Abstract

We exploit a technique, based on nonlinear optimization, to design diffractive lenses that focus optical nulls without any phase singularities. To ensure ease of fabrication, these lenses are composed of concentric circular zones. Furthermore, we show that this technique is readily extended to multiple wavelengths and can be used to improve tolerance to fabrication errors.

© 2009 Optical Society of America

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References

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    [CrossRef] [PubMed]
  8. M. P. MacDonald, L. Paterson, G. Armstrong, A. J. Bryant, W. Sibbett, and K. Dholakia, “Laguerre-gaussian laser modes for biophotonics and micromanipulation,” Proc. SPIE 5147, 48-59 (2003).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  13. H.-Y. Tsai, H. I. Smith, and R. Menon, “Fabrication of spiral-phase diffractive elements using scanning-electron-beam lithography,” J. Vac. Sci. Technol. B 25, 2068-2071 (2007).
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    [CrossRef]
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    [CrossRef]
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  27. The genetic algorithm was implemented using the Genetic Algorithm and Direct Search toolbox in MATLAB. Documentation available at http:www.mathworks.com/access/helpdesk/help/toolbox/gads/.
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    [CrossRef] [PubMed]

2007

R. Menon, H.-Y. Tsai, and S. W. Thomas, “Far-field generation of localized light fields using far-field optics via absorbance modulation,” Phys. Rev. Lett. 98, 043905 (2007).
[CrossRef] [PubMed]

H.-Y. Tsai, H. I. Smith, and R. Menon, “Fabrication of spiral-phase diffractive elements using scanning-electron-beam lithography,” J. Vac. Sci. Technol. B 25, 2068-2071 (2007).
[CrossRef]

2005

V. Westphal and S. W. Hell, “Nanoscale resolution in the focal plane of an optical microscope,” Phys. Rev. Lett. 94, 143903 (2005).
[CrossRef] [PubMed]

2004

T. Watanabe, T. Watanabe, M. Fujii, Y. Watanabe, N. Toyoma, and Y. Iketaki, “Generation of a doughnut-shaped beam with a spiral phase plate,” Rev. Sci. Instrum. 75, 5131-5135 (2004).
[CrossRef]

M. D. Levenson, T. Ebihara, G. Dai, Y. Morikawa, N. Hayashi, and S. M. Tan, “Optical vortex masks for via levels,” J. Microlithogr., Microfabr., Microsyst. 3, 293-304 (2004).
[CrossRef]

2003

J. A. O. Huguenin, B. C. dos Santos, P. A. M. dos Santos, and A. Z. Khoury, “Topological defects in moiré fringes with spiral zone plates,” J. Opt. Soc. Am. A 20, 1883-1889 (2003).
[CrossRef]

M. P. MacDonald, L. Paterson, G. Armstrong, A. J. Bryant, W. Sibbett, and K. Dholakia, “Laguerre-gaussian laser modes for biophotonics and micromanipulation,” Proc. SPIE 5147, 48-59 (2003).
[CrossRef]

D. Gil, R. Menon, and H. I. Smith, “Fabrication of high-numerical aperture phase zone plates with a single lithgraphy step and no etching,” J. Vac. Sci. Technol. B 21, 2956-2960 (2003).
[CrossRef]

2002

M. Dyba and S. W. Hell, “Focal spots of size λ/23 open up far-field fluorescence microscopy at 33 nm axial resolution,” Phys. Rev. Lett. 88, 163901 (2002).
[CrossRef] [PubMed]

A. Tavrov, R. Bohr, M. Totzeck, H. Tiziani, and M. Takeda, “Achromatic nulling interferometer based on a geometric spin-redirection phase,” Opt. Lett. 27, 2070-2072 (2002).
[CrossRef]

2001

2000

1999

1998

1997

1996

C. Paterson and R. Smith, “Higher-order bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124, 121-130 (1996).
[CrossRef]

K. T. Gahagan and G. A. Schwartzlander, Jr., “Optical vortex trapping of particles,” Opt. Lett. 21, 827-829 (1996).
[CrossRef] [PubMed]

1995

1989

1952

G. T. di Francia, “Supergain antennas and optical resolving power,” Nuovo Cimento, Suppl. 9, 426-438 (1952).
[CrossRef]

Arieli, Y.

