Abstract

Radiation force of a focused scalar twisted Gaussian Schell-model (TGSM) beam on a Rayleigh dielectric sphere is investigated. It is found that the twist phase affects the radiation force and by raising the absolute value of the twist factor it is possible to increase both transverse and longitudinal trapping ranges at the real focus where the maximum on-axis intensity is located. Numerical calculations of radiation forces induced by a focused electromagnetic TGSM beam on a Rayleigh dielectric sphere are carried out. It is found that radiation force is closely related to the twist phase, degree of polarization and correlation factors of the initial beam. The trapping stability is also discussed.

© 2009 OSA

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2009 (3)

2008 (6)

2007 (3)

2006 (4)

Y. Cai and L. Hu, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system,” Opt. Lett. 31(6), 685–687 (2006).
[CrossRef] [PubMed]

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).
[CrossRef]

E. Wolf, “Coherence and polarization properties of electromagnetic laser modes,” Opt. Commun. 265(1), 60–62 (2006).
[CrossRef]

C. Day, “Optical trap resolves the stepwise transfer of genetic information from DNA to RNA,” Phys. Today 59(1), 26–27 (2006).
[CrossRef]

2005 (3)

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7(5), 232–237 (2005).
[CrossRef]

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4-6), 379–385 (2005).
[CrossRef]

Y. Cai and S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5), 056607 (2005).
[CrossRef] [PubMed]

2004 (5)

2003 (2)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003).
[CrossRef]

Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 5(5), 453–459 (2003).
[CrossRef]

2002 (5)

2001 (5)

S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(3), 036618 (2001).
[CrossRef] [PubMed]

J. Serna and J. M. Movilla, “Orbital angular momentum of partially coherent beams,” Opt. Lett. 26(7), 405–407 (2001).
[CrossRef] [PubMed]

P. Östlund and A. T. Friberg, “Radiometry and Radiation Efficiency of Twisted Gaussian Schell-Model Sources,” Opt. Rev. 8(1), 1–8 (2001).
[CrossRef]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian schell-model beams,” J. Opt. A, Pure Appl. Opt. 3(1), 301 (2001).
[CrossRef]

Q. Lin, Z. Wang, and Z. Liu, “Radiation forces produced by standing wave trapping of non-paraxial Gaussian beams,” Opt. Commun. 198(1-3), 95–100 (2001).
[CrossRef]

2000 (2)

1999 (2)

R. Simon and N. Mukunda, “Gaussian Schell-model beams and general shape invariance,” J. Opt. Soc. Am. A 16(10), 2465–2475 (1999).
[CrossRef]

K. Okamoto and S. Kawata, “Radiation force exerted on subwavelength particles near a nanoaperture,” Phys. Rev. Lett. 83(22), 4534–4537 (1999).
[CrossRef]

1998 (2)

1997 (1)

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[CrossRef]

1996 (2)

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5-6), 529–541 (1996).
[CrossRef]

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5(3), 331–343 (1996).
[CrossRef]

1995 (1)

1994 (3)

1993 (3)

1990 (1)

S. M. Block, L. S. Goldstein, and B. J. Schnapp, “Bead movement by single kinesin molecules studied with optical tweezers,” Nature 348(6299), 348–352 (1990).
[CrossRef] [PubMed]

1988 (1)

Q. S. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67(4), 245–250 (1988).
[CrossRef]

1986 (3)

1982 (1)

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).
[CrossRef]

1979 (1)

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[CrossRef]

1978 (4)

E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,” Opt. Commun. 25(3), 293–296 (1978).
[CrossRef]

F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34(3), 301–305 (1978).
[CrossRef]

A. Ashkin, “Trapping of atoms by resonance radiation pressure,” Phys. Rev. Lett. 40(12), 729–732 (1978).
[CrossRef]

J. Kestin, M. Sokolov, and W. A. Wakeham, “Viscosity of Liquid Water in the Range - 8°C to 150° C,” J. Phys. Chem. Ref. Data 7, 941–948 (1978).
[CrossRef]

1970 (1)

A. Ashkin, “Acceleration and trapping of particles by radiation forces,” Phys. Rev. Lett. 24(4), 156–159 (1970).
[CrossRef]

Agrawal, G. P.

Ambrosini, D.

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41(7), 1391–1399 (1994).
[CrossRef]

Ansari, N. A.

N. A. Ansari and M. S. Zubairy, “Second-harmonic generation by a Gaussian Schell-model source,” Opt. Commun. 59(5-6), 385–390 (1986).
[CrossRef]

Asakura, T.

