⇦Chinese Physics Letters, 2017, Vol. 34, No. 9, Article code 094203 Leaky Modes in Ag Nanowire over Substrate Configuration * Yin-Xing Ding(丁银兴)1,2, Lu-Lu Wang(王鲁橹)1,2, Li Yu(于丽)1,2** Affiliations 1State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876 2School of Science, Beijing University of Posts and Telecommunications, Beijing 100876 Received 15 June 2017 *Supported by the Ministry of Science and Technology of China under Grant No 2016YFA0301300, the National Natural Science Foundation of China under Grant Nos 11374041, 11574035 and 11404030, and the State Key Laboratory of Information Photonics and Optical Communications.
**Corresponding author. Email: yuliyuli@bupt.edu.cn Citation Text: Ding Y X, Wang L L and Yu L 2017 Chin. Phys. Lett. 34 094203 Abstract By launching surface plasmons propagating along the Ag nanowire deposited on a substrate, we clearly observe three leaky modes using the Fourier imaging method. The effective refractive indexes, propagation lengths and electric field distributions of the modes are investigated, which indicate that the energy of the mode with a lowest effective refractive index is mainly distributed in the air, while for the other two modes, it is mainly distributed in the substrate and in the gap between the Ag nanowire and the substrate. These modes enable such a configuration to be used as a multichannel waveguide or highly directional optical antenna, which is of fundamental importance for optical device miniaturizations and photonic circuit integrations. DOI:10.1088/0256-307X/34/9/094203 PACS:42.82.Et, 42.81.Qb, 42.82.-m © 2017 Chinese Physics Society Article Text A one-dimensional (1D) metallic optical waveguide supporting surface plasmons (SPs) is of great significance for small-footprint, low power consumption, high-density, and parallel signal processing applications due to the capacity to confine light at the subwavelength scale.[1] Various types of metallic nanostructure have been proposed for guiding SPs, such as the thin metal films,[2,3] the channel waveguides,[4] the metal nanostrips on a dielectric substrate[5] and the hybrid waveguides.[6] Ag nanowires (NWs), which can be routinely chemically synthesized,[7,8] have been proven to be appealing optical waveguides due to thier high crystallinity and atomically smooth surface,[7-11] low transmission loss,[11] ultrastrong confinement of electromagnetic field,[12,13] and strong interaction with nano-emitters.[14-17] For Ag NWs with symmetric dielectric surroundings, the waveguiding properties can be routinely calculated by solving the Helmholtz equation.[18] However, in most experimental cases, a supporting substrate is usually indispensable for the Ag NWs, such as the waveguides,[13,19] resonators,[10] the optical routers,[20] and the logic gates.[21,22] Therefore, the investigation about the waveguiding properties of Ag NWs over substrate configuration is of great importance for practical applications. Investigations of the effect of a proximal substrate on the SPs propagating along a Ag NW have been carried out,[23] and the modes of a metallic nanowire with a proper substrate have been widely investigated, such as the guiding mode,[24-26] the interface mode,[24] the leaky mode[24] and the hybridized mode.[25] Among them, the leaky mode is an important category due to its ability to transmit energy along the Ag NW and highly directional leakage radiation in the substrate,[27] which enables the Ag NW to be used as an optical waveguide and directional optical antenna in nano-optics. However, most of the experimental investigations about the leaky mode are based on the fundamental mode, whose energy is mostly distributed in the air. The leaky mode whose energy is mainly distributed in the substrate or in the gap between the Ag NW and substrate has not been demonstrated experimentally. In this work, by launching SPs in the Ag NW over substrate configuration and measuring the leakage radiation using the Fourier microscopy, three leaky modes with an incremental effective refractive index are clearly observed. Theoretical investigations of the transverse electric field distributions are carried out, which indicate that the energy of the mode with the lowest effective refractive index is mainly distributed in the air, while for the other two modes, it is mainly distributed in the substrate and in the gap between the Ag NW and the substrate. Other properties such as the effective refractive indexes and the propagation lengths are also calculated, which agree well with the experimental results. Such a plasmonic waveguide is of fundamental significance for the applications in nano-optics, such as the designing of multichannel optical waveguides, directional optical antennas and optical detectors. The Ag NW over substrate configuration is illustrated in Fig. 1(a), which is constructed by depositing a Ag NW on a glass slide. The Ag NWs used in this experiment were prepared using a wet chemistry method,[7] whose length and diameter $D$ are about 10 μm and 250 nm, respectively. A scanning electron microscope (SEM) image of a Ag NW is shown in Fig. 1(b). Fourier microscopy was constructed to obtain the Fourier image of the Ag NW radiation, which could be used to analyze the angular distribution of the radiation. The relationship between the angular distribution of the radiation and the intensity distribution in the Fourier image is illustrated in Fig. 1(c). Here a rectangular coordinate system $xyz$ and a spherical coordinate frame with polar angle $\theta$ and azimuth angle $\phi$ are defined based on the Ag NW. According to the principle of the Fourier imaging method,[28,29] by using the Fourier microscopy, the radiation from the Ag NW could be collected and decomposed into individual plane waves, and each was focused at a unique position on the Fourier plane. A coordinate system ($k_{x}/k_{0}$, $k_{y}/k_{0}$) is built based on the Fourier plane, in which the intensity $I(k_{x}/k_{0}, k_{y}/k_{0})$ in the Fourier image represents the intensity of the corresponding plane wave whose $x$ and $y$ components of the wave vector are $k_{x}$ and $k_{y}$, respectively. Here $k_{0}$ is the free space wave vector.
cpl-34-9-094203-fig1.png
Fig. 1. (a) Illustration of the Ag NW over substrate configuration. (b) An SEM image of the Ag NW. (c) Principle of the Fourier imaging method.
