Abstract

Fork-shaped fringes are formed for off-axis interference between two oblique-incident vortex beams. New formulas considering various parameters [such as the angles between two vortex beams and their topological charges (TCs)] are established to describe all kinds of fork-shaped fringes. An improved Mach–Zehnder interferometer is employed to investigate these interference fringes. Experimental measurements are consistent with numerical simulations by using our formulas. Our results broaden the understanding of the off-axis interference between two vortex beams, and can be applied to detect the TCs’ sign and value of an unknown vortex beam, especially large-value TCs.

© 2020 Optical Society of America

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References

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2020 (3)

2019 (7)

D. Deng, M. C. Lin, Y. Li, and H. Zhao, “Precision measurement of fractional orbital angular momentum,” Phys. Rev. Appl. 12, 014048 (2019).
[Crossref]

Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light Sci. Appl. 8, 90 (2019).
[Crossref]

B. Lan, C. Liu, D. M. Rui, M. Chen, F. Shen, and H. Xian, “The topological charge measurement of the vortex beam based on dislocation self-reference interferometry,” Phys. Scripta 94, 055502 (2019).
[Crossref]

Y. J. Yang, Q. Zhao, L. L. Liu, Y. Liu, and Y. D. Liu, “Manipulation of orbital-angular-momentum spectrum using pinhole plates,” Phys. Rev. Appl. 12, 064007 (2019).
[Crossref]

J. P. C. Narag and N. Hermosa, “Probing higher orbital angular momentum of Laguerre-Gaussian beams via diffraction through a translated single slit,” Phys. Rev. Appl. 11, 054025 (2019).
[Crossref]

P. Kumar and N. K. Nishchal, “Modified Mach-Zehnder interferometer for determining the high-order topological charge of Laguerre-Gaussian vortex beams,” J. Opt. Soc. Am. A 36, 1447–1455 (2019).
[Crossref]

S. W. Cui, B. Xu, S. Y. Luo, H. Y. Xu, Z. P. Cai, Z. Q. Luo, J. X. Pu, and S. Chávez-Cerda, “Determining topological charge based on an improved Fizeau interferometer,” Opt. Express 27, 12774–12779 (2019).
[Crossref]

2018 (5)

S. Choomdaeng, N. Chattham, and A. Pattanapokratana, “Characteristics of fork fringes formed by two obliquely-incident vortex beams with different topological charge number,” J. Phys. Conf. Ser. 1144, 012158 (2018).
[Crossref]

K. A. Forbes and D. L. Andrews, “Optical orbital angular momentum: twisted light and chirality,” Opt. Lett. 43, 435–438 (2018).
[Crossref]

M. Babiker, D. L. Andrews, and V. E. Lembessis, “Atoms in complex twisted light,” J. Opt. 21, 013001 (2018).
[Crossref]

S. Z. Pan, C. Y. Pei, S. W. Liu, J. Wei, D. Wu, Z. O. Liu, Y. L. Yin, Y. Xia, and J. P. Yin, “Measuring orbital angular momentum of light based on petal interference patterns,” OSA Contin. 1, 451–461 (2018).
[Crossref]

M. L. Chen, L. Jiang, and W. Sha, “Orbital angular momentum generation and detection by geometric-phase based metasurfaces,” Appl. Sci. 8, 362 (2018).
[Crossref]

2017 (3)

2016 (4)

P. Panthong, S. Srisuphaphon, A. Pattanaporkratana, and S. Chiangga, “A study of optical vortices inside the Talbot interferometer,” J. Opt. 18, 035602 (2016).
[Crossref]

J. Zhou, W. H. Zhang, and L. X. Chen, “Experimental detection of high-order or fractional orbital angular momentum of light based on a robust mode converter,” Appl. Phys. Lett. 108, 111108 (2016).
[Crossref]

S. Y. Fu, S. K. Zhang, T. L. Wang, and C. Q. Gao, “Measurement of orbital angular momentum spectra of multiplexing optical vortices,” Opt. Express 24, 6240–6248 (2016).
[Crossref]

S. N. Alperin, R. D. Niederriter, J. T. Gopinath, and M. E. Siemens, “Quantitative measurement of the orbital angular momentum of light with a single, stationary lens,” Opt. Lett. 41, 5019–5022 (2016).
[Crossref]

2015 (3)

2014 (1)

