Abstract

The possibility of experimental measurement of the Poynting vector characteristics is shown. Under paraxial approximation, these characteristics may be obtained on the basis of local Stokes polarimetry and interferometry of electric field components. The experimental results for elliptically polarized Gaussian beam and heterogeneously polarized elementary fields are presented.

© 2012 Optical Society of America

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References

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  1. A. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation light beams,” J. Opt. 13, 053001 (2011).
    [CrossRef]
  2. M. V. Berry, “Optical currents,” J. Opt. A 11, 094001 (2009).
    [CrossRef]
  3. I. I. Mokhun, “Introduction to linear singular optics,” in Optical Correlation Techniques and Applications, O. V. Angelsky, ed. (SPIE, 2007), pp. 1–132.
  4. L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” in Progress in Optics, Vol. XXXIX, E. Wolf, ed. (Elsevier, 1999), pp. 291–372.
  5. L. Allen and M. J. Padgett, “The Poynting vector in Laguerre–Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67–71 (2000).
    [CrossRef]
  6. R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, 1964).
  7. M. J. Lang and S. M. Block, “Resource letter: LBOT-1: laser-based optical tweezers,” Am. J. Phys. 71, 201–215 (2003).
    [CrossRef]
  8. I. Mokhun, A. Mokhun, and Ju. Viktorovskaya, “Singularities of the Poynting vector and the structure of optical fields,” Ukr. J. Phys. Opt. 7, 129–141 (2006).
    [CrossRef]
  9. I. Mokhun and R. Khrobatin, “Shift of application point of angular momentum in the area of elementary polarization singularity,” J. Opt. A 10, 064015 (2008).
    [CrossRef]
  10. J. F. Nye, Natural Focusing and Fine Structure of Light(Institute of Physics, 1999).
  11. M. R. Dennis, “Polarization singularity anisotropy: determining monstardom,” Opt. Lett. 33, 2572–2574 (2008).
    [CrossRef]
  12. M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” in Progress in Optics, Vol. 53 (Elsevier, 2009), pp. 293–363.
    [CrossRef]
  13. K. Yu. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Singular polarimetry: evolution of polarization singularities in electromagnetic waves propagating in a weakly anisotropic medium,” Opt. Express 16, 695–709 (2008).
    [CrossRef]
  14. O. Angelsky, A. Mokhun, I. Mokhun, and M. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun. 207, 57–65 (2002).
    [CrossRef]
  15. O. Angelsky, A. Mokhun, I. Mokhun, and M. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E 65, 036602 (2002).
    [CrossRef]
  16. R. Khrobatin, I. Mokhun, and Ju. Viktorovskaya, “Potentiality of experimental analysis for characteristics of the Poynting vector components,” Ukr. J. Phys. Opt. 9, 182–186 (2008).
    [CrossRef]
  17. A. M. Beckley, T. G. Brown, and M. A. Alonso, “Full Poincare beams,” Opt. Express 18, 10777–10785 (2010).
    [CrossRef]
  18. O. Angelsky, R. Besaha, A. Mokhun, I. Mokhun, M. Sopin, M. Soskin, and M. Vasnetsov, “Singularities in vectoral fields,” Proc. SPIE 3904, 40–55 (1999).
    [CrossRef]
  19. O. V. Angelsky, N. N. Dominikov, P. P. Maksimyak, and T. Tudor, “Experimental revealing of polarization waves,” Appl. Opt. 38, 3112–3117 (1999).
    [CrossRef]
  20. I. Mokhun, Yu. Galushko, Ye. Kharitonova, Yu. Viktorovskaya, and R. Khrobatin, “Elementary heterogeneously polarized field modeling,” Opt. Lett. 36, 2137–2139(2011).
    [CrossRef]

2011 (2)

2010 (1)

2009 (1)

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

2008 (4)

I. Mokhun and R. Khrobatin, “Shift of application point of angular momentum in the area of elementary polarization singularity,” J. Opt. A 10, 064015 (2008).
[CrossRef]

M. R. Dennis, “Polarization singularity anisotropy: determining monstardom,” Opt. Lett. 33, 2572–2574 (2008).
[CrossRef]

K. Yu. Bliokh, A. Niv, V. Kleiner, and E. Hasman, “Singular polarimetry: evolution of polarization singularities in electromagnetic waves propagating in a weakly anisotropic medium,” Opt. Express 16, 695–709 (2008).
[CrossRef]

R. Khrobatin, I. Mokhun, and Ju. Viktorovskaya, “Potentiality of experimental analysis for characteristics of the Poynting vector components,” Ukr. J. Phys. Opt. 9, 182–186 (2008).
[CrossRef]

