Abstract

An approximate expression of the peak position of the point-spread function (PSF) of wavefront coding systems with a cubic phase mask (CPM) is derived and verified by simulation results. An approach called the nonaxial Strehl ratio (NASR) is used to evaluate the performance of wavefront coding systems with defocus aberrations. The characteristics of the NASR are investigated. The relationships between the NASR and the similarity of out-of-focus modulation transfer function (MTF) and recoverability of blurred encoding images are presented, and some useful guidelines are given for the optimization of phase mask parameters according to these relationships.

© 2011 Optical Society of America

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References

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2010 (2)

H. Zhao, Y. Li, H. Feng, Z. Xu, and Q. Li, “Cubic sinusoidal phase mask: another choice to extend the depth of field of incoherent imaging system,” Opt. Laser Technol. 42, 561–569(2010).
[CrossRef]

H. Zhao and Y. Li, “Optimized sinusoidal phase mask to extend the depth of field of an incoherent imaging system,” Opt. Lett. 35, 267–269 (2010).
[CrossRef] [PubMed]

2009 (2)

2008 (2)

2007 (3)

W. Zhang, Z. Ye, T. Zhao, Y. Chen, and F. Yu. “Point spread function characteristics analysis of the wavefront coding system,” Opt. Express 15, 1543–1552 (2007).
[CrossRef] [PubMed]

Y. Chen, W. Zhang, T. Zhao, F. Yu, and Z. Ye, “Imaging characteristics of wavefront coding systems within spatial domain,” Acta Opt. Sin. 27, 1425–1429 (2007).

Q. Yang, L. Liu, and J. Sun, “Optimized phase pupil masks for extended depth of field,” Opt. Commun. 272, 56–66 (2007).
[CrossRef]

2006 (4)

Q. Yang, L. Liu, J. Sun, H. Lang, Y. Zhu, and W. Lu, “Property of wave-front coding systems for extending the depth of field,” Acta Opt. Sin. 26, 1807–1812 (2006).

Q. Yang, J. Sun, and L. Liu, “Phase-space analysis of wavefront coding imaging systems,” Chin. Phys. Lett. 23, 2080–2083 (2006).
[CrossRef]

M. Somayaji and M. Christensen, “Enhancing form factor and light collection of multiplex imaging systems by using a cubic phase mask,” Appl. Opt. 45, 2911–2923 (2006).
[CrossRef] [PubMed]

Q. Yang, L. Liu, J. Sun, Y. Zhu, and W. Liu, “Analysis of optical systems with extended depth of field using the Wigner distribution function,” Appl. Opt. 45, 8586–8595 (2006).
[CrossRef] [PubMed]

2005 (1)

2004 (1)

2000 (1)

1995 (1)

Alonso, M. A.

Barwick, S.

Bleistein, N.

N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, 1986).

Cathey, W. T.

Chen, Y.

Y. Chen, W. Zhang, T. Zhao, F. Yu, and Z. Ye, “Imaging characteristics of wavefront coding systems within spatial domain,” Acta Opt. Sin. 27, 1425–1429 (2007).

W. Zhang, Z. Ye, T. Zhao, Y. Chen, and F. Yu. “Point spread function characteristics analysis of the wavefront coding system,” Opt. Express 15, 1543–1552 (2007).
[CrossRef] [PubMed]

Christensen, M.

Dowski, E. R.

Feng, H.

H. Zhao, Y. Li, H. Feng, Z. Xu, and Q. Li, “Cubic sinusoidal phase mask: another choice to extend the depth of field of incoherent imaging system,” Opt. Laser Technol. 42, 561–569(2010).
[CrossRef]

H. Zhao, Q. Li, and H. Feng, “Improved logarithmic phase mask to extend the depth of field of an incoherent imaging system,” Opt. Lett. 33, 1171–1173 (2008).
[CrossRef] [PubMed]

Finnigan, J.

Forbes, G. W.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

Handelsman, R. A.

N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, 1986).

Harvey, A. R.

Komatsu, S.

Lang, H.

Q. Yang, L. Liu, J. Sun, H. Lang, Y. Zhu, and W. Lu, “Property of wave-front coding systems for extending the depth of field,” Acta Opt. Sin. 26, 1807–1812 (2006).

