Abstract

The anisotropic nonlinear third-order susceptibility of nanocomposites consisting of aligned ellipsoidal metallic nanoparticles embedded in a dielectric matrix is modeled from the generalized Maxwell-Garnett equation involving depolarization factors. Depolarization factors take into account different anisotropic particle geometries such as flat disks, rods, or ellipsoids. The equations traditionally used to model third-order susceptibility of nanocomposites are valid only for very low metal volume fractions. Modified equations that allow metal volume fractions up to the limit of validity of the Maxwell-Garnett equation are used. The effect of the different model parameters, namely, the metal volume fraction, the real and imaginary parts of the metal dielectric constant, the matrix dielectric constant, and, finally, the ratio of the real and imaginary parts of the metal third-order susceptibility were investigated using the model gold/silica nanocomposite system. As previously reported in the literature for the isotropic particle case, counterintuitive effects such as sign reversal between the bulk metal and composite nonlinear susceptibilities have been observed. The calculations were applied to the case of gold nanorods embedded in silica that were experimentally found to exhibit anisotropic saturable absorption.

© 2008 Optical Society of America

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2008 (1)

J. M. Lamarre, F. Billard, C. H. Kerboua, M. Lequime, S. Roorda, and L. Martinu, “Anisotropic nonlinear optical absorption of gold nanorods in a silica matrix,” Opt. Commun. 281, 331-340 (2008).
[CrossRef]

2007 (1)

C. Harkati Kerboua, J.-M. Lamarre, L. Martinu, and S. Roorda, “Deformation, alignment and anisotropic optical properties of gold nanoparticles embedded in silica,” Nucl. Instrum. Methods Phys. Res. B 257, 42-46 (2007).
[CrossRef]

2006 (5)

R. Atkinson, W. R. Hendren, G. A. Wurtz, W. Dickson, A. V. Zayats, P. Evans, and R. J. Pollard, “Anisotropic properties of arrays of gold nanorods embedded in alumina,” Phys. Rev. B 73, 235402 (2006).
[CrossRef]

X. C. Jiang, A. Brioude, and M. P. Pileni, “Gold nanorods: limitation on their synthesis and optical properties,” Colloids Surf. A 277, 201-206 (2006).
[CrossRef]

M. Pelton, M. Liu, S. Park, N. F. Scherer, and P. Guyot-Sionnest, “Ultrafast resonant optical scattering from single gold nanorods: large nonlinearities and plasmon saturation,” Phys. Rev. B 73, 155419 (2006).
[CrossRef]

H. I. Elim, J. Yang, J. Y. Lee, J. Mi, and W. Ji, “Observation of saturable absorption and reverse saturable absorption at longitudinal plasmon resonance in gold nanorods,” Appl. Phys. Lett. 88, 083107 (2006).
[CrossRef]

S. Qu, Y. Zhang, H. Li, J. Qiu, and C. Zhu, “Nanosecond nonlinear absorption in Au and Ag nanoparticles precipitated glasses induced by a femtosecond laser,” Opt. Mater. 28, 259-265 (2006).
[CrossRef]

2005 (5)

H. Shen, B. Cheng, G. Lu, W. Wang, D. Guan, Z. Chen, and G. Yang, “Nonlinear optical properties of Au/PVP composite thin films,” Chin. Phys. 14, 1915-1918 (2005).
[CrossRef]

H. B. Liao, W. Lu, W. Wen, and G. K. L. Wong, “Optical characteristics of gold nanoparticles-doped multilayer thin film,” J. Opt. Soc. Am. B 22, 1923-1926 (2005).
[CrossRef]

D. J. Chen, S. Ding, J. B. Han, H. J. Zhou, S. Xiao, G. G. Xiong, and Q. Q. Wang, “A sign alternation of nonlinear absorption in gold composite films in Z-scan,” Chin. Phys. Lett. 22, 2286-2289 (2005).
[CrossRef]

J.-M. Lamarre, Z. Yu, C. Harkati, S. Roorda, and L. Martinu, “Optical and microstructural properties of nanocomposite Au/SiO2 films containing particles deformed by heavy ion irradiation,” Thin Solid Films 479, 232-237 (2005).
[CrossRef]

V. V. Kruglyak, R. J. Hicken, M. Ali, B. J. Hickey, A. T. G. Pym, and B. K. Tanner, “Ultrafast third-order optical nonlinearity of noble and transition metal thin films,” J. Opt. A, Pure Appl. Opt. 7, S235-S240 (2005).
[CrossRef]

2004 (2)

P. Wang, Y. Lu, L. Tang, J. Zhang, H. Ming, J. Xie, F. Ho, H. Chang, H. Lin, and D. Tsai, “Surface-enhanced optical nonlinearity of a gold film,” Opt. Commun. 229, 425-429 (2004).
[CrossRef]

S. Roorda, T. V. Dillen, A. Polman, C. Graf, A. van Blaaderen, and B. J. Kooi, “Aligned gold nanorods in silica made by ion irradiation of core-shell colloidal particles,” Adv. Mater. (Weinheim, Ger.) 16, 235-237 (2004).
[CrossRef]

2003 (4)

S. Giordano, “Effective medium theory for dispersions of dielectric ellipsoids,” J. Electrost. 58, 59-76 (2003).
[CrossRef]

H. B. Liao, W. Wen, and G. K. L. Wong, “Preparation and optical characterization of Au/SiO2 composite films with multilayer structure,” J. Appl. Phys. 93, 4485-4488 (2003).
[CrossRef]

S. Qu, C. Zhao, X. Jiang, G. Fang, Y. Gao, H. Zeng, Y. Song, J. Qui, C. Zhu, and K. Hirao, “Optical nonlinearities of space selectively precipated Au nanoparticles inside glasses,” Chem. Phys. Lett. 368, 352-358 (2003).
[CrossRef]

E. Cattaruzza, G. Battaglin, P. Calvelli, F. Gonella, G. Mattei, C. Maurizio, P. Mazzoldi, S. Padovani, R. Polloni, C. Sada, B. Scremin, and F. D'Acapito, “Fast nonlinear refractive index of pure and alloy metallic nanoclusters in silica glass,” Compos. Sci. Technol. 63, 1203-1208 (2003).
[CrossRef]

2002 (2)

S. Qu, C. Du, Y. Song, Y. Wang, Y. Gao, S. Liu, Y. Li, and D. Zhu, “Optical nonlinearities and optical limiting properties in gold nanoparticles protected by ligands,” Chem. Phys. Lett. 356, 403-408 (2002).
[CrossRef]

N. Pinçon, B. Palpant, D. Prot, E. Charron, and S. Debrus, “Third-order nonlinear optical responses of Au:SiO2 thin films: influence of gold nanoparticle concentration and morphologic parameters,” Eur. Phys. J. D 19, 395-402 (2002).
[CrossRef]

2000 (2)

M. Kyoung and M. Lee, “Z-scan studies on the third-order optical nonlinearity of Au nanoparticles embedded in TiO2,” Bull. Korean Chem. Soc. 21, 26-28 (2000).

