Abstract

A set of recursive formulas for diffractive optical plates design is described. The pure-phase plates simulated by this method homogeneously concentrate more than 96% of the incident laser energy in the desired focal-plane region. The intensity focal-plane profile fits a 12th-order super-Gaussian function and has a nearly perfect flat top. Its fit to the required profile measured in the mean square error is 3.576 × 10−3.

© 1996 Optical Society of America

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References

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  1. S. N. Dixit, J. K. Lawson, K. R. Manes, H. T. Powell, K. A. Nugent, Opt. Lett. 19, 417 (1994).
    [PubMed]
  2. Y. Lin, T. J. Kessler, G. N. Lawrence, Opt. Lett. 20, 764 (1995).
    [CrossRef] [PubMed]
  3. J. K. Lawson, S. N. Dixit, D. Eimerl, M. A. Henesian, K. R. Manes, A. J. Morgan, H. T. Powell, I. M. Thomas, J. B. Trenholme, B. W. Woods, Proc. SPIE 1870, 88 (1993).
    [CrossRef]
  4. T. J. Kesseler, Y. Lin, J. J. Armstrong, B. Velazquez, Proc. SPIE 1870, 95 (1993).
    [CrossRef]
  5. T. J. Kesseler, Y. Lin, J. J. Armstrong, B. Velazquez, Phys. Today 48(8), 22 (1995).
  6. G.-C. Wang, Sci. Chinese 6, 4 (1995).
  7. J. R. Fienup, Opt. Eng. 19, 297 (1980).
  8. R. W. Gerchberg, W. O. Saxton, Optik 35, 2 (1972).
  9. X. Deng, Y. Li, D. Fan, Y. Qiu, “Projection onto constraint sets algorithms for diffractive optical elements design,” submitted to Appl. Opt.
  10. X. Deng, Y. Li, Y. Qiu, D. Y. Fan, Chin. J. Lasers B 4, 447 (1995).
  11. J. R. Fienup, Appl. Opt. 21, 2758 (1982).
    [CrossRef] [PubMed]
  12. M. V. Klibanov, P. E. Stacks, A. V. Tikhonravov, Inverse Problem 11, 1 (1995).
    [CrossRef]
  13. E.g., O. Bryndahl, J. Opt. Soc. Am. 64, 1092 (1974).
    [CrossRef]
  14. X. Deng, Y. Li, Y. Qiu, D. Y. Fan, “Quantitative description of the beam quality in uniform-illumination experiments,” Chin. J. Lasers A (to be published).
  15. J. N. Mait, J. Opt. Soc. Am. A 12, 2145 (1995).
    [CrossRef]

1995 (6)

T. J. Kesseler, Y. Lin, J. J. Armstrong, B. Velazquez, Phys. Today 48(8), 22 (1995).

G.-C. Wang, Sci. Chinese 6, 4 (1995).

X. Deng, Y. Li, Y. Qiu, D. Y. Fan, Chin. J. Lasers B 4, 447 (1995).

M. V. Klibanov, P. E. Stacks, A. V. Tikhonravov, Inverse Problem 11, 1 (1995).
[CrossRef]

J. N. Mait, J. Opt. Soc. Am. A 12, 2145 (1995).
[CrossRef]

Y. Lin, T. J. Kessler, G. N. Lawrence, Opt. Lett. 20, 764 (1995).
[CrossRef] [PubMed]

1994 (1)

1993 (2)

J. K. Lawson, S. N. Dixit, D. Eimerl, M. A. Henesian, K. R. Manes, A. J. Morgan, H. T. Powell, I. M. Thomas, J. B. Trenholme, B. W. Woods, Proc. SPIE 1870, 88 (1993).
[CrossRef]

T. J. Kesseler, Y. Lin, J. J. Armstrong, B. Velazquez, Proc. SPIE 1870, 95 (1993).
[CrossRef]

1982 (1)

1980 (1)

J. R. Fienup, Opt. Eng. 19, 297 (1980).

1974 (1)

1972 (1)

R. W. Gerchberg, W. O. Saxton, Optik 35, 2 (1972).

Armstrong, J. J.

