Abstract

Bessel beams have been extensive studied to date but are always created over a finite region inside the laboratory. Means to overcome this consider multi-element refractive designs to create beams that have a longitudinal dependent cone angle, thereby allowing for a far greater quasi non–diffracting propagation region. Here we outline a generalized approach for the creation of shape-invariant Bessel-like beams with a single phase-only element, and demonstrate it experimentally with a phase-only spatial light modulator. Our experimental results are in excellent agreement with theory, suggesting an easy-to-implement approach for long range, shape-invariant Bessel-like beams.

© 2015 Optical Society of America

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References

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  1. J. Durnin, “Exact solutions for nondiffracting beams. The scalar theory,” J. Opt. Soc. Am. B 4(4), 651 (1987).
    [Crossref]
  2. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
    [Crossref] [PubMed]
  3. D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46(1), 15–28 (2005).
    [Crossref]
  4. 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]
  5. A. Dudley, M. Lavery, M. J. Padgett, and A. Forbes, “Unraveling Bessel beams,” Opt. Photonics News 24(6), 22–29 (2013).
    [Crossref]
  6. A. N. Khilo, E. G. Katranji, and A. A. Ryzhevich, “Axicon-based Bessel resonator: Analytical description and experiment,” J. Opt. Soc. Am. A 18(8), 1986–1992 (2001).
    [Crossref] [PubMed]
  7. J. Rogel-Salazar, G. H. C. New, and S. Chavez-Cerda, “Bessel-Gauss beam optical resonator,” Opt. Commun. 190(1-6), 117–122 (2001).
    [Crossref]
  8. I. A. Litvin and A. Forbes, “Bessel–Gauss resonator with internal amplitude filter,” Opt. Commun. 281(9), 2385–2392 (2008).
    [Crossref]
  9. R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, “Generating superpositions of higher-order Bessel beams,” Opt. Express 17(26), 23389–23395 (2009).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
  22. I. A. Litvin, N. A. Khilo, A. Forbes, and V. N. Belyi, “Intra-cavity generation of Bessel-like beams with longitudinally dependent cone angles,” Opt. Express 18(5), 4701–4708 (2010).
    [Crossref] [PubMed]
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2014 (2)

M. McLaren, T. Mhlanga, M. J. Padgett, F. S. Roux, and A. Forbes, “Self-healing of quantum entanglement after an obstruction,” Nat. Commun. 5, 3248 (2014).
[Crossref] [PubMed]

A. Aiello and G. S. Agarwal, “Wave-optics description of self-healing mechanism in Bessel beams,” Opt. Lett. 39(24), 6819–6822 (2014).
[Crossref] [PubMed]

2013 (1)

A. Dudley, M. Lavery, M. J. Padgett, and A. Forbes, “Unraveling Bessel beams,” Opt. Photonics News 24(6), 22–29 (2013).
[Crossref]

2012 (3)

2011 (1)

R. Bowman, N. Muller, X. Zambrana-Puyalto, O. Jedrikiewicz, P. Di Tramapani, and M. J. Padgett, “Efficient generation of Bessel beam arrays by means of an SLM,” Eur. Phys. J. Spec. Top. 199(1), 159–166 (2011).
[Crossref]

2010 (3)

2009 (2)

2008 (1)

I. A. Litvin and A. Forbes, “Bessel–Gauss resonator with internal amplitude filter,” Opt. Commun. 281(9), 2385–2392 (2008).
[Crossref]

2005 (1)

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

2003 (1)

2001 (2)

A. N. Khilo, E. G. Katranji, and A. A. Ryzhevich, “Axicon-based Bessel resonator: Analytical description and experiment,” J. Opt. Soc. Am. A 18(8), 1986–1992 (2001).
[Crossref] [PubMed]

J. Rogel-Salazar, G. H. C. New, and S. Chavez-Cerda, “Bessel-Gauss beam optical resonator,” Opt. Commun. 190(1-6), 117–122 (2001).
[Crossref]

1997 (1)

1996 (1)

C. Paterson and R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124(1-2), 121–130 (1996).
[Crossref]

1989 (1)

1988 (1)

1987 (2)

J. Durnin, “Exact solutions for nondiffracting beams. The scalar theory,” J. Opt. Soc. Am. B 4(4), 651 (1987).
[Crossref]

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

Agarwal, G. S.

Agnew, M.

Aiello, A.

Alarcon, R. R.

Aruga, T.

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]

Belyi, V. N.

Bock, M.

Bowman, R.