The Fourier microscopy was constructed based on an oil immersion objective (100$\times$, NA=1.4), as illustrated in Fig. 2. A laser beam (wavelength of 671 nm) was transmitted to the oil immersion objective and was focused at one terminal of the Ag NW by the objective to launch SPs. The entrance pupil of the objective was completely filled. The polarization of the exciting laser beam could be adjusted using a half-wave plate. Radiation of the Ag NW was collected and imaged at image plane 1 using the objective and lens 1. A charge coupled device (CCD) was mounted at the Fourier plane to record the Fourier image. By using the beam splitter and lens 2, the direct image of the Ag NW could be imaged on the image plane 2 simultaneously, where it could be recorded by CCD2. An optical image of a Ag NW that supports SPs propagating along it is shown in Fig. 3(a). The bright spot at the left terminal of the Ag NW is due to the reflection of the incident laser beam. For the SP modes, the presence of the underlying high-index substrate fulfills the phase-matching condition for a decoupling of the SP modes into photons radiating in the substrate, which is generally referred to as the leakage radiation. The bright line along the Ag NW is due to the leakage radiation of the SP mode. To image the leakage radiation on the Fourier plane and to exclude the reflection of the exciting laser, a pinhole with a diameter of $\sim$350 μm was mounted in the image plane 1, as shown in Fig. 2. The edge of the pinhole is illustrated by the green circle in Fig. 3(a), thus only the radiation in it could pass through and be imaged in the Fourier plane. A polarizer was mounted at the Fourier plane to measure the polarization of the leakage radiation, whose polarizing axis is illustrated by the red arrow in Fig. 3(a). The angle between the polarizing axis and the $x$-axis is defined as $\beta$.
cpl-34-9-094203-fig2.png
Fig. 2. Schematic diagram of the Fourier microscopy.
cpl-34-9-094203-fig3.png
Fig. 3. (a) Optical image of a Ag NW supporting SPs propagating along it. The blue arrow represents the polarization of the exciting laser beam. (b) Leakage radiation distribution in the Fourier plane. The leaky modes are recognized as three bright lines. Cross section along the $k_{x}/k_{0}$ axis is displayed by the red circles, in which the modes can be identified as the three peaks. The blue solid curves are the Lorentzian fits to the data.
The corresponding Fourier image is illustrated in Fig. 3(b). Three leaky modes can be recognized as three bright lines that are perpendicular to the axis $k_{x}/k_{0}$.[29] Here we name them as m$_{1}$, m$_{2}$ and m$_{3}$. The outer yellow circle represents the maximum angle $\theta$ of radiation that can be collected by the experimental system, which is 67$^{\circ}$. The intensity distribution along $k_{x}/k_{0}$ is extracted, as shown by the red circles in Fig. 3(b), in which the leaky modes are recognized as three peaks. For each mode, the intensity distribution around the $k_{x}/k_{0}$ axis is accurately described by a Lorentzian curve.[30] Using the least square method, the Lorentzian curves are fitted and displayed by the blue solid curves in Fig. 3(b). The effective refractive indexes of the modes can be calculated by the relation $n_{\rm eff}=k_{x, m}/k_{0}$, where $k_{x, m}$ is the value $k_{x}$ of the peak. From Fig. 3(b), $n_{\rm eff}$ of m$_{1}$, m$_{2}$ and m$_{3}$ are read to be 1.03, 1.19 and 1.26, respectively.
cpl-34-9-094203-fig4.png
Fig. 4. (a)–(e) Fourier images of the leakage radiation obtained with different polarizing axes of the polarizer. The breakages of the bright lines are marked by white arrows. (f)–(j) Intensity distributions along the bright lines extracted from (a)–(e). The position of the breakages is marked by black arrows.
