Abstract

Light-induced forces between metal nanoparticles change the geometry of the aggregates and affect their optical properties. Light absorption, scattering and scattering of a probe beam are numerically studied with Newton’s equations and the coupled dipole equations for penta-particle aggregates. The relative changes in optical responses are large compared with the linear, low-intensity limit and relatively fast with nanosecond characteristic times. Time and intensity dependencies are shown to be sensitive to the initial potential of the aggregation forces.

© 2007 Optical Society of America

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    [CrossRef]
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  27. P.B. Johnson and R.W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
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  28. C. Kittel, Introduction to Solid State Physics (Wiley: New York, 1995).
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    [CrossRef]
  32. Yu.E. Danilova and V.P. Safonov, “Absorption spectra and photomodification of silver fractal clusters,” in Fractal Reviews in the Natural and Applied Sciences, M.M. Novak, ed. (Chapman and Hall: London, 1995), pp. 101–111.
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    [CrossRef]

2007 (1)

2006 (3)

A. S. Zelenina, R. Quidant, G. Badenes, and M. Nieto-Vesperinas, “Tunable optical sorting and manipulation of nanoparticles via plasmon excitation,” Opt. Lett. 31, 2054–2056 (2006).
[CrossRef] [PubMed]

D. Pissuwan, S.M. Valenzuela, and M.B. Cortie, “Therapeutic possibilities of plasmonically heated gold nanopar-ticles,” Trends in Biotechnology 24, 62–67 (2006).
[CrossRef]

A.K. Buin, P.F. de Chatel, H. Nakotte, V.P. Drachev, and V.M. Shalaev, “Saturation effect in the optical response of Ag-nanoparticle fractal aggregates,” Phys. Rev. B 73, 035438–35449 (2006).
[CrossRef]

2005 (3)

I.H. El-Sayed, X. Huang, and M.A. El-Sayed, “Surface plasmon resonance scattering and absorption of anti EGFR antibody conjugated gold nanoparticles in cancer diagnostics: Applications in oral cancer,” Nano Lett. 5, 829–834 (2005).
[CrossRef] [PubMed]

V.P. Zharov and D.O. Lapotko, “Photothermal imaging of nanoparticles (review),” IEEE J. Sel. Top. Quantum Electron. 11, 733–751 (2005).
[CrossRef]

C. Sonnichsen, B.M. Reinhard, J. Liphardt, and A.P. Alivisatos, “A molecular ruler based on plasmon coupling of single gold and silver nanoparticles,” Nature Biotechnol. 23, 741–745 (2005).
[CrossRef]

2004 (2)

S.V. Perminov, S.G. Rautian, and V.P. Safonov, “On the Theory of Optical Properties of Fractal Clusters,” Sov. Phys. JETP 98, 691–704 (2004).
[CrossRef]

V.A. Markel, V.N. Pustovit, S.V. Karpov, A.V. Obuschenko, V.S. Gerasimov, and I.L. Isaev, “Electromagnetic density of states and absorption of radiation by aggregates of nanospheres with multipole interactions,” Phys. Rev. B 70, 054202–054221 (2004).
[CrossRef]

2003 (1)

S.V. Perminov, S.G. Rautian, and V.P. Safonov, “A model of pair interactions in the theory of optical properties of fractal clusters,” Opt. Spectrosc. 95, 416–420 (2003).
[CrossRef]

2002 (2)

H. Xu and M. Käll, “Surface-Plasmon-Enhanced Optical Forces in Silver Nanoaggregates,” Phys. Rev. Lett. 89, 246802–4 (2002).
[CrossRef] [PubMed]

V.P. Drachev, S.V. Perminov, S.G. Rautian, V.P. Safonov, and E.N. Khaliullin, “Polarization effects in nanoaggre-gates of silver caused by local and nonlocal nonlinear-optical responses,” Sov. Phys. JETP 94, 901–915 (2002).
[CrossRef]

2001 (3)

A. Hilger, M. Tenfelde, and U. Kreibig, “Silver nanoparticles deposited on dielectric surfaces,” Appl. Phys. B 73, 361–372 (2001).
[CrossRef]

A.J. Hoekstra, M. Frijlink, L.B.F.M. Waters, and P.M.A. Sloot, “Radiation forces in the discrete-dipole approximation,” J. Opt. Soc. Am. A 18, 1944–1953 (2001).
[CrossRef]

M. Quinten, “Local fields close to the surface of nanoparticles and aggregates of nanoparticles,” Appl. Phys. B 73, 245–255 (2001).
[CrossRef]

2000 (2)

A.A. Lazarides and G.C. Schatz, “DNA-Linked Metal Nanosphere Materials: Structural Basis for the Optical Properties,” J. Phys. Chem. B 104, 460–467 (2000).
[CrossRef]

I.E. Mazets, “Polarization of two close metal spheres in an external homogeneous electric field,” Sov. Phys. Tech. Phys. 451238–1240 (2000).