S. Debrus, J. Lafait, M. May, N. Pinçon, D. Prot, C. Sella, and J. Venturini, “Z-scan determination of the third-order optical nonlinearity of gold: silica nanocomposites,” J. Appl. Phys. 88, 4469-4475 (2000).
[CrossRef]

1999 (2)

D. D. Smith, Y. Yoon, R. W. Boyd, J. K. Campbell, L. A. Baker, R. M. Crooks, and M. George, “Z-scan measurement of the nonlinear absorption of a thin gold film,” J. Appl. Phys. 86, 6200-6205 (1999).
[CrossRef]

M. Kyoung and M. Lee, “Nonlinear absorption and refractive index measurements of silver nanorods by the Z-scan technique,” Opt. Commun. 171, 145-148 (1999).
[CrossRef]

1998 (1)

H. B. Liao, R. F. Xiao, H. Wang, K. S. Wong, and G. K. L. Wong, “Large third-order optical nonlinearity in Au:TiO2 composite films measured on a femtosecond time scale,” Appl. Phys. Lett. 72, 1817-1819 (1998).
[CrossRef]

1997 (1)

1996 (1)

I. Tanahashi, Y. Manabe, T. Tohda, S. Sasaki, and A. Nakamura, “Optical nonlinearities of Au/SiO2 composite thin films prepared by a sputtering method,” J. Appl. Phys. 79, 1244-1249 (1996).
[CrossRef]

1994 (1)

K. Fukumi, A. Chayahara, K. Kadono, T. Sakaguchi, Y. Horino, M. Miya, K. Fujii, K. Hayakawa, and M. Satou, “Gold nanoparticles ion implanted in glass with enhanced nonlinear optical properties,” J. Appl. Phys. 75, 3075-3080 (1994).
[CrossRef]

1989 (1)

J. W. Haus, R. Inguva, and C. M. Bowden, “Effective-medium theory of nonlinear ellipsoidal composites,” Phys. Rev. A 40, 5729-5734 (1989).
[CrossRef] [PubMed]

1985 (1)

1980 (1)

D. E. Aspnes, E. Kinsbron, and D. D. Bacon, “Optical properties of Au: sample effects,” Phys. Rev. B 21, 3290-3299 (1980).
[CrossRef]

1973 (1)

R. W. Cohen, G. D. Cody, M. D. Coutts, and B. Abeles, “Optical properties of granular silver and gold films,” Phys. Rev. B 8, 3689-3701 (1973).
[CrossRef]

1972 (1)

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

1970 (1)

M. L. Thèye, “Investigation of the optical properties of Au by means of semitransparent films,” Phys. Rev. B 2, 3060-3078 (1970).
[CrossRef]

1906 (1)

J. C. Maxwell-Garnett, 'Colours in metal glasses, in metallic films, and in metallic solutions. II,” Philos. Trans. R. Soc. London, Ser. A 205, 237-288 (1906).
[CrossRef]

1904 (1)

J. C. Maxwell-Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London, Ser. A 203, 385-420 (1904).
[CrossRef]

Abeles, B.

R. W. Cohen, G. D. Cody, M. D. Coutts, and B. Abeles, “Optical properties of granular silver and gold films,” Phys. Rev. B 8, 3689-3701 (1973).
[CrossRef]

Ali, M.

V. V. Kruglyak, R. J. Hicken, M. Ali, B. J. Hickey, A. T. G. Pym, and B. K. Tanner, “Ultrafast third-order optical nonlinearity of noble and transition metal thin films,” J. Opt. A, Pure Appl. Opt. 7, S235-S240 (2005).
[CrossRef]

Aspnes, D. E.

D. E. Aspnes, E. Kinsbron, and D. D. Bacon, “Optical properties of Au: sample effects,” Phys. Rev. B 21, 3290-3299 (1980).
[CrossRef]

Atkinson, R.

R. Atkinson, W. R. Hendren, G. A. Wurtz, W. Dickson, A. V. Zayats, P. Evans, and R. J. Pollard, “Anisotropic properties of arrays of gold nanorods embedded in alumina,” Phys. Rev. B 73, 235402 (2006).
[CrossRef]

Bacon, D. D.

D. E. Aspnes, E. Kinsbron, and D. D. Bacon, “Optical properties of Au: sample effects,” Phys. Rev. B 21, 3290-3299 (1980).
[CrossRef]

Baker, L. A.

D. D. Smith, Y. Yoon, R. W. Boyd, J. K. Campbell, L. A. Baker, R. M. Crooks, and M. George, “Z-scan measurement of the nonlinear absorption of a thin gold film,” J. Appl. Phys. 86, 6200-6205 (1999).
[CrossRef]

Battaglin, G.

E. Cattaruzza, G. Battaglin, P. Calvelli, F. Gonella, G. Mattei, C. Maurizio, P. Mazzoldi, S. Padovani, R. Polloni, C. Sada, B. Scremin, and F. D'Acapito, “Fast nonlinear refractive index of pure and alloy metallic nanoclusters in silica glass,” Compos. Sci. Technol. 63, 1203-1208 (2003).
[CrossRef]

Billard, F.

J. M. Lamarre, F. Billard, C. H. Kerboua, M. Lequime, S. Roorda, and L. Martinu, “Anisotropic nonlinear optical absorption of gold nanorods in a silica matrix,” Opt. Commun. 281, 331-340 (2008).
[CrossRef]

Bowden, C. M.

J. W. Haus, R. Inguva, and C. M. Bowden, “Effective-medium theory of nonlinear ellipsoidal composites,” Phys. Rev. A 40, 5729-5734 (1989).
[CrossRef] [PubMed]

Boyd, R. W.

D. D. Smith, Y. Yoon, R. W. Boyd, J. K. Campbell, L. A. Baker, R. M. Crooks, and M. George, “Z-scan measurement of the nonlinear absorption of a thin gold film,” J. Appl. Phys. 86, 6200-6205 (1999).
[CrossRef]

D. D. Smith, G. Fischer, R. W. Boyd, and D. A. Gregory, “Cancellation of photoinduced absorption in metal nanoparticle composites through a counterintuitive consequence of local field effects,” J. Opt. Soc. Am. B 14, 1625-1631 (1997).
[CrossRef]

Brioude, A.

