Abstract

We put forward a theory on the optical force exerted upon a dipolar particle by a stationary and ergodic partially coherent light field. We show through a rigorous analysis that the ensemble averaged electromagnetic force is given in terms of a partial gradient of the space-variable diagonal elements of the coherence tensor. Further, by following this result we characterize the conservative and nonconservative components of this force. In addition, we establish the propagation law for the optical force in terms of the coherence function of light at a diffraction plane. This permits us to evaluate the effect of the degree of coherence on the force components by using the archetypical configuration of Young’s two-aperture diffraction pattern, so often employed to characterize coherence of waves.

© 2012 Optical Society of America

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    [CrossRef]
  2. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986).
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    [CrossRef]
  5. M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express 12, 5375–5401 (2004).
    [CrossRef]
  6. M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field in magnetic media,” Opt. Express 15, 13502–13518 (2007).
    [CrossRef]
  7. B. Kemp, T. Grzegorczyk, and J. Kong, “Ab initio study of the radiation pressure on dielectric and magnetic media,” Opt. Express 13, 9280–9291 (2005).
    [CrossRef]
  8. S. M. Kim and G. Gbur, “Momentum conservation in partially coherent wave fields,” Phys. Rev. A 79, 033844 (2009).
    [CrossRef]
  9. W. Wang and M. Takeda, “Linear and angular coherence momenta in the classical second-order coherence theory of vector electromagnetic fields,” Opt. Lett. 31, 2520–2522 (2006).
    [CrossRef]
  10. P. S. Carney, E. Wolf, and G. S. Agarwal, “Statistical generalizations of the optical cross-section theorem with application to inverse scattering,” J. Opt. Soc. Am. A 14, 3366–3371(1997).
    [CrossRef]
  11. P. S. Carney, E. Wolf, and G. S. Agarwal, “Diffraction tomography using power extinction measurements,” J. Opt. Soc. Am. A 16, 2643–2648 (1999).
    [CrossRef]
  12. N. Garcia and M. Nieto-Vesperinas, “Near field inverse scattering reconstruction of reflective surfaces,” Opt. Lett. 18, 2090–2092 (1993).
    [CrossRef]
  13. N. Garcia and M. Nieto-Vesperinas, “Direct solution to the inverse scattering problem without phase retrieval,” Opt. Lett. 20, 949–951 (1995).
    [CrossRef]
  14. N. Garcia, V. Celli, and M. Nieto-Vesperinas, “Exact multiple scattering of waves from random rough surfaces,” Opt. Commun. 30, 279–281 (1979).
    [CrossRef]
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    [CrossRef]
  16. C. Zha, Y. Cai, X. Lu, and H. T. Eyyuboğlu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express 17, 1753–1765 (2009).
    [CrossRef]
  17. Ch. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams,” Opt. Express 17, 21472–21487 (2009).
    [CrossRef]
  18. M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, “Near-field photonic forces,” Phil. Trans. R. Soc. A 362, 719–737 (2004).
    [CrossRef]
  19. K. Dholakia and P. Zemánek, “Colloquium: gripped by light: optical binding,” Rev. Mod. Phys. 82, 1767–1791 (2010).
    [CrossRef]
  20. A. S. Zelenina, R. Quidant, and M. Nieto-Vesperinas, “Enhanced optical forces between coupled resonant metal nanoparticles,” Opt. Lett. 32, 1156–1158 (2007).
    [CrossRef]
  21. M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nature 3, 477–480 (2007).
  22. S. Albaladejo, M. I. Marque, M. Laroche, and J. J. Sáenz, “Scattering forces from the curl of the spin angular momentum of a light field,” Phys. Rev. Lett. 102, 113602 (2009).
    [CrossRef]
  23. P. C. Chaumet and M. Nieto-Vesperinas, “Electromagnetic force on a metallic particle in the presence of a dielectric surface,” Phys. Rev. B 62, 11185–11191 (2000).
    [CrossRef]
  24. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  25. J. Perina, Coherence of Light (Springer-Verlag, 1985).
  26. D. F. V. James and E. Wolf, “Some new aspects of young interference experiment,” Phys. Lett. 157, 6–10 (1991).
    [CrossRef]
  27. D. F. V. James and E. Wolf, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59, 771–818 (1996).
    [CrossRef]
  28. A. Dogariu and E. Wolf, “Coherence theory of pairs of correlated wave fields,” J. Mod. Opt. 50, 1791–1796 (2003).
    [CrossRef]
  29. B. J. Thompson and E. Wolf, “Two-beam interference with partially coherent light,” J. Opt. Soc. Am. A 47, 895 (1957).
    [CrossRef]
  30. M. Nieto-Vesperinas, J. J. Sáenz, R. Gómez-Medina, and L. Chantada, “Optical forces on small magnetodielectric particles,” Opt. Express 18, 11428–11443 (2010).
    [CrossRef]
  31. A. García-Etxarri, R. Gómez-Medina,, L. S. Froufe-Pérez, C. López, L. Chantada, F. Scheffold, J. Aizpurua, M. Nieto-Vesperinas, and J. J. Sáenz, “Strong magnetic response of submicron silicon particles in the infrared,” Opt. Express 19, 4815–4826 (2011).
    [CrossRef]
  32. M. Nieto-Vesperinas, R. Gomez-Medina, and J. J. Sáenz, “Angle-suppressed scattering and optical forces on submicrometer dielectric particles,” J. Opt. Soc. Am. A 28, 54–60 (2011).
    [CrossRef]
  33. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).
  34. It is well known that the Eq. (23) expression is one of the “smoothing” alternatives to estimate the cross-spectral density of the random process E(r,t) considered as truncated in time beyond |t|=T [24,25,33]. Another way is to write [37] E˜jk(r,r′,ω)=limΔω→0∫ω−Δω/2ω+Δω/2<E˜j*(r,ω)E˜k(r′,ω′)>dω′.
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    [CrossRef]
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  38. J. W. Goodmam, Introduction to Fourier Optics (McGraw-Hill, 1996).
  39. M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (World Science, 2006).
  40. J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
    [CrossRef]
  41. J. R. Arias-González and M. Nieto-Vesperinas, “Optical forces on small particles: attractive and repulsive nature and plasmon-resonance conditions,” J. Opt. Soc. Am. A 20, 1201–1209 (2003).
    [CrossRef]
  42. S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, 1989), Vol. 3.
  43. J. Ripoll, V. Ntziachristos, J. P. Culver, D. N. Pattanayak, A. G. Yodh, and M. Nieto-Vesperinas, “Recovery of optical parameters in multiple-layered diffusive media: theory and experiments,” J. Opt. Soc. Am. A 18, 821–830 (2001).
    [CrossRef]
  44. J. Riley, H. Dehghani, M. Schweiger, S. Arridge, J. Ripoll, and M. Nieto-Vesperinas, “3D optical tomography in the presence of void regions,” Opt. Express 7, 462–467 (2000).
    [CrossRef]
  45. A. García-Martin, J. A. Torres, J. J. Sáenz, and M. Nieto-Vesperinas, “Transition from diffusive to localized regimes in surface corrugated optical waveguides,” Appl. Phys. Lett. 71, 1912–1914 (1997).
    [CrossRef]