R. Bowman, N. Muller, X. Zambrana-Puyalto, O. Jedrikiewicz, P. Di Tramapani, and M. J. Padgett, “Efficient generation of Bessel beam arrays by means of an SLM,” Eur. Phys. J. Spec. Top. 199(1), 159–166 (2011).
[Crossref]

Boyd, R. W.

Chattrapiban, N.

Chavez-Cerda, S.

J. Rogel-Salazar, G. H. C. New, and S. Chavez-Cerda, “Bessel-Gauss beam optical resonator,” Opt. Commun. 190(1-6), 117–122 (2001).
[Crossref]

Cofield, D.

Das, S. K.

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]

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

Di Tramapani, P.

R. Bowman, N. Muller, X. Zambrana-Puyalto, O. Jedrikiewicz, P. Di Tramapani, and M. J. Padgett, “Efficient generation of Bessel beam arrays by means of an SLM,” Eur. Phys. J. Spec. Top. 199(1), 159–166 (2011).
[Crossref]

Dudley, A.

A. Dudley, M. Lavery, M. J. Padgett, and A. Forbes, “Unraveling Bessel beams,” Opt. Photonics News 24(6), 22–29 (2013).
[Crossref]

R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, “Generating superpositions of higher-order Bessel beams,” Opt. Express 17(26), 23389–23395 (2009).
[Crossref] [PubMed]

Durnin, J.

J. Durnin, “Exact solutions for nondiffracting beams. The scalar theory,” J. Opt. Soc. Am. B 4(4), 651 (1987).
[Crossref]

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

Forbes, A.

M. McLaren, T. Mhlanga, M. J. Padgett, F. S. Roux, and A. Forbes, “Self-healing of quantum entanglement after an obstruction,” Nat. Commun. 5, 3248 (2014).
[Crossref] [PubMed]

A. Dudley, M. Lavery, M. J. Padgett, and A. Forbes, “Unraveling Bessel beams,” Opt. Photonics News 24(6), 22–29 (2013).
[Crossref]

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]

M. McLaren, M. Agnew, J. Leach, F. S. Roux, M. J. Padgett, R. W. Boyd, and A. Forbes, “Entangled Bessel-Gaussian beams,” Opt. Express 20(21), 23589–23597 (2012).
[Crossref] [PubMed]

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]

I. A. Litvin, N. A. Khilo, A. Forbes, and V. N. Belyi, “Intra-cavity generation of Bessel-like beams with longitudinally dependent cone angles,” Opt. Express 18(5), 4701–4708 (2010).
[Crossref] [PubMed]

R. Vasilyeu, A. Dudley, N. Khilo, and A. Forbes, “Generating superpositions of higher-order Bessel beams,” Opt. Express 17(26), 23389–23395 (2009).
[Crossref] [PubMed]

I. A. Litvin and A. Forbes, “Bessel–Gauss resonator with internal amplitude filter,” Opt. Commun. 281(9), 2385–2392 (2008).
[Crossref]

Friberg, A. T.

Grunwald, R.

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]

Hill, W. T.

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]

Jedrikiewicz, O.

R. Bowman, N. Muller, X. Zambrana-Puyalto, O. Jedrikiewicz, P. Di Tramapani, and M. J. Padgett, “Efficient generation of Bessel beam arrays by means of an SLM,” Eur. Phys. J. Spec. Top. 199(1), 159–166 (2011).
[Crossref]

Katranji, E. G.

Kazak, N.

Khilo, A. N.

Khilo, N.

Khilo, N. A.

Lavery, M.

A. Dudley, M. Lavery, M. J. Padgett, and A. Forbes, “Unraveling Bessel beams,” Opt. Photonics News 24(6), 22–29 (2013).
[Crossref]

Leach, J.

Litvin, I. A.

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]

McGloin, D.

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

McLaren, M.

M. McLaren, T. Mhlanga, M. J. Padgett, F. S. Roux, and A. Forbes, “Self-healing of quantum entanglement after an obstruction,” Nat. Commun. 5, 3248 (2014).
[Crossref] [PubMed]

M. McLaren, M. Agnew, J. Leach, F. S. Roux, M. J. Padgett, R. W. Boyd, and A. Forbes, “Entangled Bessel-Gaussian beams,” Opt. Express 20(21), 23589–23597 (2012).
[Crossref] [PubMed]

Mhlanga, T.