1999 (1)

B. Cappella and G. Dietler, “Force-distance curves by atomic force microscopy,” Surface Science reports 341–104 (1999).
[CrossRef]

1998 (2)

H. Kimura and I. J. Mann “Radiation pressure cross section for fluffy aggregates,” Quant. Spectrosc. Radiat. Transf. 60, 425–438 (1998).
[CrossRef]

B.T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1998).
[CrossRef]

1997 (3)

J.F. Marco, “Supercoiled and braided DNA under tension,” Phys. Rev. E 55, 1758–1772 (1997).
[CrossRef]

M.I. Stockman, “Chaos and Spatial Correlations for Dipolar Eigenproblems,” Phys. Rev. Lett. 79, 4562–4565 (1997).
[CrossRef]

M.I. Stockman, “Inhomogeneous eigenmode localization, chaos, and correlations in large disordered clusters,” Phys. Rev. E 56, 6494–6507 (1997).
[CrossRef]

1996 (2)

V.A. Markel, V.M. Shalaev, E.B. Stechel, W. Kim, and R.L. Armstrong, “Small-particle composites. I. Linear optical properties,” Phys. Rev. B 53, 2425–2436 (1996).
[CrossRef]

B.T. Draine and J.C. Weihgarther, “Radiative torques on interstellar grains I. Superthermal spin-up,” Astrophys. J. 470, 551–565 (1996).
[CrossRef]

1995 (1)

Yu.E. Danilova and V.P. Safonov, “Absorption spectra and photomodification of silver fractal clusters,” in Fractal Reviews in the Natural and Applied Sciences, M.M. Novak, ed. (Chapman and Hall: London, 1995), pp. 101–111.

1994 (1)

F. Claro and R. Rojas, “Novel laser induced interaction profiles in clusters of mesoscopic particles,” Appl. Phys. Lett. 65, 2743–2745 (1994).
[CrossRef]

1991 (1)

V.A. Markel, L.S. Muratov, M.I. Stockman, and T.F. George, “Theory and numerical simulation of optical properties of fractal clusters,” Phys. Rev. B 43, 8183–8195 (1991).
[CrossRef]

1982 (1)

J.M. Gerardy and M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations. II. Optical properties of aggregated metal spheres,” Phys. Rev. B 25, 4204–4229 (1982).
[CrossRef]

1972 (1)

P.B. Johnson and R.W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Alivisatos, A.P.

C. Sonnichsen, B.M. Reinhard, J. Liphardt, and A.P. Alivisatos, “A molecular ruler based on plasmon coupling of single gold and silver nanoparticles,” Nature Biotechnol. 23, 741–745 (2005).
[CrossRef]

Armstrong, R.L.

V.A. Markel, V.M. Shalaev, E.B. Stechel, W. Kim, and R.L. Armstrong, “Small-particle composites. I. Linear optical properties,” Phys. Rev. B 53, 2425–2436 (1996).
[CrossRef]

Ausloos, M.

J.M. Gerardy and M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations. II. Optical properties of aggregated metal spheres,” Phys. Rev. B 25, 4204–4229 (1982).
[CrossRef]

Badenes, G.

Buin, A.K.

A.K. Buin, P.F. de Chatel, H. Nakotte, V.P. Drachev, and V.M. Shalaev, “Saturation effect in the optical response of Ag-nanoparticle fractal aggregates,” Phys. Rev. B 73, 035438–35449 (2006).
[CrossRef]

Cappella, B.

B. Cappella and G. Dietler, “Force-distance curves by atomic force microscopy,” Surface Science reports 341–104 (1999).
[CrossRef]

Christy, R.W.

P.B. Johnson and R.W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Claro, F.

F. Claro and R. Rojas, “Novel laser induced interaction profiles in clusters of mesoscopic particles,” Appl. Phys. Lett. 65, 2743–2745 (1994).
[CrossRef]

Cortie, M.B.

