Transmission of surface plasmon polaritons through a nanowire array: mechano- optical modulation and motion sensing D Yu Fedyanin and A V Arsenin Laboratory of Nanooptics and Femtosecond Electronics, Department of General Physics, MIPT e-mail: feddu du@mail.r .ru
OUTLINE ● Motivation ● Surface Plasmon Polaritons (SPPs) ● Long Range SPPs (LRSPPs) ● Nanowires ● Principle of Operation ● Coupled-Mode Equations ● Transmission of SPPs through a Nanowire Array ● Modulation of the Intensity of the SPP ● Conclusion 2
Surface Plasmon Polaritons Drude model: ε 1 = Re ε 1 i Im ε 1 Re ε 1 0 k x = Re k x i Im k x SPP wavevector SPP dispersion κ 1 ε 2 = - κ 2 ε 1 - penetration constants 2 1 1 ρ i = L spp = spp = Re κ i 2Im k x Re k x propagation length SPP wavelength penetration depths 3
Surface Plasmon Polaritons Excitation p sp = ε r ε 2 SPP wavevector is greater than the light wavevector 4
Surface Plasmon Polaritons Disadvantages: low L spp , high ρ , the size of the guide cross-section is comparable with optical components – else, high losses Solution: plasmonic waveguides Groups: S.I. Bozhevolnyi, A.V. Zayats, P. Berini … H.A. Atwater, X. Zhang, A. Polman ... 5
Long Range SPPs SPP dispersion relation where a i s a half thickness of the film 6
Long Range SPPs For IMI waveguide structures with ε 2 = ε 3 , dispersion relation can be easily simplified and rearranged as two branches: anti-symmetric mode th κ 1 a =− κ 2 ε 1 κ 1 ε 2 symmetric mode th κ 1 a =− κ 1 ε 2 κ 2 ε 1 P. Berini, Bulk and surface sensitivities of surface Long Range SPPs plasmon waveguides, New Journal of Physics 10 (2008) 105010 (37pp) 7
Nanowires Kyungsuk Yum, et.al. J. Appl. Phys. 96, 3933 Z. L. Wang, “Mechanic Properties of Nanowires and WHY? Nanobelts,” in Dekker Encyclopedia of Nanoscience and Nanotechnology, (Taylor&Francis, 2004). Typical dimensions of nanowires are 5-100 nm in diameter (width) and 1-50 μm in length. Such a small size, combined with unique electrical, mechanical and optical properties, has attracted interest in the scientific community for their potential in different applications from microelectronic to nanooptics. We focus here on the mechanical properties of nanowires. There are two main reasons for this. Firstly, the resonance frequency of mechanical oscillations is usually in the kilohertz or megahertz range and the amplitude of oscillations may exceed ten micrometers. Secondly, nanowire cantilevers are very sensitive and Fig. 3 A selected carbon nanotube at (a)stationary, (b) the first can be used even for single-atom mass sensing and tiny harmonic resonance (n1 = 1.21 MHz), and (c) the second force measurements. harmonic resonance (n2 = 5.06 MHz ). (d) The traces of a uniform one-end fixed elastic beam at the first two resonance modes, as predicted by the continuous elasticity theory. 8
Nanowires Kyungsuk Yum, et.al. J. Appl. Phys. 96, 3933 (2004) Z. L. Wang, “Mechanic Properties of Nanowires and Nanobelts,” in Dekker Encyclopedia of Nanoscience and Nanotechnology, (Taylor&Francis, 2004). Fig. 3 A selected carbon nanotube at (a)stationary, (b) the first harmonic resonance (n1 = 1.21 MHz), and (c) the second harmonic resonance (n2 = 5.06 MHz ). (d) The traces of a uniform one-end fixed elastic beam at the first two resonance modes, as predicted by the continuous elasticity theory. 9
Nanowires Z. L. Wang, “Mechanic Properties of Nanowires and Nanobelts,” in Dekker Encyclopedia of Nanoscience and Nanotechnology, (Taylor&Francis, 2004). Fig. 7 A selected ZnO nanobelt at (a,b) stationary, (c) the first harmonic resonance in the x direction, v x1 = 622 kHz, and (d) the first harmonic resonance in the y direction, v y1 = 691 kHz. (e) An enlarged image of the nanobelt and its electron diffraction pattern (inset). The projected shape of the nanobelt is apparent. (f) The FWHM of the resonance peak measured from another ZnO nanobelt. The resonance occurs at 230.9 kHz. 10
Principle of Operation A number of methods to do this have been proposed, including all-optical modulation. All those techniques are pure optical, i.e. an applied voltage, an incident optical radiation, etc. change optical properties, e.g. the refractive index or polarization tensor, of the materials used. We propose a different technique. Schematic operation of the mechano-optic modulator, β is the SPP wavevector, h 1 > h 2 . (a). If the distance h between the nanowire array and the metal film is very large, there is no interaction between the SPP and the nanowires and we do not have any effect. (b). When the distance decreases, the effect of the nanowire array may be considered as a perturbation Δ ε of the dielectric constant of the waveguide and the coupling between guided and radiation modes occurs. Changing the distance h , one can control the intensity of the transmitted SPP. 11
