1. Introduction

An electromagnetic field directed at the interface between a metal and a dielectric can cause the free electrons in the metal to oscillate at the same frequency as the field. Under the right conditions such a collective excitation of electrons, known as a surface plasmon (SP), will propagate along the interface, and can be converted back into a freely propagating field when it is scattered by a surface imperfection such as a ridge or a groove [1]. The wavelength of an SP is much smaller than the wavelength of the electromagnetic field by which it is generated. This suggests the possibility of ultra-compact “plasmonic” devices in which information-carrying electromagnetic fields generate SPs that are then processed before being turned back again into a free field [2]. Following the observation of plasmon-enhanced transmission through subwavelength-size hole arrays [3] and single subwavelength apertures [4] in metal plates, numerous research efforts to develop nanoscale plasmonic devices were triggered. Plasmonic couplers [5], waveguides [6], interferometers [7], lasers [8] and dichroic splitters [9] have already been realized. However, for the field of plasmonics to achieve its full potential, it is necessary to control the direction in which SPs are launched. Compact schemes for directional launching of SPs based on geometries such as a nanoslit with a Bragg resonator [10], an asymmetrically illuminated single nanoslit [11] and pairs of nanoslits [12], and an optimized multi-groove coupler [13] have been proposed and implemented, with extinction ratios as high as 50. These schemes all rely on some static, built-in asymmetry that favors a particular direction of SP launching. To address the important aspect of flexible directional launching of SPs, an essential feature for any kind of integrated plasmonic circuitry, we present a generic approach to dynamically switch plasmons between two channels with a constant total intensity and with a nanoscale footprint.

2. Theory

In its simplest form, our approach is depicted in Fig. 1(a), where a narrow slit in a thin gold film is illuminated from the glass substrate by three coherent beams with Transverse Magnetic (TM) polarization. The slit width is such that for an illumination wavelength λ only two TM modes–one symmetric (TM₀), the other anti-symmetric (TM₁)–are non-evanescent. Beam B is normally incident and therefore only excites the fundamental TM₀ mode. In the path of this beam a piezo element is mounted, allowing its phase to be varied. Beams A and −A, which have opposite but equal amplitudes, make an angle of +θ and −θ with B, respectively. They have the same intensity, but are π–phase shifted with respect to each other. It follows from symmetry that the combination of these two oblique beams excites only the TM₁ mode [14]. At the slit exit, both left and right travelling surface plasmons are generated. Their amplitudes are denoted by β^(l) and β^(r), respectively. A series of grooves on either side of the slit is used to evidence the steering effect by converting the launched plasmons back to freely propagating fields whose intensity is measured in the far field.

 

Fig. 1 Principle of the proposed surface plasmon steering method. (a) A subwavelength slit of width w in a gold film supports only two TM modes for λ/2 ≲ w ≲ λ: a symmetric mode (s, green curve) and an anti-symmetric mode (a, blue curve). Three coherent beams, A, −A (with opposite angle of incidence compared to A and π-phase shifted), and B are incident on the slit from the glass substrate. At the dark side of the film, SPs are launched to the left and to the right with amplitude β^(l) and β^(r), respectively. A series of grooves at 8 μm from either side of the slit converts the SPs back to freely propagating fields that are detected in the far field. (b) Illustrating how a coherent superposition of the a and s modes can lead to unidirectional SP launching at a gold-air interface. The first two panels show the intensity of the magnetic field when the slit is illuminated with either the s or the a mode. Superposed dotted blue curves show the total magnetic field scattered on the interface. The length of the white bar in the first panel indicates the illumination wavelength in vacuum (λ= 600 nm), and the slit width w is λ/2.

