Abstract

We study the propagation of a nonparaxial Gaussian beam in the isotropic Bi4Ge3O12 crystal under an external magnetic field. The analytical solutions in the paraxial limit are derived, which show that a vortex with topological charge m=2 (m=1) generates in the E (Ez) component. Numerical calculations in the nonparaxial regime confirm that, with the assistance of an external magnetic field, the spin-to-orbital angular momentum coupling effect can occur even in the isotropic crystal. We also study the Faraday rotation of the nonparaxial Gaussian beam. At the center of the transverse plane, the rotation of the plane of polarization is linearly proportional to the magnetic field. However, the rotation is not uniform on the whole transverse plane, manifesting the nonparaxial effect.

© 2014 Optical Society of America

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  1. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 1983).
  2. M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
    [CrossRef]
  3. T. Takenaka, M. Yokota, and O. Fukumitsu, J. Opt. Soc. Am. A 2, 826 (1985).
    [CrossRef]
  4. A. Wünsche, J. Opt. Soc. Am. A 9, 765 (1992).
    [CrossRef]
  5. R. Borghi and M. Santarsiero, Opt. Lett. 28, 774 (2003).
    [CrossRef]
  6. W. H. Carter, J. Opt. Soc. Am. 62, 1195 (1972).
    [CrossRef]
  7. G. P. Agrawal and D. N. Pattanayak, J. Opt. Soc. Am. 69, 575 (1979).
    [CrossRef]
  8. L. Cicchitelli, H. Hora, and R. Postle, Phys. Rev. A 41, 3727 (1990).
    [CrossRef]
  9. P. Varga and P. Török, Opt. Commun. 152, 108 (1998).
    [CrossRef]
  10. A. Ciattoni, B. Crosignani, and P. D. Porto, Opt. Commun. 177, 9 (2000).
    [CrossRef]
  11. S. M. Sepke and D. P. Umstadter, Opt. Lett. 31, 1447 (2006).
    [CrossRef]
  12. A. Cerjan and C. Cerjan, Opt. Lett. 35, 3465 (2010).
    [CrossRef]
  13. M. Couture and P.-A. Belanger, Phys. Rev. A 24, 355 (1981).
    [CrossRef]
  14. S. Yan and B. Yao, Phys. Rev. A 77, 023827 (2008).
    [CrossRef]
  15. A. Ciattoni, B. Crosignani, and P. D. Porto, Opt. Commun. 202, 17 (2002).
    [CrossRef]
  16. A. Ciattoni, B. Crosignani, and P. D. Porto, J. Opt. Soc. Am. A 18, 1656 (2001).
    [CrossRef]
  17. A. Ciattoni, G. Cincotti, and C. Palma, J. Opt. Soc. Am. A 20, 163 (2003).
    [CrossRef]
  18. M. V. Berry, J. Opt. A 6, 289 (2004).
  19. A. Ciattoni and C. Palma, J. Opt. Soc. Am. A 20, 2163 (2003).
    [CrossRef]
  20. T. A. Fadeyeva and A. V. Volyar, J. Opt. Soc. Am. A 27, 13 (2010).
    [CrossRef]
  21. Y. Zhang, Z. Wu, C. Yuan, X. Yao, K. Lu, M. Belić, and Y. Zhang, Opt. Lett. 37, 4507 (2012).
    [CrossRef]
  22. Y. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, Phys. Rev. Lett. 106, 093904 (2011).
    [CrossRef]
  23. L. Chen and W. She, Opt. Express 15, 15589 (2007).
    [CrossRef]
  24. X. Lu and L. Chen, Opt. Express 21, 8493 (2013).
    [CrossRef]
  25. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).
  26. P. A. Williams, A. H. Rose, K. S. Lee, D. C. Conrad, G. W. Day, and P. D. Hale, Appl. Opt. 35, 3562 (1996).
    [CrossRef]
  27. M. Wang, W. Zhou, P. Zhang, J. Zhao, H. Zhang, and P. Wei, Proc. SPIE 6279, 62791I (2007).
  28. M. R. Dennis, K. O’Holleran, and M. J. Padgett, Prog. Opt. 53, 293 (2009).
    [CrossRef]
  29. S. M. Barnett and L. Allen, Opt. Commun. 110, 670 (1994).
    [CrossRef]
  30. S. M. Barnett, J. Opt. B 4, S7 (2002).
  31. C.-F. Li, Phys. Rev. A 80, 063814 (2009).
    [CrossRef]
  32. A. Ciattoni, G. Cincotti, and C. Palma, Phys. Rev. E 67, 036618 (2003).
    [CrossRef]
  33. L. Chen and W. She, Opt. Lett. 33, 696 (2008).
    [CrossRef]
  34. E. Brasselet, Y. Izdebskaya, V. Shvedov, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, Opt. Lett. 34, 1021 (2009).
    [CrossRef]
  35. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  36. M. Mansuripur, Opt. Photon. News 10(11), 32 (1999).
    [CrossRef]

