Abstract

The nonlinear response of silica - gold nanoshells (SGNs) in chloroform was studied using laser pulses of 65 fs at 1560 nm. The experiments were performed using the thermally managed Z - scan technique that allows measurements of the electronic contribution for the nonlinear response, free from thermal influence. The results were analyzed using an analytical approach based on the quasi - static approximation that allowed extraction of the nonlinear susceptibility of a SGN from the data. High third - order susceptibility, χsh (3) = - 1.5 x 10−11 m2/V2, approximately four orders of magnitude larger than for gold nanospheres in the visible, and large fifth - order susceptibility, χsh (5) = - 1.4 x 10−24 m4/V4, were obtained. The present results offers new perspectives for nonlinear plasmonics in the near - infrared.

© 2010 OSA

Localized Surface Plasmon (LSP) is the quanta of collective electron charge oscillations in metallic nanoparticles. The LSP resonance frequency depends on the nanoparticle’s shape as well as the dielectric functions of the metal and the dielectric host. As a consequence of the discontinuity of the dielectric function on the metal - dielectric interface (MDI), the local electromagnetic field on the interface may be very large. Hence, the strong field may enhance the optical response of atoms or molecules located near a MDI. This is one of the reasons that make composite systems containing metallic nanoparticles attractive for technological applications. Indeed, the optical properties of such systems can be engineered by new fabrication techniques that allow a large variety of particle geometries with an excellent size - and - shape homogeneous distribution. For instance, silica - gold nanoshell deserves particular attention because their LSP resonance is tunable from the visible towards the near - infrared as the ratio between the shell thickness and the core diameter is reduced [1]. Although the usefulness of SGNs in biophotonics has been well established, relevance of the SGNs for nonlinear (NL) optics has not been fully recognized [27].

In this paper we report on the NL response of colloids containing silica - gold nanoshells (SGNs) suspended in chloroform, using a laser operating at 1560 nm. A new synthesis procedure that allows fabrication of SGNs with narrow size distribution was developed and experiments were performed to determine the NL susceptibility of a single SGN. Large values for the SGN susceptibility were measured and the results herein reported indicate possible successful uses of SGNs for applications in telecom devices.

The synthetic route was based on the method described by Pham et al [8] with modifications. The method comprises (i) Stöber silica preparation; (ii) Stöber silica functionalization with 3-aminopropyl-trimethoxysilane (APTMS); (iii) gold nanoparticles preparation; (iv) gold nanoparticles attachment to silica (to obtain nanoislands) and, (v) gold nanoshells growth.

Stöber silica nanoparticles [9] were synthesized by alkaline hydrolysis of tetraethyl-orthosilicate (TEOS) with aqueous ammonia solution, in ethanol under sonication [10] to give 120 ± 21 nm silica nanoparticles, which were washed thoroughly to remove the ammonia excess and dried. The resulting powder (0.7 g) was dispersed in toluene (100 mL) together with APTMS (200 μL), stirred at room temperature for 3 hours and refluxed for 9 hours.

Gold nanoparticles were synthesized starting from a 20 mM HAuCl4 aqueous solution. Sodium borohydride was used as reducing agent and poly(vinyl-pyrrolidone) (PVP, ~55000 g.mol−1) as the stabilizing agent. The synthesis was performed at room temperature under strong stirring to give the 2.0 ± 0.3 nm nanoparticles [11].

Nanoislands were formed by mixing the gold colloid (30 mL) and APTMS-functionalized Stöber silica colloids (3.0 mL) at room temperature under stirring for 2 hours. The nanoislands were separated from unattached gold nanoparticles by centrifugation. TEM images of the nanoislands are presented in Figs. 1 (b) ; the gold nanoparticles are very well dispersed in the silica surface.

 

Fig. 1 Electron microscope images of the nanoparticles. (a) Silica core (average diameter: ~120 nm). (b) Silica - APTMS - gold nanoparticles (average diameter: ~160 nm). (c) and (d) Silica - gold nanoshells (average diameter: ~160 nm). Figures 1(a) to 1(c) were obtained using a 100 kV transmission electron microscope. Figure 1(d) was obtained using a 200 kV scanning electron microscope.

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In order to obtain gold nanoshells, 12 μL of the nanoisland colloid (5.0×1011 particles/mL) was added to 28 mL of a K-gold solution [12]. Formaldehyde (70 μL) was then added under vigorous stirring for 4 minutes and the colloid left under rest for two hours. The resulting nanoshells are presented in Fig. 1 (c) and 1 (d). The average particle size is 160 ± 16 nm, which means that the shell thickness is 20 nm.

For the NL experiments the samples were centrifuged several times in order to eliminate residual water and the particles were re-suspended in chloroform (CHCl3), which presents small absorption at 1560 nm.

The absorption spectrum of pure chloroform is shown in Fig. 2 (red line) and the extinction spectrum of the SGNs, obtained for a colloid with filling fraction f = 3×10−5, using the CHCl3 spectrum as blank, is represented by the blue line in Fig. 2. The LSP resonance, centered at ≈830 nm, is in agreement with the relative sizes of core and shell thickness [1].

 

Fig. 2 Absorption coefficient of CHCl3 in a 10 mm cuvette (red line) and extinction coefficient of SGNs - CHCl3 colloid (blue line) using CHCl3 as blank. The SGNs filling fraction is 3×10−5.

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For the NL experiments we used a fiber laser (1560 nm; 65 fs; 50 MHz) operating in the TEM00 mode (M 2=1.08). The SGNs - CHCl3 colloid presents a linear absorption coefficient of 0.49 cm–1 at 1560 nm. Due to the large laser repetition rate thermal lensing may be induced; then, in order to obtain results not influenced by thermal effects and to determine the actual electronic NL response, we applied the Thermally Managed Z-scan (TM - Z scan) technique [13, 14]. The TM - Z scan technique is a variation of the well known Z-scan technique [15] and consists in acquiring the time evolution of the NL transmittance signal when the sample is placed in pre-focal and post-focal positions with respect to the incident beam focus (usually in the positions corresponding to the peak and valley of the Z-scan profile). The light induced refractive index is determined from measurements of ΔTPV, the peak - to - valley difference in the samples transmittance. As in the conventional Z-scan technique, using an iris placed in front of the photodetector, in the far - field, it is possible to infer sign and magnitude of the NL refractive index. The experiments made without an iris, collecting all light transmitted by the sample, allow measurements of the NL absorption coefficient [1315].

The time evolution of ΔTPV is obtained by introduction of a chopper in the conventional Z - scan setup as in refs [13, 14]. Then, the time resolution is determined by the chopper opening time, which depends on the finite size of the beam waist on the chopper wheel. The time behavior of the NL signal is determined by delaying the signal acquisition time with respect to the instant t = 0, which is determined by the opening time of the chopper. An exponential curve is used to fit the experimental data and to determine the normalized transmittance free of thermal effects at t = 0 [13,14].

The TM - Z scan results were obtained using a 1 mm thick glass cuvette. The laser was focused using a 7 cm focal - length lens and the beam waist was 32 μm. The opening time of the chopper was ≈10 μs and the iris placed in front of the far - field detector corresponds to light transmittance S = 0.002.

Figures 3(a) and 3(b) show the TM - Z scan curves obtained at different times for pure CHCl3 and for the SGNs - CHCl3 colloid, respectively. Negative NL refractive index is observed in both cases but the signal in the presence of the SGNs is larger at all times. Figures 3(c) and 3(d) show the time evolution of the transmittance signal with the sample at positions corresponding to the minimum and maximum transmittance for pure chloroform and for the SGNs - CHCl3 colloid, respectively. The red and the blue lines correspond to the experimental data and the black line represents the best numerical fit using a single exponential function. The data were taken using the maximum laser intensity, I, equal to 1.2 and 1.0 GW/cm2 for pure chloroform and for the SGNs - CHCl3 colloid, respectively. It was observed that for pure chloroform the ratio | ΔTPV | / I is constant in the range 0.1 < I < 1.0 GW/cm2, as is expected for a third - order NL material. Extrapolating the curves of Fig. 3(c) to t = 0, we get | ΔTPV | = 0.016 and using Eq. (13) of ref [15]. we obtain n 2 = – 0.8×10−18 m2/W, which corresponds to χh (3) = – 0.6×10−20 m2/V2, for pure chloroform. However, | ΔTPV | / I for the SGNs - CHCl3 colloid, for very short times, depends on the laser intensity indicating that higher order electronic nonlinearities may be contributing for the results. The extrapolated results for t = 0 are given in Fig. 4 showing a linear dependence of | ΔTPV | / I versus I. Indeed, in cases where the third- and the fifth - order contributions are present it is expected a straight line with nonzero angular coefficient for | ΔTPV | / I as a function of I [16, 17]. Figure 4 also shows the dependence of | ΔTPV | / I versus I for t = 0.95 ms. In this case, a constant ratio | ΔTPV | / I versus I is observed because the thermal contribution is dominant. Spectra of linear absorption were obtained before and after each Z-Scan measurement and no changes were observed in the results. This indicates that no relevant changes due to induced photochemical process are occurring in our samples.

