Abstract

Using vector potential and spectrum representation, we derive the expressions of the Airy evanescent field existed at the interface. Utilizing these expressions and the Arbitrary Beam Theory, the optical forces exerted on a Mie dielectric particle in the Airy evanescent field were theoretically investigated in detail. Numerical results show that the optical forces exhibit strong oscillations which are corresponding to the distributions of the evanescent field. With the increasing the size of particle radius, Morphology Dependent Resonance occurs for the particle with specific refractive index.

© 2012 OSA

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References

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2011 (1)

2010 (5)

2009 (2)

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9(10), 1334–1336 (2009).
[Crossref] [PubMed]

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy Beams,” Science 324(5924), 229–232 (2009).
[Crossref] [PubMed]

2008 (4)

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, “Efficient optical trapping and visualization of silver nanoparticles,” Nano Lett. 8(5), 1486–1491 (2008).
[Crossref] [PubMed]

H. I. Sztul and R. R. Alfano, “The Poynting vector and angular momentum of Airy beams,” Opt. Express 16(13), 9411–9416 (2008).
[Crossref] [PubMed]

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[Crossref] [PubMed]

2007 (3)

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref] [PubMed]

X. Tsampoula, V. Garces-Chavez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91(5), 053902 (2007).
[Crossref]

2006 (1)

P. H. Jones, E. Stride, and N. Saffari, “Trapping and manipulation of microscopic bubbles with a scanning optical tweezer,” Appl. Phys. Lett. 89(8), 081113 (2006).
[Crossref]

2005 (2)

T. Cizmar, V. Garces-Chavez, K. Dholakia, and P. Zemanek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86(17), 174101 (2005).
[Crossref]

R. Quidant, D. Petrov, and G. Badenes, “Radiation forces on a Rayleigh dielectric sphere in a patterned optical near field,” Opt. Lett. 30(9), 1009–1011 (2005).
[Crossref] [PubMed]

2002 (1)

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
[Crossref] [PubMed]

1995 (1)

1994 (1)

S. Chang, J. H. Jo, and S. S. Lee, “Theoretical calculations of optical force exerted on a dielectric sphere in the evanescent field generated with a totally-reflected focused gaussian beam,” Opt. Commun. 108(1-3), 133–143 (1994).
[Crossref]

1993 (1)

1992 (1)

1989 (1)

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a sphereical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4594–4602 (1989).
[Crossref]

1988 (1)

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64(4), 1632–1639 (1988).
[Crossref]

1987 (1)

A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235(4795), 1517–1520 (1987).
[Crossref] [PubMed]

1986 (1)

Aabo, T.

L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, “Efficient optical trapping and visualization of silver nanoparticles,” Nano Lett. 8(5), 1486–1491 (2008).
[Crossref] [PubMed]

Agate, B.

X. Tsampoula, V. Garces-Chavez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91(5), 053902 (2007).
[Crossref]

Alexander, D. R.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a sphereical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4594–4602 (1989).
[Crossref]

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64(4), 1632–1639 (1988).
[Crossref]

Alfano, R. R.

Almass, E.

Arie, A.

Ashkin, A.

Badenes, G.

Barton, J. P.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a sphereical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4594–4602 (1989).
[Crossref]

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64(4), 1632–1639 (1988).
[Crossref]

Baumgartl, J.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9(10), 1334–1336 (2009).
[Crossref] [PubMed]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

Bendix, P. M.

L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, “Efficient optical trapping and visualization of silver nanoparticles,” Nano Lett. 8(5), 1486–1491 (2008).
[Crossref] [PubMed]

Bjorkholm, J. E.

Bosanac, L.

L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, “Efficient optical trapping and visualization of silver nanoparticles,” Nano Lett. 8(5), 1486–1491 (2008).
[Crossref] [PubMed]

Brevik, I.

Broky, J.

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref] [PubMed]

Brown, C. T. A.

X. Tsampoula, V. Garces-Chavez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91(5), 053902 (2007).
[Crossref]

Chang, S.

S. Chang, J. H. Jo, and S. S. Lee, “Theoretical calculations of optical force exerted on a dielectric sphere in the evanescent field generated with a totally-reflected focused gaussian beam,” Opt. Commun. 108(1-3), 133–143 (1994).
[Crossref]

Cheng, H.

