Abstract

An algorithm for the design of imaging systems with circular symmetry that exhibit high resolution as well as extended depth of field for polychromatic incoherent illumination is presented. The approach provides a significant improvement over a publication [1] where the design was carried for a single wavelength. The approach is based on searching for a binary phase pupil mask that provides imaging with the highest cut-off spatial frequency, while assuring a desired contrast value over a given depth of field. Simulations followed by experimental results are provided.

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  1. E. Ben-Eliezer, N. Konforti, B. Milgrom, and E. Marom, “An optimal binary amplitude-phase mask for hybrid imaging systems that exhibit high resolution and extended depth of field,” Opt. Express 16(25), 20540–20561 (2008).
    [PubMed]
  2. J. Ojeda-Castaneda, R. Ramos, and A. Noyola-Isgleas, “High focal depth by apodization and digital restoration,” Appl. Opt. 27(12), 2583–2586 (1988).
    [PubMed]
  3. J. Ojeda-Castaneda, E. Tepichin, and A. Diaz, “Arbitrary high focal depth with a quasioptimum real and positive transmittance apodizer,” Appl. Opt. 28(13), 2666–2669 (1989).
    [PubMed]
  4. J. Ojeda-Castaneda and L. R. Berriel-Valdos, “Zone plate for arbitrary high focal depth,” Appl. Opt. 29(7), 994–997 (1990).
    [PubMed]
  5. E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34(11), 1859 (1995).
    [PubMed]
  6. J. van der Gracht, E. R. Dowski, M. G. Taylor, and D. M. Deaver, “Broadband behavior of an optical-digital focus-invariant system,” Opt. Lett. 21(13), 919–921 (1996).
    [PubMed]
  7. S. S. Sherif, W. T. Cathey, and E. R. Dowski, “Phase plate to extend the depth of field of incoherent hybrid imaging systems,” Appl. Opt. 43(13), 2709–2721 (2004).
    [PubMed]
  8. W. Chi and N. George, “Electronic imaging using a logarithmic asphere,” Opt. Lett. 26(12), 875–877 (2001).
  9. W. Chi and N. George, “Computational imaging with the logarithmic asphere: theory,” J. Opt. Soc. Am. A 20(12), 2260–2273 (2003).
  10. S. Prasad, V. P. Pauca, and J. Robert, Plemmons, Todd C. Torgersen and Joseph van der Gracht, “Pupil-phase optimization for extended focus, aberration corrected imaging systems”, Proc. SPIE 5559, 335–345 (2004).
  11. S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “High resolution imaging using integrated optical systems,” Int. J. Imaging Syst. Technol. 14(2), 67–74 (2004).
  12. J. van der Gracht, V. P. Pauca, H. Setty, R. Narayanswamy, R. Plemmons, S. Prasad, and T. Torgersen, "Iris recognition with enhanced depth-of-field image acquistion," Proc. SPIE 5438, 120–129 (2004).
  13. D. S. Barwick, “Increasing the information acquisition volume in iris recognition systems,” Appl. Opt. 47(26), 4684–4691 (2008).
    [PubMed]
  14. H. Wang and F. Gan, “High focal depth with a pure-phase apodizer,” Appl. Opt. 40(31), 5658–5662 (2001).
  15. H. Wang and F. Gan, “Phase-shifting apodizers for increasing focal depth,” Appl. Opt. 41(25), 5263–5266 (2002).
    [PubMed]
  16. X. Gao, Z. Fei, W. Xu, and F. Gan, “Tunable three-dimensional intensity distribution by a pure phase-shifting apodizer,” Appl. Opt. 44(23), 4870–4873 (2005).
    [PubMed]
  17. E. Ben-Eliezer and E. Marom, “Aberration-free superresolution imaging via binary speckle pattern encoding and processing,” J. Opt. Soc. Am. A 24(4), 1003–1010 (2007).
  18. E. Ben-Eliezer, N. Konforti, and E. Marom, “Super resolution imaging with noise reduction and aberration elimination via random structured illumination and processing,” Opt. Express 15(7), 3849–3863 (2007).
    [PubMed]
  19. E. Ben-Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, “All-optical extended depth of field imaging system,” J. Opt. A, Pure Appl. Opt. 5(5), S 164– S 169 (2003).
  20. E. Ben-Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, “All-optical extended depth of field imaging system,” Proc. SPIE 4829, 221 (2002).
  21. E. Ben-Eliezer, E. Marom, N. Konforti, and Z. Zalevsky, “Experimental realization of an imaging system with an extended depth of field,” Appl. Opt. 44(14), 2792–2798 (2005).
    [PubMed]
  22. E. Ben-Eliezer, E. Marom, N. Konforti, and Z. Zalevsky, “Radial mask for imaging systems that exhibit high resolution and extended depths of field,” Appl. Opt. 45(9), 2001–2013 (2006).
    [PubMed]
  23. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996) 126–151.
  24. H. H. Hopkins, “The Frequency response of a defocus optical system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 231(1184), 91–103 (1955).

