Abstract

We consider the oscillations of the orbital angular momentum in the vortex beams with fractional topological charges, taking into account the vortex influence, beam astigmatism, and the displacement of the center of gravity on the basis of the intensity momenta approach. We revealed also that base contribution gives the optical vortices and displacement of the center of gravity. In addition, we analyze the distribution of the current lines in the erf-G beams with the topological charge p=1/2, revealing that there are two types of the curves: (1) closed C-like lines and (2) the inner closed lines.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
    [CrossRef]
  2. J. Zagrodzinski, “Vortices in different branches of physics,” Physica C 369, 45–54 (2002).
    [CrossRef]
  3. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
    [CrossRef]
  4. Yu. A. Ushenko, O. V. Dubolazov, and A. O. Karachevtsev, “Statistical structure of skin derma Mueller matrix images in the process of cancer changes,” Opt. Mem. Neural Netw. 20, 145–154 (2011).
    [CrossRef]
  5. G. A. Swartzlander, “Achromatic optical vortex lens,” Opt. Lett. 31, 2042–2044 (2006).
    [CrossRef]
  6. L. Allen, S. M. Barnett, and M. J. Padgett, Orbital Angular Momentum (Bristol: Institute of Physics, 2003).
  7. O. V. Angelsky, A. Ya. Bekshaev, P. P. Maksimyak, A. P. Maksimyak, S. G. Hanson, and C. Yu. Zenkova, “Orbital rotation without orbital angular momentum: mechanical action of the spin part of the internal energy flow in light beams,” Opt. Express 20, 3563–3571 (2012).
    [CrossRef]
  8. I. Basisty, M. Soskin, and M. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. 119, 604–612 (1995).
    [CrossRef]
  9. M. V. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A 6, 259–268 (2004).
    [CrossRef]
  10. M. V. Berry, R. G. Chambers, M. D. Large, C. Upstill, and J. C. Walmsley, “Wavefront dislocations in the Aharonov-Bohm effect and its water wave analogue,” Eur. J. Phys. 1, 154–162 (1980).
    [CrossRef]
  11. J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
    [CrossRef]
  12. H. Garcia-Gracia and J. C. Gutiérrez-Vega, “Diffraction of plane waves by finite-radius spiral phase plates of integer and fractional topological charge,” J. Opt. Soc. Am. A 26, 794–803 (2009).
    [CrossRef]
  13. J. C. Gutiérrez-Vega and C. López-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A 10, 015009 (2008).
    [CrossRef]
  14. J. C. Gutiérrez-Vega, “Fractionalization of optical beams: I. Planar analysis,” Opt. Lett. 32, 1521–1523 (2007).
    [CrossRef]
  15. J. C. Gutiérrez-Vega, “Fractionalization of optical beams: II. Elegant Laguerre–Gaussian modes,” Opt. Express 15, 6300–6313 (2007).
    [CrossRef]
  16. S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
    [CrossRef]
  17. T. A. Fadeyeva, C. Alexeyev, A. Rubass, and A. Volyar, “Vector erf-Gaussian beams: fractional optical vortices and asymmetric TE and TM modes,” Opt. Lett. 37, 1397–1399 (2012).
    [CrossRef]
  18. A. P. Kiselev, “Localized light waves: Paraxial and exact solutions of the wave equation (a review),” Opt. Spectrosc. 102, 603–622 (2007).
    [CrossRef]
  19. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge University, 1968).
  20. A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Optical vortex symmetry breakdown and decomposition of the orbital angular momentum of light beams,” J. Opt. Soc. Am. A 20, 1635–1643 (2003).
    [CrossRef]
  21. A. Ya. Bekshaev, M. V. Vasnetsov, V. G. Denisenko, and M. S. Soskin, “Transformation of the orbital angular momentum of a beam with optical vortex in an astigmatic optical system,” JETP Lett. 75, 127–130 (2002).
    [CrossRef]
  22. Yu. A. Anan’ev and A. Ya. Bekshaev, “Theory of intensity moments for arbitrary light beams,” Opt. Spectrosc. 76, 558–568 (1994).
  23. A. Ya. Bekshaev, “Improved theory for the polarization-dependent transverse shift of a paraxial light beam in free space,” Ukr. J. Phys. Opt. 12, 10–18 (2011).
    [CrossRef]
  24. A. Ya. Bekshaev and A. I. Karamoch, “Astigmatic telescopic transformation of a high-order optical vortex,” Opt. Commun. 281, 5687–5696 (2008).
    [CrossRef]
  25. M. V. Berry, “Optical currents,” J. Opt. A 11, 094001 (2009).
    [CrossRef]

