Abstract

We have theoretically and experimentally considered transforming a single optical vortex beam into the vortex avalanche caused by weak local perturbations of the holographic grating responsible for the beam shaping. The vortex avalanche is accompanied by a sharp change of the orbital angular momentum (OAM) so that its dependence on the holographic grating perturbations forms the OAM spectrum. We revealed that the restored vortex beam has anomalous regions of the OAM spectrum in a form of sharp dips and bursts (resonances) that occur when the integer perturbation parameter coincides with a topological charge of one of the vortex modes. We found also that the intensity of the perturbed beam is nonuniformly distributed among the vortex modes with positive and negative topological charges. They form two groups of satellites with clearly marked intensity maxima. As the grating perturbation increases, the initial beam intensity is almost completely pumped over into the vortex avalanche with nearly the same energy redistribution among the satellites.

© 2019 Optical Society of America

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References

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2018 (3)

2017 (3)

N. Alperin and M. E. Siemens, “Angular momentum of topologically structured darkness,” Phys. Rev. Lett. 119, 203902 (2017).
[Crossref]

C. N. Alexeyev, Y. A. Egorov, and A. V. Volyar, “Mutual transformations of fractional-order and integer-order optical vortices,” Phys. Rev. A 96, 063807 (2017).
[Crossref]

S. M. H. Luk, N.-H. Kwong, P. Lewandowski, S. Schumacher, and R. Binder, “Optically controlled orbital angular momentum generation in a polaritonic quantum fluid,” Phys. Rev. Lett. 119, 113903 (2017).
[Crossref]

2016 (3)

2014 (2)

2012 (1)

2009 (1)

2008 (2)

J. C. Gutiérrez-Vega and C. López-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A 10, 015009 (2008).
[Crossref]

J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16, 993–1006 (2008).
[Crossref]

2007 (2)

J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54, 1723–1738 (2007).
[Crossref]

A. P. Kiselev, “Localized light waves: paraxial and exact solutions of the wave equation (a review),” Opt. Spectrosc. 102, 603–622 (2007).
[Crossref]

2004 (2)

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[Crossref]

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6, 259–268 (2004).
[Crossref]

2003 (2)

1998 (1)

M. V. Berry, “Paraxial beams of spinning light,” Proc. SPIE 3487, 6–11 (1998).
[Crossref]

Akimova, Y.

Aleksandrov, R. V.

Alexeyev, C.

Alexeyev, C. N.

C. N. Alexeyev, Y. A. Egorov, and A. V. Volyar, “Mutual transformations of fractional-order and integer-order optical vortices,” Phys. Rev. A 96, 063807 (2017).
[Crossref]

Allen, L.

L. Allen and M. Padgett, “The orbital angular momentum of light: an introduction,” in Twisted Photons: Applications of Light with Orbital Angular Momentum (Wiley, 2011), p. 288.

Alperin, N.

N. Alperin and M. E. Siemens, “Angular momentum of topologically structured darkness,” Phys. Rev. Lett. 119, 203902 (2017).
[Crossref]

Alperin, S. N.

Barnett, S. M.

J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16, 993–1006 (2008).
[Crossref]

J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54, 1723–1738 (2007).
[Crossref]

Bekshaev, A. Y.

Berry, M. V.

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6, 259–268 (2004).
[Crossref]

M. V. Berry, “Paraxial beams of spinning light,” Proc. SPIE 3487, 6–11 (1998).
[Crossref]

Binder, R.

S. M. H. Luk, N.-H. Kwong, P. Lewandowski, S. Schumacher, and R. Binder, “Optically controlled orbital angular momentum generation in a polaritonic quantum fluid,” Phys. Rev. Lett. 119, 113903 (2017).
[Crossref]

Bretsko, M.

Cui, H.

Egorov, Y.

Egorov, Y. A.