G. Vallone, V. D’Ambrosio, A. Sponselli, S. Slussarenko, L. Marrucci, F. Sciarrino, and P. Villoresi, “Free-space quantum key distribution by rotation-invariant twisted photons,” Phys. Rev. Lett. 113, 060503 (2014).
[Crossref]

2013 (3)

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref]

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padagett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341, 537–540 (2013).
[Crossref]

Y. X. Liu, S. H. Sun, J. X. Pu, and B. D. Lv, “Propagation of an optical vortex beam through a diamond-shaped aperture,” Opt. Laser. Technol. 45, 473–479 (2013).
[Crossref]

2012 (4)

M. P. J. Lavery, D. J. Robertson, G. C. G. Berkhout, B. G. D. Love, M. J. Padgett, and J. Courtial, “Refractive elements for the measurement of the orbital angular momentum of a single photon,” Opt. Express 20, 2110–2115 (2012).
[Crossref]

M. Mazilu, A. Mourka, T. Vettenburg, E. W. Wright, and K. Dholakia, “Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum,” Appl. Phys. Lett. 100, 231115 (2012).
[Crossref]

K. Toyoda, K. Miyamoto, N. Aoki, R. Morita, and T. Omatsu, “Using optical vortex to control the chirality of twisted metal nanostructures,” Nano Lett. 12, 3645–3649 (2012).
[Crossref]

H. Tao, Y. Liu, Z. Chen, and J. Pu, “Measuring the topological charge of vortex beams by using an annular ellipse aperture,” Appl. Phys. B 106, 927–932 (2012).
[Crossref]

2011 (3)

2010 (1)

2009 (2)

C. S. Guo, L. L. Lu, and H. T. Wang, “Characterizing topological charge of optical vortices by using an annular aperture,” Opt. Lett. 34, 3686–3688 (2009).
[Crossref]

C. S. Guo, S. J. Yue, and G. X. Wei, “Measuring the orbital angular momentum of optical vortices using a multipinhole plate,” Appl. Phys. Lett. 94, 231104 (2009).
[Crossref]

2008 (2)

G. Yuriy and M. Igor, “Detection of the vortices signs in the scalar fields,” Opt. Appl. 38, 705–713 (2008).

J. Vickers, M. Burch, R. Vyas, and S. Singh, “Phase and interference properties of optical vortex beams,” J. Opt. Soc. Am. A 25, 823–827 (2008).
[Crossref]

2006 (1)

2005 (1)

2004 (1)

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

2003 (1)

G. V. Bogatiryova and M. S. Soskin, “Detection and metrology of optical vortex helical wave fronts,” Semicond. Phys. Quantum Electron. Optoelectron. 6, 254–258 (2003).
[Crossref]

2002 (1)

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2002).
[Crossref]

1996 (1)

M. Padgett, J. Arlt, and N. Simpson, “An experiment to observe the intensity and phase structure of Laguerre-Gaussian laser modes,” Am. J. Phys. 64, 77–82 (1996).
[Crossref]

1995 (1)

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[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, 321–327 (1994).
[Crossref]

1992 (3)

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

V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Screw dislocations of wavefront,” J. Mod. Opt. 39, 985–990 (1992).
[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]

1991 (1)

A. G. White, C. P. Smith, N. R. Heckenberg, H. Rubinsztein-Dunlop, R. McDuff, C. O. Weiss, and C. Tamm, “Interferometric measurements of phase singularities in the output of a visible laser,” J. Mod. Opt. 38, 2531–2541 (1991).
[Crossref]

Akhlaghi, E. A.

Alfano, R. R.

Allen, L.

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]

Almazov, A. A.

Alperin, S. N.

Anderson, M. E.

Andrews, D. L.

K. A. Forbes and D. L. Andrews, “Optical orbital angular momentum: twisted light and chirality,” Opt. Lett. 43, 435–438 (2018).
[Crossref]

M. Babiker, D. L. Andrews, and V. E. Lembessis, “Atoms in complex twisted light,” J. Opt. 21, 013001 (2018).
[Crossref]

Aoki, N.

K. Toyoda, K. Miyamoto, N. Aoki, R. Morita, and T. Omatsu, “Using optical vortex to control the chirality of twisted metal nanostructures,” Nano Lett. 12, 3645–3649 (2012).
[Crossref]

Arlt, J.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2002).
[Crossref]

M. Padgett, J. Arlt, and N. Simpson, “An experiment to observe the intensity and phase structure of Laguerre-Gaussian laser modes,” Am. J. Phys. 64, 77–82 (1996).
[Crossref]

Babiker, M.

