Abstract

We study experimentally the polarisation of an asymptotically linearly polarized focused Gaussian beam. Around the focal point the polarization state is elliptical even though optical chirality is zero. As a consequence, this field allows to observe magnetic circular dichroism, but it shouldn’t give rise to natural circular dichroism. This distinction emphasizes the fundamental difference between these two forms of optical activity. This experiment, first proposed by N. Yang and A. E. Cohen [J. Phys. Chem. B 115, 5304 (2011)] is simple and sensitive. It weakly perturbs the beam propagation and probes the coherence between the field components. It is thus complementary to the existing techniques, usually only sensitive to the intensity.

© 2014 Optical Society of America

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References

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    [Crossref] [PubMed]
  2. A. E. Siegman, Lasers (University Science Books, 1986).
  3. B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. ii. structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A 253, 358–379 (1959).
    [Crossref]
  4. M. Lax, W. H. Louisell, and W. B. McKnight, “From maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [Crossref]
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  6. T. Wilson, R. Jukaitis, and P. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes,” Optics Communications 141, 298–313 (1997).
    [Crossref]
  7. Y. Fainman and J. Shamir, “Polarization of nonplanar wave fronts,” Appl. Opt. 23, 3188–3195 (1984).
    [Crossref] [PubMed]
  8. W. L. Erikson and S. Singh, “Polarization properties of maxwell-gaussian laser beams,” Phys. Rev. E 49, 5778–5786 (1994).
    [Crossref]
  9. K. Bahlmann and S. W. Hell, “Depolarization by high aperture focusing,” Applied Physics Letters77 (2000).
    [Crossref]
  10. L. Fratta, P. Debernardi, G. P. Bava, C. Degen, J. Kaiser, I. Fischer, and W. Elsäßer, “Spatially inhomogeneously polarized transverse modes in vertical-cavity surface-emitting lasers,” Phys. Rev. A 64, 031803 (2001).
    [Crossref]
  11. J. Conry, R. Vyas, and S. Singh, “Cross-polarization of linearly polarized hermite–gauss laser beams,” J. Opt. Soc. Am. A 29, 579–584 (2012).
    [Crossref]
  12. J. Conry, R. Vyas, and S. Singh, “Polarization of orbital angular momentum carrying laser beams,” J. Opt. Soc. Am. A 30, 821–824 (2013).
    [Crossref]
  13. B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485 (2000).
    [Crossref] [PubMed]
  14. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
    [Crossref] [PubMed]
  15. G. Kihara Rurimo, M. Schardt, S. Quabis, S. Malzer, C. Dotzler, A. Winkler, G. Leuchs, G. H. Dhler, D. Driscoll, M. Hanson, A. C. Gossard, and S. F. Pereira, “Using a quantum well heterostructure to study the longitudinal and transverse electric field components of a strongly focused laser beam,” Journal of Applied Physics 100, 023112 (2006).
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    [Crossref] [PubMed]
  19. N. Yang and A. E. Cohen, “Local geometry of electromagnetic fields and its role in molecular multipole transitions,” The Journal of Physical Chemistry B 115, 5304–5311 (2011). PMID: .
    [Crossref] [PubMed]
  20. J. D. Thompson, T. G. Tiecke, A. S. Zibrov, V. Vuletić, and M. D. Lukin, “Coherence and raman sideband cooling of a single atom in an optical tweezer,” Phys. Rev. Lett. 110, 133001 (2013).
    [Crossref] [PubMed]
  21. A. M. Kaufman, B. J. Lester, and C. A. Regal, “Cooling a single atom in an optical tweezer to its quantum ground state,” Phys. Rev. X 2, 041014 (2012).
  22. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, 2007).
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    [Crossref]
  24. S. Nemoto, “Waist shift of a gaussian beam by plane dielectric interfaces,” Appl. Opt. 27, 1833–1839 (1988).
    [Crossref] [PubMed]
  25. Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010).
    [Crossref] [PubMed]
  26. Y. Tang and A. Cohen, “Enhanced enantioselectivity in excitation of chiral molecules by superchiral light,” Science (New York, N.Y.) 332, 333336 (2011).
    [Crossref]
  27. D. L. Andrews and M. M. Coles, “Measures of chirality and angular momentum in the electromagnetic field,” Opt. Lett. 37, 3009–3011 (2012).
    [Crossref] [PubMed]
  28. K. Y. Bliokh and F. Nori, “Characterizing optical chirality,” Phys. Rev. A 83, 021803 (2011).
    [Crossref]

