Abstract

We investigate the binary phase filters of finite support that have a Fourier transform magnitude that is the broadest and flattest. We call this behavior maximally flat, after the electrical filters devised by Butterworth. Solutions are obtained by direct algebraic manipulation, and numerical results are given for filters of up to five elements. Their application for increasing depth of focus is described.

© 2011 Optical Society of America

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References

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  1. S. Butterworth, Exper. Wireless Wireless Engineer 536(1930).
  2. V. F. Canales, J. E. Oti, and M. P. Cagigal, Opt. Commun. 247, 11 (2005).
    [CrossRef]
  3. B. J. Thompson, J. Opt. Soc. Am. 55, 145 (1965).
    [CrossRef]
  4. G. Yang, Opt. Commun. 159, 19 (1999).
    [CrossRef]
  5. H. F. Wang and F. Gan, Appl. Opt. 40, 5658 (2001).
    [CrossRef]
  6. M. Yun, L. Liu, J. Sun, and D. Liu, J. Opt. Soc. Am. A 22, 272 (2005).
    [CrossRef]
  7. X. M. Gao, F. Gan, and W. D. Xu, Opt. Laser Technol. 39, 1074 (2007).
    [CrossRef]
  8. M. P. Cagigal, J. E. Oti, V. F. Canales, and P. J. Valle, Opt. Commun. 241, 249 (2004).
    [CrossRef]
  9. C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, Opt. Commun. 281, 913 (2008).
    [CrossRef]
  10. C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, Opt. Commun. 281, 3623 (2008).
    [CrossRef]
  11. L. Liu, C. Liu, W. Howe, C. Sheppard, and N. Chen, Opt. Lett. 32, 2375 (2007).
    [CrossRef] [PubMed]
  12. L. B. Liu, F. Diaz, L. J. Wang, B. Loiseaux, J.-P. Huignard, C. J. R. Sheppard, and N. G. Chen, J. Opt. Soc. Am. A 25, 2095 (2008).
    [CrossRef]
  13. F. Diaz, F. Goudail, B. Loiseaux, and J.-P. Huignard, Opt. Lett. 34, 1171 (2009).
    [CrossRef] [PubMed]
  14. C. W. McCutchen, J. Opt. Soc. Am. 54, 240 (1964).
    [CrossRef]
  15. J. Ojeda-Castañeda, P. Andrés, and A. Díaz, J. Opt. Soc. Am. A 5, 1233 (1988).
    [CrossRef]
  16. C. J. R. Sheppard and Z. S. Hegedus, J. Opt. Soc. Am. A 5, 643 (1988).
    [CrossRef]
  17. T. G. Jabbour and S. M. Kuebler, Opt. Express 14, 1033 (2006).
    [CrossRef] [PubMed]
  18. T. G. Jabbour, M. Petrovich, and S. M. Kuebler, Opt. Commun. 281, 2002 (2008).
    [CrossRef]
  19. H. F. Wang, L. P. Shi, G. Q. Yuan, X. S. Miao, W. L. Tan, and T. C. Chong, Appl. Phys. Lett. 89, 171102 (2006).
    [CrossRef]
  20. H. F. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, Nat. Photon. 2, 501 (2008).
    [CrossRef]

2009 (1)

2008 (5)

L. B. Liu, F. Diaz, L. J. Wang, B. Loiseaux, J.-P. Huignard, C. J. R. Sheppard, and N. G. Chen, J. Opt. Soc. Am. A 25, 2095 (2008).
[CrossRef]

C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, Opt. Commun. 281, 913 (2008).
[CrossRef]

C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, Opt. Commun. 281, 3623 (2008).
[CrossRef]

T. G. Jabbour, M. Petrovich, and S. M. Kuebler, Opt. Commun. 281, 2002 (2008).
[CrossRef]

H. F. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, Nat. Photon. 2, 501 (2008).
[CrossRef]

2007 (2)

2006 (2)

H. F. Wang, L. P. Shi, G. Q. Yuan, X. S. Miao, W. L. Tan, and T. C. Chong, Appl. Phys. Lett. 89, 171102 (2006).
[CrossRef]

T. G. Jabbour and S. M. Kuebler, Opt. Express 14, 1033 (2006).
[CrossRef] [PubMed]

2005 (2)

M. Yun, L. Liu, J. Sun, and D. Liu, J. Opt. Soc. Am. A 22, 272 (2005).
[CrossRef]

V. F. Canales, J. E. Oti, and M. P. Cagigal, Opt. Commun. 247, 11 (2005).
[CrossRef]

2004 (1)

M. P. Cagigal, J. E. Oti, V. F. Canales, and P. J. Valle, Opt. Commun. 241, 249 (2004).
[CrossRef]

2001 (1)

1999 (1)

G. Yang, Opt. Commun. 159, 19 (1999).
[CrossRef]

1988 (2)

1965 (1)

1964 (1)

1930 (1)

S. Butterworth, Exper. Wireless Wireless Engineer 536(1930).

Andrés, P.