T. J. Kesseler, Y. Lin, J. J. Armstrong, B. Velazquez, Phys. Today 48(8), 22 (1995).

T. J. Kesseler, Y. Lin, J. J. Armstrong, B. Velazquez, Proc. SPIE 1870, 95 (1993).
[CrossRef]

Bryndahl, O.

Deng, X.

X. Deng, Y. Li, Y. Qiu, D. Y. Fan, Chin. J. Lasers B 4, 447 (1995).

X. Deng, Y. Li, Y. Qiu, D. Y. Fan, “Quantitative description of the beam quality in uniform-illumination experiments,” Chin. J. Lasers A (to be published).

X. Deng, Y. Li, D. Fan, Y. Qiu, “Projection onto constraint sets algorithms for diffractive optical elements design,” submitted to Appl. Opt.

Dixit, S. N.

S. N. Dixit, J. K. Lawson, K. R. Manes, H. T. Powell, K. A. Nugent, Opt. Lett. 19, 417 (1994).
[PubMed]

J. K. Lawson, S. N. Dixit, D. Eimerl, M. A. Henesian, K. R. Manes, A. J. Morgan, H. T. Powell, I. M. Thomas, J. B. Trenholme, B. W. Woods, Proc. SPIE 1870, 88 (1993).
[CrossRef]

Eimerl, D.

J. K. Lawson, S. N. Dixit, D. Eimerl, M. A. Henesian, K. R. Manes, A. J. Morgan, H. T. Powell, I. M. Thomas, J. B. Trenholme, B. W. Woods, Proc. SPIE 1870, 88 (1993).
[CrossRef]

Fan, D.

X. Deng, Y. Li, D. Fan, Y. Qiu, “Projection onto constraint sets algorithms for diffractive optical elements design,” submitted to Appl. Opt.

Fan, D. Y.

X. Deng, Y. Li, Y. Qiu, D. Y. Fan, Chin. J. Lasers B 4, 447 (1995).

X. Deng, Y. Li, Y. Qiu, D. Y. Fan, “Quantitative description of the beam quality in uniform-illumination experiments,” Chin. J. Lasers A (to be published).

Fienup, J. R.

J. R. Fienup, Appl. Opt. 21, 2758 (1982).
[CrossRef] [PubMed]

J. R. Fienup, Opt. Eng. 19, 297 (1980).

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, Optik 35, 2 (1972).

Henesian, M. A.

J. K. Lawson, S. N. Dixit, D. Eimerl, M. A. Henesian, K. R. Manes, A. J. Morgan, H. T. Powell, I. M. Thomas, J. B. Trenholme, B. W. Woods, Proc. SPIE 1870, 88 (1993).
[CrossRef]

Kesseler, T. J.

T. J. Kesseler, Y. Lin, J. J. Armstrong, B. Velazquez, Phys. Today 48(8), 22 (1995).

T. J. Kesseler, Y. Lin, J. J. Armstrong, B. Velazquez, Proc. SPIE 1870, 95 (1993).
[CrossRef]

Kessler, T. J.

Klibanov, M. V.

M. V. Klibanov, P. E. Stacks, A. V. Tikhonravov, Inverse Problem 11, 1 (1995).
[CrossRef]

Lawrence, G. N.

Lawson, J. K.

S. N. Dixit, J. K. Lawson, K. R. Manes, H. T. Powell, K. A. Nugent, Opt. Lett. 19, 417 (1994).
[PubMed]

J. K. Lawson, S. N. Dixit, D. Eimerl, M. A. Henesian, K. R. Manes, A. J. Morgan, H. T. Powell, I. M. Thomas, J. B. Trenholme, B. W. Woods, Proc. SPIE 1870, 88 (1993).
[CrossRef]

Li, Y.

X. Deng, Y. Li, Y. Qiu, D. Y. Fan, Chin. J. Lasers B 4, 447 (1995).

X. Deng, Y. Li, Y. Qiu, D. Y. Fan, “Quantitative description of the beam quality in uniform-illumination experiments,” Chin. J. Lasers A (to be published).

X. Deng, Y. Li, D. Fan, Y. Qiu, “Projection onto constraint sets algorithms for diffractive optical elements design,” submitted to Appl. Opt.

Lin, Y.