 

Fig. 3 Thermally Managed Z-scan results. Figures 3(a) and 3(b) show Z-scan profiles obtained at t = 0.10, 0.40, 0.65, 0.95 ms; for pure chloroform and for the colloid, respectively. Figures 3(c) and 3(d) show the time evolution of the transmittance signal with the sample at positions corresponding to the minimum and maximum transmittance for pure chloroform and for the SGNs - CHCl3 colloid, respectively. The data were obtained using a 1 mm thick cuvette and intensities at the focus of 1.2 and 1.0 GW/cm2 for pure chloroform and SGNs - CHCl3 colloid, respectively.

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Fig. 4 Intensity dependence of | ΔTPV | / I as a function of I for t = 0.95 ms (black circles) and for t = 0 ms (blue squares). Red lines represent numerical fits to the data.

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The experiments made without an iris in front of the detector indicate that the NL absorption coefficient is smaller than the minimum that our apparatus can measure (0.1 cm/GW).

Considering small NL phase distortions, small iris transmittance (S << 1), and the peak - to - valley separation, ΔZPV, given by 1.4 times the Rayleigh length, the dependence of | ΔTPV | / I can be described by [16, 17]:

|ΔTPV|Ik0[0.396n2Leff(1)+0.198n4Leff(2)I],
where Leff ( m ) = [1 – exp(–m α 0 L)]/(m α 0) with m = 1,2; n 2(m 2/W) = 3Re[χeff (3)]/(4ε 0 n 0 2 c); and n 4(m 4/W 2) = 5Re[χeff (5)]/(4ε 0 2 n 0 3 c 2), with χeff (3) and χeff (5) being the effective third- and fifth - order susceptibility, respectively. The results obtained for the SGNs - CHCl3 colloid with f = 3×10−5 were n 2 = – 6.0×10−18 m2/W and n 4 = + 8.6×10−31 m4/W2 which correspond to χeff (3) = – 4.4×10−20 m2/V2 and χeff (5) = + 1.4×10−35 m4/V4.

In order to determine the nonlinearity of a single SGN we considered the quasistatic approximation [18] and developed an extension of the previous theories for metallic nanoshells [1,19], including the contribution of the fifth - order susceptibility. Considering the core - shell geometry, the filling fraction f << 1, and an external electric field, E 0, the sample polarization can be written as

P=Ph+1Vi=1NPpi,
where NP is the total number of particles in a volume V, Ph is the host polarization and pi is the induced dipole moment of each SGN that can be written as
pi=(εhαi)E0,
where εh is the dielectric function of the host medium and αi is the particle polarizability given by [19]:
αi=3υi{εsh[εc(32R)+2εshR]εh[εcR+εsh(3R)]εsh[εc(32R)+2εshR]+2εh[εcR+εsh(3R)},
where υi is the volume occupied by each SGN and R = 1 – (rc /rsh )3, where rc and rsh represent the core and the external shell radius, respectively. εsh (εc) is the shell (core) dielectric function. Due to the metallic shielding effect which reduces the magnitude of the electric field in the core and to the fact that silica presents small nonlinearity, the dominant NL contribution comes from the gold shell. Thus, neglecting the core nonlinear response, the NL contribution for the dielectric function of the SGN may be written as
εsh(NL)=34χsh(3)|Esh|2+58χsh(5)|Esh|22,
with 〈 |Esh|2 〉 representing the mean modulus squared of the electric field inside the shell, that is given by 〈 |Esh|2 〉 = |E 0|2 g(rc , rsh) / |β |2, where
β=cεsh+2dεh3aεh,
and
g(rc,rsh)=1+rc34(rshrrc)(1rc21rsh2)(ab+ab)|a|2+rc62(rshrrc)(1rc51rsh5)|b|2|a|2,
with a = εc + 2εsh ; b = εcεsh ; c = (3–2R)εc + 2sh ; and d = c + (3–R)εsh .

Introducing Eq. (5) in Eq. (4) and expanding Eq. (4) up to second order in 〈 |Esh|2 〉, we obtain

χeff(3)=χh(3)+fRg(rc,rsh)|β|2β2cεc+2dεsha2χsh(3),
and
χeff(5)=fR[g(rc,rsh)]2|β|4β2{(cεc+2dεsha2)χsh(5)           310[c(cεc+2dεsh)+18εc(R1)(εsh2+2εcεh)a2(cεsh+2dεh)][χsh(3)]2},
where χh (3) is the host third - order NL response.

The SGN susceptibilities can be determined from Eqs. (8) and (9) considering εc = 2.085 [20], εh = 2.055 [21], εsh = –115.74 + i 19.48 [1], and χh (3) = – 0.6×10−20 m2/V2. The results are χsh (3) = – 1.5×10−11 m2/V2 and χsh (5) = – 1.4×10−24 m4/V4.

An interesting point to notice is the magnitude of χsh (3) which is ≈9 orders of magnitude larger than χh (3). This giant nonlinearity is attributed to the high polarizability of gold at 1560 nm whose dielectric function, | εsh | = 117, is one order of magnitude higher than in the visible range. Hence, considering Miller’s rule [22], |χsh (3)| ∝ |χsh (1)|4, it is expected a susceptibility enhancement of 104 in comparison to the gold NL susceptibility in the visible. Indeed, from experiments in the visible range with 200 fs laser pulses [23], values of |χ (3)| = 2.1×10−15 m2/V2 were obtained for gold nanospheres and this result is ≈104 times smaller than the value of χsh (3) determined in the present work.

For a quick discussion about the different electronic contributions to the nonlinear response of the gold nanoshells, it is well know, from the band calculations, that the outermost d and s electrons of gold atoms originate the d-bands, 5 bands fairly flat which lie a few eV below the Fermi level, and the conduction band or s-p band which presents roughly a parabolic dispersion relation and exhibits almost a free-electron-like behavior. Thus, the optical properties of bulk gold are mainly associated to s-p intraband transitions and also to interband transitions between the d and s-p bands [24, 25]. Accordingly, in general, the experimental dielectric permittivity of gold is decomposed into the interband and the Drude contributions, where this last is related to the s-p intraband transitions. Due to the thickness of the gold shell (20 nm) and the fact that the electron has a mean free path of about 42 nm in gold at room temperature, the damping rate Γ in the Drude model has to be corrected by a factor proportional to d /VF, where d is the shell thickness and VF is the Fermi velocity (1.4×106 m/s) [1]. This correction introduces, for free electrons, a relaxation mechanism associated to the electron-interface scattering at the inner and outer interfaces of the metal layer. However, the major aspects related to the electronic transitions still mainly determined by the bulk band-structure of gold.

Another mechanism which may contribute to the shell nonlinearities is the hot electron contribution. The hot electron contribution is associated to the heating of the free electron gas due to the fraction of the laser light absorbed by the metal. This contribution exhibits strong dependence with pulse duration, photon energy and laser intensity. In general the hot electron contribution is more pronounced for wavelengths close to the Plasmon resonance.

Because the excitation of the samples was made with photons of 0.795 eV, we disregard the interband contributions. Indeed, in order to induce the electronic transition between the d-bands and the conduction band, the photon energy has to be larger than 1.7 eV, which corresponds to the bandgap at the X point of the first Brillouin zone in gold [24]. However, due to the density of states of gold which exhibits a large number of unoccupied states for energies above 2.38 eV, the interband transitions account significantly to the dielectric permittivity of gold typically for photon energies higher than ~2.4 eV. Thus, we attribute the NL optical response of the colloid in this experiment to contributions of intraband transitions and the hot-electrons. In particular, since that it is possible to explain the magnitude of χsh (3) in terms of the gold polarizability, i.e., by the free electron response, the dominant contribution for the nonlinear process probably is the intraband transitions. Indeed the intraband contribution should exhibits a dependence with λ 8 [25], then considering a factor of 3 relative to the difference of wavelength from the visible to the infrared, once more, we obtain a factor of ~104 of gain in magnitude for χsh (3).

Concerning the fifth order nonlinearity we recall that in colloidal systems, the presence of relevant χeff (5) was already demonstrated in silver colloids excited at 532 nm [16]. In that case the role played by the cascade process corresponding to the [χsh (3)]2 term in Eq. (9), was not relevant and the fifth - order response was dominated by the intrinsic fifth - order susceptibility, χsh (5). In the present case, although χsh (5) has large negative value the value of χeff (5), obtained from Eq. (9), have opposite sign to χsh (5) being dominated by the term associated to the local field factor, i.e., the term containing [χsh (3)]2.

The presence of relevant effective fifth - order nonlinearity in the SGN - CHCl3 colloid deserves further comments. In general, materials exhibiting cubic - quintic nonlinearity are attractive due to competitive processes associated to different NL orders that can play an important role on light propagation effects. For example, the combined contribution of the cubic - quintic nonlinearity may induce modulational instability in some regions of the parameters’ space, whereas the individual action of the cubic or quintic nonlinearity does not lead to instabilities [26]. Also, based on studies of soliton propagation in such media, the creation of entanglement states between solitons can be considered [27]. Another interesting subject is the formation of robust 2D arrays of solitons and vortices built in materials featuring the cubic - quintic nonlinearity [28]. With basis on the NL parameters reported here, we evaluate the SGNs - CHCl3 colloid as promising system for such studies.

