Abstract

We demonstrate that the conversion of spin-to-orbital angular momentum in a focused vectorial light beam can be forbidden or enhanced by the electric-field-erased or increased spatial anisotropy of a uniaxial crystal undergoing the Pockels effect. A focused right-handed circularly polarized quasi-Gaussian beam incident on an electro-optic crystal strontium barium niobate is taken to illustrate the effect, and a conversion-forbidden electric field E0=16.87kVcm is found. Of special interest is that single photons generated in this effect exhibit an entanglement of 2 deg of freedom of spin and orbital angular momentum, and it is electro-optically controllable.

© 2008 Optical Society of America

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  1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
    [CrossRef] [PubMed]
  2. A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 88, 053601 (2002).
    [CrossRef] [PubMed]
  3. M. Padgett, G. Whyte, J. Girkin, A. Wright, L. Allen, P. Ohberg, and S. M. Barnett, Opt. Lett. 31, 2205 (2006).
    [CrossRef] [PubMed]
  4. L. Allen and M. Padgett, J. Mod. Opt. 54, 487 (2007).
    [CrossRef]
  5. S. M. Barnett, J. Opt. B: Quantum Semiclassical Opt. 4, S7 (2002).
    [CrossRef]
  6. S. J. Van Enk and G. Nienhuis, J. Mod. Opt. 41, 963 (1994).
    [CrossRef]
  7. N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, Opt. Lett. 22, 52 (1997).
    [CrossRef] [PubMed]
  8. M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Nature 394, 348 (1998).
    [CrossRef]
  9. L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, Science 292, 912 (2001).
    [CrossRef] [PubMed]
  10. J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998).
    [CrossRef]
  11. I. V. Basistiy, V. V. Slyusar, M. S. Soskin, M. V. Vasnetsov, and A. Ya. Bekshaev, Opt. Lett. 28, 1185 (2003).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  13. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas'ko, S. M. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004)
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  15. G. F. Calvo and A. Picon, Opt. Lett. 32, 838 (2007).
    [CrossRef] [PubMed]
  16. A. Ciattoni, G. Cincotti, and C. Palma, Phys. Rev. E 67, 036618 (2003).
    [CrossRef]
  17. Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, Phys. Rev. Lett. 99, 073901 (2007).
    [CrossRef] [PubMed]
  18. L. Chen, G. Zheng, J. Xu, B. Zhang, and W. She, Opt. Lett. 31, 3474 (2006).
    [CrossRef] [PubMed]
  19. L. Chen, G. Zheng, and W. She, Phys. Rev. A 75, 061403 (2007).
    [CrossRef]
  20. W. She and W. Lee, Opt. Commun. 195, 303 (2001).
    [CrossRef]
  21. A. Ciattoni, B. Crosignani, and P. Di Porto, J. Opt. Soc. Am. A 18, 1656 (2001).
    [CrossRef]
  22. A. Yariv, Optical Waves in Crystals (Wiley, 1983).
  23. S. Feng and P. Kumar, Proc. SPIE 6710, 67100C (2007).
  24. A. Ciattoni, G. Cincotti, and C. Palma, J. Opt. Soc. Am. A 20, 163171 (2003).
  25. L. Marrucci, Proc. SPIE 6587, 658708 (2007).
    [CrossRef]
  26. J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
    [CrossRef] [PubMed]

2007

L. Allen and M. Padgett, J. Mod. Opt. 54, 487 (2007).
[CrossRef]

G. F. Calvo and A. Picon, Opt. Lett. 32, 838 (2007).
[CrossRef] [PubMed]

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

L. Chen, G. Zheng, and W. She, Phys. Rev. A 75, 061403 (2007).
[CrossRef]

S. Feng and P. Kumar, Proc. SPIE 6710, 67100C (2007).

L. Marrucci, Proc. SPIE 6587, 658708 (2007).
[CrossRef]

2006

2004

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas'ko, S. M. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004)
[CrossRef] [PubMed]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

