Abstract

The expression of optical forces provoked by an incident light illuminating particles can be deduced from the Lorentz law. It is shown that these forces derive from a scalar potential in the 2D problem and s-polarization, with light propagating in the cross-section plane of the particles, a fact which shows that the separation between gradient and scattering forces could be questioned. This property does not extend to the p-polarization and 3D problem. In the general case, it is shown that one of the components of the optical force is intimately linked with the reactive energy inside the particle. A possible application is given.

© 2007 Optical Society of America

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    [CrossRef]
  2. A. Ashkin,"Optical trapping and manipulation of neutral particles using lasers,"Proc. Natl. Acad. Sci. USA 94, 4853-4860 (1997).
    [CrossRef] [PubMed]
  3. M. Burns, J-M. Fournier and J. Golovshenko,"Optical Binding,"Phys. Rev. Lett. 63, 1233-1236 (1989).
    [CrossRef] [PubMed]
  4. M. Burns, J-M. Fournier and J. Golovshenko, "Lateral binding effect, due to particle's optical interaction,"Science 289, 749-754 (1990).
    [CrossRef]
  5. J-M. Fournier, G. Boer, G. Delacrétaz, P. Jacquot, J. Rohner and R. Salathé,"Building Optical Matter with Binding and Trapping Forces," Proc. SPIE 5514, 309-317 (2004).
    [CrossRef]
  6. W. Singer, M. Frick, S. Bernet and M. Ritsch-Marte,"Self-organized array of regularly spaced microbeads in a fiber-optical trap," J. Opt. Soc. Am. B 20, 1568-1574 (2003).
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  7. S. Tatarkova, A. Carruthers and K. Dholakia,"One-Dimensional Optically Bound Arrays of Microscopic Particles," Phys. Rev. Lett. 89, 283901 (2002).
    [CrossRef]
  8. N. Metzger, K. Dholakia and E. Wright,"Observation of Bistability and Hysteresis in Optical Binding of Two Dielectric Spheres," Phys. Rev. Lett. 96, 068102 (2006).
    [CrossRef] [PubMed]
  9. C. Mellor, C. Bain Chem., "Array Formation in Evanescent Waves," Phys. Chem. 7, 329-332 (2006).
    [CrossRef]
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  14. M. Povinelli, S. Johnson, M. Lonèar, M. Ibanescu, E. Smythe, F. Capasso and J. Joannopoulos,"High-Q enhancement of attractive and repulsive optical forces between coupled whispering-gallery- mode resonators,"Opt. Express 13, 8286-8295 (2005).
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  22. T. M. Grzegorczyk and Jin Au Kong, "Analytical expression of the force due to multiple TM plane wave incidences on an infinite lossless dielectric circular cylinder of arbitrary size," to be published in the J. Opt. Soc. Am. B.
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  26. B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, "Optical momentum transfer to absorbing Mie particles," Phys. Rev. Lett. 97, 133902 (2006).
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    [CrossRef]
  32. B.T. Draine, "The discrete-dipole approximation and its application to interstellar graphite grains,"Astrophys. J. 333, 848-872 (1988).
    [CrossRef]
  33. D. Maystre, "Getting effective permittivity and permeability equal to −1 in 1D dielectric photonic crystals," J. Mod. Opt. 53, 1901-1917 (2006).
    [CrossRef]
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    [CrossRef]
  35. Y. N. Obukhov and F. W. Hehl, "Electromagnetic energy-momentum and forces in matter," Phys. Lett. A 311, 277-284 (2003).
    [CrossRef]
  36. R.  Loudon, "Theory of the radiation pressure on dielectric surfaces," J. Mod. Opt.  49, 812-836 (2002).
    [CrossRef]
  37. R.  Loudon, S. M.  Barnett and C.  Baxter, "Radiation pressure and momentum transfer in dielectrics: the photon drag effect," Phys. Rev. A  71, 063802 (2005).
    [CrossRef]
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2006 (8)

