Abstract

Perfect (optical) vortex (PV) beams are fields whose radial profile is mooted to be independent of orbital angular momentum (OAM). To date, the best experimental approximation of these modes is obtained from passing Bessel–Gaussian beams through a Fourier lens. However, the OAM-dependent width of these quasi-PVs is not precisely known and is often understated. We address this here by deriving and experimentally confirming an explicit analytic expression for the second moment radius of quasi-PVs. We show that this width scales in proportion to in the best case, the same as most “regular” vortex modes, albeit with a much smaller proportionality constant. This Letter will be of interest to the large community who seek to use such structured light fields in various applications, including optical trapping, tweezing, and communications.

© 2019 Optical Society of America

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References

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2019 (1)

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A. Sabatyan and Z. Behjat, Opt. Quantum Electron. 49, 371 (2017).
[Crossref]

M. J. Padgett, Opt. Express 25, 11265 (2017).
[Crossref]

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2015 (3)

P. Vaity and L. Rusch, Opt. Lett. 40, 597 (2015).
[Crossref]

Y. Chen, Z.-X. Fang, Y.-X. Ren, L. Gong, and R.-D. Lu, Appl. Opt. 54, 8030 (2015).
[Crossref]

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, Adv. Opt. Photonics 7, 66 (2015).
[Crossref]

2014 (1)

2013 (2)

2007 (2)

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, Opt. Lett. 32, 3053 (2007).
[Crossref]

V. Arrizon, U. Ruiz, R. Carrada, and L. A. Gonzalez, J. Opt. Soc. Am. 24, 3500 (2007).
[Crossref]

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F. S. Roux, Opt. Commun. 223, 31 (2003).
[Crossref]

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L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, Phys. Rev. A 45, 8185 (1992).
[Crossref]

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F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 491 (1987).
[Crossref]

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Aadhi, A.

M. Jabir, N. A. Chaitanya, A. Aadhi, and G. Samanta, Sci. Rep. 6, 21877 (2016).
[Crossref]

Ahmed, N.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, Adv. Opt. Photonics 7, 66 (2015).
[Crossref]

Allen, L.

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, Phys. Rev. A 45, 8185 (1992).
[Crossref]

Andrews, L. C.

Arita, Y.

Arrizon, V.

R. Paez-Lopez, U. Ruiz, V. Arrizon, and R. Ramos-Garcia, Opt. Lett. 41, 4138 (2016).
[Crossref]

V. Arrizon, U. Ruiz, R. Carrada, and L. A. Gonzalez, J. Opt. Soc. Am. 24, 3500 (2007).
[Crossref]

Arrizón, V.

Ashrafi, N.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, Adv. Opt. Photonics 7, 66 (2015).
[Crossref]

Ashrafi, S.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, Adv. Opt. Photonics 7, 66 (2015).
[Crossref]

Bao, C.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, Adv. Opt. Photonics 7, 66 (2015).
[Crossref]

Behjat, Z.

A. Sabatyan and Z. Behjat, Opt. Quantum Electron. 49, 371 (2017).
[Crossref]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, Phys. Rev. A 45, 8185 (1992).
[Crossref]

Cai, Y.

Cao, Y.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, Adv. Opt. Photonics 7, 66 (2015).
[Crossref]

Carrada, R.

V. Arrizon, U. Ruiz, R. Carrada, and L. A. Gonzalez, J. Opt. Soc. Am. 24, 3500 (2007).
[Crossref]

Chaitanya, N. A.

M. Jabir, N. A. Chaitanya, A. Aadhi, and G. Samanta, Sci. Rep. 6, 21877 (2016).
[Crossref]

N. A. Chaitanya, M. Jabir, and G. K. Samanta, Opt. Lett. 41, 1348 (2016).
[Crossref]

Chen, M.

Chen, Y.

Dholakia, K.

Du, L.

C. Zhang, C. Min, L. Du, and X.-C. Yuan, Appl. Phys. Lett. 108, 201601 (2016).
[Crossref]

Dudley, A.

A. Forbes, A. Dudley, and M. McLaren, Adv. Opt. Photonics 8, 200 (2016).
[Crossref]

Fang, Z.-X.

Forbes, A.

J. Pinnell, V. Rodríguez-Fajardo, and A. Forbes, Opt. Lett. 44, 2736 (2019).
[Crossref]

A. Forbes, A. Dudley, and M. McLaren, Adv. Opt. Photonics 8, 200 (2016).
[Crossref]

Fu, S.

Gao, C.

García-García, J.

Gong, L.

Gonzalez, L. A.