2003

A. Ciattoni, G. Cincotti, and C. Palma, J. Opt. Soc. Am. A 20, 163171 (2003).

I. V. Basistiy, V. V. Slyusar, M. S. Soskin, M. V. Vasnetsov, and A. Ya. Bekshaev, Opt. Lett. 28, 1185 (2003).
[CrossRef] [PubMed]

A. Ciattoni, G. Cincotti, and C. Palma, Phys. Rev. E 67, 036618 (2003).
[CrossRef]

2002

S. M. Barnett, J. Opt. B: Quantum Semiclassical Opt. 4, S7 (2002).
[CrossRef]

A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

2001

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, Science 292, 912 (2001).
[CrossRef] [PubMed]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef] [PubMed]

W. She and W. Lee, Opt. Commun. 195, 303 (2001).
[CrossRef]

A. Ciattoni, B. Crosignani, and P. Di Porto, J. Opt. Soc. Am. A 18, 1656 (2001).
[CrossRef]

1998

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Nature 394, 348 (1998).
[CrossRef]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998).
[CrossRef]

1997

1994

S. J. Van Enk and G. Nienhuis, J. Mod. Opt. 41, 963 (1994).
[CrossRef]

1992

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

1983

A. Yariv, Optical Waves in Crystals (Wiley, 1983).

Allen, L.

L. Allen and M. Padgett, J. Mod. Opt. 54, 487 (2007).
[CrossRef]

M. Padgett, G. Whyte, J. Girkin, A. Wright, L. Allen, P. Ohberg, and S. M. Barnett, Opt. Lett. 31, 2205 (2006).
[CrossRef] [PubMed]

A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998).
[CrossRef]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, Opt. Lett. 22, 52 (1997).
[CrossRef] [PubMed]

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

Arlt, J.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, Science 292, 912 (2001).
[CrossRef] [PubMed]

Barnett, S. M.

Basistiy, I. V.

Beijersbergen, M. W.

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

Bekshaev, A. Ya.

Bryant, P. E.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, Science 292, 912 (2001).
[CrossRef] [PubMed]

Calvo, G. F.

Chen, L.

Chiu, D. T.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

Ciattoni, A.

A. Ciattoni, G. Cincotti, and C. Palma, J. Opt. Soc. Am. A 20, 163171 (2003).

A. Ciattoni, G. Cincotti, and C. Palma, Phys. Rev. E 67, 036618 (2003).
[CrossRef]

A. Ciattoni, B. Crosignani, and P. Di Porto, J. Opt. Soc. Am. A 18, 1656 (2001).
[CrossRef]

Cincotti, G.

A. Ciattoni, G. Cincotti, and C. Palma, J. Opt. Soc. Am. A 20, 163171 (2003).

A. Ciattoni, G. Cincotti, and C. Palma, Phys. Rev. E 67, 036618 (2003).
[CrossRef]

Courtial, J.

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas'ko, S. M. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004)
[CrossRef] [PubMed]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998).
[CrossRef]

Crosignani, B.

Dholakia, K.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, Science 292, 912 (2001).
[CrossRef] [PubMed]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998).
[CrossRef]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, Opt. Lett. 22, 52 (1997).
[CrossRef] [PubMed]

Di Porto, P.

Edgar, J. S.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

Feng, S.

S. Feng and P. Kumar, Proc. SPIE 6710, 67100C (2007).

Franke-Arnold, S.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas'ko, S. M. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004)
[CrossRef] [PubMed]

Friese, M. E. J.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Nature 394, 348 (1998).
[CrossRef]

Gibson, G.

Girkin, J.

Heckenberg, N. R.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Nature 394, 348 (1998).
[CrossRef]

Jeffries, G. D. M.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

Kumar, P.

S. Feng and P. Kumar, Proc. SPIE 6710, 67100C (2007).

Leach, J.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

Lee, W.

W. She and W. Lee, Opt. Commun. 195, 303 (2001).
[CrossRef]

MacDonald, M. P.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, Science 292, 912 (2001).
[CrossRef] [PubMed]

MacVicar, I.

