Abstract

We propose a novel design method for a circularly symmetric phase mask to extend the depth of focus. Using the free-form rational function as the solution space, we optimize the profile of the phase mask by analysis of the axial intensity distribution, which can be calculated efficiently by employing the fast Fourier transform algorithm. Numerical comparisons prove the resulting rational phase mask’s superiority to the existing quartic phase mask in intensity distribution and imaging performance.

© 2009 Optical Society of America

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References

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2008 (1)

2007 (2)

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]

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

2006 (1)

2005 (1)

S. Förster, H. Gross, F. Höller, and L. Höring, “Extended depth of focus as a process of pupil manipulation,” Proc. SPIE 5962, 596207 (2005).
[CrossRef]

2003 (1)

2001 (1)

2000 (1)

1998 (3)

1996 (1)

1995 (1)

1993 (2)

1992 (2)

1989 (1)

1986 (1)

1983 (1)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671-680 (1983).
[CrossRef] [PubMed]

Andres, P.

Bagheri, S.

Bara, S.

Cathey, W. T.

Chi, W.

Colautti, C.

Diaz, A.

Dowski, E. R.

Farias, D.

Fink, K. D.

J. H. Mathews and K. D. Fink, Numerical Methods Using MATLAB, 4th ed. (Prentice Hall, 2004).

Flores, A.

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]

Förster, S.

S. Förster, H. Gross, F. Höller, and L. Höring, “Extended depth of focus as a process of pupil manipulation,” Proc. SPIE 5962, 596207 (2005).
[CrossRef]

Friberg, A. T.

S. Y. Popov and A. T. Friberg “Design of diffractive axicons for partially coherent light,” Opt. Lett. 23, 1639-1641 (1998).
[CrossRef]

S. Y. Popov and A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7, 537-548 (1998).
[CrossRef]

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671-680 (1983).
[CrossRef] [PubMed]

George, N.

Gomez Reino, C.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005).

Gross, H.

S. Förster, H. Gross, F. Höller, and L. Höring, “Extended depth of focus as a process of pupil manipulation,” Proc. SPIE 5962, 596207 (2005).
[CrossRef]

Harvey, A. R.

Höller, F.

S. Förster, H. Gross, F. Höller, and L. Höring, “Extended depth of focus as a process of pupil manipulation,” Proc. SPIE 5962, 596207 (2005).
[CrossRef]

Höring, L.

S. Förster, H. Gross, F. Höller, and L. Höring, “Extended depth of focus as a process of pupil manipulation,” Proc. SPIE 5962, 596207 (2005).
[CrossRef]

Jaroszewicz, Z.

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671-680 (1983).
[CrossRef] [PubMed]

Kolodziejczyk, A.

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]

Liu, Z.

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]

Mathews, J. H.

J. H. Mathews and K. D. Fink, Numerical Methods Using MATLAB, 4th ed. (Prentice Hall, 2004).

Mezouari, S.

Muyo, G.

Ojeda-Castenada, J.

Popov, S. Y.

S. Y. Popov and A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7, 537-548 (1998).
[CrossRef]

S. Y. Popov and A. T. Friberg “Design of diffractive axicons for partially coherent light,” Opt. Lett. 23, 1639-1641 (1998).
[CrossRef]

Roman Dopazo, J. F.

Sicre, E. E.

Silveira, P.

Sochacki, J.

Staronski, L. R.

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]

Tepichin, E.

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671-680 (1983).
[CrossRef] [PubMed]

Wang, M. R.

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]

Yang, J. J.

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]

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]

Zalvidea, D.

Appl. Opt. (4)

J. Opt. Soc. Am. A (5)

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. Eng. (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]

Opt. Lett. (6)

Proc. SPIE (1)

S. Förster, H. Gross, F. Höller, and L. Höring, “Extended depth of focus as a process of pupil manipulation,” Proc. SPIE 5962, 596207 (2005).
[CrossRef]

Pure Appl. Opt. (1)

S. Y. Popov and A. T. Friberg, “Apodization of generalized axicons to produce uniform axial line images,” Pure Appl. Opt. 7, 537-548 (1998).
[CrossRef]

Science (1)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671-680 (1983).
[CrossRef] [PubMed]

Other (2)

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts, 2005).

J. H. Mathews and K. D. Fink, Numerical Methods Using MATLAB, 4th ed. (Prentice Hall, 2004).

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Figures (10)

Fig. 1
Fig. 1

Typical axial intensity distribution curve. I max denotes the maximum axial intensity value, Δ f is the defined effective focal range, and I mean denotes the mean axial intensity value in Δ f .

Fig. 2
Fig. 2

Phase profiles of the RPMs and QPMs. (a) Curve plot; (b1)–(b6) contour maps.

Fig. 3
Fig. 3

Intensity distribution of RPM0.05 and QPM0.05. (a) Axial intensity. Partial PSF of (b) RPM0.05, (c) QPM0.05.

Fig. 4
Fig. 4

Intensity distribution of RPM0.1 and QPM0.1. (a) Axial intensity. Partial PSF of (b) RPM0.1, (c) QPM0.1.

Fig. 5
Fig. 5

Intensity distribution of RPM0.25 and QPM0.25. (a) Axial intensity. Partial PSF of (b) RPM0.25, (c) QPM0.25, (d) CA.

Fig. 6
Fig. 6

Energy flow in the focal region.

Fig. 7
Fig. 7

(a) Variation of the cutoff lateral position r c in the focal region of the RPM0.1 and QPM 0.1. (b) Corresponding PSF at ψ = ± 35 .

Fig. 8
Fig. 8

Radial MTF in the focal region of (a) RPM0.1, (b) QPM0.1.

Fig. 9
Fig. 9

Raw images of a spoke target for various defocus parameters by applying the OTFs.

Fig. 10
Fig. 10

(a) Radial MTF of CA. (b) Simulated images for various defocus parameters.

Equations (13)

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P ( ρ ) = exp [ j ϕ ( ρ ) ] for 0 ρ 1 ,
I ( r , ψ ) = 4 π 2 0 1 P ( ρ ) J 0 ( 2 π r ρ ) exp [ j ψ ρ 2 ] ρ d ρ 2 ,
ψ = π L 2 4 λ ( 1 f 1 d o 1 d i ) = 2 π λ W 20 ,
I ( 0 , ψ ) = 4 π 2 0 1 P ( ρ ) exp [ j ψ ρ 2 ] ρ d ρ 2 ,
I ( 0 , ψ ) = 4 π 2 0.5 0.5 Φ ( ξ ) exp [ j 2 π ψ 2 π ξ ] d ξ 2 ,
I ( 0 , ψ ) d ψ = 4 π 2 0.5 0.5 Φ ( ξ ) exp [ j 2 π ψ 2 π ξ ] d ξ 2 d ψ = 4 π 2 0.5 0.5 Φ ( ξ ) 2 d ξ = 4 π 2 = constant ,
ϕ RPM ( ρ ) = n = 0 N a n ρ n m = 0 M b m ρ m ,
max { a n } , { b m } Δ f = λ 2 π ( ψ 2 ψ 1 ) ,
subject to:   ( 1 ) I max K ,
( 2 ) I ( ψ ) ( 1 C ) I max , for ψ 1 < ψ < ψ 2 ,
ϕ QPM ( ρ ) = 2 π α ( ρ 2 ρ 4 ) ,
r 0 = 1.22 λ f L .
E ( r m , ψ ) = 2 π 0 r m I ( r , ψ ) r d r ,

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