Abstract

The paper presents imaging properties of modified lenses with the radial and the angular modulation. We analyze three following optical elements with moderate numerical apertures: the forward logarithmic axicon and the axilens representing the radial modulation as well as the light sword optical element being a counterpart of the axilens with the angular modulation. The abilities of the elements for imaging with extended depth of focus are discussed in detail with the help of structures of output images and modulation transfer functions corresponding to them. According to the obtained results only the angular modulation of the lens makes possible to maintain the acceptable resolution, contrast and brightness of the output images for a wide range of defocusing. Therefore optical elements with angular modulations and moderate numerical apertures seem to be especially suitable for imaging with extended focal depth.

© 2007 Optical Society of America

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  1. M. Mino and Y. Okano, "Improvement in the optical transfer function of a defocused optical system through the use of shaded apertures," Appl. Opt. 10, 2219-2225 (1971).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  3. J. Ojeda-Castañeda, E. Tepichin, and A. Diaz, "Arbitrary high focal depth with a quasioptimum real and positive transmittance apodizer," Appl. Opt. 28, 2666-2670 (1989).
    [CrossRef] [PubMed]
  4. J. Ojeda-Castañeda and L. R. Berriel-Valdos, "Zone plate for arbitrarily high focal depth," Appl. Opt. 29, 994-997 (1990).
    [CrossRef] [PubMed]
  5. E. R. Dowski, Jr. and W. T. Cathey, "Extended depth of field through wave-front coding," Appl. Opt. 34, 1859-1866 (1995).
    [CrossRef] [PubMed]
  6. S. Bradburn, W. T. Cathey, and E. R. Dowski, Jr., "Realizations of focus invariance in optical-digital systems with wave-front coding," Appl. Opt. 36, 9157-9166 (1997).
    [CrossRef]
  7. H. B. Wach, E. R. Dowski, Jr., and W. T. Cathey, "Control of chromatic focal shift through wave-front coding," Appl. Opt. 37, 5359-5367 (1998).
    [CrossRef]
  8. S. C. Tucker, W. T. Cathey, and E. R. Dowski, Jr., "Extended depth of field and aberration control for inexpensive digital microscope systems," Opt. Express 4, 467-474 (1999).
    [CrossRef] [PubMed]
  9. J. Sochacki, A. Kołodziejczyk, Z. Jaroszewicz, and S. Bara, "Nonparaxial design of generalized axicons," Appl. Opt. 31, 5326-5330 (1992).
    [CrossRef] [PubMed]
  10. J. Sochacki, Z. Jaroszewicz, L.R. Staroński, and A. Kołodziejczyk, "Annular-aperture logarithmic axicon," J. Opt. Soc. Am. A 10, 1765-1768 (1993).
    [CrossRef]
  11. W. Chi, and N. George, "Electronic imaging using a logarithmic asphere," Opt. Lett. 26, 875-877 (2001).
    [CrossRef]
  12. M. A. Golub, V. Shurman, and I. Grossinger, "Extended focus diffractive optical element for Gaussian laser beams," Appl. Opt. 45, 144-150 (2006).
    [CrossRef] [PubMed]
  13. J. Ares, R. Flores, S. Bara, and Z. Jaroszewicz, "Presbyopia compensation with a quartic axicon," Optom. Vis. Sci. 82, 1071-1078 (2005).
    [CrossRef] [PubMed]
  14. G.-m. Dai, "Optical surface optimization for the correction of presbyopia," Appl. Opt. 45, 4184-4195 (2006).
    [CrossRef] [PubMed]
  15. A. Flores, M. R. Wang, and J. J. Yang, "Achromatic hybrid refractive-diffractive lens with extended depth of focus," Appl. Opt. 43, 5618-5630 (2004).
    [CrossRef] [PubMed]
  16. Z. Liu, A. Flores, M. R. Wang, and J. J. Yang, "Diffractive infrared lens with extended depth of focus," Opt. Eng.  46, 018002 (1-9) (2007).
    [CrossRef]
  17. E. E. Garcia-Guerrero, E. R. Mendez, H. M. Escamilla, T. A. Leskova, and A. A. Maradudin, "Design and fabrication of random phase diffusers for extending the depth of focus," Opt. Express 15, 910-923 (2007).
    [CrossRef] [PubMed]
  18. B.-Z. Dong, J. Liu, B.-Y. Gu, and G.-Z. Yang, "Rigorous electromagnetic analysis of a microcylindrical axilens with long focal depth and high transverse resolution," J. Opt. Soc. Am. A 18, 1465-1470 (2001).
    [CrossRef]
  19. J.-S. Ye, B.-Z. Dong, B.-Y. Gu, G.-Z. Yang, and S.-T. Liu, "Analysis of a closed-boundary axilens with long focal depth and high transverse resolution based on a rigorous electromagnetic theory," J. Opt. Soc. Am. A 19, 2030-2035 (2002).
    [CrossRef]
  20. F. Di, Y. Yingbai, J. Guofan, and W. Minxian, "Rigorous concept for the analysis of diffractive lenses with different axial resolution and high lateral resolution," Opt. Express 17, 1987-1994 (2003).
    [CrossRef]
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    [CrossRef]
  22. N. Davidson, A. A. Friesem, and E. Hasman, "Holographic axilens: high resolution and long focal depth," Opt. Lett. 16, 523-525 (1991).
    [CrossRef] [PubMed]
  23. A. Kołodziejczyk, S. Bara, Z. Jaroszewicz, and M. Sypek, "The light sword optical element - a new diffraction structure with extended depth of focus," J. Mod. Opt. 37, 1283-1286 (1990).
    [CrossRef]
  24. J. Sochacki, S. Bara, Z. Jaroszewicz, and A. Kołodziejczyk, "Phase retardation of the uniform-intensity axilens," Opt. Lett. 17, 7-9 (1992).
    [CrossRef] [PubMed]
  25. G. Mikuła, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, and M. Sypek, "Diffractive elements for imaging with extended depth of focus," Opt. Eng. 44, 058001(2005).
    [CrossRef]
  26. S. Bara, C. Frere, Z. Jaroszewicz, A. Kołodziejczyk, and D. Leseberg, "Modulated on-axis circular zone plates for a generation of three-dimensional focal curves," J. Mod. Opt. 37, 1287-1295 (1990).
  27. H. H. Emsley, Visual Optics (London: Hatton Press, 1952).
  28. M. Sypek, "Light propagation in the Fresnel region. New numerical approach," Opt. Commun. 116, 43-48 (1995).
    [CrossRef]