To obtain a full cognition about the leakage radiation, we measured the polarization of the leakage radiation by changing the polarizing axis of the polarizer. A series of Fourier images for $\beta=0^{\circ}$, 40$^{\circ}$, 90$^{\circ}$, 130$^{\circ}$ and 160$^{\circ}$ are shown in Figs. 4(a)–4(e), respectively. When $\beta=0^{\circ}$, i.e., the polarizing axis is parallel to the $k_{x}/k_{0}$ axis, the leakage radiation is mainly distributed around $k_{y}/k_{0}=0$. As $\beta$ increases to 40$^{\circ}$, a breakage occurs in each of the bright lines. As illustrated by the white arrows in Figs. 4(b)–4(d), its position moves up as $\beta$ increases, and reaches the middle of the bright lines when $\beta=90^{\circ}$. When $\beta > 90^{\circ}$, it keeps moving up until $\beta=160^{\circ}$. The corresponding intensity distributions along the bright lines are extracted and displayed in Figs. 4(f)–4(j), which give a clear demonstration about the variation of the bright lines. The red, green and blue curves are corresponding to m$_{1}$, m$_{2}$ and m$_{3}$, respectively, and the breakages are marked by the black arrows. The breakages mean the position where the radiation polarization is perpendicular to the polarizing axis. Thus the results indicate that the polarization of the leakage radiation to the direction close to the $k_{x}/k_{0}$ axis is almost parallel to the $k_{x}/k_{0}$ axis, while for the leakage radiation to the two sides of the Ag NW, the radiation has a larger component parallel to the $k_{y}/k_{0}$ axis.
cpl-34-9-094203-fig5.png
Fig. 5. (a)–(c) Normalized transverse energy density of m$_{1}$, m$_{2}$ and m$_{3}$, respectively. (d)–(f) Variation of the effective refractive indexes (red curves) and the damping lengths (blue dashed curves) of m$_{1}$, m$_{2}$ and m$_{3}$ with respect to the Ag NW diameter, respectively.
The direction-dependent polarization of the leakage radiation could be understood by considering the polarization of the SPs on the Ag NW. The electric field of the leaky modes m$_{1}$, m$_{2}$ and m$_{3}$ can be divided into component ($E_{\bot}$) perpendicular to the Ag NW surface and component ($E_{||}$) parallel to the Ag NW axis. The direction distributions of $E_{\bot}$ in the cross section are illustrated by the white arrows in Figs. 5(a)–(c). On the two sides of the Ag NW, the perpendicular components of $E_{\bot}$ of the SPs are in the opposite directions and are contributed dominantly to the light radiated to the two sides of the Ag NW. In the direction around $\phi=0^{\circ}$, the parallel component $E_{||}$ of the SPs is contributed dominantly to the observed radiation.[27] Thus the polarization varies with the direction of the leakage radiation. The properties of the modes are theoretically investigated based on the finite-difference time-domain (FDTD) method. Here the refractive index of Ag is set to be $0.0047+4.5804i$, which is obtained from the interpolation of the data measured by Johnson and Christy.[31] In reality, the cross section of the synthesized Ag NW is pentagonal. However, studying the SEM image of the Ag NWs, we find that their edges are obtuse and there are some small protuberances on the surface, which obstruct the contact of the Ag NWs and the substrate. Thus we simplify the Ag NW as a cylinder in the calculation since a pentagon cross section of the Ag NW would lead to the contact area being larger than the actual one. The diameter of the Ag NW and the refractive index of the substrate are set to be 250 nm and 1.52, respectively. All the other parameters are consistent with the experiment. A convergence test is carried out by enlarging the computation domain until the calculated parameters remain unchanged. The normalized energy density distributions of the modes are shown in Figs. 5(a)–5(c), respectively. The white arrows denote the instant electric field direction. For the m$_{1}$ mode, the energy is mainly distributed at the top of the Ag NW cross section, while for the m$_{2}$ and m$_{3}$, the energy is mainly distributed in the substrate and in the gap between the Ag NW and the substrate. The effective refractive indexes of m$_{1}$, m$_{2}$ and m$_{3}$ are calculated to be 1.04, 1.17 and 1.24, respectively, which agree well with the experimental results. The effective refractive indexes of the three modes with different Ag NW diameters (150–300 nm) are calculated as shown in Figs. 5(d)–5(f). It is seen that the effective refractive index of m$_{1}$ increases as the Ag NW diameter increases. On the contrary, the effective refractive indexes of m$_{2}$ and m$_{3}$ decrease as the Ag NW diameter increases. A damping length $r$ is defined by the distance when the mode power decays to $1/e$.[23] The damping lengths of the three modes are calculated as shown in Figs. 5(d)–5(f). It can be seen that for the m$_{1}$ mode, $r$ decreases rapidly with the increase of the diameter of the Ag NW, while for the m$_{2}$ and m$_{3}$, it increases slightly. In conclusion, by measuring the leakage radiation of the SPs in the Ag NW over substrate configuration using a Fourier imaging method, three leaky modes with an incremental effective refractive index have been observed. Their effective refractive indexes are extracted, which agree well with the theoretical results. Theoretical investigations about the modes have been carried out, which indicate that the energy of the mode with the lowest effective refractive index is mainly distributed in the air, while for the other two modes, it is mainly distributed in the substrate and in the gap between the Ag NW and the substrate. This work indicates that in addition to the fundamental mode, two other modes exist in the Ag NW over the substrate configuration, which are of fundamental significance for design of multichannel waveguides and directional optical antennas in photonic circuits integrations. 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