2013 (1)

2012 (1)

2011 (1)

Y. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, Phys. Rev. Lett. 106, 093904 (2011).
[CrossRef]

2010 (2)

2009 (3)

M. R. Dennis, K. O’Holleran, and M. J. Padgett, Prog. Opt. 53, 293 (2009).
[CrossRef]

C.-F. Li, Phys. Rev. A 80, 063814 (2009).
[CrossRef]

E. Brasselet, Y. Izdebskaya, V. Shvedov, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, Opt. Lett. 34, 1021 (2009).
[CrossRef]

2008 (2)

L. Chen and W. She, Opt. Lett. 33, 696 (2008).
[CrossRef]

S. Yan and B. Yao, Phys. Rev. A 77, 023827 (2008).
[CrossRef]

2007 (2)

M. Wang, W. Zhou, P. Zhang, J. Zhao, H. Zhang, and P. Wei, Proc. SPIE 6279, 62791I (2007).

L. Chen and W. She, Opt. Express 15, 15589 (2007).
[CrossRef]

2006 (1)

2004 (1)

M. V. Berry, J. Opt. A 6, 289 (2004).

2003 (4)

2002 (2)

S. M. Barnett, J. Opt. B 4, S7 (2002).

A. Ciattoni, B. Crosignani, and P. D. Porto, Opt. Commun. 202, 17 (2002).
[CrossRef]

2001 (1)

2000 (1)

A. Ciattoni, B. Crosignani, and P. D. Porto, Opt. Commun. 177, 9 (2000).
[CrossRef]

1999 (1)

M. Mansuripur, Opt. Photon. News 10(11), 32 (1999).
[CrossRef]

1998 (1)

P. Varga and P. Török, Opt. Commun. 152, 108 (1998).
[CrossRef]

1996 (1)

1994 (1)

S. M. Barnett and L. Allen, Opt. Commun. 110, 670 (1994).
[CrossRef]

1992 (1)

1990 (1)

L. Cicchitelli, H. Hora, and R. Postle, Phys. Rev. A 41, 3727 (1990).
[CrossRef]

1985 (1)

1981 (1)

M. Couture and P.-A. Belanger, Phys. Rev. A 24, 355 (1981).
[CrossRef]

1979 (1)

1975 (1)

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

1972 (1)

Agrawal, G. P.

Allen, L.

S. M. Barnett and L. Allen, Opt. Commun. 110, 670 (1994).
[CrossRef]

Barnett, S. M.

S. M. Barnett, J. Opt. B 4, S7 (2002).

S. M. Barnett and L. Allen, Opt. Commun. 110, 670 (1994).
[CrossRef]

Belanger, P.-A.

M. Couture and P.-A. Belanger, Phys. Rev. A 24, 355 (1981).
[CrossRef]

Belic, M.

Berry, M. V.

M. V. Berry, J. Opt. A 6, 289 (2004).

Borghi, R.

Brasselet, E.

Carter, W. H.

Cerjan, A.

Cerjan, C.

Chen, H.

Y. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, Phys. Rev. Lett. 106, 093904 (2011).
[CrossRef]

Chen, L.

Ciattoni, A.

A. Ciattoni and C. Palma, J. Opt. Soc. Am. A 20, 2163 (2003).
[CrossRef]

A. Ciattoni, G. Cincotti, and C. Palma, J. Opt. Soc. Am. A 20, 163 (2003).
[CrossRef]

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

A. Ciattoni, B. Crosignani, and P. D. Porto, Opt. Commun. 202, 17 (2002).
[CrossRef]

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

A. Ciattoni, B. Crosignani, and P. D. Porto, Opt. Commun. 177, 9 (2000).
[CrossRef]

Cicchitelli, L.

L. Cicchitelli, H. Hora, and R. Postle, Phys. Rev. A 41, 3727 (1990).
[CrossRef]

Cincotti, G.

