Abstract

We design an optical setup to extract phase vortices in which the interference intensity of reference light wave and speckle fields produced by random screens with different roughness values in the diffraction region near random screens is obtained. Random screens with different roughness are used as samples. Fourier transform is used to extract speckle phase vortices from the interference intensity, and the experimental results show that the phase vortices can be produced when the roughness of the screen is large enough, and they even may appear on the surface. The density of phase vortices would become larger with an increase of the distances in the diffraction region near the random screen. When the distance is certain, the density of phase vortices becomes larger with the increase of roughness. These results would be helpful for understanding the formation of phase vortices.

© 2012 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Ben Roberts & Company, 2007).
  2. J. F. Nye and M. V. Berry, “Dislocations in Wave Trains,” Proc. R. Soc. Lond. A Math. Phys. Sci.336(1605), 165–190 (1974).
    [CrossRef]
  3. P. S. Liu, Fundamentals of Statistical Optics of Speckles (Science Press, 1987), p.7.
  4. M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: A direct measurement of the density correlation function g(r),” Phys. Rev. Lett.85(7), 1416–1419 (2000).
    [CrossRef] [PubMed]
  5. G. M. Li, Y. S. Qiu, H. Li, Y. Huang, S. Liu, and Z. Y. Huang, “Speckle contrast in near field scattering limited by time coherence,” Opt. Express19(4), 3694–3702 (2011).
    [CrossRef] [PubMed]
  6. K. O’Holleran, M. R. Dennis, F. Flossmann, and M. J. Padgett, “Fractality of light’s darkness,” Phys. Rev. Lett.100(5), 053902 (2008).
    [CrossRef] [PubMed]
  7. W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental Investigation of Local Properties and Statistics of Optical Vortices in Random Wave Fields,” Phys. Rev. Lett.94(10), 103902 (2005).
    [CrossRef] [PubMed]
  8. F. Flossmann, K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Polarization Singularities in 2D and 3D Speckle Fields,” Phys. Rev. Lett.100(20), 203902 (2008).
    [CrossRef] [PubMed]
  9. O. V. Angelsky, A. P. Maksimyak, P. P. Maksimyak, and S. G. Hanson, “Interference diagnostics of white-light vortices,” Opt. Express13(20), 8179–8183 (2005).
    [CrossRef] [PubMed]
  10. H. S. Song, C. F. Cheng, S. Y. Teng, M. Liu, G. Y. Liu, and N. Y. Zhang, “Experimental studies on the statistical functions of speckle fields based on the extraction of the complex amplitudes by use of interference beam,” Acta Phys. Sin.58(11), 7654–7661 (2009).
  11. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer- based topography and interferometry,” J. Opt. Soc. Am.72(1), 156–160 (1982).
    [CrossRef]
  12. D. J. Bone, H. A. Bachor, and R. J. Sandeman, “Fringe-pattern analysis using a 2-D Fourier transform,” Appl. Opt.25(10), 1653–1660 (1986).
    [CrossRef] [PubMed]
  13. M. H. Zhang, J. F. Xu, X. F. Wang, and Q. Wei, “Complex-valued acquisition of the diffraction imaging by incoherent quasi-monochromatic light without a support constraint,” Phys. Rev. A82(4), 043839 (2010).
    [CrossRef]
  14. E. Wolf, “Solution of the Phase Problem in the Theory of Structure Determination of Crystals from X-Ray Diffraction Experiments,” Phys. Rev. Lett.103(7), 075501 (2009).
    [CrossRef] [PubMed]
  15. N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetsky, and V. V. Shkukov, “Dislocation of the wave-front of a speckle-inhomogeneous field,” JETP Lett.33, 195–199 (1981).
  16. N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, and B. Y. Zel’dovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. A73(5), 525–528 (1983).
    [CrossRef]
  17. M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. Lond. A456(2001), 2059–2079 (2000).
    [CrossRef]

2011 (1)

2010 (1)

M. H. Zhang, J. F. Xu, X. F. Wang, and Q. Wei, “Complex-valued acquisition of the diffraction imaging by incoherent quasi-monochromatic light without a support constraint,” Phys. Rev. A82(4), 043839 (2010).
[CrossRef]

2009 (2)

E. Wolf, “Solution of the Phase Problem in the Theory of Structure Determination of Crystals from X-Ray Diffraction Experiments,” Phys. Rev. Lett.103(7), 075501 (2009).
[CrossRef] [PubMed]

H. S. Song, C. F. Cheng, S. Y. Teng, M. Liu, G. Y. Liu, and N. Y. Zhang, “Experimental studies on the statistical functions of speckle fields based on the extraction of the complex amplitudes by use of interference beam,” Acta Phys. Sin.58(11), 7654–7661 (2009).

