In this paper, the excitations of surface plasmon polaritons (SPPs) by both perpendicular and parallel electron beam are investigated. The results of analytical theory and numerical calculation show that the mechanisms of these two excitations are essentially different, and the behavior and properties of SPPs in metal structures strongly depend on the methods of excitation. For the perpendicular excitation, SPPs contain plenty of frequency components, propagate with attenuation and are always accompanied with the transition radiation. Whereas for parallel excitation, SPPs waves are coherent, tunable, propagating without attenuation and the transition radiation does not occur. We also show that there are two modes for the parallel excited SPPs on the metal films and they all can be excited efficiently by the parallel moving electron beam. And the operating frequency of SPPs can be tuned in a large frequency range by adjusting the beam energy.
© 2014 Optical Society of America
Surface plasmon polaritons (SPPs) are slow waves due to the collective oscillations of the free electron gas in noble metals [1, 2], and become attractive research topics because of their interesting physics and important applications in various areas, such as sensors [3, 4], field imaging [2, 5], enhanced radiation [6–8], enhanced optical transmission [9, 10] and terahertz devices , etc.. SPPs can be excited by incident plane waves and electron beam (e-beam). For incident plane wave excitation, the wave vector of SPPs is larger than that of plane wave, so special experimental arrangements have to be designed to provide conservation of the wave vector, such as Krestschmann and Otto geometry, etc . Different from plane wave excitation, SPPs can be excited directly by e-beam moving both perpendicularly and parallel to the metal surface [1, 13–18]. Perpendicular excitation of SPPs has been wildly studied, and the high quality e-beam of scanning electron microscope (SEM) is often used [13–15, 19]. This study has greatly enriched the applications of SEM. Parallel excited SPPs have also potential applications in modern science and technologies. For example, it has been found that parallel excited SPPs can be transformed into coherent and tunable radiation with greatly intensity enhancement [7, 8].
The detailed investigation and comparison of SPPs excited by perpendicular and parallel e-beam are presented in this paper. The theoretical analysis and numerical calculation show that the mechanisms of these two excitation methods are essentially different, and each of them has its own unique behavior and properties.
This paper is organized as below: the theoretical and numerical investigation on the two excitations is given in section 2 and section 3, respectively. The parallel excitation of SPPs on metal films is presented in section 4. The comparison of the two excitations is presented in section 5, and section 6 is the conclusion.
2. The perpendicular excitation
The schematic of perpendicular excitation is shown in Fig. 1, and the e-beam is uniformly moving along Z direction perpendicular to the noble metal surface. Region I is vacuum, region II is metal, and its relative dielectric function is given by the modified Drude model: [20–23]. In this paper, Ag is used and we have,, .
The charge density of the perpendicular moving electron beam is , where is the beam velocity along Z direction, is the charge quantity of the beam. By means of Fourier transformation, the fields generated by the perpendicular moving e-beam can be obtained [24–26].
The e-beam together with the generated waves injects through the vacuum and metal boundary perpendicularly. It can be seen that the incident waves alone cannot satisfy the boundary conditions on the interface. Hence additional fields should be added, which can be found by using the homogenous Maxwell equations [24–26]. Accordingly, the boundary conditions should be:Eq. (3), for a pole occurs at . By means of numerical integration approach, the fields of both transition radiation (TR) and SPPs can be obtained in frequency domain. In the limit, SPPs dominate the fields on the surface, and in this case integral in Eq. (3) can also be completed by the plasmon pole approximation [26, 27].
It can be seen from Eq. (3) that, depending on the , there are two cases for perpendicular excitation. When is real, the additional fields are TR fields radiating into vacuum. When is imaginary, the additional fields are SPPs fields decaying exponentially in Z direction and propagating with attenuation in R direction. Therefore, for perpendicular excitation, SPPs are always accompanied with TR.
For the excited SPPs, the tangential fields of SPPs should be continuous to satisfy the boundary conditions on the vacuum/Ag interface. Then the dispersion equation for SPPs can be found:
It can be seen from Eq. (4) that of SPPs should be complex values for is complex. The imaginary part of determines the attenuation of SPPs propagation.
The dispersion curve of Ag excited by perpendicular e-beam is shown in Fig. 2(a), which governs all the frequency components of SPPs. As shown in the inset of Fig. 2(a), the excited SPPs contain plenty of frequency components. This can be clearly seen from Fig. 2(b) that all the frequency components of SPPs are excited by any beam energy (In this paper, the varying beam energy is descript by β, and β = u0/c). The dependence of SPPs amplitude for the fixed frequency component 800 THz on beam energy is shown in the inset of Fig. 2(b), and it shows that the SPPs amplitude increases with the increasing beam energy.