2011 (3)

2010 (2)

2009 (4)

S. Albaladejo, M. I. Marque, M. Laroche, and J. J. Sáenz, “Scattering forces from the curl of the spin angular momentum of a light field,” Phys. Rev. Lett. 102, 113602 (2009).
[CrossRef]

S. M. Kim and G. Gbur, “Momentum conservation in partially coherent wave fields,” Phys. Rev. A 79, 033844 (2009).
[CrossRef]

C. Zha, Y. Cai, X. Lu, and H. T. Eyyuboğlu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express 17, 1753–1765 (2009).
[CrossRef]

Ch. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams,” Opt. Express 17, 21472–21487 (2009).
[CrossRef]

2007 (4)

2006 (1)

2005 (1)

2004 (2)

M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express 12, 5375–5401 (2004).
[CrossRef]

M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, “Near-field photonic forces,” Phil. Trans. R. Soc. A 362, 719–737 (2004).
[CrossRef]

2003 (2)

2002 (1)

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
[CrossRef]

2001 (1)

2000 (3)

1999 (1)

1997 (2)

P. S. Carney, E. Wolf, and G. S. Agarwal, “Statistical generalizations of the optical cross-section theorem with application to inverse scattering,” J. Opt. Soc. Am. A 14, 3366–3371(1997).
[CrossRef]

A. García-Martin, J. A. Torres, J. J. Sáenz, and M. Nieto-Vesperinas, “Transition from diffusive to localized regimes in surface corrugated optical waveguides,” Appl. Phys. Lett. 71, 1912–1914 (1997).
[CrossRef]

1996 (1)

D. F. V. James and E. Wolf, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59, 771–818 (1996).
[CrossRef]

1995 (1)

1993 (1)

1991 (1)

D. F. V. James and E. Wolf, “Some new aspects of young interference experiment,” Phys. Lett. 157, 6–10 (1991).
[CrossRef]

1986 (1)

1982 (1)

1979 (1)

N. Garcia, V. Celli, and M. Nieto-Vesperinas, “Exact multiple scattering of waves from random rough surfaces,” Opt. Commun. 30, 279–281 (1979).
[CrossRef]

1970 (1)

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
[CrossRef]

1957 (1)

B. J. Thompson and E. Wolf, “Two-beam interference with partially coherent light,” J. Opt. Soc. Am. A 47, 895 (1957).
[CrossRef]

Agarwal, G. S.

