1. Introduction

Surface plasmon polaritons (SPPs) are electron density oscillations coupled to electromagnetic waves that are confined to and propagate along metal-dielectric interfaces. They provide a high degree of spatial confinement for optical radiation and hence enhance the physical processes that depend on the optical intensity [1]. Although the linear properties of SPPs have been studied extensively in the past few years [2, 3], their dynamic and nonlinear features have been less explored. Early research on SPP dynamics focused on linear propagation and measured SPP transients [4], group delay, and transit time using interferometry [5] and near-field optical microscopy [6]. Recently, researchers have studied optically-induced, reversible changes to SPP properties using pump-probe techniques, with the intent to understand short pulse SPP excitations and create high-speed SPP modulators [7–15]. Central to the modification of the optical transmission in these experiments is the coupling between SPPs and the nonlinear thermal effects of the metal induced by the pump light.

Although these experiments have provided insights into the interactions between light and electrons in metals and have demonstrated a method to modulate SPPs, they have several limitations. First, the measurements relied on SPPs that were delocalized over a periodic structure or a wide waveguide. As a result, the pump pulses were spatially broad, and did not efficiently couple the thermal energy into regions with the highest overlap with the SPPs. The inefficient coupling of the pump pulse required high energies of 0.1–10 μJ per pulse to observe the modulation. Second, the delocalized nature of the SPPs resulted in a low confinement factor of the SPPs in the metal, which necessitated the pump or probe wavelength to be at an interband transition of the metal or the optical excitation of the dielectric to adequately modulate the probe pulses. Finally, these experiments did not measure the optical phase dynamics, which are crucial for a complete characterization of the light-electron interactions.

In this paper, we report the first study of the optical phase and amplitude modulation induced by localized SPP excitations that are spatially confined in sub-wavelength rectangular apertures. The amplitude and phase of picosecond (ps) optical pulses transmitted through the apertures were retrieved from cross-correlation frequency resolved optical gating (XFROG) measurements, which resolved the dynamics of the pulses to 100 fs. Modulation was observed for film thicknesses of 80 nm, only 1/10 of the wavelength of light, at an incident energy of <0.3 nJ per pulse, 3–5 orders of magnitude lower than previous reports [9–11]. This energy was equivalent to 9 pJ per resonant aperture. The highly localized SPP resonance of the apertures enhanced the thermal response in the metal, caused by bound and free electrons, to induce optical amplitude and phase modulation even when the wavelength of the incident light at λ = 800 nm was de-tuned from the gold interband transition at 2.4 eV = 520 nm. The low energy requirements and the flexibility in the operation wavelength show the potential of these apertures for nanoscale ultra-low energy switching and modulation using SPPs.

2. Aperture design and linear properties

We designed and fabricated arrays of rectangular apertures in thin gold films deposited on a glass substrate. The lowest order resonance of the apertures arises from the hybridization of the cut-off resonance of the metallic apertures and the SPPs along the aperture sidewalls [2, 16, 17]. At this resonance, the light is strongly localized in the apertures. The apertures were sufficiently far apart such that the resonance wavelength was independent of the array period. Figure 1(a) shows the geometry of the sample. The width and height of each aperture were 80 nm and 200 nm, respectively. The apertures were arranged on a square lattice with a period of 300 nm in an 80 nm thick gold film. The gold permittivity, as described by the Drude model, is ${\varepsilon }_m/{\varepsilon }_0={\varepsilon }_b+{\omega }_p^2/\left(j\omega {\gamma }_0-{\omega }^2\right)$, where ε ₀ is the vacuum permittivity, ε _b = 9 at room temperature, and is the dielectric constant due to the bound electrons, ω _p = 1.32×10¹⁶ s⁻¹ is the plasma frequency, and γ ₀ = 1.06 × 10¹⁴ s⁻¹ is the electron damping rate at room temperature, T ₀ = 300 K [18].

 

Fig. 1 (a) A schematic of the rectangular apertures. (b) The computed electric field and (c) magnetic field amplitudes in the apertures on resonance, where red indicates the highest field intensities. (d) A scanning electron micrograph of the aperture array and (e) the measured (left axis) and calculated (right axis) normalized transmission spectra of x- and y- polarized light.

