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Plasmonic pulse shaping and velocity control via photoexcitation of electrons in a gold film Nikola E. Khokhlov,1,2,* Daria O. Ignatyeva,1,2 and Vladimir I. Belotelov1,2,3 1

Russian Quantum Center, Novaya St. 100, Skolkovo, Moscow Region, 143025, Russia 2 Lomonosov Moscow State University, Leninskie Gory 1, Moscow, 119991, Russia 3 Electron Science Research Institute, Edith Cowan University, Joondalup Drive 270, Joondalup, Western Australia, 6027 Australia * [email protected]

Abstract: We study the possibility of surface plasmon polariton (SPP) pulse shape, delay and duration manipulation on sub-picosecond timescales via a high intensity pump SPP pulse photoexciting electrons in a gold film. We present a theoretical model describing this process and show that the pump induces the phase modulation of the probe pulse leading to its compression by about 20% and the variation of the delay between two SPP pulses up to 15 fs for the incident fluence of the pump of 1.5 mJ·cm−2. ©2014 Optical Society of America OCIS codes: (240.6680) Surface plasmons; (190.7110) Ultrafast nonlinear optics; (240.4350) Nonlinear optics at surfaces; (320.5520) Pulse compression; (320.5540) Pulse shaping.

References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

11. 12. 13. 14. 15. 16. 17.

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#221986 - $15.00 USD Received 28 Aug 2014; revised 27 Oct 2014; accepted 27 Oct 2014; published 4 Nov 2014 (C) 2014 OSA 17 November 2014 | Vol. 22, No. 23 | DOI:10.1364/OE.22.028019 | OPTICS EXPRESS 28019

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1. Introduction The design of novel optical devices for light manipulation on subwavelength scales has become a subject of interest due to rapidly growing demand for miniaturization of the telecommunication devices during the last two decades. At the same time the increase of the performance is highly desirable and can be achieved with optical elements by reduction of the light switching time. Plasmonic integrated circuits with surface plasmon polariton (SPP) excitations are one of the best candidates to fulfill above mentioned requirements. Such structures have submicrometer size scales and group velocity of SPP pulses is nearly the same as one of the photons [1]. Nowadays several lab prototypes of the passive plasmonic elements have been engineered including waveguides [2, 3], beam splitters [4], antennas [5], spasers [6], lenses [7] and others. But the most interesting nanooptics elements are the active ones e.g. all-optical plasmonic switchers [8], plasmonic circulators [9], magnetooptical elements [10, 11].


