Abstract

As with a conventional lens, a Fresnel zone lens (FZL) can be used to image objects at infinity or nearby. In the latter case, the FZL converts a diverging spherical wavefront into a converging spherical wavefront. The glass substrate on which the FZL is fabricated introduces spherical aberration resulting in a shift of the image plane and blurring of the image. Two novel schemes for correction of this spherical aberration are proposed and studied in this paper. To demonstrate them, FZLs are designed with and without aberration correction. They are fabricated using electron beam direct writing. The devices are evaluated and the accuracy of the proposed aberration correction schemes is validated.

© 2013 Optical Society of America

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References

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  1. A. V. Baez, “Fresnel zone plate for optical image formation using extreme ultraviolet and soft x radiation,” J. Opt. Soc. Am. 51, 405–412 (1961).
    [CrossRef]
  2. H. H. Barret, “Fresnel zone plate imaging in nuclear medicine,” J. Nucl. Med. 13, 382–385 (1972).
  3. E. Lorenzo and A. Luque, “Fresnel lens analysis for solar energy applications,” Appl. Opt. 20, 2941–2945 (1981).
    [CrossRef]
  4. M. Ferstl and A. M. Frisch, “Static and dynamic Fresnel zone lenses for optical interconnection,” J. Mod. Opt. 43, 1451–1462 (1996).
    [CrossRef]
  5. E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
    [CrossRef]
  6. E. Schonbrun, C. Rinzler, and K. B. Crozier, “Microfabricated water immersion zone plate optical tweezer,” Appl. Phys. Lett. 92, 071112 (2008).
    [CrossRef]
  7. E. Schonbrun and K. B. Crozier, “Spring constant modulation in a zone plate tweezer using linear polarization,” Opt. Lett. 33, 2017–2019 (2008).
    [CrossRef]
  8. T. Fujita, H. Nishihara, and J. Koyama, “Blazed gratings and Fresnel lenses fabricated by electron-beam lithography,” Opt. Lett. 7, 578–580 (1982).
    [CrossRef]
  9. K. Kodate, T. Kamiya, Y. Okada, and H. Takenaka, “Focusing characteristics of high efficiency Fresnel zone plate fabricated by deep ultraviolet lithography,” Jpn. J. Appl. Phys. 25, 223–227 (1986).
    [CrossRef]
  10. T. Shiono, M. Kitagawa, K. Setsune, and T. Mitsuyu, “Reflection micro-Fresnel lenses and their use in an integrated focus sensor,” Appl. Opt. 28, 3434–3442 (1989).
    [CrossRef]
  11. T. Shiono and K. Setsune, “Blazed reflection micro-Fresnel lenses fabricated by electron-beam direct writing and dry development,” Opt. Lett. 15, 84–86 (1990).
    [CrossRef]
  12. A. Vijayakumar and S. Bhattacharya, “Analysis of versatile phase zone plates,” Proc. SPIE 8173, 817316 (2010).
    [CrossRef]
  13. B. Zhang and D. Zhao, “Focusing properties of Fresnel zone lens with spiral phase,” Opt. Express 18, 12818–12823 (2010).
    [CrossRef]
  14. A. Vijayakumar, M. Uemukai, and T. Suhara, “Phase-shifted Fresnel zone lenses for photomixing generation of coherent THz wave,” Jpn. J. Appl. Phys. 51, 070206 (2012).
    [CrossRef]
  15. A. Vijayakumar and S. Bhattacharya, “Design, fabrication, and evaluation of a multilevel spiral-phase Fresnel zone plate for optical trapping: erratum,” Appl. Opt. 52, 1148 (2013).
    [CrossRef]
  16. H. Itoh, N. Matsumoto, and T. Inoue, “Spherical aberration correction suitable for a wavefront controller,” Opt. Express 17, 14367–14373 (2009).
    [CrossRef]
  17. D. Iwaniuk, P. Rastogi, and E. Hack, “Correcting spherical aberrations induced by an unknown medium through determination of its refractive index and thickness,” Opt. Express 19, 19407–19414 (2011).
    [CrossRef]
  18. M. J. Booth and T. Wilson, “Strategies for the compensation of specimen-induced spherical aberration in confocal microscopy of skin,” J. Microsc. 200(Pt 1) 68–74 (2000).
    [CrossRef]
  19. M. J. Booth, M. Schwertner, T. Wilson, M. Nakano, Y. Kawata, M. Nakabayashi, and S. Miyata, “Predictive aberration correction for multilayer optical data storage,” Appl. Phys. Lett. 88, 031109 (2006).
    [CrossRef]
  20. B. C. Kress and P. Meyrueis, Applied Digital Optics (Wiley, 2009).
  21. M. J. Simpson and A. G. Michette, “Imaging properties of modified Fresnel zone plates,” Opt. Acta 31, 403–413 (1984).
    [CrossRef]
  22. J. Melngailis, “Focused ion beam technology and applications,” J. Vac. Sci. Technol. B 5, 469–495 (1987).
    [CrossRef]