Chong, A.

A. Chong, W. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

Christodoulides, D. N.

A. Chong, W. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy Beams,” Science 324(5924), 229–232 (2009).
[Crossref] [PubMed]

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[Crossref] [PubMed]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref] [PubMed]

Chu, S.

Cizmar, T.

T. Cizmar, V. Garces-Chavez, K. Dholakia, and P. Zemanek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86(17), 174101 (2005).
[Crossref]

Comrie, M.

X. Tsampoula, V. Garces-Chavez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91(5), 053902 (2007).
[Crossref]

Day, D.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9(10), 1334–1336 (2009).
[Crossref] [PubMed]

Dholakia, K.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9(10), 1334–1336 (2009).
[Crossref] [PubMed]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

X. Tsampoula, V. Garces-Chavez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91(5), 053902 (2007).
[Crossref]

T. Cizmar, V. Garces-Chavez, K. Dholakia, and P. Zemanek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86(17), 174101 (2005).
[Crossref]

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
[Crossref] [PubMed]

Dogariu, A.

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref] [PubMed]

Dolev, I.

Dziedzic, J. M.

Ellenbogen, T.

Garces-Chavez, V.

X. Tsampoula, V. Garces-Chavez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91(5), 053902 (2007).
[Crossref]

T. Cizmar, V. Garces-Chavez, K. Dholakia, and P. Zemanek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86(17), 174101 (2005).
[Crossref]

Garcés-Chávez, V.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
[Crossref] [PubMed]

Gu, M.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9(10), 1334–1336 (2009).
[Crossref] [PubMed]

Gunn-Moore, F.

X. Tsampoula, V. Garces-Chavez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91(5), 053902 (2007).
[Crossref]

Hannappel, G. M.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9(10), 1334–1336 (2009).
[Crossref] [PubMed]

Jo, J. H.

S. Chang, J. H. Jo, and S. S. Lee, “Theoretical calculations of optical force exerted on a dielectric sphere in the evanescent field generated with a totally-reflected focused gaussian beam,” Opt. Commun. 108(1-3), 133–143 (1994).
[Crossref]

Johnson, B. R.

Jones, P. H.

P. H. Jones, E. Stride, and N. Saffari, “Trapping and manipulation of microscopic bubbles with a scanning optical tweezer,” Appl. Phys. Lett. 89(8), 081113 (2006).
[Crossref]

Kawata, S.

Kolesik, M.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy Beams,” Science 324(5924), 229–232 (2009).
[Crossref] [PubMed]

Lee, S. S.

S. Chang, J. H. Jo, and S. S. Lee, “Theoretical calculations of optical force exerted on a dielectric sphere in the evanescent field generated with a totally-reflected focused gaussian beam,” Opt. Commun. 108(1-3), 133–143 (1994).
[Crossref]

Li, J. X.

Li, Z.-J.

Mazilu, M.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

McGloin, D.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
[Crossref] [PubMed]

Melville, H.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
[Crossref] [PubMed]

Moloney, J. V.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy Beams,” Science 324(5924), 229–232 (2009).
[Crossref] [PubMed]

Nieto-Vesperinas, M.

Oddershede, L. B.

L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, “Efficient optical trapping and visualization of silver nanoparticles,” Nano Lett. 8(5), 1486–1491 (2008).
[Crossref] [PubMed]

Petrov, D.

Polynkin, P.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy Beams,” Science 324(5924), 229–232 (2009).
[Crossref] [PubMed]

Quidant, R.

Renninger, W.

A. Chong, W. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

Saenz, J. J.

Saffari, N.

P. H. Jones, E. Stride, and N. Saffari, “Trapping and manipulation of microscopic bubbles with a scanning optical tweezer,” Appl. Phys. Lett. 89(8), 081113 (2006).
[Crossref]

Schaub, S. A.

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a sphereical particle illuminated by a focused laser beam,” J. Appl. Phys. 66(10), 4594–4602 (1989).
[Crossref]

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64(4), 1632–1639 (1988).
[Crossref]

Shang, Q.-C.

Sibbett, W.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
[Crossref] [PubMed]

Siviloglou, G. A.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy Beams,” Science 324(5924), 229–232 (2009).
[Crossref] [PubMed]

J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16(17), 12880–12891 (2008).
[Crossref] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[Crossref] [PubMed]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[Crossref] [PubMed]

Stevenson, D. J.

