Towards the nonlinear acousto-magneto-plasmonics

The influence of the external magnetic field on plasmonic properties is extensively discussed elsewhere [28]. The magnetic field control of the plasmonic properties has been demonstrated in a number of systems [29–42]. Here we shall focus on the properties of the SPPs in a hybrid (noble) metal-ferromagnet-(noble) metal multilayer structure, which has been proven as a strong playground for magneto-plasmonics [7, 43, 44] (figure 5). The geometry of SPPs at 800 nm wavelength in a Au–Co–Au multilayer structure is shown in figure 5(a). Both the real and imaginary parts of the complex SPP wave vector can be calculated within the effective medium approximation [44, 45]. The effective dielectric function

$$\begin{eqnarray}&&{\varepsilon }_{{\rm{eff}}}=\displaystyle \frac{1}{{\delta }_{{\rm{skin}}}}{\int }_{0}^{\infty }\varepsilon (z){{\rm{e}}}^{-z/{\delta }_{{\rm{skin}}}}{\rm{d}}z,\end{eqnarray} \tag{ 2 }$$

accurately reproduces the SPP dispersion relation in a magneto-plasmonic multilayer structure

$$\begin{eqnarray}&&{k}_{{\rm{spp}}}={k}_{0}\sqrt{\displaystyle \frac{{\varepsilon }_{{\rm{eff}}}}{1+{\varepsilon }_{{\rm{eff}}}}}.\end{eqnarray} \tag{ 3 }$$

Zoom In Zoom Out Reset image size

The validity of this approximation is illustrated by the dependence of SPP propagation length ${L}_{{\rm{spp}}}=1/(2{\rm{Im}}[{k}_{{\rm{spp}}}])$ in a Au–Co–Au multilayer, where the 6 nm thick cobalt layer was placed at a variable depth h below the gold–air interface (figure 5(b)). A good quantitative agreement between the effective medium approximation and exact transfer-matrix calculations [46] is obtained. A significant decrease of the SPP propagation length L_spp for small values of h is attributed to additional ohmic losses introduced when the ferromagnetic layer is in a non-negligible overlap with the spatial envelope of the intensity profile associated with the SPP mode. The influence of a thin ferromagnetic layer on the skin depth δ_skin in a hybrid noble metal-ferromagnet-noble metal multilayer is negligibly small and one can safely use its value for SPPs propagating at the noble metal-dielectric interface.

The effective medium approximation provides an adequate description of the dependence of SPP wave vector on the magnetic field as well. Assuming that the magnetization vector $\overrightarrow{M}$ is pointing along the y-direction the effective magneto-optical tensor is given by:

$$\begin{eqnarray}{\widehat{\varepsilon }}_{{\rm{eff}}}(\pm {m}_{y})=\left(\begin{array}{ccc}{\varepsilon }_{{\rm{eff}}} & 0 & \pm {\varepsilon }_{{\rm{eff}}}^{{xz}}{m}_{y}\\ 0 & {\varepsilon }_{{\rm{eff}}} & 0\\ \mp {\varepsilon }_{{\rm{eff}}}^{{xz}}{m}_{y} & 0 & {\varepsilon }_{{\rm{eff}}}\end{array}\right),\end{eqnarray} \tag{ 4 }$$

with ${m}_{y}={M}_{y}/{M}_{S}$ (normalized to the saturation magnetization M_S) and the non-diagonal components

$$\begin{eqnarray}&&{\varepsilon }_{{\rm{eff}}}^{{xz}}=\displaystyle \frac{1}{{\delta }_{{\rm{skin}}}}{\int }_{0}^{\infty }{\varepsilon }^{{xz}}(z){{\rm{e}}}^{-z/{\delta }_{{\rm{skin}}}}{\rm{d}}z,\end{eqnarray} \tag{ 5 }$$

we obtain the following dispersion relation for SPPs [47]:

$$\begin{eqnarray}&&{k}_{{\rm{spp}}}(\pm {m}_{y})={k}_{0}\sqrt{\displaystyle \frac{{\varepsilon }_{{\rm{eff}}}}{1+{\varepsilon }_{{\rm{eff}}}}}(1\pm \displaystyle \frac{{\rm{i}}{\varepsilon }_{{\rm{eff}}}^{{xz}}{m}_{y}}{(1-{\varepsilon }_{{\rm{eff}}}^{2})\sqrt{{\varepsilon }_{{\rm{eff}}}}}).\end{eqnarray} \tag{ 6 }$$

For a special case of a ferromagnet (FM) layer with thickness h₁≪δ_skin sandwiched between two layers of a noble metal, the effective medium approximation for the non-diagonal dielectric susceptibility component gives

$$\begin{eqnarray}&&{\varepsilon }_{{\rm{eff}}}^{{xz}}\simeq \displaystyle \frac{{h}_{1}}{{\delta }_{{\rm{skin}}}}{\varepsilon }_{{\rm{(FM)}}}^{{xz}}{{\rm{e}}}^{-h/{\delta }_{{\rm{skin}}}}\,\end{eqnarray} \tag{ 7 }$$

resulting in the following magneto-plasmonic modulation ${\rm{\Delta }}{k}_{{\rm{mp}}}$ of SPP wave vector:

$$\begin{eqnarray}&&{\rm{\Delta }}{k}_{{\rm{mp}}}\simeq {\rm{i}}{\varepsilon }_{{\rm{(FM)}}}^{{xz}}{m}_{y}\displaystyle \frac{2{h}_{1}{\varepsilon }_{{\rm{eff}}}{k}_{0}^{2}}{(1+{\varepsilon }_{{\rm{eff}}})(1-{\varepsilon }_{{\rm{eff}}}^{2})}{{\rm{e}}}^{-h/{\delta }_{{\rm{skin}}}}.\end{eqnarray} \tag{ 8 }$$