The contour map of SPPs and TR fields at 800 THz in the vacuum is shown in Fig. 2(c). The excited TR radiate into the vacuum and the excited SPPs are confined to the surface and propagate along the R direction. The amplitude distributions of SPPs field along the surface at 750 THz and 800 THz are shown in Fig. 2(d), respectively. It can be seen that different frequency components of SPPs have different decay lengths depending on the imagery part of .
3. The parallel excitation
The schematic of parallel excitation is shown in the inset of Fig. 3(a), and the e-beam is uniformly moving parallel to Ag surface along Z direction. The charge density of the parallel moving electron beam is , where is the beam velocity along Z direction, is the charge quantity of the beam. Based on the Maxwell equations, the fields generated by the e-beam are obtained .
Making use of the Maxwell equations and the boundary conditions, SPPs fields can be obtained:
For the phase velocity of SPPs excited by parallel e-beam equals the velocity of the beam, the dispersion equation of SPPs becomes:Eq. (8) is similar to Eq. (4) in form, but in physics they are substantially different.
As shown in Fig. 3(a), the operating frequency of parallel excited SPPs is determined by the working point, which is the intersection point of the dispersion curve and the beam line, and only SPPs at the working point can be excited. Therefore, the parallel excited SPPs waves are coherent and the operating frequency of SPPs can be tuned by adjusting the beam energy. It is illustrated in Fig. 3(b) that the operating frequency of SPPs is 820 THz for beam energy 200 keV and 870 THz for 50 keV. As shown in Fig. 3(c), the increase of the beam energy leads a lower working point and in turns a lower operating frequency. It can be seen that the operating frequency of SPPs can be tuned in a large frequency range.
The contour map of the SPPs field at 870 THz in vacuum for beam energy 50 keV is shown in Fig. 3(d). It can be seen that SPPs are excited by parallel moving e-beam without TR accompanied. Furthermore, the parallel excited SPPs propagate on the Ag surface together with the beam, and then they are able to get energy from the beam continuously to compensate the energy loss due to the metal. Accordingly, there is no attenuation for SPPs excited by the parallel moving e-beam along the propagation direction.
4. The parallel excitation for Ag film
As discussed above, for perpendicular excitation, all the SPPs components satisfying the boundary conditions on metal films will be excited. They propagate with attenuations and are always accompanied with TR. Now we study SPPs on metal films excited by parallel e-beam in detail.
A. Free-standing Ag film
As shown in Fig. 4(a), SPPs on metal films consist of two modes depending on the transverse distribution of Ez field: the symmetrical mode and the asymmetrical mode. The contour maps of the parallel excited SPPs fields for β = 0.6 (130 keV) are shown in Fig. 4(b): 781 THz for the symmetrical mode and 890 THz for the asymmetrical mode. It can be seen that both of the two modes are confined to the film surfaces, and each of two modes gets the same amplitude on the two surfaces of the film.
Compared to the evanescent waves generated by the e-beam, the excited SPPs get much higher amplitude. The dependence of the field amplitude (It is normalized by the field amplitude of evanescent waves generated by the electron beam without considering of metal at y = h in this paper.) and operating frequency of SPPs on beam energy for fixed film thickness 40 nm are shown in Fig. 4(c). For symmetrical mode, the SPPs amplitude grows with the beam energy, which reaches up to 65 for β = 0.7 (200 keV). But for asymmetrical mode, it almost remains about 25 in a large beam energy range. In general, the amplitude of symmetrical mode is larger against asymmetrical mode for the free-standing Ag film. Furthermore, the operating frequency of symmetrical mode can be tuned in a large frequency range. As shown in Fig. 4(c), it can be tuned in the frequency range from 730 THz to 870 THz by adjusting the beam energy.
The dependence of the SPPs field amplitude on the film thickness for fixed β = 0.6 is shown in Fig. 4(d). With the increase of the film thickness, the excitation of the film tends to be similar to that of semi-infinity metal. This results in the decrease of the amplitude for symmetrical mode and the increase for asymmetrical mode till to be the same.
