Two-Dimensional Electronic Spectroscopy of Gold Nanorods: Nodal Line Slope Analysis and Spectral Interference

Abstract

Two-dimensional (2D) electronic spectroscopy (ES) has been used to investigate the electronic coupling and dynamics of coupled multi-chromophore systems. In particular, off-diagonal features on 2D ES spectra provide critical information, such as on the electronic couplings, excitation transfer rates and molecular conformational changes between the involved electronic states. Recently, a 2D ES study has been performed on plasmonic gold nanorods with the surface plasmonic resonance excitation of free electrons. To gain quantitative information on the degree of dynamical inhomogeneity of polydisperse gold nanorods, nodal line slope analyses of the time-resolved 2D ES spectra with varying center wavelengths of pump and probe pulses have been carried out and shown to be of use for investigating broad distributions of dimensionally heterogeneous gold nanorods. In addition, due to the strong nonlinear surface plasmonic resonance response of gold nanorods, we observed unusual spectral fringe patterns in their 2D ES spectra, which were not observed in organic chromophores, in a reverse time-ordered pulse sequence. This was shown to be further critical evidence of the inhomogeneously broadened longitudinal surface plasmonic resonance band of gold nanorods. In this chapter, we present a detailed description of the 2D ES responses of gold nanorods that are associated with ultrafast electron heating, coherent electron dynamics, and the subsequent relaxation processes.

6.1 Introduction

Gold nanoparticles of various sizes and shapes show intriguing optical properties and have thus been widely used in a variety of applications to molecular imaging , drug delivery , photothermal therapy and optoelectronic devices [1,2,3,4]. Elucidating the general interplay between the size and shape of nanoparticles and their electronic, photothermal, and optical properties is critical not only for further understanding of the underlying physics behind nanophotonics but also for providing guidelines for the future design of nanoparticles in practical applications [1, 4, 5].

Gold nanorods (AuNRs) , which are elongated nanoparticles with a short and a long axis, have attracted a great deal of attention due to the optical tunability of their surface plasmon resonance bands via changing their aspect (length-to-width) ratio (R). Here, surface plasmon resonance (SPR) is a collective oscillation of conduction band electrons coupled to incident electromagnetic fields [1,2,3,4]. Unlike molecular excitation involving a superposition of two or multiple eigenstates, SPR leads to coherent linear oscillations (superposition of linear translation modes) of quasi-free electrons in the presence of an electric field (Fig. 6.1). Due to its elongated structure, the AuNR has two different SPR modes: transverse and longitudinal SPR (TrSPR and LgSPR) modes oscillating along its short and long axes, respectively. In particular, the LgSPR band, which appears in the longer wavelength region, strongly depends on the size and shape of the nanorod, showing a characteristic red-shift of the maximum absorption frequency with increasing aspect ratio (R) [1,2,3,4]. Their aspect ratio-dependent spectroscopic nature as well as exceptionally strong SPR responses make AuNRs promising for a variety of optical sensor applications [1, 5].

Fig. 6.1

Molecular excitation versus SPR excitation in AuNR. For a molecule, a superposition of two (or multiple) eigenstates (wavefunctions) are created by a single electron excitation. For an AuNR, however, a superposition of linear translation modes (collective and coherent linear oscillations) of many quasi-free electrons in the presence of electric field is produced by SPR excitation

Synthesized AuNRs usually have a broad size distribution (polydisperse). The polydispersity of the aspect ratio of AuNRs gives rise to a significant inhomogeneous broadening of the LgSPR band, potentially making the optical sensing of local environments with AuNRs less effective because AuNRs with different aspect ratios cause spectral congestion of overlapping SPR bands. Therefore, the monodispersity of AuNRs and their resulting narrow SPR bandwidth are important for improved optical sensitivity. Transmission electron microscopy (TEM) images are often used to examine the dimensional heterogeneity of synthesized nanorods. TEM, however, has difficulties related to not only sample preparation but also obtaining reproducible results from the limited number of images usually available for statistical analysis. As a simpler optical spectroscopic method, one can extract information on the aspect ratio distribution by fitting an experimentally measured LgSPR band to a set of calculated homogeneously broadened spectra for AuNRs with different aspect ratios [6]. However, it remains challenging or even impossible to clearly differentiate homogeneous and inhomogeneous line broadenings for an ensemble of AuNRs in steady-state or pump-probe transient absorption (TA) spectroscopy [2, 3, 7,8,9,10,11,12,13,14,15,16,17].