J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9(10), 1334–1336 (2009).
[Crossref] [PubMed]

X. Tsampoula, V. Garces-Chavez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91(5), 053902 (2007).
[Crossref]

Stride, E.

P. H. Jones, E. Stride, and N. Saffari, “Trapping and manipulation of microscopic bubbles with a scanning optical tweezer,” Appl. Phys. Lett. 89(8), 081113 (2006).
[Crossref]

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Tian, J.

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Tsampoula, X.

X. Tsampoula, V. Garces-Chavez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91(5), 053902 (2007).
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Wise, F. W.

A. Chong, W. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
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T. Cizmar, V. Garces-Chavez, K. Dholakia, and P. Zemanek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86(17), 174101 (2005).
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Appl. Phys. Lett. (3)

P. H. Jones, E. Stride, and N. Saffari, “Trapping and manipulation of microscopic bubbles with a scanning optical tweezer,” Appl. Phys. Lett. 89(8), 081113 (2006).
[Crossref]

T. Cizmar, V. Garces-Chavez, K. Dholakia, and P. Zemanek, “Optical conveyor belt for delivery of submicron objects,” Appl. Phys. Lett. 86(17), 174101 (2005).
[Crossref]

X. Tsampoula, V. Garces-Chavez, M. Comrie, D. J. Stevenson, B. Agate, C. T. A. Brown, F. Gunn-Moore, and K. Dholakia, “Femtosecond cellular transfection using a nondiffracting light beam,” Appl. Phys. Lett. 91(5), 053902 (2007).
[Crossref]

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J. Baumgartl, G. M. Hannappel, D. J. Stevenson, D. Day, M. Gu, and K. Dholakia, “Optical redistribution of microparticles and cells between microwells,” Lab Chip 9(10), 1334–1336 (2009).
[Crossref] [PubMed]

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L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, “Efficient optical trapping and visualization of silver nanoparticles,” Nano Lett. 8(5), 1486–1491 (2008).
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Nat. Photonics (2)

A. Chong, W. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010).
[Crossref]

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008).
[Crossref]

Nature (1)

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419(6903), 145–147 (2002).
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Science (2)

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy Beams,” Science 324(5924), 229–232 (2009).
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[Crossref] [PubMed]

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Figures (7)

Fig. 1
Fig. 1

Spherical particle of radius a situated in the evanescent field region z>d . An Airy beam is incident from below with an angle of incidence θ 1 ( > θ crit ) in the substrate.

Fig. 2
Fig. 2

Plots of electric field magnitude |(E)| of 2D Airy evanescent wave above the interface: (a) in the x-z plane (y = 0), (b) at the origin point, as a function of the beam center’s displacements (xc, yc) in the x-y plane (z = 0), while d = 0.8λ, zc = −2d, θ1 = 0.85 rad.

Fig. 3
Fig. 3

Plots of the source function of Airy evanescent field in the x-z plane with a dielectric spherical particle situating on the interface: (a) n3 = 1.59, a polystyrene sphere, (b) n3 = 1.5 + 3.1i, a nickel sphere. The arrows in plots represent power flow of Airy evanescent field. Particle radius a = d = 2λ, xc = yc = 0, zc = −(d + 0.8λ), θ1 = 0.85 rad.

Fig. 4
Fig. 4

Plots of optical forces as a function of beam center’s displacements: (a) xc, while yc = 0, (b) yc, while xc = 0. Particle refractive index n3 = 1.59, a = d = 0.8λ, zc = −2d, θ1 = 0.85 rad.

Fig. 5
Fig. 5

Plots of optical forces as a function of beam center’s displacements: (a) xc, while yc = 0, (b) yc, while xc = 0. Particle refractive index n3 = 1.5 + 3.1i. Other parameters are the same as in Fig. 4.

Fig. 6
Fig. 6

Plots of optical forces as a function of particle radius: (a) n3 = 1.59, (b) n3 = 1.5 + 3.1i. While d = 0.55 μm, xc = yc = 0, zc = −(d + 0.8λ), θ1 = π/3.

Fig. 7
Fig. 7

Plots of optical forces as a function of particle radius with different incident angles: (a) θ1 = 0.85 rad, (b) θ1 = π/3 rad. While n3 = 1.59, d = a, xc = yc = 0, zc = −(d + 0.8λ).