B. Substrate Supported Ag film
The dispersion curves of the Ag film supported by the dielectric substrate are shown in Fig. 5(a). When the beam energy is larger than the Cherenkov threshold, only the asymmetrical mode is excited and it will be transformed into coherent Cherenkov radiation . In this paper, we focus on the SPPs confined to the film surfaces, and the contour maps of the excited SPPs for β = 0.5 (80 keV) are shown in Fig. 5(b). It can be seen that the symmetrical mode (713 THz) dominates the SPPs fields on the substrate/Ag interface, whereas the asymmetrical mode (875 THz) dominates on another.
The dependences of the field amplitude and operating frequency of SPPs on beam energy at fixed film thickness 40 nm on the vacuum/Ag and substrate/Ag interface are shown in Figs. 6(a) and 6(b), respectively. For asymmetrical mode, the amplitude grows with beam energy till the Cherenkov threshold, whereas that of symmetrical mode increases firstly and then decreases. On the vacuum/Ag interface, the amplitude of asymmetrical mode reaches up to 108 when the beam energy is near the Cherenkov threshold, which is much larger than that of symmetrical mode. However, on the substrate/Ag interface, the symmetrical mode gets higher amplitude than asymmetrical mode, and the highest amplitude can get up to 27. In general, the asymmetrical mode gets higher amplitude against symmetrical mode for substrate supported Ag film. And it also can be seen from Figs. 6(a) and 6(b) that the operating frequency of symmetrical mode can be tuned within the large frequency range from 500 THz to 750 THz by adjusting the beam energy.
The dependences of the amplitude of SPPs field on film thickness on the vacuum/Ag and substrate/Ag interface for fixed β = 0.6 are shown in Figs. 6(c) and 6(d), respectively. On the vacuum/Ag interface, with the increase of the film thickness, the amplitude gets higher values for asymmetrical mode but lower values for symmetrical mode. However, on the substrate/Ag interface, the thicker Ag film leads lower amplitude for both asymmetrical and symmetrical mode.
5. The comparison between the two excitations
To make detailed comparison of the e-beam excitations of SPPs is of great significance for further understanding and applications of SPPs. Based on the above theoretical and numerical investigation, the mechanisms of the two excitations are essentially different. For perpendicular excitation, is induced by the momentums of the electrons in the metal transferred from the perpendicularly moving e-beam , and the dispersion equation of SPPs only shows the dependence of propagation constant () on and , and there is nothing to do with the beam energy. However, for parallel excitation, is determined by the parallel component of the wave vector of the evanescent waves generated by the e-beam, and the operating frequencies of SPPs are determined by the working points. That means the operating frequency of SPPs depend not only on the dispersion curve but also on the beam energy. Therefore, SPPs excited by the two methods have different behavior and properties. In the case of perpendicular excitation, SPPs contain plenty of frequency components, propagate with attenuation and are always accompanied with TR. Whereas in the case of parallel excitation, SPPs waves are coherent, tunable, propagating without attenuation and TR does not occur.
The results also show that SPPs on metal films can be excited efficiently by parallel e-beam. In general, the symmetrical mode gets higher amplitude for free-standing metal film, whereas for substrate supported metal film, the amplitude of asymmetrical is higher. And the operating frequency of symmetrical mode can be tuned by adjusting the beam energy within a large frequency range from visible light to ultraviolet for Ag films. By the method of parallel excitation, coherent and tunable SPPs in the frequency range from infrared to visible light can be obtained for Au films, and deep ultraviolet for Al films .
Each of SPPs excited by the two methods has its own important applications. Perpendicular excitation has been well used to study the behavior and properties of SPPs [29, 30]. For parallel excitation, SPPs are of great significance for the SPPs enhanced tunable and coherent radiation within the frequency range from infrared to ultraviolet and new particle detections [7, 8].
The results of the theoretical analyses and numerical calculations on the perpendicular and parallel e-beam excitation indicate that the mechanisms of the two excitations are essentially different, the behavior and properties of SPPs strongly depend on the excitation methods. SPPs excited by perpendicular e-beam contain plenty of frequency components, propagate with attenuation, and are always accompanied with TR. However, for parallel excitation, SPPs waves are tunable, coherent, propagating without attenuation and TR does not occur. There are two modes for the parallel excited SPPs on the metal films, and they all can be excited efficiently by the parallel moving e-beam. The operating frequency of SPPs on the film can be tuned in a large frequency range by adjusting the beam energy. Each of the e-beam excited SPPs has its own important applications.
This work is supported by the National Basic Research Program under grants No.2014CB339801, the Natural Science Foundation of China under Grant No. 61231005, No. 11305030 and No. 612111076, and National High-tech Research and Development Project under contract No. 2011AA010204.
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