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Surface plasmon modes propagating in metal nanowires are conveniently excited by focusing a laser beam on one extremity of the nanowire. We find that the precise positioning of the nanowire inside the focal region drastically influences the excitation efficiency of the different SPP modes sustained by the plasmonic waveguide. We demonstrate a spatially selective excitation of bound and leaky surface plasmon modes with excitation maps that strongly depend on the orientation of the incident linear polarization. We discuss this modal selection by considering the inhomogeneous distribution of the field components inside the focus. Our finding provides a way to discriminate the effective indices of the modes offering thus an increased coupling agility for future nanowire-based plasmonic architectures.
© 2017 Optical Society of America
1. Introduction
Structures enabling the transport of surface plasmon polaritons (SPP) have been the subject of intense interest because they provide the fundamental building blocks for developing integrated subwavelength optical routing [1,2]. In this context, metal nanowires are particularly interesting since SPP modes propagate in one direction while being strongly confined in the other two [3, 4]. Propagation losses, a general characteristics of plasmon waveguides, can be mitigated by relying on chemically synthesized metal nanowires [5, 6] and propagation distances exceeding tens of micrometer are routinely achieved. Advanced nanowire-based routing functionalities were recently developed either based on stochastic complex arrangements [3,7] or by plasmon-induced single-point nanowire welding [8–10]. These circuiteries are highly interesting because the same physical support can simultaneously carry plasmons and electrons [11], enabling thus the realization of complex architectures.
Different methods are available for optically exciting SPP in a metal nanowire including point-defect scattering [12, 13], hybride waveguide injection [14, 15], quantum emitter coupling [16] or adiabatic tapering [17]. Because of its relative simplicity, a perhaps easier and widespread approach consists at focusing a laser beam at one extremity of the nanowire [18, 19]. Light scattering by the sharp discontinuity provides a broad spectrum of in-plane momenta covering the wavevectors of the plasmon modes sustained by the nanowire. The efficiency of the SPP generation was found to depend on the nanowire cross-section [20] as well as on the polarization state of the focused laser beam, where an electric field parallel to the nanowire is the preferred orientation [21–24]. In this paper, we show that this is not necessarily the case and the optimal polarization condition strongly depends on the precise positioning of the nanowire in the focal spot. We further show that the nature of the excited SPP mode can be selectively excited by aligning the nanowire extremity within the inhomogeneous distribution of the electric field components in the waist of the focused laser.
2. Results and discussion
Silver nanowires are synthesized using a modified polyol process [25]. The synthesis forms pentagonal nanowire section with width varying from few tens of nanometer to over 300 nm [7, 26, 27]. The colloidal solution is drop-cast onto a glass substrate and left to dry, nanowires are thus randomly deposited on the interface. SPP modes are excited by focusing a laser emitting at λ = 780 nm at the apex of a nanowire with the help of a high numerical aperture (N. A.) oil-immersion objective (100× N. A.=1.49). Light scattered at the distal end of the nanowire as well as radiation of high-order SPP modes lost during the propagation [26] are collected using the same objective. Two charge-coupled device (CCD) cameras respectively placed at the conjugate image plane and at the Fourier plane of the microscope provide the intensity distribution in direct and reciprocal spaces [28, 29]. The polarization state of the incident laser beam is adjusted by a combination of a polarizer and a λ/2 waveplate. The experimental setup is schematically depicted in Fig. 1(a).
Fig. 1 (a) schematic representation of the experimental configuration used to excite the plasmon modes in a nanowire and to detect their signatures in real-plane and Fourier-plane imaging. (b) Evolution of the calculated effective indices of the bound and leaky surface plasmon modes as a function of the Ag nanowire cross-section. The section is pentagonal and the nanowire is placed at a glass/air interface. The images on the right are the calculated magnetic field intensity profiles of the leaky and bound SPP modes for a 300 nm wide pentagonal Ag nanowire.
