Abstract

It is demonstrated that the orbital angular momentum (OAM) carried by the elliptic beam without the phase-singularity can induce the anisotropic diffraction (AD). The quantitative relation between the OAM and its induced AD is analytically obtained by a comparison of two different kinds of (1+2)-dimensional beam propagations: the linear propagations of the elliptic beam without the OAM in an anisotropic medium and that with the OAM in an isotropic one. In the former case, the optical beam evolves as the fundamental mode of the eigenmodes when its ellipticity is the square root of the anisotropic parameter defined in the paper; while in the latter case, the fundamental mode exists only when the OAM carried by the optical beam equals a specific one called a critical OAM. The OAM always enhances the beam-expanding in the major-axis direction and weakens that in the minor-axis direction no matter the sign of the OAM, and the larger the OAM, the stronger the AD induced by it. Besides, the OAM can also make the elliptic beam rotate, and the absolute value of the rotation angle is no larger than π/2 during the propagation.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. M. Born and E. Wolf, Principles of Optics, 6 ed. (Pergamon Press, 1980), Chapt. 8.
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    [Crossref] [PubMed]
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    [Crossref]
  5. S. R. Seshadri, “Paraxial wave equation for the extraordinary-mode beam in a uniaxial crystal,” J. Opt. Soc. Am. A 18(10), 2628–2629 (2001).
    [Crossref]
  6. A. Ciattoni, G. Cincotti, and C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A 19(4), 792–796 (2002).
    [Crossref]
  7. A. Ciattoni, G. Cincotti, D. Provenziani, and C. Palma, “Paraxial propagation along the optical axis of a uniaxial medium,” Phy. Rev. E 66(3), 036614 (2002).
    [Crossref]
  8. S. R. Seshadri, “Basic elliptical Gaussian wave and beam in a uniaxial crystal,” J. Opt. Soc. Am. A 20(9), 1818–1826 (2003).
    [Crossref]
  9. A. Ciattoni and C. Palma, “Anisotropic beam spreading in uniaxial crystals,” Opt. Commun.,  231(1–6), 79–92 (2004).
    [Crossref]
  10. Z. X. Chen and Q. Guo, “Rotation of elliptic optical beams in anisotropic media,” Opt. Commun. 284(13), 3183–3191 (2011).
    [Crossref]
  11. I. Tinkelman and T. Melemed, “Gaussian-beam propagation in generic anisotropic wave-number profiles,” Opt. Lett. 28(13), 1081–1083 (2003).
    [Crossref] [PubMed]
  12. S. V. Polyakov and G. I. Stegeman, “Existence and properties of quadratic solitons in anisotropic media: Variational approach,” Phy. Rev. E 66(4), 046622 (2002).
    [Crossref]
  13. L. Allen, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
    [Crossref] [PubMed]
  14. S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2(4), 299–313 (2008).
    [Crossref]
  15. K. W. Madison, F. Chevy, and W. Wohlleben, “Vortex formation in a stirred Bose-Einstein condensate,” Phys. Rev. Lett. 84(5), 806–809 (2000).
    [Crossref] [PubMed]
  16. G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
    [Crossref]
  17. M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
    [Crossref]
