Abstract

In most papers about the fractional vortex continuous beams (FVCBs), the relationship between the total vortex strength Sα and the propagation distance is not analyzed since the vortex structure is not stable in the near field. In this paper, we theoretically study the fractional vortex ultrashort pulsed beams (FVUPBs) possessing non-integer topological charges α at arbitrary plane and find that the vortex structure is propagation-distance-dependent. Both the intensity and phase distributions are calculated to analyze the vortex structure. To evaluate the propagation properties of FVUPBs, we focus on the total vortex strength (TVS) of FVUPBs to investigate the number of vortex, and demonstrate that the birth of a vortex is at α = m + ɛ, where m is an integer, ɛ is a changing fraction depending on the pulse durations, peak wavelengths and propagation distances. Furthermore, we discover that the FVUPBs carry decreasing TVS along the propagation axis in free space. This special vortex structure for FVUPBs appears due to the mixture weight of vortex pulsed beam with different integer topological charges (TCs) n. However, the total orbital angular momentum is invariant during propagation. The above phenomenon presented in our paper are totally particular and intriguing compared with the FVCBs.

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

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2019 (4)

J. Wen, L. G. Wang, X. Yang, J. Zhang, and S. Y. Zhu, “Vortex strength and beam propagation factor of fractional vortex beams,” Opt. Express 27(4), 5893–5904 (2019).
[Crossref]

M. A. Porras, “Upper Bound to the Orbital Angular Momentum Carried by an Ultrashort Pulse,” Phys. Rev. Lett. 122(12), 123904 (2019).
[Crossref]

M. A. Porras, “Effects of orbital angular momentum on few-cycle and sub-cycle pulse shapes: coupling between the temporal and angular momentum degrees of freedom,” Opt. Lett. 44(10), 2538–2541 (2019).
[Crossref]

L. Rego, K. M. Dorney, N. J. Brooks, Q. L. Nguyen, C. T. Liao, J. San Roman, D. E. Couch, A. Liu, E. Pisanty, M. Lewenstein, L. Plaja, H. C. Kapteyn, M. M. Murnane, and C. Hernandez-Garcia, “Generation of extreme-ultraviolet beams with time-varying orbital angular momentum,” Science 364(6447), eaaw9486 (2019).
[Crossref]

2018 (1)

Y. Yang, X. Zhu, J. Zeng, X. Lu, C. Zhao, and Y. Cai, “Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation,” Nanophotonics 7(3), 677–682 (2018).
[Crossref]

2017 (6)

A. Turpin, L. Rego, A. Picon, J. San Roman, and C. Hernandez-Garcia, “Extreme Ultraviolet Fractional Orbital Angular Momentum Beams from High Harmonic Generation,” Sci. Rep. 7(1), 43888 (2017).
[Crossref]

Z. Qiao, L. Kong, G. Xie, Z. Qin, P. Yuan, L. Qian, X. Xu, J. Xu, and D. Fan, “Ultraclean femtosecond vortices from a tunable high-order transverse-mode femtosecond laser,” Opt. Lett. 42(13), 2547–2550 (2017).
[Crossref]

M. Miranda, M. Kotur, P. Rudawski, C. Guo, A. Harth, A. L’Huillier, and C. L. Arnold, “Spatiotemporal characterization of ultrashort optical vortex pulses,” J. Mod. Opt. 64(sup4), S1–S6 (2017).
[Crossref]

S. N. Alperin and M. E. Siemens, “Angular Momentum of Topologically Structured Darkness,” Phys. Rev. Lett. 119(20), 203902 (2017).
[Crossref]

Y. Q. Fang, Q. H. Lu, X. L. Wang, W. H. Zhang, and L. X. Chen, “Fractional-topological-charge-induced vortex birth and splitting of light fields on the submicron scale,” Phys. Rev. A 95(2), 023821 (2017).
[Crossref]

L. Ma, P. Zhang, Z. Li, C. Liu, X. Li, Y. Zhang, R. Zhang, and C. Cheng, “Spatiotemporal evolutions of ultrashort vortex pulses generated by spiral multi-pinhole plate,” Opt. Express 25(24), 29864–29873 (2017).
[Crossref]

2016 (4)

G. Gbur, “Fractional vortex Hilbert’s Hotel,” Optica 3(3), 222–225 (2016).
[Crossref]

Y. Pan, X. Z. Gao, Z. C. Ren, X. L. Wang, C. Tu, Y. Li, and H. T. Wang, “Arbitrarily tunable orbital angular momentum of photons,” Sci. Rep. 6(1), 29212 (2016).
[Crossref]

A. H. Dorrah, M. Zamboni-Rached, and M. Mojahedi, “Controlling the topological charge of twisted light beams with propagation,” Phys. Rev. A 93(6), 063864 (2016).
[Crossref]

