Abstract

Modeling of kinoforms in optical design software packages can be done in terms of the phase function ϕ(r). The required surface-relief profile t(r) is determined approximately from the relation t(r) = [λ/2π(μ − 1)][ϕ(r)]2π, where λ is the operating wavelength, μ is the refractive index of the optical material of the surface relief, and [ϕ(r)]2π is the phase function ϕ(r) modulo 2π. We present a semianalytical approach that enables one to determine the exact surface-relief profile from a given phase function. It is seen that there is a small but significant difference between the exact and the approximate surface-relief profiles. Some computational results are presented.

© 1995 Optical Society of America

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References

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  1. G. J. Swanson, “Binary optics technology: the theory and design of multilevel diffractive optical elements,” Tech. Rep. 854 (Lincoln Laboratory, MIT, Lexington, Mass., 1989).
  2. code v is a trademark of Optical Research Associates, 550 North Rosemead Boulevard, Pasadena, Calif. 91107.
  3. oslo is a trademark of Sinclair Optics, Inc., 6780 Palmyra Road, Fairport, N.Y. 14450.
  4. zemax is a trademark of Focusoft, Inc., P. O. Box 756, Pleasanton, Calif. 94566.
  5. M. W. Farn, “Modelling of diffractive optics,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 48–51.
  6. D. A. Buralli, G. M. Morris, J. R. Rogers, “Optical performance of holographic kinoforms,” Appl. Opt. 28, 976–983 (1989).
    [CrossRef] [PubMed]
  7. W. Kroninger, H.-G. Heckmann, “Taking the approximation out of diffractive optics design,” Photon. Spectra 28, 120–124 (1994).
  8. L. N. Hazra, Y. Han, C. A. Delisle, “Design of plane kinoform lenses for stigmatic imaging between two prespecified points on the axis,” Opt. Commun. 94, 203–209 (1992).
    [CrossRef]
  9. L. N. Hazra, Y. Han, C. A. Delisle, “Curved kinoform lenses for stigmatic imaging of axial objects,” Appl. Opt. 32, 4775–4794 (1993).
    [CrossRef] [PubMed]
  10. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 77–83.
  11. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), p. 191.
  12. L. N. Hazra, Y. Han, C. A. Delisle, “Effects of the thickness of the substrate on the performance and the design of planar kinoform lenses for axial stigmatism in finite conjugate imaging,” Can. J. Phys. 71, 434–441 (1993).
    [CrossRef]

1994 (1)

W. Kroninger, H.-G. Heckmann, “Taking the approximation out of diffractive optics design,” Photon. Spectra 28, 120–124 (1994).

1993 (2)

L. N. Hazra, Y. Han, C. A. Delisle, “Curved kinoform lenses for stigmatic imaging of axial objects,” Appl. Opt. 32, 4775–4794 (1993).
[CrossRef] [PubMed]

L. N. Hazra, Y. Han, C. A. Delisle, “Effects of the thickness of the substrate on the performance and the design of planar kinoform lenses for axial stigmatism in finite conjugate imaging,” Can. J. Phys. 71, 434–441 (1993).
[CrossRef]

1992 (1)

L. N. Hazra, Y. Han, C. A. Delisle, “Design of plane kinoform lenses for stigmatic imaging between two prespecified points on the axis,” Opt. Commun. 94, 203–209 (1992).
[CrossRef]

1989 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), p. 191.

Buralli, D. A.

Delisle, C. A.

L. N. Hazra, Y. Han, C. A. Delisle, “Curved kinoform lenses for stigmatic imaging of axial objects,” Appl. Opt. 32, 4775–4794 (1993).
[CrossRef] [PubMed]

L. N. Hazra, Y. Han, C. A. Delisle, “Effects of the thickness of the substrate on the performance and the design of planar kinoform lenses for axial stigmatism in finite conjugate imaging,” Can. J. Phys. 71, 434–441 (1993).
[CrossRef]

L. N. Hazra, Y. Han, C. A. Delisle, “Design of plane kinoform lenses for stigmatic imaging between two prespecified points on the axis,” Opt. Commun. 94, 203–209 (1992).
[CrossRef]

Farn, M. W.