Arlt, J.

Armstrong, G.

M. P. MacDonald, L. Paterson, G. Armstrong, A. J. Bryant, W. Sibbett, and K. Dholakia, “Laguerre-gaussian laser modes for biophotonics and micromanipulation,” Proc. SPIE 5147, 48-59 (2003).
[CrossRef]

Bohr, R.

Bryant, A. J.

M. P. MacDonald, L. Paterson, G. Armstrong, A. J. Bryant, W. Sibbett, and K. Dholakia, “Laguerre-gaussian laser modes for biophotonics and micromanipulation,” Proc. SPIE 5147, 48-59 (2003).
[CrossRef]

Calvert, G.

Colavita, M. M.

Dai, G.

M. D. Levenson, T. Ebihara, G. Dai, Y. Morikawa, N. Hayashi, and S. M. Tan, “Optical vortex masks for via levels,” J. Microlithogr., Microfabr., Microsyst. 3, 293-304 (2004).
[CrossRef]

Davidson, N.

R. Ozeri, L. Khaykovich, and N. Davidson, “Long spin relaxation times in a single-beam blue-detuned optical trap,” Phys. Rev. A 59, R1750-R1753 (1999).
[CrossRef]

Dholakia, K.

M. P. MacDonald, L. Paterson, G. Armstrong, A. J. Bryant, W. Sibbett, and K. Dholakia, “Laguerre-gaussian laser modes for biophotonics and micromanipulation,” Proc. SPIE 5147, 48-59 (2003).
[CrossRef]

di Francia, G. T.

G. T. di Francia, “Supergain antennas and optical resolving power,” Nuovo Cimento, Suppl. 9, 426-438 (1952).
[CrossRef]

dos Santos, B. C.

dos Santos, P. A. M.

Dyba, M.

M. Dyba and S. W. Hell, “Focal spots of size λ/23 open up far-field fluorescence microscopy at 33 nm axial resolution,” Phys. Rev. Lett. 88, 163901 (2002).
[CrossRef] [PubMed]

Ebihara, T.

M. D. Levenson, T. Ebihara, G. Dai, Y. Morikawa, N. Hayashi, and S. M. Tan, “Optical vortex masks for via levels,” J. Microlithogr., Microfabr., Microsyst. 3, 293-304 (2004).
[CrossRef]

Eisenberg, N.

Faklis, D.

Friberg, A. T.

Fujii, M.

T. Watanabe, T. Watanabe, M. Fujii, Y. Watanabe, N. Toyoma, and Y. Iketaki, “Generation of a doughnut-shaped beam with a spiral phase plate,” Rev. Sci. Instrum. 75, 5131-5135 (2004).
[CrossRef]

Gahagan, K. T.

Gil, D.

D. Gil, R. Menon, and H. I. Smith, “Fabrication of high-numerical aperture phase zone plates with a single lithgraphy step and no etching,” J. Vac. Sci. Technol. B 21, 2956-2960 (2003).
[CrossRef]

Goodman, J. W.

J. W. Goodman, An Introduction to Fourier Optics (McGraw-Hill, 1996), Chap. 4.

Hardy, G. J.

Hayashi, N.

M. D. Levenson, T. Ebihara, G. Dai, Y. Morikawa, N. Hayashi, and S. M. Tan, “Optical vortex masks for via levels,” J. Microlithogr., Microfabr., Microsyst. 3, 293-304 (2004).
[CrossRef]

Hell, S. W.

V. Westphal and S. W. Hell, “Nanoscale resolution in the focal plane of an optical microscope,” Phys. Rev. Lett. 94, 143903 (2005).
[CrossRef] [PubMed]

M. Dyba and S. W. Hell, “Focal spots of size λ/23 open up far-field fluorescence microscopy at 33 nm axial resolution,” Phys. Rev. Lett. 88, 163901 (2002).
[CrossRef] [PubMed]

Huguenin, J. A. O.