J. A. Davis, I. Moreno, K. Badham, M. M. Sanchez-Lopez, and D. M. Cottrell, “Nondiffracting vector beams where the charge and the polarization state vary with propagation distance,” Opt. Lett. 41(10), 2270–2273 (2016).
[Crossref]

2014 (1)

L. X. Chen, J. J. Lei, and J. Romero, “Quantum digital spiral imaging,” Light: Sci. Appl. 3(3), e153 (2014).
[Crossref]

2013 (1)

2012 (3)

2010 (2)

2009 (1)

2008 (2)

2007 (2)

I. Zeylikovich, H. I. Sztul, V. Kartazaev, T. Le, and R. R. Alfano, “Ultrashort Laguerre-Gaussian pulses with angular and group velocity dispersion compensation,” Opt. Lett. 32(14), 2025–2027 (2007).
[Crossref]

V. V. Kotlyar, S. N. Khonina, R. V. Skidanov, and V. A. Soifer, “Rotation of laser beams with zero of the orbital angular momentum,” Opt. Commun. 274(1), 8–14 (2007).
[Crossref]

2006 (2)

K. J. Moh, X. C. Yuan, D. Y. Tang, W. C. Cheong, L. S. Zhang, D. K. Y. Low, X. Peng, H. B. Niu, and Z. Y. Lin, “Generation of femtosecond optical vortices using a single refractive optical element,” Appl. Phys. Lett. 88(9), 091103 (2006).
[Crossref]

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the rayleigh criterion limit with optical vortices,” Phys. Rev. Lett. 97(16), 163903 (2006).
[Crossref]

2005 (2)

S. S. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95(24), 240501 (2005).
[Crossref]

S. Tao, X. C. Yuan, J. Lin, X. Peng, and H. Niu, “Fractional optical vortex beam induced rotation of particles,” Opt. Express 13(20), 7726–7731 (2005).
[Crossref]

2004 (6)

S. S. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, “How to observe high-dimensional two-photon entanglement with only two detectors,” Phys. Rev. Lett. 92(21), 217901 (2004).
[Crossref]

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]

W. M. Lee, X. C. Yuan, and K. Dholakia, “Experimental observation of optical vortex evolution in a Gaussian beam with an embedded fractional phase step,” Opt. Commun. 239(1-3), 129–135 (2004).
[Crossref]

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. 6(2), 259–268 (2004).
[Crossref]

K. Bezuhanov, A. Dreischuh, G. G. Paulus, M. G. Schatzel, and H. Walther, “Vortices in femtosecond laser fields,” Opt. Lett. 29(16), 1942–1944 (2004).
[Crossref]

S. N. Khonina, V. V. Kotlyar, V. A. Soifer, K. Jefimovs, and J. Turunen, “Generation and selection of laser beams represented by a superposition of two angular harmonics,” J. Mod. Opt. 51(5), 761–773 (2004).
[Crossref]

2001 (1)

S. Feng and H. G. Winful, “Higher-order transverse modes of ultrashort isodiffracting pulses,” Phys. Rev. E 63(4), 046602 (2001).
[Crossref]

1999 (1)

J. Courtial and M. J. Padgett, “Performance of a cylindrical lens mode converter for producing Laguerre–Gaussian laser modes,” Opt. Commun. 159(1-3), 13–18 (1999).
[Crossref]

1997 (1)

V. V. Kotlyar, V. A. Soifer, and S. N. Khonina, “Rotation of multimode Gauss-Laguerre light beams in free space,” Tech. Phys. Lett. 23(9), 657–658 (1997).
[Crossref]

1994 (1)

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112(5-6), 321–327 (1994).
[Crossref]

1992 (4)

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw Dislocations in Light Wavefronts,” J. Mod. Opt. 39(5), 985–990 (1992).
[Crossref]

N. R. Heckenberg, R. McDuff, C. P. Smith, and A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17(3), 221–223 (1992).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

S. N. Khonina, V. V. Kotlyar, M. V. Shinkaryev, V. A. Soifer, and G. V. Uspleniev, “The Phase Rotor Filter,” J. Mod. Opt. 39(5), 1147–1154 (1992).
[Crossref]

’t Hooft, G. W.

S. S. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95(24), 240501 (2005).
[Crossref]

Aiello, A.

S. S. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95(24), 240501 (2005).
[Crossref]

S. S. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, “How to observe high-dimensional two-photon entanglement with only two detectors,” Phys. Rev. Lett. 92(21), 217901 (2004).
[Crossref]

Alfano, R. R.

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Alperin, S. N.