Besides the high NL response of the SGN, one important point to understand is that at 1560 nm the electric field inside the shell is much smaller than for silver and gold nanospheres in the visible range, for frequencies near the LSP resonance. This is because for the infrared light the free electrons in the SGN screen more efficiently the infrared light electric field than the visible light field. As a consequence, the enhancement of the SGN susceptibility is compensated by the small electric field inside the particle providing an effective NL susceptibility for the SGNs - CHCl3 colloid, at the infrared, that is comparable to the result obtained for gold nanospheres colloids in the visible. However, we recall that the LSP resonance of a SGN can be tuned from the visible towards the near - infrared (including the telecommunication wavelength range) and the drawback of a small electric field inside the metallic shell can be overcome when performing experiments in the infrared. This may allow fabrication of composite materials with large effective nonlinearity that may be very attractive for all - optical switching applications. In this sense we consider that the present results may open new doors for applications of SGNs composites in telecom devices. For example, in order to evaluate the performance of the SGNs composites in all-optical switching devices we recall that suitable materials for such application must have n 2 values large enough to achieve switching for a sample thickness comparable to the absorption length. Accordingly, a good material for all-optical switching using the NL Fabry-Perot configuration should satisfy W = n 2 I 0 / (λ α 0) > 0.27 [29, 30]. The figure-of-merit to evaluate the material performance with respect to the two-photon absorption is T = 2 α 2 λ / n 2 which has to be smaller than 1, irrespective of the device [29, 30]. Assuming I 0 = 0.1 GW/cm2 and α 2 = 0.1 cm/GW we obtain W = 8.8×103 and T = 0.05 which are excellent values. These numbers shows that SGNs composites can be as competitive for photonic applications as Ge-As-Se-based glasses or systems containing carbon nanotubes, but it has the great capability of tuning the LSP resonance which turns SGNs composites in a very versatile system suited for different kinds of photonic applications.

Acknowledgments

We acknowledge the partial support received from the Brazilian agencies Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) and Fundação de Amparo à Ciência e Tecnologia do Estado de Pernambuco (FACEPE). The work was performed under the Photonics National Institute of Science and Technology Project (INCT de Fotônica) and the Nanophotonics Network, supported by the Brazilian Ministry of Science and Technology.

References and links

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4. D. Zhang, O. Neumann, H. Wang, V. M. Yuwono, A. Barhoumi, M. Perham, J. D. Hartgerink, P. Wittung-Stafshede, and N. J. Halas, “Gold nanoparticles can induce the formation of protein-based aggregates at physiological pH,” Nano Lett. 9(2), 666–671 (2009). [CrossRef]   [PubMed]  

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References

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  1. R. D. Averitt, S. L. Westcott, and N. J. Halas, “Linear optical properties of gold nanoshells,” J. Opt. Soc. Am. B 16(10), 1824–1832 (1999).
    [CrossRef]
  2. Y. Lu, G. L. Liu, J. Kim, Y. X. Mejia, and L. P. Lee, “Nanophotonic crescent moon structures with sharp edge for ultrasensitive biomolecular detection by local electromagnetic field enhancement effect,” Nano Lett. 5(1), 119–124 (2005).
    [CrossRef] [PubMed]
  3. M.-R. Choi, K. J. Stanton-Maxey, J. K. Stanley, C. S. Levin, R. Bardhan, D. Akin, S. Badve, J. Sturgis, J. P. Robinson, R. Bashir, N. J. Halas, and S. E. Clare, “A cellular Trojan Horse for delivery of therapeutic nanoparticles into tumors,” Nano Lett. 7(12), 3759–3765 (2007) (and references therein).
    [CrossRef] [PubMed]
  4. D. Zhang, O. Neumann, H. Wang, V. M. Yuwono, A. Barhoumi, M. Perham, J. D. Hartgerink, P. Wittung-Stafshede, and N. J. Halas, “Gold nanoparticles can induce the formation of protein-based aggregates at physiological pH,” Nano Lett. 9(2), 666–671 (2009).
    [CrossRef] [PubMed]
  5. J. J. Penninkhof, L. A. Sweatlock, A. Moroz, H. A. Atwater, A. van Blaaderen, and A. Polman, “Optical cavity modes in gold shell colloids,” J. Appl. Phys. 103(12), 123105 (2008).
    [CrossRef]
  6. J. T. Seo, Q. Yang, W. J. Kim, J. Heo, S. M. Ma, J. Austin, W. S. Yun, S. S. Jung, S. W. Han, B. Tabibi, and D. Temple, “Optical nonlinearities of Au nanoparticles and Au/Ag coreshells,” Opt. Lett. 34(3), 307–309 (2009).
    [CrossRef] [PubMed]
  7. J. F. Li, Y. F. Huang, Y. Ding, Z. L. Yang, S. B. Li, X. S. Zhou, F. R. Fan, W. Zhang, Z. Y. Zhou, Y. Wu, B. Ren, Z. L. Wang, and Z. Q. Tian, “Shell-isolated nanoparticle-enhanced Raman spectroscopy,” Nature 464(7287), 392–395 (2010).
    [CrossRef] [PubMed]
  8. T. Pham, J. B. Jackson, N. J. Halas, and T. R. Lee, “Preparation and characterization of gold nanoshells coated with self-assembled monolayers,” Langmuir 18(12), 4915–4920 (2002).
    [CrossRef]
  9. W. Stöber, A. Fink, and E. Bohn, “Controlled growth of monodisperse silica spheres in the micron size range,” J. Colloid Interface Sci. 26(1), 62–69 (1968).
    [CrossRef]
  10. C. A. R. Costa, C. A. P. Leite, and F. Galembeck, “Size dependence of Stöber silica nanoparticle microchemistry,” J. Phys. Chem. B 107(20), 4747–4755 (2003).
    [CrossRef]
  11. P. C. Lee and D. Meisel, “Adsorption and surface-enhanced Raman of dyes on silver and gold sols,” J. Phys. Chem. 86(17), 3391–3395 (1982).
    [CrossRef]
  12. S. J. Oldenburg, S. L. Westcott, R. D. Averitt, and N. J. Halas, “Surface enhanced Raman scattering in the near infrared using metal nanoshell substrates,” J. Chem. Phys. 111(10), 4729–4735 (1999).
    [CrossRef]
  13. A. Gnoli, L. Razzari, and M. Righini, “Z-scan measurements using high repetition rate lasers: how to manage thermal effects,” Opt. Express 13(20), 7976–7981 (2005).
    [CrossRef] [PubMed]
  14. L. A. Gómez, C. B. de Araújo, R. Putvinskis, S. H. Messaddeq, Y. Ledemi, and Y. Messaddeq, “Nonlinear optical properties of antimony–germanium–sulfur glasses at 1560 nm,” Appl. Phys. B 94(3), 499–502 (2009).
    [CrossRef]
  15. 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]
  16. E. L. Falcão-Filho, C. B. de Araújo, and J. J. Rodrigues, “High-order nonlinearities of aqueous colloids containing silver nanoparticles,” J. Opt. Soc. Am. B 24(12), 2948–2956 (2007).
    [CrossRef]
  17. A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9(3), 405–414 (1992).
    [CrossRef]
  18. See for example: J. D. Jackson, “Classical electrodynamics” (Wiley, New York, 1998).
  19. A. E. Neeves and M. H. Birnboim, “Composite structures for the enhancement of nonlinear-optical susceptibility,” J. Opt. Soc. Am. B 6(4), 787–796 (1989).
    [CrossRef]
  20. Corning product information data sheet for standard silica ( http://www.corning.com/docs/specialtymaterials/pisheets/H0607_hpfs_Standard_ProductSheet.pdf ).
  21. A. Samoc, “Dispersion of refractive properties of solvents: Chloroform, toluene, benzene, and carbon disulfide in ultraviolet, visible, and near-infrared,” J. Appl. Phys. 94(9), 6167–6174 (2003).
    [CrossRef]
  22. R. W. Boyd, “Nonlinear optics” (Academic, San Diego, 2003).
  23. H. B. Liao, R. F. Xiao, J. S. Fu, H. Wang, K. S. Wong, and G. K. L. Wong, “Origin of third-order optical nonlinearity in Au:SiO(2) composite films on femtosecond and picosecond time scales,” Opt. Lett. 23(5), 388–390 (1998).
    [CrossRef]
  24. N. E. Christensen and B. O. Seraphin, “Relativistic band calculation and the optical properties of gold,” Phys. Rev. B 4(10), 3321–3344 (1971).
    [CrossRef]
  25. F. Hache, D. Ricard, and C. Flytzanis, “Optical nonlinearities of small metal particles: surface-mediated resonance and quantum size effects,” J. Opt. Soc. Am. B 3(12), 1647–1655 (1986).
    [CrossRef]
  26. B. B. Baizakov, A. Bouketir, A. Messikh, and B. A. Umarov, “Modulational instability in two-component discrete media with cubic-quintic nonlinearity,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(4), 046605 (2009).
    [CrossRef] [PubMed]
  27. M. Lewenstein and B. A. Malomed, “Spatiotemporal solitons in the Ginzburg Landau model with a two-dimensional transverse grating,” N. J. Phys. 11, 113014 (2009).
    [CrossRef]
  28. D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Spatiotemporal solitons in the Ginzburg Landau model with a two-dimensional transverse grating,” Phys. Rev. A 81(2), 025801 (2010).
    [CrossRef]
  29. G. I. Stegeman, in “Nonlinear optics of organic molecules and polymers,” p.799, edited by H. S. Nalva and S. Miyata (CRC, Boca Raton, Fl., 1997).
  30. K. Wang, H. Long, M. Fu, G. Yang, and P. Lu, “Intensity-dependent reversal of nonlinearity sign in a gold nanoparticle array,” Opt. Lett. 35(10), 1560–1562 (2010).
    [CrossRef] [PubMed]

