Abstract

An experimental and theoretical study about selective photodeposition of metallic zinc nanoparticles onto an optical fiber end is presented. It is well known that metallic nanoparticles possess a high absorption coefficient and therefore trapping and manipulation is more challenging than dielectric particles. Here, we demonstrate a novel trapping mechanism that involves laser-induced convection flow (due to heat transfer from the zinc particles) that partially compensates both absorption and scattering forces in the vicinity of the fiber end. The gradient force is too small and plays no role on the deposition process. The interplay of these forces produces selective deposition of particles whose size is directly controlled by the laser power. In addition, a novel trapping mechanism termed convective-optical trapping is demonstrated.

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2012 (1)

2011 (1)

T. Liu, X. Xiao, and C. Yang, “Surfactantless Photochemical deposition of gold nanoparticles on an optical fiber core for surface-enhanced Raman scattering,” Langmuir27(8), 4623–4626 (2011).
[CrossRef] [PubMed]

2010 (2)

Z. Luo, M. Zhou, J. Weng, G. Huang, H. Xu, C. Ye, and Z. Cai, “Graphene-based passively Q-switched dualwavelength erbium-doped fiber laser,” Opt. Lett.35(21), 3709–3711 (2010).
[CrossRef] [PubMed]

L. Ming-Shan and Y. Chang-Xi, “Laser-Induced silver nanoparticles deposited on optical fiber core for surface-enhanced Raman scattering,” Chin. Phys. Lett.27(4), 044202 (2010).
[CrossRef]

2009 (1)

E. Vela, M. Hafez, and S. Régnier, “Laser-induced thermocapillary convection for mesoscale manipulation,” Int. J. of Optomechatronics3(4), 289–302 (2009).
[CrossRef]

2008 (3)

M. Dienerowitz, M. Mazilu, P. J. Reece, T. F. Krauss, and K. Dholakia, “Optical vortex trap for resonant confinement of metal nanoparticles,” Opt. Express16(7), 4991–4999 (2008).
[CrossRef] [PubMed]

M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophotonics2(1), 1–32 (2008).
[CrossRef]

Y. Chu, E. Schonbrun, T. Yang, and K. B. Crozier, “Experimental observation of narrow surface plasmon resonances in gold nanoparticle arrays,” Appl. Phys. Lett.93(18), 181108 (2008).
[CrossRef]

2007 (3)

H. Amekura, N. Umeda, K. Kono, Y. Takeda, N. Kishimoto, Ch. Buchal, and S. Mantl, “Dual surface plasmon resonances in Zn nanoparticles in SiO2: an experimental study based on optical absorption and thermal stability,” Nanotechnology18(39), 1–6 (2007).
[CrossRef]

J. W. Nicholson, R. S. Windeler, and D. J. Digiovanni, “Optically driven deposition of single-walled carbon-nanotube saturable absorbers on optical fiber end-faces,” Opt. Express15(15), 9176–9183 (2007).
[CrossRef] [PubMed]

K. Kashiwagi, S. Yamashita, and S. Y. Set, “Optically manipulated deposition of carbon nanotubes onto optical fiber end,” Jpn. J. Appl. Phys.46(40), L988–L990 (2007).
[CrossRef]

2004 (2)

H. Amekura, H. Kitazawa, N. Umeda, Y. Takeda, and N. Kishimoto, “Nickel nanoparticles in silica glass fabricated by 60 keV negative-ion implantation,” Nucl. Instrum. Meth. B222(1-2), 114–122 (2004).
[CrossRef]

R. S. Taylor and C. Hnatovsky, “Trapping and mixing of particles in water using a microbubble attached to an NSOM fiber probe,” Opt. Express12(5), 916–928 (2004).
[CrossRef] [PubMed]

2003 (2)

I. O. Sosa, C. Noguez, and R. G. Barrera, “Optical properties of metal nanoparticles with arbitrary shapes,” J. Phys. Chem. B107(26), 6269–6275 (2003).
[CrossRef]

I. Tanahashi, H. Inouye, and A. Mito, “Femtosecond nonlinear optical properties of Au/SiO2 composite thin films prepared by a sputtering method,” Jpn. J. Appl. Phys.42(Part 1, No. 6A), 3467–3468 (2003).
[CrossRef]

