Abstract

In a previous Letter [Opt. Lett. 33, 1171 (2008)], we proposed an improved logarithmic phase mask by making modifications to the original one designed by Sherif. However, further studies in another paper [Appl. Opt. 49, 229 (2010)] show that even when the Sherif mask and the improved one are optimized, their corresponding defocused modulation transfer functions (MTFs) are still not stable with respect to focus errors. So, by further modifying their phase profiles, we design another two logarithmic phase masks that exhibit more stable defocused MTF. However, with the defocus-induced phase effect considered, we find that the performance of the two masks proposed in this Letter is better than the Sherif mask, but worse than our previously proposed phase mask, according to the Hilbert space angle.

© 2010 Optical Society of America

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References

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2010

2009

2008

2007

2004

2003

2001

S. S. Sherif, E. R. Dowski, and W. Thomas Cathey, Proc. SPIE 4471, 4471 (2001).

W. Chi and N. George, Opt. Lett. 26, 875 (2001).
[CrossRef]

1995

Caron, N.

N. Caron and Y. Sheng, Proc. SPIE 6832, 68321G (2007).
[CrossRef]

Cathey, W. T.

Cathey, W. Thomas

S. S. Sherif, E. R. Dowski, and W. Thomas Cathey, Proc. SPIE 4471, 4471 (2001).

Chi, W.

Demenikov, M.

Dowski, E. R.

S. S. Sherif, E. R. Dowski, and W. Thomas Cathey, Proc. SPIE 4471, 4471 (2001).

E. R. Dowski, Jr. and W. T. Cathey, Appl. Opt. 34, 1859 (1995).
[CrossRef] [PubMed]

Fainman, Y.

Feng, H.

Ford, J. E.

Franks, L. E.

L. E. Franks, Signal Theory, rev. ed. (Dowden & Culver, 1981).

George, N.

Harvey, A. R.

Komatsu, S.

Li, G.

Li, Q.

Li, Y.

Liu, L.

Q. Yang, L. Liu, and J. Sun, Opt. Commun. 272, 56 (2007).
[CrossRef]

Mezouari, S.

Morrison, R. L.

Neifeld, M. A.

Ojeda-Castañeda, J.

Rutkowski, J.

Sauceda, A.

Sheng, Y.

N. Caron and Y. Sheng, Proc. SPIE 6832, 68321G (2007).
[CrossRef]

Sherif, S. S.

S. S. Sherif, E. R. Dowski, and W. Thomas Cathey, Proc. SPIE 4471, 4471 (2001).

Silveira, P. E. X.

Stack, R. A.

Sun, J.

Q. Yang, L. Liu, and J. Sun, Opt. Commun. 272, 56 (2007).
[CrossRef]

Takahashi, Y.

Tamayo, I.

Tremblay, E. J.

Wang, D.

Yang, Q.

Q. Yang, L. Liu, and J. Sun, Opt. Commun. 272, 56 (2007).
[CrossRef]

Ye, R.

Zhang, H.

Zhao, H.

Zhou, F.

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Figures (6)

Fig. 1
Fig. 1

Defocused MTF comparison between the mask f Sherif (first row of images) and f Sherif _ new (second row of images) with defocus parameter equaling 0, 3, 9, 18, and 25.

Fig. 2
Fig. 2

Defocused MTF comparison between the mask f improved (first row of images) and f improved _ new (second row of images) with defocus parameter equaling 0, 3, 9, 18, and 25.

Fig. 3
Fig. 3

Difference between phase angle of defocused OTF (ψ equaling 3, 9, 18, and 25) and that of in-focus OTF. f Sherif , first row of images; f Sherif _ new , second row of images.

Fig. 4
Fig. 4

Difference between phase angle of defocused OTF (ψ equaling 3, 9, 18, and 25) and that of in-focus OTF. f improved , first row of images; f improved _ new , second row of images.

Fig. 5
Fig. 5

Comparison of Hilbert space angle among four phase masks.

Fig. 6
Fig. 6

Two-dimensional phase contours (left image) and one-dimensional phase profiles (right image) for each mask. [Considering the odd symmetry of phase masks, the α of Eq. (2, 3) is changed into α to better visualize the contours, and the parameters used correspond to Th equaling 0.21.]

Tables (1)

Tables Icon

Table 1 Optimum Parameters for Each Mask Denoted by Eq. (1, 2, 3, 4), and the Range [ 25 , 25 ] of Defocus Parameter Is Used While Doing the Optimization

Equations (6)

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f Sherif ( x , y ) = α · sgn ( x ) · x 2 · ( log | x | + β ) + α · sgn ( y ) · y 2 · ( log | y | + β ) ,
f improved ( x , y ) = α · sgn ( x ) · x 2 · ( log | | x | + β | ) + α · sgn ( y ) · y 2 · ( log | | y | + β | ) ,
f Sherif _ new ( x , y ) = α · sgn ( x ) · x 4 · ( log | x | + β ) + α · sgn ( y ) · y 4 · ( log | y | + β ) ,
f improved _ new ( x , y ) = α · sgn ( x ) · x 4 · ( log | | x | + β | ) + α · sgn ( y ) · y 4 · ( log | | y | + β | ) .
θ = cos 1 { f · g f · g } ,
θ ( ψ ) = cos 1 { h ( x , 0 ) · h ( x , ψ ) · d x [ h ( x , 0 ) · h ( x , 0 ) · d x ] 1 / 2 · [ h ( x , ψ ) · h ( x , ψ ) · d x ] 1 / 2 } ,

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