Abstract

A general analytical formula for the propagation of the new kind of power-exponent-phase vortex beam through a paraxial ABCD optical system is derived. With two different calculation methods, the evolution of the intensity distribution and phase contour of such a beam in free space is investigated. Some experiments are carried out to verify the theoretical predictions. Both of the theoretical and experimental results show that the beam’s profile can be modulated by the topological charge and the power order. In addition, the orbital angular momentum (OAM) density and the normalized OAM of such a beam are also studied.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]

2019 (3)

2018 (1)

2016 (1)

2015 (1)

Y. S. Rumala, “Propagation of structured light beams after multiple reflections in a spiral phase plate,” Opt. Eng. 54, 111306 (2015).
[Crossref]

2014 (1)

2011 (1)

2010 (1)

J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104, 103601 (2010).
[Crossref] [PubMed]

2009 (1)

2008 (2)

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] [PubMed]

Q. Chen, B. Shi, Y. Zhang, and G. Guo, “Entanglement of the orbital angular momentum states of the photon pairs generated in a hot atomic ensemble,” Phys. Rev. A 78, 053810 (2008).
[Crossref]

2007 (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305 (2007).
[Crossref]

2004 (3)

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004).
[Crossref] [PubMed]

T. J. Alexander, A. A. Sukhorukov, and Y. S. Kivshar, “Asymmetric vortex solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 93, 063901 (2004).
[Crossref] [PubMed]

I. V. Basistiy, V. A. P. ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A: Pure Appl. Opt. 6, S166–S169 (2004).
[Crossref]

2003 (3)

J. E. Curtis and D. G. Grier, “Modulated optical vortices,” Opt. Lett. 28, 872–874 (2003).
[Crossref] [PubMed]

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

A. Vaziri, J. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, “Concentration of higher dimensional entanglement: Qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91, 227902 (2003).
[Crossref] [PubMed]

2002 (1)

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[Crossref]

2001 (1)

1999 (1)

L. Allen, M. J. Padgett, and M. Babiker, “Iv the orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[Crossref]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

1974 (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. Royal Soc. London. Ser. A, Math. Phys. Sci. 336, 165–190 (1974).
[Crossref]

1970 (1)

Alda, J.

J. Alda, “Laser and gaussian beam propagation and transformation,” in Encyclopedia of Optical Engineering, (Dekker, 2003), pp. 999–1013.

Alexander, T. J.

T. J. Alexander, A. A. Sukhorukov, and Y. S. Kivshar, “Asymmetric vortex solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 93, 063901 (2004).
[Crossref] [PubMed]

Allen, L.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[Crossref]

L. Allen, M. J. Padgett, and M. Babiker, “Iv the orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Babiker, M.

L. Allen, M. J. Padgett, and M. Babiker, “Iv the orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[Crossref]

Barnett, S. M.

Basistiy, I. V.

I. V. Basistiy, V. A. P. ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A: Pure Appl. Opt. 6, S166–S169 (2004).
[Crossref]

Baumann, S. M.

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Berry, M. V.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. Royal Soc. London. Ser. A, Math. Phys. Sci. 336, 165–190 (1974).
[Crossref]

Cai, Y.

Chan, C. T.

J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104, 103601 (2010).
[Crossref] [PubMed]

Chen, Q.

Q. Chen, B. Shi, Y. Zhang, and G. Guo, “Entanglement of the orbital angular momentum states of the photon pairs generated in a hot atomic ensemble,” Phys. Rev. A 78, 053810 (2008).
[Crossref]

Chen, Z.

Collins, S. A.

Courtial, J.

Curtis, J. E.

Fan, C.

Flossmann, F.

Franke-Arnold, S.

Galvez, E. J.

Gan, X.

Gibson, G.

Götte, J. B.

Grier, D. G.

Guo, G.

Q. Chen, B. Shi, Y. Zhang, and G. Guo, “Entanglement of the orbital angular momentum states of the photon pairs generated in a hot atomic ensemble,” Phys. Rev. A 78, 053810 (2008).
[Crossref]

Jennewein, T.

A. Vaziri, J. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, “Concentration of higher dimensional entanglement: Qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91, 227902 (2003).
[Crossref] [PubMed]

Kalb, D. M.

Kivshar, Y. S.

T. J. Alexander, A. A. Sukhorukov, and Y. S. Kivshar, “Asymmetric vortex solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 93, 063901 (2004).
[Crossref] [PubMed]

ko, V. A. P.

