Abstract

The photonic interaction generally responsible for the electromagnetic trapping of molecules is forward-Rayleigh scattering, a process that is mediated by transition electric dipoles connecting the ground electronic state and virtual excited states. Higher order electric and magnetic multipole contributions to the scattering amplitude are usually negligible. However, on consideration of chiral discrimination effects (in which an input light of left-handed circular polarization can present different observables compared to right-handed polarization, or molecules of opposite enantiomeric form respond differently to a set circular polarization), the mechanism must be extended to specifically accommodate transition magnetic dipoles. Moreover, it is important to account for the fact that chiral molecules are necessarily nonspherical, so that their interactions with a laser beam will have an orientational dependence. Using quantum electrodynamics, this article quantifies the extent of the energetic discrimination that arises when chiral molecules are optically trapped, placing particular emphasis on the orientational effects of the trapping beam. An in-depth description of the intricate ensemble-weighted method used to incorporate the latter is presented. It is thus shown that, when a mixture of molecular enantiomers is irradiated by a continuous beam of circularly polarized light, a difference arises in the relative rates of migration of each enantiomer in and out of the most intense regions of the beam. As a consequence, optical trapping can be used as a means of achieving enantiomer separation.

© 2015 Optical Society of America

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  1. G. Cipparrone, A. Mazzulla, A. Pane, R. J. Hernandez, and R. Bartolino, “Chiral self-assembled solid microspheres: a novel multifunctional microphotonic device,” Adv. Mater. 23, 5773–5778 (2011).
    [Crossref]
  2. R. J. Hernandez, A. Mazzulla, A. Pane, K. Volke-Sepulveda, and G. Cipparrone, “Attractive-repulsive dynamics on light-responsive chiral microparticles induced by polarized tweezers,” Lab Chip 13, 459–467 (2013).
    [Crossref]
  3. M. G. Donato, J. Hernandez, A. Mazzulla, C. Provenzano, R. Saija, R. Sayed, S. Vasi, A. Magazzù, P. Pagliusi, R. Bartolino, P. G. Gucciardi, O. M. Maragò, and G. Cipparrone, “Polarization-dependent optomechanics mediated by chiral microresonators,” Nat. Commun. 5, 3656 (2014).
    [Crossref]
  4. G. Tkachenko and E. Brasselet, “Helicity-dependent three-dimensional optical trapping of chiral microparticles,” Nat. Commun. 5, 4491 (2014).
  5. M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophoton. 2, 021875 (2008).
    [Crossref]
  6. T. Čižmár, L. C. Dávila Romero, K. Dholakia, and D. L. Andrews, “Multiple optical trapping and binding: new routes to self-assembly,” J. Phys. B 43, 102001 (2010).
    [Crossref]
  7. O. M. Marago, P. H. Jones, P. G. Gucciardi, G. Volpe, and A. C. Ferrari, “Optical trapping and manipulation of nanostructures,” Nat. Nanotechnol. 8, 807–819 (2013).
    [Crossref]
  8. R. W. Bowman and M. J. Padgett, “Optical trapping and binding,” Rep. Prog. Phys. 76, 026401 (2013).
    [Crossref]
  9. D. S. Bradshaw and D. L. Andrews, “Interparticle interactions: energy potentials, energy transfer, and nanoscale mechanical motion in response to optical radiation,” J. Phys. Chem. A 117, 75–82 (2013).
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    [Crossref]
