Abstract

Points within a fully coherent complex scalar optical field, where the amplitude is identically zero but the optical phase has a jump discontinuity, have been widely investigated by the singular-optics community. More recent researches have extended the domain of singular optics to include partially coherent fields. For example, in coherence vortices the phase of the two-point spectral degree of coherence of a partially coherent field exhibits vortex structure around a point where the magnitude of the spectral degree of coherence vanishes. We show that the spectral degree of coherence of Mie scattered partially coherent statistically stationary electromagnetic fields exhibits a rich set of coherence vortices in both the internal and external fields. Specifically, we look at Mie scattering of a stationary beam from a dielectric sphere and study the formation of coherence vortices and their evolution with both the properties of the scattering sphere, and of the incident partially coherent beam.

© 2010 Optical Society of America

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References

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  2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).
  3. H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28, 968–970 (2003).
    [Crossref] [PubMed]
  4. G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003).
    [Crossref]
  5. M. V. Berry, “Making waves in physics: three wave singularities from the miraculous 1830s,” Nature 403, 21 (2000).
    [Crossref] [PubMed]
  6. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. Roy. Soc. Lond. A 336, 165–190 (1974).
    [Crossref]
  7. M. Soskin and M. Vasnetsov, “Singular optics,” in Progress in Optics, E. Wolf, ed., (North-Holland, Amsterdam, 2001)  42, 219–276.
    [Crossref]
  8. G. C. G. Berkhout and M. W. Beijersbergen, “Using a multipoint interferometer to measure the orbital angular momentum of light in astrophysics,” J. Opt. A: Pure Appl. Opt. 11, 094021 (2009).
    [Crossref]
  9. G. A. Swartzlander, “The optical vortex coronagraph,” J. Opt. A: Pure Appl. Opt. 11, 094022 (2009).
    [Crossref]
  10. Maurer C., S. Bernet, and M. Ritsch-Marte, “Refining common path interferometry with a spiral-phase Fourier filter,” J. Opt. A: Pure Appl. Opt. 11, 094023 (2009).
    [Crossref]
  11. W. A. Woźniak and M. Banach, “Measurements of linearly birefringent media parameters using the optical vortex interferometer with the Wollaston compensator,” J. Opt. A: Pure Appl. Opt. 11, 094024 (2009).
    [Crossref]
  12. P. A. M. Dirac, “Quantised singularities in the electromagnetic field,” Proc. Roy. Soc. Lond. A 133, 60–72 (1931).
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    [Crossref]
  14. G. Gbur and T. D. Visser, “Phase singularities and coherence vortices in linear optical systems,” Opt. Commun. 259, 428–435 (2006).
    [Crossref]
  15. G. V. Bogatyryova, C. V. Felde, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett. 28, 878–880 (2003).
    [Crossref] [PubMed]
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  17. Y. Gu and G. Gbur, “Topological reactions of optical correlation vortices,” Opt. Commun. 282, 709–716 (2009).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  26. M. Morin, P. Bernard, and P. Galarneau, “Moment definition of the pointing stability of a laser beam,” Opt. Lett. 19, 1379–1381 (1994).
    [Crossref] [PubMed]
  27. M. Levesque, A. Mailloux, M. Morin, P. Galarneau, Y. Champagne, O. Plomteux, and M. Tiedtke, “Laser pointing stability measurements,” Proc. SPIE 2870, 216–224 (1996).
    [Crossref]
  28. P. Kwee, F. Seifert, B. Willke, and K. Danzmann, “Laser beam quality and pointing measurement with an optical resonator,” Rev. Sci. Instrum. 78, 073103 (2007).
    [Crossref] [PubMed]
  29. C. Mätlzer, “MATLAB functions for Mie scattering and absorption,” IAP Res. Rep. No. 02–08, June (2002).
  30. M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities” in Progress in Optics, E. Wolf, ed., (North-Holland, Amsterdam, 2009)  53, 293–363.
  31. J. F. Nye, Natural Focusing and Fine Structure of Light (Institute of Physics Publishing, 1999).
  32. W. Gardner, A. Napolitano, and L. Paura, “Cyclostationarity: Half a Century of Research,” Signal Process. 86, 639–697 (2006).
    [Crossref]
  33. B. J. Davis, “Observable coherence theory for statistically periodic fields,” Phys. Rev. A 76, 043843 (2007).
    [Crossref]
  34. V. Manea, “General interference law for nonstationary, separable optical fields,” J. Opt. Soc. Am. A 26, 1907–1914 (2009).
    [Crossref]
  35. R. W. Schoonover, B. J. Davis, and P. S. Carney, “The generalized Wolf shift for cyclostationary fields,” Opt. Express 17, 4705–4711 (2009).
    [Crossref] [PubMed]
  36. C. R. Fernández-Pousa, “Intensity spectra after first-order dispersion of composite models of scalar cyclostationary light,” J. Opt. Soc. Am. A 26, 993–1007 (2009).
    [Crossref]