2013 (2)

J. D. Thompson, T. G. Tiecke, A. S. Zibrov, V. Vuletić, and M. D. Lukin, “Coherence and raman sideband cooling of a single atom in an optical tweezer,” Phys. Rev. Lett. 110, 133001 (2013).
[Crossref] [PubMed]

J. Conry, R. Vyas, and S. Singh, “Polarization of orbital angular momentum carrying laser beams,” J. Opt. Soc. Am. A 30, 821–824 (2013).
[Crossref]

2012 (3)

2011 (3)

N. Yang and A. E. Cohen, “Local geometry of electromagnetic fields and its role in molecular multipole transitions,” The Journal of Physical Chemistry B 115, 5304–5311 (2011). PMID: .
[Crossref] [PubMed]

Y. Tang and A. Cohen, “Enhanced enantioselectivity in excitation of chiral molecules by superchiral light,” Science (New York, N.Y.) 332, 333336 (2011).
[Crossref]

K. Y. Bliokh and F. Nori, “Characterizing optical chirality,” Phys. Rev. A 83, 021803 (2011).
[Crossref]

2010 (1)

Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010).
[Crossref] [PubMed]

2008 (1)

2007 (1)

2006 (1)

G. Kihara Rurimo, M. Schardt, S. Quabis, S. Malzer, C. Dotzler, A. Winkler, G. Leuchs, G. H. Dhler, D. Driscoll, M. Hanson, A. C. Gossard, and S. F. Pereira, “Using a quantum well heterostructure to study the longitudinal and transverse electric field components of a strongly focused laser beam,” Journal of Applied Physics 100, 023112 (2006).
[Crossref]

2003 (1)

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90, 013903 (2003).
[Crossref] [PubMed]

2001 (2)

L. Fratta, P. Debernardi, G. P. Bava, C. Degen, J. Kaiser, I. Fischer, and W. Elsäßer, “Spatially inhomogeneously polarized transverse modes in vertical-cavity surface-emitting lasers,” Phys. Rev. A 64, 031803 (2001).
[Crossref]

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref] [PubMed]

2000 (1)

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485 (2000).
[Crossref] [PubMed]

1997 (1)

T. Wilson, R. Jukaitis, and P. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes,” Optics Communications 141, 298–313 (1997).
[Crossref]

1994 (1)

W. L. Erikson and S. Singh, “Polarization properties of maxwell-gaussian laser beams,” Phys. Rev. E 49, 5778–5786 (1994).
[Crossref]

1988 (1)

1984 (1)

1975 (1)

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

1966 (1)

1959 (1)

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. ii. structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A 253, 358–379 (1959).
[Crossref]

Andrews, D. L.

Bahlmann, K.

K. Bahlmann and S. W. Hell, “Depolarization by high aperture focusing,” Applied Physics Letters77 (2000).
[Crossref]

Barron, L. D.

L. D. Barron, Molecular Light Scattering and Optical Activity (Cambridge University Press, 2004).
[Crossref]

Bava, G. P.

L. Fratta, P. Debernardi, G. P. Bava, C. Degen, J. Kaiser, I. Fischer, and W. Elsäßer, “Spatially inhomogeneously polarized transverse modes in vertical-cavity surface-emitting lasers,” Phys. Rev. A 64, 031803 (2001).
[Crossref]

Beversluis, M.

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90, 013903 (2003).
[Crossref] [PubMed]

Beversluis, M. R.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref] [PubMed]

Bliokh, K. Y.

K. Y. Bliokh and F. Nori, “Characterizing optical chirality,” Phys. Rev. A 83, 021803 (2011).
[Crossref]

Bouhelier, A.

