Abstract

Tracing rays through arbitrary diffraction gratings (including holographic gratings of the second generation fabricated on a curved substrate) by the vector form is somewhat complicated. Vector ray tracing utilizes the local groove density, the calculation of which highly depends on how the grooves are generated. Characterizing a grating by its groove function, available for almost arbitrary gratings, is much simpler than doing so by its groove density, essentially being a vector. Applying the concept of Riemann geometry, we give an expression of the groove density by the groove function. The groove function description of a grating can thus be incorporated into vector ray tracing, which is beneficial especially at the design stage. A unified explicit grating ray-tracing formalism is given as well.

© 2005 Optical Society of America

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References

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  1. G. R. Rosendahl, “Contributions to the optics of mirror systems and gratings with oblique incidence. I. Ray tracing formulas for the meridional plane,” J. Opt. Soc. Am. 51, 1–3 (1961).
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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1995 (1)

1994 (1)

1989 (1)

1983 (1)

A. Takahashi, T. Katayama, “A generalized diffraction grating equation,” Opt. Acta 30, 1735–1742 (1983).
[CrossRef]

1981 (2)

H. W. Holloway, R. A. Ferrante, “Computer analysis of holographic systems by means of vector ray tracing,” Appl. Opt. 20, 2081–2089 (1981).
[CrossRef] [PubMed]

C. J. Mitchell, “Generalized ray-tracing for diffraction gratings of arbitrary form,” J. Opt. (Paris) 12, 301–308 (1981).
[CrossRef]

1974 (1)

1973 (1)

1966 (1)

1963 (1)

1962 (2)

G. H. Spencer, M. V. R. K. Murty, “General ray-tracing procedure,” J. Opt. Soc. Am. 52, 672–678 (1962).
[CrossRef]

W. T. Welford, “Tracing skew rays through concave diffraction grating,” Opt. Acta 9, 389–394 (1962).
[CrossRef]

1961 (1)

Content, D.

Ferrante, R. A.

Holloway, H. W.

Kastner, S. O.

Katayama, T.

A. Takahashi, T. Katayama, “A generalized diffraction grating equation,” Opt. Acta 30, 1735–1742 (1983).
[CrossRef]

Koike, M.

Ludwig, U. W.

Mitchell, C. J.

C. J. Mitchell, “Generalized ray-tracing for diffraction gratings of arbitrary form,” J. Opt. (Paris) 12, 301–308 (1981).
[CrossRef]

Murty, M. V. R. K.

Namioka, T.

Neupert, W. M.

Noda, H.

Offner, A.

Palmer, C.

Rosendahl, G. R.

Seya, M.

Spencer, G. H.

Takahashi, A.

A. Takahashi, T. Katayama, “A generalized diffraction grating equation,” Opt. Acta 30, 1735–1742 (1983).
[CrossRef]

Welford, W. T.

W. T. Welford, “Tracing skew rays through concave diffraction grating,” Opt. Acta 9, 389–394 (1962).
[CrossRef]

W. T. Welford, Aberration of Optical Systems (Hilger, Bristol, UK, 1986).

Appl. Opt. (3)

J. Opt. (Paris) (1)

C. J. Mitchell, “Generalized ray-tracing for diffraction gratings of arbitrary form,” J. Opt. (Paris) 12, 301–308 (1981).
[CrossRef]

J. Opt. Soc. Am. (6)

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

Opt. Acta (2)

A. Takahashi, T. Katayama, “A generalized diffraction grating equation,” Opt. Acta 30, 1735–1742 (1983).
[CrossRef]

W. T. Welford, “Tracing skew rays through concave diffraction grating,” Opt. Acta 9, 389–394 (1962).
[CrossRef]

Other (1)

W. T. Welford, Aberration of Optical Systems (Hilger, Bristol, UK, 1986).

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Figures (1)

Fig. 1
Fig. 1

Diffraction of light at incident point P on a grating surface. ri is the incident ray, ro is the diffracted ray, s is the grating normal, and p is the groove lines; all are unit vectors. σ is the groove line normal whose magnitude corresponds to the local groove density value.

Equations (16)

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(gij)=1+ξw2ξwξlξwξl1+ξl2.
(gij)=1+ξl2-ξwξl-ξwξl1+ξw2(1+ξw2+ξl2).
σ=-[(1+ξl2)nw-ξwξlnl] w+[(1+ξw2)nl-ξwξlnw] l(1+ξw2+ξl2).
σ=-(uξw+vξl, u, v)/(1+ξw2+ξl2),
u=(1+ξl2)nw-ξwξlnl, v=(1+ξw2)nl-ξwξlnw,
|σ|=nw2+nl2+(nlξw-nwξl)21+ξw2+ξl21/2.
|σ|p×s=-σ.
s=(-1, ξw, ξl)/[(1+ξw2+ξl2)]1/2.
ri=(L, M, N),  ro=(L, M, N), ro-μri=(a, b, c).
ro×s=μri×s+m λν2 |σ|p,
(ro-μri)×s×s=m λν2 |σ|p×s=-m λν2 σ.
-ξlc-aξl2-aξw2-ξwb1+ξw2+ξl2,  -aξw-b-bξl2+ξlcξw1+ξw2+ξl2,  ξwbξl-cξw2-c-aξl1+ξw2+ξl2  =mλ(ξwnw+ξlnl)ν2(1+ξw2+ξl2),  -mλ(-nw-nwξl2+ξwξlnl)ν2(1+ξw2+ξl2),  mλ(nl+nlξw2-ξwξlnw)ν2(1+ξw2+ξl2).
b=mλnwν2-aξw, c=mλnlν2-aξl.
L=μL+a,  M=μM+b,  N=μN+c.
q-2pa+ea2=0a=p±(p2-eq)1/2e,
e=1+ξw2+ξl2, p=mλnlν2+Nμξl+mλnwν2+Mμξw-Lμ, q=mλnwν2+Mμ2-1+L2μ2+mλnlν2+Nμ2.

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