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5-6), 529–541 (1996).
[CrossRef]

Ashkin, A.

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(5), 288–290 (1986).
[CrossRef] [PubMed]

A. Ashkin, “Trapping of atoms by resonance radiation pressure,” Phys. Rev. Lett. 40(12), 729–732 (1978).
[CrossRef]

A. Ashkin, “Acceleration and trapping of particles by radiation forces,” Phys. Rev. Lett. 24(4), 156–159 (1970).
[CrossRef]

Bagini, V.

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41(7), 1391–1399 (1994).
[CrossRef]

Baker, J. R.

Bastiaans, M. J.

Baykal, Y.

Bjorkholm, J. E.

Block, S. M.

S. M. Block, L. S. Goldstein, and B. J. Schnapp, “Bead movement by single kinesin molecules studied with optical tweezers,” Nature 348(6299), 348–352 (1990).
[CrossRef] [PubMed]

Borghi, R.

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25(5), 1016–1021 (2008).
[CrossRef]

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
[CrossRef]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian schell-model beams,” J. Opt. A, Pure Appl. Opt. 3(1), 301 (2001).
[CrossRef]

Cai, Y.

Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96(2-3), 499–507 (2009).
[CrossRef]

C. L. Zhao, 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(3), 1753–1765 (2009).
[CrossRef] [PubMed]

Y. Cai, Q. Lin, and O. Korotkova, “Ghost imaging with twisted Gaussian Schell-model beam,” Opt. Express 17(4), 2453–2464 (2009).
[CrossRef] [PubMed]

M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “The evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Lett. 33(19), 2266–2268 (2008).
[CrossRef] [PubMed]

O. Korotkova, M. Yao, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “The state of polarization of a stochastic electromagnetic beam in an optical resonator,” J. Opt. Soc. Am. A 25(11), 2710–2720 (2008).
[CrossRef]

Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834 (2008).
[CrossRef] [PubMed]

F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24(7), 1937–1944 (2007).
[CrossRef]

Y. Cai and U. Peschel, “Second-harmonic generation by an astigmatic partially coherent beam,” Opt. Express 15(23), 15480–15492 (2007).
[CrossRef] [PubMed]

Y. Cai and L. Hu, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system,” Opt. Lett. 31(6), 685–687 (2006).
[CrossRef] [PubMed]

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).
[CrossRef]

Y. Cai and S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5), 056607 (2005).
[CrossRef] [PubMed]

Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 5(5), 453–459 (2003).
[CrossRef]

Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002).
[CrossRef] [PubMed]

Y. Cai, Q. Lin, and D. Ge, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams in dispersive and absorbing media,” J. Opt. Soc. Am. A 19(10), 2036–2042 (2002).
[CrossRef]

Q. Lin and Y. Cai, “Fractional Fourier transform for partially coherent Gaussian-Schell model beams,” Opt. Lett. 27(19), 1672–1674 (2002).
[CrossRef] [PubMed]

Chang, G.

Chen, C. H.

C. H. Chen, P. T. Tai, and W. F. Hsieh, “Bottle beam from a bare laser for single-beam trapping,” Appl. Phys. Lett. 43, 6001–6006 (2004).

Chu, S.

Collett, E.

E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,” Opt. Commun. 25(3), 293–296 (1978).
[CrossRef]

Davidson, F. M.

Day, C.

C. Day, “Optical trap resolves the stepwise transfer of genetic information from DNA to RNA,” Phys. Today 59(1), 26–27 (2006).
[CrossRef]

De Santis, P.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[CrossRef]

Dziedzic, J. M.

Eyyuboglu, H. T.

Friberg, A. T.

P. Östlund and A. T. Friberg, “Radiometry and Radiation Efficiency of Twisted Gaussian Schell-Model Sources,” Opt. Rev. 8(1), 1–8 (2001).
[CrossRef]

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5(3), 331–343 (1996).
[CrossRef]

A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11(6), 1818–1826 (1994).
[CrossRef]

Q. S. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67(4), 245–250 (1988).
[CrossRef]

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).
[CrossRef]

Ge, D.

Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 5(5), 453–459 (2003).
[CrossRef]

Y. Cai, Q. Lin, and D. Ge, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams in dispersive and absorbing media,” J. Opt. Soc. Am. A 19(10), 2036–2042 (2002).
[CrossRef]

Goldstein, L. S.

S. M. Block, L. S. Goldstein, and B. J. Schnapp, “Bead movement by single kinesin molecules studied with optical tweezers,” Nature 348(6299), 348–352 (1990).
[CrossRef] [PubMed]

Gori, F.

