Abstract

Nonlinear optical absorption of Au-Ag nanoshuttles (NSs) was studied using an open-aperture Z-scan experiment with a 532 nm nanosecond laser at different energies. It was found that, when the laser energy is relatively low, the Au-Ag NSs exhibit saturated absorption (SA). When the laser energy is high, a conversion from SA to reverse saturated absorption (RSA) occurs. The ultrafast dynamic process of Au-Ag NSs was also investigated by using a femtosecond pump-probe technique. It is found that the process contains a fast and slow decay component that depends strongly on the laser intensity. Furthermore, when the probe wavelength is far away from the plasma resonance peak, the decay shows modulation due to the vibration mode of the coherent excitation.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Due to the strong local surface plasmon resonance (LSPR) effect [15], gold and silver nanoparticles have shown potential applications in drug research, biological detection, cell labeling, site-specific diagnosis, molecular dynamics research and disease diagnosis in recent years. Their optical nonlinearity has aroused great interest. Current reports show that people have carried out extensive research on nanospheres [68], nanorods [918], nanotriangles [1921], nanorings, nanostars, nanocages [2224], film [25], and achieved gratifying results, such as green synthesis method, gas phase detection, application of photothermal therapy and an anti-inflammatory prodrug, biosensor, combined cancer photothermal-chemotherapy, etc. Current studies have shown that silver nanoparticles have high refractive index sensitivity, unstable chemical properties, easy oxidation and vulcanization; gold nanoparticles have slightly lower refractive index sensitivity, but their chemical properties are very stable. How to control the SPR absorption peak with high refractive index sensitivity is of great significance. Because gold nanorods have two absorption peaks, their longitudinal absorption peaks can be controlled from visible to infrared bands. Nanoparticles with tip structure can cause significant enhancement of electric field, therefore it is important to study the optical properties of nanorods and spindle-like structures with tip structure. In 2014, Tingting Bai et al. developed a convenient and reliable approach for the synthesis of Au/Ag NSs, and they found that, these Au/Ag NSs possessed higher refractive index sensitivity as well as excellent SERS activity compared with the original Au nanorods [26]. Up to now, there are few reports on the optical nonlinearity of NSs. In 2008, X. Zhang et al. researched the optical properties of Au/Ag core/shell NSs using a Ti: sapphire laser (Mira 900,Coherent) with a pulse width of 2.5 ps and a repetition rate of 76 MHz. They found that, at the corresponding LSPR wavelengths, the extinction cross section and nonlinear refraction of the Au/Ag NSs are about 1.5 and 8.0 times of those of original Au nanorods, respectively [27]. However, the nonlinear absorption (NLA) of Au-Ag NSs has not been studied yet. In the paper, we investigated the nonlinear absorption of Au-Ag NSs using Z-scan experiments, measured the SA intensity and RSA coefficient of materials, and analyzed the origins of material nonlinearity. Moreover, we investigate the ultrafast dynamics process of Au-Ag NSs with femtosecond transient absorption measurements.

2. Sample and experiments

The Au-Ag NSs aqueous solution with a concentration of 0.05 mg/ml used in our experiments were obtained from Nanjing XFNANO Materials Tech Co. Ltd. The Au-Ag NSs were examined by scanning electron microscopy (SEM). The linear absorption spectra of Au-Ag NSs were measured by dual-beam ultraviolet-visible spectrophotometer.

The nonlinear absorption of Au-Ag NSs was studied by using open-aperture Z-scan technique [28,29]. For Z-scan measurements, a nanosecond Nd:YAG laser (6 ns pulse duration operated at 10 Hz) with a wavelength of 532 nm was used. The thickness of the sample cell used in the experiment is 2 mm, the linear transmittance of the sample is 44.6%, laser energy varies from 50 to 750 uJ. The beam waist radius at focal point is about 68 um obtained by using blade method. The incident and transmitted laser pulses for each z point were recorded by a computer.

The ultrafast dynamics of Au-Ag NSs were studied using pump-probe technique. The experiments were carried out using Yb: KGW femtosecond laser with repetition frequency of 6 KHz and pulse width of 190 fs under different laser power. The output of the laser was split into two beams. The main part of the output passed through a 1 mm thick BBO crystal, and the generated 400 nm wavelength pulses were used as the pump beam. The other beam through the delay system passed through a 2 mm thick Ti: sapphire plate to produce supercontinuum white light (450 nm-755 nm) used as the probe beam.The output signal of the photodetector was input into the Lock-in amplifier. After the Lock-in processing the signal was input to the computer for direct observation.

3. Result and discussion

The characteristics of the sample are shown in Fig. 1. As shown in Fig. 1(a), the sample shows khaki color, which is due to the selective absorption of visible light by the sample. The SEM image is shown in Fig. 1(b), we can observe that each end of the Au-Ag NSs particle has a sharp angle, the middle is rod-shaped, the whole is shuttle-shaped, the size distribution is relatively uniform, the diameter of the Au-Ag NSs is 16.6 nm, and the length of the shuttle is 49 nm. The linear absorption spectrum of Au-Ag NSs is shown in Fig. 1(c). We can find that there are two SPR absorption peaks at the wavelength of 500 nm and 800 nm, respectively.

 figure: Fig. 1.

Fig. 1. The characteristics of Au-Ag NSs. (a) Appearance color, (b) SEM image, (c) Linear absorption spectra.

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Open-aperture nanosecond Z-scan measurements were conducted to investigate the nonlinear absorption of Au-Ag NSs. In the experiment, a low repetition rate of 10 Hz is chosen, thus the thermal effect can be ignored [30,31].

As shown in Fig. 2(a), when laser energy is moderate and increases from50 uJ to 450uJ, the normalized transmittance of Au-Ag NSs at four different energies increases gradually when sample comes towards focal point (z=0), indicating that the sample shows SA. At moderate irradiance, electrons are pumped to the excited state, resulting in a smaller number of the ground state, which is called bleaching of ground state plasma resulting in the relatively weak absorption ability of the samples to external photons at relatively low energies. The SA is relatively strong under higher energies (E = 150 uJ, 350 uJ, 450 uJ). This is because with the increase of laser energy, the number of particles excited to the excited state increases, even all of them are excited to the excited state, which leads to the decrease of the absorption of incident photons. When laser energy is moderate, the bleaching of ground-state plasmon band results in SA. The same phenomenon has also been found in the nonlinear measurement of gold nanorods [32].As shown in Fig. 2(b), under relatively higher intensities (700 uJ and 750 uJ), the sample shows the characteristic of conversion from SA to RSA, and when the laser energy is 750 uJ, the conversion is more obvious. The SA and RSA are related to the interplay of plasmon band bleaching and free-carrier absorption, respectively.

 figure: Fig. 2.