M. W. Farn, “Modelling of diffractive optics,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 48–51.

Goodman, J. W.

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

Han, Y.

L. N. Hazra, Y. Han, C. A. Delisle, “Curved kinoform lenses for stigmatic imaging of axial objects,” Appl. Opt. 32, 4775–4794 (1993).
[CrossRef] [PubMed]

L. N. Hazra, Y. Han, C. A. Delisle, “Effects of the thickness of the substrate on the performance and the design of planar kinoform lenses for axial stigmatism in finite conjugate imaging,” Can. J. Phys. 71, 434–441 (1993).
[CrossRef]

L. N. Hazra, Y. Han, C. A. Delisle, “Design of plane kinoform lenses for stigmatic imaging between two prespecified points on the axis,” Opt. Commun. 94, 203–209 (1992).
[CrossRef]

Hazra, L. N.

L. N. Hazra, Y. Han, C. A. Delisle, “Effects of the thickness of the substrate on the performance and the design of planar kinoform lenses for axial stigmatism in finite conjugate imaging,” Can. J. Phys. 71, 434–441 (1993).
[CrossRef]

L. N. Hazra, Y. Han, C. A. Delisle, “Curved kinoform lenses for stigmatic imaging of axial objects,” Appl. Opt. 32, 4775–4794 (1993).
[CrossRef] [PubMed]

L. N. Hazra, Y. Han, C. A. Delisle, “Design of plane kinoform lenses for stigmatic imaging between two prespecified points on the axis,” Opt. Commun. 94, 203–209 (1992).
[CrossRef]

Heckmann, H.-G.

W. Kroninger, H.-G. Heckmann, “Taking the approximation out of diffractive optics design,” Photon. Spectra 28, 120–124 (1994).

Kroninger, W.

W. Kroninger, H.-G. Heckmann, “Taking the approximation out of diffractive optics design,” Photon. Spectra 28, 120–124 (1994).

Morris, G. M.

Rogers, J. R.

Swanson, G. J.

G. J. Swanson, “Binary optics technology: the theory and design of multilevel diffractive optical elements,” Tech. Rep. 854 (Lincoln Laboratory, MIT, Lexington, Mass., 1989).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), p. 191.

Appl. Opt. (2)

Can. J. Phys. (1)

L. N. Hazra, Y. Han, C. A. Delisle, “Effects of the thickness of the substrate on the performance and the design of planar kinoform lenses for axial stigmatism in finite conjugate imaging,” Can. J. Phys. 71, 434–441 (1993).
[CrossRef]

Opt. Commun. (1)

L. N. Hazra, Y. Han, C. A. Delisle, “Design of plane kinoform lenses for stigmatic imaging between two prespecified points on the axis,” Opt. Commun. 94, 203–209 (1992).
[CrossRef]

Photon. Spectra (1)

W. Kroninger, H.-G. Heckmann, “Taking the approximation out of diffractive optics design,” Photon. Spectra 28, 120–124 (1994).

Other (7)

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

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), p. 191.

G. J. Swanson, “Binary optics technology: the theory and design of multilevel diffractive optical elements,” Tech. Rep. 854 (Lincoln Laboratory, MIT, Lexington, Mass., 1989).

code v is a trademark of Optical Research Associates, 550 North Rosemead Boulevard, Pasadena, Calif. 91107.

oslo is a trademark of Sinclair Optics, Inc., 6780 Palmyra Road, Fairport, N.Y. 14450.

zemax is a trademark of Focusoft, Inc., P. O. Box 756, Pleasanton, Calif. 94566.

M. W. Farn, “Modelling of diffractive optics,” in Diffractive Optics, Vol. 11 of 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), pp. 48–51.

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Figures (9)

Fig. 1
Fig. 1

(a) Kinoform AP transforming an incident spherical wave front AB into another spherical wave front AB′. (b) Kinoform AP transforming an incident nonspherical wave front AB into a spherical wave front AB′.