Download Full Size | PDF

Accounting for the rounding of the corners, the apertures were designed to support a fundamental resonance at λ ≈ 800 nm. The computed electric and magnetic field amplitudes on resonance are shown in Fig. 1(b) and 1(c) respectively, illustrating the strongly localized nature of the mode. The apertures were fabricated by focused ion beam milling of a 80 nm gold on a 5 nm chromium layer evaporated on a glass substrate. Figure 1(d) shows a scanning electron micrograph of a sample. Following the coordinate convention of Fig. 1(a), x and y-polarized transmission spectra of the array measured with a continuous-wave broadband light source are shown in Fig. 1(e). Since the rectangular slits are anisotropic, x-polarized light exhibited a resonance near λ = 800 nm, which was absent for y-polarized light in the spectral window of interest. The experimental results are in good agreement with the computed spectra shown in Fig. 1(e). The measured spectra were broadened and slightly red-shifted compared to the numerical simulations due to fabrication inhomogeneities of the apertures.

3. Dynamics model and simulations

Over ps time scales, the absorption of an optical pulse leaves the metal in an non-equilibrium state, where the electron temperature, T _e, is vastly different from the lattice temperature, T _l [19]. The elevated T _e and T _l change the metal permittivity which in turn alters the optical pulse transmitted through the apertures.

To model the phenomenon, we accounted for the evolution of both the light and the metal properties and combined electromagnetic wave simulations with thermal transient modeling. At each time step, we first solved the Helmholtz wave equation in the three dimensional (3D) space. Second, we calculated the increase in T _e and T _l in the metal at every spatial point from the absorption of the incident light using a two temperature model. Third, we calculated the change in the metal permittivity as a result of the elevated T _e and T _l. Finally, the modified metal permittivity was used in the next time step to solve the Helmholtz wave equation assuming that the changes in permittivity occurred much faster than the changes in the envelope of the incident light. This model is summarized in Fig. 2.

 

Fig. 2 A summary of the model used to compute the evolution of light and the properties of the metal. 1. The Helmholtz wave equation was solved at time t. The absorption of light modified the lattice and electron temperatures. 2. The modified temperatures altered the behavior of the electrons to 3. change the metal permittivity. The Helmholtz wave equation was then solved at the next time step, t + Δt, with the new value of metal permittivity assuming the envelope of the light adiabatically followed the changes in permittivity.

Download Full Size | PDF

3.1. Optical propagation

Due to the sub-ps resolution of the measurements, we simplified the electromagnetic modeling and assumed the electromagnetic fields adiabatically followed the permittivity. We simulated pulse propagation and solved, at each time step, the 3D, spatial Helmholtz wave equation for the magnetic field :

Since the thermal relaxation time of the metal was much shorter than the repetition period of the laser (a few ns [20] vs. 12.5 ns), we model the transmission dynamics with a single incident pulse. The incident pulse was assumed to be Gaussian and x-polarized, with an electric field of E⃗ _i = x̂E ₀(t) exp(jω ₀ t), where ω ₀ is the carrier frequency, and E ₀(t) is a slowly-varying envelope of the form

Using the above incident field as the source of electromagnetic radiation in our simulations, Eq. (1) was solved at 0.2 ps time steps with a spatial resolution of 6 nm in three dimensions in the metal to properly account for the thermally induced inhomogeneities in the metal permittivity.

3.2. Metal thermal response

The absorption of the incident field modifies the properties of the metal. The response of the metal to the incident field can be described by two steps. First, the absorption of light elevates both T _e and T _l. The increased temperatures modify the behavior of free and bound electrons inside the metal which in turn changes the metal permittivity, ε _m. For the computations, T _e, T _l, and ε _m were calculated for 10 ps using the same time step and spatial resolution as those for the electromagnetic calculations.

3.2.1. Electron and lattice temperatures

We described the evolution of T_e and T_l by the two temperature model [19]:

For the computations, C_e = C ₀ T_e with C ₀ = 71 Jm⁻³K⁻², C_l = 2.48×10⁶ Jm⁻³K⁻² and g = 2.5 × 10¹⁶ Wm⁻¹K⁻¹. The electron thermal conductivity was calculated via the free electron gas model with γ(T_l, T_e) as the electron scattering rate, such that $K_e=K_0{T}_e/(T_l+\frac{B}{A^{\prime }}{T}_e^2)$, where K ₀ = 318 Wm⁻¹K⁻¹ and B = 1.2 × 10⁷ s⁻¹K⁻². The value of A ^′ for thermal conductivity was evaluated at a zero frequency limit from the constant-current electrical resistivity and was A ^′ = 1.23 × 10¹¹ s⁻¹K⁻¹ [21]. The electron scattering rate was also used to calculate the free electron contribution to the permittivity as discussed in Section 3.2.2.B.