2009 (9)

G. C. G. Berkhout and M. W. Beijersbergen, “Using a multipoint interferometer to measure the orbital angular momentum of light in astrophysics,” J. Opt. A: Pure Appl. Opt. 11, 094021 (2009).
[Crossref]

G. A. Swartzlander, “The optical vortex coronagraph,” J. Opt. A: Pure Appl. Opt. 11, 094022 (2009).
[Crossref]

Maurer C., S. Bernet, and M. Ritsch-Marte, “Refining common path interferometry with a spiral-phase Fourier filter,” J. Opt. A: Pure Appl. Opt. 11, 094023 (2009).
[Crossref]

W. A. Woźniak and M. Banach, “Measurements of linearly birefringent media parameters using the optical vortex interferometer with the Wollaston compensator,” J. Opt. A: Pure Appl. Opt. 11, 094024 (2009).
[Crossref]

Y. Gu and G. Gbur, “Topological reactions of optical correlation vortices,” Opt. Commun. 282, 709–716 (2009).
[Crossref]

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities” in Progress in Optics, E. Wolf, ed., (North-Holland, Amsterdam, 2009)  53, 293–363.

V. Manea, “General interference law for nonstationary, separable optical fields,” J. Opt. Soc. Am. A 26, 1907–1914 (2009).
[Crossref]

R. W. Schoonover, B. J. Davis, and P. S. Carney, “The generalized Wolf shift for cyclostationary fields,” Opt. Express 17, 4705–4711 (2009).
[Crossref] [PubMed]

C. R. Fernández-Pousa, “Intensity spectra after first-order dispersion of composite models of scalar cyclostationary light,” J. Opt. Soc. Am. A 26, 993–1007 (2009).
[Crossref]

2008 (1)

G. Gbur, “Optical and coherence vortices and their relationships,” in Eighth International Conference on Correlation Optics, M. Kujawinska and O. V. Angelsky, eds. Proc. SPIE 7008, 70080N-1–70080N-7 (2008).

2007 (2)

P. Kwee, F. Seifert, B. Willke, and K. Danzmann, “Laser beam quality and pointing measurement with an optical resonator,” Rev. Sci. Instrum. 78, 073103 (2007).
[Crossref] [PubMed]

B. J. Davis, “Observable coherence theory for statistically periodic fields,” Phys. Rev. A 76, 043843 (2007).
[Crossref]

2006 (2)

W. Gardner, A. Napolitano, and L. Paura, “Cyclostationarity: Half a Century of Research,” Signal Process. 86, 639–697 (2006).
[Crossref]

G. Gbur and T. D. Visser, “Phase singularities and coherence vortices in linear optical systems,” Opt. Commun. 259, 428–435 (2006).
[Crossref]

2004 (3)

G. Gbur, T. Visser, and E. Wolf, “‘Hidden’ singularities in partially coherent wavefields,” J. Opt. A: Pure Appl. Opt. 6, S239–S242 (2004).
[Crossref]

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[Crossref] [PubMed]

D. G. Fischer and T. D. Visser, “Spatial correlation properties of focused partially coherent fields,” J. Opt. Soc. Am. A 21, 2097–2102 (2004).
[Crossref]

2003 (3)

2001 (2)

M. Soskin and M. Vasnetsov, “Singular optics,” in Progress in Optics, E. Wolf, ed., (North-Holland, Amsterdam, 2001)  42, 219–276.
[Crossref]

