Abstract

We discuss a method to transform any optical field, with finite frequency bandwidth, into a shape invariant beam with transverse scaling, dependent on the propagation distance. The method consists in modulating the field with a quadratic phase of appropriate curvature radius. As a particular application, we employ the method to extend the existence region of a finite non-diffracting field to an unlimited propagation range.

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References

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  1. J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
    [Crossref] [PubMed]
  2. D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
    [Crossref]
  3. F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4(11), 780–785 (2010).
    [Crossref]
  4. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987).
    [Crossref]
  5. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25(20), 1493–1495 (2000).
    [Crossref] [PubMed]
  6. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29(1), 44–46 (2004).
    [Crossref] [PubMed]
  7. S. N. Khonina, V. V. Kotlyar, V. A. Soifer, J. Lautanen, M. Honkanen, and J. Turunen, “Generating a couple of rotating nondiffracting beams using a binary-phase DOE,” Optik (Stuttg.) 110, 137–144 (1999).
  8. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
    [Crossref] [PubMed]
  9. M. A. Bandres and J. C. Gutiérrez-Vega, “Airy-Gauss beams and their transformation by paraxial optical systems,” Opt. Express 15(25), 16719–16728 (2007).
    [Crossref] [PubMed]
  10. S. N. Khonina, “Specular and vortical Airy beams,” Opt. Commun. 284(19), 4263–4271 (2011).
    [Crossref]
  11. A. E. Siegman, Lasers (Universe Science Books, 1986) pp. 642–652.
  12. Z. Jaroszewicz and J. Morales, “Lens axicons: systems composed of a diverging aberrated lens and a perfect converging lens,” J. Opt. Soc. Am. A 15(9), 2383–2390 (1998).
    [Crossref]
  13. T. Aruga, S. W. Li, S. Yoshikado, M. Takabe, and R. Li, “Nondiffracting narrow light beam with small atmospheric turbulence-influenced propagation,” Appl. Opt. 38(15), 3152–3156 (1999).
    [Crossref] [PubMed]
  14. V. Belyi, A. Forbes, N. Kazak, N. Khilo, and P. Ropot, “Bessel-like beams with z-dependent cone angles,” Opt. Express 18(3), 1966–1973 (2010).
    [Crossref] [PubMed]
  15. Y. Ismail, N. Khilo, V. Belyi, and A. Forbes, “Shape invariant higher-order Bessel-like beams carrying orbital angular momentum,” J. Opt. 14(8), 085703 (2012).
    [Crossref]
  16. J. W. Goodman, Introduction to Fourier Optics, 2 ed. (McGraw-Hill, 1988) pp. 73–75.
  17. M. J. Caola, “Self-Fourier functions,” J. Phys. Math. Gen. 24(19), L1143–L1144 (1991).
    [Crossref]
  18. A. W. Lohmann and D. Mendlovic, “Self-Fourier objects and other self-transform objects,” J. Opt. Soc. Am. A 9(11), 2009–2012 (1992).
    [Crossref]
  19. S. G. Lipson, “Self-Fourier objects and other self-transform objects: comment,” J. Opt. Soc. Am. A 10(9), 2088–2089 (1993).
    [Crossref]

2012 (1)

Y. Ismail, N. Khilo, V. Belyi, and A. Forbes, “Shape invariant higher-order Bessel-like beams carrying orbital angular momentum,” J. Opt. 14(8), 085703 (2012).
[Crossref]

2011 (1)

S. N. Khonina, “Specular and vortical Airy beams,” Opt. Commun. 284(19), 4263–4271 (2011).
[Crossref]

2010 (2)

V. Belyi, A. Forbes, N. Kazak, N. Khilo, and P. Ropot, “Bessel-like beams with z-dependent cone angles,” Opt. Express 18(3), 1966–1973 (2010).
[Crossref] [PubMed]

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4(11), 780–785 (2010).
[Crossref]

2007 (2)

2005 (1)

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

2004 (1)

2000 (1)

1999 (2)

T. Aruga, S. W. Li, S. Yoshikado, M. Takabe, and R. Li, “Nondiffracting narrow light beam with small atmospheric turbulence-influenced propagation,” Appl. Opt. 38(15), 3152–3156 (1999).
[Crossref] [PubMed]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, J. Lautanen, M. Honkanen, and J. Turunen, “Generating a couple of rotating nondiffracting beams using a binary-phase DOE,” Optik (Stuttg.) 110, 137–144 (1999).

1998 (1)

1993 (1)

1992 (1)

1991 (1)

M. J. Caola, “Self-Fourier functions,” J. Phys. Math. Gen. 24(19), L1143–L1144 (1991).
[Crossref]

1987 (2)

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

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

Aruga, T.

