Dispersion relation of surface plasmon wave propagating along a curved metal-dielectric interface

Transcription

1 Disersion relation of surface lasmon wave roagating along a curved metal-dielectric interface Jiunn-Woei Liaw * and Po-Tsang Wu Deartment of Mechanical Engineering, Chang Gung University 59 Wen-Hwa 1 st Rd., Kwei-Shan, Tao-Yuan, 333, Taiwan * Corresonding author: markliaw@mail.cgu.edu.tw Abstract: The disersion relations of surface lasmon wave (SPW) roagating along a convex or concave metal-dielectric interface with a radius of curvature are studied by solving the root of a characteristic equation in terms of Bessel and Hankel functions of comlex order numerically. For the convex geometry, a metallic circular cylinder embedded in a dielectric host is modeled, whereas for the concave one, a dielectric cylinder in a metallic host is modeled. We found that the hase velocity of SPW along a convex interface is always less than that of SPW along a lanar one. On the contrary, the hase velocity of a concave case is faster than that of a lanar one. For both cases, the attenuation constants are larger than a lanar one, due to the radial radiation of the energy into the surrounding medium, excet the dissiation in the metal. 008 Otical Society of America OCIS codes: ( ) Surface lasmons; ( ) Metal otics; ( ) Surface waves; ( ) Waveguides, lanar. References and links 1. B. E. Sernelius, Surface Modes in Physics (Wiley-Vch, 001).. R. H. Ritchie, Plasma losses by fast electrons in thin films, Phys. Rev. 106, (1957). 3. K. Hasegawa, J. U. Nockel, and M. Deutsch, Surface lasmon olariton roagation around bends at a metal-dielectric interface, Al. Phys. Lett. 84, (004). 4. K. Hasegawa, J. U. Nockel, and M. Deutsch, Curvature-induced radiation of surface lasmon olaritons roagating around bends, Phys. Rev. A 75, (007). 5. Z. Sun, Vertical dielectric-sandwiched thin metal layer for comact, low-loss long range surface lasmon waveguiding, Al. Phys. Lett. 91, (007). 6. P. Berini and J. Lu, Curved long-range surface lasmon-olariton waveguides, Ot. Exress 14, (006). 7. W.-K. Kim, W.-S. Yang, H.-M. Lee, H.-Y. Lee, M.-H. Lee, and Woo-Jin-Jung, Leaky modes of curved long-range surface lasmon-olariton waveguide, Ot. exress 14, (006). 8. J.-W. Liaw, Simulation of surface lasmon resonance of metallic nanoarticles by boundary-element method, J. Ot. Soc. Am. A 3, (006). 9. A. Viktorov, Rayleigh and Lamb Waves (Plenum, New York, 1967). 10. B. Johnson and R. W. Christy, Otical constants of the noble metals, Phys. Rev. B 6, 4370 (197). 11. B. Wang, G. P. Wang, Plasmonic waveguide ring resonator at terahert frequencies, Al. Phys. Lett. 89, (006). 1. P. L. Rochon, and L. Levesque, Standing wave surface lasmon mediated forward and backward scattering, Ot. Exress 14, (006). 1. Introduction For some metals, esecially the gold and silver, the real art of the ermittivity is negative within a certain frequency range of UV to NIR. The behavior is due to the collective motion of the free electrons in the metal, oscillating with the incident electromagnetic field. Because of that, there exists a unique surface electromagnetic wave roagating along a metaldielectric interface, known as surface lasmon wave (SPW), or called surface lasmon (C) 008 OSA 31 March 008 / Vol. 16, No. 7 / OPTICS EXPRESS 4945

