On the modeling of spectral map of glass-metal nanocomposite optical nonlinearity

Transcription

1 On the modeling of spectral map of glass-metal nanocomposite optical nonlinearity A.A. Lipovskii, 1, O.V. Shustova, 1 V.V. Zhurikhina, 1,* and Yu. Svirko 3 1 St.-Petersburg State Polytechnical University, Polytechnicheskaya 9, St.Petersburg, 19551, Russia St.-Petersburg Academic University, Khlopina 8/3, St.Petersburg, 1950, Russia 3 University of Eastern Finland, Yliopistokatu 7, P.O. Box 111, Joensuu, FI-80101, Finland * jourikhina@mail.ru Abstract: The spectral map of the nonlinear absorption coefficient of glasscopper nanocomposite in the pump-probe scheme constructed with the use of a simple anharmonic oscillator model reproduced well the spectral map obtained in the experiment. It is shown that spectral features in nonlinear response of glass-metal nanocomposites () can be engineered by varying the size of nanoparticles. The pronounced dependence of the magnitude of the third-order nonlinearity on the particles size explains the diversity of experimental data related to nonlinear optical response of s in different experiments. Performed modeling proves that silver demonstrate much sharper spectral dependence than copper ones due to strong frequency dependence of local field enhancement factor for silver nanoparticles. 01 Optical Society of America OCIS codes: ( ) icroparticle nonlinear optics; ( ) Plasmonics. References and links 1. S. A. aier, Plasmonics: Fundamentals and Applications (Springer, 007).. W. L. Barnes, A. Dereux, and T. W. Ebbesen, Surface plasmon subwavelength optics, Nature 44(6950), (003). 3. A. V. Zayats, I. I. Smolyaninov, and A. A. aradudin, Nano-optics of surface plasmon polaritons, Phys. Rep. 408(3-4), (005). 4.. L. Brongersma and V.. Shalaev, Applied physics. The case for plasmonics, Science 38(5977), (010). 5. J. A. Schuller, E. S. Barnard, W. Cai, Y. Ch. Jun, J. S. White, and. L. Brongersma, Plasmonics for extreme light concentration and manipulation, Nat. ater. 9, (010). 6. D. Lu, J. Kan, E. E. Fullerton, and Z. Liu, Tunable surface plasmon polaritons in Ag composite films by adding dielectrics or semiconductors, Appl. Phys. Lett. 98(4), (011). 7. Z. Shi, G. Piredda, A. C. Liapis,. A. Nelson, L. Novotny, and R. W. Boyd, Surface-plasmon polaritons on metal-dielectric nanocomposite films, Opt. Lett. 34(), (009). 8. A. V. Krasavin, K. F. acdonald, A. S. Schwanecke, and N. I. Zheludev, Gallium/aluminum nanocomposite material for nonlinear optics and nonlinear plasmonics, Appl. Phys. Lett. 89, (006). 9. E. Ozbay, Plasmonics: merging photonics and electronics at nanoscale dimensions, Science 311(5758), (006). 10. R. Zia, J. A. Schuller, A. Chandran, and. L. Brongersma, Plasmonics: the next chip-scale technology, ater. Today 9(7-8), 0 7 (006). 11. S. A. aier, Plasmonics-towards subwavelength optical devices, Current Nanosci. 1(1), 17 (005). 1. S. ukamel, Principles of Nonlinear Optical Spectroscopy (Oxford University Press, New York, 1995) 13. L. Pálfalvi, B. C. Tóth, G. Almási, J. A. Fülöp, and J. Hebling, A general Z-scan theory, Appl. Phys. B 97, (009). 14. J. Wang,. Sheik-Bahae, A. A. Said, D. J. Hagan, and E. W. Van Stryland, Time-resolved Z-scan measurements of optical nonlinearities, J. Opt. Soc. Am. B 11(6), (1994). 15. J.-Y. Bigot, V. Halte, J.-C. erle, and A. Daunois, Electron dynamics in metallic nanoparticles, Chem. Phys. 51(1-3), (000) Halonen, A. A. Lipovskii, and Y. P. Svirko, Femtosecond absorption dynamics in glass-metal nanocomposites, Opt. Express 15(11), (007). 17. J. C. axwell Garnett, Colours in metal glasses and metal films, Philos. Trans. R. Soc. London Ser. A 03( ), (1904). (C) 01 OSA 1 ay 01 / Vol. 0, No. 11 / OPTICS EXPRESS 1040