A. Ciattoni, G. Cincotti, and C. Palma, J. Opt. Soc. Am. A 20, 163 (2003).
[CrossRef]

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

Conrad, D. C.

Couture, M.

M. Couture and P.-A. Belanger, Phys. Rev. A 24, 355 (1981).
[CrossRef]

Crosignani, B.

A. Ciattoni, B. Crosignani, and P. D. Porto, Opt. Commun. 202, 17 (2002).
[CrossRef]

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

A. Ciattoni, B. Crosignani, and P. D. Porto, Opt. Commun. 177, 9 (2000).
[CrossRef]

Day, G. W.

Dennis, M. R.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, Prog. Opt. 53, 293 (2009).
[CrossRef]

Desyatnikov, A. S.

Fadeyeva, T. A.

Fukumitsu, O.

Hale, P. D.

Hora, H.

L. Cicchitelli, H. Hora, and R. Postle, Phys. Rev. A 41, 3727 (1990).
[CrossRef]

Izdebskaya, Y.

Kivshar, Y. S.

Krolikowski, W.

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Lee, K. S.

Li, C.

Y. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, Phys. Rev. Lett. 106, 093904 (2011).
[CrossRef]

Li, C.-F.

C.-F. Li, Phys. Rev. A 80, 063814 (2009).
[CrossRef]

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Lu, K.

Y. Zhang, Z. Wu, C. Yuan, X. Yao, K. Lu, M. Belić, and Y. Zhang, Opt. Lett. 37, 4507 (2012).
[CrossRef]

Y. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, Phys. Rev. Lett. 106, 093904 (2011).
[CrossRef]

Lu, X.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Mansuripur, M.

M. Mansuripur, Opt. Photon. News 10(11), 32 (1999).
[CrossRef]

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Nie, Z.

Y. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, Phys. Rev. Lett. 106, 093904 (2011).
[CrossRef]

O’Holleran, K.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, Prog. Opt. 53, 293 (2009).
[CrossRef]

Padgett, M. J.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, Prog. Opt. 53, 293 (2009).
[CrossRef]

Palma, C.

Pattanayak, D. N.

Porto, P. D.

A. Ciattoni, B. Crosignani, and P. D. Porto, Opt. Commun. 202, 17 (2002).
[CrossRef]

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

A. Ciattoni, B. Crosignani, and P. D. Porto, Opt. Commun. 177, 9 (2000).
[CrossRef]

Postle, R.

L. Cicchitelli, H. Hora, and R. Postle, Phys. Rev. A 41, 3727 (1990).
[CrossRef]

Rose, A. H.

Santarsiero, M.

Sepke, S. M.

She, W.

Shvedov, V.

Takenaka, T.

Török, P.

P. Varga and P. Török, Opt. Commun. 152, 108 (1998).
[CrossRef]

Umstadter, D. P.

Varga, P.

P. Varga and P. Török, Opt. Commun. 152, 108 (1998).
[CrossRef]

Volyar, A. V.

Wang, M.

M. Wang, W. Zhou, P. Zhang, J. Zhao, H. Zhang, and P. Wei, Proc. SPIE 6279, 62791I (2007).

Wang, Z.

Y. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, Phys. Rev. Lett. 106, 093904 (2011).
[CrossRef]

Wei, P.

M. Wang, W. Zhou, P. Zhang, J. Zhao, H. Zhang, and P. Wei, Proc. SPIE 6279, 62791I (2007).

Williams, P. A.

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Wu, Z.

Wünsche, A.

Xiao, M.

Y. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, Phys. Rev. Lett. 106, 093904 (2011).
[CrossRef]

Yan, S.

S. Yan and B. Yao, Phys. Rev. A 77, 023827 (2008).
[CrossRef]

Yao, B.

S. Yan and B. Yao, Phys. Rev. A 77, 023827 (2008).
[CrossRef]

Yao, X.

Yariv, A.

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

Yeh, P.

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

Yokota, M.

Yuan, C.

Zhang, H.

M. Wang, W. Zhou, P. Zhang, J. Zhao, H. Zhang, and P. Wei, Proc. SPIE 6279, 62791I (2007).

Zhang, P.

M. Wang, W. Zhou, P. Zhang, J. Zhao, H. Zhang, and P. Wei, Proc. SPIE 6279, 62791I (2007).

Zhang, Y.

Zhao, J.

M. Wang, W. Zhou, P. Zhang, J. Zhao, H. Zhang, and P. Wei, Proc. SPIE 6279, 62791I (2007).

Zhou, W.