L. J. Allen, H. M. L. Faulkner, M. P. Oxley, and D. Paganin, “Phase retrieval and aberration correction in the presence of vortices in high-resolution transmission electron microscopy,” Ultramicroscopy 88, 85–97 (2001).
[Crossref] [PubMed]

2000 (1)

M. V. Berry, “Making waves in physics: three wave singularities from the miraculous 1830s,” Nature 403, 21 (2000).
[Crossref] [PubMed]

1996 (1)

M. Levesque, A. Mailloux, M. Morin, P. Galarneau, Y. Champagne, O. Plomteux, and M. Tiedtke, “Laser pointing stability measurements,” Proc. SPIE 2870, 216–224 (1996).
[Crossref]

1994 (1)

1974 (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. Roy. Soc. Lond. A 336, 165–190 (1974).
[Crossref]

1931 (1)

P. A. M. Dirac, “Quantised singularities in the electromagnetic field,” Proc. Roy. Soc. Lond. A 133, 60–72 (1931).
[Crossref]

1908 (1)

G. Mie, “Beiträge zur Optik Trüber Medien speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–442 (1908).
[Crossref]

Allen, L. J.

L. J. Allen, H. M. L. Faulkner, M. P. Oxley, and D. Paganin, “Phase retrieval and aberration correction in the presence of vortices in high-resolution transmission electron microscopy,” Ultramicroscopy 88, 85–97 (2001).
[Crossref] [PubMed]

Banach, M.

W. A. Woźniak and M. Banach, “Measurements of linearly birefringent media parameters using the optical vortex interferometer with the Wollaston compensator,” J. Opt. A: Pure Appl. Opt. 11, 094024 (2009).
[Crossref]

Beijersbergen, M. W.

G. C. G. Berkhout and M. W. Beijersbergen, “Using a multipoint interferometer to measure the orbital angular momentum of light in astrophysics,” J. Opt. A: Pure Appl. Opt. 11, 094021 (2009).
[Crossref]

Berkhout, G. C. G.

G. C. G. Berkhout and M. W. Beijersbergen, “Using a multipoint interferometer to measure the orbital angular momentum of light in astrophysics,” J. Opt. A: Pure Appl. Opt. 11, 094021 (2009).
[Crossref]

Bernard, P.

Bernet, S.

Maurer C., S. Bernet, and M. Ritsch-Marte, “Refining common path interferometry with a spiral-phase Fourier filter,” J. Opt. A: Pure Appl. Opt. 11, 094023 (2009).
[Crossref]

Berry, M. V.

M. V. Berry, “Making waves in physics: three wave singularities from the miraculous 1830s,” Nature 403, 21 (2000).
[Crossref] [PubMed]

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. Roy. Soc. Lond. A 336, 165–190 (1974).
[Crossref]

Bogatyryova, G. V.

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) edition (Cambridge University Press, Cambridge, 1999).

C., Maurer

Maurer C., S. Bernet, and M. Ritsch-Marte, “Refining common path interferometry with a spiral-phase Fourier filter,” J. Opt. A: Pure Appl. Opt. 11, 094023 (2009).
[Crossref]

Carney, P. S.

Champagne, Y.

M. Levesque, A. Mailloux, M. Morin, P. Galarneau, Y. Champagne, O. Plomteux, and M. Tiedtke, “Laser pointing stability measurements,” Proc. SPIE 2870, 216–224 (1996).
[Crossref]

Danzmann, K.

P. Kwee, F. Seifert, B. Willke, and K. Danzmann, “Laser beam quality and pointing measurement with an optical resonator,” Rev. Sci. Instrum. 78, 073103 (2007).
[Crossref] [PubMed]

Davis, B. J.

Dennis, M. R.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities” in Progress in Optics, E. Wolf, ed., (North-Holland, Amsterdam, 2009)  53, 293–363.

Dirac, P. A. M.

P. A. M. Dirac, “Quantised singularities in the electromagnetic field,” Proc. Roy. Soc. Lond. A 133, 60–72 (1931).
[Crossref]

Faulkner, H. M. L.