Equations (54)

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k 3 = n 3 k 0 , k 2 = n 2 k 0 .
Y lm ( θ,ϕ )= [ 2l+1 4π ( lm )! ( l+m )! ] 1/2 P l m ( cosθ )exp( imϕ ),
ψ l ( x )=x j l ( x )= ( πx 2 ) 1/2 J v ( x ),
E r ( i ) = 1 r ˜ 2 l=1 m=l l l( l+1 ) A lm ψ l ( α r ˜ ) Y lm ( θ,ϕ ), H r ( i ) = 1 r ˜ 2 l=1 m=l l l( l+1 ) B lm ψ l ( α r ˜ ) Y lm ( θ,ϕ ),
α= k 2 a, r ˜ =r/a,
A lm = 1 l( l+1 ) ψ l ( α ) 0 2π 0 π E r ( i ) ( a,θ,ϕ ) Y lm * ( θ,ϕ )sinθdθdϕ ,
B lm = 1 l( l+1 ) ψ l ( α ) 0 2π 0 π H r ( i ) ( a,θ,ϕ ) Y lm * ( θ,ϕ )sinθdθdϕ ,
F = S n ^ T dS ,
F x +i F y = i α 4 16π k 2 2 l=1 m=l l { [ ( l+m+2 )( l+m+1 ) ( 2l+1 )( 2l+3 ) ] 1/2 l( l+2 )( 2 n 2 2 a lm a l+1,m+1 * + n 2 2 a lm A l+1,m+1 * + n 2 2 A lm a l+1,m+1 * +2 b lm b l+1,m+1 * + b lm B l+1,m+1 * + B lm b l+1,m+1 * )+ [ ( lm+1 )( lm+2 ) ( 2l+1 )( 2l+3 ) ] 1/2 ×l( l+2 )( 2 n 2 2 a l+1,m1 a lm * + n 2 2 a l+1,m1 A lm * + n 2 2 A l+1,m1 a lm * +2 b l+1,m1 b lm * + b l+1,m1 B lm * + B l+1,m1 b lm * ) [ ( l+m+1 )( lm ) ] 1/2 n 2 ( 2 a lm b l,m+1 * +2 b lm a l,m+1 * a lm B l,m+1 * + b lm A l,m+1 * + B lm a l,m+1 * A lm b l,m+1 * ) },
F z = α 4 8π k 2 2 l=1 m=l l { [ ( l+m+2 )( l+m+1 ) ( 2l+1 )( 2l+3 ) ] 1/2 l( l+2 )Im [ 2 n 2 2 a l+1,m a lm * + n 2 2 a l+1,m A lm * + n 2 2 A l+1,m a lm * +2 b l+1,m b lm * + b l+1,m B lm * + B l+1,m b lm * + n 2 m( 2 a lm b lm * + a lm B lm * + A lm b lm * ) ].
E= i n 1 2 k 0 ××A,H=×A,
Φ( s x , s y )exp[ i 3 ( x ¯ 0 3 s x 3 + y ¯ 0 3 s y 3 3 a 0 2 x ¯ 0 s x 3 a 0 2 y ¯ 0 s y ) a 0 ( x ¯ 0 2 s x 2 + y ¯ 0 2 s y 2 ) ].
s x = n x cos θ 1 n z sin θ 1 , s y = n y , s z = n x sin θ 1 + n z cos θ 1 , x ¯ 0 = n 1 k 0 x 0 , y ¯ 0 = n 1 k 0 y 0 .
A= y ^ C i k 0 d s x d s y Φ( s x , s y )exp[ i f i ( x,y,z ) ],
E ( i ) =C d s x d s y n x s ^ i + n y n z p ^ i 1 n z 2 Φ( s x , s y )exp[ i f i ( x,y,z ) ],
H ( i ) = n 1 C d s x d s y n y n z s ^ i n x p ^ i 1 n z 2 Φ( s x , s y )exp[ i f i ( x,y,z ) ],
