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Abstract
Surface phonon polaritons (SPhPs) supported by polar dielectrics have been a promising platform for nanophotonics in mid-infrared spectral range. In this work, the temporal dynamic behavior of polar dielectric nanoparticles without (or with) spatial dispersion/nonlocality driven by the ultrashort Gaussian pulses is carried out. We demonstrate that three possible scenarios for the temporal evolutions of the dipole moment including ultrafast oscillations with the decay, exponential decay, and keeping a Gaussian shape exist, when the pulse duration of the incident field is much shorter than, similar to, and much longer than the localized SPhP lifetime. Once the nonlocal effect is considered, the oscillation period becomes large slightly, and the exponential decay turns fast. Furthermore, nonlocality-induced novel temporal behavior is found such as the decay with long-period oscillations when the center frequency of the incident pulse lies at the frequency of adjacent longitudinal resonant modes. The positive and negative time-shifts of the dielectric response reveal that the excitation of the dipole moment will be delayed or advanced. These temporal evolutions can pave the way towards potential applications in the modulation of ultrafast signals for the mid-infrared optoelectronic nanodevices.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Nanophotonics has great advantages on controlling light in nanostructures and even breaking the diffraction limit [1]. Due to the high loss in conventional materials adopted in nanodevices, searching for the low-loss optical nanomaterials has become a new research hotspot. Polar dielectrics which could excite surface phonon polariton (SPhP) due to the coupling of the electromagnetic fields and the transverse optical phonons in Reststrahlen region (between the transverse and longitudinal phonon frequencies) with low optical loss is currently a promising material for this field. Notably, the polar dielectrics support longitudinal optical (LO) phonons as well [2], which is analogous to longitudinal plasma waves in noble metals [3–5]. It has been demonstrated that the excellent properties of polar dielectrics have led to a wide range of applications in many fields, including nonlinear optics [6,7], optical imaging [8,9], and optoelectronic devices [10–12]. The study of polar dielectrics not only optimizes conventional optical devices but also reveals new perspectives and phenomena in nanophotonics.
In recent years, significant progress has been made in the study of polar dielectrics. SPhPs have special optical properties in the mid-infrared spectral region, including enhanced near-field light-matter interaction [13], effect of charges on the SPhP modes [14], hyperbolic phonon polaritons in van der Waals materials [15–17], and using proton irradiation to tailor the SPhPs [18]. As SPhP devices grow smaller, when the scale approaches the incident wavelength, the nonlocal effect becomes prominent, i.e., the dielectric response of the polar dielectrics exhibits spatial dispersion. The nonlocality will affect the optical response of polar nanoparticles [19,20], polariton evolution of polar metasurfaces [21], as well as the field enhancements of polar resonators [22,23]. This enables mid-infrared light-generation [24] due to the existence of longitudinal-transverse polariton (LTP) that arises from the strong coupling of SPhPs and LO phonons [25–27]. As one knows, most of these researches on polar dielectrics are limited to the monochromatic continuous incident wave, and the combination of ultrashort incident pulses with polar dielectrics is still lacking. Actually, temporal dynamic response leads to a wealth of new phenomena, and many studies have been carried out in plasmonic systems. Some typical examples are the use of ultrashort pulses to realize nanoparticle shape transformations [28], plasmon and magnetic resonance dynamics [29], Fano and Rabi spectral responses [30] and so on. Moreover, complicated dynamic features may bring many photoelectric applications, such as plasmonic nanowire lasers [31], nonlinear circuitry [32], optical nanoantennas [33,34] and near-field imaging [35]. More recently, Jiang et al investigated the nonlocal effects on the temporal dynamics of plasmonic nanoparticles [36]. However, to the best of our acknowledge, the study of the temporal dynamics in polar dielectrics nanoparticles has not been carried out yet.
In this work, we shall investigate the optical response of polar dielectric nanoparticles illuminated by an ultrashort Gaussian pulse when the nonlocality is introduced. We show that the far-field and near-field behavior of polar dielectric nanoparticles is dramatically different from that of conventional plasmonic materials. The trends of dipole moment enhancement with time under variable central frequency and duration time of pulse are theoretically calculated. Specially, we highlight the impacts of nonlocal effects on temporal dynamics of the proposed polar dielectric nanoparticles, and nonlocality-induced novel dynamic behavior is predicted in polar dielectric nanoparticles with spatial dispersion/nonlocality. We believe that the SPhP resonance dynamics in polar dielectric nanomaterials may lead to a wealth of new phenomena and applications in nanophononics.