Here we have used the expression for SPP skin depth ${\delta }_{{\rm{skin}}}=\tfrac{1}{2{k}_{0}}{\rm{Im}}\tfrac{\sqrt{1+{\varepsilon }_{{\rm{eff}}}}}{{\varepsilon }_{{\rm{eff}}}}$ introduced the magnetic modulation of SPP wave vector $2{\rm{\Delta }}{k}_{{\rm{mp}}}={k}_{{\rm{spp}}}(+{m}_{y})-{k}_{{\rm{spp}}}(-{m}_{y})$. It is worth noting that this result is identical to the analytical expression obtained from transfer-matrix calculations for the limit of an infinitely thin ferromagnetic layer [44, 46]. Figure 5(c) compares the results of this analytical approach with the experimentally observed magneto-plasmonic modulation of SPP wave vector in a Au/Co/Au trilayer structure induced by a weak periodic external magnetic field with an amplitude of 20 mT [44], i.e. upon the in-plane magnetization switching: ${k}_{{\rm{spp}}}(\pm M)\equiv {k}_{{\rm{spp}}}({m}_{y}=\pm 1)$.

The developed theoretical approach appears to be useful not only in the interferometric measurements with SPPs, but also in the conventional Kretschmann geometry. In a Au/Co/Ag multilayer the SPP resonance can be excited at both 800 and 400 nm wavelengths, figure 6(a). The SPP dispersion in figure 6(b) is displayed by the shaded area where the width is proportional to SPP damping (imaginary part of SPP wave vector), which strongly increases when approaching the blue part of the visible spectral range. Whereas SPP dispersion results in the shift of the SPP resonance angle θ₀ for 400 nm light (as compared to 800 nm), the increased damping makes the Kretschmann reflectivity dip much broader (figure 6(c)).

Zoom In Zoom Out Reset image size

The effect of the external magnetic field is the same for both frequencies and leads to a small shift of the resonance angles upon magnetization reversal. The positions of the Kretschmann reflectivity minima precisely correspond to the zeros on magneto-plasmonic modulation curves, confirming the physical interpretation suggesting that the magneto-plasmonic modulation of SPP wave vector should result in an angular shift of the reflectivity minima (see figure 6(c)).

Here we would like to note that the reflectivity minimum in Kretschmann configuration cannot always be attributed to the excitation of SPPs. The dispersion of surface electromagnetic waves obeying the boundary conditions may depend on the direction of the perpendicular energy flow inside a thin metal film [19]. Whereas the energy flow from the top to the bottom (i.e. from the metal-air to the metal-dielectric interface) corresponds to the propagation of damped SPP modes, in the opposite case (energy flow from the bottom to the top) results in the excitation of the so-called perfectly absorbed (PA) mode. The difference in dispersion relations between an SPP and a PA mode (the latter corresponds to the reflectivity minimum in Kretschmann configuration) has been experimentally observed only recently [48]. In the forthcoming discussion of the nonlinear magneto-plasmonic response we leave the question about possible contributions of PA modes at the frequency 2ω open.

As we are going to show in the forthcoming sections, the effective medium approximation for SPPs, which was originally introduced in the linear magneto-plasmonics, appears to be also useful for the nonlinear magneto-plasmonics and acousto-plasmonics.

Nonlinear magneto-plasmonics is a relatively young field of magneto-photonics, which utilizes SPP excitations for tailoring the magnetization-induced contributions to the nonlinear-optical response of systems. The motivation for the emergence of this field of research is essentially twofold. First, as compared with nonlinear plasmonics [4, 5, 27], magnetic field conveniently offers a universal tool for controlling the nonlinear-optical response. Second, nonlinear magneto-optical effects are much stronger as compared to their linear counterparts, where magnetization-induced effects stay relatively small despite of being enhanced by SPPs [45, 49].

As it has been discussed in the previous section, in linear magneto-plasmonics the magnetic effects in the transversal Kerr geometry (magnetization is perpendicular to the incidence plane) are usually ascribed to the magnetization-induced modulation of SPP wave vector k_spp. However, in the nonlinear magneto-optics the response is more complex. In what follows, we shall restrict our considerations of nonlinear optical effects to SHG only, although the general principles of nonlinear magneto-plasmonics also apply to other effects such as difference frequency (often in THz domain), third harmonic generation, etc. Here we focus on the advantages of the SPP-induced SHG in magnetic media, including both enhancement of the magnetic effects and unraveling new SPP-assisted mechanisms to control optical nonlinearities.