2007 (2)

2006 (2)

2005 (2)

J. Ares, R. Flores, S. Bara, and Z. Jaroszewicz, "Presbyopia compensation with a quartic axicon," Optom. Vis. Sci. 82, 1071-1078 (2005).
[CrossRef] [PubMed]

G. Mikuła, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, and M. Sypek, "Diffractive elements for imaging with extended depth of focus," Opt. Eng. 44, 058001(2005).
[CrossRef]

2004 (1)

2003 (1)

F. Di, Y. Yingbai, J. Guofan, and W. Minxian, "Rigorous concept for the analysis of diffractive lenses with different axial resolution and high lateral resolution," Opt. Express 17, 1987-1994 (2003).
[CrossRef]

2002 (1)

2001 (2)

1999 (1)

1998 (1)

1997 (1)

1995 (2)

M. Sypek, "Light propagation in the Fresnel region. New numerical approach," Opt. Commun. 116, 43-48 (1995).
[CrossRef]

E. R. Dowski, Jr. and W. T. Cathey, "Extended depth of field through wave-front coding," Appl. Opt. 34, 1859-1866 (1995).
[CrossRef] [PubMed]

1993 (1)

1992 (2)

1991 (1)

1990 (3)

A. Kołodziejczyk, S. Bara, Z. Jaroszewicz, and M. Sypek, "The light sword optical element - a new diffraction structure with extended depth of focus," J. Mod. Opt. 37, 1283-1286 (1990).
[CrossRef]

S. Bara, C. Frere, Z. Jaroszewicz, A. Kołodziejczyk, and D. Leseberg, "Modulated on-axis circular zone plates for a generation of three-dimensional focal curves," J. Mod. Opt. 37, 1287-1295 (1990).

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

1989 (1)

1986 (1)

1971 (1)

Andres, P.

Ares, J.

J. Ares, R. Flores, S. Bara, and Z. Jaroszewicz, "Presbyopia compensation with a quartic axicon," Optom. Vis. Sci. 82, 1071-1078 (2005).
[CrossRef] [PubMed]

Bara, S.

