Abstract

We report the observation of optical limiting in Fe3O4-Ag nanocomposites in solution. With these nanocomposites, we demonstrate that broad temporal optical limiting can be accomplished with low limiting threshold. Due to the presence of Ag nanoparticles, nonlinear scattering gives rise to enhanced optical limiting responses to 532-nm nanosecond laser pulses, with a limiting threshold comparable to carbon nanotubes. As exposed to 780-nm femtosecond laser pulses, the largest value (~10−44 cm4s photon−1 or 106 GM) for two-photon absorption cross-sections reported to date results in superior limiting responses with a limiting threshold as low as 0.04 J/cm2 or 100 GW/cm2 for Fe3O4–Ag (7nm) solution in 1 cm quartz cell. The limiting threshold can be further reduced by increasing Ag particle size through plasmon enhancement or taking advantage of self-defocusing.

© 2010 OSA

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    [CrossRef]
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    [CrossRef]
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  5. N. Izard, P. Billaud, D. Riehl, E. Anglaret, C. Mioskowski, and E. Anglaret, “Influence of structure on the optical limiting properties of nanotubes,” Opt. Lett. 30(12), 1509–1511 (2005).
    [CrossRef] [PubMed]
  6. Y. P. Sun, J. E. Riggs, H. W. Rollins, and R. Guduru, “Strong optical limiting of silver-containing nanocrystalline particles in stable suspensions,” J. Phys. Chem. B 103(1), 77–82 (1999).
    [CrossRef]
  7. C. Liu, X. Wang, Q. Gong, K. Tang, X. Jin, H. Yan, and P. Cui, “Nanosecond optical limiting property of a novel octanuclear silver cluster complex containing arylselenolate ligands,” Adv. Mater. 13(22), 1687–1690 (2001).
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    [CrossRef] [PubMed]
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  11. M. P. Joshi, J. Swiatkiewicz, F. M. Xu, P. N. Prasad, B. A. Reinhardt, and R. Kannan, “Energy transfer coupling of two-photon absorption and reverse saturable absorption for enhanced optical power limiting,” Opt. Lett. 23(22), 1742–1744 (1998).
    [CrossRef]
  12. S. Webster, M. Reyes-Reyes, X. Pedron, R. López-Sandoval, M. Terrones, and D. L. Carroll, “Enhanced nonlinear transmittance by complementary nonlinear mechanisms: A reverse-saturable absorbing dye blended with nonlinear-scattering carbon nanotubes,” Adv. Mater. 17(10), 1239–1243 (2005).
    [CrossRef]
  13. Z. B. Liu, J. G. Tian, Z. Guo, D. M. Ren, F. Du, J. Y. Zheng, and Y. S. Chen, “Enhanced optical limiting effects in porphyrin-covalently functionalized single-walled carbon nanotubes,” Adv. Mater. 20(3), 511–515 (2008).
    [CrossRef]
  14. S. S. Nair, J. Thomas, C. S. Suchand Sandeep, M. R. Anantharaman, and R. Philip, “An optical limiter based on ferrofluids,” Appl. Phys. Lett. 92(17), 171908 (2008).
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    [CrossRef]
  23. J. Park, K. An, Y. Hwang, J. G. Park, H. J. Noh, J. Y. Kim, J. H. Park, N. M. Hwang, and T. Hyeon, “Ultra-large-scale syntheses of monodisperse nanocrystals,” Nat. Mater. 3(12), 891–895 (2004).
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    [CrossRef] [PubMed]
  25. J. Jiang, H. W. Gu, H. L. Shao, E. Devlin, G. C. Papaefthymiou, and J. Y. Ying, “Bifunctional Fe3O4–Ag heterodimer nanoparticles for two-photon fluorescence imaging and magnetic manipulation,” Adv. Mater. 20(23), 4403 (2008).
    [CrossRef]
  26. Y. P. He, Y. M. Miao, C. R. Li, S. Q. Wang, L. Cao, S. S. Xie, G. Z. Yang, B. S. Zou, and C. Burda, “Size and structure effect on optical transitions of iron oxide nanocrystals,” Phys. Rev. B 71(12), 125411 (2005).
    [CrossRef]
  27. C. Voisin, N. Del Fatti, D. Christofilos, and F. Vallée, “Ultrafast electron dynamics and optical nonlinearities in metal nanoparticles,” J. Phys. Chem. B 105(12), 2264–2280 (2001).
    [CrossRef]
  28. Y. Hamanaka, A. Nakamura, N. Hayashi, and S. Omi, “Dispersion curves of complex third-order optical susceptibilities around the surface plasmon resonance in Ag nanocrystal-glass composites,” J. Opt. Soc. Am. B 20(6), 1227–1232 (2003).
    [CrossRef]
  29. N. J. Cherepy, D. B. Liston, J. A. Lovejoy, H. Deng, and J. Z. Zhang, “Ultrafast studies of photoexcited electron dynamics in γ- and α-Fe2O3 semiconductor nanoparticles,” J. Phys. Chem. B 102(5), 770–776 (1998).
    [CrossRef]
  30. M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
    [CrossRef]
  31. D. R. Larson, W. R. Zipfel, R. M. Williams, S. W. Clark, M. P. Bruchez, F. W. Wise, and W. W. W. Webb, “Water-soluble quantum dots for multiphoton fluorescence imaging in vivo,” Science 300(5624), 1434–1436 (2003).
    [CrossRef] [PubMed]
  32. Y. L. Qu, W. Ji, Y. G. Zheng, and J. Y. Ying, “Auger recombination and intraband absorption of two-photon-excited carriers in colloidal CdSe quantum dots,” Appl. Phys. Lett. 90(13), 133112 (2007).
    [CrossRef]
  33. L. A. Padilha, J. Fu, D. J. Hagan, E. W. Van Stryland, C. L. Cesar, L. C. Barbosa, C. H. B. Cruz, D. Buso, and A. Martucci, “Frequency degenerate and nondegenerate two-photon absorption spectra of semiconductor quantum dots,” Phys. Rev. B 75(7), 075325 (2007).
    [CrossRef]
  34. B. Gu, W. Ji, P. S. Patil, S. M. Dharmaprakash, and H. T. Wang, “Two-photon-induced excited-state absorption: Theory and experiment,” Appl. Phys. Lett. 92(9), 091118 (2008).
    [CrossRef]
  35. M. S. Liao, J. D. Watts, and M. J. Huang, “Effects of peripheral substituents and axial ligands on the electronic structure and properties of cobalt porphyrins,” J. Phys. Chem. A 109(51), 11996–12005 (2005).
    [CrossRef] [PubMed]
  36. I. Cohanoschi, S. Yao, K. D. Belfield, and F. E. Hernandez, “Effect of the concentration of organic dyes on their surface plasmon enhanced two-photon absorption cross section using activated Au nanoparticles,” J. Appl. Phys. 101(8), 086112 (2007).
    [CrossRef]

