1. Introduction

Surface plasmon polariton (SPP), with the capability of breaking though the traditional diffraction limit [1], provides nanoscale spatial resolution and tremendous localized electromagnetic field enhancement [2], and thus has formed a platform for designing nanophotonic devices for many applications including optical storage [3], optical sensing [4], optical tweezers [5], Raman scattering [6,7], and others. Following the rapid development of plasmonic devices, the methods of manipulating SPP in space and time domain are urgently needed. Most previous researches on SPP manipulation were performed in space domain by designing various metallic nanostructures, which can achieve modulation functions on SPP such as on-chip focusing [8–10], bend propagation [11–14], and directional propagation [15]. These plasmonic nanostructures have widely been applied in the fields of nanofocusing [16–18], nanoconfined light source [19,20], spin-selective metasurfaces [21], photonic Fourier transform [22], plasmonic non-diffractive beams [11–14,23,24] and vortex beams [25].

By contrast, the researches on the temporal modulation of SPP are less, due to the fact that measuring spatiotemporal properties of SPP pulse is difficult, which usually requires large experimental systems such as near-field scanning optical microscopy (NSOM) combining spectral interferometry [17,26,27]. Currently, the studies on modulation of SPP in time domain can be classified into three groups. The first is to use several dynamically controlled devices, such as spatial light modulator (SLM), to temporally modulate the incident light for SPP excitation [28–30]). The second is directly embedding actively controlled materials into the plasmonic nanostructures, such as electro-optical or nonlinear materials [31,32], which could bring much faster operation speed (picosecond level for nonlinear materials [33]), but it also brings the disadvantages of complex external modulation systems and difficulty in fabrication of material composited nanostructures. The last is to employ the plasmonic resonance response of metallic nanostructures (such as nano-antennas and waveguides) to shape the SPP pulse in time domain [28,34,35], which also requires precision fabrication of nanostructures and has limited modulation functions with each structure. Therefore, a simple method for spatiotemporal manipulation of SPP under nanoscale and ultrafast conditions is still urgently needed.

Recently, it has been reported that metasurfaces can be used to spatiotemporally manipulate the light pulses with frequency-domain modulation methods [36,37]. Although these frequency-domain methods are proposed for light pulses in free space, they bring possibility for control the near-field behavior of SPP pulse especially in time domain. Here, we propose and numerically study a new method of spatiotemporal manipulation on focusing and propagation of SPP pulses, based on frequency-domain modulation through an angularly-dispersed femtosecond incident light. In a plasmonic focusing structure illuminated by the dispersed femtosecond light, we achieve focused femtosecond SPP pulses with dynamic wavefront rotation (WFR) effect [38–40] and the corresponding redirection of SPP propagation. We build an analytical model to explain such behaviors of SPP pulses, and reveal that the spatiotemporal properties of SPP pulses can be controlled by changing the incident conditions and structural parameters. Finally, we investigate the dynamic coupling of modulated SPP pulses to dielectric waveguides, and demonstrate an ultrafast turning of propagation direction to the waveguides, presenting the potential of this method in on-chip optical signal transmission and processing.

2. Results and discussion

 Figure 1(a) shows the schematic of the structure for excitation and spatiotemporal modulation of SPP pulses. The structure is composed of a glass substrate (n=1.515) and a gold film (200 nm in thickness) etched with an arc-shaped slit (100 nm in width) in xy plane. To excite the SPP pulse and modulate it in frequency domain, a y-polarized angularly dispersed femtosecond laser (central wavelength λ₀=800 nm) is obliquely incident in xz plane from the bottom with an angle θ to the z-axis. Here, the SPP is mainly excited by the scattering from the edge of the arc-shaped slit, rather than the wave vector matching caused by the glass substrate. As the arc-shaped nano-slit acts like a focusing lens with focal length R (the radius of the arc), the excited dispersive SPP pulses could propagate towards the center and then form a strong SPP focus [8].