2008 (2)

F. Flossmann, K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Polarization Singularities in 2D and 3D Speckle Fields,” Phys. Rev. Lett.100(20), 203902 (2008).
[CrossRef] [PubMed]

K. O’Holleran, M. R. Dennis, F. Flossmann, and M. J. Padgett, “Fractality of light’s darkness,” Phys. Rev. Lett.100(5), 053902 (2008).
[CrossRef] [PubMed]

2005 (2)

W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental Investigation of Local Properties and Statistics of Optical Vortices in Random Wave Fields,” Phys. Rev. Lett.94(10), 103902 (2005).
[CrossRef] [PubMed]

O. V. Angelsky, A. P. Maksimyak, P. P. Maksimyak, and S. G. Hanson, “Interference diagnostics of white-light vortices,” Opt. Express13(20), 8179–8183 (2005).
[CrossRef] [PubMed]

2000 (2)

M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: A direct measurement of the density correlation function g(r),” Phys. Rev. Lett.85(7), 1416–1419 (2000).
[CrossRef] [PubMed]

M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. Lond. A456(2001), 2059–2079 (2000).
[CrossRef]

1986 (1)

1983 (1)

N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, and B. Y. Zel’dovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. A73(5), 525–528 (1983).
[CrossRef]

1982 (1)

1981 (1)

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetsky, and V. V. Shkukov, “Dislocation of the wave-front of a speckle-inhomogeneous field,” JETP Lett.33, 195–199 (1981).

1974 (1)

J. F. Nye and M. V. Berry, “Dislocations in Wave Trains,” Proc. R. Soc. Lond. A Math. Phys. Sci.336(1605), 165–190 (1974).
[CrossRef]

Angelsky, O. V.

Bachor, H. A.

Baranova, N. B.

N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, and B. Y. Zel’dovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. A73(5), 525–528 (1983).
[CrossRef]

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetsky, and V. V. Shkukov, “Dislocation of the wave-front of a speckle-inhomogeneous field,” JETP Lett.33, 195–199 (1981).

Berry, M. V.

M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. Lond. A456(2001), 2059–2079 (2000).
[CrossRef]

J. F. Nye and M. V. Berry, “Dislocations in Wave Trains,” Proc. R. Soc. Lond. A Math. Phys. Sci.336(1605), 165–190 (1974).
[CrossRef]

Bone, D. J.

Carpineti, M.

M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: A direct measurement of the density correlation function g(r),” Phys. Rev. Lett.85(7), 1416–1419 (2000).
[CrossRef] [PubMed]

Cheng, C. F.

H. S. Song, C. F. Cheng, S. Y. Teng, M. Liu, G. Y. Liu, and N. Y. Zhang, “Experimental studies on the statistical functions of speckle fields based on the extraction of the complex amplitudes by use of interference beam,” Acta Phys. Sin.58(11), 7654–7661 (2009).

Dennis, M. R.

F. Flossmann, K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Polarization Singularities in 2D and 3D Speckle Fields,” Phys. Rev. Lett.100(20), 203902 (2008).
[CrossRef] [PubMed]

K. O’Holleran, M. R. Dennis, F. Flossmann, and M. J. Padgett, “Fractality of light’s darkness,” Phys. Rev. Lett.100(5), 053902 (2008).
[CrossRef] [PubMed]

M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. Lond. A456(2001), 2059–2079 (2000).
[CrossRef]

Flossmann, F.

K. O’Holleran, M. R. Dennis, F. Flossmann, and M. J. Padgett, “Fractality of light’s darkness,” Phys. Rev. Lett.100(5), 053902 (2008).
[CrossRef] [PubMed]

F. Flossmann, K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Polarization Singularities in 2D and 3D Speckle Fields,” Phys. Rev. Lett.100(20), 203902 (2008).
[CrossRef] [PubMed]

Giglio, M.

M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: A direct measurement of the density correlation function g(r),” Phys. Rev. Lett.85(7), 1416–1419 (2000).
[CrossRef] [PubMed]

Hanson, S. G.