N. Metzger, K. Dholakia and E. Wright,"Observation of Bistability and Hysteresis in Optical Binding of Two Dielectric Spheres," Phys. Rev. Lett. 96, 068102 (2006).
[CrossRef] [PubMed]

C. Mellor, C. Bain Chem., "Array Formation in Evanescent Waves," Phys. Chem. 7, 329-332 (2006).
[CrossRef]

T. Grzegorczyk, B. Kemp, and J. Kong, "Stable optical trapping based on optical binding forces, "Phys. Rev. Lett. 96, 113903 (2006).
[CrossRef] [PubMed]

B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, "Optical momentum transfer to absorbing Mie particles," Phys. Rev. Lett. 97, 133902 (2006).
[CrossRef] [PubMed]

D. Maystre, "Getting effective permittivity and permeability equal to −1 in 1D dielectric photonic crystals," J. Mod. Opt. 53, 1901-1917 (2006).
[CrossRef]

D. Maystre and P. Vincent, "Making photonic crystals using trapping and binding optical forces on particles," J. Opt. A: Pure Appl. Opt. 8, 1059-1066 (2006).
[CrossRef]

N. Metzger, E. Wright, W. Sibbett and K. Dholakia," Visualization of optical binding of microparticles using a femtosecond fiber optical trap," Opt. Express 14, 3677-3687 (2006).
[CrossRef] [PubMed]

T. Grzegorczyk, B. Kemp, and J. Kong, "Trapping and binding of an arbitrary number of cylindrical particles in an in-plane electromagnetic field," J. Opt. Soc. Am. A 23, 2324-2330 (2006).
[CrossRef]

2005 (6)

2004 (3)

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

J-M. Fournier, G. Boer, G. Delacrétaz, P. Jacquot, J. Rohner and R. Salathé,"Building Optical Matter with Binding and Trapping Forces," Proc. SPIE 5514, 309-317 (2004).
[CrossRef]

D. McGloin, A. Carruthers, K. Dholakia and E. Wright,"Optically bound microscopic particles in one. dimension," Phys. Rev. E 69,021403 (2004).
[CrossRef]

2003 (2)

2002 (2)

R.  Loudon, "Theory of the radiation pressure on dielectric surfaces," J. Mod. Opt.  49, 812-836 (2002).
[CrossRef]

S. Tatarkova, A. Carruthers and K. Dholakia,"One-Dimensional Optically Bound Arrays of Microscopic Particles," Phys. Rev. Lett. 89, 283901 (2002).
[CrossRef]

2001 (1)

2000 (1)

1999 (1)

M. Antonoyiannakis and J. Pendry, "Electromagnetic forces in photonic crystals, "Phys. Rev. B 60, 2363-2374 (1999).
[CrossRef]

1997 (2)

E. Lidorikis, Q. Li and C. Soukoulis, "Optical Bistability in Colloidal Crystals," Phys. Rev. E 55, 3613-3618 (1997).
[CrossRef]

A. Ashkin,"Optical trapping and manipulation of neutral particles using lasers,"Proc. Natl. Acad. Sci. USA 94, 4853-4860 (1997).
[CrossRef] [PubMed]

1990 (1)

M. Burns, J-M. Fournier and J. Golovshenko, "Lateral binding effect, due to particle's optical interaction,"Science 289, 749-754 (1990).
[CrossRef]

1989 (1)

M. Burns, J-M. Fournier and J. Golovshenko,"Optical Binding,"Phys. Rev. Lett. 63, 1233-1236 (1989).
[CrossRef] [PubMed]

1988 (1)

B.T. Draine, "The discrete-dipole approximation and its application to interstellar graphite grains,"Astrophys. J. 333, 848-872 (1988).
[CrossRef]

1973 (1)

J. P. Gordon, "Radiation Forces and Momenta in Dielectric Media," Phys. Rev. A 8,14-21 (1973).
[CrossRef]

1972 (1)

Ch. Imbert, "Calculation and Experimental Proof of the Transverse Shift Induced by Total Internal Reflection of a Circularly Polarized Light Beam," Phys. Rev. D 5, 787 - 796 (1972).
[CrossRef]

1970 (1)

A. Ashkin,"Acceleration and Trapping of Particles by Radiation Pressure," Phys. Rev. Lett. 24, 156-159 (1970).
[CrossRef]

Antonoyiannakis, M.