A. V. Volyar and Y. A. Egorov, “Super pulses of orbital angular momentum in fractional-order spiroid vortex beams,” Opt. Lett. 43, 74–77 (2018).
[Crossref]

C. N. Alexeyev, Y. A. Egorov, and A. V. Volyar, “Mutual transformations of fractional-order and integer-order optical vortices,” Phys. Rev. A 96, 063807 (2017).
[Crossref]

Fadeyeva, T.

Fadeyeva, T. A.

Flossmann, F.

Flusser, J.

J. Flusser, B. Zitova, and T. Suk, Moments and Moment Invariants in Pattern Recognition (Wiley, 2009).

Franke-Arnold, S.

J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16, 993–1006 (2008).
[Crossref]

J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54, 1723–1738 (2007).
[Crossref]

Garcia, H.

Gbur, G. J.

G. J. Gbur, Singular Optics (CRC Press, 2017).

Golub, M. A.

V. A. Soifer and M. A. Golub, Laser Beam Mode Selection by Computer-Generated Holograms (CRC Press, 1994).

Gopinath, J. T.

Götte, J. B.

J. B. Götte, K. O’Holleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, “Light beams with fractional orbital angular momentum and their vortex structure,” Opt. Express 16, 993–1006 (2008).
[Crossref]

J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54, 1723–1738 (2007).
[Crossref]

Grier, D. G.

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

Gupta, D. L.

H. Singh, D. L. Gupta, and A. K. Singh, “Quantum key distribution protocols: a review,” J. Comput. Eng. 16, 1–9 (2014).
[Crossref]

Gutierrez-Vega, J.

Gutiérrez-Vega, J. C.

J. C. Gutiérrez-Vega and C. López-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A 10, 015009 (2008).
[Crossref]

Hu, Z.-A.

Huang, Y.-Q.

Kazanskiy, N. L.

S. N. Khonina, N. L. Kazanskiy, and V. A. Soifer, “Optical vortices in a fiber: mode division multiplexing and multimode selfimaging,” in Recent Progress in Optical Fiber Research (Intech, 2012), p. 450.

Khonina, S. N.

S. N. Khonina, N. L. Kazanskiy, and V. A. Soifer, “Optical vortices in a fiber: mode division multiplexing and multimode selfimaging,” in Recent Progress in Optical Fiber Research (Intech, 2012), p. 450.

Kiselev, A. P.

A. P. Kiselev, “Localized light waves: paraxial and exact solutions of the wave equation (a review),” Opt. Spectrosc. 102, 603–622 (2007).
[Crossref]

Kotlyar, V. V.

Kovalev, A. A.

Kwong, N.-H.

S. M. H. Luk, N.-H. Kwong, P. Lewandowski, S. Schumacher, and R. Binder, “Optically controlled orbital angular momentum generation in a polaritonic quantum fluid,” Phys. Rev. Lett. 119, 113903 (2017).
[Crossref]

Leach, J.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[Crossref]

Lewandowski, P.

S. M. H. Luk, N.-H. Kwong, P. Lewandowski, S. Schumacher, and R. Binder, “Optically controlled orbital angular momentum generation in a polaritonic quantum fluid,” Phys. Rev. Lett. 119, 113903 (2017).
[Crossref]

López-Mariscal, C.

J. C. Gutiérrez-Vega and C. López-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A 10, 015009 (2008).
[Crossref]

Luk, S. M. H.

S. M. H. Luk, N.-H. Kwong, P. Lewandowski, S. Schumacher, and R. Binder, “Optically controlled orbital angular momentum generation in a polaritonic quantum fluid,” Phys. Rev. Lett. 119, 113903 (2017).
[Crossref]

Luo, A.-P.

Luo, Z.-C.

Masajada, J.

P. Senthilkumaran, J. Masajada, and S. Sato, Singular Optics (International Journal of Optics, 2012).

Molina-Terriza, G.

G. Molina-Terriza and A. Zeilinger, “Experimental control of the orbital angular momentum of single and entangled photons,” in Twisted Photons: Applications of Light with Orbital Angular Momentum (Wiley, 2011), p. 280.

Niederiter, R. D.

O’Holleran, K.