W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental Investigation of Local Properties and Statistics of Optical Vortices in Random Wave Fields,” Phys. Rev. Lett.94(10), 103902 (2005).
[CrossRef] [PubMed]

O. V. Angelsky, A. P. Maksimyak, P. P. Maksimyak, and S. G. Hanson, “Interference diagnostics of white-light vortices,” Opt. Express13(20), 8179–8183 (2005).
[CrossRef] [PubMed]

Huang, Y.

Huang, Z. Y.

Ina, H.

Kobayashi, S.

Li, G. M.

Li, H.

Liu, G. Y.

H. S. Song, C. F. Cheng, S. Y. Teng, M. Liu, G. Y. Liu, and N. Y. Zhang, “Experimental studies on the statistical functions of speckle fields based on the extraction of the complex amplitudes by use of interference beam,” Acta Phys. Sin.58(11), 7654–7661 (2009).

Liu, M.

H. S. Song, C. F. Cheng, S. Y. Teng, M. Liu, G. Y. Liu, and N. Y. Zhang, “Experimental studies on the statistical functions of speckle fields based on the extraction of the complex amplitudes by use of interference beam,” Acta Phys. Sin.58(11), 7654–7661 (2009).

Liu, S.

Maksimyak, A. P.

Maksimyak, P. P.

Mamaev, A. V.

N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, and B. Y. Zel’dovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. A73(5), 525–528 (1983).
[CrossRef]

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetsky, and V. V. Shkukov, “Dislocation of the wave-front of a speckle-inhomogeneous field,” JETP Lett.33, 195–199 (1981).

Miyamoto, Y.

W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental Investigation of Local Properties and Statistics of Optical Vortices in Random Wave Fields,” Phys. Rev. Lett.94(10), 103902 (2005).
[CrossRef] [PubMed]

Nye, J. F.

J. F. Nye and M. V. Berry, “Dislocations in Wave Trains,” Proc. R. Soc. Lond. A Math. Phys. Sci.336(1605), 165–190 (1974).
[CrossRef]

O’Holleran, K.

K. O’Holleran, M. R. Dennis, F. Flossmann, and M. J. Padgett, “Fractality of light’s darkness,” Phys. Rev. Lett.100(5), 053902 (2008).
[CrossRef] [PubMed]

F. Flossmann, K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Polarization Singularities in 2D and 3D Speckle Fields,” Phys. Rev. Lett.100(20), 203902 (2008).
[CrossRef] [PubMed]

Padgett, M. J.

F. Flossmann, K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Polarization Singularities in 2D and 3D Speckle Fields,” Phys. Rev. Lett.100(20), 203902 (2008).
[CrossRef] [PubMed]

K. O’Holleran, M. R. Dennis, F. Flossmann, and M. J. Padgett, “Fractality of light’s darkness,” Phys. Rev. Lett.100(5), 053902 (2008).
[CrossRef] [PubMed]

Pilipetsky, N. F.

N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, and B. Y. Zel’dovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. A73(5), 525–528 (1983).
[CrossRef]

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetsky, and V. V. Shkukov, “Dislocation of the wave-front of a speckle-inhomogeneous field,” JETP Lett.33, 195–199 (1981).

Qiu, Y. S.

Sandeman, R. J.

Shkukov, V. V.

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetsky, and V. V. Shkukov, “Dislocation of the wave-front of a speckle-inhomogeneous field,” JETP Lett.33, 195–199 (1981).

Shkunov, V. V.

N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, and B. Y. Zel’dovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. A73(5), 525–528 (1983).
[CrossRef]

Song, H. S.

H. S. Song, C. F. Cheng, S. Y. Teng, M. Liu, G. Y. Liu, and N. Y. Zhang, “Experimental studies on the statistical functions of speckle fields based on the extraction of the complex amplitudes by use of interference beam,” Acta Phys. Sin.58(11), 7654–7661 (2009).

Takeda, M.

W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental Investigation of Local Properties and Statistics of Optical Vortices in Random Wave Fields,” Phys. Rev. Lett.94(10), 103902 (2005).
[CrossRef] [PubMed]

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer- based topography and interferometry,” J. Opt. Soc. Am.72(1), 156–160 (1982).
[CrossRef]

Teng, S. Y.

H. S. Song, C. F. Cheng, S. Y. Teng, M. Liu, G. Y. Liu, and N. Y. Zhang, “Experimental studies on the statistical functions of speckle fields based on the extraction of the complex amplitudes by use of interference beam,” Acta Phys. Sin.58(11), 7654–7661 (2009).

