Abstract

We unveil a relationship between generating a point spread function with a pair of conjugate phase elements and visualizing the modulation transfer function (MTF) of a single phase element for a variable focus error, at a tunable spatial frequency. We show that the defocused MTF of a pair of conjugate phase elements can be expressed as the modulus of the second order ambiguity function of a single phase element. Finally, we propose a tunable wavefront coding technique with a pair of quartic (4th power) conjugate phase elements.

© 2008 Optical Society of America

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References

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  1. J. Ojeda-Castañeda, R. Ramos, and A. Noyola-Isgleas, “High focal depth by apodization and digital restoration,” Appl. Opt. 27, 2583-2586 (1988).
    [CrossRef] [PubMed]
  2. J. Ojeda-Castañeda and L. R. Berriel-Valdos, "Arbitrarily high focal depth with finite apertures," Opt. Lett. 13, 183-185(1988).
    [CrossRef] [PubMed]
  3. J. Ojeda-Castañeda and L. R. Berriel-Valdos, “Zone plate for arbitrarily high focal depth,” Appl. Opt. 29, 994-997 (1990).
    [CrossRef] [PubMed]
  4. E. R. Dowski, Jr., and T. W. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34, 1859-1865 (1995).
    [CrossRef] [PubMed]
  5. H. Wang and F. Gan, “High focal depth with a pure-phase apodizer,” Appl. Opt. 40, 5658-5662 (2001).
    [CrossRef]
  6. W. Chi and N. George, “Electronic imaging using a logarithmic asphere,” Opt. Lett. 26, 875-877 (2001).
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  7. A. Sauceda and J. Ojeda-Castañeda, “High focal depth with fractional-power wave fronts,” Opt. Lett. 29, 560-562 (2004).
    [CrossRef] [PubMed]
  8. A. Castro and J. Ojeda-Castañeda, “Asymmetric phase masks for extended depth of field,” Appl. Opt. 43, 3474-3479 (2004).
    [CrossRef] [PubMed]
  9. E. Ben-Eliezer, E. Maron, N. Konforti, and Z. Zalevsky, “Experimental realization of an imaging system with an extended depth of field,” Appl. Opt. 44, 2792-2798 (2005).
    [CrossRef] [PubMed]
  10. K. Brenner, A. Lohmann, and J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323-326 (1983).
    [CrossRef]
  11. J. Ojeda- Castañeda, L. R. Berriel-Valdos, and E. Montes, “Ambiguity function as a design tool for high focal depth,” Appl. Opt. 27, 790-795 (1988).
    [CrossRef]
  12. J. Ojeda-Castañeda, J. E. A. Landgrave, and H. M. Escamilla, “Annular phase-only mask for high focal depth,” Opt. Lett. 30, 1647-1649 (2005).
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  13. A. Castro, J. Ojeda-Castañeda, and A. W. Lohmann, “Bow-tie effect: differential operator,” Appl. Opt. 45, 7878-7884 (2006).
    [CrossRef] [PubMed]
  14. L. W. Alvarez, “Two-element variable-power spherical lens,” U.S. patent 3,305,294 (3 December 1964).
  15. A. W. Lohmann, “Lente focale variabile,” Italian patent 727,848 (19 June 1964).
  16. A. W. Lohmann, “Improvements relating to lenses and to variable optical lens systems formed by such lenses,” Patent Specification 998,191, The Patent Office, London (1965).
  17. A. W. Lohmann, “A new class of varifocal lenses,” Appl. Opt. 9, 1669-1671 (1970).
    [CrossRef] [PubMed]
  18. I. A. Palusinski, J. M. Sasián, and J. E. Greivenkamp, “Lateral shift variable aberrations generators,” Appl. Opt. 38, 86-90(1999).
    [CrossRef]
  19. A. W. Rihaczek, Principles of High Resolution Radar (McGraw-Hill, 1969), p. 120.
  20. M. Somayaji and M. P. 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]
  21. N. López-Gil, H. C. Howland, B. Howland, N. Charman, and R. Applegate, “Generation of third-order spherical and coma aberrations by the use of radially symmetrical fourth-order lenses,” J. Opt. Soc. Am. A 15, 2563-2571 (1998).
    [CrossRef]

2006 (2)

2005 (2)

2004 (2)

2001 (2)

1999 (1)

1998 (1)

1995 (1)

1990 (1)

1988 (3)

1983 (1)

K. Brenner, A. Lohmann, and J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323-326 (1983).
[CrossRef]

1970 (1)

1969 (1)

A. W. Rihaczek, Principles of High Resolution Radar (McGraw-Hill, 1969), p. 120.

1965 (1)

A. W. Lohmann, “Improvements relating to lenses and to variable optical lens systems formed by such lenses,” Patent Specification 998,191, The Patent Office, London (1965).

Alvarez, L. W.

L. W. Alvarez, “Two-element variable-power spherical lens,” U.S. patent 3,305,294 (3 December 1964).

Applegate, R.

Ben-Eliezer, E.

Berriel-Valdos, L. R.

Brenner, K.

K. Brenner, A. Lohmann, and J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323-326 (1983).
[CrossRef]

Castañeda, J. Ojeda-

Castro, A.