Butterworth, S.

S. Butterworth, Exper. Wireless Wireless Engineer 536(1930).

Cagigal, M. P.

V. F. Canales, J. E. Oti, and M. P. Cagigal, Opt. Commun. 247, 11 (2005).
[CrossRef]

M. P. Cagigal, J. E. Oti, V. F. Canales, and P. J. Valle, Opt. Commun. 241, 249 (2004).
[CrossRef]

Campos, J.

C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, Opt. Commun. 281, 3623 (2008).
[CrossRef]

C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, Opt. Commun. 281, 913 (2008).
[CrossRef]

Canales, V. F.

V. F. Canales, J. E. Oti, and M. P. Cagigal, Opt. Commun. 247, 11 (2005).
[CrossRef]

M. P. Cagigal, J. E. Oti, V. F. Canales, and P. J. Valle, Opt. Commun. 241, 249 (2004).
[CrossRef]

Chen, N.

Chen, N. G.

Chong, C. T.

H. F. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, Nat. Photon. 2, 501 (2008).
[CrossRef]

Chong, T. C.

H. F. Wang, L. P. Shi, G. Q. Yuan, X. S. Miao, W. L. Tan, and T. C. Chong, Appl. Phys. Lett. 89, 171102 (2006).
[CrossRef]

Diaz, F.

Díaz, A.

Escalera, J. C.

C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, Opt. Commun. 281, 913 (2008).
[CrossRef]

C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, Opt. Commun. 281, 3623 (2008).
[CrossRef]

Gan, F.

X. M. Gao, F. Gan, and W. D. Xu, Opt. Laser Technol. 39, 1074 (2007).
[CrossRef]

H. F. Wang and F. Gan, Appl. Opt. 40, 5658 (2001).
[CrossRef]

Gao, X. M.

X. M. Gao, F. Gan, and W. D. Xu, Opt. Laser Technol. 39, 1074 (2007).
[CrossRef]

Goudail, F.

Hegedus, Z. S.

Howe, W.

Huignard, J.-P.

Jabbour, T. G.

T. G. Jabbour, M. Petrovich, and S. M. Kuebler, Opt. Commun. 281, 2002 (2008).
[CrossRef]

T. G. Jabbour and S. M. Kuebler, Opt. Express 14, 1033 (2006).
[CrossRef] [PubMed]

Kuebler, S. M.

T. G. Jabbour, M. Petrovich, and S. M. Kuebler, Opt. Commun. 281, 2002 (2008).
[CrossRef]

T. G. Jabbour and S. M. Kuebler, Opt. Express 14, 1033 (2006).
[CrossRef] [PubMed]

Ledesma, S.

C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, Opt. Commun. 281, 3623 (2008).
[CrossRef]

C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, Opt. Commun. 281, 913 (2008).
[CrossRef]

Liu, C.

Liu, D.

Liu, L.

Liu, L. B.

Loiseaux, B.

Lukyanchuk, B.

H. F. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, Nat. Photon. 2, 501 (2008).
[CrossRef]

McCutchen, C. W.

Miao, X. S.

H. F. Wang, L. P. Shi, G. Q. Yuan, X. S. Miao, W. L. Tan, and T. C. Chong, Appl. Phys. Lett. 89, 171102 (2006).
[CrossRef]

Ojeda-Castañeda, J.

Oti, J. E.

V. F. Canales, J. E. Oti, and M. P. Cagigal, Opt. Commun. 247, 11 (2005).
[CrossRef]

M. P. Cagigal, J. E. Oti, V. F. Canales, and P. J. Valle, Opt. Commun. 241, 249 (2004).
[CrossRef]

Petrovich, M.

T. G. Jabbour, M. Petrovich, and S. M. Kuebler, Opt. Commun. 281, 2002 (2008).
[CrossRef]

Sheppard, C.

H. F. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, Nat. Photon. 2, 501 (2008).
[CrossRef]

L. Liu, C. Liu, W. Howe, C. Sheppard, and N. Chen, Opt. Lett. 32, 2375 (2007).
[CrossRef] [PubMed]

Sheppard, C. J. R.

C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, Opt. Commun. 281, 913 (2008).
[CrossRef]

C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, Opt. Commun. 281, 3623 (2008).
[CrossRef]

L. B. Liu, F. Diaz, L. J. Wang, B. Loiseaux, J.-P. Huignard, C. J. R. Sheppard, and N. G. Chen, J. Opt. Soc. Am. A 25, 2095 (2008).
[CrossRef]

C. J. R. Sheppard and Z. S. Hegedus, J. Opt. Soc. Am. A 5, 643 (1988).
[CrossRef]

Shi, L.

H. F. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, Nat. Photon. 2, 501 (2008).
[CrossRef]

Shi, L. P.

H. F. Wang, L. P. Shi, G. Q. Yuan, X. S. Miao, W. L. Tan, and T. C. Chong, Appl. Phys. Lett. 89, 171102 (2006).
[CrossRef]

Sun, J.

Tan, W. L.