Coupled-Mode Equations ● H. Kogelnik, “Theory of dielectric waveguides,” in Integrated Optics , T. Tamir, ed. (Springer, Berlin, 1979). ● D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974). ● D. Marcuse, Light Transmission Opti cs (Van Nostrand Reinhold, New York, 1982). ● Wei-Ping Huang, “Coupled-mode theory for optical waveguides: an overview”, J. Opt. Soc. Am. A, 11 , 963(1994). ALGORITHM 1. Find all modes 2. Normalize modes 3. Write coupled-mode equations 4. Calculate coupling coefficients 12
Coupled-Mode Equations (Modes) AS radiation mode AS guided mode S guided mode S radiation mode 13
Coupled-Mode Equations (Mode Normalization) Since the radiation losses interest us, the amplitude coefficients should be related to the power carried by the mode. For each pair ν and µ of guided modes, we require and for for each pair of radiation modes 14
Coupled-Mode Equations Assuming β to be positive, introduce now amplitude coefficients A (for the forward- running mode ( β >0)) and B (for the backward-running mode ( β <0)), so that In general case, the system of coupled mode equations is written as 15
Coupled-Mode Equations We have an infinite number of differential equations, since we have an infinite number of radiation modes. To solve the problem, we have to simplify the above system. Assume the relative change of amplitude A g AS of the transmitted SPP to be much less than unity, i.e. assume a weak coupling regime (|A g AS | A ν ,B ν ), and only ≫ one amplitude coefficient remains in the right-hand parts Integration of Eq. (13) gives the exact expressions for the amplitude coefficients A and B: 16
Coupled-Mode Equations 17
Transmission of SPPs through a Nanowire Array ZnO nanobelts R =6.8% 18
Transmission of SPPs through a Nanowire Array ZnO nanowires R =4.3% 19
Modulation of the Intensity of the SPP We have an exponentially decaying term exp(− κ 2 gAS h ), consequently, the radiation loss power decreases as h increases. However, there are oscillating terms in braces η (h, β ) as a function of h are expected. The form of the and therefore oscillations in K z dependence can be approximately expressed as 20
Modulation of the Intensity of the SPP Dependence of the normalized total radiation loss power on the gap between the nanobelt array and the metal film for different values of the nanobelt transverse size, l = 500 nm, N = 10, 2 π c/ ω = 800 nm. 21
Modulation of the Intensity of the SPP Dependence of the normalized total radiation loss power on the gap between the nanobelt array and the metal film for different values of the nanobelt transverse size, l = 500 nm, N = 10, 2 π c/ ω = 800 nm. Results of direct calculations show that it is possible to achieve a modulation depth of 5% with only 60 nm amplitude of mechanical oscillations for u = b =80 nm. Such a high sensitivity of the system can be used for the detection of the mechanical motion of nanowires. 22
Modulation of the Intensity of the SPP Dependence of the normalized total radiation loss power on the number of nanobelts, h =500 nm, l =500 nm, N =10, 2πc/ ω =800 nm. (b). Dependence of the normalized total radiation loss power on the permittivity of the nanobelts, h =500 nm, l =500 nm, N =5, 2πc/ ω =800 nm. 23
Conclusion We have brought together the mechanics of nanowires and the guiding properties of plasmonic waveguides, proposed a compact machano-optical modulator and characterized it analytically with the help of the coupled-mode theory. The modulator is based on a nanowire array placed above a thin metal film. The intensity of the SPP is modulated by changing the distance between the nanowire array and the film. The maximum possible modulation depth depends strongly on the number of nanowires and their parameters and thus it can be varied in a wide range. The longitudinal size of the device is of the order of a few micrometers and depends on the number of nanowires and the distance between them, while the transverse size is dictated by the length of nanowires. Moreover, the proposed technique can be used for the detection of the mechanical motion of nanowires and for the measurement of their oscillation amplitude. 24
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