Abstract

Recent studies have shown that quadratic-power-exponent-phase (QPEP) vortex and modified QPEP vortex have some novel properties and potential applications in optical manipulation, orbital angular momentum (OAM) communication, OAM multicasting and so on. In these applications, there may be potential need of processing these kinds of beams by using uniaxial crystals. In this paper, the analytical propagation equations of Gaussian QPEP vortex and modified QPEP vortex propagating in uniaxial crystals are derived and the evolution of the angular momentum via spin-orbital coupling during the propagation is investigated. This may be meaningful for guiding and promoting the applications of the QPEP vortex and modified QPEP vortex.

© 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]
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    [Crossref]
  3. G. Zheng, S. Xu, Q. Wu, Q. Wang, and Z. Ouyang, “Electro-optical coupling of a circular Airy beam in a uniaxial crystal,” Opt. Express 25(13), 14654–14667 (2017).
    [Crossref]
  4. G. Zheng, X. Deng, S. Xu, and Q. Wu, “Propagation dynamics of a circular Airy beam in a uniaxial crystal,” Appl. Opt. 56(9), 2444–2448 (2017).
    [Crossref]
  5. J. Xie, J. Zhang, X. Zheng, J. Ye, and D. Deng, “Paraxial propagation dynamics of the radially polarized Airy beams in uniaxial crystals orthogonal to the optical axis,” Opt. Express 26(9), 11309–11320 (2018).
    [Crossref]
  6. G. Q. Zhou, R. P. Chen, and X. X. Chu, “Propagation of Airy beams in uniaxial crystals orthogonal to the optical axis,” Opt. Express 20(3), 2196–2205 (2012).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  13. X. Yan, L. Guo, M. Cheng, J. Li, Q. Huang, and R. Sun, “Probability density of orbital angular momentum mode of autofocusing Airy beam carrying power-exponent-phase vortex through weak anisotropic atmosphere turbulence,” Opt. Express 25(13), 15286–15298 (2017).
    [Crossref]
  14. G. Zhou, Z. Ji, Y. Zhou, and R. Chen, “Focusing of radially polarized Lorentz–Gauss beams with the power–exponent–phase vortex,” J. Mod. Opt. 65(7), 796–802 (2018).
    [Crossref]
  15. Y. Zhang, Y. Xue, Z. Zhu, G. Rui, Y. Cui, and B. Gu, “Theoretical investigation on asymmetrical spinning and orbiting motions of particles in a tightly focused power-exponent azimuthal-variant vector field,” Opt. Express 26(4), 4318–4329 (2018).
    [Crossref]
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    [Crossref]
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    [Crossref]
  18. M. Chen, W. Gao, H. Liu, C. Teng, S. Deng, H. Deng, and L. Yuan, “Polarization controllable generation of flat superimposed OAM states based on metasurface,” Opt. Express 27(15), 20133–20144 (2019).
    [Crossref]
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    [Crossref]
  20. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, 2007).
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    [Crossref]
  22. W. Zhu and W. She, “Electrically controlling spin and orbital angular momentum of a focused light beam in a uniaxial crystal,” Opt. Express 20(23), 25876–25883 (2012).
    [Crossref]

2019 (1)

2018 (5)

2017 (3)

2016 (2)

W. Yu, R. Zhao, F. Deng, J. Huang, C. Chen, X. Yang, Y. Zhao, and D. Deng, “Propagation of Airy Gaussian vortex beams in uniaxial crystals,” Chin. Phys. B 25(4), 044201 (2016).
[Crossref]

G. Lao, Z. Zhang, and D. Zhao, “Propagation of the power-exponent-phase vortex beam in paraxial ABCD system,” Opt. Express 24(16), 18082–18094 (2016).
[Crossref]

2015 (1)

M. L. Zhou, C. D. Chen, B. Chen, X. Peng, Y. L. Peng, and D. M. Deng, “Propagation of an Airy-Gaussian beam in uniaxial crystals,” Chin. Phys. B 24(12), 124102 (2015).
[Crossref]

2014 (1)

2012 (2)

2010 (1)

2008 (1)

D. Liu and Z. Zhou, “Various Dark Hollow Beams Propagating in Uniaxial Crystals Orthogonal to the Optical Axis,” J. Opt. A: Pure Appl. Opt. 10(9), 095005 (2008).
[Crossref]

2003 (1)

2002 (1)

2001 (2)

A. Ciattoni, B. Crosignani, and P. D. Porto, “Vectorial theory of propagation in uniaxially anisotropic media,” J. Opt. Soc. Am. A 18(7), 1656–1661 (2001).
[Crossref]

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001).
[Crossref]

Chen, B.