I. V. Basistiy, V. A. P. ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A: Pure Appl. Opt. 6, S166–S169 (2004).
[Crossref]

Lao, G.

Li, H.

Li, P.

Li, X.

Lin, Z.

J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104, 103601 (2010).
[Crossref] [PubMed]

Liu, S.

Liu, Y.

Ma, H.

MacMillan, L. H.

MacVicar, I.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[Crossref]

Molina-Terriza, G.

Ng, J.

J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104, 103601 (2010).
[Crossref] [PubMed]

Nye, J. F.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. Royal Soc. London. Ser. A, Math. Phys. Sci. 336, 165–190 (1974).
[Crossref]

O’Holleran, K.

O’Neil, A. T.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[Crossref]

Padgett, M. J.

Pan, J.

A. Vaziri, J. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, “Concentration of higher dimensional entanglement: Qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91, 227902 (2003).
[Crossref] [PubMed]

Pas’ko, V.

Peng, T.

Preece, D.

Pu, J.

Rumala, Y. S.

Y. S. Rumala, “Propagation of structured light beams after multiple reflections in a spiral phase plate,” Opt. Eng. 54, 111306 (2015).
[Crossref]

Shen, D.

Shi, B.

Q. Chen, B. Shi, Y. Zhang, and G. Guo, “Entanglement of the orbital angular momentum states of the photon pairs generated in a hot atomic ensemble,” Phys. Rev. A 78, 053810 (2008).
[Crossref]

Slyusar, V. V.

I. V. Basistiy, V. A. P. ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A: Pure Appl. Opt. 6, S166–S169 (2004).
[Crossref]

Soskin, M. S.

I. V. Basistiy, V. A. P. ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A: Pure Appl. Opt. 6, S166–S169 (2004).
[Crossref]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Sukhorukov, A. A.

T. J. Alexander, A. A. Sukhorukov, and Y. S. Kivshar, “Asymmetric vortex solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 93, 063901 (2004).
[Crossref] [PubMed]

Tang, J.

Tang, M.

Torner, L.

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305 (2007).
[Crossref]

Vasnetsov, M.

Vasnetsov, M. V.

I. V. Basistiy, V. A. P. ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A: Pure Appl. Opt. 6, S166–S169 (2004).
[Crossref]

Vaziri, A.

A. Vaziri, J. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, “Concentration of higher dimensional entanglement: Qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91, 227902 (2003).
[Crossref] [PubMed]

Wang, J.

Wang, L.

Wang, S.

S. Wang and D. Zhao, in Matrix Optics, (CHEP-Springer, 2000).

Wang, X.

Weihs, G.

A. Vaziri, J. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, “Concentration of higher dimensional entanglement: Qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91, 227902 (2003).
[Crossref] [PubMed]

Wen, J.

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Wright, E. M.

Xie, G.

Yang, X.

Yin, J.

Zeilinger, A.

A. Vaziri, J. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, “Concentration of higher dimensional entanglement: Qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91, 227902 (2003).
[Crossref] [PubMed]

Zhang, H.

Zhang, J.

Zhang, Y.

Q. Chen, B. Shi, Y. Zhang, and G. Guo, “Entanglement of the orbital angular momentum states of the photon pairs generated in a hot atomic ensemble,” Phys. Rev. A 78, 053810 (2008).
[Crossref]

Zhang, Z.

Zhao, D.

Zhao, J.

Zhu, S.

J. Opt. A: Pure Appl. Opt. (1)

I. V. Basistiy, V. A. P. ko, V. V. Slyusar, M. S. Soskin, and M. V. Vasnetsov, “Synthesis and analysis of optical vortices with fractional topological charges,” J. Opt. A: Pure Appl. Opt. 6, S166–S169 (2004).
[Crossref]

J. Opt. Soc. Am. (1)

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

Nat. Phys. (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305 (2007).
[Crossref]

Nature (1)

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

Opt. Eng. (1)

Y. S. Rumala, “Propagation of structured light beams after multiple reflections in a spiral phase plate,” Opt. Eng. 54, 111306 (2015).
[Crossref]

Opt. Express (6)

Opt. Lett. (5)

Phys. Rev. A (2)