A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

Mair, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef] [PubMed]

Manzo, C.

L. Marrucci, C. Manzo, and D. Paparo, Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef] [PubMed]

Marrucci, L.

L. Marrucci, Proc. SPIE 6587, 658708 (2007).
[CrossRef]

L. Marrucci, C. Manzo, and D. Paparo, Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef] [PubMed]

McGloin, D.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

Nieminen, T. A.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Nature 394, 348 (1998).
[CrossRef]

Nienhuis, G.

S. J. Van Enk and G. Nienhuis, J. Mod. Opt. 41, 963 (1994).
[CrossRef]

Ohberg, P.

O'Neil, A. T.

A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

Padgett, M.

Padgett, M. J.

G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas'ko, S. M. Barnett, and S. Franke-Arnold, Opt. Express 12, 5448 (2004)
[CrossRef] [PubMed]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998).
[CrossRef]

N. B. Simpson, K. Dholakia, L. Allen, and M. J. Padgett, Opt. Lett. 22, 52 (1997).
[CrossRef] [PubMed]

Palma, C.

A. Ciattoni, G. Cincotti, and C. Palma, Phys. Rev. E 67, 036618 (2003).
[CrossRef]

A. Ciattoni, G. Cincotti, and C. Palma, J. Opt. Soc. Am. A 20, 163171 (2003).

Paparo, D.

L. Marrucci, C. Manzo, and D. Paparo, Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef] [PubMed]

Pas'ko, V.

Paterson, L.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, Science 292, 912 (2001).
[CrossRef] [PubMed]

Picon, A.

Robertson, D. A.

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998).
[CrossRef]

Rubinsztein-Dunlop, H.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Nature 394, 348 (1998).
[CrossRef]

She, W.

L. Chen, G. Zheng, and W. She, Phys. Rev. A 75, 061403 (2007).
[CrossRef]

L. Chen, G. Zheng, J. Xu, B. Zhang, and W. She, Opt. Lett. 31, 3474 (2006).
[CrossRef] [PubMed]

W. She and W. Lee, Opt. Commun. 195, 303 (2001).
[CrossRef]

Sibbett, W.

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, Science 292, 912 (2001).
[CrossRef] [PubMed]

Simpson, N. B.

Skeldon, K.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

Slyusar, V. V.

Soskin, M. S.

Spreeuw, R. J. C.

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

Van Enk, S. J.

S. J. Van Enk and G. Nienhuis, J. Mod. Opt. 41, 963 (1994).
[CrossRef]

Vasnetsov, M.

Vasnetsov, M. V.

Vaziri, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef] [PubMed]

Weihs, G.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef] [PubMed]

Whyte, G.

Woerdman, J. P.

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

Wright, A.

Xu, J.

Yariv, A.

A. Yariv, Optical Waves in Crystals (Wiley, 1983).

Zeilinger, A.

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef] [PubMed]

Zhang, B.

Zhao, Y.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

Zheng, G.

J. Mod. Opt.

L. Allen and M. Padgett, J. Mod. Opt. 54, 487 (2007).
[CrossRef]

S. J. Van Enk and G. Nienhuis, J. Mod. Opt. 41, 963 (1994).
[CrossRef]

J. Opt. B: Quantum Semiclassical Opt.

S. M. Barnett, J. Opt. B: Quantum Semiclassical Opt. 4, S7 (2002).
[CrossRef]

J. Opt. Soc. Am. A

A. Ciattoni, B. Crosignani, and P. Di Porto, J. Opt. Soc. Am. A 18, 1656 (2001).
[CrossRef]

A. Ciattoni, G. Cincotti, and C. Palma, J. Opt. Soc. Am. A 20, 163171 (2003).

Nature

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, Nature 394, 348 (1998).
[CrossRef]

A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, Nature 412, 313 (2001).
[CrossRef] [PubMed]

Opt. Commun.