M. Babiker, D. L. Andrews, and V. E. Lembessis, “Atoms in complex twisted light,” J. Opt. 21, 013001 (2018).
[Crossref]

Bai, Y. H.

Barnett, S. M.

M. P. J. Lavery, F. C. Speirits, S. M. Barnett, and M. J. Padagett, “Detection of a spinning object using light’s orbital angular momentum,” Science 341, 537–540 (2013).
[Crossref]

Basistiy, I. V.

I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
[Crossref]

Bazhenov, V. Y.

V. Y. Bazhenov, M. V. Vasnetsov, and M. S. Soskin, “Screw dislocations of wavefront,” J. Mod. Opt. 39, 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, 321–327 (1994).
[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]

Berkhout, G. C. G.

Berry, M. V.

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

Bogatiryova, G. V.

G. V. Bogatiryova and M. S. Soskin, “Detection and metrology of optical vortex helical wave fronts,” Semicond. Phys. Quantum Electron. Optoelectron. 6, 254–258 (2003).
[Crossref]

Bozinovic, N.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref]

Bryant, P. E.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, “Controlled rotation of optically trapped microscopic particles,” Science 292, 912–914 (2002).
[Crossref]

Burch, M.

Cai, Z. P.

Chattham, N.

S. Choomdaeng, N. Chattham, and A. Pattanapokratana, “Characteristics of fork fringes formed by two obliquely-incident vortex beams with different topological charge number,” J. Phys. Conf. Ser. 1144, 012158 (2018).
[Crossref]

Chávez-Cerda, S.

Chen, D. X.

Chen, L. X.

J. Zhou, W. H. Zhang, and L. X. Chen, “Experimental detection of high-order or fractional orbital angular momentum of light based on a robust mode converter,” Appl. Phys. Lett. 108, 111108 (2016).
[Crossref]

Chen, M.

B. Lan, C. Liu, D. M. Rui, M. Chen, F. Shen, and H. Xian, “The topological charge measurement of the vortex beam based on dislocation self-reference interferometry,” Phys. Scripta 94, 055502 (2019).
[Crossref]

Chen, M. L.

M. L. Chen, L. Jiang, and W. Sha, “Orbital angular momentum generation and detection by geometric-phase based metasurfaces,” Appl. Sci. 8, 362 (2018).
[Crossref]

Chen, Z.

H. Tao, Y. Liu, Z. Chen, and J. Pu, “Measuring the topological charge of vortex beams by using an annular ellipse aperture,” Appl. Phys. B 106, 927–932 (2012).
[Crossref]

Chiangga, S.

P. Panthong, S. Srisuphaphon, A. Pattanaporkratana, and S. Chiangga, “A study of optical vortices inside the Talbot interferometer,” J. Opt. 18, 035602 (2016).
[Crossref]

Choomdaeng, S.

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Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light Sci. Appl. 8, 90 (2019).
[Crossref]

Wang, Y. L.

Wang, Y. S.

Wei, G. X.

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[Crossref]

Wei, J.

S. Z. Pan, C. Y. Pei, S. W. Liu, J. Wei, D. Wu, Z. O. Liu, Y. L. Yin, Y. Xia, and J. P. Yin, “Measuring orbital angular momentum of light based on petal interference patterns,” OSA Contin. 1, 451–461 (2018).
[Crossref]

Weiss, C. O.

A. G. White, C. P. Smith, N. R. Heckenberg, H. Rubinsztein-Dunlop, R. McDuff, C. O. Weiss, and C. Tamm, “Interferometric measurements of phase singularities in the output of a visible laser,” J. Mod. Opt. 38, 2531–2541 (1991).
[Crossref]

White, A. G.

A. G. White, C. P. Smith, N. R. Heckenberg, H. Rubinsztein-Dunlop, R. McDuff, C. O. Weiss, and C. Tamm, “Interferometric measurements of phase singularities in the output of a visible laser,” J. Mod. Opt. 38, 2531–2541 (1991).
[Crossref]

Willner, A. E.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
[Crossref]

Woerdman, J. P.

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, 321–327 (1994).
[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]

Wright, E. W.

