Abstract

We present a multiplane algorithm for three-dimensional uniform illumination. The large-diameter diffractive optical element simulated by this algorithm homogeneously concentrates more than 86.5% of the incident energy into a 200μm length of columnar space around the focal plane. The intensity profile in the whole space is nearly flattop, and the beam's quality measured by the root mean square is less than 20.6%. The algorithm is very useful if a great deal of tolerance is required for the installation error of the optical system or if it is used for some particular application, such as uniform illumination on an incline plane.

© 2007 Optical Society of America

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References

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  1. M. A. Kutay and H. W. Ozoktas, "Optimal filtering in fractional Fourier domains," IEEE Trans. Signal Process. 45, 1129-1143 (1997).
    [CrossRef]
  2. W. Wang and T. Li, "Design of large-caliber phase element used in ICF," Chin. J. Lasers A26, 395-399 (1999).
  3. W. Wang and T. Li, "A hybrid algorithm for the design of DOE in uniform illumination," Opt. Commun. 181, 261-265 (2000).
    [CrossRef]
  4. R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).
  5. J. R. Fienup, "Iterative method applied to image reconstruction and to computer-generated holograms," Opt. Eng. 19, 297-305 (1980).
  6. M. P. Chang and O. K. Erosy. "The modified input-output algorithm for the synthesis of computer generated holograms," Optik 95, 155-160 (1994).
  7. X. G. Deng and Y. P. Li, "Phase-mixture algorithm applied to the design of pure phase elements," Chin. J. Lasers B4, 447-454 (1995).
  8. K. Matsushima, H. Schimmel, and F. Wyrowski, "Fast calculation method for optical diffraction on tilted planes by use of the angular spectrum of plane waves," J. Opt. Soc. Am. A 20, 1755-1762 (2003).
    [CrossRef]
  9. S. De Nicola, A. Finizio, and G. Pierattini, "Angular spectrum method with correction of anamorphism for numerical reconstruction of digital holograms on tilted planes," Opt. Express 13, 9935-9940 (2005).
    [CrossRef] [PubMed]
  10. Z. Liu, A. Flores, M. R. Wang, and J. J. Yang, "Diffractive infrared lens with extended depth of focus," Opt. Eng. 46, 018002 (2007).
    [CrossRef]
  11. R. S. Denis, N. Passilly, and M. Laroche, "Beam-shaping longitudinal range of a binary diffractive optical element," Appl. Opt. 45, 8136-8141 (2006).
    [CrossRef]
  12. Y. S. Chen, J. J. Zhang, and N. K. Bai, "An adjacent sequence iteration method for designing a diffractive element with function of large depth of focus," Acta Photonica Sin. 32, 269-271 (2003).
  13. O. F. Tan, Y. B. Yan, and G. F. Jing, "Diffractive optical elements to realize uniform illumination in inclined planes at different angles," Acta Photonica Sin. 29, 431-435 (2000).
  14. W. Zhang, X. B. Zhang, and F. J. Shu, "Design of diffractive optical elements by step iterative algorithm," High Power Laser Part. Beams 17, 1665-1668 (2005).
  15. Y. P. Li, R. Wu, and X. B. Zhang, "Beam smoothing on oblique surface after focus plane," Proc. SPIE 5876, 121-127 (2005).
    [CrossRef]
  16. J. W. Goodman, Introduction to Fourier Optics, 1st ed. (McGraw-Hill, 1968), pp. 54-59.

2007 (1)

Z. Liu, A. Flores, M. R. Wang, and J. J. Yang, "Diffractive infrared lens with extended depth of focus," Opt. Eng. 46, 018002 (2007).
[CrossRef]

2006 (1)

2005 (3)

S. De Nicola, A. Finizio, and G. Pierattini, "Angular spectrum method with correction of anamorphism for numerical reconstruction of digital holograms on tilted planes," Opt. Express 13, 9935-9940 (2005).
[CrossRef] [PubMed]

W. Zhang, X. B. Zhang, and F. J. Shu, "Design of diffractive optical elements by step iterative algorithm," High Power Laser Part. Beams 17, 1665-1668 (2005).

Y. P. Li, R. Wu, and X. B. Zhang, "Beam smoothing on oblique surface after focus plane," Proc. SPIE 5876, 121-127 (2005).
[CrossRef]

2003 (2)

K. Matsushima, H. Schimmel, and F. Wyrowski, "Fast calculation method for optical diffraction on tilted planes by use of the angular spectrum of plane waves," J. Opt. Soc. Am. A 20, 1755-1762 (2003).
[CrossRef]

Y. S. Chen, J. J. Zhang, and N. K. Bai, "An adjacent sequence iteration method for designing a diffractive element with function of large depth of focus," Acta Photonica Sin. 32, 269-271 (2003).

2000 (2)

O. F. Tan, Y. B. Yan, and G. F. Jing, "Diffractive optical elements to realize uniform illumination in inclined planes at different angles," Acta Photonica Sin. 29, 431-435 (2000).

W. Wang and T. Li, "A hybrid algorithm for the design of DOE in uniform illumination," Opt. Commun. 181, 261-265 (2000).
[CrossRef]

1999 (1)

W. Wang and T. Li, "Design of large-caliber phase element used in ICF," Chin. J. Lasers A26, 395-399 (1999).

1997 (1)

M. A. Kutay and H. W. Ozoktas, "Optimal filtering in fractional Fourier domains," IEEE Trans. Signal Process. 45, 1129-1143 (1997).
[CrossRef]

1995 (1)

X. G. Deng and Y. P. Li, "Phase-mixture algorithm applied to the design of pure phase elements," Chin. J. Lasers B4, 447-454 (1995).

1994 (1)

M. P. Chang and O. K. Erosy. "The modified input-output algorithm for the synthesis of computer generated holograms," Optik 95, 155-160 (1994).