3.2.2. Permittivity

We describe the temperature dependence of the metal permittivity to link the modified properties of the metal to the changes in the evolution of light. We divided the changes to the permittivity resultant from T_e and T_l into bound electron, ε_b, and free electron, ε_f, contributions. The total permittivity of the metal, ε_m, can be written as:

3.2.2.A. Bound electron contribution

The bound electron contribution is due to the optically induced electron interband absorption from the d-band of the gold to its Fermi level, which is at about 2.4 eV (520 nm). Since the wavelengths used in this experiment (λ ≈ 800 nm) were detuned from this transition, the probability of interband transition of bound electrons at room temperature, T ₀, was negligible. However, as we shall show, in our experiment, T_e could exceed 1000 K, sufficient to promote more conduction electrons to states above the Fermi level. This would increase the number of vacant states below the Fermi level, to which bound electrons in the lower d-band could be optically excited. The interband transition probability could thus be effectively increased even at detuned wavelengths to affect the bound electron permittivity [22].

To evaluate ε_b, we first calculated the change in the imaginary part of ε_b, ΔIm{ε_b}, over a broad range of frequencies by calculating the change in the Fermi distribution of electrons as the temperature rose from T ₀ to T_e. Then ΔRe{ε_b} was calculated from ΔIm{ε_b} using Kramers-Kronig relations. At specific frequencies, ΔRe{ε_b} and ΔIm{ε_b} were fitted with polynomials to interpolate their values at an arbitrary temperatures.

Figures 3(a) and 3(b) respectively show –ΔIm{ε_b} and ΔRe{ε_b} for several temperatures and frequencies. The total bound electron permittivity is given by ε_b(T_e) = ε_b(T ₀) + ΔRe{ε_b} + jΔIm{ε_b}. The wavelengths used for the simulations were slightly different from the experimental values because the calculated and measured resonance peak wavelengths (see Fig. 1(e)) were not identical. The simulation wavelengths were chosen to have the same detuning from the resonance peak as the wavelengths used in the experiments.

 

Fig. 3 (a) –ΔIm{ε_b} and (b) ΔRe{ε_b} calculated at different temperatures for several wavelengths λ = 780 nm, λ = 850 nm, and λ = 750 nm (symbols) and the polynomial fitting for the data at λ = 780 nm (solid lines).

Download Full Size | PDF

3.2.2.B. Free electron contribution

The elevated temperatures also affect the free electron permittivity. This contribution is primarily due to increases in the electron-electron and electron-lattice scattering rates, which increase the electron damping constant and reduce the thermal conductivity of the metal. Using the Drude model with a modified electron damping rate, γ(T_e, T_l), the temperature-dependent permittivity due to free electrons becomes:

Since we are interested in ps dynamics, we neglected thermal expansion which would make ω_p temperature-dependent. We approximated the electron damping rate at optical frequencies as $\gamma =A{T}_l+B{T}_e^2$, where A = γ ₀/T ₀ = 3.54 × 10¹¹ s⁻¹K⁻¹ and B = 1.20 × 10⁷ s⁻¹K⁻² [23].

3.3. Simulation results

The coupled equations above were solved in the time domain using a finite-element method software (COMSOL Multiphysics) to compute the temporal evolutions of E⃗, T_e, and T_l at all spatial points. Figure 4 shows the results of our model for a 2.7 ps FWHM pulse centered at the SPP resonance wavelength. The evolution of T_e and T_l at the spatial point of maximum temperature are shown in Fig. 4(a). The magnitude and rate of the increase in T_l are lower than those of T_e because of the larger heat capacity of the lattice compared to the electrons. Figure 4(b) shows the large change in Im{ε_m} has a similar time-dependence as T_e, which implies the T_e contribution to the permittivity dominates over T_l.

 

Fig. 4 The calculated temporal evolution of (a) T_e, T_l, and (b) ε_m for a 2.7 ps FWHM incident pulse with a fluence of 8 mJcm⁻² on resonance at the spatial point indicated by the arrows in (e) and (f), which is the point of maximum temperature. The corresponding temporal evolution of (c) the transmission intensity and (d) the transmission phase at various fluences. The spatial distributions of the (e) electron and (f) lattice temperatures at 5 ps.