M. Mazilu, A. Mourka, T. Vettenburg, E. W. Wright, and K. Dholakia, “Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum,” Appl. Phys. Lett. 100, 231115 (2012).
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Wu, D.

S. Z. Pan, C. Y. Pei, S. W. Liu, J. Wei, D. Wu, Z. O. Liu, Y. L. Yin, Y. Xia, and J. P. Yin, “Measuring orbital angular momentum of light based on petal interference patterns,” OSA Contin. 1, 451–461 (2018).
[Crossref]

Xia, Y.

S. Z. Pan, C. Y. Pei, S. W. Liu, J. Wei, D. Wu, Z. O. Liu, Y. L. Yin, Y. Xia, and J. P. Yin, “Measuring orbital angular momentum of light based on petal interference patterns,” OSA Contin. 1, 451–461 (2018).
[Crossref]

Xian, H.

B. Lan, C. Liu, D. M. Rui, M. Chen, F. Shen, and H. Xian, “The topological charge measurement of the vortex beam based on dislocation self-reference interferometry,” Phys. Scripta 94, 055502 (2019).
[Crossref]

Xie, Z.

Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light Sci. Appl. 8, 90 (2019).
[Crossref]

Xu, B.

Xu, H. Y.

Yang, Y. J.

Q. Zhao, M. Dong, Y. H. Bai, and Y. J. Yang, “Measuring high orbital angular momentum of vortex beams with an improved multipoint interferometer,” Photon. Res. 8, 745–749 (2020).
[Crossref]

Y. J. Yang, Q. Zhao, L. L. Liu, Y. Liu, and Y. D. Liu, “Manipulation of orbital-angular-momentum spectrum using pinhole plates,” Phys. Rev. Appl. 12, 064007 (2019).
[Crossref]

Yin, J. P.

S. Z. Pan, C. Y. Pei, S. W. Liu, J. Wei, D. Wu, Z. O. Liu, Y. L. Yin, Y. Xia, and J. P. Yin, “Measuring orbital angular momentum of light based on petal interference patterns,” OSA Contin. 1, 451–461 (2018).
[Crossref]

Yin, Y. L.

S. Z. Pan, C. Y. Pei, S. W. Liu, J. Wei, D. Wu, Z. O. Liu, Y. L. Yin, Y. Xia, and J. P. Yin, “Measuring orbital angular momentum of light based on petal interference patterns,” OSA Contin. 1, 451–461 (2018).
[Crossref]

Yuan, X.

Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light Sci. Appl. 8, 90 (2019).
[Crossref]

Yue, S. J.

C. S. Guo, S. J. Yue, and G. X. Wei, “Measuring the orbital angular momentum of optical vortices using a multipinhole plate,” Appl. Phys. Lett. 94, 231104 (2009).
[Crossref]

Yue, Y.

N. Bozinovic, Y. Yue, Y. Ren, M. Tur, P. Kristensen, H. Huang, A. E. Willner, and S. Ramachandran, “Terabit-scale orbital angular momentum mode division multiplexing in fibers,” Science 340, 1545–1548 (2013).
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G. Yuriy and M. Igor, “Detection of the vortices signs in the scalar fields,” Opt. Appl. 38, 705–713 (2008).

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Zhang, S. K.

Zhang, W. H.

J. Zhou, W. H. Zhang, and L. X. Chen, “Experimental detection of high-order or fractional orbital angular momentum of light based on a robust mode converter,” Appl. Phys. Lett. 108, 111108 (2016).
[Crossref]

Zhao, G. H.

Zhao, H.

D. Deng, M. C. Lin, Y. Li, and H. Zhao, “Precision measurement of fractional orbital angular momentum,” Phys. Rev. Appl. 12, 014048 (2019).
[Crossref]

Zhao, Q.

Q. Zhao, M. Dong, Y. H. Bai, and Y. J. Yang, “Measuring high orbital angular momentum of vortex beams with an improved multipoint interferometer,” Photon. Res. 8, 745–749 (2020).
[Crossref]

Y. J. Yang, Q. Zhao, L. L. Liu, Y. Liu, and Y. D. Liu, “Manipulation of orbital-angular-momentum spectrum using pinhole plates,” Phys. Rev. Appl. 12, 064007 (2019).
[Crossref]

Zhou, J.