2006 (1)

I. Mokhun, A. Mokhun, and Ju. Viktorovskaya, “Singularities of the Poynting vector and the structure of optical fields,” Ukr. J. Phys. Opt. 7, 129–141 (2006).
[CrossRef]

2003 (1)

M. J. Lang and S. M. Block, “Resource letter: LBOT-1: laser-based optical tweezers,” Am. J. Phys. 71, 201–215 (2003).
[CrossRef]

2002 (2)

O. Angelsky, A. Mokhun, I. Mokhun, and M. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun. 207, 57–65 (2002).
[CrossRef]

O. Angelsky, A. Mokhun, I. Mokhun, and M. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E 65, 036602 (2002).
[CrossRef]

2000 (1)

L. Allen and M. J. Padgett, “The Poynting vector in Laguerre–Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67–71 (2000).
[CrossRef]

1999 (2)

O. Angelsky, R. Besaha, A. Mokhun, I. Mokhun, M. Sopin, M. Soskin, and M. Vasnetsov, “Singularities in vectoral fields,” Proc. SPIE 3904, 40–55 (1999).
[CrossRef]

O. V. Angelsky, N. N. Dominikov, P. P. Maksimyak, and T. Tudor, “Experimental revealing of polarization waves,” Appl. Opt. 38, 3112–3117 (1999).
[CrossRef]

Allen, L.

L. Allen and M. J. Padgett, “The Poynting vector in Laguerre–Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67–71 (2000).
[CrossRef]

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” in Progress in Optics, Vol. XXXIX, E. Wolf, ed. (Elsevier, 1999), pp. 291–372.

Alonso, M. A.

Angelsky, O.

O. Angelsky, A. Mokhun, I. Mokhun, and M. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun. 207, 57–65 (2002).
[CrossRef]

O. Angelsky, A. Mokhun, I. Mokhun, and M. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E 65, 036602 (2002).
[CrossRef]

O. Angelsky, R. Besaha, A. Mokhun, I. Mokhun, M. Sopin, M. Soskin, and M. Vasnetsov, “Singularities in vectoral fields,” Proc. SPIE 3904, 40–55 (1999).
[CrossRef]

Angelsky, O. V.

Babiker, M.

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” in Progress in Optics, Vol. XXXIX, E. Wolf, ed. (Elsevier, 1999), pp. 291–372.

Beckley, A. M.

Bekshaev, A.

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

Berry, M. V.

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

Besaha, R.

O. Angelsky, R. Besaha, A. Mokhun, I. Mokhun, M. Sopin, M. Soskin, and M. Vasnetsov, “Singularities in vectoral fields,” Proc. SPIE 3904, 40–55 (1999).
[CrossRef]

Bliokh, K.

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

Bliokh, K. Yu.

Block, S. M.

M. J. Lang and S. M. Block, “Resource letter: LBOT-1: laser-based optical tweezers,” Am. J. Phys. 71, 201–215 (2003).
[CrossRef]

Brown, T. G.

Dennis, M. R.

M. R. Dennis, “Polarization singularity anisotropy: determining monstardom,” Opt. Lett. 33, 2572–2574 (2008).
[CrossRef]

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” in Progress in Optics, Vol. 53 (Elsevier, 2009), pp. 293–363.
[CrossRef]

Dominikov, N. N.

Feynman, R. P.

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, 1964).

Galushko, Yu.

Hasman, E.

Kharitonova, Ye.

Khrobatin, R.

I. Mokhun, Yu. Galushko, Ye. Kharitonova, Yu. Viktorovskaya, and R. Khrobatin, “Elementary heterogeneously polarized field modeling,” Opt. Lett. 36, 2137–2139(2011).
[CrossRef]

R. Khrobatin, I. Mokhun, and Ju. Viktorovskaya, “Potentiality of experimental analysis for characteristics of the Poynting vector components,” Ukr. J. Phys. Opt. 9, 182–186 (2008).
[CrossRef]

I. Mokhun and R. Khrobatin, “Shift of application point of angular momentum in the area of elementary polarization singularity,” J. Opt. A 10, 064015 (2008).
[CrossRef]

Kleiner, V.

Lang, M. J.

M. J. Lang and S. M. Block, “Resource letter: LBOT-1: laser-based optical tweezers,” Am. J. Phys. 71, 201–215 (2003).
[CrossRef]

Leighton, R. B.

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, 1964).

Maksimyak, P. P.

Mokhun, A.