J. Ares, R. Flores, S. Bara, and Z. Jaroszewicz, "Presbyopia compensation with a quartic axicon," Optom. Vis. Sci. 82, 1071-1078 (2005).
[CrossRef] [PubMed]

J. Sochacki, S. Bara, Z. Jaroszewicz, and A. Kołodziejczyk, "Phase retardation of the uniform-intensity axilens," Opt. Lett. 17, 7-9 (1992).
[CrossRef] [PubMed]

J. Sochacki, A. Kołodziejczyk, Z. Jaroszewicz, and S. Bara, "Nonparaxial design of generalized axicons," Appl. Opt. 31, 5326-5330 (1992).
[CrossRef] [PubMed]

A. Kołodziejczyk, S. Bara, Z. Jaroszewicz, and M. Sypek, "The light sword optical element - a new diffraction structure with extended depth of focus," J. Mod. Opt. 37, 1283-1286 (1990).
[CrossRef]

S. Bara, C. Frere, Z. Jaroszewicz, A. Kołodziejczyk, and D. Leseberg, "Modulated on-axis circular zone plates for a generation of three-dimensional focal curves," J. Mod. Opt. 37, 1287-1295 (1990).

Berriel-Valdos, L. R.

Bradburn, S.

Cathey, W. T.

Chi, W.

Dai, G.-m.

Davidson, N.

Di, F.

F. Di, Y. Yingbai, J. Guofan, and W. Minxian, "Rigorous concept for the analysis of diffractive lenses with different axial resolution and high lateral resolution," Opt. Express 17, 1987-1994 (2003).
[CrossRef]

Diaz, A.

Dong, B.-Z.

Dowski, E. R.

Escamilla, H. M.

Flores, A.

Flores, R.

J. Ares, R. Flores, S. Bara, and Z. Jaroszewicz, "Presbyopia compensation with a quartic axicon," Optom. Vis. Sci. 82, 1071-1078 (2005).
[CrossRef] [PubMed]

Frere, C.

S. Bara, C. Frere, Z. Jaroszewicz, A. Kołodziejczyk, and D. Leseberg, "Modulated on-axis circular zone plates for a generation of three-dimensional focal curves," J. Mod. Opt. 37, 1287-1295 (1990).

Friesem, A. A.

Garcia-Guerrero, E. E.

George, N.

Golub, M. A.

Grossinger, I.

Gu, B.-Y.

Guofan, J.

F. Di, Y. Yingbai, J. Guofan, and W. Minxian, "Rigorous concept for the analysis of diffractive lenses with different axial resolution and high lateral resolution," Opt. Express 17, 1987-1994 (2003).
[CrossRef]

Hasman, E.

Jaroszewicz, Z.

J. Ares, R. Flores, S. Bara, and Z. Jaroszewicz, "Presbyopia compensation with a quartic axicon," Optom. Vis. Sci. 82, 1071-1078 (2005).
[CrossRef] [PubMed]

J. Sochacki, Z. Jaroszewicz, L.R. Staroński, and A. Kołodziejczyk, "Annular-aperture logarithmic axicon," J. Opt. Soc. Am. A 10, 1765-1768 (1993).
[CrossRef]

J. Sochacki, A. Kołodziejczyk, Z. Jaroszewicz, and S. Bara, "Nonparaxial design of generalized axicons," Appl. Opt. 31, 5326-5330 (1992).
[CrossRef] [PubMed]

J. Sochacki, S. Bara, Z. Jaroszewicz, and A. Kołodziejczyk, "Phase retardation of the uniform-intensity axilens," Opt. Lett. 17, 7-9 (1992).
[CrossRef] [PubMed]

S. Bara, C. Frere, Z. Jaroszewicz, A. Kołodziejczyk, and D. Leseberg, "Modulated on-axis circular zone plates for a generation of three-dimensional focal curves," J. Mod. Opt. 37, 1287-1295 (1990).

A. Kołodziejczyk, S. Bara, Z. Jaroszewicz, and M. Sypek, "The light sword optical element - a new diffraction structure with extended depth of focus," J. Mod. Opt. 37, 1283-1286 (1990).
[CrossRef]

Kolodziejczyk, A.

