Abstract

Edge diffraction of a circular Laguerre–Gaussian beam represents an example of the optical vortex symmetry breakdown in which the hidden “vortex” energy circulation is partially transformed into the visible “asymmetry” form. The diffracted beam evolution is studied in terms of the irradiance moments and the moment-based parameters. In spite of the limited applicability of the moment-based formalism, we show that the “vortex” and “asymmetry” parts of the orbital angular momentum can still be reasonably defined for the hard-edge diffracted beams and retain their physical role of quantifying the corresponding forms of the transverse energy circulation.

© 2014 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  8. V. Garcés-Chavéz, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003).
    [CrossRef]
  9. M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophotonics 2, 021875 (2008).
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    [CrossRef]
  24. A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241, 237–247 (2004).
    [CrossRef]
  25. T. Alieva and M. J. Bastiaans, “Evolution of the vortex and the asymmetrical parts of orbital angular momentum in separable first-order optical systems,” Opt. Lett. 29, 1587–1589 (2004).
    [CrossRef]
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    [CrossRef]
  36. A. Ya. Bekshaev and S. V. Sviridova, “Effects of misalignments in the optical vortex transformation performed by holograms with embedded phase singularity,” Opt. Commun. 283, 4866–4876 (2010).
    [CrossRef]
  37. Z. Mei and D. Zhao, “Generalized beam propagation factor of hard-edged diffracted controllable dark-hollow beams,” Opt. Commun. 263, 261–266 (2006).
    [CrossRef]
  38. Z. Lu, H. Jiang, X. Du, and D. Zhao, “Generalized M2 factor of truncated partially coherent controllable dark-hollow beams,” J. Mod. Opt. 55, 2381–2390 (2008).
    [CrossRef]
  39. Z. Mei and D. Zhao, “Approximate method for the generalized M2 factor of rotationally symmetric hard-edged diffracted flattened Gaussian beams,” Appl. Opt. 44, 1381–1386 (2005).
    [CrossRef]
  40. G. Zhou, “Generalized M2 factors of truncated partially coherent Lorentz and Lorentz–Gauss beams,” J. Opt. 12, 015701 (2010).
    [CrossRef]
  41. A. Bekshaev and K. A. Mohammed, “Transverse energy redistribution upon edge diffraction of a paraxial laser beam with optical vortex,” Proc. SPIE 9066, 906602 (2013).
    [CrossRef]

2013

R. W. Bowman and M. J. Padgett, “Optical trapping and binding,” Rep. Prog. Phys. 76, 026401 (2013).
[CrossRef]

A. Bekshaev and K. A. Mohammed, “Transverse energy redistribution upon edge diffraction of a paraxial laser beam with optical vortex,” Proc. SPIE 9066, 906602 (2013).
[CrossRef]

Y. Luo, Z. Gao, B. Tang, and B. Lü, “Electric and magnetic polarization singularities of first-order Laguerre-Gaussian Beams diffracted at a half-plane screen,” J. Opt. Soc. Am. A 30, 1646–1653 (2013).
[CrossRef]

2012

H. X. Cui, X. L. Wang, B. Gu, Y. N. Li, J. Chen, and H. T. Wang, “Angular diffraction of an optical vortex induced by the Gouy phase,” J. Opt. 14, 055707 (2012).
[CrossRef]

2011

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

2010

G. Zhou, “Generalized M2 factors of truncated partially coherent Lorentz and Lorentz–Gauss beams,” J. Opt. 12, 015701 (2010).
[CrossRef]

A. Ya. Bekshaev and S. V. Sviridova, “Effects of misalignments in the optical vortex transformation performed by holograms with embedded phase singularity,” Opt. Commun. 283, 4866–4876 (2010).
[CrossRef]

2009

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[CrossRef]

D. P. Ghai, P. Senthilkumaran, and R. S. Sirohi, “Single-slit diffraction of an optical beam with phase singularity,” Opt. Lasers Eng. 47, 123–126 (2009).
[CrossRef]

J. Masajada, M. Leniec, S. Drobczyński, H. Thienpont, and B. Kress, “Micro-step localization using double charge optical vortex interferometer,” Opt. Express 17, 16144–16159 (2009).
[CrossRef]