S. N. Alperin and M. E. Siemens, “Angular Momentum of Topologically Structured Darkness,” Phys. Rev. Lett. 119(20), 203902 (2017).
[Crossref]

Anzolin, G.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the rayleigh criterion limit with optical vortices,” Phys. Rev. Lett. 97(16), 163903 (2006).
[Crossref]

Arnold, C. L.

M. Miranda, M. Kotur, P. Rudawski, C. Guo, A. Harth, A. L’Huillier, and C. L. Arnold, “Spatiotemporal characterization of ultrashort optical vortex pulses,” J. Mod. Opt. 64(sup4), S1–S6 (2017).
[Crossref]

Atencia, J.

Badham, K.

Bañares, L.

Barbieri, C.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the rayleigh criterion limit with optical vortices,” Phys. Rev. Lett. 97(16), 163903 (2006).
[Crossref]

Barnett, S. M.

Baumann, S. M.

Bazhenov, V. Y.

V. Y. Bazhenov, M. S. Soskin, and M. V. Vasnetsov, “Screw Dislocations in Light Wavefronts,” J. Mod. Opt. 39(5), 985–990 (1992).
[Crossref]

Beijersbergen, M. W.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112(5-6), 321–327 (1994).
[Crossref]

L. Allen, M. W. Beijersbergen, R. J. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref]

Berry, M. V.

M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. 6(2), 259–268 (2004).
[Crossref]

Bezuhanov, K.

Bianchini, A.

F. Tamburini, G. Anzolin, G. Umbriaco, A. Bianchini, and C. Barbieri, “Overcoming the rayleigh criterion limit with optical vortices,” Phys. Rev. Lett. 97(16), 163903 (2006).
[Crossref]

Brooks, N. J.

L. Rego, K. M. Dorney, N. J. Brooks, Q. L. Nguyen, C. T. Liao, J. San Roman, D. E. Couch, A. Liu, E. Pisanty, M. Lewenstein, L. Plaja, H. C. Kapteyn, M. M. Murnane, and C. Hernandez-Garcia, “Generation of extreme-ultraviolet beams with time-varying orbital angular momentum,” Science 364(6447), eaaw9486 (2019).
[Crossref]

Cai, Y.

Y. Yang, X. Zhu, J. Zeng, X. Lu, C. Zhao, and Y. Cai, “Anomalous Bessel vortex beam: modulating orbital angular momentum with propagation,” Nanophotonics 7(3), 677–682 (2018).
[Crossref]

Calvo, M. L.

Cheben, P.

Chen, L. X.

Y. Q. Fang, Q. H. Lu, X. L. Wang, W. H. Zhang, and L. X. Chen, “Fractional-topological-charge-induced vortex birth and splitting of light fields on the submicron scale,” Phys. Rev. A 95(2), 023821 (2017).
[Crossref]

L. X. Chen, J. J. Lei, and J. Romero, “Quantum digital spiral imaging,” Light: Sci. Appl. 3(3), e153 (2014).
[Crossref]

Cheng, C.

Cheong, W. C.

K. J. Moh, X. C. Yuan, D. Y. Tang, W. C. Cheong, L. S. Zhang, D. K. Y. Low, X. Peng, H. B. Niu, and Z. Y. Lin, “Generation of femtosecond optical vortices using a single refractive optical element,” Appl. Phys. Lett. 88(9), 091103 (2006).
[Crossref]

Coerwinkel, R. P. C.

M. W. Beijersbergen, R. P. C. Coerwinkel, M. Kristensen, and J. P. Woerdman, “Helical-wavefront laser beams produced with a spiral phaseplate,” Opt. Commun. 112(5-6), 321–327 (1994).
[Crossref]

Collados, M. V.

Cottrell, D. M.

Couch, D. E.