X. C. Jiang, A. Brioude, and M. P. Pileni, “Gold nanorods: limitation on their synthesis and optical properties,” Colloids Surf. A 277, 201-206 (2006).
[CrossRef]

Calvelli, P.

E. Cattaruzza, G. Battaglin, P. Calvelli, F. Gonella, G. Mattei, C. Maurizio, P. Mazzoldi, S. Padovani, R. Polloni, C. Sada, B. Scremin, and F. D'Acapito, “Fast nonlinear refractive index of pure and alloy metallic nanoclusters in silica glass,” Compos. Sci. Technol. 63, 1203-1208 (2003).
[CrossRef]

Campbell, J. K.

D. D. Smith, Y. Yoon, R. W. Boyd, J. K. Campbell, L. A. Baker, R. M. Crooks, and M. George, “Z-scan measurement of the nonlinear absorption of a thin gold film,” J. Appl. Phys. 86, 6200-6205 (1999).
[CrossRef]

Cattaruzza, E.

E. Cattaruzza, G. Battaglin, P. Calvelli, F. Gonella, G. Mattei, C. Maurizio, P. Mazzoldi, S. Padovani, R. Polloni, C. Sada, B. Scremin, and F. D'Acapito, “Fast nonlinear refractive index of pure and alloy metallic nanoclusters in silica glass,” Compos. Sci. Technol. 63, 1203-1208 (2003).
[CrossRef]

Chang, H.

P. Wang, Y. Lu, L. Tang, J. Zhang, H. Ming, J. Xie, F. Ho, H. Chang, H. Lin, and D. Tsai, “Surface-enhanced optical nonlinearity of a gold film,” Opt. Commun. 229, 425-429 (2004).
[CrossRef]

Charron, E.

N. Pinçon, B. Palpant, D. Prot, E. Charron, and S. Debrus, “Third-order nonlinear optical responses of Au:SiO2 thin films: influence of gold nanoparticle concentration and morphologic parameters,” Eur. Phys. J. D 19, 395-402 (2002).
[CrossRef]

Chayahara, A.

K. Fukumi, A. Chayahara, K. Kadono, T. Sakaguchi, Y. Horino, M. Miya, K. Fujii, K. Hayakawa, and M. Satou, “Gold nanoparticles ion implanted in glass with enhanced nonlinear optical properties,” J. Appl. Phys. 75, 3075-3080 (1994).
[CrossRef]

Chen, D. J.

D. J. Chen, S. Ding, J. B. Han, H. J. Zhou, S. Xiao, G. G. Xiong, and Q. Q. Wang, “A sign alternation of nonlinear absorption in gold composite films in Z-scan,” Chin. Phys. Lett. 22, 2286-2289 (2005).
[CrossRef]

Chen, Z.

H. Shen, B. Cheng, G. Lu, W. Wang, D. Guan, Z. Chen, and G. Yang, “Nonlinear optical properties of Au/PVP composite thin films,” Chin. Phys. 14, 1915-1918 (2005).
[CrossRef]

Cheng, B.

H. Shen, B. Cheng, G. Lu, W. Wang, D. Guan, Z. Chen, and G. Yang, “Nonlinear optical properties of Au/PVP composite thin films,” Chin. Phys. 14, 1915-1918 (2005).
[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]

Cody, G. D.

R. W. Cohen, G. D. Cody, M. D. Coutts, and B. Abeles, “Optical properties of granular silver and gold films,” Phys. Rev. B 8, 3689-3701 (1973).
[CrossRef]

Cohen, R. W.

R. W. Cohen, G. D. Cody, M. D. Coutts, and B. Abeles, “Optical properties of granular silver and gold films,” Phys. Rev. B 8, 3689-3701 (1973).
[CrossRef]

Coutts, M. D.

R. W. Cohen, G. D. Cody, M. D. Coutts, and B. Abeles, “Optical properties of granular silver and gold films,” Phys. Rev. B 8, 3689-3701 (1973).
[CrossRef]

Crooks, R. M.

D. D. Smith, Y. Yoon, R. W. Boyd, J. K. Campbell, L. A. Baker, R. M. Crooks, and M. George, “Z-scan measurement of the nonlinear absorption of a thin gold film,” J. Appl. Phys. 86, 6200-6205 (1999).
[CrossRef]

D'Acapito, F.

E. Cattaruzza, G. Battaglin, P. Calvelli, F. Gonella, G. Mattei, C. Maurizio, P. Mazzoldi, S. Padovani, R. Polloni, C. Sada, B. Scremin, and F. D'Acapito, “Fast nonlinear refractive index of pure and alloy metallic nanoclusters in silica glass,” Compos. Sci. Technol. 63, 1203-1208 (2003).
[CrossRef]

Debrus, S.

N. Pinçon, B. Palpant, D. Prot, E. Charron, and S. Debrus, “Third-order nonlinear optical responses of Au:SiO2 thin films: influence of gold nanoparticle concentration and morphologic parameters,” Eur. Phys. J. D 19, 395-402 (2002).
[CrossRef]

S. Debrus, J. Lafait, M. May, N. Pinçon, D. Prot, C. Sella, and J. Venturini, “Z-scan determination of the third-order optical nonlinearity of gold: silica nanocomposites,” J. Appl. Phys. 88, 4469-4475 (2000).
[CrossRef]

Dickson, W.

R. Atkinson, W. R. Hendren, G. A. Wurtz, W. Dickson, A. V. Zayats, P. Evans, and R. J. Pollard, “Anisotropic properties of arrays of gold nanorods embedded in alumina,” Phys. Rev. B 73, 235402 (2006).
[CrossRef]

Dillen, T. V.

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S. Qu, Y. Zhang, H. Li, J. Qiu, and C. Zhu, “Nanosecond nonlinear absorption in Au and Ag nanoparticles precipitated glasses induced by a femtosecond laser,” Opt. Mater. 28, 259-265 (2006).
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S. Qu, Y. Zhang, H. Li, J. Qiu, and C. Zhu, “Nanosecond nonlinear absorption in Au and Ag nanoparticles precipitated glasses induced by a femtosecond laser,” Opt. Mater. 28, 259-265 (2006).
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[CrossRef]

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[CrossRef]

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S. Qu, C. Zhao, X. Jiang, G. Fang, Y. Gao, H. Zeng, Y. Song, J. Qui, C. Zhu, and K. Hirao, “Optical nonlinearities of space selectively precipated Au nanoparticles inside glasses,” Chem. Phys. Lett. 368, 352-358 (2003).
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Roorda, S.