2008

J. Masajada, M. Leniec, E. Jankowska, H. Thienpont, H. Ottevaere, and V. Gomez, “Deep microstructure topography characterization with optical vortex interferometer,” Opt. Express 16, 19179–19191 (2008).
[CrossRef]

P. Liu and B. Lü, “Propagation of Gaussian background vortex beams diffracted at a half-plane screen,” Opt. Laser Technol. 40, 227–234 (2008).
[CrossRef]

M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophotonics 2, 021875 (2008).
[CrossRef]

Z. Lu, H. Jiang, X. Du, and D. Zhao, “Generalized M2 factor of truncated partially coherent controllable dark-hollow beams,” J. Mod. Opt. 55, 2381–2390 (2008).
[CrossRef]

2006

Z. Mei and D. Zhao, “Generalized beam propagation factor of hard-edged diffracted controllable dark-hollow beams,” Opt. Commun. 263, 261–266 (2006).
[CrossRef]

A. Bekshaev and M. Soskin, “Rotational transformations and transverse energy flow in paraxial light beams: linear azimuthons,” Opt. Lett. 31, 2199–2201 (2006).
[CrossRef]

2005

2004

T. Alieva and M. J. Bastiaans, “Evolution of the vortex and the asymmetrical parts of orbital angular momentum in separable first-order optical systems,” Opt. Lett. 29, 1587–1589 (2004).
[CrossRef]

Ch. V. Felde, “Diffraction diagnostics of phase singularities in optical fields,” Proc. SPIE 5477, 67–76 (2004).
[CrossRef]

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241, 237–247 (2004).
[CrossRef]

2003

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]

V. Garcés-Chavéz, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef]

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

H. V. Bogatyryova, Ch. V. Felde, and P. V. Polyanskii, “Referenceless testing of vortex optical beams,” Opt. Appl. 33, 695–708 (2003).

2002

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef]

P. M. Mejias, R. Martinez-Herrero, G. Piquero, and J. M. Movilla, “Parametric characterization of the spatial structure of non-uniformly polarized laser beams,” Prog. Quantum Electron. 26, 65–130 (2002).
[CrossRef]

2001

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

2000

M. V. Vasnetsov, I. G. Marienko, and M. S. Soskin, “Self-reconstruction of an optical vortex,” J. Exp. Theor. Phys. Lett. 71, 130–133 (2000).
[CrossRef]

J. Masajada, “Gaussian beams with optical vortex of charge 2- and 3-diffraction by a half-plane and slit,” Opt. Appl. 30, 248–256 (2000).

J. Masajada, “Half-plane diffraction in the case of Gaussian beams containing an optical vortex,” Opt. Commun. 175, 289–294 (2000).
[CrossRef]

R. Simon and N. J. Mukunda, “Optical phase space, Wigner representation, and invariant quality parameters,” J. Opt. Soc. Am. A 17, 2440–2463 (2000).
[CrossRef]

1999

L. Allen, M. J. Padgett, and M. Babiker, “Orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[CrossRef]

1998

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
[CrossRef]

1997

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” Prog. Opt. 37, 1–56 (1997).
[CrossRef]

1995

1994

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

1993

1991

M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik 88, 163–168 (1991).

Alieva, T.

Allen, L.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef]

L. Allen, M. J. Padgett, and M. Babiker, “Orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[CrossRef]

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).

Yu. A. Anan’ev, Laser Resonators and the Beam Divergence Problem (Adam Hilger, 1992).

Arias, M.

Arlt, J.

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

Babiker, M.

L. Allen, M. J. Padgett, and M. Babiker, “Orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[CrossRef]

Bastiaans, M. J.

T. Alieva and M. J. Bastiaans, “Evolution of the vortex and the asymmetrical parts of orbital angular momentum in separable first-order optical systems,” Opt. Lett. 29, 1587–1589 (2004).
[CrossRef]

M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik 88, 163–168 (1991).

Bekshaev, A.

