1. Introduction

In the last ten years, periodic arrays of sub-wavelength patterns in metallic films have lead to numerous experimental and theoretical works because of their interesting optical properties [1–8] and their potential applications in nanodevices [9–11]. In such nanostructured films surface plasmon polaritons (SPP) can be resonantly excited for particular optical wavelengths depending on the geometrical parameters of the nanostructures as well as on the dielectric function of the metal [12–16]. From a general point of view one can envisage using plasmonic devices either by taking advantage of their enhanced local field effects or by propagating the surface plasmons over distances large enough to make active integrated devices. In that latter case it is necessary that the (SPP) be preserved from intrinsic or extrinsic sources of damping. It is therefore an important issue to study the decay of surface plasmons in a patterned array and to model their corresponding dynamical behavior using a realistic metallic dielectric function. It is the purpose of this article to perform such dynamical studies inspired by the large number of works that have been performed in metallic nanoparticles using femtosecond optical spectroscopy.

Noble metal nanoparticles are archetype systems to study the static and dynamical properties of surface plasmons [17–25]. Even though in the dipolar approximation propagating modes are not relevant for nanoparticles, the analogy with the (SPP) dynamics in patterned arrays is straightforward. Indeed, for both systems the intraband and interband optical transitions are determinant to explain the detailed behavior of the plasmon resonances. For patterned structures, in addition to the characteristic properties of the metal supporting the nanostructure, one needs to incorporate a tractable diffraction model to explain the spectral (SPP) resonances as we have recently shown to explain the static properties of patterned arrays made in gold, silver and copper films [26–27]. In the present article, we focus our attention on the spectro-temporal dynamics of such periodic metallic nanostructures performing time-resolved femtosecond spectroscopy experiments. We also extend our preceding theoretical investigations, inspired from Darmanyan's work [28], to the dynamical case.

2. Experimental details

The time-resolved pump-probe measurements have been performed with a femtosecond Titanium Sapphire laser amplified at a repetition rate of 5 kHz and delivering pulses with 150 fs duration. The 400 nm pump is obtained by frequency doubling in a β-BaB₂O₄ crystal. The probe is a broadband continuum generated in a sapphire crystal. The frequency chirp of the probe is mainly compensated by a double pass in a pair of prisms and is less than 1fs/nm in the spectral region 500–600 nm. The residual chirp in the 600–800 nm range is de-convoluted from the spectro-temporal signals by using an appropriate conform transformation in the (λ, τ) plane, the coordinates λ and τ referring to the wavelength and delay between the pump and probe pulses [29]. The two beams are focused at normal incidence on the sample with a mean diameter of 50 μm, so that a given nanostructure (array of sub-wavelength holes with total dimensions of a few microns) is excited and probed uniformly. Each patterned array is then imaged on the entrance slit of a spectrometer, corrected from optical aberrations. The static light transmission T(λ) as well as the time dependent differential transmission spectra ∆T(λ, τ)/T(λ), defined as ∆T/T = (T_pump-on - T_pump-off)/T_pump-off are measured with a nitrogen cooled CCD camera.

The samples consist in arrays of subwavelength holes designed in a 250 nm gold film evaporated either on glass or Al₂O₃ substrates. The arrays have various periods a ranging from 150 to 350 nm. Their transmission spectra all exhibit two main peaks as shown for example in the inset of Fig. 1(a) for an array of gold on a glass substrate with the period a=350 nm. The peak at 660 nm corresponds to the (1, 0) surface plasmon polariton mode on the metal/substrate interface named in the following (1, 0)SS. The broader peak extending in the spectral region 500–600 nm results from a combination of both the (1, 0) order (SPP) on the air side and to the transmission properties specific to gold situated at 510 nm. As shown in our previous works [26–27], a transparency window occurs in that spectral range due to the interplay between the light absorption at the interband threshold (527 nm in gold), related to the excitation of the core level 3d electrons, and from the large reflectivity associated to the Drude-like metallic conductivity due to the conduction states.

 

Fig. 1. (a)Differential transmission of a gold nanostructures with the period a=350 nm. The density of excitation is 0.5 mJ.cm-². Inset: Linear transmission of the corresponding nanostructure. (b)Spectrally resolved differential transmission for a pump-probe delay τ=0.9 ps.

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3. Surface plasmon dynamics near the interband threshold

Figure 1(a) displays the differential transmission as a function of the delay τ for the probe wavelength λ_o=557 nm (vertical dotted line in the inset). For this measurement, a narrow spectral band is pre-selected in the continuum and is detected with a photomultiplier. The observed dynamics reveals the characteristic relaxation processes of metallic systems excited by femtosecond optical pulses. Initially the electrons thermalize to a hot distribution involving the Coulomb scattering between quasi-particles (electrons and holes). Correspondingly, the pump-probe signal builds up with a time constant which depends on the excess energy relative to the Fermi level [30–33]. In the present case, it is faster than the pump pulse duration. The exponential relaxation during the next few picoseconds is related to the cooling of the electrons distribution and the simultaneous heating of the lattice assisted by electron-phonon interactions. As it is well known, this dynamical process can be described by a two temperatures model [30, 34–35] for the electrons and the lattice. The corresponding relaxation time τ_e-l depends both on the strength of the electron-phonon coupling and on the density of laser excitation via the temperature dependent electronic specific heat. For the present density of excitation (I_pump=0.5 mJ.cm^-2), the relaxation time is 1.4 ps. It increases linearly with I_pump in the explored density range (not shown in Fig. 1).