2. Theoretical model
As shown in Fig. 1, a Gaussian pulse is lighting upon the 3C-SiC sphere with radius a. Without loss of generality, we solve all problems in the quasistatic approximation since the incident wavelength $\lambda $ and particle size a satisfy the condition $ka \ll 1$, where $k = 2\pi /\lambda $ is the wave vector. The transverse permittivity of the polar dielectrics reads [37],
Fig. 1. Schematic of a 3C-SiC polar dielectric nanoparticle with radius a illuminated by an incident Gaussian pulse.
For the Gaussian pulse incidence, we assume that it is polarized along the $x$ axis and propagates along $z$ axis, and the incoming pulse satisfies the Helmholtz wave equation
Under the paraxial approximation, the pulse can be written as the product of a spatial and a temporal part,
Here, ${\textbf E}({\textbf r})$ represents the spatial part of the solution of Eq. (3), which can be simplified as a uniform field within the quasistatic approximation. ${E_0}$ is the pulse amplitude with a constant value due to the fact that the size of the nanoparticle is much smaller than the beam spot of pulse. The second term $E(t)$ is the temporal part, whose form is given by a Gaussian pulse time profile
where ${\alpha _p} = 4\ln (2)/{\tau _{\textrm{pulse}}}$ is related to the pulse duration ${\tau _{\textrm{pulse}}}$ and ${\omega _c}$ is the central pulse frequency.In order to derive the dipole response to the Gaussian pulse, we adopt the Fourier transform to transform the temporal part of the pulse into frequency domain as follows,
In frequency domain, the incident field Eq. (4) can be written as
Based on the semiclassical infinite barrier (SCIB) model [39], we can obtain the electric potentials inside and outside nanoparticle with appropriate boundary conditions as
Actually, if one neglects the nonlocal nature of the polar dielectrics, i.e., ${\varepsilon _L}(k,\omega ) = {\varepsilon _T}(\omega )$, ${\varepsilon _{eq}}$ is naturally reduced to ${\varepsilon _T}(\omega )$ in Eq. (9). This is just for the local case.
Next, we mainly focus on the dipole moment ${\textbf p}(\omega )= \alpha (\omega ){\textbf E}(\omega )$ induced by the external field, where $\alpha (\omega )$ is the electric polarizability with the form,
Then, the temporal dipole moment in time domain is given by,
For simplicity, we adopt the dimensionless temporal dipole moment $\tilde{p}(t )= p(t)/(4\pi {\varepsilon _0}{\varepsilon _2}{a^3})$ in the following calculations.
Then, the extinction cross section for the polar dielectric nanosphere can be written as [41]
3. Results and discussion
In what follows, we have carried out some numerical calculations based on the proposed theoretical model. The relevant physical parameters for nonlocal 3C-SiC nanosphere are high-frequency permittivity ${\varepsilon _\infty } = 6.52$, transverse optical (TO) phonon frequency $\omega _T^{} = 796.1\textrm{ }c{m^{ - 1}}$, longitudinal optical (LO) phonon frequency $\omega _L^{} = 973\textrm{ }c{m^{ - 1}}$ [2], damping rate $\gamma = 4\textrm{ }c{m^{ - 1}}$, and nonlocal velocity $\beta _L^{} = 15.39 \times {10^5}\textrm{ }cm/s$. For comparison, the parameters of the metal can be referred to [29] and the radius of the sphere is 5 nm. For simplicity, the nanoparticles are assumed to be placed in vacuum with the permittivity ${\varepsilon _2} = 1$.
Figure 2 reveals the extinction efficiency of the 3C-SiC nanosphere with the radius $a = 5nm$. In the local case, it exhibits a dipolar Fröhlich resonance at the Fröhlich resonant frequency ${\omega _F}$ with ${\varepsilon _T}(\omega = {\omega _F}) ={-} 2$, and the main resonant peak is due to the transverse SPhP resonance within the polar nanosphere. When the nonlocality is taken into account, the longitudinal modes inside the polar dielectric nanosphere lead to smaller and red-shifted main peaks, as well as a series of discrete additional peaks that appear for $\omega < \omega _L^{}$, which is different from the one for the plasmonic systems (see the insert in Fig. 2). Here, the predicted behavior is quite in good agreement with that predicted based on full-wave Mie theory [19]. Since the propagative optical longitudinal phonons coexist with the region in which the dipolar Fröhlich resonance takes place, one observes discrete propagating modes around the same spectral region as SPhPs.
Fig. 2. The extinction efficiency ${Q_{ext}}$ of 3C-SiC nanoparticle as a function of the incident frequency $\omega$ in nonlocal (blue) and local (red) cases respectively. The inset illustrates the case of Ag nanoparticles.