2008 (2)

2007 (2)

2006 (1)

2005 (2)

2004 (2)

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “High resolution imaging using integrated optical systems,” Int. J. Imaging Syst. Technol. 14(2), 67–74 (2004).

S. S. Sherif, W. T. Cathey, and E. R. Dowski, “Phase plate to extend the depth of field of incoherent hybrid imaging systems,” Appl. Opt. 43(13), 2709–2721 (2004).
[PubMed]

2003 (2)

W. Chi and N. George, “Computational imaging with the logarithmic asphere: theory,” J. Opt. Soc. Am. A 20(12), 2260–2273 (2003).

E. Ben-Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, “All-optical extended depth of field imaging system,” J. Opt. A, Pure Appl. Opt. 5(5), S 164– S 169 (2003).

2002 (2)

E. Ben-Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, “All-optical extended depth of field imaging system,” Proc. SPIE 4829, 221 (2002).

H. Wang and F. Gan, “Phase-shifting apodizers for increasing focal depth,” Appl. Opt. 41(25), 5263–5266 (2002).
[PubMed]

2001 (2)

1996 (1)

1995 (1)

1990 (1)

1989 (1)

1988 (1)

1955 (1)

H. H. Hopkins, “The Frequency response of a defocus optical system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 231(1184), 91–103 (1955).

Barwick, D. S.

Ben-Eliezer, E.

Berriel-Valdos, L. R.

Cathey, W. T.

Chi, W.

Deaver, D. M.

Diaz, A.

Dowski, E. R.

Fei, Z.

Gan, F.

Gao, X.

George, N.

Hopkins, H. H.

H. H. Hopkins, “The Frequency response of a defocus optical system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 231(1184), 91–103 (1955).

Konforti, N.

Marom, E.

Milgrom, B.

Noyola-Isgleas, A.

Ojeda-Castaneda, J.

Pauca, V. P.

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “High resolution imaging using integrated optical systems,” Int. J. Imaging Syst. Technol. 14(2), 67–74 (2004).

Plemmons, R. J.

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “High resolution imaging using integrated optical systems,” Int. J. Imaging Syst. Technol. 14(2), 67–74 (2004).

Prasad, S.

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “High resolution imaging using integrated optical systems,” Int. J. Imaging Syst. Technol. 14(2), 67–74 (2004).

Ramos, R.

Sherif, S. S.

Taylor, M. G.

Tepichin, E.

Torgersen, T. C.

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “High resolution imaging using integrated optical systems,” Int. J. Imaging Syst. Technol. 14(2), 67–74 (2004).

van der Gracht, J.

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “High resolution imaging using integrated optical systems,” Int. J. Imaging Syst. Technol. 14(2), 67–74 (2004).

J. van der Gracht, E. R. Dowski, M. G. Taylor, and D. M. Deaver, “Broadband behavior of an optical-digital focus-invariant system,” Opt. Lett. 21(13), 919–921 (1996).
[PubMed]

Wang, H.

Xu, W.

Zalevsky, Z.

