Abstract

It is known that the orbital angular momentum of a paraxial light beam is related to the rotational features of the instantaneous optical-frequency oscillation pattern within the beam cross section [J. Opt. A 11, 094004 (2009)]. Now this conclusion is generalized: any identifiable directed motion of the instantaneous two-dimensional pattern of the field oscillations (“running” behavior of the instant oscillatory pattern) corresponds to the transverse energy flow in the experimentally observable time-averaged field. The transverse orbital flow density can be treated as a natural geometric and kinematic characteristic of this running behavior.

© 2012 Optical Society of America

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References

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  1. J. Lekner, “Phase and transport velocities in particle and electromagnetic beams,” J. Opt. A 4, 491–499 (2002).
    [CrossRef]
  2. J. Lekner, “Polarization of tightly focused laser beams,” J. Opt. A 5, 6–14 (2003).
    [CrossRef]
  3. A. Y. Bekshaev and M. S. Soskin, “Rotational transformations and transverse energy flow in paraxial light beams: linear azimuthons,” Opt. Lett. 31, 2199–2201 (2006).
    [CrossRef]
  4. I. Mokhun, A. Mokhun, and J. Viktorovskaya, “Singularities of the Poynting vector and the structure of optical field,” Proc. SPIE 6254, 625409 (2006).
    [CrossRef]
  5. I. I. Mokhun, “Introduction to linear singular optics,” in Optical Correlation Techniques and Applications (SPIE, 2007), pp. 1–132.
  6. A. Y. Bekshaev and M. S. Soskin, “Transverse energy flows in vectorial fields of paraxial beams with singularities,” Opt. Commun. 271, 332–348 (2007).
    [CrossRef]
  7. M. V. Berry, “Optical currents,” J. Opt. A 11, 094001 (2009).
    [CrossRef]
  8. A. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13, 053001 (2011).
    [CrossRef]
  9. A. Y. Bekshaev, “Spin angular momentum of inhomogeneous and transversely limited light beams,” Proc. SPIE 6254, 625407 (2006).
    [CrossRef]
  10. A. Y. Bekshaev, “Oblique section of a paraxial light beam: criteria for azimuthal energy flow and orbital angular momentum,” J. Opt. A 11, 094003 (2009).
    [CrossRef]
  11. A. Ya. Bekshaev, “Transverse rotation of the momentary field distribution and the orbital angular momentum of a light beam,” http://arXiv.org/abs/0812.0888 (accessed Feb. 17, 2012).
  12. A. Y. Bekshaev, “Transverse rotation of the instantaneous field distribution and the orbital angular momentum of a light beam,” J. Opt. A 11, 094004 (2009).
    [CrossRef]
  13. M. Lax, W. H. Louisell, and B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  14. A. Bekshaev, M. Soskin, and M. Vasnetsov, Paraxial Light Beams with Angular Momentum (Nova Science, 2008).
  15. L. Allen, M. V. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian modes,” Phys. Rev. A 45, 8185–8189 (1992).
    [CrossRef]

2011 (1)

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

2009 (3)

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

A. Y. Bekshaev, “Oblique section of a paraxial light beam: criteria for azimuthal energy flow and orbital angular momentum,” J. Opt. A 11, 094003 (2009).
[CrossRef]

A. Y. Bekshaev, “Transverse rotation of the instantaneous field distribution and the orbital angular momentum of a light beam,” J. Opt. A 11, 094004 (2009).
[CrossRef]

2007 (1)

A. Y. Bekshaev and M. S. Soskin, “Transverse energy flows in vectorial fields of paraxial beams with singularities,” Opt. Commun. 271, 332–348 (2007).
[CrossRef]

2006 (3)

A. Y. Bekshaev, “Spin angular momentum of inhomogeneous and transversely limited light beams,” Proc. SPIE 6254, 625407 (2006).
[CrossRef]

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

I. Mokhun, A. Mokhun, and J. Viktorovskaya, “Singularities of the Poynting vector and the structure of optical field,” Proc. SPIE 6254, 625409 (2006).
[CrossRef]

2003 (1)

J. Lekner, “Polarization of tightly focused laser beams,” J. Opt. A 5, 6–14 (2003).
[CrossRef]

2002 (1)

J. Lekner, “Phase and transport velocities in particle and electromagnetic beams,” J. Opt. A 4, 491–499 (2002).
[CrossRef]

1992 (1)

L. Allen, M. V. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef]

1975 (1)

M. Lax, W. H. Louisell, and B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Allen, L.

L. Allen, M. V. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef]

Beijersbergen, M. V.

L. Allen, M. V. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef]

Bekshaev, A.

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

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

Bekshaev, A. Y.

A. Y. Bekshaev, “Oblique section of a paraxial light beam: criteria for azimuthal energy flow and orbital angular momentum,” J. Opt. A 11, 094003 (2009).
[CrossRef]

A. Y. Bekshaev, “Transverse rotation of the instantaneous field distribution and the orbital angular momentum of a light beam,” J. Opt. A 11, 094004 (2009).
[CrossRef]

A. Y. Bekshaev and M. S. Soskin, “Transverse energy flows in vectorial fields of paraxial beams with singularities,” Opt. Commun. 271, 332–348 (2007).
[CrossRef]

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

A. Y. Bekshaev, “Spin angular momentum of inhomogeneous and transversely limited light beams,” Proc. SPIE 6254, 625407 (2006).
[CrossRef]

Berry, M. V.

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

Bliokh, K.

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

Lax, M.

M. Lax, W. H. Louisell, and B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Lekner, J.

J. Lekner, “Polarization of tightly focused laser beams,” J. Opt. A 5, 6–14 (2003).
[CrossRef]

J. Lekner, “Phase and transport velocities in particle and electromagnetic beams,” J. Opt. A 4, 491–499 (2002).
[CrossRef]

Louisell, W. H.

