Abstract

A new method for encoding high-efficiency radially symmetric computer-generated holograms is formulated. An iterative approach is used to determine fringe widths and phase values for phase-only holographic optical elements. Computer-simulation results indicate that this approach can be used to increase the efficiency of low-f-number elements significantly. The experimental results for an f/1 element with a diffraction efficiency of 90% are presented.

© 1993 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 multi-level diffractive optical elements,” Lincoln Lab. Tech. Rep. 854 (MIT Lincoln Laboratory, Lexington, Mass., 1989).
  2. J. R. Leger, M. L. Scott, P. Bundman, M. P. Griswold, “Astigmatic wavefront correction of a gain-guided laser diode array using anamorphic diffractive microlenses,” in Computer-Generated Holography II, S. H. Lee, ed., Proc. Soc. Photo-Opt. Instrum. Eng.82–89 (1988).
    [CrossRef]
  3. K. S. Urquhart, S. H. Lee, C. C. Guest, M. R. Feldman, H. Farhoosh, “Computer aided design of computer generated holograms for electron beam fabrication,” Appl. Opt. 28, 3387–3396 (1989).
    [CrossRef] [PubMed]
  4. J. Jahns, S. J. Walker, “Two-dimensional array of diffractive microlenses fabricated by thin film deposition,” Appl. Opt. 29, 931–936 (1990).
    [CrossRef] [PubMed]
  5. M. R. Feldman, C. C. Guest, “Iterative encoding of high efficiency hologram encoding for generation of spot arrays,” Opt. Lett. 14, 479–481 (1989).
    [CrossRef] [PubMed]
  6. J. D. Stack, M. R. Feldman, “Recursive mean-squared-error algorithm for iterative discrete on-axis encoded holograms,” Appl. Opt. 31, 4839–4846 (1991).
    [CrossRef]
  7. S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
    [CrossRef] [PubMed]
  8. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1988), p. 45.

1991

1990

1989

1983

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Bundman, P.

J. R. Leger, M. L. Scott, P. Bundman, M. P. Griswold, “Astigmatic wavefront correction of a gain-guided laser diode array using anamorphic diffractive microlenses,” in Computer-Generated Holography II, S. H. Lee, ed., Proc. Soc. Photo-Opt. Instrum. Eng.82–89 (1988).
[CrossRef]

Farhoosh, H.

Feldman, M. R.

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1988), p. 45.

Griswold, M. P.

J. R. Leger, M. L. Scott, P. Bundman, M. P. Griswold, “Astigmatic wavefront correction of a gain-guided laser diode array using anamorphic diffractive microlenses,” in Computer-Generated Holography II, S. H. Lee, ed., Proc. Soc. Photo-Opt. Instrum. Eng.82–89 (1988).
[CrossRef]

Guest, C. C.

Jahns, J.

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Lee, S. H.

Leger, J. R.

J. R. Leger, M. L. Scott, P. Bundman, M. P. Griswold, “Astigmatic wavefront correction of a gain-guided laser diode array using anamorphic diffractive microlenses,” in Computer-Generated Holography II, S. H. Lee, ed., Proc. Soc. Photo-Opt. Instrum. Eng.82–89 (1988).
[CrossRef]

Scott, M. L.

J. R. Leger, M. L. Scott, P. Bundman, M. P. Griswold, “Astigmatic wavefront correction of a gain-guided laser diode array using anamorphic diffractive microlenses,” in Computer-Generated Holography II, S. H. Lee, ed., Proc. Soc. Photo-Opt. Instrum. Eng.82–89 (1988).
[CrossRef]

Stack, J. D.

Swanson, G. J.

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

Urquhart, K. S.

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Walker, S. J.

Appl. Opt.

Opt. Lett.

Science

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Other

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1988), p. 45.

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

J. R. Leger, M. L. Scott, P. Bundman, M. P. Griswold, “Astigmatic wavefront correction of a gain-guided laser diode array using anamorphic diffractive microlenses,” in Computer-Generated Holography II, S. H. Lee, ed., Proc. Soc. Photo-Opt. Instrum. Eng.82–89 (1988).
[CrossRef]

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Figures (15)

Fig. 1
Fig. 1

System configuration for a single radially symmetric computer-generated hologram.

Fig. 2
Fig. 2

Radial coordinate system in the hologram plane for a hologram with radius R.

Fig. 3
Fig. 3

Element representation for the RSIDO algorithm. The element is divided into K annularly shaped rings of equal width Δρ. The rings are then grouped into P fringes of constant phase. One can then change fringe widths by changing the ring groupings, provided that no fringe becomes smaller than a specified minimum feature size δ.

Fig. 4
Fig. 4

Cross section of a CGH. Seventy-four rings are grouped into eight fringes such that no fringe has fewer than seven rings.