V. Arrizon, U. Ruiz, R. Carrada, and L. A. Gonzalez, J. Opt. Soc. Am. 24, 3500 (2007).
[Crossref]

Gori, F.

F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 491 (1987).
[Crossref]

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 491 (1987).
[Crossref]

Huang, H.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, Adv. Opt. Photonics 7, 66 (2015).
[Crossref]

Huang, S.

W. Shao, S. Huang, X. Liu, and M. Chen, Opt. Commun. 427, 545 (2018).
[Crossref]

Jabir, M.

M. Jabir, N. A. Chaitanya, A. Aadhi, and G. Samanta, Sci. Rep. 6, 21877 (2016).
[Crossref]

N. A. Chaitanya, M. Jabir, and G. K. Samanta, Opt. Lett. 41, 1348 (2016).
[Crossref]

Karimi, E.

Kotlyar, V. V.

Kovalev, A. A.

Lavery, M. P. J.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, Adv. Opt. Photonics 7, 66 (2015).
[Crossref]

Lei, M.

Li, H.

Li, L.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, Adv. Opt. Photonics 7, 66 (2015).
[Crossref]

Li, M.

Li, X.

Liang, Y.

Liu, X.

W. Shao, S. Huang, X. Liu, and M. Chen, Opt. Commun. 427, 545 (2018).
[Crossref]

Lu, R.-D.

Ma, H.

Marrucci, L.

Mazilu, M.

McLaren, M.

A. Forbes, A. Dudley, and M. McLaren, Adv. Opt. Photonics 8, 200 (2016).
[Crossref]

Min, C.

C. Zhang, C. Min, L. Du, and X.-C. Yuan, Appl. Phys. Lett. 108, 201601 (2016).
[Crossref]

Molisch, A. F.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, Adv. Opt. Photonics 7, 66 (2015).
[Crossref]

Ostrovsky, A. S.

Padgett, M. J.

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 491 (1987).
[Crossref]

Paez-Lopez, R.

Phillips, R. L.

Piccirillo, B.

Pinnell, J.

Porfirev, A. P.

Ramachandran, S.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, Adv. Opt. Photonics 7, 66 (2015).
[Crossref]

Ramos-Garcia, R.

Ramos-García, R.

Ren, Y.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, Adv. Opt. Photonics 7, 66 (2015).
[Crossref]

Ren, Y.-X.

Rickenstorff-Parrao, C.

Rodríguez-Fajardo, V.

Roux, F. S.

F. S. Roux, Opt. Commun. 223, 31 (2003).
[Crossref]

Ruiz, U.

R. Paez-Lopez, U. Ruiz, V. Arrizon, and R. Ramos-Garcia, Opt. Lett. 41, 4138 (2016).
[Crossref]

V. Arrizon, U. Ruiz, R. Carrada, and L. A. Gonzalez, J. Opt. Soc. Am. 24, 3500 (2007).
[Crossref]

Rusch, L.

Sabatyan, A.

A. Sabatyan and Z. Behjat, Opt. Quantum Electron. 49, 371 (2017).
[Crossref]

Samanta, G.

M. Jabir, N. A. Chaitanya, A. Aadhi, and G. Samanta, Sci. Rep. 6, 21877 (2016).
[Crossref]

Samanta, G. K.

Santamato, E.

Shao, W.

W. Shao, S. Huang, X. Liu, and M. Chen, Opt. Commun. 427, 545 (2018).
[Crossref]

Siegman, A. E.

A. E. Siegman, in DPSS (Diode Pumped Solid State) Lasers: Applications and Issues (Optical Society of America, 1998), paper MQ1.

Spreeuw, R.

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, Phys. Rev. A 45, 8185 (1992).
[Crossref]

Tai, Y.

Tang, J.

Tang, M.

Tur, M.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, Adv. Opt. Photonics 7, 66 (2015).
[Crossref]

Vaity, P.

Wang, J.

X. Li, H. Ma, C. Yin, J. Tang, H. Li, M. Tang, J. Wang, Y. Tai, X. Li, and Y. Wang, Opt. Express 26, 651 (2018).
[Crossref]

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, Adv. Opt. Photonics 7, 66 (2015).
[Crossref]

Wang, T.

Wang, Y.

Wang, Z.

Willner, A. E.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, Adv. Opt. Photonics 7, 66 (2015).
[Crossref]

Woerdman, J.

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, Phys. Rev. A 45, 8185 (1992).
[Crossref]

Wright, E. M.

Xie, G.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, Adv. Opt. Photonics 7, 66 (2015).
[Crossref]

Yan, S.