Li, G.

Li, Q.

H. Zhao, Y. Li, H. Feng, Z. Xu, and Q. Li, “Cubic sinusoidal phase mask: another choice to extend the depth of field of incoherent imaging system,” Opt. Laser Technol. 42, 561–569(2010).
[CrossRef]

H. Zhao, Q. Li, and H. Feng, “Improved logarithmic phase mask to extend the depth of field of an incoherent imaging system,” Opt. Lett. 33, 1171–1173 (2008).
[CrossRef] [PubMed]

Li, Y.

H. Zhao and Y. Li, “Optimized sinusoidal phase mask to extend the depth of field of an incoherent imaging system,” Opt. Lett. 35, 267–269 (2010).
[CrossRef] [PubMed]

H. Zhao, Y. Li, H. Feng, Z. Xu, and Q. Li, “Cubic sinusoidal phase mask: another choice to extend the depth of field of incoherent imaging system,” Opt. Laser Technol. 42, 561–569(2010).
[CrossRef]

Liu, L.

Q. Yang, L. Liu, and J. Sun, “Optimized phase pupil masks for extended depth of field,” Opt. Commun. 272, 56–66 (2007).
[CrossRef]

Q. Yang, L. Liu, J. Sun, H. Lang, Y. Zhu, and W. Lu, “Property of wave-front coding systems for extending the depth of field,” Acta Opt. Sin. 26, 1807–1812 (2006).

Q. Yang, L. Liu, J. Sun, Y. Zhu, and W. Liu, “Analysis of optical systems with extended depth of field using the Wigner distribution function,” Appl. Opt. 45, 8586–8595 (2006).
[CrossRef] [PubMed]

Q. Yang, J. Sun, and L. Liu, “Phase-space analysis of wavefront coding imaging systems,” Chin. Phys. Lett. 23, 2080–2083 (2006).
[CrossRef]

Liu, W.

Lu, W.

Q. Yang, L. Liu, J. Sun, H. Lang, Y. Zhu, and W. Lu, “Property of wave-front coding systems for extending the depth of field,” Acta Opt. Sin. 26, 1807–1812 (2006).

Muyo, G.

Sherif, S. S.

Somayaji, M.

Sun, J.

Q. Yang, L. Liu, and J. Sun, “Optimized phase pupil masks for extended depth of field,” Opt. Commun. 272, 56–66 (2007).
[CrossRef]

Q. Yang, L. Liu, J. Sun, H. Lang, Y. Zhu, and W. Lu, “Property of wave-front coding systems for extending the depth of field,” Acta Opt. Sin. 26, 1807–1812 (2006).

Q. Yang, L. Liu, J. Sun, Y. Zhu, and W. Liu, “Analysis of optical systems with extended depth of field using the Wigner distribution function,” Appl. Opt. 45, 8586–8595 (2006).
[CrossRef] [PubMed]

Q. Yang, J. Sun, and L. Liu, “Phase-space analysis of wavefront coding imaging systems,” Chin. Phys. Lett. 23, 2080–2083 (2006).
[CrossRef]

Takahashi, Y.

Wang, D.

Xu, Z.

H. Zhao, Y. Li, H. Feng, Z. Xu, and Q. Li, “Cubic sinusoidal phase mask: another choice to extend the depth of field of incoherent imaging system,” Opt. Laser Technol. 42, 561–569(2010).
[CrossRef]

Yang, Q.

Q. Yang, L. Liu, and J. Sun, “Optimized phase pupil masks for extended depth of field,” Opt. Commun. 272, 56–66 (2007).
[CrossRef]

Q. Yang, L. Liu, J. Sun, H. Lang, Y. Zhu, and W. Lu, “Property of wave-front coding systems for extending the depth of field,” Acta Opt. Sin. 26, 1807–1812 (2006).

Q. Yang, L. Liu, J. Sun, Y. Zhu, and W. Liu, “Analysis of optical systems with extended depth of field using the Wigner distribution function,” Appl. Opt. 45, 8586–8595 (2006).
[CrossRef] [PubMed]

Q. Yang, J. Sun, and L. Liu, “Phase-space analysis of wavefront coding imaging systems,” Chin. Phys. Lett. 23, 2080–2083 (2006).
[CrossRef]

Ye, Z.