2013 (1)

2012 (1)

A. Vijayakumar, M. Uemukai, and T. Suhara, “Phase-shifted Fresnel zone lenses for photomixing generation of coherent THz wave,” Jpn. J. Appl. Phys. 51, 070206 (2012).
[CrossRef]

2011 (1)

2010 (2)

B. Zhang and D. Zhao, “Focusing properties of Fresnel zone lens with spiral phase,” Opt. Express 18, 12818–12823 (2010).
[CrossRef]

A. Vijayakumar and S. Bhattacharya, “Analysis of versatile phase zone plates,” Proc. SPIE 8173, 817316 (2010).
[CrossRef]

2009 (1)

2008 (2)

E. Schonbrun and K. B. Crozier, “Spring constant modulation in a zone plate tweezer using linear polarization,” Opt. Lett. 33, 2017–2019 (2008).
[CrossRef]

E. Schonbrun, C. Rinzler, and K. B. Crozier, “Microfabricated water immersion zone plate optical tweezer,” Appl. Phys. Lett. 92, 071112 (2008).
[CrossRef]

2006 (1)

M. J. Booth, M. Schwertner, T. Wilson, M. Nakano, Y. Kawata, M. Nakabayashi, and S. Miyata, “Predictive aberration correction for multilayer optical data storage,” Appl. Phys. Lett. 88, 031109 (2006).
[CrossRef]

2000 (2)

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[CrossRef]

M. J. Booth and T. Wilson, “Strategies for the compensation of specimen-induced spherical aberration in confocal microscopy of skin,” J. Microsc. 200(Pt 1) 68–74 (2000).
[CrossRef]

1996 (1)

M. Ferstl and A. M. Frisch, “Static and dynamic Fresnel zone lenses for optical interconnection,” J. Mod. Opt. 43, 1451–1462 (1996).
[CrossRef]

1990 (1)

1989 (1)

1987 (1)

J. Melngailis, “Focused ion beam technology and applications,” J. Vac. Sci. Technol. B 5, 469–495 (1987).
[CrossRef]

1986 (1)

K. Kodate, T. Kamiya, Y. Okada, and H. Takenaka, “Focusing characteristics of high efficiency Fresnel zone plate fabricated by deep ultraviolet lithography,” Jpn. J. Appl. Phys. 25, 223–227 (1986).
[CrossRef]

1984 (1)

M. J. Simpson and A. G. Michette, “Imaging properties of modified Fresnel zone plates,” Opt. Acta 31, 403–413 (1984).
[CrossRef]

1982 (1)

1981 (1)

1972 (1)

H. H. Barret, “Fresnel zone plate imaging in nuclear medicine,” J. Nucl. Med. 13, 382–385 (1972).

1961 (1)

Anderson, E. H.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[CrossRef]

Attwood, D.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[CrossRef]

Baez, A. V.

Barret, H. H.

H. H. Barret, “Fresnel zone plate imaging in nuclear medicine,” J. Nucl. Med. 13, 382–385 (1972).

Bhattacharya, S.