T. J. Kesseler, Y. Lin, J. J. Armstrong, B. Velazquez, Phys. Today 48(8), 22 (1995).

Y. Lin, T. J. Kessler, G. N. Lawrence, Opt. Lett. 20, 764 (1995).
[CrossRef] [PubMed]

T. J. Kesseler, Y. Lin, J. J. Armstrong, B. Velazquez, Proc. SPIE 1870, 95 (1993).
[CrossRef]

Mait, J. N.

Manes, K. R.

S. N. Dixit, J. K. Lawson, K. R. Manes, H. T. Powell, K. A. Nugent, Opt. Lett. 19, 417 (1994).
[PubMed]

J. K. Lawson, S. N. Dixit, D. Eimerl, M. A. Henesian, K. R. Manes, A. J. Morgan, H. T. Powell, I. M. Thomas, J. B. Trenholme, B. W. Woods, Proc. SPIE 1870, 88 (1993).
[CrossRef]

Morgan, A. J.

J. K. Lawson, S. N. Dixit, D. Eimerl, M. A. Henesian, K. R. Manes, A. J. Morgan, H. T. Powell, I. M. Thomas, J. B. Trenholme, B. W. Woods, Proc. SPIE 1870, 88 (1993).
[CrossRef]

Nugent, K. A.

Powell, H. T.

S. N. Dixit, J. K. Lawson, K. R. Manes, H. T. Powell, K. A. Nugent, Opt. Lett. 19, 417 (1994).
[PubMed]

J. K. Lawson, S. N. Dixit, D. Eimerl, M. A. Henesian, K. R. Manes, A. J. Morgan, H. T. Powell, I. M. Thomas, J. B. Trenholme, B. W. Woods, Proc. SPIE 1870, 88 (1993).
[CrossRef]

Qiu, Y.

X. Deng, Y. Li, Y. Qiu, D. Y. Fan, Chin. J. Lasers B 4, 447 (1995).

X. Deng, Y. Li, Y. Qiu, D. Y. Fan, “Quantitative description of the beam quality in uniform-illumination experiments,” Chin. J. Lasers A (to be published).

X. Deng, Y. Li, D. Fan, Y. Qiu, “Projection onto constraint sets algorithms for diffractive optical elements design,” submitted to Appl. Opt.

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, Optik 35, 2 (1972).

Stacks, P. E.

M. V. Klibanov, P. E. Stacks, A. V. Tikhonravov, Inverse Problem 11, 1 (1995).
[CrossRef]

Thomas, I. M.

J. K. Lawson, S. N. Dixit, D. Eimerl, M. A. Henesian, K. R. Manes, A. J. Morgan, H. T. Powell, I. M. Thomas, J. B. Trenholme, B. W. Woods, Proc. SPIE 1870, 88 (1993).
[CrossRef]

Tikhonravov, A. V.

M. V. Klibanov, P. E. Stacks, A. V. Tikhonravov, Inverse Problem 11, 1 (1995).
[CrossRef]

Trenholme, J. B.

J. K. Lawson, S. N. Dixit, D. Eimerl, M. A. Henesian, K. R. Manes, A. J. Morgan, H. T. Powell, I. M. Thomas, J. B. Trenholme, B. W. Woods, Proc. SPIE 1870, 88 (1993).
[CrossRef]

Velazquez, B.

T. J. Kesseler, Y. Lin, J. J. Armstrong, B. Velazquez, Phys. Today 48(8), 22 (1995).

T. J. Kesseler, Y. Lin, J. J. Armstrong, B. Velazquez, Proc. SPIE 1870, 95 (1993).
[CrossRef]

Wang, G.-C.

G.-C. Wang, Sci. Chinese 6, 4 (1995).

Woods, B. W.

