Abstract

We propose a novel method for all-optical modulation. Our method is based on tuning the dimensions of nanoparticles by manipulating thermal energy. The excitation characteristics of coated nanoparticles change following minor variations in their dimensions. This is because incident radiation will encounter particles with an altered extinction cross section. We considered nanoparticles comprising gold cores with silica coating. We found distinct nanoparticle dimensions that exhibit steep changes in their optical extinction cross section under thermal expansion. Based on these dimensions we calculated the optical contrast. The results showed that optical contrast reaches 1.86 dB for radii of 900 nm.

© 2012 Optical Society of America

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    [CrossRef]
  32. P. A. Martin, Multiple Scattering: Interaction Of Time-Harmonic Waves with N Obstacles (Cambridge University, 2006).
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    [CrossRef]
  34. G. Domingues, S. Volz, K. Joulain, and J. J. Greffet, “Heat transfer between two nanoparticles through near field interaction,” Phys. Rev. Lett. 94, 085901 (2005).
    [CrossRef]

2009

G. Baffou, R. Quidant, and C. Girard, “Heat generation in plasmonic nanostructures: influence of morphology,” Appl. Phys. Lett. 94, 153109 (2009).
[CrossRef]

N. Zeng and A. B. Murphy, “Heat generation by optically and thermally interacting aggregates of gold nanoparticles under illumination,” Nanotechnology 20, 375702 (2009).
[CrossRef]

2008

O. Ekici, R. Harrison, N. Durr, D. Eversole, M. Lee, and A. Ben-Yakar, “Thermal analysis of gold nanorods heated with femtosecond laser pulses,” J. Phys. D 41, 185501(2008).
[CrossRef]

A. Perez-Madrid, J. Rubi, and L. C. Lapas, “Heat transfer between nanoparticles: thermal conductance for near-field interactions,” Phys. Rev. B 77, 155417 (2008).
[CrossRef]

2006

J. Hoyland and D. Sands, “Temperature dependent refractive index of amorphous silicon determined by time-resolved reflectivity during low fluence excimer laser heating,” J. Appl. Phys. 99, 063516 (2006).
[CrossRef]

T. Hao and R. E. Riman, “Calculation of interparticle spacing in colloidal systems,” J. Colloid Interface Sci. 297, 374–377 (2006).
[CrossRef]

2005

G. Domingues, S. Volz, K. Joulain, and J. J. Greffet, “Heat transfer between two nanoparticles through near field interaction,” Phys. Rev. Lett. 94, 085901 (2005).
[CrossRef]

W. S. Rabinovich, R. Mahon, H. Burris, G. C. Gilbreath, P. G. Goetz, C. I. Moore, M. Stell, M. J. Vilcheck, J. L. Witkowsky, and L. Swingen, “Free-space optical communications link at 1550 nm using multiple-quantum-well modulating retroreflectors in a marine environment,” Opt. Eng. 44, 056001 (2005).
[CrossRef]

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435, 325–327 (2005).
[CrossRef]

A. Dawes, L. Illing, S. M. Clark, and D. J. Gauthier, “All-optical switching in rubidium vapor,” Science 308, 672–674 (2005).
[CrossRef]

2004

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084 (2004).
[CrossRef]

M. Rashidi-Huyeh and B. Palpant, “Thermal response of nanocomposite materials under pulsed laser excitation,” J. Appl. Phys. 96, 4475–4482 (2004).
[CrossRef]

2002

J. Mock, M. Barbic, D. Smith, D. Schultz, and S. Schultz, “Shape effects in plasmon resonance of individual colloidal silver nanoparticles,” J. Chem. Phys. 116, 6755–6759 (2002).
[CrossRef]

2000

S. Link and M. A. El-Sayed, “Shape and size dependence of radiative, non-radiative and photothermal properties of gold nanocrystals,” Int. Rev. Phys. Chem. 19, 409–453(2000).
[CrossRef]

T. Anderson, R. Magruder, J. Wittig, D. Kinser, and R. Zuhr, “Fabrication of Cu-coated Ag nanocrystals in silica by sequential ion implantation,” Nucl. Instrum. Methods. Phys. Res. B: Beam Interact. Mater. Atoms 171, 401–405 (2000).
[CrossRef]