  14. V. V. Klimov and V. S. Letokhov, “Gradient optical force on atoms: beyond dipole approximation,” Opt. Commun. 126, 45–48 (1996).
    [Crossref]
  15. E. A. Power and T. Thirunamachandran, “On the nature of Hamiltonian for interaction of radiation with atoms and molecules: (e/mc)p.A, -μ.E, and all that,” Am. J. Phys. 46, 370–378 (1978).
    [Crossref]
  16. E. A. Power and T. Thirunamachandran, “The multipolar Hamiltonian in radiation theory,” Proc. R. Soc. A 372, 265–273 (1980).
    [Crossref]
  17. R. G. Woolley, “Gauge invariance in non-relativistic electrodynamics,” Proc. R. Soc. A 456, 1803–1819 (2000).
    [Crossref]
  18. D. S. Bradshaw and D. L. Andrews, “Chiral discrimination in optical trapping and manipulation,” New J. Phys. 16, 103021 (2014).
    [Crossref]
  19. D. T. Butcher, S. Y. Buhmann, and S. Scheel, “Casimir–Polder forces between chiral objects,” New J. Phys. 14, 113013 (2012).
    [Crossref]
  20. Q.-C. Shang, Z.-S. Wu, T. Qu, Z.-J. Li, L. Bai, and L. Gong, “Analysis of the radiation force and torque exerted on a chiral sphere by a Gaussian beam,” Opt. Express 21, 8677–8688 (2013).
    [Crossref]
  21. K. Ding, J. Ng, L. Zhou, and C. T. Chan, “Realization of optical pulling forces using chirality,” Phys. Rev. A 89, 063825 (2014).
    [Crossref]
  22. S. B. Wang and C. T. Chan, “Lateral optical force on chiral particles near a surface,” Nat. Commun. 5, 3307 (2014).
  23. D. Patterson, M. Schnell, and J. M. Doyle, “Enantiomer-specific detection of chiral molecules via microwave spectroscopy,” Nature 497, 475–477 (2013).
    [Crossref]
  24. D. Patterson and J. M. Doyle, “Sensitive chiral analysis via microwave three-wave mixing,” Phys. Rev. Lett. 111, 023008 (2013).
    [Crossref]
  25. M. H. M. Janssen and I. Powis, “Detecting chirality in molecules by imaging photoelectron circular dichroism,” Phys. Chem. Chem. Phys. 16, 856–871 (2014).
    [Crossref]
  26. D. Patterson and M. Schnell, “New studies on molecular chirality in the gas phase: enantiomer differentiation and determination of enantiomeric excess,” Phys. Chem. Chem. Phys. 16, 11114–11123 (2014).
    [Crossref]
  27. M. Okuno and T.-A. Ishibashi, “Chirality discriminated by heterodyne-detected vibrational sum frequency generation,” J. Phys. Chem. Lett. 5, 2874–2878 (2014).
    [Crossref]
  28. V. A. Shubert, D. Schmitz, D. Patterson, J. M. Doyle, and M. Schnell, “Identifying enantiomers in mixtures of chiral molecules with broadband microwave spectroscopy,” Angew. Chem. Int. Ed. 53, 1152–1155 (2014).
    [Crossref]
  29. P. Král, I. Thanopulos, M. Shapiro, and D. Cohen, “Two-step enantio-selective optical switch,” Phys. Rev. Lett. 90, 033001 (2003).
    [Crossref]
  30. Y. Li, C. Bruder, and C. P. Sun, “Generalized Stern–Gerlach effect for chiral molecules,” Phys. Rev. Lett. 99, 130403 (2007).
    [Crossref]
  31. W. Z. Jia and L. F. Wei, “Distinguishing left- and right-handed molecules using two-step coherent pulses,” J. Phys. B 43, 185402 (2010).
    [Crossref]
  32. A. Eilam and M. Shapiro, “Spatial separation of dimers of chiral molecules,” Phys. Rev. Lett. 110, 213004 (2013).
    [Crossref]
  33. X. Li and M. Shapiro, “Spatial separation of enantiomers by coherent optical means,” J. Chem. Phys. 132, 041101 (2010).
    [Crossref]
  34. X. Li and M. Shapiro, “Theory of the optical spatial separation of racemic mixtures of chiral molecules,” J. Chem. Phys. 132, 194315 (2010).
    [Crossref]
  35. A. Canaguier-Durand, J. A. Hutchison, C. Genet, and T. W. Ebbesen, “Mechanical separation of chiral dipoles by chiral light,” New J. Phys. 15, 123037 (2013).