Figure 2(a) displays a wide-field optical transmission image of a ~17μm long nanowire. Figure 2(b) shows the same nanowire locally excited by the laser beam focused on its left extremity (intensity-saturated area). The polarization is linearly oriented with the long axis of the nanowire. Light emitted from the other extremity as well as the presence of luminuous scattering defects distributed along the nanowire confirm the excitation and the propagation of a surface plasmon in this waveguide. The corresponding Fourier image (not shown) does not exhibit the signature of a higher-order leaky mode [26] indicating that only the fundamental bound mode is excited in this nanowire while all the other modes are at cutoff [26, 30]. Figure 1(b) displays the calculated evolution of the effective indices of the modes existing in a pentagonal Ag nanowire deposited on a glass/air interface as a function of its width. The effective indices are numerically computed with a finite-element mode solver (Comsol). For the considered widths, two modes are existing resulting from the hybridization of the excitations located at the corner of the pentagon [26, 27]. The fundamental mode is referred as the bound mode and is characterized by a diverging effective index with reducing section. A second mode is also existing for nanowire’s width larger than 125 nm. This mode has an effective index lower than the glass medium is therefore loosing radiation in the substrate. This mode is identified as a leaky mode. The magnetic field distribution of the modes are shown in the computed cross-sectional maps in Fig. 1(b) for a 300 nm wide pentagonal nanowire. While we do not have a precise measurement of the width of the nanowire pictured in Fig. 2(a), the graph helps us at placing an upper limit at the section since the only mode present is the fundamental mode.
Fig. 2 (a) Optical transmission image of a Ag nanowire deposited on a glass substrate. (b) Optical image when the left apex is illuminated with a strongly focused λ = 780 nm laser. Scattering at the distal end and point-defects distributed along the nanowire confirm the propagation of a surface plasmon. The incident polarization is aligned with the nanowire.
In the following we demonstrate that the efficiency of the plasmon excitation strongly depends on the position of the nanowire apex inside the focal region. To this aim, the apex is raster scanned through the focus by a linearized piezoelectric stage with a step size of about 94 nm. For each (x, y) position, an image such as the one depicted in Fig. 2(b) is recorded. We estimate the scattered intensity at the nanowire extremity by integrating the response over a selected area [red box in Fig. 2(a)] for each (x, y) coordinates. The position of the measurement box in each frame follows the nanowire displacement to insure that only the light scattered by the extremity is integrated. We then reconstruct an intensity map of the scattered light at the output end as a function of the position of the in-coupling apex. Such a representation reveals the focal regions where the nanowire’s termination should be positioned to efficiently excite the fundamental SPP bound mode.
Figure 3(a) shows a 3μm × 3μm reconstructed SPP excitation map when the incident polarization is parallel to the nanowire. The cross-hair in the image approximately coincides with the (0,0) position, i.e. an extremity aligned at the center of the focus. It is clear that the maximum coupling is achieved when the nanowire extremity is precisely placed at this location.
Fig. 3 (a) Reconstructed SPP excitation map when the polarization is (a) along the nanowire (x-direction) and (b) orthogonal to the nanowire (y-direction). The background intensity in the experimental images arises from the residual spill out of the intense laser signal at the focal region. Insets: Corresponding numerical maps evaluated 10 nm below the glass interface for a 1.2μm × 1.2μm scan window. The calculation maps are scaled to the experimental images.