One of the basic nonlinear-optical processes, SHG is governed by the second-order nonlinear-optical susceptibility tensor χ⁽²⁾. Various mechanisms of the anharmonicity in the optical response to electromagnetic field E(ω) lead to the emergence of the second-order nonlinear polarization at the double frequency 2ω [50]:

$$\begin{eqnarray}&&{P}_{i}(2\omega )={\chi }_{{ijk}}^{(2)}(-2\omega ;\omega ,\omega )\,:{E}_{j}(\omega ){E}_{k}(\omega ).\end{eqnarray} \tag{ 9 }$$

This polarization then emits an electromagnetic wave. One of the key properties of the χ⁽²⁾ tensor is its strong sensitivity to the inversion symmetry. It can be shown that dipolar χ⁽²⁾ vanishes in the centrosymmetric media such as, for instance, bulk metals. In such systems efficient SHG is generated predominantly at the interfaces, where the inversion symmetry is broken. Under resonant excitation conditions in Kretschmann configuration SPPs boost up the local electromagnetic fields and can strongly enhance the SHG output [51]. In certain cases it is also convenient to take into account the influence of SPPs in the form of a resonant contribution to the χ⁽²⁾-tensor [50, 52]. Importantly, since ${\chi }^{(2)}(-2\omega ;\omega ,\omega )$ is sensitive to resonances (of various nature) not only at the fundamental ω, but also at the double frequency 2ω, SPPs at both of these frequencies contribute to the SHG output [53, 54].

Within this formalism, the magnetization-induced modulation of the SHG output is described in the following way. In magnetized media, the magnetization-dependent χ⁽²⁾ tensor can be represented as a sum of the non-magnetic (crystallographic) χ⁽²⁾ and magnetization-induced ±χ^(2,m) contributions [55]. The latter changes sign upon reversing the magnetization, whereas the interference of these two contributions results in a magnetization-induced modulation of the SHG intensity (figure 7(a)). The magnitude of the effect can be conveniently characterized by the so-called magnetic SHG (mSHG) contrast ρ_2ω:

$$\begin{eqnarray}&&{\rho }_{2\omega }=\displaystyle \frac{{I}_{2\omega }(+M)-{I}_{2\omega }(-M)}{{I}_{2\omega }(+M)+{I}_{2\omega }(-M)},\end{eqnarray} \tag{ 10 }$$

where ${I}_{2\omega }(+M)$ and I_2ω(−M) are the SHG intensities measured for the two opposite directions of magnetization. It should be noted that in a complex multilayer system, where multiple SHG sources are located at each interface, the rigorous description of the SHG output becomes increasingly complicated.

Zoom In Zoom Out Reset image size

Due to a large number of the non-zero χ⁽²⁾ components, whose magnitudes and phases are a priori unknown, the approach we propose here refrains from taking into account all of them. Instead, the effective interface approximation treats effective χ⁽²⁾ at the two interfaces as unknown fit parameters (figure 7(b)). Within this approach, the role of SPP excitation in altering the magnetization-induced effects in SHG can be considered twofold. Upon such an excitation, either the relative magnitude of the χ⁽²⁾ contributions or the relative phase between them can be modulated. The former, realized by Krutyanskiy et al [56] relies on the tensorial nature of χ⁽²⁾. Here, the crystallographic and magnetic SHG contributions are provided by the different projections of the fundamental field E(ω). These projections experience unequal SPP-assisted enhancement resulting in a modulation of the magnetization-induced SHG output. The second option [57, 58] highlighted the importance of the SPP-induced variations of the phase of the electromagnetic field. The physical origin of such phase modulations beyond the usual field enhancement remains largely unexplored.

A number of systems has been successfully investigated with nonlinear magneto-plasmonics. We note that nonlinear magneto-optics of nanoparticles and planar nanostructures [59–62] supporting localized surface plasmon resonances are beyond the scope of this work. The most straightforward approach utilizing SPP excitation at the interface of a ferromagnetic (Fe, Co, Ni) rather than conventional plasmonic (Al, Ag, Au, Co) metals faces a few challenges. Apart from dealing with overdamped plasmonic excitations (L_spp < λ_spp), these metals possess a relatively low second-order nonlinearity usually attributed to the localized d-band electrons. Nevertheless, magnetic SHG studies both on periodically perforated films [57] and in Kretschmann geometry [63] demonstrated significant SPP-assisted modulation of the magnetic contrast ρ_2ω. In the systems discussed above, the incident electromagnetic wave excited SPPs at the fundamental frequency ω and the plasmonic enhancement of the local electromagnetic field E(ω) boosted the efficiency of the nonlinear-optical conversion.

Multilayer structures consisting of a ferromagnetic layer sandwiched between the two layers of noble metals offer a considerable expansion of opportunities in nonlinear magneto-plasmonics. For example, photons initially up-converted (frequency-doubled) at the interfaces can excite SPPs at 2ω and thus contribute to the enhancement of the SHG output [54, 64]. This mechanism requires the sample to support propagating SPP modes at the double frequency 2ω. It can be especially challenging in the case of SPP excitation on periodically corrugated surfaces, where the spatial periodicity was designed to achieve resonant conditions for SPP excitation at the fundamental frequency ω only. Another drawback is related to increased optical losses caused by the spectral overlap of SPP at 2ω with interband transitions, which is the case for Au at photon energies above 2.4 eV. This shifted the focus of attention toward other plasmonic metals such as Al or Ag which are capable of sustaining SPPs across the whole visible spectral range [65–68].

In the following we discuss a showcase of the proposed nonlinear magneto-plasmonic mechanism where a significant improvement of the magnetization-induced modulation reaching up to 33% has been achieved. We consider a thin gold/cobalt/silver trilayer grown on a glass substrate by means of the magnetron sputtering. A 5 nm thin magneto-optically active layer of ferromagnetic cobalt was protected from oxidation by a 3 nm thin layer of gold. A 25 nm thick silver layer acted as the main constituent in this hybrid plasmonic nanostructure, which was excited by 100 fs short laser pulses through the glass prism. The laser beam was focused into a spot of approximately 50 μm  in diameter by means of a lens with 150 mm focal distance, resulting in the angular excitation width of about 1°. The reflected SHG intensity was recorded as a function of the incidence angle θ for the two opposite directions of magnetization in cobalt in the transverse Kerr geometry (see figure 8(a)).