J. M. Lamarre, F. Billard, C. H. Kerboua, M. Lequime, S. Roorda, and L. Martinu, “Anisotropic nonlinear optical absorption of gold nanorods in a silica matrix,” Opt. Commun. 281, 331-340 (2008).
[CrossRef]

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[CrossRef]

J.-M. Lamarre, Z. Yu, C. Harkati, S. Roorda, and L. Martinu, “Optical and microstructural properties of nanocomposite Au/SiO2 films containing particles deformed by heavy ion irradiation,” Thin Solid Films 479, 232-237 (2005).
[CrossRef]

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Sada, C.

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[CrossRef]

Sakaguchi, T.

K. Fukumi, A. Chayahara, K. Kadono, T. Sakaguchi, Y. Horino, M. Miya, K. Fujii, K. Hayakawa, and M. Satou, “Gold nanoparticles ion implanted in glass with enhanced nonlinear optical properties,” J. Appl. Phys. 75, 3075-3080 (1994).
[CrossRef]

Sasaki, S.

I. Tanahashi, Y. Manabe, T. Tohda, S. Sasaki, and A. Nakamura, “Optical nonlinearities of Au/SiO2 composite thin films prepared by a sputtering method,” J. Appl. Phys. 79, 1244-1249 (1996).
[CrossRef]

Satou, M.

K. Fukumi, A. Chayahara, K. Kadono, T. Sakaguchi, Y. Horino, M. Miya, K. Fujii, K. Hayakawa, and M. Satou, “Gold nanoparticles ion implanted in glass with enhanced nonlinear optical properties,” J. Appl. Phys. 75, 3075-3080 (1994).
[CrossRef]

Scherer, N. F.

M. Pelton, M. Liu, S. Park, N. F. Scherer, and P. Guyot-Sionnest, “Ultrafast resonant optical scattering from single gold nanorods: large nonlinearities and plasmon saturation,” Phys. Rev. B 73, 155419 (2006).
[CrossRef]

Scremin, B.

E. Cattaruzza, G. Battaglin, P. Calvelli, F. Gonella, G. Mattei, C. Maurizio, P. Mazzoldi, S. Padovani, R. Polloni, C. Sada, B. Scremin, and F. D'Acapito, “Fast nonlinear refractive index of pure and alloy metallic nanoclusters in silica glass,” Compos. Sci. Technol. 63, 1203-1208 (2003).
[CrossRef]

Sella, C.

S. Debrus, J. Lafait, M. May, N. Pinçon, D. Prot, C. Sella, and J. Venturini, “Z-scan determination of the third-order optical nonlinearity of gold: silica nanocomposites,” J. Appl. Phys. 88, 4469-4475 (2000).
[CrossRef]

Shen, H.

H. Shen, B. Cheng, G. Lu, W. Wang, D. Guan, Z. Chen, and G. Yang, “Nonlinear optical properties of Au/PVP composite thin films,” Chin. Phys. 14, 1915-1918 (2005).
[CrossRef]

Smith, D. D.

D. D. Smith, Y. Yoon, R. W. Boyd, J. K. Campbell, L. A. Baker, R. M. Crooks, and M. George, “Z-scan measurement of the nonlinear absorption of a thin gold film,” J. Appl. Phys. 86, 6200-6205 (1999).
[CrossRef]

D. D. Smith, G. Fischer, R. W. Boyd, and D. A. Gregory, “Cancellation of photoinduced absorption in metal nanoparticle composites through a counterintuitive consequence of local field effects,” J. Opt. Soc. Am. B 14, 1625-1631 (1997).
[CrossRef]

Song, Y.

S. Qu, C. Zhao, X. Jiang, G. Fang, Y. Gao, H. Zeng, Y. Song, J. Qui, C. Zhu, and K. Hirao, “Optical nonlinearities of space selectively precipated Au nanoparticles inside glasses,” Chem. Phys. Lett. 368, 352-358 (2003).
[CrossRef]

S. Qu, C. Du, Y. Song, Y. Wang, Y. Gao, S. Liu, Y. Li, and D. Zhu, “Optical nonlinearities and optical limiting properties in gold nanoparticles protected by ligands,” Chem. Phys. Lett. 356, 403-408 (2002).
[CrossRef]

Tanahashi, I.

I. Tanahashi, Y. Manabe, T. Tohda, S. Sasaki, and A. Nakamura, “Optical nonlinearities of Au/SiO2 composite thin films prepared by a sputtering method,” J. Appl. Phys. 79, 1244-1249 (1996).
[CrossRef]

Tang, L.

P. Wang, Y. Lu, L. Tang, J. Zhang, H. Ming, J. Xie, F. Ho, H. Chang, H. Lin, and D. Tsai, “Surface-enhanced optical nonlinearity of a gold film,” Opt. Commun. 229, 425-429 (2004).
[CrossRef]

Tanner, B. K.

V. V. Kruglyak, R. J. Hicken, M. Ali, B. J. Hickey, A. T. G. Pym, and B. K. Tanner, “Ultrafast third-order optical nonlinearity of noble and transition metal thin films,” J. Opt. A, Pure Appl. Opt. 7, S235-S240 (2005).
[CrossRef]

Thèye, M. L.

M. L. Thèye, “Investigation of the optical properties of Au by means of semitransparent films,” Phys. Rev. B 2, 3060-3078 (1970).
[CrossRef]

Tohda, T.

I. Tanahashi, Y. Manabe, T. Tohda, S. Sasaki, and A. Nakamura, “Optical nonlinearities of Au/SiO2 composite thin films prepared by a sputtering method,” J. Appl. Phys. 79, 1244-1249 (1996).
[CrossRef]

Tsai, D.

P. Wang, Y. Lu, L. Tang, J. Zhang, H. Ming, J. Xie, F. Ho, H. Chang, H. Lin, and D. Tsai, “Surface-enhanced optical nonlinearity of a gold film,” Opt. Commun. 229, 425-429 (2004).
[CrossRef]

van Blaaderen, A.

S. Roorda, T. V. Dillen, A. Polman, C. Graf, A. van Blaaderen, and B. J. Kooi, “Aligned gold nanorods in silica made by ion irradiation of core-shell colloidal particles,” Adv. Mater. (Weinheim, Ger.) 16, 235-237 (2004).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

Venturini, J.

S. Debrus, J. Lafait, M. May, N. Pinçon, D. Prot, C. Sella, and J. Venturini, “Z-scan determination of the third-order optical nonlinearity of gold: silica nanocomposites,” J. Appl. Phys. 88, 4469-4475 (2000).
[CrossRef]

Vollmer, M.