M. Antonoyiannakis and J. Pendry, "Electromagnetic forces in photonic crystals, "Phys. Rev. B 60, 2363-2374 (1999).
[CrossRef]

Ashkin, A.

A. Ashkin,"Optical trapping and manipulation of neutral particles using lasers,"Proc. Natl. Acad. Sci. USA 94, 4853-4860 (1997).
[CrossRef] [PubMed]

A. Ashkin,"Acceleration and Trapping of Particles by Radiation Pressure," Phys. Rev. Lett. 24, 156-159 (1970).
[CrossRef]

Bain Chem, C.

C. Mellor, C. Bain Chem., "Array Formation in Evanescent Waves," Phys. Chem. 7, 329-332 (2006).
[CrossRef]

Barnett, S. M.

R.  Loudon, S. M.  Barnett and C.  Baxter, "Radiation pressure and momentum transfer in dielectrics: the photon drag effect," Phys. Rev. A  71, 063802 (2005).
[CrossRef]

Baxter, C.

R.  Loudon, S. M.  Barnett and C.  Baxter, "Radiation pressure and momentum transfer in dielectrics: the photon drag effect," Phys. Rev. A  71, 063802 (2005).
[CrossRef]

Bernet, S.

Boer, G.

J-M. Fournier, G. Boer, G. Delacrétaz, P. Jacquot, J. Rohner and R. Salathé,"Building Optical Matter with Binding and Trapping Forces," Proc. SPIE 5514, 309-317 (2004).
[CrossRef]

Burns, M.

M. Burns, J-M. Fournier and J. Golovshenko, "Lateral binding effect, due to particle's optical interaction,"Science 289, 749-754 (1990).
[CrossRef]

M. Burns, J-M. Fournier and J. Golovshenko,"Optical Binding,"Phys. Rev. Lett. 63, 1233-1236 (1989).
[CrossRef] [PubMed]

Capasso, F.

Carruthers, A.

D. McGloin, A. Carruthers, K. Dholakia and E. Wright,"Optically bound microscopic particles in one. dimension," Phys. Rev. E 69,021403 (2004).
[CrossRef]

S. Tatarkova, A. Carruthers and K. Dholakia,"One-Dimensional Optically Bound Arrays of Microscopic Particles," Phys. Rev. Lett. 89, 283901 (2002).
[CrossRef]

Chan, J.

Chaumet, P.

Delacrétaz, G.

J-M. Fournier, G. Boer, G. Delacrétaz, P. Jacquot, J. Rohner and R. Salathé,"Building Optical Matter with Binding and Trapping Forces," Proc. SPIE 5514, 309-317 (2004).
[CrossRef]

Dholakia, K.

N. Metzger, K. Dholakia and E. Wright,"Observation of Bistability and Hysteresis in Optical Binding of Two Dielectric Spheres," Phys. Rev. Lett. 96, 068102 (2006).
[CrossRef] [PubMed]

N. Metzger, E. Wright, W. Sibbett and K. Dholakia," Visualization of optical binding of microparticles using a femtosecond fiber optical trap," Opt. Express 14, 3677-3687 (2006).
[CrossRef] [PubMed]

D. McGloin, A. Carruthers, K. Dholakia and E. Wright,"Optically bound microscopic particles in one. dimension," Phys. Rev. E 69,021403 (2004).
[CrossRef]

S. Tatarkova, A. Carruthers and K. Dholakia,"One-Dimensional Optically Bound Arrays of Microscopic Particles," Phys. Rev. Lett. 89, 283901 (2002).
[CrossRef]

Draine, B.T.

B.T. Draine, "The discrete-dipole approximation and its application to interstellar graphite grains,"Astrophys. J. 333, 848-872 (1988).
[CrossRef]

Fournier, J-M.