Two-dimensional (2D) electronic spectroscopy (ES) [18,19,20,21,22,23,24,25,26,27] has often been used to investigate the electronic coupling and dynamics of coupled multi-chromophore systems such as photosynthetic light-harvesting complexes [19, 20]. By analyzing the off-diagonal signals on 2D ES spectra, one can obtain information, such as on the electronic couplings, excitation transfer rates and molecular conformational changes between the involved electronic states of the molecule or molecular complexes of interest [20, 21]. So far, 2D ES studies have mainly focused on molecular systems involving transitions between the discrete quantum states of electrons bound to nuclei. In contrast, 2D ES studies on plasmonic metal nanoparticle systems involving SPR excitation of free electrons have been hardly performed. This is probably because 2D ES of plasmonic AuNR systems does not show any distinct features, such as off-diagonal peaks, that can provide useful information on the structure or electronic couplings of molecular systems. Other than off-diagonal signals, however, there remain useful observables extractable from 2D ES spectra. One of them is the nodal line slope (NLS) [21, 22], which has proven to be of critical use for gaining information on the degree of dynamic inhomogeneity of solvated molecular systems in condensed phases. This NLS analysis can additionally be used to examine the aspect ratio inhomogeneity of AuNRs in a broad distribution because nodal lines, created by rapid electron heating after SPR excitation of AuNRs, are easily observed on AuNR 2D ES spectra. Furthermore, due to the strong nonlinear SPR response of AuNR [7, 8], unusual spectral fringe patterns that are not often observed in organic chromophores are observed in the 2D ES spectra of AuNRs in a reverse time-ordered pulse sequence, which can be further critical evidence of the inhomogeneously broadened LgSPR band of AuNRs [28].

6.2 Photo-Induced Electron Dynamics

The electron dynamics of AuNRs excited by ultrashort pulses have been extensively studied and well characterized [7,8,9,10,11,12,13,14,15,16,17]. The involved relaxation processes show somewhat different aspects from those of excited molecules. In molecular excitation, a single electron is usually excited in such a way that no electron-electron collisions occur within each molecule. However, the electronic oscillations of multiple two-level systems (chromophores) in condensed phases start to decohere due to electron-bath interactions , followed by electronic population relaxation to the bath degree of freedom [21]. In the LgSPR excitation of AuNRs, however, electron-electron (e-e) collisions (or scatterings) dominantly occur within each AuNR, and lead to electron heating (0.1–1 ps) as well as decoherence (5–20 fs) in the LgSPR mode. The hot electrons are then thermally equilibrated with lattice phonons via electron-phonon (e-ph) scatterings (2–5 ps), and then the dissipation of the excited lattice phonon modes to the surrounding bath degree of freedom via phonon-phonon (ph-ph) scatterings (~100 ps) [9,10,11,12,13,14,15,16,17]. Figure 6.2 displays a typical TA time trace showing the relaxation processes of photo-excited AuNR. In addition to the stepwise TA signal decay behavior, a slowly oscillating component is often observed during the phonon-phonon relaxation process, which can be attributed to the coherent excitation of extensional vibrational modes along the longitudinal axis of AuNR [2, 16].

Fig. 6.2

TA time profile showing typical photo-induced relaxation processes of AuNRs. The TA trace was obtained at the probe wavelength of 850 nm after excitation at the pump center wavelength of λ_{pump_c} = 800 nm for silica-coated AuNRs with the average R = 3.84 (see Fig. 6.4). Note that a slowly oscillating component observed together during the phonon-phonon relaxation process is due to a coherent excitation of extensional vibrational modes along the long axis of AuNR

Pump-probe TA spectroscopy of molecules involves transitions between discrete eigenstates (Fig. 6.3a). Pump photons initially excite molecules from the ground state (g) to an excited state (e). Then, the probe photons interact with the excited molecules causing further absorption (TA > 0) to higher excited states (f) and stimulated emission (TA < 0) back to the ground state (g). In this case, the TA signal may show various spectral features depending on the electronic structure of the molecule under investigation. However, the TA signal of AuNR upon LgSPR excitation can simply be explained by the change in dielectric constant (i.e., changes in both absorption and refractive index) from a rapid increase of electronic temperature by electron heating (Fig. 6.3b), which leads to spectral red-shift and line broadening of the LgSPR band (Fig. 6.3c). As a result, a negative TA signal (ΔA < 0) appears in the higher frequency (shorter wavelength) region while a positive TA signal (ΔA > 0) appears in the lower frequency (longer wavelength) region (Fig. 6.3d), which is the characteristic spectral signature of the TA signal observed in the LgSPR band of AuNR.

