Abstract

A novel, to our knowledge, configuration for the design and fabrication of zero-order computer-generated holograms in which each pixel is split into two centrosymmetric equal-sized regions is proposed and tested. In a manner similar to other approaches this configuration also permits the encoding and the reconstruction of a complex function that exhibits phase as well as amplitude variations by use of a phase-only filter. A detailed mathematical analysis is followed by evaluation of the error of the encoding approach, which is calculated and compared with the error exhibited by other approaches. Computer simulations as well as optical experiments demonstrate the capabilities of this novel configuration.

© 2001 Optical Society of America

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References

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  1. R. Gruhlke, L. Giamonna, C. Langhorn, K. Kanzler, J. Kedmi, “Medical and industrial laser beam shaping by diffractive optical elements,” in Laser Coherence Control: Technology and Applications, T. K. Kessler, H. T. Powell, eds., Proc. SPIE1870, 120–129 (1993).
  2. J. L. Horner, J. R. Leger, “Pattern recognition with binary phase-only filter,” Appl. Opt. 24, 609–611 (1985).
    [CrossRef] [PubMed]
  3. S. J. Walker, J. Jahns, “Optical clock distribution using integrated free-space optics,” Opt. Commun. 90, 359–371 (1992).
    [CrossRef]
  4. T. R. M. Sales, G. M. Morris, “Diffractive–refractive behavior of kinoform lenses,” Appl. Opt. 36, 253–257 (1997).
    [CrossRef] [PubMed]
  5. B. R. Brown, A. W. Lohmann, “Complex spatial filtering with binary masks,” Appl. Opt. 5, 967–969 (1966).
    [CrossRef] [PubMed]
  6. A. W. Lohmann, D. P. Paris, “Binary Fraunhofer holograms generated by computer,” Appl. Opt. 6, 1739–1749 (1967).
    [CrossRef] [PubMed]
  7. W. H. Lee, “Sampled Fourier transform holograms generated by computer,” Appl. Opt. 9, 639–643 (1970).
    [CrossRef] [PubMed]
  8. C. B. Burkhardt, “A simplification of Lee’s method of generating holograms by computer,” Appl. Opt. 9, 1949 (1970).
  9. C. K. Hsueh, A. A. Sawchuck, “Computer-generated double-phase holograms,” Appl. Opt. 17, 3874–3883 (1978).
    [CrossRef] [PubMed]
  10. N. C. Gallagher, J. A. Bucklew, “Nondetour phase digital holograms: an analysis: an errata,” Appl. Opt. 19, 4266–4272 (1980).
    [CrossRef] [PubMed]
  11. R. M. Matic, E. W. Hensen, “Nondetour computer-generated holograms: an improvement variation,” Appl. Opt. 21, 2304–2305 (1982).
    [CrossRef]
  12. W. H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, 16th ed., E. Wolf, ed. (North-Holland, Amsterdam, 1978), pp. 121–232.
  13. U. Levy, E. Marom, D. Mendlovic, “Modifications of detour phase computer-generated holograms,” Appl. Opt. 37, 3044–3052 (1998).
    [CrossRef]
  14. J. P. Kirk, A. L. Joens, “Phase-only complex-valued spatial filter,” J. Opt. Soc. Am. 61, 1024–1028 (1971).
    [CrossRef]
  15. J. Turunen, P. Vahimaa, M. Honkanen, O. Salminen, E. Noponen, “Zeroth-order complex-amplitude modulation with dielectric Fourier-type diffractive elements,” J. Mod. Opt. 43, 1389–1398 (1996).
  16. Y. Danziger, E. Hasman, A. A. Friesem, A. W. Lohmann, “Multilevel diffractive elements for generalized wave-front shaping,” Opt. Eng. 35, 2556–2565 (1996).
    [CrossRef]
  17. U. Levy, D. Mendlovic, Z. Zalevsky, “On-axis phase-only computer-generated holograms based on a minimal etching process,” J. Mod. Opt. 45, 1437–1449 (1998).
    [CrossRef]
  18. J. M. Florence, R. D. Juday, “Full complex spatial filtering with a phase-mostly DMD,” in Wave Propagation and Scattering in Varied Media II, V. K. Varadan, ed., Proc. SPIE1558, 487–498 (1991).
    [CrossRef]
  19. D. Mendlovic, G. Shabtay, U. Levy, Z. Zalevsky, E. Marom, “Encoding technique for design of zero-order (on-axis) Fraunhofer computer-generated holograms,” Appl. Opt. 36, 8427–8434 (1997).
    [CrossRef]