2010 (3)

J. F. Li, Y. F. Huang, Y. Ding, Z. L. Yang, S. B. Li, X. S. Zhou, F. R. Fan, W. Zhang, Z. Y. Zhou, Y. Wu, B. Ren, Z. L. Wang, and Z. Q. Tian, “Shell-isolated nanoparticle-enhanced Raman spectroscopy,” Nature 464(7287), 392–395 (2010).
[CrossRef] [PubMed]

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Spatiotemporal solitons in the Ginzburg Landau model with a two-dimensional transverse grating,” Phys. Rev. A 81(2), 025801 (2010).
[CrossRef]

K. Wang, H. Long, M. Fu, G. Yang, and P. Lu, “Intensity-dependent reversal of nonlinearity sign in a gold nanoparticle array,” Opt. Lett. 35(10), 1560–1562 (2010).
[CrossRef] [PubMed]

2009 (5)

J. T. Seo, Q. Yang, W. J. Kim, J. Heo, S. M. Ma, J. Austin, W. S. Yun, S. S. Jung, S. W. Han, B. Tabibi, and D. Temple, “Optical nonlinearities of Au nanoparticles and Au/Ag coreshells,” Opt. Lett. 34(3), 307–309 (2009).
[CrossRef] [PubMed]

B. B. Baizakov, A. Bouketir, A. Messikh, and B. A. Umarov, “Modulational instability in two-component discrete media with cubic-quintic nonlinearity,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(4), 046605 (2009).
[CrossRef] [PubMed]

M. Lewenstein and B. A. Malomed, “Spatiotemporal solitons in the Ginzburg Landau model with a two-dimensional transverse grating,” N. J. Phys. 11, 113014 (2009).
[CrossRef]

L. A. Gómez, C. B. de Araújo, R. Putvinskis, S. H. Messaddeq, Y. Ledemi, and Y. Messaddeq, “Nonlinear optical properties of antimony–germanium–sulfur glasses at 1560 nm,” Appl. Phys. B 94(3), 499–502 (2009).
[CrossRef]

D. Zhang, O. Neumann, H. Wang, V. M. Yuwono, A. Barhoumi, M. Perham, J. D. Hartgerink, P. Wittung-Stafshede, and N. J. Halas, “Gold nanoparticles can induce the formation of protein-based aggregates at physiological pH,” Nano Lett. 9(2), 666–671 (2009).
[CrossRef] [PubMed]

2008 (1)

J. J. Penninkhof, L. A. Sweatlock, A. Moroz, H. A. Atwater, A. van Blaaderen, and A. Polman, “Optical cavity modes in gold shell colloids,” J. Appl. Phys. 103(12), 123105 (2008).
[CrossRef]

2007 (2)

M.-R. Choi, K. J. Stanton-Maxey, J. K. Stanley, C. S. Levin, R. Bardhan, D. Akin, S. Badve, J. Sturgis, J. P. Robinson, R. Bashir, N. J. Halas, and S. E. Clare, “A cellular Trojan Horse for delivery of therapeutic nanoparticles into tumors,” Nano Lett. 7(12), 3759–3765 (2007) (and references therein).
[CrossRef] [PubMed]

E. L. Falcão-Filho, C. B. de Araújo, and J. J. Rodrigues, “High-order nonlinearities of aqueous colloids containing silver nanoparticles,” J. Opt. Soc. Am. B 24(12), 2948–2956 (2007).
[CrossRef]

2005 (2)

A. Gnoli, L. Razzari, and M. Righini, “Z-scan measurements using high repetition rate lasers: how to manage thermal effects,” Opt. Express 13(20), 7976–7981 (2005).
[CrossRef] [PubMed]

Y. Lu, G. L. Liu, J. Kim, Y. X. Mejia, and L. P. Lee, “Nanophotonic crescent moon structures with sharp edge for ultrasensitive biomolecular detection by local electromagnetic field enhancement effect,” Nano Lett. 5(1), 119–124 (2005).
[CrossRef] [PubMed]

2003 (2)

C. A. R. Costa, C. A. P. Leite, and F. Galembeck, “Size dependence of Stöber silica nanoparticle microchemistry,” J. Phys. Chem. B 107(20), 4747–4755 (2003).
[CrossRef]

A. Samoc, “Dispersion of refractive properties of solvents: Chloroform, toluene, benzene, and carbon disulfide in ultraviolet, visible, and near-infrared,” J. Appl. Phys. 94(9), 6167–6174 (2003).
[CrossRef]

2002 (1)

T. Pham, J. B. Jackson, N. J. Halas, and T. R. Lee, “Preparation and characterization of gold nanoshells coated with self-assembled monolayers,” Langmuir 18(12), 4915–4920 (2002).
[CrossRef]

1999 (2)

S. J. Oldenburg, S. L. Westcott, R. D. Averitt, and N. J. Halas, “Surface enhanced Raman scattering in the near infrared using metal nanoshell substrates,” J. Chem. Phys. 111(10), 4729–4735 (1999).
[CrossRef]

R. D. Averitt, S. L. Westcott, and N. J. Halas, “Linear optical properties of gold nanoshells,” J. Opt. Soc. Am. B 16(10), 1824–1832 (1999).
[CrossRef]

1998 (1)

1992 (1)

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]

1989 (1)

1986 (1)

1982 (1)

P. C. Lee and D. Meisel, “Adsorption and surface-enhanced Raman of dyes on silver and gold sols,” J. Phys. Chem. 86(17), 3391–3395 (1982).
[CrossRef]

1971 (1)

N. E. Christensen and B. O. Seraphin, “Relativistic band calculation and the optical properties of gold,” Phys. Rev. B 4(10), 3321–3344 (1971).
[CrossRef]

1968 (1)

W. Stöber, A. Fink, and E. Bohn, “Controlled growth of monodisperse silica spheres in the micron size range,” J. Colloid Interface Sci. 26(1), 62–69 (1968).
[CrossRef]

Akin, D.

M.-R. Choi, K. J. Stanton-Maxey, J. K. Stanley, C. S. Levin, R. Bardhan, D. Akin, S. Badve, J. Sturgis, J. P. Robinson, R. Bashir, N. J. Halas, and S. E. Clare, “A cellular Trojan Horse for delivery of therapeutic nanoparticles into tumors,” Nano Lett. 7(12), 3759–3765 (2007) (and references therein).
[CrossRef] [PubMed]

Atwater, H. A.

J. J. Penninkhof, L. A. Sweatlock, A. Moroz, H. A. Atwater, A. van Blaaderen, and A. Polman, “Optical cavity modes in gold shell colloids,” J. Appl. Phys. 103(12), 123105 (2008).
[CrossRef]

Austin, J.

Averitt, R. D.

R. D. Averitt, S. L. Westcott, and N. J. Halas, “Linear optical properties of gold nanoshells,” J. Opt. Soc. Am. B 16(10), 1824–1832 (1999).
[CrossRef]

S. J. Oldenburg, S. L. Westcott, R. D. Averitt, and N. J. Halas, “Surface enhanced Raman scattering in the near infrared using metal nanoshell substrates,” J. Chem. Phys. 111(10), 4729–4735 (1999).
[CrossRef]

Badve, S.

M.-R. Choi, K. J. Stanton-Maxey, J. K. Stanley, C. S. Levin, R. Bardhan, D. Akin, S. Badve, J. Sturgis, J. P. Robinson, R. Bashir, N. J. Halas, and S. E. Clare, “A cellular Trojan Horse for delivery of therapeutic nanoparticles into tumors,” Nano Lett. 7(12), 3759–3765 (2007) (and references therein).
[CrossRef] [PubMed]

Baizakov, B. B.

B. B. Baizakov, A. Bouketir, A. Messikh, and B. A. Umarov, “Modulational instability in two-component discrete media with cubic-quintic nonlinearity,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(4), 046605 (2009).
[CrossRef] [PubMed]

Bardhan, R.