2002 (3)

N. Pincon-Roetzinger, D. Prot, B. Palpant, E. Charron, and S. Debrus, “Large optical Kerr effect in matrix-embedded metal nanoparticles,” Mater. Sci. Eng. C19(1-2), 51–54 (2002).
[CrossRef]

R. R. Agayan, F. Gittes, R. Kopelman, and C. F. Schmidt, “Optical trapping near resonance absorption,” Appl. Opt.41(12), 2318–2327 (2002).
[CrossRef] [PubMed]

D. Braun and A. Libchaber, “Trapping of DNA by thermophoretic depletion and convection,” Phys. Rev. Lett.89(18), 188103 (2002).
[CrossRef] [PubMed]

2000 (1)

H. Gleiter, “Nanostructured materials: basic concepts and microstructure,” Acta Mater.48(1), 1–29 (2000).
[CrossRef]

1996 (1)

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]

1994 (2)

R. F. Haglund, L. Yang, R. H. Magruder, C. W. White, R. A. Zuhr, L. Yang, R. Dorsinville, and R. R. Alfano, “Nonlinear optical properties of metal-quantum-dot composites synthesized by ion implantation,” Nucl. Instrum. Meth. B91(1-4), 493–504 (1994).
[CrossRef]

K. Svoboda and S. M. Block, “Optical trapping of metallic Rayleigh particles,” Opt. Lett.19(13), 930–932 (1994).
[CrossRef] [PubMed]

1993 (1)

1986 (1)

1974 (1)

L. A. Spielman and S. K. Friedlander, “Role of the electrical double layer in particle deposition by convective diffusion,” J. Colloid Interface Sci.46(1), 22–31 (1974).
[CrossRef]

1917 (1)

E. Ehrenhaft, “On the physics of millionth of centimeters,” Phys. Z.18, 352–368 (1917).

Agayan, R. R.

Alfano, R. R.

R. F. Haglund, L. Yang, R. H. Magruder, C. W. White, R. A. Zuhr, L. Yang, R. Dorsinville, and R. R. Alfano, “Nonlinear optical properties of metal-quantum-dot composites synthesized by ion implantation,” Nucl. Instrum. Meth. B91(1-4), 493–504 (1994).
[CrossRef]

Amekura, H.

H. Amekura, N. Umeda, K. Kono, Y. Takeda, N. Kishimoto, Ch. Buchal, and S. Mantl, “Dual surface plasmon resonances in Zn nanoparticles in SiO2: an experimental study based on optical absorption and thermal stability,” Nanotechnology18(39), 1–6 (2007).
[CrossRef]

H. Amekura, H. Kitazawa, N. Umeda, Y. Takeda, and N. Kishimoto, “Nickel nanoparticles in silica glass fabricated by 60 keV negative-ion implantation,” Nucl. Instrum. Meth. B222(1-2), 114–122 (2004).
[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.

Barrera, R. G.

I. O. Sosa, C. Noguez, and R. G. Barrera, “Optical properties of metal nanoparticles with arbitrary shapes,” J. Phys. Chem. B107(26), 6269–6275 (2003).
[CrossRef]

Bjorkholm, J. E.

Block, S. M.

Braun, D.

D. Braun and A. Libchaber, “Trapping of DNA by thermophoretic depletion and convection,” Phys. Rev. Lett.89(18), 188103 (2002).
[CrossRef] [PubMed]

Buchal, Ch.

H. Amekura, N. Umeda, K. Kono, Y. Takeda, N. Kishimoto, Ch. Buchal, and S. Mantl, “Dual surface plasmon resonances in Zn nanoparticles in SiO2: an experimental study based on optical absorption and thermal stability,” Nanotechnology18(39), 1–6 (2007).
[CrossRef]

Cai, Z.

Chang-Xi, Y.

L. Ming-Shan and Y. Chang-Xi, “Laser-Induced silver nanoparticles deposited on optical fiber core for surface-enhanced Raman scattering,” Chin. Phys. Lett.27(4), 044202 (2010).
[CrossRef]

Charron, E.

N. Pincon-Roetzinger, D. Prot, B. Palpant, E. Charron, and S. Debrus, “Large optical Kerr effect in matrix-embedded metal nanoparticles,” Mater. Sci. Eng. C19(1-2), 51–54 (2002).
[CrossRef]

Chu, S.