s ^ i = n y x ^ + n x y ^ , p ^ i =( n x x ^ + n y y ^ ) n z ( 1 n z 2 ) z ^ .
T p = 2 n 1 n z n 1 ξ+ n 2 n z , T s = 2 n 1 n z n 1 n z + n 2 ξ .
E ( t ) =C d s x d s y n x T s s ^ t + n y n z T p p ^ t 1 n z 2 Φ( s x , s y )exp[ i f t ( x,y,z ) ],
H ( t ) = n 2 C d s x d s y n y n z T p s ^ t n x T s p ^ t 1 n z 2 Φ( s x , s y )exp[ i f t ( x,y,z ) ],
s ^ t = n y x ^ + n x y ^ , p ^ t =( n x x ^ + n y y ^ )ξ( n 1 / n 2 )( 1 n z 2 ) z ^ .
f t ( x,y,z )= n 1 k 0 [ n x ( x x c )+ n y ( y y c ) n z ( z c +d ) ]+ n 2 k 0 ( z+d )ξ.
E ( t ) = E x ( t ) x ^ + E y ( t ) y ^ + E z ( t ) z ^ ,
H ( t ) = H x ( t ) x ^ + H y ( t ) y ^ + H z ( t ) z ^ ,
E r ( t ) = E x ( t ) sinθcosϕ+ E y ( t ) sinθsinϕ+ E z ( t ) cosθ,
H r ( t ) = H x ( t ) sinθcosϕ+ H y ( t ) sinθsinϕ+ H z ( t ) cosθ.
A lm = A c d s x d s y Φ( s x , s y )exp[ ig( r 0 ) ] ( n x i n y ) m ( 1 n z 2 ) ( m+1 )/2 [ i n x β lm ( ξ ) T s + n y n z α lm ( ξ ) T p ] ,
B lm = B c d s x d s y Φ( s x , s y )exp[ ig( r 0 ) ] ( n x i n y ) m ( 1 n z 2 ) ( m+1 )/2 [ i n x α lm ( ξ ) T s + n y n z β lm ( ξ ) T p ] ,
A c =C i l+1 l( l+1 ) α 2 π( 2l+1 ) ( lm )! ( l+m )! , B c = n 2 C i l l( l+1 ) α 2 π( 2l+1 ) ( lm )! ( l+m )! ,
g( r 0 )= n 1 k 0 [ n x x c + n y y c + n z ( z c +d ) ]+ n 2 k 0 ξd.
α lm ( ξ )=ξ[ P l m+1 ( ξ )+( l+m )( lm+1 ) P l m1 ( ξ ) ]2m ( 1 ξ 2 ) 1/2 P l m ( ξ ),
β lm ( ξ )= P l m+1 ( ξ )( l+m )( lm+1 ) P l m1 ( ξ ).
a lm = ψ l ( n ˜ α ) ψ l ( α ) n ˜ ψ l ( n ˜ α ) ψ l ( α ) n ˜ ψ l ( n ˜ α ) ξ l ( 1 ) ( α ) ψ l ( n ˜ α ) ξ l ( 1 ) ( α ) A lm ,
b lm = n ˜ ψ l ( n ˜ α ) ψ l ( α ) ψ l ( n ˜ α ) ψ l ( α ) ψ l ( n ˜ α ) ξ l ( 1 ) ( α ) n ˜ ψ l ( n ˜ α ) ξ l ( 1 ) ( α ) B lm ,
c lm = ξ l ( 1 ) ( α ) ψ l ( α ) ξ l ( 1 ) ( α ) ψ l ( α ) n ˜ 2 ψ l ( n ˜ α ) ξ l ( 1 ) ( α ) n ˜ ψ l ( n ˜ α ) ξ l ( 1 ) ( α ) A lm ,
d lm = ξ l ( 1 ) ( α ) ψ l ( α ) ξ l ( 1 ) ( α ) ψ l ( α ) ψ l ( n ˜ α ) ξ l ( 1 ) ( α ) n ˜ ψ l ( n ˜ α ) ξ l ( 1 ) ( α ) B lm .
E r ( i ) = 1 r ˜ 2 l=1 m=l l [ l( l+1 ) A lm ψ l ( α r ˜ ) Y lm ( θ,ϕ ) ] ,