G. Mikuła, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, and M. Sypek, "Diffractive elements for imaging with extended depth of focus," Opt. Eng. 44, 058001(2005).
[CrossRef]

J. Sochacki, Z. Jaroszewicz, L.R. Staroński, and A. Kołodziejczyk, "Annular-aperture logarithmic axicon," J. Opt. Soc. Am. A 10, 1765-1768 (1993).
[CrossRef]

J. Sochacki, A. Kołodziejczyk, Z. Jaroszewicz, and S. Bara, "Nonparaxial design of generalized axicons," Appl. Opt. 31, 5326-5330 (1992).
[CrossRef] [PubMed]

J. Sochacki, S. Bara, Z. Jaroszewicz, and A. Kołodziejczyk, "Phase retardation of the uniform-intensity axilens," Opt. Lett. 17, 7-9 (1992).
[CrossRef] [PubMed]

S. Bara, C. Frere, Z. Jaroszewicz, A. Kołodziejczyk, and D. Leseberg, "Modulated on-axis circular zone plates for a generation of three-dimensional focal curves," J. Mod. Opt. 37, 1287-1295 (1990).

A. Kołodziejczyk, S. Bara, Z. Jaroszewicz, and M. Sypek, "The light sword optical element - a new diffraction structure with extended depth of focus," J. Mod. Opt. 37, 1283-1286 (1990).
[CrossRef]

Leseberg, D.

S. Bara, C. Frere, Z. Jaroszewicz, A. Kołodziejczyk, and D. Leseberg, "Modulated on-axis circular zone plates for a generation of three-dimensional focal curves," J. Mod. Opt. 37, 1287-1295 (1990).

Leskova, T. A.

Lin, J.

Liu, J.

Liu, S.

Liu, S.-T.

Makowski, M.

G. Mikuła, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, and M. Sypek, "Diffractive elements for imaging with extended depth of focus," Opt. Eng. 44, 058001(2005).
[CrossRef]

Maradudin, A. A.

Mendez, E. R.

Mikula, G.

G. Mikuła, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, and M. Sypek, "Diffractive elements for imaging with extended depth of focus," Opt. Eng. 44, 058001(2005).
[CrossRef]

Mino, M.

Minxian, W.

F. Di, Y. Yingbai, J. Guofan, and W. Minxian, "Rigorous concept for the analysis of diffractive lenses with different axial resolution and high lateral resolution," Opt. Express 17, 1987-1994 (2003).
[CrossRef]

Ojeda-Castañeda, J.

Okano, Y.

Prokopowicz, C.

G. Mikuła, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, and M. Sypek, "Diffractive elements for imaging with extended depth of focus," Opt. Eng. 44, 058001(2005).
[CrossRef]

Shurman, V.

Sochacki, J.

Staronski, L.R.

Sypek, M.

G. Mikuła, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, and M. Sypek, "Diffractive elements for imaging with extended depth of focus," Opt. Eng. 44, 058001(2005).
[CrossRef]

M. Sypek, "Light propagation in the Fresnel region. New numerical approach," Opt. Commun. 116, 43-48 (1995).
[CrossRef]

A. Kołodziejczyk, S. Bara, Z. Jaroszewicz, and M. Sypek, "The light sword optical element - a new diffraction structure with extended depth of focus," J. Mod. Opt. 37, 1283-1286 (1990).
[CrossRef]

Tepichin, E.

Tucker, S. C.

Wach, H. B.

Wang, M. R.

Yang, G.-Z.

Yang, J. J.

Ye, J.

Ye, J.-S.

Yingbai, Y.

F. Di, Y. Yingbai, J. Guofan, and W. Minxian, "Rigorous concept for the analysis of diffractive lenses with different axial resolution and high lateral resolution," Opt. Express 17, 1987-1994 (2003).
[CrossRef]

Appl. Opt. (10)

J. Sochacki, A. Kołodziejczyk, Z. Jaroszewicz, and S. Bara, "Nonparaxial design of generalized axicons," Appl. Opt. 31, 5326-5330 (1992).
[CrossRef] [PubMed]

S. Bradburn, W. T. Cathey, and E. R. Dowski, Jr., "Realizations of focus invariance in optical-digital systems with wave-front coding," Appl. Opt. 36, 9157-9166 (1997).
[CrossRef]

H. B. Wach, E. R. Dowski, Jr., and W. T. Cathey, "Control of chromatic focal shift through wave-front coding," Appl. Opt. 37, 5359-5367 (1998).
[CrossRef]