2 18. G. A. Niklasson, C. G. Granqvist, and O. Hunderi, Effective medium models for the optical properties of inhomogeneous materials, Appl. Opt. 0(1), 6 30 (1981). 19. H. a, R. Xiao, and P. Sheng, Third-order optical nonlinearity enhancement through composite microstructures, J. Opt. Soc. Am. B 15, (1998). 0. D. A. G. Bruggeman, Berechnung verschiedener physikalischer konstanten von heterogenen substanzen. I. dielektrizitatskonstanten und leitfahigkeiten der mischk.orper aus isotropen sub-stanzen, Ann. Phys. 416(7), (1935). 1.. Halonen, A. Lipovskii, V. Zhurikhina, D. Lyashenko, and Yu. Svirko, Spectral mapping of the third-order optical nonlinearity of glass-metal nanocomposites, Opt. Express 17(19), (009)... Halonen, A. A. Lipovskii, and Yu. P. Svirko, Femtosecond transmission control in glass-metal nanocomposites, in International Conference on Coherent and Nonlinear Optics (insk, Belarus, 007) 3. L. D. Landau and E.. Lifshitz, echanics (Pergamon Press, 1976). 4. H. Wormeester, E. Kooij, and B. Poelsema, Effective dielectric response of nanostructured layers, Phys. Status Solidi 05(4), (008) (a). 5. R. Boyd, Introduction to Nonlinear Optics (Academic Press, Boston, ass., 199). 6.. G. Papadopoulos, et al., eds., Non-Linear Optical Properties of atter (Springer, 006). 7. L. D. Landau, E.. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous edia (Butterworth- Heinemann, 1984). 8. N. W. Ashcroft and D. N. ermin, Solid State Physics (Holt, Rinehart and Winston, 1987). 9. P. B. Johnson and R. W. Christy, Optical constants of the noble metals, Phys. Rev. B 6(1), (197). 30. U. Kreibig, Electronic properties of small silver particles: the optical constants and their temperature dependence, J. Phys. F et. Phys. 4(7), (1974). 31. Y. Takeda, H. omida,. Ohnuma, T. Ohno, and N. Kishimoto, Wavelength dispersion of nonlinear dielectric function of Cu nanoparticle materials, Opt. Express 16(10), (008). 3. Y. Takeda, O. A. Plaksin, and N. Kishimoto, Dispersion of nonlinear dielectric function of Au nanoparticles in silica glass, Opt. Express 15(10), (007). 33. N. Del Fatti, F. Vallee, C. Flytzanis, Y. Hamanaka, and A. Nakamura, Electron dynamics and surface plasmon resonance nonlinearities in metal nanoparticles, Chem. Phys. 51(1-3), 15 6 (000). 34. B. Karthikeyan, J. Thomas, and R. Philip, Optical nonlinearity in glass-embedded silver nanoclusters under ultrafast laser excitation, Chem. Phys. Lett. 414(4-6), (005). 35. P. P. Kiran, G. De, and D. N. Rao, Nonlinear optical properties of copper and silver nanoclusters in SiO sol-gel films, IEEE Proc. Circuits Dev. and Syst. 150(6), (003). 36. X. C. Yang, Z. H. Li, W. J. Li, J. X. Xu, Z. W. Dong, and S. X. Qian, Optical nonlinearity and ultrafast dynamics of ion exchanged silver nanoparticles embedded in soda-lime silicate glass, Chin. Sci. Bull. 53(5), (008). 1. Introduction Recent advances in nanotechnology gave birth to plasmonics [1 5], a new branch of the optical and material science that studies optical phenomena in metal nanostructures. Optical properties of such systems dominate surface plasmon modes capable of concentration and enhancement of the electromagnetic field in the proximity of the metal-dielectric interface. In glass-metal nanocomposites () comprising of nanosized metal inclusions in glassy matrix, the spectral position and strength of the surface plasmon resonance (SPR) is determined by properties of both metal particles and host matrix. This enables tailoring of the optical properties by varying size, shape and packing density of the particles and by changing the matrix [6 8]. Such a tunability of optical properties along with strong optical nonlinearity of metals and compatibility with all-solid-state opto-electronic circuits makes a key material for various plasmonic devices [9 11]. A combination of the strong but featureless optical nonlinearity of metal and SPRenriched optical response of the composite has also made a unique playground to study ensembles of highly localized hot electrons. The dynamics of hot electron ensembles can be visualized by using ultrafast nonlinear spectroscopy techniques [1]. In the time domain, the dynamics of light-excited electrons is usually visualized by pump-probe measurements, i. e. by studying temporal evolution of light-induced transmission change of the at the excitation with ultrashort light pulses tuned to the SPR. In the frequency domain, the study of the nonlinear optical properties of is often restricted to the measurements of the nonlinear refraction and absorption coefficients (i.e. to the measurements of the real and imaginary parts of the frequency degenerate third-order susceptibility, respectively) by Z-scan (C) 01 OSA 1 ay 01 / Vol. 0, No. 11 / OPTICS EXPRESS 1041