W. She and W. Lee, Opt. Commun. 195, 303 (2001).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

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

L. Chen, G. Zheng, and W. She, Phys. Rev. A 75, 061403 (2007).
[CrossRef]

Phys. Rev. E

A. Ciattoni, G. Cincotti, and C. Palma, Phys. Rev. E 67, 036618 (2003).
[CrossRef]

Phys. Rev. Lett.

Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin, and D. T. Chiu, Phys. Rev. Lett. 99, 073901 (2007).
[CrossRef] [PubMed]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 80, 3217 (1998).
[CrossRef]

L. Marrucci, C. Manzo, and D. Paparo, Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef] [PubMed]

A. T. O'Neil, I. MacVicar, L. Allen, and M. J. Padgett, Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef] [PubMed]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

Proc. SPIE

L. Marrucci, Proc. SPIE 6587, 658708 (2007).
[CrossRef]

Science

L. Paterson, M. P. MacDonald, J. Arlt, W. Sibbett, P. E. Bryant, and K. Dholakia, Science 292, 912 (2001).
[CrossRef] [PubMed]

Other

A. Yariv, Optical Waves in Crystals (Wiley, 1983).

S. Feng and P. Kumar, Proc. SPIE 6710, 67100C (2007).

Cited By

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Figures (1)

Fig. 1
Fig. 1

Electro-optically controlled yields (normalized) of photons having an entanglement of SAM and OAM within a focused vectorial Gaussian beam of waist w = 6 μ m . E 0 is in kilovolts per centimeter.

Equations (11)

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

2 E ( r ) [ E ( r ) ] + k 0 2 T ε T t E ( r ) + μ 0 ω 2 P E O ( r ) = 0 ,
2 E ̃ x z 2 i k x E ̃ z z + k x k y E ̃ y + [ k 0 2 n 0 2 ( 1 n e 2 γ 13 E 0 ) k y 2 ] E ̃ x = 0 ,
2 E ̃ y z 2 i k y E ̃ z z + k x k y E ̃ x + [ k 0 2 n 0 2 ( 1 n e 2 γ 13 E 0 ) k x 2 ] E ̃ y = 0 ,
[ k 0 2 n e 2 ( 1 n e 2 γ 33 E 0 ) k 2 ] E ̃ z i k x z E ̃ x i k y z E ̃ y = 0 .
g = [ g x g y g z ] = [ T y z T z y T z x T x z T x y T y x ] = ε 0 [ ( n o 2 n e 2 ) + n e 2 E 0 ( n e 2 γ 33 n o 2 γ 13 ) ] E z [ E y E x 0 ] .
E + ( r , ϕ , z ) = π 0 d k [ exp ( i z k 2 2 k 0 n 0 1 n e 2 γ 13 E 0 ) + exp ( i z n 0 k 2 1 n e 2 γ 13 E 0 2 k 0 n e 2 ( 1 n e 2 γ 33 E 0 ) ) ] k J 0 ( k r ) E ̃ ( k ) ,
E ( r , ϕ , z ) = π exp ( i 2 ϕ ) 0 d k [ exp ( i k 2 z 2 k 0 n 0 1 n e 2 γ 13 E 0 ) exp ( i n 0 k 2 z 1 n e 2 γ 13 E 0 2 k 0 n e 2 ( 1 n e 2 γ 33 E 0 ) ) ] k J 2 ( k r ) E ̃ ( k ) ,
E ̃ ( k ) = 1 ( 2 π ) 2 0 r d r J 0 ( k r ) 0 2 π d ϕ E ( r , ϕ , 0 ) = A 0 w 2 2 π exp ( k 2 w 2 2 ) ,
SAM = N / [ 1 + ( z z 0 ) 2 ] ,
OAM = { 1 1 / [ 1 + ( z z 0 ) 2 ] } ,
z 0 = 2 k 0 n o w 2 n e 2 ( 1 n e 2 γ 33 E 0 ) [ n o 2 ( 1 n e 2 γ 13 E 0 ) n e 2 ( 1 n e 2 γ 33 E 0 ) ] ,

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