Cathey, T. W.

Charman, N.

Chi, W.

Christensen, M. P.

Dowski, E. R.

Escamilla, H. M.

Gan, F.

George, N.

Greivenkamp, J. E.

Howland, B.

Howland, H. C.

Konforti, N.

Landgrave, J. E. A.

Lohmann, A.

K. Brenner, A. Lohmann, and J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323-326 (1983).
[CrossRef]

Lohmann, A. W.

A. Castro, J. Ojeda-Castañeda, and A. W. Lohmann, “Bow-tie effect: differential operator,” Appl. Opt. 45, 7878-7884 (2006).
[CrossRef] [PubMed]

A. W. Lohmann, “A new class of varifocal lenses,” Appl. Opt. 9, 1669-1671 (1970).
[CrossRef] [PubMed]

A. W. Lohmann, “Improvements relating to lenses and to variable optical lens systems formed by such lenses,” Patent Specification 998,191, The Patent Office, London (1965).

A. W. Lohmann, “Lente focale variabile,” Italian patent 727,848 (19 June 1964).

López-Gil, N.

Maron, E.

Montes, E.

Noyola-Isgleas, A.

Ojeda-Castañeda, J.

Palusinski, I. A.

Ramos, R.

Rihaczek, A. W.

A. W. Rihaczek, Principles of High Resolution Radar (McGraw-Hill, 1969), p. 120.

Sasián, J. M.

Sauceda, A.

Somayaji, M.

Wang, H.

Zalevsky, Z.

Appl. Opt. (11)

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

E. R. Dowski, Jr., and T. W. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34, 1859-1865 (1995).
[CrossRef] [PubMed]

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

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

A. Castro and J. Ojeda-Castañeda, “Asymmetric phase masks for extended depth of field,” Appl. Opt. 43, 3474-3479 (2004).
[CrossRef] [PubMed]

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

A. Castro, J. Ojeda-Castañeda, and A. W. Lohmann, “Bow-tie effect: differential operator,” Appl. Opt. 45, 7878-7884 (2006).
[CrossRef] [PubMed]

J. Ojeda- Castañeda, L. R. Berriel-Valdos, and E. Montes, “Ambiguity function as a design tool for high focal depth,” Appl. Opt. 27, 790-795 (1988).
[CrossRef]

A. W. Lohmann, “A new class of varifocal lenses,” Appl. Opt. 9, 1669-1671 (1970).
[CrossRef] [PubMed]

I. A. Palusinski, J. M. Sasián, and J. E. Greivenkamp, “Lateral shift variable aberrations generators,” Appl. Opt. 38, 86-90(1999).
[CrossRef]

M. Somayaji and M. P. 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]

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

Opt. Commun. (1)

K. Brenner, A. Lohmann, and J. Ojeda-Castañeda, “The ambiguity function as a polar display of the OTF,” Opt. Commun. 44, 323-326 (1983).
[CrossRef]

Opt. Lett. (4)

Other (4)

A. W. Rihaczek, Principles of High Resolution Radar (McGraw-Hill, 1969), p. 120.

L. W. Alvarez, “Two-element variable-power spherical lens,” U.S. patent 3,305,294 (3 December 1964).

A. W. Lohmann, “Lente focale variabile,” Italian patent 727,848 (19 June 1964).

A. W. Lohmann, “Improvements relating to lenses and to variable optical lens systems formed by such lenses,” Patent Specification 998,191, The Patent Office, London (1965).

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Figures (4)

Fig. 1
Fig. 1

Schematics depicting the use of (a) cubic and (b) quartic conjugate phase plates. The vertical scale has been grossly exaggerated.

Fig. 2
Fig. 2

4 f coherent optical processor with conjugate phase plates as a spatial filter. This setup was chosen to analyze the frequency response of imaging systems designed for an extended depth of field.

Fig. 3
Fig. 3

MTFs of a single phase element with cubic phase profile of strength p = 12 . They are plotted as functions of the normalized spatial frequency and the defocus coefficient, given in wavelengths.

Fig. 4
Fig. 4

MTFs of a tunable wavefront coding system with quartic conjugate phase elements of strength p = 12 . They are plotted as functions of the normalized spatial frequency and the defocus coefficient, given in wavelengths, for various lateral shifts of the conjugate phase elements: (a)  μ / Ω = 0.00 , (b)  μ / Ω = 0.02 , (c)  μ / Ω = 0.06 , and (d)  μ / Ω = 0.30 .