I. Mokhun, A. Mokhun, and Ju. Viktorovskaya, “Singularities of the Poynting vector and the structure of optical fields,” Ukr. J. Phys. Opt. 7, 129–141 (2006).
[CrossRef]

O. Angelsky, A. Mokhun, I. Mokhun, and M. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E 65, 036602 (2002).
[CrossRef]

O. Angelsky, A. Mokhun, I. Mokhun, and M. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun. 207, 57–65 (2002).
[CrossRef]

O. Angelsky, R. Besaha, A. Mokhun, I. Mokhun, M. Sopin, M. Soskin, and M. Vasnetsov, “Singularities in vectoral fields,” Proc. SPIE 3904, 40–55 (1999).
[CrossRef]

Mokhun, I.

I. Mokhun, Yu. Galushko, Ye. Kharitonova, Yu. Viktorovskaya, and R. Khrobatin, “Elementary heterogeneously polarized field modeling,” Opt. Lett. 36, 2137–2139(2011).
[CrossRef]

R. Khrobatin, I. Mokhun, and Ju. Viktorovskaya, “Potentiality of experimental analysis for characteristics of the Poynting vector components,” Ukr. J. Phys. Opt. 9, 182–186 (2008).
[CrossRef]

I. Mokhun and R. Khrobatin, “Shift of application point of angular momentum in the area of elementary polarization singularity,” J. Opt. A 10, 064015 (2008).
[CrossRef]

I. Mokhun, A. Mokhun, and Ju. Viktorovskaya, “Singularities of the Poynting vector and the structure of optical fields,” Ukr. J. Phys. Opt. 7, 129–141 (2006).
[CrossRef]

O. Angelsky, A. Mokhun, I. Mokhun, and M. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun. 207, 57–65 (2002).
[CrossRef]

O. Angelsky, A. Mokhun, I. Mokhun, and M. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E 65, 036602 (2002).
[CrossRef]

O. Angelsky, R. Besaha, A. Mokhun, I. Mokhun, M. Sopin, M. Soskin, and M. Vasnetsov, “Singularities in vectoral fields,” Proc. SPIE 3904, 40–55 (1999).
[CrossRef]

Mokhun, I. I.

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

Niv, A.

O’Holleran, K.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” in Progress in Optics, Vol. 53 (Elsevier, 2009), pp. 293–363.
[CrossRef]

Padgett, M. J.

L. Allen and M. J. Padgett, “The Poynting vector in Laguerre–Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67–71 (2000).
[CrossRef]

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” in Progress in Optics, Vol. XXXIX, E. Wolf, ed. (Elsevier, 1999), pp. 291–372.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” in Progress in Optics, Vol. 53 (Elsevier, 2009), pp. 293–363.
[CrossRef]

Sands, M.

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, 1964).

Sopin, M.

O. Angelsky, R. Besaha, A. Mokhun, I. Mokhun, M. Sopin, M. Soskin, and M. Vasnetsov, “Singularities in vectoral fields,” Proc. SPIE 3904, 40–55 (1999).
[CrossRef]

Soskin, M.

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

O. Angelsky, A. Mokhun, I. Mokhun, and M. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E 65, 036602 (2002).
[CrossRef]

O. Angelsky, A. Mokhun, I. Mokhun, and M. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun. 207, 57–65 (2002).
[CrossRef]

O. Angelsky, R. Besaha, A. Mokhun, I. Mokhun, M. Sopin, M. Soskin, and M. Vasnetsov, “Singularities in vectoral fields,” Proc. SPIE 3904, 40–55 (1999).
[CrossRef]

Tudor, T.

Vasnetsov, M.

O. Angelsky, R. Besaha, A. Mokhun, I. Mokhun, M. Sopin, M. Soskin, and M. Vasnetsov, “Singularities in vectoral fields,” Proc. SPIE 3904, 40–55 (1999).
[CrossRef]

Viktorovskaya, Ju.

R. Khrobatin, I. Mokhun, and Ju. Viktorovskaya, “Potentiality of experimental analysis for characteristics of the Poynting vector components,” Ukr. J. Phys. Opt. 9, 182–186 (2008).
[CrossRef]

I. Mokhun, A. Mokhun, and Ju. Viktorovskaya, “Singularities of the Poynting vector and the structure of optical fields,” Ukr. J. Phys. Opt. 7, 129–141 (2006).
[CrossRef]

Viktorovskaya, Yu.