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To illustrate how plasmon beam steering may be achieved with a two-mode nanoslit, Fig. 1(b) shows the interference pattern generated by an appropriate linear combination of the TM₀ and TM₁ modes in the slit. These modes scatter at the slit exit and a complete extinction of SPs in one launching direction is predicted. The fields are calculated with a frequency-domain, aperiodic Fourier modal method, incorporating perfectly-matched layers (method MM3 in the benchmark article Ref. [15]). The distributions of the magnetic field intensity |H|² in the near field of the metal-air interface are first shown for the cases where the same slit is illuminated with either the symmetric TM₀ or the anti-symmetric TM₁ mode. To illustrate the phase relationship between the excited SP fields, the total magnetic field Re(H) on the gold-air interface is superimposed as a dotted red curve. On each side, the oscillating wave is composed of an SP mode, and a quasi-cylindrical wave that rapidly decays within a few-wavelengths from the slit [16]. For illumination with either the TM₀ or the TM₁ mode, the fields on opposite sides of the slit are in phase or π–phase shifted, respectively. Let us adjust the (complex) amplitude of the TM₀ mode such that it excites SPs on the right side of the slit with the same phase and intensity as the TM₁ mode. It is apparent that the linear combination of the two modes (Fig. 1(b), right panel) then gives rise to complete destructive interference on the left side of the slit whereas constructive interference takes place on the right side. In the specific example of Fig. 1(b) it is taken that λ = 600 nm, w = λ/2, and the refractive index of gold n_Au = 0.23 + i2.98 (see Ref. [17]). The right-most panel of Fig. 1(b) clearly shows that the total field at the gold-air interface is almost null to the left of the slit, indicating that not only the SP excitation is zero, but also that the excitation of the accompanying quasi-cylindrical waves is very weak.

As explained, beams A and −A together excite only the TM₁ mode whereas beam B excites only the TM₀ mode. At the slit exit, each mode can be scattered into radiation that propagates to the far field, or can be dissipated as absorption loss on the gold surface, or be reflected as a backward propagating mode in the slit. Part of the absorption loss is carried by the SPs launched on both sides of the slit. Let the SP scattering coefficients at the left- and right-hand side of the slit for the two-beam system {A, −A} be A_a and −A_a, respectively. For beam B we denote the SP scattering coefficient on each side by B_s. To calculate these coefficients, we use the mode orthogonality of translational-invariant lossy waveguides [16], which yields $A_a={\int }_{-\infty }^{\infty }\left[E_z^{(a)}\left(x,z\right)H^{\text{SP}}\left(x,z\right)-H^{(a)}\left(x,z\right)E_z^{\text{SP}}\left(x,z\right)\right]dz$, and $B_s={\int }_{-\infty }^{\infty }\left[E_z^{(s)}\left(x,z\right)H^{\text{SP}}\left(x,z\right)-H^{(s)}\left(x,z\right)E_z^{\text{SP}}\left(x,z\right)\right]dz$, where the field components $\left[H^{\text{SP}}\left(x,z\right),E_z^{\text{SP}}\left(x,z\right)\right]$, corresponding to an SP propagating in the negative x-direction with a unit power-flow at x = 0, are calculated analytically [1]. Also, $\left[H^{(a)}\left(x,z\right),E_z^{(a)}\left(x,z\right)\right]$ and $\left[H^{(s)}\left(x,z\right),E_z^{(s)}\left(x,z\right)\right]$ are the scattered field components of the combined incident field of beams A and −A, and of the incident beam B alone, respectively. Note that the integral over z is independent of x, provided that x corresponds to an abscissa on the right side of the slit (see details in Ref. [16]).

We define

Figure 2 summarizes the main results obtained for the cross-sections at λ = 632.8 nm and for a gold-film thickness of 200 nm. In Fig. 2(a) the influence of the slit width is shown. The calculation of σ_a is performed by assuming that the angle of incidence of the plane waves with amplitude A and −A is θ = 30°. Starting from w = 0, the SP cross section of the symmetric case (circles) increases gradually to a maximum value ∼ 60 nm at w ≈ 300 nm, and then decreases as the slit width is further increased. The overall behavior is consistent with earlier works on the ability of isolated slits or grooves to launch SPs [16]. More interesting is the anti-symmetric case (crosses) for which the slit is placed at an anti-node of the interference pattern formed by the two incident plane waves. For very small slit widths, the incident field on the slit is virtually null and the TM₁ mode is weakly excited. In addition, since the TM₁ mode is below cutoff, the energy transfer towards the upper slit aperture is inefficient, and it is only when this mode becomes propagating (for w ≈ 300 nm) that a rising behavior is observed. Then σ_a rapidly becomes significantly larger than σ_s. This remarkable result (note that the incident field is null at the slit center for the anti-symmetric case) attests to the great potential of the TM₁ mode to deliver large and steady SP flows, a property that is rarely used in plasmonic devices [21] whose operation mostly rely on the fundamental TM₀ mode and on tiny slits or grooves. [1–5,9–13]