M.-R. Choi, K. J. Stanton-Maxey, J. K. Stanley, C. S. Levin, R. Bardhan, D. Akin, S. Badve, J. Sturgis, J. P. Robinson, R. Bashir, N. J. Halas, and S. E. Clare, “A cellular Trojan Horse for delivery of therapeutic nanoparticles into tumors,” Nano Lett. 7(12), 3759–3765 (2007) (and references therein).
[CrossRef] [PubMed]

Barhoumi, A.

D. Zhang, O. Neumann, H. Wang, V. M. Yuwono, A. Barhoumi, M. Perham, J. D. Hartgerink, P. Wittung-Stafshede, and N. J. Halas, “Gold nanoparticles can induce the formation of protein-based aggregates at physiological pH,” Nano Lett. 9(2), 666–671 (2009).
[CrossRef] [PubMed]

Bashir, R.

M.-R. Choi, K. J. Stanton-Maxey, J. K. Stanley, C. S. Levin, R. Bardhan, D. Akin, S. Badve, J. Sturgis, J. P. Robinson, R. Bashir, N. J. Halas, and S. E. Clare, “A cellular Trojan Horse for delivery of therapeutic nanoparticles into tumors,” Nano Lett. 7(12), 3759–3765 (2007) (and references therein).
[CrossRef] [PubMed]

Birnboim, M. H.

Bohn, E.

W. Stöber, A. Fink, and E. Bohn, “Controlled growth of monodisperse silica spheres in the micron size range,” J. Colloid Interface Sci. 26(1), 62–69 (1968).
[CrossRef]

Bouketir, A.

B. B. Baizakov, A. Bouketir, A. Messikh, and B. A. Umarov, “Modulational instability in two-component discrete media with cubic-quintic nonlinearity,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(4), 046605 (2009).
[CrossRef] [PubMed]

Choi, M.-R.

M.-R. Choi, K. J. Stanton-Maxey, J. K. Stanley, C. S. Levin, R. Bardhan, D. Akin, S. Badve, J. Sturgis, J. P. Robinson, R. Bashir, N. J. Halas, and S. E. Clare, “A cellular Trojan Horse for delivery of therapeutic nanoparticles into tumors,” Nano Lett. 7(12), 3759–3765 (2007) (and references therein).
[CrossRef] [PubMed]

Christensen, N. E.

N. E. Christensen and B. O. Seraphin, “Relativistic band calculation and the optical properties of gold,” Phys. Rev. B 4(10), 3321–3344 (1971).
[CrossRef]

Clare, S. E.

M.-R. Choi, K. J. Stanton-Maxey, J. K. Stanley, C. S. Levin, R. Bardhan, D. Akin, S. Badve, J. Sturgis, J. P. Robinson, R. Bashir, N. J. Halas, and S. E. Clare, “A cellular Trojan Horse for delivery of therapeutic nanoparticles into tumors,” Nano Lett. 7(12), 3759–3765 (2007) (and references therein).
[CrossRef] [PubMed]

Costa, C. A. R.

C. A. R. Costa, C. A. P. Leite, and F. Galembeck, “Size dependence of Stöber silica nanoparticle microchemistry,” J. Phys. Chem. B 107(20), 4747–4755 (2003).
[CrossRef]

de Araújo, C. B.

L. A. Gómez, C. B. de Araújo, R. Putvinskis, S. H. Messaddeq, Y. Ledemi, and Y. Messaddeq, “Nonlinear optical properties of antimony–germanium–sulfur glasses at 1560 nm,” Appl. Phys. B 94(3), 499–502 (2009).
[CrossRef]

E. L. Falcão-Filho, C. B. de Araújo, and J. J. Rodrigues, “High-order nonlinearities of aqueous colloids containing silver nanoparticles,” J. Opt. Soc. Am. B 24(12), 2948–2956 (2007).
[CrossRef]

Ding, Y.

J. F. Li, Y. F. Huang, Y. Ding, Z. L. Yang, S. B. Li, X. S. Zhou, F. R. Fan, W. Zhang, Z. Y. Zhou, Y. Wu, B. Ren, Z. L. Wang, and Z. Q. Tian, “Shell-isolated nanoparticle-enhanced Raman spectroscopy,” Nature 464(7287), 392–395 (2010).
[CrossRef] [PubMed]

Falcão-Filho, E. L.

Fan, F. R.

J. F. Li, Y. F. Huang, Y. Ding, Z. L. Yang, S. B. Li, X. S. Zhou, F. R. Fan, W. Zhang, Z. Y. Zhou, Y. Wu, B. Ren, Z. L. Wang, and Z. Q. Tian, “Shell-isolated nanoparticle-enhanced Raman spectroscopy,” Nature 464(7287), 392–395 (2010).
[CrossRef] [PubMed]

Fink, A.

W. Stöber, A. Fink, and E. Bohn, “Controlled growth of monodisperse silica spheres in the micron size range,” J. Colloid Interface Sci. 26(1), 62–69 (1968).
[CrossRef]

Flytzanis, C.

Fu, J. S.

Fu, M.

Galembeck, F.

C. A. R. Costa, C. A. P. Leite, and F. Galembeck, “Size dependence of Stöber silica nanoparticle microchemistry,” J. Phys. Chem. B 107(20), 4747–4755 (2003).
[CrossRef]

Gnoli, A.

Gómez, L. A.

L. A. Gómez, C. B. de Araújo, R. Putvinskis, S. H. Messaddeq, Y. Ledemi, and Y. Messaddeq, “Nonlinear optical properties of antimony–germanium–sulfur glasses at 1560 nm,” Appl. Phys. B 94(3), 499–502 (2009).
[CrossRef]

Hache, F.

Hagan, D. J.

A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9(3), 405–414 (1992).
[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]

Halas, N. J.

D. Zhang, O. Neumann, H. Wang, V. M. Yuwono, A. Barhoumi, M. Perham, J. D. Hartgerink, P. Wittung-Stafshede, and N. J. Halas, “Gold nanoparticles can induce the formation of protein-based aggregates at physiological pH,” Nano Lett. 9(2), 666–671 (2009).
[CrossRef] [PubMed]

M.-R. Choi, K. J. Stanton-Maxey, J. K. Stanley, C. S. Levin, R. Bardhan, D. Akin, S. Badve, J. Sturgis, J. P. Robinson, R. Bashir, N. J. Halas, and S. E. Clare, “A cellular Trojan Horse for delivery of therapeutic nanoparticles into tumors,” Nano Lett. 7(12), 3759–3765 (2007) (and references therein).
[CrossRef] [PubMed]

T. Pham, J. B. Jackson, N. J. Halas, and T. R. Lee, “Preparation and characterization of gold nanoshells coated with self-assembled monolayers,” Langmuir 18(12), 4915–4920 (2002).
[CrossRef]

R. D. Averitt, S. L. Westcott, and N. J. Halas, “Linear optical properties of gold nanoshells,” J. Opt. Soc. Am. B 16(10), 1824–1832 (1999).
[CrossRef]

S. J. Oldenburg, S. L. Westcott, R. D. Averitt, and N. J. Halas, “Surface enhanced Raman scattering in the near infrared using metal nanoshell substrates,” J. Chem. Phys. 111(10), 4729–4735 (1999).
[CrossRef]

Han, S. W.

Hartgerink, J. D.

D. Zhang, O. Neumann, H. Wang, V. M. Yuwono, A. Barhoumi, M. Perham, J. D. Hartgerink, P. Wittung-Stafshede, and N. J. Halas, “Gold nanoparticles can induce the formation of protein-based aggregates at physiological pH,” Nano Lett. 9(2), 666–671 (2009).
[CrossRef] [PubMed]

Heo, J.

Huang, Y. F.

J. F. Li, Y. F. Huang, Y. Ding, Z. L. Yang, S. B. Li, X. S. Zhou, F. R. Fan, W. Zhang, Z. Y. Zhou, Y. Wu, B. Ren, Z. L. Wang, and Z. Q. Tian, “Shell-isolated nanoparticle-enhanced Raman spectroscopy,” Nature 464(7287), 392–395 (2010).
[CrossRef] [PubMed]

Jackson, J. B.

T. Pham, J. B. Jackson, N. J. Halas, and T. R. Lee, “Preparation and characterization of gold nanoshells coated with self-assembled monolayers,” Langmuir 18(12), 4915–4920 (2002).
[CrossRef]

Jung, S. S.

Kim, J.

Y. Lu, G. L. Liu, J. Kim, Y. X. Mejia, and L. P. Lee, “Nanophotonic crescent moon structures with sharp edge for ultrasensitive biomolecular detection by local electromagnetic field enhancement effect,” Nano Lett. 5(1), 119–124 (2005).
[CrossRef] [PubMed]

Kim, W. J.

Leblond, H.

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Spatiotemporal solitons in the Ginzburg Landau model with a two-dimensional transverse grating,” Phys. Rev. A 81(2), 025801 (2010).
[CrossRef]

Ledemi, Y.

L. A. Gómez, C. B. de Araújo, R. Putvinskis, S. H. Messaddeq, Y. Ledemi, and Y. Messaddeq, “Nonlinear optical properties of antimony–germanium–sulfur glasses at 1560 nm,” Appl. Phys. B 94(3), 499–502 (2009).
[CrossRef]

Lederer, F.

