Abstract

We propose a method for designing multifocal diffractive lenses generating prescribed sets of foci with fixed positions at several different wavelengths. The method is based on minimizing the difference between the complex amplitudes of the beams generated by the lens microrelief at the design wavelengths, and the functions of the complex transmission of multifocal lenses calculated for these wavelengths. As an example, a zone plate generating three fixed foci at three different wavelengths was designed, fabricated, and experimentally investigated. The proof-of-concept experimental results confirm the formation of foci with fixed positions at the design wavelengths. The obtained results may find applications in the design and fabrication of novel multifocal contact and intraocular lenses with reduced chromatic effects.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (1)

2016 (2)

U. Hinze, A. El-Tamer, L. L. Doskolovich, E. A. Bezus, S. Reiß, H. Stolz, R. F. Guthoff, O. Stachs, and B. Chichkov, “Additive manufacturing of a trifocal diffractive-refractive lens,” Opt. Commun. 372, 235–240 (2016).

S. I. Kharitonov, S. G. Volotovsky, and S. N. Khonina, “Geometic-optical calculation of the focal spot of a harmonic diffractive lens,” Comput. Opt. 40(3), 331–337 (2016).

2015 (2)

V. Osipov, L. L. Doskolovich, E. A. Bezus, T. Drew, K. Zhou, K. Sawalha, G. Swadener, and J. S. W. Wolffsohn, “Application of nanoimprinting technique for fabrication of trifocal diffractive lens with sine-like radial profile,” J. Biomed. Opt. 20(2), 25008 (2015).
[PubMed]

T. Kohnen, “First implantation of a diffractive quadrafocal (trifocal) intraocular lens,” J. Cataract Refract. Surg. 41(10), 2330–2332 (2015).
[PubMed]

2014 (1)

S. Ravikumar, A. Bradley, and L. N. Thibos, “Chromatic aberration and polychromatic image quality with diffractive multifocal intraocular lenses,” J. Cataract Refract. Surg. 40(7), 1192–1204 (2014).
[PubMed]

2011 (1)

D. Gatinel, C. Pagnoulle, Y. Houbrechts, and L. Gobin, “Design and qualification of a diffractive trifocal optical profile for intraocular lenses,” J. Cataract Refract. Surg. 37(11), 2060–2067 (2011).
[PubMed]

2006 (1)

S. S. Lane, M. Morris, L. Nordan, M. Packer, N. Tarantino, and R. B. Wallace, “Multifocal intraocular lenses,” Ophthalmol. Clin. North Am. 19(1), 89–105 (2006).
[PubMed]

2005 (1)

2004 (1)

M. A. Golub, “Laser beam splitting by diffractive optics,” Opt. Photonics News 15(2), 36–41 (2004).

2002 (1)

V. M. Gurenko, L. B. Kastorsky, V. P. Kiryanov, A. V. Kiryanov, S. A. Kokarev, V. M. Vedernikov, and A. G. Verkhogliad, “Laser writing system CLWS-300/C-M for microstructure synthesis on axisymmetric 3D surfaces,” Proc. SPIE 4900, 320–325 (2002).

2000 (1)

L. L. Doskolovich and P. M. Repetto, “Design of DOEs for wavelength demultiplexing and spatial focusing,” J. Opt. A, Pure Appl. Opt. 2, 488–493 (2000).

1998 (1)

1997 (1)

1996 (1)

1995 (1)

1992 (1)

M. A. Golub, L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, and V. A. Soifer, “Computer generated diffractive multi-focal lens,” J. Mod. Opt. 39(6), 1245–1251 (1992).

1978 (1)

Akondi, V.

Bengtsson, J.

Bezus, E. A.

U. Hinze, A. El-Tamer, L. L. Doskolovich, E. A. Bezus, S. Reiß, H. Stolz, R. F. Guthoff, O. Stachs, and B. Chichkov, “Additive manufacturing of a trifocal diffractive-refractive lens,” Opt. Commun. 372, 235–240 (2016).

V. Osipov, L. L. Doskolovich, E. A. Bezus, T. Drew, K. Zhou, K. Sawalha, G. Swadener, and J. S. W. Wolffsohn, “Application of nanoimprinting technique for fabrication of trifocal diffractive lens with sine-like radial profile,” J. Biomed. Opt. 20(2), 25008 (2015).
[PubMed]

Bradley, A.