  18. K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011).
    [Crossref]
  19. K. Volke-Sepulveda, V. Garcés-Chéz, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B 4(2), S82–S89 (2002).
    [Crossref]
  20. J. C. Gutiérrez-Vega, “Characterization of elliptic dark hollow beams,” Proc. SPIE 7062, 706207 (2008).
    [Crossref]
  21. J. Courtial, K. Dholakia, L. Allen, and M.J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. 144(4–6), 210–213 (1997).
    [Crossref]
  22. A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Suppression of collapse for spiraling elliptic solitons,” Phys. Rev. Lett. 104(5), 053902 (2010).
    [Crossref] [PubMed]
  23. G. Liang and Q. Quo, “Spiraling elliptic solitons in nonlocal nonlinear media without anisotropy,” Phys. Rev. A 88(4), 043825 (2013).
    [Crossref]
  24. V. V. Kotlyar, A. A. Kovalev, and A. P. Porfirev, “Astigmatic laser beams with a large orbital angular momentum,” Opt. Express 26(1), 141–156 (2018).
    [Crossref] [PubMed]
  25. They obtained such a beam by making an elliptical Gaussian beam with a plane phase pass through a tilted cylindrical lens, so it is named. But the physical nature of such a beam is that the beam has the cross-phase term, therefore we tend to name it an “elliptical Gaussian beam with the cross-phase.”
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    [Crossref]
  27. Q. S. Ferreira, A. J. Jesus-Silva, E. J. S. Fonseca, and J. M. Hickmann, “Fraunhofer diffraction of light with orbital angular momentum by a slit,” Opt. Lett. 36(16), 3106–3108 (2011).
    [Crossref] [PubMed]
  28. K. Saitoh, Y. Hasegawa, K. Hirakawa, N. Tanaka, and M. Uchida, “Measuring the orbital angular momentum of electron vortex beams using a forked grating,” Phys. Rev. Lett. 111(7), 074801 (2013).
    [Crossref] [PubMed]
  29. K. Dai, C. Gao, L. Zhong, Q. Na, and Q. Wang, “Measuring OAM states of light beams with gradually-changing-period gratings,” Opt. Lett. 40(4), 562–565 (2015).
    [Crossref] [PubMed]
  30. J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105(5), 053904 (2010).
    [Crossref] [PubMed]
  31. J. G. Silva, A. J. Jesus-Silva, M. A. R. C. Alencar, J. M. Hickmann, and E. J. S. Fonseca, “Unveiling square and triangular optical lattices: a comparative study,” Opt. Lett. 39(4), 949–952 (2014).
    [Crossref] [PubMed]
  32. C. S. Guo, L. L. Lu, and H. T. Wang, “Characterizing topological charge of optical vortices by using an annular aperture,” Opt. Lett. 34(23), 3686–3688 (2009).
    [Crossref] [PubMed]
  33. A. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011).
    [Crossref]
  34. Q. Guo and S. Chi, “Nonlinear light beam propagation in uniaxial crystals: nonlinear refractive index, self-trapping and self-focusing,” J. Opt. A 2(1), 5–15 (2000).
    [Crossref]
  35. S. Chi and Q. Guo, “Vector theory of self-focusing of an optical beam in Kerr media,” Opt. Lett. 20(15), 1598–1600 (1995).
    [Crossref] [PubMed]
  36. The eigenmodes are the modes of propagation which are all of the solutions of Eq. (3) that form a complete and orthogonal set of functions.
  37. A. S. Desyatnikov and Y. S. Kivshar, “Rotating optical soliton clusters,” Phys. Rev. Lett. 88(5), 053901 (2002).
    [Crossref] [PubMed]