Fig. 2. Normalized transmission of Au-Ag NSs for open-aperture Z-scan at different energies of (a) 50 uJ, 150 uJ, 350 uJ, 450 uJ and (b) 700uJ, 750 uJ. The dots are experimental data while the solid lines are theoretical fit.

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From the results above, we can find that, there two composite nonlinear absorptions with opposite signs in Au-Ag NSs. The absorption can be defined as:

$$\alpha (I )= \frac{{{\alpha _0}}}{{1 + ({I/{I_s}} )}} + \beta I$$
where ${\alpha _0}$ is the linear absorption coefficient of the material, I is the laser intensity, IS is the saturation intensity and β is the nonlinear absorption coefficient. As we know that I can be expressed as:
$$I = \frac{{{I_0}}}{{1 + {{{z^2}} / {z_0^2}}}}$$
I0 is the laser intensity at focal point. So Eq. (1) can be denoted further as:
$$\alpha ({I_0}) = \frac{{{\alpha _0}}}{{1 + \frac{{{I_0}}}{{(1 + {{{z^2}} / {z_0^2}}){I_s}}}}} + \frac{{\beta {I_0}}}{{1 + {{{z^2}} / {z_0^2}}}}$$

When there is only two-photon absorption in open-aperture Z-scan experiment, the normalized transmittance is [28]:

$$T(z) = \sum\limits_{m = 0}^\infty {\frac{{{{[\frac{{ - \beta {I_0}{L_{eff}}}}{{(1 + {{{z^2}} / {z_0^2}})}}]}^m}}}{{{{(m + 1)}^{{3 / 2}}}}}}$$
where ${L_{eff}} = (1 - {e^{ - {\alpha _0}l}})/{\alpha _0}$, ${L_{eff}}$ is the effective interaction length. Thus, a theoretical fit to the experimental data could be conducted by replacing ${{\beta I} / {(1 + {{{z^2}} / {z_0^2}}}})$ in Eq. (4) by Eq. (3). The solid lines in Fig. 2 are theoretical fit. We can find that the theoretical fit agrees well with the experimental results. As shown in Table 1, the saturation strength ${I_s}$ and nonlinear absorption coefficient β can be obtained by theoretical fit.

Tables Icon

Table 1. Fitting results of experimental data

The transient absorption spectra obtained under the action of 400nm femtosecond pump laser pulse with energy of 13mW are present in Fig. 3(a) using surface plot. It can be seen from Fig. 3(a) that the region with obvious changes in absorbance is concentrated in the wavelength range of 475-525nm, and the strongest absorption peak is at about 500nm, corresponding to the linear absorption peak of the Au-Ag NSs in Fig. 1(c). Figure 3(b) shows the transient absorption spectra at different delay times after the excitation at 400nm. Immediately, after the excitation, the bleaching signal of ground state plasma is seen at about 493nm. Also, a transient absorption at about 562nm is observed, which is a photo-induced absorption signal. In Fig. 3(b), the transient absorption peak at 562nm have a spectral shift in respect to time delay, it is a photo-induced absorption signal, in the transient spectra may be that the spectral shift around 592nm is due to the plasmon band of hot electrons [31]. After absorption of a short laser pulse, only a little part of plasma electrons are excited to higher energy states, most of the plasma electrons are still in the ground state, which leads to the increase of their selective absorption of incident photons. While for bleaching signal at 493nm, almost all of the plasmon band of hot electrons are excited to the higher energy state, and the few hot electrons left in the ground state absorb very little incident photons, thus, the bleaching signal at 493nm has no spectral shift. Then with the increase of time delay between the pump and probe beams, the bleached spectra recovers and transient absorption decays.

 figure: Fig. 3.

Fig. 3. (a) Two-dimensional contour map of Au-Ag NSs pump-probe data.(b)Time resolved transient absorption data of Au-Ag NSs.(c) Dynamic traces of Au-Ag NSs at wavelength of 493 nm.(d) Dynamic traces of Au-Ag NSs at wavelength of 562 nm. (Note:ΔmOD is the optical density difference

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Figure 3(c) shows the dynamic traces of recovering plasmon maximum probed at peak wavelength of 493 nm under different pump powers of 8 mW, 10 mW and 13 mW. All curves in Fig. 3(c) display a rapid descent and decay processes. The rapid descent process is due to the ground state plasma bleaching of the system, which is also called electron-electron scattering with time-scale of a few hundred fs. Because of the limited time resolution (about ps) of our system, the process cannot be accurately determined. The decay includes a fast process and a slow one. The former results from the balance of excited electrons with the nanoparticle lattice through electron-phonon interaction. The latter (subsequent slower decay) is ascribed to phonon-phonon interaction with the surrounding medium. The latter (subsequent slower decay) was ascribed to phonon-phonon interaction

In order to analyse the process of decay of Au-Ag NSs quantitatively, we use double-exponential functions shown in Eq. (5) to fit the photodynamic curves.

$$\frac{{\Delta T}}{T} = {A_1}\exp ( - \frac{t}{{{\tau _1}}}) + {A_2}\exp ( - \frac{t}{{{\tau _2}}})$$
where ${A_1}$, ${A_2}$ are the amplitudes of two decay components, and ${\tau _1}$ and ${\tau _2}$ represent time constants of the two decay components, respectively. The fitting results to the experimental data are shown in Fig. 3(c) using solid lines. From the fitting, initial fast relaxation and slow relaxation times at different powers are obtained and shown in Table 2. We can find that, the relaxation times in Au-Ag NSs is longer than that reported in Au nanospheres (${\tau _1}$=2.5 ps, ${\tau _2}$=50 ps) because Au-Ag NSs have higher initial temperatures of hot electrons [33,34], which results from different SPR depending on shape and metal.

Tables Icon

Table 2. Fitting results for the decay processes for Au-Ag NSs at different powers.