2012 (2)

2011 (2)

Yu. A. Ushenko, O. V. Dubolazov, and A. O. Karachevtsev, “Statistical structure of skin derma Mueller matrix images in the process of cancer changes,” Opt. Mem. Neural Netw. 20, 145–154 (2011).
[CrossRef]

A. Ya. Bekshaev, “Improved theory for the polarization-dependent transverse shift of a paraxial light beam in free space,” Ukr. J. Phys. Opt. 12, 10–18 (2011).
[CrossRef]

2009 (2)

2008 (2)

A. Ya. Bekshaev and A. I. Karamoch, “Astigmatic telescopic transformation of a high-order optical vortex,” Opt. Commun. 281, 5687–5696 (2008).
[CrossRef]

J. C. Gutiérrez-Vega and C. López-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A 10, 015009 (2008).
[CrossRef]

2007 (3)

2006 (1)

2005 (1)

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef]

2004 (2)

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

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[CrossRef]

2003 (2)

2002 (2)

A. Ya. Bekshaev, M. V. Vasnetsov, V. G. Denisenko, and M. S. Soskin, “Transformation of the orbital angular momentum of a beam with optical vortex in an astigmatic optical system,” JETP Lett. 75, 127–130 (2002).
[CrossRef]

J. Zagrodzinski, “Vortices in different branches of physics,” Physica C 369, 45–54 (2002).
[CrossRef]

2001 (1)

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[CrossRef]

1995 (1)

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

1994 (1)

Yu. A. Anan’ev and A. Ya. Bekshaev, “Theory of intensity moments for arbitrary light beams,” Opt. Spectrosc. 76, 558–568 (1994).

1980 (1)

M. V. Berry, R. G. Chambers, M. D. Large, C. Upstill, and J. C. Walmsley, “Wavefront dislocations in the Aharonov-Bohm effect and its water wave analogue,” Eur. J. Phys. 1, 154–162 (1980).
[CrossRef]

’t Hooft, G. W.

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef]

Aiello, A.

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef]

Alexeyev, C.

Allen, L.

L. Allen, S. M. Barnett, and M. J. Padgett, Orbital Angular Momentum (Bristol: Institute of Physics, 2003).

Anan’ev, Yu. A.

Yu. A. Anan’ev and A. Ya. Bekshaev, “Theory of intensity moments for arbitrary light beams,” Opt. Spectrosc. 76, 558–568 (1994).

Angelsky, O. V.

Barnett, S. M.

L. Allen, S. M. Barnett, and M. J. Padgett, Orbital Angular Momentum (Bristol: Institute of Physics, 2003).

Basisty, I.

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

Bekshaev, A. Ya.

O. V. Angelsky, A. Ya. Bekshaev, P. P. Maksimyak, A. P. Maksimyak, S. G. Hanson, and C. Yu. Zenkova, “Orbital rotation without orbital angular momentum: mechanical action of the spin part of the internal energy flow in light beams,” Opt. Express 20, 3563–3571 (2012).
[CrossRef]

A. Ya. Bekshaev, “Improved theory for the polarization-dependent transverse shift of a paraxial light beam in free space,” Ukr. J. Phys. Opt. 12, 10–18 (2011).
[CrossRef]

A. Ya. Bekshaev and A. I. Karamoch, “Astigmatic telescopic transformation of a high-order optical vortex,” Opt. Commun. 281, 5687–5696 (2008).
[CrossRef]

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Optical vortex symmetry breakdown and decomposition of the orbital angular momentum of light beams,” J. Opt. Soc. Am. A 20, 1635–1643 (2003).
[CrossRef]

A. Ya. Bekshaev, M. V. Vasnetsov, V. G. Denisenko, and M. S. Soskin, “Transformation of the orbital angular momentum of a beam with optical vortex in an astigmatic optical system,” JETP Lett. 75, 127–130 (2002).
[CrossRef]

Yu. A. Anan’ev and A. Ya. Bekshaev, “Theory of intensity moments for arbitrary light beams,” Opt. Spectrosc. 76, 558–568 (1994).