M. L. Zhou, C. D. Chen, B. Chen, X. Peng, Y. L. Peng, and D. M. Deng, “Propagation of an Airy-Gaussian beam in uniaxial crystals,” Chin. Phys. B 24(12), 124102 (2015).
[Crossref]

Chen, C.

W. Yu, R. Zhao, F. Deng, J. Huang, C. Chen, X. Yang, Y. Zhao, and D. Deng, “Propagation of Airy Gaussian vortex beams in uniaxial crystals,” Chin. Phys. B 25(4), 044201 (2016).
[Crossref]

Chen, C. D.

M. L. Zhou, C. D. Chen, B. Chen, X. Peng, Y. L. Peng, and D. M. Deng, “Propagation of an Airy-Gaussian beam in uniaxial crystals,” Chin. Phys. B 24(12), 124102 (2015).
[Crossref]

Chen, M.

Chen, R.

G. Zhou, Z. Ji, Y. Zhou, and R. Chen, “Focusing of radially polarized Lorentz–Gauss beams with the power–exponent–phase vortex,” J. Mod. Opt. 65(7), 796–802 (2018).
[Crossref]

Chen, R. P.

Chen, Z.

Cheng, M.

Chu, X. X.

Ciattoni, A.

Cincotti, G.

Crosignani, B.

Cui, Y.

Deng, D.

J. Xie, J. Zhang, X. Zheng, J. Ye, and D. Deng, “Paraxial propagation dynamics of the radially polarized Airy beams in uniaxial crystals orthogonal to the optical axis,” Opt. Express 26(9), 11309–11320 (2018).
[Crossref]

W. Yu, R. Zhao, F. Deng, J. Huang, C. Chen, X. Yang, Y. Zhao, and D. Deng, “Propagation of Airy Gaussian vortex beams in uniaxial crystals,” Chin. Phys. B 25(4), 044201 (2016).
[Crossref]

Deng, D. M.

M. L. Zhou, C. D. Chen, B. Chen, X. Peng, Y. L. Peng, and D. M. Deng, “Propagation of an Airy-Gaussian beam in uniaxial crystals,” Chin. Phys. B 24(12), 124102 (2015).
[Crossref]

Deng, F.

W. Yu, R. Zhao, F. Deng, J. Huang, C. Chen, X. Yang, Y. Zhao, and D. Deng, “Propagation of Airy Gaussian vortex beams in uniaxial crystals,” Chin. Phys. B 25(4), 044201 (2016).
[Crossref]

Deng, H.

Deng, S.

Deng, X.

Fadeyeva, T. A.

Fan, C.

Gan, X.

Gao, W.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, 2007).

Gu, B.

Guo, L.

Huang, J.

W. Yu, R. Zhao, F. Deng, J. Huang, C. Chen, X. Yang, Y. Zhao, and D. Deng, “Propagation of Airy Gaussian vortex beams in uniaxial crystals,” Chin. Phys. B 25(4), 044201 (2016).
[Crossref]

Huang, Q.

Ji, Z.

G. Zhou, Z. Ji, Y. Zhou, and R. Chen, “Focusing of radially polarized Lorentz–Gauss beams with the power–exponent–phase vortex,” J. Mod. Opt. 65(7), 796–802 (2018).
[Crossref]

Lao, G.

Li, J.

Li, P.

Liu, D.

D. Liu and Z. Zhou, “Various Dark Hollow Beams Propagating in Uniaxial Crystals Orthogonal to the Optical Axis,” J. Opt. A: Pure Appl. Opt. 10(9), 095005 (2008).
[Crossref]

Liu, H.

Liu, S.

Liu, Y.

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001).
[Crossref]

Ouyang, Z.

Palma, C.

Peng, T.

Peng, X.

