Magneto-opticall-controlled surface plasmon excitation Dmitr A. Bkov, 1,,a) and Leonid L. Doskolovich 1, 1 Image Processing Sstems Institute of the Russian Academ of Sciences, Molodogvardeskaa 151, 443001 Samara, Russia Samara State Aerospace Universit, Molodogvardeskaa 151, 443001 Samara, Russia The diffraction of a plane wave b a magneto-optical cavit located on a metal interface is investigated. We show that the excitation of localized eigenmodes of the cavit allows one to efficientl excite the surface plasmon polariton (SPP). We exploit the smmetr properties of the cavit modes to propose a new method for controlling the SPP intensit through an external magnetic field. Our theoretical predictions are in good agreement with the rigorous computations based on the generalized Lorentz reciprocit theorem and aperiodic Fourier modal method. a) Author to whom correspondence should be addressed. Electronic mail: bkovd@gmail.com. 1
In recent ears, considerable attention has been given to the investigation of nanoscale structures for the excitation and manipulation of surface plasmon polaritons (SPP). In a number of papers, a variet of geometries for the highl efficient excitation of the SPP 1 and SPP steering,3 were proposed and analzed. Simple theoretical models 4 and efficient numerical methods 1 to design the corresponding structures have been proposed. The hbrid plasmonic structures containing magneto-optical materials open new possibilities for controlling the SPP characteristics on sub-nanosecond time scales. In particular, the multilaer magneto-plasmonic structures to control the SPP dispersion through an external magnetic field were proposed and investigated 5,6. In this work we, for the first time, look into the possibilit of controlling the SPP s intensit b means of optical cavities made of magnetooptical materials (Fig. 1). In our approach the same optical cavit is used both to excite the SPP and to control the excitation efficienc. FIG. 1. (Color online) Magneto-optical cavit located on metal interface: the excitation of two SPPs b normall incident plane wave (cavit height is h 1040 nm, cavit width is w 1900nm ). Consider the problem of the excitation of an SPP using a normall incident plane wave (Fig. 1). Note that the problem in question is smmetrical, with both the incident wave and the cavit showing smmetr. Because of this, as a starting point of our analsis, we need to stud the smmetr properties of the cavit modes. It can be easil shown that non-magnetized smmetric structure modes are either even or odd 7,8. Hereafter, we consider the smmetr of the -
component of the field, i.e. we sa that TE (TM) mode is even (or smmetric) if E x E x ( H x H x ). Correspondingl, TE (TM) mode is odd (or asmmetric) if E x E x ( H x H x ). Note that the odd modes are unable to be b a (smmetric) normall incident plane wave. In this work, we stud magnetized structures, with the magnetization vector being perpendicular to the structure s smmetr plane (see Fig. 1). Rather than violating the smmetr properties, the said magnetization direction just modifies them as follows 7,8. Following the magnetization, the odd TE-modes (TM-modes) of the non-magnetized structure retain odd TEcomponents (TM-components), while respectivel acquiring even TM-components (TEcomponents) (Fig. ). This statement remains to be valid if the mode smmetr is reversed (if we interchange the words odd and even ) 7. To excite the mode its smmetr must coincide with smmetr of the incident wave. Thus, onl modes with even TE-components can be b normall incident plane TE-wave and onl modes with even TM-components can be b normall incident plane TM-wave. FIG.. (Color online) Mode smmetr: field distribution of (a) odd TE-mode of a nonmagnetized structure, (b) the same mode in the magnetized structure: odd TE-component ( E ) and even TM-component ( H ). Let us now analze the SPP excitation in the non-magnetized structure b a normall incident plane TM-wave. If there are no eigenmodes supported b the structure, there occurs the non-resonant scattering of light b the structure and the excitation of low-amplitude SPPs. If, 3