1999

S. Link and M. A. El-Sayed, “Size and temperature dependence of the plasmon absorption of colloidal gold nanoparticles,” J. Phys. Chem. B 103, 4212–4217 (1999).
[CrossRef]

1998

T. Klar, M. Perner, S. Grosse, G. Von Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett. 80, 4249–4252 (1998).
[CrossRef]

1997

M. Asobe, “Nonlinear optical properties of chalcogenide glass fibers and their application to all-optical switching,” Opt. Fiber Technol. 3, 142–148 (1997).
[CrossRef]

1996

R. Takahashi, Y. Kawamura, and H. Iwamura, “Ultrafast 1.55 μm all-optical switching using low-temperature-grown multiple quantum wells,” Appl. Phys. Lett. 68, 153–155 (1996).
[CrossRef]

1994

S. Arnon, D. Sadot, and N. Kopeika, “Simple mathematical models for temporal, spatial, angular, and attenuation characteristics of light propagating through the atmosphere for space optical communication,” J. Mod. Opt. 41, 1955–1972 (1994).
[CrossRef]

1989

1984

Y. Okada and Y. Tokumaru, “Precise determination of lattice parameter and thermal expansion coefficient of silicon between 300 and 1500 K,” J. Appl. Phys. 56, 314–320(1984).
[CrossRef]

1983

G. W. Scherer, “Viscoelastic analysis of thermal stresses in a composite sphere,” J. Am. Ceram. Soc. 66, 59–65 (1983).
[CrossRef]

1980

H. Li, “Refractive index of silicon and germanium and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 9, 561–658 (1980).
[CrossRef]

1970

A. Fahmy and A. Ragai, “Thermal-expansion behavior of two-phase solids,” J. Appl. Phys. 41, 5108–5111 (1970).
[CrossRef]

1951

A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

1908

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330, 377–445 (1908).
[CrossRef]

Aden, A. L.

A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Almeida, V. R.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084 (2004).
[CrossRef]

Anderson, T.

T. Anderson, R. Magruder, J. Wittig, D. Kinser, and R. Zuhr, “Fabrication of Cu-coated Ag nanocrystals in silica by sequential ion implantation,” Nucl. Instrum. Methods. Phys. Res. B: Beam Interact. Mater. Atoms 171, 401–405 (2000).
[CrossRef]

Andrejco, M.

Arnon, S.

S. Arnon, D. Sadot, and N. Kopeika, “Simple mathematical models for temporal, spatial, angular, and attenuation characteristics of light propagating through the atmosphere for space optical communication,” J. Mod. Opt. 41, 1955–1972 (1994).
[CrossRef]

Asobe, M.

M. Asobe, “Nonlinear optical properties of chalcogenide glass fibers and their application to all-optical switching,” Opt. Fiber Technol. 3, 142–148 (1997).
[CrossRef]

Baffou, G.

G. Baffou, R. Quidant, and C. Girard, “Heat generation in plasmonic nanostructures: influence of morphology,” Appl. Phys. Lett. 94, 153109 (2009).
[CrossRef]

Barbic, M.

J. Mock, M. Barbic, D. Smith, D. Schultz, and S. Schultz, “Shape effects in plasmon resonance of individual colloidal silver nanoparticles,” J. Chem. Phys. 116, 6755–6759 (2002).
[CrossRef]

Barrios, C. A.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084 (2004).
[CrossRef]

Ben-Yakar, A.

O. Ekici, R. Harrison, N. Durr, D. Eversole, M. Lee, and A. Ben-Yakar, “Thermal analysis of gold nanorods heated with femtosecond laser pulses,” J. Phys. D 41, 185501(2008).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Burris, H.

W. S. Rabinovich, R. Mahon, H. Burris, G. C. Gilbreath, P. G. Goetz, C. I. Moore, M. Stell, M. J. Vilcheck, J. L. Witkowsky, and L. Swingen, “Free-space optical communications link at 1550 nm using multiple-quantum-well modulating retroreflectors in a marine environment,” Opt. Eng. 44, 056001 (2005).
[CrossRef]

Chen, G. C. K.