D. Pissuwan, S.M. Valenzuela, and M.B. Cortie, “Therapeutic possibilities of plasmonically heated gold nanopar-ticles,” Trends in Biotechnology 24, 62–67 (2006).
[CrossRef]

Danilova, Yu.E.

Yu.E. Danilova and V.P. Safonov, “Absorption spectra and photomodification of silver fractal clusters,” in Fractal Reviews in the Natural and Applied Sciences, M.M. Novak, ed. (Chapman and Hall: London, 1995), pp. 101–111.

de Chatel, P.F.

A.K. Buin, P.F. de Chatel, H. Nakotte, V.P. Drachev, and V.M. Shalaev, “Saturation effect in the optical response of Ag-nanoparticle fractal aggregates,” Phys. Rev. B 73, 035438–35449 (2006).
[CrossRef]

Dietler, G.

B. Cappella and G. Dietler, “Force-distance curves by atomic force microscopy,” Surface Science reports 341–104 (1999).
[CrossRef]

Drachev, V.P.

A.K. Buin, P.F. de Chatel, H. Nakotte, V.P. Drachev, and V.M. Shalaev, “Saturation effect in the optical response of Ag-nanoparticle fractal aggregates,” Phys. Rev. B 73, 035438–35449 (2006).
[CrossRef]

V.P. Drachev, S.V. Perminov, S.G. Rautian, V.P. Safonov, and E.N. Khaliullin, “Polarization effects in nanoaggre-gates of silver caused by local and nonlocal nonlinear-optical responses,” Sov. Phys. JETP 94, 901–915 (2002).
[CrossRef]

Draine, B.T.

B.T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1998).
[CrossRef]

B.T. Draine and J.C. Weihgarther, “Radiative torques on interstellar grains I. Superthermal spin-up,” Astrophys. J. 470, 551–565 (1996).
[CrossRef]

El-Sayed, I.H.

I.H. El-Sayed, X. Huang, and M.A. El-Sayed, “Surface plasmon resonance scattering and absorption of anti EGFR antibody conjugated gold nanoparticles in cancer diagnostics: Applications in oral cancer,” Nano Lett. 5, 829–834 (2005).
[CrossRef] [PubMed]

El-Sayed, M.A.

I.H. El-Sayed, X. Huang, and M.A. El-Sayed, “Surface plasmon resonance scattering and absorption of anti EGFR antibody conjugated gold nanoparticles in cancer diagnostics: Applications in oral cancer,” Nano Lett. 5, 829–834 (2005).
[CrossRef] [PubMed]

Frijlink, M.

George, T.F.

V.A. Markel, L.S. Muratov, M.I. Stockman, and T.F. George, “Theory and numerical simulation of optical properties of fractal clusters,” Phys. Rev. B 43, 8183–8195 (1991).
[CrossRef]

Gerardy, J.M.

J.M. Gerardy and M. Ausloos, “Absorption spectrum of clusters of spheres from the general solution of Maxwell’s equations. II. Optical properties of aggregated metal spheres,” Phys. Rev. B 25, 4204–4229 (1982).
[CrossRef]

Gerasimov, V.S.

V.A. Markel, V.N. Pustovit, S.V. Karpov, A.V. Obuschenko, V.S. Gerasimov, and I.L. Isaev, “Electromagnetic density of states and absorption of radiation by aggregates of nanospheres with multipole interactions,” Phys. Rev. B 70, 054202–054221 (2004).
[CrossRef]

Hilger, A.

A. Hilger, M. Tenfelde, and U. Kreibig, “Silver nanoparticles deposited on dielectric surfaces,” Appl. Phys. B 73, 361–372 (2001).
[CrossRef]

Hoekstra, A.J.

Huang, X.

I.H. El-Sayed, X. Huang, and M.A. El-Sayed, “Surface plasmon resonance scattering and absorption of anti EGFR antibody conjugated gold nanoparticles in cancer diagnostics: Applications in oral cancer,” Nano Lett. 5, 829–834 (2005).
[CrossRef] [PubMed]

Isaev, I.L.

V.A. Markel, V.N. Pustovit, S.V. Karpov, A.V. Obuschenko, V.S. Gerasimov, and I.L. Isaev, “Electromagnetic density of states and absorption of radiation by aggregates of nanospheres with multipole interactions,” Phys. Rev. B 70, 054202–054221 (2004).
[CrossRef]

Johnson, P.B.