Figure 3(d) shows the transient absorption dynamic traces of Au-Ag NSs probed at 562 nm under different pump powers of 8 mW, 10 mW and 13 mW. Similarly, the decay includes a fast process and a slow one which can be described quantitatively by fitting the data using Eq. (5). Correspondingly, the fast and slow decay times obtained are shown in Table 3.

Tables Icon

Table 3. Fitting results for the decay processes for Au-Ag NSs at different powers.

From Table 2 and 3, we can find that the decay time is intensity-dependent, and increases with the laser intensity. This is because the hot electron exchanges energy with phonon sub-systems in a way related to their temperature differences [35]. The higher the energy, the more electrons are in the higher electronic states, and the longer the time required for electrons to transfer energy to the phonons.

We also extracted dynamics curves at a non-resonant wavelength of 510 nm from the contour plot shown in Fig. 3(a). As seen in Fig. 4, we found that the dynamics curve show a modulation response on the basis of exponential decay. The phenomenon can be analyzed as follows. After ultrafast excitation, energy flows out of the electrons and enters the lattice within a few picoseconds, causing a rise in the lattice temperature, resulting in a small amount of expansion. We consider the transient absorption process as the microscopic process of nano particles, the micro mechanism has to be explained as a rapid descent and decay processes. The rapid descent process is electron-electron scattering. The decay includes a fast process and a slow one. The former results was electron-phonon interaction. The latter (subsequent slower decay) was ascribed to phonon-phonon interaction [33]. The heating time is faster than the vibration mode period associated with the expansion coordinate, so it can coherently excite the modes of the particles, which produces modulations in transient absorption traces [36,37].

 figure: Fig. 4.

Fig. 4. Dynamic traces of Au-Ag NSs at 510 nm (The dashed line is experimental data while the solid line is theoretical fit generated).

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The transient absorption traces is fitted using a damped cosine function plus an exponential decaying background as [38]:

$$S(t )= A\cos \left( {\frac{{2\pi t}}{T} + \varphi } \right){e^{ - \frac{t}{{{\tau _v}}}}} + {A_1}{e^{ - \frac{t}{{{\tau _1}}}}} + {A_2}{e^{ - \frac{t}{{{\tau _2}}}}} + B$$
where $T$ is the vibrational period, $\varphi$ is the phase for the vibration, and ${\tau _v}$ is the vibrational damping time. Where${A_1}$, ${A_2}$ are the amplitudes of two decay components, ${\tau _1}$ and ${\tau _2}$ represent time constants of the two decay components, respectively. The fitting to the experimental data is shown in Fig. 4. The vibrational period obtained for this trace was 18.5 ps. The electron-phonon coupling and heat dissipation investigation are very important for the applications of thermally or electrically conductive particles. .

4. Conclusion

In summary, we have investigated the nonlinear absorption of Au-Ag NSs. The results show that the nonlinear absorption is intensity-dependent. At low laser intensities, Au-Ag NSs exhibit SA, while RSA at high laser intensities. Besides, we have investigated the ultrafast dynamics process of Au-Ag NSs. We found that the relaxation processes depend on the laser intensity. When the power of pump laser is 10mW, the fast decay time and slow decay time are 5.6 ps and 180 ps, respectively. When probe wavelength is away from the plasma resonance peak, the decay of relaxation also shows a modulation due to the vibration mode of the coherent excitation. The vibrational period is about 18.5 ps.

Funding

Harbin University of Commerce (17XN035); Natural Science Foundation of Heilongjiang Province (F2018027).

Acknowledgments

Statistical support was provided by Shuang Chen.

Disclosures

The authors declare no conflicts of interest.

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References

  • View by:

  1. A. Agrawal, S. H. Cho, O. Zandi, S. Ghosh, R. W. Johns, and D. J. Milliron, “Localized Surface Plasmon Resonance in Semiconductor Nanocrystals,” Chem. Rev. 118(6), 3121–3207 (2018).
    [Crossref]
  2. O. Nicoletti, F. D. L. Pena, R. K. Leary, D. J. Holland, C. Ducati, and P. A. Midgley, “Three-dimensional imaging of localized surface plasmon resonances of metal nanoparticles,” Nature 502(7469), 80–84 (2013).
    [Crossref]
  3. L. F. Qiao, D. Wang, L. J. Zuo, Y. Q. Ye, J. Qian, H. Z. Chen, and S. L. He, “Localized surface plasmon resonance enhanced organic solar cell with gold nanospheres,” Appl. Energy 88(3), 848–852 (2011).
    [Crossref]
  4. B. Sepúlveda, P. C. Angelomé, L. M. Lechuga, and L. M. Liz-Marzán, “LSPR-based nanobiosensors,” Nano Today 4(3), 244–251 (2009).
    [Crossref]
  5. K. M. Mayer, S. Lee, H. Liao, B. C. Rostro, A. Fuentes, P. T. Scully, C. L. Nehl, and J. H. Hafner, “A label-free immunoassay based upon localized surface plasmon resonance of gold nanorods,” ACS Nano 2(4), 687–692 (2008).
    [Crossref]
  6. N. Jayaprakash, J. J. Vijaya, K. Kaviyarasu, K. Kombaiah, L. J. Kennedy, R. J. Ramalingam, M. A. Munusamy, and H. A. Al-Lohedan, ““Green synthesis of Ag nanoparticles using Tamarind fruit extract for the antibacterial studies” J,” J. Photochem. Photobiol., B 169, 178–185 (2017).
    [Crossref]
  7. D. Mandler, T. Shahar, G. Feldheim, and S. Marx, “Core-shell nanoparticles for gas phase detection based on silver nanospheres coated with a thin molecularly imprinted polymer adsorbed on a chemiresistor”,” Nanoscale 10(37), 17593–17602 (2018).
    [Crossref]
  8. P. Z. El-Khoury, E. Khon, Y. Gong, A. G. Joly, P. Abellan, and J. E. Evans, “Electric field enhancement in a self-assembled 2D array of silver nanospheres,” J. Chem. Phys. 141(21), 214308 (2014).
    [Crossref]
  9. Q. Dong, X. W. Wang, X. X. Hu, L. Q. Xiao, L. Zhang, L. J. Song, M. L. Xu, Y. X. Zou, L. Chen, Z. Chen, and W. H. Tan, ““Simultaneous Application of Photothermal Therapy and an Anti-inflammatory Prodrug using Pyrene-Aspirin-Loaded Gold Nanorod Graphitic Nanocapsules” Angew,” Angew. Chem., Int. Ed. 57(1), 177–181 (2018).
    [Crossref]
  10. M. Azimzadeh, M. Rahaie, N. Nasirizadeh, K. Ashtari, and H. Naderi-Manesh, “An electrochemical nanobiosensor for plasma miRNA-155, based on graphene oxide and gold nanorod, for early detection of breast cancer,” Biosens. Bioelectron. 77, 99–106 (2016).
    [Crossref]
  11. J. F. Liao, W. T. Li, J. R. Peng, Q. Yang, H. Li, Y. Q. Wei, X. N. Zhang, and Z. Y. Qian, “Combined Cancer Photothermal-Chemotherapy Based on Doxorubicin/Gold Nanorod-Loaded Polymersomes,” Theranostics 5(4), 345–356 (2015).
    [Crossref]
  12. M. F. Tsai, S. H. G. Chang, F. Y. Cheng, V. Shanmugam, Y. S. Cheng, C. H. Su, and C. S. Yeh, “Au Nanorod Design as Light-Absorber in the First and Second Biological Near-Infrared Windows for in Vivo Photothermal Therapy”,” ACS Nano 7(6), 5330–5342 (2013).
    [Crossref]
  13. P. Zijlstra, P. M. R. Paulo, and M. Orrit, “Optical detection of single non-absorbing molecules using the surface plasmon resonance of a gold nanorod,” Nat. Nanotechnol. 7(6), 379–382 (2012).
    [Crossref]
  14. B. M. Ma, P. Li, X. H. Chen, L. L. Wang, B. H. Liu, and T. Song, “Gold nano-triangles as saturable absorbers for a dual-wavelength passively Q-switched Nd:GYSGG laser,” Laser Phys. 28(7), 075802 (2018).
    [Crossref]
  15. J. F. Huang, Y. H. Zhu, C. X. Liu, Y. F. Zhao, Z. H. Liu, M. N. Hedhili, A. Fratalocchi, and Y. Han, “Fabricating a homogeneously alloyed AuAg shell on Au nanorods to achieve strong, stable, and tunable surface plasmon resonances,” Small 11(39), 5214–5221 (2015).
    [Crossref]
  16. H. W. Dai, L. M. Zhang, Z. W. Wang, X. Wang, J. P. Zhang, H. M. Gong, J. Han, and Y. B. Han, “Linear and nonlinear optical properties of silver-coated gold nanorods,” J. Phys. Chem. C 121(22), 12358–12364 (2017).
    [Crossref]
  17. T. Gan, J. B. Li, H. X. Li, Y. X. Liu, and Z. H. Xu, “Synthesis of Au nanorod-embedded and graphene oxide-wrapped microporous ZIF-8 with high electrocatalytic activity for the sensing of pesticides,” Nanoscale 11(16), 7839–7849 (2019).
    [Crossref]
  18. Y. Liu, F. Zhou, H. C. Wang, and Y. D. Wei, “Au-nanorod-clusters patterned optical fiber SERS probes fabricated by laser-induced evaporation self-assembly method,” Opt. Express 28(5), 6648 (2020).
    [Crossref]
  19. G. Fletcher, M. D. Arnold, T. Pedersen, V. J. Keast, and M. B. Cortie, ““Multipolar and dark-mode plasmon resonances on drilled silver nano-triangles” Opt,” Opt. Express 23(14), 18002 (2015).
    [Crossref]
  20. R. Dharmatti, C. Phadke, A. Mewada, M. Thakur, S. Pandey, and M. Sharon, ““Biogenic gold nano-triangles: Cargos for anticancer drug delivery” Mat,” Mater. Sci. Eng., C 44, 92–98 (2014).
    [Crossref]
  21. L. E. Hennemann, A. Kolloch, A. Kern, J. Mihaljevic, J. Boneberg, P. Leiderer, A. J. Meixner, and D. Zhang, “Assessing the plasmonics of gold nano-triangles with higher order laser modes,” Beilstein J. Nanotechnol. 3, 674–683 (2012).
    [Crossref]
  22. S. E. Skrabalak, J. Y. Chen, Y. G. Sun, X. M. Lu, L. Au, C. M. Cobley, and Y. N. Xia, ““Gold Nanocages: Synthesis, Properties, and Applications” Acc,” Acc. Chem. Res. 41(12), 1587–1595 (2008).
    [Crossref]
  23. H. P. Sun, J. H. Su, Q. S. Meng, Q. Yin, L. L. Chen, W. W. Gu, Z. W. Zhang, H. J. Yu, P. C. Zhang, S. L. Wang, and Y. P. Li, “Cancer Cell Membrane-Coated Gold Nanocages with Hyperthermia-Triggered Drug Release and Homotypic Target Inhibit Growth and Metastasis of Breast Cancer,” Adv. Funct. Mater. 27(3), 1604300 (2017).
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  24. C. Zheng, J. X. Huang, L. Lei, W. Z. Chen, H. Y. Wang, and W. Li, “Nanosecond nonlinear optical and optical limiting properties of hollow gold nanocages,” Appl. Phys. B 124(1), 17 (2018).
    [Crossref]
  25. S. Abedin, J. Kenison, C. Vargas, and E. O. Potma, “Sensing biomolecular interactions by the luminescence of a planar gold film,” Anal. Chem. 91(24), 15883–15889 (2019).
    [Crossref]
  26. T. T. Bai, J. F. Sun, R. C. Che, L. N. Xu, C. Y. Yin, Z. R. Guo, and N. Gu, “Controllable Preparation of Core-Shell Au-Ag Nanoshuttles with Improved Refractive Index Sensitivity and SERS Activity,” ACS Appl. Mater. Interfaces 6(5), 3331–3340 (2014).
    [Crossref]
  27. M. Li, Z. S. Zhang, X. Zhang, K. Y. Li, and X. F. Yu, “Optical properties of Au/Ag core/shell nanoshuttles” Opt,” Opt. Express 16(18), 14288–14293 (2008).
    [Crossref]
  28. 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]
  29. D. D. Smith, Y. Yoon, R. W. Boyd, J. K. Campbell, L. A. Baker, R. M. Crooks, and M. A. George, “Z-scan measurement of the nonlinear absorption of a thin gold film,” J. Appl. Phys. 86(11), 6200–6205 (1999).
    [Crossref]
  30. R. West, Y. Wang, and T. Goodson, “Nonlinear absorption properties in novel gold nanostructured topologies,” J. Phys. Chem. B 107(15), 3419–3426 (2003).
    [Crossref]
  31. J. Li, S. Liu, Y. Liu, F. Zhou, and Z. Y. Li, “Anisotropic and enhanced absorptive nonlinearities in a macroscopic film induced by aligned gold nanorods,” Appl. Phys. Lett. 96(26), 263103 (2010).
    [Crossref]
  32. Y. C. Gao, X. R. Zhang, Y. L. Li, H. F. Liu, Y. X. Wang, Q. C. Chang, W. Y. Jiao, and Y. L. Song, “Saturable absorption and reverse saturable absorption in platinum nanoparticles,” Opt. Commun. 251(4-6), 429–433 (2005).
    [Crossref]
  33. T. S. Ahmadi, S. L. Logunov, L. Stephan, and M. A. El-Sayed, ““Picosecond Dynamics of Colloidal Gold Nanoparticles” J,” J. Phys. Chem. 100(20), 8053–8056 (1996).
    [Crossref]
  34. G. V. Hartland, “Measurements of the material properties of metal nanoparticles by time-resolved spectroscopy,” Phys. Chem. Chem. Phys. 6(23), 5263–5274 (2004).
    [Crossref]
  35. G. V. Hartland, “Coherent vibrational motion in metal particles: Determination of the vibrational amplitude and excitation mechanism,” J. Chem. Phys. 116(18), 8048–8055 (2002).
    [Crossref]
  36. G. V. Hartland, “Optical studies of dynamics in noble metal nanostructures,” Chem. Rev. 111(6), 3858–3887 (2011).
    [Crossref]
  37. J. H. Hodak, A. Henglein, and G. V. Hartland, “Photophysics of nanometer sized metal particles: electron-phonon coupling and coherent excitation of breathing vibrational modes,” J. Phys. Chem. B 104(43), 9954–9965 (2000).
    [Crossref]
  38. M. Hu, H. Petrova, J. Y. Chen, J. M. McLellan, A. R. Siekkinen, M. Marquez, X. D. Li, Y. N. Xia, and G. V. Hartland, “Ultrafast laser studies of the photothermal properties of gold nanocages,” J. Phys. Chem. B 110(4), 1520–1524 (2006).
    [Crossref]

