Abstract

We show the possibility of arbitrary longitudinal spatial modeling of non-diffracting light beams over micrometric regions. The resulting beams, which are highly non-paraxial, possess subwavelength spots and can acquire multiple intensity peaks at predefined locations over regions that are few times larger than the wavelength. The formulation we present here provides exact solutions to the Maxwell’s equations where the linear, radial, and azimuthal beam polarizations are all considered. Modeling the longitudinal intensity pattern at small scale can address many challenges in three-dimensional optical trapping and micromanipulation.

© 2017 Optical Society of America

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References

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  1. H. E. Hernandez Figueroa, E. Recami, and Michel Zamboni-Rached, Non-Diffracting Waves (Wiley-VCH Verlag, 2014).
  2. T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2014).
    [Crossref]
  3. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
    [Crossref] [PubMed]
  4. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
    [Crossref]
  5. M. Zamboni-Rached, “Stationary optical wave fields with arbitrary longitudinal shape by superposing equal frequency Bessel beams: Frozen Waves,” Opt. Express 12(17), 4001–4006 (2004).
    [Crossref] [PubMed]
  6. M. Zamboni-Rached and M. Mojahedi, “Shaping finite-energy diffraction-and attenuation-resistant beams through Bessel-Gauss beam superposition,” Phys. Rev. A 92, 043839 (2015).
    [Crossref]
  7. M. Corato Zanarella and M. Zamboni-Rached, “Electromagnetic frozen waves with radial, azimuthal, linear, circular, and elliptical polarizations,” Phys. Rev. A 94, 053802 (2016).
    [Crossref]
  8. T. A. Vieira, M. R. R. Gesualdi, and M. Zamboni-Rached, “Frozen waves: experimental generation,” Opt. Lett. 37(11), 2034–2036 (2012).
    [Crossref] [PubMed]
  9. A. H. Dorrah, M. Zamboni-Rached, and M. Mojahedi, “Generating attenuation-resistant frozen waves in absorbing fluid,” Opt. Lett. 41(16), 3702–3705 (2015).
    [Crossref]
  10. L. A. Ambrosio and M. Zamboni-Rached, “Analytical approach of ordinary frozen waves for optical trapping and micromanipulation,” Appl. Opt. 54(10), 2584–2593 (2015).
    [Crossref] [PubMed]
  11. E. G. P. Pachon, M. Zamboni-Rached, A. H. Dorrah, M. Mojahedi, M. R. R. Gesualdi, and G. G. Cabrera, “Architecting new diffraction-resistant light structures and their possible applications in atom guidance,” Opt. Express 24(22), 25403–25408 (2016).
    [Crossref] [PubMed]
  12. M. Zamboni-Rached and E. Recami, “Subluminal wave bullets: Exact localized subluminal solutions to the wave equations,” Phys. Rev. A 77, 033824 (2008).
    [Crossref]
  13. R. L. Garay-Avendaño and M. Zamboni-Rached, “Exact analytic solutions of Maxwell’s equations describing propagating nonparaxial electromagnetic beams,” Appl. Opt. 53, 4524–4531 (2014).
    [Crossref]

2016 (2)

2015 (3)

2014 (2)

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2014).
[Crossref]

R. L. Garay-Avendaño and M. Zamboni-Rached, “Exact analytic solutions of Maxwell’s equations describing propagating nonparaxial electromagnetic beams,” Appl. Opt. 53, 4524–4531 (2014).
[Crossref]

2012 (1)

2011 (1)

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
[Crossref]

2008 (1)

M. Zamboni-Rached and E. Recami, “Subluminal wave bullets: Exact localized subluminal solutions to the wave equations,” Phys. Rev. A 77, 033824 (2008).
[Crossref]

2004 (1)

2003 (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[Crossref] [PubMed]

Ambrosio, L. A.

Betzig, E.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2014).
[Crossref]

Bowman, R.

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
[Crossref]

Cabrera, G. G.

Corato Zanarella, M.

M. Corato Zanarella and M. Zamboni-Rached, “Electromagnetic frozen waves with radial, azimuthal, linear, circular, and elliptical polarizations,” Phys. Rev. A 94, 053802 (2016).
[Crossref]

Davidson, M. W.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2014).
[Crossref]

Dorrah, A. H.