P.B. Johnson and R.W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Käll, M.

H. Xu and M. Käll, “Surface-Plasmon-Enhanced Optical Forces in Silver Nanoaggregates,” Phys. Rev. Lett. 89, 246802–4 (2002).
[CrossRef] [PubMed]

Karpov, S.V.

V.A. Markel, V.N. Pustovit, S.V. Karpov, A.V. Obuschenko, V.S. Gerasimov, and I.L. Isaev, “Electromagnetic density of states and absorption of radiation by aggregates of nanospheres with multipole interactions,” Phys. Rev. B 70, 054202–054221 (2004).
[CrossRef]

Khaliullin, E.N.

V.P. Drachev, S.V. Perminov, S.G. Rautian, V.P. Safonov, and E.N. Khaliullin, “Polarization effects in nanoaggre-gates of silver caused by local and nonlocal nonlinear-optical responses,” Sov. Phys. JETP 94, 901–915 (2002).
[CrossRef]

Kim, W.

V.A. Markel, V.M. Shalaev, E.B. Stechel, W. Kim, and R.L. Armstrong, “Small-particle composites. I. Linear optical properties,” Phys. Rev. B 53, 2425–2436 (1996).
[CrossRef]

Kimura, H.

H. Kimura and I. J. Mann “Radiation pressure cross section for fluffy aggregates,” Quant. Spectrosc. Radiat. Transf. 60, 425–438 (1998).
[CrossRef]

Kittel, C.

C. Kittel, Introduction to Solid State Physics (Wiley: New York, 1995).

Kreibig, U.

A. Hilger, M. Tenfelde, and U. Kreibig, “Silver nanoparticles deposited on dielectric surfaces,” Appl. Phys. B 73, 361–372 (2001).
[CrossRef]

U. Kreibig and M. Vollmer, Optical properties of metal clusters (Springer Verlag: Berlin Heidelberg New-York, 1995).

Landau, L.D.

L.D. Landau and E.M. Lifshitz, Classical Theory of Fields (3rd ed.) (Pergamon: London, 1971).

Lapotko, D.O.

V.P. Zharov and D.O. Lapotko, “Photothermal imaging of nanoparticles (review),” IEEE J. Sel. Top. Quantum Electron. 11, 733–751 (2005).
[CrossRef]

Lazarides, A.A.

A.A. Lazarides and G.C. Schatz, “DNA-Linked Metal Nanosphere Materials: Structural Basis for the Optical Properties,” J. Phys. Chem. B 104, 460–467 (2000).
[CrossRef]

Lifshitz, E.M.

L.D. Landau and E.M. Lifshitz, Classical Theory of Fields (3rd ed.) (Pergamon: London, 1971).

Liphardt, J.

C. Sonnichsen, B.M. Reinhard, J. Liphardt, and A.P. Alivisatos, “A molecular ruler based on plasmon coupling of single gold and silver nanoparticles,” Nature Biotechnol. 23, 741–745 (2005).
[CrossRef]

Mann, I. J.

H. Kimura and I. J. Mann “Radiation pressure cross section for fluffy aggregates,” Quant. Spectrosc. Radiat. Transf. 60, 425–438 (1998).
[CrossRef]

Marco, J.F.

J.F. Marco, “Supercoiled and braided DNA under tension,” Phys. Rev. E 55, 1758–1772 (1997).
[CrossRef]

Markel, V.A.

V.A. Markel, V.N. Pustovit, S.V. Karpov, A.V. Obuschenko, V.S. Gerasimov, and I.L. Isaev, “Electromagnetic density of states and absorption of radiation by aggregates of nanospheres with multipole interactions,” Phys. Rev. B 70, 054202–054221 (2004).
[CrossRef]

V.A. Markel, V.M. Shalaev, E.B. Stechel, W. Kim, and R.L. Armstrong, “Small-particle composites. I. Linear optical properties,” Phys. Rev. B 53, 2425–2436 (1996).
[CrossRef]

V.A. Markel, L.S. Muratov, M.I. Stockman, and T.F. George, “Theory and numerical simulation of optical properties of fractal clusters,” Phys. Rev. B 43, 8183–8195 (1991).
[CrossRef]

Mazets, I.E.

I.E. Mazets, “Polarization of two close metal spheres in an external homogeneous electric field,” Sov. Phys. Tech. Phys. 451238–1240 (2000).

Muratov, L.S.

