Abstract

The conventional design of phase gratings or kinoforms with a paraxial transmission function is restricted to the paraxial domain and thin elements. Therefore, the design and analysis of thick phase-relief structures require a nonparaxial theory, as given by the Born approximation. The Born approximation is derived as an extension of the scalar thin-element theory, which is applicable for thick elements with large propagation angles. As an example, general prism gratings on curved surfaces are treated.

© 1998 Optical Society of America

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References

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  1. J. A. Jordan, P. M. Hirsch, L. B. Lesem, D. L. Van Rooy, “Kinoform lenses,” Appl. Opt. 9, 1883–1887 (1970).
    [PubMed]
  2. M. T. Gale, M. Rossi, J. Pedersen, H. Schütz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresist,” Opt. Eng. 33, 3556–3566 (1994).
    [CrossRef]
  3. M. Rossi, C. G. Blough, D. H. Raguin, E. K. Popov, D. Maystre, “Diffraction efficiency of high-NA continuous-relief diffractive lenses,” in Diffractive Optics and Micro-Optics, Vol. 5 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D. C., 1996), pp. 233–236.
  4. S. Sinzinger, M. Testorf, “Transition between diffractive and refractive micro-optical components,” Appl. Opt. 34, 5970–5976 (1995);M. Rossi, R. E. Kunz, H. P. Herzig, “Refractive and diffractive properties of planar micro-optical elements,” Appl. Opt. 34, 5996–6007 (1995);see also Ref. 12.
    [CrossRef] [PubMed]
  5. Ch. Hofman, Die Optische Abbildung (Akademische Verlagsgesellschaft, Leipzig, 1980).
  6. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 77–83.
  7. D. W. Sweeney, G. E. Sommargreen, “Harmonic diffractive lenses,” Appl. Opt. 34, 2469–2475 (1995).
    [CrossRef] [PubMed]
  8. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
    [CrossRef]
  9. W. Singer, K.-H. Brenner, “Transition of the scalar field at a refracting surface in the generalized Kirchhoff diffraction theory,” J. Opt. Soc. Am. A 12, 1913–1919 (1995).
    [CrossRef]
  10. K.-H. Brenner, W. Singer, “Light propagation through microlenses: a new simulation method,” Appl. Opt. 32, 4984–4988 (1993).
    [CrossRef] [PubMed]
  11. W. T. Cathey, Optical Information Processing (Wiley, New York, 1974), p. 5 and Ref. 11.
  12. T. R. M. Sales, G. M. Morris, “Diffractive-refractive behaviour of kinoform lenses,” Appl. Opt. 36, 253–257 (1997).
    [CrossRef] [PubMed]
  13. L.-J. Gelius, J. J. Stamnes, “Diffraction tomography: potentials and problems,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (Elsevier, Amsterdam, 1990), pp. 91–109.
  14. T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
  15. N. Abramson, “Principle of least wave change,” J. Opt. Soc. Am. A 6, 627–629 (1989).
    [CrossRef] [PubMed]

1997 (1)

1995 (3)

1994 (1)

M. T. Gale, M. Rossi, J. Pedersen, H. Schütz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresist,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

1993 (1)

1989 (1)

1985 (1)

T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

1970 (1)

1969 (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

Abramson, N.

Blough, C. G.

M. Rossi, C. G. Blough, D. H. Raguin, E. K. Popov, D. Maystre, “Diffraction efficiency of high-NA continuous-relief diffractive lenses,” in Diffractive Optics and Micro-Optics, Vol. 5 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D. C., 1996), pp. 233–236.

Brenner, K.-H.

Cathey, W. T.

W. T. Cathey, Optical Information Processing (Wiley, New York, 1974), p. 5 and Ref. 11.

Gale, M. T.

M. T. Gale, M. Rossi, J. Pedersen, H. Schütz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresist,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

Gaylord, T. K.

T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Gelius, L.-J.

L.-J. Gelius, J. J. Stamnes, “Diffraction tomography: potentials and problems,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (Elsevier, Amsterdam, 1990), pp. 91–109.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 77–83.

Hirsch, P. M.

Hofman, Ch.

Ch. Hofman, Die Optische Abbildung (Akademische Verlagsgesellschaft, Leipzig, 1980).

Jordan, J. A.

Lesem, L. B.

Maystre, D.

M. Rossi, C. G. Blough, D. H. Raguin, E. K. Popov, D. Maystre, “Diffraction efficiency of high-NA continuous-relief diffractive lenses,” in Diffractive Optics and Micro-Optics, Vol. 5 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D. C., 1996), pp. 233–236.

Moharam, M. G.

T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Morris, G. M.

Pedersen, J.

M. T. Gale, M. Rossi, J. Pedersen, H. Schütz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresist,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

Popov, E. K.

M. Rossi, C. G. Blough, D. H. Raguin, E. K. Popov, D. Maystre, “Diffraction efficiency of high-NA continuous-relief diffractive lenses,” in Diffractive Optics and Micro-Optics, Vol. 5 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D. C., 1996), pp. 233–236.

Raguin, D. H.

M. Rossi, C. G. Blough, D. H. Raguin, E. K. Popov, D. Maystre, “Diffraction efficiency of high-NA continuous-relief diffractive lenses,” in Diffractive Optics and Micro-Optics, Vol. 5 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D. C., 1996), pp. 233–236.

Rossi, M.

M. T. Gale, M. Rossi, J. Pedersen, H. Schütz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresist,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

M. Rossi, C. G. Blough, D. H. Raguin, E. K. Popov, D. Maystre, “Diffraction efficiency of high-NA continuous-relief diffractive lenses,” in Diffractive Optics and Micro-Optics, Vol. 5 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D. C., 1996), pp. 233–236.

