Abstract

A novel axially-symmetric filter for increasing focal depth and generating an approximation to a Bessel beam is proposed. It consists of an array of rings of strength –1,0 and 1. The design is based on an analytic solution, and combines high resolution in the transverse direction with good efficiency. One presented design increases the depth of focus compared with a standard lens by more than 30 times, with a very flat axial intensity distribution over this range. Effects of discretization are discussed. Various different approaches to increasing depth of focus are compared, to put the new design into perspective.

© 2012 OSA

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2011 (3)

2010 (3)

2009 (2)

T. Čižmár and K. Dholakia, “Tunable Bessel light modes: engineering the axial propagation,” Opt. Express17(18), 15558–15570 (2009).
[CrossRef] [PubMed]

C. J. R. Sheppard, S. Rehman, N. K. Balla, E. Y. S. Yew, and T. W. Teng, “Bessel beams: Effects of polarization,” Opt. Commun.282(24), 4647–4656 (2009).
[CrossRef]

2008 (3)

M. R. Foreman, S. S. Sherif, P. R. T. Munro, and P. Török, “Inversion of the Debye-Wolf diffraction integral using an eigenfunction representation of the electric fields in the focal region,” Opt. Express16(7), 4901–4917 (2008).
[CrossRef] [PubMed]

H. F. Wang, L. Shi, B. Luk`yanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008).
[CrossRef]

C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, “Three-zone pupil filters,” Opt. Commun.281(14), 3623–3630 (2008).
[CrossRef]

2007 (3)

2005 (1)

V. F. Canales, J. E. Oti, and M. P. Cagigal, “Three-dimensional control of the focal light intensity distribution by analytically designed phase masks,” Opt. Commun.247(1-3), 11–18 (2005).
[CrossRef]

2001 (1)

1999 (1)

G. Yang, “An optical pickup using a diffractive optical element for a high-density optical disc,” Opt. Commun.159(1-3), 19–22 (1999).
[CrossRef]

1996 (1)

C. J. R. Sheppard, “Synthesis of filters for specified axial properties,” J. Mod. Opt.43(3), 525–536 (1996).
[CrossRef]

1995 (2)

Z. Bouchal, J. Wagner, and M. Olivik, “Bessel beams in the focal region,” Opt. Eng.34(6), 1680–1688 (1995).
[CrossRef]

E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt.34(11), 1859–1866 (1995).
[CrossRef] [PubMed]

1992 (1)

1988 (1)

1987 (2)

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun.64(6), 491–495 (1987).
[CrossRef]

J. Durnin, “Exact solutions for nondiffracting beams. I. the scalar theory,” J. Opt. Soc. Am. A4(4), 651–654 (1987).
[CrossRef]

1980 (1)

R. Boivin and A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field - I. Achievement of maximum central irradiance under an energy constraint,” Opt. Acta (Lond.)27, 857 (1980).

1979 (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.47(3), 264–267 (1979).
[CrossRef]

1978 (1)

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves Opt. Acoust.2(4), 105–112 (1978).
[CrossRef]

1965 (1)

1958 (1)

J. Dyson, “Circular and spiral diffraction gratings,” Proc. R. Soc. Lond. A Math. Phys. Sci.248(1252), 93–106 (1958).
[CrossRef]

1954 (1)

1953 (1)

E. H. Linfoot and E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. B66(2), 145–149 (1953).
[CrossRef]

1926 (1)

G. C. Steward, “IV Aberration diffraction effects,” Phil. Trans. Roy. Soc. London Ser. A225(626-635), 131–198 (1926).
[CrossRef]

1872 (1)

J. W. S. Rayleigh, “On the diffraction of object glasses,” Mon. Not. R. Astron. Soc.33, 59–63 (1872).

1841 (1)

G. B. Airy, “The diffraction of an annular aperture,” Phil. Mag. Ser. 318, 1–10 (1841).

Airy, G. B.

G. B. Airy, “The diffraction of an annular aperture,” Phil. Mag. Ser. 318, 1–10 (1841).

Balazs, N. L.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.47(3), 264–267 (1979).
[CrossRef]

Balla, N. K.

C. J. R. Sheppard, S. Rehman, N. K. Balla, E. Y. S. Yew, and T. W. Teng, “Bessel beams: Effects of polarization,” Opt. Commun.282(24), 4647–4656 (2009).
[CrossRef]

Berry, M. V.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.47(3), 264–267 (1979).
[CrossRef]

Boivin, A.

R. Boivin and A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field - I. Achievement of maximum central irradiance under an energy constraint,” Opt. Acta (Lond.)27, 857 (1980).

Boivin, R.

R. Boivin and A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field - I. Achievement of maximum central irradiance under an energy constraint,” Opt. Acta (Lond.)27, 857 (1980).