Zoom In Zoom Out Reset image size

Due to the dispersion in equation (1), the SPP excitations at fundamental and double frequencies in Kretschmann geometry occur at slightly different angles. However, nonlinear-optical considerations discussed above suggest a possibility of nonlinear phase-matching between the second harmonic SPP at the gold–air interface with the k-vector ${k}_{{\rm{spp}}}^{2\omega }$ and the excitation source at the silver-glass interface characterized by the in-plane component of the k-vector $n(\omega ){k}_{0}(\omega )\sin \theta$. This phase-matching occurs at an angle θ_nl given by:

$$\begin{eqnarray}&&{k}_{{\rm{spp}}}^{2\omega }=2n(\omega ){k}_{0}(\omega )\sin {\theta }_{{\rm{nl}}}.\end{eqnarray} \tag{ 11 }$$

Being just one of several possible SPP frequency conversion pathways [64, 69], this phase-matching condition is of paramount importance as it determines the resonant SPP-induced enhancement of the nonlinear susceptibility ${\chi }^{(2)}={\chi }_{{\rm{nr}}}^{(2)}+{\chi }_{{\rm{res}}}^{(2)}(\theta )$ with SPP-mediated resonant contribution [50, 52]:

$$\begin{eqnarray}&&{\chi }_{{\rm{res}}}^{(2)}(\theta )\propto \displaystyle \frac{1}{\theta -{\theta }_{{\rm{nl}}}+{\rm{i}}{\rm{\Gamma }}}\,\end{eqnarray} \tag{ 12 }$$

with ${\rm{\Gamma }}={\rm{Im}}[{k}_{{\rm{spp}}}^{2\omega }]/({k}_{0}(\omega )n(\omega ))$.

Whereas the linear reflectivity at both frequencies ω and 2ω shows only small modulations of the order of 1%, as discussed in the previous section (figure 6(c)), the angular dependence of the nonlinear SHG spectra in figure 8(b) displays drastic changes. Contrary to the previously reported results on a gold film [54], the angular positions of SPP resonances for the fundamental and SHG frequencies in our multilayer structure correspond to the pronounced minima in the SHG intensity. A strong dependence of the total SHG intensity on the magnetization direction ${I}_{2\omega }(\pm M)$ is quantified by the magnetic SHG contrast ρ_2ω shown in figure 8(c).

It is seen that the largest magnetization-induced modulation of the SHG intensity is accompanied by the SPP excitation at the SHG frequency and not the fundamental one. We note that the largest mSHG contrast is observed in between of the linear and nonlinear SPP excitation angles, while decreasing almost to zero at the two resonances. The latter is likely to happen due to the 90° phase shift between the magnetic and non-magnetic SHG contributions at the resonances, driven by the SPP electromagnetic field distribution. Angular dependence of mSHG contrast displays a large modulation amplitude reaching 33% at θ = 44°. The data for other excitation wavelengths can be found elsewhere [45]. For the shortest wavelength (760 nm) the mSHG maximum is only about 20%, since in this case the SPP damping at the SHG frequency becomes so large that the system approaches the region with the non-propagating (overdamped) SPPs.

Note that the mSHG contrast at the fundamental SPP resonance is barely reaching 10%. This observation is in line with the most recent results by Zheng et al [63], who reported similar values of the mSHG contrast on a 10 nm thin iron film on glass, as well as with the results by Pavlov et al [70] obtained on Au/Co/Au multilayer, the structures not supporting SPPs at the SHG frequency. This fact, along with the dispersive shift of mSHG maximum reinforces our conclusion that the 33% large mSHG contrast (which is equivalent to the increase of the SHG intensity by a factor of 2 upon magnetization reversal) is dominated by the nonlinear SPP resonance at the SHG frequency.

Following Palomba and Novotny [54], the complex angular dependence of the SHG intensity from a thin gold film on glass is explained by an interference of two contributions coming from the metal-air and metal-glass interfaces. In our case (figure 7(b)) the silver-glass interface acts as a source of the nonmagnetic SHG ${\vec{E}}_{1}^{2\omega }$, and the upper part consisting of Au and Co layers is assumed to generate the electric field ${\vec{E}}_{2}^{2\omega }$ containing both magnetic χ^(2m) and non-magnetic χ⁽²⁾ contributions. Thus the total SHG intensity I_2ω is described by:

$$\begin{eqnarray}{I}_{2\omega } & \propto & | {\vec{E}}_{1}^{2\omega }+{\vec{E}}_{2}^{2\omega }\pm {\vec{E}}_{2m}^{2\omega }| \\ & = & | {\chi }_{1}^{(2)}\,:{\vec{E}}_{1}{\vec{E}}_{1}+({\chi }_{2}^{(2)}\pm {\chi }_{2}^{(2m)})\,:{\vec{E}}_{2}{\vec{E}}_{2}{| }^{2}\mathrm{.}\end{eqnarray} \tag{ 13 }$$