M. Lax, W. H. Louisell, and B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

McKnight, B.

M. Lax, W. H. Louisell, and B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Mokhun, A.

I. Mokhun, A. Mokhun, and J. Viktorovskaya, “Singularities of the Poynting vector and the structure of optical field,” Proc. SPIE 6254, 625409 (2006).
[CrossRef]

Mokhun, I.

I. Mokhun, A. Mokhun, and J. Viktorovskaya, “Singularities of the Poynting vector and the structure of optical field,” Proc. SPIE 6254, 625409 (2006).
[CrossRef]

Mokhun, I. I.

I. I. Mokhun, “Introduction to linear singular optics,” in Optical Correlation Techniques and Applications (SPIE, 2007), pp. 1–132.

Soskin, M.

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

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

Soskin, M. S.

A. Y. Bekshaev and M. S. Soskin, “Transverse energy flows in vectorial fields of paraxial beams with singularities,” Opt. Commun. 271, 332–348 (2007).
[CrossRef]

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

Spreeuw, R. J. C.

L. Allen, M. V. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef]

Vasnetsov, M.

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

Viktorovskaya, J.

I. Mokhun, A. Mokhun, and J. Viktorovskaya, “Singularities of the Poynting vector and the structure of optical field,” Proc. SPIE 6254, 625409 (2006).
[CrossRef]

Woerdman, J. P.

L. Allen, M. V. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef]

J. Opt. (1)

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

J. Opt. A (5)

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

J. Lekner, “Phase and transport velocities in particle and electromagnetic beams,” J. Opt. A 4, 491–499 (2002).
[CrossRef]

J. Lekner, “Polarization of tightly focused laser beams,” J. Opt. A 5, 6–14 (2003).
[CrossRef]

A. Y. Bekshaev, “Oblique section of a paraxial light beam: criteria for azimuthal energy flow and orbital angular momentum,” J. Opt. A 11, 094003 (2009).
[CrossRef]

A. Y. Bekshaev, “Transverse rotation of the instantaneous field distribution and the orbital angular momentum of a light beam,” J. Opt. A 11, 094004 (2009).
[CrossRef]

Opt. Commun. (1)

A. Y. Bekshaev and M. S. Soskin, “Transverse energy flows in vectorial fields of paraxial beams with singularities,” Opt. Commun. 271, 332–348 (2007).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (2)

M. Lax, W. H. Louisell, and B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

L. Allen, M. V. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre–Gaussian modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef]

Proc. SPIE (2)

I. Mokhun, A. Mokhun, and J. Viktorovskaya, “Singularities of the Poynting vector and the structure of optical field,” Proc. SPIE 6254, 625409 (2006).
[CrossRef]

A. Y. Bekshaev, “Spin angular momentum of inhomogeneous and transversely limited light beams,” Proc. SPIE 6254, 625407 (2006).
[CrossRef]

Other (3)

I. I. Mokhun, “Introduction to linear singular optics,” in Optical Correlation Techniques and Applications (SPIE, 2007), pp. 1–132.

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

A. Ya. Bekshaev, “Transverse rotation of the momentary field distribution and the orbital angular momentum of a light beam,” http://arXiv.org/abs/0812.0888 (accessed Feb. 17, 2012).

Supplementary Material (4)

» Media 1: AVI (555 KB)     
» Media 2: AVI (744 KB)     
» Media 3: AVI (704 KB)     
» Media 4: AVI (660 KB)     

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Figures (1)

Fig. 1.
Fig. 1.

Temporal evolution of the instantaneous intensity distribution [E(r,t)]2 (see Eq. (4)) in cross sections of paraxial beams calculated for He-Ne laser radiation, k=105cm1 (views against the beam propagation). Column 1 is the immediately observed time-average spot pattern; columns 2–5 are snapshots of the instantaneous field distributions at the time moments specified above each column; time-average OFD maps (arrows) calculated from Eq. (3) are added as a background. Each panel represents a square 0.4×0.4cm in the beam cross section; calculations are made under the following conditions: First row: diverging Gaussian beam of Eqs. (6) and (7), b=0.1cm, R=330cm (Media 1). Second row: inclined plane wave of Eq. (8), ky/k=104 (Media 2). Third row: circular Laguerre–Gaussian beam of Eq. (9), l=1, b=0.1cm (Media 3). Fourth row: rotating Gaussian beam of Eqs. (13), bx=0.1cm, by=0.15cm, αxx=αyy=103cm, αxy=3·103cm (Media 4).

Equations (16)

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E(r,z,t)=u(r,z)exp[i(kzωt)],
u(r)=A(r)exp[ikφ(r)],
SO(r)=c8πA2(r)φ(r),
E(r,t)=Re[E(r,t)]=A(r)cos[kφ(r)ωt]
φ(rc)=ct,
A(r)A0(r)=exp(r22b2),φ(r)=r22R,
E(r,t)cos(kr22Rωt).
E(r,t)cos(kyyωt),
E(r,t)=Al(r)cos(lϕωt),
E(r,t)t=ωkyE(r,t)y,E(r,t)t=ωlE(r,t)ϕ,
E(r,t)tandE=exE(r,t)x+eyE(r,t)y
rEtEE(r,t)E(r,t)t=1T0TE(r,t)E(r,t)tdt,
rEtE=kω2A2(r)φ(r),
rEtE=4πk2SO.
A(r)exp(x22bx2y22by2),φ(r)=12(αxxx2+αyyy2+2αxyxy),(αxy0).
rEXtEX+rEYtEY=4πk2SOT.

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