W. Zhang, Z. Ye, T. Zhao, Y. Chen, and F. Yu. “Point spread function characteristics analysis of the wavefront coding system,” Opt. Express 15, 1543–1552 (2007).
[CrossRef] [PubMed]

Y. Chen, W. Zhang, T. Zhao, F. Yu, and Z. Ye, “Imaging characteristics of wavefront coding systems within spatial domain,” Acta Opt. Sin. 27, 1425–1429 (2007).

Yu, F.

Y. Chen, W. Zhang, T. Zhao, F. Yu, and Z. Ye, “Imaging characteristics of wavefront coding systems within spatial domain,” Acta Opt. Sin. 27, 1425–1429 (2007).

W. Zhang, Z. Ye, T. Zhao, Y. Chen, and F. Yu. “Point spread function characteristics analysis of the wavefront coding system,” Opt. Express 15, 1543–1552 (2007).
[CrossRef] [PubMed]

Zhang, H.

Zhang, W.

W. Zhang, Z. Ye, T. Zhao, Y. Chen, and F. Yu. “Point spread function characteristics analysis of the wavefront coding system,” Opt. Express 15, 1543–1552 (2007).
[CrossRef] [PubMed]

Y. Chen, W. Zhang, T. Zhao, F. Yu, and Z. Ye, “Imaging characteristics of wavefront coding systems within spatial domain,” Acta Opt. Sin. 27, 1425–1429 (2007).

Zhao, H.

Zhao, T.

W. Zhang, Z. Ye, T. Zhao, Y. Chen, and F. Yu. “Point spread function characteristics analysis of the wavefront coding system,” Opt. Express 15, 1543–1552 (2007).
[CrossRef] [PubMed]

Y. Chen, W. Zhang, T. Zhao, F. Yu, and Z. Ye, “Imaging characteristics of wavefront coding systems within spatial domain,” Acta Opt. Sin. 27, 1425–1429 (2007).

Zhou, F.

Zhu, Y.

Q. Yang, L. Liu, J. Sun, H. Lang, Y. Zhu, and W. Lu, “Property of wave-front coding systems for extending the depth of field,” Acta Opt. Sin. 26, 1807–1812 (2006).

Q. Yang, L. Liu, J. Sun, Y. Zhu, and W. Liu, “Analysis of optical systems with extended depth of field using the Wigner distribution function,” Appl. Opt. 45, 8586–8595 (2006).
[CrossRef] [PubMed]

Acta Opt. Sin. (2)

Q. Yang, L. Liu, J. Sun, H. Lang, Y. Zhu, and W. Lu, “Property of wave-front coding systems for extending the depth of field,” Acta Opt. Sin. 26, 1807–1812 (2006).

Y. Chen, W. Zhang, T. Zhao, F. Yu, and Z. Ye, “Imaging characteristics of wavefront coding systems within spatial domain,” Acta Opt. Sin. 27, 1425–1429 (2007).

Appl. Opt. (4)

Chin. Phys. Lett. (1)

Q. Yang, J. Sun, and L. Liu, “Phase-space analysis of wavefront coding imaging systems,” Chin. Phys. Lett. 23, 2080–2083 (2006).
[CrossRef]

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

Opt. Commun. (1)

Q. Yang, L. Liu, and J. Sun, “Optimized phase pupil masks for extended depth of field,” Opt. Commun. 272, 56–66 (2007).
[CrossRef]

Opt. Express (1)

Opt. Laser Technol. (1)

H. Zhao, Y. Li, H. Feng, Z. Xu, and Q. Li, “Cubic sinusoidal phase mask: another choice to extend the depth of field of incoherent imaging system,” Opt. Laser Technol. 42, 561–569(2010).
[CrossRef]

Opt. Lett. (6)

Other (2)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).

N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals (Dover, 1986).

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Figures (8)

Fig. 1
Fig. 1

Peak position of the PSF with α of (a)  15 π , (b)  20 π , (c)  30 π , (d)  40 π . Solid curves denote peak positions of the PSFs obtained by FFT according to Eq. (4). Dashed curves denote peak positions of the PSFs obtained according to Eq. (9).

