison-Wesley Publishing Co., Reading, MA. 1987).

Hutley, M.C.

M.C. Hutley, Diffraction Gratings (Academic Press, London, UK., 1982).

Klein, M.V.

M.V. Klein, Optics (John Wiley Sons Inc., New York, NY, 1970).

Moharam, M.G.

Nevis, E.A.

Pedrotti, F.L.

F.L. Pedrotti, L. S. Petrotti, Introduction to Optics (Prentice-Hall Inc., Englewood Cliffs, NJ. 1987).

Petrotti, L. S.

F.L. Pedrotti, L. S. Petrotti, Introduction to Optics (Prentice-Hall Inc., Englewood Cliffs, NJ. 1987).

Saleh, B.E.A.

B.E.A. Saleh, M.C. Teich, Fundamentals of Photonics (John Wiley Sons Inc., New York, NY, 1994).

Teich, M.C.

B.E.A. Saleh, M.C. Teich, Fundamentals of Photonics (John Wiley Sons Inc., New York, NY, 1994).

Wolf, E.

M. Born, E. Wolf, Principles of Optics, sixth ed. (Pergamon Press, Oxford, U.K., 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. (1)

J. Opt. Soc. Am. (1)

Other (8)

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

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

F.L. Pedrotti, L. S. Petrotti, Introduction to Optics (Prentice-Hall Inc., Englewood Cliffs, NJ. 1987).

M.V. Klein, Optics (John Wiley Sons Inc., New York, NY, 1970).

M. Born, E. Wolf, Principles of Optics, sixth ed. (Pergamon Press, Oxford, U.K., 1980).

M.C. Hutley, Diffraction Gratings (Academic Press, London, UK., 1982).

E. Hecht, Optics, second ed., (Addison-Wesley Publishing Co., Reading, MA. 1987).

B.E.A. Saleh, M.C. Teich, Fundamentals of Photonics (John Wiley Sons Inc., New York, NY, 1994).

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Figures (5)

Figure 1
Figure 1

A simple classroom diffraction grating demonstration: (a) paraxial (course grating) diffraction grating bohavior, (b) non-paraxial (fino grating) planar diffraction grating behavior, (c) non-paraxial “conical” (fine grating tilted about x-axis) behavior.

Figure 2
Figure 2

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

Figure 3
Figure 3

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

Figure 4
Figure 4

(a) Illustration of the relationship between diffracted orders in real space, and (b) direction cosine space for planar diffraction with θi = 15° and grating period d = 3λ.

Figure 5
Figure 5

Direction cosine diagrams for four different orientations of a diffraction grating with period d = 3λ Illuminated with an obliquely Incident beam (αi = −0.3, βi = −0.4).

Equations (6)

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sin θ m + sin θ i = m λ / d , m = 0 ,   = ± 1 ,   ± 2 ,   ± 3 ,
x ^ = x / λ ,   y ^ = y / λ ,   z ^ = z / λ ,   r ^ = r / λ ,   etc .
α = x ^ / r ^ ,   β = y ^ / r ^ ,   and   γ = z ^ / r ^ .
α m + α i = m λ / d β m + β i = 0
α m = sin θ m   cos ϕ o α i = sin θ o   cos ϕ o β i = sin ϕ o .
α m + α i = ( m λ / d )   sin ψ β m + β i = ( m λ / d )   cos ψ

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