Berry, M. V.

M. V. Berry, “Optical currents,” J. Opt. A 11, 094001 (2009).
[CrossRef]

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

M. V. Berry, R. G. Chambers, M. D. Large, C. Upstill, and J. C. Walmsley, “Wavefront dislocations in the Aharonov-Bohm effect and its water wave analogue,” Eur. J. Phys. 1, 154–162 (1980).
[CrossRef]

Chambers, R. G.

M. V. Berry, R. G. Chambers, M. D. Large, C. Upstill, and J. C. Walmsley, “Wavefront dislocations in the Aharonov-Bohm effect and its water wave analogue,” Eur. J. Phys. 1, 154–162 (1980).
[CrossRef]

Denisenko, V. G.

A. Ya. Bekshaev, M. V. Vasnetsov, V. G. Denisenko, and M. S. Soskin, “Transformation of the orbital angular momentum of a beam with optical vortex in an astigmatic optical system,” JETP Lett. 75, 127–130 (2002).
[CrossRef]

Dubolazov, O. V.

Yu. A. Ushenko, O. V. Dubolazov, and A. O. Karachevtsev, “Statistical structure of skin derma Mueller matrix images in the process of cancer changes,” Opt. Mem. Neural Netw. 20, 145–154 (2011).
[CrossRef]

Eliel, E. R.

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef]

Fadeyeva, T. A.

Garcia-Gracia, H.

Grier, D. G.

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[CrossRef]

Gutiérrez-Vega, J. C.

Hanson, S. G.

Karachevtsev, A. O.

Yu. A. Ushenko, O. V. Dubolazov, and A. O. Karachevtsev, “Statistical structure of skin derma Mueller matrix images in the process of cancer changes,” Opt. Mem. Neural Netw. 20, 145–154 (2011).
[CrossRef]

Karamoch, A. I.

A. Ya. Bekshaev and A. I. Karamoch, “Astigmatic telescopic transformation of a high-order optical vortex,” Opt. Commun. 281, 5687–5696 (2008).
[CrossRef]

Kiselev, A. P.

A. P. Kiselev, “Localized light waves: Paraxial and exact solutions of the wave equation (a review),” Opt. Spectrosc. 102, 603–622 (2007).
[CrossRef]

Large, M. D.

M. V. Berry, R. G. Chambers, M. D. Large, C. Upstill, and J. C. Walmsley, “Wavefront dislocations in the Aharonov-Bohm effect and its water wave analogue,” Eur. J. Phys. 1, 154–162 (1980).
[CrossRef]

Leach, J.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[CrossRef]

López-Mariscal, C.

J. C. Gutiérrez-Vega and C. López-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A 10, 015009 (2008).
[CrossRef]

Ma, X.

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef]

Maksimyak, A. P.

Maksimyak, P. P.

Oemrawsingh, S. S. R.

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef]

Padgett, M. J.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[CrossRef]

L. Allen, S. M. Barnett, and M. J. Padgett, Orbital Angular Momentum (Bristol: Institute of Physics, 2003).

Rubass, A.

Soskin, M.

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

Soskin, M. S.

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Optical vortex symmetry breakdown and decomposition of the orbital angular momentum of light beams,” J. Opt. Soc. Am. A 20, 1635–1643 (2003).
[CrossRef]

A. Ya. Bekshaev, M. V. Vasnetsov, V. G. Denisenko, and M. S. Soskin, “Transformation of the orbital angular momentum of a beam with optical vortex in an astigmatic optical system,” JETP Lett. 75, 127–130 (2002).
[CrossRef]

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[CrossRef]

Swartzlander, G. A.

Upstill, C.

M. V. Berry, R. G. Chambers, M. D. Large, C. Upstill, and J. C. Walmsley, “Wavefront dislocations in the Aharonov-Bohm effect and its water wave analogue,” Eur. J. Phys. 1, 154–162 (1980).
[CrossRef]

Ushenko, Yu. A.