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90, 013903 (2003).
[Crossref] [PubMed]

Brown, T. G.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref] [PubMed]

Cohen, A.

Y. Tang and A. Cohen, “Enhanced enantioselectivity in excitation of chiral molecules by superchiral light,” Science (New York, N.Y.) 332, 333336 (2011).
[Crossref]

Cohen, A. E.

N. Yang and A. E. Cohen, “Local geometry of electromagnetic fields and its role in molecular multipole transitions,” The Journal of Physical Chemistry B 115, 5304–5311 (2011). PMID: .
[Crossref] [PubMed]

Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010).
[Crossref] [PubMed]

Coles, M. M.

Conry, J.

Debernardi, P.

L. Fratta, P. Debernardi, G. P. Bava, C. Degen, J. Kaiser, I. Fischer, and W. Elsäßer, “Spatially inhomogeneously polarized transverse modes in vertical-cavity surface-emitting lasers,” Phys. Rev. A 64, 031803 (2001).
[Crossref]

Degen, C.

L. Fratta, P. Debernardi, G. P. Bava, C. Degen, J. Kaiser, I. Fischer, and W. Elsäßer, “Spatially inhomogeneously polarized transverse modes in vertical-cavity surface-emitting lasers,” Phys. Rev. A 64, 031803 (2001).
[Crossref]

Dhler, G. H.

G. Kihara Rurimo, M. Schardt, S. Quabis, S. Malzer, C. Dotzler, A. Winkler, G. Leuchs, G. H. Dhler, D. Driscoll, M. Hanson, A. C. Gossard, and S. F. Pereira, “Using a quantum well heterostructure to study the longitudinal and transverse electric field components of a strongly focused laser beam,” Journal of Applied Physics 100, 023112 (2006).
[Crossref]

Dotzler, C.

G. Kihara Rurimo, M. Schardt, S. Quabis, S. Malzer, C. Dotzler, A. Winkler, G. Leuchs, G. H. Dhler, D. Driscoll, M. Hanson, A. C. Gossard, and S. F. Pereira, “Using a quantum well heterostructure to study the longitudinal and transverse electric field components of a strongly focused laser beam,” Journal of Applied Physics 100, 023112 (2006).
[Crossref]

Driscoll, D.

G. Kihara Rurimo, M. Schardt, S. Quabis, S. Malzer, C. Dotzler, A. Winkler, G. Leuchs, G. H. Dhler, D. Driscoll, M. Hanson, A. C. Gossard, and S. F. Pereira, “Using a quantum well heterostructure to study the longitudinal and transverse electric field components of a strongly focused laser beam,” Journal of Applied Physics 100, 023112 (2006).
[Crossref]

Elsäßer, W.

L. Fratta, P. Debernardi, G. P. Bava, C. Degen, J. Kaiser, I. Fischer, and W. Elsäßer, “Spatially inhomogeneously polarized transverse modes in vertical-cavity surface-emitting lasers,” Phys. Rev. A 64, 031803 (2001).
[Crossref]

Erikson, W. L.

W. L. Erikson and S. Singh, “Polarization properties of maxwell-gaussian laser beams,” Phys. Rev. E 49, 5778–5786 (1994).
[Crossref]

Fainman, Y.

Fischer, I.

L. Fratta, P. Debernardi, G. P. Bava, C. Degen, J. Kaiser, I. Fischer, and W. Elsäßer, “Spatially inhomogeneously polarized transverse modes in vertical-cavity surface-emitting lasers,” Phys. Rev. A 64, 031803 (2001).
[Crossref]

Fratta, L.

L. Fratta, P. Debernardi, G. P. Bava, C. Degen, J. Kaiser, I. Fischer, and W. Elsäßer, “Spatially inhomogeneously polarized transverse modes in vertical-cavity surface-emitting lasers,” Phys. Rev. A 64, 031803 (2001).
[Crossref]

Gossard, A. C.