3 technique [13,14]. However study of the pronounced spectral features in the ultrafast nonlinear optical response [15,16] of the in the vicinity of the SPR requires combining both approaches, i.e. the time-resolved measurements of the essentially non-degenerate nonlinear optical susceptibility at the excitation tunable over a wide frequency range. If the volume fraction of metal in does not exceed 10-15%, the optical properties of are well described within the framework of the axwell Garnett (G) effective medium approximation [17]. It is worth noting however that at higher volume fraction of metal another approaches should be used [18 0]. In particular G approximation allows one to reproduce the observed in the experiment SPR-dominated linear absorption spectra of at the metal volume fraction less than ~15% provided that dielectric functions of metal and glass matrix are known. Thus one may expect that G approximation can also be employed for description of the nonlinear optical effects in such. These effects originate from the anharmonic oscillations of conduction electrons in nanoparticles, while the spectral properties of nonlinear optical response of being strongly influenced by the surface plasmon modes. In this paper, we calculate the third-order nonlinear optical susceptibility χ of as a function of the pump and probe wavelengths using G approximation. The obtained results allow us to interpret recently obtained spectral map of the imaginary part of χ for copperbased [1] and copper film []. We show in particular that conventional model of the anharmonic oscillator [3] satisfactory describes the wavelength dependence of χ of the copper based, while the spectral features of nonlinear absorption are strongly influenced by the volume concentration and size of the metal nanoparticles. This implies that comparison of the results obtained in different experiments requires detailed information on constituencies and composition of the involved.. χ of bulk copper and copper-based At relatively low (up to 15% [4]) volume concentration of metal, the dielectric coefficient of the ε can be obtained in the framework of the axwell-garnett relation [17] in the following form: ε ( ε ε ) ( ε ε ) ε G + ε + f a = εg. ε G ε f + + G Here f is volume fraction of the metal particles, ε G and ε are permittivity of glass matrix and metal, respectively. Linear optical absorption coefficient of the [5] at frequency ω is then given by: ( ) ω α0( ω) = Im ε ( ω). () c When the pump wave at frequency ω 1 propagates through the nonlinear media, the optical absorption coefficient of the at frequency ω can be described as [5] ( ) ( ) ( ) α ω, ω = α ω + α ω, ω I, where I is intensity of the pump wave and α (ω 1,ω ) is the so-called nonlinear absorption coefficient. The nonlinear absorption coefficient of is described by the imaginary part of the relevant third-order susceptibility. If the pump wave at frequency ω 1 and probe wave at frequency ω are co-linearly polarized, α (ω 1,ω ) can be written as [5] 48π α( ω1, ω) = Im { ( ; 1, 1, ) }, Re c ω χ ω ω ω ω (4) ( ε ) (1) (C) 01 OSA 1 ay 01 / Vol. 0, No. 11 / OPTICS EXPRESS 104