Bouchal, Z.

Z. Bouchal, J. Wagner, and M. Olivik, “Bessel beams in the focal region,” Opt. Eng.34(6), 1680–1688 (1995).
[CrossRef]

Broky, J.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett.99(21), 213901 (2007).
[CrossRef] [PubMed]

Cagigal, M. P.

V. F. Canales, J. E. Oti, and M. P. Cagigal, “Three-dimensional control of the focal light intensity distribution by analytically designed phase masks,” Opt. Commun.247(1-3), 11–18 (2005).
[CrossRef]

Campos, J.

C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, “Three-zone pupil filters,” Opt. Commun.281(14), 3623–3630 (2008).
[CrossRef]

Canales, V. F.

V. F. Canales, J. E. Oti, and M. P. Cagigal, “Three-dimensional control of the focal light intensity distribution by analytically designed phase masks,” Opt. Commun.247(1-3), 11–18 (2005).
[CrossRef]

Cathey, W. T.

Chen, N. G.

Chen, W.

Chong, C. T.

H. F. Wang, L. Shi, B. Luk`yanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008).
[CrossRef]

Christodoulides, D. N.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett.99(21), 213901 (2007).
[CrossRef] [PubMed]

Cižmár, T.

Cox, A. J.

D’Anna, J.

Dholakia, K.

M. Mazilu, D. J. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light beats the spread: “non-diffracting” beams,” Laser Photon. Rev.4(4), 529–547 (2010).
[CrossRef]

T. Čižmár and K. Dholakia, “Tunable Bessel light modes: engineering the axial propagation,” Opt. Express17(18), 15558–15570 (2009).
[CrossRef] [PubMed]

Dogariu, A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett.99(21), 213901 (2007).
[CrossRef] [PubMed]

Dowski, E. R.

Durnin, J.

Dyson, J.

J. Dyson, “Circular and spiral diffraction gratings,” Proc. R. Soc. Lond. A Math. Phys. Sci.248(1252), 93–106 (1958).
[CrossRef]

Escalera, J. C.

C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, “Three-zone pupil filters,” Opt. Commun.281(14), 3623–3630 (2008).
[CrossRef]

Foreman, M. R.

Gan, F.

Gori, F.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun.64(6), 491–495 (1987).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun.64(6), 491–495 (1987).
[CrossRef]

Gunn-Moore, F.

M. Mazilu, D. J. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light beats the spread: “non-diffracting” beams,” Laser Photon. Rev.4(4), 529–547 (2010).
[CrossRef]

Hao, X.

Hashimoto, N.

Hegedus, Z. S.

Hibi, T.

Horanai, H.

Kozawa, Y.

Kuang, C. F.

Kurihara, M.

Ledesma, S.

C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, “Three-zone pupil filters,” Opt. Commun.281(14), 3623–3630 (2008).
[CrossRef]

Linfoot, E. H.

E. H. Linfoot and E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. B66(2), 145–149 (1953).
[CrossRef]

Liu, X.

Luk`yanchuk, B.

H. F. Wang, L. Shi, B. Luk`yanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008).
[CrossRef]

Mazilu, M.

M. Mazilu, D. J. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light beats the spread: “non-diffracting” beams,” Laser Photon. Rev.4(4), 529–547 (2010).
[CrossRef]

McLeod, J. H.

Munro, P. R. T.

Nemoto, T.

Olivik, M.

Z. Bouchal, J. Wagner, and M. Olivik, “Bessel beams in the focal region,” Opt. Eng.34(6), 1680–1688 (1995).
[CrossRef]

Oti, J. E.

V. F. Canales, J. E. Oti, and M. P. Cagigal, “Three-dimensional control of the focal light intensity distribution by analytically designed phase masks,” Opt. Commun.247(1-3), 11–18 (2005).
[CrossRef]

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun.64(6), 491–495 (1987).
[CrossRef]

Rayleigh, J. W. S.

J. W. S. Rayleigh, “On the diffraction of object glasses,” Mon. Not. R. Astron. Soc.33, 59–63 (1872).

Rehman, S.

C. J. R. Sheppard and S. Rehman, “Highly convergent focusing of light based on rotating dipole polarization,” Appl. Opt.50(22), 4463–4467 (2011).
[CrossRef] [PubMed]

C. J. R. Sheppard, S. Rehman, N. K. Balla, E. Y. S. Yew, and T. W. Teng, “Bessel beams: Effects of polarization,” Opt. Commun.282(24), 4647–4656 (2009).
[CrossRef]

Sato, A.

Sato, S.

Sheppard, C.