G. Kihara Rurimo, M. Schardt, S. Quabis, S. Malzer, C. Dotzler, A. Winkler, G. Leuchs, G. H. Dhler, D. Driscoll, M. Hanson, A. C. Gossard, and S. F. Pereira, “Using a quantum well heterostructure to study the longitudinal and transverse electric field components of a strongly focused laser beam,” Journal of Applied Physics 100, 023112 (2006).
[Crossref]

Hanson, M.

G. Kihara Rurimo, M. Schardt, S. Quabis, S. Malzer, C. Dotzler, A. Winkler, G. Leuchs, G. H. Dhler, D. Driscoll, M. Hanson, A. C. Gossard, and S. F. Pereira, “Using a quantum well heterostructure to study the longitudinal and transverse electric field components of a strongly focused laser beam,” Journal of Applied Physics 100, 023112 (2006).
[Crossref]

Hao, B.

Hartschuh, A.

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90, 013903 (2003).
[Crossref] [PubMed]

Hecht, B.

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485 (2000).
[Crossref] [PubMed]

Hell, S. W.

K. Bahlmann and S. W. Hell, “Depolarization by high aperture focusing,” Applied Physics Letters77 (2000).
[Crossref]

Higdon, P.

T. Wilson, R. Jukaitis, and P. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes,” Optics Communications 141, 298–313 (1997).
[Crossref]

Jukaitis, R.

T. Wilson, R. Jukaitis, and P. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes,” Optics Communications 141, 298–313 (1997).
[Crossref]

Kaiser, J.

L. Fratta, P. Debernardi, G. P. Bava, C. Degen, J. Kaiser, I. Fischer, and W. Elsäßer, “Spatially inhomogeneously polarized transverse modes in vertical-cavity surface-emitting lasers,” Phys. Rev. A 64, 031803 (2001).
[Crossref]

Kaufman, A. M.

A. M. Kaufman, B. J. Lester, and C. A. Regal, “Cooling a single atom in an optical tweezer to its quantum ground state,” Phys. Rev. X 2, 041014 (2012).

Kihara Rurimo, G.

G. Kihara Rurimo, M. Schardt, S. Quabis, S. Malzer, C. Dotzler, A. Winkler, G. Leuchs, G. H. Dhler, D. Driscoll, M. Hanson, A. C. Gossard, and S. F. Pereira, “Using a quantum well heterostructure to study the longitudinal and transverse electric field components of a strongly focused laser beam,” Journal of Applied Physics 100, 023112 (2006).
[Crossref]

Kogelnik, H.

Kozawa, Y.

Lax, M.

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

Leger, J.

Lester, B. J.

A. M. Kaufman, B. J. Lester, and C. A. Regal, “Cooling a single atom in an optical tweezer to its quantum ground state,” Phys. Rev. X 2, 041014 (2012).

Leuchs, G.

G. Kihara Rurimo, M. Schardt, S. Quabis, S. Malzer, C. Dotzler, A. Winkler, G. Leuchs, G. H. Dhler, D. Driscoll, M. Hanson, A. C. Gossard, and S. F. Pereira, “Using a quantum well heterostructure to study the longitudinal and transverse electric field components of a strongly focused laser beam,” Journal of Applied Physics 100, 023112 (2006).
[Crossref]

Li, T.

Louisell, W. H.

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

Lukin, M. D.

J. D. Thompson, T. G. Tiecke, A. S. Zibrov, V. Vuletić, and M. D. Lukin, “Coherence and raman sideband cooling of a single atom in an optical tweezer,” Phys. Rev. Lett. 110, 133001 (2013).
[Crossref] [PubMed]

Malzer, S.

G. Kihara Rurimo, M. Schardt, S. Quabis, S. Malzer, C. Dotzler, A. Winkler, G. Leuchs, G. H. Dhler, D. Driscoll, M. Hanson, A. C. Gossard, and S. F. Pereira, “Using a quantum well heterostructure to study the longitudinal and transverse electric field components of a strongly focused laser beam,” Journal of Applied Physics 100, 023112 (2006).
[Crossref]

McKnight, W. B.

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

Nemoto, S.

Nori, F.

K. Y. Bliokh and F. Nori, “Characterizing optical chirality,” Phys. Rev. A 83, 021803 (2011).
[Crossref]

Novotny, L.