Here the complex tensor components ${\chi }_{1}^{(2)}$ and ${\chi }_{2}^{(2)}\pm {\chi }_{2}^{(2m)}$ represent the effective optical nonlinearities at both interfaces. Owing to the resonant ${\chi }_{{\rm{res}}}^{(2)}$ contribution, the SHG field enhanced at the top interface destructively interferes with the one generated at the bottom interfaces, which explains the experimentally observed SHG intensity minima. Within this approach, using equation (13) and assuming the resonant ${\chi }_{{\rm{res}}}^{(2)}$ given by equation (12) we were able to fit the experimental angular spectra of both SHG intensity and magnetic contrast [45]. The frequency dependence over the dispersive SPP spectral range revealed that the maximum value of 33% mSHG contrast remained constant whereas its angular width decreased as the nonlinear SPP resonance became narrower (Γ decreased) and shifted towards the fundamental one. As such, the dominant role of the nonlinear SPP resonance in Kretschmann geometry, which is responsible for the increase of the mSHG magnetic contrast, was identified.

In conclusion, the nonlinear magneto-plasmonics offers a strong enhancement of the magnetization-induced effects as compared to its linear counterpart. In the given spectral range one would expect to reach further enhancement of mSHG contrast by systematically varying the individual thicknesses in this multilayer structure along the lines discussed in [69–76].

In previous sections we have investigated stationary properties of metal-ferromagnet multilayer structures. Femtosecond light pulses were used to achieve high peak intensities of electromagnetic radiation facilitating efficient nonlinear optical frequency conversion. However, irradiation of metallic samples with femtosecond optical pulses is known to trigger the complex dynamics of electronic, acoustic and magnetic excitations [77–80]. In this section we will discuss transient effects of ultrashort acoustic pulses on SPPs.

Among different mechanisms of acoustic generation the so-called thermo-elastic mechanism is the most common one [81, 82]. Absorption of light leads to fast rise of lattice temperature on a sub-picosecond time scale and the build-up of thermo-elastic stress followed by thermal expansion. This process results in the emission of coherent acoustic pulses with the shape resembling the initial energy deposition profile with a characteristic spatial scale δ_heat. The acoustic pulse duration is given by ${\delta }_{{\rm{heat}}}/{c}_{s}$, where ${c}_{{\rm{s}}}=\sqrt{{C}_{2}/\rho }$ denotes the longitudinal speed of sound, ρ is material density and C₂ is determined by the elastic tensor and depends on the acoustic propagation direction [83].

The heat deposition depth δ_heat is determined not only by the optical skin depth, but also by the diffusion depth of laser-excited hot electrons during the thermalization time with initially cold lattice. This depth is given by ${\delta }_{{\rm{hot}}}\sim \sqrt{\kappa /g}$, where κ and g denote the thermal conductivity and electron-phonon coupling constant in the material, respectively [84]. In plasmonic metals characterized by relatively high thermal conductivity and weak electron-phonon coupling [85] the heat deposition depth ${\delta }_{{\rm{heat}}}\simeq {\delta }_{{\rm{hot}}}\,\sim$ 100 nm largely exceeds the optical skin depth resulting in the generation of relatively long acoustic pulses with the duration of a few tens of picoseconds [86].

For noble metal films of thickness smaller than δ_heat, the temperature distribution is spatially homogeneous and the thermo-elastic dynamics can be reduced to breathing motion of the entire film consisting of the alternating expansion and contraction. This behavior was evidenced in femtosecond time-resolved pump-probe experiments in Kretschmann configuration [3, 87, 88].

Hot electron diffusion in ferromagnetic metals appears to be less efficient (${\delta }_{{\rm{hot}}}\sim {\delta }_{{\rm{skin}}}\,\sim \,$ 10 nm) and results in the overall heat penetration depth δ_heat ∼ 15–20 nm [89, 90]. This short heat penetration depth in combination with a larger speed of sound enables the generation of large-amplitude ultrashort acoustic pulses in ferromagnetic transition metals like Ni, Co and Fe. However, SPPs do not propagate on ferromagnet surfaces thereby rendering the plasmonic detection schemes extremely challenging. Up to now, there are no acousto-plasmonic investigations on thin ferromagnetic films.

The use of hybrid (noble metal)-ferromagnet bilayer structures on a dielectric substrate allows for the generation of ultrashort acoustic pulses in the ferromagnetic transducer and their plasmonic detection at the (noble metal)/air interface, see figure 9(a).

Zoom In Zoom Out Reset image size

As the duration of the acoustic pulses is conserved upon the transmission from one elastic medium into another one, acoustic pulses generated in cobalt are spatially compressed by a factor of 1.8 (ratio of sound velocities in cobalt and gold) down to 10 nm upon injection in softer gold [90].

Femtosecond time-resolved plasmonic interferometry [23] measures transient phase shifts of the plasmonic interferogram at the gold–air interface (figure 9(b)) caused by thermal and acoustic effects triggered by the absorption of an intense femtosecond pump pulse in cobalt layer. The reconstructed pump-induced modulation ${\rm{\Delta }}{\varepsilon }_{{\rm{eff}}}(t)={\varepsilon }_{{\rm{eff}}}(t)-{\varepsilon }_{{\rm{Au}}}$ in figure 9(c) displays the strong acoustic modulation capturing the reflection of the acoustic pulse η(z, t) from the gold–air interface at 70 ps acoustic delay superimposed on a slowly varying thermal background (which will be neglected in the following discussion). The much weaker acousto-plasmonic signals observed at 197 ps and 208 ps are caused by the secondary 10% acoustic reflections from the gold–cobalt and cobalt–sapphire interfaces, respectively, and indicate a good acoustic impedance matching at these interfaces.