Since the SPP dispersion depends on the permittivities of its host media, the SPPs can be controlled by modulation of the permittivities [1]. As an external impact for the modulation it may be used external electric [12] or magnetic field [13–17], acoustic waves [18, 19], thermal heating [20], etc. Central frequency and time delay of the SPP pulse can be controlled via collision of this pulse with the other one of higher energy at interface of a metal and Kerr dielectric [21]. The major disadvantage of that approach is the requirement of very high pump intensities that may cause in dielectric not only nonlinear refraction but also other nonlinear phenomena such as nonlinear absorption, self-action, Raman scattering, thermal effects etc. In this respect, dealing with metal – air interface and photoexcitation of electrons in metal as the nonlinear mechanism might prevent from such problems of experimental realization. In this paper we describe and theoretically investigate this approach. In particular, we demonstrate the SPP pulse shaping due its interaction with another high energy SPP pulse (pump) that photoexcites electrons in a metal. The process of the electron photoexcitation produced by laser pulses was demonstrated experimentally as the way for manipulation of the optical properties of the plasmonic structures [22, 23], of the SPP coupling to gold films [24] and of the SPP-pulse shape and phase [25]. However, the possibility for the control of plasmonic pulse by one another with higher energy causing photoexcitation of electrons in a metal has not been considered yet. 2. Theoretical approach The electron temperature Te determines the metal permittivity [21, 26, 27] and may be changed via photoexcitation of electrons by short laser or SPP pulse [26–31]. On a time scale of about 50 fs the light energy absorption is due to the photon-electron excitation in the skin layer and a large change of Te is induced without changes of the lattice temperature. After excitation the energy is redistributed among the hot and nonthermalized electrons through electron-electron interactions with few hundred femtoseconds time scales. After thermalization, and partly during it, the electron gas loses its energy and externally thermalizes with the lattice through electron-phonon scattering. After a few picoseconds a local equilibrium is reached at a slightly higher temperature than the initial condition. As a final step, thermal diffusion into the bulk metal or the substrate occurs on a 100-ps time scale. The photoexcitation process and relaxation dynamics of the electrons in metals were studied intensively in the last two decades [26–33]. The changes of the electron temperature Te during the thermalization result in the temporal metal permittivity change Δεm. Therefore, |Δεm| rises up to its maximum value at the hundred femtosecond time scale and exponentially decreases with characteristic time τrelax of about 500 fs [26–33]. The electron temperature changes ΔTe and corresponding values of Δεm are linear in the first approximation with the absorbed optical power and with the pulse fluence Ф0. Typical values of optical pump fluences Ф0 are about 4 mJ·cm−2 for relative permittivity changes Δεm/εm up to 6% [28]. Such high fluences are needed due to the small absorption of light because of the very high reflectance from the metallic surfaces. In the case of the SPP excitation by a laser pulse large amount of the incident light energy is transferred to the SPP that leads to the increase of the optical energy absorbed by a metal [1]. Consequently, for the same laser pulse fluences Ф0 SPP provides much higher changes of the electron temperature ΔTe. For example ΔTe ~1700 K is achieved with Ф0 ~1.3 mJ·cm−2 in non-SPP-pump experiments [29] and ΔTe ~3500 K – with Ф0 ~0.5 mJ·cm−2 in SPP-pump experiments at λpump ≈800 nm for a gold films [30]. Here we consider SPP-SPP interaction at a smooth metal-air interface. Figure 1 presents the principal scheme of the possible experimental realization: a pump-probe optical technique with two spatially separated metal gratings for laser-SPP coupling and decoupling back as it was performed by K.F. MacDonald et al. in [25].


Fig. 1. The scheme of the control of the SPP-probe with the SPP-pump via the photoexcitation of electrons in a metal. Red and green curves show pump and probe positions respectively. SPP pulses propagate along positive direction of x-axis. Dark color gradient of the metal represents schematically the spatial variation of the metal permittivity induced by the pump SPP.

When the pump-SPP pulse propagates along the metal interface, the induced change in the permittivity Δεm has some spatial and temporal distribution that can be described by: Δε m ( x, t ) = Δε max (ω , ΔTe ) f (τ ),

(1)

where the temporal coordinate τ = t -x/up is associated with the group velocity of the pump pulse u p = ( ∂ω ∂β ) ω with τ = 0 corresponding to the center of the pump, and Δεmax is the p

magnitude of the induced inhomogeneity. Here we take ƒ(τ) in the form of:  τ σ f (τ ) = ξ 1 + erf  −   σ 2τ relax

 σ    exp   2τ relax   

2

−

τ 

, 

τ relax 

(2)

where erf is the Gaussian error function, σ = τ pump / (2 ln 2) , τpump is a full width at half maximum of the pump pulse that is considered to have Gaussian profile, ξ is a normalization constant to obtain maximum value of ƒ(τ) equals to unity. This form of ƒ(τ) corresponds to the approximation of the experimental data used in [30] by M. Pohl et al. It is modified for the pump-SPP pulse moving along the x-axis. Complex dependence (2) is associated with the nonuniform photoexcitation of the electrons within the pump pulse. The function f(τ) represents the profile of ΔTe: f(τ) is almost zero ahead of the pump’s front as the electrons in the metal film have not been heated yet; f(τ) increases on the pulse spatial-time scales as the electrons are “warmed” by the pump and almost not “cooled” by the heat transfer since τrelax>>τpump, and f(τ) decreases exponentially behind the pump tail with the characteristic time τrelax. The induced change Δεm influences on the SPP probe dispersion that depends on the metal permittivity [1] and the modification of the dispersion is responsible for the effects of the pulse compression, shape and velocity changes. For the description of the SPP-SPP interaction we use the method of slowly varying envelope similar to the one presented in [34] by G.P. Agrawal for the waveguide modes of the optical fibers and in [35] by D.O. Ignatyeva et al. for the SPP modes at the interface between a metal and a nonlinear dielectric. We neglect the influence of the weak probe on the photoexcitation process. The electric field of both SPP pulses can be written as: 1 E j (r, t ) = [F j ( z ) Aj ( x, t ) exp(i β 0 j x − iω j t ) + c.c.], 2