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L. Rego, K. M. Dorney, N. J. Brooks, Q. L. Nguyen, C. T. Liao, J. San Roman, D. E. Couch, A. Liu, E. Pisanty, M. Lewenstein, L. Plaja, H. C. Kapteyn, M. M. Murnane, and C. Hernandez-Garcia, “Generation of extreme-ultraviolet beams with time-varying orbital angular momentum,” Science 364(6447), eaaw9486 (2019).
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L. Rego, K. M. Dorney, N. J. Brooks, Q. L. Nguyen, C. T. Liao, J. San Roman, D. E. Couch, A. Liu, E. Pisanty, M. Lewenstein, L. Plaja, H. C. Kapteyn, M. M. Murnane, and C. Hernandez-Garcia, “Generation of extreme-ultraviolet beams with time-varying orbital angular momentum,” Science 364(6447), eaaw9486 (2019).
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L. Rego, K. M. Dorney, N. J. Brooks, Q. L. Nguyen, C. T. Liao, J. San Roman, D. E. Couch, A. Liu, E. Pisanty, M. Lewenstein, L. Plaja, H. C. Kapteyn, M. M. Murnane, and C. Hernandez-Garcia, “Generation of extreme-ultraviolet beams with time-varying orbital angular momentum,” Science 364(6447), eaaw9486 (2019).
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L. Rego, K. M. Dorney, N. J. Brooks, Q. L. Nguyen, C. T. Liao, J. San Roman, D. E. Couch, A. Liu, E. Pisanty, M. Lewenstein, L. Plaja, H. C. Kapteyn, M. M. Murnane, and C. Hernandez-Garcia, “Generation of extreme-ultraviolet beams with time-varying orbital angular momentum,” Science 364(6447), eaaw9486 (2019).
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Y. Pan, X. Z. Gao, Z. C. Ren, X. L. Wang, C. Tu, Y. Li, and H. T. Wang, “Arbitrarily tunable orbital angular momentum of photons,” Sci. Rep. 6(1), 29212 (2016).
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K. Yamane, Y. Toda, and R. Morita, “Generation of ultrashort optical vortex pulses using optical parametric amplification,” in Conference on Lasers and Electro-Optics 2012, OSA Technical Digest Series (Optical Society of America, 2012), paper JTu1K.4.

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Figures (5)

Fig. 1.
Fig. 1. The normalized intensity and phase distributions of FVUPBs at the propagation distances of $0.5{z_R}$, ${z_R}$ and $2{z_R}$. (a) and (b) are the intensity and phase at the time when the intensity is maximum (${t_m}$). (c) and (d) represent the intensity and phase at the time of ${t_m} + 4{\tau _0}$.
Fig. 2.
Fig. 2. The intensity (a) and phase (b) distributions of FVUPBs at the propagation distance of $5{z_R}$, $8{z_R}$ and $9{z_R}$ with $t = {t_m}$.
Fig. 3.
Fig. 3. The propagation properties of the FVUPBs for different durations and peak wavelengths. The solid curves represent the normalized pulse shape ${\mathop{\rm Re}\nolimits} (E )$, while the dotted curves are its modulus $|E |$. (a) s = 1, $z = 0.5{z_R} \, (\lambda = 800nm)$, (b) s = 10, $z = 0.5{z_R} \, (\lambda = 800nm)$, (c) s = 1, $z = 2.5{z_R} \, (\lambda = 800nm)$, (d) s = 10, $z = 2.5{z_R} \, (\lambda = 800nm)$.
Fig. 4.
Fig. 4. The TVS varies as a function of the propagation distance z for the FVUPBs with different fraction TCs α.
Fig. 5.
Fig. 5. The TVS varies as a function of α for the FVUPBs (a) with different s, (b) with different peak wavelengths, (c) at different propagation distances.

Equations (10)

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E n , C W ( ρ , φ , z ) = 2 π | n | ! 1 w ξ | n | L n ( ξ 2 ) e x p ( i k ρ 2 2 R ρ 2 w 2 ) e x p [ i n φ i ( | n | + 1 ) a r c t a n ( z z R ) ] ,
f ( ω ) = ( ω τ 0 ) s e x p ( ω τ 0 ) ,
E n , U P ( ρ , φ , z , t ) = 2 π | n | ! ξ T | n | L 0 n ( s , n ) 1 + z 2 / z R 2 1 ( 1 + i T ) s + 1 β s + 1 e x p [ i n φ i ( | n | + 1 ) a r c t a n ( z z R ) ] .
ξ T = 2 β ( 1 + i T ) ρ a .
T = 1 τ 0 β [ t 1 c ( z + ρ 2 2 R ) ] .
e x p ( i α φ ) = e x p ( i π α ) s i n ( π α ) π n = + e x p ( i n φ ) α n .
E α , U P ( ρ , φ , z , t ) = e x p ( i π α ) s i n ( π α ) π n = + E n , U P α n .
S α = l i m ρ [ 1 2 π 0 2 π d φ φ a r g E α , U P ] = l i m ρ { 1 2 π 0 2 π d φ R e [ ( i ) E α , U P / φ E α , U P ] } .
E n , U P ( ρ , φ ) 2 π | n | ! 2 | n | 2 L 0 n ( s , n ) ( ρ 2 a 2 ) ( s + 1 ) e x p ( i n φ ) .
E n , U P ( ρ , φ ) 2 π | n | ! 2 | n | 2 L 0 n ( s , n ) ( z / z R ) ( 2 + s ) ( z / z R ) | n | / 2 ( ρ 2 a 2 ) ( s + 1 ) e x p [ i n φ i ( | n | 2 s ) a r c t a n ( z z R ) ] .