Download Full Size | PDF

Figures 4(c) and 4(d) show the change in transmission intensity and phase of the pulse at several fluences. At low fluences, the thermal transients are negligible. At higher fluences, thermal effects become more pronounced and the transmission amplitude is reduced at t = 0 due to the change in ε_m. At high fluences, the phase change follows the time evolution of Re{ε_m}. This is because the phase change is related to the propagation constant of the SPPs in the apertures, which is strongly influenced by the ΔRe{ε_m}.

4. Experimental setup and results

4.1. XFROG measurements

To measure the temporal evolution of the phase and amplitude of ps pulses coupled out of the apertures, we used a sum frequency generation (SFG) XFROG [24]. Figure 5(a) shows the experimental setup. Pulses with a FWHM of 2.7 ps from a wavelength tuneable Ti:Sapphire mode-locked laser (Coherent Mira) were divided into two paths. The pump pulses passed through the sample, while the gating pulses traversed through a delay line. The pump pulses were incident from the glass side to keep the spot area small. Although the optical resonance did not depend on the coupling between the apertures, approximately 30 apertures were excited (spot area of 2.7 μm²) to attain a sufficient signal-to-noise ratio for the measurement. For this spot area, about 57% of the incident light was transmitted and 10% reflected at λ = 800 nm. The time-averaged transmitted power varied almost linearly with the pump fluence. The pump and the gating pulses were combined in a nonlinear β-BaB₂O₄ (BBO) crystal to generate a SFG signal that was spectrally resolved with an optical spectrum analyzer (OSA) as a function of the delay time between the pump and gating pulses, τ. A typical XFROG trace is shown in Fig. 5(b).

 

Fig. 5 (a) The XFROG measurement setup. (b) An example of an XFROG trace at λ = 775 nm.

Download Full Size | PDF

The XFROG traces were analyzed by a commercially available software (FemtoSoft XFROG), which retrieved the amplitude and phase of the transmitted pump pulses given the phase and amplitude of the gating pulses. The amplitude and phase of the gating pulses were measured using the identical setup but with the sample removed. The XFROG data were retrieved with a 512× 512 grid with errors G ≲ 0.0001, where G is the root mean square average of the difference between the measured and retrieved XFROG traces [25]. To check the validity of the setup and the retrieval algorithm, we removed the sample, and then compared the frequency marginals, defined as M = ∫ dτI _F, where I _F(ω, τ) is the resulting FROG trace, with the frequency self-convolution, [I(ω)*I(ω)], of the time-averaged spectrum of the pulse, I(ω), measured by the OSA. The results showed a good agreement between the two, confirming the validity of the retrieval [25].

4.2. Optical modulation

Since the spectral bandwidth of ps pulses was much narrower than the transmission bandwidth of the apertures, the linear dispersive effects of the apertures on the pulses were negligible. Any substantial changes to the amplitude and phase of the transmitted pulses were thus dominated by the transient thermal effects. By adjusting the intensity of the incident pump pulses, we observed significant changes to the transmitted pulse shapes.

4.3. Pulse amplitude and phase

Figures 6(a) and 6(b) respectively show the experimentally measured and numerically calculated normalized electric field intensities of the pulses on resonance at several pump fluences. The fronts of the pulses were aligned in time for comparison. The measurements showed that with increasing fluence, the tails of the pulses became more attenuated, consistent with simulations in Fig. 6(b) and Fig. 4(c). The discrepancy between the simulated and measured fluence values was attributed to the difference between the measured and simulated linear transmission as shown in Fig. 1(e). The amplitude change of the falling edge of the pulse exhibited two components: a rapid initial attenuation over about 0.5 ps followed by a more gradual decay. The fast initial decay was due to the T_e contribution to ε_m. As shown in Fig. 4(c), the transmission peaks as T_e reaches a maximum and then decays in a time-scale comparable to the T_e decay time. The gradual attenuation occurring at a later time was due to the rise in T_l. As a consequence of the different attenuation mechanisms, the transmitted pulses appeared to be narrowed and the peaks appeared to be advanced in time as the fluence increased.

 

Fig. 6 (Left column) The retrieved normalized electric field intensity, |E|², and phase on resonance at λ = 800 nm, negatively detuned at λ = 775 nm, and positively detuned at λ = 860 nm from the resonance. (Right column) The numerically calculated normalized electric field intensity, |E|², and phase on resonance, positively detuned, and negatively detuned from the resonance.