Yan, Y.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, Adv. Opt. Photonics 7, 66 (2015).
[Crossref]

Yao, B.

Yin, C.

Yu, X.

Yuan, X.-C.

C. Zhang, C. Min, L. Du, and X.-C. Yuan, Appl. Phys. Lett. 108, 201601 (2016).
[Crossref]

Zhang, C.

C. Zhang, C. Min, L. Du, and X.-C. Yuan, Appl. Phys. Lett. 108, 201601 (2016).
[Crossref]

Zhao, Z.

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, Adv. Opt. Photonics 7, 66 (2015).
[Crossref]

Zito, G.

Adv. Opt. Photonics (2)

A. E. Willner, H. Huang, Y. Yan, Y. Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y. Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi, Adv. Opt. Photonics 7, 66 (2015).
[Crossref]

A. Forbes, A. Dudley, and M. McLaren, Adv. Opt. Photonics 8, 200 (2016).
[Crossref]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

C. Zhang, C. Min, L. Du, and X.-C. Yuan, Appl. Phys. Lett. 108, 201601 (2016).
[Crossref]

J. Opt. Soc. Am. (1)

V. Arrizon, U. Ruiz, R. Carrada, and L. A. Gonzalez, J. Opt. Soc. Am. 24, 3500 (2007).
[Crossref]

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

Opt. Commun. (3)

W. Shao, S. Huang, X. Liu, and M. Chen, Opt. Commun. 427, 545 (2018).
[Crossref]

F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 491 (1987).
[Crossref]

F. S. Roux, Opt. Commun. 223, 31 (2003).
[Crossref]

Opt. Express (2)

Opt. Lett. (8)

Opt. Quantum Electron. (1)

A. Sabatyan and Z. Behjat, Opt. Quantum Electron. 49, 371 (2017).
[Crossref]

Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. Spreeuw, and J. Woerdman, Phys. Rev. A 45, 8185 (1992).
[Crossref]

Sci. Rep. (1)

M. Jabir, N. A. Chaitanya, A. Aadhi, and G. Samanta, Sci. Rep. 6, 21877 (2016).
[Crossref]

Other (1)

A. E. Siegman, in DPSS (Diode Pumped Solid State) Lasers: Applications and Issues (Optical Society of America, 1998), paper MQ1.

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Figures (4)

Fig. 1.
Fig. 1. Schematic of the experimental set-up; Li are lenses of focal length fi, SLMi are SLMs, and CCDi are CCD cameras. A beam splitter was placed after L3 so that the OAM content of the PV could be verified with SLM2 and CCD2, while the width is calculated from the image taken with CCD1.
Fig. 2.
Fig. 2. PV width plotted over [0,50] for (a) fixed kr=10mm1 and (b) fixed wG=1mm. The lines denote theoretical widths calculated using Eq. (6), and the symbols denote experimental widths computed from the images taken with CCD1 and using Eq. (9). The corresponding curves in (a) and (b) have the same value for the product krwG (or equivalently R/T) and thus, have the same width scaling.
Fig. 3.
Fig. 3. Comparison of the normalized OAM-dependent width between PVs and LG beams for different values of R/T. For small values of R/T, the scaling of the width with approaches that of an LG beam. Lines (symbols) denote theoretical (experimental) widths.
Fig. 4.
Fig. 4. Example of the process of correcting for the width increase by adjusting R individually for each . This is equivalent to adjusting the axicon parameter that generates the corresponding BG mode. In (a), three examples of the numerical inversion of Eq. (13) are given for different R(0). In (b), we show the uncorrected and corrected widths for the case of R(0)/T=5.

Equations (15)

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

PVideal(r,ϕ)δ(rR)exp(iϕ),
BB(r,ϕ)J(krr)exp(iϕ),
PV(r,ϕ)exp(r2+R2T2)I(2RrT2)exp(iϕ),
PV(r,ϕ)exp((rR)2T2)exp(iϕ).
ω2=20rdr02πdϕr2|U(r,ϕ)|20rdr02πdϕ|U(r,ϕ)|2,
ω2()=T2(+1)+R2[1+I+1(R2T2)/I(R2T2)].
ω2()T2(+1)+2R2,
ω2()=ω2=2R2,
ωx24i=1Hj=1Vxi2I(xi,yj)ΔxΔyi=1Hj=1VI(xi,yj)ΔxΔy,
xi=(x0+i)Δx,
yj=(y0+j)Δy,
ω2()T2(+1)+R2.
x2(I1(x2)I0(x2)I+1(x2)I(x2))=,
||RkNA,
Rc=||kNA.