The quantitative reconstruction of the acoustic pulse η (z, t) propagating in gold relies on the effective medium approximation. The compressional acoustic pulse in gold $\eta (z,t)=({n}_{{\rm{i}}}(z,t)-{n}_{{\rm{i}}}^{0})/{n}_{{\rm{i}}}^{0}$ creates a layer of higher ion density ${n}_{{\rm{i}}}(z,t)\gt {n}_{{\rm{i}}}^{0}$, which moves at the sound velocity c_s = 3.45 nm ps^–1 in gold in the (111) direction. Since the stationary charge separation between electrons and ions in a metal occurs only within the Thomas–Fermi radius r_TF ∼ 10⁻³ nm, the spatial profile of electron (charge) density n_e(z, t) follows the ionic one: n_e(z, t) = n_i(z, t). Evaluated at the probe wavelength of 800 nm, the dielectric function of gold, ${\varepsilon }_{{\rm{Au}}}={\varepsilon }^{\prime }+{\rm{i}}{\varepsilon }^{\prime\prime }=-24.8+1.5{\rm{i}}$, is dominated by the free-carrier contribution with ${\varepsilon }^{\prime }\simeq -{\omega }_{{\rm{p}}}^{2}/{\omega }^{2}\propto -{n}_{{e}}$. An ultrashort acoustic strain pulse creates a time-dependent spatial profile of the dielectric function ${\varepsilon }^{\prime }(z,t)={\varepsilon }^{\prime }[1+\eta (z,t)]$ inside the metal, which modulates the SPP wave vector ${k}_{{\rm{spp}}}(t)={k}_{0}\sqrt{{\varepsilon }_{{\rm{eff}}}(t)/(1+{\varepsilon }_{{\rm{eff}}}(t))}$, when the strain pulse arrives within the SPP skin depth δ_skin = 13 nm at the gold–air interface:

$$\begin{eqnarray}&&{\varepsilon }_{{\rm{eff}}}(t)=\displaystyle \frac{{\varepsilon }_{{\rm{Au}}}}{{\delta }_{{\rm{skin}}}}{\int }_{0}^{\infty }[1+\eta (z,t)]{\rm{\exp }}(-z/{\delta }_{{\rm{skin}}}){\rm{d}}z.\end{eqnarray} \tag{ 14 }$$

Using the explicit equation for the acoustic reflection at the (free) gold–air interface, $\eta (z,t)={\eta }_{0}(t+z/{c}_{{\rm{s}}})-{\eta }_{0}(t-z/{c}_{{\rm{s}}})$, the integral over space can be converted into the integral over time. The real part of the acoustic modulation

$$\begin{eqnarray}&&{\rm{\Delta }}{\varepsilon }_{{\rm{eff}}}^{\prime }(\tau )=\displaystyle \frac{| {\varepsilon }_{{\rm{Au}}}| }{{\tau }_{{\rm{skin}}}}{\int }_{-\infty }^{\infty }{\eta }_{0}(t)G(\tau -t){\rm{d}}t\,\end{eqnarray} \tag{ 15 }$$

of the effective dielectric function can be used to reconstruct the acoustic pulse η₀(t). Here τ_skin = δ_skin/c_s = 3.8 ps denotes the acoustic travel time through the skin depth in gold (at 800 nm wavelength) and the acousto-plasmonic response function G(t), defined as

$$\begin{eqnarray}&&G(t)={\rm{\exp }}(-| t| /{\tau }_{{\rm{skin}}}){\rm{sign}}(t),\end{eqnarray} \tag{ 16 }$$

is shown in the inset in figure 9(c). Application of the Fourier-based algorithm [91] to ${\rm{\Delta }}{\varepsilon }_{{\rm{eff}}}^{\prime }(\tau )$ allows to accurately reconstruct the acoustic pulse shape η₀(t) with duration of 3 ps and spatial extent of 10 nm, as shown in figure 9(a). It is remarkable that plasmonic measurements provide the detailed shape of an acoustic pulse with the spatial dimensions smaller than the skin depth δ_skin.

Under certain conditions such shorter-than-skin-depth acoustic pulses can be also measured by simple time-resolved reflectivity measurements at the metal-dielectric interface, as shown for the gold–air interface at 400 nm optical probe wavelength [91].

The most striking advantage of the plasmonic interferometry as compared to the conventional pump-probe reflectivity analysis, is that it provides direct measurement of the absolute strain values. Upon increase of the peak pump fluence up to 30 mJ cm^–2, the acoustic pulses reach very large amplitude of 1% and also change their shape, see figure 10(a). The acoustic reshaping at high pump fluences becomes more pronounced in thicker gold films, which allowed us to establish its close relation to the acoustic propagation effects in gold [90]. The peak of the acoustic pulse, where the density of gold is higher, propagates through the gold layer at a higher speed of sound leading to the steepening of the acoustic pulse.