Am. J. Phys. (1)

M. J. Lang and S. M. Block, “Resource letter: LBOT-1: laser-based optical tweezers,” Am. J. Phys. 71, 201–215 (2003).
[CrossRef]

Appl. Opt. (1)

J. Opt. (1)

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

J. Opt. A (2)

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

I. Mokhun and R. Khrobatin, “Shift of application point of angular momentum in the area of elementary polarization singularity,” J. Opt. A 10, 064015 (2008).
[CrossRef]

Opt. Commun. (2)

O. Angelsky, A. Mokhun, I. Mokhun, and M. Soskin, “The relationship between topological characteristics of component vortices and polarization singularities,” Opt. Commun. 207, 57–65 (2002).
[CrossRef]

L. Allen and M. J. Padgett, “The Poynting vector in Laguerre–Gaussian beams and the interpretation of their angular momentum density,” Opt. Commun. 184, 67–71 (2000).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. E (1)

O. Angelsky, A. Mokhun, I. Mokhun, and M. Soskin, “Interferometric methods in diagnostics of polarization singularities,” Phys. Rev. E 65, 036602 (2002).
[CrossRef]

Proc. SPIE (1)

O. Angelsky, R. Besaha, A. Mokhun, I. Mokhun, M. Sopin, M. Soskin, and M. Vasnetsov, “Singularities in vectoral fields,” Proc. SPIE 3904, 40–55 (1999).
[CrossRef]

Ukr. J. Phys. Opt. (2)

R. Khrobatin, I. Mokhun, and Ju. Viktorovskaya, “Potentiality of experimental analysis for characteristics of the Poynting vector components,” Ukr. J. Phys. Opt. 9, 182–186 (2008).
[CrossRef]

I. Mokhun, A. Mokhun, and Ju. Viktorovskaya, “Singularities of the Poynting vector and the structure of optical fields,” Ukr. J. Phys. Opt. 7, 129–141 (2006).
[CrossRef]

Other (5)

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley, 1964).

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

L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” in Progress in Optics, Vol. XXXIX, E. Wolf, ed. (Elsevier, 1999), pp. 291–372.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities,” in Progress in Optics, Vol. 53 (Elsevier, 2009), pp. 293–363.
[CrossRef]

J. F. Nye, Natural Focusing and Fine Structure of Light(Institute of Physics, 1999).

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Figures (7)

Fig. 1.
Fig. 1.

Typical behavior of the transversal component of the Poynting vector in the area of P-singularity: (a) energy currents around vortex P-singularity, (b)–(d) energy currents around passive P-singularity.

Fig. 2.
Fig. 2.

Experimental arrangement for measurement of Poynting vector components of Gaussian beam. 1, He–Ne laser; 2, polarizer; 3, λ/4-plate; 4–6, beam expander with pinhole; 5, 7, 8, Stokes polarimeter; 9, CCD camera.

Fig. 3.
Fig. 3.

Energy flows around the center of elliptically polarized Gaussian beam: (a) intensity distribution of laser beam after collimator and (b) behavior of the Poynting vector transversal component.

Fig. 4.
Fig. 4.

Typical polarization modulation of resulting field, obtained as superposition of two practically plane orthogonally linearly polarized waves with close intensities.

Fig. 5.
Fig. 5.

Experimental arrangement for the formation of elementary fields with heterogeneous polarization. 1, 15: λ/4-plates; 2–4: beam expander; 5, 6, 11, 13: beam splitters; 7, 10: crossed polarizers; 14: output polarizer; 8, 9, 12: mirrors.

Fig. 6.
Fig. 6.

The results of superposition of linearly polarized orthogonal Gaussian beams and corresponding interferometry of resulting field components. (a), (b) intensity distributions of Gaussian beams. (c) intensity distributions of resulting field (without output polarizer 14). 5 C-points, obtained by the analysis of experimental data are indicated by open and solid squares. Open and solid squares, C-points with positive and negative indices correspondingly, (d), (e) interferograms of the linearly polarized orthogonal components of the resulting field, (f) superposition of orthogonal polarized components when angle between axis of output polarizer 14 and direction of field vector vibration is 45°.

Fig. 7.
Fig. 7.

Behavior of the Poynting vector transversal component (energy flows) of resulting field. Experimental data. The area of resulting field with five C-points is presented. The magnitude of modulus of transversal component is indicated by shades of gray. Additionally magnitude and azimuth of component is illustrated by length and orientation of white arrows. The Poynting singularities are the points, where transversal component is exact zero are denoted. Open and solid squares, positive and negative C-points correspondingly. Open and solid squares with X in the middle, vortex and passive P-singularities correspondingly.

Equations (3)

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

{P¯xc16πk{[(s0+s1)Φxx+(s0s1)Φyx]s3y}P¯yc16πk{[(s0+s1)Φxy+(s0s1)Φyy]+s3x}P¯zc8πs0,
Φx=Δ+Φy,
{P¯xc16πks3y}P¯yc16πks3x}P¯zc8πs0.

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