Fig. 6.3

Photo-induced processes of a molecule and b AuNR in pump-probe experiment. In a molecule, a single electron excited from the ground (g) to first excited (e) states by the pump can undergo either a further excitation to the higher excited state (f) (excited state absorption, ESA) or a de-excitation to the ground state (stimulated emission, SE). Usually, complicated TA spectra by overlapping ESA and SE signals are observed depending on the electronic structure of the involved excited states. In an AuNR, a SPR excitation by the pump induces a coherent and collective oscillation of conduction electrons. Then, electron dephasing and heating occur by electron-electron collisions. Hot electrons generated by the electron heating (e-heating) further undergo a heat dissipation to the phonon and bath modes. c Spectral red-shift and line broadening of the LgSPR band of AuNRs by electron heating. d TA spectrum (ΔA(t)) obtained at a pump-probe delay time t after the electron heating is done. Due to the spectral red-shift and broadening, ΔA has positive and negative values at the longer and shorter wavelengths, respectively, which is the characteristic TA spectral feature for the LgSPR band of AuNRs

6.3 Transient Absorption

For ideally monodisperse AuNRs, the LgSPR bandwidth is dominated by homogeneous line broadening. All the nanorods therefore undergo the same SPR excitation-and-subsequent relaxation processes, giving rise to the same TA spectral change regardless of pump (excitation) frequency. However, for an ensemble of highly polydisperse AuNRs with different aspect ratios, the LgSPR band is inhomogeneously broadened so that the AuNRs with smaller R are dominantly excited by higher frequency excitation fields while those with larger R are excited by lower frequency fields (Fig. 6.4a). In this case of an inhomogeneous line broadening limit , the entire LgSPR band can be represented by a collection of homogeneously broadened LgSPR bands with different center frequencies. If a pump (excitation) beam with a bandwidth narrower than that of the LgSPR band is used in a pump-probe TA experiment, as shown in Fig. 6.4a, selective excitations of just subensembles of the AuNRs are achieved [28].

Fig. 6.4

This figure is adapted from Fig. 6.1 in [28]

a LgSPR absorption band (black solid line) of silica-coated AuNRs with the average R = 3.84 and the pump laser spectra (dashed line) of different center wavelengths (blue: 720 nm, green: 760 nm, red: 800 nm) used for TA measurement. b TA spectra measured at 1 ps of pump-probe delay time (T_w) with different pump center wavelengths in (a).

Figure 6.4b shows the TA spectra of AuNRs excited with different pump center wavelengths (λ_pump = 720, 760, 800 nm in Fig. 6.4a) at T_w = 1 ps of pump-probe delay (waiting) time. Note that the pump spectral bandwidth is narrower than the LgSPR bandwidth of the silica-coated AuNRs (length: 35–41 nm, width: 9–11 nm, average aspect ratio: 3.84) used in the TA measurement [28]. All the individual TA spectra exhibit negative and positive TA signals at the higher and lower frequency regions, respectively, with the nodal point red-shifted from the excitation wavelength, which is the characteristic pump-probe TA spectral signature created by electron heating after a femtosecond excitation of LgSPR modes as shown in Fig. 6.3. It is noteworthy that the nodal point of the TA spectrum red-shifts as the excitation wavelength increases. This indicates that the LgSPR band is significantly inhomogeneously broadened so that the subensembles of AuNRs with different aspect ratios, which are selectively excited with varying pump wavelength, produce their own TA spectra with different nodal points. Another notable feature of Fig. 6.4b is that the positive TA signal is much smaller than the negative one when the sample is excited at the blue edge (λ_pump = 720 nm) of the LgSPR band. This implies that inhomogeneous broadening is still dominant even within a pump bandwidth narrower than the entire LgSPR band. That is to say, the positive TA signal from a subensemble of AuNRs excited at the blue edge of the pump spectrum interferes with the negative TA signal from another subensemble of AuNRs excited at the red edge, leading to a significant cancellation of the TA signal [28]. It should be noted that one-dimensional (1D) pump-probe TA spectroscopy , however, does not provide detailed information on how significantly the LgSPR band is inhomogeneously broadened within the pump bandwidth. The next section will show how the degree of aspect ratio inhomogeneity of AuNRs can be investigated by analyzing the slope of the nodal line on each 2D ES spectrum.

6.4 Two-Dimensional Electronic Spectroscopy

6.4.1 Nodal Line Slope Analysis: Basic Concept

In principle, a 2D ES spectrum can be represented by a collection of TA spectra (y-axis) of the probe beam as the excitation wavelength or frequency (x-axis) is varied. Due to electron heating after LgSPR excitation by the pump beam (Fig. 6.3b), the band undergoes both the spectral red-shift and line broadening , producing positive and negative TA signals at lower and higher probe frequency regions, respectively, as shown in Fig. 6.3c, d. For a homogeneous system with AuNRs of uniform aspect ratio, the TA spectra (y-axis) created by varying the pump wavelength (x-axis) would, regardless of pump wavelength, all show the same spectral shape and nodal point wavelength (NPW) , which is the probe wavelength at which the nodal line crosses the vertical line at the pump center wavelength [28]. As a result, the nodal line connecting all the nodal points of the TA spectra would be parallel to the pump wavelength axis (x-axis), giving NLS = 0. However, as inhomogeneous broadening becomes significant, the nodal line slope (NLS) increases depending on the aspect ratio inhomogeneity of AuNRs and should have finite values larger than zero.