Vailati, A.

M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: A direct measurement of the density correlation function g(r),” Phys. Rev. Lett.85(7), 1416–1419 (2000).
[CrossRef] [PubMed]

Wang, W.

W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental Investigation of Local Properties and Statistics of Optical Vortices in Random Wave Fields,” Phys. Rev. Lett.94(10), 103902 (2005).
[CrossRef] [PubMed]

Wang, X. F.

M. H. Zhang, J. F. Xu, X. F. Wang, and Q. Wei, “Complex-valued acquisition of the diffraction imaging by incoherent quasi-monochromatic light without a support constraint,” Phys. Rev. A82(4), 043839 (2010).
[CrossRef]

Wei, Q.

M. H. Zhang, J. F. Xu, X. F. Wang, and Q. Wei, “Complex-valued acquisition of the diffraction imaging by incoherent quasi-monochromatic light without a support constraint,” Phys. Rev. A82(4), 043839 (2010).
[CrossRef]

Wolf, E.

E. Wolf, “Solution of the Phase Problem in the Theory of Structure Determination of Crystals from X-Ray Diffraction Experiments,” Phys. Rev. Lett.103(7), 075501 (2009).
[CrossRef] [PubMed]

Xu, J. F.

M. H. Zhang, J. F. Xu, X. F. Wang, and Q. Wei, “Complex-valued acquisition of the diffraction imaging by incoherent quasi-monochromatic light without a support constraint,” Phys. Rev. A82(4), 043839 (2010).
[CrossRef]

Zel’dovich, B. Y.

N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, and B. Y. Zel’dovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. A73(5), 525–528 (1983).
[CrossRef]

Zel’dovich, B. Ya.

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetsky, and V. V. Shkukov, “Dislocation of the wave-front of a speckle-inhomogeneous field,” JETP Lett.33, 195–199 (1981).

Zhang, M. H.

M. H. Zhang, J. F. Xu, X. F. Wang, and Q. Wei, “Complex-valued acquisition of the diffraction imaging by incoherent quasi-monochromatic light without a support constraint,” Phys. Rev. A82(4), 043839 (2010).
[CrossRef]

Zhang, N. Y.

H. S. Song, C. F. Cheng, S. Y. Teng, M. Liu, G. Y. Liu, and N. Y. Zhang, “Experimental studies on the statistical functions of speckle fields based on the extraction of the complex amplitudes by use of interference beam,” Acta Phys. Sin.58(11), 7654–7661 (2009).

Acta Phys. Sin. (1)

H. S. Song, C. F. Cheng, S. Y. Teng, M. Liu, G. Y. Liu, and N. Y. Zhang, “Experimental studies on the statistical functions of speckle fields based on the extraction of the complex amplitudes by use of interference beam,” Acta Phys. Sin.58(11), 7654–7661 (2009).

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

N. B. Baranova, A. V. Mamaev, N. F. Pilipetsky, V. V. Shkunov, and B. Y. Zel’dovich, “Wave-front dislocations: topological limitations for adaptive systems with phase conjugation,” J. Opt. Soc. Am. A73(5), 525–528 (1983).
[CrossRef]

JETP Lett. (1)

N. B. Baranova, B. Ya. Zel’dovich, A. V. Mamaev, N. F. Pilipetsky, and V. V. Shkukov, “Dislocation of the wave-front of a speckle-inhomogeneous field,” JETP Lett.33, 195–199 (1981).

Opt. Express (2)

Phys. Rev. A (1)

M. H. Zhang, J. F. Xu, X. F. Wang, and Q. Wei, “Complex-valued acquisition of the diffraction imaging by incoherent quasi-monochromatic light without a support constraint,” Phys. Rev. A82(4), 043839 (2010).
[CrossRef]

Phys. Rev. Lett. (5)

E. Wolf, “Solution of the Phase Problem in the Theory of Structure Determination of Crystals from X-Ray Diffraction Experiments,” Phys. Rev. Lett.103(7), 075501 (2009).
[CrossRef] [PubMed]

K. O’Holleran, M. R. Dennis, F. Flossmann, and M. J. Padgett, “Fractality of light’s darkness,” Phys. Rev. Lett.100(5), 053902 (2008).
[CrossRef] [PubMed]