(3)


where F j ( z ) = (e x + i sgn( z ) β 0 j γ −j 1e z ) exp(−γ j z ) describes the profile of the SPP, ex and ez are unit vectors along x and z-axis, respectively, Aj(x, t) is the slowly varying envelope of the corresponding pulse, β0j is the wavenumber at the central frequency ωj in the absence of the thermalization and γj is the SPP localization coefficient and index j denotes the pump (index “p”) or the probe (index “s”) pulse. Using the assumptions mentioned above we obtain following equations describing the SPP pump and probe dynamics: ∂Ap ∂x

+ iD p

∂ 2 Ap ∂τ 2

− iΔβ p ( Ap , x,τ ) + Γ p Ap = 0,

∂As ∂A ∂ 2 As + ν s + iDs − iΔβ s ( Ap , x,τ ) + Γ s As = 0, ∂x ∂τ ∂τ 2

1 ∂ βj 2 ∂ω 2

(4) (5)

2

where D j =

is the group velocity dispersion coefficient; ν = 1/us - 1/up is the group ω0 j

velocity mismatch between the pulses and Γj is the imaginary part of the SPP’s wavenumber βj and Δβj is the pump-induced change of the SPP propagation constant. Since dispersion of the SPP depends on the dispersion of εm described by Drude model [1], the dispersion coefficient Dj > 0 and both phase and group velocities of the SPP increase with the increase of wavelength. It should be noted that the damping and the self-action of the pump SPP is taken into account in our model in accordance to Eq. (4). The localization coefficients γj are modified by the pump as well producing Δγj, but this effect could be neglected since Δγj produces only minor changes in the pump-probe field overlapping in comparison to the one order of magnitude damping of the pump during its propagation. Modification of both real and imaginary parts of the SPP’s wavenumber Δβj at the smooth metal-dielectric interface is calculated as a small perturbation of the βj in analogy with [35]: Δβ j (ω j , x,τ ) =

k02 j 2β0 j



+∞

0

2

Δε m ( Ap , ω j , x,τ ) F ( z ) dz



+∞

−∞

2

F ( z ) dz

=

Δε m ( Ap , ω j , x,τ ) β 03 j 2ε m2 (ω j )k02 j

, (6)

where k0 = ω/c; с is the speed of light in vacuum and term F(z) is the envelope of unperturbed probe. Here we also ignore Δγj since Δγj influences on the pulse dispersion in the second order of the perturbation theory only [34], but here we consider the first order approximation. 3. Results and discussion Though the concept of the SPP pulse control via another SPP pulse of larger energy is applicable to any plasmonic metals including silver and aluminum, here we make quantitative analysis for the case of gold-air interface. Permittivity of gold εm is taken from experimental data [35] and its photoinduced changes Δεmax(ω,ΔTe) = ∂Teε(ω)ΔTe are described with the theoretical model taking into account interband transitions of the electrons in gold [33]. Δεm depends on ΔTe and, therefore, on the absorbed pump energy which is determined by the pump’s central frequency ωp and pump fluence Ф0. Heating of electrons by the pump at ωp affects εm across the whole spectrum. Thereby, the probe SPP central frequency ωs may be chosen independent on ωp. It is seen from Fig. 2 that Δβ can be positive or negative depending on the SPP frequency and gets maximal at wavelengths around 500-600 nm [33]. Small SPP propagation lengths at these wavelengths imply that the pump and the probe SPPs propagate in close proximity to each other. First of all, let us consider the details of the pump SPP propagation in the presence of the thermalization process. According to Eq. (5) the SPP propagates along the interface inside the