2018 (1)

2015 (1)

2014 (1)

2013 (2)

G. Liang and Q. Quo, “Spiraling elliptic solitons in nonlocal nonlinear media without anisotropy,” Phys. Rev. A 88(4), 043825 (2013).
[Crossref]

K. Saitoh, Y. Hasegawa, K. Hirakawa, N. Tanaka, and M. Uchida, “Measuring the orbital angular momentum of electron vortex beams using a forked grating,” Phys. Rev. Lett. 111(7), 074801 (2013).
[Crossref] [PubMed]

2011 (5)

Z. X. Chen and Q. Guo, “Rotation of elliptic optical beams in anisotropic media,” Opt. Commun. 284(13), 3183–3191 (2011).
[Crossref]

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
[Crossref]

K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011).
[Crossref]

Q. S. Ferreira, A. J. Jesus-Silva, E. J. S. Fonseca, and J. M. Hickmann, “Fraunhofer diffraction of light with orbital angular momentum by a slit,” Opt. Lett. 36(16), 3106–3108 (2011).
[Crossref] [PubMed]

A. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011).
[Crossref]

2010 (2)

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105(5), 053904 (2010).
[Crossref] [PubMed]

A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Suppression of collapse for spiraling elliptic solitons,” Phys. Rev. Lett. 104(5), 053902 (2010).
[Crossref] [PubMed]

2009 (1)

2008 (2)

J. C. Gutiérrez-Vega, “Characterization of elliptic dark hollow beams,” Proc. SPIE 7062, 706207 (2008).
[Crossref]

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2(4), 299–313 (2008).
[Crossref]

2007 (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[Crossref]

2004 (1)

A. Ciattoni and C. Palma, “Anisotropic beam spreading in uniaxial crystals,” Opt. Commun.,  231(1–6), 79–92 (2004).
[Crossref]

2003 (3)

2002 (5)

A. S. Desyatnikov and Y. S. Kivshar, “Rotating optical soliton clusters,” Phys. Rev. Lett. 88(5), 053901 (2002).
[Crossref] [PubMed]

A. Ciattoni, G. Cincotti, and C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A 19(4), 792–796 (2002).
[Crossref]

A. Ciattoni, G. Cincotti, D. Provenziani, and C. Palma, “Paraxial propagation along the optical axis of a uniaxial medium,” Phy. Rev. E 66(3), 036614 (2002).
[Crossref]

S. V. Polyakov and G. I. Stegeman, “Existence and properties of quadratic solitons in anisotropic media: Variational approach,” Phy. Rev. E 66(4), 046622 (2002).
[Crossref]

K. Volke-Sepulveda, V. Garcés-Chéz, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B 4(2), S82–S89 (2002).
[Crossref]

2001 (1)

2000 (2)

Q. Guo and S. Chi, “Nonlinear light beam propagation in uniaxial crystals: nonlinear refractive index, self-trapping and self-focusing,” J. Opt. A 2(1), 5–15 (2000).
[Crossref]

K. W. Madison, F. Chevy, and W. Wohlleben, “Vortex formation in a stirred Bose-Einstein condensate,” Phys. Rev. Lett. 84(5), 806–809 (2000).
[Crossref] [PubMed]

1997 (1)

J. Courtial, K. Dholakia, L. Allen, and M.J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. 144(4–6), 210–213 (1997).
[Crossref]

1995 (1)

1992 (1)

L. Allen, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

1983 (1)

1966 (1)

Alencar, M. A. R. C.

Allen, L.

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2(4), 299–313 (2008).
[Crossref]

J. Courtial, K. Dholakia, L. Allen, and M.J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. 144(4–6), 210–213 (1997).
[Crossref]

L. Allen, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992).
[Crossref] [PubMed]

Arlt, J.

J. Arlt, “Handedness and azimuthal energy flow of optical vortex beams,” J. Mod. Opt. 50(10), 1573–1580 (2003).
[Crossref]

K. Volke-Sepulveda, V. Garcés-Chéz, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B 4(2), S82–S89 (2002).
[Crossref]

Bekshaev, A.

A. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011).
[Crossref]

Bliokh, K.

A. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 6 ed. (Pergamon Press, 1980), Chapt. 8.

Bowman, R.

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
[Crossref]

Buccoliero, D.

A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Suppression of collapse for spiraling elliptic solitons,” Phys. Rev. Lett. 104(5), 053902 (2010).
[Crossref] [PubMed]

Chávez-Cerda, S.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105(5), 053904 (2010).
[Crossref] [PubMed]

K. Volke-Sepulveda, V. Garcés-Chéz, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B 4(2), S82–S89 (2002).
[Crossref]

Chen, Z. X.

Z. X. Chen and Q. Guo, “Rotation of elliptic optical beams in anisotropic media,” Opt. Commun. 284(13), 3183–3191 (2011).
[Crossref]

Chevy, F.

K. W. Madison, F. Chevy, and W. Wohlleben, “Vortex formation in a stirred Bose-Einstein condensate,” Phys. Rev. Lett. 84(5), 806–809 (2000).
[Crossref] [PubMed]

Chi, S.