In order to explain this result, we recall that the polarization in the focal region of a strongly focused beam is highly inhomogeneous. With the numerical aperture used in the work, the field polarization is spatially distributed within the focal region in E0,x and E0,y in-plane components and an out-of-plane E0,z contribution [31]. The field focused on the glass/air interface is expanded as two-dimensional plane waves with a gaussian apodization weight where f0 = 0.5 refers to the experimental filling factor on the high NA objective aperture and θmax = arcsin(NA/n1) [31,32]. n1 is the refractive index of the glass substrate. For a -linearly y-polarized incident electromagnetic field, it comes
where and the integrals I00, I01 and I02 write τs and τp are the transmission Fresnel coefficient for s and p polarized plane waves, k║ = n1k0 sin θ1 and w3 = n3k0 cos θ3. The angle θ3 in the superstrate is linked to θ1 by the Snell-Descartes law (according to the definition of the cos θ3 and sin θ3 in the complex plane above the critical angle). The calculated field distributions of Eq. 1 are illustrated in Fig. 4. The main component is the field aligned with the incident polarization E0,y (x, y, z = 0) and is shown in Fig. 4(a). The large numerical aperture objective used in this work generates an orthogonal in-plane component of the electromagnetic field E0,x (x, y, z = 0) forming a clover-like distribution with out-phase oscillating lobes. The distribution is displayed in Fig. 4(b). Figure 4(c) shows the focal distribution of the out-of-plane component E0,z (x, y, z = 0), which also results from the tight focusing of the laser beam.Fig. 4 Calculated spatial distribution of field components of a focused Gaussian beam polarized along the y-direction. The calculation is done considering an objective with a numerical aperture of 1.49. (a), (b) and (c) are the field distributions of the in-plane components E0,y, E0,x and the out-of-plane contribution E0,z, respectively. Note the phase variations between the lobes in (b) and (c). The distributions are extracted from a plane corresponding at the glass/air interface z = 0. The distributions are rotated by 90° if the incident polarization is aligned along the x-direction.
Having calculated the excitation conditions, we proceed to compute the reconstructed maps of Fig. 3. The radius of the model nanowire is 50 nm, and to reduce the computational cost its length is limited to 2.5 μm. The electric field is calculated thanks to the Lippmann-Schwinger [Eq. (3)] and Dyson’s [Eq. (4)] equations sequence
where E0 is the focused beam [Eq. (1)] and ϵAg is the permittivity of the silver. G0 is the reference Green’s function taking into account the susbtrate and G is the Green’s function modified by the presence of the nanowire. The integration over the elongated nanowire is performed using a cuboidal meshing as described in [33]. The inset of Fig. 3(a) shows the intensity calculated at the extremity of the nanowire when it is raster scanned into the focused beam. The computed position-dependent map calculated at a plane 10 nm below the interface shows that the largest out-coupling response is obtained from a central lobe that is in fair agreement with the measured response. For the experimental condition of Fig. 3(a), the E0,x component of the field is predominant and is precisely located at the center of the focal region. We thus explain the reconstructed map by the nature of the SPP mode, which requires a component of the incident electric field aligned with the in-plane SPP wavevector. For the bound mode considered here the effective index lies outside the wavevector spread allowed by the numerical aperture. Coupling of the incident photons to the SPP mode can only be obtained by increasing the wavector distribution via scattering at a sharp discontinuity. This is precisely achieved by the end facet of the nanowire when placed in the E0,x component of the focal field. This interpretation confirms that a polarization aligned with the nanowire leads to the strongest injection [21–24].The reconstructed map is significantly different for an orthogonal polarization. In Fig. 3(b), the maximum coupling efficiency is distributed in four lobes. When the apex is perfectly centered at (0,0) turning the polarization from an electric field aligned with the nanowire (maximum SPP coupling) to an orthogonal direction effectively hinders the SPP excitation as consistently reported in the literature. This condition is pictured by the cross-hairs in the reconstructed maps. However, when the extremity is slightly displaced from the focus center and