however, the structure supports even TM-mode, a resonant scattering of the plane wave from the structure will take place, which ma result, as we show below, in the resonant growth of the SPP intensit. The magnetization of a structure leads to a change in the mode parameters (smmetr, frequenc, qualit factor, coupling coefficient), making it possible to control the SPP intensit. Consider a cavit whose eigenmode cannot be in the absence of magnetization due to violation of smmetr and/or polarization conditions. For instance, assume that a plane TM-wave strikes a cavit that supports an odd TE-mode. In the absence of magnetization, the said mode is unable to be due to the polarization mismatch. However, in that case, there occurs a non-resonant excitation of a low-amplitude SPP (see Fig. 3(a)). If the cavit gets magnetized, the eigenmode with an even TM-component will be able to be, which will cause the resonant scattering of light and, consequentl, the excitation of a resonance-enhanced largeamplitude SPP should be expected (Fig. 3(b)). FIG. 3. (Color online) Excitation of an SPP b a normall incident plane wave with (a) TM polarization in a non-magnetized structure, (b) TM polarization in a magnetized structure; (c) TE polarization in a non-magnetized structure; (d) TE polarization in a magnetized structure. Of greater interest is the situation when the cavit supporting odd TM-eigenmode is illuminated b a plane TE-wave. In this case, the non-magnetized structure will be unable to excite a 4
plasmon, i.e. the intensit of the plasmon will be strictl zero (Fig. 3(c)). If, however, the structure is magnetized, the mode will acquire even TE-components, and therefore can be b a TE-wave, thus providing a resonance-enhanced SPP excitation (Fig. 3(d)). In this case, the magnetization/demagnetization of the structure enables the SPP to be entirel switched on/off. Similar analsis can be carried if even TE-mode is b the TE-wave or even TMmode b the TM-wave. In total, there are four cases corresponding to different mode smmetr and polarization combinations (Table I). Note that, due to smmetr, the two SPPs (see Fig. 1) alwas have equal intensit. However, the SPPs b TM-wave (cases 1 and 3) are in phase, whereas the SPPs b TE-wave (cases and 4) are out of phase. TABLE I. SPP excitation using eigenmodes of different polarization and smmetr. Incident Non-magnetized structure Magnetized structure Case wave Eigenmode SPP excitation Eigenmode SPP excitation odd-te, odd TE: 1 TM non-resonant even-tm: resonant not TE 3 TM 4 TE odd TM: not even TM: even TE: no excitation resonant no excitation odd-tm, even-te: even-tm, odd-te: even-te, odd-tm: resonant resonant resonant For a proof-of-principle of this approach we considered the simplest cavit geometr: a rectangular block located on metal interface (geometrical parameters are presented in the caption to Fig. 1). In the course of calculations, we considered silver substrate with its permittivit described b a Drude Lorentz model 9. The permittivit of the magnetized material was described b a tensor 5
where 4 5.06 4.3 10 i, g 0 0 ε 0 i g, (1) 0 ig 5 0.015 3 10 i at wavelength 100nm. The said parameters are characteristic to the material Bi.D 0.8Fe5O 10 1. The SPP s excitation efficienc was estimated using an approach proposed in Ref. 1. Instead of solving the direct problem of diffraction of a plane wave and calculating the SPP s intensit, the inverse problem was solved, with the surface plasmon-polariton considered to be an incident wave and the plane wave interpreted as a scattered wave. For reciprocal (nonmagnetic) materials, the scattering coefficient from the SPP into a plane wave (inverse problem) is proportional to the excitation coefficient of the SPP b the plane wave (direct problem). Strictl speaking, this fact is described b the Lorentz reciprocit theorem, which allows one to derive the following relationship for the complex amplitude of a SPP b a normall incident TM-polarized plane wave 1 : where H, TM A K H x, z dx, () 0 x z is the magnetic field distribution derived from the solution of the inverse problem of diffraction of the SPP b a cavit, K is the normalization coefficient defined b the SPP field distribution at a given wavelength 1. The integral in Eq. () is taken along a straight line z z0 marked b a dashed line in Fig. 1, with z 0 assumed to be sufficientl large 1. Note that the integral in Eq. () defines the transformation coefficient from the SPP to a plane wave propagating along the Oz-axis. The inverse problem was solved using Aperiodic Fourier Modal Method 11. 6
Because the structure under analsis contains nonreciprocal (magneto-optical) materials, a generalized Lorentz reciprocit theorem needs to be used. This means that the inverse problem should be solved for a modified structure described b a transposed permittivit tensor. For the magneto-optical materials, the transposition of the tensor (1) means that the structure is reversel magnetized. In this case, Eq. () remains valid, describing the SPP s excitation efficienc for a normall incident plane TM-wave. As we indicated above, in a magnetized structure, the SPP can be using not onl a TM-wave but also a TE-wave. In the latter case, the complex amplitude of the SPP b the plane TE-wave is given b where E, TE A K E x, z d x, (3) x z is the electric field distribution derived b solving the inverse problem of diffraction of the SPP b a cavit (rectangular block) magnetized in the opposite direction to the Ox-axis. Shown in Fig. 4(a) are the intensities of the SPPs b normall incident TM- and TE-waves in non-magnetized structure ( g 0 ). The intensities were calculated numericall using Eqs. () and (3). Fig. 4(b) presents the magnetization-induced SPP intensit variation for different polarizations of the incident plane wave. The spectra are seen to contain pronounced resonance peaks denoted as A, B, C, D. We calculated the complex frequencies (wavelengths) along with the field patterns of the localized modes of the structure through calculating poles of the scattering matrix analtical continuation 1. The spectral positions and the FWHM of the intensit peaks are in good agreement with the complex frequencies (wavelengths) of the structure eigenmodes. 0 7