2009 (1)

2008 (7)

C. P. Singh, K. S. Bindra, G. M. Bhalerao, and S. M. Oak, “Investigation of optical limiting in iron oxide nanoparticles,” Opt. Express 16(12), 8440–8450 (2008).
[CrossRef] [PubMed]

H. Zeng and S. Sun, “Syntheses, properties, and potential applications of multicomponent magnetic nanoparticles,” Adv. Funct. Mater. 18(3), 391–400 (2008).
[CrossRef]

J. Jiang, H. W. Gu, H. L. Shao, E. Devlin, G. C. Papaefthymiou, and J. Y. Ying, “Bifunctional Fe3O4–Ag heterodimer nanoparticles for two-photon fluorescence imaging and magnetic manipulation,” Adv. Mater. 20(23), 4403 (2008).
[CrossRef]

B. Gu, W. Ji, P. S. Patil, S. M. Dharmaprakash, and H. T. Wang, “Two-photon-induced excited-state absorption: Theory and experiment,” Appl. Phys. Lett. 92(9), 091118 (2008).
[CrossRef]

Z. B. Liu, J. G. Tian, Z. Guo, D. M. Ren, F. Du, J. Y. Zheng, and Y. S. Chen, “Enhanced optical limiting effects in porphyrin-covalently functionalized single-walled carbon nanotubes,” Adv. Mater. 20(3), 511–515 (2008).
[CrossRef]

S. S. Nair, J. Thomas, C. S. Suchand Sandeep, M. R. Anantharaman, and R. Philip, “An optical limiter based on ferrofluids,” Appl. Phys. Lett. 92(17), 171908 (2008).
[CrossRef]

G. S. He, L. S. Tan, Q. Zheng, and P. N. Prasad, “Multiphoton absorbing materials: molecular designs, characterizations, and applications,” Chem. Rev. 108(4), 1245–1330 (2008).
[CrossRef] [PubMed]

2007 (7)

D. Soga, S. Alves, A. Campos, F. A. Tourinho, J. Depeyrot, and A. M. F. Neto, “Nonlinear optical properties of ionic magnetic colloids in the femto- and millisecond time scales: Change from convergent-to-divergent lens-type behaviors,” J. Opt. Soc. Am. B 24(1), 49–55 (2007).
[CrossRef]

G. S. He, K. T. Yong, Q. Zheng, Y. Sahoo, A. Baev, A. I. Ryasnyanskiy, and P. N. Prasad, “Multi-photon excitation properties of CdSe quantum dots solutions and optical limiting behavior in infrared range,” Opt. Express 15(20), 12818–12833 (2007).
[CrossRef] [PubMed]

I. Cohanoschi, S. Yao, K. D. Belfield, and F. E. Hernandez, “Effect of the concentration of organic dyes on their surface plasmon enhanced two-photon absorption cross section using activated Au nanoparticles,” J. Appl. Phys. 101(8), 086112 (2007).
[CrossRef]

M. V. Kovalenko, M. I. Bodnarchuk, R. T. Lechner, G. Hesser, F. Schäffler, and W. Heiss, “Fatty acid salts as stabilizers in size- and shape-controlled nanocrystal synthesis: the case of inverse spinel iron oxide,” J. Am. Chem. Soc. 129(20), 6352–6353 (2007).
[CrossRef] [PubMed]

Y. L. Qu, W. Ji, Y. G. Zheng, and J. Y. Ying, “Auger recombination and intraband absorption of two-photon-excited carriers in colloidal CdSe quantum dots,” Appl. Phys. Lett. 90(13), 133112 (2007).
[CrossRef]

L. A. Padilha, J. Fu, D. J. Hagan, E. W. Van Stryland, C. L. Cesar, L. C. Barbosa, C. H. B. Cruz, D. Buso, and A. Martucci, “Frequency degenerate and nondegenerate two-photon absorption spectra of semiconductor quantum dots,” Phys. Rev. B 75(7), 075325 (2007).
[CrossRef]

P. Gong, H. Li, X. He, K. Wang, J. Hu, W. Tan, S. Zhang, and X. Yang, “Preparation and antibacterial activity of Fe3O4@Ag nanoparticles,” Nanotechnology 18(28), 285604 (2007).
[CrossRef]

2006 (1)

H. Pan, W. Chen, Y. P. Feng, W. Ji, and J. Lin, “Optical limiting properties of metal nanowires,” Appl. Phys. Lett. 88(22), 223106 (2006).
[CrossRef]

2005 (5)

H. Gu, Z. Yang, J. Gao, C. K. Chang, and B. Xu, “Heterodimers of nanoparticles: formation at a liquid-liquid interface and particle-specific surface modification by functional molecules,” J. Am. Chem. Soc. 127(1), 34–35 (2005).
[CrossRef] [PubMed]

Y. P. He, Y. M. Miao, C. R. Li, S. Q. Wang, L. Cao, S. S. Xie, G. Z. Yang, B. S. Zou, and C. Burda, “Size and structure effect on optical transitions of iron oxide nanocrystals,” Phys. Rev. B 71(12), 125411 (2005).
[CrossRef]

S. Webster, M. Reyes-Reyes, X. Pedron, R. López-Sandoval, M. Terrones, and D. L. Carroll, “Enhanced nonlinear transmittance by complementary nonlinear mechanisms: A reverse-saturable absorbing dye blended with nonlinear-scattering carbon nanotubes,” Adv. Mater. 17(10), 1239–1243 (2005).
[CrossRef]