4 while the third-order susceptibility of, χ, is given by the following equation [6]: ( ) f L( ) L( ) ( ) χ ω ; ω, ω, ω = ω ω χ ω ; ω, ω, ω. (5) Here χ is nonlinear susceptibility of metal, and L(ω 1, ) are local field factors that describe enhancement of the light waves at the frequencies ω 1, in the vicinity of a spherical metal nanoparticle [7]: ε G + ε L=. (6) ε G + ε One can observe from Eq. (5) that the dependence of the light-induced absorption in on the frequencies of the pump and probe is governed by the third-order susceptibility of the metal and the local field factors. The local field factors can be readily obtained from Eqs. (1), (6). In this paper, we will calculate nonlinear susceptibility of metal using anharmonic oscillator model [3]. The frequency dispersion of the dielectric function in noble metals can be described by combining the contributions to the permittivity from both free conduction electrons and interband transitions due to bound d-electrons [8]: ω pf ω pb ε = ε +, ω + iγ ω ω ω iγ ω f 0 b (7) where ε is the background high-frequency dielectric constant, ω pf (ω pb ) and Γ f (Γ b ) are plasma frequency and damping rate for free (bound) electrons, respectively. One may expect that the interplay of the free and bound electrons in copper may result in interesting spectral features in the dielectric function of the copper-based. Fig. 1. Imaginary (a) and real (b) components of the permittivity of copper calculated from Eq. (5) at ω 0 = s 1, ω pf = s 1, ω pb = s 1, Γ f = s 1, and Γ b = s 1 (solid lines) and plotted according to the handbook [9] (dash lines). Figure 1 shows the real and imaginary parts of the copper permittivity calculated using Eq. (7) (red solid line) and obtained from the standard handbook data [9]. One can observe that at ω 0 = s 1, ω pf = s 1, ω pb = s 1, Γ f = s 1, and Γ b = s 1, permittivity obtained from Eq. (7) well corresponds to the experimental data for light wavelength longer than 570nm. This indicates that in this spectral range, Eq. (7) can be employed for the modeling the optical properties of copper-based. In order to describe frequency dependence of the third-order susceptibility of bulk copper we assume that optical response of the bound electrons can be described in terms of the conventional anharmonic oscillator model [3]. Generally, the consideration is valid for (C) 01 OSA 1 ay 01 / Vol. 0, No. 11 / OPTICS EXPRESS 1043

5 longer, NIR and IR, wavelengths, however, the nonlinearity of noble metal nanoparticles at corresponding frequencies is low, for their resonance falls in visible range. Since the bulk copper possesses the inversion centre we present the potential energy of the bound electron with mass m in the following form: U = mω0 x + mξ x, (8) 4 where x is the electron displacement from equilibrium position and ξ is the anharmonicity parameter. The motion of the bound electrons is described by the following equation of motion: where { ω } { ω } e ɺɺ x+ Гbxɺ + ω0 x+ ξ x = E( t), (9) m E( t) = E exp i t + E exp i t + c. c. is the electric field in the medium due to presence of the pump and probe waves at the frequencies of ω 1 and ω, respectively. The perturbative solution on Eq. (9) results in the following equation for the third-order nonlinear optical susceptibility of copper: ξe ω χ ω ; ω, ω, ω. Cu pb = 4π m ω0 iωг ω ω0 + iω1г b ω1 ( ) ( b ) Thus the imaginary part of the third-order susceptibility of, ( ) { } ( ) { ( )} { } { ( )} χ = f L ω 1 L ω χcu + L ω { χcu} (10) Im Im Re Re Im, (11) (frequency arguments in the χ and χ are omitted) can be also presented in terms of the 0,8 Im(χ 3 Cu )108, esu 0,6 0,4 0, λ 1 =60 nm λ 1 =580 nm wavelength λ, nm Fig.. Probe wavelength dependence of the nonlinear optical susceptibility of copper measured [] (solid lines) and calculated from Eq. (4) for pump wavelengths λ 1 = 580nm (upper curves) and λ 1 = 60nm (lower curves). Anharmonicity parameter ξ = nm s. anharmonicity parameter ξ. By using experimental data [1,] on Im{ χ Cu } and Im{ } χ, the anharmonicity parameter, associated with bound electrons in copper, is ξ = (4.3 ± 0.) 10 3 nm s. Figure shows the calculated (dash lines) and experimentally measured (solid lines) imaginary part of the third-order optical susceptibility of copper for pump (C) 01 OSA 1 ay 01 / Vol. 0, No. 11 / OPTICS EXPRESS 1044