Booth, M. J.

M. J. Booth, M. Schwertner, T. Wilson, M. Nakano, Y. Kawata, M. Nakabayashi, and S. Miyata, “Predictive aberration correction for multilayer optical data storage,” Appl. Phys. Lett. 88, 031109 (2006).
[CrossRef]

M. J. Booth and T. Wilson, “Strategies for the compensation of specimen-induced spherical aberration in confocal microscopy of skin,” J. Microsc. 200(Pt 1) 68–74 (2000).
[CrossRef]

Chao, W.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[CrossRef]

Crozier, K. B.

E. Schonbrun and K. B. Crozier, “Spring constant modulation in a zone plate tweezer using linear polarization,” Opt. Lett. 33, 2017–2019 (2008).
[CrossRef]

E. Schonbrun, C. Rinzler, and K. B. Crozier, “Microfabricated water immersion zone plate optical tweezer,” Appl. Phys. Lett. 92, 071112 (2008).
[CrossRef]

Denbeaux, G.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[CrossRef]

Ferstl, M.

M. Ferstl and A. M. Frisch, “Static and dynamic Fresnel zone lenses for optical interconnection,” J. Mod. Opt. 43, 1451–1462 (1996).
[CrossRef]

Frisch, A. M.

M. Ferstl and A. M. Frisch, “Static and dynamic Fresnel zone lenses for optical interconnection,” J. Mod. Opt. 43, 1451–1462 (1996).
[CrossRef]

Fujita, T.

Hack, E.

Harteneck, B.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[CrossRef]

Inoue, T.

Itoh, H.

Iwaniuk, D.

Johnson, L.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[CrossRef]

Kamiya, T.

K. Kodate, T. Kamiya, Y. Okada, and H. Takenaka, “Focusing characteristics of high efficiency Fresnel zone plate fabricated by deep ultraviolet lithography,” Jpn. J. Appl. Phys. 25, 223–227 (1986).
[CrossRef]

Kawata, Y.

M. J. Booth, M. Schwertner, T. Wilson, M. Nakano, Y. Kawata, M. Nakabayashi, and S. Miyata, “Predictive aberration correction for multilayer optical data storage,” Appl. Phys. Lett. 88, 031109 (2006).
[CrossRef]

Kitagawa, M.

Kodate, K.

K. Kodate, T. Kamiya, Y. Okada, and H. Takenaka, “Focusing characteristics of high efficiency Fresnel zone plate fabricated by deep ultraviolet lithography,” Jpn. J. Appl. Phys. 25, 223–227 (1986).
[CrossRef]

Koyama, J.

Kress, B. C.

B. C. Kress and P. Meyrueis, Applied Digital Optics (Wiley, 2009).

Lorenzo, E.

Lucero, A.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[CrossRef]

Luque, A.

Matsumoto, N.

Melngailis, J.

J. Melngailis, “Focused ion beam technology and applications,” J. Vac. Sci. Technol. B 5, 469–495 (1987).
[CrossRef]

Meyrueis, P.

B. C. Kress and P. Meyrueis, Applied Digital Optics (Wiley, 2009).

Michette, A. G.

M. J. Simpson and A. G. Michette, “Imaging properties of modified Fresnel zone plates,” Opt. Acta 31, 403–413 (1984).
[CrossRef]

Mitsuyu, T.

Miyata, S.

M. J. Booth, M. Schwertner, T. Wilson, M. Nakano, Y. Kawata, M. Nakabayashi, and S. Miyata, “Predictive aberration correction for multilayer optical data storage,” Appl. Phys. Lett. 88, 031109 (2006).
[CrossRef]

Nakabayashi, M.

M. J. Booth, M. Schwertner, T. Wilson, M. Nakano, Y. Kawata, M. Nakabayashi, and S. Miyata, “Predictive aberration correction for multilayer optical data storage,” Appl. Phys. Lett. 88, 031109 (2006).
[CrossRef]

Nakano, M.