M. S. Liao, J. D. Watts, and M. J. Huang, “Effects of peripheral substituents and axial ligands on the electronic structure and properties of cobalt porphyrins,” J. Phys. Chem. A 109(51), 11996–12005 (2005).
[CrossRef] [PubMed]

N. Izard, P. Billaud, D. Riehl, E. Anglaret, C. Mioskowski, and E. Anglaret, “Influence of structure on the optical limiting properties of nanotubes,” Opt. Lett. 30(12), 1509–1511 (2005).
[CrossRef] [PubMed]

2004 (2)

N. R. Jana, Y. Chen, and X. Peng, “Size- and shape-controlled magnetic (Cr, Mn, Fe, Co, Ni) oxide nanocrystals via a simple and general approach,” Chem. Mater. 16(20), 3931–3935 (2004).
[CrossRef]

J. Park, K. An, Y. Hwang, J. G. Park, H. J. Noh, J. Y. Kim, J. H. Park, N. M. Hwang, and T. Hyeon, “Ultra-large-scale syntheses of monodisperse nanocrystals,” Nat. Mater. 3(12), 891–895 (2004).
[CrossRef] [PubMed]

2003 (2)

Y. Hamanaka, A. Nakamura, N. Hayashi, and S. Omi, “Dispersion curves of complex third-order optical susceptibilities around the surface plasmon resonance in Ag nanocrystal-glass composites,” J. Opt. Soc. Am. B 20(6), 1227–1232 (2003).
[CrossRef]

D. R. Larson, W. R. Zipfel, R. M. Williams, S. W. Clark, M. P. Bruchez, F. W. Wise, and W. W. W. Webb, “Water-soluble quantum dots for multiphoton fluorescence imaging in vivo,” Science 300(5624), 1434–1436 (2003).
[CrossRef] [PubMed]

2001 (2)

C. Voisin, N. Del Fatti, D. Christofilos, and F. Vallée, “Ultrafast electron dynamics and optical nonlinearities in metal nanoparticles,” J. Phys. Chem. B 105(12), 2264–2280 (2001).
[CrossRef]

C. Liu, X. Wang, Q. Gong, K. Tang, X. Jin, H. Yan, and P. Cui, “Nanosecond optical limiting property of a novel octanuclear silver cluster complex containing arylselenolate ligands,” Adv. Mater. 13(22), 1687–1690 (2001).
[CrossRef]

1999 (2)

P. Chen, X. Wu, X. Sun, J. Lin, W. Ji, and K. L. Tan, “Electronic structure and optical limiting behavior of carbon nanotubes,” Phys. Rev. Lett. 82(12), 2548–2551 (1999).
[CrossRef]

Y. P. Sun, J. E. Riggs, H. W. Rollins, and R. Guduru, “Strong optical limiting of silver-containing nanocrystalline particles in stable suspensions,” J. Phys. Chem. B 103(1), 77–82 (1999).
[CrossRef]

1998 (3)

X. Sun, R. Q. Yu, G. Q. Xu, T. S. A. Hor, and W. Ji, “Broadband optical limiting with multiwalled carbon nanotubes,” Appl. Phys. Lett. 73(25), 3632 (1998).
[CrossRef]

N. J. Cherepy, D. B. Liston, J. A. Lovejoy, H. Deng, and J. Z. Zhang, “Ultrafast studies of photoexcited electron dynamics in γ- and α-Fe2O3 semiconductor nanoparticles,” J. Phys. Chem. B 102(5), 770–776 (1998).
[CrossRef]

M. P. Joshi, J. Swiatkiewicz, F. M. Xu, P. N. Prasad, B. A. Reinhardt, and R. Kannan, “Energy transfer coupling of two-photon absorption and reverse saturable absorption for enhanced optical power limiting,” Opt. Lett. 23(22), 1742–1744 (1998).
[CrossRef]

1996 (1)

J. W. Perry, K. Mansour, I. Y. S. Lee, X. L. Wu, P. V. Bedworth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear absorptive response,” Science 273(5281), 1533–1536 (1996).
[CrossRef]

1995 (1)

M. Ando, K. Kadono, M. Haruta, T. Sakaguchi, and M. Miya, “Large third-order optical nonlinearities in transition-metal oxides,” Nature 374(6523), 625–627 (1995).
[CrossRef]

1992 (1)

L. W. Tutt and A. Kost, “Optical limiting performance of C60 and C70 solutions,” Nature 356(6366), 225–226 (1992).
[CrossRef]

1990 (1)

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[CrossRef]

Alves, S.

An, K.

J. Park, K. An, Y. Hwang, J. G. Park, H. J. Noh, J. Y. Kim, J. H. Park, N. M. Hwang, and T. Hyeon, “Ultra-large-scale syntheses of monodisperse nanocrystals,” Nat. Mater. 3(12), 891–895 (2004).
[CrossRef] [PubMed]

Anantharaman, M. R.

S. S. Nair, J. Thomas, C. S. Suchand Sandeep, M. R. Anantharaman, and R. Philip, “An optical limiter based on ferrofluids,” Appl. Phys. Lett. 92(17), 171908 (2008).
[CrossRef]

Ando, M.

M. Ando, K. Kadono, M. Haruta, T. Sakaguchi, and M. Miya, “Large third-order optical nonlinearities in transition-metal oxides,” Nature 374(6523), 625–627 (1995).
[CrossRef]

Anglaret, E.

Baev, A.

Barbosa, L. C.

L. A. Padilha, J. Fu, D. J. Hagan, E. W. Van Stryland, C. L. Cesar, L. C. Barbosa, C. H. B. Cruz, D. Buso, and A. Martucci, “Frequency degenerate and nondegenerate two-photon absorption spectra of semiconductor quantum dots,” Phys. Rev. B 75(7), 075325 (2007).
[CrossRef]

Bedworth, P. V.

J. W. Perry, K. Mansour, I. Y. S. Lee, X. L. Wu, P. V. Bedworth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear absorptive response,” Science 273(5281), 1533–1536 (1996).
[CrossRef]

Belfield, K. D.

