Abstract

Accelerating beams are wave packets that preserve their shape while propagating along curved trajectories. In this article, we extend the ray-based treatment in Part I of this series to nonparaxial accelerating fields in three dimensions, whose intensity maxima trace circular or helical paths. We also describe a simple procedure for finding mirror shapes that convert collimated beams into fields whose intensity features trace arcs that can extend well beyond 180 degrees.

© 2014 Optical Society of America

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References

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  1. M. A. Alonso and M. A. Bandres, “Generation of nonparaxial accelerating fields through mirrors. I: Two dimensions,” Opt. Express 22, 7124–7132 (2014).
    [CrossRef] [PubMed]
  2. M. P. Do-Carmo, Differential Geometry of Curves and Surfaces (Prentice Hall, Englewood Cliffs, 1976), pp. 16–22.
  3. C. Patterson and R. Smith, “Helicon waves: propagation-invariant waves in a rotating coordinate system,” Opt. Commun. 124, 131–140 (1996).
    [CrossRef]
  4. R. Piestun and J. Shamir, “Generalized propagation-invariant wave fields,” J. Opt. Soc. Am. A 15, 3039–3044 (1998).
    [CrossRef]
  5. S. R. P. Pavani and R. Piestun, “High-efficiency rotating point spread functions,” Opt. Express 16, 3484–3489 (2008).
    [CrossRef] [PubMed]
  6. S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Nat. Acad. Sci. 106, 2995–2999 (2009).
    [CrossRef] [PubMed]
  7. V. R. Daria, D. Z. Palima, and J. Glückstad, “Optical twists in phase and amplitude,” Opt. Express 19, 476–481 (2012).
    [CrossRef]
  8. D. B. Ruffner and D. G. Grier, “Optical conveyors: A class of active tractor beams,” Phys. Rev. Lett. 109, 163903 (2012).
    [CrossRef] [PubMed]
  9. M. A. Alonso and M. A. Bandres, “Spherical fields as nonparaxial accelerating waves,” Opt. Lett. 37, 5175–5177 (2012).
    [CrossRef] [PubMed]
  10. M. A. Bandres, M. A. Alonso, I. Kaminer, and M. Segev, “Three-dimensional accelerating electromagnetic waves,” Opt. Express 21, 13917–13929 (2013).
    [CrossRef] [PubMed]
  11. P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109, 203902 (2012).
    [CrossRef] [PubMed]
  12. A. Mathis, F. Courvoisier, R. Giust, L. Furfaro, M. Jacquot, L. Froehly, and J. M. Dudley, “Arbitrary nonparaxial accelerating periodic beams and spherical shaping of light,” Opt. Lett. 38, 2218–2220 (2013).
    [CrossRef] [PubMed]
  13. I. D. Chremmos and N. K. Efremidis, “Nonparaxial accelerating Bessel-like beams,” Phys. Rev. A 88, 063816 (2013).
    [CrossRef]
  14. S. Vo, K. Fuerschbach, K. P. Thompson, M. A. Alonso, and J. P. Rolland, “Airy beams: a geometric optics perspective,” J. Opt. Soc. Am. A 27, 2574–2582 (2010).
    [CrossRef]
  15. Y. Kaganovsky and E. Heyman, “Nonparaxial wave analysis of three-dimensional Airy beams,” J. Opt. Soc. Am. A 29, 671–688 (2012).
    [CrossRef]
  16. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999), 7, pp. 484–498.

2014 (1)

2013 (3)

2012 (5)

D. B. Ruffner and D. G. Grier, “Optical conveyors: A class of active tractor beams,” Phys. Rev. Lett. 109, 163903 (2012).
[CrossRef] [PubMed]

P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef] [PubMed]

V. R. Daria, D. Z. Palima, and J. Glückstad, “Optical twists in phase and amplitude,” Opt. Express 19, 476–481 (2012).
[CrossRef]

Y. Kaganovsky and E. Heyman, “Nonparaxial wave analysis of three-dimensional Airy beams,” J. Opt. Soc. Am. A 29, 671–688 (2012).
[CrossRef]

M. A. Alonso and M. A. Bandres, “Spherical fields as nonparaxial accelerating waves,” Opt. Lett. 37, 5175–5177 (2012).
[CrossRef] [PubMed]

2010 (1)

2009 (1)

S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Nat. Acad. Sci. 106, 2995–2999 (2009).
[CrossRef] [PubMed]

2008 (1)

1998 (1)

1996 (1)

C. Patterson and R. Smith, “Helicon waves: propagation-invariant waves in a rotating coordinate system,” Opt. Commun. 124, 131–140 (1996).
[CrossRef]

Aleahmad, P.