J. Zhou, W. H. Zhang, and L. X. Chen, “Experimental detection of high-order or fractional orbital angular momentum of light based on a robust mode converter,” Appl. Phys. Lett. 108, 111108 (2016).
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Appl. Opt. (2)

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J. Zhou, W. H. Zhang, and L. X. Chen, “Experimental detection of high-order or fractional orbital angular momentum of light based on a robust mode converter,” Appl. Phys. Lett. 108, 111108 (2016).
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M. Mazilu, A. Mourka, T. Vettenburg, E. W. Wright, and K. Dholakia, “Simultaneous determination of the constituent azimuthal and radial mode indices for light fields possessing orbital angular momentum,” Appl. Phys. Lett. 100, 231115 (2012).
[Crossref]

C. S. Guo, S. J. Yue, and G. X. Wei, “Measuring the orbital angular momentum of optical vortices using a multipinhole plate,” Appl. Phys. Lett. 94, 231104 (2009).
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Light Sci. Appl. (1)

Y. Shen, X. Wang, Z. Xie, C. Min, X. Fu, Q. Liu, M. Gong, and X. Yuan, “Optical vortices 30 years on: OAM manipulation from topological charge to multiple singularities,” Light Sci. Appl. 8, 90 (2019).
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Opt. Appl. (1)

G. Yuriy and M. Igor, “Detection of the vortices signs in the scalar fields,” Opt. Appl. 38, 705–713 (2008).

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OSA Contin. (1)

S. Z. Pan, C. Y. Pei, S. W. Liu, J. Wei, D. Wu, Z. O. Liu, Y. L. Yin, Y. Xia, and J. P. Yin, “Measuring orbital angular momentum of light based on petal interference patterns,” OSA Contin. 1, 451–461 (2018).
[Crossref]

Photon. Res. (1)

Phys. Rev. A (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]

Phys. Rev. Appl. (3)

Y. J. Yang, Q. Zhao, L. L. Liu, Y. Liu, and Y. D. Liu, “Manipulation of orbital-angular-momentum spectrum using pinhole plates,” Phys. Rev. Appl. 12, 064007 (2019).
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J. P. C. Narag and N. Hermosa, “Probing higher orbital angular momentum of Laguerre-Gaussian beams via diffraction through a translated single slit,” Phys. Rev. Appl. 11, 054025 (2019).
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Phys. Rev. Lett. (1)

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Phys. Scripta (1)

B. Lan, C. Liu, D. M. Rui, M. Chen, F. Shen, and H. Xian, “The topological charge measurement of the vortex beam based on dislocation self-reference interferometry,” Phys. Scripta 94, 055502 (2019).
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Figures (9)

Fig. 1.
Fig. 1. Schematic diagram showing orientation relationship between two off-axis vortex beams (with inclination $\alpha$ and azimuth $\beta$) for interference; $| {{l_1}} \rangle$ denotes vortex beam 1, and $| {{l_2}}\rangle$ denotes vortex beam 2. The centers of two beams coincide at the coordinate origin, and the measurement plane is $z = {{0}}$.
Fig. 2.
Fig. 2. Numerical results of off-axis interference patterns between a vortex beam ($| {{l_1} = 2}\rangle$) and a ${{\rm{TEM}}_{00}}$ Gaussian beam ($| {{l_2} = 0} \rangle$) versus the phase difference $\delta$. The color bar in calculation is shown on right side.
Fig. 3.
Fig. 3. Schematic of the experimental setup. ${{\rm{M}}_1}$ and ${{\rm{M}}_2}$, high reflection mirrors; L, lens; HWP, half-wave plate; SPP, spiral phase plate; PBS, polarizing beam splitter; BS, 50/50 non-polarizing beam splitter.
Fig. 4.
Fig. 4. Off-axis interference patterns between the vortex beam ($| {{l_{{1}}}= 1} \rangle$) and ${{\rm{TEM}}_{00}}$ Gaussian beams ($| {{l_{{2}}}= 0} \rangle$) versus inclination $\alpha$. (a1)–(e1) Experimental results and (a2)–(e2) corresponding simulations. Color bar shown on right side is the simulation case.
Fig. 5.
Fig. 5. Off-axis interference patterns between a vortex beam ($| {l_{{1}}}= 2 \rangle$) and ${{\rm{TEM}}_{00}}$ Gaussian beams ($| {l_{{2}}}= 0 \rangle$) for changing azimuth $\beta$. (a1)–(h1) Experimental results and (a2)–(h2) corresponding simulations. The inclination $\alpha$ during simulations is ${0.0015}\pi$. Color bars are simulation cases.
Fig. 6.
Fig. 6. Off-axis interference patterns between a vortex beam ($| {{l_{{1}}}= m} \rangle, m = 0, \pm 1, \pm 2, \pm 3$) and a ${{\rm{TEM}}_{00}}$ Gaussian beam ($| {{l_{{2}}}{= 0}} \rangle$). (a1)–(g1) Experimental results and (a2)–(g2) corresponding simulation results. Color bar shown on right side is the simulation case.
Fig. 7.
Fig. 7. Off-axis interference patterns between two vortex beams with TCs of the same sign. (a1)–(e1) Experimental results and (a2)–(e2) simulation results. Color bar is the simulation case.
Fig. 8.
Fig. 8. Off-axis interference patterns between two optical vortices with TCs of the opposite sign. (a1)–(e1) Experimental results and (a2)–(e2) simulation results. Color bar shown on right side is the simulation case.
Fig. 9.
Fig. 9. Simulation results of off-axis interference patterns between two vortex beams carrying TCs of large numbers ($| {{l_{{1}}}= 100} \rangle + | {{l_{{2}}}}=m \rangle ,m = 100,100 \pm 100,100 \pm 50,100 \pm 20,100 \pm 10,100 \pm 5,100 \pm 2,100 \pm 1$).