I. Cohanoschi, S. Yao, K. D. Belfield, and F. E. Hernandez, “Effect of the concentration of organic dyes on their surface plasmon enhanced two-photon absorption cross section using activated Au nanoparticles,” J. Appl. Phys. 101(8), 086112 (2007).
[CrossRef]

Bhalerao, G. M.

Billaud, P.

Bindra, K. S.

Bodnarchuk, M. I.

M. V. Kovalenko, M. I. Bodnarchuk, R. T. Lechner, G. Hesser, F. Schäffler, and W. Heiss, “Fatty acid salts as stabilizers in size- and shape-controlled nanocrystal synthesis: the case of inverse spinel iron oxide,” J. Am. Chem. Soc. 129(20), 6352–6353 (2007).
[CrossRef] [PubMed]

Bruchez, M. P.

D. R. Larson, W. R. Zipfel, R. M. Williams, S. W. Clark, M. P. Bruchez, F. W. Wise, and W. W. W. Webb, “Water-soluble quantum dots for multiphoton fluorescence imaging in vivo,” Science 300(5624), 1434–1436 (2003).
[CrossRef] [PubMed]

Burda, C.

Y. P. He, Y. M. Miao, C. R. Li, S. Q. Wang, L. Cao, S. S. Xie, G. Z. Yang, B. S. Zou, and C. Burda, “Size and structure effect on optical transitions of iron oxide nanocrystals,” Phys. Rev. B 71(12), 125411 (2005).
[CrossRef]

Buso, D.

L. A. Padilha, J. Fu, D. J. Hagan, E. W. Van Stryland, C. L. Cesar, L. C. Barbosa, C. H. B. Cruz, D. Buso, and A. Martucci, “Frequency degenerate and nondegenerate two-photon absorption spectra of semiconductor quantum dots,” Phys. Rev. B 75(7), 075325 (2007).
[CrossRef]

Campos, A.

Cao, L.

Y. P. He, Y. M. Miao, C. R. Li, S. Q. Wang, L. Cao, S. S. Xie, G. Z. Yang, B. S. Zou, and C. Burda, “Size and structure effect on optical transitions of iron oxide nanocrystals,” Phys. Rev. B 71(12), 125411 (2005).
[CrossRef]

Carroll, D. L.

S. Webster, M. Reyes-Reyes, X. Pedron, R. López-Sandoval, M. Terrones, and D. L. Carroll, “Enhanced nonlinear transmittance by complementary nonlinear mechanisms: A reverse-saturable absorbing dye blended with nonlinear-scattering carbon nanotubes,” Adv. Mater. 17(10), 1239–1243 (2005).
[CrossRef]

Cesar, C. L.

L. A. Padilha, J. Fu, D. J. Hagan, E. W. Van Stryland, C. L. Cesar, L. C. Barbosa, C. H. B. Cruz, D. Buso, and A. Martucci, “Frequency degenerate and nondegenerate two-photon absorption spectra of semiconductor quantum dots,” Phys. Rev. B 75(7), 075325 (2007).
[CrossRef]

Chang, C. K.

H. Gu, Z. Yang, J. Gao, C. K. Chang, and B. Xu, “Heterodimers of nanoparticles: formation at a liquid-liquid interface and particle-specific surface modification by functional molecules,” J. Am. Chem. Soc. 127(1), 34–35 (2005).
[CrossRef] [PubMed]

Chen, C. T.

J. W. Perry, K. Mansour, I. Y. S. Lee, X. L. Wu, P. V. Bedworth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear absorptive response,” Science 273(5281), 1533–1536 (1996).
[CrossRef]

Chen, P.

P. Chen, X. Wu, X. Sun, J. Lin, W. Ji, and K. L. Tan, “Electronic structure and optical limiting behavior of carbon nanotubes,” Phys. Rev. Lett. 82(12), 2548–2551 (1999).
[CrossRef]

Chen, W.

H. Pan, W. Chen, Y. P. Feng, W. Ji, and J. Lin, “Optical limiting properties of metal nanowires,” Appl. Phys. Lett. 88(22), 223106 (2006).
[CrossRef]

Chen, Y.

N. R. Jana, Y. Chen, and X. Peng, “Size- and shape-controlled magnetic (Cr, Mn, Fe, Co, Ni) oxide nanocrystals via a simple and general approach,” Chem. Mater. 16(20), 3931–3935 (2004).
[CrossRef]

Chen, Y. S.

Z. B. Liu, J. G. Tian, Z. Guo, D. M. Ren, F. Du, J. Y. Zheng, and Y. S. Chen, “Enhanced optical limiting effects in porphyrin-covalently functionalized single-walled carbon nanotubes,” Adv. Mater. 20(3), 511–515 (2008).
[CrossRef]

Cherepy, N. J.

N. J. Cherepy, D. B. Liston, J. A. Lovejoy, H. Deng, and J. Z. Zhang, “Ultrafast studies of photoexcited electron dynamics in γ- and α-Fe2O3 semiconductor nanoparticles,” J. Phys. Chem. B 102(5), 770–776 (1998).
[CrossRef]

Christofilos, D.

C. Voisin, N. Del Fatti, D. Christofilos, and F. Vallée, “Ultrafast electron dynamics and optical nonlinearities in metal nanoparticles,” J. Phys. Chem. B 105(12), 2264–2280 (2001).
[CrossRef]

Clark, S. W.

D. R. Larson, W. R. Zipfel, R. M. Williams, S. W. Clark, M. P. Bruchez, F. W. Wise, and W. W. W. Webb, “Water-soluble quantum dots for multiphoton fluorescence imaging in vivo,” Science 300(5624), 1434–1436 (2003).
[CrossRef] [PubMed]

Cohanoschi, I.

I. Cohanoschi, S. Yao, K. D. Belfield, and F. E. Hernandez, “Effect of the concentration of organic dyes on their surface plasmon enhanced two-photon absorption cross section using activated Au nanoparticles,” J. Appl. Phys. 101(8), 086112 (2007).
[CrossRef]

Cruz, C. H. B.