X. Cong and G. C. K. Chen, “Design of micro-lens array for indoor optical wireless communication,” in Proceedings of IEEE PhotonicsGlobal@Singapore, 2008 (IEEE, 2008), pp. 1–4.

Clark, S. M.

A. Dawes, L. Illing, S. M. Clark, and D. J. Gauthier, “All-optical switching in rubidium vapor,” Science 308, 672–674 (2005).
[CrossRef]

Cong, X.

X. Cong and G. C. K. Chen, “Design of micro-lens array for indoor optical wireless communication,” in Proceedings of IEEE PhotonicsGlobal@Singapore, 2008 (IEEE, 2008), pp. 1–4.

Dawes, A.

A. Dawes, L. Illing, S. M. Clark, and D. J. Gauthier, “All-optical switching in rubidium vapor,” Science 308, 672–674 (2005).
[CrossRef]

DeLong, K.

Domingues, G.

G. Domingues, S. Volz, K. Joulain, and J. J. Greffet, “Heat transfer between two nanoparticles through near field interaction,” Phys. Rev. Lett. 94, 085901 (2005).
[CrossRef]

Durr, N.

O. Ekici, R. Harrison, N. Durr, D. Eversole, M. Lee, and A. Ben-Yakar, “Thermal analysis of gold nanorods heated with femtosecond laser pulses,” J. Phys. D 41, 185501(2008).
[CrossRef]

Ekici, O.

O. Ekici, R. Harrison, N. Durr, D. Eversole, M. Lee, and A. Ben-Yakar, “Thermal analysis of gold nanorods heated with femtosecond laser pulses,” J. Phys. D 41, 185501(2008).
[CrossRef]

El-Sayed, M. A.

S. Link and M. A. El-Sayed, “Shape and size dependence of radiative, non-radiative and photothermal properties of gold nanocrystals,” Int. Rev. Phys. Chem. 19, 409–453(2000).
[CrossRef]

S. Link and M. A. El-Sayed, “Size and temperature dependence of the plasmon absorption of colloidal gold nanoparticles,” J. Phys. Chem. B 103, 4212–4217 (1999).
[CrossRef]

Eversole, D.

O. Ekici, R. Harrison, N. Durr, D. Eversole, M. Lee, and A. Ben-Yakar, “Thermal analysis of gold nanorods heated with femtosecond laser pulses,” J. Phys. D 41, 185501(2008).
[CrossRef]

Fahmy, A.

A. Fahmy and A. Ragai, “Thermal-expansion behavior of two-phase solids,” J. Appl. Phys. 41, 5108–5111 (1970).
[CrossRef]

Feldmann, J.

T. Klar, M. Perner, S. Grosse, G. Von Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett. 80, 4249–4252 (1998).
[CrossRef]

Gauthier, D. J.

A. Dawes, L. Illing, S. M. Clark, and D. J. Gauthier, “All-optical switching in rubidium vapor,” Science 308, 672–674 (2005).
[CrossRef]

Gilbreath, G. C.

W. S. Rabinovich, R. Mahon, H. Burris, G. C. Gilbreath, P. G. Goetz, C. I. Moore, M. Stell, M. J. Vilcheck, J. L. Witkowsky, and L. Swingen, “Free-space optical communications link at 1550 nm using multiple-quantum-well modulating retroreflectors in a marine environment,” Opt. Eng. 44, 056001 (2005).
[CrossRef]

Girard, C.

G. Baffou, R. Quidant, and C. Girard, “Heat generation in plasmonic nanostructures: influence of morphology,” Appl. Phys. Lett. 94, 153109 (2009).
[CrossRef]

Girifalco, L. A.

L. A. Girifalco, Statistical Mechanics of Solids (Oxford University, 2003).

Goetz, P. G.