H. F. Wang, L. P. Shi, G. Q. Yuan, X. S. Miao, W. L. Tan, and T. C. Chong, Appl. Phys. Lett. 89, 171102 (2006).
[CrossRef]

Thompson, B. J.

Valle, P. J.

M. P. Cagigal, J. E. Oti, V. F. Canales, and P. J. Valle, Opt. Commun. 241, 249 (2004).
[CrossRef]

Wang, H. F.

H. F. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, Nat. Photon. 2, 501 (2008).
[CrossRef]

H. F. Wang, L. P. Shi, G. Q. Yuan, X. S. Miao, W. L. Tan, and T. C. Chong, Appl. Phys. Lett. 89, 171102 (2006).
[CrossRef]

H. F. Wang and F. Gan, Appl. Opt. 40, 5658 (2001).
[CrossRef]

Wang, L. J.

Xu, W. D.

X. M. Gao, F. Gan, and W. D. Xu, Opt. Laser Technol. 39, 1074 (2007).
[CrossRef]

Yang, G.

G. Yang, Opt. Commun. 159, 19 (1999).
[CrossRef]

Yuan, G. Q.

H. F. Wang, L. P. Shi, G. Q. Yuan, X. S. Miao, W. L. Tan, and T. C. Chong, Appl. Phys. Lett. 89, 171102 (2006).
[CrossRef]

Yun, M.

Appl. Opt. (1)

Appl. Phys. Lett. (1)

H. F. Wang, L. P. Shi, G. Q. Yuan, X. S. Miao, W. L. Tan, and T. C. Chong, Appl. Phys. Lett. 89, 171102 (2006).
[CrossRef]

Exper. Wireless Wireless Engineer (1)

S. Butterworth, Exper. Wireless Wireless Engineer 536(1930).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

Nat. Photon. (1)

H. F. Wang, L. Shi, B. Lukyanchuk, C. Sheppard, and C. T. Chong, Nat. Photon. 2, 501 (2008).
[CrossRef]

Opt. Commun. (6)

T. G. Jabbour, M. Petrovich, and S. M. Kuebler, Opt. Commun. 281, 2002 (2008).
[CrossRef]

G. Yang, Opt. Commun. 159, 19 (1999).
[CrossRef]

V. F. Canales, J. E. Oti, and M. P. Cagigal, Opt. Commun. 247, 11 (2005).
[CrossRef]

M. P. Cagigal, J. E. Oti, V. F. Canales, and P. J. Valle, Opt. Commun. 241, 249 (2004).
[CrossRef]

C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, Opt. Commun. 281, 913 (2008).
[CrossRef]

C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, Opt. Commun. 281, 3623 (2008).
[CrossRef]

Opt. Express (1)

Opt. Laser Technol. (1)

X. M. Gao, F. Gan, and W. D. Xu, Opt. Laser Technol. 39, 1074 (2007).
[CrossRef]

Opt. Lett. (2)

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Figures (3)

Fig. 1
Fig. 1

Squared modulus of the Fourier transform of the MF filters: upper, first solution; lower, second solution.

Fig. 2
Fig. 2

Variation of S with G for n-element MF filters. The loci for a general two-element filter and a four- element AMF are also shown.

Fig. 3
Fig. 3

Transverse point-spread function for annular filters based on the MF solutions: upper, first solution; lower, second solution.

Equations (9)

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f ( x ) = 0 1 F ( m ) e i 2 π m x d m .
q p = 0 1 F ( m ) m p d m = 1 p + 1 [ 1 2 m n 1 p + 1 + 2 m n 2 p + 1 ... ( ) n 1 2 m 1 p + 1 ] .
S = q 0 2 = [ 1 2 m n 1 + 2 m n 2 ... ( ) n 1 2 m 1 ] 2 .
G = 2 q 1 q 0 ,
m 1 = 1 2 [ 1 + ( 2 3 3 ) 1 / 2 ] = 0.840625 , or m 1 = 1 2 [ 1 ( 2 3 3 ) 1 / 2 ] = 0 . 159375 .
m 1 = 0.0820326 , m 2 = 0.319769 , S = 0.275129 , G = 1.54225 , or m 1 = 0.680231 , m 2 = 0.917967 , S = 0.275129 , G = 0.457748 .
m 1 = 0.05045809 , m 2 = 0.19751478 , m 3 = 0.432293027 , S = 0.184494 , G = 1.62778 , or m 1 = 0.07619715 , m 2 = 0.25178446 , m 3 = 0.89838308 , S = 0.198552 , G = 1.11987 .
m 1 = 0.034269568 , m 2 = 0.13487973 , m 3 = 0.29607105 , m 4 = 0.51280495 , S = 0.133452 , G = 1.68443 , or m 1 = 0.05514951 , m 2 = 0.23140226 , m 3 = 0.79844190 , m 4 = 0.92980499 , S = 0.148047 , G = 1.15637 .
U ( v , u ) = 0 1 P ( m ) J 0 ( v m ) exp ( i u m / 2 ) d m ,

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