Iketaki, Y.

T. Watanabe, T. Watanabe, M. Fujii, Y. Watanabe, N. Toyoma, and Y. Iketaki, “Generation of a doughnut-shaped beam with a spiral phase plate,” Rev. Sci. Instrum. 75, 5131-5135 (2004).
[CrossRef]

Khaykovich, L.

R. Ozeri, L. Khaykovich, and N. Davidson, “Long spin relaxation times in a single-beam blue-detuned optical trap,” Phys. Rev. A 59, R1750-R1753 (1999).
[CrossRef]

Khoury, A. Z.

Kress, B.

B. Kress and P. Meyrueis, Digital Diffractive Optics: An Introduction to Planar Diffractive Optics and Related Technology (Wiley, 2000).

Levenson, M. D.

M. D. Levenson, T. Ebihara, G. Dai, Y. Morikawa, N. Hayashi, and S. M. Tan, “Optical vortex masks for via levels,” J. Microlithogr., Microfabr., Microsyst. 3, 293-304 (2004).
[CrossRef]

MacDonald, M. P.

M. P. MacDonald, L. Paterson, G. Armstrong, A. J. Bryant, W. Sibbett, and K. Dholakia, “Laguerre-gaussian laser modes for biophotonics and micromanipulation,” Proc. SPIE 5147, 48-59 (2003).
[CrossRef]

Menon, R.

R. Menon, H.-Y. Tsai, and S. W. Thomas, “Far-field generation of localized light fields using far-field optics via absorbance modulation,” Phys. Rev. Lett. 98, 043905 (2007).
[CrossRef] [PubMed]

H.-Y. Tsai, H. I. Smith, and R. Menon, “Fabrication of spiral-phase diffractive elements using scanning-electron-beam lithography,” J. Vac. Sci. Technol. B 25, 2068-2071 (2007).
[CrossRef]

D. Gil, R. Menon, and H. I. Smith, “Fabrication of high-numerical aperture phase zone plates with a single lithgraphy step and no etching,” J. Vac. Sci. Technol. B 21, 2956-2960 (2003).
[CrossRef]

Meyrueis, P.

B. Kress and P. Meyrueis, Digital Diffractive Optics: An Introduction to Planar Diffractive Optics and Related Technology (Wiley, 2000).

Mitchell, M.

M. Mitchell, An Introduction to Genetic Algorithms (MIT Press, 1996).

Morikawa, Y.

M. D. Levenson, T. Ebihara, G. Dai, Y. Morikawa, N. Hayashi, and S. M. Tan, “Optical vortex masks for via levels,” J. Microlithogr., Microfabr., Microsyst. 3, 293-304 (2004).
[CrossRef]

Morris, G. M.

Nabiev, R.

Nguyen, H. T.

Noach, S.

Ozeri, R.

R. Ozeri, L. Khaykovich, and N. Davidson, “Long spin relaxation times in a single-beam blue-detuned optical trap,” Phys. Rev. A 59, R1750-R1753 (1999).
[CrossRef]

Ozeri, S.

Padgett, M. J.

Paterson, C.

C. Paterson and R. Smith, “Higher-order bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124, 121-130 (1996).
[CrossRef]

Paterson, L.

M. P. MacDonald, L. Paterson, G. Armstrong, A. J. Bryant, W. Sibbett, and K. Dholakia, “Laguerre-gaussian laser modes for biophotonics and micromanipulation,” Proc. SPIE 5147, 48-59 (2003).
[CrossRef]

Prather, D. W.

Sales, T. R. M.

Schmidtlin, E. G. H.

Schwartzlander, G. A.

Serabyn, E.

Sheppard, C. J. R.

Shi, S.

Sibbett, W.

M. P. MacDonald, L. Paterson, G. Armstrong, A. J. Bryant, W. Sibbett, and K. Dholakia, “Laguerre-gaussian laser modes for biophotonics and micromanipulation,” Proc. SPIE 5147, 48-59 (2003).
[CrossRef]

Smith, H. I.