Bandres, M. A.

Belyi, V.

Y. Ismail, N. Khilo, V. Belyi, and A. Forbes, “Shape invariant higher-order Bessel-like beams carrying orbital angular momentum,” J. Opt. 14(8), 085703 (2012).
[Crossref]

V. Belyi, A. Forbes, N. Kazak, N. Khilo, and P. Ropot, “Bessel-like beams with z-dependent cone angles,” Opt. Express 18(3), 1966–1973 (2010).
[Crossref] [PubMed]

Caola, M. J.

M. J. Caola, “Self-Fourier functions,” J. Phys. Math. Gen. 24(19), L1143–L1144 (1991).
[Crossref]

Chávez-Cerda, S.

Christodoulides, D. N.

Dholakia, K.

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

Durnin, J.

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

Eberly, J. H.

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

Fahrbach, F. O.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4(11), 780–785 (2010).
[Crossref]

Forbes, A.

Y. Ismail, N. Khilo, V. Belyi, and A. Forbes, “Shape invariant higher-order Bessel-like beams carrying orbital angular momentum,” J. Opt. 14(8), 085703 (2012).
[Crossref]

V. Belyi, A. Forbes, N. Kazak, N. Khilo, and P. Ropot, “Bessel-like beams with z-dependent cone angles,” Opt. Express 18(3), 1966–1973 (2010).
[Crossref] [PubMed]

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]

Gutiérrez-Vega, J. C.

Honkanen, M.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, J. Lautanen, M. Honkanen, and J. Turunen, “Generating a couple of rotating nondiffracting beams using a binary-phase DOE,” Optik (Stuttg.) 110, 137–144 (1999).

Ismail, Y.

Y. Ismail, N. Khilo, V. Belyi, and A. Forbes, “Shape invariant higher-order Bessel-like beams carrying orbital angular momentum,” J. Opt. 14(8), 085703 (2012).
[Crossref]

Iturbe-Castillo, M. D.

Jaroszewicz, Z.

Kazak, N.

Khilo, N.

Y. Ismail, N. Khilo, V. Belyi, and A. Forbes, “Shape invariant higher-order Bessel-like beams carrying orbital angular momentum,” J. Opt. 14(8), 085703 (2012).
[Crossref]

V. Belyi, A. Forbes, N. Kazak, N. Khilo, and P. Ropot, “Bessel-like beams with z-dependent cone angles,” Opt. Express 18(3), 1966–1973 (2010).
[Crossref] [PubMed]

Khonina, S. N.

S. N. Khonina, “Specular and vortical Airy beams,” Opt. Commun. 284(19), 4263–4271 (2011).
[Crossref]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, J. Lautanen, M. Honkanen, and J. Turunen, “Generating a couple of rotating nondiffracting beams using a binary-phase DOE,” Optik (Stuttg.) 110, 137–144 (1999).

Kotlyar, V. V.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, J. Lautanen, M. Honkanen, and J. Turunen, “Generating a couple of rotating nondiffracting beams using a binary-phase DOE,” Optik (Stuttg.) 110, 137–144 (1999).

Lautanen, J.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, J. Lautanen, M. Honkanen, and J. Turunen, “Generating a couple of rotating nondiffracting beams using a binary-phase DOE,” Optik (Stuttg.) 110, 137–144 (1999).

Li, R.

Li, S. W.

Lipson, S. G.

Lohmann, A. W.

McGloin, D.

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

Mendlovic, D.

Miceli, J.

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

Morales, J.

Padovani, C.

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

Rohrbach, A.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4(11), 780–785 (2010).
[Crossref]

Ropot, P.

Simon, P.

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4(11), 780–785 (2010).
[Crossref]

Siviloglou, G. A.

Soifer, V. A.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, J. Lautanen, M. Honkanen, and J. Turunen, “Generating a couple of rotating nondiffracting beams using a binary-phase DOE,” Optik (Stuttg.) 110, 137–144 (1999).

Takabe, M.

Turunen, J.

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, J. Lautanen, M. Honkanen, and J. Turunen, “Generating a couple of rotating nondiffracting beams using a binary-phase DOE,” Optik (Stuttg.) 110, 137–144 (1999).

Yoshikado, S.

