Abstract

An algorithm for the design of diffractive optical phase elements (kinoforms) that give rise to fan-out (i.e., spot) patterns was developed and tested. The algorithm is based on the Helmholtz–Kirchhoff rigorous scalar diffraction integral for the evaluation of the electric field behind the kinoform. The optimization of the kinoform phase modulation is performed with an efficient optimal-rotation-angle method. The algorithm permits any spatial configuration of the locations of the desired spots. For example, the spots (all or some) can be located at large angles to the optical axis (nonparaxial case) or they can be located in the near near field of the kinoform, i.e., where the Fresnel approximation is no longer valid. Two examples of fabricated kinoforms designed with this algorithm are presented.

© 1997 Optical Society of America

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References

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  1. M. Duparré, B. Lüdge, R. Kowarschik, M. A. Golub, V. A. Soifer, “Investigation of Computer-generated diffractive beam-shapers for diverse tasks of laser-beam transformation,” in Photodynamic Therapy of Cancer II, D. Brault, G. Jori, J. Moan, eds., Proc. SPIE2325, 118–128 (1995).
  2. T. Dresel, M. Beyerlein, J. Schwider, “Design of computer-generated beam-shaping holograms by iterative finite-element mesh adaption,” Appl. Opt. 35, 6865–6874 (1996).
    [CrossRef] [PubMed]
  3. J. Bengtsson, “Kinoform-only Gaussian-to-rectangle beam shaper for a semiconductor laser,” Appl. Opt. 35, 3807–3814 (1996).
    [CrossRef] [PubMed]
  4. D. A. Gremaux, N. C. Gallagher, “Limits of scalar diffraction theory for conducting gratings,” Appl. Opt. 32, 1948–1953 (1993).
    [CrossRef] [PubMed]
  5. D. A. Pommet, M. G. Moharam, E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A 11, 1827–1834 (1994).
    [CrossRef]
  6. J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
    [CrossRef]
  7. J. Bengtsson, N. Eriksson, A. Larsson, “Small-feature-size fan-out kinoform etched in GaAs,” Appl. Opt. 35, 801–806 (1996).
    [CrossRef] [PubMed]
  8. J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1167 (1989).
    [CrossRef]
  9. B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Efficient design of direct-binary-search computer-generated holograms,” J. Opt. Soc. Am. A 8, 652–660 (1991).
    [CrossRef]
  10. N. Yoshikawa, T. Yatagai, “Phase optimization of a kinoform by simulated annealing,” Appl. Opt. 33, 863–868 (1994).
    [CrossRef] [PubMed]
  11. H. Lüpken, T. Peter, F. Wyrowski, O. Bryngdahl, “Phase synthesis for an array illuminator,” Opt. Commun. 91, 163–167 (1992).
    [CrossRef]
  12. R. W. Gerchberg, W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237–246 (1972).
  13. F. Wyrowski, O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A 5, 1058–1065 (1988).
    [CrossRef]
  14. M. W. Farn, “New iterative algorithm for the design of phase-only gratings,” in Holographic Optics: Optically and Computer Generated, I. Cindrich, S. H. Lee, eds., Proc. SPIE1052, 142–149 (1989).
  15. D. Prongué, H. P. Herzig, R. Dändliker, M. T. Gale, “Optimized kinoform structures for highly efficient fan-out elements,” Appl. Opt. 31, 5706–5711 (1992).
    [CrossRef] [PubMed]
  16. M. T. Gale, M. Rossi, H. Schütz, P. Ehbets, H. P. Herzig, D. Prongué, “Continuous-relief diffractive optical elements for two-dimensional array generation,” Appl. Opt. 32, 2526–2533 (1993).
    [CrossRef] [PubMed]
  17. J. Bengtsson, “Kinoform design with an optimal-rotation-angle method,” Appl. Opt. 33, 6879–6884 (1994).
    [CrossRef] [PubMed]
  18. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

1996 (3)

1994 (3)

1993 (3)

1992 (2)

H. Lüpken, T. Peter, F. Wyrowski, O. Bryngdahl, “Phase synthesis for an array illuminator,” Opt. Commun. 91, 163–167 (1992).
[CrossRef]

D. Prongué, H. P. Herzig, R. Dändliker, M. T. Gale, “Optimized kinoform structures for highly efficient fan-out elements,” Appl. Opt. 31, 5706–5711 (1992).
[CrossRef] [PubMed]

1991 (1)

1989 (1)

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1167 (1989).
[CrossRef]

1988 (1)

1972 (1)

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

Allebach, J. P.

Bengtsson, J.

Beyerlein, M.

Bryngdahl, O.