 

Fig. 1. Schematic of excitation and spatiotemporal modulation of SPP pulses. (a) The structure is composed of a glass substrate (n=1.515) and a gold film (200 nm in thickness) etched with an arc-shaped slit (100 nm in width) in xy plane. R and α represent the radius and opening angle of the arc slit, respectively. A y-polarized angularly dispersed femtosecond laser (central wavelength λ₀=800 nm) is obliquely incident in xz plane from the bottom with an angle θ to the z-axis. (b) Schematic diagram of angular dispersion of femtosecond laser pulses after passing through a prism (left), and the temporal wavefron of the angularly dispersed laser source on the incident plane (right). (c) Schematics of in-plane SPP focusing. The field at any point N(x,y) in the vicinity of focus O(0,0) can be calculated by summing up the contributions from all the points, e.g., M(ξ, ζ) on a convergent wavefront. γ measures the inclination angle of the distance with respect to the normal of the arc ∑. β represents the angle between MO and the y-axis. (d) Spatiotemporal evolution of the SPP pulse’s wavefront in the focal region. Different frequency components (shown as blue/green/red colors) focus on different positions of focal plane, and the pulse wavefront (dashed black line) rotates as well as the pulse's propagation direction (purple arrow). (e) The timing diagram of the evolution process of the SPP pulse in the focal region calculated by the 3-D FDTD method under the conditions of θ = 10°, R = 10 µm, α = 160° and τ = 10 fs (pulse duration). The upper right corner gives the moment of the pulse, and the green arrow indicates the pulse propagation direction at that moment. (f) The analytical results obtained by Eq. (10) under the same conditions as (e).

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To observe the spatial distribution and temporal evolution of SPP pulses in the structure, we use a 3-dimentional finite-difference time domain (3-D FDTD) simulation (Lumerical FDTD Solutions) method. In the FDTD simulation, the time-domain expression of the incident laser pulse is expressed as:

In order to explain the basic principles of this WFR phenomenon of SPP pulse, we build a simple theoretical model. For the SPP pulse excited by the arc-shaped slit. The dominated electric-field component E_z perpendicular to the metal surface satisfies the two-dimensional (2-D) Helmholtz equation,

As shown in Fig. 1(d), the field at any point N(x,y) in the vicinity of focus O(0,0) can be calculated by summing up the contributions from all the points, e.g., M(ξ, ζ) on a convergent wavefront. When the focal length R is much larger than the SPP wavelength ${\lambda _{SPP}}$, by using the 2-D Helmholtz and Kirchhoff integral theorem [42,43], the perturbation at any point in the focal region can be approximated as [22],

Then, based on this, we introduce the above-mentioned angular dispersion light source, and substitute Eq. (5) into Eq. (8), then we have:

The ${\boldsymbol{ E^{\prime}}_{\textbf Z}}$ in Eq. (9) gives the SPP focal filed, which is a function of spatial position (x, y), incident frequency ($\omega$) and the time ($\textrm{t}$). When the ${\boldsymbol{ E^{\prime}}_{\textbf Z}}$ in Eq. (9) is integrated for all frequency components $\omega$ of the pulse, the spatiotemporal evolution process of SPP pulse can be obtained as:

To study such dynamic behavior of SPP pulse, in Fig. 2 we compare the spatiotemporal properties of SPP excited with different source conditions. We choose R = 10 µm, α=120° and τ = 10 fs, and change the incident angle θ that affects the dispersion property according to Eq. (2) for simulating and analysing. Firstly, we consider the case of normal incidence (θ = 0°) in Fig. 2(a), where the frequency-dependent incident angle ${\theta _f}$ is always zero without angular dispersion according to Eq. (2). Thus, different frequency components of SPP pulse are all focused at the center (right part of Fig. 2(a)) with no dispersion. Accordingly, the wavefront of SPP pulse at the focal plane (green dotted line in left part of Fig. 2(a)) remains unchanged during its temporal evolution of the focusing process, as shown in Fig. 2(d). By contrast, in the case of oblique incidence (θ = 10°) with angular dispersion, both the FDTD simulation results (Fig. 2(b)) and the analytical calculation results (Fig. 2(c)) show that, the focal point shifts to the right side of center, and the different frequency components of SPP pulse focus at different positions in the focal plane as predicted in Fig. 1(d). So in time domain (Fig. 2(e)), the wavefront of SPP pulse gradually rotates during the focusing process (WFR effect), leading to the temporal redirection of its propagation as shown in Figs. 1(e). For comparison, we also consider the case of an 800 nm continuous wave (CW) source with same oblique incidence (θ = 10°) that has no dispersion, and find that the temporal WFR effect of the SPP pulse disappears (Fig. 2(f)), which confirms that the spatiotemporal WFR effect of SPP is generated by the dispersion.