E. Ben-Eliezer, E. Marom, N. Konforti, and Z. Zalevsky, “Radial mask for imaging systems that exhibit high resolution and extended depths of field,” Appl. Opt. 45(9), 2001–2013 (2006).
[PubMed]

E. Ben-Eliezer, E. Marom, N. Konforti, and Z. Zalevsky, “Experimental realization of an imaging system with an extended depth of field,” Appl. Opt. 44(14), 2792–2798 (2005).
[PubMed]

E. Ben-Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, “All-optical extended depth of field imaging system,” J. Opt. A, Pure Appl. Opt. 5(5), S 164– S 169 (2003).

E. Ben-Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, “All-optical extended depth of field imaging system,” Proc. SPIE 4829, 221 (2002).

Appl. Opt. (11)

J. Ojeda-Castaneda, R. Ramos, and A. Noyola-Isgleas, “High focal depth by apodization and digital restoration,” Appl. Opt. 27(12), 2583–2586 (1988).
[PubMed]

J. Ojeda-Castaneda, E. Tepichin, and A. Diaz, “Arbitrary high focal depth with a quasioptimum real and positive transmittance apodizer,” Appl. Opt. 28(13), 2666–2669 (1989).
[PubMed]

J. Ojeda-Castaneda and L. R. Berriel-Valdos, “Zone plate for arbitrary high focal depth,” Appl. Opt. 29(7), 994–997 (1990).
[PubMed]

E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34(11), 1859 (1995).
[PubMed]

S. S. Sherif, W. T. Cathey, and E. R. Dowski, “Phase plate to extend the depth of field of incoherent hybrid imaging systems,” Appl. Opt. 43(13), 2709–2721 (2004).
[PubMed]

D. S. Barwick, “Increasing the information acquisition volume in iris recognition systems,” Appl. Opt. 47(26), 4684–4691 (2008).
[PubMed]

H. Wang and F. Gan, “High focal depth with a pure-phase apodizer,” Appl. Opt. 40(31), 5658–5662 (2001).

H. Wang and F. Gan, “Phase-shifting apodizers for increasing focal depth,” Appl. Opt. 41(25), 5263–5266 (2002).
[PubMed]

X. Gao, Z. Fei, W. Xu, and F. Gan, “Tunable three-dimensional intensity distribution by a pure phase-shifting apodizer,” Appl. Opt. 44(23), 4870–4873 (2005).
[PubMed]

E. Ben-Eliezer, E. Marom, N. Konforti, and Z. Zalevsky, “Experimental realization of an imaging system with an extended depth of field,” Appl. Opt. 44(14), 2792–2798 (2005).
[PubMed]

E. Ben-Eliezer, E. Marom, N. Konforti, and Z. Zalevsky, “Radial mask for imaging systems that exhibit high resolution and extended depths of field,” Appl. Opt. 45(9), 2001–2013 (2006).
[PubMed]

Int. J. Imaging Syst. Technol. (1)

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gracht, “High resolution imaging using integrated optical systems,” Int. J. Imaging Syst. Technol. 14(2), 67–74 (2004).

J. Opt. A, Pure Appl. Opt. (1)

E. Ben-Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, “All-optical extended depth of field imaging system,” J. Opt. A, Pure Appl. Opt. 5(5), S 164– S 169 (2003).

J. Opt. Soc. Am. A (2)

Opt. Express (2)

Opt. Lett. (2)

Proc. R. Soc. Lond. A Math. Phys. Sci. (1)

H. H. Hopkins, “The Frequency response of a defocus optical system,” Proc. R. Soc. Lond. A Math. Phys. Sci. 231(1184), 91–103 (1955).

Proc. SPIE (1)

E. Ben-Eliezer, Z. Zalevsky, E. Marom, and N. Konforti, “All-optical extended depth of field imaging system,” Proc. SPIE 4829, 221 (2002).

Other (3)

J. van der Gracht, V. P. Pauca, H. Setty, R. Narayanswamy, R. Plemmons, S. Prasad, and T. Torgersen, "Iris recognition with enhanced depth-of-field image acquistion," Proc. SPIE 5438, 120–129 (2004).

S. Prasad, V. P. Pauca, and J. Robert, Plemmons, Todd C. Torgersen and Joseph van der Gracht, “Pupil-phase optimization for extended focus, aberration corrected imaging systems”, Proc. SPIE 5559, 335–345 (2004).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996) 126–151.