Zoom In Zoom Out Reset image size

The results of the acousto-plasmonic measurements are found to be in an excellent quantitative agreement with the solutions of the nonlinear Korteveg-de Vries (KdV) equation [92]

$$\begin{eqnarray}&&\displaystyle \frac{\partial \eta }{\partial t}+{c}_{{\rm{s}}}\displaystyle \frac{\partial \eta }{\partial z}+\gamma \displaystyle \frac{{\partial }^{3}\eta }{\partial {z}^{3}}+\displaystyle \frac{{C}_{3}}{2\rho {c}_{s}}\eta \displaystyle \frac{\partial \eta }{\partial z}=0.\end{eqnarray} \tag{ 17 }$$

These dynamics are governed by the interplay between the acoustic broadening due to the phonon dispersion ω(q) = c_sq − γq³ with γ = 7.41 × 10⁻¹⁸ m³ s^–1 and self-steepening and reshaping due to the elastic nonlinearity ${C}_{3}=-2.63\times {10}^{12}$ kg m^–1 s^–2. See [83] for a definition of C₃ and [93, 94] for linear and higher order elastic constants in gold; $\rho =19.2$ g cm^–3 is the density of gold. The agreement between the nonlinear theory and the experiment is achieved by setting a single fit parameter, the initial heat penetration depth in cobalt to ${\delta }_{{\rm{heat}}}=20\,{\rm{nm}}$ (corresponding to the initial pulse duration ${\tau }_{0}=3.2$ ps). This value is 50% larger than the optical skin depth in cobalt of 13 nm, indicating the importance of the electronic transport phenomena for establishing the initial distribution of thermal stress driving the thermo-elastic generation.

The quantitative agreement between theory and experiment for samples with different gold thickness demonstrated that nonlinear reshaping was dominated by acoustic nonlinearities in gold, whereas a possible dependence of the initial strain shape in cobalt on pump power played a minor role [90]. The amplitude of the acoustic strain reached peak values ${\eta }_{{\rm{\max }}}\simeq 1 \%$, corresponding to a compressional stress ${\rm{d}}p={c}_{{\rm{s}}}^{2}\rho {\eta }_{{\rm{\max }}}=2.3$ GPa before acoustic reflection and negative −2.3 GPa tensile stress afterwards. Control measurements at lower pump fluences confirmed that the nonlinear reshaping presented in figure 10(a) was fully reversible. At even higher pump fluences in the range of 50 mJ cm^–2, strain pulses with amplitudes reaching 1.5% were obtained. However, under such strong optical pumping consistent with elevating the lattice temperature in cobalt close to its melting point of 1768 K, irreversible degradation of the samples was observed.

Intrinsic damping of longitudinal phonons in gold caused by anharmonic phonon-phonon interactions [95] becomes increasingly important for frequencies exceeding 1 THz. However, in this frequency range and for our experimental geometry, the effect of nano-scale surface roughness entirely masks possible contributions due to phonon attenuation.

In order to illustrate this effect we have studied the topography of the gold–air interface at the position on the sample where acousto-plasmonic measurements were performed, by means of the atomic force microscopy (AFM). The inset in figure 10(b) shows a 3D representation of surface as seen by the acoustic pulses: given the tiny 3.45 nm wavelength of longitudinal acoustic phonons in (111) gold at 1 THz frequency, the nanoscale surface roughness results in the substantial distribution of acoustic arrival times at the gold–air interface. This effect is quantified in the histogram of the acoustic arrival times at the gold–air interface, which we denote as an acoustic delay spread function with a characteristic width of 0.6 ps. Therefore, all possible high-frequency components exceeding 1 THz forming ultrafast acoustic transients on a sub-picosecond time scale (such as, for example, acoustic solitons [83, 92, 96, 97]) are smeared out by the convolution with the response function (compare KdV and KdV+SR curves in figure 10(a)).

The nonlinearity of acoustic propagation can be used to calibrate the conventional pump-probe reflectivity measurements, when the latter are compared with a quantitative experimental technique capable to measure the absolute values of strain amplitudes. Both ultrafast x-ray diffraction [98] and acousto-plasmonic interferometry [90] are reliable techniques to investigate this kind of systems. For example, the comparison of the KdV solutions with the fingerprints of the nonlinear acoustic reshaping observed in femtosecond pump-probe reflectivity measurements on the same structures [99] provides the values for photo-elastic coefficients ${\rm{d}}n/{\rm{d}}\eta =2\pm 0.7$ and ${\rm{d}}k/{\rm{d}}\eta =1\pm 0.3$ in gold at 400 nm probe wavelength, where $n+{\rm{i}}k=1.47+1.95{\rm{i}}$ denotes the complex index of refraction. Accurately calibrated time-resolved reflectivity measurements carry the same physical information while offering considerable advantage over more complex quantitative techniques based on SPPs or x-rays.

When thinking in terms of the Kretschmann configuration, the nonlinearity of the acoustic propagation is unlikely to play a role due to the much smaller thickness, which typically does not exceed 50 nm for the entire multilayer structure. In contrast, the entire excitement about the nonlinear magneto-plasmonics is triggered by a much larger effect in the optical nonlinearities as compared to conventional plasmonics. It would be very instructive to learn if SHG intensity can be modulated by ultrashort acoustic pulses as well.

Here we illustrate a novel and poorly understood phenomenon of SHG modulation by acoustic pulses [100–102] with an example of femtosecond pump-probe measurements on a Au/Fe/Au/Fe multilayer structure grown epitaxially on a MgO(001) substrate. This multilayer structure has been designed to study ultrafast transport of spin-polarized hot carriers in gold injected from laser-excited ferromagnetic iron [103].