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25(5), 1016–1021 (2008).
[CrossRef]

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
[CrossRef]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian schell-model beams,” J. Opt. A, Pure Appl. Opt. 3(1), 301 (2001).
[CrossRef]

F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998).
[CrossRef] [PubMed]

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41(7), 1391–1399 (1994).
[CrossRef]

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[CrossRef]

F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34(3), 301–305 (1978).
[CrossRef]

Guattari, G.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[CrossRef]

Harada, Y.

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5-6), 529–541 (1996).
[CrossRef]

He, Q. S.

Q. S. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67(4), 245–250 (1988).
[CrossRef]

He, S.

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).
[CrossRef]

Hirano, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[CrossRef]

Hollman, K. W.

Hsieh, W. F.

C. H. Chen, P. T. Tai, and W. F. Hsieh, “Bottle beam from a bare laser for single-beam trapping,” Appl. Phys. Lett. 43, 6001–6006 (2004).

Hu, L.

James, D. F. V.

Kandpal, H. C.

Kanseri, B.

Kawata, S.

K. Okamoto and S. Kawata, “Radiation force exerted on subwavelength particles near a nanoaperture,” Phys. Rev. Lett. 83(22), 4534–4537 (1999).
[CrossRef]

Kestin, J.

J. Kestin, M. Sokolov, and W. A. Wakeham, “Viscosity of Liquid Water in the Range - 8°C to 150° C,” J. Phys. Chem. Ref. Data 7, 941–948 (1978).
[CrossRef]

Korotkova, O.

Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96(2-3), 499–507 (2009).
[CrossRef]

Y. Cai, Q. Lin, and O. Korotkova, “Ghost imaging with twisted Gaussian Schell-model beam,” Opt. Express 17(4), 2453–2464 (2009).
[CrossRef] [PubMed]

Y. Cai, O. Korotkova, H. T. Eyyuboğlu, and Y. Baykal, “Active laser radar systems with stochastic electromagnetic beams in turbulent atmosphere,” Opt. Express 16(20), 15834 (2008).
[CrossRef] [PubMed]

M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “The evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Lett. 33(19), 2266–2268 (2008).
[CrossRef] [PubMed]

O. Korotkova, M. Yao, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “The state of polarization of a stochastic electromagnetic beam in an optical resonator,” J. Opt. Soc. Am. A 25(11), 2710–2720 (2008).
[CrossRef]

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7(5), 232–237 (2005).
[CrossRef]

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4-6), 379–385 (2005).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. 29(11), 1173–1175 (2004).
[CrossRef] [PubMed]

Kuga, T.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[CrossRef]

Lin, Q.

Liu, Z.

Q. Lin, Z. Wang, and Z. Liu, “Radiation forces produced by standing wave trapping of non-paraxial Gaussian beams,” Opt. Commun. 198(1-3), 95–100 (2001).
[CrossRef]

Lu, X.

Lu, X. H.

Mondello, A.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
[CrossRef]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian schell-model beams,” J. Opt. A, Pure Appl. Opt. 3(1), 301 (2001).
[CrossRef]

Movilla, J. M.

Mukunda, N.

Norris, T. B.

O’Donnell, M.

Okamoto, K.

K. Okamoto and S. Kawata, “Radiation force exerted on subwavelength particles near a nanoaperture,” Phys. Rev. Lett. 83(22), 4534–4537 (1999).
[CrossRef]

Östlund, P.

P. Östlund and A. T. Friberg, “Radiometry and Radiation Efficiency of Twisted Gaussian Schell-Model Sources,” Opt. Rev. 8(1), 1–8 (2001).
[CrossRef]

Palma, C.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[CrossRef]

Peschel, U.

Piquero, G.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
[CrossRef]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian schell-model beams,” J. Opt. A, Pure Appl. Opt. 3(1), 301 (2001).
[CrossRef]

Ponomarenko, S. A.

S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(3), 036618 (2001).
[CrossRef] [PubMed]

Ramírez-Sánchez, V.

Ricklin, J. C.

Romanini, P.

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
[CrossRef]

Roychowdhury, H.

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4-6), 379–385 (2005).
[CrossRef]

Salem, M.

Santarsiero, M.

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25(5), 1016–1021 (2008).
[CrossRef]

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
[CrossRef]

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian schell-model beams,” J. Opt. A, Pure Appl. Opt. 3(1), 301 (2001).
[CrossRef]

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41(7), 1391–1399 (1994).
[CrossRef]

Sasada, H.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[CrossRef]

Schnapp, B. J.