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H. Zeng and S. Sun, “Syntheses, properties, and potential applications of multicomponent magnetic nanoparticles,” Adv. Funct. Mater. 18(3), 391–400 (2008).
[CrossRef]

Sun, X.

P. Chen, X. Wu, X. Sun, J. Lin, W. Ji, and K. L. Tan, “Electronic structure and optical limiting behavior of carbon nanotubes,” Phys. Rev. Lett. 82(12), 2548–2551 (1999).
[CrossRef]

X. Sun, R. Q. Yu, G. Q. Xu, T. S. A. Hor, and W. Ji, “Broadband optical limiting with multiwalled carbon nanotubes,” Appl. Phys. Lett. 73(25), 3632 (1998).
[CrossRef]

Sun, Y. P.

Y. P. Sun, J. E. Riggs, H. W. Rollins, and R. Guduru, “Strong optical limiting of silver-containing nanocrystalline particles in stable suspensions,” J. Phys. Chem. B 103(1), 77–82 (1999).
[CrossRef]

Swiatkiewicz, J.

Tan, K. L.

P. Chen, X. Wu, X. Sun, J. Lin, W. Ji, and K. L. Tan, “Electronic structure and optical limiting behavior of carbon nanotubes,” Phys. Rev. Lett. 82(12), 2548–2551 (1999).
[CrossRef]

Tan, L. S.

G. S. He, L. S. Tan, Q. Zheng, and P. N. Prasad, “Multiphoton absorbing materials: molecular designs, characterizations, and applications,” Chem. Rev. 108(4), 1245–1330 (2008).
[CrossRef] [PubMed]

Tan, W.

P. Gong, H. Li, X. He, K. Wang, J. Hu, W. Tan, S. Zhang, and X. Yang, “Preparation and antibacterial activity of Fe3O4@Ag nanoparticles,” Nanotechnology 18(28), 285604 (2007).
[CrossRef]

Tang, K.

C. Liu, X. Wang, Q. Gong, K. Tang, X. Jin, H. Yan, and P. Cui, “Nanosecond optical limiting property of a novel octanuclear silver cluster complex containing arylselenolate ligands,” Adv. Mater. 13(22), 1687–1690 (2001).
[CrossRef]

Terrones, M.

S. Webster, M. Reyes-Reyes, X. Pedron, R. López-Sandoval, M. Terrones, and D. L. Carroll, “Enhanced nonlinear transmittance by complementary nonlinear mechanisms: A reverse-saturable absorbing dye blended with nonlinear-scattering carbon nanotubes,” Adv. Mater. 17(10), 1239–1243 (2005).
[CrossRef]

Thomas, J.

S. S. Nair, J. Thomas, C. S. Suchand Sandeep, M. R. Anantharaman, and R. Philip, “An optical limiter based on ferrofluids,” Appl. Phys. Lett. 92(17), 171908 (2008).
[CrossRef]

Tian, J. G.

Z. B. Liu, J. G. Tian, Z. Guo, D. M. Ren, F. Du, J. Y. Zheng, and Y. S. Chen, “Enhanced optical limiting effects in porphyrin-covalently functionalized single-walled carbon nanotubes,” Adv. Mater. 20(3), 511–515 (2008).
[CrossRef]

Tian, M.

J. W. Perry, K. Mansour, I. Y. S. Lee, X. L. Wu, P. V. Bedworth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear absorptive response,” Science 273(5281), 1533–1536 (1996).
[CrossRef]

Tourinho, F. A.

Tutt, L. W.

L. W. Tutt and A. Kost, “Optical limiting performance of C60 and C70 solutions,” Nature 356(6366), 225–226 (1992).
[CrossRef]

Vallée, F.

C. Voisin, N. Del Fatti, D. Christofilos, and F. Vallée, “Ultrafast electron dynamics and optical nonlinearities in metal nanoparticles,” J. Phys. Chem. B 105(12), 2264–2280 (2001).
[CrossRef]

Van Stryland, E. W.

L. A. Padilha, J. Fu, D. J. Hagan, E. W. Van Stryland, C. L. Cesar, L. C. Barbosa, C. H. B. Cruz, D. Buso, and A. Martucci, “Frequency degenerate and nondegenerate two-photon absorption spectra of semiconductor quantum dots,” Phys. Rev. B 75(7), 075325 (2007).
[CrossRef]

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[CrossRef]

Voisin, C.

C. Voisin, N. Del Fatti, D. Christofilos, and F. Vallée, “Ultrafast electron dynamics and optical nonlinearities in metal nanoparticles,” J. Phys. Chem. B 105(12), 2264–2280 (2001).
[CrossRef]

Wada, T.

J. W. Perry, K. Mansour, I. Y. S. Lee, X. L. Wu, P. V. Bedworth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear absorptive response,” Science 273(5281), 1533–1536 (1996).
[CrossRef]

Wang, H. T.

B. Gu, W. Ji, P. S. Patil, S. M. Dharmaprakash, and H. T. Wang, “Two-photon-induced excited-state absorption: Theory and experiment,” Appl. Phys. Lett. 92(9), 091118 (2008).
[CrossRef]

Wang, K.

P. Gong, H. Li, X. He, K. Wang, J. Hu, W. Tan, S. Zhang, and X. Yang, “Preparation and antibacterial activity of Fe3O4@Ag nanoparticles,” Nanotechnology 18(28), 285604 (2007).
[CrossRef]

Wang, S. Q.

Y. P. He, Y. M. Miao, C. R. Li, S. Q. Wang, L. Cao, S. S. Xie, G. Z. Yang, B. S. Zou, and C. Burda, “Size and structure effect on optical transitions of iron oxide nanocrystals,” Phys. Rev. B 71(12), 125411 (2005).
[CrossRef]

Wang, X.

C. Liu, X. Wang, Q. Gong, K. Tang, X. Jin, H. Yan, and P. Cui, “Nanosecond optical limiting property of a novel octanuclear silver cluster complex containing arylselenolate ligands,” Adv. Mater. 13(22), 1687–1690 (2001).
[CrossRef]

Watts, J. D.