FIG. 4. (Color online) (a) SPP excitation efficienc with a normall incident plane TE-wave ( TE A, blue dashed curve) and TM-wave ( TM A, green solid curve) in a non-magnetized structure. (b) Efficienc variation in response to the structure magnetization. (c) Field distribution of eigenmode with 1198.4 3.39inm (resonance C). (d) Field distribution of eigenmode with 1090. 7.79i nm (resonance D). The resonance С at a wavelength of 1197nm corresponds to the excitation of a mode with the following complex wavelength 1198.4 3.39i nm. An analsis of the mode s field distribution (Fig. 4(c)) suggests that this mode is odd TM-mode (no antinodes of H on the smmetr line), whereas mode's TE components are even (two antinodes of E on the smmetr line). In the non-magnetized structure this mode is an odd TM-mode. In full compliance with the above-specified smmetr conditions, the said mode is b a TE-wave, leading to a resonance of the TE A g magnitude. The resonance A at a 135-nm wavelength also corresponds to the excitation of an even- TE odd-tm mode ( 137.3 1.56 i nm ) of the magnetized structure, but as distinct from the previous mode, the mode in question in the non-magnetized structure represents an even TEmode. 8
The resonances B and D of magnitude TM A ( 1113 нм, 1090 нм in Fig. 4(a),(b)) are associated with the excitation of eigenmodes with the complex wavelengths 1116.1 8.15i nm and 1090. 7.79i nm. An analsis of the modes field distribution (see Fig. 4(d)) suggests that while being an even TM-modes in a non-magnetized structure the are converted into an even-tm odd-te modes following the magnetization. According to Table I, resonances A, B, C and D correspond to the cases 4, 3, and 3, respectivel. It is seen from Figs. 4(a), (b) that a larger-intensit SPP is b the TM-wave. Moreover, with the incident TM-wave, the magnetization of the structure makes it possible to achieve a larger value of the SPP intensit modulation. However, if we consider relative intensit modulation A 0 we get the following maximal value for TM polarization A g 100%, (4) Ag TM %. At the same time for TE polarization A TE 0 0 and we obtain TE 100%. Thus, in the case of TE-polarized incident wave the SPP can be totall switched off b the cavit demagnetization and switched on b magnetization. At the same time the intensit of the SPP is also quite high because of localized mode excitation. In conclusion, based on the analsis of smmetr properties of the MO-cavities, we have proposed a new method for magneto-opticall-controlled SPP excitation. The excitation of magneto-optical cavit localized modes using a TE-polarized plane wave has been shown to enable attaining a relative magnitude of SPP intensit modulation equal to 100 per cent. 9
ACKNOWLEDGMENTS The work was financiall supported b Russian Foundation for Basic Research (RFBR) grants 1-07-31116, 1-07-00495, 13-07-00464, 11-07-00153, 14-07-97005, b the ministr of education and science of the Russian Federation (RF), and b RF Presidential grant NSh- 3970.014.9 and scholarship SP-1665.01.5. REFERENCES 1 H. Liu, P. Lalanne, X. Yang, and J.-P. Hugonin, J. Sel. Top. Quant. Electr. 14, 15 (008). B. Bai, X. Meng, J. Laukkanen, T. Sfez, L. Yu, W. Nakagawa, H. P. Herzig, L. Li, and J. Turunen, Phs. Rev. B 80, 035407 (009). 3 X. Li, Q. Tan, B. Bai, and G. Jin, Appl. Phs. Lett. 98, 51109 (011). 4 P. Lalanne, J.-P. Hugonin, and J. C. Rodier, J. Opt. Soc. Am. A 3, 1608, (006). 5 V. V. Temnov, G. Armelles, U. Woggon, D. Guzatov, A. Cebollada, A. Garcia-Martin, J.-M. Garcia-Martin, T. Thoma, A. Leitenstorfer, and R. Bratschitsch, Nat. Photon. 4, 107 (010). 6 J. F. Torrado, J. B. González-Díaz, A. García-Martín, and G. Armelles, New J. Phs. 15, 07505 (013). 7 D. A. Bkov, and L. L. Doskolovich, J. Mod. Opt. 57, 1611 (010). 8 V. I. Belotelov, L. E. Kreilkamp, I. A. Akimov, A. N. Kalish, D. A. Bkov, S. Kasture, V. J. Yallapragada, Achanta Venu Gopal, A. M. Grishin, S. I. Khartsev, M. Nur-E-Alam, M. Vasiliev, L. L. Doskolovich, D. R. Yakovlev, K. Alameh, A. K. Zvezdin, and M. Baer, Nat. Commun. 4, 18 (013). 9 A. D. Rakic, A. B. Djurišic, J. M. Elazar, and M. L. Majewski, Appl. Opt. 37, 571 (1998). 10
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