1998 (2)

U. Levy, D. Mendlovic, Z. Zalevsky, “On-axis phase-only computer-generated holograms based on a minimal etching process,” J. Mod. Opt. 45, 1437–1449 (1998).
[CrossRef]

U. Levy, E. Marom, D. Mendlovic, “Modifications of detour phase computer-generated holograms,” Appl. Opt. 37, 3044–3052 (1998).
[CrossRef]

1997 (2)

1996 (2)

J. Turunen, P. Vahimaa, M. Honkanen, O. Salminen, E. Noponen, “Zeroth-order complex-amplitude modulation with dielectric Fourier-type diffractive elements,” J. Mod. Opt. 43, 1389–1398 (1996).

Y. Danziger, E. Hasman, A. A. Friesem, A. W. Lohmann, “Multilevel diffractive elements for generalized wave-front shaping,” Opt. Eng. 35, 2556–2565 (1996).
[CrossRef]

1992 (1)

S. J. Walker, J. Jahns, “Optical clock distribution using integrated free-space optics,” Opt. Commun. 90, 359–371 (1992).
[CrossRef]

1985 (1)

1982 (1)

1980 (1)

1978 (1)

1971 (1)

J. P. Kirk, A. L. Joens, “Phase-only complex-valued spatial filter,” J. Opt. Soc. Am. 61, 1024–1028 (1971).
[CrossRef]

1970 (2)

1967 (1)

1966 (1)

Brown, B. R.

Bucklew, J. A.

Burkhardt, C. B.

Danziger, Y.

Y. Danziger, E. Hasman, A. A. Friesem, A. W. Lohmann, “Multilevel diffractive elements for generalized wave-front shaping,” Opt. Eng. 35, 2556–2565 (1996).
[CrossRef]

Florence, J. M.

J. M. Florence, R. D. Juday, “Full complex spatial filtering with a phase-mostly DMD,” in Wave Propagation and Scattering in Varied Media II, V. K. Varadan, ed., Proc. SPIE1558, 487–498 (1991).
[CrossRef]

Friesem, A. A.

Y. Danziger, E. Hasman, A. A. Friesem, A. W. Lohmann, “Multilevel diffractive elements for generalized wave-front shaping,” Opt. Eng. 35, 2556–2565 (1996).
[CrossRef]

Gallagher, N. C.

Giamonna, L.

R. Gruhlke, L. Giamonna, C. Langhorn, K. Kanzler, J. Kedmi, “Medical and industrial laser beam shaping by diffractive optical elements,” in Laser Coherence Control: Technology and Applications, T. K. Kessler, H. T. Powell, eds., Proc. SPIE1870, 120–129 (1993).

Gruhlke, R.

R. Gruhlke, L. Giamonna, C. Langhorn, K. Kanzler, J. Kedmi, “Medical and industrial laser beam shaping by diffractive optical elements,” in Laser Coherence Control: Technology and Applications, T. K. Kessler, H. T. Powell, eds., Proc. SPIE1870, 120–129 (1993).

Hasman, E.

Y. Danziger, E. Hasman, A. A. Friesem, A. W. Lohmann, “Multilevel diffractive elements for generalized wave-front shaping,” Opt. Eng. 35, 2556–2565 (1996).
[CrossRef]

Hensen, E. W.

Honkanen, M.

J. Turunen, P. Vahimaa, M. Honkanen, O. Salminen, E. Noponen, “Zeroth-order complex-amplitude modulation with dielectric Fourier-type diffractive elements,” J. Mod. Opt. 43, 1389–1398 (1996).

Horner, J. L.

Hsueh, C. K.

Jahns, J.