A. Bekshaev and K. A. Mohammed, “Transverse energy redistribution upon edge diffraction of a paraxial laser beam with optical vortex,” Proc. SPIE 9066, 906602 (2013).
[CrossRef]

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

A. Bekshaev and M. Soskin, “Rotational transformations and transverse energy flow in paraxial light beams: linear azimuthons,” Opt. Lett. 31, 2199–2201 (2006).
[CrossRef]

Bekshaev, A. Ya.

A. Ya. Bekshaev and S. V. Sviridova, “Effects of misalignments in the optical vortex transformation performed by holograms with embedded phase singularity,” Opt. Commun. 283, 4866–4876 (2010).
[CrossRef]

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, “Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241, 237–247 (2004).
[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]

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

A. Ya. Bekshaev, M. S. Soskin, and M. V. Vasnetsov, Paraxial Light Beams with Angular Momentum (Nova Science, 2008).

Bliokh, K.

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

Bogatyryova, H. V.

H. V. Bogatyryova, Ch. V. Felde, and P. V. Polyanskii, “Referenceless testing of vortex optical beams,” Opt. Appl. 33, 695–708 (2003).

Bowman, R. W.

R. W. Bowman and M. J. Padgett, “Optical trapping and binding,” Rep. Prog. Phys. 76, 026401 (2013).
[CrossRef]

Chen, J.

H. X. Cui, X. L. Wang, B. Gu, Y. N. Li, J. Chen, and H. T. Wang, “Angular diffraction of an optical vortex induced by the Gouy phase,” J. Opt. 14, 055707 (2012).
[CrossRef]

Cui, H. X.

H. X. Cui, X. L. Wang, B. Gu, Y. N. Li, J. Chen, and H. T. Wang, “Angular diffraction of an optical vortex induced by the Gouy phase,” J. Opt. 14, 055707 (2012).
[CrossRef]

Dennis, M. R.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[CrossRef]

Dholakia, K.

M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophotonics 2, 021875 (2008).
[CrossRef]

V. Garcés-Chavéz, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef]

Dienerowitz, M.

M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophotonics 2, 021875 (2008).
[CrossRef]

Dragoman, D.

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” Prog. Opt. 37, 1–56 (1997).
[CrossRef]

Drobczynski, S.

Du, X.

Z. Lu, H. Jiang, X. Du, and D. Zhao, “Generalized M2 factor of truncated partially coherent controllable dark-hollow beams,” J. Mod. Opt. 55, 2381–2390 (2008).
[CrossRef]

Dultz, W.

V. Garcés-Chavéz, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef]

Felde, Ch. V.

Ch. V. Felde, “Diffraction diagnostics of phase singularities in optical fields,” Proc. SPIE 5477, 67–76 (2004).
[CrossRef]

H. V. Bogatyryova, Ch. V. Felde, and P. V. Polyanskii, “Referenceless testing of vortex optical beams,” Opt. Appl. 33, 695–708 (2003).

Friese, M. E. J.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
[CrossRef]

Gao, Z.

Garcés-Chavéz, V.

V. Garcés-Chavéz, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef]

Ghai, D. P.

D. P. Ghai, P. Senthilkumaran, and R. S. Sirohi, “Single-slit diffraction of an optical beam with phase singularity,” Opt. Lasers Eng. 47, 123–126 (2009).
[CrossRef]

Gomez, V.

Gu, B.

H. X. Cui, X. L. Wang, B. Gu, Y. N. Li, J. Chen, and H. T. Wang, “Angular diffraction of an optical vortex induced by the Gouy phase,” J. Opt. 14, 055707 (2012).
[CrossRef]

Heckenberg, N. R.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
[CrossRef]

Jankowska, E.

Jiang, H.

Z. Lu, H. Jiang, X. Du, and D. Zhao, “Generalized M2 factor of truncated partially coherent controllable dark-hollow beams,” J. Mod. Opt. 55, 2381–2390 (2008).
[CrossRef]

Kress, B.

Leniec, M.

Li, Y. N.

H. X. Cui, X. L. Wang, B. Gu, Y. N. Li, J. Chen, and H. T. Wang, “Angular diffraction of an optical vortex induced by the Gouy phase,” J. Opt. 14, 055707 (2012).
[CrossRef]

Liu, P.