J-M. Fournier, G. Boer, G. Delacrétaz, P. Jacquot, J. Rohner and R. Salathé,"Building Optical Matter with Binding and Trapping Forces," Proc. SPIE 5514, 309-317 (2004).
[CrossRef]

M. Burns, J-M. Fournier and J. Golovshenko, "Lateral binding effect, due to particle's optical interaction,"Science 289, 749-754 (1990).
[CrossRef]

M. Burns, J-M. Fournier and J. Golovshenko,"Optical Binding,"Phys. Rev. Lett. 63, 1233-1236 (1989).
[CrossRef] [PubMed]

Frick, M.

Golovshenko, J.

M. Burns, J-M. Fournier and J. Golovshenko, "Lateral binding effect, due to particle's optical interaction,"Science 289, 749-754 (1990).
[CrossRef]

M. Burns, J-M. Fournier and J. Golovshenko,"Optical Binding,"Phys. Rev. Lett. 63, 1233-1236 (1989).
[CrossRef] [PubMed]

Gordon, J. P.

J. P. Gordon, "Radiation Forces and Momenta in Dielectric Media," Phys. Rev. A 8,14-21 (1973).
[CrossRef]

Grzegorczyk, T.

Grzegorczyk, T. M.

B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, "Optical momentum transfer to absorbing Mie particles," Phys. Rev. Lett. 97, 133902 (2006).
[CrossRef] [PubMed]

B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, "Ab initio study of the radiation pressure on dielectric and magnetic media," Opt. Express 13, 9280-9291 (2005).
[CrossRef] [PubMed]

Hehl, F. W.

Y. N. Obukhov and F. W. Hehl, "Electromagnetic energy-momentum and forces in matter," Phys. Lett. A 311, 277-284 (2003).
[CrossRef]

Ibanescu, M.

Imbert, Ch.

Ch. Imbert, "Calculation and Experimental Proof of the Transverse Shift Induced by Total Internal Reflection of a Circularly Polarized Light Beam," Phys. Rev. D 5, 787 - 796 (1972).
[CrossRef]

Jacquot, P.

J-M. Fournier, G. Boer, G. Delacrétaz, P. Jacquot, J. Rohner and R. Salathé,"Building Optical Matter with Binding and Trapping Forces," Proc. SPIE 5514, 309-317 (2004).
[CrossRef]

Joannopoulos, J.

Johnson, S.

Kemp, B.

Kemp, B. A.

B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, "Optical momentum transfer to absorbing Mie particles," Phys. Rev. Lett. 97, 133902 (2006).
[CrossRef] [PubMed]

B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, "Ab initio study of the radiation pressure on dielectric and magnetic media," Opt. Express 13, 9280-9291 (2005).
[CrossRef] [PubMed]

Kong, J.

Kong, J. A.

B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, "Optical momentum transfer to absorbing Mie particles," Phys. Rev. Lett. 97, 133902 (2006).
[CrossRef] [PubMed]

B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, "Ab initio study of the radiation pressure on dielectric and magnetic media," Opt. Express 13, 9280-9291 (2005).
[CrossRef] [PubMed]

Li, Q.

E. Lidorikis, Q. Li and C. Soukoulis, "Optical Bistability in Colloidal Crystals," Phys. Rev. E 55, 3613-3618 (1997).
[CrossRef]

Lidorikis, E.

E. Lidorikis, Q. Li and C. Soukoulis, "Optical Bistability in Colloidal Crystals," Phys. Rev. E 55, 3613-3618 (1997).
[CrossRef]

Lonèar, M.

Loudon, R.

R.  Loudon, S. M.  Barnett and C.  Baxter, "Radiation pressure and momentum transfer in dielectrics: the photon drag effect," Phys. Rev. A  71, 063802 (2005).
[CrossRef]

R.  Loudon, "Theory of the radiation pressure on dielectric surfaces," J. Mod. Opt.  49, 812-836 (2002).
[CrossRef]

Mansuripur, M.

Maystre, D.