The simulation results in Fig. 6.5 show that 2D ES is capable of distinguishing between ensembles of AuNRs with different aspect ratio distributions (ΔR: standard deviation of R) and how the NLS on the 2D ES spectra varies with the degree of aspect ratio inhomogeneity (β). For simplicity, we assume that the absorption spectra (Figs. 6.5a–c) can be given by a Voigt function , which is the convolution of homogeneously (Lorentzian) and inhomogeneously (Gaussian) broadened line shape functions. It should be noted that β is defined as the ratio of the inhomogeneous (Δω_in) to homogeneous (Δω_h) line broadenings, i.e., β = Δω_in/Δω_h, where Δω_h and Δω_in are the full widths at half maximum (FWHM) of the corresponding Lorentzian and Gaussian functions, respectively. Three cases of AuNR systems with large (β = 3), intermediate (β = 1) and small (β = 0.33) ΔR are compared in Fig. 6.5. For β = 0.33, where homogeneous broadening dominates inhomogeneous broadening , the nodal line is almost parallel to the pump frequency axis (x-axis) so that the NLS value is close to zero (NLS = 0.11). However, the NLS sensitively varies with increasing β and approaches unity as Δω_in becomes much larger than Δω_h (NLS = 0.87 for β = 3). This indicates that the NLS of 2D ES can be a decisive measure of the aspect ratio inhomogeneity of AuNRs [28].

Fig. 6.5

This figure is from Fig. 6.3 in [28], which was published under a Creative Commons license

Simulation on a–c LgSPR bands, d–f 2D ES spectra and nodal line slopes of AuNRs with different aspect ratio inhomogeneity: Large (β = 3), intermediate (β = 1) and small (β = 0.33) ΔR. The homogeneous (blue solid) and inhomogeneous (red dashed) line broadenings contributing to the LgSPR band are assumed to be Lorentzian (Δω_h: FWHM) and Gaussian (Δω_in: FWHM) functions, respectively. The degree of inhomogeneity is defined as the ratio of Δω_in to Δω_h, i.e., β = Δω_in/Δω_h. Calculating 2D ES spectra, it is assumed that each homogeneous band undergoes a red-shift of the peak by 30 cm⁻¹ and a spectral broadening by 2% of the original bandwidth by the electron heating of AuNRs. g Slice spectra (black dashed lines: 1–5 in (d)) taken at different pump frequencies of the 2D ES spectrum for β = 3. h Projected (integrated) spectra of the 2D ES in d–f onto the probe frequency axis (y-axis), which are essentially equivalent to the pump-probe TA spectra, for β = 3 (solid), 1 (dashed), 0.33 (dash-dotted).

One of the advantages of 2D ES over 1D pump-probe TA spectroscopy is its capability of resolving each individual spectral signature and electron dynamics of a given homogeneous subensemble in a heterogeneous system. In 2D ES, AuNRs with different aspect ratios are selectively photo-excited by varying pump frequency, so their pump-induced TA spectra (y-axis) can spread out along the pump frequency axis (x-axis). The slice probe spectra of AuNR under β = 3 (a highly inhomogeneous case) taken at different pump frequencies (1–5 in Fig. 6.5d) of the 2D ES spectrum represent the individual TA spectra of AuNR subensembles and are plotted together in Fig. 6.5g. The nodal points extracted from the individual TA spectra of AuNRs with different aspect ratios are clearly separated and red-shift with pump frequency. However, such spectral features disappear in 1D pump-probe TA spectra because the contributions from different-sized AuNRs are averaged out. Indeed, in the spectra projected onto the probe frequency axis, which are equivalent to the pump-probe TA spectra, the nodal point frequencies hardly depend on the degree of aspect ratio inhomogeneity at all and are almost insensitive to β (Fig. 6.5h). This indicates that using pump-probe spectroscopy it is difficult to separately obtain the spectral signatures of different-sized AuNRs or investigate their relaxation dynamics simultaneously. In Sect. 6.4.3 we will show that the time evolutions of the NLS and nodal point wavelength (NPW) extracted from the 2D ES spectra enable us to investigate the homogeneous (heterogeneity-free) dephasing or relaxation of photo-excited AuNRs.

6.4.2 Pulse Shaper-Based 2D ES Setup (Pump-Probe Geometry)

By varying waiting times (T_w), two-dimensional electronic spectra are plotted with respect to two (independently controllable) frequencies that correspond to the excitation (ω_pu) and emission (ω_pr) axes. In general, there are two ways to plot a 2D spectrum in the frequency and time domains: one (frequency domain) is to measure broadband probe (emission) spectra while varying the frequency (ω_pu) of the narrowband pump (excitation) beams, and the other (time domain) is to measure broadband probe spectra while scanning the time delay (τ) between two broadband excitation pulses (twin pulses generated by a pulse shaper in our case). Current 2D spectroscopy experiments usually employ the two-dimensional Fourier transform (FT) method because the frequency domain method, which utilizes temporally broad pump pulses, inevitably loses the time resolution with respect to T_w.