S. J. Walker, J. Jahns, “Optical clock distribution using integrated free-space optics,” Opt. Commun. 90, 359–371 (1992).
[CrossRef]

Joens, A. L.

J. P. Kirk, A. L. Joens, “Phase-only complex-valued spatial filter,” J. Opt. Soc. Am. 61, 1024–1028 (1971).
[CrossRef]

Juday, R. D.

J. M. Florence, R. D. Juday, “Full complex spatial filtering with a phase-mostly DMD,” in Wave Propagation and Scattering in Varied Media II, V. K. Varadan, ed., Proc. SPIE1558, 487–498 (1991).
[CrossRef]

Kanzler, K.

R. Gruhlke, L. Giamonna, C. Langhorn, K. Kanzler, J. Kedmi, “Medical and industrial laser beam shaping by diffractive optical elements,” in Laser Coherence Control: Technology and Applications, T. K. Kessler, H. T. Powell, eds., Proc. SPIE1870, 120–129 (1993).

Kedmi, J.

R. Gruhlke, L. Giamonna, C. Langhorn, K. Kanzler, J. Kedmi, “Medical and industrial laser beam shaping by diffractive optical elements,” in Laser Coherence Control: Technology and Applications, T. K. Kessler, H. T. Powell, eds., Proc. SPIE1870, 120–129 (1993).

Kirk, J. P.

J. P. Kirk, A. L. Joens, “Phase-only complex-valued spatial filter,” J. Opt. Soc. Am. 61, 1024–1028 (1971).
[CrossRef]

Langhorn, C.

R. Gruhlke, L. Giamonna, C. Langhorn, K. Kanzler, J. Kedmi, “Medical and industrial laser beam shaping by diffractive optical elements,” in Laser Coherence Control: Technology and Applications, T. K. Kessler, H. T. Powell, eds., Proc. SPIE1870, 120–129 (1993).

Lee, W. H.

W. H. Lee, “Sampled Fourier transform holograms generated by computer,” Appl. Opt. 9, 639–643 (1970).
[CrossRef] [PubMed]

W. H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, 16th ed., E. Wolf, ed. (North-Holland, Amsterdam, 1978), pp. 121–232.

Leger, J. R.

Levy, U.

Lohmann, A. W.

Marom, E.

Matic, R. M.

Mendlovic, D.

Morris, G. M.

Noponen, E.

J. Turunen, P. Vahimaa, M. Honkanen, O. Salminen, E. Noponen, “Zeroth-order complex-amplitude modulation with dielectric Fourier-type diffractive elements,” J. Mod. Opt. 43, 1389–1398 (1996).

Paris, D. P.

Sales, T. R. M.

Salminen, O.

J. Turunen, P. Vahimaa, M. Honkanen, O. Salminen, E. Noponen, “Zeroth-order complex-amplitude modulation with dielectric Fourier-type diffractive elements,” J. Mod. Opt. 43, 1389–1398 (1996).

Sawchuck, A. A.

Shabtay, G.

Turunen, J.

J. Turunen, P. Vahimaa, M. Honkanen, O. Salminen, E. Noponen, “Zeroth-order complex-amplitude modulation with dielectric Fourier-type diffractive elements,” J. Mod. Opt. 43, 1389–1398 (1996).

Vahimaa, P.

J. Turunen, P. Vahimaa, M. Honkanen, O. Salminen, E. Noponen, “Zeroth-order complex-amplitude modulation with dielectric Fourier-type diffractive elements,” J. Mod. Opt. 43, 1389–1398 (1996).

Walker, S. J.

S. J. Walker, J. Jahns, “Optical clock distribution using integrated free-space optics,” Opt. Commun. 90, 359–371 (1992).
[CrossRef]

Zalevsky, Z.