P. Liu and B. Lü, “Propagation of Gaussian background vortex beams diffracted at a half-plane screen,” Opt. Laser Technol. 40, 227–234 (2008).
[CrossRef]

Lu, Z.

Z. Lu, H. Jiang, X. Du, and D. Zhao, “Generalized M2 factor of truncated partially coherent controllable dark-hollow beams,” J. Mod. Opt. 55, 2381–2390 (2008).
[CrossRef]

Lü, B.

Luo, Y.

MacVicar, I.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef]

Marienko, I. G.

M. V. Vasnetsov, I. G. Marienko, and M. S. Soskin, “Self-reconstruction of an optical vortex,” J. Exp. Theor. Phys. Lett. 71, 130–133 (2000).
[CrossRef]

Martinez-Herrero, R.

Masajada, J.

Mazilu, M.

M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophotonics 2, 021875 (2008).
[CrossRef]

McGloin, D.

V. Garcés-Chavéz, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef]

Mei, Z.

Z. Mei and D. Zhao, “Generalized beam propagation factor of hard-edged diffracted controllable dark-hollow beams,” Opt. Commun. 263, 261–266 (2006).
[CrossRef]

Z. Mei and D. Zhao, “Approximate method for the generalized M2 factor of rotationally symmetric hard-edged diffracted flattened Gaussian beams,” Appl. Opt. 44, 1381–1386 (2005).
[CrossRef]

Mejias, P. M.

Mohammed, K. A.

A. Bekshaev and K. A. Mohammed, “Transverse energy redistribution upon edge diffraction of a paraxial laser beam with optical vortex,” Proc. SPIE 9066, 906602 (2013).
[CrossRef]

Movilla, J. M.

P. M. Mejias, R. Martinez-Herrero, G. Piquero, and J. M. Movilla, “Parametric characterization of the spatial structure of non-uniformly polarized laser beams,” Prog. Quantum Electron. 26, 65–130 (2002).
[CrossRef]

Mukunda, N. J.

Nieminen, T. A.

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
[CrossRef]

O’Holleran, K.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[CrossRef]

O’Neil, A. T.

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef]

Ottevaere, H.

Padgett, M. J.

R. W. Bowman and M. J. Padgett, “Optical trapping and binding,” Rep. Prog. Phys. 76, 026401 (2013).
[CrossRef]

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” Prog. Opt. 53, 293–363 (2009).
[CrossRef]

V. Garcés-Chavéz, D. McGloin, M. J. Padgett, W. Dultz, H. Schmitzer, and K. Dholakia, “Observation of the transfer of the local angular momentum density of a multiringed light beam to an optically trapped particle,” Phys. Rev. Lett. 91, 093602 (2003).
[CrossRef]

A. T. O’Neil, I. MacVicar, L. Allen, and M. J. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[CrossRef]

L. Allen, M. J. Padgett, and M. Babiker, “Orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999).
[CrossRef]

Piquero, G.

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H. X. Cui, X. L. Wang, B. Gu, Y. N. Li, J. Chen, and H. T. Wang, “Angular diffraction of an optical vortex induced by the Gouy phase,” J. Opt. 14, 055707 (2012).
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H. X. Cui, X. L. Wang, B. Gu, Y. N. Li, J. Chen, and H. T. Wang, “Angular diffraction of an optical vortex induced by the Gouy phase,” J. Opt. 14, 055707 (2012).
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Z. Lu, H. Jiang, X. Du, and D. Zhao, “Generalized M2 factor of truncated partially coherent controllable dark-hollow beams,” J. Mod. Opt. 55, 2381–2390 (2008).
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A. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13, 053001 (2011).
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H. X. Cui, X. L. Wang, B. Gu, Y. N. Li, J. Chen, and H. T. Wang, “Angular diffraction of an optical vortex induced by the Gouy phase,” J. Opt. 14, 055707 (2012).
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[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

(a) Intensity profile and (b) equiphase lines of the LG beam with complex amplitude distribution (1a); (c) geometrical conditions of the diffraction. In (a), the green circumference is the intensity ellipse (17), and the TEC calculated along Eq. (9) is shown by arrows; in (b), the blue to brown radial lines correspond to the phase increment 1 (concentration of the radial lines near the negative x half-axis reflects the cut of the screw wavefront); black (blue) circumferences are contours with the intensity level 0.36 (0.1) of the maximum. See text for further explanation.