Our pulse shaper -based FT 2D ES setup in a two-beam pump-probe geometry is depicted in Fig. 6.6a, where a pair of co-propagating replica pulses (E_pu1 and E_pu2) from the pulse shaper is used as two pump pulses and another time-delayed (noncollinearly propagating with the pump pulses) pulse (E_pr) from E_pu2 by T_w (waiting time) is used as a probe pulse [26,27,28]. The role of scanning τ, controlled by the pulse shaper, is essentially to give the same effect as varying the excitation frequency (ω_pu) as follows. Each broadband excitation pulse, E_pu1(ω_pu) and E_pu2(ω_pu), can be decomposed into constituting electric field components with different optical frequencies (ω_n). As τ is scanned, τ-varying interference between these electric field waves from the two excitation pulses results in intensity modulations I(ω_n,τ) with the corresponding frequencies. Consequently, the individual AuNRs with different aspect ratios (R_n) are selectively excited with each I(ω_n,τ). That is to say, they are individually tagged with the intensity modulations at different temporal periods with respect to τ. After waiting time T_w, the excited hot electrons of the AuNRs are probed by the time-delayed E_pr(ω_pr), which is spectrally dispersed by a monochromator and then detected with a CCD (charge-coupled device) detector. Finally, the FT of the τ-dependent interferogram S(τ,T_w,ω_pr) with respect to τ provides the 2D correlation spectrum, S(ω_pu,T_w,ω_pr), between ω_pu (excitation or pump) and ω_pr (emission or probe) frequencies [21]. In summary, the role of the τ-scan-and-FT of the signal in our method is to provide a way (1) to spectrally resolve broadband pump beams, then (2) to individually excite the AuNRs with different aspect ratios, and finally (3) to spread the correlated emission spectra (ω_pr) with each excitation frequency (ω_pu) on the 2D frequency axes [28].

Fig. 6.6

This figure is adapted from Fig. 6.4 in [28]

a Pulse shaper-based 2D ES setup in pump-probe two-beam geometry. E_pu1,2: twin pump pulses generated from the pulse shaper, E_pr: probe pulse, CCD: charge-coupled device detector. b 2D ES spectra of the AuNRs measured at T_w = 1 ps varying the center wavelengths of both the pump and probe pulses from 700 nm to 820 nm. The thick black line in each panel represents a collection of nodal points at which the 2D signal is zero and the red line is the linear fit to the nodal points. c Wavelength dependence of the NLS (black circles) of the 2DES spectra of the AuNRs, which well fits to a Gaussian line shape with an FWHM of 1320 cm⁻¹ approximately matching the LgSPR band shown in Fig. 6.4a. The individual homogeneous LgSPR peaks (gray lines) for AuNRs with different aspect ratios are drawn together to show an unequal distribution of aspect ratio inhomogeneity across the LgSPR band. A more diverse and denser distribution of AuNRs with different aspect ratios around the center of the LgSPR band (red shaded area) leads to a more heterogeneous excitation of AuNRs by the relatively narrowband pump than at the band edges (blue shaded area), producing a higher NLS (NLS ~1 at 780 nm, NLS ~0 at 700 nm).

6.4.3 2D ES Spectra and Nodal Line Slope Analysis: Experiment

In general, it is difficult to obtain detailed information on the dimensional distribution of the AuNR system only from the 1D absorption or TA spectra. As shown in the simulations (Fig. 6.5), however, the 2D ES provides a way to judge the degree of aspect ratio inhomogeneity of AuNRs using the nodal line slope (NLS) of the 2D ES spectrum measured at a particular waiting time. Figure 6.6b displays the 2D ES spectra and the NLS of the silica-coated AuNRs measured at various pump center wavelengths (λ_{pump_c} = 700 ~ 820 nm) at T_w = 1 ps, which corresponds to the waiting time after the completion of electron heating . Note that the bandwidth of the pump beam used is narrower than the LgSPR band under investigation, allowing a selective examination of the aspect ratio inhomogeneity of the subsets of the AuNRs (parts of the LgSPR band) excited by the pump. The NLS values (black circles) in Fig. 6.6c, which are obtained from a linear fit (red line) to a number of nodal points (black points) around λ_{pump_c} in Fig. 6.6b, vary with the pump center wavelength according to a Gaussian-shaped dependence, which approximately follows the absorption line shape of the LgSPR band. The NLS value has a maximum (close to unity) around 780 nm, which indeed corresponds to the center wavelength of the LgSPR band while it decreases as λ_{pump_c} approaches the red or blue edge of the LgSPR band. These unequal NLS values across the LgSPR band indicate that the degree and nature of the aspect ratio distributions of the individual AuNR subensembles are different significantly at both the center and the two edges of the LgSPR band. The NLS value being close to unity at the LgSPR band maximum means that the inhomogeneous broadening of the AuNR system investigated is much larger than the homogeneous one. That is to say, a subset of the AuNRs around the LgSPR band maximum (red shaded area in Fig. 6.6c), which is selectively excited by the relatively narrowband pump, has a diverse and dense distribution of homogeneous subensembles of AuNRs (gray lines in Fig. 6.6c), so that the pump causes a considerably heterogeneous excitation of the AuNRs. On the other hand, a less heterogeneous excitation of the AuNRs is induced by the pump tuned to the band edge (blue shaded area in Fig. 6.6c) because the aspect ratio distribution of the AuNRs is more discrete and less dense at either edge of the LgSPR band.