M. S. Liao, J. D. Watts, and M. J. Huang, “Effects of peripheral substituents and axial ligands on the electronic structure and properties of cobalt porphyrins,” J. Phys. Chem. A 109(51), 11996–12005 (2005).
[CrossRef] [PubMed]

Webb, W. W. W.

D. R. Larson, W. R. Zipfel, R. M. Williams, S. W. Clark, M. P. Bruchez, F. W. Wise, and W. W. W. Webb, “Water-soluble quantum dots for multiphoton fluorescence imaging in vivo,” Science 300(5624), 1434–1436 (2003).
[CrossRef] [PubMed]

Webster, S.

S. Webster, M. Reyes-Reyes, X. Pedron, R. López-Sandoval, M. Terrones, and D. L. Carroll, “Enhanced nonlinear transmittance by complementary nonlinear mechanisms: A reverse-saturable absorbing dye blended with nonlinear-scattering carbon nanotubes,” Adv. Mater. 17(10), 1239–1243 (2005).
[CrossRef]

Wei, T.-H.

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[CrossRef]

Williams, R. M.

D. R. Larson, W. R. Zipfel, R. M. Williams, S. W. Clark, M. P. Bruchez, F. W. Wise, and W. W. W. Webb, “Water-soluble quantum dots for multiphoton fluorescence imaging in vivo,” Science 300(5624), 1434–1436 (2003).
[CrossRef] [PubMed]

Wise, F. W.

D. R. Larson, W. R. Zipfel, R. M. Williams, S. W. Clark, M. P. Bruchez, F. W. Wise, and W. W. W. Webb, “Water-soluble quantum dots for multiphoton fluorescence imaging in vivo,” Science 300(5624), 1434–1436 (2003).
[CrossRef] [PubMed]

Wu, X.

P. Chen, X. Wu, X. Sun, J. Lin, W. Ji, and K. L. Tan, “Electronic structure and optical limiting behavior of carbon nanotubes,” Phys. Rev. Lett. 82(12), 2548–2551 (1999).
[CrossRef]

Wu, X. L.

J. W. Perry, K. Mansour, I. Y. S. Lee, X. L. Wu, P. V. Bedworth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear absorptive response,” Science 273(5281), 1533–1536 (1996).
[CrossRef]

Xie, S. S.

Y. P. He, Y. M. Miao, C. R. Li, S. Q. Wang, L. Cao, S. S. Xie, G. Z. Yang, B. S. Zou, and C. Burda, “Size and structure effect on optical transitions of iron oxide nanocrystals,” Phys. Rev. B 71(12), 125411 (2005).
[CrossRef]

Xu, B.

H. Gu, Z. Yang, J. Gao, C. K. Chang, and B. Xu, “Heterodimers of nanoparticles: formation at a liquid-liquid interface and particle-specific surface modification by functional molecules,” J. Am. Chem. Soc. 127(1), 34–35 (2005).
[CrossRef] [PubMed]

Xu, F. M.

Xu, G. Q.

X. Sun, R. Q. Yu, G. Q. Xu, T. S. A. Hor, and W. Ji, “Broadband optical limiting with multiwalled carbon nanotubes,” Appl. Phys. Lett. 73(25), 3632 (1998).
[CrossRef]

Yan, H.

C. Liu, X. Wang, Q. Gong, K. Tang, X. Jin, H. Yan, and P. Cui, “Nanosecond optical limiting property of a novel octanuclear silver cluster complex containing arylselenolate ligands,” Adv. Mater. 13(22), 1687–1690 (2001).
[CrossRef]

Yang, G. Z.

Y. P. He, Y. M. Miao, C. R. Li, S. Q. Wang, L. Cao, S. S. Xie, G. Z. Yang, B. S. Zou, and C. Burda, “Size and structure effect on optical transitions of iron oxide nanocrystals,” Phys. Rev. B 71(12), 125411 (2005).
[CrossRef]

Yang, X.

P. Gong, H. Li, X. He, K. Wang, J. Hu, W. Tan, S. Zhang, and X. Yang, “Preparation and antibacterial activity of Fe3O4@Ag nanoparticles,” Nanotechnology 18(28), 285604 (2007).
[CrossRef]

Yang, Z.

H. Gu, Z. Yang, J. Gao, C. K. Chang, and B. Xu, “Heterodimers of nanoparticles: formation at a liquid-liquid interface and particle-specific surface modification by functional molecules,” J. Am. Chem. Soc. 127(1), 34–35 (2005).
[CrossRef] [PubMed]

Yao, S.

I. Cohanoschi, S. Yao, K. D. Belfield, and F. E. Hernandez, “Effect of the concentration of organic dyes on their surface plasmon enhanced two-photon absorption cross section using activated Au nanoparticles,” J. Appl. Phys. 101(8), 086112 (2007).
[CrossRef]

Ying, J. Y.

J. Jiang, H. W. Gu, H. L. Shao, E. Devlin, G. C. Papaefthymiou, and J. Y. Ying, “Bifunctional Fe3O4–Ag heterodimer nanoparticles for two-photon fluorescence imaging and magnetic manipulation,” Adv. Mater. 20(23), 4403 (2008).
[CrossRef]

Y. L. Qu, W. Ji, Y. G. Zheng, and J. Y. Ying, “Auger recombination and intraband absorption of two-photon-excited carriers in colloidal CdSe quantum dots,” Appl. Phys. Lett. 90(13), 133112 (2007).
[CrossRef]

Yong, K. T.

Yu, R. Q.

X. Sun, R. Q. Yu, G. Q. Xu, T. S. A. Hor, and W. Ji, “Broadband optical limiting with multiwalled carbon nanotubes,” Appl. Phys. Lett. 73(25), 3632 (1998).
[CrossRef]

Zeng, H.

H. Zeng and S. Sun, “Syntheses, properties, and potential applications of multicomponent magnetic nanoparticles,” Adv. Funct. Mater. 18(3), 391–400 (2008).
[CrossRef]

Zhang, J. Z.