2 olariton (SPP) [1]. The existence of SPW was first redicted by Ref. in a transversemagnetic (TM) mode, where the magnetic field is along the -axis (out of the x-y lane) and the electric field is in the x-y lane. If the metal is a nonmagnetic medium, the disersion relation of a SPW roagating along a lanar metal-dielectric interface was derived as [1,] ε1ε ε1 ε k = ω s c + where k s is the wavenumber of the SPW in terms of the relative ermittivity of the medium-1 (ε 1 ) and medium- (ε ), ω is the angular frequency, and c is the light seed in vacuum. The medium-1 and medium- are the metal and dielectric, resectively. The characteristic of a SPW is that the amlitude of EM field decays exonentially as the distance from the interface. Recently, the characteristic equation of SPW roagating along a bend metal-dielectric interface was derived, and an asymtotic method was used to study the reflection and transmission coefficients of an incident SPW along a flat metal-dielectric interface encountering a bended corner [3, 4]. Moreover, SPW along a bended metal late was studied [5], and FDTD method was used to show the wave roagation along the late. The longrange SPP along a 3D finite-width stri and its bending loss were also investigated [6, 7]. In this aer, we will directly solve the characteristic equation numerically to identify the disersion relation of SPW roagating along a curved interface quantitatively. Two tyes of SPW along curved metal-dielectric interfaces are studied by comaring their hase velocity and attenuation constant with that of SPW along a lanar one; one is along the convex interface and the other along the concave. In order to identify the relation of the SPW s disersion relation with the radius of curvature of the metal-dielectric interface, we assume that the SPW roagates along the surface of a circular cylinder in the transverse-magnetic (TM) mode; i.e., the magnetic field is along the -axis (out of the lane) and the electric field is in the x-y lane. For a convex case, a metallic circular cylinder with radius a embedded in a dielectric host is considered, and vice versa, a dielectric cylinder in a metal is considered for a concave one, as shown in Fig. 1, where the circular cylinder (medium-) is embedded in the host (medium-1). Both media are assumed nonmagnetic materials. The relative ermittivity of each medium is denoted by ε j, and the relative ermeability by μ j, j=1,. y y a metal x a dielectric x 1 dielectric 1 metal 1(a) 1(b) Fig. 1. The configurations of SPW roagating along (a) a convex and (b) a concave metal-dielectric interfaces with radius of curvature a. On the other hand, two SPWs were redicted [8] to be generated simultaneously to roagate along the circumference of a circular metallic cylinder of submicron-radius embedded in a dielectric host clockwise and counterclockwise, when the cylinder is irradiated by an incident lane wave. Therefore this model will be used to generate two SPWs roagating along a convex metal-dielectric interface. Utiliing a series solution of D Mie theory, this roblem can be solved exactly, and calculated numerically to obtain the (C) 008 OSA 31 March 008 / Vol. 16, No. 7 / OPTICS EXPRESS 4946

3 electromagnetic field distribution for each frequency. Furthermore, since the resonse of this scattering roblem is time-harmonic steady-state, a dynamic animation of the roagation of SPW can be reconstructed. Using this technique, we can trace the migration of SPW, and its hase velocity will be measured to comare with the result of disersion relation.. Characteristic equation of disersion relation Consider the electromagnetic field of a harmonic resonse for a two-dimensional (D) TMmode roblem, where the time harmonic factor is ex( iωt). In this aer, the characteristic equation [3] of a SPW roagating along a curved (convex or concave) metal-dielectric interface of radius of curvature a is rearranged as [ ] 1 0 ( ) H ( )/ J '( ) H ( ) J ' = () where J is the Bessel function of first kind of order and H is the Hankel function of first kind of order in terms of = 1 k1a, k a = and the comlex wavenumber k j = ω ε μ / c, j = 1, j j. In Eq. (), the rimes of J and H denote the differentiation with resect to the argument. Here, the ermeability of the medium-1 is the same with that of medium-, μ =μ 1. The arguments 1 and are given values (if a is given), and the order is an unknown comlex number, which is the root of Eq. (). The magnetic fields of SPW in the iθ iθ two adjoining media are in the forms of H = AH k r) e and H = BJ k r) e, ( 1 ( iθ i( / a) aθ resectively, where A and B are the amlitudes. The term e can be regarded as e [9]. Physically, the meaning of /a is the wavenumber of SPW roagating along the circumference of a circular cylinder with radius a, and aθ is the roagating distance of SPW. If the SPW crees along the circumference of the circular cylinder counterclockwise, the real art of the comlex wavenumber (/a) should be ositive because the hase shift should increase with the roagating direction, and the imaginary art should be ositive, too, because of the attenuation of the amlitude; i.e., Re(/a) >0, and Im(/a) >0. Conversely, If the SPW crees along the circumference clockwise, then Re(/a) <0, and Im(/a) <0. Since Eq. () is a transcendental equation, there is no analytical solution for the root, which is the comlex order. Therefore the numerical method is used for the calculation. When the radius of curvature, a, aroaches infinite, the wavenumber (/a) of SPW of the convex and concave cases will aroach the value k s of a lanar interface, as shown in Eq.. In order to comare the hase velocity and the attenuation constant of a SPW along a curved interface with those of a lanar one, the relative wavenumber α and the relative attenuation constant β are defined as α = Re( / a) / Re( k ), s β = Im( / a) / Im( k ), where Re is the real art and Im is the imaginary art. The relative wavenumber α is the ratio of the wavenumber of SPW along a curved interface to that of a lanar one, and the relative attenuation constant β is the ratio of the attenuation constant of SPW along a curved interface to that of a lanar one at the same frequency. Here the relative wavenumber α is also regarded as the ratio of the hase velocity of SPW along a lanar interface to that of a curved one. s 3. Numerical Results and Discussion The terms in the left-hand side of Eq. () is the residue corresonding to any comlex order for the given 1 and. The roots of the comlex order, which enable the real and imaginary (C) 008 OSA 31 March 008 / Vol. 16, No. 7 / OPTICS EXPRESS 4947