In this study, we consider the influence of the acoustic pulses generated in the first 16 nm thin iron layer on the SHG output in the second one, see figure 11. The elementary and oversimplified understanding of photo-acoustic generation suggests that femtosecond pump pulses are primarily absorbed in iron which then generates acoustic pulses propagating in both directions. A 3 nm thin layer of gold aiming to protect the upper iron layer from oxidation acts as an acoustic delay line and results in the bi-polar acoustic pulse emitted into the thicker (49 nm) epitaxial Au(001) layer. An approximate pulse shape is shown in figure 11(a) for three distinct time moments ${\tau }_{1},{\tau }_{2}$ and ${\tau }_{3}$ corresponding to the arrival times of the sharp acoustic fronts at the second Fe/Au interface. This schematic helps to understand the effect of acoustic pulses on the linear reflectivity and SHG output from the second iron layer with the thickness ${d}_{{\rm{Fe}}}=15\,{\rm{nm}}$, see figure 11(b) and their high-resolution temporal zoom in figure 11(c). A few remarkable observations can be inferred from these measurements. First, the nonlinear acoustically-induced modulation ${{\rm{d}}I}_{2\omega }/{I}_{2\omega }$ appears to be almost an order of magnitude larger as compared to the linear acoustic modulation ${\rm{d}}R(t)/R$. Second, both optical signals are precisely correlated with the arrival times of acoustic fronts at the interfaces.

Zoom In Zoom Out Reset image size

Interestingly, transient ${{\rm{d}}I}_{2\omega }/{I}_{2\omega }$ (dSHG/SHG in figure 11) is a continuous function of time. A comparison of the SHG output variations with the linear reflectivity measurements in figure 11(c) clearly shows that the maximum compression (dilution) of the second iron layer correspond to the minimum (maximum) of both SHG intensity and reflectivity modulation. However, the fact that the kinks in the transient variations of the SHG signal are significantly narrower than those in linear reflectivity data demonstrates the extreme sharpness of the fronts of the acoustic pulses.

In centrosymmetric media SHG may arise due to electro-dipole (interface) or magneto-dipole and electro-quadrupole (bulk) contributions. It is commonly believed that the bulk contributions in metals are small as compared to the interface terms [50, 52, 104]. On the other hand, inversion symmetry in the bulk can be broken, for example, by a gradient of strain resulting in the allowed volumetric electro-dipole contribution. Whereas static strain gradients emerge from the growth on a lattice-mismatched substrate, dynamic strain gradients can be induced by ultrashort acoustic pulses. Another possible mechanism relies upon transient acoustic modulation of the surface properties of Fe, including the susceptibility tensor ${\chi }^{(2)}$. In any case, the total SHG output is produced by an interference of multiple SHG sources, whereas the interference conditions are strongly affected by the propagating acoustic pulses.

Despite these ambiguities, the shape of acoustically-induced SHG output modulation can be described within a simple model which considers an interference of the two surface SHG contributions ${E}_{1}^{2\omega }{\rm{\exp }}({\rm{i}}{\phi }_{0})$ and ${E}_{2}^{2\omega }$ generated at the Au/Fe and Fe/MgO interfaces. Here ${\phi }_{0}$ denotes the phase difference between these two interfering terms suggesting that the total SHG intensity is proportional to ${I}_{2\omega }\propto | {E}_{1}^{2\omega }{\rm{\exp }}({\rm{i}}{\phi }_{0})+{E}_{2}^{2\omega }{| }^{2}$. An acoustic pulse $\eta (z,\tau )$ propagating through the iron layer modulates this optical phase $\phi (\tau )={\phi }_{0}+{\rm{\Delta }}\phi (\tau )$ according to

$$\begin{eqnarray}&&{\rm{\Delta }}\phi (\tau )\propto \displaystyle \frac{2\pi }{\lambda }{\int }_{0}^{{d}_{{\rm{Fe}}}}\eta (z,\tau ){\rm{d}}z,\end{eqnarray} \tag{ 18 }$$

where the integration is performed over the entire Fe layer. As such, we obtain ${{\rm{d}}I}_{2\omega }=-(2{E}_{1}^{2\omega }{E}_{2}^{2\omega }\sin {\phi }_{0}){\rm{\Delta }}\phi (\tau )$. Using the acoustic pulse shape shown in figure 11(a) and equation (18) we have calculated transient phase variations ${\rm{\Delta }}\phi (\tau )$.

With an appropriate scaling factor (which depends on the photo-elastic coefficients of iron at fundamental and/or SHG wavelengths) and vertical offset (thermal background), this simulated phase shift demonstrates surprising similarity to the experimental SHG signal in figure 11(c). We note, however, that at present the underlying origin of this phase modulation remains unclear. Due to the nonlinear character of the discussed interference, this phase modulation is acquired by either the driving fundamental fields E₁, E₂ or the emitted SHG field at the double frequency $2\omega$.

From a practical perspective, these preliminary experimental observations and theoretical modeling hold high potential in application to the nonlinear plasmonics and magneto-plasmonics. They can serve as a first step to test the effective interface approximation for complex multilayer structures, where a simple interference model can be generalized to include strain-induced modification of bulk SHG terms dominating in semiconductor and dielectic media [100, 102].

A more general phenomenological description using a single time-dependent nonlinear effective susceptibility ${\chi }_{{\rm{eff}}}^{(2)}(t)$ for the upper interface in figure 7(b) would be also useful to understand the physical limits of ultrafast acoustic modulation in the nonlinear magneto-plasmonic switch sketched in figure 1.