Q. Guo and S. Chi, “Nonlinear light beam propagation in uniaxial crystals: nonlinear refractive index, self-trapping and self-focusing,” J. Opt. A 2(1), 5–15 (2000).
[Crossref]

S. Chi and Q. Guo, “Vector theory of self-focusing of an optical beam in Kerr media,” Opt. Lett. 20(15), 1598–1600 (1995).
[Crossref] [PubMed]

Ciattoni, A.

A. Ciattoni and C. Palma, “Anisotropic beam spreading in uniaxial crystals,” Opt. Commun.,  231(1–6), 79–92 (2004).
[Crossref]

A. Ciattoni, G. Cincotti, D. Provenziani, and C. Palma, “Paraxial propagation along the optical axis of a uniaxial medium,” Phy. Rev. E 66(3), 036614 (2002).
[Crossref]

A. Ciattoni, G. Cincotti, and C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A 19(4), 792–796 (2002).
[Crossref]

Cincotti, G.

A. Ciattoni, G. Cincotti, and C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A 19(4), 792–796 (2002).
[Crossref]

A. Ciattoni, G. Cincotti, D. Provenziani, and C. Palma, “Paraxial propagation along the optical axis of a uniaxial medium,” Phy. Rev. E 66(3), 036614 (2002).
[Crossref]

Cizmar, T.

K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011).
[Crossref]

Courtial, J.

J. Courtial, K. Dholakia, L. Allen, and M.J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. 144(4–6), 210–213 (1997).
[Crossref]

Dai, K.

Dennis, M. R.

A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Suppression of collapse for spiraling elliptic solitons,” Phys. Rev. Lett. 104(5), 053902 (2010).
[Crossref] [PubMed]

Desyatnikov, A. S.

A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Suppression of collapse for spiraling elliptic solitons,” Phys. Rev. Lett. 104(5), 053902 (2010).
[Crossref] [PubMed]

A. S. Desyatnikov and Y. S. Kivshar, “Rotating optical soliton clusters,” Phys. Rev. Lett. 88(5), 053901 (2002).
[Crossref] [PubMed]

Dholakia, K.

K. Dholakia and T. Cizmar, “Shaping the future of manipulation,” Nat. Photonics 5(6), 335–342 (2011).
[Crossref]

K. Volke-Sepulveda, V. Garcés-Chéz, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B 4(2), S82–S89 (2002).
[Crossref]

J. Courtial, K. Dholakia, L. Allen, and M.J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. 144(4–6), 210–213 (1997).
[Crossref]

Feit, M. D.

Ferreira, Q. S.

Fleck, J. A.

Fonseca, E. J. S.

Franke-Arnold, S.

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2(4), 299–313 (2008).
[Crossref]

Gao, C.

Garcés-Chéz, V.

K. Volke-Sepulveda, V. Garcés-Chéz, S. Chávez-Cerda, J. Arlt, and K. Dholakia, “Orbital angular momentum of a high-order Bessel light beam,” J. Opt. B 4(2), S82–S89 (2002).
[Crossref]

Guo, C. S.

Guo, Q.

Z. X. Chen and Q. Guo, “Rotation of elliptic optical beams in anisotropic media,” Opt. Commun. 284(13), 3183–3191 (2011).
[Crossref]

Q. Guo and S. Chi, “Nonlinear light beam propagation in uniaxial crystals: nonlinear refractive index, self-trapping and self-focusing,” J. Opt. A 2(1), 5–15 (2000).
[Crossref]

S. Chi and Q. Guo, “Vector theory of self-focusing of an optical beam in Kerr media,” Opt. Lett. 20(15), 1598–1600 (1995).
[Crossref] [PubMed]

Gutiérrez-Vega, J. C.

J. C. Gutiérrez-Vega, “Characterization of elliptic dark hollow beams,” Proc. SPIE 7062, 706207 (2008).
[Crossref]

Hasegawa, Y.

K. Saitoh, Y. Hasegawa, K. Hirakawa, N. Tanaka, and M. Uchida, “Measuring the orbital angular momentum of electron vortex beams using a forked grating,” Phys. Rev. Lett. 111(7), 074801 (2013).
[Crossref] [PubMed]

Haus, H. A.