M. J. Booth, M. Schwertner, T. Wilson, M. Nakano, Y. Kawata, M. Nakabayashi, and S. Miyata, “Predictive aberration correction for multilayer optical data storage,” Appl. Phys. Lett. 88, 031109 (2006).
[CrossRef]

Nishihara, H.

Okada, Y.

K. Kodate, T. Kamiya, Y. Okada, and H. Takenaka, “Focusing characteristics of high efficiency Fresnel zone plate fabricated by deep ultraviolet lithography,” Jpn. J. Appl. Phys. 25, 223–227 (1986).
[CrossRef]

Olynick, D. L.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[CrossRef]

Rastogi, P.

Rinzler, C.

E. Schonbrun, C. Rinzler, and K. B. Crozier, “Microfabricated water immersion zone plate optical tweezer,” Appl. Phys. Lett. 92, 071112 (2008).
[CrossRef]

Schonbrun, E.

E. Schonbrun, C. Rinzler, and K. B. Crozier, “Microfabricated water immersion zone plate optical tweezer,” Appl. Phys. Lett. 92, 071112 (2008).
[CrossRef]

E. Schonbrun and K. B. Crozier, “Spring constant modulation in a zone plate tweezer using linear polarization,” Opt. Lett. 33, 2017–2019 (2008).
[CrossRef]

Schwertner, M.

M. J. Booth, M. Schwertner, T. Wilson, M. Nakano, Y. Kawata, M. Nakabayashi, and S. Miyata, “Predictive aberration correction for multilayer optical data storage,” Appl. Phys. Lett. 88, 031109 (2006).
[CrossRef]

Setsune, K.

Shiono, T.

Simpson, M. J.

M. J. Simpson and A. G. Michette, “Imaging properties of modified Fresnel zone plates,” Opt. Acta 31, 403–413 (1984).
[CrossRef]

Suhara, T.

A. Vijayakumar, M. Uemukai, and T. Suhara, “Phase-shifted Fresnel zone lenses for photomixing generation of coherent THz wave,” Jpn. J. Appl. Phys. 51, 070206 (2012).
[CrossRef]

Takenaka, H.

K. Kodate, T. Kamiya, Y. Okada, and H. Takenaka, “Focusing characteristics of high efficiency Fresnel zone plate fabricated by deep ultraviolet lithography,” Jpn. J. Appl. Phys. 25, 223–227 (1986).
[CrossRef]

Uemukai, M.

A. Vijayakumar, M. Uemukai, and T. Suhara, “Phase-shifted Fresnel zone lenses for photomixing generation of coherent THz wave,” Jpn. J. Appl. Phys. 51, 070206 (2012).
[CrossRef]

Veklerov, E.

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[CrossRef]

Vijayakumar, A.

A. Vijayakumar and S. Bhattacharya, “Design, fabrication, and evaluation of a multilevel spiral-phase Fresnel zone plate for optical trapping: erratum,” Appl. Opt. 52, 1148 (2013).
[CrossRef]

A. Vijayakumar, M. Uemukai, and T. Suhara, “Phase-shifted Fresnel zone lenses for photomixing generation of coherent THz wave,” Jpn. J. Appl. Phys. 51, 070206 (2012).
[CrossRef]

A. Vijayakumar and S. Bhattacharya, “Analysis of versatile phase zone plates,” Proc. SPIE 8173, 817316 (2010).
[CrossRef]

Wilson, T.

M. J. Booth, M. Schwertner, T. Wilson, M. Nakano, Y. Kawata, M. Nakabayashi, and S. Miyata, “Predictive aberration correction for multilayer optical data storage,” Appl. Phys. Lett. 88, 031109 (2006).
[CrossRef]

M. J. Booth and T. Wilson, “Strategies for the compensation of specimen-induced spherical aberration in confocal microscopy of skin,” J. Microsc. 200(Pt 1) 68–74 (2000).
[CrossRef]

Zhang, B.

Zhao, D.