S. M. Block, L. S. Goldstein, and B. J. Schnapp, “Bead movement by single kinesin molecules studied with optical tweezers,” Nature 348(6299), 348–352 (1990).
[CrossRef] [PubMed]

Serna, J.

Shimizu, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[CrossRef]

Shiokawa, N.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[CrossRef]

Shirai, T.

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7(5), 232–237 (2005).
[CrossRef]

Simon, R.

Sokolov, M.

J. Kestin, M. Sokolov, and W. A. Wakeham, “Viscosity of Liquid Water in the Range - 8°C to 150° C,” J. Phys. Chem. Ref. Data 7, 941–948 (1978).
[CrossRef]

Sudol, R. J.

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).
[CrossRef]

Sundar, K.

Tai, P. T.

C. H. Chen, P. T. Tai, and W. F. Hsieh, “Bottle beam from a bare laser for single-beam trapping,” Appl. Phys. Lett. 43, 6001–6006 (2004).

Tervonen, E.

Torii, Y.

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[CrossRef]

Tse, C.

Turunen, J.

A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11(6), 1818–1826 (1994).
[CrossRef]

Q. S. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67(4), 245–250 (1988).
[CrossRef]

Wakeham, W. A.

J. Kestin, M. Sokolov, and W. A. Wakeham, “Viscosity of Liquid Water in the Range - 8°C to 150° C,” J. Phys. Chem. Ref. Data 7, 941–948 (1978).
[CrossRef]

Wang, F.

Wang, L. G.

Wang, L. Q.

Wang, Z.

Q. Lin, Z. Wang, and Z. Liu, “Radiation forces produced by standing wave trapping of non-paraxial Gaussian beams,” Opt. Commun. 198(1-3), 95–100 (2001).
[CrossRef]

Wolf, E.

M. Salem and E. Wolf, “Coherence-induced polarization changes in light beams,” Opt. Lett. 33(11), 1180–1182 (2008).
[CrossRef] [PubMed]

E. Wolf, “Coherence and polarization properties of electromagnetic laser modes,” Opt. Commun. 265(1), 60–62 (2006).
[CrossRef]

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7(5), 232–237 (2005).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004).
[CrossRef]

O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. 29(11), 1173–1175 (2004).
[CrossRef] [PubMed]

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003).
[CrossRef]

G. P. Agrawal and E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A 17(11), 2019–2023 (2000).
[CrossRef]

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5(3), 331–343 (1996).
[CrossRef]

E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,” Opt. Commun. 25(3), 293–296 (1978).
[CrossRef]

Yao, M.

Ye, J. Y.

Zhan, Q.

Zhao, C. L.

Zhu, S. Y.

L. G. Wang, C. L. Zhao, L. Q. Wang, X. H. Lu, and S. Y. Zhu, “Effect of spatial coherence on radiation forces acting on a Rayleigh dielectric sphere,” Opt. Lett. 32(11), 1393–1395 (2007).
[CrossRef] [PubMed]

Y. Cai and S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5), 056607 (2005).
[CrossRef] [PubMed]

Zohdy, M. J.

Zubairy, M. S.

N. A. Ansari and M. S. Zubairy, “Second-harmonic generation by a Gaussian Schell-model source,” Opt. Commun. 59(5-6), 385–390 (1986).
[CrossRef]

Appl. Phys. B (1)

Y. Cai and O. Korotkova, “Twist phase-induced polarization changes in electromagnetic Gaussian Schell-model beams,” Appl. Phys. B 96(2-3), 499–507 (2009).
[CrossRef]

Appl. Phys. Lett. (2)

C. H. Chen, P. T. Tai, and W. F. Hsieh, “Bottle beam from a bare laser for single-beam trapping,” Appl. Phys. Lett. 43, 6001–6006 (2004).

Y. Cai and S. He, “Propagation of a partially coherent twisted anisotropic Gaussian Schell-model beam in a turbulent atmosphere,” Appl. Phys. Lett. 89(4), 041117 (2006).
[CrossRef]

J. Mod. Opt. (1)

D. Ambrosini, V. Bagini, F. Gori, and M. Santarsiero, “Twisted Gaussian Schell-model beams: a superposition model,” J. Mod. Opt. 41(7), 1391–1399 (1994).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (3)

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, and R. Simon, “Partially polarized Gaussian schell-model beams,” J. Opt. A, Pure Appl. Opt. 3(1), 301 (2001).
[CrossRef]

Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 5(5), 453–459 (2003).
[CrossRef]

T. Shirai, O. Korotkova, and E. Wolf, “A method of generating electromagnetic Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 7(5), 232–237 (2005).
[CrossRef]