W. S. Rabinovich, R. Mahon, H. Burris, G. C. Gilbreath, P. G. Goetz, C. I. Moore, M. Stell, M. J. Vilcheck, J. L. Witkowsky, and L. Swingen, “Free-space optical communications link at 1550 nm using multiple-quantum-well modulating retroreflectors in a marine environment,” Opt. Eng. 44, 056001 (2005).
[CrossRef]

Greffet, J. J.

G. Domingues, S. Volz, K. Joulain, and J. J. Greffet, “Heat transfer between two nanoparticles through near field interaction,” Phys. Rev. Lett. 94, 085901 (2005).
[CrossRef]

Grosse, S.

T. Klar, M. Perner, S. Grosse, G. Von Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett. 80, 4249–4252 (1998).
[CrossRef]

Hao, T.

T. Hao and R. E. Riman, “Calculation of interparticle spacing in colloidal systems,” J. Colloid Interface Sci. 297, 374–377 (2006).
[CrossRef]

Harrison, R.

O. Ekici, R. Harrison, N. Durr, D. Eversole, M. Lee, and A. Ben-Yakar, “Thermal analysis of gold nanorods heated with femtosecond laser pulses,” J. Phys. D 41, 185501(2008).
[CrossRef]

Heavens, O. S.

O. S. Heavens, Optical Properties of Thin Solid Films (Dover, 1991).

Hoyland, J.

J. Hoyland and D. Sands, “Temperature dependent refractive index of amorphous silicon determined by time-resolved reflectivity during low fluence excimer laser heating,” J. Appl. Phys. 99, 063516 (2006).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Illing, L.

A. Dawes, L. Illing, S. M. Clark, and D. J. Gauthier, “All-optical switching in rubidium vapor,” Science 308, 672–674 (2005).
[CrossRef]

Iwamura, H.

R. Takahashi, Y. Kawamura, and H. Iwamura, “Ultrafast 1.55 μm all-optical switching using low-temperature-grown multiple quantum wells,” Appl. Phys. Lett. 68, 153–155 (1996).
[CrossRef]

Joulain, K.

G. Domingues, S. Volz, K. Joulain, and J. J. Greffet, “Heat transfer between two nanoparticles through near field interaction,” Phys. Rev. Lett. 94, 085901 (2005).
[CrossRef]

Kawamura, Y.

R. Takahashi, Y. Kawamura, and H. Iwamura, “Ultrafast 1.55 μm all-optical switching using low-temperature-grown multiple quantum wells,” Appl. Phys. Lett. 68, 153–155 (1996).
[CrossRef]

Kerker, M.

A. L. Aden and M. Kerker, “Scattering of electromagnetic waves from two concentric spheres,” J. Appl. Phys. 22, 1242–1246 (1951).
[CrossRef]

Kinser, D.

T. Anderson, R. Magruder, J. Wittig, D. Kinser, and R. Zuhr, “Fabrication of Cu-coated Ag nanocrystals in silica by sequential ion implantation,” Nucl. Instrum. Methods. Phys. Res. B: Beam Interact. Mater. Atoms 171, 401–405 (2000).
[CrossRef]

Klar, T.

T. Klar, M. Perner, S. Grosse, G. Von Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett. 80, 4249–4252 (1998).
[CrossRef]

Kopeika, N.

S. Arnon, D. Sadot, and N. Kopeika, “Simple mathematical models for temporal, spatial, angular, and attenuation characteristics of light propagating through the atmosphere for space optical communication,” J. Mod. Opt. 41, 1955–1972 (1994).
[CrossRef]

Lapas, L. C.

A. Perez-Madrid, J. Rubi, and L. C. Lapas, “Heat transfer between nanoparticles: thermal conductance for near-field interactions,” Phys. Rev. B 77, 155417 (2008).
[CrossRef]

Lee, M.

O. Ekici, R. Harrison, N. Durr, D. Eversole, M. Lee, and A. Ben-Yakar, “Thermal analysis of gold nanorods heated with femtosecond laser pulses,” J. Phys. D 41, 185501(2008).
[CrossRef]

Li, H.

H. Li, “Refractive index of silicon and germanium and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 9, 561–658 (1980).
[CrossRef]

Link, S.