Padgett, M.

L. Allen and M. Padgett, “The orbital angular momentum of light: an introduction,” in Twisted Photons: Applications of Light with Orbital Angular Momentum (Wiley, 2011), p. 288.

Padgett, M. J.

Porfirev, A. P.

Preece, D.

Rubass, A.

Rubass, A. F.

Sato, S.

P. Senthilkumaran, J. Masajada, and S. Sato, Singular Optics (International Journal of Optics, 2012).

Schumacher, S.

S. M. H. Luk, N.-H. Kwong, P. Lewandowski, S. Schumacher, and R. Binder, “Optically controlled orbital angular momentum generation in a polaritonic quantum fluid,” Phys. Rev. Lett. 119, 113903 (2017).
[Crossref]

Senthilkumaran, P.

P. Senthilkumaran, J. Masajada, and S. Sato, Singular Optics (International Journal of Optics, 2012).

Siemens, M. E.

N. Alperin and M. E. Siemens, “Angular momentum of topologically structured darkness,” Phys. Rev. Lett. 119, 203902 (2017).
[Crossref]

Siements, K. E.

Singh, A. K.

H. Singh, D. L. Gupta, and A. K. Singh, “Quantum key distribution protocols: a review,” J. Comput. Eng. 16, 1–9 (2014).
[Crossref]

Singh, H.

H. Singh, D. L. Gupta, and A. K. Singh, “Quantum key distribution protocols: a review,” J. Comput. Eng. 16, 1–9 (2014).
[Crossref]

Soifer, V. A.

S. N. Khonina, N. L. Kazanskiy, and V. A. Soifer, “Optical vortices in a fiber: mode division multiplexing and multimode selfimaging,” in Recent Progress in Optical Fiber Research (Intech, 2012), p. 450.

V. A. Soifer and M. A. Golub, Laser Beam Mode Selection by Computer-Generated Holograms (CRC Press, 1994).

Soskin, M. S.

Suk, T.

J. Flusser, B. Zitova, and T. Suk, Moments and Moment Invariants in Pattern Recognition (Wiley, 2009).

Vasnetsov, M. V.

Volyar, A.

Volyar, A. V.

Wang, J.

Xu, W.-C.

Yao, E.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[Crossref]

Zambrini, R.

J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54, 1723–1738 (2007).
[Crossref]

Zeilinger, A.

G. Molina-Terriza and A. Zeilinger, “Experimental control of the orbital angular momentum of single and entangled photons,” in Twisted Photons: Applications of Light with Orbital Angular Momentum (Wiley, 2011), p. 280.

Zitova, B.

J. Flusser, B. Zitova, and T. Suk, Moments and Moment Invariants in Pattern Recognition (Wiley, 2009).

J. Comput. Eng. (1)

H. Singh, D. L. Gupta, and A. K. Singh, “Quantum key distribution protocols: a review,” J. Comput. Eng. 16, 1–9 (2014).
[Crossref]

J. Mod. Opt. (1)

J. B. Götte, S. Franke-Arnold, R. Zambrini, and S. M. Barnett, “Quantum formulation of fractional orbital angular momentum,” J. Mod. Opt. 54, 1723–1738 (2007).
[Crossref]

J. Opt. A (2)

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6, 259–268 (2004).
[Crossref]

J. C. Gutiérrez-Vega and C. López-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A 10, 015009 (2008).
[Crossref]

J. Opt. Soc. Am. A (2)

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

Nature (1)

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

New J. Phys. (1)

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[Crossref]

Opt. Express (3)

Opt. Lett. (4)

Opt. Spectrosc. (1)

A. P. Kiselev, “Localized light waves: paraxial and exact solutions of the wave equation (a review),” Opt. Spectrosc. 102, 603–622 (2007).
[Crossref]

Photon. Res. (1)

Phys. Rev. A (1)

C. N. Alexeyev, Y. A. Egorov, and A. V. Volyar, “Mutual transformations of fractional-order and integer-order optical vortices,” Phys. Rev. A 96, 063807 (2017).
[Crossref]