In addition to the spectral dependence of NLS due to an unequal aspect ratio distribution across the LgSPR band, it is noteworthy that the NLS and NPW are waiting time-dependent, as is apparent in Fig. 6.7. Since the NLS and NPW are free from inhomogeneous line broadening contributions, their time-dependent changes are direct signatures of homogeneous relaxation dynamics after photo-excitation of the LgSPR modes of AuNRs. The transient behaviors of the NLS and NPW at T_w < 300 fs can be attributed to ultrafast electron dynamics, such as the dephasing of the coherent oscillations of plasmonic electrons and rapid electron heating in each AuNR, which are induced by the e-e scattering (collision) process [17, 28]. On the other hand, the subsequent monotonic decay of the NLS and the slow blue-shift of the NPW over similar time scales (a few ps) result from homogeneous dephasing due to e-ph couplings. These heterogeneity-free electron-electron (e-e) or electron-phonon (e-ph) relaxation dynamics of AuNRs cannot be observed in conventional pump-probe spectroscopy because the corresponding pump-probe signals result from averaging all the transient dynamics and relaxations of the entire heterogeneous AuNRs.

Fig. 6.7

This figure is adapted from Fig. 6.5 in [28]

a Time evolution of 2D ES spectra measured with the pump center wavelength of λ_{pump_c} = 760 nm at different waiting times (T_w = 0.05 ps, 0.1 ps, 0.2 ps and 1 ps). b Time-dependent nodal line slope (NLS, red circle) and nodal point wavelength (NPW, blue square) changes of the 2D ES spectra in (a) as a function of waiting time (T_w). Here, the NPW is the probe wavelength at which the nodal line and vertical dashed line corresponding to λ_{pump_c} = 760 nm in (a) are crossing with each other.

6.4.4 Comparison Between Nonlinear Signals in Positive and Negative Waiting Times

2D ES is a four-wave-mixing (4WM) process involving three field-matter interactions with incident pulses and one field-matter interaction with the signal field. In the photon echo geometry often used in 2D ES experiments, where three laser beams are noncollinearly incident to the sample, a third-order signal field is generated in a specific direction different from those of the incident beams due to the corresponding phase-matching condition . Then, the thus generated 2D ES signal field is heterodyne-detected by allowing it to interfere with another laser field called a local oscillator (LO) [19,20,21]. The amplitude and phase of the 2D ES signal electric field can be measured with respect to the delay times between the involved laser pulses. In pump-probe-type 2D ES, on the other hand, the generated signal field interferes with the collinearly propagating incident laser field (probe) after the sample and, consequently, only the absorptive part of the interference term is detected. This technique is often called self-heterodyne-detection [26,27,28,29]. Below we compare the observables in normal (positive waiting time, T_w > 0) and reverse (negative waiting time, T_w < 0) time-orderings of the twin pump pulses and one probe pulse, and explain why spectral fringe patterns are observed only in the negative waiting time domain.

Normal 2D ES measurement (T_w > 0). In the pump-probe-type 2D ES experimental setup shown in Fig. 6.8a, collinearly propagating twin pump pulses (pu1 and pu2) originate from the pulse shaper , which controls the time delay (τ) between them, and are injected into the sample. For a normal time-ordered pulse sequence, the probe pulse (pr) is delayed from pu2 by the waiting time T_w, i.e., T_w> 0. After three field-matter interactions with the sample in the order of pu1→pu2→pr with wavevectors of k_pu1, k_pu2 and k_pr, respectively, the 2D ES signal field (k_s) is generated in the same direction as the probe wavevector (k_s = −k_pu1 + k_pu2 + k_pr = k_pr) and interferes with the probe beam itself (Fig. 6.8b). The interference term between the 2D ES signal field and the probe beam is then detected by a spectrometer . The 2D ES spectra measured at positive waiting times (T_w > 0) exhibit both positive and negative peaks with a nodal line separating the two, as shown in Fig. 6.6b.