E. R. Dowski, Jr. and W. T. Cathey, "Extended depth of field through wave-front coding," Appl. Opt. 34, 1859-1866 (1995).
[CrossRef] [PubMed]

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

J. Ojeda-Castañeda, E. Tepichin, and A. Diaz, "Arbitrary high focal depth with a quasioptimum real and positive transmittance apodizer," Appl. Opt. 28, 2666-2670 (1989).
[CrossRef] [PubMed]

M. Mino and Y. Okano, "Improvement in the optical transfer function of a defocused optical system through the use of shaded apertures," Appl. Opt. 10, 2219-2225 (1971).
[CrossRef] [PubMed]

A. Flores, M. R. Wang, and J. J. Yang, "Achromatic hybrid refractive-diffractive lens with extended depth of focus," Appl. Opt. 43, 5618-5630 (2004).
[CrossRef] [PubMed]

M. A. Golub, V. Shurman, and I. Grossinger, "Extended focus diffractive optical element for Gaussian laser beams," Appl. Opt. 45, 144-150 (2006).
[CrossRef] [PubMed]

G.-m. Dai, "Optical surface optimization for the correction of presbyopia," Appl. Opt. 45, 4184-4195 (2006).
[CrossRef] [PubMed]

J. Mod. Opt. (2)

S. Bara, C. Frere, Z. Jaroszewicz, A. Kołodziejczyk, and D. Leseberg, "Modulated on-axis circular zone plates for a generation of three-dimensional focal curves," J. Mod. Opt. 37, 1287-1295 (1990).

A. Kołodziejczyk, S. Bara, Z. Jaroszewicz, and M. Sypek, "The light sword optical element - a new diffraction structure with extended depth of focus," J. Mod. Opt. 37, 1283-1286 (1990).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Commun. (1)

M. Sypek, "Light propagation in the Fresnel region. New numerical approach," Opt. Commun. 116, 43-48 (1995).
[CrossRef]

Opt. Eng. (1)

G. Mikuła, A. Kolodziejczyk, M. Makowski, C. Prokopowicz, and M. Sypek, "Diffractive elements for imaging with extended depth of focus," Opt. Eng. 44, 058001(2005).
[CrossRef]

Opt. Express (3)

Opt. Lett. (4)

Optom. Vis. Sci. (1)

J. Ares, R. Flores, S. Bara, and Z. Jaroszewicz, "Presbyopia compensation with a quartic axicon," Optom. Vis. Sci. 82, 1071-1078 (2005).
[CrossRef] [PubMed]

Other (2)

Z. Liu, A. Flores, M. R. Wang, and J. J. Yang, "Diffractive infrared lens with extended depth of focus," Opt. Eng.  46, 018002 (1-9) (2007).
[CrossRef]

H. H. Emsley, Visual Optics (London: Hatton Press, 1952).

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Figures (5)

Fig. 1.
Fig. 1.

Geometry of focusing by the LSOE. The infinitesimal angular sector of the element focuses an incident plane wave into a segment PP1 oriented perpendicularly to the sector.

Fig. 2.
Fig. 2.

The form of input object used in numerical simulations and experiments. Each letter E has the same angular dimension 5 minutes of arc. Singular strips of the letter have an angular width 1 minute of arc.

Fig. 3.
Fig. 3.

Intensity distributions of the output images formed by the AXL, FLA and LSOE for different object distances p given in milimeters. The columns AXL-s, FLA-s and LSOE-s includes results of numerical simulations. The remaining distributions correspond to experimental verifications.

Fig. 4.
Fig. 4.

MTFs calculated for the AXL and the FLA for different object distances p given in millimeters.

Fig. 5.
Fig. 5.

MTFs calculated for the LSOE and different object distances p given in millimeters. The column LSOE H corresponds to a direction νx in the spatial frequencies domain and the column LSOE V to a direction νy.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

Φ ( r ) = k 2 a ln ( 1 + ar 2 f 1 ) ,
Φ ( r ) = kr 2 2 [ f 1 + ( Δ fr 2 R 2 ) ] .
Φ ( r ) = kr 2 2 [ f 1 + ( Δ f θ 2 π ) ] ,
ρ = r Δ f ( 4 π f 1 + 2 Δ f θ ) , φ = θ + π 2 .
L = R Δ f ( 4 π f 1 + 2 Δ f θ ) ,

Metrics