L. J. Allen, H. M. L. Faulkner, M. P. Oxley, and D. Paganin, “Phase retrieval and aberration correction in the presence of vortices in high-resolution transmission electron microscopy,” Ultramicroscopy 88, 85–97 (2001).
[Crossref] [PubMed]

Felde, C. V.

Fernández-Pousa, C. R.

Fischer, D. G.

Galarneau, P.

M. Levesque, A. Mailloux, M. Morin, P. Galarneau, Y. Champagne, O. Plomteux, and M. Tiedtke, “Laser pointing stability measurements,” Proc. SPIE 2870, 216–224 (1996).
[Crossref]

M. Morin, P. Bernard, and P. Galarneau, “Moment definition of the pointing stability of a laser beam,” Opt. Lett. 19, 1379–1381 (1994).
[Crossref] [PubMed]

Gardner, W.

W. Gardner, A. Napolitano, and L. Paura, “Cyclostationarity: Half a Century of Research,” Signal Process. 86, 639–697 (2006).
[Crossref]

Gbur, G.

Y. Gu and G. Gbur, “Topological reactions of optical correlation vortices,” Opt. Commun. 282, 709–716 (2009).
[Crossref]

G. Gbur, “Optical and coherence vortices and their relationships,” in Eighth International Conference on Correlation Optics, M. Kujawinska and O. V. Angelsky, eds. Proc. SPIE 7008, 70080N-1–70080N-7 (2008).

G. Gbur and T. D. Visser, “Phase singularities and coherence vortices in linear optical systems,” Opt. Commun. 259, 428–435 (2006).
[Crossref]

G. Gbur, T. Visser, and E. Wolf, “‘Hidden’ singularities in partially coherent wavefields,” J. Opt. A: Pure Appl. Opt. 6, S239–S242 (2004).
[Crossref]

G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003).
[Crossref]

H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28, 968–970 (2003).
[Crossref] [PubMed]

Gu, Y.

Y. Gu and G. Gbur, “Topological reactions of optical correlation vortices,” Opt. Commun. 282, 709–716 (2009).
[Crossref]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983).

Kwee, P.

P. Kwee, F. Seifert, B. Willke, and K. Danzmann, “Laser beam quality and pointing measurement with an optical resonator,” Rev. Sci. Instrum. 78, 073103 (2007).
[Crossref] [PubMed]

Lacis, A.

M. Mishchenko, L. Travis, and A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles, 3rd electronic release (Cambridge University Press, United Kingdom, 2002).

Levesque, M.

M. Levesque, A. Mailloux, M. Morin, P. Galarneau, Y. Champagne, O. Plomteux, and M. Tiedtke, “Laser pointing stability measurements,” Proc. SPIE 2870, 216–224 (1996).
[Crossref]

Mailloux, A.

M. Levesque, A. Mailloux, M. Morin, P. Galarneau, Y. Champagne, O. Plomteux, and M. Tiedtke, “Laser pointing stability measurements,” Proc. SPIE 2870, 216–224 (1996).
[Crossref]

Maleev, I. D.

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[Crossref] [PubMed]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).

Manea, V.

Marathay, A. S.

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[Crossref] [PubMed]

Mätlzer, C.

C. Mätlzer, “MATLAB functions for Mie scattering and absorption,” IAP Res. Rep. No. 02–08, June (2002).

Mie, G.

G. Mie, “Beiträge zur Optik Trüber Medien speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–442 (1908).
[Crossref]

Mishchenko, M.

M. Mishchenko, L. Travis, and A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles, 3rd electronic release (Cambridge University Press, United Kingdom, 2002).

Morin, M.

M. Levesque, A. Mailloux, M. Morin, P. Galarneau, Y. Champagne, O. Plomteux, and M. Tiedtke, “Laser pointing stability measurements,” Proc. SPIE 2870, 216–224 (1996).
[Crossref]

M. Morin, P. Bernard, and P. Galarneau, “Moment definition of the pointing stability of a laser beam,” Opt. Lett. 19, 1379–1381 (1994).
[Crossref] [PubMed]

Napolitano, A.