D. Maystre, "Getting effective permittivity and permeability equal to −1 in 1D dielectric photonic crystals," J. Mod. Opt. 53, 1901-1917 (2006).
[CrossRef]

D. Maystre and P. Vincent, "Making photonic crystals using trapping and binding optical forces on particles," J. Opt. A: Pure Appl. Opt. 8, 1059-1066 (2006).
[CrossRef]

McGloin, D.

D. McGloin, A. Carruthers, K. Dholakia and E. Wright,"Optically bound microscopic particles in one. dimension," Phys. Rev. E 69,021403 (2004).
[CrossRef]

Mellor, C.

C. Mellor, C. Bain Chem., "Array Formation in Evanescent Waves," Phys. Chem. 7, 329-332 (2006).
[CrossRef]

Metzger, N.

N. Metzger, K. Dholakia and E. Wright,"Observation of Bistability and Hysteresis in Optical Binding of Two Dielectric Spheres," Phys. Rev. Lett. 96, 068102 (2006).
[CrossRef] [PubMed]

N. Metzger, E. Wright, W. Sibbett and K. Dholakia," Visualization of optical binding of microparticles using a femtosecond fiber optical trap," Opt. Express 14, 3677-3687 (2006).
[CrossRef] [PubMed]

Moloney, J. V.

Ng, J.

Nieto-Vesperinas, M.

Obukhov, Y. N.

Y. N. Obukhov and F. W. Hehl, "Electromagnetic energy-momentum and forces in matter," Phys. Lett. A 311, 277-284 (2003).
[CrossRef]

Pendry, J.

M. Antonoyiannakis and J. Pendry, "Electromagnetic forces in photonic crystals, "Phys. Rev. B 60, 2363-2374 (1999).
[CrossRef]

Povinelli, M.

Raabe, C.

C. Raabe and D. G. Welsch, "Casimir force acting on magnetodielectric bodies embedded in media," Phys. Rev. A 71, 013814 (2005).
[CrossRef]

Ritsch-Marte, M.

Rohner, J.

J-M. Fournier, G. Boer, G. Delacrétaz, P. Jacquot, J. Rohner and R. Salathé,"Building Optical Matter with Binding and Trapping Forces," Proc. SPIE 5514, 309-317 (2004).
[CrossRef]

Rohrbach, A.

Salathé, R.

J-M. Fournier, G. Boer, G. Delacrétaz, P. Jacquot, J. Rohner and R. Salathé,"Building Optical Matter with Binding and Trapping Forces," Proc. SPIE 5514, 309-317 (2004).
[CrossRef]

Sibbett, W.

Singer, W.

Smythe, E.

Soukoulis, C.

E. Lidorikis, Q. Li and C. Soukoulis, "Optical Bistability in Colloidal Crystals," Phys. Rev. E 55, 3613-3618 (1997).
[CrossRef]

Stelzer, E.

Tatarkova, S.

S. Tatarkova, A. Carruthers and K. Dholakia,"One-Dimensional Optically Bound Arrays of Microscopic Particles," Phys. Rev. Lett. 89, 283901 (2002).
[CrossRef]

Vincent, P.

D. Maystre and P. Vincent, "Making photonic crystals using trapping and binding optical forces on particles," J. Opt. A: Pure Appl. Opt. 8, 1059-1066 (2006).
[CrossRef]

Welsch, D. G.

C. Raabe and D. G. Welsch, "Casimir force acting on magnetodielectric bodies embedded in media," Phys. Rev. A 71, 013814 (2005).
[CrossRef]

Wright, E.

N. Metzger, K. Dholakia and E. Wright,"Observation of Bistability and Hysteresis in Optical Binding of Two Dielectric Spheres," Phys. Rev. Lett. 96, 068102 (2006).
[CrossRef] [PubMed]

N. Metzger, E. Wright, W. Sibbett and K. Dholakia," Visualization of optical binding of microparticles using a femtosecond fiber optical trap," Opt. Express 14, 3677-3687 (2006).
[CrossRef] [PubMed]

D. McGloin, A. Carruthers, K. Dholakia and E. Wright,"Optically bound microscopic particles in one. dimension," Phys. Rev. E 69,021403 (2004).
[CrossRef]

Zakharian, A. R.