J. Opt. Soc. Am. A (16)

F. Gori, M. Santarsiero, R. Borghi, and V. Ramírez-Sánchez, “Realizability condition for electromagnetic Schell-model sources,” J. Opt. Soc. Am. A 25(5), 1016–1021 (2008).
[CrossRef]

O. Korotkova, M. Yao, Y. Cai, H. T. Eyyuboğlu, and Y. Baykal, “The state of polarization of a stochastic electromagnetic beam in an optical resonator,” J. Opt. Soc. Am. A 25(11), 2710–2720 (2008).
[CrossRef]

G. P. Agrawal and E. Wolf, “Propagation-induced polarization changes in partially coherent optical beams,” J. Opt. Soc. Am. A 17(11), 2019–2023 (2000).
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M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3(8), 1227–1238 (1986).
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D. F. V. James, “Changes of polarization of light beams on propagation in free space,” J. Opt. Soc. Am. A 11(5), 1641–1643 (1994).
[CrossRef]

Y. Cai, Q. Lin, and D. Ge, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams in dispersive and absorbing media,” J. Opt. Soc. Am. A 19(10), 2036–2042 (2002).
[CrossRef]

M. J. Bastiaans, “Wigner-distribution function applied to twisted Gaussian light propagating in first-order optical systems,” J. Opt. Soc. Am. A 17(12), 2475–2480 (2000).
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K. Sundar, N. Mukunda, and R. Simon, “Coherent-mode decomposition of general anisotropic Gaussian Schell-model beams,” J. Opt. Soc. Am. A 12(3), 560–569 (1995).
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R. Simon and N. Mukunda, “Gaussian Schell-model beams and general shape invariance,” J. Opt. Soc. Am. A 16(10), 2465–2475 (1999).
[CrossRef]

R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10(1), 95–109 (1993).
[CrossRef]

R. Simon, K. Sundar, and N. Mukunda, “Twisted Gaussian Schell-model beams. I. Symmetry structure and normal-mode spectrum,” J. Opt. Soc. Am. A 10(9), 2008–2016 (1993).
[CrossRef]

K. Sundar, R. Simon, and N. Mukunda, “Twisted Gaussian Schell-model beams. II. Spectrum analysis and propagation characteristics,” J. Opt. Soc. Am. A 10(9), 2017–2023 (1993).
[CrossRef]

R. Simon and N. Mukunda, “Twist phase in Gaussian-beam optics,” J. Opt. Soc. Am. A 15(9), 2373–2382 (1998).
[CrossRef]

A. T. Friberg, E. Tervonen, and J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11(6), 1818–1826 (1994).
[CrossRef]

F. Wang and Y. Cai, “Experimental observation of fractional Fourier transform for a partially coherent optical beam with Gaussian statistics,” J. Opt. Soc. Am. A 24(7), 1937–1944 (2007).
[CrossRef]

J. C. Ricklin and F. M. Davidson, “Atmospheric turbulence effects on a partially coherent Gaussian beam: implications for free-space laser communication,” J. Opt. Soc. Am. A 19(9), 1794–1802 (2002).
[CrossRef]

J. Phys. Chem. Ref. Data (1)

J. Kestin, M. Sokolov, and W. A. Wakeham, “Viscosity of Liquid Water in the Range - 8°C to 150° C,” J. Phys. Chem. Ref. Data 7, 941–948 (1978).
[CrossRef]

Nature (1)

S. M. Block, L. S. Goldstein, and B. J. Schnapp, “Bead movement by single kinesin molecules studied with optical tweezers,” Nature 348(6299), 348–352 (1990).
[CrossRef] [PubMed]

Opt. Commun. (12)

O. Korotkova, M. Salem, and E. Wolf, “The far-zone behavior of the degree of polarization of electromagnetic beams propagating through atmospheric turbulence,” Opt. Commun. 233(4-6), 225–230 (2004).
[CrossRef]

H. Roychowdhury and O. Korotkova, “Realizability conditions for electromagnetic Gaussian Schell-model sources,” Opt. Commun. 249(4-6), 379–385 (2005).
[CrossRef]

Q. Lin, Z. Wang, and Z. Liu, “Radiation forces produced by standing wave trapping of non-paraxial Gaussian beams,” Opt. Commun. 198(1-3), 95–100 (2001).
[CrossRef]

Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124(5-6), 529–541 (1996).
[CrossRef]

E. Wolf, “Coherence and polarization properties of electromagnetic laser modes,” Opt. Commun. 265(1), 60–62 (2006).
[CrossRef]