overlaps one of the four lobes, a SPP is readily excited in spite of an incident polarization state perpendicularly aligned to the nanowire. The pattern observed in Fig. 3(b) for a polarization aligned along the y axis strongly resembles the four-lobe distribution of the depolarized in-plane field component created by the high N. A. objective [see Fig. 4(b)]. These depolarized fields are now aligned with the nanowire, and are thus favoring the excitation of the SPP. Because their field intensity is significantly weaker than the main polarization component, the intensity of the scattered light recorded at the distal end must be weaker than a properly aligned polarization. We do observe such reduction experimentally. The maximum intensity scattered at the nanowire distal end in Fig. 3(b) is reduced by six-fold compared Fig. 3(a), which is consistent with the above interpretation. The simulated map is capturing most of the experimental data including the excitation restricted to the focal area as well as the asymmetric intensities between the left lobes and the right ones. Since the effective index of the mode is not contained within the excitation wave-vector spectrum, scattering of the focal field at the in-coupling facet is thus necessary for providing the SPP in-plane momentum. The excitation asymmetry between the lobes is understood from the SPP mode and the parity of the excitation fields. Although the in-focal component aligned with the SPP propagation, here E0,x, is the major contributing field, the out-of-plane focal field component E0,z may also contribute to the SPP excitation. These two contributions may add up in phase resulting in a higher coupling efficiency or inversely, may reduce the SPP coupling yield when they oscillate out of phase. These two situations are depicted in the focal field map of Fig. 4(b) and (c). The left lobes of E0,x are oscillating in phase with the E0,z lobes whereas the right E0,x lobes are out of phase with the E0,z pattern.
We now turn our attention to a nanowire with a cross-section large enough to accommodate a higher-order SPP mode in addition to the fundamental SPP bound mode. A wide-field optical transmission image of such a nanowire is displayed in Fig. 5(a). Because the effective index of the high-order mode is contained in the light cone of the susbtrate, this SPP is referred as a leaky mode [26, 27, 34] and its progagation along the nanowire can be visualized by leakage radiation microscopy [29]. Depending on the position of the extremity inside the focus, the leaky mode or the bound mode can be selectively excited. Figure 5(b) shows an image of the intensity recorded when the nanowire is excited at the position P1 in the focus. The image reveals the propagation of the low-index leaky SPP mode identified by the presence of continous light emission (leakage) located on both sides of the nanowire. By slightly displacing the nanowire by 300 nm to position P2 [Fig. 5(c)], only the bound mode is excited. Comparing the images, a discrimination of the bound mode from the leaky mode is univocal. The reverse situation is however more difficult. In Fig. 5(b), both modes may contribute to the intensity scattered by the terminal end of the nanowire and a polarization-resolved image may help at evaluating the relative contributions of the two SPPs [24]. Simulated intensity profiles of the leaky and bound modes discussed in Fig. 1(b) are recalled on the right.
Fig. 5 (a) Transmission image of a 18μm-long Ag nanowire. The boxes are integration areas used for reconstructing the excitation maps discussed in Fig. 6 (b) and (c). Visualisation of the leaky and bound SPP modes for two positions P1 and P2 of the nanowire extremity in the focus. The polarization is aligned along the nanowire. The images of the right are the corresponding distributions of the magnetic field and are recalled from Fig.1(b).
Fig. 6 (a) and (b) are reconstructed SPP excitation maps obtained by integrating the intensity emitted in the dotted and solid red boxes indicated in the optical image of the nanowire recalled in the top inset, respectively. The incident light is linearly polarized along the nanowire. The insets are calculated maps obtained by integrating the computed intensity taken 10 nm above the nanowire for the leaky mode and 10 nm below the nanowire in the substrate to account for the bound mode. The calculation windows are scaled to the experimental images. (c) and (d) depict the situation for an incident light polarized perpendicularly from the nanowire. In (b) and (d) the integrated signal may contain a mix contribution of bound and leaky modes.
