Abstract

Fraunhofer diffraction formula cannot be applied to calculate the diffraction wave energy distribution of concave gratings like plane gratings because their grooves are distributed on a concave spherical surface. In this paper, a method based on the Kirchhoff diffraction theory is proposed to calculate the diffraction efficiency on concave gratings by considering the curvature of the whole concave spherical surface. According to this approach, each groove surface is divided into several limited small planes, on which the Kirchhoff diffraction field distribution is calculated, and then the diffraction field of whole concave grating can be obtained by superimposition. Formulas to calculate the diffraction efficiency of Rowland-type and flat-field concave gratings are deduced from practical applications. Experimental results showed strong agreement with theoretical computations. With the proposed method, light energy can be optimized to the expected diffraction wave range while implementing aberration-corrected design of concave gratings, particularly for the concave blazed gratings.

© 2013 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Macmillan, 1964), Chap. 8.
  2. M. C. Hutley, Diffraction Gratings (Academic, 1982).
  3. R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, 1980).
  4. M. G. Moharam, E. B. Grann, and D. A. Pomment, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. 12, 1068–1076 (1995).
    [CrossRef]
  5. M. G. Moharam, D. A. Pomment, and E. B. Grann, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. 12, 1077–1086 (1995).
    [CrossRef]
  6. M. G. Moharam and T. K. Gaylord, “Planar dielectric diffraction theories,” Appl. Phys. B 28, 1–14 (1982).
    [CrossRef]
  7. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [CrossRef]
  8. M. G. Moharam and T. K. Gaylord, “Coupled-wave analysis of reflection gratings,” Appl. Opt. 20, 240–244 (1981).
    [CrossRef]
  9. H. A. Rowland, “Preliminary notice of the results accomplished in the manufacture and theory of gratings for optical purposes,” Phil. Mag. 13(84), 469–474 (1882).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  15. H. Noda, T. Namioka, and M. Seya, “Geometric theory of the grating,” J. Opt. Soc. Am. 64, 1031–1036 (1974).
    [CrossRef]
  16. D. Pi, Y. Huang, D. Zhang, Z. Ni, and S. Zhuang, “Optimization of the flat-field holographic concave grating in wide spectral range,” Acta Phys. Sinica 59, 1009–1016 (2010).

2010 (1)

D. Pi, Y. Huang, D. Zhang, Z. Ni, and S. Zhuang, “Optimization of the flat-field holographic concave grating in wide spectral range,” Acta Phys. Sinica 59, 1009–1016 (2010).

2007 (1)

2006 (1)

2002 (1)

1995 (2)

M. G. Moharam, E. B. Grann, and D. A. Pomment, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. 12, 1068–1076 (1995).
[CrossRef]

M. G. Moharam, D. A. Pomment, and E. B. Grann, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. 12, 1077–1086 (1995).
[CrossRef]

1982 (1)

M. G. Moharam and T. K. Gaylord, “Planar dielectric diffraction theories,” Appl. Phys. B 28, 1–14 (1982).
[CrossRef]

1981 (3)

1980 (1)

1974 (1)

1882 (1)

H. A. Rowland, “Preliminary notice of the results accomplished in the manufacture and theory of gratings for optical purposes,” Phil. Mag. 13(84), 469–474 (1882).
[CrossRef]

Bazhanov, Yu. V.

Born, M.

M. Born and E. Wolf, Principles of Optics (Macmillan, 1964), Chap. 8.

Bugaenko, A. G.

Chen, N.-P.

Gaylord, T. K.

Grann, E. B.

M. G. Moharam, D. A. Pomment, and E. B. Grann, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. 12, 1077–1086 (1995).
[CrossRef]

M. G. Moharam, E. B. Grann, and D. A. Pomment, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. 12, 1068–1076 (1995).
[CrossRef]

Huang, Y.

D. Pi, Y. Huang, D. Zhang, Z. Ni, and S. Zhuang, “Optimization of the flat-field holographic concave grating in wide spectral range,” Acta Phys. Sinica 59, 1009–1016 (2010).

Hunter, W. R.

Hutley, M. C.

Ko, C.-H.

Kulakova, N. A.

Lin, J.-S.

Liu, W.-C.

Mirumyants, S. O.

Moharam, M. G.

M. G. Moharam, E. B. Grann, and D. A. Pomment, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. 12, 1068–1076 (1995).
[CrossRef]

M. G. Moharam, D. A. Pomment, and E. B. Grann, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. 12, 1077–1086 (1995).
[CrossRef]

M. G. Moharam and T. K. Gaylord, “Planar dielectric diffraction theories,” Appl. Phys. B 28, 1–14 (1982).
[CrossRef]

M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
[CrossRef]

M. G. Moharam and T. K. Gaylord, “Coupled-wave analysis of reflection gratings,” Appl. Opt. 20, 240–244 (1981).
[CrossRef]

Namioka, T.

Neviere, M.

Ni, Z.

