Abstract

Optical components are usually classified into diffractive and refractive elements. In this classification, refractive components are defined as elements that are sufficiently described by geometrical optics. For micro-optics this distinction is very often not applicable. Our goal is to understand which parameters control the transition from elements that can be interpreted as refractive to those elements that are called diffractive. We investigate the linear blazed grating and focus on the wavelength dependence of its properties. For this we adopt an approach well known from the theory of echelette gratings. Our results can easily be transferred to other blazed components, such as Fresnel lenses.

© 1995 Optical Society of America

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References

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  1. F. Sauer, J. Jahns, C. R. Nijander, A. Y. Feldblum, W. P. Townsend, “Refractive-diffractive micro-optics for permutation interconnects,” Opt. Eng. 33, 1550–1560 (1994).
    [Crossref]
  2. N. Davidson, A. A. Friesem, E. Hasman, “Analytic design of hybrid diffractive–refractive achromates,” Appl. Opt. 32, 4770–4774 (1993).
    [Crossref] [PubMed]
  3. D. Kubalak, G. M. Morris, “A hybrid diffractive/refractive lens for use in optical data storage,” in Diffractive Optics: Design, Fabrication and Applications, Vol. 9 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 93–95.
  4. A. Lohmann, “Scaling laws for lenses,” Appl. Opt. 28, 4996–4998 (1989).
    [Crossref] [PubMed]
  5. H. M. Ozaktas, H. Urey, A. Lohmann, “Scaling of diffractive and refractive lenses for optical computing and interconnections,” Appl. Opt. 33, 3782–3789 (1994).
    [Crossref] [PubMed]
  6. E. Bonet, P. Andres, J. C. Barreiro, A. Pons, “Self-imaging of a periodic microlens array: versatile array illuminator realization,” Opt. Commun. 106, 39–44 (1994).
    [Crossref]
  7. M. Kovatchev, R. Ilieva, “Aberration characteristics of optical elements,” in Holography '89, Y. N. Denisyuk, T. H. Jeong, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1183, 643–652 (1990).
  8. D. W. Sweeney, G. Sommargren, “Single element achromatic diffractive lens,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 26–29.
  9. G. M. Morris, D. Faklis, “Achromatic and apochromatic diffractive singlets,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 53–56.
  10. S. Sinzinger, M. Testorf, W. Singer, “The transition between diffractive and refractive micro-optical components,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 143–146.
  11. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1980), Chap. VIII.
  12. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 2.

1994 (3)

F. Sauer, J. Jahns, C. R. Nijander, A. Y. Feldblum, W. P. Townsend, “Refractive-diffractive micro-optics for permutation interconnects,” Opt. Eng. 33, 1550–1560 (1994).
[Crossref]

H. M. Ozaktas, H. Urey, A. Lohmann, “Scaling of diffractive and refractive lenses for optical computing and interconnections,” Appl. Opt. 33, 3782–3789 (1994).
[Crossref] [PubMed]

E. Bonet, P. Andres, J. C. Barreiro, A. Pons, “Self-imaging of a periodic microlens array: versatile array illuminator realization,” Opt. Commun. 106, 39–44 (1994).
[Crossref]

1993 (1)

1989 (1)

Andres, P.

E. Bonet, P. Andres, J. C. Barreiro, A. Pons, “Self-imaging of a periodic microlens array: versatile array illuminator realization,” Opt. Commun. 106, 39–44 (1994).
[Crossref]

Barreiro, J. C.

E. Bonet, P. Andres, J. C. Barreiro, A. Pons, “Self-imaging of a periodic microlens array: versatile array illuminator realization,” Opt. Commun. 106, 39–44 (1994).
[Crossref]

Bonet, E.

E. Bonet, P. Andres, J. C. Barreiro, A. Pons, “Self-imaging of a periodic microlens array: versatile array illuminator realization,” Opt. Commun. 106, 39–44 (1994).
[Crossref]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1980), Chap. VIII.

Davidson, N.

Faklis, D.

G. M. Morris, D. Faklis, “Achromatic and apochromatic diffractive singlets,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 53–56.

