We introduce a new metric to characterise spatial variations occurring in a time varying vector field, and we derive a rotationally invariant formula that quantifies temporal fluctuations within a consistent framework. So as to highlight the physics behind these metrics, both are derived from a well–known experiment in polarimetry. The derivation yields a set of expressions in a two–dimensional space, which is subsequently expanded to n–dimensions for special cases. The resulting expressions of the temporal and spatial metrics are incorporated into the electromagnetic theory of coherence and polarisation. Examples are given in the context of single molecule detection when measuring asymmetrically and radially polarised beams.

© 2011 OSA

1. J. Ellis and A. Dogariu, “Optical polarimetry of random fields,” Phys. Rev. Lett.95(20), 203905 (2004). [CrossRef]
2. A. F. Abouraddy and K. C. Toussaint, “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett.96(15), 153901 (2006). [CrossRef] [PubMed]
3. K. Serrels, E. Ramsay, R. Warburton, and D. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nature Phot.2(5), 311–314 (2008). [CrossRef]
4. T. Ha, T. Enderle, D. S. Chemla, P. R. Selvin, and S. Weiss, “Single Molecule Dynamics Studied by Polarization Modulation,” Phys. Rev. Lett.77(19), 3979–3982 (1996). [CrossRef] [PubMed]
5. E. Wolf, “Optics in terms of observable quantities,” Il Nuovo Cimento12(6), 884–888 (1954). [CrossRef]
6. P. Munro and P. Török, “Properties of high-numerical-aperture Müller-matrix polarimeters.” Opt. Lett.33, 2428–2430 (2008). [CrossRef] [PubMed]
7. T. Shirai and E. Wolf, “Correlations between intensity fluctuations in stochastic electromagnetic beams of any state of coherence and polarization,” Opt. Commun.272(2), 289–292 (2007). [CrossRef]
8. H. C. Jacks and O. Korotkova, “Intensity–intensity fluctuations of stochastic fields produced upon weak scattering,” J. Opt. Soc. Am. A28(6), 1139–1144 (2011). [CrossRef]
9. E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Il Nuovo Cimento13(6), 1165–1181 (1959). [CrossRef]
10. T. Setälä, A. Shevchenko, M. Kaivola, and A. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E66(1), 16615 (2002). [CrossRef]
11. U. Fano, “A Stokes-Parameter Technique for the Treatment of Polarization in Quantum Mechanics,” Phys. Rev.93(1), 121–123 (1954). [CrossRef]
12. M. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microscopy197, 219–223 (2000). [CrossRef]
13. J. Gil, “Polarimetric characterization of light and media,” Europ. Phys. J. App. Phys.40(1), 1–47 (2007). [CrossRef]
14. J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun.248(4–6), 333–337 (2005). [CrossRef]
15. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).
16. J. Movilla, G. Piquero, R. Martínez-Herrero, and P. Mejías, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun.149(4–6), 230–234 (1998). [CrossRef]
17. J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett.29(6), 536–538 (2004). [CrossRef] [PubMed]
18. H. Moya-Cessa, J. R. Moya-Cessa, J. Landgrave, A. Martinez-Niconoff, G. Perez-Leija, and A. T. Friberg, “Degree of polarization and quantum-mechanical purity,” J. Europ. Opt. Soc. Rap. Public.3(5), 08014 (2008). [CrossRef]
19. P. Török, P. Higdon, and T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Comm.148(4–6), 300–315 (1998). [CrossRef]

Europ. Phys. J. App. Phys.

• J. Gil, “Polarimetric characterization of light and media,” Europ. Phys. J. App. Phys.40(1), 1–47 (2007). [CrossRef]

Il Nuovo Cimento

• E. Wolf, “Coherence properties of partially polarized electromagnetic radiation,” Il Nuovo Cimento13(6), 1165–1181 (1959). [CrossRef]
• E. Wolf, “Optics in terms of observable quantities,” Il Nuovo Cimento12(6), 884–888 (1954). [CrossRef]

J. Europ. Opt. Soc. Rap. Public.

• H. Moya-Cessa, J. R. Moya-Cessa, J. Landgrave, A. Martinez-Niconoff, G. Perez-Leija, and A. T. Friberg, “Degree of polarization and quantum-mechanical purity,” J. Europ. Opt. Soc. Rap. Public.3(5), 08014 (2008). [CrossRef]

J. Microscopy

• M. Neil, T. Wilson, and R. Juskaitis, “A wavefront generator for complex pupil function synthesis and point spread function engineering,” J. Microscopy197, 219–223 (2000). [CrossRef]

J. Opt. Soc. Am. A

• H. C. Jacks and O. Korotkova, “Intensity–intensity fluctuations of stochastic fields produced upon weak scattering,” J. Opt. Soc. Am. A28(6), 1139–1144 (2011). [CrossRef]

Nature Phot.

• K. Serrels, E. Ramsay, R. Warburton, and D. Reid, “Nanoscale optical microscopy in the vectorial focusing regime,” Nature Phot.2(5), 311–314 (2008). [CrossRef]

Opt. Comm.

• P. Török, P. Higdon, and T. Wilson, “On the general properties of polarised light conventional and confocal microscopes,” Opt. Comm.148(4–6), 300–315 (1998). [CrossRef]

Opt. Commun.

• T. Shirai and E. Wolf, “Correlations between intensity fluctuations in stochastic electromagnetic beams of any state of coherence and polarization,” Opt. Commun.272(2), 289–292 (2007). [CrossRef]
• J. Ellis, A. Dogariu, S. Ponomarenko, and E. Wolf, “Degree of polarization of statistically stationary electromagnetic fields,” Opt. Commun.248(4–6), 333–337 (2005). [CrossRef]
• J. Movilla, G. Piquero, R. Martínez-Herrero, and P. Mejías, “Parametric characterization of non-uniformly polarized beams,” Opt. Commun.149(4–6), 230–234 (1998). [CrossRef]

Opt. Lett.

• J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett.29(6), 536–538 (2004). [CrossRef] [PubMed]
• P. Munro and P. Török, “Properties of high-numerical-aperture Müller-matrix polarimeters.” Opt. Lett.33, 2428–2430 (2008). [CrossRef] [PubMed]

Phys. Rev.

• U. Fano, “A Stokes-Parameter Technique for the Treatment of Polarization in Quantum Mechanics,” Phys. Rev.93(1), 121–123 (1954). [CrossRef]

Phys. Rev. E

• T. Setälä, A. Shevchenko, M. Kaivola, and A. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E66(1), 16615 (2002). [CrossRef]

Phys. Rev. Lett.

• T. Ha, T. Enderle, D. S. Chemla, P. R. Selvin, and S. Weiss, “Single Molecule Dynamics Studied by Polarization Modulation,” Phys. Rev. Lett.77(19), 3979–3982 (1996). [CrossRef] [PubMed]
• J. Ellis and A. Dogariu, “Optical polarimetry of random fields,” Phys. Rev. Lett.95(20), 203905 (2004). [CrossRef]
• A. F. Abouraddy and K. C. Toussaint, “Three-dimensional polarization control in microscopy,” Phys. Rev. Lett.96(15), 153901 (2006). [CrossRef] [PubMed]

Other

• R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977).

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