Aizpurua, J.

Albaladejo, S.

S. Albaladejo, M. I. Marque, M. Laroche, and J. J. Sáenz, “Scattering forces from the curl of the spin angular momentum of a light field,” Phys. Rev. Lett. 102, 113602 (2009).
[CrossRef]

Arias-González, J. R.

Arridge, S.

Ashkin, A.

Bjorkholm, J. E.

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).

Cai, Y.

Carney, P. S.

Celli, V.

N. Garcia, V. Celli, and M. Nieto-Vesperinas, “Exact multiple scattering of waves from random rough surfaces,” Opt. Commun. 30, 279–281 (1979).
[CrossRef]

Chantada, L.

Chaumet, P. C.

M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, “Near-field photonic forces,” Phil. Trans. R. Soc. A 362, 719–737 (2004).
[CrossRef]

P. C. Chaumet and M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagnetic field,” Opt. Lett. 25, 1065–1067 (2000).
[CrossRef]

P. C. Chaumet and M. Nieto-Vesperinas, “Electromagnetic force on a metallic particle in the presence of a dielectric surface,” Phys. Rev. B 62, 11185–11191 (2000).
[CrossRef]

Chu, S.

Culver, J. P.

Curtis, J. E.

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
[CrossRef]

Dehghani, H.

Deych, L.

Dholakia, K.

K. Dholakia and P. Zemánek, “Colloquium: gripped by light: optical binding,” Rev. Mod. Phys. 82, 1767–1791 (2010).
[CrossRef]

Dogariu, A.

A. Dogariu and E. Wolf, “Coherence theory of pairs of correlated wave fields,” J. Mod. Opt. 50, 1791–1796 (2003).
[CrossRef]

Dziedzic, J. M.

Eyyuboglu, H. T.

Froufe-Pérez, L. S.

Garcia, N.

García-Etxarri, A.

García-Martin, A.

A. García-Martin, J. A. Torres, J. J. Sáenz, and M. Nieto-Vesperinas, “Transition from diffusive to localized regimes in surface corrugated optical waveguides,” Appl. Phys. Lett. 71, 1912–1914 (1997).
[CrossRef]

Gbur, G.

S. M. Kim and G. Gbur, “Momentum conservation in partially coherent wave fields,” Phys. Rev. A 79, 033844 (2009).
[CrossRef]

Girard, C.

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nature 3, 477–480 (2007).

Gomez-Medina, R.

Gómez-Medina, R.

Gómez-Medina,, R.

Goodmam, J. W.

J. W. Goodmam, Introduction to Fourier Optics (McGraw-Hill, 1996).

Grier, D. G.

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
[CrossRef]

Grzegorczyk, T.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

James, D. F. V.

D. F. V. James and E. Wolf, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59, 771–818 (1996).
[CrossRef]

D. F. V. James and E. Wolf, “Some new aspects of young interference experiment,” Phys. Lett. 157, 6–10 (1991).
[CrossRef]

Kemp, B.

Kim, S. M.

S. M. Kim and G. Gbur, “Momentum conservation in partially coherent wave fields,” Phys. Rev. A 79, 033844 (2009).
[CrossRef]

Kong, J.

Korotkova, O.

Koss, B. A.

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
[CrossRef]

Kravtsov, Y. A.

S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, 1989), Vol. 3.

Laroche, M.

S. Albaladejo, M. I. Marque, M. Laroche, and J. J. Sáenz, “Scattering forces from the curl of the spin angular momentum of a light field,” Phys. Rev. Lett. 102, 113602 (2009).
[CrossRef]

López, C.

Lu, X.

Lu, X. H.

Mandel, L.

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

Mansuripur, M.

Marque, M. I.