Fig. 2
Fig. 2

PSF results when the defocus parameter ω 20 is (a) 0, (b)  ω 20 (nonoffset position), (c) 10. The normalized image plane distance ranges from 10 to 10. Approximate: PSFs are calculated according to Eq. (8), FFT: PSFs are calculated by FFT according to Eq. (4).

Fig. 3
Fig. 3

SR of the incoherent diffraction-limited systems with ω 20 ranging from 10 to 10. The value of SR decreases rapidly as the defocus error increases.

Fig. 4
Fig. 4

NASR of the CPM wavefront coding systems when α is (a)  15 π , (b)  20 π , (c)  30 π , (d)  40 π . The increase in phase mask parameters can reduce the amplitudes of oscillation and magnitudes of NASR.

Fig. 5
Fig. 5

Relationship between the phase plate parameter and the magnitude of NASR; solid curve is calculated by Eq. (13), dashed curve is calculated by Eq. (12).

Fig. 6
Fig. 6

MTF results when the phase mask system parameter α is (a)  15 π , (b)  20 π , (c)  30 π , (d)  40 π . The defocus parameter ω 20 is 0, 5, and 10. The Hilbert space angle θ and the integral area of MTF S decrease as the phase mask parameters increase.

Fig. 7
Fig. 7

Plot of magnitude of SR NA C versus Hilbert space angle for various phase plate parameters, with α ranging from 15 π to 40 π and the extended ranges of depth of field ranging from 2.5 to 2.5, 5 to 5, 7.5 to 7.5, and 10 to 10.

Fig. 8
Fig. 8

Plot of magnitude of SR NA C versus the integral area of MTF S for various phase plate parameters, with α ranging from 15 π to 40 π and the extended ranges of depth of field ranging from 2.5 to 2.5, 5 to 5, 7.5 to 7.5, and 10 to 10.

Equations (15)

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f ( u ) = α u 3 ,
P ( u ) = q ( u ) exp [ j f ( u ) ] ,
h ( x , ω 20 ) = | P ( u ) exp [ j 2 π ( ω 20 u 2 x u ) ] d u | 2 ,
h ( x , ω 20 ) = 1 2 | 2 π ω 20 / ( 3 α ) 1 2 π ω 20 / ( 3 α ) + 1 exp { j [ α u 3 2 π ( x + 2 π ω 20 2 3 α ) u ] } d u | 2 .
h ( x , ω 20 ) 1 2 | cos [ α u 3 2 π ( x + 2 π ω 20 2 3 α ) u ] + j sin [ α u 3 2 π ( x + 2 π ω 20 2 3 α ) u ] d u | 2 .
h ( x , ω 20 ) 2 | 0 cos [ α u 3 2 π ( x + 2 π ω 20 2 3 α ) u ] d u | 2 .
Ai ( v ) = 1 π 0 cos ( z 3 3 + v z ) d z ,
h ( x , ω 20 ) 2 π 2 ( 3 α ) 2 / 3 | Ai [ 2 π ( 3 α ) 1 / 3 ( x + 2 π ω 20 2 3 α ) ] | 2 .
x p 1.019 ( 3 α ) 1 / 3 2 π 2 π ω 20 2 3 α .
SR NA = PSF ( x p , y p ) WC with aberration PSF ( 0 , 0 ) without aberration ,
SR NA = 1 4 | 2 π ω 20 2 / ( 3 α ) 1 2 π ω 20 2 / ( 3 α ) + 1 exp { j [ α u 3 1.019 ( 3 α ) 1 3 u ] } d u | 1 2 .
SR NA C = π 2 ( 3 α ) 2 / 3 | Ai ( 1.019 ) | 2 0.287 π 2 ( 3 α ) 2 / 3 .
SR NA 0 = SR NA ( α , ω 20 = 0 ) = 1 4 | 1 1 exp { j [ α u 3 1.019 ( 3 α ) 1 3 u ] } d u | 1 2 .
cos θ = MTF ( u , α , ω 20 = 0 ) , MTF ( u , α , ω 20 = max ( ω 20 ) ) MTF ( u , α , ω 20 = 0 ) MTF ( u , α , ω 20 = max ( ω 20 ) ) ,
S = MTF ( u , α , max ( ω 20 ) ) d u .

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