Feldblum, A. Y.

F. Sauer, J. Jahns, C. R. Nijander, A. Y. Feldblum, W. P. Townsend, “Refractive-diffractive micro-optics for permutation interconnects,” Opt. Eng. 33, 1550–1560 (1994).
[Crossref]

Friesem, A. A.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 2.

Hasman, E.

Ilieva, R.

M. Kovatchev, R. Ilieva, “Aberration characteristics of optical elements,” in Holography '89, Y. N. Denisyuk, T. H. Jeong, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1183, 643–652 (1990).

Jahns, J.

F. Sauer, J. Jahns, C. R. Nijander, A. Y. Feldblum, W. P. Townsend, “Refractive-diffractive micro-optics for permutation interconnects,” Opt. Eng. 33, 1550–1560 (1994).
[Crossref]

Kovatchev, M.

M. Kovatchev, R. Ilieva, “Aberration characteristics of optical elements,” in Holography '89, Y. N. Denisyuk, T. H. Jeong, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1183, 643–652 (1990).

Kubalak, D.

D. Kubalak, G. M. Morris, “A hybrid diffractive/refractive lens for use in optical data storage,” in Diffractive Optics: Design, Fabrication and Applications, Vol. 9 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 93–95.

Lohmann, A.

Morris, G. M.

D. Kubalak, G. M. Morris, “A hybrid diffractive/refractive lens for use in optical data storage,” in Diffractive Optics: Design, Fabrication and Applications, Vol. 9 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 93–95.

G. M. Morris, D. Faklis, “Achromatic and apochromatic diffractive singlets,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 53–56.

Nijander, C. R.

F. Sauer, J. Jahns, C. R. Nijander, A. Y. Feldblum, W. P. Townsend, “Refractive-diffractive micro-optics for permutation interconnects,” Opt. Eng. 33, 1550–1560 (1994).
[Crossref]

Ozaktas, H. M.

Pons, A.

E. Bonet, P. Andres, J. C. Barreiro, A. Pons, “Self-imaging of a periodic microlens array: versatile array illuminator realization,” Opt. Commun. 106, 39–44 (1994).
[Crossref]

Sauer, F.

F. Sauer, J. Jahns, C. R. Nijander, A. Y. Feldblum, W. P. Townsend, “Refractive-diffractive micro-optics for permutation interconnects,” Opt. Eng. 33, 1550–1560 (1994).
[Crossref]

Singer, W.

S. Sinzinger, M. Testorf, W. Singer, “The transition between diffractive and refractive micro-optical components,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 143–146.

Sinzinger, S.

S. Sinzinger, M. Testorf, W. Singer, “The transition between diffractive and refractive micro-optical components,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 143–146.

Sommargren, G.

D. W. Sweeney, G. Sommargren, “Single element achromatic diffractive lens,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 26–29.

Sweeney, D. W.

D. W. Sweeney, G. Sommargren, “Single element achromatic diffractive lens,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 26–29.

Testorf, M.

S. Sinzinger, M. Testorf, W. Singer, “The transition between diffractive and refractive micro-optical components,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 143–146.

Townsend, W. P.

F. Sauer, J. Jahns, C. R. Nijander, A. Y. Feldblum, W. P. Townsend, “Refractive-diffractive micro-optics for permutation interconnects,” Opt. Eng. 33, 1550–1560 (1994).
[Crossref]

Urey, H.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1980), Chap. VIII.

Appl. Opt. (3)

Opt. Commun. (1)

E. Bonet, P. Andres, J. C. Barreiro, A. Pons, “Self-imaging of a periodic microlens array: versatile array illuminator realization,” Opt. Commun. 106, 39–44 (1994).
[Crossref]

Opt. Eng. (1)

F. Sauer, J. Jahns, C. R. Nijander, A. Y. Feldblum, W. P. Townsend, “Refractive-diffractive micro-optics for permutation interconnects,” Opt. Eng. 33, 1550–1560 (1994).
[Crossref]

Other (7)

D. Kubalak, G. M. Morris, “A hybrid diffractive/refractive lens for use in optical data storage,” in Diffractive Optics: Design, Fabrication and Applications, Vol. 9 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 93–95.