N. J. Cherepy, D. B. Liston, J. A. Lovejoy, H. Deng, and J. Z. Zhang, “Ultrafast studies of photoexcited electron dynamics in γ- and α-Fe2O3 semiconductor nanoparticles,” J. Phys. Chem. B 102(5), 770–776 (1998).
[CrossRef]

Zhang, S.

P. Gong, H. Li, X. He, K. Wang, J. Hu, W. Tan, S. Zhang, and X. Yang, “Preparation and antibacterial activity of Fe3O4@Ag nanoparticles,” Nanotechnology 18(28), 285604 (2007).
[CrossRef]

Zheng, J. Y.

Z. B. Liu, J. G. Tian, Z. Guo, D. M. Ren, F. Du, J. Y. Zheng, and Y. S. Chen, “Enhanced optical limiting effects in porphyrin-covalently functionalized single-walled carbon nanotubes,” Adv. Mater. 20(3), 511–515 (2008).
[CrossRef]

Zheng, Q.

Zheng, Y. G.

Y. L. Qu, W. Ji, Y. G. Zheng, and J. Y. Ying, “Auger recombination and intraband absorption of two-photon-excited carriers in colloidal CdSe quantum dots,” Appl. Phys. Lett. 90(13), 133112 (2007).
[CrossRef]

Zipfel, W. R.

D. R. Larson, W. R. Zipfel, R. M. Williams, S. W. Clark, M. P. Bruchez, F. W. Wise, and W. W. W. Webb, “Water-soluble quantum dots for multiphoton fluorescence imaging in vivo,” Science 300(5624), 1434–1436 (2003).
[CrossRef] [PubMed]

Zou, B. S.

Y. P. He, Y. M. Miao, C. R. Li, S. Q. Wang, L. Cao, S. S. Xie, G. Z. Yang, B. S. Zou, and C. Burda, “Size and structure effect on optical transitions of iron oxide nanocrystals,” Phys. Rev. B 71(12), 125411 (2005).
[CrossRef]

Adv. Funct. Mater. (1)

H. Zeng and S. Sun, “Syntheses, properties, and potential applications of multicomponent magnetic nanoparticles,” Adv. Funct. Mater. 18(3), 391–400 (2008).
[CrossRef]

Adv. Mater. (4)

J. Jiang, H. W. Gu, H. L. Shao, E. Devlin, G. C. Papaefthymiou, and J. Y. Ying, “Bifunctional Fe3O4–Ag heterodimer nanoparticles for two-photon fluorescence imaging and magnetic manipulation,” Adv. Mater. 20(23), 4403 (2008).
[CrossRef]

C. Liu, X. Wang, Q. Gong, K. Tang, X. Jin, H. Yan, and P. Cui, “Nanosecond optical limiting property of a novel octanuclear silver cluster complex containing arylselenolate ligands,” Adv. Mater. 13(22), 1687–1690 (2001).
[CrossRef]

S. Webster, M. Reyes-Reyes, X. Pedron, R. López-Sandoval, M. Terrones, and D. L. Carroll, “Enhanced nonlinear transmittance by complementary nonlinear mechanisms: A reverse-saturable absorbing dye blended with nonlinear-scattering carbon nanotubes,” Adv. Mater. 17(10), 1239–1243 (2005).
[CrossRef]

Z. B. Liu, J. G. Tian, Z. Guo, D. M. Ren, F. Du, J. Y. Zheng, and Y. S. Chen, “Enhanced optical limiting effects in porphyrin-covalently functionalized single-walled carbon nanotubes,” Adv. Mater. 20(3), 511–515 (2008).
[CrossRef]

Appl. Phys. Lett. (5)

S. S. Nair, J. Thomas, C. S. Suchand Sandeep, M. R. Anantharaman, and R. Philip, “An optical limiter based on ferrofluids,” Appl. Phys. Lett. 92(17), 171908 (2008).
[CrossRef]

H. Pan, W. Chen, Y. P. Feng, W. Ji, and J. Lin, “Optical limiting properties of metal nanowires,” Appl. Phys. Lett. 88(22), 223106 (2006).
[CrossRef]

X. Sun, R. Q. Yu, G. Q. Xu, T. S. A. Hor, and W. Ji, “Broadband optical limiting with multiwalled carbon nanotubes,” Appl. Phys. Lett. 73(25), 3632 (1998).
[CrossRef]

Y. L. Qu, W. Ji, Y. G. Zheng, and J. Y. Ying, “Auger recombination and intraband absorption of two-photon-excited carriers in colloidal CdSe quantum dots,” Appl. Phys. Lett. 90(13), 133112 (2007).
[CrossRef]

B. Gu, W. Ji, P. S. Patil, S. M. Dharmaprakash, and H. T. Wang, “Two-photon-induced excited-state absorption: Theory and experiment,” Appl. Phys. Lett. 92(9), 091118 (2008).
[CrossRef]

Chem. Mater. (1)

N. R. Jana, Y. Chen, and X. Peng, “Size- and shape-controlled magnetic (Cr, Mn, Fe, Co, Ni) oxide nanocrystals via a simple and general approach,” Chem. Mater. 16(20), 3931–3935 (2004).
[CrossRef]

Chem. Rev. (1)

G. S. He, L. S. Tan, Q. Zheng, and P. N. Prasad, “Multiphoton absorbing materials: molecular designs, characterizations, and applications,” Chem. Rev. 108(4), 1245–1330 (2008).
[CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

M. Sheik-Bahae, A. A. Said, T.-H. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26(4), 760–769 (1990).
[CrossRef]

J. Am. Chem. Soc. (2)

H. Gu, Z. Yang, J. Gao, C. K. Chang, and B. Xu, “Heterodimers of nanoparticles: formation at a liquid-liquid interface and particle-specific surface modification by functional molecules,” J. Am. Chem. Soc. 127(1), 34–35 (2005).
[CrossRef] [PubMed]

M. V. Kovalenko, M. I. Bodnarchuk, R. T. Lechner, G. Hesser, F. Schäffler, and W. Heiss, “Fatty acid salts as stabilizers in size- and shape-controlled nanocrystal synthesis: the case of inverse spinel iron oxide,” J. Am. Chem. Soc. 129(20), 6352–6353 (2007).
[CrossRef] [PubMed]