W. Gardner, A. Napolitano, and L. Paura, “Cyclostationarity: Half a Century of Research,” Signal Process. 86, 639–697 (2006).
[Crossref]

Nye, J. F.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. Roy. Soc. Lond. A 336, 165–190 (1974).
[Crossref]

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

O’Holleran, K.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities” in Progress in Optics, E. Wolf, ed., (North-Holland, Amsterdam, 2009)  53, 293–363.

Oxley, M. P.

L. J. Allen, H. M. L. Faulkner, M. P. Oxley, and D. Paganin, “Phase retrieval and aberration correction in the presence of vortices in high-resolution transmission electron microscopy,” Ultramicroscopy 88, 85–97 (2001).
[Crossref] [PubMed]

Padgett, M. J.

M. R. Dennis, K. O’Holleran, and M. J. Padgett, “Singular optics: optical vortices and polarization singularities” in Progress in Optics, E. Wolf, ed., (North-Holland, Amsterdam, 2009)  53, 293–363.

Paganin, D.

L. J. Allen, H. M. L. Faulkner, M. P. Oxley, and D. Paganin, “Phase retrieval and aberration correction in the presence of vortices in high-resolution transmission electron microscopy,” Ultramicroscopy 88, 85–97 (2001).
[Crossref] [PubMed]

Paganin, D. M.

D. M. Paganin, Coherent X-ray Optics (Oxford University Press, New York, 2006).

Palacios, D. M.

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[Crossref] [PubMed]

Paura, L.

W. Gardner, A. Napolitano, and L. Paura, “Cyclostationarity: Half a Century of Research,” Signal Process. 86, 639–697 (2006).
[Crossref]

Plomteux, O.

M. Levesque, A. Mailloux, M. Morin, P. Galarneau, Y. Champagne, O. Plomteux, and M. Tiedtke, “Laser pointing stability measurements,” Proc. SPIE 2870, 216–224 (1996).
[Crossref]

Polyanskii, P. V.

Ponomarenko, S. A.

Ritsch-Marte, M.

Maurer C., S. Bernet, and M. Ritsch-Marte, “Refining common path interferometry with a spiral-phase Fourier filter,” J. Opt. A: Pure Appl. Opt. 11, 094023 (2009).
[Crossref]

Schoonover, R. W.

Schouten, H. F.

Seifert, F.

P. Kwee, F. Seifert, B. Willke, and K. Danzmann, “Laser beam quality and pointing measurement with an optical resonator,” Rev. Sci. Instrum. 78, 073103 (2007).
[Crossref] [PubMed]

Soskin, M.

M. Soskin and M. Vasnetsov, “Singular optics,” in Progress in Optics, E. Wolf, ed., (North-Holland, Amsterdam, 2001)  42, 219–276.
[Crossref]

Soskin, M. S.

Swartzlander, G. A.

G. A. Swartzlander, “The optical vortex coronagraph,” J. Opt. A: Pure Appl. Opt. 11, 094022 (2009).
[Crossref]

D. M. Palacios, I. D. Maleev, A. S. Marathay, and G. A. Swartzlander, “Spatial correlation singularity of a vortex field,” Phys. Rev. Lett. 92, 143905 (2004).
[Crossref] [PubMed]

Tiedtke, M.

M. Levesque, A. Mailloux, M. Morin, P. Galarneau, Y. Champagne, O. Plomteux, and M. Tiedtke, “Laser pointing stability measurements,” Proc. SPIE 2870, 216–224 (1996).
[Crossref]

Travis, L.

M. Mishchenko, L. Travis, and A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles, 3rd electronic release (Cambridge University Press, United Kingdom, 2002).

Vasnetsov, M.

M. Soskin and M. Vasnetsov, “Singular optics,” in Progress in Optics, E. Wolf, ed., (North-Holland, Amsterdam, 2001)  42, 219–276.
[Crossref]

Visser, T.

G. Gbur, T. Visser, and E. Wolf, “‘Hidden’ singularities in partially coherent wavefields,” J. Opt. A: Pure Appl. Opt. 6, S239–S242 (2004).
[Crossref]

Visser, T. D.

