Surface plasmon polariton induced optical amplitude and phase modulation in sub-wavelength apertures

We report on the amplitude and phase modulation of picosecond optical pulses, near λ = 800 nm, transmitted through sub-wavelength rectangular apertures in thin gold films with thicknesses of λ/10 at per-pulse energies of <0.3 nJ or 9 pJ per aperture. Due to the excitation and strong confinement of surface plasmon polaritons in the apertures, the leading edge of a pulse causes a rapid heating of the electrons and lattice to modulate its falling edge. By comparing cross-correlation frequency resolved optical gating measurements with simulations, the thermal effects responsible for the induced pulse dynamics are identified. © 2011 Optical Society of America OCIS codes: (240.6680) Surface plasmons; (300.6430) Spectroscopy, photothermal. References and links 1. S. Lal, S. Link, and N. J. Halas, “Nano-optics from sensing to waveguiding,” Nat. Photonics 1, 641–648 (2007). 2. F. J. Garcia-Vidal, L. Martin-Moreno, T. W. Ebbesen, and L. 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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][8][9][10][11][12][13][14][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.(C) 2011 OSA pulse, 3-5 orders of magnitude lower than previous reports [9][10][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 detuned 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.

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 , 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 = 1.32×10 16 s −1 is the plasma frequency, and γ 0 = 1.06 × 10 14 s −1 is the electron damping rate at room temperature, T 0 = 300 K [18].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.1(e).The measured spectra were broadened and slightly red-shifted compared to the numerical simulations due to fabrication inhomogeneities of the apertures.

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. Solve T e and T l in 3D at t 3. Solve permittivity in 3D at t 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.

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 : where ε( r,t) is the permittivity at position r and time t.The simulation domain included the metallic apertures, the glass substrate, and the air surrounding the structure.Using symmetry, only one quadrant of the unit cell was modeled.
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 E i = xE 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 where τ p is the full-width at half-maximum (FWHM) temporal width of the pulse intensity, and the phase is φ (t) = 2(ln 2)α(t/τ p ) 2 , where α is the chirp parameter.
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.

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.

Electron and lattice temperatures
We described the evolution of T e and T l by the two temperature model [19]: where C e(l) is the electron (lattice) heat capacity, K e is the electron thermal conductivity, g is the electron-phonon coupling constant, and E the electric field in the metal.The last term in Eq.
(3)(a) results from the ohmic and dielectric dissipation rates in the metal and increases T e by the absorption of light.This term relates the thermal properties of the metal to the magnitude of the electric field.
For the computations, C e = C 0 T e with C 0 = 71 Jm −3 K −2 , C l = 2.48 × 10 6 Jm −3 K −2 and g = 2.5 × 10 16 Wm −1 K −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 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 × 10 11 s −1 K −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.

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 (4)

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

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 γ = AT l + BT 2 e , where A = γ 0 /T 0 = 3.54 × 10 11 s −1 K −1 and B = 1.20 × 10 7 s −1 K −2 [23].

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 E, 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.

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 modelocked 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 μm 2 ) 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 timeaveraged transmitted power varied almost linearly with the pump fluence.The pump and the gating pulses were combined in a nonlinear β -BaB 2 O 4 (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).
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].

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.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.

Pulse amplitude and phase
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 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.

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.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 the effects in the metal, leading to a smaller change in the FWHM.

Δτ
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.

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 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 ultrahigh-speed optical modulators.

Fig. 1 .
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.

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 #142188 -$15.00USD Received 7 Feb 2011; revised 29 Mar 2011; accepted 31 Mar 2011; published 15 Apr 2011 (C) 2011 OSA of interest.The experimental results are in good agreement with the computed spectra shown in Fig.

#
142188 -$15.00USD Received 7 Feb 2011; revised 29 Mar 2011; accepted 31 Mar 2011; published 15 Apr 2011 (C) 2011 OSA total permittivity of the metal, ε m , can be written as: 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.(b)

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 .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 ΔRe{ε m }.

Fig. 4 .
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.

Figures 6 ((
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
(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 }.

pFig. 7 .
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.