S. Albaladejo, M. I. Marque, M. Laroche, and J. J. Sáenz, “Scattering forces from the curl of the spin angular momentum of a light field,” Phys. Rev. Lett. 102, 113602 (2009).
[CrossRef]

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, R. Gomez-Medina, and J. J. Sáenz, “Angle-suppressed scattering and optical forces on submicrometer dielectric particles,” J. Opt. Soc. Am. A 28, 54–60 (2011).
[CrossRef]

A. García-Etxarri, R. Gómez-Medina,, L. S. Froufe-Pérez, C. López, L. Chantada, F. Scheffold, J. Aizpurua, M. Nieto-Vesperinas, and J. J. Sáenz, “Strong magnetic response of submicron silicon particles in the infrared,” Opt. Express 19, 4815–4826 (2011).
[CrossRef]

M. Nieto-Vesperinas, J. J. Sáenz, R. Gómez-Medina, and L. Chantada, “Optical forces on small magnetodielectric particles,” Opt. Express 18, 11428–11443 (2010).
[CrossRef]

A. S. Zelenina, R. Quidant, and M. Nieto-Vesperinas, “Enhanced optical forces between coupled resonant metal nanoparticles,” Opt. Lett. 32, 1156–1158 (2007).
[CrossRef]

M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, “Near-field photonic forces,” Phil. Trans. R. Soc. A 362, 719–737 (2004).
[CrossRef]

J. R. Arias-González and M. Nieto-Vesperinas, “Optical forces on small particles: attractive and repulsive nature and plasmon-resonance conditions,” J. Opt. Soc. Am. A 20, 1201–1209 (2003).
[CrossRef]

J. Ripoll, V. Ntziachristos, J. P. Culver, D. N. Pattanayak, A. G. Yodh, and M. Nieto-Vesperinas, “Recovery of optical parameters in multiple-layered diffusive media: theory and experiments,” J. Opt. Soc. Am. A 18, 821–830 (2001).
[CrossRef]

P. C. Chaumet and M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagnetic field,” Opt. Lett. 25, 1065–1067 (2000).
[CrossRef]

P. C. Chaumet and M. Nieto-Vesperinas, “Electromagnetic force on a metallic particle in the presence of a dielectric surface,” Phys. Rev. B 62, 11185–11191 (2000).
[CrossRef]

J. Riley, H. Dehghani, M. Schweiger, S. Arridge, J. Ripoll, and M. Nieto-Vesperinas, “3D optical tomography in the presence of void regions,” Opt. Express 7, 462–467 (2000).
[CrossRef]

A. García-Martin, J. A. Torres, J. J. Sáenz, and M. Nieto-Vesperinas, “Transition from diffusive to localized regimes in surface corrugated optical waveguides,” Appl. Phys. Lett. 71, 1912–1914 (1997).
[CrossRef]

N. Garcia and M. Nieto-Vesperinas, “Direct solution to the inverse scattering problem without phase retrieval,” Opt. Lett. 20, 949–951 (1995).
[CrossRef]

N. Garcia and M. Nieto-Vesperinas, “Near field inverse scattering reconstruction of reflective surfaces,” Opt. Lett. 18, 2090–2092 (1993).
[CrossRef]

N. Garcia, V. Celli, and M. Nieto-Vesperinas, “Exact multiple scattering of waves from random rough surfaces,” Opt. Commun. 30, 279–281 (1979).
[CrossRef]

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (World Science, 2006).

Ntziachristos, V.

Pattanayak, D. N.

Perina, J.

J. Perina, Coherence of Light (Springer-Verlag, 1985).

Quidant, R.

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nature 3, 477–480 (2007).

A. S. Zelenina, R. Quidant, and M. Nieto-Vesperinas, “Enhanced optical forces between coupled resonant metal nanoparticles,” Opt. Lett. 32, 1156–1158 (2007).
[CrossRef]

Rahmani, A.

M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, “Near-field photonic forces,” Phil. Trans. R. Soc. A 362, 719–737 (2004).
[CrossRef]

Righini, M.

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nature 3, 477–480 (2007).

Riley, J.

Ripoll, J.

Rubin, J. T.

Rytov, S. M.

S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, 1989), Vol. 3.

Sáenz, J. J.

Scheffold, F.

Schweiger, M.

Takeda, M.

Tatarskii, V. I.

S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, 1989), Vol. 3.

Thompson, B. J.

B. J. Thompson and E. Wolf, “Two-beam interference with partially coherent light,” J. Opt. Soc. Am. A 47, 895 (1957).
[CrossRef]

Torres, J. A.

A. García-Martin, J. A. Torres, J. J. Sáenz, and M. Nieto-Vesperinas, “Transition from diffusive to localized regimes in surface corrugated optical waveguides,” Appl. Phys. Lett. 71, 1912–1914 (1997).
[CrossRef]

Wang, L. G.

Wang, L. Q.

Wang, W.

Wolf, E.