S. Link and M. A. El-Sayed, “Shape and size dependence of radiative, non-radiative and photothermal properties of gold nanocrystals,” Int. Rev. Phys. Chem. 19, 409–453(2000).
[CrossRef]

S. Link and M. A. El-Sayed, “Size and temperature dependence of the plasmon absorption of colloidal gold nanoparticles,” J. Phys. Chem. B 103, 4212–4217 (1999).
[CrossRef]

Lipson, M.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435, 325–327 (2005).
[CrossRef]

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084 (2004).
[CrossRef]

Magruder, R.

T. Anderson, R. Magruder, J. Wittig, D. Kinser, and R. Zuhr, “Fabrication of Cu-coated Ag nanocrystals in silica by sequential ion implantation,” Nucl. Instrum. Methods. Phys. Res. B: Beam Interact. Mater. Atoms 171, 401–405 (2000).
[CrossRef]

Mahon, R.

W. S. Rabinovich, R. Mahon, H. Burris, G. C. Gilbreath, P. G. Goetz, C. I. Moore, M. Stell, M. J. Vilcheck, J. L. Witkowsky, and L. Swingen, “Free-space optical communications link at 1550 nm using multiple-quantum-well modulating retroreflectors in a marine environment,” Opt. Eng. 44, 056001 (2005).
[CrossRef]

Martin, P. A.

P. A. Martin, Multiple Scattering: Interaction Of Time-Harmonic Waves with N Obstacles (Cambridge University, 2006).

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330, 377–445 (1908).
[CrossRef]

Mizrahi, V.

Mock, J.

J. Mock, M. Barbic, D. Smith, D. Schultz, and S. Schultz, “Shape effects in plasmon resonance of individual colloidal silver nanoparticles,” J. Chem. Phys. 116, 6755–6759 (2002).
[CrossRef]

Moore, C. I.

W. S. Rabinovich, R. Mahon, H. Burris, G. C. Gilbreath, P. G. Goetz, C. I. Moore, M. Stell, M. J. Vilcheck, J. L. Witkowsky, and L. Swingen, “Free-space optical communications link at 1550 nm using multiple-quantum-well modulating retroreflectors in a marine environment,” Opt. Eng. 44, 056001 (2005).
[CrossRef]

Murphy, A. B.

N. Zeng and A. B. Murphy, “Heat generation by optically and thermally interacting aggregates of gold nanoparticles under illumination,” Nanotechnology 20, 375702 (2009).
[CrossRef]

Okada, Y.

Y. Okada and Y. Tokumaru, “Precise determination of lattice parameter and thermal expansion coefficient of silicon between 300 and 1500 K,” J. Appl. Phys. 56, 314–320(1984).
[CrossRef]

Palpant, B.

M. Rashidi-Huyeh and B. Palpant, “Thermal response of nanocomposite materials under pulsed laser excitation,” J. Appl. Phys. 96, 4475–4482 (2004).
[CrossRef]

Panepucci, R. R.

V. R. Almeida, C. A. Barrios, R. R. Panepucci, and M. Lipson, “All-optical control of light on a silicon chip,” Nature 431, 1081–1084 (2004).
[CrossRef]

Perez-Madrid, A.

A. Perez-Madrid, J. Rubi, and L. C. Lapas, “Heat transfer between nanoparticles: thermal conductance for near-field interactions,” Phys. Rev. B 77, 155417 (2008).
[CrossRef]

Perner, M.

T. Klar, M. Perner, S. Grosse, G. Von Plessen, W. Spirkl, and J. Feldmann, “Surface-plasmon resonances in single metallic nanoparticles,” Phys. Rev. Lett. 80, 4249–4252 (1998).
[CrossRef]

Pradhan, S.

Q. Xu, B. Schmidt, S. Pradhan, and M. Lipson, “Micrometre-scale silicon electro-optic modulator,” Nature 435, 325–327 (2005).
[CrossRef]

Quidant, R.

G. Baffou, R. Quidant, and C. Girard, “Heat generation in plasmonic nanostructures: influence of morphology,” Appl. Phys. Lett. 94, 153109 (2009).
[CrossRef]

Rabinovich, W. S.