Q. Chen, B. Shi, Y. Zhang, and G. Guo, “Entanglement of the orbital angular momentum states of the photon pairs generated in a hot atomic ensemble,” Phys. Rev. A 78, 053810 (2008).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[Crossref] [PubMed]

Phys. Rev. Lett. (4)

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[Crossref]

J. Ng, Z. Lin, and C. T. Chan, “Theory of optical trapping by an optical vortex beam,” Phys. Rev. Lett. 104, 103601 (2010).
[Crossref] [PubMed]

A. Vaziri, J. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, “Concentration of higher dimensional entanglement: Qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91, 227902 (2003).
[Crossref] [PubMed]

T. J. Alexander, A. A. Sukhorukov, and Y. S. Kivshar, “Asymmetric vortex solitons in nonlinear periodic lattices,” Phys. Rev. Lett. 93, 063901 (2004).
[Crossref] [PubMed]

Proc. Royal Soc. London. Ser. A, Math. Phys. Sci. (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. Royal Soc. London. Ser. A, Math. Phys. Sci. 336, 165–190 (1974).
[Crossref]

Prog. Opt. (1)

L. Allen, M. J. Padgett, and M. Babiker, “Iv the orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[Crossref]

Other (2)

S. Wang and D. Zhao, in Matrix Optics, (CHEP-Springer, 2000).

J. Alda, “Laser and gaussian beam propagation and transformation,” in Encyclopedia of Optical Engineering, (Dekker, 2003), pp. 999–1013.

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Figures (10)

Fig. 1
Fig. 1 (a) Phase of CV beam with m = 3. (b) Phase of PEPV beam with m = 3, n = 3. (c) Phase of NPEPV beam with m = 3, n = 3.
Fig. 2
Fig. 2 Experimental setup for generating a NPEPV beam and measuring its intensity properties and phase singularity location in free space. HWP, half-wave plate; PBS, polarized beam splitter; BE, beam expander; SLM, spatial light modulator; MR, mirror reflector; CCD, charge-coupled device; PC, personal computer.
Fig. 3
Fig. 3 The holographs for the generation of the NPEPV beam with different power orders n and topological charges m. (a) n = 2, m = 1 ; (b) n = 2, m = 2 ; (c) n = 2, m = 5 ; (d) n = 3, m = 2.
Fig. 4
Fig. 4 Theoretical and experimental results of the intensity distributions of the NPEPV beam with m = 3 at the propagation distance z = 1.75m for different values of power order n. (a) theoretical results calculating with Eq.(12); (b) theoretical results calculating with Eq.(14); (c) experimental results.
Fig. 5
Fig. 5 Theoretical and experimental results of the phase contour of the NPEPV beam with m = 3 at the propagation distance z = 1.75m for different values of power order n. (a) theoretical results calculating with Eq.(12); (b) theoretical results calculating with Eq.(14); (c) experimental results of the interference pattern. The locations of singularities are labeled by white circles.
Fig. 6
Fig. 6 Theoretical results of the intensity distribution and phase contour of the NPEPV beam with m = 3 at the propagation distance z = 1.75m for large power order n. (a) intensity distribution; (b) phase contour.
Fig. 7
Fig. 7 Theoretical and experimental results of the intensity distribution and phase contour of the NPEPV beam with n = 2 at the propagation distance z = 1.75m for different values of TC m. (a) theoretical results of the intensity distribution; (b) experimental results of the intensity distribution; (c) theoretical results of the phase contour; (d) experimental results of the interference pattern.
Fig. 8
Fig. 8 Theoretical results for the optical intensity distribution (a–d) and the phase contour (e–h) of the NPEPV beam at different propagation distances z with m = n = 3. (a,e) z = 1m; (b,f) z = 2m; (c,g) z = 4m; (d,h) z = 8m.
Fig. 9
Fig. 9 Numerical results of the OAM density distribution (normalized) of the NPEPV beam with different phases at distance z = 1.75m. (a) m = n = 1; (b) m = n = 2; (c) m = 3, n = 2; (d) m = n = 3.
Fig. 10
Fig. 10 Influence of m and n on the OAM of NPEPV beam.