Galbraith, C. G.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2014).
[Crossref]

Galbraith, J. A.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2014).
[Crossref]

Gao, L.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2014).
[Crossref]

Garay-Avendaño, R. L.

Gesualdi, M. R. R.

Grier, D. G.

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[Crossref] [PubMed]

Hernandez Figueroa, H. E.

H. E. Hernandez Figueroa, E. Recami, and Michel Zamboni-Rached, Non-Diffracting Waves (Wiley-VCH Verlag, 2014).

Milkie, D. E.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2014).
[Crossref]

Mojahedi, M.

Pachon, E. G. P.

Padgett, M.

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
[Crossref]

Planchon, T. A.

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2014).
[Crossref]

Recami, E.

M. Zamboni-Rached and E. Recami, “Subluminal wave bullets: Exact localized subluminal solutions to the wave equations,” Phys. Rev. A 77, 033824 (2008).
[Crossref]

H. E. Hernandez Figueroa, E. Recami, and Michel Zamboni-Rached, Non-Diffracting Waves (Wiley-VCH Verlag, 2014).

Vieira, T. A.

Zamboni-Rached, M.

E. G. P. Pachon, M. Zamboni-Rached, A. H. Dorrah, M. Mojahedi, M. R. R. Gesualdi, and G. G. Cabrera, “Architecting new diffraction-resistant light structures and their possible applications in atom guidance,” Opt. Express 24(22), 25403–25408 (2016).
[Crossref] [PubMed]

M. Corato Zanarella and M. Zamboni-Rached, “Electromagnetic frozen waves with radial, azimuthal, linear, circular, and elliptical polarizations,” Phys. Rev. A 94, 053802 (2016).
[Crossref]

M. Zamboni-Rached and M. Mojahedi, “Shaping finite-energy diffraction-and attenuation-resistant beams through Bessel-Gauss beam superposition,” Phys. Rev. A 92, 043839 (2015).
[Crossref]

A. H. Dorrah, M. Zamboni-Rached, and M. Mojahedi, “Generating attenuation-resistant frozen waves in absorbing fluid,” Opt. Lett. 41(16), 3702–3705 (2015).
[Crossref]

L. A. Ambrosio and M. Zamboni-Rached, “Analytical approach of ordinary frozen waves for optical trapping and micromanipulation,” Appl. Opt. 54(10), 2584–2593 (2015).
[Crossref] [PubMed]

R. L. Garay-Avendaño and M. Zamboni-Rached, “Exact analytic solutions of Maxwell’s equations describing propagating nonparaxial electromagnetic beams,” Appl. Opt. 53, 4524–4531 (2014).
[Crossref]

T. A. Vieira, M. R. R. Gesualdi, and M. Zamboni-Rached, “Frozen waves: experimental generation,” Opt. Lett. 37(11), 2034–2036 (2012).
[Crossref] [PubMed]

M. Zamboni-Rached and E. Recami, “Subluminal wave bullets: Exact localized subluminal solutions to the wave equations,” Phys. Rev. A 77, 033824 (2008).
[Crossref]

M. Zamboni-Rached, “Stationary optical wave fields with arbitrary longitudinal shape by superposing equal frequency Bessel beams: Frozen Waves,” Opt. Express 12(17), 4001–4006 (2004).
[Crossref] [PubMed]

Zamboni-Rached, Michel

H. E. Hernandez Figueroa, E. Recami, and Michel Zamboni-Rached, Non-Diffracting Waves (Wiley-VCH Verlag, 2014).

Appl. Opt. (2)

Nat. Methods (1)

T. A. Planchon, L. Gao, D. E. Milkie, M. W. Davidson, J. A. Galbraith, C. G. Galbraith, and E. Betzig, “Rapid three-dimensional isotropic imaging of living cells using Bessel beam plane illumination,” Nat. Methods 8(5), 417–423 (2014).
[Crossref]

Nat. Photonics (1)

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
[Crossref]

Nature (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[Crossref] [PubMed]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. A (3)

M. Zamboni-Rached and M. Mojahedi, “Shaping finite-energy diffraction-and attenuation-resistant beams through Bessel-Gauss beam superposition,” Phys. Rev. A 92, 043839 (2015).
[Crossref]

M. Corato Zanarella and M. Zamboni-Rached, “Electromagnetic frozen waves with radial, azimuthal, linear, circular, and elliptical polarizations,” Phys. Rev. A 94, 053802 (2016).
[Crossref]