M. L. Zhou, C. D. Chen, B. Chen, X. Peng, Y. L. Peng, and D. M. Deng, “Propagation of an Airy-Gaussian beam in uniaxial crystals,” Chin. Phys. B 24(12), 124102 (2015).
[Crossref]

Peng, Y. L.

M. L. Zhou, C. D. Chen, B. Chen, X. Peng, Y. L. Peng, and D. M. Deng, “Propagation of an Airy-Gaussian beam in uniaxial crystals,” Chin. Phys. B 24(12), 124102 (2015).
[Crossref]

Porto, P. D.

Pu, J.

Rui, G.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, 2007).

She, W.

Sun, R.

Teng, C.

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001).
[Crossref]

Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001).
[Crossref]

Volyar, A. V.

Wang, Q.

Wang, X.

Wu, Q.

Xie, G.

Xie, J.

Xu, R.

Xu, S.

Xue, Y.

Yan, X.

Yang, H.

Yang, X.

W. Yu, R. Zhao, F. Deng, J. Huang, C. Chen, X. Yang, Y. Zhao, and D. Deng, “Propagation of Airy Gaussian vortex beams in uniaxial crystals,” Chin. Phys. B 25(4), 044201 (2016).
[Crossref]

Ye, J.

Yu, W.

W. Yu, R. Zhao, F. Deng, J. Huang, C. Chen, X. Yang, Y. Zhao, and D. Deng, “Propagation of Airy Gaussian vortex beams in uniaxial crystals,” Chin. Phys. B 25(4), 044201 (2016).
[Crossref]

Yuan, L.

Zhang, J.

Zhang, Y.

Zhang, Z.

Zhao, D.

Zhao, J.

Zhao, R.

W. Yu, R. Zhao, F. Deng, J. Huang, C. Chen, X. Yang, Y. Zhao, and D. Deng, “Propagation of Airy Gaussian vortex beams in uniaxial crystals,” Chin. Phys. B 25(4), 044201 (2016).
[Crossref]

Zhao, Y.

W. Yu, R. Zhao, F. Deng, J. Huang, C. Chen, X. Yang, Y. Zhao, and D. Deng, “Propagation of Airy Gaussian vortex beams in uniaxial crystals,” Chin. Phys. B 25(4), 044201 (2016).
[Crossref]

Zheng, G.

Zheng, X.

Zhou, G.

G. Zhou, Z. Ji, Y. Zhou, and R. Chen, “Focusing of radially polarized Lorentz–Gauss beams with the power–exponent–phase vortex,” J. Mod. Opt. 65(7), 796–802 (2018).
[Crossref]

Zhou, G. Q.

Zhou, M. L.

M. L. Zhou, C. D. Chen, B. Chen, X. Peng, Y. L. Peng, and D. M. Deng, “Propagation of an Airy-Gaussian beam in uniaxial crystals,” Chin. Phys. B 24(12), 124102 (2015).
[Crossref]

Zhou, Y.

G. Zhou, Z. Ji, Y. Zhou, and R. Chen, “Focusing of radially polarized Lorentz–Gauss beams with the power–exponent–phase vortex,” J. Mod. Opt. 65(7), 796–802 (2018).
[Crossref]

Zhou, Z.

D. Liu and Z. Zhou, “Various Dark Hollow Beams Propagating in Uniaxial Crystals Orthogonal to the Optical Axis,” J. Opt. A: Pure Appl. Opt. 10(9), 095005 (2008).
[Crossref]

Zhu, W.

Zhu, Z.

Appl. Opt. (1)

Chin. Phys. B (2)

M. L. Zhou, C. D. Chen, B. Chen, X. Peng, Y. L. Peng, and D. M. Deng, “Propagation of an Airy-Gaussian beam in uniaxial crystals,” Chin. Phys. B 24(12), 124102 (2015).
[Crossref]

W. Yu, R. Zhao, F. Deng, J. Huang, C. Chen, X. Yang, Y. Zhao, and D. Deng, “Propagation of Airy Gaussian vortex beams in uniaxial crystals,” Chin. Phys. B 25(4), 044201 (2016).
[Crossref]

J. Mod. Opt. (1)

G. Zhou, Z. Ji, Y. Zhou, and R. Chen, “Focusing of radially polarized Lorentz–Gauss beams with the power–exponent–phase vortex,” J. Mod. Opt. 65(7), 796–802 (2018).
[Crossref]