S. Ravikumar, A. Bradley, and L. N. Thibos, “Chromatic aberration and polychromatic image quality with diffractive multifocal intraocular lenses,” J. Cataract Refract. Surg. 40(7), 1192–1204 (2014).
[PubMed]

Cagigal, M.

Canales, V.

Chen, M.-L.

Chen, Y.-S.

Chichkov, B.

U. Hinze, A. El-Tamer, L. L. Doskolovich, E. A. Bezus, S. Reiß, H. Stolz, R. F. Guthoff, O. Stachs, and B. Chichkov, “Additive manufacturing of a trifocal diffractive-refractive lens,” Opt. Commun. 372, 235–240 (2016).

Cui, X.-M.

Dammann, H.

Dong, B.-Z.

Dorronsoro, C.

Doskolovich, L. L.

U. Hinze, A. El-Tamer, L. L. Doskolovich, E. A. Bezus, S. Reiß, H. Stolz, R. F. Guthoff, O. Stachs, and B. Chichkov, “Additive manufacturing of a trifocal diffractive-refractive lens,” Opt. Commun. 372, 235–240 (2016).

V. Osipov, L. L. Doskolovich, E. A. Bezus, T. Drew, K. Zhou, K. Sawalha, G. Swadener, and J. S. W. Wolffsohn, “Application of nanoimprinting technique for fabrication of trifocal diffractive lens with sine-like radial profile,” J. Biomed. Opt. 20(2), 25008 (2015).
[PubMed]

L. L. Doskolovich and P. M. Repetto, “Design of DOEs for wavelength demultiplexing and spatial focusing,” J. Opt. A, Pure Appl. Opt. 2, 488–493 (2000).

M. A. Golub, L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, and V. A. Soifer, “Computer generated diffractive multi-focal lens,” J. Mod. Opt. 39(6), 1245–1251 (1992).

Drew, T.

V. Osipov, L. L. Doskolovich, E. A. Bezus, T. Drew, K. Zhou, K. Sawalha, G. Swadener, and J. S. W. Wolffsohn, “Application of nanoimprinting technique for fabrication of trifocal diffractive lens with sine-like radial profile,” J. Biomed. Opt. 20(2), 25008 (2015).
[PubMed]

El-Tamer, A.

U. Hinze, A. El-Tamer, L. L. Doskolovich, E. A. Bezus, S. Reiß, H. Stolz, R. F. Guthoff, O. Stachs, and B. Chichkov, “Additive manufacturing of a trifocal diffractive-refractive lens,” Opt. Commun. 372, 235–240 (2016).

Gambra, E.

Gatinel, D.

D. Gatinel, C. Pagnoulle, Y. Houbrechts, and L. Gobin, “Design and qualification of a diffractive trifocal optical profile for intraocular lenses,” J. Cataract Refract. Surg. 37(11), 2060–2067 (2011).
[PubMed]

Gobin, L.

D. Gatinel, C. Pagnoulle, Y. Houbrechts, and L. Gobin, “Design and qualification of a diffractive trifocal optical profile for intraocular lenses,” J. Cataract Refract. Surg. 37(11), 2060–2067 (2011).
[PubMed]

Golub, M. A.

M. A. Golub, “Laser beam splitting by diffractive optics,” Opt. Photonics News 15(2), 36–41 (2004).

M. A. Golub, L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, and V. A. Soifer, “Computer generated diffractive multi-focal lens,” J. Mod. Opt. 39(6), 1245–1251 (1992).

Gu, B.-Y.

Gurenko, V. M.

V. M. Gurenko, L. B. Kastorsky, V. P. Kiryanov, A. V. Kiryanov, S. A. Kokarev, V. M. Vedernikov, and A. G. Verkhogliad, “Laser writing system CLWS-300/C-M for microstructure synthesis on axisymmetric 3D surfaces,” Proc. SPIE 4900, 320–325 (2002).

Guthoff, R. F.

U. Hinze, A. El-Tamer, L. L. Doskolovich, E. A. Bezus, S. Reiß, H. Stolz, R. F. Guthoff, O. Stachs, and B. Chichkov, “Additive manufacturing of a trifocal diffractive-refractive lens,” Opt. Commun. 372, 235–240 (2016).

Hinze, U.

U. Hinze, A. El-Tamer, L. L. Doskolovich, E. A. Bezus, S. Reiß, H. Stolz, R. F. Guthoff, O. Stachs, and B. Chichkov, “Additive manufacturing of a trifocal diffractive-refractive lens,” Opt. Commun. 372, 235–240 (2016).