 

Fig. 2. Spatiotemporal properties of SPP pulses with different source conditions. (a) FDTD time-averaged field |E_z| at pulse center wavelength (800 nm) with incident angle θ = 0° (left), and the enlarged field |E_z| in the focal region (white dashed square in (a)) at 756/800/850 nm wavelength of the SPP pulse (right). Results are normalized by the maximum value at the focal point. Other parameters are R = 10 µm, α = 120° and τ = 10 fs. (b) Same as (a) except with θ = 10°. (c) The analytical results obtained by Eq. (8) under the same conditions as (b). (d)-(f) Temporal evolution of the SPP pulse across the focus plane in x direction (green dotted line in (a) and (b)) with different source conditions: (d) θ = 0°, pulsed light (τ = 10 fs), (e) θ = 10°, pulsed light (τ = 10 fs), (f) θ = 10°, CW (800 nm wavelength). (g) The spatial position of the SPP focus at 800 nm wavelength as a function of θ. (h) The WFR velocities of SPP in focal region as a function of θ. $V_r^{cal}$ and $V_r^{sim}$ represent the analytical result obtained by Eq. (12) and the FDTD simulation result, respectively.

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As the incident angle θ changes, the spatiotemporal properties of SPP could be modulated accordingly. In Fig. 2(g) the spatial position (x, y) of the SPP focus at 800 nm wavelength is shown as a function of the incident angle θ. It can be seen that the SPP focal position shifts farther away from the center (θ = 0° position) with larger θ in space domain, due to the increase of the gradient phase factor caused by the oblique incidence. In time domain, to quantitatively study the influence of incident angle θ on the WFR effect of SPP, we calculate the corresponding WFR velocity ${V_r}$ [40]:

According to Eq. (12), there are four key parameters that affect $V_r^{cal}$: θ, R, α and τ. Besides the above studied incident angle θ, the influence of the other three parameters (radius (R), opening angle (α), and pulse duration (τ)) on the spatiotemporal properties of SPP are investigated in Fig. 3. Here, the propagation length of the SPP corresponding to the center wavelength is 47.18 µm. Figures 3(a1) and 3(a2) show the distribution of the time-averaged field |E_z| at pulse center wavelength with R=5 and 15 µm, respectively. The two cases have almost same SPP field distribution in the focal region. Their corresponding temporal evolution of the SPP pulse across the focus plane (green dotted line in Figs. 3(a1) and 3(a2)) are shown in Figs. 3(a4) and 3(a5), respectively, where the case of R=15 µm presents a more obvious temporal wavefront rotation than that of R=5 µm during the focusing process. In Figs. 3(a3) and 3(a6), as the radius R increases, in space domain the SPP focus moves approximately linearly to the lower right side caused by the linear amplification of the overall structure, meanwhile the increase of the optical path enhances the dispersion effect, leading to the increase of the WFR velocities in time domain (Fig. 3(a6)). Figures 3(b1)-(b6) show the effect of opening angle α. In space domain, when the opening angle α increases, the weakly focused SPP gradually becomes tightly focused, resulting in a decrease in its focal depth (Figs. 3(b1) and 3(b2)), at the same time, the focal position of SPP only has a small shift in y direction towards the theoretical focal point (Fig. 3(b3)). In time domain, as α increases, the rotation of the temporal wavefront becomes more obvious (Figs. 3(b4)-3(b5)), which means that, the WFR velocities also increase (Fig. 3(b6)). This is equivalent to the effect of increasing the beam diameter w in Eq. (11). We finally study the effect of pulse duration τ in Figs. 3(c1)-(c6). Since the pulse duration only affects the spectral bandwidth rather than the center frequency, the spatial focal position remains unchanged as τ increases (Figs. 3(c1)-(c3)), but stronger WFR effect appears with shorter pulse duration (Fig. 3(c4)-(c6)), because the broader bandwidth of shorter pulse induces a larger angular dispersion, which is also consistent with the prediction of Eq. (12). Combining the effects of these parameters on SPP pulses in both space and time domain, our proposed method is proved to flexibly manipulate the spatiotemporal properties of SPP pulses. It is noted that, in our FDTD simulation the dispersion effect of the glass is very small and negligible, however, in experiment the glass's dispersion effect could be large depending on its thickness. For example, a femtosecond pulse with a center wavelength of 800 nm and a pulse duration of 10 fs could be stretched to about 20 fs after passing through a fused silica glass with a thickness of 1 mm, so that the SPP temporal property could be influenced according to Fig. 3(c6). In addition, in experiment the peak power of the input femtosecond laser pulse should be very low to avoid the annihilation of gold film and the nonlinearity-induced dispersion effect.

 

Fig. 3. (a1-a6) Effect of radius (R) on spatiotemporal properties of SPP pulses. (a1) and (a2) show FDTD time-averaged field |E_z| normalized by the maximum value at the focal point at pulse center wavelength (800 nm) with R=5 and 15 µm, respectively. (a3) Spatial position of the SPP focus at 800 nm wavelength as a function of R. (a4) and (a5) Temporal evolution of the SPP pulse across the focal plane corresponding to the green dotted line in (a1) and (a2), respectively. (a6) The calculated WFR velocities of SPP in focal region as a function of R. (b1-b6) Similar as (a1-a6) but for the effect of opening angle (α). (c1-c6) Similar as (a1-a6) but for the effect of pulse duration (τ). The white solid line indicates a length of 5 µm in all figures. Unless specified in the figure, the parameters are chosen as: θ = 10°, R = 10 µm, α = 120° and τ = 10 fs.

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To demonstrate the potential of such spatiotemporal manipulation of SPP pulses in on-chip devices, we study the coupling of the modulated SPP pulses to waveguides on metal surface. As shown in Fig. 4(a), two dielectric waveguides are placed on the metal surface near the SPP focus with the included angles of −50° and 50°, respectively. Figures 4(b) and 4(c) show the spatial distribution of the SPP field at 756 nm and 850 nm wavelength of the SPP pulse, respectively, where the 756/850 nm component is mainly coupled to the left/right waveguide depending on their different focal positions (Fig. 2(b)), indicating that the ultrashort SPP pulse can be controlled to couple into waveguides with different frequency components. The timing diagram of the evolution process of the SPP pulse coupling into the waveguides are shown in Fig. 4(d), which demonstrates that the SPP pulse could couple into the left and right waveguides successively in time domain due to the WFR effect (Fig. 1(e)). In order to observe this coupling process more intuitively, we compare the time signal (Fig. 4(f)), spectral amplitude (Fig. 4(g)) and spectral phase (Fig. 4(h)) of the SPP pulse at the focal point (F) and two sampling points (L and R) inside the two waveguides. It is found that the SPP pulse is coupled into the two waveguides at different time during the WFR process (Fig. 4(f)), which agrees with the results in Fig. 4(d) and proves that the pulse coupling time can be tuned by the WFR effect. The spectral bandwidth of the pulse at L and R points are also different from that at the focal point (F), consistent with the results shown in Figs. 4(b) and 4(c). We also calculated the waveguide coupling efficiency ($\Psi \textrm{(}\omega \textrm{)}$) for different frequency components (Fig. 4(i)). Here, the coupling efficiency is defined as the ratio of the total energy coupled into the waveguide (${I_\textrm{WG}}(\omega )$) to the total energy at the focus plane along the x direction (${I_\textrm{foci}}(\omega )$), that is $\Psi \textrm{(}\omega \textrm{) = }{{{I_\textrm{WG}}(\omega )} / {{I_\textrm{foci}}(\omega )}}$. And the coupling efficiency of the 756/850 nm component to the left and right waveguides are apparently different, in accord with the spectral amplitude shown in Fig. 4(g). Further, in Fig. 4(e) we consider the case of a single waveguide and change the included angle ϕ of the waveguide around the SPP focus, and then obtain the dependence of the coupling strength on the time and the inclined angle ϕ. It shows that the propagation direction of SPP pulse coupled into the waveguide changes over time during the WFR process, and the effective coupling region covers an angular range of −55° to 70° within a time window of ∼20 fs. Therefore, ultrafast beam splitting of SPP pulse at different time and different propagation directions could be realized. Compared to the dynamic beam steering in free space [37], here the dynamic steering of SPP pulse shows broader angular range and shorter time, which offers great potential to ultrafast on-chip photonic information processing.