    [Crossref]
  36. R. P. Cameron, S. M. Barnett, and A. M. Yao, “Discriminatory optical force for chiral molecules,” New J. Phys. 16, 013020 (2014).
    [Crossref]
  37. R. P. Cameron, A. M. Yao, and S. M. Barnett, “Diffraction gratings for chiral molecules and their applications,” J. Phys. Chem. A 118, 3472–3478 (2014).
    [Crossref]
  38. R. P. Cameron and S. Barnett, “Optical activity in the scattering of structured light,” Phys. Chem. Chem. Phys. 16, 25819–25829 (2014).
  39. A. Canaguier-Durand and C. Genet, “Chiral near fields generated from plasmonic optical lattices,” Phys. Rev. A 90, 023842 (2014).
    [Crossref]
  40. G. Tkachenko and E. Brasselet, “Optofluidic sorting of material chirality by chiral light,” Nat. Commun. 5, 3577 (2014).
  41. S. N. A. Smith and D. L. Andrews, “Three-dimensional ensemble averages for tensorial interactions in partially oriented, multi-particle systems,” J. Phys. A 44, 395001 (2011).
    [Crossref]
  42. D. L. Andrews and M. J. Harlow, “Phased and Boltzmann-weighted rotational averages,” Phys. Rev. A 29, 2796–2806 (1984).
    [Crossref]
  43. C. E. Hecht, Statistical Thermodynamics and Kinetic Theory (W. H. Freeman, 1990).
  44. D. L. Andrews and P. Allcock, Optical Harmonics in Molecular Systems (Wiley-VCH, 2002).
  45. D. L. Andrews and T. Thirunamachandran, “On three-dimensional rotational averages,” J. Chem. Phys. 67, 5026–5033 (1977).
    [Crossref]
  46. D. H. Friese, M. T. P. Beerepoot, and K. Ruud, “Rotational averaging of multiphoton absorption cross sections,” J. Chem. Phys. 141, 204103 (2014).
    [Crossref]
  47. J. S. Ford and D. L. Andrews, “One- and two-photon absorption in solution: the effects of a passive auxiliary beam,” J. Chem. Phys. 141, 034504 (2014).
    [Crossref]
  48. D. L. Andrews and W. A. Ghoul, “Eighth-rank isotropic tensors and rotational averages,” J. Phys. A 14, 1281–1290 (1981).
    [Crossref]
  49. J. M. Leeder, D. S. Bradshaw, and D. L. Andrews, “Laser-controlled fluorescence in two-level systems,” J. Phys. Chem. B 115, 5227–5233 (2011).
    [Crossref]
  50. J. M. Leeder and D. L. Andrews, “A molecular theory for two-photon and three-photon fluorescence polarization,” J. Chem. Phys. 134, 094503 (2011).
    [Crossref]
  51. D. V. Guzatov and V. V. Klimov, “Chiral particles in a circularly polarised light field: new effects and applications,” Quantum Electron. 41, 526–533 (2011).
    [Crossref]
  52. V. V. Klimov, D. V. Guzatov, and M. Ducloy, “Engineering of radiation of optically active molecules with chiral nano-meta-particles,” Europhys. Lett. 97, 47004 (2012).
    [Crossref]
  53. D. V. Guzatov and V. V. Klimov, “The influence of chiral spherical particles on the radiation of optically active molecules,” New J. Phys. 14, 123009 (2012).
    [Crossref]
  54. Z.-S. Wu, Q.-C. Shang, and Z.-J. Li, “Calculation of electromagnetic scattering by a large chiral sphere,” Appl. Opt. 51, 6661–6668 (2012).
    [Crossref]
  55. Q.-C. Shang, Z.-S. Wu, T. Qu, Z.-J. Li, L. Bai, and L. Gong, “Analysis of rainbow scattering by a chiral sphere,” Opt. Express 21, 21879–21888 (2013).
    [Crossref]
  56. V. V. Klimov, I. V. Zabkov, A. A. Pavlov, and D. V. Guzatov, “Eigen oscillations of a chiral sphere and their influence on radiation of chiral molecules,” Opt. Express 22, 18564–18578 (2014).
    [Crossref]
  57. P. W. Atkins and J. De Paula, Atkins’ Physical Chemistry (Oxford University, 2010).
  58. D. L. Andrews and M. Babiker, The Angular Momentum of Light (Cambridge University, 2013).