We plot the spatial distribution of electric field intensity inside the polar dielectric nanoparticles in Fig. 3 for both local and nonlocal cases with different frequencies, at which the additional and main resonant peaks take place, as denoted in Fig. 2. It is evident that the field is enhanced at the nanosphere edge due to the excitation of surface phonon polariton in the local case (see Fig. 3(a) and 3(b)). The dipolar Fröhlich resonance [Fig. 3(b) and 3(d)] brings much higher near-field enhancements than the other discrete resonances [Fig. 3(a) and 3(c)]. Moreover, the fields outside the nanoparticles are quite similar in both cases, displaying a pattern that decreases uniformly around it. Note that the field enhancement in the polar sphere is oscillating due to the excitation of discrete longitudinal modes as the result of the nonlocal nature of the polar dielectrics. When the frequency is set to be ${\omega _{n1}} = 923\textrm{ }c{m^{ - 1}}$, at which the additional peak takes place, the field intensity oscillates more strongly than that in dipolar Fröhlich resonance due to the existence of the propagating longitudinal modes (see Fig. 3(c) and 3(d)). Such an extraordinary phenomenon may directly lead to the design of high-efficient mid-IR enhance emitters.
Fig. 3. Spatial distribution of near-field intensity for the 3C-SiC nanoparticles under the monochromatic wave with (a),(c) ${\omega_{n1}} = 923\textrm{ }c{m^{ - 1}}$ and (b),(d) ${\omega_F} = 934\textrm{ }c{m^{ - 1}}$. The color scale denotes the electric-field enhancement, with the upper panel for local cases and the lower panel for nonlocal cases.
Next, we would like to investigate the temporal dipole moment in time domain and characterize the variations of the dipole moment with respect to the incident pulse. It is known that the lifetime of SPhP resonance can be extracted by fitting ${Q_{ext}}$ with a Lorentz-type curve. In this connection, one obtains that the resonance lifetime is double the reciprocal of the full wave at half maximum (FWHM), i.e., ${\tau _{SPhP}} = 2.5\textrm{ ps}$, which is orders of magnitude larger than LSPR lifetime [29]. Figure 4 illustrates the reduced dipole moment $\tilde{p}(t)$ with different durations of the incident Gaussian pulse ${\tau _{\textrm{pulse}}} = 0.02,2.5,25\textrm{ ps}$, which respectively correspond to much smaller than, similar to, and much larger than the SPhP resonance lifetime. We can observe that the excitation of the dipole response has almost no delay, but there are oscillations with small periodicity in the decay when the duration of the driving pulse is much shorter than the lifetime of SPhP resonance as shown in Fig. 4(a). To reveal the characteristics of oscillation, we analytically solve the temporal response of the dipole moment in the local case, and Eq. (11) can be represented as a convolution of functions $u$ and v,
Fig. 4. The time evolution of the reduced dipole moment for local (red line) and nonlocal (blue line) cases. The central frequency equals the Fröhlich resonant frequency ${\omega _F} = 934\textrm{ }c{m^{ - 1}}$ and durations of pulse are set as (a) 0.02 ps, (b) 2.5 ps, and (c) 25 ps respectively. The grey line denotes the pulse whose center is located at 0 ps.
Adopting Fourier transform on Eq. (14), we have
We then obtain the period of small oscillations in the local case as $T = \pi /\tilde{\omega } = \textrm{17}\textrm{.8 fs}$ when substituting the adopted parameters into Eq. (16), and it is consistent with the period numerically measured in Fig. 4(a). In addition, Eq. (16) reveals that the oscillation is the intrinsic feature of polar dielectrics, and is independent of the incident field. It is evident that if the damping parameter $\gamma $ is small enough to be neglected, $\tilde{\omega }$ is just the Fröhlich resonant frequency. For the nonlocal case, the introduction of the nonlocal term $\beta _L^{}k$ leads to $\tilde{\omega }$ a slight red-shift as shown in Fig. 2. As a consequence, one expects the oscillation period increasing to $T = \textrm{18 fs}$ in the nonlocal case.
In Fig. 4(b), when ${\tau _{\textrm{pulse}}}$ is similar to the SPhP resonance lifetime, the delays of dipole response are remarkable compared with the driving Gaussian pulse. When the driving pulse decays quickly, the dipole moment presents slow exponential decay with time. Note that the dipole moment of the plasmonic nanostructures decreases by two orders of magnitude within 30 fs [29], however the decay of polar dielectric is much smaller than their plasmonic counterparts, due to the low optical loss and long lifetime of phonons in polar crystals. For the case of ${\tau _{\textrm{pulse}}} = 25\textrm{ ps}$, there is no time-delayed decay holding a Gaussian shape. Incidentally, nonlocal effects in polar dielectric systems will lead to smaller magnitude of the dipole moment, as expected (see Fig. 4(b) and 4(c)).