Phys. Rev. Lett. (2)

N. Alperin and M. E. Siemens, “Angular momentum of topologically structured darkness,” Phys. Rev. Lett. 119, 203902 (2017).
[Crossref]

S. M. H. Luk, N.-H. Kwong, P. Lewandowski, S. Schumacher, and R. Binder, “Optically controlled orbital angular momentum generation in a polaritonic quantum fluid,” Phys. Rev. Lett. 119, 113903 (2017).
[Crossref]

Proc. SPIE (1)

M. V. Berry, “Paraxial beams of spinning light,” Proc. SPIE 3487, 6–11 (1998).
[Crossref]

Other (8)

J. Flusser, B. Zitova, and T. Suk, Moments and Moment Invariants in Pattern Recognition (Wiley, 2009).

G. Molina-Terriza and A. Zeilinger, “Experimental control of the orbital angular momentum of single and entangled photons,” in Twisted Photons: Applications of Light with Orbital Angular Momentum (Wiley, 2011), p. 280.

P. Senthilkumaran, J. Masajada, and S. Sato, Singular Optics (International Journal of Optics, 2012).

G. J. Gbur, Singular Optics (CRC Press, 2017).

L. Allen and M. Padgett, “The orbital angular momentum of light: an introduction,” in Twisted Photons: Applications of Light with Orbital Angular Momentum (Wiley, 2011), p. 288.

V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, Vortex Laser Beams (Taylor & Francis, 2018).

V. A. Soifer and M. A. Golub, Laser Beam Mode Selection by Computer-Generated Holograms (CRC Press, 1994).

S. N. Khonina, N. L. Kazanskiy, and V. A. Soifer, “Optical vortices in a fiber: mode division multiplexing and multimode selfimaging,” in Recent Progress in Optical Fiber Research (Intech, 2012), p. 450.

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Figures (13)

Fig. 1.
Fig. 1. (a) OAM spectrum z ( p ) as a function of the perturbation parameter p for two types of combined beams: curve 1 is obtained in accordance with Eq. (4); curve 2 is plotted via Eq. (2). The callouts correspond to the intensity distributions of the beam in Eqs. (1) and (3), respectively; (b) and (c) holographic grating corresponding to the beams in the callouts Ω = 10 , p = 15.5 .
Fig. 2.
Fig. 2. Intensity I ( r , ϕ , z ) (upper row) and phase ϕ ( r , ϕ , z ) (lower row) distributions of the combined beam in Eq. (5) with parameters p = 20.5 and Ω = 5 .
Fig. 3.
Fig. 3. Spectral curves z ( p ) as a dependence of the OAM z on the perturbed topological parameter p = M + δ p for different scale parameters Ω .
Fig. 4.
Fig. 4. Dependence of the contour width Δ p on the scale parameter Ω for OAM Eq. (7).
Fig. 5.
Fig. 5. Computer simulation of the squared amplitude | C m | 2 of the vortex-beam [Eq. (6)] in the state | 10 (red color) perturbed by (a)  δ p = 0.1 ; (b)  δ p = 0.3 ; (c)  δ p = 0.4 ; (d)  δ p = 0.5 ; Ω = 10 .
Fig. 6.
Fig. 6. Computer simulation of the intensity redistribution | C m ( m ) | 2 over the beam modes m in the state | 20 (red color) for different values of the perturbations δ p , Ω = 10 . (a) δ p = 0.01 . (b) δ p = 0.03 , (c) δ p = 0.1 , (d) δ p = 0.5 .
Fig. 7.
Fig. 7. Shift of the of the satellite maximum m max with a change of the scale parameter Ω , p = 100.5 .
Fig. 8.
Fig. 8. Intensity redistribution | C m ( m ) | 2 among the combined beam modes for the perturbation δ p = 0.5 and (a)  M = | 50 , Ω = 10 ; (b)  M = | 200 , Ω = 10 ; (c)  M = | 50 , Ω = 5 ; (d)  M = | 50 , Ω = 1 .
Fig. 9.
Fig. 9. Local perturbation of the holographic grating for two vortex beam states (a)  | 10 , δ p = 0.001 , (b)  | 20 , δ p = 0.001 , Ω = 10 ; callout, the enlarged central domain of the local perturbation.
Fig. 10.
Fig. 10. Experimental setup for real-time measurement of the vortex and the OAM spectrum, P, polarizer; SLF, space light filter; SLM, space light modulator; L1, L2, spherical lenses with a focal length f s h ; BS, beam splitter; CL, cylindrical lens with a focal length f cyl ; CCD1, 2, CCD cameras.
Fig. 11.
Fig. 11. Experimental detection of the avalanche of vortices in the state | 20 caused by a weak perturbation δ p as the squared vortex amplitudes | C m | 2 evolution. (a)  δ β = 0.05 ; (b)  δ β = 0.1 ; (c)  δ β = 0.3 ; (d)  δ β = 0.5 , Ω = 10 ; the initial state, red.
Fig. 12.
Fig. 12. Breakdown of the OAM induced by the vortex avalanche of the state | 20 in the form of the spectral curves z ( p ) . Curve 1, the basic experiment; curve 2, the control experiment; curve 3, the computer simulation of Eq. (7); the scale parameter Ω = 10 .
Fig. 13.
Fig. 13. Intensity distributions of the initial and restored combined beams in the state | 20 subjected to weak perturbations δ p and their correlation degree η . (a) Initial beam with δ p = 0.05 , (b) restored beam with η = 0.93 ; (c) initial beam with δ p = 0.1 , (d) restored beam with η = 0.91 .