Houbrechts, Y.

D. Gatinel, C. Pagnoulle, Y. Houbrechts, and L. Gobin, “Design and qualification of a diffractive trifocal optical profile for intraocular lenses,” J. Cataract Refract. Surg. 37(11), 2060–2067 (2011).
[PubMed]

Kastorsky, L. B.

V. M. Gurenko, L. B. Kastorsky, V. P. Kiryanov, A. V. Kiryanov, S. A. Kokarev, V. M. Vedernikov, and A. G. Verkhogliad, “Laser writing system CLWS-300/C-M for microstructure synthesis on axisymmetric 3D surfaces,” Proc. SPIE 4900, 320–325 (2002).

Kazanskiy, N. L.

M. A. Golub, L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, and V. A. Soifer, “Computer generated diffractive multi-focal lens,” J. Mod. Opt. 39(6), 1245–1251 (1992).

Kharitonov, S. I.

S. I. Kharitonov, S. G. Volotovsky, and S. N. Khonina, “Geometic-optical calculation of the focal spot of a harmonic diffractive lens,” Comput. Opt. 40(3), 331–337 (2016).

M. A. Golub, L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, and V. A. Soifer, “Computer generated diffractive multi-focal lens,” J. Mod. Opt. 39(6), 1245–1251 (1992).

Khonina, S. N.

S. I. Kharitonov, S. G. Volotovsky, and S. N. Khonina, “Geometic-optical calculation of the focal spot of a harmonic diffractive lens,” Comput. Opt. 40(3), 331–337 (2016).

Kiryanov, A. V.

V. M. Gurenko, L. B. Kastorsky, V. P. Kiryanov, A. V. Kiryanov, S. A. Kokarev, V. M. Vedernikov, and A. G. Verkhogliad, “Laser writing system CLWS-300/C-M for microstructure synthesis on axisymmetric 3D surfaces,” Proc. SPIE 4900, 320–325 (2002).

Kiryanov, V. P.

V. M. Gurenko, L. B. Kastorsky, V. P. Kiryanov, A. V. Kiryanov, S. A. Kokarev, V. M. Vedernikov, and A. G. Verkhogliad, “Laser writing system CLWS-300/C-M for microstructure synthesis on axisymmetric 3D surfaces,” Proc. SPIE 4900, 320–325 (2002).

Kohnen, T.

T. Kohnen, “First implantation of a diffractive quadrafocal (trifocal) intraocular lens,” J. Cataract Refract. Surg. 41(10), 2330–2332 (2015).
[PubMed]

Kokarev, S. A.

V. M. Gurenko, L. B. Kastorsky, V. P. Kiryanov, A. V. Kiryanov, S. A. Kokarev, V. M. Vedernikov, and A. G. Verkhogliad, “Laser writing system CLWS-300/C-M for microstructure synthesis on axisymmetric 3D surfaces,” Proc. SPIE 4900, 320–325 (2002).

Lane, S. S.

S. S. Lane, M. Morris, L. Nordan, M. Packer, N. Tarantino, and R. B. Wallace, “Multifocal intraocular lenses,” Ophthalmol. Clin. North Am. 19(1), 89–105 (2006).
[PubMed]

Li, D.-H.

Liu, H.-D.

Marcos, S.

Morris, M.

S. S. Lane, M. Morris, L. Nordan, M. Packer, N. Tarantino, and R. B. Wallace, “Multifocal intraocular lenses,” Ophthalmol. Clin. North Am. 19(1), 89–105 (2006).
[PubMed]

Nordan, L.

S. S. Lane, M. Morris, L. Nordan, M. Packer, N. Tarantino, and R. B. Wallace, “Multifocal intraocular lenses,” Ophthalmol. Clin. North Am. 19(1), 89–105 (2006).
[PubMed]

Osipov, V.

V. Osipov, L. L. Doskolovich, E. A. Bezus, T. Drew, K. Zhou, K. Sawalha, G. Swadener, and J. S. W. Wolffsohn, “Application of nanoimprinting technique for fabrication of trifocal diffractive lens with sine-like radial profile,” J. Biomed. Opt. 20(2), 25008 (2015).
[PubMed]

Oti, J.

Packer, M.

S. S. Lane, M. Morris, L. Nordan, M. Packer, N. Tarantino, and R. B. Wallace, “Multifocal intraocular lenses,” Ophthalmol. Clin. North Am. 19(1), 89–105 (2006).
[PubMed]

Pagnoulle, C.