region with the thermalization-induced Δβ moving with the group velocity of the pump SPP. The front and the tail of the SPP pulse experience different impact of the inhomogeneity. The pulse’s tail propagates in the region with Re[β] larger than in the region of the front (such situation corresponds to the pump SPP wavelength larger than 560 nm so that Re[∆β] > 0, see Fig. 2) it experiences the pump-induced phase modulation. The increase of the tail wavenumber diminishes its frequency in analogy with the Doppler effect in a moving dielectric or moving pump-induced permittivity inhomogeneity in a nonlinear crystal [37]. As a result, group velocity of this part of the pulse increases. The acceleration of the SPP tail along with the non-disturbed front propagation leads to the compression of the whole pulse. At the same time, the effective velocity of the pump-SPP center increases compared to the low-pump intensity case when the photoexcitation process is negligible. The second phenomenon that plays the crucial role in the SPP dynamics is the inhomogeneous damping. In the case of Im[∆β] > 0 the tail experiences stronger attenuation that the front. This leads to the additional decay of the SPP pulse, and, which is more important, to the additional decrease of the SPP duration and shift of its center. Analogous impact of the thermalization process on the dynamics of the SPP probe pulse can be predicted if its central frequency ωs corresponds to the case of Re[Δβ]>0 and Im[∆β] > 0 and the probe propagates slightly ahead of the pump SPP. This process can be quantified by the compression coefficient η = (τfs - τ0s)/τ0s, where τ0s and τfs are the initial and the resulting duration of the probe SPP pulse, respectively. The effective duration of the pulse was determined as:

τ fs =

(

+∞

−∞

2

τ 2 A( L,τ ) dτ

)( 

+∞

−∞



+∞

−∞

2

) (

A( L,τ ) dτ −

+∞

−∞

2

τ A( L,τ ) dτ

)

2

2

A( L,τ ) dτ

(7)

The compression coefficient can be estimated by the time difference required for the nondisturbed front and accelerated tail of the pulse to pass the propagation distance L:  

1 2

1 2

 

η = 2 Ds us Lτ 0−s1  Δβ (τ c + τ 0 s ) − Δβ (τ c − τ 0 s )  ,

(8)

where τc corresponds to the initial position of the probe-SPP center. The conducted numerical simulations show that values of η predicted by Eq. (8) are a bit lower than obtained via the direct numerical solution of Eqs. (4) and (5). However, Eq. (8) is a good starting point for the evaluation of the typical parameters required for the observation of the SPP pulse compression.

Fig. 2. The dispersion of the real (solid lines) and imaginary (dashed lines) parts of Δβ for ΔTe = 1000 K (black line), ΔTe = 5000 K (green line), ΔTe = 8000 K (orange line). The dispersion of ∆βc required for 10% compression is depicted by gray solid line.

Dispersion curves of ∆β and its critical value ∆βc necessary for 10% compression of the SPP pulse calculated according to Eqs. (6) and (8) are presented in Fig. 2. The absorption peak in the 520-550 nm range results from the interband electron transition in gold near 520


nm [28]. Although the extremum of the Im[Δεmax] coincides with the transition spectrally, the spectral maximum of Re[Δεmax] is shifted to ~550 nm [33]. Since in accordance to Eq. (6) ∆β depends on the real and imaginary parts of both Δεmax and εmax its dispersion acquires rather complex shapes. Figure 2 demonstrates that the wavelength range around 600 nm is optimal for the experimental realization since shorter wavelengths experience too large absorption enhanced by the thermalization process and longer wavelengths experience only a very slight impact of the photoexcitation effect. Therefore, we choose both the pump and the probe wavelengths of 600 nm. The separation of the two SPP pulses with equal central wavelengths can be realized if the pulses propagate under small angle to each other. Since the propagation distance is limited by a strong damping, we assume that the probe-SPP passes the distance L = 1.5lprop = 1.5/(2Γ), corresponding to 75%-magnitude decay of its intensity for the nonthermalized case. At the chosen wavelength of 600 nm we have D = 0.19 fs2/µm, Γ = 0.015 µm−1 and Δβ = 0.18 + i·0.06 µm−1 for ΔTe ~8000 K (pump fluence Ф0 ~1.5 mJ·cm−2).