Fig. 2.
Fig. 2.

Intensity distributions and transverse energy flows for the LG beam with σ=1 after diffraction at the half-plane screen of Fig. 1(c). The screen edge position a is indicated above each column, and rows are marked by the corresponding propagation distances z. Horizontal straight line is the screen edge projection, and circumferences are the contours of constant intensity (0.36 of the maximum) for the hypothetical LG beam that propagates without the screen. The intensity ellipses of Eq. (17) are shown by solid (absolute moments) and dashed (central moments) green lines; the orientation angle θ in Eq. (18) is illustrated in the panel of the second row, fourth column. The far field images (bottom row) represent the steady field patterns where the energy redistribution stops, so the transverse energy flows are not shown.

Fig. 3.
Fig. 3.

(a) Central and (b) absolute OAM constituents [Eqs. (19), (20)] as functions of a for the diffracted LG beam of Fig. 2. Thick lines represent the theoretical quantities calculated by Eqs. (30)–(32) (independent of z); thin lines describe the empirical quantities calculated via the moments obtained by Eqs. (14)–(16), where the integration domain is defined by Eq. (24). Blue lines, Λ; green lines, ΛV; red lines, ΛA. Values of z are indicated near the curves.

Fig. 4.
Fig. 4.

Absolute Λ and central ΛC OAM constituents for the diffracted LG beam of Fig. 2. Blue, total Λ, ΛC; green, vortex ΛV, ΛVC; red, asymmetry ΛA, ΛAC for a=0 calculated via the empirical moments defined as in the caption to Fig. 3. Thick horizontal lines mark the corresponding z-independent values of the theoretically defined central OAM constituents calculated by Eqs. (30)–(32). Blue, Λ; green, ΛVC; and red, ΛAC. According to Fig. 3, for a=0, theoretically defined Λ=ΛV=1, ΛA=0.

Fig. 5.
Fig. 5.

(a) Intensity profile and (b) equiphase lines of the diffracted LG beam of Fig. 2 for a=0.5, z=0.35. The bright horizontal line is the screen edge projection, and concentric circumferences are the constant-intensity contours of the hypothetical LG beam that would propagate without the screen. The y-elongated blue lines in (b) are the contours of constant intensity at the level 0.1 of the maximum for the diffracted beam. Highlighted points are secondary OVs V1 and V2, and the sense of the TEC is indicated by arrows.

Equations (52)