U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer, 1995).

Wang, H.

H. B. Liao, R. F. Xiao, H. Wang, K. S. Wong, and G. K. L. Wong, “Large third-order optical nonlinearity in Au:TiO2 composite films measured on a femtosecond time scale,” Appl. Phys. Lett. 72, 1817-1819 (1998).
[CrossRef]

Wang, P.

P. Wang, Y. Lu, L. Tang, J. Zhang, H. Ming, J. Xie, F. Ho, H. Chang, H. Lin, and D. Tsai, “Surface-enhanced optical nonlinearity of a gold film,” Opt. Commun. 229, 425-429 (2004).
[CrossRef]

Wang, Q. Q.

D. J. Chen, S. Ding, J. B. Han, H. J. Zhou, S. Xiao, G. G. Xiong, and Q. Q. Wang, “A sign alternation of nonlinear absorption in gold composite films in Z-scan,” Chin. Phys. Lett. 22, 2286-2289 (2005).
[CrossRef]

Wang, W.

H. Shen, B. Cheng, G. Lu, W. Wang, D. Guan, Z. Chen, and G. Yang, “Nonlinear optical properties of Au/PVP composite thin films,” Chin. Phys. 14, 1915-1918 (2005).
[CrossRef]

Wang, Y.

S. Qu, C. Du, Y. Song, Y. Wang, Y. Gao, S. Liu, Y. Li, and D. Zhu, “Optical nonlinearities and optical limiting properties in gold nanoparticles protected by ligands,” Chem. Phys. Lett. 356, 403-408 (2002).
[CrossRef]

Weber, M. J.

M. J. Weber, Handbook of Optical Materials (CRC, 2003).

Wen, W.

H. B. Liao, W. Lu, W. Wen, and G. K. L. Wong, “Optical characteristics of gold nanoparticles-doped multilayer thin film,” J. Opt. Soc. Am. B 22, 1923-1926 (2005).
[CrossRef]

H. B. Liao, W. Wen, and G. K. L. Wong, “Preparation and optical characterization of Au/SiO2 composite films with multilayer structure,” J. Appl. Phys. 93, 4485-4488 (2003).
[CrossRef]

Wong, G. K. L.

H. B. Liao, W. Lu, W. Wen, and G. K. L. Wong, “Optical characteristics of gold nanoparticles-doped multilayer thin film,” J. Opt. Soc. Am. B 22, 1923-1926 (2005).
[CrossRef]

H. B. Liao, W. Wen, and G. K. L. Wong, “Preparation and optical characterization of Au/SiO2 composite films with multilayer structure,” J. Appl. Phys. 93, 4485-4488 (2003).
[CrossRef]

H. B. Liao, R. F. Xiao, H. Wang, K. S. Wong, and G. K. L. Wong, “Large third-order optical nonlinearity in Au:TiO2 composite films measured on a femtosecond time scale,” Appl. Phys. Lett. 72, 1817-1819 (1998).
[CrossRef]

Wong, K. S.

H. B. Liao, R. F. Xiao, H. Wang, K. S. Wong, and G. K. L. Wong, “Large third-order optical nonlinearity in Au:TiO2 composite films measured on a femtosecond time scale,” Appl. Phys. Lett. 72, 1817-1819 (1998).
[CrossRef]

Wurtz, G. A.

R. Atkinson, W. R. Hendren, G. A. Wurtz, W. Dickson, A. V. Zayats, P. Evans, and R. J. Pollard, “Anisotropic properties of arrays of gold nanorods embedded in alumina,” Phys. Rev. B 73, 235402 (2006).
[CrossRef]

Xiao, R. F.

H. B. Liao, R. F. Xiao, H. Wang, K. S. Wong, and G. K. L. Wong, “Large third-order optical nonlinearity in Au:TiO2 composite films measured on a femtosecond time scale,” Appl. Phys. Lett. 72, 1817-1819 (1998).
[CrossRef]

Xiao, S.

D. J. Chen, S. Ding, J. B. Han, H. J. Zhou, S. Xiao, G. G. Xiong, and Q. Q. Wang, “A sign alternation of nonlinear absorption in gold composite films in Z-scan,” Chin. Phys. Lett. 22, 2286-2289 (2005).
[CrossRef]

Xie, J.

P. Wang, Y. Lu, L. Tang, J. Zhang, H. Ming, J. Xie, F. Ho, H. Chang, H. Lin, and D. Tsai, “Surface-enhanced optical nonlinearity of a gold film,” Opt. Commun. 229, 425-429 (2004).
[CrossRef]

Xiong, G. G.

D. J. Chen, S. Ding, J. B. Han, H. J. Zhou, S. Xiao, G. G. Xiong, and Q. Q. Wang, “A sign alternation of nonlinear absorption in gold composite films in Z-scan,” Chin. Phys. Lett. 22, 2286-2289 (2005).
[CrossRef]

Yang, G.

H. Shen, B. Cheng, G. Lu, W. Wang, D. Guan, Z. Chen, and G. Yang, “Nonlinear optical properties of Au/PVP composite thin films,” Chin. Phys. 14, 1915-1918 (2005).
[CrossRef]

Yang, J.

H. I. Elim, J. Yang, J. Y. Lee, J. Mi, and W. Ji, “Observation of saturable absorption and reverse saturable absorption at longitudinal plasmon resonance in gold nanorods,” Appl. Phys. Lett. 88, 083107 (2006).
[CrossRef]

Yoon, Y.

D. D. Smith, Y. Yoon, R. W. Boyd, J. K. Campbell, L. A. Baker, R. M. Crooks, and M. George, “Z-scan measurement of the nonlinear absorption of a thin gold film,” J. Appl. Phys. 86, 6200-6205 (1999).
[CrossRef]

Yu, Z.

J.-M. Lamarre, Z. Yu, C. Harkati, S. Roorda, and L. Martinu, “Optical and microstructural properties of nanocomposite Au/SiO2 films containing particles deformed by heavy ion irradiation,” Thin Solid Films 479, 232-237 (2005).
[CrossRef]

Zayats, A. V.

R. Atkinson, W. R. Hendren, G. A. Wurtz, W. Dickson, A. V. Zayats, P. Evans, and R. J. Pollard, “Anisotropic properties of arrays of gold nanorods embedded in alumina,” Phys. Rev. B 73, 235402 (2006).
[CrossRef]

Zeng, H.