Fig. 2
Fig. 2

Phase function ϕ(r) versus r curves. Solid curves ϕ(r); dashed curves, ϕp(r) = a1r. (a) a1 < 0, (b) a1 > 0, (c) a1 < 0 but dominating higher-order terms.

Fig. 3
Fig. 3

Surface-relief profile over the mth full period zone of the kinoform for transforming the nonspherical incident wave front ∑ into a spherical wave front ∑′ converging to axial point O′.

Fig. 4
Fig. 4

(a) Planar kinoform lens AP forming a stigmatic image of axial point O at O′. l = −120 mm, l′ = 600 mm, f = 100 mm. (b) Planar kinoform lens with a substrate of thickness D. The relief profile is on face à P ˜.

Fig. 5
Fig. 5

First five blazed zones of the plane kinoform lens of the setup of Fig. 4 providing a stigmatic image of O at O′: (left) D = 0, (center) D = 2 mm, (right) D = 4 mm.

Fig. 6
Fig. 6

Refractive–diffractive hybrid lens making a stigmatic image of axial point O at O′. The wave front incident upon plane AP is nonspherical owing to aberrations introduced by the refractive lens.

Fig. 7
Fig. 7

Required phase function ϕ(r) of the kinoform lens at AP of Fig. 6.

Fig. 8
Fig. 8

Full period zone radii corresponding to ϕ(r).

Fig. 9
Fig. 9

Surface-relief profiles over (a) the 1st, (b) the 14th, (c) the 54th, and (d) the 124th zones of the kinoform. Solid curves, exact profiles; dashed curves, approximate profiles.

Equations (27)

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ϕ ( r ) = 2 π λ i = 1 I a i r 2 i ,
t ( r ) = { [ ϕ ( r ) ] 2 π } λ 2 π ( μ - 1 ) ,
ϕ ( r ) = 2 π λ ( a 1 r 2 + a 2 r 4 + a 3 r 6 + a 4 r 8 + a 5 r 10 ) .
ϕ p ( r ) = - ( π λ f ) r 2 .
ϕ ( r m ) = - S m 2 π ,
S = { + 1 if a 1 < 0 or f > 0 - 1 if a 1 > 0 or f < 0 .
δ m ( r ) = λ 2 π { [ ϕ ( r ) ] 2 π } m ,
Δ m ( r ) = δ m ( r ) + s λ ,
s = { + 1 for δ m ( r m + r m - 1 2 ) < 0 0 for δ m ( r m + r m - 1 2 ) > 0 .
δ m ( r m + r m - 1 2 ) 0.
t m ( r ) = Δ m ( r ) μ - 1 ,
( OPL ) 1 = Q R + Δ m ( r ) + R O .
( OPL ) 2 = Q R + [ R S ] + S O = Q R + μ t m + S O ,
Δ m ( r ) + R O = μ t m + S O
Δ m ( r ) + r sin U = μ t m + r - t m sin U ˜ sin U ,
sin U = r + t m sin I [ ( l - t m cos I ) 2 + ( r + t m sin I ) 2 ] 1 / 2 = ( 1 l ) r + t m sin I [ ( 1 - t m cos I l ) + ( r + t m sin I l ) 2 ] 1 / 2 ,
sin U = r r 2 + l 2 = ( 1 l ) r [ 1 + ( r l ) 2 ] 1 / 2 .
U = - I ,             U ˜ = - I ,
sin U ˜ = - sin I = ( 1 / μ ) sin U ,
cos I = ( 1 / μ ) μ 2 - sin 2 U .
A t m 2 + B t m + C = 0 ,
A = 1 - μ 2 ,
B = 2 { μ [ Δ m ( r ) + l ( 1 + r 2 l 2 ) 1 / 2 ] + ( r sin I - l cos I ) } ,
C = - Δ m ( r ) [ Δ m ( r ) + 2 l ( 1 + r 2 l ) 1 / 2 ] .
μ t m = Δ m ( r ) + t m cos I ,
t m = Δ m ( r ) μ - cos I .
[ t m cos I , ( r + t m sin I ) ] [ t m μ ( μ 2 - sin 2 U ) 1 / 2 , ( r - t m μ sin U ) ] .

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