Figure 1(b) shows the spectrally resolved differential transmission, for a pump-probe delay τ = 0.9 ps. The energy density of the pump is 0.5 mJ.cm^-2. Several interesting features are observed. First, the amplitude of the differential signal is larger than 20% near the interband transition wavelength whereas it is only of 3% in the spectral region corresponding to the (1, 0) order (SPP). This result clearly indicates that the induced change of transmission is mostly due to the modification of the dielectric function of gold near the interband transition from the d band to the Fermi level. In that spectral range it is therefore essential to consider the role played by the core electrons and not only the Drude-like metal conductivity if one is interested in describing correctly the propagation of the surface plasmon polaritons. The spectral shape of the differential transmission in the range 500–600 nm is characteristic of a spectral broadening of the (SPP) resonance. Such broadening mostly comes from the modification of the imaginary part of the complex dielectric function of the metal. In contrast, the spectral shape of ∆T/T in the range 620–750 nm is characteristic of a red shift as well as a weak broadening of the resonance associated to the excitation of the (SPP) on the substrate side. This behavior mainly implies a dynamical increase of the real part of the dielectric function in this spectral range.

The above experimental results stress out the key role played by the dielectric function of the metal if one is interested in propagating surface plasmon polaritons in active integrated plasmonic devices. To emphasize this statement, we have developed a comprehensive description of our dynamical results modeling the patterned arrays with different dielectric functions of gold. The main steps are the following. First we simplify the complicated two dimensional nanostructure considering a one dimensional array and we model the light diffraction within the so-called RCWA (rigorous coupled wave approximation). We previously reported such procedure to describe the static properties of gold, copper and silver arrays [26–27]. The next step is to introduce the time dependent dielectric function ε(λ,τ) of gold. We have used two different approaches. The first one, named hereafter ε_RPA, is based on the Lindhard theory [36] and the second one, named ε_exp, is based on the experimental determination of the dielectric function of gold.

The Lindhard dielectric function ε_RPA which we use is derived following our previous investigations of noble metal nanoparticles [18, 33]. It consists in considering two terms corresponding respectively to the conduction electrons, modeled by a Drude like dielectric function, and to the core electrons which can undergo interband transitions from the filled d-bands to the Fermi level. The temporal dependency of the dielectric function comes from the occupied electronic states at a given electronic temperature, which itself is calculated with the two temperatures model. In addition, a temperature dependent electron-electron scattering is provided by the Fermi liquid theory derived by Pines and Nozières [37].

The second time-dependent dielectric function ε_exp that we have used to model the dynamical spectrum of our nanostructures is obtained experimentally from the time-resolved differential spectra of the reflectivity and transmission measured on a 29 nm thin gold film. This standard approach is similar to the one developed by Rosei [38], describing thermo-modulation experiments on thin metallic films, except that the temperature variation stands for the one of the electrons excited by the femtosecond optical pulses. The real and imaginary parts of the differential dielectric functions that we have calculated and obtained experimentally are shown on Fig. 2. The experimental curves are given for a pump-probe time delay of 1.4 ps. In the case of the Lindhard dielectric function, it corresponds to an increase of the electronic temperature of 395 K.

 

Fig. 2. Differential spectra of the real part (open symbols) and imaginary part (closed symbols) of the two dielectric functions ε_RPA (red color) and ε_exp (blue color). The experimental delay is fixed at τ=1.4 ps which corresponds to an electronic temperature of 695 K.

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The two time-dependent dielectric functions of gold ε_RPA(λ,τ) and ε_exp(λ,τ) have been introduced in our analytical model of diffraction to model the femtosecond dynamics of the gold nanostructures. Figure 3 represents differential spectra for the simplified one dimensional gold nanostructure with a period a = 350 nm calculated with ε_RPA (straight line) and ε_exp (dotted line). The pump-probe delay is τ=1.4 ps. The open circles corresponds to the measured differential transmission obtained with a two dimensional nanostructure having the same characteristics as in the modeling (periodicity and filling factor of the holes). The main spectral features are reproduced both with ε_RPA and ε_exp : the broadening in the 500–600 nm spectral region and the shift in the vicinity of the (1,0) surface plasmon polariton resonance. As mentioned above, it stresses the respective role of the imaginary and real part of the dielectric functions. Let us notice that the use of ε_exp gives a better result. It emphasizes the important role of extrinsic sources of damping, leading to electron scattering, absent in the modeling of ε_RPA.

 

Fig. 3. Differential transmission spectrally resolved for a fixed delay τ=1.4 ps. The open circles represent the experimental measurement on the gold nanostructure. Theoretical dynamical spectrum calculated with the Lindhard dielectric function ε_RPA (dashed dotted line) and the experimental dielectric function ε_exp (full line).

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4. Conclusion

In conclusion, our experimental study and modeling of the surface plasmon polariton dynamics in metallic films patterned with subwavelength holes demonstrates the key role played by the scattering of electrons. Intrinsically, such scattering events are particularly important near the interband optical transitions of the metal but they can also be accentuated by extrinsic sources of scattering like the structural inhomogeneities of the metal film or by defects in the patterned nanostructure induced by the Focused Ion Beam processing. These results are important for the development of active plamonic devices where the electron scattering, leading to a loss of coherence of the surface plasmon polaritons, should be minimized.