Appl. Opt. (1)

Contemp. Phys. (1)

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
[Crossref]

J. Opt. (1)

Y. Ismail, N. Khilo, V. Belyi, and A. Forbes, “Shape invariant higher-order Bessel-like beams carrying orbital angular momentum,” J. Opt. 14(8), 085703 (2012).
[Crossref]

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

J. Phys. Math. Gen. (1)

M. J. Caola, “Self-Fourier functions,” J. Phys. Math. Gen. 24(19), L1143–L1144 (1991).
[Crossref]

Nat. Photonics (1)

F. O. Fahrbach, P. Simon, and A. Rohrbach, “Microscopy with self-reconstructing beams,” Nat. Photonics 4(11), 780–785 (2010).
[Crossref]

Opt. Commun. (2)

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

S. N. Khonina, “Specular and vortical Airy beams,” Opt. Commun. 284(19), 4263–4271 (2011).
[Crossref]

Opt. Express (2)

Opt. Lett. (3)

Optik (Stuttg.) (1)

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, J. Lautanen, M. Honkanen, and J. Turunen, “Generating a couple of rotating nondiffracting beams using a binary-phase DOE,” Optik (Stuttg.) 110, 137–144 (1999).

Phys. Rev. Lett. (1)

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[Crossref] [PubMed]

Other (2)

A. E. Siegman, Lasers (Universe Science Books, 1986) pp. 642–652.

J. W. Goodman, Introduction to Fourier Optics, 2 ed. (McGraw-Hill, 1988) pp. 73–75.

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Figures (10)

Fig. 1
Fig. 1 Generation of the propagated fields in the far field domain of the field G0(−x/λR,y/λR).
Fig. 2
Fig. 2 (a) Transverse amplitudes of the SG field in Eq. (10) and (b) its Fourier spectrum. In (c) we show the transverse amplitude of the Fourier transform of the PMSG field defined in Eq. (11).
Fig. 3
Fig. 3 Transverse amplitude of the propagated PMSG field, at the range [0, 2R] for z0.
Fig. 4
Fig. 4 (a) Transverse amplitudes of the BG field in Eq. (12) and (b) its Fourier spectrum. In (c) we show the transverse amplitude of the Fourier transform of the PMBG field g(x,y) obtained with A = 1/4.
Fig. 5
Fig. 5 Transverse amplitudes of the propagated (a) BG beam and (b) PMBG beam (for A = 1/4) at the propagation range [0,1.5zC]. The transverse amplitudes at three specific propagation distances for the BG beam and the PMBG beam are displayed in (c, d), respectively.
Fig. 6
Fig. 6 Transverse amplitude of the propagated PMBG beam (for A = 1/4) at the propagation range [0, zC/4]. In this case the transverse coordinate normalization is the same for every z0.
Fig. 7
Fig. 7 Transverse amplitude of the Fourier transform of the PMBG field [Eq. (13)] obtained with A = 1, 2, 3, and 4.
Fig. 8
Fig. 8 Normalized transverse amplitudes of the propagated PMBG beams for A = 3 and 4, at z0 = 6zC.
Fig. 9
Fig. 9 Transverse intensities for (a) the BG beam and for the Fourier spectra of the PMBG beams, obtained with A equal to (b) 1, (c) 2, and (d)3.
Fig. 10
Fig. 10 Intensity deviations (a) E and (b) ln(E) for the propagated BG beam (black) and for the propagated PMBG beams with parameter A equal to 1/4 (red), 1 (blue), and 2 (green).

Equations (14)

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E(x,y, z 0 )=exp[ ik( x 2 + y 2 )/(2 z 0 ) ]
g(x,y)=E( x,y,R ) g 0 ( x,y ).
G(u,v)= Ω G 0 ( u 0 , v 0 ) exp(iπλR|V V 0 | 2 )d u 0 d v 0 ,
λR ρ C 2 =A<<1.
G(u,v)=exp[ iπλR( u 2 + v 2 ) ] g 0 (λRu,λRv).
g p (x,y)= 1 {G(u,v)exp[iπλ z 0 ( u 2 + v 2 )]},
g p (x,y)=E( x,y,MR ) g 0 ( x M , y M ),
M=1+ z 0 /R.
g 1 (x)=sinc(x/p)exp[ (x/ w 0 ) 2 ],
g 0 (x,y)=sinc( x p )sinc( y p )exp( x 2 + y 2 w 0 2 )
g(x,y)=exp[i10π( x 2 + y 2 )/ p 2 ] g 0 (x,y).
g 0 (x,y)= J 0 (2π ρ 0 r)exp( r 2 / w 0 2 ),
g(x,y)=exp(iπ ρ 0 2 r 2 /A) g 0 (x,y).
E= 0 r p r [ I 0 (r)αI(r)] 2 dr 0 r p r [ I 0 (r)] 2 dr .

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