Fig. 5
Fig. 5

Limits on F/# with the AQ method. The minimum F/# is plotted as a function of minimum feature size δ. The wavelength is 0.8 μm.

Fig. 6
Fig. 6

Simulation results of diffraction efficiency versus minimum feature size δ for an F/1 collimating element with the RSIDO method.

Fig. 7
Fig. 7

Comparison of RSIDO and AQ methods for a focusing element. Sixteen phase levels were used with the RSIDO method. The maximum number of phase levels that will fit sequentially into the smallest grating period of the element were used for the AQ method. This value varies with F/# according to relations (1) and (2).

Fig. 8
Fig. 8

Power incident upon a detector in the focal plane versus detector diameter. The wavelength is 0.8 μm. Shaded bars: AQ, N = 3; hatched bars: RSIDO, N = 16.

Fig. 9
Fig. 9

Quantized and continuous phase versus radius near the center of the hologram, with large grating period relative to minimum feature size δ. A continuous-phase function with 100% diffraction efficiency is overlaid on the results from RSIDO. When the grating period is relatively large compared with the minimum feature size, the RSIDO method behaves similarly to the AQ method.

Fig. 10
Fig. 10

Quantized and continuous phase versus radius at approximately half the radius of the hologram, with moderate grating period relative to minimum feature size δ. A continuous-phase function with 100% diffraction efficiency is overlaid on the results from RSIDO. When the grating period becomes too small for all eight phase levels to fit within the grating period, the AQ method no longer can be used. The RSIDO method determines which phase levels result in the best performance.

Fig. 11
Fig. 11

Quantized and continuous phase versus radius near the edge of the hologram, with small grating period relative to minimum feature size δ. A continuous-phase function with 100% diffraction efficiency is overlaid on the results from RSIDO. The grating period is so small that only two of the eight phase levels fit within the grating period. The RSIDO method determines which two phase levels result in the best performance.

Fig. 12
Fig. 12

Scanning electron microscope photograph of an F/1 eight-phase-level hologram with a diameter of 1.5 mm.

Fig. 13
Fig. 13

Scanning electron microscope photograph of a cross section of an F/1 hologram. Five of the eight possible phase levels appear at this radius.

Fig. 14
Fig. 14

Photograph of an F/1 eight-phase-level hologram. Near the center all eight phase levels appear in sequential order within each grating period. At a point approximately halfway out along the radius phase level, skipping begins, which manifests itself in the coarser appearance of the element.

Fig. 15
Fig. 15

Photograph of a laser diode, a hologram, a collimated beam, and a detector on an optics table.

Tables (5)

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Table 1 Comparison of Best Overall Results of AQ and RSIDO Methodsa

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Table 2 Comparison of Best Overall Results of AQ and RSIDO Methodsa

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Table 3 Comparison of Best Overall Results of AQ and RSIDO Methodsa

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Table 4 Comparison of Best Overall Results of AQ and RSIDO Methodsa

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Table 5 Diffraction Efficiency and Possible Number of Phase Levels versus Minimum Feature Size

Equations (20)

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N T min δ ,
T min = λ [ 1 + ( 2 F / # ) 2 ] 1 / 2 .
η = [ N sin ( π / N ) π ] 2 .
U ( P 0 ) = 1 j λ U ( P 1 ) exp ( j κ r 01 ) r 01 cos ( n ^ , r 01 ¯ ) d s ,
U ( P 1 ) = H ( P 1 ) A ( r 21 ) .
U ( P 0 ) = z 01 j λ 0 R ρ H ( ρ ) 0 2 π A ( ρ , θ ) exp ( j κ r 01 ) r 01 2 d θ d ρ ,
U m n = z 01 j λ k = 0 K k Δ ρ H k L = 0 2 π / Δ θ A k L exp ( j κ r k L m n ) r k L m n 2 Δ θ Δ ρ .
U m n = k H k C k m n ,
C k m n = z 01 Δ θ ( Δ ρ ) 2 j λ k L A k L exp ( j κ r k L m n ) r k L m n 2 .
ϕ k { 0 , 2 π N , 2 2 π N , 3 2 π N , , ( N - 1 2 π N ) } .
H k = exp ( j ϕ k ) .
U m n = k = 0 K exp ( j ϕ k ) C k m n .
U m n = p = 0 P exp ( j ϕ p ) S p m n ,
S p m n = k = k s ( p ) k s ( p + 1 ) - 1 C k m n .
e = 1 - η ,
η = m = 1 M n = 1 N P m n P inc ,
e = q = 1 Q a q ( 1 - η q )
U m n = U m n + [ exp ( j ϕ ) - exp ( j ϕ p - 1 ) ] Δ S p m n ,
Δ S p m n = S p m n - S p m n
U m n = U m n + [ exp ( j ϕ p ) - exp ( j ϕ p ) ] S p m n ,

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