D. Pi, Y. Huang, D. Zhang, Z. Ni, and S. Zhuang, “Optimization of the flat-field holographic concave grating in wide spectral range,” Acta Phys. Sinica 59, 1009–1016 (2010).

Noda, H.

Petit, R.

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

Pi, D.

D. Pi, Y. Huang, D. Zhang, Z. Ni, and S. Zhuang, “Optimization of the flat-field holographic concave grating in wide spectral range,” Acta Phys. Sinica 59, 1009–1016 (2010).

Pomment, D. A.

M. G. Moharam, D. A. Pomment, and E. B. Grann, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. 12, 1077–1086 (1995).
[CrossRef]

M. G. Moharam, E. B. Grann, and D. A. Pomment, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. 12, 1068–1076 (1995).
[CrossRef]

Rowland, H. A.

H. A. Rowland, “Preliminary notice of the results accomplished in the manufacture and theory of gratings for optical purposes,” Phil. Mag. 13(84), 469–474 (1882).
[CrossRef]

Seya, M.

Shen, J.-L.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Macmillan, 1964), Chap. 8.

Zhang, D.

D. Pi, Y. Huang, D. Zhang, Z. Ni, and S. Zhuang, “Optimization of the flat-field holographic concave grating in wide spectral range,” Acta Phys. Sinica 59, 1009–1016 (2010).

Zhuang, S.

D. Pi, Y. Huang, D. Zhang, Z. Ni, and S. Zhuang, “Optimization of the flat-field holographic concave grating in wide spectral range,” Acta Phys. Sinica 59, 1009–1016 (2010).

Acta Phys. Sinica (1)

D. Pi, Y. Huang, D. Zhang, Z. Ni, and S. Zhuang, “Optimization of the flat-field holographic concave grating in wide spectral range,” Acta Phys. Sinica 59, 1009–1016 (2010).

Appl. Opt. (3)

Appl. Phys. B (1)

M. G. Moharam and T. K. Gaylord, “Planar dielectric diffraction theories,” Appl. Phys. B 28, 1–14 (1982).
[CrossRef]

J. Opt. Soc. Am. (4)

M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
[CrossRef]

M. G. Moharam, E. B. Grann, and D. A. Pomment, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. 12, 1068–1076 (1995).
[CrossRef]

M. G. Moharam, D. A. Pomment, and E. B. Grann, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. 12, 1077–1086 (1995).
[CrossRef]

H. Noda, T. Namioka, and M. Seya, “Geometric theory of the grating,” J. Opt. Soc. Am. 64, 1031–1036 (1974).
[CrossRef]

J. Opt. Technol. (2)

Opt. Express (1)

Phil. Mag. (1)

H. A. Rowland, “Preliminary notice of the results accomplished in the manufacture and theory of gratings for optical purposes,” Phil. Mag. 13(84), 469–474 (1882).
[CrossRef]

Other (3)

M. Born and E. Wolf, Principles of Optics (Macmillan, 1964), Chap. 8.

M. C. Hutley, Diffraction Gratings (Academic, 1982).

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

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Figures (11)

Fig. 1.
Fig. 1.

Structure of Rowland-type concave grating.

Fig. 2.
Fig. 2.

Principal section of concave grating.

Fig. 3.
Fig. 3.

Structure of flat-field concave grating.

Fig. 4.
Fig. 4.

Diffraction efficiency versus wavelength at different incident angles.

Fig. 5.
Fig. 5.

Diffraction efficiency versus wavelength at different F numbers with incident angle equal to 10°.

Fig. 6.
Fig. 6.

Diffraction efficiency versus wavelength at different densities of grating grooves with incident angle equal to 10°.

Fig. 7.
Fig. 7.

Diffraction efficiency versus wavelength at different F numbers with incident angle equal to 6°.

Fig. 8.
Fig. 8.

Diffraction efficiency versus wavelength at different densities of grating grooves with incident angle equal to 6°.

Fig. 9.
Fig. 9.

Diffraction efficiency versus wavelength for flat-field concave grating.

Fig. 10.
Fig. 10.

Diffraction efficiency curves of both theoretical calculation and experimental measurement for Rowland-type concave grating.

Fig. 11.
Fig. 11.

Diffraction efficiency curves of both theoretical calculation and experimental measurement for flat-field concave grating.

Equations (12)

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(xR)2+y2+z2=R2.
sinθsinθ=1e0kλ,
E˜i(A)=BiλΣexp(ikr)rexp(ikr)r[cos(np,r)cos(np,r)2]dσ,
r=|AP|=[(xrcosθ)2+(yrsinθ)2+z2]12,
r=|AP|=[(xrcosθ)2+(yrsinθ)2+z2]12,
cos(np,r)=n⃗p·r⃗|n⃗p|·|r⃗|,
cos(np,r)=n⃗p·r⃗|n⃗p|·|r⃗|.
n⃗p(cosαRxzR,sinαcosθRxzR+sinαsinθR,cosαR).
E˜j(A)=i=1nE˜i(A).
E˜(A)=j=1NE˜j(A).
y+12u20y2=e0m,
y+0.5×0.0056y2=e0m,

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