The intensities inferred from the integration boxes placed along the nanowire and at the out-coupling extremity [red boxes Fig. 5(a)] are recorded for each position (x, y) of the nanowire scanning the focus. The intensity collected by the dashed red box is reminiscent of the excitation of the leaky mode whereas the integration box placed at the extremity may contain the contributions of both modes. Figure 6(a) represents the excitation map for the leaky mode and Fig. 6(b) is an excitation map with a probable mix of the leaky and bound modes. To reconstruct these maps, the left apex is raster scanned from right to left. In other words, the nanowire completely overlap the focus on the left hand side of the images, while the extremity of the nanowire is away from the focus on the right hand side of the images.
The two reconstructed maps for an incident polarization aligned with nanowire are displaying the same sensitivity to the main in-plane component of the field in the focus with the presence of an intense lobe [Figs. 6(a) and 6(b)]. However, there are differences providing a mean to distinguish the order of the excited modes. For instance, the excitation maxima between Fig. 6(a) and Fig. 6(b) are spatially shifted by about 300 nm as indicated by the dotted vertical lines. The relative intensities are also significantly different. The scattered intensity at the extremity of the nanowire is typically more intense than the leakage radiation collected along the nanowire. The images feature a slight asymmetry with respect to the center indicating that the modes are weakly excited when the focal spot no longer irradiates the sole extremity but the entire section of the nanowire [left side of Figs. 6(a) and 6(b)]. Like in the previous section we numerically compute the intensity distribution of the model nanowire when its apex is scanned in the focus. To discriminate between the bound and leaky modes, we use their different field distributions. The intensity of the leaky mode is evaluated by integrating the field intensity located in a plane 10 nm above the nanowire, where the modal distribution is predominant [see inset of Fig. 5(b)]. The computed excitation map of the bound mode is calculated by extracting the field intensity at the extremity of the nanowire 10 nm below the glass interface. The insets of Fig. 6(a) and (b) display the computed excitation maps for the leaky and bound modes, respectively. The maps indicate a strong excitation for both modes when the coupling end is placed on the main field component of the focal fields. We numerically retrieve the displacement of the maxima between the two modes. Subtle discrepancies are noted in the shape of the distributions. Since the intensity experimentally recorded at the nanowire extremity (solid red box) contains a contribution of both modes the reconstructed map in Fig. 6(a) may contribute to the distribution of Fig. 6(b).
Figures 6(c) and 6(d) are excitation maps reconstructed obtained from the intensities recorded by the red boxes for an incident polarization oriented perpendicularly to the nanowire. The insets are the computed maps. As for a polarization aligned with the nanowire, the calculations are reproducing the salient experimental features. The maxima are located into two main excitation lobes aligned with the polarization (y axis). Following our argumentation about the sensitivity of the surface plasmon excitation to the inhomogeneous field distribution inside the focus, the two-lobe pattern ressembles the distribution of the out-of-plane E0,z component of the focal fields depicted in Fig. 4(c) [31]. Both SPP modes are characterized by an out-of-plane component of the electric field [26]. When the out-of-plane focal fields are overlapping the facet of the nanowire the scattered in-plane momentum spectrum contains the effective indices of the two SPP modes and are thus readily excited. Figure 6(c) suggests that the leaky mode can also be launched from the edges of the nanowire albeit with a low excitation efficiency (see color scale). The effective index of this mode is contained within the in-plane momentum provided by the numerical aperture of the objective. The only requirement for exciting the mode is thus the need to have an non-vanishing field overlap between the modal structure of the leaky SPP and the E0,z fields.
We complete our analysis by visualizing the corresponding Fourier planes. Imaging the Fourier transform of the intensity distribution of the type of Figs. 5(a) or 5(b) gives access to the angular distribution of the emitted light in the substrate. By construction the in-plane momenta or effective index of the modes contained within the light cone defined by the numerical aperture of the objective are readily retrieved [26, 35, 36]. For both bound and leaky surface plasmon modes, the light emitted in the substrate bears the character of the excited mode [7, 26, 27] and there is no need to detect the angular distribution of the light scattered by the end face along the propagation direction as recently performed for studing the modal emission pattern of integrated dielectric waveguides [37].