Astrophys. J. (1)

B.T. Draine, "The discrete-dipole approximation and its application to interstellar graphite grains,"Astrophys. J. 333, 848-872 (1988).
[CrossRef]

J. Mod. Opt. (2)

D. Maystre, "Getting effective permittivity and permeability equal to −1 in 1D dielectric photonic crystals," J. Mod. Opt. 53, 1901-1917 (2006).
[CrossRef]

R.  Loudon, "Theory of the radiation pressure on dielectric surfaces," J. Mod. Opt.  49, 812-836 (2002).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

D. Maystre and P. Vincent, "Making photonic crystals using trapping and binding optical forces on particles," J. Opt. A: Pure Appl. Opt. 8, 1059-1066 (2006).
[CrossRef]

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

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

Opt. Express (5)

Opt. Lett. (2)

Phys. Chem. (1)

C. Mellor, C. Bain Chem., "Array Formation in Evanescent Waves," Phys. Chem. 7, 329-332 (2006).
[CrossRef]

Phys. Lett. A (1)

Y. N. Obukhov and F. W. Hehl, "Electromagnetic energy-momentum and forces in matter," Phys. Lett. A 311, 277-284 (2003).
[CrossRef]

Phys. Rev. A (3)

J. P. Gordon, "Radiation Forces and Momenta in Dielectric Media," Phys. Rev. A 8,14-21 (1973).
[CrossRef]

R.  Loudon, S. M.  Barnett and C.  Baxter, "Radiation pressure and momentum transfer in dielectrics: the photon drag effect," Phys. Rev. A  71, 063802 (2005).
[CrossRef]

C. Raabe and D. G. Welsch, "Casimir force acting on magnetodielectric bodies embedded in media," Phys. Rev. A 71, 013814 (2005).
[CrossRef]

Phys. Rev. B (1)

M. Antonoyiannakis and J. Pendry, "Electromagnetic forces in photonic crystals, "Phys. Rev. B 60, 2363-2374 (1999).
[CrossRef]

Phys. Rev. D (1)

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Phys. Rev. E (2)

D. McGloin, A. Carruthers, K. Dholakia and E. Wright,"Optically bound microscopic particles in one. dimension," Phys. Rev. E 69,021403 (2004).
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[CrossRef]

Phys. Rev. Lett. (6)

B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, "Optical momentum transfer to absorbing Mie particles," Phys. Rev. Lett. 97, 133902 (2006).
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T. Grzegorczyk, B. Kemp, and J. Kong, "Stable optical trapping based on optical binding forces, "Phys. Rev. Lett. 96, 113903 (2006).
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N. Metzger, K. Dholakia and E. Wright,"Observation of Bistability and Hysteresis in Optical Binding of Two Dielectric Spheres," Phys. Rev. Lett. 96, 068102 (2006).
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Proc. SPIE (1)

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D. Maystre and P. Vincent, "Phenomenological study of binding in optically trapped photonic crystals," submitted to the J. Opt. Soc. Am. A.

T. M. Grzegorczyk and Jin Au Kong, "Analytical expression of the force due to multiple TM plane wave incidences on an infinite lossless dielectric circular cylinder of arbitrary size," to be published in the J. Opt. Soc. Am. B.

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Figures (3)

Fig.1
Fig.1

Notations

Fig.2.
Fig.2.

he dielectric slab

Fig. 3.
Fig. 3.