N. A. Ansari and M. S. Zubairy, “Second-harmonic generation by a Gaussian Schell-model source,” Opt. Commun. 59(5-6), 385–390 (1986).
[CrossRef]

E. Wolf and E. Collett, “Partially coherent sources which produce same far-field intensity distribution as a laser,” Opt. Commun. 25(3), 293–296 (1978).
[CrossRef]

F. Gori, “Collet-Wolf sources and multimode lasers,” Opt. Commun. 34(3), 301–305 (1978).
[CrossRef]

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett-Wolf source,” Opt. Commun. 29(3), 256–260 (1979).
[CrossRef]

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41(6), 383–387 (1982).
[CrossRef]

Q. S. He, J. Turunen, and A. T. Friberg, “Propagation and imaging experiments with Gaussian Schell-model beams,” Opt. Commun. 67(4), 245–250 (1988).
[CrossRef]

G. Piquero, F. Gori, P. Romanini, M. Santarsiero, R. Borghi, and A. Mondello, “Synthesis of partially polarized Gaussian Schell-model sources,” Opt. Commun. 208(1-3), 9–16 (2002).
[CrossRef]

Opt. Express (5)

Opt. Lett. (12)

B. Kanseri and H. C. Kandpal, “Experimental determination of electric cross-spectral density matrix and generalized Stokes parameters for a laser beam,” Opt. Lett. 33(20), 2410–2412 (2008).
[CrossRef] [PubMed]

M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “The evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Lett. 33(19), 2266–2268 (2008).
[CrossRef] [PubMed]

O. Korotkova, M. Salem, and E. Wolf, “Beam conditions for radiation generated by an electromagnetic Gaussian Schell-model source,” Opt. Lett. 29(11), 1173–1175 (2004).
[CrossRef] [PubMed]

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(5), 288–290 (1986).
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J. Y. Ye, G. Chang, T. B. Norris, C. Tse, M. J. Zohdy, K. W. Hollman, M. O’Donnell, and J. R. Baker., “Trapping cavitation bubbles with a self-focused laser beam,” Opt. Lett. 29(18), 2136–2138 (2004).
[CrossRef] [PubMed]

L. G. Wang, C. L. Zhao, L. Q. Wang, X. H. Lu, and S. Y. Zhu, “Effect of spatial coherence on radiation forces acting on a Rayleigh dielectric sphere,” Opt. Lett. 32(11), 1393–1395 (2007).
[CrossRef] [PubMed]

M. Salem and E. Wolf, “Coherence-induced polarization changes in light beams,” Opt. Lett. 33(11), 1180–1182 (2008).
[CrossRef] [PubMed]

Y. Cai and L. Hu, “Propagation of partially coherent twisted anisotropic Gaussian Schell-model beams through an apertured astigmatic optical system,” Opt. Lett. 31(6), 685–687 (2006).
[CrossRef] [PubMed]

Q. Lin and Y. Cai, “Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams,” Opt. Lett. 27(4), 216–218 (2002).
[CrossRef] [PubMed]

Q. Lin and Y. Cai, “Fractional Fourier transform for partially coherent Gaussian-Schell model beams,” Opt. Lett. 27(19), 1672–1674 (2002).
[CrossRef] [PubMed]

J. Serna and J. M. Movilla, “Orbital angular momentum of partially coherent beams,” Opt. Lett. 26(7), 405–407 (2001).
[CrossRef] [PubMed]

F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23(4), 241–243 (1998).
[CrossRef] [PubMed]

Opt. Rev. (1)

P. Östlund and A. T. Friberg, “Radiometry and Radiation Efficiency of Twisted Gaussian Schell-Model Sources,” Opt. Rev. 8(1), 1–8 (2001).
[CrossRef]

Phys. Lett. A (1)

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312(5-6), 263–267 (2003).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (2)

Y. Cai and S. Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5), 056607 (2005).
[CrossRef] [PubMed]

S. A. Ponomarenko, “Twisted Gaussian Schell-model solitons,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(3), 036618 (2001).
[CrossRef] [PubMed]

Phys. Rev. Lett. (4)

T. Kuga, Y. Torii, N. Shiokawa, T. Hirano, Y. Shimizu, and H. Sasada, “Novel optical trap of atoms with a doughnut beam,” Phys. Rev. Lett. 78(25), 4713–4716 (1997).
[CrossRef]

K. Okamoto and S. Kawata, “Radiation force exerted on subwavelength particles near a nanoaperture,” Phys. Rev. Lett. 83(22), 4534–4537 (1999).
[CrossRef]

A. Ashkin, “Acceleration and trapping of particles by radiation forces,” Phys. Rev. Lett. 24(4), 156–159 (1970).
[CrossRef]