Then, let’s focus our attention on the discrete longitudinal modes caused by the nonlocal nature of the polar dielectrics. In Fig. 5, we plot the temporal dipole moment evolutions when the center frequency of the pulse is taken to be ${\omega _{n1}} = 923\textrm{ }c{m^{ - 1}}$ (grey line in Fig. 2). As can be seen in Fig. 5(a) and 5(d), when pulse durations are much smaller or larger than the SPhP resonance lifetime, as similarly discussed in Fig. 4, they present small oscillations and a Gaussian shape respectively. For the case of ${\tau _{\textrm{pulse}}} = 0.25\textrm{ ps}$, the temporal evolutions $\tilde{p}(t )$ present an exponential decay for both local and nonlocal situations. According to our fitting, a pure decay with rate $0.16\textrm{ }p{s^{ - 1}}$ is found for the local case, while the decay rate becomes $0.29\textrm{ }p{s^{ - 1}}$ for the nonlocal one. This is due to the fact that at such a central frequency, the nonlocality becomes quite important and the discrete longitudinal propagating modes take place, the equivalent damping term is found to be larger than the damping term $\gamma $, resulting in the fast decay.
Fig. 5. Dipole moment excited by Gaussian pulses centered at ${\omega _{n1}} = 923\textrm{ }c{m^{ - 1}}$ with (a) ${\tau _{\textrm{pulse}}} = 0.02\textrm{ ps}$, (b) ${\tau _{\textrm{pulse}}} = 0.25\textrm{ ps}$, (c) ${\tau _{\textrm{pulse}}} = 2.5\textrm{ ps}$, and (d) ${\tau _{\textrm{pulse}}} = 25\textrm{ ps}$ in nonlocal (blue) and local (red) cases.
To one’s interest, when the nonlocality is considered, one further observes a unique oscillatory decay with long period $T = \textrm{3}\textrm{.12 ps}$ and large amplitude when the pulse duration matches ${\tau _{SPhP}}$ as shown in Fig. 5(c). The period of the oscillation may be related to the difference between the frequencies of the Fröhlich resonance and discrete longitudinal modes $|{{\omega_F} - {\omega_{n1}}} |$, and we believe this is worthy of further investigation [30]. Moreover, such oscillations are more intense and slower than those for the case of ${\tau _{\textrm{pulse}}} = 0.02\textrm{ ps}$ [Fig. 5(a)], and one can call it nonlocality-induced oscillatory decay, which is absent for the nonlocal plasmonic systems indeed [36].
In order to get a deep understanding of the nonlocality-induced oscillatory decay with the long-period and large-magnitude, we plot the temporal dipole moment for different radii of nanoparticles and different center frequencies in Fig. 6. The left and right columns in Fig. 6 correspond to the left and right adjacent longitudinal modes near the SPhP modes for different radii of nanoparticles. These oscillations arise from the coupling between discrete longitudinal modes and the Fröhlich resonance in the polar dielectric materials [19]. The adjacent (left and right) longitudinal modes are strongly coupled to the Fröhlich resonance so that intense oscillations occur as shown in Fig. 6(a) and 6(b). When the radius is increased, it is clearly shown that the period of the oscillation changes from 3.12 ps to 4.97 ps and the relative amplitude reduces from 0.5 to 0.2 by comparing Fig. 6(a) and 6(c). And the right panels display similar variations as expected with the period becoming longer and the relative amplitude becoming smaller. Further increasing the radius to 10 nm will result in smoother oscillation curve as shown in Fig. 6(e) and 6(f). Actually, with increasing the radius, the nonlocal effect becomes less strong, and hence such novel phenomena weaken. These unique properties also illustrate the potential advantages of tunable frequencies to modulate the pulse signal.
Fig. 6. Comparison of dipole resonance excited by Gaussian pulses with ${\tau _{\textrm{pulse}}} = 2.5\textrm{ ps}$. The three rows show data obtained from nanoparticles with radii 5 nm (a),(b), 7 nm (c),(d), and 10 nm (e),(f). In the left column we set the center frequency to the left adjacent longitudinal modes and take ${\omega _{n1}} = 923,928,929\textrm{ }c{m^{ - 1}}$, while in the right column we set ${\omega _{n2}} = 940,940,939\textrm{ }c{m^{ - 1}}$, respectively.