H.-Y. Tsai, H. I. Smith, and R. Menon, “Fabrication of spiral-phase diffractive elements using scanning-electron-beam lithography,” J. Vac. Sci. Technol. B 25, 2068-2071 (2007).
[CrossRef]

D. Gil, R. Menon, and H. I. Smith, “Fabrication of high-numerical aperture phase zone plates with a single lithgraphy step and no etching,” J. Vac. Sci. Technol. B 21, 2956-2960 (2003).
[CrossRef]

Smith, R.

C. Paterson and R. Smith, “Higher-order bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124, 121-130 (1996).
[CrossRef]

Sommargren, G. E.

Sweeney, D. W.

Takeda, M.

Tan, S. M.

M. D. Levenson, T. Ebihara, G. Dai, Y. Morikawa, N. Hayashi, and S. M. Tan, “Optical vortex masks for via levels,” J. Microlithogr., Microfabr., Microsyst. 3, 293-304 (2004).
[CrossRef]

Tavrov, A.

Thomas, S. W.

R. Menon, H.-Y. Tsai, and S. W. Thomas, “Far-field generation of localized light fields using far-field optics via absorbance modulation,” Phys. Rev. Lett. 98, 043905 (2007).
[CrossRef] [PubMed]

Tiziani, H.

Totzeck, M.

Toyoma, N.

T. Watanabe, T. Watanabe, M. Fujii, Y. Watanabe, N. Toyoma, and Y. Iketaki, “Generation of a doughnut-shaped beam with a spiral phase plate,” Rev. Sci. Instrum. 75, 5131-5135 (2004).
[CrossRef]

Tsai, H.-Y.

H.-Y. Tsai, H. I. Smith, and R. Menon, “Fabrication of spiral-phase diffractive elements using scanning-electron-beam lithography,” J. Vac. Sci. Technol. B 25, 2068-2071 (2007).
[CrossRef]

R. Menon, H.-Y. Tsai, and S. W. Thomas, “Far-field generation of localized light fields using far-field optics via absorbance modulation,” Phys. Rev. Lett. 98, 043905 (2007).
[CrossRef] [PubMed]

Turunen, J.

Vasara, A.

Wallace, J. K.

Watanabe, T.

T. Watanabe, T. Watanabe, M. Fujii, Y. Watanabe, N. Toyoma, and Y. Iketaki, “Generation of a doughnut-shaped beam with a spiral phase plate,” Rev. Sci. Instrum. 75, 5131-5135 (2004).
[CrossRef]

T. Watanabe, T. Watanabe, M. Fujii, Y. Watanabe, N. Toyoma, and Y. Iketaki, “Generation of a doughnut-shaped beam with a spiral phase plate,” Rev. Sci. Instrum. 75, 5131-5135 (2004).
[CrossRef]

Watanabe, Y.

T. Watanabe, T. Watanabe, M. Fujii, Y. Watanabe, N. Toyoma, and Y. Iketaki, “Generation of a doughnut-shaped beam with a spiral phase plate,” Rev. Sci. Instrum. 75, 5131-5135 (2004).
[CrossRef]

Westphal, V.

V. Westphal and S. W. Hell, “Nanoscale resolution in the focal plane of an optical microscope,” Phys. Rev. Lett. 94, 143903 (2005).
[CrossRef] [PubMed]

Wheatland, M.

Yeh, P.

Zhou, S.

Appl. Opt.

J. Microlithogr., Microfabr., Microsyst.

M. D. Levenson, T. Ebihara, G. Dai, Y. Morikawa, N. Hayashi, and S. M. Tan, “Optical vortex masks for via levels,” J. Microlithogr., Microfabr., Microsyst. 3, 293-304 (2004).
[CrossRef]

J. Opt. Soc. Am. A

J. Vac. Sci. Technol. B

D. Gil, R. Menon, and H. I. Smith, “Fabrication of high-numerical aperture phase zone plates with a single lithgraphy step and no etching,” J. Vac. Sci. Technol. B 21, 2956-2960 (2003).
[CrossRef]

H.-Y. Tsai, H. I. Smith, and R. Menon, “Fabrication of spiral-phase diffractive elements using scanning-electron-beam lithography,” J. Vac. Sci. Technol. B 25, 2068-2071 (2007).
[CrossRef]

Nuovo Cimento, Suppl.