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Spatiotemporal solitons in the Ginzburg Landau model with a two-dimensional transverse grating,” Phys. Rev. A 81(2), 025801 (2010).
[CrossRef]

Lee, L. P.

Y. Lu, G. L. Liu, J. Kim, Y. X. Mejia, and L. P. Lee, “Nanophotonic crescent moon structures with sharp edge for ultrasensitive biomolecular detection by local electromagnetic field enhancement effect,” Nano Lett. 5(1), 119–124 (2005).
[CrossRef] [PubMed]

Lee, P. C.

P. C. Lee and D. Meisel, “Adsorption and surface-enhanced Raman of dyes on silver and gold sols,” J. Phys. Chem. 86(17), 3391–3395 (1982).
[CrossRef]

Lee, T. R.

T. Pham, J. B. Jackson, N. J. Halas, and T. R. Lee, “Preparation and characterization of gold nanoshells coated with self-assembled monolayers,” Langmuir 18(12), 4915–4920 (2002).
[CrossRef]

Leite, C. A. P.

C. A. R. Costa, C. A. P. Leite, and F. Galembeck, “Size dependence of Stöber silica nanoparticle microchemistry,” J. Phys. Chem. B 107(20), 4747–4755 (2003).
[CrossRef]

Levin, C. S.

M.-R. Choi, K. J. Stanton-Maxey, J. K. Stanley, C. S. Levin, R. Bardhan, D. Akin, S. Badve, J. Sturgis, J. P. Robinson, R. Bashir, N. J. Halas, and S. E. Clare, “A cellular Trojan Horse for delivery of therapeutic nanoparticles into tumors,” Nano Lett. 7(12), 3759–3765 (2007) (and references therein).
[CrossRef] [PubMed]

Lewenstein, M.

M. Lewenstein and B. A. Malomed, “Spatiotemporal solitons in the Ginzburg Landau model with a two-dimensional transverse grating,” N. J. Phys. 11, 113014 (2009).
[CrossRef]

Li, J. F.

J. F. Li, Y. F. Huang, Y. Ding, Z. L. Yang, S. B. Li, X. S. Zhou, F. R. Fan, W. Zhang, Z. Y. Zhou, Y. Wu, B. Ren, Z. L. Wang, and Z. Q. Tian, “Shell-isolated nanoparticle-enhanced Raman spectroscopy,” Nature 464(7287), 392–395 (2010).
[CrossRef] [PubMed]

Li, S. B.

J. F. Li, Y. F. Huang, Y. Ding, Z. L. Yang, S. B. Li, X. S. Zhou, F. R. Fan, W. Zhang, Z. Y. Zhou, Y. Wu, B. Ren, Z. L. Wang, and Z. Q. Tian, “Shell-isolated nanoparticle-enhanced Raman spectroscopy,” Nature 464(7287), 392–395 (2010).
[CrossRef] [PubMed]

Liao, H. B.

Liu, G. L.

Y. Lu, G. L. Liu, J. Kim, Y. X. Mejia, and L. P. Lee, “Nanophotonic crescent moon structures with sharp edge for ultrasensitive biomolecular detection by local electromagnetic field enhancement effect,” Nano Lett. 5(1), 119–124 (2005).
[CrossRef] [PubMed]

Long, H.

Lu, P.

Lu, Y.

Y. Lu, G. L. Liu, J. Kim, Y. X. Mejia, and L. P. Lee, “Nanophotonic crescent moon structures with sharp edge for ultrasensitive biomolecular detection by local electromagnetic field enhancement effect,” Nano Lett. 5(1), 119–124 (2005).
[CrossRef] [PubMed]

Ma, S. M.

Malomed, B. A.

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Spatiotemporal solitons in the Ginzburg Landau model with a two-dimensional transverse grating,” Phys. Rev. A 81(2), 025801 (2010).
[CrossRef]

M. Lewenstein and B. A. Malomed, “Spatiotemporal solitons in the Ginzburg Landau model with a two-dimensional transverse grating,” N. J. Phys. 11, 113014 (2009).
[CrossRef]

Mazilu, D.

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Spatiotemporal solitons in the Ginzburg Landau model with a two-dimensional transverse grating,” Phys. Rev. A 81(2), 025801 (2010).
[CrossRef]

Meisel, D.

P. C. Lee and D. Meisel, “Adsorption and surface-enhanced Raman of dyes on silver and gold sols,” J. Phys. Chem. 86(17), 3391–3395 (1982).
[CrossRef]

Mejia, Y. X.

Y. Lu, G. L. Liu, J. Kim, Y. X. Mejia, and L. P. Lee, “Nanophotonic crescent moon structures with sharp edge for ultrasensitive biomolecular detection by local electromagnetic field enhancement effect,” Nano Lett. 5(1), 119–124 (2005).
[CrossRef] [PubMed]

Messaddeq, S. H.

L. A. Gómez, C. B. de Araújo, R. Putvinskis, S. H. Messaddeq, Y. Ledemi, and Y. Messaddeq, “Nonlinear optical properties of antimony–germanium–sulfur glasses at 1560 nm,” Appl. Phys. B 94(3), 499–502 (2009).
[CrossRef]

Messaddeq, Y.

L. A. Gómez, C. B. de Araújo, R. Putvinskis, S. H. Messaddeq, Y. Ledemi, and Y. Messaddeq, “Nonlinear optical properties of antimony–germanium–sulfur glasses at 1560 nm,” Appl. Phys. B 94(3), 499–502 (2009).
[CrossRef]

Messikh, A.

B. B. Baizakov, A. Bouketir, A. Messikh, and B. A. Umarov, “Modulational instability in two-component discrete media with cubic-quintic nonlinearity,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(4), 046605 (2009).
[CrossRef] [PubMed]

Mihalache, D.

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Spatiotemporal solitons in the Ginzburg Landau model with a two-dimensional transverse grating,” Phys. Rev. A 81(2), 025801 (2010).
[CrossRef]

Moroz, A.

J. J. Penninkhof, L. A. Sweatlock, A. Moroz, H. A. Atwater, A. van Blaaderen, and A. Polman, “Optical cavity modes in gold shell colloids,” J. Appl. Phys. 103(12), 123105 (2008).
[CrossRef]

Neeves, A. E.

Neumann, O.

D. Zhang, O. Neumann, H. Wang, V. M. Yuwono, A. Barhoumi, M. Perham, J. D. Hartgerink, P. Wittung-Stafshede, and N. J. Halas, “Gold nanoparticles can induce the formation of protein-based aggregates at physiological pH,” Nano Lett. 9(2), 666–671 (2009).
[CrossRef] [PubMed]

Oldenburg, S. J.

S. J. Oldenburg, S. L. Westcott, R. D. Averitt, and N. J. Halas, “Surface enhanced Raman scattering in the near infrared using metal nanoshell substrates,” J. Chem. Phys. 111(10), 4729–4735 (1999).
[CrossRef]

Penninkhof, J. J.

J. J. Penninkhof, L. A. Sweatlock, A. Moroz, H. A. Atwater, A. van Blaaderen, and A. Polman, “Optical cavity modes in gold shell colloids,” J. Appl. Phys. 103(12), 123105 (2008).
[CrossRef]

Perham, M.

D. Zhang, O. Neumann, H. Wang, V. M. Yuwono, A. Barhoumi, M. Perham, J. D. Hartgerink, P. Wittung-Stafshede, and N. J. Halas, “Gold nanoparticles can induce the formation of protein-based aggregates at physiological pH,” Nano Lett. 9(2), 666–671 (2009).
[CrossRef] [PubMed]

Pham, T.

T. Pham, J. B. Jackson, N. J. Halas, and T. R. Lee, “Preparation and characterization of gold nanoshells coated with self-assembled monolayers,” Langmuir 18(12), 4915–4920 (2002).
[CrossRef]

Polman, A.

J. J. Penninkhof, L. A. Sweatlock, A. Moroz, H. A. Atwater, A. van Blaaderen, and A. Polman, “Optical cavity modes in gold shell colloids,” J. Appl. Phys. 103(12), 123105 (2008).
[CrossRef]

Putvinskis, R.

L. A. Gómez, C. B. de Araújo, R. Putvinskis, S. H. Messaddeq, Y. Ledemi, and Y. Messaddeq, “Nonlinear optical properties of antimony–germanium–sulfur glasses at 1560 nm,” Appl. Phys. B 94(3), 499–502 (2009).
[CrossRef]

Razzari, L.

Ren, B.

J. F. Li, Y. F. Huang, Y. Ding, Z. L. Yang, S. B. Li, X. S. Zhou, F. R. Fan, W. Zhang, Z. Y. Zhou, Y. Wu, B. Ren, Z. L. Wang, and Z. Q. Tian, “Shell-isolated nanoparticle-enhanced Raman spectroscopy,” Nature 464(7287), 392–395 (2010).
[CrossRef] [PubMed]

Ricard, D.

Righini, M.