Appl. Opt. (3)

Appl. Phys. Lett. (2)

M. J. Booth, M. Schwertner, T. Wilson, M. Nakano, Y. Kawata, M. Nakabayashi, and S. Miyata, “Predictive aberration correction for multilayer optical data storage,” Appl. Phys. Lett. 88, 031109 (2006).
[CrossRef]

E. Schonbrun, C. Rinzler, and K. B. Crozier, “Microfabricated water immersion zone plate optical tweezer,” Appl. Phys. Lett. 92, 071112 (2008).
[CrossRef]

J. Microsc. (1)

M. J. Booth and T. Wilson, “Strategies for the compensation of specimen-induced spherical aberration in confocal microscopy of skin,” J. Microsc. 200(Pt 1) 68–74 (2000).
[CrossRef]

J. Mod. Opt. (1)

M. Ferstl and A. M. Frisch, “Static and dynamic Fresnel zone lenses for optical interconnection,” J. Mod. Opt. 43, 1451–1462 (1996).
[CrossRef]

J. Nucl. Med. (1)

H. H. Barret, “Fresnel zone plate imaging in nuclear medicine,” J. Nucl. Med. 13, 382–385 (1972).

J. Opt. Soc. Am. (1)

J. Vac. Sci. Technol. B (2)

J. Melngailis, “Focused ion beam technology and applications,” J. Vac. Sci. Technol. B 5, 469–495 (1987).
[CrossRef]

E. H. Anderson, D. L. Olynick, B. Harteneck, E. Veklerov, G. Denbeaux, W. Chao, A. Lucero, L. Johnson, and D. Attwood, “Nanofabrication and diffractive optics for high-resolution x-ray applications,” J. Vac. Sci. Technol. B 18, 2970–2975 (2000).
[CrossRef]

Jpn. J. Appl. Phys. (2)

A. Vijayakumar, M. Uemukai, and T. Suhara, “Phase-shifted Fresnel zone lenses for photomixing generation of coherent THz wave,” Jpn. J. Appl. Phys. 51, 070206 (2012).
[CrossRef]

K. Kodate, T. Kamiya, Y. Okada, and H. Takenaka, “Focusing characteristics of high efficiency Fresnel zone plate fabricated by deep ultraviolet lithography,” Jpn. J. Appl. Phys. 25, 223–227 (1986).
[CrossRef]

Opt. Acta (1)

M. J. Simpson and A. G. Michette, “Imaging properties of modified Fresnel zone plates,” Opt. Acta 31, 403–413 (1984).
[CrossRef]

Opt. Express (3)

Opt. Lett. (3)

Proc. SPIE (1)

A. Vijayakumar and S. Bhattacharya, “Analysis of versatile phase zone plates,” Proc. SPIE 8173, 817316 (2010).
[CrossRef]

Other (1)

B. C. Kress and P. Meyrueis, Applied Digital Optics (Wiley, 2009).

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Figures (14)

Fig. 1.
Fig. 1.

uv configuration for converting a diverging spherical wavefront into a converging spherical wavefront using an FZL.

Fig. 2.
Fig. 2.

Plot of radii of FZL as a function of the 2π period zone number for f-configuration (dashed line) and uv configuration (solid line) for case 1 (u=1mm, v=5mm, and f=0.83mm) (bottom) and for case 2 (u=5mm, v=30mm, and f=4.3mm) (top).

Fig. 3.
Fig. 3.

uv configuration for an FZL fabricated on a glass substrate of thickness t.

Fig. 4.
Fig. 4.

Plot of shape of wavefront (a) just before light enters the glass substrate (dotted line), (b) at the FZL plane in the absence of glass substrate (dashed line), (c) at the FZL plane in the presence of glass substrate (solid line), and (d) plot of the aberration function U (dotted–dashed line).

Fig. 5.
Fig. 5.

(a) Plot of the position of virtual source for different radial distances. (b) Ray tracing of the rays emanating from the real source (dashed line) and virtual source (solid line) generated due to the glass substrate.

Fig. 6.
Fig. 6.