2020 (1)

2019 (2)

T. Gan, J. B. Li, H. X. Li, Y. X. Liu, and Z. H. Xu, “Synthesis of Au nanorod-embedded and graphene oxide-wrapped microporous ZIF-8 with high electrocatalytic activity for the sensing of pesticides,” Nanoscale 11(16), 7839–7849 (2019).
[Crossref]

S. Abedin, J. Kenison, C. Vargas, and E. O. Potma, “Sensing biomolecular interactions by the luminescence of a planar gold film,” Anal. Chem. 91(24), 15883–15889 (2019).
[Crossref]

2018 (5)

C. Zheng, J. X. Huang, L. Lei, W. Z. Chen, H. Y. Wang, and W. Li, “Nanosecond nonlinear optical and optical limiting properties of hollow gold nanocages,” Appl. Phys. B 124(1), 17 (2018).
[Crossref]

D. Mandler, T. Shahar, G. Feldheim, and S. Marx, “Core-shell nanoparticles for gas phase detection based on silver nanospheres coated with a thin molecularly imprinted polymer adsorbed on a chemiresistor”,” Nanoscale 10(37), 17593–17602 (2018).
[Crossref]

B. M. Ma, P. Li, X. H. Chen, L. L. Wang, B. H. Liu, and T. Song, “Gold nano-triangles as saturable absorbers for a dual-wavelength passively Q-switched Nd:GYSGG laser,” Laser Phys. 28(7), 075802 (2018).
[Crossref]

A. Agrawal, S. H. Cho, O. Zandi, S. Ghosh, R. W. Johns, and D. J. Milliron, “Localized Surface Plasmon Resonance in Semiconductor Nanocrystals,” Chem. Rev. 118(6), 3121–3207 (2018).
[Crossref]

Q. Dong, X. W. Wang, X. X. Hu, L. Q. Xiao, L. Zhang, L. J. Song, M. L. Xu, Y. X. Zou, L. Chen, Z. Chen, and W. H. Tan, ““Simultaneous Application of Photothermal Therapy and an Anti-inflammatory Prodrug using Pyrene-Aspirin-Loaded Gold Nanorod Graphitic Nanocapsules” Angew,” Angew. Chem., Int. Ed. 57(1), 177–181 (2018).
[Crossref]

2017 (3)

H. P. Sun, J. H. Su, Q. S. Meng, Q. Yin, L. L. Chen, W. W. Gu, Z. W. Zhang, H. J. Yu, P. C. Zhang, S. L. Wang, and Y. P. Li, “Cancer Cell Membrane-Coated Gold Nanocages with Hyperthermia-Triggered Drug Release and Homotypic Target Inhibit Growth and Metastasis of Breast Cancer,” Adv. Funct. Mater. 27(3), 1604300 (2017).
[Crossref]

N. Jayaprakash, J. J. Vijaya, K. Kaviyarasu, K. Kombaiah, L. J. Kennedy, R. J. Ramalingam, M. A. Munusamy, and H. A. Al-Lohedan, ““Green synthesis of Ag nanoparticles using Tamarind fruit extract for the antibacterial studies” J,” J. Photochem. Photobiol., B 169, 178–185 (2017).
[Crossref]

H. W. Dai, L. M. Zhang, Z. W. Wang, X. Wang, J. P. Zhang, H. M. Gong, J. Han, and Y. B. Han, “Linear and nonlinear optical properties of silver-coated gold nanorods,” J. Phys. Chem. C 121(22), 12358–12364 (2017).
[Crossref]

2016 (1)

M. Azimzadeh, M. Rahaie, N. Nasirizadeh, K. Ashtari, and H. Naderi-Manesh, “An electrochemical nanobiosensor for plasma miRNA-155, based on graphene oxide and gold nanorod, for early detection of breast cancer,” Biosens. Bioelectron. 77, 99–106 (2016).
[Crossref]

2015 (3)