Tables (1)

Tables Icon

Table 1 Expressions for the Pupil Function, the In-Focus PSF, and the Defocused MTF for Pupil Masks Consisting of a Single Element and a Pair of Elements

Equations (27)

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P ( α , β ) = T ( α ) rect [ β / ( 2 Ω ) ] ,
T ( α ) = exp [ i φ ( α ) ] rect [ α / ( 2 Ω ) ] .
Q ( α , β ; μ ) = T ( α + μ / 2 ) T * ( α μ / 2 ) rect [ β / ( 2 Ω ) ] .
q ( x , y ; μ ) = sinc ( 2 Ω y ) [ T ( α + μ / 2 ) T * ( α μ / 2 ) exp ( i 2 π x α ) d α ] .
| q ( x , y ; μ ) | = | sinc ( 2 Ω y ) | | A T ( μ , x ) | .
| A T [ μ , 2 W 2 , 0 μ / ( λ Ω 2 ) ] | = | H ( μ ; W 2 , 0 ) | .
| q ( x , 0 ; μ ) | = | H ( μ ; W 2 , 0 ) | .
T ( α ) = exp [ i 2 π p ( α / Ω ) 3 ] rect [ α / ( 2 Ω ) ] .
Q ( α , β ; μ ) = exp [ i π ( p / 2 ) ( μ / Ω ) 3 ] exp [ i 2 π ( 3 p ) ( μ / Ω ) ( α / Ω ) 2 ] × rect [ α / ( 2 Ω | μ | ) ] rect [ β / ( 2 Ω ) ] .
| q ( x , y ; μ ) | = | sinc ( 2 Ω y ) | | exp [ i 2 π ( 3 p ) ( μ / Ω ) ( α / Ω ) 2 ] × rect [ α / ( 2 Ω | μ | ) ] exp ( i 2 π x α ) d α | .
| H ( μ ; W 2 , 0 ) | = | exp { i 2 π [ 3 p ( α / Ω ) 2 + 2 ( W 2 , 0 / λ ) ( α / Ω ) ] ( μ / Ω ) } × rect [ α / ( 2 Ω | μ | ) ] d α | .
Q ( α ; μ ) = T ( α + μ / 2 ) T * ( α μ / 2 ) .
R ( α ; W 2 , 0 , μ ) = T ( α + μ / 2 ) T * ( α μ / 2 ) exp [ i 2 π ( W 2 , 0 / λ ) ( α / Ω ) 2 ] .
| H ( ν ; W 2.0 , μ ) | = | R ( α + ν / 2 ; W 2 , 0 , μ ) R * ( α ν / 2 ; W 2 , 0 , μ ) d α | .
| H ( ν ; W 2 , 0 , μ ) | = | T ( α + ν / 2 + μ / 2 ) T * ( α + ν / 2 μ / 2 ) T * ( α ν / 2 + μ / 2 ) T ( α ν / 2 μ / 2 ) × exp { i 2 π [ 2 W 2 , 0 ν / ( λ Ω 2 ) ] α } d α | .
| H ( ν ; W 2 , 0 , μ ) | = | A T { μ , x + [ ν / ( λ Ω 2 ) ] W 2 , 0 } A * T { μ , x [ ν / ( λ Ω 2 ) ] W 2 , 0 } exp ( i 2 π ν x ) d x | = | A A T { ν , [ 2 ν / ( λ Ω 2 ) ] W 2 , 0 ; μ } | .
T ( α ) = exp [ i 2 π p ( α / Ω ) 4 ] rect [ α / ( 2 Ω ) ] .
R ( α ; W 2 , 0 , μ ) = exp [ i 2 π ( 4 p ) ( μ / Ω ) ( α / Ω ) 3 ] exp [ i 2 π p ( μ / Ω ) 3 ( α / Ω ) ] × rect [ α / ( 2 Ω | μ | ) ] exp [ i 2 π ( W 2 , 0 / λ ) ( α / Ω ) 2 ] .
| H ( ν ; W 2 , 0 , μ ) | = | exp { i 2 π [ 12 p ( μ / Ω ) ( α / Ω ) 2 + 2 ( W 2 , 0 / λ ) ( α / Ω ) ] ( ν / Ω ) } × rect [ α / ( 2 Ω | μ | | ν | ) ] d α | .
P ( α ; W 2 , 0 ) = T ( α ) exp [ i 2 π ( W 2 , 0 / λ ) ( α / Ω ) 2 ] ,
| H ( μ ; W 2 , 0 ) | = | T ( α + μ / 2 ) T * ( α μ / 2 ) exp { i 2 π [ 2 μ / ( λ Ω 2 ) ] W 2 , 0 α } d α | .
A T ( μ , y ) = T ( α + μ / 2 ) T * ( α μ / 2 ) exp ( i 2 π y α ) d α ,
A T ( μ , x ) = Q ( α ; μ ) exp ( i 2 π x α ) d α .
A A T ( ν , y ; μ ) = A Q ( ν , y ; μ ) ,
A A T ( ν , y ; μ ) = A T ( μ , x + y / 2 ) A * T ( μ , x y / 2 ) exp ( i 2 π ν x ) d x ,
A Q ( ν , y ; μ ) = Q ( α + ν / 2 ; μ ) Q * ( α ν / 2 ; μ ) exp ( i 2 π y α ) d α .
| T ( α + ν / 2 + μ / 2 ) T * ( α + ν / 2 μ / 2 ) T * ( α ν / 2 + μ / 2 ) T ( α ν / 2 μ / 2 ) exp ( i 2 π y α ) d α | = | A T ( μ , x + y / 2 ) A * T ( μ , x y / 2 ) exp ( i 2 π ν x ) d x | = | A A T ( ν , y ; μ ) | .

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