Robinson, J. P.

M.-R. Choi, K. J. Stanton-Maxey, J. K. Stanley, C. S. Levin, R. Bardhan, D. Akin, S. Badve, J. Sturgis, J. P. Robinson, R. Bashir, N. J. Halas, and S. E. Clare, “A cellular Trojan Horse for delivery of therapeutic nanoparticles into tumors,” Nano Lett. 7(12), 3759–3765 (2007) (and references therein).
[CrossRef] [PubMed]

Rodrigues, J. J.

Said, A. A.

A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9(3), 405–414 (1992).
[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]

Samoc, A.

A. Samoc, “Dispersion of refractive properties of solvents: Chloroform, toluene, benzene, and carbon disulfide in ultraviolet, visible, and near-infrared,” J. Appl. Phys. 94(9), 6167–6174 (2003).
[CrossRef]

Seo, J. T.

Seraphin, B. O.

N. E. Christensen and B. O. Seraphin, “Relativistic band calculation and the optical properties of gold,” Phys. Rev. B 4(10), 3321–3344 (1971).
[CrossRef]

Sheik-Bahae, M.

A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9(3), 405–414 (1992).
[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]

Stanley, J. K.

M.-R. Choi, K. J. Stanton-Maxey, J. K. Stanley, C. S. Levin, R. Bardhan, D. Akin, S. Badve, J. Sturgis, J. P. Robinson, R. Bashir, N. J. Halas, and S. E. Clare, “A cellular Trojan Horse for delivery of therapeutic nanoparticles into tumors,” Nano Lett. 7(12), 3759–3765 (2007) (and references therein).
[CrossRef] [PubMed]

Stanton-Maxey, K. J.

M.-R. Choi, K. J. Stanton-Maxey, J. K. Stanley, C. S. Levin, R. Bardhan, D. Akin, S. Badve, J. Sturgis, J. P. Robinson, R. Bashir, N. J. Halas, and S. E. Clare, “A cellular Trojan Horse for delivery of therapeutic nanoparticles into tumors,” Nano Lett. 7(12), 3759–3765 (2007) (and references therein).
[CrossRef] [PubMed]

Stöber, W.

W. Stöber, A. Fink, and E. Bohn, “Controlled growth of monodisperse silica spheres in the micron size range,” J. Colloid Interface Sci. 26(1), 62–69 (1968).
[CrossRef]

Sturgis, J.

M.-R. Choi, K. J. Stanton-Maxey, J. K. Stanley, C. S. Levin, R. Bardhan, D. Akin, S. Badve, J. Sturgis, J. P. Robinson, R. Bashir, N. J. Halas, and S. E. Clare, “A cellular Trojan Horse for delivery of therapeutic nanoparticles into tumors,” Nano Lett. 7(12), 3759–3765 (2007) (and references therein).
[CrossRef] [PubMed]

Sweatlock, L. A.

J. J. Penninkhof, L. A. Sweatlock, A. Moroz, H. A. Atwater, A. van Blaaderen, and A. Polman, “Optical cavity modes in gold shell colloids,” J. Appl. Phys. 103(12), 123105 (2008).
[CrossRef]

Tabibi, B.

Temple, D.

Tian, Z. Q.

J. F. Li, Y. F. Huang, Y. Ding, Z. L. Yang, S. B. Li, X. S. Zhou, F. R. Fan, W. Zhang, Z. Y. Zhou, Y. Wu, B. Ren, Z. L. Wang, and Z. Q. Tian, “Shell-isolated nanoparticle-enhanced Raman spectroscopy,” Nature 464(7287), 392–395 (2010).
[CrossRef] [PubMed]

Umarov, B. A.

B. B. Baizakov, A. Bouketir, A. Messikh, and B. A. Umarov, “Modulational instability in two-component discrete media with cubic-quintic nonlinearity,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(4), 046605 (2009).
[CrossRef] [PubMed]

van Blaaderen, A.

J. J. Penninkhof, L. A. Sweatlock, A. Moroz, H. A. Atwater, A. van Blaaderen, and A. Polman, “Optical cavity modes in gold shell colloids,” J. Appl. Phys. 103(12), 123105 (2008).
[CrossRef]

Van Stryland, E. W.

A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9(3), 405–414 (1992).
[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]

Wang, H.

D. Zhang, O. Neumann, H. Wang, V. M. Yuwono, A. Barhoumi, M. Perham, J. D. Hartgerink, P. Wittung-Stafshede, and N. J. Halas, “Gold nanoparticles can induce the formation of protein-based aggregates at physiological pH,” Nano Lett. 9(2), 666–671 (2009).
[CrossRef] [PubMed]

H. B. Liao, R. F. Xiao, J. S. Fu, H. Wang, K. S. Wong, and G. K. L. Wong, “Origin of third-order optical nonlinearity in Au:SiO(2) composite films on femtosecond and picosecond time scales,” Opt. Lett. 23(5), 388–390 (1998).
[CrossRef]

Wang, J.

Wang, K.

Wang, Z. L.

J. F. Li, Y. F. Huang, Y. Ding, Z. L. Yang, S. B. Li, X. S. Zhou, F. R. Fan, W. Zhang, Z. Y. Zhou, Y. Wu, B. Ren, Z. L. Wang, and Z. Q. Tian, “Shell-isolated nanoparticle-enhanced Raman spectroscopy,” Nature 464(7287), 392–395 (2010).
[CrossRef] [PubMed]

Wei, T. H.

A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, “Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe,” J. Opt. Soc. Am. B 9(3), 405–414 (1992).
[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]

Westcott, S. L.

S. J. Oldenburg, S. L. Westcott, R. D. Averitt, and N. J. Halas, “Surface enhanced Raman scattering in the near infrared using metal nanoshell substrates,” J. Chem. Phys. 111(10), 4729–4735 (1999).
[CrossRef]

R. D. Averitt, S. L. Westcott, and N. J. Halas, “Linear optical properties of gold nanoshells,” J. Opt. Soc. Am. B 16(10), 1824–1832 (1999).
[CrossRef]

Wittung-Stafshede, P.

D. Zhang, O. Neumann, H. Wang, V. M. Yuwono, A. Barhoumi, M. Perham, J. D. Hartgerink, P. Wittung-Stafshede, and N. J. Halas, “Gold nanoparticles can induce the formation of protein-based aggregates at physiological pH,” Nano Lett. 9(2), 666–671 (2009).
[CrossRef] [PubMed]

Wong, G. K. L.

Wong, K. S.

Wu, Y.

J. F. Li, Y. F. Huang, Y. Ding, Z. L. Yang, S. B. Li, X. S. Zhou, F. R. Fan, W. Zhang, Z. Y. Zhou, Y. Wu, B. Ren, Z. L. Wang, and Z. Q. Tian, “Shell-isolated nanoparticle-enhanced Raman spectroscopy,” Nature 464(7287), 392–395 (2010).
[CrossRef] [PubMed]

Xiao, R. F.

Yang, G.

Yang, Q.

Yang, Z. L.

J. F. Li, Y. F. Huang, Y. Ding, Z. L. Yang, S. B. Li, X. S. Zhou, F. R. Fan, W. Zhang, Z. Y. Zhou, Y. Wu, B. Ren, Z. L. Wang, and Z. Q. Tian, “Shell-isolated nanoparticle-enhanced Raman spectroscopy,” Nature 464(7287), 392–395 (2010).
[CrossRef] [PubMed]

Young, J.

Yun, W. S.

Yuwono, V. M.

D. Zhang, O. Neumann, H. Wang, V. M. Yuwono, A. Barhoumi, M. Perham, J. D. Hartgerink, P. Wittung-Stafshede, and N. J. Halas, “Gold nanoparticles can induce the formation of protein-based aggregates at physiological pH,” Nano Lett. 9(2), 666–671 (2009).
[CrossRef] [PubMed]

Zhang, D.

D. Zhang, O. Neumann, H. Wang, V. M. Yuwono, A. Barhoumi, M. Perham, J. D. Hartgerink, P. Wittung-Stafshede, and N. J. Halas, “Gold nanoparticles can induce the formation of protein-based aggregates at physiological pH,” Nano Lett. 9(2), 666–671 (2009).
[CrossRef] [PubMed]

Zhang, W.

J. F. Li, Y. F. Huang, Y. Ding, Z. L. Yang, S. B. Li, X. S. Zhou, F. R. Fan, W. Zhang, Z. Y. Zhou, Y. Wu, B. Ren, Z. L. Wang, and Z. Q. Tian, “Shell-isolated nanoparticle-enhanced Raman spectroscopy,” Nature 464(7287), 392–395 (2010).
[CrossRef] [PubMed]

Zhou, X. S.

J. F. Li, Y. F. Huang, Y. Ding, Z. L. Yang, S. B. Li, X. S. Zhou, F. R. Fan, W. Zhang, Z. Y. Zhou, Y. Wu, B. Ren, Z. L. Wang, and Z. Q. Tian, “Shell-isolated nanoparticle-enhanced Raman spectroscopy,” Nature 464(7287), 392–395 (2010).
[CrossRef] [PubMed]

Zhou, Z. Y.