S. Qu, C. Zhao, X. Jiang, G. Fang, Y. Gao, H. Zeng, Y. Song, J. Qui, C. Zhu, and K. Hirao, “Optical nonlinearities of space selectively precipated Au nanoparticles inside glasses,” Chem. Phys. Lett. 368, 352-358 (2003).
[CrossRef]

Zhang, J.

P. Wang, Y. Lu, L. Tang, J. Zhang, H. Ming, J. Xie, F. Ho, H. Chang, H. Lin, and D. Tsai, “Surface-enhanced optical nonlinearity of a gold film,” Opt. Commun. 229, 425-429 (2004).
[CrossRef]

Zhang, Y.

S. Qu, Y. Zhang, H. Li, J. Qiu, and C. Zhu, “Nanosecond nonlinear absorption in Au and Ag nanoparticles precipitated glasses induced by a femtosecond laser,” Opt. Mater. 28, 259-265 (2006).
[CrossRef]

Zhao, C.

S. Qu, C. Zhao, X. Jiang, G. Fang, Y. Gao, H. Zeng, Y. Song, J. Qui, C. Zhu, and K. Hirao, “Optical nonlinearities of space selectively precipated Au nanoparticles inside glasses,” Chem. Phys. Lett. 368, 352-358 (2003).
[CrossRef]

Zhou, H. J.

D. J. Chen, S. Ding, J. B. Han, H. J. Zhou, S. Xiao, G. G. Xiong, and Q. Q. Wang, “A sign alternation of nonlinear absorption in gold composite films in Z-scan,” Chin. Phys. Lett. 22, 2286-2289 (2005).
[CrossRef]

Zhu, C.

S. Qu, Y. Zhang, H. Li, J. Qiu, and C. Zhu, “Nanosecond nonlinear absorption in Au and Ag nanoparticles precipitated glasses induced by a femtosecond laser,” Opt. Mater. 28, 259-265 (2006).
[CrossRef]

S. Qu, C. Zhao, X. Jiang, G. Fang, Y. Gao, H. Zeng, Y. Song, J. Qui, C. Zhu, and K. Hirao, “Optical nonlinearities of space selectively precipated Au nanoparticles inside glasses,” Chem. Phys. Lett. 368, 352-358 (2003).
[CrossRef]

Zhu, D.

S. Qu, C. Du, Y. Song, Y. Wang, Y. Gao, S. Liu, Y. Li, and D. Zhu, “Optical nonlinearities and optical limiting properties in gold nanoparticles protected by ligands,” Chem. Phys. Lett. 356, 403-408 (2002).
[CrossRef]

Adv. Mater. (Weinheim, Ger.) (1)

S. Roorda, T. V. Dillen, A. Polman, C. Graf, A. van Blaaderen, and B. J. Kooi, “Aligned gold nanorods in silica made by ion irradiation of core-shell colloidal particles,” Adv. Mater. (Weinheim, Ger.) 16, 235-237 (2004).
[CrossRef]

Appl. Phys. Lett. (2)

H. I. Elim, J. Yang, J. Y. Lee, J. Mi, and W. Ji, “Observation of saturable absorption and reverse saturable absorption at longitudinal plasmon resonance in gold nanorods,” Appl. Phys. Lett. 88, 083107 (2006).
[CrossRef]

H. B. Liao, R. F. Xiao, H. Wang, K. S. Wong, and G. K. L. Wong, “Large third-order optical nonlinearity in Au:TiO2 composite films measured on a femtosecond time scale,” Appl. Phys. Lett. 72, 1817-1819 (1998).
[CrossRef]

Bull. Korean Chem. Soc. (1)

M. Kyoung and M. Lee, “Z-scan studies on the third-order optical nonlinearity of Au nanoparticles embedded in TiO2,” Bull. Korean Chem. Soc. 21, 26-28 (2000).

Chem. Phys. Lett. (2)

S. Qu, C. Du, Y. Song, Y. Wang, Y. Gao, S. Liu, Y. Li, and D. Zhu, “Optical nonlinearities and optical limiting properties in gold nanoparticles protected by ligands,” Chem. Phys. Lett. 356, 403-408 (2002).
[CrossRef]

S. Qu, C. Zhao, X. Jiang, G. Fang, Y. Gao, H. Zeng, Y. Song, J. Qui, C. Zhu, and K. Hirao, “Optical nonlinearities of space selectively precipated Au nanoparticles inside glasses,” Chem. Phys. Lett. 368, 352-358 (2003).
[CrossRef]

Chin. Phys. (1)

H. Shen, B. Cheng, G. Lu, W. Wang, D. Guan, Z. Chen, and G. Yang, “Nonlinear optical properties of Au/PVP composite thin films,” Chin. Phys. 14, 1915-1918 (2005).
[CrossRef]

Chin. Phys. Lett. (1)

D. J. Chen, S. Ding, J. B. Han, H. J. Zhou, S. Xiao, G. G. Xiong, and Q. Q. Wang, “A sign alternation of nonlinear absorption in gold composite films in Z-scan,” Chin. Phys. Lett. 22, 2286-2289 (2005).
[CrossRef]

Colloids Surf. A (1)

X. C. Jiang, A. Brioude, and M. P. Pileni, “Gold nanorods: limitation on their synthesis and optical properties,” Colloids Surf. A 277, 201-206 (2006).
[CrossRef]

Compos. Sci. Technol. (1)

E. Cattaruzza, G. Battaglin, P. Calvelli, F. Gonella, G. Mattei, C. Maurizio, P. Mazzoldi, S. Padovani, R. Polloni, C. Sada, B. Scremin, and F. D'Acapito, “Fast nonlinear refractive index of pure and alloy metallic nanoclusters in silica glass,” Compos. Sci. Technol. 63, 1203-1208 (2003).
[CrossRef]

Eur. Phys. J. D (1)

N. Pinçon, B. Palpant, D. Prot, E. Charron, and S. Debrus, “Third-order nonlinear optical responses of Au:SiO2 thin films: influence of gold nanoparticle concentration and morphologic parameters,” Eur. Phys. J. D 19, 395-402 (2002).
[CrossRef]

J. Appl. Phys. (5)

S. Debrus, J. Lafait, M. May, N. Pinçon, D. Prot, C. Sella, and J. Venturini, “Z-scan determination of the third-order optical nonlinearity of gold: silica nanocomposites,” J. Appl. Phys. 88, 4469-4475 (2000).
[CrossRef]

H. B. Liao, W. Wen, and G. K. L. Wong, “Preparation and optical characterization of Au/SiO2 composite films with multilayer structure,” J. Appl. Phys. 93, 4485-4488 (2003).
[CrossRef]