M. McLaren, T. Mhlanga, M. J. Padgett, F. S. Roux, and A. Forbes, “Self-healing of quantum entanglement after an obstruction,” Nat. Commun. 5, 3248 (2014).
[Crossref] [PubMed]

Miceli, J. J.

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

Morelos, F. J.

Muller, N.

R. Bowman, N. Muller, X. Zambrana-Puyalto, O. Jedrikiewicz, P. Di Tramapani, and M. J. Padgett, “Efficient generation of Bessel beam arrays by means of an SLM,” Eur. Phys. J. Spec. Top. 199(1), 159–166 (2011).
[Crossref]

New, G. H. C.

J. Rogel-Salazar, G. H. C. New, and S. Chavez-Cerda, “Bessel-Gauss beam optical resonator,” Opt. Commun. 190(1-6), 117–122 (2001).
[Crossref]

Padgett, M. J.

M. McLaren, T. Mhlanga, M. J. Padgett, F. S. Roux, and A. Forbes, “Self-healing of quantum entanglement after an obstruction,” Nat. Commun. 5, 3248 (2014).
[Crossref] [PubMed]

A. Dudley, M. Lavery, M. J. Padgett, and A. Forbes, “Unraveling Bessel beams,” Opt. Photonics News 24(6), 22–29 (2013).
[Crossref]

M. McLaren, M. Agnew, J. Leach, F. S. Roux, M. J. Padgett, R. W. Boyd, and A. Forbes, “Entangled Bessel-Gaussian beams,” Opt. Express 20(21), 23589–23597 (2012).
[Crossref] [PubMed]

R. Bowman, N. Muller, X. Zambrana-Puyalto, O. Jedrikiewicz, P. Di Tramapani, and M. J. Padgett, “Efficient generation of Bessel beam arrays by means of an SLM,” Eur. Phys. J. Spec. Top. 199(1), 159–166 (2011).
[Crossref]

Paterson, C.

C. Paterson and R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124(1-2), 121–130 (1996).
[Crossref]

Ramrez, H. C.

Rogel-Salazar, J.

J. Rogel-Salazar, G. H. C. New, and S. Chavez-Cerda, “Bessel-Gauss beam optical resonator,” Opt. Commun. 190(1-6), 117–122 (2001).
[Crossref]

Rogers, E. A.

Ropot, P.

Roux, F. S.

M. McLaren, T. Mhlanga, M. J. Padgett, F. S. Roux, and A. Forbes, “Self-healing of quantum entanglement after an obstruction,” Nat. Commun. 5, 3248 (2014).
[Crossref] [PubMed]

M. McLaren, M. Agnew, J. Leach, F. S. Roux, M. J. Padgett, R. W. Boyd, and A. Forbes, “Entangled Bessel-Gaussian beams,” Opt. Express 20(21), 23589–23597 (2012).
[Crossref] [PubMed]

Roy, R.

Ryzhevich, A. A.

Smith, R.

C. Paterson and R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124(1-2), 121–130 (1996).
[Crossref]

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]

Su, P. A. Q.

Turunen, J.

U’Ren, A. B.

Vasara, A.

Vasilyeu, R.

Vega, J. C. G.

Zambrana-Puyalto, X.

R. Bowman, N. Muller, X. Zambrana-Puyalto, O. Jedrikiewicz, P. Di Tramapani, and M. J. Padgett, “Efficient generation of Bessel beam arrays by means of an SLM,” Eur. Phys. J. Spec. Top. 199(1), 159–166 (2011).
[Crossref]

Appl. Opt. (2)

Contemp. Phys. (1)

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

Eur. Phys. J. Spec. Top. (1)

R. Bowman, N. Muller, X. Zambrana-Puyalto, O. Jedrikiewicz, P. Di Tramapani, and M. J. Padgett, “Efficient generation of Bessel beam arrays by means of an SLM,” Eur. Phys. J. Spec. Top. 199(1), 159–166 (2011).
[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 (2)

J. Opt. Soc. Am. B (1)

J. Durnin, “Exact solutions for nondiffracting beams. The scalar theory,” J. Opt. Soc. Am. B 4(4), 651 (1987).
[Crossref]

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]

Nat. Commun. (1)

M. McLaren, T. Mhlanga, M. J. Padgett, F. S. Roux, and A. Forbes, “Self-healing of quantum entanglement after an obstruction,” Nat. Commun. 5, 3248 (2014).
[Crossref] [PubMed]

Opt. Commun. (3)

J. Rogel-Salazar, G. H. C. New, and S. Chavez-Cerda, “Bessel-Gauss beam optical resonator,” Opt. Commun. 190(1-6), 117–122 (2001).
[Crossref]