M. Wang, W. Zhou, P. Zhang, J. Zhao, H. Zhang, and P. Wei, Proc. SPIE 6279, 62791I (2007).

Appl. Opt. (1)

J. Opt. A (1)

M. V. Berry, J. Opt. A 6, 289 (2004).

J. Opt. B (1)

S. M. Barnett, J. Opt. B 4, S7 (2002).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (6)

Opt. Commun. (4)

P. Varga and P. Török, Opt. Commun. 152, 108 (1998).
[CrossRef]

A. Ciattoni, B. Crosignani, and P. D. Porto, Opt. Commun. 177, 9 (2000).
[CrossRef]

A. Ciattoni, B. Crosignani, and P. D. Porto, Opt. Commun. 202, 17 (2002).
[CrossRef]

S. M. Barnett and L. Allen, Opt. Commun. 110, 670 (1994).
[CrossRef]

Opt. Express (2)

Opt. Lett. (6)

Opt. Photon. News (1)

M. Mansuripur, Opt. Photon. News 10(11), 32 (1999).
[CrossRef]

Phys. Rev. A (5)

C.-F. Li, Phys. Rev. A 80, 063814 (2009).
[CrossRef]

L. Cicchitelli, H. Hora, and R. Postle, Phys. Rev. A 41, 3727 (1990).
[CrossRef]

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

M. Couture and P.-A. Belanger, Phys. Rev. A 24, 355 (1981).
[CrossRef]

S. Yan and B. Yao, Phys. Rev. A 77, 023827 (2008).
[CrossRef]

Phys. Rev. E (1)

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

Phys. Rev. Lett. (1)

Y. Zhang, Z. Wang, Z. Nie, C. Li, H. Chen, K. Lu, and M. Xiao, Phys. Rev. Lett. 106, 093904 (2011).
[CrossRef]

Proc. SPIE (1)

M. Wang, W. Zhou, P. Zhang, J. Zhao, H. Zhang, and P. Wei, Proc. SPIE 6279, 62791I (2007).

Prog. Opt. (1)

M. R. Dennis, K. O’Holleran, and M. J. Padgett, Prog. Opt. 53, 293 (2009).
[CrossRef]

Other (3)

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

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

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Figures (2)

Fig. 1.
Fig. 1.

Nonparaxial field with z=500d and Bz=0.996T, where, in the first and second rows, f=0.4 and 0.1, respectively. (a) and (d) plot the moduli of E+ and Ez as a function of radius r, where the paraxial results, |E+pa| and |Ezpa| calculated from Eqs. (11) and (13), are also shown for comparison. (b), (e), (c), (f) Phases of Ez and E, respectively.

Fig. 2.
Fig. 2.

(a) Rotation of polarization θ as a function of Bz at the center of transverse plane (x=0, y=0) with z=500d. (b) and (c) Polarization ellipse and azimuth on the whole transverse plane, with f=0.3, Bz=3.98T, and z=2500d.

Equations (13)

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

εeff=[n02iγBz0iγBzn02000n02].
2E(·E)+k02εeffE=0.
E(r,z)=d2kexp(ik·r)E˜(k,z),
E˜(k,z)=[c1c3c5]exp(λ1z)+[c2c4c6]exp(λ2z).
[c1c3]=P1·E˜(0),[c2c4]=P2·E˜(0),
[c5c6]=1αk2[λ1(kxc1+kyc3)λ2(kxc2+kyc4)],
P1,2=12[1001]±iα(αk2)[kxky(αkx2)αky2kxky]
E+(r,ϕ,z)=2πdkkexp(izαk2)A(σ,k)E˜+(0)J0(kr),
E(r,ϕ,z)=2πexp(i2ϕ)dkkexp(izαk2)B(σ,k)E˜+(0)J2(kr),
Ez(r,ϕ,z)=2πiexp(iϕ)dkexp(izαk2)k2αk2C(σ,k)E˜+(0)J1(kr),
E+(r,ϕ,z)=E0s2s2(z)exp(ik0n0z)exp(iσ)exp[r22s2(z)],
E(r,ϕ,z)=E0s22αexp(ik0n0z)(isinσ)r2[s2(z)]3exp[r22s2(z)]exp(i2ϕ),
Ez(r,ϕ,z)=irs22αexp(ik0n0z)exp(r22s2)(1σzα)exp(iσ)exp(iϕ),

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