J. F. Liao, W. T. Li, J. R. Peng, Q. Yang, H. Li, Y. Q. Wei, X. N. Zhang, and Z. Y. Qian, “Combined Cancer Photothermal-Chemotherapy Based on Doxorubicin/Gold Nanorod-Loaded Polymersomes,” Theranostics 5(4), 345–356 (2015).
[Crossref]

J. F. Huang, Y. H. Zhu, C. X. Liu, Y. F. Zhao, Z. H. Liu, M. N. Hedhili, A. Fratalocchi, and Y. Han, “Fabricating a homogeneously alloyed AuAg shell on Au nanorods to achieve strong, stable, and tunable surface plasmon resonances,” Small 11(39), 5214–5221 (2015).
[Crossref]

G. Fletcher, M. D. Arnold, T. Pedersen, V. J. Keast, and M. B. Cortie, ““Multipolar and dark-mode plasmon resonances on drilled silver nano-triangles” Opt,” Opt. Express 23(14), 18002 (2015).
[Crossref]

2014 (3)

R. Dharmatti, C. Phadke, A. Mewada, M. Thakur, S. Pandey, and M. Sharon, ““Biogenic gold nano-triangles: Cargos for anticancer drug delivery” Mat,” Mater. Sci. Eng., C 44, 92–98 (2014).
[Crossref]

P. Z. El-Khoury, E. Khon, Y. Gong, A. G. Joly, P. Abellan, and J. E. Evans, “Electric field enhancement in a self-assembled 2D array of silver nanospheres,” J. Chem. Phys. 141(21), 214308 (2014).
[Crossref]

T. T. Bai, J. F. Sun, R. C. Che, L. N. Xu, C. Y. Yin, Z. R. Guo, and N. Gu, “Controllable Preparation of Core-Shell Au-Ag Nanoshuttles with Improved Refractive Index Sensitivity and SERS Activity,” ACS Appl. Mater. Interfaces 6(5), 3331–3340 (2014).
[Crossref]

2013 (2)

M. F. Tsai, S. H. G. Chang, F. Y. Cheng, V. Shanmugam, Y. S. Cheng, C. H. Su, and C. S. Yeh, “Au Nanorod Design as Light-Absorber in the First and Second Biological Near-Infrared Windows for in Vivo Photothermal Therapy”,” ACS Nano 7(6), 5330–5342 (2013).
[Crossref]

O. Nicoletti, F. D. L. Pena, R. K. Leary, D. J. Holland, C. Ducati, and P. A. Midgley, “Three-dimensional imaging of localized surface plasmon resonances of metal nanoparticles,” Nature 502(7469), 80–84 (2013).
[Crossref]

2012 (2)

P. Zijlstra, P. M. R. Paulo, and M. Orrit, “Optical detection of single non-absorbing molecules using the surface plasmon resonance of a gold nanorod,” Nat. Nanotechnol. 7(6), 379–382 (2012).
[Crossref]

L. E. Hennemann, A. Kolloch, A. Kern, J. Mihaljevic, J. Boneberg, P. Leiderer, A. J. Meixner, and D. Zhang, “Assessing the plasmonics of gold nano-triangles with higher order laser modes,” Beilstein J. Nanotechnol. 3, 674–683 (2012).
[Crossref]

2011 (2)

L. F. Qiao, D. Wang, L. J. Zuo, Y. Q. Ye, J. Qian, H. Z. Chen, and S. L. He, “Localized surface plasmon resonance enhanced organic solar cell with gold nanospheres,” Appl. Energy 88(3), 848–852 (2011).
[Crossref]

G. V. Hartland, “Optical studies of dynamics in noble metal nanostructures,” Chem. Rev. 111(6), 3858–3887 (2011).
[Crossref]

2010 (1)

J. Li, S. Liu, Y. Liu, F. Zhou, and Z. Y. Li, “Anisotropic and enhanced absorptive nonlinearities in a macroscopic film induced by aligned gold nanorods,” Appl. Phys. Lett. 96(26), 263103 (2010).
[Crossref]

2009 (1)

B. Sepúlveda, P. C. Angelomé, L. M. Lechuga, and L. M. Liz-Marzán, “LSPR-based nanobiosensors,” Nano Today 4(3), 244–251 (2009).
[Crossref]

2008 (3)

K. M. Mayer, S. Lee, H. Liao, B. C. Rostro, A. Fuentes, P. T. Scully, C. L. Nehl, and J. H. Hafner, “A label-free immunoassay based upon localized surface plasmon resonance of gold nanorods,” ACS Nano 2(4), 687–692 (2008).
[Crossref]

S. E. Skrabalak, J. Y. Chen, Y. G. Sun, X. M. Lu, L. Au, C. M. Cobley, and Y. N. Xia, ““Gold Nanocages: Synthesis, Properties, and Applications” Acc,” Acc. Chem. Res. 41(12), 1587–1595 (2008).
[Crossref]

M. Li, Z. S. Zhang, X. Zhang, K. Y. Li, and X. F. Yu, “Optical properties of Au/Ag core/shell nanoshuttles” Opt,” Opt. Express 16(18), 14288–14293 (2008).
[Crossref]

2006 (1)

M. Hu, H. Petrova, J. Y. Chen, J. M. McLellan, A. R. Siekkinen, M. Marquez, X. D. Li, Y. N. Xia, and G. V. Hartland, “Ultrafast laser studies of the photothermal properties of gold nanocages,” J. Phys. Chem. B 110(4), 1520–1524 (2006).
[Crossref]

2005 (1)

Y. C. Gao, X. R. Zhang, Y. L. Li, H. F. Liu, Y. X. Wang, Q. C. Chang, W. Y. Jiao, and Y. L. Song, “Saturable absorption and reverse saturable absorption in platinum nanoparticles,” Opt. Commun. 251(4-6), 429–433 (2005).
[Crossref]

2004 (1)

G. V. Hartland, “Measurements of the material properties of metal nanoparticles by time-resolved spectroscopy,” Phys. Chem. Chem. Phys. 6(23), 5263–5274 (2004).
[Crossref]

2003 (1)

R. West, Y. Wang, and T. Goodson, “Nonlinear absorption properties in novel gold nanostructured topologies,” J. Phys. Chem. B 107(15), 3419–3426 (2003).
[Crossref]

2002 (1)