 

Fig. 4. Coupling of the modulated SPP pulses to waveguides. (a) Schematic of the waveguide coupling structure. Two rectangular glass waveguides (L and R) are placed on the gold surface with a distance of 0.5 µm from the focus, and their included angles to the y-axis are −50° and 50°, respectively. Both waveguides have refractive index of 1.515 and 300 nm in both width and height. Other parameters are: α = 160°, R = 10 µm, θ = 10°, τ = 10 fs. (b) and (c) Spatial distribution of time-averaged field |E_z| normalized by the maximum value at the focal point at 756/850 nm wavelength, respectively. (d) The timing diagram of the evolution process of the SPP pulse coupling into the left and the right waveguides. The red line marks the position of the two waveguides. (e) The coupling strength of SPP pulse to single waveguide as functions of the included angle ϕ of the waveguide and the time progress. The time signal (f), spectral amplitude (g) and spectral phase (h) of SPP pulses at the focus (F) and at the two sampling points inside the two waveguides (L and R, both 2 µm away from the focus F). The time signals and spectral amplitudes are normalized to the maximum value at the focus. (i) Coupling efficiency of waveguides L and R at different wavelengths.

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3. Conclusion

In summary, we propose a novel method for spatiotemporal manipulation of SPP pulses based on frequency-domain modulation. In this method, an obliquely incident femtosecond laser with angular dispersion is used to control the dynamic behavior of SPP pulses in time domain, meanwhile, an arc-shaped nano-slit is used to manipulate the spatial focusing of SPP, and thereby achieving the spatiotemporal manipulation of SPP pulses. We demonstrate the ultrafast WFR effect in SPP focusing and the corresponding redirection of SPP propagation, and verify that such spatiotemporal properties of SPP pulses can be dynamically controlled by changing the parameters of structure and light source. A simple analytical model is built and agrees well with the FDTD simulation results. By studying the on-chip coupling of the SPP pulse to the waveguides, we show that the dynamic steering of SPP pulses can achieve a large-angle rotation in propagation direction within tens of femtoseconds. It is worth noting that, theoretically, this method is suitable not only for SPP, but also for other surface waves such as Bloch surface waves [45]. Although these are simulation results, one can use photoemission electron microscopy (PEEM) [8,9] system, or combine near-field optical probe microscopy (such as NSOM or AFM) with spectral interferometry [17,26,27] to experimentally verify the temporal and spatial characteristics of femtosecond SPP pulses. This work could provide new ideas for spatiotemporal manipulation of optical pulses, and has great potential in applications of ultrafast photonic information processing [25], ultrafast beam shaping [46] and splitting [47], and attosecond pulses generation [39].