A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90, 013903 (2003).
[Crossref] [PubMed]

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref] [PubMed]

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485 (2000).
[Crossref] [PubMed]

Pereira, S. F.

G. Kihara Rurimo, M. Schardt, S. Quabis, S. Malzer, C. Dotzler, A. Winkler, G. Leuchs, G. H. Dhler, D. Driscoll, M. Hanson, A. C. Gossard, and S. F. Pereira, “Using a quantum well heterostructure to study the longitudinal and transverse electric field components of a strongly focused laser beam,” Journal of Applied Physics 100, 023112 (2006).
[Crossref]

Quabis, S.

G. Kihara Rurimo, M. Schardt, S. Quabis, S. Malzer, C. Dotzler, A. Winkler, G. Leuchs, G. H. Dhler, D. Driscoll, M. Hanson, A. C. Gossard, and S. F. Pereira, “Using a quantum well heterostructure to study the longitudinal and transverse electric field components of a strongly focused laser beam,” Journal of Applied Physics 100, 023112 (2006).
[Crossref]

Regal, C. A.

A. M. Kaufman, B. J. Lester, and C. A. Regal, “Cooling a single atom in an optical tweezer to its quantum ground state,” Phys. Rev. X 2, 041014 (2012).

Richards, B.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. ii. structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A 253, 358–379 (1959).
[Crossref]

Sato, S.

Schardt, M.

G. Kihara Rurimo, M. Schardt, S. Quabis, S. Malzer, C. Dotzler, A. Winkler, G. Leuchs, G. H. Dhler, D. Driscoll, M. Hanson, A. C. Gossard, and S. F. Pereira, “Using a quantum well heterostructure to study the longitudinal and transverse electric field components of a strongly focused laser beam,” Journal of Applied Physics 100, 023112 (2006).
[Crossref]

Shamir, J.

Sick, B.

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485 (2000).
[Crossref] [PubMed]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Singh, S.

Tang, Y.

Y. Tang and A. Cohen, “Enhanced enantioselectivity in excitation of chiral molecules by superchiral light,” Science (New York, N.Y.) 332, 333336 (2011).
[Crossref]

Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010).
[Crossref] [PubMed]

Thompson, J. D.

J. D. Thompson, T. G. Tiecke, A. S. Zibrov, V. Vuletić, and M. D. Lukin, “Coherence and raman sideband cooling of a single atom in an optical tweezer,” Phys. Rev. Lett. 110, 133001 (2013).
[Crossref] [PubMed]

Tiecke, T. G.

J. D. Thompson, T. G. Tiecke, A. S. Zibrov, V. Vuletić, and M. D. Lukin, “Coherence and raman sideband cooling of a single atom in an optical tweezer,” Phys. Rev. Lett. 110, 133001 (2013).
[Crossref] [PubMed]

Vuletic, V.

J. D. Thompson, T. G. Tiecke, A. S. Zibrov, V. Vuletić, and M. D. Lukin, “Coherence and raman sideband cooling of a single atom in an optical tweezer,” Phys. Rev. Lett. 110, 133001 (2013).
[Crossref] [PubMed]

Vyas, R.

Wilson, T.

T. Wilson, R. Jukaitis, and P. Higdon, “The imaging of dielectric point scatterers in conventional and confocal polarisation microscopes,” Optics Communications 141, 298–313 (1997).
[Crossref]

Winkler, A.

G. Kihara Rurimo, M. Schardt, S. Quabis, S. Malzer, C. Dotzler, A. Winkler, G. Leuchs, G. H. Dhler, D. Driscoll, M. Hanson, A. C. Gossard, and S. F. Pereira, “Using a quantum well heterostructure to study the longitudinal and transverse electric field components of a strongly focused laser beam,” Journal of Applied Physics 100, 023112 (2006).
[Crossref]

Wolf, E.

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. ii. structure of the image field in an aplanatic system,” Proc. R. Soc. Lond. A 253, 358–379 (1959).
[Crossref]

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

Yang, N.