Yu. A. Ushenko, O. V. Dubolazov, and A. O. Karachevtsev, “Statistical structure of skin derma Mueller matrix images in the process of cancer changes,” Opt. Mem. Neural Netw. 20, 145–154 (2011).
[CrossRef]

Vasnetsov, M.

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

Vasnetsov, M. V.

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Optical vortex symmetry breakdown and decomposition of the orbital angular momentum of light beams,” J. Opt. Soc. Am. A 20, 1635–1643 (2003).
[CrossRef]

A. Ya. Bekshaev, M. V. Vasnetsov, V. G. Denisenko, and M. S. Soskin, “Transformation of the orbital angular momentum of a beam with optical vortex in an astigmatic optical system,” JETP Lett. 75, 127–130 (2002).
[CrossRef]

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[CrossRef]

Voigt, D.

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef]

Volyar, A.

Walmsley, J. C.

M. V. Berry, R. G. Chambers, M. D. Large, C. Upstill, and J. C. Walmsley, “Wavefront dislocations in the Aharonov-Bohm effect and its water wave analogue,” Eur. J. Phys. 1, 154–162 (1980).
[CrossRef]

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge University, 1968).

Woerdman, J. P.

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef]

Yao, E.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[CrossRef]

Zagrodzinski, J.

J. Zagrodzinski, “Vortices in different branches of physics,” Physica C 369, 45–54 (2002).
[CrossRef]

Zenkova, C. Yu.

Eur. J. Phys. (1)

M. V. Berry, R. G. Chambers, M. D. Large, C. Upstill, and J. C. Walmsley, “Wavefront dislocations in the Aharonov-Bohm effect and its water wave analogue,” Eur. J. Phys. 1, 154–162 (1980).
[CrossRef]

J. Opt. A (3)

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

J. C. Gutiérrez-Vega and C. López-Mariscal, “Nondiffracting vortex beams with continuous orbital angular momentum order dependence,” J. Opt. A 10, 015009 (2008).
[CrossRef]

M. V. Berry, “Optical currents,” J. Opt. A 11, 094001 (2009).
[CrossRef]

J. Opt. Soc. Am. A (2)

JETP Lett. (1)

A. Ya. Bekshaev, M. V. Vasnetsov, V. G. Denisenko, and M. S. Soskin, “Transformation of the orbital angular momentum of a beam with optical vortex in an astigmatic optical system,” JETP Lett. 75, 127–130 (2002).
[CrossRef]

Nature (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[CrossRef]

New J. Phys. (1)

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[CrossRef]

Opt. Commun. (2)

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

A. Ya. Bekshaev and A. I. Karamoch, “Astigmatic telescopic transformation of a high-order optical vortex,” Opt. Commun. 281, 5687–5696 (2008).
[CrossRef]

Opt. Express (2)

Opt. Lett. (3)

Opt. Mem. Neural Netw. (1)

Yu. A. Ushenko, O. V. Dubolazov, and A. O. Karachevtsev, “Statistical structure of skin derma Mueller matrix images in the process of cancer changes,” Opt. Mem. Neural Netw. 20, 145–154 (2011).
[CrossRef]

Opt. Spectrosc. (2)

Yu. A. Anan’ev and A. Ya. Bekshaev, “Theory of intensity moments for arbitrary light beams,” Opt. Spectrosc. 76, 558–568 (1994).

A. P. Kiselev, “Localized light waves: Paraxial and exact solutions of the wave equation (a review),” Opt. Spectrosc. 102, 603–622 (2007).
[CrossRef]

Phys. Rev. Lett. (1)

S. S. R. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. ’t Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef]

Physica C (1)

J. Zagrodzinski, “Vortices in different branches of physics,” Physica C 369, 45–54 (2002).
[CrossRef]

Prog. Opt. (1)

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[CrossRef]

Ukr. J. Phys. Opt. (1)

A. Ya. Bekshaev, “Improved theory for the polarization-dependent transverse shift of a paraxial light beam in free space,” Ukr. J. Phys. Opt. 12, 10–18 (2011).
[CrossRef]

Other (2)

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge University, 1968).