M. Zamboni-Rached and E. Recami, “Subluminal wave bullets: Exact localized subluminal solutions to the wave equations,” Phys. Rev. A 77, 033824 (2008).
[Crossref]

Other (1)

H. E. Hernandez Figueroa, E. Recami, and Michel Zamboni-Rached, Non-Diffracting Waves (Wiley-VCH Verlag, 2014).

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Figures (2)

Fig. 1
Fig. 1

The calculated spectra S(kz) and S′(kz) for the case of linear (a), radial (b), and azimuthal (c) polarization states. Figures (d–f) show the corresponding longitudinal intensity patterns, where the desired profile |F(z)|2 is shown in the continuous blue line and the actual field profile in red dashed line exhibiting a very good agreement.

Fig. 2
Fig. 2

The 3D intensities of: (a) Ex, for the case of linear polarization and (b) the corresponding axial field component Ez; (c) Eϕ, for the azimuthal polarization; (d) Eρ for the radial polarization and (e) the correspondent axial field component Ez.

Equations (15)

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Ψ ( ρ , z , t ) = exp ( i ω t ) ω / c ω / c d k z S ( k z ) J 0 ( ρ ω 2 c 2 k z 2 ) exp ( i k z z ) ,
Ψ ( ρ , z , t ) = exp ( i ω t ) n = F ( 2 n π K ) sinc ( ω 2 c 2 ρ 2 + ( ω c z + n π ) 2 ) ,
Ψ d z = exp ( i ω t ) ω / c ω / c d k z S ( k z ) i k z J 0 ( ρ ω 2 c 2 k z 2 ) exp ( i k z z ) .
S ( k z ) i k z = 1 K n = g ( 2 n π K ) exp ( in 2 π K k z ) , where , g ( 2 n π K ) = [ F ( z ) d z ] z = 2 n π / K ,
E z = E 0 exp ( i ω t ) cos ϕ n = g ( 2 n π K ) ρ sinc ( ω 2 c 2 ρ 2 + ( ω c z + n π ) 2 ) ,
F ( z ) = exp ( i Q z ) exp ( 1 2 ( z Z ) 8 ) Ai ( 3 π z 0.8 Z Z ) .
E ϕ ( ρ , z , t ) = c ω E 0 exp ( i ω t ) ω / c ω / c d k z S ( k z ) J 1 ( ρ ω 2 c 2 k z 2 ) exp ( i k z z ) .
E ϕ ( ρ , z , t ) = c ω E 0 exp ( i ω t ) n = F ( 2 n π K ) ρ sinc ( ω 2 c 2 ρ 2 + ( ω c z + n π ) 2 ) ,
B ρ ( ρ , z , t ) = i c ω 2 E 0 exp ( i ω t ) n = F ( 2 n π K ) 2 z ρ sinc ( ω 2 c 2 ρ 2 + ( ω c z + n π ) 2 ) ,
and B z ( ρ , z , t ) = i c ω 2 E 0 exp ( i ω t ) n = F ( 2 n π K ) 1 ρ ρ [ ρ ρ sinc ( ω 2 c 2 ρ 2 + ( ω c z + n π ) 2 ) ] ,
F ( z ) = exp ( i Q z ) [ exp ( 1 2 ( z + Z / 2 Z / 4 ) 8 ) + exp ( 1 2 ( z Z / 2 Z / 4 ) 8 ) ] ,
E ρ ( ρ , z , t ) = i c 2 ω 2 E 0 exp ( i ω t ) n = F ( 2 n π K ) 2 z ρ sinc ( ω 2 c 2 ρ 2 + ( ω c z + n π ) 2 ) ,
E ρ ( ρ , z , t ) = i c 2 ω 2 E 0 exp ( i ω t ) n = F ( 2 n π K ) 1 ρ ρ [ ρ ρ sinc ( ω 2 c 2 ρ 2 + ( ω c z + n π ) 2 ) ] ,
and B ϕ ( ρ , z , t ) = E 0 ω exp ( i ω t ) n = F ( 2 n π K ) ρ sinc ( ω 2 c 2 ρ 2 + ( ω c z + n π ) 2 ) .
F ( z ) = exp ( i Q z ) exp ( 1 2 ( z Z ) 8 ) cos ( 7 π z 2 Z ) ,

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