P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef] [PubMed]

Alonso, M. A.

Bandres, M. A.

Biteen, J. S.

S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Nat. Acad. Sci. 106, 2995–2999 (2009).
[CrossRef] [PubMed]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999), 7, pp. 484–498.

Chremmos, I. D.

I. D. Chremmos and N. K. Efremidis, “Nonparaxial accelerating Bessel-like beams,” Phys. Rev. A 88, 063816 (2013).
[CrossRef]

Christodoulides, D. N.

P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef] [PubMed]

Courvoisier, F.

Daria, V. R.

Do-Carmo, M. P.

M. P. Do-Carmo, Differential Geometry of Curves and Surfaces (Prentice Hall, Englewood Cliffs, 1976), pp. 16–22.

Dudley, J. M.

Efremidis, N. K.

I. D. Chremmos and N. K. Efremidis, “Nonparaxial accelerating Bessel-like beams,” Phys. Rev. A 88, 063816 (2013).
[CrossRef]

Froehly, L.

Fuerschbach, K.

Furfaro, L.

Giust, R.

Glückstad, J.

Grier, D. G.

D. B. Ruffner and D. G. Grier, “Optical conveyors: A class of active tractor beams,” Phys. Rev. Lett. 109, 163903 (2012).
[CrossRef] [PubMed]

Heyman, E.

Jacquot, M.

Kaganovsky, Y.

Kaminer, I.

M. A. Bandres, M. A. Alonso, I. Kaminer, and M. Segev, “Three-dimensional accelerating electromagnetic waves,” Opt. Express 21, 13917–13929 (2013).
[CrossRef] [PubMed]

P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef] [PubMed]

Liu, N.

S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Nat. Acad. Sci. 106, 2995–2999 (2009).
[CrossRef] [PubMed]

Lord, S. J.

S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Nat. Acad. Sci. 106, 2995–2999 (2009).
[CrossRef] [PubMed]

Mathis, A.

Mills, M. S.

P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef] [PubMed]

Miri, M.-A.

P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef] [PubMed]

Moerner, W. E.

S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Nat. Acad. Sci. 106, 2995–2999 (2009).
[CrossRef] [PubMed]

Palima, D. Z.

Patterson, C.

C. Patterson and R. Smith, “Helicon waves: propagation-invariant waves in a rotating coordinate system,” Opt. Commun. 124, 131–140 (1996).
[CrossRef]

Pavani, S. R. P.

S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Nat. Acad. Sci. 106, 2995–2999 (2009).
[CrossRef] [PubMed]

S. R. P. Pavani and R. Piestun, “High-efficiency rotating point spread functions,” Opt. Express 16, 3484–3489 (2008).
[CrossRef] [PubMed]

Piestun, R.

S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Nat. Acad. Sci. 106, 2995–2999 (2009).
[CrossRef] [PubMed]

S. R. P. Pavani and R. Piestun, “High-efficiency rotating point spread functions,” Opt. Express 16, 3484–3489 (2008).
[CrossRef] [PubMed]

R. Piestun and J. Shamir, “Generalized propagation-invariant wave fields,” J. Opt. Soc. Am. A 15, 3039–3044 (1998).
[CrossRef]

Rolland, J. P.

Ruffner, D. B.

D. B. Ruffner and D. G. Grier, “Optical conveyors: A class of active tractor beams,” Phys. Rev. Lett. 109, 163903 (2012).
[CrossRef] [PubMed]

Segev, M.

M. A. Bandres, M. A. Alonso, I. Kaminer, and M. Segev, “Three-dimensional accelerating electromagnetic waves,” Opt. Express 21, 13917–13929 (2013).
[CrossRef] [PubMed]

P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef] [PubMed]

Shamir, J.

Smith, R.

C. Patterson and R. Smith, “Helicon waves: propagation-invariant waves in a rotating coordinate system,” Opt. Commun. 124, 131–140 (1996).
[CrossRef]

Thompson, K. P.

Thompson, M. A.

S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Nat. Acad. Sci. 106, 2995–2999 (2009).
[CrossRef] [PubMed]

Twieg, R. J.

S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Nat. Acad. Sci. 106, 2995–2999 (2009).
[CrossRef] [PubMed]

Vo, S.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999), 7, pp. 484–498.