D. Gatinel, C. Pagnoulle, Y. Houbrechts, and L. Gobin, “Design and qualification of a diffractive trifocal optical profile for intraocular lenses,” J. Cataract Refract. Surg. 37(11), 2060–2067 (2011).
[PubMed]

Ravikumar, S.

S. Ravikumar, A. Bradley, and L. N. Thibos, “Chromatic aberration and polychromatic image quality with diffractive multifocal intraocular lenses,” J. Cataract Refract. Surg. 40(7), 1192–1204 (2014).
[PubMed]

Reiß, S.

U. Hinze, A. El-Tamer, L. L. Doskolovich, E. A. Bezus, S. Reiß, H. Stolz, R. F. Guthoff, O. Stachs, and B. Chichkov, “Additive manufacturing of a trifocal diffractive-refractive lens,” Opt. Commun. 372, 235–240 (2016).

Repetto, P. M.

L. L. Doskolovich and P. M. Repetto, “Design of DOEs for wavelength demultiplexing and spatial focusing,” J. Opt. A, Pure Appl. Opt. 2, 488–493 (2000).

Sawalha, K.

V. Osipov, L. L. Doskolovich, E. A. Bezus, T. Drew, K. Zhou, K. Sawalha, G. Swadener, and J. S. W. Wolffsohn, “Application of nanoimprinting technique for fabrication of trifocal diffractive lens with sine-like radial profile,” J. Biomed. Opt. 20(2), 25008 (2015).
[PubMed]

Soifer, V. A.

M. A. Golub, L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, and V. A. Soifer, “Computer generated diffractive multi-focal lens,” J. Mod. Opt. 39(6), 1245–1251 (1992).

Sommargren, G. E.

Stachs, O.

U. Hinze, A. El-Tamer, L. L. Doskolovich, E. A. Bezus, S. Reiß, H. Stolz, R. F. Guthoff, O. Stachs, and B. Chichkov, “Additive manufacturing of a trifocal diffractive-refractive lens,” Opt. Commun. 372, 235–240 (2016).

Stolz, H.

U. Hinze, A. El-Tamer, L. L. Doskolovich, E. A. Bezus, S. Reiß, H. Stolz, R. F. Guthoff, O. Stachs, and B. Chichkov, “Additive manufacturing of a trifocal diffractive-refractive lens,” Opt. Commun. 372, 235–240 (2016).

Swadener, G.

V. Osipov, L. L. Doskolovich, E. A. Bezus, T. Drew, K. Zhou, K. Sawalha, G. Swadener, and J. S. W. Wolffsohn, “Application of nanoimprinting technique for fabrication of trifocal diffractive lens with sine-like radial profile,” J. Biomed. Opt. 20(2), 25008 (2015).
[PubMed]

Sweeney, D. W.

Tarantino, N.

S. S. Lane, M. Morris, L. Nordan, M. Packer, N. Tarantino, and R. B. Wallace, “Multifocal intraocular lenses,” Ophthalmol. Clin. North Am. 19(1), 89–105 (2006).
[PubMed]

Thibos, L. N.

S. Ravikumar, A. Bradley, and L. N. Thibos, “Chromatic aberration and polychromatic image quality with diffractive multifocal intraocular lenses,” J. Cataract Refract. Surg. 40(7), 1192–1204 (2014).
[PubMed]

Valle, P.

Vedernikov, V. M.

V. M. Gurenko, L. B. Kastorsky, V. P. Kiryanov, A. V. Kiryanov, S. A. Kokarev, V. M. Vedernikov, and A. G. Verkhogliad, “Laser writing system CLWS-300/C-M for microstructure synthesis on axisymmetric 3D surfaces,” Proc. SPIE 4900, 320–325 (2002).

Verkhogliad, A. G.

V. M. Gurenko, L. B. Kastorsky, V. P. Kiryanov, A. V. Kiryanov, S. A. Kokarev, V. M. Vedernikov, and A. G. Verkhogliad, “Laser writing system CLWS-300/C-M for microstructure synthesis on axisymmetric 3D surfaces,” Proc. SPIE 4900, 320–325 (2002).

Volotovsky, S. G.

S. I. Kharitonov, S. G. Volotovsky, and S. N. Khonina, “Geometic-optical calculation of the focal spot of a harmonic diffractive lens,” Comput. Opt. 40(3), 331–337 (2016).

Wallace, R. B.