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Figures (10)

Fig. 1
Fig. 1

Flow chart of the algorithm used for finding the best mask parameters (the radii of the two phase rings and of the opaque center).

Fig. 2
Fig. 2

Phase of the PPM (Polychromatic Phase mask). The light throughput is 100%.

Fig. 3
Fig. 3

MTF curves of the MPM. designed for optimal response at a wavelength of 550nm, for several defocus positions, for 3 different wavelengths: 450, 550 and 650nm, (a) in-focus Ψ = 0; (b) Ψ = −2; (c) Ψ = −5; (d) Ψ = −8; The contrast value of 5% is marked with a black horizontal line.

Fig. 4
Fig. 4

MTF curves for imaging with a PPM mask for several defocus positions, for 3 different wavelengths – 450, 550 and 650nm, (a)- infocus Ψ = 0; (b)- Ψ = −2; (c)- Ψ = −5; (d)- Ψ = −8; The contrast value of 5% is marked with a black horizontal line. The PPM provides improved results for low values of Ψ in particular.

Fig. 5
Fig. 5

MTF curves for an out of focus condition of Ψ = 4 for all wavelengths between 400 to 700nm (a) provided by a clear aperture (the various colors represent respective wavelengths) (b) PPM for comparison.

Fig. 6
Fig. 6

Spatial frequency limit at which the MTF response falls below a contrast level of 0.05 as a function of Ψ. Red, Green, Blue curves provide the limit for the respective wavelengths. (a): monochromatic phase mask (MPM). (b): polychromatic phase mask (PPM.) response.

Fig. 8
Fig. 8

“Real life” images acquired with a Mustek5200 digital camera equipped with MPM and PPM mask respectively, (a) with MPM mask, (b) with PPM mask. Notice image of the red car that is out of focus (ψ ~-4) with respect to the far away building.

Fig. 7
Fig. 7

Simulation results: (a) original “spiral” object ; (b) image at ψ = 5 with a clear aperture pupil; (c) same but MPM filter inserted in the pupil and (d) same but using a PPM filter.

Fig. 9
Fig. 9

Acquisition of images by a digital camera incorporating a mask for ψ = 4, (a) - image obtained with MPM, (b) – image obtained with PPM

Fig. 10
Fig. 10

RGB target with various spatial frequency bands (a), original object (b)- monochromatic representation when image is obtained with the MPM, (c) – same as (b), but image is obtained with PPM.

Equations (15)

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U i m g ( x , y ) = h ( x , y ; u , v ) U o b j ( u , v ) d u d v ,
U i m g ( x , y ) = e j π λ d i m g ( x 2 + y 2 ) h p ( x u , y v ) U o b j ( u / | M | , v / | M | ) d u d v ,
H ( ν x , ν y ) p ˜ ( λ d i m g ν x , λ d i m g ν y ) ,
p ^ ( x , y ) = p ˜ ( x , y ) exp { j π λ ( 1 d o b j + 1 d i m g 1 f ) ( x 2 + y 2 ) } ,
ψ = π D 2 4 λ ( 1 d o b j + 1 d i m g 1 f ) .
I i m g ( x , y ) = κ | h ( x u ; y v ) | 2 I g ( u , v ) d u d v ,
O T F ( ν x ; ν y ) = H ( ξ + ν x 2 ; η + ν y 2 ) H * ( ξ ν x 2 ; η ν y 2 ) d ξ d η | H ( ξ , η ) | 2 d ξ d η .
ψ ( 1 λ ) .
ψ λ 1 ψ λ 2 = λ 2 λ 1 .
ϕ = 2 π h [ n ( λ ) 1 ] λ .
h = λ 2 [ n ( λ ) 1 ] .
ϕ λ 1 ϕ λ 2 = λ 2 λ 1 n ( λ 1 ) 1 n ( λ 2 ) 1 λ 2 λ 1 .
C O F = 1 λ F # .
C O F λ 1 C O F λ 2 = λ 2 λ 1 .
P P M = max c , 1 r 1 , r 2 , r 3 , r 4 { min λ , ψ C O F ( M T F ( c , 1 r 1 , r 2 , r 3 , r 4 , λ , ψ ) } .

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