J. F. Li, Y. F. Huang, Y. Ding, Z. L. Yang, S. B. Li, X. S. Zhou, F. R. Fan, W. Zhang, Z. Y. Zhou, Y. Wu, B. Ren, Z. L. Wang, and Z. Q. Tian, “Shell-isolated nanoparticle-enhanced Raman spectroscopy,” Nature 464(7287), 392–395 (2010).
[CrossRef] [PubMed]

Appl. Phys. B (1)

L. A. Gómez, C. B. de Araújo, R. Putvinskis, S. H. Messaddeq, Y. Ledemi, and Y. Messaddeq, “Nonlinear optical properties of antimony–germanium–sulfur glasses at 1560 nm,” Appl. Phys. B 94(3), 499–502 (2009).
[CrossRef]

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. Appl. Phys. (2)

A. Samoc, “Dispersion of refractive properties of solvents: Chloroform, toluene, benzene, and carbon disulfide in ultraviolet, visible, and near-infrared,” J. Appl. Phys. 94(9), 6167–6174 (2003).
[CrossRef]

J. J. Penninkhof, L. A. Sweatlock, A. Moroz, H. A. Atwater, A. van Blaaderen, and A. Polman, “Optical cavity modes in gold shell colloids,” J. Appl. Phys. 103(12), 123105 (2008).
[CrossRef]

J. Chem. Phys. (1)

S. J. Oldenburg, S. L. Westcott, R. D. Averitt, and N. J. Halas, “Surface enhanced Raman scattering in the near infrared using metal nanoshell substrates,” J. Chem. Phys. 111(10), 4729–4735 (1999).
[CrossRef]

J. Colloid Interface Sci. (1)

W. Stöber, A. Fink, and E. Bohn, “Controlled growth of monodisperse silica spheres in the micron size range,” J. Colloid Interface Sci. 26(1), 62–69 (1968).
[CrossRef]

J. Opt. Soc. Am. B (5)

J. Phys. Chem. (1)

P. C. Lee and D. Meisel, “Adsorption and surface-enhanced Raman of dyes on silver and gold sols,” J. Phys. Chem. 86(17), 3391–3395 (1982).
[CrossRef]

J. Phys. Chem. B (1)

C. A. R. Costa, C. A. P. Leite, and F. Galembeck, “Size dependence of Stöber silica nanoparticle microchemistry,” J. Phys. Chem. B 107(20), 4747–4755 (2003).
[CrossRef]

Langmuir (1)

T. Pham, J. B. Jackson, N. J. Halas, and T. R. Lee, “Preparation and characterization of gold nanoshells coated with self-assembled monolayers,” Langmuir 18(12), 4915–4920 (2002).
[CrossRef]

N. J. Phys. (1)

M. Lewenstein and B. A. Malomed, “Spatiotemporal solitons in the Ginzburg Landau model with a two-dimensional transverse grating,” N. J. Phys. 11, 113014 (2009).
[CrossRef]

Nano Lett. (3)

Y. Lu, G. L. Liu, J. Kim, Y. X. Mejia, and L. P. Lee, “Nanophotonic crescent moon structures with sharp edge for ultrasensitive biomolecular detection by local electromagnetic field enhancement effect,” Nano Lett. 5(1), 119–124 (2005).
[CrossRef] [PubMed]

M.-R. Choi, K. J. Stanton-Maxey, J. K. Stanley, C. S. Levin, R. Bardhan, D. Akin, S. Badve, J. Sturgis, J. P. Robinson, R. Bashir, N. J. Halas, and S. E. Clare, “A cellular Trojan Horse for delivery of therapeutic nanoparticles into tumors,” Nano Lett. 7(12), 3759–3765 (2007) (and references therein).
[CrossRef] [PubMed]

D. Zhang, O. Neumann, H. Wang, V. M. Yuwono, A. Barhoumi, M. Perham, J. D. Hartgerink, P. Wittung-Stafshede, and N. J. Halas, “Gold nanoparticles can induce the formation of protein-based aggregates at physiological pH,” Nano Lett. 9(2), 666–671 (2009).
[CrossRef] [PubMed]

Nature (1)

J. F. Li, Y. F. Huang, Y. Ding, Z. L. Yang, S. B. Li, X. S. Zhou, F. R. Fan, W. Zhang, Z. Y. Zhou, Y. Wu, B. Ren, Z. L. Wang, and Z. Q. Tian, “Shell-isolated nanoparticle-enhanced Raman spectroscopy,” Nature 464(7287), 392–395 (2010).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. A (1)

D. Mihalache, D. Mazilu, F. Lederer, H. Leblond, and B. A. Malomed, “Spatiotemporal solitons in the Ginzburg Landau model with a two-dimensional transverse grating,” Phys. Rev. A 81(2), 025801 (2010).
[CrossRef]

Phys. Rev. B (1)

N. E. Christensen and B. O. Seraphin, “Relativistic band calculation and the optical properties of gold,” Phys. Rev. B 4(10), 3321–3344 (1971).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

B. B. Baizakov, A. Bouketir, A. Messikh, and B. A. Umarov, “Modulational instability in two-component discrete media with cubic-quintic nonlinearity,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(4), 046605 (2009).
[CrossRef] [PubMed]

Other (4)

G. I. Stegeman, in “Nonlinear optics of organic molecules and polymers,” p.799, edited by H. S. Nalva and S. Miyata (CRC, Boca Raton, Fl., 1997).

R. W. Boyd, “Nonlinear optics” (Academic, San Diego, 2003).

See for example: J. D. Jackson, “Classical electrodynamics” (Wiley, New York, 1998).

Corning product information data sheet for standard silica ( http://www.corning.com/docs/specialtymaterials/pisheets/H0607_hpfs_Standard_ProductSheet.pdf ).

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Figures (4)

Fig. 1
Fig. 1

Electron microscope images of the nanoparticles. (a) Silica core (average diameter: ~120 nm). (b) Silica - APTMS - gold nanoparticles (average diameter: ~160 nm). (c) and (d) Silica - gold nanoshells (average diameter: ~160 nm). Figures 1(a) to 1(c) were obtained using a 100 kV transmission electron microscope. Figure 1(d) was obtained using a 200 kV scanning electron microscope.

Fig. 2
Fig. 2

Absorption coefficient of CHCl3 in a 10 mm cuvette (red line) and extinction coefficient of SGNs - CHCl3 colloid (blue line) using CHCl3 as blank. The SGNs filling fraction is 3×10−5.

Fig. 3
Fig. 3

Thermally Managed Z-scan results. Figures 3(a) and 3(b) show Z-scan profiles obtained at t = 0.10, 0.40, 0.65, 0.95 ms; for pure chloroform and for the colloid, respectively. Figures 3(c) and 3(d) show the time evolution of the transmittance signal with the sample at positions corresponding to the minimum and maximum transmittance for pure chloroform and for the SGNs - CHCl3 colloid, respectively. The data were obtained using a 1 mm thick cuvette and intensities at the focus of 1.2 and 1.0 GW/cm2 for pure chloroform and SGNs - CHCl3 colloid, respectively.

Fig. 4
Fig. 4

Intensity dependence of | ΔTPV | / I as a function of I for t = 0.95 ms (black circles) and for t = 0 ms (blue squares). Red lines represent numerical fits to the data.

Equations (9)

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

| Δ T P V | I k 0 [ 0.396 n 2 L e f f ( 1 ) + 0.198 n 4 L e f f ( 2 ) I ] ,
P = P h + 1 V i = 1 N P p i ,
p i = ( ε h α i ) E 0 ,
α i = 3 υ i { ε s h [ ε c ( 3 2 R ) + 2 ε s h R ] ε h [ ε c R + ε s h ( 3 R ) ] ε s h [ ε c ( 3 2 R ) + 2 ε s h R ] + 2 ε h [ ε c R + ε s h ( 3 R ) } ,
ε s h ( N L ) = 3 4 χ s h ( 3 ) | E s h | 2 + 5 8 χ s h ( 5 ) | E s h | 2 2 ,
β = c ε s h + 2 d ε h 3 a ε h ,
g ( r c , r s h ) = 1 + r c 3 4 ( r s h r r c ) ( 1 r c 2 1 r s h 2 ) ( a b + a b ) | a | 2 + r c 6 2 ( r s h r r c ) ( 1 r c 5 1 r s h 5 ) | b | 2 | a | 2 ,
χ e f f ( 3 ) = χ h ( 3 ) + f R g ( r c , r s h ) | β | 2 β 2 c ε c + 2 d ε s h a 2 χ s h ( 3 ) ,
χ e f f ( 5 ) = f R [ g ( r c , r s h ) ] 2 | β | 4 β 2 { ( c ε c + 2 d ε s h a 2 ) χ s h ( 5 )             3 10 [ c ( c ε c + 2 d ε s h ) + 18 ε c ( R 1 ) ( ε s h 2 + 2 ε c ε h ) a 2 ( c ε s h + 2 d ε h ) ] [ χ s h ( 3 ) ] 2 } ,

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