Square modulus of the field (SI) at the boundary of a circular cylinder illuminated by a plane wave

Equations (54)

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× E i ω μ 0 H = 0
× H + i ω ε E = 0
× H + i ω ε 0 E = i ω ( ε 0 ε ) E
× H + i ω ε 0 E = j
j = i ω ( ε 0 ε ) E
. j = i ω ρ
ρ = ε 0 . E
f M V = μ 0 2 Re { j × H * }
= ω ( ε ε 0 ) μ 0 2 Im { E × H * }
f E V = 1 2 Re { ρ E * } = ε 0 2 Re { E * . E }
f M = ω μ 0 ( ε r ε 0 ) 2 Ω Im { E × H * } d V
E = E t + n E n
f E , t * = ε 0 2 Re { E t * . E }
. E = n . [ E ] δ Ω
all space u δ Ω d V = Ω u ( M ) d S
f E , t = all space ε 0 2 Re { E t * ( n . [ E ] ) } δ Ω d V = ε 0 2 Ω Re { [ E n ] E t * } d s
ε 0 E n out = ε r E n in
f E , t = ( ε r ε 0 ) 2 Ω Re { E n in E t * } d S
f E , n V = ε 0 2 Re { n E n * . ( E t + n E n ) }
f E , n V = ε 0 2 Re { n E n * . ( n E n ) }
f E , n V = ε 0 2 Re { n ( ε E n * ) 1 ε . ( n ε E n ε ) }
f E , n V = ε 0 2 Re { n ( ε E n * ) [ 1 ε ( 1 ε ) . n ε E n ] }
= ε 0 4 Re { n ( ε E n * ) [ ( 1 ε 2 ) . n ε E n ] }
( 1 ε 2 ) ( 1 ε 0 2 1 ε r 2 ) n δ Ω
f E , n V = ε 0 ε r 2 4 ( 1 ε 0 2 1 ε r 2 ) E n in 2 . n δ Ω
= ε 0 4 ( ε r 2 ε 0 2 1 ) E n in 2 n δ Ω
f E , n = ε 0 4 ( ε r 2 ε 0 2 1 ) Ω E n in 2 n d s
f E , n = ε 0 4 Ω ( ε r ε 0 + 1 ) n E n * in ( ε r ε 0 1 ) E n in d s
f E , n = 1 2 Ω n E n * ε 0 ( E n out E n in ) d s
f E , n = 1 2 Ω n E n * σ d s
f = f M + f E , t + f E , n = ω μ 0 ( ε r ε 0 ) 2 Ω Im { E × H * } d V
+ ( ε r ε 0 ) 2 Ω Re { E n in E t * } d s + ( ε r 2 ε 0 2 4 ε 0 ) Ω E n in 2 n d s
E xy = i ω ε r z ̂ × H z
H xy = i ω μ 0 z ̂ × E z
f = f xy + z ̂ f z
f xy = Ω UdV + ( ε r ε 0 ) 2 Ω Re { E n in E t , xy * } d s
+ ( ε r s ε 0 2 4 ε 0 ) Ω E n in 2 n d S
U = 1 4 ( 1 ε 0 ε r ) ( μ 0 H z 2 ε r E z 2 )
f z = 1 2 ω ( ε r ε 0 ε r ) Ω Im [ z ̂ . ( H z × E z * ) ] d V
+ ( ε r ε 0 ) 2 Ω Re { E n in E z * } d S
f xy = Ω U n d S + ( ε r ε 0 ) 2 Ω Re { E n in E t , xy * } d S
+ ( ε r 2 ε 0 2 4 ε 0 ) Ω E n in 2 n d S
b = ρ a
ρ = v r 1 v r + 1
E z = a [ exp ( i k 0 v r y ) + ρ exp ( + i k 0 v r y ) ]
E z 2 = E 2 = a 2 exp ( i k 0 v r y ) + ρ exp ( + i k 0 v r y ) 2
= a 2 1 + ρ exp ( + 2 i k 0 v r y ) 2
y = p λ 0 2 v r , p = 0 , 1 , 2 ,
J = Ω U n d S
J = ε r ε 0 4 Ω R cos ( θ ) E z 2 d θ
Φ = 1 2 Ω n . ( E × H * ) d S
Φ = 1 2 Ω . ( E × H * ) d V
= 1 2 Ω ( H * . × E E . × H * ) d V
Φ = 2 i ω Ω ( μ 0 4 H 2 ε r 4 E 2 ) d V = 2 i ω Ω ( W M W E ) d V

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