V.A. Markel, L.S. Muratov, M.I. Stockman, and T.F. George, “Theory and numerical simulation of optical properties of fractal clusters,” Phys. Rev. B 43, 8183–8195 (1991).
[CrossRef]

Nakotte, H.

A.K. Buin, P.F. de Chatel, H. Nakotte, V.P. Drachev, and V.M. Shalaev, “Saturation effect in the optical response of Ag-nanoparticle fractal aggregates,” Phys. Rev. B 73, 035438–35449 (2006).
[CrossRef]

Nieto-Vesperinas, M.

Obuschenko, A.V.

V.A. Markel, V.N. Pustovit, S.V. Karpov, A.V. Obuschenko, V.S. Gerasimov, and I.L. Isaev, “Electromagnetic density of states and absorption of radiation by aggregates of nanospheres with multipole interactions,” Phys. Rev. B 70, 054202–054221 (2004).
[CrossRef]

Perminov, S.V.

S.V. Perminov, S.G. Rautian, and V.P. Safonov, “On the Theory of Optical Properties of Fractal Clusters,” Sov. Phys. JETP 98, 691–704 (2004).
[CrossRef]

S.V. Perminov, S.G. Rautian, and V.P. Safonov, “A model of pair interactions in the theory of optical properties of fractal clusters,” Opt. Spectrosc. 95, 416–420 (2003).
[CrossRef]

V.P. Drachev, S.V. Perminov, S.G. Rautian, V.P. Safonov, and E.N. Khaliullin, “Polarization effects in nanoaggre-gates of silver caused by local and nonlocal nonlinear-optical responses,” Sov. Phys. JETP 94, 901–915 (2002).
[CrossRef]

Pissuwan, D.

D. Pissuwan, S.M. Valenzuela, and M.B. Cortie, “Therapeutic possibilities of plasmonically heated gold nanopar-ticles,” Trends in Biotechnology 24, 62–67 (2006).
[CrossRef]

Pustovit, V.N.

V.A. Markel, V.N. Pustovit, S.V. Karpov, A.V. Obuschenko, V.S. Gerasimov, and I.L. Isaev, “Electromagnetic density of states and absorption of radiation by aggregates of nanospheres with multipole interactions,” Phys. Rev. B 70, 054202–054221 (2004).
[CrossRef]

Quidant, R.

Quinten, M.

M. Quinten, “Local fields close to the surface of nanoparticles and aggregates of nanoparticles,” Appl. Phys. B 73, 245–255 (2001).
[CrossRef]

Rautian, S.G.

S.V. Perminov, S.G. Rautian, and V.P. Safonov, “On the Theory of Optical Properties of Fractal Clusters,” Sov. Phys. JETP 98, 691–704 (2004).
[CrossRef]

S.V. Perminov, S.G. Rautian, and V.P. Safonov, “A model of pair interactions in the theory of optical properties of fractal clusters,” Opt. Spectrosc. 95, 416–420 (2003).
[CrossRef]

V.P. Drachev, S.V. Perminov, S.G. Rautian, V.P. Safonov, and E.N. Khaliullin, “Polarization effects in nanoaggre-gates of silver caused by local and nonlocal nonlinear-optical responses,” Sov. Phys. JETP 94, 901–915 (2002).
[CrossRef]

Reinhard, B.M.

C. Sonnichsen, B.M. Reinhard, J. Liphardt, and A.P. Alivisatos, “A molecular ruler based on plasmon coupling of single gold and silver nanoparticles,” Nature Biotechnol. 23, 741–745 (2005).
[CrossRef]

Rojas, R.

F. Claro and R. Rojas, “Novel laser induced interaction profiles in clusters of mesoscopic particles,” Appl. Phys. Lett. 65, 2743–2745 (1994).
[CrossRef]

Safonov, V.P.

S.V. Perminov, S.G. Rautian, and V.P. Safonov, “On the Theory of Optical Properties of Fractal Clusters,” Sov. Phys. JETP 98, 691–704 (2004).
[CrossRef]

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M.I. Stockman, “Chaos and Spatial Correlations for Dipolar Eigenproblems,” Phys. Rev. Lett. 79, 4562–4565 (1997).
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[CrossRef]

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U. Kreibig and M. Vollmer, Optical properties of metal clusters (Springer Verlag: Berlin Heidelberg New-York, 1995).