2014 (17)

K. Ding, J. Ng, L. Zhou, and C. T. Chan, “Realization of optical pulling forces using chirality,” Phys. Rev. A 89, 063825 (2014).
[Crossref]

S. B. Wang and C. T. Chan, “Lateral optical force on chiral particles near a surface,” Nat. Commun. 5, 3307 (2014).

M. H. M. Janssen and I. Powis, “Detecting chirality in molecules by imaging photoelectron circular dichroism,” Phys. Chem. Chem. Phys. 16, 856–871 (2014).
[Crossref]

D. Patterson and M. Schnell, “New studies on molecular chirality in the gas phase: enantiomer differentiation and determination of enantiomeric excess,” Phys. Chem. Chem. Phys. 16, 11114–11123 (2014).
[Crossref]

M. Okuno and T.-A. Ishibashi, “Chirality discriminated by heterodyne-detected vibrational sum frequency generation,” J. Phys. Chem. Lett. 5, 2874–2878 (2014).
[Crossref]

V. A. Shubert, D. Schmitz, D. Patterson, J. M. Doyle, and M. Schnell, “Identifying enantiomers in mixtures of chiral molecules with broadband microwave spectroscopy,” Angew. Chem. Int. Ed. 53, 1152–1155 (2014).
[Crossref]

R. P. Cameron, S. M. Barnett, and A. M. Yao, “Discriminatory optical force for chiral molecules,” New J. Phys. 16, 013020 (2014).
[Crossref]

R. P. Cameron, A. M. Yao, and S. M. Barnett, “Diffraction gratings for chiral molecules and their applications,” J. Phys. Chem. A 118, 3472–3478 (2014).
[Crossref]

R. P. Cameron and S. Barnett, “Optical activity in the scattering of structured light,” Phys. Chem. Chem. Phys. 16, 25819–25829 (2014).

A. Canaguier-Durand and C. Genet, “Chiral near fields generated from plasmonic optical lattices,” Phys. Rev. A 90, 023842 (2014).
[Crossref]

G. Tkachenko and E. Brasselet, “Optofluidic sorting of material chirality by chiral light,” Nat. Commun. 5, 3577 (2014).

D. H. Friese, M. T. P. Beerepoot, and K. Ruud, “Rotational averaging of multiphoton absorption cross sections,” J. Chem. Phys. 141, 204103 (2014).
[Crossref]

J. S. Ford and D. L. Andrews, “One- and two-photon absorption in solution: the effects of a passive auxiliary beam,” J. Chem. Phys. 141, 034504 (2014).
[Crossref]

M. G. Donato, J. Hernandez, A. Mazzulla, C. Provenzano, R. Saija, R. Sayed, S. Vasi, A. Magazzù, P. Pagliusi, R. Bartolino, P. G. Gucciardi, O. M. Maragò, and G. Cipparrone, “Polarization-dependent optomechanics mediated by chiral microresonators,” Nat. Commun. 5, 3656 (2014).
[Crossref]

G. Tkachenko and E. Brasselet, “Helicity-dependent three-dimensional optical trapping of chiral microparticles,” Nat. Commun. 5, 4491 (2014).

D. S. Bradshaw and D. L. Andrews, “Chiral discrimination in optical trapping and manipulation,” New J. Phys. 16, 103021 (2014).
[Crossref]

V. V. Klimov, I. V. Zabkov, A. A. Pavlov, and D. V. Guzatov, “Eigen oscillations of a chiral sphere and their influence on radiation of chiral molecules,” Opt. Express 22, 18564–18578 (2014).
[Crossref]

2013 (10)

Q.-C. Shang, Z.-S. Wu, T. Qu, Z.-J. Li, L. Bai, and L. Gong, “Analysis of the radiation force and torque exerted on a chiral sphere by a Gaussian beam,” Opt. Express 21, 8677–8688 (2013).
[Crossref]

Q.-C. Shang, Z.-S. Wu, T. Qu, Z.-J. Li, L. Bai, and L. Gong, “Analysis of rainbow scattering by a chiral sphere,” Opt. Express 21, 21879–21888 (2013).
[Crossref]

A. Eilam and M. Shapiro, “Spatial separation of dimers of chiral molecules,” Phys. Rev. Lett. 110, 213004 (2013).
[Crossref]

A. Canaguier-Durand, J. A. Hutchison, C. Genet, and T. W. Ebbesen, “Mechanical separation of chiral dipoles by chiral light,” New J. Phys. 15, 123037 (2013).
[Crossref]

R. J. Hernandez, A. Mazzulla, A. Pane, K. Volke-Sepulveda, and G. Cipparrone, “Attractive-repulsive dynamics on light-responsive chiral microparticles induced by polarized tweezers,” Lab Chip 13, 459–467 (2013).
[Crossref]

O. M. Marago, P. H. Jones, P. G. Gucciardi, G. Volpe, and A. C. Ferrari, “Optical trapping and manipulation of nanostructures,” Nat. Nanotechnol. 8, 807–819 (2013).
[Crossref]