A. Ashkin, “Trapping of atoms by resonance radiation pressure,” Phys. Rev. Lett. 40(12), 729–732 (1978).
[CrossRef]

Phys. Today (1)

C. Day, “Optical trap resolves the stepwise transfer of genetic information from DNA to RNA,” Phys. Today 59(1), 26–27 (2006).
[CrossRef]

Pure Appl. Opt. (1)

R. Simon, A. T. Friberg, and E. Wolf, “Transfer of radiance by twisted Gaussian Schell-model beams in paraxial systems,” Pure Appl. Opt. 5(3), 331–343 (1996).
[CrossRef]

Other (3)

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

C. Brosseau, Fundamentals of polarized light-a statistical approach (Wiley, New York, 1998).

E. Wolf, Introduction to the theory of coherence and polarization of light (Cambridge U. Press, 2007).

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Figures (12)

Fig. 1
Fig. 1

Schematic diagram of a focusing optical system

Fig. 2
Fig. 2

(a) Intensity distribution of a scalar twisted GSM beam at the real focal plane for different absolute values of the initial twist factor normalized with γ = [ k 2 δ 4 ] 1 / 2 = 0.1 m 1 . The inset shows scaled down intensity distribution for the case | μ 0 / γ | = 1 . (b) Focus shift f z max of a scalar twisted GSM beam behind the thin lens versus the normalized initial twist factor | μ 0 / γ |

Fig. 3
Fig. 3

Intensity distribution of a twisted EGSM beam at the real focal plane for different absolute values of the initial twist factors μ x x and μ y y normalized with γ x x = [ k 2 δ x x 4 ] 1 / 2 = 0.1 m 1 and γ y y = [ k 2 δ y y 4 ] 1 / 2 = 0.025 m 1 , respectively.

Fig. 4
Fig. 4

Intensity distribution of a twisted EGSM beam at the real focal plane for different values of the initial degree of polarization

Fig. 5
Fig. 5

Intensity distribution of an electromagnetic twisted GSM beam at the real focal plane for different values of the initial correlation coefficients δ x x and δ y y . In Fig. 5 The inset shows the scaled down intensity distribution for the case δ x x =1mm and δ y y = 2 mm

Fig. 6
Fig. 6

(a) Scattering force (cross-section y=0) at the real focal plane, (b) transverse gradient force (cross-section y=0) at the real focal plane, and (c) longitudinal gradient force at r = 0 of a scalar twisted GSM beam for different absolute values of the initial twist factor μ 0 normalized with γ = [ k 2 δ 4 ] 1 / 2 = 0.1 m 1 . The insets in Fig. 6(a) and 6(b) show scaled down radiation forces for the case | μ 0 / γ | = 1 . The inset in Fig. 6(c) shows the zoomed region where the crossing of the radiation forces with zero occurs.

Fig. 7
Fig. 7

(a) the scattering force (cross-section y=0) at the real focal plane, in Fig. 7(b) the transverse gradient force (cross-section y=0) at the real focal plane, and in Fig. 7(c) the longitudinal gradient force at r = 0 of a scalar twisted GSM beam for different values of χ = n p / n m with n m = 1.33 , | μ 0 / γ | = 0.05 and γ = [ k 2 δ 4 ] 1 / 2 = 0.1 m 1 . The inset in Fig. 7 (c) shows the zoomed region where the crossing of the radiation forces with zero occurs.

Fig. 8
Fig. 8

(a) Scattering force (cross-section y = 0) at the real focal plane, (b) transverse gradient force (cross-section y=0) at the real focal plane, and (c) longitudinal gradient force at r = 0 of an electromagnetic twisted GSM beam for different absolute values of the initial twist factors μ x x and μ y y normalized with γ x x = [ k 2 δ x x 4 ] 1 / 2 = 0.1 m 1 and γ y y = [ k 2 δ y y 4 ] 1 / 2 = 0.025 m 1 , respectively. The inset in Fig. 8(c) shows the zoomed region where the crossing of the radiation forces with zero occurs.

Fig. 9
Fig. 9

(a) Scattering force (cross-section y=0) at the real focal plane, (b) transverse gradient force (cross-section y=0) at the real focal plane, and (c) longitudinal gradient force at r = 0 of an electromagnetic twisted GSM beam for different values of the initial degree of polarization. In Fig. 9(c) the inset shows the zoomed region where the crossing of the radiation forces with zero occurs.