H. A. Haus, Waves and fields in optoelectronics (Prentice-Hall, 1984), Chapt. 4 and 5.

Hickmann, J. M.

Hirakawa, K.

K. Saitoh, Y. Hasegawa, K. Hirakawa, N. Tanaka, and M. Uchida, “Measuring the orbital angular momentum of electron vortex beams using a forked grating,” Phys. Rev. Lett. 111(7), 074801 (2013).
[Crossref] [PubMed]

Jesus-Silva, A. J.

Kivshar, Y. S.

A. S. Desyatnikov, D. Buccoliero, M. R. Dennis, and Y. S. Kivshar, “Suppression of collapse for spiraling elliptic solitons,” Phys. Rev. Lett. 104(5), 053902 (2010).
[Crossref] [PubMed]

A. S. Desyatnikov and Y. S. Kivshar, “Rotating optical soliton clusters,” Phys. Rev. Lett. 88(5), 053901 (2002).
[Crossref] [PubMed]

Kogelnik, H.

Kotlyar, V. V.

Kovalev, A. A.

Li, T.

Liang, G.

G. Liang and Q. Quo, “Spiraling elliptic solitons in nonlocal nonlinear media without anisotropy,” Phys. Rev. A 88(4), 043825 (2013).
[Crossref]

Lu, L. L.

Madison, K. W.

K. W. Madison, F. Chevy, and W. Wohlleben, “Vortex formation in a stirred Bose-Einstein condensate,” Phys. Rev. Lett. 84(5), 806–809 (2000).
[Crossref] [PubMed]

Melemed, T.

Molina-Terriza, G.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[Crossref]

Na, Q.

Padgett, M.

M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics 5(6), 343–348 (2011).
[Crossref]

S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser Photon. Rev. 2(4), 299–313 (2008).
[Crossref]

Padgett, M.J.

J. Courtial, K. Dholakia, L. Allen, and M.J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. 144(4–6), 210–213 (1997).
[Crossref]

Palma, C.

A. Ciattoni and C. Palma, “Anisotropic beam spreading in uniaxial crystals,” Opt. Commun.,  231(1–6), 79–92 (2004).
[Crossref]

A. Ciattoni, G. Cincotti, D. Provenziani, and C. Palma, “Paraxial propagation along the optical axis of a uniaxial medium,” Phy. Rev. E 66(3), 036614 (2002).
[Crossref]

A. Ciattoni, G. Cincotti, and C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A 19(4), 792–796 (2002).
[Crossref]

Polyakov, S. V.

S. V. Polyakov and G. I. Stegeman, “Existence and properties of quadratic solitons in anisotropic media: Variational approach,” Phy. Rev. E 66(4), 046622 (2002).
[Crossref]

Porfirev, A. P.

Provenziani, D.

A. Ciattoni, G. Cincotti, D. Provenziani, and C. Palma, “Paraxial propagation along the optical axis of a uniaxial medium,” Phy. Rev. E 66(3), 036614 (2002).
[Crossref]

Quo, Q.

G. Liang and Q. Quo, “Spiraling elliptic solitons in nonlocal nonlinear media without anisotropy,” Phys. Rev. A 88(4), 043825 (2013).
[Crossref]

Saitoh, K.

K. Saitoh, Y. Hasegawa, K. Hirakawa, N. Tanaka, and M. Uchida, “Measuring the orbital angular momentum of electron vortex beams using a forked grating,” Phys. Rev. Lett. 111(7), 074801 (2013).
[Crossref] [PubMed]

Seshadri, S. R.

Silva, J. G.

Soares, W. C.

J. M. Hickmann, E. J. S. Fonseca, W. C. Soares, and S. Chávez-Cerda, “Unveiling a truncated optical lattice associated with a triangular aperture using light’s orbital angular momentum,” Phys. Rev. Lett. 105(5), 053904 (2010).
[Crossref] [PubMed]

Soskin, M.

A. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011).
[Crossref]

Stegeman, G. I.

S. V. Polyakov and G. I. Stegeman, “Existence and properties of quadratic solitons in anisotropic media: Variational approach,” Phy. Rev. E 66(4), 046622 (2002).
[Crossref]

Tanaka, N.

K. Saitoh, Y. Hasegawa, K. Hirakawa, N. Tanaka, and M. Uchida, “Measuring the orbital angular momentum of electron vortex beams using a forked grating,” Phys. Rev. Lett. 111(7), 074801 (2013).
[Crossref] [PubMed]

Tinkelman, I.

Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3(5), 305–310 (2007).
[Crossref]

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K. Saitoh, Y. Hasegawa, K. Hirakawa, N. Tanaka, and M. Uchida, “Measuring the orbital angular momentum of electron vortex beams using a forked grating,” Phys. Rev. Lett. 111(7), 074801 (2013).
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Other (4)

They obtained such a beam by making an elliptical Gaussian beam with a plane phase pass through a tilted cylindrical lens, so it is named. But the physical nature of such a beam is that the beam has the cross-phase term, therefore we tend to name it an “elliptical Gaussian beam with the cross-phase.”

The eigenmodes are the modes of propagation which are all of the solutions of Eq. (3) that form a complete and orthogonal set of functions.

H. A. Haus, Waves and fields in optoelectronics (Prentice-Hall, 1984), Chapt. 4 and 5.

M. Born and E. Wolf, Principles of Optics, 6 ed. (Pergamon Press, 1980), Chapt. 8.

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Figures (4)

Fig. 1
Fig. 1 Ellipticity functions ρ(z) of elliptic beams propagating in anisotropic media for four different initial ρ0s: w0x = 1.5 and w0y = 1, w 0 x = 1.2 and w0y = 1, w0x = 1.03 and w0y = 1 (ρ0 > 1), and w0x = 1 and w0y = 1.5 (ρ0 < 1). Parameters are taken to be δxx = 1.2, and δyy = 1.

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Fig. 2
Fig. 2 Evolutions of the elliptic beams with and without the OAM in the isotropic medium. (a1–a3): zero OAM Θ = 0, (b1–b3): with OAM Θ = 1/2, (c–d): evolutions of two semi-axes, and (e): evolutions of the ellipticities. w 0 x = 2 and w0y = 1. Red dashed line: with the OAM, and green solid line: without the OAM.

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Fig. 3
Fig. 3 Evolutions of the elliptic beams for two cases. (a1–a3): with the OAM M̃ = 1/8 in the isotropic medium, (b1–b3): without the OAM in the fictitious anisotropic medium of δxx = 3/2 and δyy = 3/4. The beam widthes are w 0 x = 2 and w0y = 1.

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Fig. 4
Fig. 4 Rotation angle θ (a) and angular velocity ω (b) of elliptic beams with OAM propagating in isotropic media. Parameters are taken to be w 0 x = 2, w0y = 1 and Θ = 1/4.

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Equations (32)