 

Fig. 2 The calculated SP cross sections σ_a and σ_s as defined in Eqs. (1) and (2) for a unit Poynting vector of each of the incident beams. (a) Variation of σ_a and σ_s with slit width w. The angles of incidence of the plane waves A, −A are taken to be θ = ±30°. The two insets indicate the setup for calculating σ_a and σ_s, respectively. (b) Variation of σ_a with angle of incidence θ for slit widths w = 300, 320, 350, and 450 nm. The refractive index of gold n_Au = 0.18 + i2.99 for λ = 632.8 nm, is taken from Ref. [17], and the thickness of the gold film is 200 nm.

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Figure 2(b) shows the influence of the angle θ on the SP cross section σ_a. Starting from θ = 0 (a degenerate asymptotic case for which the incident field is null), the general trend is an increase of σ_a to a peak value for an intermediate angle of incidence, followed by a monotonic decrease to null for θ = 90°. This behavior depends only weakly on the slit width, although we note that as w increases, the angle for maximum SP excitation is gradually shifted to a less oblique angle of incidence, ranging from 46° > θ > 36° for the range of slit widths considered from 300 nm to 450 nm. For θ ≈ 20° and w = 450 nm as used in the experiment hereafter, σ_a = 100 nm, implying that 22% of the energy directly incident onto the slit is converted into SPs launched on the upper interface.

Turning our attention back to the plasmon switch (Fig. 1), it is clear that the SP amplitudes β^(l) and β^(r) of the left and right traveling surface plasmons may be represented as a linear combination of the SPs excited by the TM₀ and the TM₁ mode. It follows that β^(l) and β^(r) are given by the expressions

We note that for wider slits, that allow more than two TM modes, one could use the same scheme to obtain plasmon steering. In such a multi-mode slit the combination of beams A and −A only excites odd modes, whereas beam B excites only even modes. Cancellation of the combined odd modes by the combined even modes at one side of the slit can be achieved by balancing the amplitudes of the beams. The SPs are then launched from the other side of the slit. However, in a wider slit the conversion of incident light into SPs will be less efficient as more light is directly transmitted.

3. Experiment

Figure 3 shows the experimental setup with which the proposed steering of the SP intensities was realized. The linearly-polarized output of a 16 mW He-Ne laser operating at λ = 632.8 nm is first divided into three beams. Each beam is passed through a combination of quarter-wave plates and a linear polarizer such that the field at the sample is TM polarized. To ensure coherent mode excitation, the path difference between the arms was minimized by use of delay lines in arms B and A. Arm B is normally incident, whereas arms A and −A are obliquely incident in air at +30° and −30°, respectively. By mounting the last mirror in arm −A on a micrometer linear translator the two oblique arms are made to have a π phase difference with respect to each other. The last mirror in Arm B is mounted on top of a piezo element which is connected to a DC voltage source with a range of 0 – 300 V. The voltage across the piezo determines the phase δ of beam B. In a separate interference experiment with the same laser, the piezo voltage scale was calibrated in terms of phase, yielding that a 120 V ramp corresponds to a π–phase shift in δ.

 

Fig. 3 Sketch of the experimental setup. The sample is illuminated from the glass-substrate side.

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Scanning electron microscopy (SEM) images of the fabricated sample are shown in the insets of Fig. 4. Subwavelength slits with widths varying between 250 and 650 nm were etched by electron-beam lithography followed by ion-beam etching in a 200 nm thick gold layer evaporated onto a 0.5 mm thick fused-silica substrate. On either side of this “central slit”, at a distance of 8 μm, there is a set of 6 grooves with a 600 nm center-to-center spacing. Due to their tiny widths, only the central slits are etched all throughout the gold film, whereas the grooves are only partially engraved. To ease the alignment procedure, there is a reference slit located at a distance of around 25 μm to the left of each central slit. A 450 nm wide slit, which supports both the TM₀ and TM₁ modes, was used for the experimental results reported hereafter, but similar results have been obtained with other widths. The 8 μm slit-grooves separation helps suppress the amplitude of the quasi-cylindrical waves so that the grooves serve only to scatter the SPs. This ensures that the line trace pattern of the CCD camera in the far-field is effectively proportional to the intensity of the SPs and is not contaminated with additional direct-wave contributions [22].