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

Opt. Commun. (1)

C. Patterson and R. Smith, “Helicon waves: propagation-invariant waves in a rotating coordinate system,” Opt. Commun. 124, 131–140 (1996).
[CrossRef]

Opt. Express (4)

Opt. Lett. (2)

Phys. Rev. A (1)

I. D. Chremmos and N. K. Efremidis, “Nonparaxial accelerating Bessel-like beams,” Phys. Rev. A 88, 063816 (2013).
[CrossRef]

Phys. Rev. Lett. (2)

D. B. Ruffner and D. G. Grier, “Optical conveyors: A class of active tractor beams,” Phys. Rev. Lett. 109, 163903 (2012).
[CrossRef] [PubMed]

P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109, 203902 (2012).
[CrossRef] [PubMed]

Proc. Nat. Acad. Sci. (1)

S. R. P. Pavani, M. A. Thompson, J. S. Biteen, S. J. Lord, N. Liu, R. J. Twieg, R. Piestun, and W. E. Moerner, “Three-dimensional, single-molecule fluorescence imaging beyond the diffraction limit by using a double-helix point spread function,” Proc. Nat. Acad. Sci. 106, 2995–2999 (2009).
[CrossRef] [PubMed]

Other (2)

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1999), 7, pp. 484–498.

M. P. Do-Carmo, Differential Geometry of Curves and Surfaces (Prentice Hall, Englewood Cliffs, 1976), pp. 16–22.

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Figures (7)

Fig. 1
Fig. 1

Transverse cross-sections of the intensity profile and corresponding caustics (white lines) for accelerated fields whose caustic sheets are sections of (a) two confocal paraboloids, (b) a two-sheeted cone and a sphere, (c) a two-sheeted hyperboloid and a prolate spheroid, and (d) a one-sheeted hyperboloid and an oblate spheroid. Also shown in (e) are the corresponding intensity and caustic cross sections for a paraxial Airy beam and (f) paraxial accelerating parabolic beam.

Fig. 2
Fig. 2

(a) View of the projection of the ray onto the xz plane. (b) View of the projection onto a plane parallel to the ray and to the y axis. In both parts, the black dot indicates r0 = (ρ̄ cosϕ̄, ȳ, −ρ̄ sinϕ̄).

Fig. 3
Fig. 3

View of the two caustic sheets and a ray with ϕ̄ = 0 and given .

Fig. 4
Fig. 4

Mirrors that generate symmetric paraboloidal accelerated fields and the resulting caustic segment (shown in orange) following illumination by a collimated beam that is (a) normal and (b) parallel to the caustic’s axis of symmetry, for T = 6.2R, and −π/4 ≤ π/4, and (a) −3π/4 ≤ ϕ̄ ≤ 3π/4, (b) −πϕ̄π.

Fig. 5
Fig. 5

(a,b) Mirrors that generate helical accelerated fields and the resulting caustic segment (shown in orange) following illumination by a collimated beam that is (a) normal and (b) parallel to the caustic’s axis of symmetry, for T = 6.2R, and −π/4 ≤ π/4, and (a) −3π/4 ≤ ϕ̄ ≤ 3π/4, (b) −πϕ̄π.

Fig. 6
Fig. 6

(a) Transverse and (b) longitudinal intensity patterns for a helical version of a symmetric paraboloidal field (a = 0) for kρ̄0 = 50 and γ = π/4. The dashed orange circle in (a) and sinusoidal in (b) show the projections onto the corresponding planes of the helical path followed by the caustic’s ridge. Note that both figures contain the same information, and that varying y would simply cause the transverse profile in (a) to rotate, while varying ϕ would cause the radial section in (b) to move laterally.

Fig. 7
Fig. 7

(a–d) Ray and (e,f) intensity cross sections over the z = 0 plane within −80 ≤ ky ≤ 80 and 48 ≤ kx ≤ 150, for fields with ȳ(u) = ρ̄0 sinh au2/(1 − u2) for kρ̄0 = 50 and different values of γ and a. In (a–d), blue (green) dots correspond to rays for which > (<)π/2 − γ.

Equations (26)