N. Yang and A. E. Cohen, “Local geometry of electromagnetic fields and its role in molecular multipole transitions,” The Journal of Physical Chemistry B 115, 5304–5311 (2011). PMID: .
[Crossref] [PubMed]

Youngworth, K. S.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
[Crossref] [PubMed]

Zibrov, A. S.

J. D. Thompson, T. G. Tiecke, A. S. Zibrov, V. Vuletić, and M. D. Lukin, “Coherence and raman sideband cooling of a single atom in an optical tweezer,” Phys. Rev. Lett. 110, 133001 (2013).
[Crossref] [PubMed]

Appl. Opt. (3)

J. Opt. Soc. Am. A (2)

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

Journal of Applied Physics (1)

G. Kihara Rurimo, M. Schardt, S. Quabis, S. Malzer, C. Dotzler, A. Winkler, G. Leuchs, G. H. Dhler, D. Driscoll, M. Hanson, A. C. Gossard, and S. F. Pereira, “Using a quantum well heterostructure to study the longitudinal and transverse electric field components of a strongly focused laser beam,” Journal of Applied Physics 100, 023112 (2006).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Optics Communications (1)

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[Crossref]

Phys. Rev. A (3)

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[Crossref]

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[Crossref]

K. Y. Bliokh and F. Nori, “Characterizing optical chirality,” Phys. Rev. A 83, 021803 (2011).
[Crossref]

Phys. Rev. E (1)

W. L. Erikson and S. Singh, “Polarization properties of maxwell-gaussian laser beams,” Phys. Rev. E 49, 5778–5786 (1994).
[Crossref]

Phys. Rev. Lett. (5)

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485 (2000).
[Crossref] [PubMed]

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001).
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A. Bouhelier, M. Beversluis, A. Hartschuh, and L. Novotny, “Near-field second-harmonic generation induced by local field enhancement,” Phys. Rev. Lett. 90, 013903 (2003).
[Crossref] [PubMed]

Y. Tang and A. E. Cohen, “Optical chirality and its interaction with matter,” Phys. Rev. Lett. 104, 163901 (2010).
[Crossref] [PubMed]

J. D. Thompson, T. G. Tiecke, A. S. Zibrov, V. Vuletić, and M. D. Lukin, “Coherence and raman sideband cooling of a single atom in an optical tweezer,” Phys. Rev. Lett. 110, 133001 (2013).
[Crossref] [PubMed]

Phys. Rev. X (1)

A. M. Kaufman, B. J. Lester, and C. A. Regal, “Cooling a single atom in an optical tweezer to its quantum ground state,” Phys. Rev. X 2, 041014 (2012).

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

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Science (New York, N.Y.) (1)

Y. Tang and A. Cohen, “Enhanced enantioselectivity in excitation of chiral molecules by superchiral light,” Science (New York, N.Y.) 332, 333336 (2011).
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The Journal of Physical Chemistry B (1)

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See e. g. H. C. Kim and Y. H. Lee [Opt. Comm. 169, 9 (1999)] and reference therein for a detailed historical overview, K. Duan, B. Wang and B. Lu [J. Opt. Soc. Am. A 22, 1976 (2005)] for an alternate approach; recent extensions to other families of Gaussian beams are not considered here.

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[Crossref]

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Figures (6)