R. W. Bowman and M. J. Padgett, “Optical trapping and binding,” Rep. Prog. Phys. 76, 026401 (2013).
[Crossref]

D. S. Bradshaw and D. L. Andrews, “Interparticle interactions: energy potentials, energy transfer, and nanoscale mechanical motion in response to optical radiation,” J. Phys. Chem. A 117, 75–82 (2013).
[Crossref]

D. Patterson, M. Schnell, and J. M. Doyle, “Enantiomer-specific detection of chiral molecules via microwave spectroscopy,” Nature 497, 475–477 (2013).
[Crossref]

D. Patterson and J. M. Doyle, “Sensitive chiral analysis via microwave three-wave mixing,” Phys. Rev. Lett. 111, 023008 (2013).
[Crossref]

2012 (4)

V. V. Klimov, D. V. Guzatov, and M. Ducloy, “Engineering of radiation of optically active molecules with chiral nano-meta-particles,” Europhys. Lett. 97, 47004 (2012).
[Crossref]

D. V. Guzatov and V. V. Klimov, “The influence of chiral spherical particles on the radiation of optically active molecules,” New J. Phys. 14, 123009 (2012).
[Crossref]

D. T. Butcher, S. Y. Buhmann, and S. Scheel, “Casimir–Polder forces between chiral objects,” New J. Phys. 14, 113013 (2012).
[Crossref]

Z.-S. Wu, Q.-C. Shang, and Z.-J. Li, “Calculation of electromagnetic scattering by a large chiral sphere,” Appl. Opt. 51, 6661–6668 (2012).
[Crossref]

2011 (5)

G. Cipparrone, A. Mazzulla, A. Pane, R. J. Hernandez, and R. Bartolino, “Chiral self-assembled solid microspheres: a novel multifunctional microphotonic device,” Adv. Mater. 23, 5773–5778 (2011).
[Crossref]

J. M. Leeder, D. S. Bradshaw, and D. L. Andrews, “Laser-controlled fluorescence in two-level systems,” J. Phys. Chem. B 115, 5227–5233 (2011).
[Crossref]

J. M. Leeder and D. L. Andrews, “A molecular theory for two-photon and three-photon fluorescence polarization,” J. Chem. Phys. 134, 094503 (2011).
[Crossref]

D. V. Guzatov and V. V. Klimov, “Chiral particles in a circularly polarised light field: new effects and applications,” Quantum Electron. 41, 526–533 (2011).
[Crossref]

S. N. A. Smith and D. L. Andrews, “Three-dimensional ensemble averages for tensorial interactions in partially oriented, multi-particle systems,” J. Phys. A 44, 395001 (2011).
[Crossref]

2010 (4)

X. Li and M. Shapiro, “Spatial separation of enantiomers by coherent optical means,” J. Chem. Phys. 132, 041101 (2010).
[Crossref]

X. Li and M. Shapiro, “Theory of the optical spatial separation of racemic mixtures of chiral molecules,” J. Chem. Phys. 132, 194315 (2010).
[Crossref]

W. Z. Jia and L. F. Wei, “Distinguishing left- and right-handed molecules using two-step coherent pulses,” J. Phys. B 43, 185402 (2010).
[Crossref]

T. Čižmár, L. C. Dávila Romero, K. Dholakia, and D. L. Andrews, “Multiple optical trapping and binding: new routes to self-assembly,” J. Phys. B 43, 102001 (2010).
[Crossref]

2008 (1)

M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophoton. 2, 021875 (2008).
[Crossref]

2007 (1)

Y. Li, C. Bruder, and C. P. Sun, “Generalized Stern–Gerlach effect for chiral molecules,” Phys. Rev. Lett. 99, 130403 (2007).
[Crossref]

2004 (1)

D. L. Andrews and D. S. Bradshaw, “Virtual photons, dipole fields and energy transfer: a quantum electrodynamical approach,” Eur. J. Phys. 25, 845–858 (2004).
[Crossref]

2003 (1)

P. Král, I. Thanopulos, M. Shapiro, and D. Cohen, “Two-step enantio-selective optical switch,” Phys. Rev. Lett. 90, 033001 (2003).
[Crossref]

2000 (1)

R. G. Woolley, “Gauge invariance in non-relativistic electrodynamics,” Proc. R. Soc. A 456, 1803–1819 (2000).
[Crossref]

1996 (1)