In the end, we show the variation of the time-shift (the temporal shift between the peak value of the dipole moment with regard to that of the freely propagating Gaussian pulse) along with the wave number for local and nonlocal cases (see Fig. 7). First of all, with increasing the duration time ${\tau _{\textrm{pulse}}}$, the magnitude of the time-shift becomes much larger due to the long interaction time between the external field and nanoparticle. In addition, the nonlocality reduces the magnitude of this time-shift, accompanied by the red-shift of peak value as well. In the meantime, nonlocality will lead to fluctuations on the left side of the curve in Fig. 7(a), and in Fig. 7(b) many positive and negative oscillations appear due to the excitations of the propagating longitudinal modes which are not present in the metal case [36]. The negative or positive time-shifts in Fig. 7 demonstrate that the peak of the dipole moment can be either advanced or retarded with respect to the one of external stimulation. Such behavior is quite similar to the detection of superluminal or subluminal transmission in the Fabry-Perot cavity [42,43]. The maximal retardation corresponds to the resonant peak of the scattering cross section, i.e., Fröhlich resonance, while the maximal advance is near $1000\textrm{ c}{\textrm{m}^{ - 1}}$, where the valley of the scattering cross section appears. Moreover, we numerically calculate the phase time ${t_{ph}} = { {\partial \arg [{\alpha (\omega )} ]/\partial \omega } |_{\omega = {\omega _0}}}$ of the monochromatic incident wave scattering by this nanoparticle [42], as shown in Fig. 7(b). Good agreement between ${t_{ph}}$ and the time-shift when ${\tau _{\textrm{pulse}}}$ is much larger than the lifetime of SPhP resonance. This is due to the fact that when the bandwidth of a Gaussian pulse is narrow, its excitation is similar to that of a monochromatic incident wave. Our results provide us the information on the key role of anomalous dispersion in the temporal dynamics.
Fig. 7. Comparison of the local (red line) and nonlocal (blue line) time-shift of the peak value of the electric dipole moment with different ${\tau _{\textrm{pulse}}}$ (a) 2.5 ps, (b) 25 ps. The phase time ${t_{ph}}$ of monochromatic incident wave varied with frequency labeled with corresponding graphics.
4. Conclusion
In summary, we give a detailed investigation on the temporal dynamics of nonlocal polar dielectric nanoparticles with an ultrashort Gaussian pulse excitation. The lifetime of the SPhP resonance of the proposed polar dielectric nanoparticle can be achieved by using extinction spectroscopy, and it demonstrates that a large near-field enhancement with oscillatory decay occurs inside the nanoparticles. In time domain, the dipole moment at Fröhlich frequency exhibits different responses at three durations: oscillation, pure exponential decay, and Gaussian shape. To one’s interest, we predict the nonlocality-induced long-periodic oscillation decay with the form of fluctuation in the temporal dipole moment evolutions in the case of the central frequency of the incident pulse being set at the position of the discrete longitudinal modes. Finally, we check the time-shift of the dielectric response with respect to the incident pulse and observe both advanced and retarded behaviors. This indicates that the maximal value of the dipole moment can appear earlier or later than the incident pulse. The phase time is defined to describe the time-shift, and good agreement is found when duration time is large.
Here some comments are in order. In this study, in order to emphasize the effect of spatial dispersion on the ultrafast dynamics of phonon polaritons resonances, we only consider the temporal dynamics of the single nanoparticle of polar dielectrics. Actually, understanding the dynamics of SPhP resonances in symmetric or asymmetric dimers is essential to the understanding of fundamental physics and emerging phonon polaritons-based applications of polar materials. The coherent time dynamics of the strong interacting dimers are relevant to the spectral properties of the isolated polar dielectric materials constituting the dimers. In addition, our study takes into account the nonlocality or spatial dispersion based on semiclassic model and the case of weak laser intensity, but ignores some sub-nanometer quantum effects as well as optical nonlinearities. We hope the work can motivate further theoretical and experimental studies devoted to modulating the response of polar dielectric nanoparticles on a time scale, as well as paving a new way for promising temporal coherent phonon polaritons.
Funding
National Natural Science Foundation of China (12274314); Natural Science Foundation of Jiangsu Province (BK20221240); Suzhou Basic Research Project (SJC2023003).
Acknowledgments
The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China (Grant No. 12274314), Natural Science Foundation of Jiangsu Province (Grant No. BK20221240), and Suzhou Basic Research Project (SJC2023003).
Disclosures
The authors declare no conflicts of interest regarding this article.
Data availability
The data supporting the findings of this study are available from the corresponding author upon reasonable request.
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