A. Dogariu and E. Wolf, “Coherence theory of pairs of correlated wave fields,” J. Mod. Opt. 50, 1791–1796 (2003).
[CrossRef]

P. S. Carney, E. Wolf, and G. S. Agarwal, “Diffraction tomography using power extinction measurements,” J. Opt. Soc. Am. A 16, 2643–2648 (1999).
[CrossRef]

P. S. Carney, E. Wolf, and G. S. Agarwal, “Statistical generalizations of the optical cross-section theorem with application to inverse scattering,” J. Opt. Soc. Am. A 14, 3366–3371(1997).
[CrossRef]

D. F. V. James and E. Wolf, “Correlation-induced spectral changes,” Rep. Prog. Phys. 59, 771–818 (1996).
[CrossRef]

D. F. V. James and E. Wolf, “Some new aspects of young interference experiment,” Phys. Lett. 157, 6–10 (1991).
[CrossRef]

B. J. Thompson and E. Wolf, “Two-beam interference with partially coherent light,” J. Opt. Soc. Am. A 47, 895 (1957).
[CrossRef]

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

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).

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

Yodh, A. G.

Zelenina, A. S.

A. S. Zelenina, R. Quidant, and M. Nieto-Vesperinas, “Enhanced optical forces between coupled resonant metal nanoparticles,” Opt. Lett. 32, 1156–1158 (2007).
[CrossRef]

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nature 3, 477–480 (2007).

Zemánek, P.

K. Dholakia and P. Zemánek, “Colloquium: gripped by light: optical binding,” Rev. Mod. Phys. 82, 1767–1791 (2010).
[CrossRef]

Zha, C.

Zhao, C. L.

Zhao, Ch.

Zhu, S. Y.

Appl. Phys. Lett. (1)

A. García-Martin, J. A. Torres, J. J. Sáenz, and M. Nieto-Vesperinas, “Transition from diffusive to localized regimes in surface corrugated optical waveguides,” Appl. Phys. Lett. 71, 1912–1914 (1997).
[CrossRef]

J. Mod. Opt. (1)

A. Dogariu and E. Wolf, “Coherence theory of pairs of correlated wave fields,” J. Mod. Opt. 50, 1791–1796 (2003).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

Nature (1)

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nature 3, 477–480 (2007).

Opt. Commun. (2)

N. Garcia, V. Celli, and M. Nieto-Vesperinas, “Exact multiple scattering of waves from random rough surfaces,” Opt. Commun. 30, 279–281 (1979).
[CrossRef]

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
[CrossRef]

Opt. Express (9)

M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express 12, 5375–5401 (2004).
[CrossRef]

B. Kemp, T. Grzegorczyk, and J. Kong, “Ab initio study of the radiation pressure on dielectric and magnetic media,” Opt. Express 13, 9280–9291 (2005).
[CrossRef]

M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field in magnetic media,” Opt. Express 15, 13502–13518 (2007).
[CrossRef]

C. Zha, Y. Cai, X. Lu, and H. T. Eyyuboğlu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express 17, 1753–1765 (2009).
[CrossRef]

Ch. Zhao, Y. Cai, and O. Korotkova, “Radiation force of scalar and electromagnetic twisted Gaussian Schell-model beams,” Opt. Express 17, 21472–21487 (2009).
[CrossRef]

M. Nieto-Vesperinas, J. J. Sáenz, R. Gómez-Medina, and L. Chantada, “Optical forces on small magnetodielectric particles,” Opt. Express 18, 11428–11443 (2010).
[CrossRef]

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J. T. Rubin and L. Deych, “On optical forces in spherical whispering gallery mode resonators,” Opt. Express 19, 22337–22349 (2011).
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J. Riley, H. Dehghani, M. Schweiger, S. Arridge, J. Ripoll, and M. Nieto-Vesperinas, “3D optical tomography in the presence of void regions,” Opt. Express 7, 462–467 (2000).
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M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, “Near-field photonic forces,” Phil. Trans. R. Soc. A 362, 719–737 (2004).
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M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics (World Science, 2006).

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It is well known that the Eq. (23) expression is one of the “smoothing” alternatives to estimate the cross-spectral density of the random process E(r,t) considered as truncated in time beyond |t|=T [24,25,33]. Another way is to write [37] E˜jk(r,r′,ω)=limΔω→0∫ω−Δω/2ω+Δω/2<E˜j*(r,ω)E˜k(r′,ω′)>dω′.

S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, 1989), Vol. 3.

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

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

J. Perina, Coherence of Light (Springer-Verlag, 1985).

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Figures (5)

Fig. 1.
Fig. 1.

Schematics of the configuration for observing interference at points P=(x,y,d) of a screen B by diffraction of light, propagated from an incoherent source, at two apertures in A, centered at points r1=(q1,0) and r2=(q2,0), respectively. q1=(x1,y1), q2=(x2,y2). 2h=|q1q2|.