Chu, Y.

Y. Chu, E. Schonbrun, T. Yang, and K. B. Crozier, “Experimental observation of narrow surface plasmon resonances in gold nanoparticle arrays,” Appl. Phys. Lett.93(18), 181108 (2008).
[CrossRef]

Constable, A.

Crozier, K. B.

Y. Chu, E. Schonbrun, T. Yang, and K. B. Crozier, “Experimental observation of narrow surface plasmon resonances in gold nanoparticle arrays,” Appl. Phys. Lett.93(18), 181108 (2008).
[CrossRef]

Debrus, S.

N. Pincon-Roetzinger, D. Prot, B. Palpant, E. Charron, and S. Debrus, “Large optical Kerr effect in matrix-embedded metal nanoparticles,” Mater. Sci. Eng. C19(1-2), 51–54 (2002).
[CrossRef]

Dholakia, K.

Dienerowitz, M.

Digiovanni, D. J.

Dorsinville, R.

R. F. Haglund, L. Yang, R. H. Magruder, C. W. White, R. A. Zuhr, L. Yang, R. Dorsinville, and R. R. Alfano, “Nonlinear optical properties of metal-quantum-dot composites synthesized by ion implantation,” Nucl. Instrum. Meth. B91(1-4), 493–504 (1994).
[CrossRef]

Dziedzic, J. M.

Ehrenhaft, E.

E. Ehrenhaft, “On the physics of millionth of centimeters,” Phys. Z.18, 352–368 (1917).

Friedlander, S. K.

L. A. Spielman and S. K. Friedlander, “Role of the electrical double layer in particle deposition by convective diffusion,” J. Colloid Interface Sci.46(1), 22–31 (1974).
[CrossRef]

Gittes, F.

Gleiter, H.

H. Gleiter, “Nanostructured materials: basic concepts and microstructure,” Acta Mater.48(1), 1–29 (2000).
[CrossRef]

Hafez, M.

E. Vela, M. Hafez, and S. Régnier, “Laser-induced thermocapillary convection for mesoscale manipulation,” Int. J. of Optomechatronics3(4), 289–302 (2009).
[CrossRef]

Haglund, R. F.

R. F. Haglund, L. Yang, R. H. Magruder, C. W. White, R. A. Zuhr, L. Yang, R. Dorsinville, and R. R. Alfano, “Nonlinear optical properties of metal-quantum-dot composites synthesized by ion implantation,” Nucl. Instrum. Meth. B91(1-4), 493–504 (1994).
[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]

Hernández-Cordero, J.

Hnatovsky, C.

Huang, G.

Inouye, H.

I. Tanahashi, H. Inouye, and A. Mito, “Femtosecond nonlinear optical properties of Au/SiO2 composite thin films prepared by a sputtering method,” Jpn. J. Appl. Phys.42(Part 1, No. 6A), 3467–3468 (2003).
[CrossRef]

Kashiwagi, K.

K. Kashiwagi, S. Yamashita, and S. Y. Set, “Optically manipulated deposition of carbon nanotubes onto optical fiber end,” Jpn. J. Appl. Phys.46(40), L988–L990 (2007).
[CrossRef]

Kim, J.

Kishimoto, N.

H. Amekura, N. Umeda, K. Kono, Y. Takeda, N. Kishimoto, Ch. Buchal, and S. Mantl, “Dual surface plasmon resonances in Zn nanoparticles in SiO2: an experimental study based on optical absorption and thermal stability,” Nanotechnology18(39), 1–6 (2007).
[CrossRef]

H. Amekura, H. Kitazawa, N. Umeda, Y. Takeda, and N. Kishimoto, “Nickel nanoparticles in silica glass fabricated by 60 keV negative-ion implantation,” Nucl. Instrum. Meth. B222(1-2), 114–122 (2004).
[CrossRef]

Kitazawa, H.

H. Amekura, H. Kitazawa, N. Umeda, Y. Takeda, and N. Kishimoto, “Nickel nanoparticles in silica glass fabricated by 60 keV negative-ion implantation,” Nucl. Instrum. Meth. B222(1-2), 114–122 (2004).
[CrossRef]

Kono, K.