U. Levy, D. Mendlovic, Z. Zalevsky, “On-axis phase-only computer-generated holograms based on a minimal etching process,” J. Mod. Opt. 45, 1437–1449 (1998).
[CrossRef]

D. Mendlovic, G. Shabtay, U. Levy, Z. Zalevsky, E. Marom, “Encoding technique for design of zero-order (on-axis) Fraunhofer computer-generated holograms,” Appl. Opt. 36, 8427–8434 (1997).
[CrossRef]

Appl. Opt. (11)

T. R. M. Sales, G. M. Morris, “Diffractive–refractive behavior of kinoform lenses,” Appl. Opt. 36, 253–257 (1997).
[CrossRef] [PubMed]

B. R. Brown, A. W. Lohmann, “Complex spatial filtering with binary masks,” Appl. Opt. 5, 967–969 (1966).
[CrossRef] [PubMed]

A. W. Lohmann, D. P. Paris, “Binary Fraunhofer holograms generated by computer,” Appl. Opt. 6, 1739–1749 (1967).
[CrossRef] [PubMed]

W. H. Lee, “Sampled Fourier transform holograms generated by computer,” Appl. Opt. 9, 639–643 (1970).
[CrossRef] [PubMed]

C. B. Burkhardt, “A simplification of Lee’s method of generating holograms by computer,” Appl. Opt. 9, 1949 (1970).

C. K. Hsueh, A. A. Sawchuck, “Computer-generated double-phase holograms,” Appl. Opt. 17, 3874–3883 (1978).
[CrossRef] [PubMed]

N. C. Gallagher, J. A. Bucklew, “Nondetour phase digital holograms: an analysis: an errata,” Appl. Opt. 19, 4266–4272 (1980).
[CrossRef] [PubMed]

R. M. Matic, E. W. Hensen, “Nondetour computer-generated holograms: an improvement variation,” Appl. Opt. 21, 2304–2305 (1982).
[CrossRef]

J. L. Horner, J. R. Leger, “Pattern recognition with binary phase-only filter,” Appl. Opt. 24, 609–611 (1985).
[CrossRef] [PubMed]

U. Levy, E. Marom, D. Mendlovic, “Modifications of detour phase computer-generated holograms,” Appl. Opt. 37, 3044–3052 (1998).
[CrossRef]

D. Mendlovic, G. Shabtay, U. Levy, Z. Zalevsky, E. Marom, “Encoding technique for design of zero-order (on-axis) Fraunhofer computer-generated holograms,” Appl. Opt. 36, 8427–8434 (1997).
[CrossRef]

J. Mod. Opt. (2)

J. Turunen, P. Vahimaa, M. Honkanen, O. Salminen, E. Noponen, “Zeroth-order complex-amplitude modulation with dielectric Fourier-type diffractive elements,” J. Mod. Opt. 43, 1389–1398 (1996).

U. Levy, D. Mendlovic, Z. Zalevsky, “On-axis phase-only computer-generated holograms based on a minimal etching process,” J. Mod. Opt. 45, 1437–1449 (1998).
[CrossRef]

J. Opt. Soc. Am. (1)

J. P. Kirk, A. L. Joens, “Phase-only complex-valued spatial filter,” J. Opt. Soc. Am. 61, 1024–1028 (1971).
[CrossRef]

Opt. Commun. (1)

S. J. Walker, J. Jahns, “Optical clock distribution using integrated free-space optics,” Opt. Commun. 90, 359–371 (1992).
[CrossRef]

Opt. Eng. (1)

Y. Danziger, E. Hasman, A. A. Friesem, A. W. Lohmann, “Multilevel diffractive elements for generalized wave-front shaping,” Opt. Eng. 35, 2556–2565 (1996).
[CrossRef]

Other (3)

J. M. Florence, R. D. Juday, “Full complex spatial filtering with a phase-mostly DMD,” in Wave Propagation and Scattering in Varied Media II, V. K. Varadan, ed., Proc. SPIE1558, 487–498 (1991).
[CrossRef]

W. H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics, 16th ed., E. Wolf, ed. (North-Holland, Amsterdam, 1978), pp. 121–232.

R. Gruhlke, L. Giamonna, C. Langhorn, K. Kanzler, J. Kedmi, “Medical and industrial laser beam shaping by diffractive optical elements,” in Laser Coherence Control: Technology and Applications, T. K. Kessler, H. T. Powell, eds., Proc. SPIE1870, 120–129 (1993).

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Figures (11)

Fig. 1
Fig. 1

Basic macropixel cell configurations.