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u(xd,yd,0)=uLG(xd,yd)=(xd+iσydb)exp(xd2+yd22b2),σ=±1,
uLG(x,y)=(x+iσy)exp(x2+y22),
u(x,y,0)={0,y<auaLG(x,y),ya.
u(x,y,z)=12πizu(xa,ya,0)exp{i2z[(xxa)2+(yya)2]}dxadya,
u(x,y,z)=12πizexp[i2z(x2+y2)]×{xaexp[12(iz1)xa2izxax]dxaaexp[12(iz1)ya2izyay]dya+iσexp[12(iz1)xa2izxax]dxaayaexp[12(iz1)ya2izyay]dya}.
erfc(x)=2πxet2dt,
u(x,y,z)=1(1+iz)2exp[x2+y22(1+iz)][x+iσy2erfc(τ)+iσiz(1+iz)2πexp(τ2)],
τ=1+iz2iz(ay1+iz).
u(xa,ya,0)u(xa,ya,0)exp(ixa2+ya22R),
exp{i2[(xa2+ya2)(1z+1R)2xxa+yyaz+x2+y2z]}=exp(ix2+y22zixe2+ye22ze)exp{i2ze[(xaxe)2+(yaye)2]},
ze=z1+z/R;xe=x1+z/Rye=y1+z/R.
ue(x,y,z)=zezexp(ix2+y22zixe2+ye22ze)u(xe,ye,ze)=11+zRexp[ix2+y22(z+R)]u(x1+z/R,y1+z/R,z1+z/R).
S=(SxSy)=ic16πkb[u(xu*yu*)u*(xuyu)].
PC=(rCpC),
rC=(xCyC)=1I(xy)|u|2dxdy,pC=(pxCpyC)=1IImu*(xuyu)dxdy,
I=|u|2dxdy,
M=(M11M12M˜12M22),
M11=(axxaxyaxyayy)=1I(xy)(xy)|u|2dxdy,
M12=(mxxmxymyxmyy)=1IIm(xy)(xu*yu*)udxdy,
M22=(qxxqxyqxyqyy)=1I(xu*yu*)(xuyu)dxdy.
(r·M111r)=ayyx2+axxy22axyxydetM11=const
θ=12arccot(axxayy2axy)
Λ=Sp(M12J),
ΛV=2Sp(M12M11J)SpM11,ΛA=ΛΛV.
ΛV=2ayymxyaxxmyx+axy(mxxmyy)axx+ayy,
ΛA=(axxayy)(mxy+myx)2axy(mxxmyy)axx+ayy.
MC=M(rCpC)(r˜Cp˜C).
4b1+z2<x<4b1+z2,4b1+z2<y<4b1+z2.
M11(z)=M11(0)+[M12(0)+M˜12(0)]z+M22(0)z2,
M12(z)=M12(0)+M22(0)z,M22(z)=M22(0).
M11=M110,M12=M120,andM22=M220+qyyb,
axx=axx0+2mxx0z+qxx0z2,ayy=ayy0+2myy0z+(qyy0+qyyb)z2,axy=(mxy0+myx0)z,
mxx=mxx0+qxx0z,myy=myy0+(qyy0+qyyb)z,mxy=mxy0,myx=myx0.
Λ=mxymyx=mxy0myx0
ΛA=mxymyx+2axymyyayy=mxy0myx0+2(mxy0+myx0)zmyy0+(qyy0+qyyb)zayy0+2myy0z+(qyy0+qyyb)z2qyybmxy0+myx0,
ΛV=ΛΛA=2myx0.
u(x,y)={w(x,y),y>a0,ya.
u(x,y)={w(x,y),ya+δya+δ2δw(x,y),aδ<y<a+δ0,yaδ.
yu(x,y)={yw(x,y),ya+δ12δw(x,y)+ya+δ2δyw(x,y),aδ<y<a+δ0,yaδ.
rC0=1I0adydx(xy)|w|2,pC0=1I0Imadydxw*(xwyw),
M110=1I0adydx(xy)(xy)|w|2,
M120=1I0Imadydx(xy)(xw*yw*)w,
M220=1I0adydx(xw*yw*)(xwyw),
I0=adydx|w(x,y)|2.
M11b=1I01(2δ)2aδa+δ(ya+δ)2dydx(xy)(xy)|w(x,y)|2.
1(2δ)2aδa+δ(ya+δ)2dy=2δ3,
M11b1I02δ3dx(xa)(xa)|w(x,a)|2δ00.
M12b=1I0Imaδa+δ(ya+δ)2(2δ)2dydx(xy)(xw*yw*)w+1I0Imaδa+δya+δ(2δ)2dydx(xy)(0w*)w1I02δ3Imdx(xa)[(xw*yw*)w]y=aδ00,
M22b=1I0aδa+δ(ya+δ)2(2δ)2dydx(xw*yw*)(xwyw)+1I0aδa+δya+δ(2δ)2dydx(0wxw*w*xwy|w|2)+1I0aδa+δ1(2δ)2dydx(000|w|2).
qyyb=12I0[(y|w|2)y=adx+1δ|w(x,a)|2dx]=12δI0|w(x,a)|2dx.
rCb1I02δ3dx(xa)|w(x,a)|2δ00
pCb=1I0(2δ)2Imaδa+δ(ya+δ)w*((ya+δ)xww+(ya+δ)yw)dydxδ00.

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