H. Amekura, N. Umeda, K. Kono, Y. Takeda, N. Kishimoto, Ch. Buchal, and S. Mantl, “Dual surface plasmon resonances in Zn nanoparticles in SiO2: an experimental study based on optical absorption and thermal stability,” Nanotechnology18(39), 1–6 (2007).
[CrossRef]

Kopelman, R.

Krauss, T. F.

Libchaber, A.

D. Braun and A. Libchaber, “Trapping of DNA by thermophoretic depletion and convection,” Phys. Rev. Lett.89(18), 188103 (2002).
[CrossRef] [PubMed]

Liu, T.

T. Liu, X. Xiao, and C. Yang, “Surfactantless Photochemical deposition of gold nanoparticles on an optical fiber core for surface-enhanced Raman scattering,” Langmuir27(8), 4623–4626 (2011).
[CrossRef] [PubMed]

Luo, Z.

Magruder, R. H.

R. F. Haglund, L. Yang, R. H. Magruder, C. W. White, R. A. Zuhr, L. Yang, R. Dorsinville, and R. R. Alfano, “Nonlinear optical properties of metal-quantum-dot composites synthesized by ion implantation,” Nucl. Instrum. Meth. B91(1-4), 493–504 (1994).
[CrossRef]

Mantl, S.

H. Amekura, N. Umeda, K. Kono, Y. Takeda, N. Kishimoto, Ch. Buchal, and S. Mantl, “Dual surface plasmon resonances in Zn nanoparticles in SiO2: an experimental study based on optical absorption and thermal stability,” Nanotechnology18(39), 1–6 (2007).
[CrossRef]

Mazilu, M.

Mervis, J.

Ming-Shan, L.

L. Ming-Shan and Y. Chang-Xi, “Laser-Induced silver nanoparticles deposited on optical fiber core for surface-enhanced Raman scattering,” Chin. Phys. Lett.27(4), 044202 (2010).
[CrossRef]

Mito, A.

I. Tanahashi, H. Inouye, and A. Mito, “Femtosecond nonlinear optical properties of Au/SiO2 composite thin films prepared by a sputtering method,” Jpn. J. Appl. Phys.42(Part 1, No. 6A), 3467–3468 (2003).
[CrossRef]

Nicholson, J. W.

Noguez, C.

I. O. Sosa, C. Noguez, and R. G. Barrera, “Optical properties of metal nanoparticles with arbitrary shapes,” J. Phys. Chem. B107(26), 6269–6275 (2003).
[CrossRef]

Palpant, B.

N. Pincon-Roetzinger, D. Prot, B. Palpant, E. Charron, and S. Debrus, “Large optical Kerr effect in matrix-embedded metal nanoparticles,” Mater. Sci. Eng. C19(1-2), 51–54 (2002).
[CrossRef]

Pimentel-Domínguez, R.

Pincon-Roetzinger, N.

N. Pincon-Roetzinger, D. Prot, B. Palpant, E. Charron, and S. Debrus, “Large optical Kerr effect in matrix-embedded metal nanoparticles,” Mater. Sci. Eng. C19(1-2), 51–54 (2002).
[CrossRef]

Prentiss, M.

Prot, D.

N. Pincon-Roetzinger, D. Prot, B. Palpant, E. Charron, and S. Debrus, “Large optical Kerr effect in matrix-embedded metal nanoparticles,” Mater. Sci. Eng. C19(1-2), 51–54 (2002).
[CrossRef]

Reece, P. J.

Régnier, S.

E. Vela, M. Hafez, and S. Régnier, “Laser-induced thermocapillary convection for mesoscale manipulation,” Int. J. of Optomechatronics3(4), 289–302 (2009).
[CrossRef]

Schmidt, C. F.

Schonbrun, E.

Y. Chu, E. Schonbrun, T. Yang, and K. B. Crozier, “Experimental observation of narrow surface plasmon resonances in gold nanoparticle arrays,” Appl. Phys. Lett.93(18), 181108 (2008).
[CrossRef]

Set, S. Y.

K. Kashiwagi, S. Yamashita, and S. Y. Set, “Optically manipulated deposition of carbon nanotubes onto optical fiber end,” Jpn. J. Appl. Phys.46(40), L988–L990 (2007).
[CrossRef]

Sosa, I. O.