Equations (19)

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

E ( 0 ) ( r , ϕ ) = A 0 exp ( r 2 w 2 ) exp ( i ψ ) = A 0 exp ( r 2 w 2 ) exp ( i 2 π [ rem ( m ϕ , 2 π ) 2 π ] n ) ,
TC 1 2 π C ψ ( s ) d s ,
ψ 1 = m ϕ , ψ 2 = 2 m π ( ϕ 2 π ) n , ψ 3 = 2 π [ rem ( m ϕ , 2 π ) 2 π ] n ,
exp ( i ψ ) = exp [ i rem ( ψ , 2 π ) ] .
E ( ρ , θ , z ) = i k 2 π B exp ( i k L 0 ) 0 0 2 π E ( 0 ) ( r , ϕ ) exp ( i k 2 B [ A r 2 2 r ρ cos ( θ ϕ ) + D ρ 2 ] ) r d r d ϕ ,
E ( ρ , θ , z ) = i k 2 π B exp ( i k L 0 ) exp ( i k D 2 B ρ 2 ) 0 0 2 π exp ( r 2 w 2 ) exp ( i k A r 2 2 B ) × exp [ i k ρ r B cos ( θ ϕ ) ] exp ( i 2 π [ rem ( m φ , 2 π ) 2 π ] n ) r d r d ϕ .
exp [ i k r ρ B cos ( φ θ ) ] = h = i h J h ( k r ρ B ) exp [ i h ( φ θ ) ] , J l ( x ) = ( 1 ) l J l ( x ) , J l ( x ) = p = 0 ( 1 ) p 1 p ! Γ ( l + p + 1 ) ( x 2 ) l + 2 p , Γ ( x ) = 0 exp ( t ) t x 1 d t , exp ( s x n ) = j = 0 s j x n j j ! , 0 u x ν 1 e μ x d x = μ ν γ ( ν , μ u ) , γ ( α , x ) = x α α Φ ( α , α + 1 ; x ) , Φ ( α , γ ; z ) = 1 + α γ z 1 ! + α ( α + 1 ) γ ( γ + 1 ) z 2 2 ! + α ( α + 1 ) ( α + 2 ) z 3 3 ! + ,
E ( ρ , θ , z ) = ( i 2 λ B R ) exp ( i k D 2 B ρ 2 ) { exp ( k 2 ρ 2 4 B 2 R ) M 0 + l = 1 [ p = 0 Γ ( p + l / 2 + 1 ) p ! Γ ( p + l + 1 ) ( k 2 ρ 2 4 B 2 R ) p + l / 2 ] × [ exp ( i l θ ) M l + exp ( i l θ ) M l ] } ,
M l = 0 2 π exp ( i 2 π [ rem ( m φ , 2 π ) 2 π ] n + i l φ ) d φ = { j = 0 h = 0 q = 0 m 1 i j + h l h ( 2 π ) j + h + 1 e i q l 2 π m j ! h ! m h + 1 ( n j + h + 1 ) , l 0 , j = 0 i j ( 2 π ) j + 1 j ! ( n j + 1 ) , l = 0 .
I ( ρ , θ , z ) = E * ( ρ , θ , z ) E ( ρ , θ , z ) , ψ ( ρ , θ , z ) = Arg [ E ( ρ , θ , z ) ] .
( A B C D ) = ( 1 z 0 1 ) .
E ( ρ , θ , z ) = ( i 2 λ z R ) exp ( i k 2 z ρ 2 ) { exp ( k 2 ρ 2 4 z 2 R ) M 0 + l = 1 [ p = 0 Γ ( p + l / 2 + 1 ) p ! Γ ( p + l + 1 ) ( k 2 ρ 2 4 z 2 R ) p + l / 2 ] × [ exp ( i l θ ) M l + exp ( i l θ ) M l ] } ,
E ( 0 ) ( x , y ) = A 0 exp ( x 2 + y 2 w 2 ) exp ( i 2 π [ rem ( m ϕ , 2 π ) 2 π ] n ) ,
E ( x , y , z ) = F 1 { F { E 0 ( x , y ) } exp [ i ( ω x 2 + ω y 2 ) z 2 k ] } exp ( i k z ) ,
F { E 0 ( x , y ) } = + + E 0 ( x , y ) e i ( ω x x + ω y y ) d x d y .
I ( x , y , z ) = E * ( x , y , z ) E ( x , y , z ) , ψ ( x , y , z ) = Arg [ E ( x , y , z ) ] .
x = ρ cos θ , y = ρ sin θ , z = z , E ( x , y , z ) = E ( ρ , θ , z ) .
j z = ( r × 0 E × B ) z = x S y y S x ,
J z W = d x d y ( r × E × B ) z c d x d y E × B z .

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