Download Full Size | PDF

The phase shift was also characterized by a rapid change followed by a more gradual change. However, the relative magnitude of the gradual phase change was much smaller than that of the amplitude. This is because the phase change depends primarily on ΔRe{ε_m}, but free carrier absorption mainly modifies Im{ε_m}, and as shown in Fig. 4(b) ΔRe{ε_m} ≪ ΔIm{ε_m}, which results in a small phase change. The rapid phase change near the peak of the pulse coincided with the maximum of T_e and was dominated by the change in ε_b(T_e), which primarily modified Re{ε_m}.

The electric field intensity and phase of the transmitted pulses were also simulated and experimentally retrieved at wavelengths detuned from the center of the aperture resonance. These results are shown in Fig. 6(c)–6(f). The net changes in the phase and amplitude of the pulse for the off-resonance wavelengths were smaller than the on-resonance case, because higher fluences were required to couple the same amount of energy into the apertures. The measurements are also in good qualitative agreement with the simulation results.

The discrepancy between the simulated and measured data can be attributed to several factors. Our expressions for C_e, K_e, and g were derived from the free electron gas model. At high electron temperatures (T_e ≈ 6000 K), the bound electrons can also be thermally excited, so the free electron gas model we have used is not as accurate. At high incident fluences in our experiments, T_e did approach this limit. Also, the measured phase of the incident pulse was not exactly quadratic as modeled in our simulations, and the phase retrieval was most accurate near the center of the pulse, when the amplitude was the highest.

We emphasize that none of the constants in our model were free parameters, which could be adjusted based on the experimental results. It is known that the dielectric constant, interband transition matrix element, and the plasma frequency depend on the deposition and fabrication conditions, so using fitting parameters can improve the agreement between the simulations and measurements [8]. Even though the material parameters in the model were completely independent of our experiments, our model nonetheless achieved good qualitative agreement with our measurements and successfully predicted the trends in the evolution of the optical amplitude and phase.

4.4. Pulse width modulation

We have also characterized the effects of the incident pulse fluence on the magnitude of the intensity modulation using the FWHM of transmitted pulse as a metric. Figure 7 shows the percentage change in the FWHM of the retrieved electric field intensity, Δτ_p, as a function of incident pulse fluence at several wavelengths where Δτ_p = (τ_{p, pump} – τ_{p, gate})/τ_{p, gate} and τ_{p, gate(pump)} is the FWHM of the electric field intensity before(after) it propagated through the apertures.

 

Fig. 7 The percentage change in the FWHM of the retrieved electric field intensity, |E|², for pump wavelengths centered on resonance at λ = 800 nm and detuned at λ = 775 nm and λ = 860 nm.

Download Full Size | PDF

The largest change in FWHM was observed for pump wavelengths which were on the aperture resonance at λ = 800 nm. As shown in the linear transmission spectrum in Fig. 1(e), detuning the pump wavelength from the peak of the resonance reduces the amount of optical power coupled into the apertures and the resultant thermal effects in the metal, leading to a smaller change in the FWHM.

Figure 7 also shows that Δτ_p saturated at high fluences. At these fluences, the electron and lattice temperatures in the metal were raised sufficiently high to reduce the coupling of the incident pump pulse to the aperture, resulting in a saturation of the modulation. Increasing the fluence beyond these values typically resulted in a permanent damage to the apertures.

5. Conclusion

In summary, we have measured and explained the amplitude and phase evolution of ps pulses transmitted through sub-wavelength apertures supporting strongly localized resonances. The observed effects were made possible by the excitation and strong confinement of SPPs in the rectangular apertures, which enhanced the fluence inside the apertures by more than a factor of 5 compared to the pulse fluence in vacuum. The XFROG technique enabled the characterization of the amplitude and phase of the ps pulses with temporal resolution of 100 fs, allowing us to identify the thermal mechanisms responsible for the change in the real and imaginary parts of the metal permittivity.

Even though the incident light was detuned from the metal interband transition and was at modest fluences ≈ 10 mJcm⁻², corresponding to an incident energy of 0.275 nJ or 9 pJ per aperture, sub-ps pulse modulation was observed for a propagation length of only 80 nm, 1/10 of the optical wavelength in free-space. Localized resonances circumvent the need for optical pulses at wavelengths determined by the interband transitions of the metal. These apertures can also be used to realize rapid, sub-ps thermally-induced transients in materials incorporated inside the sub-wavelength apertures for applications in low energy, ultra-compact, and ultra-high-speed optical modulators.