Sales, T. R. M.

Schütz, H.

M. T. Gale, M. Rossi, J. Pedersen, H. Schütz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresist,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

Singer, W.

Sinzinger, S.

Sommargreen, G. E.

Stamnes, J. J.

L.-J. Gelius, J. J. Stamnes, “Diffraction tomography: potentials and problems,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (Elsevier, Amsterdam, 1990), pp. 91–109.

Sweeney, D. W.

Testorf, M.

Van Rooy, D. L.

Wolf, E.

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

Appl. Opt. (5)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[CrossRef]

Opt. Eng. (1)

M. T. Gale, M. Rossi, J. Pedersen, H. Schütz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresist,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

Proc. IEEE (1)

T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Other (5)

L.-J. Gelius, J. J. Stamnes, “Diffraction tomography: potentials and problems,” in Scattering in Volumes and Surfaces, M. Nieto-Vesperinas, J. C. Dainty, eds. (Elsevier, Amsterdam, 1990), pp. 91–109.

M. Rossi, C. G. Blough, D. H. Raguin, E. K. Popov, D. Maystre, “Diffraction efficiency of high-NA continuous-relief diffractive lenses,” in Diffractive Optics and Micro-Optics, Vol. 5 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D. C., 1996), pp. 233–236.

Ch. Hofman, Die Optische Abbildung (Akademische Verlagsgesellschaft, Leipzig, 1980).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 77–83.

W. T. Cathey, Optical Information Processing (Wiley, New York, 1974), p. 5 and Ref. 11.

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Figures (12)

Fig. 1
Fig. 1

Scalar approximation for thin elements.

Fig. 2
Fig. 2

Transmission function at a blazed prism grating.

Fig. 3
Fig. 3

Visual interpretation of Eq. (10) in the spatial frequency domain: (a) nonzero contributions of a surface relief; (b) spectrum shifted by convolution; (c) transmitted field distribution; (d) coordinate system of the object spectrum.

Fig. 4
Fig. 4

Condition for a single refracted plane wave with an arbitrary spectrum of surface relief.

Fig. 5
Fig. 5

Frequency spectrum representation of a plane interface after longitudinal Fourier transformation (real part).

Fig. 6
Fig. 6

Frequency spectrum representation and Ewald construction for a plane interface.

Fig. 7
Fig. 7

Prism grating with blaze order m = 4.

Fig. 8
Fig. 8

Longitudinal spectrum of a prism grating with blaze order m = 4.

Fig. 9
Fig. 9

Frequency spectrum of a prism grating with blaze order m = 4 and Ewald construction.

Fig. 10
Fig. 10

Prism grating on a positive lens (f = 2100; blaze order m = 4).

Fig. 11
Fig. 11

Longitudinal spectrum of a prism grating on the positive lens of Fig. 10.

Fig. 12
Fig. 12

Spectrum of a prism grating on a lens (f = 2100; m = 4).

Equations (18)

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U t x ,   z = 0 + = U i x ,   z = 0 - T x .
Ũ t ν x ,   z = 0 + = Ũ i ν x ,   z = 0 -     T ˜ ν x .
T x = A x exp i 2 π h x n - λ cos   α i - n + λ cos   α t = A x exp i 2 π h x ν z i - ν z t ,
ν z i = n - λ 2 - ν x i 2 1 / 2 ,   ν z t = n + λ 2 - ν x t 2 1 / 2 .
T x A x exp i   2 π λ n - - n + h x .
h m x = h x + m λ n - - n + ,   m N
exp i 2 π h x ν z i - ν z t =   exp - i 2 π h x ν z   δ ν z + ν z i δ ν z - ν z t d ν z .
Ũ t ν x = T ˜ ν x ,   ν z   ν x   δ ν x + ν x i =     exp - i 2 π h x ν z exp - i 2 π x ν x d x ν x ,   ν z   δ ν x + ν x i ,   ν z + ν z i δ ν z - ν z t d ν z .
Δ ˜ ν x ,   ν z = B ˜ ν z     exp - i 2 π h x ν z exp - i 2 π x ν x d x .
Ũ t ν x t =   δ ν z - ν z t Δ ˜ ν x ,   ν z   ν x ,   ν z   Ũ i ν x ,   ν z d ν z ,
U t x = ν x t ν z x ,   z   Δ x ,   z U i x ,   z exp - i 2 π ν x x + ν z z d x d z δ ν z - ν z t exp i 2 π ν x t x d ν z d ν x t = x ,   z   Δ x ,   z U i x ,   z exp - i 2 π ν x t x - x + ν z t z d x d z ,
Δ x ,   z = 1 , 0 , z < h x z h x .
Δ ν x ,   ν z δ ν x - ν x ,   ν z =   exp - i 2 π h x ν z exp - i 2 π x ν x d x .
h x = ax + ζ = x   tan   β + ζ ,
Δ ν x ,   ν z = B ˜ ν z exp - i 2 π ζ ν z     exp - i 2 π   tan   β ν z x × exp - i 2 π x ν x d x = B ˜ ν z exp - i 2 π ζ ν z δ ν x - tan   β ν z ,   ν z .
B ˜ ν z cos 2 π tan   β x + ζ ν z = B ˜ ν z cos 2 π   tan   β ν z x + 2 π ζ ν z ,
ζ m = m ν z = m ν z t - ν z i = m λ n + cos   α t - n - cos   α i .
ν Bragg ,   z p = p ζ m ;   ν Bragg ,   x p = tan   β ν Bragg ,   z p = p   tan   β ζ m .

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