H. Lüpken, T. Peter, F. Wyrowski, O. Bryngdahl, “Phase synthesis for an array illuminator,” Opt. Commun. 91, 163–167 (1992).
[CrossRef]

F. Wyrowski, O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A 5, 1058–1065 (1988).
[CrossRef]

Dändliker, R.

Dresel, T.

Duparré, M.

M. Duparré, B. Lüdge, R. Kowarschik, M. A. Golub, V. A. Soifer, “Investigation of Computer-generated diffractive beam-shapers for diverse tasks of laser-beam transformation,” in Photodynamic Therapy of Cancer II, D. Brault, G. Jori, J. Moan, eds., Proc. SPIE2325, 118–128 (1995).

Ehbets, P.

Eriksson, N.

Farn, M. W.

M. W. Farn, “New iterative algorithm for the design of phase-only gratings,” in Holographic Optics: Optically and Computer Generated, I. Cindrich, S. H. Lee, eds., Proc. SPIE1052, 142–149 (1989).

Gale, M. T.

Gallagher, N. C.

Gerchberg, R. W.

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

Golub, M. A.

M. Duparré, B. Lüdge, R. Kowarschik, M. A. Golub, V. A. Soifer, “Investigation of Computer-generated diffractive beam-shapers for diverse tasks of laser-beam transformation,” in Photodynamic Therapy of Cancer II, D. Brault, G. Jori, J. Moan, eds., Proc. SPIE2325, 118–128 (1995).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Grann, E. B.

Gremaux, D. A.

Herzig, H. P.

Jennison, B. K.

Kowarschik, R.

M. Duparré, B. Lüdge, R. Kowarschik, M. A. Golub, V. A. Soifer, “Investigation of Computer-generated diffractive beam-shapers for diverse tasks of laser-beam transformation,” in Photodynamic Therapy of Cancer II, D. Brault, G. Jori, J. Moan, eds., Proc. SPIE2325, 118–128 (1995).

Larsson, A.

Lüdge, B.

M. Duparré, B. Lüdge, R. Kowarschik, M. A. Golub, V. A. Soifer, “Investigation of Computer-generated diffractive beam-shapers for diverse tasks of laser-beam transformation,” in Photodynamic Therapy of Cancer II, D. Brault, G. Jori, J. Moan, eds., Proc. SPIE2325, 118–128 (1995).

Lüpken, H.

H. Lüpken, T. Peter, F. Wyrowski, O. Bryngdahl, “Phase synthesis for an array illuminator,” Opt. Commun. 91, 163–167 (1992).
[CrossRef]

Miller, J. M.

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

Moharam, M. G.

Noponen, E.

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

Peter, T.

H. Lüpken, T. Peter, F. Wyrowski, O. Bryngdahl, “Phase synthesis for an array illuminator,” Opt. Commun. 91, 163–167 (1992).
[CrossRef]

Pommet, D. A.

Prongué, D.

Ross, N.

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

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 (Stuttgart) 35, 237–246 (1972).

Schütz, H.

Schwider, J.

Soifer, V. A.

M. Duparré, B. Lüdge, R. Kowarschik, M. A. Golub, V. A. Soifer, “Investigation of Computer-generated diffractive beam-shapers for diverse tasks of laser-beam transformation,” in Photodynamic Therapy of Cancer II, D. Brault, G. Jori, J. Moan, eds., Proc. SPIE2325, 118–128 (1995).

Sweeney, D. W.

Taghizadeh, M. R.

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

Turunen, J.

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1167 (1989).
[CrossRef]

Vasara, A.

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1167 (1989).
[CrossRef]

Westerholm, J.

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1167 (1989).
[CrossRef]

Wyrowski, F.

H. Lüpken, T. Peter, F. Wyrowski, O. Bryngdahl, “Phase synthesis for an array illuminator,” Opt. Commun. 91, 163–167 (1992).
[CrossRef]

F. Wyrowski, O. Bryngdahl, “Iterative Fourier-transform algorithm applied to computer holography,” J. Opt. Soc. Am. A 5, 1058–1065 (1988).
[CrossRef]

Yatagai, T.

Yoshikawa, N.

Appl. Opt. (8)

J. Mod. Opt. (1)

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

H. Lüpken, T. Peter, F. Wyrowski, O. Bryngdahl, “Phase synthesis for an array illuminator,” Opt. Commun. 91, 163–167 (1992).
[CrossRef]

Opt. Eng. (1)

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1167 (1989).
[CrossRef]

Optik (Stuttgart) (1)

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

Other (3)

M. W. Farn, “New iterative algorithm for the design of phase-only gratings,” in Holographic Optics: Optically and Computer Generated, I. Cindrich, S. H. Lee, eds., Proc. SPIE1052, 142–149 (1989).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

M. Duparré, B. Lüdge, R. Kowarschik, M. A. Golub, V. A. Soifer, “Investigation of Computer-generated diffractive beam-shapers for diverse tasks of laser-beam transformation,” in Photodynamic Therapy of Cancer II, D. Brault, G. Jori, J. Moan, eds., Proc. SPIE2325, 118–128 (1995).