Fig. 1
Fig. 1 a) Longitudinal cross-section of a light beam asymptotically linearly polarized. (b) Sketch of the polarization evolution across the beam. (c) Beam transverse cross-section in the waist plane. Color scale measures the circular intensity V from green to magenta. Black arrows point toward the extrema of V where the circular fraction is ±θ.
Fig. 2
Fig. 2 Experimental setup. LD: laser diode, FC1,2: monomode fiber couplers, D: iris diaphgram, BS: beam splitter, HWP: half-wave plate, P: dichroic polarizer, L1–3: lenses, TS: translation stage, PEM: photo-elastic modulator, PhMCD: photodiode, Ph2Q: split photodiode.
Fig. 3
Fig. 3 a) Spectroscopy of the Nd:YAG MCD response. ΔA/B is the differential absorption of the right and left handed circular polarisation in a unit magnetic field. b) Top panel: reference MCD signal recorded as the laser wavelength is modulated in the linear part of the MCD lineshape mark out by vertical dotted lines in the left panel. Bottom panel: raw data from lock-in amplifier with 1 s integration time, horizontal (red) or vertical (blue) polarization.
Fig. 4
Fig. 4 a) Signal as a function of the focus position inside the crystal. Solid line results from a simple model taking into account the refraction of the beam (see right panel) and the fact that the signal builds up over the Rayleigh range. b) Refraction of the beam as it enters the crystal: the divergence angle θ0 is reduced to θ0/n and the focal point is displaced by Δz = (1 − 1/n)L = 0.9 mm. The actual length of the crystal L = 2 mm is reduced to L/n = 1.1 mm in the model.
Fig. 5
Fig. 5 Beam divergence θ is varied using diaphragm (red points) and signal is normalized to the full aperture one (blue point). Black line is a fit corresponding to Eq. (8).
Fig. 6
Fig. 6 Beam widths as a fonction of the position beyond the focusing lens labeled L1 in Fig. 2 (arbitrary origin). Black squares/red circles: vertical/horizontal cross sections. Straight lines are fits from which the beam parameters can be extracted.

Equations (21)

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E x = E 0 z R q ( z ) exp ( i k ρ 2 2 q ( z ) ) exp ( i ω t ) ,
E = 0 = E x x + E z z ,
E z x q ( z ) E x = x z + i z R E x .
p = α ( E × E ˙ ) B QS ,
E × E ˙ t = ( 0 , ω E 0 2 ( r ) Im ( x z + i z R ) , 0 ) ,
p ( r ) t = η P 0 x / z R 1 + ( z / z R ) 2 w 0 2 w 2 ( z ) exp ( 2 r 2 w 2 ( z ) )
d P + d z = 0 + d x + d y n v p ( r ) = π 32 η P 0 n v w 0 2 w 0 z R ( 1 + ( z / z R ) 2 ) 1 / 2 ,
Δ A = Δ P / P 0 = π 2 η n v w 0 3 sinh 1 ( L / 2 z R ) = κ θ 1 sinh 1 ( ( θ / θ 0 ) 2 )
Δ A = w 0 L sinh 1 ( L / 2 z R ) Δ A MCD .
Δ A ( λ 2 ) Δ A ( λ 1 ) = s B 0 Δ λ = 190 ± 25 × 10 6 .
χ = 1 2 ε 0 ( E . ( × E ) + c 2 B . ( × B ) ) .
φ = 1 2 ε 0 c 2 ( E × ( × B ) B × ( × E ) )
B y = B 0 z R q ( z ) exp ( i k ρ 2 2 q ( z ) ) exp ( i ω t ) ,
φ 1 1 + ( z / z R ) 2 w 0 2 w 2 ( z ) exp ( 2 r 2 w 2 ( z ) ) ( y / z R , x / z R , 0 )
ρ ( z ) = ( x ( z ) , y ( z ) ) = ( x 0 , y 0 ) ( 1 + ( z / z R ) 2 ) 1 2
I Γ ( r ) = I 0 ( w 0 w ( z ) ) 2 exp ( 2 r 2 / w ( z ) 2 ) ,
d ( I Γ d S ) d l = n v σ ( I Γ d S ) ,
d ( I Γ ) d l = n v σ ( I Γ ) .
I Γ ( + ) = I Γ ( ) exp [ L / 2 + L / 2 d u n v σ 0 x 0 ( 1 + u 2 ) 1 / 2 1 + u 2 ] = I Γ ( ) exp [ 2 n v σ 0 x 0 sinh 1 ( L / 2 z R ) ] ,
Δ I ( x d , y d ; z d ) = κ x d 1 + ( z d / z R ) 2 I 0 .
P + = 0 + d x d + d y d ( Δ I ( x d , y d ; z d ) ) = κ 1 + ( z d / z R ) 2 π 32 w 0 3 1 + ( z d / z R ) 2 I 0 = π 32 κ w 0 3 I 0 ,

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