M. Kovatchev, R. Ilieva, “Aberration characteristics of optical elements,” in Holography '89, Y. N. Denisyuk, T. H. Jeong, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1183, 643–652 (1990).

D. W. Sweeney, G. Sommargren, “Single element achromatic diffractive lens,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 26–29.

G. M. Morris, D. Faklis, “Achromatic and apochromatic diffractive singlets,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 53–56.

S. Sinzinger, M. Testorf, W. Singer, “The transition between diffractive and refractive micro-optical components,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 143–146.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1980), Chap. VIII.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 2.

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Figures (9)

Fig. 1
Fig. 1

Examples for diffractive and refractive components.

Fig. 2
Fig. 2

Deflection by a prism and a diffraction grating (wavelength dependence).

Fig. 3
Fig. 3

Blazed grating with different blaze depths q, as an array of microprisms.

Fig. 4
Fig. 4

Fourier setup for the reconstruction of the effect of a blazed grating.

Fig. 5
Fig. 5

Reconstruction with the design wavelength λ i = λ D : (a) p prism ( ν ) and p periodic ( ν ) ; (b) p ( ν ) = p prism ( ν ) p periodic ( ν ) .

Fig. 6
Fig. 6

Reconstruction with a wavelength λ i ≠ λ D with J ≈ λ D i = 0.74: (a) p prism ( ν ) and p periodic ( ν ) ; (b) p ( ν ) = p prism ( ν ) p periodic ( ν ) .

Fig. 7
Fig. 7

Pseudo-three-dimensional plot of the amplitude in the Fourier plane of a blazed grating for varying illumination wavelength (q = 1): (a) wide envelope of and the very sharp peaks of p prism ( ν ) and the very sharp peaks of p periodic ( ν ) ; (b) p prism ( ν ) p periodic ( ν )

Fig. 8
Fig. 8

Rastered gray-scale pictures of the intensity distributions [p(ν)] in the Fourier plane of blazed gratings with varying blaze depth q for varying illumination wavelength λ i

Fig. 9
Fig. 9

Light intensity at the position of the blaze order (design wavelength λ D = 480 nm): (a) Blaze depth q = 4; (b) q = 8.

Equations (18)

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α prism λ n ( λ ) λ 0.0695 μ m 1
α grat λ m d 0.182 μ m 1
h = q λ D n D 1 d = λ D q ( n D 1 ) .
P ( x ) = p prism ( x ) p periodic ( x ) = exp [ 2 π iX ( n i 1 ) λ i ] REF × rect ( x d ) m δ ( x md ) rect ( x B ) , DIF
p ( ν ) = p prism ( ν ) p periodic ( ν ) = sinc { [ ν ( n i 1 ) λ i ] d } × [ m δ ( ν m d ) sinc ( ν B ) ] .
sinc ( x ) = sin ( π x ) π x .
p prism ( ν ) = sinc { [ ν λ D ( n D 1 ) J ] q } , p periodic ( ν ) = m sinc { [ ν m q ( n D 1 ) λ D ] B } .
J = ( n i 1 ) λ D ( n D 1 ) λ i .
p prism ( ν ) = sinc { [ ν λ D ( n D 1 ) 1 ] q } .
ũ ( ν ) = δ [ ν ( n D 1 ) λ ] .
ν m = m d = m q n D 1 λ D .
P periodic ( ν ) = m δ [ ν m q ( n D 1 ) λ D ] .
C m = sinc { [ ( n i 1 ) ( n D 1 ) λ D λ i ] q m } = sinc ( Jq m ) .
λ i = λ D , n i = n D J = 1 .
Δ λ = | λ i λ D | = [ q m ± 1 ( n i 1 ) ( n D 1 ) 1 λ D ( q m ± 1 1 ) λ D .
X v q = λ i f ν q = f ( n i 1 ) 1 J
f ( n i 1 ) q q + 1 x ν q f ( n i 1 ) q q 1 .
Δ x detector f ( n i 1 ) 2 q q 2 1 .

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