Fig. 2.
Fig. 2.

Spatial distributions in the XY plane of the mean intensity and the normalized mean gradient force components for |μ(q1,q2,ω¯)|=1. (a) Normalized mean intensity I. (b) F˜xgrad. (c) F˜ygrad. (d) F˜zgrad. All values are calculated on a dipolar particle at the screen plane B, placed at distance z=d=1.5m from the aperture screen A. The force components are normalized to αe and to the magnitude of the total mean force |F˜tot|=|F˜grad+F˜sc|.

Fig. 3.
Fig. 3.

(top) Normalized mean intensity I. (bottom) Normalized mean gradient force component F˜ygrad for different values of |μ(q1,q2,ω¯)|.

Fig. 4.
Fig. 4.

Spatial distribution in the XY plane of the normalized averaged scattering force components for |μ(q1,q2,ω¯)|=1. The normalization factor is αe|F˜tot|. (a) Fxsc. (b) Fysc. (c) Fzsc. All values are calculated on a dipolar particle in the screen plane B, placed at distance z=d=1.5m from the aperture screen A.

Fig. 5.
Fig. 5.

Spatial distribution, in pN, of the averaged total force Cartesian components on a dielectric particle with r0=25nm and εp=2.25, in the screen plane B, placed at distance z=d=1.5m from the aperture mask A, |μ(q1,q2,ω¯)|=1. No normalization is done. (a) Fxtot. (b) Fytot. (c) Fztot.

Equations (60)