H. F. Wang, L. Shi, B. Luk`yanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard and S. Rehman, “Highly convergent focusing of light based on rotating dipole polarization,” Appl. Opt.50(22), 4463–4467 (2011).
[CrossRef] [PubMed]

C. J. R. Sheppard, “Binary phase filters with a maximally-flat response,” Opt. Lett.36(8), 1386–1388 (2011).
[CrossRef] [PubMed]

C. J. R. Sheppard, S. Rehman, N. K. Balla, E. Y. S. Yew, and T. W. Teng, “Bessel beams: Effects of polarization,” Opt. Commun.282(24), 4647–4656 (2009).
[CrossRef]

C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, “Three-zone pupil filters,” Opt. Commun.281(14), 3623–3630 (2008).
[CrossRef]

Y. S. Xu, J. Singh, C. J. R. Sheppard, and N. G. Chen, “Ultra long high resolution beam by multi-zone rotationally symmetrical complex pupil filter,” Opt. Express15(10), 6409–6413 (2007).
[CrossRef] [PubMed]

C. J. R. Sheppard, “Synthesis of filters for specified axial properties,” J. Mod. Opt.43(3), 525–536 (1996).
[CrossRef]

C. J. R. Sheppard and Z. S. Hegedus, “Axial behaviour of pupil plane filters,” J. Opt. Soc. Am. A5(5), 643–647 (1988).
[CrossRef]

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves Opt. Acoust.2(4), 105–112 (1978).
[CrossRef]

Sherif, S. S.

Shi, L.

H. F. Wang, L. Shi, B. Luk`yanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008).
[CrossRef]

Singh, J.

Siviloglou, G. A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett.99(21), 213901 (2007).
[CrossRef] [PubMed]

Stevenson, D. J.

M. Mazilu, D. J. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light beats the spread: “non-diffracting” beams,” Laser Photon. Rev.4(4), 529–547 (2010).
[CrossRef]

Steward, G. C.

G. C. Steward, “IV Aberration diffraction effects,” Phil. Trans. Roy. Soc. London Ser. A225(626-635), 131–198 (1926).
[CrossRef]

Teng, T. W.

C. J. R. Sheppard, S. Rehman, N. K. Balla, E. Y. S. Yew, and T. W. Teng, “Bessel beams: Effects of polarization,” Opt. Commun.282(24), 4647–4656 (2009).
[CrossRef]

Thompson, B. J.

Török, P.

Wagner, J.

Z. Bouchal, J. Wagner, and M. Olivik, “Bessel beams in the focal region,” Opt. Eng.34(6), 1680–1688 (1995).
[CrossRef]

Wang, H. F.

H. F. Wang, L. Shi, B. Luk`yanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008).
[CrossRef]

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

Wang, J.

Wang, T. T.

Wilson, T.

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves Opt. Acoust.2(4), 105–112 (1978).
[CrossRef]

Wolf, E.

E. H. Linfoot and E. Wolf, “Diffraction images in systems with an annular aperture,” Proc. Phys. Soc. B66(2), 145–149 (1953).
[CrossRef]

Xu, Y. S.

Yang, G.

G. Yang, “An optical pickup using a diffractive optical element for a high-density optical disc,” Opt. Commun.159(1-3), 19–22 (1999).
[CrossRef]

Yew, E. Y. S.

C. J. R. Sheppard, S. Rehman, N. K. Balla, E. Y. S. Yew, and T. W. Teng, “Bessel beams: Effects of polarization,” Opt. Commun.282(24), 4647–4656 (2009).
[CrossRef]

Yokoyama, H.

Zhan, Q. W.

Am. J. Phys. (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.47(3), 264–267 (1979).
[CrossRef]

Appl. Opt. (3)

IEE J. Microwaves Opt. Acoust. (1)

C. J. R. Sheppard and T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” IEE J. Microwaves Opt. Acoust.2(4), 105–112 (1978).
[CrossRef]

J. Mod. Opt. (1)

C. J. R. Sheppard, “Synthesis of filters for specified axial properties,” J. Mod. Opt.43(3), 525–536 (1996).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

Laser Photon. Rev. (1)

M. Mazilu, D. J. Stevenson, F. Gunn-Moore, and K. Dholakia, “Light beats the spread: “non-diffracting” beams,” Laser Photon. Rev.4(4), 529–547 (2010).
[CrossRef]

Mon. Not. R. Astron. Soc. (1)

J. W. S. Rayleigh, “On the diffraction of object glasses,” Mon. Not. R. Astron. Soc.33, 59–63 (1872).

Nat. Photonics (1)

H. F. Wang, L. Shi, B. Luk`yanchuk, C. Sheppard, and C. T. Chong, “Creation of a needle of longitudinally polarized light in vacuum using binary optics,” Nat. Photonics2(8), 501–505 (2008).
[CrossRef]

Opt. Acta (Lond.) (1)

R. Boivin and A. Boivin, “Optimized amplitude filtering for superresolution over a restricted field - I. Achievement of maximum central irradiance under an energy constraint,” Opt. Acta (Lond.)27, 857 (1980).

