Abstract

Kinoforms (i.e., computer-generated phase holograms) are designed with a new algorithm, the optimal-rotation-angle method, in the paraxial domain. This is a direct Fourier method (i.e., no inverse transform is performed) in which the height of the kinoform relief in each discrete point is chosen so that the diffraction efficiency is increased. The optimal-rotation-angle algorithm has a straightforward geometrical interpretation. It yields excellent results close to, or better than, those obtained with other state-of-the-art methods. The optimal-rotation-angle algorithm can easily be modified to take different restraints into account; as an example, phase-swing-restricted kinoforms, which distribute the light into a number of equally bright spots (so called fan-outs), were designed. The phase-swing restriction lowers the efficiency, but the uniformity can still be made almost perfect.

© 1994 Optical Society of America

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References

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  1. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,”; Optik 35, 237–246 (1972).
  2. M. W. Farn, “New iterative algorithm for the design of phase-only gratings,” in Computer and Optically Generated Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1555, 34–42 (1991).
  3. M. Gale, M. Rossi, H. Schütz, P. Ehbets, H. P. Herzig, D. Prongué, “Continous-relief diffractive optical elements for two-dimensional array generation,” Appl. Opt. 32, 2526–2533 (1993).
    [CrossRef] [PubMed]
  4. F. Nikolajeff, M. Ekberg, M. Larsson, J. Bengtsson, S. Hård, “Shape distortion of diffractive optical elements, directly written with electron beam lithography,” IEE Conf. Publ. 379, 60–61 (1993).

1993 (2)

M. Gale, M. Rossi, H. Schütz, P. Ehbets, H. P. Herzig, D. Prongué, “Continous-relief diffractive optical elements for two-dimensional array generation,” Appl. Opt. 32, 2526–2533 (1993).
[CrossRef] [PubMed]

F. Nikolajeff, M. Ekberg, M. Larsson, J. Bengtsson, S. Hård, “Shape distortion of diffractive optical elements, directly written with electron beam lithography,” IEE Conf. Publ. 379, 60–61 (1993).

1972 (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,”; Optik 35, 237–246 (1972).

Bengtsson, J.

F. Nikolajeff, M. Ekberg, M. Larsson, J. Bengtsson, S. Hård, “Shape distortion of diffractive optical elements, directly written with electron beam lithography,” IEE Conf. Publ. 379, 60–61 (1993).

Ehbets, P.

Ekberg, M.

F. Nikolajeff, M. Ekberg, M. Larsson, J. Bengtsson, S. Hård, “Shape distortion of diffractive optical elements, directly written with electron beam lithography,” IEE Conf. Publ. 379, 60–61 (1993).

Farn, M. W.

M. W. Farn, “New iterative algorithm for the design of phase-only gratings,” in Computer and Optically Generated Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1555, 34–42 (1991).

Gale, M.

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,”; Optik 35, 237–246 (1972).

Hård, S.

F. Nikolajeff, M. Ekberg, M. Larsson, J. Bengtsson, S. Hård, “Shape distortion of diffractive optical elements, directly written with electron beam lithography,” IEE Conf. Publ. 379, 60–61 (1993).

Herzig, H. P.

Larsson, M.

F. Nikolajeff, M. Ekberg, M. Larsson, J. Bengtsson, S. Hård, “Shape distortion of diffractive optical elements, directly written with electron beam lithography,” IEE Conf. Publ. 379, 60–61 (1993).

Nikolajeff, F.

F. Nikolajeff, M. Ekberg, M. Larsson, J. Bengtsson, S. Hård, “Shape distortion of diffractive optical elements, directly written with electron beam lithography,” IEE Conf. Publ. 379, 60–61 (1993).

Prongué, D.

Rossi, M.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,”; Optik 35, 237–246 (1972).

Schütz, H.

Appl. Opt. (1)

IEE Conf. Publ. (1)

F. Nikolajeff, M. Ekberg, M. Larsson, J. Bengtsson, S. Hård, “Shape distortion of diffractive optical elements, directly written with electron beam lithography,” IEE Conf. Publ. 379, 60–61 (1993).

Optik (1)

R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,”; Optik 35, 237–246 (1972).

Other (1)

M. W. Farn, “New iterative algorithm for the design of phase-only gratings,” in Computer and Optically Generated Holographic Optics, I. Cindrich, S. H. Lee, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1555, 34–42 (1991).

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Figures (8)

Fig. 1
Fig. 1

Kinoform setup and schematic complex-plane representation of the amplitude in one pixel in the diffraction plane (the number of links is reduced for clarity).

Fig. 2
Fig. 2

Amplitude vectors for three points in the diffraction plane.

Fig. 3
Fig. 3

Elongation Δln of amplitude vector n when its ith link is rotated.

Fig. 4
Fig. 4

ORA algorithm: FFT's, fast Fourier transforms.

Fig. 5
Fig. 5

200°-phase-swing kinoform.

Fig. 6
Fig. 6

Intensity distribution from the 200° kinoform.

Fig. 7
Fig. 7

90°-phase-swing kinoform.

Fig. 8
Fig. 8

Intensity distribution from the 90° kinoform (the zeroth-order central spike is attenuated by a factor of 100).

Tables (1)

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Table 1 Efficiency a Results (%) from the Optimal-Rotation-Angle, the Farn, and the Herzig Methods

Equations (12)

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à ( n 1 , n 2 ) = ( i 1 , i 2 ) A ( i 1 , i 2 ) exp [ j ϕ ( i 1 , i 2 ) ] × exp [ j 2 π N ( n 1 i 1 + n 2 i 2 ) ] ,
α = ϕ ( i 1 , i 2 ) + 2 π N ( n 1 i 1 + n 2 i 2 ) .
Δ l n = A i cos ( ω n i γ i ) A i cos ω n i ,
n Δ l n = n A i cos ω n i + cos γ i n A i cos ω n i + sin γ i n A i sin ω n i
= C 0 + C 1 cos γ i + C 2 sin γ i = C 0 + C 3 cos ( γ i φ ) ,
C 3 = ( C 1 2 + C 2 2 ) 1 / 2 > 0 ,
φ = { arctan ( C 1 / C 2 ) for C 1 > 0 arctan ( C 1 / C 2 ) + π for C 1 < 0 ( π / 2 ) sign ( C 2 ) for C 1 = 0 .
C 1 = n A i weight n cos ω n i , C 2 = n A i weight n sin ω n i .
weight n new = weight n old ( desired intensity in spot n obtained intensity in spot n ) 0.25 ,
efficiency = n I n total intensity in all matrix pixels in the diffraction plane ,
uniformity error = I n max I n min I n max + I n min ,
C 1 = n ( nonzeroth ) A i weight n cos ω n i A i weight zeroth cos ω zeroth , i ,

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