Fig. 6.8

Comparison between pulse sequences and generated third-order signals (2D ES and TG signals) before and after field-matter interaction in a–b normal (T_w> 0) and c–d reverse (T_w < 0) time orderings of twin pump and probe pulses. At positive waiting times (T_w > 0), the 2D ES signal interfering with the probe pulse is normally detected. In contrast, another third-order signal called transient grating (TG), which results from a diffraction of the time-delayed second pump pulse (pu2) due to a TG in the sample created by the first pump (pu1) and probe (pr) pulses, is heterodyned detected with the preceding probe pulse at negative waiting times (T_w < 0)

Transient grating at negative waiting times (T_w < 0). Now, let us consider another pulse sequence and 4WM field-matter interactions at negative waiting times (T_w < 0), as schematically shown in Figs. 6.8c–d. From the definition of the waiting time, which is the delay time of the probe pulse (pr) from the second pump pulse (pu2), the probe pulse (pr) precedes the second pump pulse (pu2) by |T_w| at negative waiting times. When the delay time τ is scanned from 0 to a finite time for the 2D ES measurement, the pump 1 (pu1) and probe (pr), which propagate differently in space, can overlap in time and thus interact with the AuNRs to generate a transient spatial grating (TG) inside the sample due to a non-parallel overlap of the wave fronts of the pu1 and pr fields [30]. The following second pump pulse (pu2) is then diffracted by the created TG (k_TG) in the same direction as the probe wavevector (k_TG =  k_pr), and consequently the TG signal field can interfere with the preceding probe field by |T_w|. The time delay between the probe pulse and the generated TG signal field is fixed at |T_w| regardless of the scanning delay time τ. Therefore, one can measure the spectral interference between those time-separated fields (TG signal and probe), which produce a spectral fringe pattern with a fringe spacing of 1/|T_w| along the probe frequency axis (y-axis) in a given 2D ES spectrum. The following section gives further detailed descriptions of how the TG can create a spectral fringe pattern on the 2D ES spectrum and explain how the spectral interference is related to the aspect ratio inhomogeneity of AuNRs.

6.4.5 Spectral Interference by Transient Grating (TG) in 2D ES

The TG is a temporal and spatial grating created by spatially regular and interferometric SPR excitation across the sample, which is induced by noncollinear mixing of the pu1 and pr fields into the sample at T_w < 0, as shown in Figs. 6.8c–d. During scanning the delay time (τ) between the pu1 and pu2, two pulse sequences, pr → pu1 → pu2 (k_TG =  k_pr − k_pu1 + k_pu2) and pu1 → pr → pu2 (k_TG = −k_pu1 + k_pr +  k_pu2), are involved, which have the same phase-matching condition for the TG, i.e., k_TG =  k_pr at a fixed negative waiting time, T_w. Consequently, the TG signal field spectrally interferes with the preceding probe pulse by |T_w| in the probe frequency domain (ω_pr). Since the spatial phase of the TG across the sample sensitively varies with the time delay (τ′  =  |T_w| − |τ|) between pu1 and pr and oscillates at optical frequencies (ω_τ′) of the incident electric fields (pu1 and pr) with respect to τ′, scanning τ′ results in the temporal modulation of the spectral interference signal between the pu1 and pr fields. Fourier transformation of this modulated spectral interferogram , S(τ′, T_w, ω_pr), with respect to τ′ yields the spectral interferometric 2D ES spectra, S(ω_τ′, T_w, ω_pr).

Figure 6.9a shows the spectral fringe patterns on the experimentally measured 2D ES spectra of the silica-coated AuNRs with an inhomogeneous aspect ratio distribution (length: 35–41 nm, width: 9–11 nm, average aspect ratio: 3.84) at T_w = −150 fs, −50 and 0 fs. One of the characteristic features of those interference patterns is the fringe spacing increasing according to 1/|T_w| as |T_w| goes to zero, which is also clearly shown in the slice spectra (Fig. 6.9b) taken at the center frequencies (vertical solid lines) of the 2D ES spectra in Fig. 6.9a. To further investigate the |T_w|-dependence of the spectral interference , one can perform a Fourier transformation of the spectral fringe patterns in Fig. 6.9b to analyze the resulting time domain signals with respect to the waiting time (Fig. 6.9d). Note that when two pulses are separated in time by t_d, the Fourier transformed time domain signal from the spectral fringe created by the two signal pulses peaks at t = t_d [31]. Although the Fourier transformed signals are relatively broad and do not show ideally sharp peaks in time due to the finite chirp of the laser pulses used in experiment, the first moment of the Fourier transformed time domain signal can give meaningful information on the average delay time between the signals involved in spectral interference . As expected, the t_d-versus-T_w plot in the inset of Fig. 6.9d shows that the average delay time (t_d) decreases in linear proportion to |T_w| at negative waiting times. This experimental observation is in good agreement with the explanation about the origin of spectral interference above, which is based on the TG-induced diffraction of the second pump pulse (pu2) that is time-delayed from the probe (pr) by |T_w|.