6 wavelengths λ 1 = 580nm and λ 1 = 60nm. One may observe that at ξ = nm s, ω 0 = s 1, ω pf = s 1, ω pb = s 1, Γ f = s 1, and Γ b = s 1 { Cu 1 1 } (see Fig. 1) the calculated from Eq. (10) Im χ ( ω ; ω, ω, ω ) measured in experiment [] at λ >570nm. well corresponds to that 3. Dependence of the optical nonlinearity of on the size of nanoparticles In order to visualize the effect of the nanoparticles size on the nonlinearity we calculated the real and imaginary part of the third-order susceptibility of the copper-based with different parameters using Johnson and Christy data [9] for copper permittivity. The confinement of the conduction electrons in nanoparticle results in the increase of the momentum relaxation rate Γ f as [30] ( R) v / R, Γ =Γ + (1) f f F where Γ f is the damping rate for bulk metal, ν F and R are Fermi velocity in the metal and the nanoparticle radius. This results in the dependence of the linear and nonlinear optical Fig. 3. Calculated (a) and experimental (b) χ spectral map [1] for glass copper ( 3) Im( ) nanocomposite, anharmonicity parameter ξ = nm s. properties of on the nanoparticle size. Results of the calculations of Im{ } χ for copper-based are presented in Fig. 4. One can observe that the spectral map of Im { } χ is qualitatively different for s composed of nanoparticles with radius 1nm and 5nm. Specifically the probe wavelength at which the third-order susceptibility of the changes sign depends on the size of copper nanoparticles. This dependence is pronounced in silver-based because for silver nanoparticles the local field factor L(ω) shows much sharper frequency dependence (see Fig. 5). The obtained results explain the diversity of experimental data on the nonlinearity of [15, 6, 31 36], because even relatively small variations in the particle size may produce a considerable change in the spectral properties of the nonlinear response. The broadening of particle size distribution will result in smoothing resonant features and respective decrease in the magnitude of nonlinearity. In contrary the variation of the metal volume fraction, which is often considered as the most important characteristic of (C) 01 OSA 1 ay 01 / Vol. 0, No. 11 / OPTICS EXPRESS 1045

7 Fig. 4. Imaginary part of glass-copper nanocomposite third order susceptibility, Im( χ ) 10 (esu), metal volume fraction f = 10 5, particles size is marked in the figures. ( 3) 14 The spectral position where changes its sign is shown with dashed line. nanoparticles, can only scale the magnitude of the nonlinearity leaving the spectral map unchanged. Thus comparative analysis of data obtained in different experiments is possible only if the comprehensive information on parameters is available. It should be noted that such information is especially important for silver-based because the local field enhancement factor for silver nanoparticles is about four orders of magnitude higher than that of copper nanoparticles and has very sharp frequency dependence. It is worth to mention, that used G approximation is valid for containing below 15 vol.% of metal. For higher metal content the model developed can be broaden using the Sheng theory [19], which contrary to wide-spread Bruggeman theory [0] allows consideration of in the vicinity of resonance frequency. (C) 01 OSA 1 ay 01 / Vol. 0, No. 11 / OPTICS EXPRESS 1046

8 4. Conclusion Fig. 5. Calculated spectral maps of real and imaginary part of local field enhancement factor for embedded in glass silver and copper nanoparticles of the same 15 nm radius. We developed simple anharmonic oscillator model to describe dependence of the third-order nonlinearity for the on the frequencies of the light waves involved in the nonlinear interaction. The model is valid for which can be described in the frames of axwell Garnett effective media approximation, that is for up to ~15 vol.% metal content in the. The calculated spectral map of the nonlinear absorption coefficient in the pump-probe scheme reproduced well that obtained in the experiment for the cooper-based. The dependence of the nonlinearity on the frequencies of the light waves involved implies that spectral features in nonlinear response of is governed by both spectral dependence of the metal dielectric function and local field enhancement factor, and hence it can be engineered by varying the size of nanoparticles. The pronounced dependence of the magnitude of the thirdorder nonlinearity on the particles size explains the diversity of sometimes contradictory experimental data obtained in the investigations of the nonlinear optical response of made using different manufacturing methods and experimental techniques. It is essential that silver demonstrate much sharper spectral dependence than copper ones, and this could be explained by strong frequency dependence of local field enhancement factor for silver nanoparticles. Acknowledgments This study was supported by Russian foundation for Basic Research (project# ), Joensuu University Foundation, Academy of Finland (projects # and ), and EU (FP7 project "Nanocom"). (C) 01 OSA 1 ay 01 / Vol. 0, No. 11 / OPTICS EXPRESS 1047