I. Tanahashi, Y. Manabe, T. Tohda, S. Sasaki, and A. Nakamura, “Optical nonlinearities of Au/SiO2 composite thin films prepared by a sputtering method,” J. Appl. Phys. 79, 1244-1249 (1996).
[CrossRef]

K. Fukumi, A. Chayahara, K. Kadono, T. Sakaguchi, Y. Horino, M. Miya, K. Fujii, K. Hayakawa, and M. Satou, “Gold nanoparticles ion implanted in glass with enhanced nonlinear optical properties,” J. Appl. Phys. 75, 3075-3080 (1994).
[CrossRef]

D. D. Smith, Y. Yoon, R. W. Boyd, J. K. Campbell, L. A. Baker, R. M. Crooks, and M. George, “Z-scan measurement of the nonlinear absorption of a thin gold film,” J. Appl. Phys. 86, 6200-6205 (1999).
[CrossRef]

J. Electrost. (1)

S. Giordano, “Effective medium theory for dispersions of dielectric ellipsoids,” J. Electrost. 58, 59-76 (2003).
[CrossRef]

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

V. V. Kruglyak, R. J. Hicken, M. Ali, B. J. Hickey, A. T. G. Pym, and B. K. Tanner, “Ultrafast third-order optical nonlinearity of noble and transition metal thin films,” J. Opt. A, Pure Appl. Opt. 7, S235-S240 (2005).
[CrossRef]

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

Nucl. Instrum. Methods Phys. Res. B (1)

C. Harkati Kerboua, J.-M. Lamarre, L. Martinu, and S. Roorda, “Deformation, alignment and anisotropic optical properties of gold nanoparticles embedded in silica,” Nucl. Instrum. Methods Phys. Res. B 257, 42-46 (2007).
[CrossRef]

Opt. Commun. (3)

M. Kyoung and M. Lee, “Nonlinear absorption and refractive index measurements of silver nanorods by the Z-scan technique,” Opt. Commun. 171, 145-148 (1999).
[CrossRef]

J. M. Lamarre, F. Billard, C. H. Kerboua, M. Lequime, S. Roorda, and L. Martinu, “Anisotropic nonlinear optical absorption of gold nanorods in a silica matrix,” Opt. Commun. 281, 331-340 (2008).
[CrossRef]

P. Wang, Y. Lu, L. Tang, J. Zhang, H. Ming, J. Xie, F. Ho, H. Chang, H. Lin, and D. Tsai, “Surface-enhanced optical nonlinearity of a gold film,” Opt. Commun. 229, 425-429 (2004).
[CrossRef]

Opt. Lett. (1)

Opt. Mater. (1)

S. Qu, Y. Zhang, H. Li, J. Qiu, and C. Zhu, “Nanosecond nonlinear absorption in Au and Ag nanoparticles precipitated glasses induced by a femtosecond laser,” Opt. Mater. 28, 259-265 (2006).
[CrossRef]

Philos. Trans. R. Soc. London, Ser. A (2)

J. C. Maxwell-Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London, Ser. A 203, 385-420 (1904).
[CrossRef]

J. C. Maxwell-Garnett, 'Colours in metal glasses, in metallic films, and in metallic solutions. II,” Philos. Trans. R. Soc. London, Ser. A 205, 237-288 (1906).
[CrossRef]

Phys. Rev. A (1)

J. W. Haus, R. Inguva, and C. M. Bowden, “Effective-medium theory of nonlinear ellipsoidal composites,” Phys. Rev. A 40, 5729-5734 (1989).
[CrossRef] [PubMed]

Phys. Rev. B (6)

R. W. Cohen, G. D. Cody, M. D. Coutts, and B. Abeles, “Optical properties of granular silver and gold films,” Phys. Rev. B 8, 3689-3701 (1973).
[CrossRef]

D. E. Aspnes, E. Kinsbron, and D. D. Bacon, “Optical properties of Au: sample effects,” Phys. Rev. B 21, 3290-3299 (1980).
[CrossRef]

M. L. Thèye, “Investigation of the optical properties of Au by means of semitransparent films,” Phys. Rev. B 2, 3060-3078 (1970).
[CrossRef]

M. Pelton, M. Liu, S. Park, N. F. Scherer, and P. Guyot-Sionnest, “Ultrafast resonant optical scattering from single gold nanorods: large nonlinearities and plasmon saturation,” Phys. Rev. B 73, 155419 (2006).
[CrossRef]

R. Atkinson, W. R. Hendren, G. A. Wurtz, W. Dickson, A. V. Zayats, P. Evans, and R. J. Pollard, “Anisotropic properties of arrays of gold nanorods embedded in alumina,” Phys. Rev. B 73, 235402 (2006).
[CrossRef]

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[CrossRef]

Thin Solid Films (1)

J.-M. Lamarre, Z. Yu, C. Harkati, S. Roorda, and L. Martinu, “Optical and microstructural properties of nanocomposite Au/SiO2 films containing particles deformed by heavy ion irradiation,” Thin Solid Films 479, 232-237 (2005).
[CrossRef]

Other (5)

J.-M. Lamarre and L. Martinu, in Proceedings of the 47th Annual Technical Conference of the Society of Vacuum Coaters, (SVC, 2004), p. 343.

M. J. Weber, Handbook of Optical Materials (CRC, 2003).

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

U. Kreibig and M. Vollmer, Optical Properties of Metal Clusters (Springer, 1995).

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Figures (12)

Fig. 1
Fig. 1

Normalized I m { χ eff , j ( 3 ) } and R e { χ eff , j ( 3 ) } as a function of L j . Dotted curves: low p model [Eqs. (14, 15)]. Solid curves: high p model [Eqs. (22, 23)] for p values of 1, 2, 5, and 10 vol. % . For p values exceeding 1%, the two models’ predictions are clearly different.

Fig. 2
Fig. 2

Model predictions for normalized I m { χ eff , j ( 3 ) } and R e { χ eff , j ( 3 ) } as a function of L j for p values varying from 0 to 20 vol. % by increments of 2 vol. % .

Fig. 3
Fig. 3

Normalized I m { χ eff , j ( 3 ) } and R e { χ eff , j ( 3 ) } as a function of p for different L j values.

Fig. 4
Fig. 4

Model predictions for normalized I m { χ eff , j ( 3 ) } and R e { χ eff , j ( 3 ) } as a function of L j for R e { ϵ m } values varying from 7 to 3 by increments of 0.4.

Fig. 5
Fig. 5

Normalized I m { χ eff , j ( 3 ) } and R e { χ eff , j ( 3 ) } as a function of R e { ϵ m } for different L j values.