L. Allen, S. M. Barnett, and M. J. Padgett, Orbital Angular Momentum (Bristol: Institute of Physics, 2003).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1.
Fig. 1.

Specific OAM lz as a function of the topological charge p for different values of the K parameters [the green curve corresponds to Eq. (9) while the red curve is associated with Eq. (8)]. The abscissa in Fig. 1 is of the topological charge of the FV beams.

Fig. 2.
Fig. 2.

Lateral shift y0/w0 as a function of the topological charge p.

Fig. 3.
Fig. 3.

Component kx/k of the wave vector as a function of the topological charge p.

Fig. 4.
Fig. 4.

(a) Extrinsic OAM lextr for small k parameter k=104m1. (b) Comparison of the total lz (1) with the extrinsic (2) OAM lextr for the large k parameter k=105m1; w0=30μm.

Fig. 5.
Fig. 5.

Intrinsic lintr(p) OAM with the parameters defined by Fig. 4.

Fig. 6.
Fig. 6.

Contribution of the astigmatism last(p) (a) and the optical vortices lV to the intrinsic OAM (b) for k=105m1 and w0=30μm.

Fig. 7.
Fig. 7.

Lines of the transverse optical currents J with p=1/2 for (a) k=105m1 and (b) k=3·104m1, w0=200μm at z=0, p=0.5. The thin lines in (a) are the equi-phase lines.

Fig. 8.
Fig. 8.

Longitudinal Jz (a) and transverse J current lines for k=105m1, w0=200μm along the beam length z. Thin lines in (b) are the equi-phase lines.

Equations (54)