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i ( ∂ ϕ ∂ ζ − δ ξ ∂ ϕ ∂ ξ − δ η ∂ ϕ ∂ η ) + 1 2 k 0 ( δ ξ ξ ∂ 2 ϕ ∂ ξ 2 + δ ξ η ∂ 2 ϕ ∂ ξ ∂ η + δ η η ∂ 2 ϕ ∂ η 2 ) = 0 ,
i ( ∂ ϕ ∂ ζ − δ ∂ ϕ ∂ ξ ) + 1 2 k 0 ( δ ξ ξ ∂ 2 ϕ ∂ ξ 2 + δ η η ∂ 2 ϕ ∂ η 2 ) = 0 .
i ∂ ϕ ∂ z + 1 2 ( δ x x ∂ 2 ϕ ∂ x 2 + δ y y ∂ 2 ϕ ∂ y 2 ) = 0 ,
| δ x x ∂ 2 ϕ ∂ x 2 | ~ δ x x w x 2 | ϕ | ,                       | δ y y ∂ 2 ϕ ∂ y 2 | ~ δ y y w y 2 | ϕ | ,
ϕ m n ( x , y , z ) = C m n w H m ( x w x h g ) H n ( y w y h g ) exp [ − ( x 2 2 w x h g 2 + y 2 2 w y h g 2 ) ] × exp [ − i 1 2 R ( x 2 δ x x + y 2 δ y y ) + i ( m + n + 1 ) ψ ] ,
| ϕ 00 ( x , y , z ) | = 1 π δ x x 1 / 2 δ y y 1 / 2 w exp ( − x 2 2 w x h g 2 − y 2 2 w y h g 2 ) ,
ϕ ( x , y , 0 ) = P 0 π w 0 x w 0 y exp ( − x 2 2 w 0 x 2 − y 2 2 w 0 y 2 ) exp ( i Θ x y ) ,
M = Im ∫ − ∞ ∞ ∫ − ∞ ∞ ϕ * ( x ∂ ϕ ∂ y − y ∂ ϕ ∂ x ) d x d y = P 0 2 ( w 0 x 2 − w 0 y 2 ) Θ .
ϕ ( x , y , z ) = − i exp ( i z ) 2 π z ∫ − ∞ ∞ ∫ − ∞ ∞ ϕ ( x ′ , y ′ , 0 ) exp [ i ( x − x ′ ) 2 / δ x x + ( y − y ′ ) 2 / δ y y 2 z ] d x ′ d y ′ ,
ϕ ( x , y , z ) = ϕ 0 exp ( − x 2 2 w x 2 − y 2 2 w y 2 − x y 2 w x y + i φ ) ,
ϕ 0 = w 0 y P 0 w 0 x w 0 y π α r ,                 φ = c x y x y + c x x 2 + c y y 2 + arctan κ − π 2 ,
w x = δ x x δ y y w 0 x α r w 0 y 4 + δ y y 2 ( w 0 x 2 + w 0 y 2 Θ 2 + 1 ) z 2 ,     w y = δ x x δ y y w 0 y α r w 0 x 4 + δ x x 2 ( w 0 x 2 + w 0 y 2 Θ 2 + 1 ) z 2 ,
w x y = − α r δ x x 2 δ y y 2 2 w 0 x 2 w 0 y 2 ( w 0 x 2 δ y y + w 0 y 2 δ x x ) Θ z ,     κ = ( δ y y w 0 x 2 + δ x x w 0 y 2 ) z w 0 x 2 w 0 y 2 ( 1 − δ x x δ y y Θ 2 z 2 ) − δ x x δ y y z 2 ,
c x = z [ δ x x δ y y 2 z 2 ( 2 w 0 x 2 w 0 y 2 Θ 2 + w 0 x 4 w 0 y 4 Θ 4 ) + w 0 y 4 ( δ x x − δ y y w 0 x 4 Θ 2 ) ] 2 δ x x 2 δ y y 2 α r ,
c y = z [ δ x x 2 δ y y z 2 ( 2 w 0 x 2 w 0 y 2 Θ 2 + w 0 x 4 w 0 y 4 Θ 4 ) + w 0 x 4 ( δ y y − δ x x w 0 y 4 Θ 2 ) ] 2 δ x x 2 δ y y 2 α r ,
c x y = w 0 x 2 w 0 y 2 Θ [ w 0 x 2 w 0 y 2 − δ x x δ y y z 2 ( w 0 x 2 w 0 y 2 Θ 2 + 1 ) ] α r δ x x 2 δ y y 2 ,