Fig. 6
Fig. 6

Model predictions for normalized I m { χ eff , j ( 3 ) } and R e { χ eff , j ( 3 ) } as a function of L j for I m { ϵ m } values varying from 1 to 3 by increments of 0.2.

Fig. 7
Fig. 7

Model predictions for normalized I m { χ eff , j ( 3 ) } and R e { χ eff , j ( 3 ) } as a function of L j for n d values varying from 1.33 to 1.63 by increments of 0.02.

Fig. 8
Fig. 8

Normalized I m { χ eff , j ( 3 ) } and R e { χ eff , j ( 3 ) } as a function of n d for different L j values.

Fig. 9
Fig. 9

Normalized I m { χ eff , j ( 3 ) } and R e { χ eff , j ( 3 ) } plotted as a function of L j for different values of the I m { χ m ( 3 ) } R e { χ m ( 3 ) } ratio.

Fig. 10
Fig. 10

Depolarization factors for the LA ( L LA ) and SA ( L SA ) as a function of the axis ratio.

Fig. 11
Fig. 11

Normalized I m { χ eff , j ( 3 ) } and R e { χ eff , j ( 3 ) } plotted as a function of L j according to Eqs. (22, 23) using ϵ d = 2.1316 , R e { ϵ m } = 6.508 , I m { ϵ m } = 1.71 , p = 0.2 , I m { χ m ( 3 ) } = 5 , and R e { χ m ( 3 ) } = 1 . The NLS for L LA and L SA values optimized to experimentally determined ratios are represented by the points. The NLS corresponding to nanospheres ( L 1 3 ) is also shown. The coordinates of each point are indicated on the top left.

Fig. 12
Fig. 12

Difference between normalized χ eff , j ( 3 ) values for the long and short axes as a function of the axis ratio. R e and I m labels represent the real and imaginary parts of the difference. The maximum anisotropy is obtained for slightly elliptical particles, namely, when the axis ratio is close to 1.25.

Tables (3)

Tables Icon

Table 1 Gold Dielectric Permittivity, ϵ m , Measured at 532 nm by Several Authors a

Tables Icon

Table 2 List of the Simulation Parameters a

Tables Icon

Table 3 Calculated and Experimental Nonlinear Absorption Ratios

Equations (28)

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

ϵ eff ϵ d ϵ eff + 2 ϵ d = p ϵ m ϵ d ϵ m + 2 ϵ d .
χ eff ( 3 ) = p f 2 ( ω ) f ( ω ) 2 χ m ( 3 ) ,
f ( ω ) = 3 ϵ d ϵ m + 2 ϵ d .
ϵ eff j ϵ d L j ϵ eff j + ( 1 L j ) ϵ d = p ϵ m ϵ d L j ϵ m + ( 1 L j ) ϵ d ,
L z = L LA = 1 e 2 e 2 ( 1 2 e ln ( 1 + e 1 e ) 1 ) ,
L x = L y = L SA = 1 L LA 2 .
ϵ eff j ϵ d + p ϵ d ( ϵ m ϵ d ) L j ϵ m + ( 1 L j ) ϵ d .
d ϵ eff j p ( ϵ d L j ϵ m + ( 1 L j ) ϵ d ) 2 d ϵ m j = p f j 2 ( ω , L j ) d ϵ m j ,
f j ( ω , L j ) = ϵ d L j ϵ m + ( 1 L j ) ϵ d .
d ϵ m j = P m NL ϵ 0 E loc j = 3 4 χ m ( 3 ) E loc j 2 ,
d ϵ eff j = P eff , j NL ϵ 0 E j = 3 4 χ eff , j ( 3 ) E j 2 ,
E loc j = ϵ d L j ϵ m + ( 1 L j ) ϵ d E j = f j ( ω , L j ) E j .
χ eff , j ( 3 ) = p f j ( ω , L j ) 2 f j ( ω , L j ) 2 χ m ( 3 ) ,
R e { χ eff , j ( 3 ) } = p ϵ d 4 1 ( A 0 2 + B 0 2 ) 3 ( ( A 0 2 B 0 2 ) R e { χ m ( 3 ) } + 2 A 0 B 0 I m { χ m ( 3 ) } ) ,
I m { χ eff , j ( 3 ) } = p ϵ d 4 1 ( A 0 2 + B 0 2 ) 3 ( 2 A 0 B 0 R e { χ m ( 3 ) } + ( A 0 2 B 0 2 ) I m { χ m ( 3 ) } ) .
A 0 = ϵ d + L j ( R e { ϵ m } ϵ d ) ,
B 0 = L j I m { ϵ m } ,
ϵ eff j = ϵ d 1 + ( 1 L j ) p G 1 p L j G ,
G = ϵ m ϵ d L j ϵ m + ( 1 L j ) ϵ d .
d ϵ eff j = p ( ϵ d L j ϵ m + ( 1 L j ) ϵ d p L j ( ϵ m ϵ d ) ) 2 d ϵ m = p H j 2 ( ω , L j ) d ϵ m ,
χ eff , j ( 3 ) = p H j 2 ( ω , L j ) f j ( ω , L j ) 2 χ m ( 3 ) .
R e { χ eff , j ( 3 ) } = p ϵ d 4 1 ( A 0 2 + B 0 2 ) ( A p 2 + B p 2 ) 2 × ( ( A p 2 B p 2 ) R e { χ m ( 3 ) } + 2 A p B p I m { χ m ( 3 ) } ) ,
I m { χ eff , j ( 3 ) } = p ϵ d 4 1 ( A 0 2 + B 0 2 ) ( A p 2 + B p 2 ) 2 × ( 2 A p B p R e { χ m ( 3 ) } + ( A p 2 B p 2 ) I m { χ m ( 3 ) } ) .
A p = ϵ d + L j ( 1 p ) ( R e { ϵ m } ϵ d ) ,
B p = L j ( 1 p ) I m { ϵ m } ,
R χ m = I m { χ m ( 3 ) } R e { χ m ( 3 ) } .
I m { χ eff , LA ( 3 ) } I m { χ eff , SA ( 3 ) } = f LA 2 f SA 2 ( I m { H LA 2 } R e { χ m ( 3 ) } + R e { H LA 2 } I m { χ m ( 3 ) } I m { H SA 2 } R e { χ m ( 3 ) } + R e { H SA 2 } I m { χ m ( 3 ) } ) .
χ eff , random ( 3 ) = 1 3 ( 2 χ eff , SA ( 3 ) + χ eff , LA ( 3 ) ) ,

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