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W. S. Rabinovich, R. Mahon, H. Burris, G. C. Gilbreath, P. G. Goetz, C. I. Moore, M. Stell, M. J. Vilcheck, J. L. Witkowsky, and L. Swingen, “Free-space optical communications link at 1550 nm using multiple-quantum-well modulating retroreflectors in a marine environment,” Opt. Eng. 44, 056001 (2005).
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G. Domingues, S. Volz, K. Joulain, and J. J. Greffet, “Heat transfer between two nanoparticles through near field interaction,” Phys. Rev. Lett. 94, 085901 (2005).
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Figures (5)

Fig. 1.
Fig. 1.

Optical modulator based on coated NPs illustrating the principal mode of operation: (a) a signal-carrying beam (black) encounters weak extinction at a coated NP. A control beam (red) generates heat at the plasmonic frequency of the NP, (b) the NP absorbs the energy and undergoes thermal expansion as a result, (c) the dimensions of the NP change, and the signal beam (black) now encounters strong extinction due to a change in the optical extinction cross section. Finally, (d) the NP relaxes back to its original dimension.

Fig. 2.
Fig. 2.

Coated NP: a metal core with refractive index Ncore and radius Rcore coated with a dielectric coating with refractive index Ncoat making the particle radius Rcoat.

Fig. 3.
Fig. 3.

Effective thermal expansion coefficient, αeff, as a function of the ratio of the core and coating radii, ρ=Rcore/Rcoat. αeff is depicted for various NP core materials: silver (squares), copper (triangles), gold (solid curve), and nickel (circles).

Fig. 4.
Fig. 4.

Extinction efficiency (solid) and its derivative (dotted) as a function of the coating radii, calculated forρ=0.6. Vertical lines indicate extremum gradients according to the derivative.

Fig. 5.
Fig. 5.

Optical contrast for ρ=0.6, z=1cm, and N=5×1015m3.

Tables (1)

Tables Icon

Table 1. Extrema Points Resulting from the Extinction Efficiency Derivative

Equations (19)

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

QscaCscaπRcoat2=2x2n=1(2n+1)(|an|2+|bn|2),
Qext=CextπRcoat2=2x2n=1(2n+1)Re{an+bn},
Qabs=QextQsca,
an=(D˜n/m2+n/x2)ψn(x2)ψn1(x2)(D˜n/m2+n/x2)ξn(x2)ξn1(x2),
bn=(m2G˜n+n/x2)ψn(x2)ψn1(x2)(m2G˜n+n/x2)ξn(x2)ξn1(x2),
D˜n=Dn(m2x2)Anχn(m2x2)/ψn(m2x2)1Anχn(m2x2)/ψn(m2x2),
G˜n=Dn(m2x2)Bnχn(m2x2)/ψn(m2x2)1Bnχn(m2x2)/ψn(m2x2),
An=ψn(m2x1)mDn(m1x1)Dn(m2x1)mDn(m1x1)χn(m2x1)χn(m2x1),
Bn=ψn(m2x1)mDn(m2x1)Dn(m1x1)mχn(m2x1)Dn(m1x1)χn(m2x1),
lm=1N·Cext.
OD=zlm,
P=P0exp(OD).
Cm=τ(T2)τ(T1)=eOD(T2)eOD(T1)=eOD(T1)OD(T2),
ΔRR=α·ΔT,
αeff=αcore13[σ1Γcore(ρ+1)+σ1σ3Γcore+Γcoat(ρ31)ρ3+σ2σ3(ρ3σ2(σ21)ρ)]+αcoat13[Γcoat(1ρ3)],
σ1=Ecore(2μcoat1)(5μcore1)(1+μcore)[Ecore(2μcoat1)+Ecoat(12μcore)],
σ2=2Ecoat1+μcoat+Ecore12μcoreEcoat2μcore1+Ecore12μcore,
σ3=1+μcoat24μcoat,
Γcore=1+μcore1μcore,Γcoat=1+μcoat1μcoat.

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