S. S. Lane, M. Morris, L. Nordan, M. Packer, N. Tarantino, and R. B. Wallace, “Multifocal intraocular lenses,” Ophthalmol. Clin. North Am. 19(1), 89–105 (2006).
[PubMed]

Wolffsohn, J. S. W.

V. Osipov, L. L. Doskolovich, E. A. Bezus, T. Drew, K. Zhou, K. Sawalha, G. Swadener, and J. S. W. Wolffsohn, “Application of nanoimprinting technique for fabrication of trifocal diffractive lens with sine-like radial profile,” J. Biomed. Opt. 20(2), 25008 (2015).
[PubMed]

Yang, G.-Z.

Zhang, G.-Q.

Zheng, S.-H.

Zhou, K.

V. Osipov, L. L. Doskolovich, E. A. Bezus, T. Drew, K. Zhou, K. Sawalha, G. Swadener, and J. S. W. Wolffsohn, “Application of nanoimprinting technique for fabrication of trifocal diffractive lens with sine-like radial profile,” J. Biomed. Opt. 20(2), 25008 (2015).
[PubMed]

Appl. Opt. (4)

Biomed. Opt. Express (1)

Comput. Opt. (1)

S. I. Kharitonov, S. G. Volotovsky, and S. N. Khonina, “Geometic-optical calculation of the focal spot of a harmonic diffractive lens,” Comput. Opt. 40(3), 331–337 (2016).

J. Biomed. Opt. (1)

V. Osipov, L. L. Doskolovich, E. A. Bezus, T. Drew, K. Zhou, K. Sawalha, G. Swadener, and J. S. W. Wolffsohn, “Application of nanoimprinting technique for fabrication of trifocal diffractive lens with sine-like radial profile,” J. Biomed. Opt. 20(2), 25008 (2015).
[PubMed]

J. Cataract Refract. Surg. (3)

S. Ravikumar, A. Bradley, and L. N. Thibos, “Chromatic aberration and polychromatic image quality with diffractive multifocal intraocular lenses,” J. Cataract Refract. Surg. 40(7), 1192–1204 (2014).
[PubMed]

D. Gatinel, C. Pagnoulle, Y. Houbrechts, and L. Gobin, “Design and qualification of a diffractive trifocal optical profile for intraocular lenses,” J. Cataract Refract. Surg. 37(11), 2060–2067 (2011).
[PubMed]

T. Kohnen, “First implantation of a diffractive quadrafocal (trifocal) intraocular lens,” J. Cataract Refract. Surg. 41(10), 2330–2332 (2015).
[PubMed]

J. Mod. Opt. (1)

M. A. Golub, L. L. Doskolovich, N. L. Kazanskiy, S. I. Kharitonov, and V. A. Soifer, “Computer generated diffractive multi-focal lens,” J. Mod. Opt. 39(6), 1245–1251 (1992).

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

L. L. Doskolovich and P. M. Repetto, “Design of DOEs for wavelength demultiplexing and spatial focusing,” J. Opt. A, Pure Appl. Opt. 2, 488–493 (2000).

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

Ophthalmol. Clin. North Am. (1)

S. S. Lane, M. Morris, L. Nordan, M. Packer, N. Tarantino, and R. B. Wallace, “Multifocal intraocular lenses,” Ophthalmol. Clin. North Am. 19(1), 89–105 (2006).
[PubMed]

Opt. Commun. (1)

U. Hinze, A. El-Tamer, L. L. Doskolovich, E. A. Bezus, S. Reiß, H. Stolz, R. F. Guthoff, O. Stachs, and B. Chichkov, “Additive manufacturing of a trifocal diffractive-refractive lens,” Opt. Commun. 372, 235–240 (2016).

Opt. Express (1)

Opt. Photonics News (1)

M. A. Golub, “Laser beam splitting by diffractive optics,” Opt. Photonics News 15(2), 36–41 (2004).

Proc. SPIE (1)

V. M. Gurenko, L. B. Kastorsky, V. P. Kiryanov, A. V. Kiryanov, S. A. Kokarev, V. M. Vedernikov, and A. G. Verkhogliad, “Laser writing system CLWS-300/C-M for microstructure synthesis on axisymmetric 3D surfaces,” Proc. SPIE 4900, 320–325 (2002).