1980 (1)

J. R. Fienup, "Iterative method applied to image reconstruction and to computer-generated holograms," Opt. Eng. 19, 297-305 (1980).

1972 (1)

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

Acta Photonica Sin. (2)

Y. S. Chen, J. J. Zhang, and N. K. Bai, "An adjacent sequence iteration method for designing a diffractive element with function of large depth of focus," Acta Photonica Sin. 32, 269-271 (2003).

O. F. Tan, Y. B. Yan, and G. F. Jing, "Diffractive optical elements to realize uniform illumination in inclined planes at different angles," Acta Photonica Sin. 29, 431-435 (2000).

Appl. Opt. (1)

Chin. J. Lasers (2)

X. G. Deng and Y. P. Li, "Phase-mixture algorithm applied to the design of pure phase elements," Chin. J. Lasers B4, 447-454 (1995).

W. Wang and T. Li, "Design of large-caliber phase element used in ICF," Chin. J. Lasers A26, 395-399 (1999).

High Power Laser Part. Beams (1)

W. Zhang, X. B. Zhang, and F. J. Shu, "Design of diffractive optical elements by step iterative algorithm," High Power Laser Part. Beams 17, 1665-1668 (2005).

IEEE Trans. Signal Process. (1)

M. A. Kutay and H. W. Ozoktas, "Optimal filtering in fractional Fourier domains," IEEE Trans. Signal Process. 45, 1129-1143 (1997).
[CrossRef]

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

Opt. Commun. (1)

W. Wang and T. Li, "A hybrid algorithm for the design of DOE in uniform illumination," Opt. Commun. 181, 261-265 (2000).
[CrossRef]

Opt. Eng. (2)

J. R. Fienup, "Iterative method applied to image reconstruction and to computer-generated holograms," Opt. Eng. 19, 297-305 (1980).

Z. Liu, A. Flores, M. R. Wang, and J. J. Yang, "Diffractive infrared lens with extended depth of focus," Opt. Eng. 46, 018002 (2007).
[CrossRef]

Opt. Express (1)

Optik (2)

M. P. Chang and O. K. Erosy. "The modified input-output algorithm for the synthesis of computer generated holograms," Optik 95, 155-160 (1994).

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

Proc. SPIE (1)

Y. P. Li, R. Wu, and X. B. Zhang, "Beam smoothing on oblique surface after focus plane," Proc. SPIE 5876, 121-127 (2005).
[CrossRef]

Other (1)

J. W. Goodman, Introduction to Fourier Optics, 1st ed. (McGraw-Hill, 1968), pp. 54-59.

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Figures (3)

Fig. 1
Fig. 1

Optical scheme for the 3D UIP amplified around the focal plane. IN is the plane of the DOE and lens, φ is the diameter of the DOE, and f is the focal length. We divide the 3D focal space into total 2 n + 1 parallel planes, u n u n 1 , … ,  u 1 , u 0 , u 1 , … ,  u n 1 , u n , around the focal plane u 0 , and let Δd be the distance between the two neighboring planes and d = 2 n × Δ d be the total length of the space.

Fig. 2
Fig. 2

Traditional design results on focal plane u 0 and variations of the DE and rms. (a) One-dimensional normalized intensity profile on u 0 ; (b) variations of the DE (dotted curve) and rms (solid curve) along with d j .

Fig. 3
Fig. 3

Variations of the DE (dotted curve) and rms (solid curve) along with d j designed by the multiplane algorithm. (a) First parameter set S 1 ; (b) second parameter set S 2 .

Tables (1)

Tables Icon

Table 1 Specific rms and DE of S 1 and S 2 along with Defocusing Distance d j

Equations (18)

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U o ( x o , y o ) = C U i ( x i , y i ) exp [ i Φ ( x i , y i ) ] × exp [ i 2 π λ f ( x o y o + x i y i ) ] d x i d y i ,
I ( x , y ) = U ( x , y ) U * ( x , y ) = | U ( x , y ) | 2 ,
I i d e a l ( x o , y o ) C | F { I i ( x i , y i ) exp [ i Φ ( x i , y i ) ] } | 2 ,
U o ( k ) = F { I i   exp [ Φ ( k ) ] } ,
U o ( k ) = ( I o + β | I o | U o ( k ) | | ) exp [ i   arg { U o ( k ) } ] ,
U i ( k + 1 ) = F 1 { U o ( k ) } ,
Φ ( k + 1 ) = arg { U i ( k + 1 ) } ,
U o j ( k ) = T { U i j ( k ) } = T { I i   exp [ Φ ( k ) ] } ,
U o j ( k ) = ( I o + β j | I o | U o j ( k ) | | ) exp [ i   arg ( U o j ( k ) ) ] ,
U i j ( k + 1 ) = T 1 { U o j ( k ) } ,
U i j ( k + 1 ) = | I i | exp { i [ α j   arg ( U i j ( k + 1 ) ) + ( 1 α j ) arg ( U i j ( k ) ) ] } ,
T = AST [ FFT ( IN , u 0 ) , u j ] ,
T 1 = FFT 1 [ AST 1 ( u j , u 0 ) , IN ] ,
A ( f x , f y ; 0 ) = U ( x , y ; 0 ) exp [ i 2 π ( f x x + f y y ) ] d x d y ,
A ( f x , f y ; d j ) = A ( f x , f y ; 0 ) exp ( i 2 π 1 μ 2 ν 2 d j / λ ) ,
U ( x , y ; d j ) = F 1 { A ( f x , f y ; d j ) } ,
rms = ( t o p | I r e a l I i d e a l | 2 / I i d e a l 2 ) n ,
DE = t o p I r e a l t o t a l I r e a l ,

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