Opt. Commun. (5)

C. J. R. Sheppard, S. Rehman, N. K. Balla, E. Y. S. Yew, and T. W. Teng, “Bessel beams: Effects of polarization,” Opt. Commun.282(24), 4647–4656 (2009).
[CrossRef]

V. F. Canales, J. E. Oti, and M. P. Cagigal, “Three-dimensional control of the focal light intensity distribution by analytically designed phase masks,” Opt. Commun.247(1-3), 11–18 (2005).
[CrossRef]

G. Yang, “An optical pickup using a diffractive optical element for a high-density optical disc,” Opt. Commun.159(1-3), 19–22 (1999).
[CrossRef]

C. J. R. Sheppard, J. Campos, J. C. Escalera, and S. Ledesma, “Three-zone pupil filters,” Opt. Commun.281(14), 3623–3630 (2008).
[CrossRef]

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun.64(6), 491–495 (1987).
[CrossRef]

Opt. Eng. (1)

Z. Bouchal, J. Wagner, and M. Olivik, “Bessel beams in the focal region,” Opt. Eng.34(6), 1680–1688 (1995).
[CrossRef]

Opt. Express (5)

Opt. Lett. (3)

Phil. Mag. Ser. 3 (1)

G. B. Airy, “The diffraction of an annular aperture,” Phil. Mag. Ser. 318, 1–10 (1841).

Phil. Trans. Roy. Soc. London Ser. A (1)

G. C. Steward, “IV Aberration diffraction effects,” Phil. Trans. Roy. Soc. London Ser. A225(626-635), 131–198 (1926).
[CrossRef]

Phys. Rev. Lett. (1)

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Figures (8)

Fig.
       1
Fig. 1

The intensity in a meridional plane for: (a) – (d) the four different solutions for a maximally-flat pupil (MF) of 5 elements, (e) a circular pupil, calculated using fast Fourier transform with a sampling distance of 0.004 of the pupil radius. Solution (a) gives the smallest focal spot. Pairs of solutions (a), (b) and (c), (d) each give the same axial intensity.

Fig.
       2
Fig. 2

Parameters for different strategies to achieve increased depth of focus. (a) The parameters S and F as functions of G, the transverse gain, for an annular pupil, 2-zone binary phase filters, and maximally-flat (MF) filters of 2-5 elements. (b) The parameter G plotted against W 90 , the 90% depth of focus. MF and MFA solutions are compared with an annular pupil. (c) The parameters S and F as functions of W 90 for MF filters, MFA filters, and annular pupils.

Fig.
       3
Fig. 3

The intensity in the focal region for MFA pupils of 2 and 5 elements (total 3 or 6 rings). (Solution (a) for 5 elements.) (a) Axial intensity. (b) Transverse intensity. The behavior for a circular aperture is shown in green for comparison.

Fig.
       4
Fig. 4

The intensity in a meridional plane for (a-d) the four solutions for an annular maximally-flat pupil (MFA) of 5 elements (total 6 rings) ( t 0 = 0.75 ), (e) a plain annular pupil ( t 0 = 0.75 ). These are calculated using fast Fourier transform with a sampling distance of 0.004 of the pupil radius. Solution (a) gives the smallest focal spot, but all solutions give close to a Bessel beam in the transverse direction. Pairs of solutions (a), (b) and (c), (d) each give the same axial intensity. Note the different axial scale from Fig. 1.

Fig.
       5
Fig. 5

The axial intensity for five element MF filters, with ring radii truncated to 3, 2, 1.5 and 1 decimal place.

Fig.
       6
Fig. 6

The pixelated filters used in our simulations for MF filters. The four solutions for five elements are shown.

Fig.
       7
Fig. 7

The pixelated filters used in our simulations for MFA filters. The four solutions for five elements are shown.

Fig.
       8
Fig. 8

The transverse (a), (b) and axial (c), (d) intensities (un-normalized) for: (a), (c) pixelated 5 ring MF (labeled biphase), and (b), (d) pixilated MFA filters (labeled 3 level). Results for a circular pupil and an annular pupil are also shown. 5a and 5b correspond to the two different axial intensity variations, while 5amirror, 5bmirror correspond to the filters that are mirror images in coordinate t of 5a and 5b.

Tables (1)

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Table 1 Performance Parameters for Different Filter Strategies

Equations (4)

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q p = 0 1 Q ( t ) t p d t .
F = q 0 2 0 1 | Q ( t ) | 2 d t .
G = 2 q 1 q 0 .
U ( v , u ) = 0 1 Q ( t ) J 0 ( v t ) exp ( i u t / 2 ) d t ,

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