α r = ( 2 w 0 x 2 w 0 y 2 Θ 2 + 1 ) z 4 + ( w 0 x 4 δ x x 2 + w 0 y 4 δ y y 2 ) z 2 + w 0 x 4 w 0 y 4 ( 1 δ x x δ y y − Θ 2 z 2 ) 2 .
| ϕ ( x , y , z ) | = ϕ 0 exp ( − x 2 2 w x a 2 − y 2 2 w y a 2 ) ,
w x a 2 = w 0 x 2 + ( δ x x w 0 x ) 2 z 2 ,             w y a 2 = w 0 y 2 + ( δ y y w 0 y ) 2 z 2 .
ρ 0 = δ x x δ y y ,
tan ( 2 θ ) = 2 w 0 x 2 w 0 y 2 ( w 0 x 2 + w 0 y 2 ) Θ z ( w 0 x 2 − w 0 y 2 ) ( w 0 x 2 w 0 y 2 − w 0 x 2 w 0 y 2 Θ 2 z 2 − z 2 ) .
w b 2 = 2 w x m 2 w y m 2 w x m 2 + w y m 2 − ( w x m 2 − w y m 2 ) 2 + w x m 2 w y m 2 / w x y m 2 , w c 2 = 2 w x m 2 w y m 2 w x m 2 + w y m 2 + ( w x m 2 − w y m 2 ) 2 + w x m 2 w y m 2 / w x y m 2 ,
δ x x eff = w 0 x w b 2 − w 0 x 2 z ,     δ y y eff = w 0 y w c 2 − w 0 y 2 z ,
w b 2 ≈ w 0 x 2 + 1 w 0 x 2 [ 1 + w 0 x 4 ( w 0 x 2 + 3 w 0 y 2 ) Θ 2 w 0 x 2 − w 0 y 2 ] z 2 , w c 2 ≈ w 0 y 2 + 1 w 0 y 2 [ 1 − w 0 y 4 ( w 0 y 2 + 3 w 0 x 2 ) Θ 2 w 0 x 2 − w 0 y 2 ] z 2 ,
δ x x eff ≈ 1 + w 0 x 4 ( w 0 x 2 + 3 w 0 y 2 ) Θ 2 w 0 x 2 − w 0 y 2 ,     δ y y eff ≈ 1 − w 0 y 4 ( w 0 y 2 + 3 w 0 x 2 ) Θ 2 w 0 x 2 − w 0 y 2 .
Θ c = ± w 0 x 2 − w 0 y 2 2 w 0 x 2 w 0 y 2 .
( w b cri ) 2 = w 0 x 2 + ( δ x x c eff w 0 x ) 2 z 2 ,                 ( w c cri ) 2 = w 0 y 2 + ( δ y y c eff w 0 y ) 2 z 2 ,
δ x x c eff = | M ˜ | + 1 + | M ˜ | ( | M ˜ | + 1 ) ,             δ y y c eff = | M ˜ | + 1 − | M ˜ | ( | M ˜ | + 1 ) ,
M c P 0 = 1 2 ( w 0 x 2 − w 0 y 2 ) Θ c = ± 1 4 ( w 0 x 2 − w 0 y 2 ) 2 w 0 x 2 w 0 y 2 .
w b cri w c cri = ( δ x x c eff δ y y c eff ) 1 / 2 = | M ˜ | + 1 + | M ˜ | .
θ = { − 1 2 arctan [ 2 w 0 x 2 w 0 y 2 ( w 0 x 2 + w 0 y 2 ) Θ z ( w 0 x 2 + w 0 y 2 ) [ z 2 + w 0 x 2 w 0 y 2 ( − 1 + z 2 Θ 2 ) ] ] , z < w 0 x w 0 y w 0 x 2 w 0 y 2 Θ 2 + 1 − 1 2 arctan [ 2 w 0 x 2 w 0 y 2 ( w 0 x 2 + w 0 y 2 ) Θ z ( w 0 x 2 + w 0 y 2 ) [ z 2 + w 0 x 2 w 0 y 2 ( − 1 + z 2 Θ 2 ) ] ] + π 2 , z > w 0 x w 0 y w 0 x 2 w 0 y 2 Θ 2 + 1 .
ω ≡ d θ d z = 1 2 ( 1 + tan 2 θ ) 2 d d z [ w x m 2 w y m 2 w x y m ( w y m 2 − w x m 2 ) ] .

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