G. Gbur and T. D. Visser, “Phase singularities and coherence vortices in linear optical systems,” Opt. Commun. 259, 428–435 (2006).
[Crossref]

D. G. Fischer and T. D. Visser, “Spatial correlation properties of focused partially coherent fields,” J. Opt. Soc. Am. A 21, 2097–2102 (2004).
[Crossref]

H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28, 968–970 (2003).
[Crossref] [PubMed]

G. Gbur and T. D. Visser, “Coherence vortices in partially coherent beams,” Opt. Commun. 222, 117–125 (2003).
[Crossref]

Willke, B.

P. Kwee, F. Seifert, B. Willke, and K. Danzmann, “Laser beam quality and pointing measurement with an optical resonator,” Rev. Sci. Instrum. 78, 073103 (2007).
[Crossref] [PubMed]

Wolf, E.

G. Gbur, T. Visser, and E. Wolf, “‘Hidden’ singularities in partially coherent wavefields,” J. Opt. A: Pure Appl. Opt. 6, S239–S242 (2004).
[Crossref]

H. F. Schouten, G. Gbur, T. D. Visser, and E. Wolf, “Phase singularities of the coherence functions in Young’s interference pattern,” Opt. Lett. 28, 968–970 (2003).
[Crossref] [PubMed]

G. V. Bogatyryova, C. V. Felde, P. V. Polyanskii, S. A. Ponomarenko, M. S. Soskin, and E. Wolf, “Partially coherent vortex beams with a separable phase,” Opt. Lett. 28, 878–880 (2003).
[Crossref] [PubMed]

E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, Cambridge, 2007).

M. Born and E. Wolf, Principles of Optics, 7th (expanded) edition (Cambridge University Press, Cambridge, 1999).

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1995).

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Figures (7)

Fig. 1.
Fig. 1.

Schematic illustration of the dielectric sphere with its center at the origin of the co-ordinate system. The incident x-polarized plane wave propagates in the positive z-direction.

Fig. 2.
Fig. 2.

Coherence vortices associated with spectral degree of coherence.

Fig. 3.
Fig. 3.

Pointing stability of stationary laser source.

Fig. 4.
Fig. 4.

Internal spectral degree of coherence and spectral density plots in xz (left column) and yz (right column) planes. R = λ 0 and pointing stability is π psr. All dimensions are in nm.

Fig. 5.
Fig. 5.

Topological reactions of coherence vortices: annihilation and creation of coherence vortices in the internal spectral degree of coherence in xz (left column) and yz (right column) planes with varying cone angle ΔΩ, and R = λ 0 . All dimensions are in nm.

Fig. 6.
Fig. 6.

Singularities of the ensemble averaged Poynting vector and coherence vortices in the field inside the scattering particle in xz plane. R = λ 0 and pointing stability is π psr. All dimensions are given in nm, with field-of-view indicated via the x and z axes.

Fig. 7.
Fig. 7.

Phase of spectral degree of coherence of scattered field over planes (a) z = R, (b) z = 1.17R, (c) z = 1.2R, (d) z = 1.37R. Here R = 2λ 0 and pointing stability is π psr. All dimensions are given in μm.

Equations (9)

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E inc = E ( t ) e ikz iωt e ̂ x ,
E int = E ( t ) e iωt n = 1 i n 2 n + 1 n ( n + 1 ) ( c n M o 1 n 1 id n N e 1 n 1 ) ,
E sca = E ( t ) e iωt n = 1 i n 2 n + 1 n ( n + 1 ) ( ia n N e 1 n 3 b n M o 1 n 1 ) ,
W ij ( r 1 , r 2 , ω ) = E i * ( r 1 , ω ) E j ( r 2 , ω ) ,
μ ( r 1 , r 2 ; ω ) = Tr [ W ij ( r 1 , r 2 ; ω ) ] S ( r 1 , ω ) S ( r 2 , ω ) ,
C ( Γ ) Γ arg [ μ ( r 1 r 2 ; ω ) ] d r 1 = 2 ,
C ( Γ + δ Γ ) = C ( Γ ) .
{ e ik s ̂ 0 ·r e ̂ S 0 } .
{ A ( r , t ) e i ( k s ˜ ̂ 0 ( t ) ·r ωt ) e ˜ ̂ S 0 ( t ) } .

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