G. T. di Francia, “Supergain antennas and optical resolving power,” Nuovo Cimento, Suppl. 9, 426-438 (1952).
[CrossRef]

Opt. Commun.

C. Paterson and R. Smith, “Higher-order bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124, 121-130 (1996).
[CrossRef]

Opt. Lett.

Phys. Rev. A

R. Ozeri, L. Khaykovich, and N. Davidson, “Long spin relaxation times in a single-beam blue-detuned optical trap,” Phys. Rev. A 59, R1750-R1753 (1999).
[CrossRef]

Phys. Rev. Lett.

V. Westphal and S. W. Hell, “Nanoscale resolution in the focal plane of an optical microscope,” Phys. Rev. Lett. 94, 143903 (2005).
[CrossRef] [PubMed]

R. Menon, H.-Y. Tsai, and S. W. Thomas, “Far-field generation of localized light fields using far-field optics via absorbance modulation,” Phys. Rev. Lett. 98, 043905 (2007).
[CrossRef] [PubMed]

M. Dyba and S. W. Hell, “Focal spots of size λ/23 open up far-field fluorescence microscopy at 33 nm axial resolution,” Phys. Rev. Lett. 88, 163901 (2002).
[CrossRef] [PubMed]

Proc. SPIE

M. P. MacDonald, L. Paterson, G. Armstrong, A. J. Bryant, W. Sibbett, and K. Dholakia, “Laguerre-gaussian laser modes for biophotonics and micromanipulation,” Proc. SPIE 5147, 48-59 (2003).
[CrossRef]

Rev. Sci. Instrum.

T. Watanabe, T. Watanabe, M. Fujii, Y. Watanabe, N. Toyoma, and Y. Iketaki, “Generation of a doughnut-shaped beam with a spiral phase plate,” Rev. Sci. Instrum. 75, 5131-5135 (2004).
[CrossRef]

Other

B. Kress and P. Meyrueis, Digital Diffractive Optics: An Introduction to Planar Diffractive Optics and Related Technology (Wiley, 2000).

J. W. Goodman, An Introduction to Fourier Optics (McGraw-Hill, 1996), Chap. 4.

M. Mitchell, An Introduction to Genetic Algorithms (MIT Press, 1996).

The genetic algorithm was implemented using the Genetic Algorithm and Direct Search toolbox in MATLAB. Documentation available at http:www.mathworks.com/access/helpdesk/help/toolbox/gads/.

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Figures (5)

Fig. 1
Fig. 1

Schematic of the optic and design methodology. The design parameters are the radii of the zones, r 1 , r 2 , , r M , and the height of the zones, h. Stylized intensity distributions in the focal plane illustrate the design requirement for a dichromat that focuses a bright spot at λ 1 and a ring-shaped spot at λ 2 . The remaining variables are described in the text.

Fig. 2
Fig. 2

Dichromat, all lengths are in units of λ 1 . Transmission functions of dichromats of (a) NA = 0.56 and (c) NA = 0.7 , both with focal length = 100 λ 1 , λ 1 = 400 nm , λ 2 = 532 nm , n ( λ 1 ) = 1.501 , and n ( λ 2 ) = 1.487 . The remaining optimization parameters are described in the text. Within the aperture of the dichromat, the gray rings are phase shifted with respect to the white ones; their relative height difference being (a) 0.98 λ 1 and (c) 0.74 λ 1 . Outside the dichromat is opaque. (b) and (d) Radial intensity distributions of the focal spots. The data calculated using FDTD are plotted with x’s while the scalar data are plotted as solid curves.

Fig. 3
Fig. 3

Intensity distributions calculated using FDTD of the dichromat shown in Fig. 2c at various transverse planes near the focus of the dichromat for (a) λ 1 and (b) λ 2 .