J. Appl. Phys. (1)

I. Cohanoschi, S. Yao, K. D. Belfield, and F. E. Hernandez, “Effect of the concentration of organic dyes on their surface plasmon enhanced two-photon absorption cross section using activated Au nanoparticles,” J. Appl. Phys. 101(8), 086112 (2007).
[CrossRef]

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

J. Phys. Chem. A (1)

M. S. Liao, J. D. Watts, and M. J. Huang, “Effects of peripheral substituents and axial ligands on the electronic structure and properties of cobalt porphyrins,” J. Phys. Chem. A 109(51), 11996–12005 (2005).
[CrossRef] [PubMed]

J. Phys. Chem. B (3)

N. J. Cherepy, D. B. Liston, J. A. Lovejoy, H. Deng, and J. Z. Zhang, “Ultrafast studies of photoexcited electron dynamics in γ- and α-Fe2O3 semiconductor nanoparticles,” J. Phys. Chem. B 102(5), 770–776 (1998).
[CrossRef]

Y. P. Sun, J. E. Riggs, H. W. Rollins, and R. Guduru, “Strong optical limiting of silver-containing nanocrystalline particles in stable suspensions,” J. Phys. Chem. B 103(1), 77–82 (1999).
[CrossRef]

C. Voisin, N. Del Fatti, D. Christofilos, and F. Vallée, “Ultrafast electron dynamics and optical nonlinearities in metal nanoparticles,” J. Phys. Chem. B 105(12), 2264–2280 (2001).
[CrossRef]

Nanotechnology (1)

P. Gong, H. Li, X. He, K. Wang, J. Hu, W. Tan, S. Zhang, and X. Yang, “Preparation and antibacterial activity of Fe3O4@Ag nanoparticles,” Nanotechnology 18(28), 285604 (2007).
[CrossRef]

Nat. Mater. (1)

J. Park, K. An, Y. Hwang, J. G. Park, H. J. Noh, J. Y. Kim, J. H. Park, N. M. Hwang, and T. Hyeon, “Ultra-large-scale syntheses of monodisperse nanocrystals,” Nat. Mater. 3(12), 891–895 (2004).
[CrossRef] [PubMed]

Nature (2)

L. W. Tutt and A. Kost, “Optical limiting performance of C60 and C70 solutions,” Nature 356(6366), 225–226 (1992).
[CrossRef]

M. Ando, K. Kadono, M. Haruta, T. Sakaguchi, and M. Miya, “Large third-order optical nonlinearities in transition-metal oxides,” Nature 374(6523), 625–627 (1995).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Phys. Rev. B (2)

Y. P. He, Y. M. Miao, C. R. Li, S. Q. Wang, L. Cao, S. S. Xie, G. Z. Yang, B. S. Zou, and C. Burda, “Size and structure effect on optical transitions of iron oxide nanocrystals,” Phys. Rev. B 71(12), 125411 (2005).
[CrossRef]

L. A. Padilha, J. Fu, D. J. Hagan, E. W. Van Stryland, C. L. Cesar, L. C. Barbosa, C. H. B. Cruz, D. Buso, and A. Martucci, “Frequency degenerate and nondegenerate two-photon absorption spectra of semiconductor quantum dots,” Phys. Rev. B 75(7), 075325 (2007).
[CrossRef]

Phys. Rev. Lett. (1)

P. Chen, X. Wu, X. Sun, J. Lin, W. Ji, and K. L. Tan, “Electronic structure and optical limiting behavior of carbon nanotubes,” Phys. Rev. Lett. 82(12), 2548–2551 (1999).
[CrossRef]

Science (2)

J. W. Perry, K. Mansour, I. Y. S. Lee, X. L. Wu, P. V. Bedworth, C. T. Chen, D. Ng, S. R. Marder, P. Miles, T. Wada, M. Tian, and H. Sasabe, “Organic optical limiter with a strong nonlinear absorptive response,” Science 273(5281), 1533–1536 (1996).
[CrossRef]

D. R. Larson, W. R. Zipfel, R. M. Williams, S. W. Clark, M. P. Bruchez, F. W. Wise, and W. W. W. Webb, “Water-soluble quantum dots for multiphoton fluorescence imaging in vivo,” Science 300(5624), 1434–1436 (2003).
[CrossRef] [PubMed]

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Figures (5)

Fig. 1
Fig. 1

(A) Low-resolution and (B) high-resolution TEM images of Fe3O4–Ag. (C) Schematic of a Fe3O4–Ag nanohybrid composite with a capping layer.

Fig. 2
Fig. 2

(a) Absorption spectra of Fe3O4–Ag heterodimers (red) and Fe3O4 nanocubes (green) in toluene. The difference between the two spectra is shown by the black line. The inset displays the size distributions of Fe3O4 nanocubes and Ag nanoparticles. (b) Powder XRD pattern of Fe3O4–Ag heterodimers.

Fig. 3
Fig. 3

Fluence-dependent transmittance measured for toluene solution of (▪) Fe3O4, nanocubes, (●) Fe3O4–Ag (7 nm) heterodimers, (▲) Fe3O4–Ag (10 nm) heterodimers, and (•) carbon nanotubes in water. The data (▼) were measured from Fe3O4–Ag (10 nm) heterodimers with an aperture of 90% transmittance placing in front of the transmission detector.

Fig. 4
Fig. 4

Scattered light measured with 532-nm, 7-ns laser pulses (a) at an angle of 45۫ (b) at various angles with an input energy of 1 mJ.

Fig. 5
Fig. 5

(a) Open-aperture Z-scans (open circles) with theoretical fits (solid lines). The inset shows the dependence of effective TPA coefficients on input irradiance. (b) Transient absorption measurements with two-exponential fittings (black solid lines). The measurements were conducted on the 1-mm-thick toluene solution of Fe3O4–Ag (7 nm) heterodimers with 780-nm, 330-fs laser pulses.

Tables (1)

Tables Icon

Table 1 Effective nonlinear refractive and TPA cross-sections

Equations (1)

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1 r r ( r E r ) 2 i k E z i k α E + 2 k 2 n 0 n 2 | E 2 | E = 0

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