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

Ψ˜=NGeikz02πeipϕ+iKRcos(ϕφ)dϕ=eikzΨ,
Ψ=NG02πeipϕ+iKRcos(ϕφ)dϕ=NGeipπsin(πp)πm=eimφpm02πeimϕ+iKRcosϕdϕ=NG2eipπsin(πp)m=imJm(KR)pmeimφ,
02πeimϕ+iKRcosϕdϕ=2πimJm(KR).
Lz=iΨ*|φ|Ψ.
Lz=4|N|2sin2(πp)j,m=imjm(pm)(pj)×02πei(mj)φdφ0rJm(Kr)Jj(K*r)dr,
Lz=4π|N|2sin2(πp)w02×exp([K2+K*2]w02/8)m=mIm(|K|2w02/4)(pm)2,
lz=LzI=m=mIm(|K|2w02/4)(pm)2m=Im(|K|2w02/4)(pm)2.
Im(K2w02/4)exp(K2w02/4),K2w02/41
lz=m=m(pm)2m=1(pm)2.
P0=(rp)=c8πΦΨ*(r)(r(i/k))Ψ(r)d2r,
|Ψ|2=|Np|2{J0p+m=1(1)mimJmp+meimφ+m=1imJmpmeimφ}×{J0p+j=1(1)jijJjp+jeijφ+m=1ijJjpjeijφ}=(J0p)2+m=1[imeimφp+m+imeimφpm+imeimφp+m+imeimφpm]J0,mp+m,j=1i(mj)(1)m+j(p+m)(p+j)ei(mj)φJm,j+m,j=1im+j(1)j(pm)(p+j)ei(m+j)φJm,j.
x0=1Φ0r2dr02πcosφ|Ψ|2dφ.
x0=0.
y0=1Φ0r2dr02πsinφ|Ψ|2dφ.
02πsinφe±imφdφ=±iπ2,m=1.
i(i)p+m+i·ipm+i·ip+m+i(i)pm=4pp21.
02πsinφei(mj)φdφ=i02πsinφsin(mj)φdφ=iπ{1,mj=1.j=1+m,m=1,2,3,1,mj=1.j=m1,m=2,3,4,
02πsinφei(mj)φdφ=i02πsinφsin(mj)φdφ=iπ{1,mj=1.j=1+m,m=1,2,3,1,mj=1.j=m1,m=2,3,4,
m,j=1i(mj)(1)m+j(p+m)(p+j)ei(mj)φsinφJm,jiπm=2i1(1)(p+m)(p+m1)Jm,m1+iπm=1i(1)(p+m)(p+m+1)Jm,m+1=2πm=1Jm,m+1(p+m)(p+m+1),
m,j=1imjJm,j(pm)(pj)ei(mj)φsinφiπm=1i1(pm)(pm1)Jm,m1iπm=2i1(pm)(pm+1)Jm,m+1=2πm=1Jm,m1(pm)(pm1),
Jm,m+1=Jm+1,m=0r2Jm(kr)Jm+1(kr)er2/w02dr=w02ek2w02422m+3m!kM12,m+12(k2w022),
0r2J0(kr)J1(kr)e2r2/w02dr=kw0416ek2w024[I0(k2w024)I1(k2w024)],
Φ=w024exp(K2w02/4)m=Im(K2w02/4)(pm)2.
Y0=y0w0={kw0[I0(k2w024)I1(k2w024)]m=1(2m+1)M12,m+12(k2w022)22m+3m!kw02(p2m2)(p2(m+1)2)}×(p(p21)m=Im(K2w02/4)(pm)2)1.
k0x=1Φ0k2dk0cosϕ|Uq(k,ϕ)|2dϕ,
k0y=1Φ0k2dk0sinϕ|Uq(k,ϕ)|2dϕ,
U(p,z)=k2πΨ(r,z)exp[ik(p·r)]d2r.
Uq(kp,ϕ)=kNqexp{(k2+kp2)w024}m=eimϕpmIm(kkpw022).
|Uq|2=Kexp{(k2+kp2)w022}×{I02p2+m=1[(1)meimϕp+m+eimϕpm+(1)meimϕp+meimϕpm]Im+m,j=1(1)m+jei(mj)ϕ(p+m)(q+j)Im,j+m,j=1(1)mei(m+j)ϕ(pm)(q+j)Im,j+m,j=1(1)jei(m+j)ϕ(p+m)(pj)Im,j+m,j=1ei(mj)ϕ(pm)(pj)Im,j}.
(1)meimϕp+m+eimϕpm+(1)meimϕp+meimϕpm4qp21.
kxk=1Φ4pp21I0,1,
lextr=k(x0pyy0px)=ky0px.
lintr=lzlextr.
M=(M11M12M˜12M22),
M11=c8πΦrr˜|Ψ(r)|2d2r,
M12=M˜21=i2kc8πΦr[Ψ(r)˜Ψ*(r)Ψ*(r)˜Ψ(r)]d2r,
M22=1k2c8πΦΨ*(r)˜Ψ(r)d2r.
lz=kSp(M12J),
lV=2kSp(M12M11J)SpM11
last=lintrlv.
J=Re(E×H*),
Ex=ikΨ˜,Ey=kΨ˜,EzikE,
HxikzEy,HyzEx,HzikH,
02πei(mj)φdφ0rJm(Kr)Jj(K*r)eikr22Zdr.
02πei(mj)φdφ=2π,m=j.
0xJm(bx)Jm(cx)exp(px2)dx=12pexp(b2+c24p)Im(bc2p).
Lz=4π|N|2sin2(πp)w02×exp(K2w02/4)m=mIm(K2w02/4)(pm)2.
02πcosφe±imφdφ=π,m=1,
ip+m+ipm+ip+m+ipm=0.
02πcosφe±i(m+j)φdφ=02πcosφcos(m+j)φdφ=π{j=1m,dnot exist,sincej,m>0j=1m,dnot exist,sincej>0,
02πcosφcos(m+j)φdφ=πδ1,j+m.
02πcosφe±i(mj)φdφ=02πcosφcos(mj)φdφ=π{mj=1.j=1+m,m=1,2,3,mj=1.j=m1,m=2,3,4,,J>0.
02πsinφei(mj)φdφ=i02πsinφsin(mj)φdφ=iπ{1,mj=1.j=1+m,m=1,2,3,1,mj=1.j=m1,m=2,3,4,
02πsinφei(mj)φdφ=i02πsinφsin(mj)φdφ=iπ{1,mj=1.j=1+m,m=1,2,3,1,mj=1.j=m1,m=2,3,4,.

Metrics