Yu.E. Danilova and V.P. Safonov, “Absorption spectra and photomodification of silver fractal clusters,” in Fractal Reviews in the Natural and Applied Sciences, M.M. Novak, ed. (Chapman and Hall: London, 1995), pp. 101–111.

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Figures (6)

Fig. 1.
Fig. 1.

The model potential describing the aggregation forces for a pair of nanoparticles.

Fig. 2.
Fig. 2.

The structure (left) and the linear absorption spectrum (right) of the Ag nano-aggregate. The light wave is polarized along the y axis. The dashed lines on the spectrum mark three frequencies, 2.83 eV, 2.865 eV, and 2.89 eV, which are specifically considered in the text.

Fig. 3.
Fig. 3.

The normalized nonlinear absorption PNL as a function of light frequency: (a) the peak values PNL (t 0) (t 0 corresponds to the maximum of the incident light pulse); (b) the pulse-average values. The peak intensity Ip of the incoming light amounts to 1.75 MW/cm2 for square data points; 17.5 MW/cm2 for the open circles, and 57.8 MW/cm2 for the solid circles.

Fig. 4.
Fig. 4.

Relative cross section of the nonlinear forward scattering: FNL (t 0) = [F(t 0)-F 0]/F 0 vs. the peak intensity of the excitation pulse. F(t 0) is given by (8), and F 0 is the linear (low intensity) scattering cross section. Circles, triangles, and diamonds correspond to different frequencies of the incident light at 2.83 eV, 2.865 eV, and 2.89 eV, respectively.

Fig. 5.
Fig. 5.

Relative scattering cross section of the probe beam in the forward direction vs. the probe frequency ω at pump intensity Ip = 17.5 MW/cm2 for (a) pulse-peak values and (b) pulse-average values. The three curves correspond to different frequencies of the pump radiation (the values indicated are in eV), and are the same frequencies as for Fig. 4

Fig. 6.
Fig. 6.

Temporal dependence of the relative displacement of particle #3 (dashed line, right axis) and #5 (solid line, left axis). The excitation is a 20-ns rectangular pulse with an intensity Ip = 17.5 MW/cm2.

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

E 0 ( r , t ) = E 0 ( r ) exp ( i ω t ) + c . c .
d i = α i ε h [ E 0 ( r i ) + i j N 3 n i j ( n i j d j ) φ i j d j ψ i j ε h r i j 3 ] , i = 1,2 , , N ,
r i j r i r j , r i j r i j , n i j = r i j / r i j ,
φ i j = [ 1 ikr i j ( kr ij ) 2 3 ] exp ( ikr i j ) , ψ i j = [ 1 ikr i j ( kr i j ) 2 ] exp ( ikr i j ) ,
m i r ̈ i + γ i r ˙ i = r i [ U 0 ( r 1 , , r N ) + U EM ( r 1 , , r N ) ] , i = 1,2 , , N ,
U EM ( r 1 , , r N ) = i = 1 N Re [ ( d i ( r 1 , , r N ) E 0 * ( r i ) ) ε h α i E 0 ( r i ) 2 ] .
U 0 = j 1 N U 0 i j ( r i , r j ) ,
U 0 i j = ( 1 2 ξ i j 2 3 ξ i j ) , ξ i j = Δ i j / a 1 ,
I ( t ) = c ε h 2 π E 0 2 = I p exp [ ( t t 0 τ p / 2 ) 2 ] ,
the absorbed power , P ( t ) = 2 ω ε h i = 1 N d t ( t ) 2 Im α i α i 2 ,
the scattering cross sec tion , F ( t ) f ( k / k ; t ) 2 E 0 2 ,
f ( s ; t ) = k 2 i = 1 N [ d i s ( d i s ) ] exp [ i k ( s r i ) ] .
F probe ( t ) = f P P ( k / k ; t ) 2 E 0 2 ,
f P P ( s ; t ) = k 2 i = 1 N { d i [ ω ; r i ( t , ω pump ) ] s ( d i [ ω ; r i ( t , ω pump ) s ] ) } exp [ i k ( s r i ) ]
F EM [ pN ] = 6 × 10 10 ε h ( a 1 a 2 ) 3 2 R 4 { a 1 3 + a 2 3 2 ( a 1 a 2 ) 3 2 2 ξ Re [ κ 2 κ ξ + 1 ]
( a 1 3 2 + a 2 3 2 ) 2 2 ξ Re [ κ 2 κ ξ 1 ] } I 0 [ W cm 2 ] ,

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