V. V. Klimov and V. S. Letokhov, “Gradient optical force on atoms: beyond dipole approximation,” Opt. Commun. 126, 45–48 (1996).
[Crossref]

1984 (1)

D. L. Andrews and M. J. Harlow, “Phased and Boltzmann-weighted rotational averages,” Phys. Rev. A 29, 2796–2806 (1984).
[Crossref]

1981 (1)

D. L. Andrews and W. A. Ghoul, “Eighth-rank isotropic tensors and rotational averages,” J. Phys. A 14, 1281–1290 (1981).
[Crossref]

1980 (1)

E. A. Power and T. Thirunamachandran, “The multipolar Hamiltonian in radiation theory,” Proc. R. Soc. A 372, 265–273 (1980).
[Crossref]

1978 (1)

E. A. Power and T. Thirunamachandran, “On the nature of Hamiltonian for interaction of radiation with atoms and molecules: (e/mc)p.A, -μ.E, and all that,” Am. J. Phys. 46, 370–378 (1978).
[Crossref]

1977 (1)

D. L. Andrews and T. Thirunamachandran, “On three-dimensional rotational averages,” J. Chem. Phys. 67, 5026–5033 (1977).
[Crossref]

Allcock, P.

D. L. Andrews and P. Allcock, Optical Harmonics in Molecular Systems (Wiley-VCH, 2002).

Andrews, D. L.

D. S. Bradshaw and D. L. Andrews, “Chiral discrimination in optical trapping and manipulation,” New J. Phys. 16, 103021 (2014).
[Crossref]

J. S. Ford and D. L. Andrews, “One- and two-photon absorption in solution: the effects of a passive auxiliary beam,” J. Chem. Phys. 141, 034504 (2014).
[Crossref]

D. S. Bradshaw and D. L. Andrews, “Interparticle interactions: energy potentials, energy transfer, and nanoscale mechanical motion in response to optical radiation,” J. Phys. Chem. A 117, 75–82 (2013).
[Crossref]

J. M. Leeder, D. S. Bradshaw, and D. L. Andrews, “Laser-controlled fluorescence in two-level systems,” J. Phys. Chem. B 115, 5227–5233 (2011).
[Crossref]

S. N. A. Smith and D. L. Andrews, “Three-dimensional ensemble averages for tensorial interactions in partially oriented, multi-particle systems,” J. Phys. A 44, 395001 (2011).
[Crossref]

J. M. Leeder and D. L. Andrews, “A molecular theory for two-photon and three-photon fluorescence polarization,” J. Chem. Phys. 134, 094503 (2011).
[Crossref]

T. Čižmár, L. C. Dávila Romero, K. Dholakia, and D. L. Andrews, “Multiple optical trapping and binding: new routes to self-assembly,” J. Phys. B 43, 102001 (2010).
[Crossref]

D. L. Andrews and D. S. Bradshaw, “Virtual photons, dipole fields and energy transfer: a quantum electrodynamical approach,” Eur. J. Phys. 25, 845–858 (2004).
[Crossref]

D. L. Andrews and M. J. Harlow, “Phased and Boltzmann-weighted rotational averages,” Phys. Rev. A 29, 2796–2806 (1984).
[Crossref]

D. L. Andrews and W. A. Ghoul, “Eighth-rank isotropic tensors and rotational averages,” J. Phys. A 14, 1281–1290 (1981).
[Crossref]

D. L. Andrews and T. Thirunamachandran, “On three-dimensional rotational averages,” J. Chem. Phys. 67, 5026–5033 (1977).
[Crossref]

D. L. Andrews and M. Babiker, The Angular Momentum of Light (Cambridge University, 2013).

D. L. Andrews and P. Allcock, Optical Harmonics in Molecular Systems (Wiley-VCH, 2002).

Atkins, P. W.

P. W. Atkins and J. De Paula, Atkins’ Physical Chemistry (Oxford University, 2010).

Babiker, M.

D. L. Andrews and M. Babiker, The Angular Momentum of Light (Cambridge University, 2013).

Bai, L.

Barnett, S.

R. P. Cameron and S. Barnett, “Optical activity in the scattering of structured light,” Phys. Chem. Chem. Phys. 16, 25819–25829 (2014).

Barnett, S. M.