Fig. 4
Fig. 4

Tolerance to fabrication errors. Fabrication errors were simulated by adding random noise to the radii of the zones with zero mean and standard deviation, σ r , and to the height of the zones, also with zero mean and standard deviation, σ h . The focusing efficiencies at λ 1 and λ 2 as defined by Eqs. (10, 11) were computed. The standard deviations of the focusing efficiencies at λ 1 and λ 2 are shown in (a) and (b), respectively. The corresponding data for the newly optimized dichromat are shown in (c) and (d). A sample size of 25 was used for these calculations. Focal intensity distributions calculated using FDTD for the new dichromat at (e) λ 1 (left) and λ 2 (right). (f) Radial intensity distributions in the focal plane. FDTD data are plotted with x’s while the scalar data are plotted with solid curves.

Fig. 5
Fig. 5

Transmission function of a trichromat, a lens that focuses λ 1 and λ 3 to bright spots and λ 2 to a ring-shaped spot containing (a) 40 zones and (c) 80 zones. The parameters of the trichromat were λ 3 = 633 nm , and n ( λ 3 ) = 1.482 . The remaining parameters were the same as for the corresponding dichromat. The height difference between the gray and white zones is (a) λ 1 and (c) 0.6101 λ 1 . (b) and(d) Intensity distributions in the focal plane for the trichromats with 40 and 80 zones, respectively. FDTD data are plotted using x’s while the scalar data are plotted using solid curves.

Tables (1)

Tables Icon

Table 1 Radii and Height of the Zones in Units of λ 1

Equations (13)

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

T ( ρ ) = { e i ψ r 2 m < ρ r 2 m + 1 , m { 0 , 1 , 2 , 3 , ... } 0 ρ > r M 1 elsewhere } ,
ψ = 2 π h λ [ Re ( n ( λ ) ) 1 ] ,
Υ ( ρ , z , λ , { r , h } ) = 1 i λ 0 ρ d ρ 0 2 π d ϕ T ( ρ , { r , h } ) exp ( i 2 π λ ρ 2 + ρ 2 2 ρ ρ cos ( ϕ ) + z 2 ) ρ 2 + ρ 2 2 ρ ρ cos ( ϕ ) + z 2 ( 1 + z ρ 2 + ρ 2 2 ρ ρ cos ( ϕ ) + z 2 ) 2 ,
E ( { r , h } ) = w 1 0 ρ 1 Υ ( ρ , λ 1 , { r , h } ) ρ d ρ w 2 ρ 2 ρ 3 Υ ( ρ , λ 2 , { r , h } ) ρ d ρ + w 3 Υ ( ρ = 0 , λ 2 , { r , h } ) ,
r p > r q M p > q 1 , { p , q } I ,
r p r p 1 > Δ > 0 M p > 1 , p I ,
r 1 > 2 Δ .
r = { r 1 , r 2 , , r M } r ̃ = { r 1 + δ r 1 ˜ , r 2 + δ r 2 ˜ , , r M + δ r M ˜ } ,
h h ̃ = h + δ h ˜ ,
η 1 ( { r , h } ) = 0 ρ 1 Υ ( ρ , λ 1 , { r , h } ) ρ d ρ 0 Υ ( ρ , λ 1 , { r , h } ) ρ d ρ ,
η 2 ( { r , h } ) = ρ 2 ρ 3 Υ ( ρ , λ 2 , { r , h } ) ρ d ρ 0 Υ ( ρ , λ 2 , { r , h } ) ρ d ρ .
E r ( { r , h } ) = μ { E ( { r ̃ , h ̃ } ) } + σ { E ( { r ̃ , h ̃ } ) } ,
E ( { r , h } ) = w 1 0 ρ 1 Υ ( ρ , λ 1 , { r , h } ) ρ d ρ w 2 ρ 2 ρ 3 Υ ( ρ , λ 2 , { r , h } ) ρ d ρ + w 3 Υ ( ρ = 0 , λ 2 , { r , h } ) w 4 0 ρ 4 Υ ( ρ , λ 3 , { r , h } ) ρ d ρ ,

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