E θ ( i ) = α r ˜ l=1 m=l l ( A lm ψ l ( α r ˜ ) Y lm ( θ,ϕ ) θ m n 2 B lm ψ l ( α r ˜ ) Y lm ( θ,ϕ ) sinθ ) ,
E ϕ ( i ) = α r ˜ l=1 m=l l ( im A lm ψ l ( α r ˜ ) Y lm ( θ,ϕ ) sinθ i n 2 B lm ψ l ( α r ˜ ) Y lm ( θ,ϕ ) θ ) .
H r ( i ) = 1 r ˜ 2 l=1 m=l l [ l( l+1 ) B lm ψ l ( α r ˜ ) Y lm ( θ,ϕ ) ] ,
H θ ( i ) = α r ˜ l=1 m=l l ( B lm ψ l ( α r ˜ ) Y lm ( θ,ϕ ) θ +m n 2 A lm ψ l ( α r ˜ ) Y lm ( θ,ϕ ) sinθ ) ,
H ϕ ( i ) = α r ˜ l=1 m=l l ( im B lm ψ l ( α r ˜ ) Y lm ( θ,ϕ ) sinθ +i n 2 A lm ψ l ( α r ˜ ) Y lm ( θ,ϕ ) θ ) .
E r ( s ) = 1 r ˜ 2 l=1 m=l l [ l( l+1 ) a lm ξ l ( 1 ) ( α r ˜ ) Y lm ( θ,ϕ ) ] ,
E θ ( s ) = α r ˜ l=1 m=l l ( a lm ξ l ( 1 ) ( α r ˜ ) Y lm ( θ,ϕ ) θ m n 2 b lm ξ l ( 1 ) ( α r ˜ ) Y lm ( θ,ϕ ) sinθ ) ,
E ϕ ( s ) = α r ˜ l=1 m=l l ( im a lm ξ l ( 1 ) ( α r ˜ ) Y lm ( θ,ϕ ) sinθ i n 2 b lm ξ l ( 1 ) ( α r ˜ ) Y lm ( θ,ϕ ) θ ) .
H r ( s ) = 1 r ˜ 2 l=1 m=l l [ l( l+1 ) b lm ξ l ( 1 ) ( α r ˜ ) Y lm ( θ,ϕ ) ] ,
H θ ( s ) = α r ˜ l=1 m=l l ( b lm ξ l ( 1 ) ( α r ˜ ) Y lm ( θ,ϕ ) θ +m n 2 a lm ξ l ( 1 ) ( α r ˜ ) Y lm ( θ,ϕ ) sinθ ) ,
H ϕ ( s ) = α r ˜ l=1 m=l l ( im b lm ξ l ( 1 ) ( α r ˜ ) Y lm ( θ,ϕ ) sinθ +i n 2 a lm ξ l ( 1 ) ( α r ˜ ) Y lm ( θ,ϕ ) θ ) .
E r ( w ) = 1 r ˜ 2 l=1 m=l l [ l( l+1 ) c lm ψ l ( n ˜ α r ˜ ) Y lm ( θ,ϕ ) ] ,
E θ ( w ) = α r ˜ l=1 m=l l ( n ˜ c lm ψ l ( n ˜ α r ˜ ) Y lm ( θ,ϕ ) θ m n 2 d lm ψ l ( n ˜ α r ˜ ) Y lm ( θ,ϕ ) sinθ ) ,
E ϕ ( w ) = α r ˜ l=1 m=l l ( im n ˜ c lm ψ l ( n ˜ α r ˜ ) Y lm ( θ,ϕ ) sinθ i n 2 d lm ψ l ( n ˜ α r ˜ ) Y lm ( θ,ϕ ) θ ) .
H r ( w ) = 1 r ˜ 2 l=1 m=l l [ l( l+1 ) d lm ψ l ( n ˜ α r ˜ ) Y lm ( θ,ϕ ) ] ,
H θ ( w ) = α r ˜ l=1 m=l l ( n ˜ d lm ψ l ( n ˜ α r ˜ ) Y lm ( θ,ϕ ) θ +m n 2 n ˜ 2 c lm ψ l ( n ˜ α r ˜ ) Y lm ( θ,ϕ ) sinθ ) ,
H ϕ ( w ) = α r ˜ l=1 m=l l ( im n ˜ d lm ψ l ( n ˜ α r ˜ ) Y lm ( θ,ϕ ) sinθ +i n 2 n ˜ 2 c lm ψ l ( n ˜ α r ˜ ) Y lm ( θ,ϕ ) θ ) .

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