R. P. Cameron, S. M. Barnett, and A. M. Yao, “Discriminatory optical force for chiral molecules,” New J. Phys. 16, 013020 (2014).
[Crossref]

R. P. Cameron, A. M. Yao, and S. M. Barnett, “Diffraction gratings for chiral molecules and their applications,” J. Phys. Chem. A 118, 3472–3478 (2014).
[Crossref]

Bartolino, R.

M. G. Donato, J. Hernandez, A. Mazzulla, C. Provenzano, R. Saija, R. Sayed, S. Vasi, A. Magazzù, P. Pagliusi, R. Bartolino, P. G. Gucciardi, O. M. Maragò, and G. Cipparrone, “Polarization-dependent optomechanics mediated by chiral microresonators,” Nat. Commun. 5, 3656 (2014).
[Crossref]

G. Cipparrone, A. Mazzulla, A. Pane, R. J. Hernandez, and R. Bartolino, “Chiral self-assembled solid microspheres: a novel multifunctional microphotonic device,” Adv. Mater. 23, 5773–5778 (2011).
[Crossref]

Beerepoot, M. T. P.

D. H. Friese, M. T. P. Beerepoot, and K. Ruud, “Rotational averaging of multiphoton absorption cross sections,” J. Chem. Phys. 141, 204103 (2014).
[Crossref]

Bowman, R. W.

R. W. Bowman and M. J. Padgett, “Optical trapping and binding,” Rep. Prog. Phys. 76, 026401 (2013).
[Crossref]

Bradshaw, D. S.

D. S. Bradshaw and D. L. Andrews, “Chiral discrimination in optical trapping and manipulation,” New J. Phys. 16, 103021 (2014).
[Crossref]

D. S. Bradshaw and D. L. Andrews, “Interparticle interactions: energy potentials, energy transfer, and nanoscale mechanical motion in response to optical radiation,” J. Phys. Chem. A 117, 75–82 (2013).
[Crossref]

J. M. Leeder, D. S. Bradshaw, and D. L. Andrews, “Laser-controlled fluorescence in two-level systems,” J. Phys. Chem. B 115, 5227–5233 (2011).
[Crossref]

D. L. Andrews and D. S. Bradshaw, “Virtual photons, dipole fields and energy transfer: a quantum electrodynamical approach,” Eur. J. Phys. 25, 845–858 (2004).
[Crossref]

Brasselet, E.

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Figures (1)

Fig. 1.
Fig. 1. Diagram depicting the relative positions, within a circularly polarized trapping beam of Gaussian profile, of two enantiomers: ( L )-hexahelicene (yellow) and ( R )-hexahelicene (green) in a solution. Both enantiomers are attracted to the center of the Gaussian beam, although, as denoted by the differential chiral force Δ F (the optical force acting on the left-handed enantiomer compared with the right-handed one), a greater tendency arises for left-handed molecules to be positioned toward the beam center and right-handed enantiomers toward the periphery. Black wavy arrows denote the direction of the throughput light beam.

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Equations (24)