 

Fig. 4 Typical line trace of the CCD camera screen, perpendicular to the slits. The first low peak on the left (near pixel 70) is the signal from the indicator slit. The second and fourth peak are the intensities I^(l)(δ) and I^(r)(δ) from surface plasmons scattered by the left-hand grooves and right-hand grooves, respectively. The highest peak is the intensity transmitted by the central slit. The insets show sample details of the the central slit and the plasmon grooves made by a scanning electron microscope.

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Experimental results for the 450 nm wide slit are shown in Fig. 5, where the intensities of the left- and right-travelling surface plasmons, I^(l)(δ) and I^(r)(δ) are plotted as a function of the voltage across the piezo element that regulates the phase δ of the normally incident beam. The agreement with Eq. (4) is very good. It is seen that more than 94% of the surface plasmons are launched to the left when the piezo voltage is 80 V, whereas for a voltage of 200 V about 92% is launched to the right. For intermediate voltage settings, arbitrary ratios of I^(l)(δ)/I^(r)(δ) can be obtained, which makes the device act as a variable beam splitter. The average total intensity I^(l)(δ) + I^(r)(δ) = 36.6 (dotted grey line). The attained visibility of 0.82 is limited by several factors, viz. a) the three beams not being spatially fully coherent due to the relatively low coherence length of the He-Ne laser, b) the amplitudes of arms A and −A being different by about 2 – 4%, and c) the amplitude |B_s| differing from |A_a| by about 2 – 5%. Notice however, that the sum of the two SP intensities is rather constant, with a mean value of 36.6 and a relative standard deviation less than 7% over the entire voltage sweep. Also the peak to minimum distance of 120 V is in excellent agreement with the independently observed π change in the phase δ of arm B. Additionally, by performing far-field measurements of the intensity radiated by the slit, we have observed a “lighthouse effect”, i.e. the maximum in the far-field intensity distribution can be continuously shifted from the left to the right and vice versa, as one varies the voltage across the piezo element. This suggests that one may also achieve beam steering in the far-field of a two-mode nanoslit by controlling the linear combination of the two modes.

 

Fig. 5 Experimental results of the proposed plasmon switching method. The SP intensities I^(l)(δ) (red curve) and I^(r)(δ) (blue curve) are shown as a function of the phase δ of arm B or, equivalently, as a function of the voltage across the piezo element. The total intensity I^(l)(δ) + I^(r)(δ) is shown as a dotted grey curve. The error bars indicate the standard deviation of ten independent measurements.

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4. Conclusion

In conclusion, we have demonstrated that the selective coherent excitation of the two fundamental TM modes in a sub-wavelength slit allows us to launch an approximately constant intensity of surface plasmons either to the left or to the right of the slit; or to distribute them in any desired ratio over these two directions. This gives, for the first time, dynamic control over the directionality of surface plasmons. Our theoretical analysis shows that, although its excitation requires a null illumination at the slit center, the TM₁ mode above cutoff offers the potential of higher SP conversion efficiencies compared to narrow sub-wavelength apertures that support only the TM₀ mode. Note that the radially polarized TM₀₁ mode of subwavelength circular holes [23] presents an axial field singularity and is likely, just as the TM₁ mode of slits, to efficiently generate SPs.

The present work illustrates how the combination of a static symmetric structure with a versatile illumination scheme may lead to the controlled launching of surface or guided waves at the nanoscale, and as such it may be considered as a generic demonstration. Indeed, further work is needed to realize a competitive device. With additional calculations, we have checked for λ = 0.6 μm that as high as 40% and 55% of the TM₀ and TM₁ modes are scattered into SPs at the slit exit aperture. Therefore, the throughput of our experimental system is presently limited by the coupling between the incident beams and the slit modes. This coupling can be further improved by increasing either the refractive index of the substrate, or the cross-section of the slit aperture. Different approaches that preserve the symmetry are possible, for example surrounding the slit with an array of optimized phased grooves [13,24,25], or placing a nano-antenna at the near-field of the slit entrance [26]. This would keep the transverse size of the switching device below the diffraction limit. A drastic miniaturization of our table-top illumination setup can be achieved with micro-optical components and gratings, and thanks to the very fast development of active plasmonics technologies [27], it would be interesting to investigate architectures for full on-chip integration. Such an ultra-compact plasmonic switch would have potential application in telecommunications and optical sensing.