Fig. 6.9

a Spectral interference patterns in the 2D ES spectra of AuNRs measured at waiting times of T_w = −150, −50, and 0 fs. The center wavelengths of both the pump and probe pulses are 770 nm. b Slice spectra (vertical lines) taken at ω_pump = 13,000 cm⁻¹ (λ_pump = 770 nm) in the 2D ES spectra of (a). c Comparison between the pump-probe TA (black lines) and projected (red lines) spectra onto the probe frequency axis (y-axis) in the 2D ES spectra measured at the corresponding waiting times (T_w). d Fourier transformed signals of the slice spectra in (b) at ω_pump = 13,000 cm⁻¹ (λ_pump = 770 nm). Inset: First moment (t_d) of the time-domain signal in (d) as a function of waiting time (T_w)

Another notable feature of the spectral interference on the 2D ES spectra of the AuNR is the diagonal elongation of the fringe pattern (Figs. 6.9a and 6.10a), which is in close connection to the aspect ratio inhomogeneity of the AuNRs. The strong SPR effect of the AuNRs, created by the interfering pu1 and pr fields, significantly modulates the refractive index in a regular spacing across the sample. This spatially modulated refractive index not only gives rise to the diffraction of the pu2 beam in space, but can also further time-delay the TG signal field from the probe pulse by modulating its velocity at the sample. Furthermore, since with varied pump frequencies (ω_pump) the TG-inducing beams (pu1 and pr) can individually excite AuNR subensembles with different aspect ratios (R_i) and therefore different effective refractive indices (n_eff,_i), the delay time (T_w,i) between the TG and probe pulse is dependent on pump frequency (ω_pump). Thus, T_w,i is larger at lower ω_pump (Fig. 6.10c). As a consequence, as shown in Fig. 6.10b, the spectral interferogram along the probe frequency axis is phase-shifted depending on ω_pump, i.e., the aspect ratio (R_i) of the AuNRs, producing the diagonally elongated fringe pattern on the 2D ES spectrum [28]. However, this spectral fringe, which contains information on the inhomogeneous dimensional distribution of the AuNRs investigated, disappears in the pump-probe spectra (black lines in Fig. 6.9c), which is equivalent to the average spectra (red lines in Fig. 6.9c) projected onto the probe axis, because the oscillating positive and negative signals on the 2D spectrum along the pump frequency axis are averaged out.

Fig. 6.10

This figure is adapted from Fig. 6.6 in [28]

a 2D ES spectrum of the AuNRs measured at T_w = −100 fs. b Slice spectra taken at different pump frequencies, ω_pump = 12,860, 13,000 and 13,175 cm⁻¹ (vertical dashed lines in (a)), which excite different-sized AuNRs with R_j, in the 2D ES spectrum at T_w = −100 fs. c Pump frequency dependence of time delay between the TG signal and probe pulse at negative (reverse time ordering in Fig. 6.8c–d) waiting times (T_w < 0). The spatially modulated refractive index by the strong SPR effect of AuNRs creates a TG in the sample, which can lead to a further time delay (T_w,j) of the TG signal from the preceding probe pulse by modulating the velocity of the TG signal field. This time delay (T_w,j) is pump frequency dependent because the effective refractive indices (n_eff,j) induced by the TG change with the size of the AuNRs with different aspect ratios (R_j), which are individually excited by different pump frequencies.

6.5 A Few Concluding Remarks

In the present chapter, we presented our recent 2D ES results for AuNRs featuring a broad dimensional distribution. Although a vast quantity of literature has been published reporting the optical properties of AuNR with respect to size, aspect ratio, surface ligand, shape, and so on, measured using linear absorption spectroscopy and the time-resolved pump-probe (or transient absorption) measurement method, these spectroscopic techniques cannot provide incisive information about homogeneous dephasing processes involving e-e, e-ph, and ph-ph scatterings. This is because those spectroscopic observables are intrinsically not free from the dimensional heterogeneity of AuNRs in condensed phases. Our 2D ES studies of AuNRs with selective excitations of LgSPR modes provide quantitative information about the homogeneous relaxation dynamics of gold nanorods excited by femtosecond laser pulses. Despite the success of 2D ES application to the study of heterogeneous AuNR samples, due to the pump pulse bandwidth being narrower than the entire LgSPR bandwidth, we could not obtain the entire 2D electronic spectra of the AuNRs used. It will be of great interest to carry out a 2D ES study of the same sample with continuum pump and probe pulses so that the entire ensembles of the AuNRs are excited simultaneously and their ultrafast electron dynamics can be probed with sub-10 fs pulses in real time. In the future, a two-quantum 2D ES technique that utilizes a slightly different phase-matching condition would be useful for studying the large-amplitude longitudinal electronic oscillations of quasi-free electrons in AuNRs and other metallic nanoparticles, and could reveal unexpected electron correlations in such multi-electronic excitation systems.