I. O. Sosa, C. Noguez, and R. G. Barrera, “Optical properties of metal nanoparticles with arbitrary shapes,” J. Phys. Chem. B107(26), 6269–6275 (2003).
[CrossRef]

Spielman, L. A.

L. A. Spielman and S. K. Friedlander, “Role of the electrical double layer in particle deposition by convective diffusion,” J. Colloid Interface Sci.46(1), 22–31 (1974).
[CrossRef]

Svoboda, K.

Takeda, Y.

H. Amekura, N. Umeda, K. Kono, Y. Takeda, N. Kishimoto, Ch. Buchal, and S. Mantl, “Dual surface plasmon resonances in Zn nanoparticles in SiO2: an experimental study based on optical absorption and thermal stability,” Nanotechnology18(39), 1–6 (2007).
[CrossRef]

H. Amekura, H. Kitazawa, N. Umeda, Y. Takeda, and N. Kishimoto, “Nickel nanoparticles in silica glass fabricated by 60 keV negative-ion implantation,” Nucl. Instrum. Meth. B222(1-2), 114–122 (2004).
[CrossRef]

Tanahashi, I.

I. Tanahashi, H. Inouye, and A. Mito, “Femtosecond nonlinear optical properties of Au/SiO2 composite thin films prepared by a sputtering method,” Jpn. J. Appl. Phys.42(Part 1, No. 6A), 3467–3468 (2003).
[CrossRef]

Taylor, R. S.

Umeda, N.

H. Amekura, N. Umeda, K. Kono, Y. Takeda, N. Kishimoto, Ch. Buchal, and S. Mantl, “Dual surface plasmon resonances in Zn nanoparticles in SiO2: an experimental study based on optical absorption and thermal stability,” Nanotechnology18(39), 1–6 (2007).
[CrossRef]

H. Amekura, H. Kitazawa, N. Umeda, Y. Takeda, and N. Kishimoto, “Nickel nanoparticles in silica glass fabricated by 60 keV negative-ion implantation,” Nucl. Instrum. Meth. B222(1-2), 114–122 (2004).
[CrossRef]

Vela, E.

E. Vela, M. Hafez, and S. Régnier, “Laser-induced thermocapillary convection for mesoscale manipulation,” Int. J. of Optomechatronics3(4), 289–302 (2009).
[CrossRef]

Weng, J.

White, C. W.

R. F. Haglund, L. Yang, R. H. Magruder, C. W. White, R. A. Zuhr, L. Yang, R. Dorsinville, and R. R. Alfano, “Nonlinear optical properties of metal-quantum-dot composites synthesized by ion implantation,” Nucl. Instrum. Meth. B91(1-4), 493–504 (1994).
[CrossRef]

Windeler, R. S.

Xiao, X.

T. Liu, X. Xiao, and C. Yang, “Surfactantless Photochemical deposition of gold nanoparticles on an optical fiber core for surface-enhanced Raman scattering,” Langmuir27(8), 4623–4626 (2011).
[CrossRef] [PubMed]

Xu, H.

Yamashita, S.

K. Kashiwagi, S. Yamashita, and S. Y. Set, “Optically manipulated deposition of carbon nanotubes onto optical fiber end,” Jpn. J. Appl. Phys.46(40), L988–L990 (2007).
[CrossRef]

Yang, C.

T. Liu, X. Xiao, and C. Yang, “Surfactantless Photochemical deposition of gold nanoparticles on an optical fiber core for surface-enhanced Raman scattering,” Langmuir27(8), 4623–4626 (2011).
[CrossRef] [PubMed]

Yang, L.

R. F. Haglund, L. Yang, R. H. Magruder, C. W. White, R. A. Zuhr, L. Yang, R. Dorsinville, and R. R. Alfano, “Nonlinear optical properties of metal-quantum-dot composites synthesized by ion implantation,” Nucl. Instrum. Meth. B91(1-4), 493–504 (1994).
[CrossRef]

R. F. Haglund, L. Yang, R. H. Magruder, C. W. White, R. A. Zuhr, L. Yang, R. Dorsinville, and R. R. Alfano, “Nonlinear optical properties of metal-quantum-dot composites synthesized by ion implantation,” Nucl. Instrum. Meth. B91(1-4), 493–504 (1994).
[CrossRef]

Yang, T.