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Figures (14)

Fig. 1
Fig. 1

Kinoform plane with pixel k indicated and one of the locations m in space.

Fig. 2
Fig. 2

Construction of the components of the k-vector in the center of pixel k.

Fig. 3
Fig. 3

Complex-number plane showing the amplitude change Δl of the field in spot m from changing the phase modulation of pixel k by Δφk. Re, real; Im, imaginary.

Fig. 4
Fig. 4

Flowchart of the complete design algorithm.

Fig. 5
Fig. 5

Uniformity error versus the number of iterations for the design of the near near-field kinoform.

Fig. 6
Fig. 6

Designed phase modulation of the near near-field kinoform.

Fig. 7
Fig. 7

Measured intensity distribution from the fabricated kinoform with the designed phase modulation of Fig. 6. The two arrows indicate line-scan endpoints for Figs. 8 and 9.

Fig. 8
Fig. 8

Calculated intensity along the line whose endpoints are indicated by the white arrows in Fig. 7.

Fig. 9
Fig. 9

Measured intensity reconstructed along the same line as for Fig. 8. The slight defocusing leads to the spots being broader than those shown in Fig. 8.

Fig. 10
Fig. 10

Uniformity error versus the number of iterations for the three-dimensional fan-out kinoform.

Fig. 11
Fig. 11

Designed phase modulation of the kinoform giving a three-dimensional fan-out.

Fig. 12
Fig. 12

Measured intensity distribution in a plane 5 cm behind the kinoform.

Fig. 13
Fig. 13

Measured intensity distribution in a plane 10 cm behind the kinoform.

Fig. 14
Fig. 14

Measured intensity distribution in a plane 20 cm behind the kinoform.

Equations (27)

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Ukm=Akm expjφkmexpjφinc+φk,
Ukm=14πpixel kUknG-UkGndxdy,
Gx, y, z, u, v, L|z=0=expjkr01r01,
r01=u-x2+v-y2+L-z21/2|z=0,
k=2πλ0.
Ukξ, η, ζζ=0=Ak expjφinc+φk×expjkxξ+kyη+kzζζ=0.
Ukn=-Ukζ=-jkzUk.
k¯k=R¯kR¯k=xcxˆ+ycŷ+Rzˆxc2+yc2+R21/2.
k¯=kxxˆ+kyŷ+kzzˆ,
Gn=-Gz=Lr01jk-1r01G,
Ukm=14πpixel k-jkzUkG-UkLr01jk-1r01Gdξdη14π-jkz-Lr01cjk-1r01cpixel k UkGdξdη.
r01c=u-xc2+v-yc2+L21/2.
pixel k UkGdξdη=Ak expjφinc+φkr01c×pixel kexpjkxξ+kyη×expjkr01dξdη,
r01=u-xc+ξ2+v-yc+η2+L21/2r01c+xcξ-uξ+ycη-vηr01c,
k˜x=kx+kxc-ur01c, k˜y=ky+kyc-vr01c,
pixel kUkGdξdη=4Ak expjφinc+φkr01c×expjkr01csink˜xa2k˜xsink˜yb2k˜y.
Akm expjφkm=14π-jkz-Lr01cjk-1r01c×4Akr01cexpjkr01csink˜xa2k˜xsink˜yb2k˜y,
φkm=φm-φkm+φinc+φk,
Δl=Akm cosφkm-Δφk-Akm cos φkm.
m Δl=mAkm cosφkm-Δφk-Akm cos φkm=S1 cos Δφk+S2 sin Δφk-S1=S3 cosΔφk-αk-S1if S1>0-S3 cosΔφk-αk-S1if S1<0
S1=mAkm cos ϕkm, S2=m Akm sin ϕkm, S3=S12+S221/2, αk=arctanS2S1.
Δφk=αk,if S1>0,Δφk=αk+π,if S1<0,Δφk=π/2,if S1=0 and S2>0,Δφk=-π/2,if S1=0 and S2<0.
S1=m wmAkm cos ϕkm, S2=m wmAkm sin ϕkm.
wmnew=wmoldImdesiredIm0.35.
Um=k Ukm=kAkm expjφkmexpjφinc+φk.
unif. err.=Im/Imdesiredmax-Im/ImdesiredminIm/Imdesiredmax+Im/Imdesiredmin
depthk=φk2πλ0nresist-1,

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