Figures 7(a) and 7(b) are back-focal plane images recorded at the nanowire positions P1 and P2 respectively. These Fourier images are truncated to the half-space of interest. The signature of the leaky mode in Fig. 7(a) is recognized as a bright vertical line at an effective index of neff=1.07.
Fig. 7 (a) and (b) are partial Fourier plane images showing the wave-vector distributions for the nanowire positions P1 and P2 indicated in Fig. 5(a) and (b), respectively. The vertical bright line at an effective index of 1.07 (arrow) is recognized as the signature of the leaky mode. The vertical fringe pattern observed in both distributions originates from the bound mode sustained by the nanowire. (c) Fourier transform of the modal intensity calculated using Eq.6.
As already observed by N. Hartmann and co-workers [38], we detect a fringe pattern in the wavevector distributions when the bound mode is excited [Figs. 7(a) and 7 (b)]. With the nanowires investigated here, we never observed the Fourier signature of a back-reflected mode at negative kx/k0 suggesting that the reflection coefficient at the extremity is small [39]. A standing wave pattern is thus unlikely to explain this Fourier distribution. The oscillations in Fourier space originates from the finite length LNW of the waveguide [40]. The signal recorded in the Fourier plane can be approximated by the truncated Fourier transform of a guided mode propagating along the x-direction. Using the mode solver described by G. Colas des Francs [41], we determine the effective indices of the leaky and bound modes at nspp=1.03 and 1.52, respectively. The discrepancy between the measured effective index of the leaky mode and the calculated one is probably arising from the indeterminate experimental estimation of the nanowire section. The propagation lengths Lspp are respectively 3.3 μm and 36 μm for the two modes supported by a 300 nm-wide nanowire. The bound plasmon mode is TM polarized and the magnetic field follows roughly
where the lateral extension of the mode is approximated by a gaussian profile with a width w0 ≃ 200 nm and ko = 2π/λ. The Fourier plane image is numerically simulated as Figure 7(c) shows the results of the Fourier transform. We observe a good agreement of the fringe spacing with the measured Fourier plane [Fig. 7(b)]. The angular dependence of the collection efficiency of the objective is not taken into account in our simulation and may be responsible for the decay of the visibility for low wave-vectors [42].In the case of a nanowire length LNW smaller or close to the mode propagation length Lspp, the Fourier image presents Gibbs oscillations according to the expression [40]
The oscillation period is therefore Δkx = 2π/LNW or Δn = λ/LNW. In the case of Fig. 7, Δn is evaluated at 0.043 corresponding to a nanowire length LNW=18μm. Within the experimental errors and the limited resolution of the optical microscope this value coincides with the length measured from the optical image in Fig. 5(a). The mode confinement (w0 ≃ 200 nm) is clearly demonstrated by the extension of the intensity distribution of the fringes along the (ky) dimension. Since the fringes measured in the Fourier plane fill the entire (ky) detection window, the mode extension must be smaller than 200 nm in agreement with the calculated profile in the inset of Fig. 5(c).3. Conclusion
In summary, we show that the excitation of surface plasmon in Ag nanowires obtained by focusing a high-numerical aperture objective at one extremity is sensitive to the inhomogeneous distribution of the focal field components. Depending on the nanowire cross-section and incoming linear polarization, the extremity of the nanowire must be correctly positioned in the focal area to optimize the excitation of the different SPP modes sustained by the metal nanowire.
Funding
European Research Council (ERC) under the European Community’s Seventh Framework Program FP7/2007–2013 Grant Agreement no 306772; Agence Nationale de la Recherche (grants PLASTIPS ANR-09-BLAN-0049 and PLACORE ANR-13-BS10-0007); Labex ACTION program (contract ANR-11-LABX-01-01); National Natural Science Foundation of China (NSFC) Grants Nos. 11305091, 11374286, and 61427818.
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