G. V. Hartland, “Coherent vibrational motion in metal particles: Determination of the vibrational amplitude and excitation mechanism,” J. Chem. Phys. 116(18), 8048–8055 (2002).
[Crossref]

2000 (1)

J. H. Hodak, A. Henglein, and G. V. Hartland, “Photophysics of nanometer sized metal particles: electron-phonon coupling and coherent excitation of breathing vibrational modes,” J. Phys. Chem. B 104(43), 9954–9965 (2000).
[Crossref]

1999 (1)

D. D. Smith, Y. Yoon, R. W. Boyd, J. K. Campbell, L. A. Baker, R. M. Crooks, and M. A. George, “Z-scan measurement of the nonlinear absorption of a thin gold film,” J. Appl. Phys. 86(11), 6200–6205 (1999).
[Crossref]

1996 (1)

T. S. Ahmadi, S. L. Logunov, L. Stephan, and M. A. El-Sayed, ““Picosecond Dynamics of Colloidal Gold Nanoparticles” J,” J. Phys. Chem. 100(20), 8053–8056 (1996).
[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]

Abedin, S.

S. Abedin, J. Kenison, C. Vargas, and E. O. Potma, “Sensing biomolecular interactions by the luminescence of a planar gold film,” Anal. Chem. 91(24), 15883–15889 (2019).
[Crossref]

Abellan, P.

P. Z. El-Khoury, E. Khon, Y. Gong, A. G. Joly, P. Abellan, and J. E. Evans, “Electric field enhancement in a self-assembled 2D array of silver nanospheres,” J. Chem. Phys. 141(21), 214308 (2014).
[Crossref]

Agrawal, A.

A. Agrawal, S. H. Cho, O. Zandi, S. Ghosh, R. W. Johns, and D. J. Milliron, “Localized Surface Plasmon Resonance in Semiconductor Nanocrystals,” Chem. Rev. 118(6), 3121–3207 (2018).
[Crossref]

Ahmadi, T. S.

T. S. Ahmadi, S. L. Logunov, L. Stephan, and M. A. El-Sayed, ““Picosecond Dynamics of Colloidal Gold Nanoparticles” J,” J. Phys. Chem. 100(20), 8053–8056 (1996).
[Crossref]

Al-Lohedan, H. A.

N. Jayaprakash, J. J. Vijaya, K. Kaviyarasu, K. Kombaiah, L. J. Kennedy, R. J. Ramalingam, M. A. Munusamy, and H. A. Al-Lohedan, ““Green synthesis of Ag nanoparticles using Tamarind fruit extract for the antibacterial studies” J,” J. Photochem. Photobiol., B 169, 178–185 (2017).
[Crossref]

Angelomé, P. C.

B. Sepúlveda, P. C. Angelomé, L. M. Lechuga, and L. M. Liz-Marzán, “LSPR-based nanobiosensors,” Nano Today 4(3), 244–251 (2009).
[Crossref]

Arnold, M. D.

Ashtari, K.

M. Azimzadeh, M. Rahaie, N. Nasirizadeh, K. Ashtari, and H. Naderi-Manesh, “An electrochemical nanobiosensor for plasma miRNA-155, based on graphene oxide and gold nanorod, for early detection of breast cancer,” Biosens. Bioelectron. 77, 99–106 (2016).
[Crossref]

Au, L.

S. E. Skrabalak, J. Y. Chen, Y. G. Sun, X. M. Lu, L. Au, C. M. Cobley, and Y. N. Xia, ““Gold Nanocages: Synthesis, Properties, and Applications” Acc,” Acc. Chem. Res. 41(12), 1587–1595 (2008).
[Crossref]

Azimzadeh, M.

M. Azimzadeh, M. Rahaie, N. Nasirizadeh, K. Ashtari, and H. Naderi-Manesh, “An electrochemical nanobiosensor for plasma miRNA-155, based on graphene oxide and gold nanorod, for early detection of breast cancer,” Biosens. Bioelectron. 77, 99–106 (2016).
[Crossref]

Bai, T. T.

T. T. Bai, J. F. Sun, R. C. Che, L. N. Xu, C. Y. Yin, Z. R. Guo, and N. Gu, “Controllable Preparation of Core-Shell Au-Ag Nanoshuttles with Improved Refractive Index Sensitivity and SERS Activity,” ACS Appl. Mater. Interfaces 6(5), 3331–3340 (2014).
[Crossref]

Baker, L. A.

D. D. Smith, Y. Yoon, R. W. Boyd, J. K. Campbell, L. A. Baker, R. M. Crooks, and M. A. George, “Z-scan measurement of the nonlinear absorption of a thin gold film,” J. Appl. Phys. 86(11), 6200–6205 (1999).
[Crossref]

Boneberg, J.

L. E. Hennemann, A. Kolloch, A. Kern, J. Mihaljevic, J. Boneberg, P. Leiderer, A. J. Meixner, and D. Zhang, “Assessing the plasmonics of gold nano-triangles with higher order laser modes,” Beilstein J. Nanotechnol. 3, 674–683 (2012).
[Crossref]

Boyd, R. W.

D. D. Smith, Y. Yoon, R. W. Boyd, J. K. Campbell, L. A. Baker, R. M. Crooks, and M. A. George, “Z-scan measurement of the nonlinear absorption of a thin gold film,” J. Appl. Phys. 86(11), 6200–6205 (1999).
[Crossref]

Campbell, J. K.

D. D. Smith, Y. Yoon, R. W. Boyd, J. K. Campbell, L. A. Baker, R. M. Crooks, and M. A. George, “Z-scan measurement of the nonlinear absorption of a thin gold film,” J. Appl. Phys. 86(11), 6200–6205 (1999).
[Crossref]

Chang, Q. C.

Y. C. Gao, X. R. Zhang, Y. L. Li, H. F. Liu, Y. X. Wang, Q. C. Chang, W. Y. Jiao, and Y. L. Song, “Saturable absorption and reverse saturable absorption in platinum nanoparticles,” Opt. Commun. 251(4-6), 429–433 (2005).
[Crossref]

Chang, S. H. G.

M. F. Tsai, S. H. G. Chang, F. Y. Cheng, V. Shanmugam, Y. S. Cheng, C. H. Su, and C. S. Yeh, “Au Nanorod Design as Light-Absorber in the First and Second Biological Near-Infrared Windows for in Vivo Photothermal Therapy”,” ACS Nano 7(6), 5330–5342 (2013).
[Crossref]

Che, R. C.