I. A. Litvin and A. Forbes, “Bessel–Gauss resonator with internal amplitude filter,” Opt. Commun. 281(9), 2385–2392 (2008).
[Crossref]

C. Paterson and R. Smith, “Higher-order Bessel waves produced by axicon-type computer-generated holograms,” Opt. Commun. 124(1-2), 121–130 (1996).
[Crossref]

Opt. Express (6)

Opt. Lett. (2)

Opt. Photonics News (1)

A. Dudley, M. Lavery, M. J. Padgett, and A. Forbes, “Unraveling Bessel beams,” Opt. Photonics News 24(6), 22–29 (2013).
[Crossref]

Phys. Rev. Lett. (1)

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

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Figures (8)

Fig. 1
Fig. 1 Illustration of the difference between a conventional and shape-invariant Bessel beam: (a) A conventional Bessel beam is created with a digitally implemented axicon, resulting in a finite region where the Bessel field exists. (b) A novel phase screen is implemented to create a shape-invariant Bessel beam that exists in all space.
Fig. 2
Fig. 2 We trace ray between two planes and demand that they arrive with the same cone angle at the two extreme radii (origin and at rI).
Fig. 3
Fig. 3 Dependence of the cone angle from the geometrical optics and the wave optics approaches for different parameters of n and m. Design parameters where a12 = 2 × 10−3; a23 = 1 a13 = 1.2 × 10−3 where the subscripts refer to orders n and m, and rI = 3 mm.
Fig. 4
Fig. 4 Examples of the propagation characteristics of Bessel-like beams over an extended distance for rI = 1 mm, λ = 633 nm and (a) a12 = 2.8 10−3, (b) a13 = 8.7 10−3, (c) a23 = 0.2. Note that at the plane z = 0 the beam is a phase modulated Gaussian beam and not a Bessel function.
Fig. 5
Fig. 5 The far-field intensity pattern for two examples of n and m overlaid with this is a fit of an ideal Bessel beam intensity (red). The following parameters for the initial field were used: (a) (n = 1, m = 2): a12 = 2.8 10−3; (b) (n = 2, m = 3): a = 0.2, for rI = 1 mm, λ = 633 nm.
Fig. 6
Fig. 6 (a) Schematic of the experimental set-up. (b) Some example holograms for creating Bessel-like beams and the resulting measured intensity profiles. The laser was continuous wave at a wavelength of λ = 632.8 nm. It was expanded through a 3x magnification telescope (f1 = 100 mm and f2 = 300 mm) and directed to the SLM. The holograms on the SLM have a depicted range from 0 (white) through to 2π (black) phase shift in 256 steps. The beam after the SLM was Fourier transformed with lens L3 (focal length f3 = 200 mm) and the resulting beam measured on the CCD camera at the focal plane of this lens.
Fig. 7
Fig. 7 (a) Near-field and (b) Far-field images of the Bessel-like beam, both theoretically (TOP) and experimentally (BOTTOM). The near-field experimental image was at a distance of 62 cm from the optical element, while the far-field was measured at the focal plane of a lens with a focal length of 40 cm.
Fig. 8
Fig. 8 (a)The experimental verification of the z-dependent radial wave number k r (z)=k α c (z) , as given in Table 1 for following parameters of the phase screen and laser beam: λ = 633 nm, a = 0.01, rI = 0.85 mm. (b) Correspondingly obtained experimental intensity distributions at distances 0.8 m (1 and 2) and 1.6 m (3 and 4) and for n = 1, m = 2 (1 and 3) ; n = 2, m = 3 (2 and 4).

Tables (1)

Tables Icon

Table 1 Dependence of αcn,m angle (the cone angle of obtained Bessel beam transversal distribution) on the propagation distance z and the coefficients a and rI.

Equations (8)

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

φ(r)=exp[ i k 0 ( a r n +b r m ) ],
an r 0 n1 +bm r 0 m1 + r 0 z r s z =0.
b=a n m ( r I ) nm .
u(r) J 0 [k α c (z)r],
2 α c = α b .
{ an r c n1 +bm r c m1 + r c z =0 r c z α c ,
{ an r b n1 +bm r b m1 + r b z r I z =0 r b + r I z α b .
α c +an ( r I ) nm ( z α c ) m1 +an( r I nm ( r I +2z α c ) m1 + ( r I +2z α c ) n1 )an ( z α c ) n1 =0.

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