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E(r)(r,t)=E˜(r)(r,ω)eiωtdω,
B(r)(r,t)=B˜(r)(r,ω)eiωtdω.
E(r,t)=E˜(r,ω)eiωtdω,
B(r,t)=B˜(r,ω)eiωtdω,
E˜(r,ω)=E˜(r)(r,ω),ω0=0,ω<0,
B˜(r,ω)=B˜(r)(r,ω),ω0=0,ω<0,
E(r,t)=12[E(r)(r,t)+iE(i)(r,t)],
B(r,t)=12[B(r)(r,t)+iB(i)(r,t)].
F(r,t)=(p(r)(r,t)·)E(r)(r,t)+1cp(r)(r,t)t×B(r)(r,t),
p(r,t)=αeE(r,t).
pj(r)jEi(r)=(pj+pj*)j(Ei+Ei*)=LimT12TTTdt[ei(ω1+ω2)tp˜j(r,ω1)jE˜i(r,ω2)dω1dω2+ei(ω1ω2)tpj˜(r,ω1)jEi˜*(r,ω2)dω1dω2+ei(ω1+ω2)tpj˜*(r,ω1)jEi˜(r,ω2)dω1dω2+e+i(ω1+ω2)tpj˜*(r,ω1)jEi˜*(r,ω2)dω1dω2],
pj(r)jEi(r)=2π[gi(p,E)(r,ω1,ω2)δ(ω1+ω2)dω1dω2+gi(p,E*)(r,ω1,ω2)δ(ω1ω2)dω1dω2+gi(p*,E)(r,ω1,ω2)δ(ω1ω2)dω1dω2+gi(p*,E*)(r,ω1,ω2)δ(ω1+ω2)dω1dω2],
gi(U,V)(r,ω1,ω2)=LimT12TU˜j(r,ω1)jV˜i(r,ω2),
gi(U,V)(r,ω1,ω1)=0,
pj(r)jEi(r)=4πLimT12Tpj˜(r,ω1)jEi˜*(r,ω1)dω1=4πgi(p*,E)(r,ω1,ω1)dω1,
1cϵijkp˙j(r)Bk(r)=1cϵijk(p˙j+p˙j*)(Bk+Bk*).
1cϵijk(p˙j+p˙j*)(Bk+Bk*)=ϵijkcLimT12TTTdt[(iω1)ei(ω1+ω2)tp˜j(r,ω1)B˜k(r,ω2)dω1dω2+(iω1)ei(ω1ω2)tpj˜(r,ω1)Bk˜*(r,ω2)dω1dω2+iω1ei(ω1+ω2)tpj˜*(r,ω1)Bk˜(r,ω2)dω1dω2+iω1e+i(ω1+ω2)tpj˜*(r,ω1)Bk˜*(r,ω2)dω1dω2].
1cϵijk(p˙j+p˙j*)(Bk+Bk*)=2πcϵijk[(iω1)Wjk(p,B)(r,ω1,ω2)δ(ω1+ω2)dω1dω2+(iω1)Wjk(p,B*)(r,ω1,ω2)δ(ω1ω2)dω1dω2+iω1Wjk(p*,B)(r,ω1,ω2)δ(ω1ω2)dω1dω2+iω1Wjk(p*,B*)(r,ω1,ω2)δ(ω1+ω2)dω1dω2],
Wjk(U,V)(r,ω1,ω2)=LimT12TU˜j(r,ω1)V˜k(r,ω2).
Wjk(U,V)(r,ω1,ω1)=0,
1cϵijkp˙j(r)Bk(r)=4πLimT12T[pj˜(r,ω1)iEj˜*(r,ω1)pj˜(r,ω1)jEi˜*(r,ω1)]dω1.
Ejk(r,r,τ)=E˜jk(r,r,ω)eiωτdω,
E˜jk(r,r,ω)=LimT12TE˜j*(r,ω)E˜k(r,ω).
Fi(r,t)=4πLimT12Tpj˜(r,ω)iEj˜*(r,ω)dω=4παei(*)LimT12TEj˜(r,ω)Ej˜*(r,ω)dω=4παei(*)TrE˜jk(r,r,ω)dω,
Fi(r,t)=4π{pj(r,t)iEj*(r,t)}=4π{αei(*)Ej(r,t)Ej*(r,t)}=4π{αei(*)TrEjk(r,r,0)}.
Fi(r,t)=2πF˜i(r,ω)dω,
F˜i(r,ω)=2[αei(*)TrE˜jk(r,r,ω)].
i(*)TrE˜jk(r,r,ω)=[iTrE˜jk(r,r,ω)]r=r,
i(*)TrEjk(r,r,τ)=[iTrEjk(r,r,τ)]r=r,
p˙j(r)Bk(r)=2p˙jBk*,pj(r)jEi(r)=2EjjEi*.
F(r,t)=2παeLimT12T|E˜(r,ω)|2dω+4παe{LimT12TkE˜(r,ω)×B˜*(r,ω)dω}+4παe{LimT12T(E˜*(r,ω)·)E˜(r,ω)dω},
F˜(r,ω)=αe|E˜(r,ω)|2+2kαe{E˜(r,ω)×B˜*(r,ω)}+2αe{(E˜*(r,ω)·)E˜(r,ω)}.
F˜m(r,ω)=αm|B˜(r,ω)|2+2kαm{E˜(r,ω)×B˜*(r,ω)}+2αm{(B˜*(r,ω)·)B˜(r,ω)}.
F˜em=4k43{(αeαm*)E˜×B˜*(αeαm*)E˜×B˜*}=4k43(αeαm*)E˜×B˜*+4k33(αeαm*)[12|E˜|2(E˜*·)E˜].
F˜i(r,ω)=2|e(ω)|2{αeiU*(r,ω)U(r,ω)}=2|e(ω)|2[αeU(r,ω)iU*(r,ω)αeU(r,ω)iU*(r,ω)].
F˜igrad(r,ω)=|e(ω)|2αeiW(r,r,ω)=|e(ω)|2αei|U(r,ω)|2.
Si(r,ω)=1kU(r,ω)iU*(r,ω).