Equations (7)

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E 0 l ( x , y , z ) = P 0 l π n c ε 0 | l | ! w 00 2 ( z ) [ 2 ( x 2 + y 2 ) w 00 2 ( z ) ] | l | / 2 × exp ( x 2 + y 2 w 00 2 ( z ) ) exp [ i ψ 0 l ( x , y , z ) ] = P 0 l ( | l | + 1 ) π n c ε 0 | l | ! w 0 l 2 ( z ) [ 2 ( x 2 + y 2 ) ( | l | + 1 ) w 0 l 2 ( z ) ] | l | / 2 × exp [ ( x 2 + y 2 ) ( | l | + 1 ) w 0 l 2 ( z ) ] exp [ i ψ 0 l ( x , y , z ) ] ,
ψ 0 l ( x , y , z ) = ψ 0 l ( ρ , φ , z ) = ( | l | + 1 ) arctan ( z / z R ) + k ρ 2 / 2 R ( z ) + l φ + k z .
E 0 l 1 ( x , y , z ) = P ( | l 1 | + 1 ) π n c ε 0 | l 1 | ! w 2 [ 2 ( x 2 + y 2 ) ( | l 1 | + 1 ) w 2 ] | l 1 | / 2 × exp [ ( x 2 + y 2 ) ( | l 1 | + 1 ) w 2 ] exp ( i l 1 φ + i k z ) .
E 0 l 2 ( x , y , z ) = P ( | l 2 | + 1 ) π n c ε 0 | l 2 | ! w 2 × { 2 [ ( x x 0 ) 2 + ( y y 0 ) 2 ] ( | l 2 | + 1 ) w 2 } | l 2 | / 2 × exp { [ ( x x 0 ) 2 + ( y y 0 ) 2 ] ( | l 2 | + 1 ) w 2 } × exp [ i l 2 φ + i k ( x x 0 ) sin α cos β + i k ( y y 0 ) sin α sin β + i k z cos α ] ,
I ( x , y , z 0 ) = 2 n c ε 0 | E 0 l 1 ( x , y , z 0 ) + E 0 l 2 ( x , y , z 0 ) exp ( i δ ) | 2 ,
I ( x , y , 0 ) = 2 P ( | l 1 | + 1 ) π | l 1 | ! w 2 [ 2 ( x 2 + y 2 ) ( | l 1 | + 1 ) w 2 ] | l 1 | × exp [ 2 ( x 2 + y 2 ) ( | l 1 | + 1 ) w 2 ] × { 1 + Q 2 ( x , y ) + 2 Q ( x , y ) cos [ l φ 2 π sin α ( x cos β + y sin β ) / λ + δ ] } ,
Q ( x , y ) = ( | l 2 | + 1 ) | l 2 | + 1 | l 1 | ! ( | l 1 | + 1 ) | l 1 | + 1 | l 2 | ! [ 2 ( x 2 + y 2 ) w 2 ] | l 2 | | l 1 | 2 × exp [ ( | l 1 | | l 2 | ) ( x 2 + y 2 ) w 2 ] .

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