4 arts of the residue of Eq. () to be ero, reresent the SPW modes. Using a mathematic ackage (Male) to calculate the Bessel and Hankel functions of comlex order with comlex argument, the root of Eq. (), the comlex order, can be solved numerically. For simlicity, we only consider the case of counterclockwise SPW, so that the root of Eq. () in the first quadrant of the comlex domain is searched. For examle, consider a SPW roagating along an Ag circular cylinder of radius 400nm in air at.88ev (λ= nm). The relative ermittivity of Ag is ( , ) [10] for this frequency. The distribution of the absolute value of the residue of Eq. () in the first quadrant of is lotted in Fig.. Figure indicates that there are several discrete ero-oints of Eq. (), where the first ero with the minimum imaginary art of reresents the fundamental mode (the first mode) of SPW, and the others are the higher-order modes of SPW [3]. For this tyical case, the comlex order of the fundamental mode is = (7.1936, ), where the real art of /a (the wavenumber of SPW along a convex interface) is (1/m); i.e., the wavelength of SPW is λ sw =349.4 nm. The relative wavenumber of the fundamental mode is α = , and the relative attenuation constant is β = These higher-order modes of SPW decay very fast due to their larger attenuation constants, so that the fundamental mode becomes relatively essential, because it can roagate a longer distance. Therefore, in the following calculations, only the fundamental mode of SPW is searched. 3rd mode nd mode 1st mode Fig.. The distribution of the absolute value of the residue in the first quadrant of the order. For a SPW along a convex or a concave Ag-air interface, the disersion curves of the relative wavenumber α and the relative attenuation constant β are lotted in Fig. 3 for different radii a (00, 400, and 1000 nm). The frequency-deendent ermittivity of silver is cited from Ref. 10. The curves of α and β versus radius of curvature a are also lotted in Fig. 4 for different frequencies (.13,.63, and 3 ev). These curves show that the relative wavenumber α of a convex case is always larger than one, and that of a concave case is always less than one. This is to say that the hase velocity of a convex case is always slower than that of a lanar case, and the hase velocity of a convex case is always faster than that of a lanar case. Generally, the larger the radius of curvature, the less the value of α for the convex case but the larger the relative wavenumber α for the concave case. In addition the higher the frequency is, the less the relative wavenumber α will be for the convex case but the larger the value of α for the concave case. On the other hand, the relative attenuation constant β for both cases (convex and concave) is always larger than one; i.e., the attenuation constant of SPW along a curved interface is always larger than that of a lanar one for all frequencies and radii of curvature. This is because that the radial radiation of the electromagnetic energy (C) 008 OSA 31 March 008 / Vol. 16, No. 7 / OPTICS EXPRESS 4948

5 into the host accomanies the roagation of SPW along a curved interface, excet the dissiation in the metal. In addition, the larger the radius of curvature is, the less the relative attenuation constant β will be for both cases; the attenuation constant of SPW along a curved interface aroaches the value of Eq., as the radius of curvature increases. Moreover, when the frequency increases, the value of β also becomes less for both cases. 3(a) 3(b) Fig. 3. The curves of (a) the relative wavenumber α vs. frequency, and (b) the relative attenuation constant β vs. frequency of SPW at Ag-air interface of different radii a (00, 400, 1000 nm), where the solid oints: convex, and the void oints: concave. 4(a) Fig. 4. The curves of (a) the relative wavenumber α vs. radius a, and (b) the relative attenuation constant β vs. radius a of SPW at Ag-air interface for different frequencies (.13,.63, 3 ev), where the solid oints: convex, and the void oints: concave. Furthermore, the curves of the relative wavenumber α and the relative attenuation constant β versus the relative ermittivity of the dielectric medium are lotted in Fig. 5 for different radii a (00, 400, and 1000 nm) at.63 ev, where the metal is silver. These curves show that the larger the ermittivity of the dielectric medium, the less the relative wavenumber α for the convex case and the larger the value of α for the concave case. However, these hase velocities of the convex and concave cases will aroach the value of a lanar case by increasing the ermittivity of the dielectric medium. Moreover, the larger the ermittivity of the dielectric medium is, the less the relative attenuation constant β will be for both the convex and concave cases, if the frequency and the radius of curvature are fixed. 4(b) (C) 008 OSA 31 March 008 / Vol. 16, No. 7 / OPTICS EXPRESS 4949