T. T. Bai, J. F. Sun, R. C. Che, L. N. Xu, C. Y. Yin, Z. R. Guo, and N. Gu, “Controllable Preparation of Core-Shell Au-Ag Nanoshuttles with Improved Refractive Index Sensitivity and SERS Activity,” ACS Appl. Mater. Interfaces 6(5), 3331–3340 (2014).
[Crossref]

Chen, H. Z.

L. F. Qiao, D. Wang, L. J. Zuo, Y. Q. Ye, J. Qian, H. Z. Chen, and S. L. He, “Localized surface plasmon resonance enhanced organic solar cell with gold nanospheres,” Appl. Energy 88(3), 848–852 (2011).
[Crossref]

Chen, J. Y.

S. E. Skrabalak, J. Y. Chen, Y. G. Sun, X. M. Lu, L. Au, C. M. Cobley, and Y. N. Xia, ““Gold Nanocages: Synthesis, Properties, and Applications” Acc,” Acc. Chem. Res. 41(12), 1587–1595 (2008).
[Crossref]

M. Hu, H. Petrova, J. Y. Chen, J. M. McLellan, A. R. Siekkinen, M. Marquez, X. D. Li, Y. N. Xia, and G. V. Hartland, “Ultrafast laser studies of the photothermal properties of gold nanocages,” J. Phys. Chem. B 110(4), 1520–1524 (2006).
[Crossref]

Chen, L.

Q. Dong, X. W. Wang, X. X. Hu, L. Q. Xiao, L. Zhang, L. J. Song, M. L. Xu, Y. X. Zou, L. Chen, Z. Chen, and W. H. Tan, ““Simultaneous Application of Photothermal Therapy and an Anti-inflammatory Prodrug using Pyrene-Aspirin-Loaded Gold Nanorod Graphitic Nanocapsules” Angew,” Angew. Chem., Int. Ed. 57(1), 177–181 (2018).
[Crossref]

Chen, L. L.

H. P. Sun, J. H. Su, Q. S. Meng, Q. Yin, L. L. Chen, W. W. Gu, Z. W. Zhang, H. J. Yu, P. C. Zhang, S. L. Wang, and Y. P. Li, “Cancer Cell Membrane-Coated Gold Nanocages with Hyperthermia-Triggered Drug Release and Homotypic Target Inhibit Growth and Metastasis of Breast Cancer,” Adv. Funct. Mater. 27(3), 1604300 (2017).
[Crossref]

Chen, W. Z.

C. Zheng, J. X. Huang, L. Lei, W. Z. Chen, H. Y. Wang, and W. Li, “Nanosecond nonlinear optical and optical limiting properties of hollow gold nanocages,” Appl. Phys. B 124(1), 17 (2018).
[Crossref]

Chen, X. H.

B. M. Ma, P. Li, X. H. Chen, L. L. Wang, B. H. Liu, and T. Song, “Gold nano-triangles as saturable absorbers for a dual-wavelength passively Q-switched Nd:GYSGG laser,” Laser Phys. 28(7), 075802 (2018).
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Chen, Z.

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M. F. Tsai, S. H. G. Chang, F. Y. Cheng, V. Shanmugam, Y. S. Cheng, C. H. Su, and C. S. Yeh, “Au Nanorod Design as Light-Absorber in the First and Second Biological Near-Infrared Windows for in Vivo Photothermal Therapy”,” ACS Nano 7(6), 5330–5342 (2013).
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T. T. Bai, J. F. Sun, R. C. Che, L. N. Xu, C. Y. Yin, Z. R. Guo, and N. Gu, “Controllable Preparation of Core-Shell Au-Ag Nanoshuttles with Improved Refractive Index Sensitivity and SERS Activity,” ACS Appl. Mater. Interfaces 6(5), 3331–3340 (2014).
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H. P. Sun, J. H. Su, Q. S. Meng, Q. Yin, L. L. Chen, W. W. Gu, Z. W. Zhang, H. J. Yu, P. C. Zhang, S. L. Wang, and Y. P. Li, “Cancer Cell Membrane-Coated Gold Nanocages with Hyperthermia-Triggered Drug Release and Homotypic Target Inhibit Growth and Metastasis of Breast Cancer,” Adv. Funct. Mater. 27(3), 1604300 (2017).
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Figures (4)

Fig. 1.
Fig. 1. The characteristics of Au-Ag NSs. (a) Appearance color, (b) SEM image, (c) Linear absorption spectra.
Fig. 2.
Fig. 2. Normalized transmission of Au-Ag NSs for open-aperture Z-scan at different energies of (a) 50 uJ, 150 uJ, 350 uJ, 450 uJ and (b) 700uJ, 750 uJ. The dots are experimental data while the solid lines are theoretical fit.
Fig. 3.
Fig. 3. (a) Two-dimensional contour map of Au-Ag NSs pump-probe data.(b)Time resolved transient absorption data of Au-Ag NSs.(c) Dynamic traces of Au-Ag NSs at wavelength of 493 nm.(d) Dynamic traces of Au-Ag NSs at wavelength of 562 nm. (Note:ΔmOD is the optical density difference
Fig. 4.
Fig. 4. Dynamic traces of Au-Ag NSs at 510 nm (The dashed line is experimental data while the solid line is theoretical fit generated).

Tables (3)

Tables Icon

Table 1. Fitting results of experimental data

Tables Icon

Table 2. Fitting results for the decay processes for Au-Ag NSs at different powers.

Tables Icon

Table 3. Fitting results for the decay processes for Au-Ag NSs at different powers.

Equations (6)

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

α ( I ) = α 0 1 + ( I / I s ) + β I
I = I 0 1 + z 2 / z 0 2
α ( I 0 ) = α 0 1 + I 0 ( 1 + z 2 / z 0 2 ) I s + β I 0 1 + z 2 / z 0 2
T ( z ) = m = 0 [ β I 0 L e f f ( 1 + z 2 / z 0 2 ) ] m ( m + 1 ) 3 / 2
Δ T T = A 1 exp ( t τ 1 ) + A 2 exp ( t τ 2 )
S ( t ) = A cos ( 2 π t T + φ ) e t τ v + A 1 e t τ 1 + A 2 e t τ 2 + B

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