Other (2)

V. A. Soifer, V. V. Kotlyar, and L. L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Taylor & Francis, 1997).

V. A. Soifer, L. L. Doskolovich, D. L. Golovashkin, N. L. Kazanskiy, S. I. Kharitonov, S. N. Khonina, V. V. Kotlyar, V. S. Pavelyev, R. V. Skidanov, V. S. Solovyev, G. V. Uspleniev, and A. V. Volkov, Methods for Computer Design of Diffractive Optical Elements (Wiley, 2002).

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Figures (7)

Fig. 1
Fig. 1 Radial profile h s m f ( ρ ) of the designed three-wavelength spectral trifocal zone plate (a) and the half-tone image of the zone plate microrelief used for its fabrication in resist with the laser-writing system CLWS-300 (b).
Fig. 2
Fig. 2 The normalized intensity distributions along the optical axis generated by the spectral trifocal lens for the three design wavelengths 450 nm, 540 nm and 580 nm (a) and by the conventional trifocal lens calculated for one central wavelength 540 nm.
Fig. 3
Fig. 3 (a) Radial profile h s m f ( ρ ) of the four-wavelength spectral zone plate designed for the wavelengths 450 nm, 540 nm, 580 nm and 640 nm. (b) The normalized intensity distributions along the optical axis generated by the four-wavelength spectral zone plate.
Fig. 4
Fig. 4 (a) Measured profile of the fabricated optical element; (b) Comparison of fragments of theoretical (blue) and measured (red) profiles. The green curve corresponds to the convolution of the theoretical profile with a 10 μm-wide rectangular window.
Fig. 5
Fig. 5 Optical setup of the experiment: 1 — laser, 2 — collimator (two lenses and a 20-μm pinhole), 3 — trifocal spectral zone plate, 4 — collecting lens, 5 — CCD camera, 6 — motorized stage.
Fig. 6
Fig. 6 Normalized registered longitudinal intensity distributions generated by the fabricated lens (blue solid lines) for the three design wavelengths 450 nm (a), 540nm (b), and 580 nm (c), and the normalized theoretical distributions calculated for the spectral trifocal lens of Fig. 1(a) (red dashed lines).
Fig. 7
Fig. 7 Normalized registered transverse distributions generated by the fabricated lens at the three design wavelengths 450 nm (a), 540 nm (b), and 580 nm (c) at the three foci. Red circles depict the theoretical diffraction spot sizes.

Equations (15)

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φ m f ( ρ ; λ ) = Φ [ φ d ( ρ ; λ ) ] = Φ [ mod 2 π ( π ρ 2 λ f d ) ] , ρ [ 0 , R ]
F m = f 0 f d f d m f 0 , m = 0 , ± 1 , ± 2 , ...
c m = 1 2 π 0 2 π exp ( i Φ [ ξ ] i m ξ ) d ξ
Φ [ ξ ] = { 0 , ξ [ 0 , π ) , φ , ξ [ π , 2 π ) ,
Φ [ ξ ] = 1.435 sin ( ξ ) .
λ ± 1 = λ 0 N N ± 1 ,
h j = λ 0 n 1 j , j = 0 , N 1 ¯ ,
φ ( h j ; λ ) = 2 π λ ( n 1 ) h j , j = 0 , N 1 ¯ .
φ ( h j ; λ 0 ) = 2 π j , φ ( h j ; λ ± 1 ) = 2 π j ( 1 ± 1 N ) , j = 0 , N 1 ¯ .
φ ( h j ; λ 0 ) = 0 , φ ( h j ; λ ± 1 ) = ± 2 π j N , j = 0 , N 1 ¯ .
F ( h 1 , ... , h N ) = l = 1 1 j = 0 N 1 | exp [ i φ ( h j ; λ l ) ] exp ( i l 2 π N j ) | 2 ,
F ( h 0 , ... , h N 1 ) = l = 1 L w l j = 0 N 1 | P s m f ( h j ; λ l ) P m f ( ρ j ; λ l ) | 2 min ,
F j ( h j ) = l = 1 L w l | P s m f ( h j ; λ l ) P m f ( ρ j ; λ l ) | 2 min .
h j = h max m j M , m j = arg min m { 0 , ... , M 1 } [ l = 1 L w l | P s m f ( h max m M ; λ l ) P m f ( ρ j ; λ l ) | 2 ] .
I ( r , z ; λ ) = | 2 π λ z 0 R A ( ρ ) P s m f ( ρ ; λ ) P l e n s ( ρ ; λ ) exp ( i π ρ 2 λ z ) J 0 ( 2 π r ρ λ z ) r d r | 2 ,

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