Fig. 10
Fig. 10

(a) Scattering force (cross-section y=0) at the real focal plane, (b) transverse gradient force (cross-section y=0) at the real focal plane, and (c) longitudinal gradient force at r = 0 of an electromagnetic twisted GSM beam for different values of the initial correlation coefficients δ x x and δ y y . In Fig. 10(c), the inset shows the zoomed region where the crossing of the radiation forces with zero occurs.

Fig. 11
Fig. 11

Dependence of the radiation forces F Scat Max , F Grad-x Max and F Grad-z Max induced by a scalar twisted GSM beam on the absolute value of the normalized initial twist factor | μ 0 / γ | at the real focal plane. F B is the Brownian force

Fig. 12
Fig. 12

(a) Dependence of the radiation forces F Scat Max , F Grad-x Max and F Grad-z Max a induced by an electromagnetic twisted GSM beam on the initial degree of polarization at z 1 = 0, (b) dependencies of the radiation forces F Scat Max , F Grad-x Max and F Grad-z Max on the initial correlation coefficients with ξ = δ x x / σ x x and δ y y = 2.5 δ x x . In Fig. 12 (b) the inset shows the zoomed region where the crossing of the three radiation forces with the Brownian force occurs.

Tables (2)

Tables Icon

Table 1 The trapping ranges from Fig. 6(b) for different absolute values of μ 0

Tables Icon

Table 2 The trapping ranges from Fig. 6(c) for different absolute values of μ 0

Equations (18)

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W ( r 1 , r 2 ; 0 ) = W ( r ˜ ; 0 ) = G 0 exp ( i k 2 r ˜ T M 0 1 r ˜ ) ,
M 0 1 = ( ( i 2 k σ 2 i k δ 2 ) I i k δ 2 I + μ 0 J i k δ 2 I + μ 0 J T ( i 2 k σ 2 i k δ 2 ) I ) ,
J = ( 0 1 1 0 ) .
W ( u ˜ ; z ) = G 0 [ det ( A ˜ + B ˜ M 0 1 ) ] 1 / 2 exp ( i k 2 u ˜ T M 1 1 u ˜ ) ,
M 1 1 = ( C ˜ + D ˜ M 0 1 ) ( A ˜ + B ˜ M 0 1 ) 1 ,
A ˜ = ( A 0 I 0 I A * ) , B ˜ = ( B 0 I 0 I B * ) , C ˜ = ( C 0 I 0 I C * ) , D ˜ = ( D 0 I 0 I D * ) ,
W α β ( r 1 , r 2 ; 0 ) = W α β ( r ˜ ; 0 ) = E α * ( r 1 ; 0 ) E β ( r 2 ; 0 ) = A α A β B α β exp [ i k 2 r ˜ T M 0 α β 1 r ˜ ] , ( α = x , y ; β = x , y )
M 0 α β 1 = ( 1 i k ( 1 2 σ a β 2 + 1 δ α β 2 ) I i k δ α β 2 I + μ α β J i k δ α β 2 I + μ α β J T 1 i k ( 1 2 σ α β 2 + 1 δ α β 2 ) I ) ,
I ( r ; 0 ) = T r W ( r , r ; 0 ) = A x exp [ i k 2 r ˜ T M 0 x x 1 r ˜ ] + A y exp [ i k 2 r ˜ T M 0 y y 1 r ˜ ] .
Q = A x π ( 1 + η ) ( σ x x 2 + σ y y 2 ) 2 .
P 0 ( r ; 0 ) = 1 4 D e t W ( r , r ; 0 ) [ T r W ( r , r ; 0 ) ] 2 ,
W ( r 1 , r 2 ; 0 ) = ( W x x ( 0 ) ( r 1 , r 2 ; 0 ) 0 0 W y y ( 0 ) ( r 1 , r 2 ; 0 ) ) .
P 0 ( r ; 0 ) = | 1 η | 1 + η .
W α β ( u 1 , u 2 ; z ) = W α β ( u ˜ ; z ) = A α A β B α β [ D e t ( A ¯ + B ¯ M 0 α β 1 ) ] 1 / 2 exp [ i k 2 u ˜ T M 1 α β 1 u ˜ ] ,
M 1 α β 1 = ( C ˜ + D ˜ M 0 α β 1 ) ( A ˜ + B ˜ M 0 α β 1 ) 1 .
( A B C D ) = ( I z I 0 I I ) ( I 0 I ( 1 / f ) I I ) = ( ( 1 z / f ) I z I ( 1 / f ) I I ) .
F Scat ( r ; z ) = e z n m α I ( r ; z ) / c ,
F G r a d ( r ; z ) = 2 π n m β I ( r ; z ) / c ,

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