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M F I ≡ 〈 F | M | I 〉 = 〈 F | ∑ p = 0 ∞ H int ( T 0 H int ) p | I 〉 ,
U = Re   M I I ≃ Re   { ∑ R 〈 I | H int | R 〉 〈 R | H int | I 〉 E I − E R } ,
H int = − ε 0 − 1 μ i d i ⊥ − ε 0 − 1 Q i j ∇ j d i ⊥ − m i b i + … ,
U ( r ) = U E 1 − E 1 ( r ) + U E 1 − E 2 ( r ) + U E 1 − M 1 ( r ) ,
U E 1 − E 1 ( L | R ) ( r ) = Re { − ( I ( r ) 2 ε 0 c ) e ¯ i ( L | R ) e j ( L | R ) α i j } ,
U E 1 − E 2 ( L | R ) ( r ) = Re { − ( i k k I ( r ) 2 ε 0 c ) e ¯ i ( L | R ) e j ( L | R ) ( A i j k − A j i k ) } ,
U E 1 − M 1 ( L | R ) ( r ) = Re { − ( I ( r ) 2 ε 0 c 2 ) ( e ¯ i ( L | R ) b j ( L | R ) G i j + b ¯ i ( L | R ) e j ( L | R ) G ¯ j i ) } .
A i j k = ∑ s { μ i 0 s Q j k s 0 E s 0 − ℏ ω + Q j k 0 s μ i s 0 E s 0 + ℏ ω } ,
G i j = ∑ s { μ i 0 s m j s 0 E s 0 − ℏ ω + m j 0 s μ i s 0 E s 0 + ℏ ω } .
U ( L − R ) ( r ) = i ( I ( r ) 2 ε 0 c 2 ) ( e ¯ i ( L ) e j ( L ) + e ¯ i ( R ) e j ( R ) ) ( G i j − G ¯ j i ) ,
U ( L − R ) ( r ) = − ( I ( r ) ε 0 c 2 ) ( δ i j − k ^ i k ^ j ) G i j ′ ,
〈 U ( L − R ) 〉 ≡ − ( I ε 0 c 2 ) 〈 A : B   exp ( C : D ) 〉 〈 exp ( C : D ) 〉 ,
〈 U ( L − R ) 〉 = − ( I ε 0 c 2 ) { 〈 A ( 0 ) : B ( 0 ) 〉 + ∑ n = 0 3 1 n ! 〈 ( A ( 2 ) : B ( 2 ) ) ( C ( 2 ) : D ( 2 ) ) n 〉 ∑ m = n + 1 4 1 m ! 〈 ( C ( 2 ) : D ( 2 ) ) m 〉 } ,
〈 A ( 0 ) : B ( 0 ) 〉 = 2 G ,
〈 A ( 2 ) : B ( 2 ) : C ( 2 ) : D ( 2 ) 〉 = β 5 ( G α − G α ) ,
1 2 〈 C ( 2 ) : D ( 2 ) : C ( 2 ) : D ( 2 ) 〉 = β 10 ( α α − α 2 ) ,
1 2 〈 A ( 2 ) : B ( 2 ) : C ( 2 ) : D ( 2 ) : C ( 2 ) : D ( 2 ) 〉 = − β 2 35 ( G α α + G ( 2 α 2 − α α ) − 2     G α α ) ,
1 6 〈 C ( 2 ) : D ( 2 ) : C ( 2 ) : D ( 2 ) : C ( 2 ) : D ( 2 ) 〉 = − β 2 210 ( α α α − 3 α α α + 2 α 3 ) ,
1 6 〈 A ( 2 ) : B ( 2 ) : C ( 2 ) : D ( 2 ) : C ( 2 ) : D ( 2 ) : C ( 2 ) : D ( 2 ) 〉 = β 3 7250 ( G α ( 39 α α − 9 α 2 ) + 9 G α ( α 2 − α α ) + 10 G α α α − 10 G α α α ) ,
1 24 〈 C ( 2 ) : D ( 2 ) : C ( 2 ) : D ( 2 ) : C ( 2 ) : D ( 2 ) : C ( 2 ) : D ( 2 ) 〉 = β 3 58000 ( 3 α α ( 13 α α − 6 α 2 ) + 10 α α α α + 40 α α α α + 9 α 4 ) .
〈 U ( L − R ) 〉 = − ( I ε 0 c 2 ) { 2 G + 2 β [ G α − G α − β 21 ( G α α + 3 G × ( 2 α 2 − α α ) − 6     G α α ) + β 2 1450 ( 3     G α ( 13 α α − 3 α 2 ) + 9 G α × ( α 2 − α α ) + 10 ( G α α α − 3     G α α α − 2 G α α α ) ) ] × ( 10 + β 2 [ α α − α 2 − β 21 ( α α α + α ( 2 α 2 − 3 α α ) ) + β 2 58000 ( 3 α α ( 13 α α − 6 α 2 ) + 10 α α α α − 40 α α α α + 9 α 4 ) ] ) − 1 } .
〈 U ( L − R ) 〉 = − ( I g ε 0 c 2 ) { 1 + β ( 2 a + 4 7 β a 2 + 63 725 β 2 a 3 ) × ( 5 + β 2 ( a 2 − 1 21 β a 3 + 27 14500 β 2 a 4 ) ) − 1 } ,
〈 U ( L − R ) 〉 = − ( 80 I ε 0 c 2 ) { 3 G α ( 13 α α − 3 α 2 ) + 9 G α ( α 2 − α α ) + 10 ( G α α α − 3     G α α α − 2 G α α α ) } ( 3 α α ( 13 α α − 6 α 2 ) + 10 α α α α − 40 α α α α + 9 α 4 ) − 1 ,
〈 U ( L − R ) 〉 = − ( 140 I 3 ε 0 c 2 ) ( g β a ) ,

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