F˜isc(r,ω)=2|e(ω)|2αe{i(*)W(r,r,ω)}=2|e(ω)|2αe{[iW(r,r,ω)]r=r}=2k|e(ω)|2αeSi(r,ω).
F˜icurl(r,ω)=2αe{ej*(ω)ei(ω)jW(r,r,ω)},
U(r,ω)=ik2πAU(r,ω)eikRRd2r,
F˜i(r,ω)=2(k2π)2|e(ω)|2{αeAA[ik+1R1]R1R1W(r1,r2,ω)×eikR1R1eikR2R2d2r1d2r2};
F˜grad(r,ω)=2(k2π)2|e(ω)|2αe{|U(q1,ω)|2R1R14+|U(q2,ω)|2R2R24+|W(q1,q2,ω)|R1R2[(R1R12+R2R22)cos(k(R1R2)+α(q1,q2,ω))+(R1R1R2R2)ksin(k(R1R2)+α(q1,q2,ω))]},
F˜sc(r,ω)=4(k2π)2|e(ω)|2αe{|U(q1,ω)|2kR1R13+|U(q2,ω)|2kR2R23+|W(q1,q2,ω)|R1R2[R1R12(kR1cos(k(R1R2)+α(q1,q2,ω))+sin(k(R1R2)+α(q1,q2,ω)))+R2R22(kR2cos(k(R1R2)+α(q1,q2,ω))sin(k(R1R2)+α(q1,q2,ω)))]},
E˜(r,ω)=ieikr2πrk×An×E˜(i)(r,ω)eik·rds,
E˜(r,ω)=U(r,ω)e(ω),
U(r,ω)=eikzeik2z(x2+y2)(πa2iλz)(2J1(v0)v0).
U(r,ω¯)=eik¯deik¯2d(x2+y2)(πa2iλ¯d)(2J1(v¯0)v¯0)[U(q1,ω¯)eik¯hyd+U(q2,ω¯)eik¯hyd],
I(r,ω¯)=E˜(r,ω¯)·E˜*(r,ω¯)=2I0(πa2|e(ω¯)|λ¯d)2(2J1(v¯0)v¯0)2×[1+|μ(q1,q2,ω¯)|cos(ϕ(q1,q2,ω¯)+2k¯hyd)].
F˜ygrad4αeI0(πa2|e(ω¯)|λ¯d)2(2J1(v0)v0)2hk¯d×|μ(q1,q2,ω¯)|sin(ϕ(q1,q2,ω¯)+2k¯hyd),
F˜zsc4k¯αeI0(πa2|e(ω¯)|λ¯d)2(2J1(v¯0)v¯0)2×[1+|μ(q1,q2,ω¯)|cos(ϕ(q1,q2,ω¯)+2k¯hyd)]2k¯αeI(r,ω¯),
αe=αe(0)(1i23k¯3αe(0))1,
αe(0)=r03εp1εp+2.
F˜xgrad=4αeI0(ak¯zv¯0)2(πa2|e(ω¯)|2λ¯z)2x[(2J1(v¯0)v¯0)2+2J1(v¯0)v¯0×[J0(v¯0)J2(v¯0)]][1+|μ(q1,q2,ω¯)|cos(ϕ(q1,q2,ω¯)+2k¯hyz)].
F˜ygrad=4αeI0(ak¯zv¯0)2(πa2|e(ω¯)|2λ¯z)2y[(2J1(v¯0)v¯0)2+2J1(v¯0)v¯0×[J0(v¯0)J2(v¯0)]][1+|μ(q1,q2,ω¯)|cos(ϕ(q1,q2,ω¯)+2k¯hyz)]4αeI0(πa2|e(ω¯)|2λ¯z)2hk¯z(2J1(v¯0)v¯0)2×|μ(q1,q2,ω¯)|sin(ϕ(q1,q2,ω¯)+2k¯hyz)=F˜xgradyx4αeI0(πa2|e(ω¯)|2λ¯z)2hk¯z(2J1(v¯0)v¯0)2×|μ(q1,q2,ω¯)|sin(ϕ(q1,q2,ω¯)+2k¯hyz).
F˜ygrad4αeI0(πa2|e(ω¯)|2λ¯z)2hk¯z(2J1(v¯0)v¯0)2×|μ(q1,q2,ω¯)|sin(ϕ(q1,q2,ω¯)+2k¯hyz).
F˜zgrad=4αeI0(πa2|e(ω¯)|2λ¯z)21z(2J1(v¯0)v¯0)2[J0(v¯0)J2(v¯0)]×[1+|μ(q1,q2,ω¯)|cos(ϕ(q1,q2,ω¯)+2k¯hyz)]+4αeI0(πa2|e(ω¯)|2λ¯z)2hk¯yz2(2J1(v¯0)v¯0)2×|μ(q1,q2,ω¯)|sin(ϕ(q1,q2,ω¯)+2k¯hyz).
F˜zgrad4αeI0(πa2|e(ω¯)|2λ¯z)hk¯yz2(2J1(v¯0)v¯0)2×|μ(q1,q2,ω¯)|sin(ϕ(q1,q2,ω¯)+2k¯hyz)=yzF˜ygrad.
F˜xsc=4αeI0(πa2|e(ω¯)|2λ¯z)2k¯xz(2J1(v¯0)v¯0)2×[1+|μ(q1,q2,ω¯)|cos(ϕ(q1,q2,ω¯)+2k¯hyz)].
F˜xsc=4αeI0(πa2|e(ω¯)|2λ¯z)2k¯yz(2J1(v¯0)v¯0)2×[1+|μ(q1,q2,ω¯)|cos(ϕ(q1,q2,ω¯)+2k¯hyz)].
F˜zsc=4k¯αeI0(πa2|e(ω¯)|2λ¯z)2(2z2x2y2)(2J1(v¯0)v¯0)2×[1+|μ(q1,q2,ω¯)|cos(ϕ(q1,q2,ω¯)+2k¯hyz)]4k¯αeI0(πa2|e(ω¯)|2λ¯z)2(2J1(v¯0)v¯0)2×[1+|μ(q1,q2,ω¯)|cos(ϕ(q1,q2,ω¯)+2k¯hyz)]=2k¯αeI(r,ω¯).

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