Equations (14)

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Ψ p ( R , ϕ , z = 0 ) = e i p π sin p π m = i | m | ψ | m | p m ,
z ( p ) = i S Ψ p ϕ Ψ p * d S S Ψ p Ψ p * d S = m = m I m ( Ω 2 ) / ( p m ) 2 m = I m ( Ω 2 ) / ( p m ) 2 ,
Ψ p = sin ( π p ) m = i | m | ψ | m | ( r , ϕ ) / [ ( p m ) N | m | ] ,
z ( p ) = p 1 2 π sin ( 2 p π ) .
Ψ ( r , ϕ , z ) = e i p π sin π p π m = ( Ω R ) | m | | m | ! e i m ϕ p m e R 2 σ | m | + 1 ,
C m = e i p π sin π p π Ω | m | | m | ! ( p m ) .
z ( p ) = m = m Ω 2 | m | | m | ! 2 ( p m ) 2 m = Ω 2 | m | | m | ! 2 ( p m ) 2 .
1 2 M = m = m Ω 2 | m | | m | ! 2 ( M Δ p m ) 2 m = Ω 2 | m | | m | ! 2 ( M Δ p m ) 2 .
| C m ( p , Ω ) | 2 = Ω 2 | m | | m | ! 2 ( M m δ p ) 2 m = Ω 2 | m | | m | ! 2 ( M m δ p ) 2 .
m max = ( round ) p I 1 ( 2 Ω ) + 4 Ω 2 I 0 ( 2 Ω ) 4 Ω I 1 ( 2 Ω ) + 2 p I 0 ( 2 Ω ) ,
I p ( r , ϕ ) = Ψ p * Ψ p = m = 0 N 1 | C m | 2 r 2 m G 2 + 2 m , j = 0 , m > j v C m C j * r m + j cos [ ( m j ) ϕ ] G 2 2 m , j = 0 , m > j N 1 C j C m * r m + j sin [ ( m j ) ϕ ] G 2 ,
J s = m = 0 N 1 ( s + m + 1 ) ! m ! | C m | 2 , s = 0 , 1 , 2 , , N 1 ,
J s , q = S μ s , q ( x , y ) I p ( x , y ) d x d y / J 00 ,
T = signum [ cos ( arg Ψ p Q r cos ϕ ) ] ,

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