Y. Chu, E. Schonbrun, T. Yang, and K. B. Crozier, “Experimental observation of narrow surface plasmon resonances in gold nanoparticle arrays,” Appl. Phys. Lett.93(18), 181108 (2008).
[CrossRef]

Ye, C.

Zarinetchi, F.

Zenit, R.

Zhou, M.

Zuhr, R. A.

R. F. Haglund, L. Yang, R. H. Magruder, C. W. White, R. A. Zuhr, L. Yang, R. Dorsinville, and R. R. Alfano, “Nonlinear optical properties of metal-quantum-dot composites synthesized by ion implantation,” Nucl. Instrum. Meth. B91(1-4), 493–504 (1994).
[CrossRef]

Acta Mater. (1)

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Supplementary Material (2)

» Media 1: MOV (4026 KB)     
» Media 2: MOV (2427 KB)     

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Figures (10)

Fig. 1
Fig. 1

Experimental setup for depositing Zn nanoparticles onto optical fiber end.

Fig. 2
Fig. 2

a) SEM micrograph of zinc powder and the b) corresponding XRD spectrum, confirming the metallic structure of the particles.

Fig. 3
Fig. 3

Images of the optical fiber end obtained with a SEM. a) Optical fiber was immersed into solution for up to 15 minutes with laser light off, b) laser was turned on for 180 seconds and power = 5 mW, c) closer view of b) around the fiber core, d) laser was turned on for 60 seconds and P = 50 mW, e) closer view of the fiber core showed in d).

Fig. 4
Fig. 4

Consecutive frames show that particles move in opposite direction to the beam propagation. Circles indicate the position of the clusters at different frames (P = 5 mW).

Fig. 5
Fig. 5

a) Velocity field illustrating the motion of particles obtained from a real-time movie of convection currents (Media 1). b) Flow velocity as determined by particle tracking for two concentrations using a power of 10 mW.

Fig. 6
Fig. 6

Schematic representation of the proposed model. Light is propagating along + z axis. The scattering and absorption forces are directed along the + z axis while the Stokes force is directed in the opposite direction.

Fig. 7
Fig. 7

Longitudinal component of optical forces exerted on a zinc nanoparticle (50 nm) in the photodeposition process as a function of z position. a) Gradient force, b) scattering and absorption forces.

Fig. 8
Fig. 8

Forces exerted on a zinc nanoparticle in the photodeposition process as a function of its size for two power beam (5 and 50 mW). a) Scattering and absorption forces when the nanoparticle is at position (0, 0, 0), b) Stokes force as a function of the particle radius.

Fig. 9
Fig. 9

Net force exerted on zinc nanoparticle in the photodeposition process as a function of its size for beam powers of 5 and 50 mW.

Fig. 10
Fig. 10

Movie of convective-optical particle trapped by approximately two seconds (Media 2).

Equations (10)

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F Stokes =6πψR v s ,
F net = F abs + F scat + F Stokes + F grad .
F grad = 1 2 α'(ω) E 2 , 
F scat = n c I(r) C scat ,
F abs = n c I(r) C abs ,
I( r )=( 2P π w 0 2 ) k 2 w 0 4 4 z 2 + ( k w 0 2 ) 2 exp[ 2k w 0 ( x 2 + y 2 ) 4 z 2 + ( k w 0 2 ) 2 ],
Q scat = 2 x 2 l=1 (2l+1)(| a l | 2 +| b l | 2 ) ,
Q ext = 2 x 2 l=1 (2l+1)Re( a l + b l ) ,
a l = μ m 2 j l ( mx ) [ x j l ( x ) ] μ 1 j l (x)[mx j l (mx) ] μ m 2 j l ( mx ) [ x h l (1) ( x ) ] μ 1 h l (1) (x)[mx j l (mx) ] ,
b l = μ 1 j l ( mx ) [ x j l ( x ) ] μ j l (x)[mx j l (mx) ] μ 1 j l ( mx ) [ x h l (1) ( x ) ] μ h l (1) (x)[mx j l (mx) ] ,

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