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[ r × ( k u ) ] y = k ρ ¯ cos α ¯ = m ,
Φ ( α ¯ , ϕ ¯ ) ϕ = m k .
Φ ( α ¯ , ϕ ¯ ) α ¯ = y ¯ cos α ¯
Φ ( α ¯ , ϕ ¯ ) = m k ϕ ¯ Y ¯ ( sin α ¯ ) , Y ¯ ( u ) = u y ¯ ( u ) d u ,
A ( α ¯ , ϕ ¯ ) = 𝒜 ( α ¯ , ϕ ¯ ) exp [ i k Φ ( α ¯ , ϕ ¯ ) ] ,
ρ ¯ 0 = m k .
tan α ¯ = h n ( x 0 2 + z n 2 ) y ¯ z n , tan α ¯ = d h n ( x 0 2 + z 2 ) d z | z = z n ,
tan α ¯ = h n ( ρ n ) y ¯ ρ n 2 x 0 2 , tan α ¯ = ρ n 2 x 0 2 ρ n h n ( ρ n ) .
h n ( ρ n ) = y ¯ ( sin α ¯ ) + tan α ¯ ρ n 2 ρ ¯ 0 2 sec 2 α ¯ ,
h n ( ρ n ) = ρ n tan α ¯ ρ n 2 ρ ¯ 0 2 sec 2 α ¯ .
h 1 , 2 ( ρ ) = e a ρ 2 ρ ¯ 0 2 2 ρ ¯ 0 , y ¯ ( sin α ¯ ) = ρ ¯ 0 sinh a tan 2 α ¯ .
k Φ = k ρ ¯ 0 sinh a [ sin α ¯ ln ( 1 + sin α ¯ cos α ¯ ) ] + m ϕ ¯ ,
h n ( ρ ¯ 0 ) Δ n y ¯ ( 0 ) α ¯ 2 2 + α ¯ 2 ρ ¯ 0 Δ n ρ ¯ 0 2 α ¯ 2 ,
h n ( ρ ¯ 0 ) 2 ρ ¯ 0 Δ n ρ ¯ 0 2 α ¯ 2 ( ρ ¯ 0 + Δ n ) α ¯ ,
h 1 , 2 ( ρ ¯ 0 ) = y ¯ ( 0 ) 2 ρ ¯ 0 ± [ y ¯ ( 0 ) 2 ρ ¯ 0 ] 2 + 1 .
n R u R = T + Φ ( α ¯ , ϕ ¯ ) ,
τ ( α ¯ , ϕ ¯ ) = n r 0 u r 0 T Φ ( α ¯ , ϕ ¯ ) ( 1 n u ) = n r 0 T ρ ¯ 0 ϕ ¯ + Y ¯ ( sin α ¯ ) y ¯ sin α ¯ ( 1 n u ) .
R ( α ¯ , ϕ ¯ ) = r 0 ( α ¯ , ϕ ¯ ) + τ ( α ¯ , ϕ ¯ ) u ( α ¯ , ϕ ¯ ) .
Y ¯ γ ( sin α ¯ , ϕ ¯ ) = Y ¯ [ cos ( α ¯ + γ ) ] + ρ ¯ 0 ϕ ¯ cot γ sin α ¯ .
Φ γ ( α ¯ , ϕ ¯ ) = ρ ¯ 0 ϕ ¯ 1 sin α ¯ cos γ sin γ Y ¯ [ cos ( α ¯ + γ ) ] .
ρ ¯ ( α ¯ ) = 1 sin α ¯ cos γ sin γ cos α ¯ ρ ¯ 0 .
U ( r ) = 0 2 π π / 2 π / 2 𝒜 ( α ¯ , ϕ ¯ ) exp { i k [ Φ γ ( α ¯ , ϕ ¯ ) + u r ] } cos α ¯ d α ¯ d ϕ ¯ = π / 2 π / 2 0 2 π 𝒜 ( α ¯ , ϕ ¯ ) exp { i k [ ρ ¯ 0 ϕ ¯ 1 sin α ¯ cos γ sin γ + ρ cos ( ϕ ¯ ϕ ) cos α ¯ ] } d ϕ ¯ × exp ( i k { y sin α ¯ Y ¯ [ cos ( α ¯ + γ ) ] } ) cos α ¯ d α ¯ .
α ¯ n = arcsin [ ( 1 n sin γ k ρ ¯ 0 ) sec γ ] .
U H ( r ) = n = n min n max 𝒜 ( α ¯ n ) cos 2 α ¯ n J n ( k ρ cos α ¯ n ) × exp [ i ( n ϕ + { y sin α ¯ n Y ¯ [ cos ( α ¯ + γ ) ] } ) ] ,
h n ( ρ n ) = sin ( α ¯ + γ ) cos α ¯ y ¯ [ cos ( α ¯ + γ ) ] + tan α ¯ ρ n 2 κ 2 ρ ¯ 0 2 + ρ ¯ 0 cot γ arctan [ ρ n 2 κ 2 ρ ¯ 0 2 κ ρ ¯ 0 ] ,
h n ( ρ n ) = ρ n 2 tan α ¯ κ ρ ¯ 0 2 cot γ ρ n ρ n 2 κ 2 ρ ¯ 0 2 ,

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