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

D. Liu and Z. Zhou, “Various Dark Hollow Beams Propagating in Uniaxial Crystals Orthogonal to the Optical Axis,” J. Opt. A: Pure Appl. Opt. 10(9), 095005 (2008).
[Crossref]

J. Opt. Soc. Am. A (5)

Opt. Express (10)

H. Liu, C. Teng, H. Yang, H. Deng, R. Xu, S. Deng, M. Chen, and L. Yuan, “Proposed phase plate for superimposed orbital angular momentum state generation,” Opt. Express 26(11), 14792–14799 (2018).
[Crossref]

M. Chen, W. Gao, H. Liu, C. Teng, S. Deng, H. Deng, and L. Yuan, “Polarization controllable generation of flat superimposed OAM states based on metasurface,” Opt. Express 27(15), 20133–20144 (2019).
[Crossref]

G. Q. Zhou, R. P. Chen, and X. X. Chu, “Propagation of Airy beams in uniaxial crystals orthogonal to the optical axis,” Opt. Express 20(3), 2196–2205 (2012).
[Crossref]

W. Zhu and W. She, “Electrically controlling spin and orbital angular momentum of a focused light beam in a uniaxial crystal,” Opt. Express 20(23), 25876–25883 (2012).
[Crossref]

P. Li, S. Liu, T. Peng, G. Xie, X. Gan, and J. Zhao, “Spiral autofocusing Airy beams carrying power-exponent phase vortices,” Opt. Express 22(7), 7598–7606 (2014).
[Crossref]

G. Lao, Z. Zhang, and D. Zhao, “Propagation of the power-exponent-phase vortex beam in paraxial ABCD system,” Opt. Express 24(16), 18082–18094 (2016).
[Crossref]

G. Zheng, S. Xu, Q. Wu, Q. Wang, and Z. Ouyang, “Electro-optical coupling of a circular Airy beam in a uniaxial crystal,” Opt. Express 25(13), 14654–14667 (2017).
[Crossref]

X. Yan, L. Guo, M. Cheng, J. Li, Q. Huang, and R. Sun, “Probability density of orbital angular momentum mode of autofocusing Airy beam carrying power-exponent-phase vortex through weak anisotropic atmosphere turbulence,” Opt. Express 25(13), 15286–15298 (2017).
[Crossref]

Y. Zhang, Y. Xue, Z. Zhu, G. Rui, Y. Cui, and B. Gu, “Theoretical investigation on asymmetrical spinning and orbiting motions of particles in a tightly focused power-exponent azimuthal-variant vector field,” Opt. Express 26(4), 4318–4329 (2018).
[Crossref]

J. Xie, J. Zhang, X. Zheng, J. Ye, and D. Deng, “Paraxial propagation dynamics of the radially polarized Airy beams in uniaxial crystals orthogonal to the optical axis,” Opt. Express 26(9), 11309–11320 (2018).
[Crossref]

Phys. Rev. Lett. (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Management of the angular momentum of light: preparation of photons in multidimensional vector states of angular momentum,” Phys. Rev. Lett. 88(1), 013601 (2001).
[Crossref]

Other (1)

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic Press, 2007).

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Figures (8)