J. K. Lawson, S. N. Dixit, D. Eimerl, M. A. Henesian, K. R. Manes, A. J. Morgan, H. T. Powell, I. M. Thomas, J. B. Trenholme, B. W. Woods, Proc. SPIE 1870, 88 (1993).
[CrossRef]

Appl. Opt. (1)

Chin. J. Lasers B (1)

X. Deng, Y. Li, Y. Qiu, D. Y. Fan, Chin. J. Lasers B 4, 447 (1995).

Inverse Problem (1)

M. V. Klibanov, P. E. Stacks, A. V. Tikhonravov, Inverse Problem 11, 1 (1995).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

J. R. Fienup, Opt. Eng. 19, 297 (1980).

Opt. Lett. (2)

Optik (1)

R. W. Gerchberg, W. O. Saxton, Optik 35, 2 (1972).

Phys. Today (1)

T. J. Kesseler, Y. Lin, J. J. Armstrong, B. Velazquez, Phys. Today 48(8), 22 (1995).

Proc. SPIE (2)

J. K. Lawson, S. N. Dixit, D. Eimerl, M. A. Henesian, K. R. Manes, A. J. Morgan, H. T. Powell, I. M. Thomas, J. B. Trenholme, B. W. Woods, Proc. SPIE 1870, 88 (1993).
[CrossRef]

T. J. Kesseler, Y. Lin, J. J. Armstrong, B. Velazquez, Proc. SPIE 1870, 95 (1993).
[CrossRef]

Sci. Chinese (1)

G.-C. Wang, Sci. Chinese 6, 4 (1995).

Other (2)

X. Deng, Y. Li, D. Fan, Y. Qiu, “Projection onto constraint sets algorithms for diffractive optical elements design,” submitted to Appl. Opt.

X. Deng, Y. Li, Y. Qiu, D. Y. Fan, “Quantitative description of the beam quality in uniform-illumination experiments,” Chin. J. Lasers A (to be published).

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Figures (2)

Fig. 1
Fig. 1

(a) Two-dimensional projected isograms and (b) pseudo-three-dimensional view of the simulated intensity on the focal plane. The intensity-homogeneous region has the correct dimensions of 500 μm × 500 μm, and there is hardly any noticeable energy loss owing to such factors as higher angular diffractions. The intensity distributions decrease abruptly to 0 outside the target region. The MSE value of this distribution is 3.576 × 10−3.

Fig. 2
Fig. 2

Energy in the bucket versus the dimension Db of the bucket. The fractional energy of the theoretical prediction and that of the simulated intensity are denoted by FEt and FEs, respectively, scaled on the left vertical axis. The curve of the difference between FEs and FEt is scaled in accordance with the right vertical axis.

Tables (1)

Tables Icon

Table 1 First and Second Moments of the Target Intensity Profilea

Equations (8)

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U 0 ( r 2 ) = C F { U i ( r 1 ) exp [ i Φ i ( r 1 ) ] } ,
U 0 ( r 2 ) = I 0 ( r 2 ) exp [ i Φ 0 ( r 2 ) ] .
Φ i ( n + 1 ) ( r 1 ) = α Φ i ( n ) ( r 1 ) + ( 1 - α ) A × { g ( n ) ( r 1 ) exp [ i A [ U i ( r 1 ) ] } ,             0 < α < 1 , g ( n ) ( r 1 ) 1 C F - 1 { I 0 ( r 2 ) exp [ i Φ 0 ( n ) ( r 2 ) ] } × exp { - i A [ U i ( r 1 ) ] } ,
MSE = C F { U i ( r 1 ) exp [ i Φ i ( n + 1 ) ( r 1 ) ] } - I 0 ( r 2 ) 2 I 0 ( r 2 ) 2
{ U i ( r 1 ) exp [ i Φ i ( n ) ( r 1 ) ] } [ 1 + i δ Φ i ( n + 1 ) ] g ( n ) ( r 1 ) exp { i A [ U i ( r 1 ) ] } .
Φ i ( n + 1 ) ( r 1 ) = Φ i ( n ) ( r 1 ) + Im { g ( n ) ( r 1 ) U m i ( n ) ( r 1 ) } , U m i ( n ) ( r 1 ) = { I i ( r 1 ) + α f [ | x 1 - x c σ x | 2 s + | y 1 - y c σ y | 2 s ] + ɛ } exp [ i Φ i ( n ) ( r 1 ) ] ,
exp ( - | x l σ l | 2 N l - | y l σ l | 2 N l ) ,             l = 1 , 2.
Φ i ( 0 ) ( r 1 ) = ± ( π σ 2 λ f σ 1 ) r 1 2 ,

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