W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental Investigation of Local Properties and Statistics of Optical Vortices in Random Wave Fields,” Phys. Rev. Lett.94(10), 103902 (2005).
[CrossRef] [PubMed]

F. Flossmann, K. O’Holleran, M. R. Dennis, and M. J. Padgett, “Polarization Singularities in 2D and 3D Speckle Fields,” Phys. Rev. Lett.100(20), 203902 (2008).
[CrossRef] [PubMed]

M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: A direct measurement of the density correlation function g(r),” Phys. Rev. Lett.85(7), 1416–1419 (2000).
[CrossRef] [PubMed]

Proc. R. Soc. Lond. A (1)

M. V. Berry and M. R. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. Lond. A456(2001), 2059–2079 (2000).
[CrossRef]

Proc. R. Soc. Lond. A Math. Phys. Sci. (1)

J. F. Nye and M. V. Berry, “Dislocations in Wave Trains,” Proc. R. Soc. Lond. A Math. Phys. Sci.336(1605), 165–190 (1974).
[CrossRef]

Other (2)

P. S. Liu, Fundamentals of Statistical Optics of Speckles (Science Press, 1987), p.7.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Ben Roberts & Company, 2007).

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 (11)

Fig. 1
Fig. 1

AFM images of the four random screen samples grinded with silicon carbide powders with grain sizes of (a) 3.5μm , (b) 5μm , (c) 7μm and (d) 40μm , respectively, with scanning areas of 40μm×40μm .

Fig. 2
Fig. 2

Schematic diagram of the experimental setup for recording interference patterns.

Fig. 3
Fig. 3

(a) The interference intensity pattern of a reference light and a speckle field produced by random screen sample No. 2 at distance Z=20μm . (b) Local interference intensity pattern of the white squared part in Fig. 3(a). The gray-scale values take arbitrary units and the number of gray-scale level is 32.

Fig. 4
Fig. 4

(a) The gray-scale image of the spatial frequency spectrum of the image in Fig. 3(a) after Fourier transform. The gray-scale images from (b) to (e), extracted from the image in Fig. 3(a), are the distributions of the phase, the real part, the imaginary part and the intensity, respectively. (f) The same intensity distribution directly recorded by CCD.

Fig. 5
Fig. 5

Distributions of zero lines of the real and the imaginary parts. The red solid lines and the black dashed lines are respectively the zero lines of the real parts and the imaginary parts. Figures (a1)-(a3), (b1)-(b3), (c1)-(c3) and (d1)-(d3) are for sample No.1, No.2, No.3 and No.4, respectively. Dimensions of each image are 8000μm×8000μm and distances are given under each image.

Fig. 6
Fig. 6

Local phase distributions one-to-one corresponding to the images in Fig. 5. Dimensions of each image are 2000μm×2000μm .

Fig. 7
Fig. 7

Phase distributions of (a) sample No. 1, (b) sample No. 2, (c) sample No. 3 and (d) sample No. 4 with distance z=60μm z=60μm and dimensions of each image are 8000μm×8000μm .

Fig. 8
Fig. 8

Phase probability density distributions of four samples when z=60μm .

Fig. 9
Fig. 9

Phase distributions of (a) incident light and (b) sample No. 1.

Fig. 10
Fig. 10

Phase distributions of sample No. 2 when (a) z=0μm (b) z=30μm and (c) z=60μm and dimensions of each image are 8000μm×8000μm .

Fig. 11
Fig. 11

Phase probability density distributions at different distances of sample No. 2.

Tables (1)

Tables Icon

Table 1 The Number and the Average Core Eccentricity of Phase Vortices of
Different Samples at Different Distances

Equations (4)

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

U(x,y)=A(x,y)exp[jφ(x,y)]= U r (x,y)+j U i (x,y),
r(x,y)=exp[j2π( f 0x x+ f 0y y)],
I 1 (x,y)= | U(x,y)+r(x,y) | 2 =U(x,y) U (x,y)+r(x,y) r * (x,y)+U(x,y) r * (x,y)+ U * (x,y)r(x,y),
I f ( f x , f y )= B f ( f x , f y )+ U f ( f x , f y )δ( f x + f 0x , f y + f 0y ) + U f * ( f x , f y )δ( f x f 0x , f y f 0y ) = B f ( f x , f y )+ U f ( f x + f 0x , f y + f 0y )+ U f * ( f x f 0x , f y f 0y ).

Metrics