6 5(a) Fig. 5. The curves of (a) the relative wavenumber α, and (b) the relative attenuation constant β vs. the relative ermittivity of the dielectric medium for different radii a (00, 400, and 1000 nm) at.63 ev, where the solid oints: convex, and the void oints: concave. 5(b) 4. Numerical Exeriment Consider the revious case, SPW roagating along an Ag circular cylinder of radius 400nm in air at.88ev. From the characteristic equation, the value of is = (7.1936, ); i.e., the wavelength of SPW is λ sw =349.4 nm. From the other asect, when the Ag circular cylinder is irradiated by an incident lane wave of -olariation, two SPWs are redicted to be generated simultaneously; one roagates along the circumference clockwise and the other counterclockwise [8]. The henomenon is very similar to the SPP along a ring resonator [1]. Moreover, a standing wave will be formed on the backside of the cylinder due to the interference of the two oosite-direction SPWs [13]. According to this henomenon, we can estimate the wavelength of SPW along a convex metal-dielectric interface of a secific radius of curvature for a secific frequency by measuring the arc length of two adjacent nodal oints of the standing wave. Utiliing an analytic solution of D Mie theory in series form, this roblem can be solved exactly and calculated numerically to obtain the distributions of the electric and magnetic fields for each frequency. Since the field values we obtain are comlex values of the time-harmonic resonses of steady state in terms of the hase difference, they can be reconstructed in time domain by using the formulation, Lωt g( x, t) = Re( G( x) e ), 0 t < T (3) where the field function G(x) can be the magnetic field H and the electric fields E x, E y. We divide the motion of a eriod T into 64 frames. The time resonse of each frame at t= nt/64, n=0, 1, 63, can be calculated by using Eq. (3), and then an animation of the steady-state wave roagating can be obtained by laying these frames in sequence. In Figs. 6(a) and 6(b), only the results of the magnetic field of t=5t/64 and the electric field of t= 13T/64 at.88ev are shown resectively, in which the incident wave roagates from the left-hand side to the right-hand side. Obviously, there is a standing wave on the backside of the Ag-cylinder, no matter from the electric field or the magnetic field. Since the attenuation of SPW is associated with the roagation due to the existence of the imaginary art of the wavenumber, the amlitudes of the two SPW decay as they roagate along the circumference. The attenuation makes the attern of the standing wave blurred. However on the right backside, which is around the region of θ=0 o, the amlitudes of both SPWs are almost the same, because their roagating distance are almost identical. Therefore, we only measure the angle between the two marked nodal oints, as shown in Fig. 6(b). The angle is 50 o, which covers a wavelength, (C) 008 OSA 31 March 008 / Vol. 16, No. 7 / OPTICS EXPRESS 4950

7 so that the estimated wavelength is λ sw =349 nm, which is in agreement with the theoretical value of λ sw =349.4 nm obtained from the characteristic equation. Using this method, the wavelengths of the other frequencies are also checked, and they are all consistent with the theoretical ones. 6(a) 6(b) Fig. 6. (a). The distribution of the magnetic field at t=5t/64. (b) The distribution of the absolute of the electric field at t=13t/64 of Ag cylinder of a=400nm irradiated by a lane wave at.88ev. All the values are normalied with the amlitudes of the incident fields. 5. Conclusion The characteristic equation of a SPW creeing along a convex or concave Ag-dielectric interface was solved numerically to obtain its disersion relation. The results indicate that the hase velocity of SPW roagating along a convex interface is always less than that along a lanar one, and will aroach the value of the lanar one as the radius of curvature of the convex interface, the ermittivity of the dielectric host, or the frequency increases. In contrast, the hase velocity of SPW roagating along a concave interface is always faster than that along a lanar one, and will close to the lanar one as the radius of curvature of the concave interface, the ermittivity of the dielectric cylinder, or the frequency increases. In addition, the attenuation constants of SPW of the convex and concave cases are always larger than that of a lanar one, due to the radial radiation of the electromagnetic energy into the host, excet the dissiation in the metal. These disersion relations will be useful to interret the bend-induced loss. Using a model of an incident lane wave irradiating an Ag circular cylinder in air for generating two SPWs, the wavelength of SPW along the convex interface can be obtained from the nodal oints of the standing wave. The wavelength of this model is in agreement with that of the disersion relation. Acknowledgment This research was suorted by National Science Council, Taiwan, R.O.C. (Grant No. NSC 95-1-E ). (C) 008 OSA 31 March 008 / Vol. 16, No. 7 / OPTICS EXPRESS 4951