Fig. 2
Fig. 2

Deviations from the desired complex value within the cell of interest along the horizontal axis: (a) amplitude deviations, (b) phase deviations. Results are plotted for input with padding (P = 2) of the same size as the object and without padding (P = 1).

Fig. 3
Fig. 3

Vector description of the distance function [Eq. (19)] between phasors.

Fig. 4
Fig. 4

Effect of zero padding on the quality of the reconstructed image of the letter H: (a) desired reconstruction, (b) reconstructed image for the case of P = 4, (c) reconstructed image for the case of P = 8.

Fig. 5
Fig. 5

Cross section of the reconstructed image [Figs. 4(a) and 4(b), respectively] through the center axis: (a) P = 4, (b) P = 8.

Fig. 6
Fig. 6

Quantization effect on image quality. The simulation was done for the P = 4 case [Fig. 4(a)] with 32 phase levels.

Fig. 7
Fig. 7

Image quality for the case of the desired construction’s being multiplied by a quadratic phase. The simulation was carried out for P = 4.

Fig. 8
Fig. 8

Intensity variations along the central cross section shown in Fig. 7: (a) the desired reconstruction, (b) the simulation results.

Fig. 9
Fig. 9

Optical experimental setup: f 1 and f 2 are the focal lengths of L1 and L2, respectively.

Fig. 10
Fig. 10

Obtained optical reconstruction from the setup shown in Fig. 9.

Fig. 11
Fig. 11

Intensity variations along the horizontal cross section shown in Fig. 10.

Tables (3)

Tables Icon

Table 1 Maximum-Error Evaluation for the Proposed Technique

Tables Icon

Table 2 Maximum-Error Evaluation for Ref. 19 (1-D Configuration)

Tables Icon

Table 3 Maximum-Error Evaluation for Ref. 17

Equations (28)

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Hνx, νy=nm Hn,m,
Hn,mνx, νy=rectνxδνx, νyδνy-rectνxδνx2, νyδνy2expiϕn,m1+rectνxδνx2, νyδνy2expiϕn,m2* δνx-nδν, νy-mδν.
rectα, β=rectαrectβ=1-0.5<α, β<0.50elsewhere.
hx, y=-- Hνx, νyexpi2πxνx+yνydνxdνy.
hx, y=δνxδνynmsin cδνxxsin cδνyy-12 sin cδνx2 xsin cδνy2 yexpiϕn,m1+12 sin cδνx2 xsin cδνy2 y×expiϕn,m2expi2πxnδνx+ymδνy,
sin cx=sinπxπx.
xδνx  1,  yδνy  1.
sin cδνxx, sin cδνyy, sin cδνx2 x,  sin cδνy2 y1.
hx, y=δνxδνy2nmexpiϕn,m1+expiϕn,m2×expi2πxnδνx+ymδνy.
Hnδνx, mδνy=An,m expiϕn,m.
hx, yδνxδνynm Hnδνx, mδνy×expi2πxnδνx+ymδνy.
hx, y=δνxδνynm cosϕn,m1-ϕn,m22×expϕn,m1+ϕn,m22expi2πxnδνx+ymδνy.
An,m=cosϕn,m1-ϕn,m22,
ϕn,m=ϕn,m1+ϕn,m22,
ϕn,m1=ϕn,m+cos-1An,m,
ϕn,m2=ϕn,m-cos-1An,m.
12 sin cδνx2 xsin cδνy2 y,  sincδνxxsincδνyy-12 sincδνx2xsincδνy2y.
P=Umaxumax.
Percent error=100 DistanceAdesired,
Distance=Aobtained2+Adesired2-2AobtainedAdesired cosϕobtained-ϕdesired1/2.
MSE=nm |hn,mdesired-hn,mobtained|2,
nm |hn,mdesired|2=nm |hn,mobtained|2=1.
Idesired=nm |An,m|2,
I=kp |Ak,pfilter|2=kp |1|2=Δ2.
ρdesired=nm |An,mdesired|2Δ2.
η=ρobtainedρdesired,
Izero orderobtainedρobtained=Izero orderobtainedItotalobtained
ϕn,m=4π1-n-n02+m-m02n02+m02,

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