(a) Plot of the position of image for different radial distances. (b) Ray tracing of the rays from FZL plane without glass plate (dashed line) and with glass plate (solid line).

Fig. 7.
Fig. 7.

Radii of FZL calculated with (solid line) and without (dashed line) aberration correction for glass substrate.

Fig. 8.
Fig. 8.

Ray tracing of rays emanating from the FZL plane to the image plane for DOE1.

Fig. 9.
Fig. 9.

Phase aberration function plotted as a function of radial distance for radial magnifications Mρ=0.9 (dotted), 0.95 (dashed), 1.05 (dashed–dotted), and 1.1 (solid line). Mρ=1 has zero phase aberration ρ=0 line.

Fig. 10.
Fig. 10.

Phase of wave plotted as a function of radial distance for radial magnifications Mρ=0.9 (dotted), 0.95 (dashed), 1 (thick line), 1.05 (dashed–dotted), and 1.1 (solid line).

Fig. 11.
Fig. 11.

Ray tracing of rays emanating from the FZL plane to the image plane for DOE2.

Fig. 12.
Fig. 12.

(a) Optical microscope image of the entire 2 mm device without aberration correction. (b) Optical microscope image of the central part of the device without aberration correction. (c) Optical microscope image of outermost part of the FZL without aberration correction.

Fig. 13.
Fig. 13.

(a)–(c) are the images of the beam at a distance of 5 mm after the image plane, for FZL without aberration correction, DOE1 and DOE2, respectively; (d)–(f) are the images of the beam at the image plane, at the image plane for FZL without aberration correction, DOE1 and DOE2, respectively; (g)–(i) are the images of the beam at a distance of 5 mm before the image plane for FZL without aberration correction, DOE1 and DOE2, respectively.

Fig. 14.
Fig. 14.

Intensity profile of Figs. 13(d)13(f) at the focal plane for the FZL without aberration correction (dotted line), DOE1 (solid line), and DOE2 (dashed line).

Equations (24)

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

OPD=(un+vn)(u+v)=nλ,
un=(u2+ρn2)1/2,
vn=(v2+ρn2)1/2.
ρn=[Cρ24u2v24(u2+v2+Cρ)]1/2,
ΦBFZL(ρ)={Φ1ρn(x2+y2)1/2ρn+120elsewhere,n=0,1,2,3,p.
1f=1u+1v.
ρfn=(n2λ2+2nfλ)1/2.
tanθ1n=ρ1nut=ρ2nu.
nasinθ1n=ngsinθ2n.
ρ2n=ρ1n+ttanθ2n.
ρ2n=ρ2n[utu]+ttan[sin1(nang(1+u2/ρ2n2)1/2)].
Δρ=ρ2nρ2n.
u(ρ2n)=ρ2nρ2nu.
Ltθ1n0u(ρ2n)=ut+tnang.
U(ρ2n)=|uu(ρ2n)+(u2+ρ2n2)1/2(u2+ρ2n2)1/2|.
v=B±(B24AC)1/22A,
A=4n2λ2+8unλ4ρn2,B=12un2λ2+4n3λ3+8u2nλ8ρn2u8ρn2λn,C=n4λ4+4u2n2λ2+4un3λ34ρn2n2λ28ρn2unλ4ρn2u2.
ρn=[Cρ24u2v24(u2+v2+Cρ)]1/2,
ΦFZP=ΦoutΦin=k(ρ2+v2)1/2+k(ρ2+u2)1/2=const+2mπ,
ΦFZP=k(ρ2+v2)1/2+k(ρ2+u2)1/2=k(u+v)+2mπ.
ΦR(x,y)=k(ρ2+v2)1/2.
ΦA=ΦoutΦR=ΦFZP+ΦinΦR=k[(ρMρ)2+v2]1/2+k[(ρMρ)2+u2]1/2k(ρ2+u2)1/2k(ρ2+v2)1/2,
2w(z)=2w0[1+(zλπw02)2]1/2.
2wi=[(2Mwo)2+(4λvDπ)2]1/2,

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