Fig. 1.
Fig. 1. The OAM spectra, i.e., the relative power of the OAM modes of the Gaussian QPEP vortex at several z plane. Where (a) - (d) are the OAM spectra of the LHCP component of the light at z = 0 mm, z = 4 mm, z = 15 mm and z = 30 mm, respectively; (e) - (h) are the OAM spectra of the RHCP component of the light at z = 0 mm, z = 4 mm, z = 15 mm and z = 30 mm, respectively. The unit in (e) is 1 and, the units in (a) - (d) and (f) - (h) are dB.
Fig. 2.
Fig. 2. The dependence of $W_n^ + (z)$ and $W_n^ - (z)$ on z.
Fig. 3.
Fig. 3. The OAM spectra of (a) LHCP and (b) RHCP components of the Gaussian QPEP vortex at z = 3.85 mm.
Fig. 4.
Fig. 4. Plots of ${\Phi _s}(z )$ (dash line) and ${\Phi _o}(z )$ (solid line) of the Gaussian QPEP vortex vs propagation distance with different m.
Fig. 5.
Fig. 5. The OAM spectra the modified Gaussian QPEP vortex at several z plane. Where (a) - (d) are the OAM spectra of the LHCP component of the light at z = 0 mm, z = 4 mm, z = 15 mm and z = 30 mm, respectively; (e) - (h) are the OAM spectra of the RHCP component of the light at z = 0 mm, z = 4 mm, z = 15 mm and z = 30 mm, respectively. The unit in (e) is 1 and, the units in (a) - (d) and (f) - (h) are dB.
Fig. 6.
Fig. 6. The dependence of $W_n^ - (z)$ and $W_n^ + (z)$ of the modified Gaussian QPEP vortex on z.
Fig. 7.
Fig. 7. The OAM spectra of (a) LHCP and (b) RHCP components of the modified Gaussian QPEP vortex at z = 13.67 mm.
Fig. 8.
Fig. 8. Plots of ${\Phi _s}(z )$ (dash line) and ${\Phi _o}(z )$ (solid line) of the modified Gaussian QPEP vortex vs propagation distance with different m.

Equations (16)

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

2 E ( r , z ) [ E ( r , z ) ] + k 0 2 ε E ( r , z ) = 0 ,
ε = [ n o 2 0 0 0 n o 2 0 0 0 n e 2 ] ,
E ( r , φ , z ) = exp ( i k 0 n o z ) n exp ( i n φ ) [ F ( n ) ( r , z ) e ^ + + G ( n ) ( r , z ) e ^ ] ,
F ( n ) ( r , z ) = π 0 d k k [ exp ( i k 2 z 2 k 0 n o ) + exp ( i n o k 2 z 2 k 0 n e 2 ) ] J n ( k r ) E ~ ( n ) ( k ) ,
G ( n ) ( r , z ) = π 0 d k k [ exp ( i k 2 z 2 k 0 n o ) exp ( i n o k 2 z 2 k 0 n e 2 ) ] J n ( k r ) E ~ ( n 2 ) ( k ) ,
E ~ ( n ) ( k ) = 1 ( 2 π ) 2 0 d r r J n ( k r ) 0 2 π d φ exp ( i n φ ) E ( r , φ , 0 ) .
W n ( z ) = 4 π 3 W i n 0 d k k [ 1 cos ( k 2 z Δ 2 k 0 n o ) ] | E ~ ( n 2 ) ( k ) | 2 ,
W n + ( z ) = 4 π 3 W i n 0 d k k [ 1 + cos ( k 2 z Δ 2 k 0 n o ) ] | E ~ ( n ) ( k ) | 2 ,
Φ s ( z ) = n [ W n + ( z ) W n ( z ) ] 1 ,
Φ o ( z ) = 2 n W n ( z ) .
E ( r , φ , 0 ) = E 0 exp ( r 2 s 2 ) exp [ i 2 m π ( φ α 2 π ) 2 ] ,
E ( r , φ , 0 ) = E 0 exp ( r 2 s 2 ) β H β ( α ) e i β φ ,
H β ( α ) = 1 2 π 0 2 π exp [ i 2 m π ( φ α 2 π ) 2 ] e i β φ d φ .
E ~ ( n ) ( k ) = 1 2 π E 0 H n ( α ) 0 d r r J n ( k r ) exp ( r 2 s 2 ) .
E ~ ( n ) ( k ) = E 0 H n ( α ) π k s 3 16 π e k 2 s 2 8 { I ( n 1 ) / ( n 1 ) 2 2 ( k 2 s 2 / k 2 s 2 8 8 ) I ( n + 1 ) / ( n + 1 ) 2 2 ( k 2 s 2 / k 2 s 2 8 8 ) ; n 0 ( 1 ) n [ I ( n 1 ) / ( n 1 ) 2 2 ( k 2 s 2 / k 2 s 2 8 8 ) I ( n + 1 ) / ( n + 1 ) 2 2 ( k 2 s 2 / k 2 s 2 8 8 ) ] ; n < 0
E ( r , φ , 0 ) = E 0 exp ( r 2 s 2 ) [ 1 + i exp ( i π cos φ 2 ) ] exp [ i 2 m π ( φ α 2 π ) 2 ] .

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