Fig. 3. The dependences of (a) the probe compression η and (b) the shift of the probe center on the initial SPP-probe position for different probe and pump durations (black line: 30 fs pump and 30 fs probe; red line: 30 fs pump and 45 fs probe; blue line: 45 fs pump and 30 fs probe; green line: 45 fs pump and 45 fs probe). The dependences of (c) compression coefficient and (d) probe center shift on the pump SPP duration (green line: τc = -10 fs; black line: τc = 0 fs; red line: τc = 10 fs; 45 fs probe). The decay of the probe SPP in the corresponding cases is shown by dashed lines. (e) The schemes of the probe and pump SPPs relative initial positions in the cases of different τc.

In order to achieve higher compression one can tune the initial delay and the duration of the both pulses. The shorter pump pulse provides more pronounced gradient of Δβ along with greater differences in the velocities of the front and tail parts of the probe pulse and leads to larger changes of the probe-pulse duration. Figure 3 demonstrates that a 30-fs pump pulse #221986 - $15.00 USD Received 28 Aug 2014; revised 27 Oct 2014; accepted 27 Oct 2014; published 4 Nov 2014 (C) 2014 OSA 17 November 2014 | Vol. 22, No. 23 | DOI:10.1364/OE.22.028019 | OPTICS EXPRESS 28025

causes compression and pulse center shift higher than a 45-fs pump does. At the same time, the increase of the probe duration also makes the difference between the front and the tail propagation constants and damping higher. Figure 3 illustrates the evident enhancement of the observed impact for the case of the shorter pump and longer probe SPPs. The decay of the probe SPP intensity was also analyzed since it increases in the region with the photoexcited electrons. The additional decrease of the probe SPP intensity does not exceed 2 times at τc = −15 fs which corresponds to the maximal probe compression of 20%. However, if the delay is selected so that the whole probe propagates in the region of the photoexcited electrons (τc > τ0s/2) its intensity lowers by more than 50 times. Evolution of the 45-fs probe SPP profile during its propagation in the presence of the inhomogeneity induced by the 30-fs SPP pump pulse is depicted in Fig. 4.

Fig. 4. Evolution of the probe SPP profile (black solid line corresponds to the initial and red solid line to the resulting probe-SPP profiles) during the propagation along the interface with moving inhomogeneity (its initial and resulting shapes are shown by dashed lines of the corresponding color). For the convenience the intensities of the probe SPP and the magnitude of Δβ are normalized.

Since the gradient of Δβ is smooth the shape of the probe SPP does not experience significant changes while the shift of its center and the compression are clearly seen. Therefore, the dynamics of the probe, its resulting intensity, delay and duration can be efficiently controlled via the femtosecond pump SPP. 4. Conclusion To conclude, active SPP control can be produced by the pump SPP on the gold surface via the process of the electron photoexcitation. Usage of the SPP pump instead of the bulk laser radiation results in the increase of the absorption and electron temperature. The compression of the probe SPP pulse as well as manipulation of its temporal delay can be performed in the considered plasmonic system at the spatial scales of tens of microns and sub-picosecond times. Interaction of the SPP pulses considered here might be significantly increased in periodic and microresonator plasmonic structures [38, 39] or using gain medium to increase the SPPs interaction length [40]. Acknowledgments The work was supported by RFBR grants Nº 14-02-01012, 13-02-01122, 13-02-91334. V.I. Belotelov acknowledges support by the Alexander von Humboldt Foundation.
