Voigt Airy surface magneto plasmons

We present a basic theory on Airy surface magneto plasmons (SMPs) at the interface between a dielectric layer and a metal layer (or a doped semiconductor layer) under an external static magnetic field in the Voigt configuration. It is shown that, in the paraxial approximation, the Airy SMPs can propagate along the surface without violating the nondiffracting characteristics, while the ballistic trajectory of the Airy SMPs can be tuned by the applied magnetic field. In addition, the selfdeflection-tuning property of the Airy SMPs depends on the direction of the external magnetic field applied, owing to the nonreciprocal effect. ©2012 Optical Society of America OCIS codes: (240.6680) Surface plasmons; (230.3810) Magneto-optic systems. References and links 1. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979). 2. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007). 3. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007). 4. T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009). 5. A. Rudnick and D. M. Marom, “Airy-soliton interactions in Kerr media,” Opt. Express 19(25), 25570–25582 (2011). 6. G. Zhou, R. Chen, and X. Chu, “Propagation of Airy beams in uniaxial crystals orthogonal to the optical axis,” Opt. Express 20(3), 2196–2205 (2012). 7. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008). 8. D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105(25), 253901 (2010). 9. C. J. Zapata-Rodríguez, S. Vuković, M. R. Belić, D. Pastor, and J. J. Miret, “Nondiffracting Bessel plasmons,” Opt. Express 19(20), 19572–19581 (2011). 10. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25(20), 1493–1495 (2000). 11. A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett. 35(12), 2082–2084 (2010). 12. W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar, “Plasmonic Airy beam manipulation in linear optical potentials,” Opt. Lett. 36(7), 1164–1166 (2011). 13. A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, “Generation and near-field imaging of Airy surface plasmons,” Phys. Rev. Lett. 107(11), 116802 (2011). 14. L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic Airy beam generated by in-plane diffraction,” Phys. Rev. Lett. 107(12), 126804 (2011). 15. J. J. Brion, R. F. Wallis, A. Hartstein, and E. Burstein, “Theory of surface magnetoplasmons in semiconductors,” Phys. Rev. Lett. 28(22), 1455–1458 (1972). 16. M. S. Kushwaha, “Plasmons and magnetoplasmons in semiconductor heterostructures,” Surf. Sci. Rep. 41(1-8), 1–416 (2001). 17. Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100(2), 023902 (2008). 18. B. Hu, Q. J. Wang, and Y. Zhang, “Broadly tunable one-way terahertz plasmonic waveguide based on nonreciprocal surface magneto plasmons,” Opt. Lett. 37(11), 1895–1897 (2012). 19. E. D. Palik and J. K. Furdyna, “Infrared and microwave magnetoplasma effects in semiconductors,” Rep. Prog. Phys. 33(3), 1193–1322 (1970). #172155 $15.00 USD Received 9 Jul 2012; revised 19 Aug 2012; accepted 19 Aug 2012; published 31 Aug 2012 (C) 2012 OSA 10 September 2012 / Vol. 20, No. 19/ OPTICS EXPRESS 21187 20. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33(3), 207–209 (2008). 21. I. L. Tyler, B. Fischer, and R. J. Bell, “On the observation of surface magnetoplasmons,” Opt. Commun. 8(2), 145–146 (1973). 22. L. Remer, E. Mohler, W. Grill, and B. Lüthi, “Nonreciprocity in the optical reflection of magnetoplasmas,” Phys. Rev. B 30(6), 3277–3282 (1984). 23. J. Gómez Rivas, C. Janke, P. H. Bolivar, and H. Kurz, “Transmission of THz radiation through InSb gratings of subwavelength apertures,” Opt. Express 13(3), 847–859 (2005). 24. M. S. Kushwaha and P. Halevi, “Magnetoplasmons in thin films in the Voigt configuration,” Phys. Rev. B Condens. Matter 36(11), 5960–5967 (1987).


Introduction
Airy wave packets, which were first proposed as nonspreading beams by Berry and Balaz [1], have attracted a surge of interest since their experimental observation in free space [2,3]. Airy beams are well known by the intriguing properties of nondiffracting, asymmetric field profile, self-bending, and self-healing [3]. This kind of novel light beam has been widely studied in various materials, such as nonlinear mediums [4,5] and uniaxial crystals [6], and broadly applied in particle cleaning [7] and light bullet generation [8] applications.
Compared with other diffraction-free wave packets, e.g. Bessel [9] and Mathieu [10] beams, Airy beams have a unique property such that they are the only nonspreading solution to the one-dimensional potential-free Schrödinger equation. This suggests that only Airy surface plasmon (SP) beams can propagate on a metal surface without diffraction. It is found in both theoretical studies [11,12] and experimental observations [13,14] that Airy SPs remain the properties of both SPs and free space-propagating Airy beams, including the energy confinement at a subwavelength scale and self-bending property of the Airy beams.
On the other hand, it is known that when an external static magnetic field is applied on a metal or a semiconductor, propagation of the SP wave (also called surface magneto plasmons (SMPs) [15]) can be changed, due to the existence of the Lorentz force which alters the response of free carriers. In this situation, the resonant oscillation of free carriers (which causes SP waves) is not only characterized by the plasma frequency ω p and the incident frequency ω, but also by the cyclotron frequency ω c , which is a function of the external magnetic field. In consequence, the medium becomes highly anisotropic (the permittivity of the conductor becomes a tensor) under an external magnetic field -even though the medium is isotropic. Therefore, SMPs have some unique and intriguing features, compared with general SP waves [15,16]. For example, in the Voigt configuration (the applied magnetic field is parallel to the surface and perpendicular to the propagating direction of SMPs), the nonreciprocal effect can be observed, i.e. SMP waves propagating in two opposite directions have different propagating constants and cutoff frequencies [15,17,18].
In this paper, we analytically investigate TM-polarized paraxial Airy SMPs in the Voigt configuration. It is found that unlike the Airy SPs on a metal surface, the electromagnetic field components of the Airy SMPs are coupled in the wave equation due to the anisotropy of the metal or semiconductor when a magnetic field is applied. Thus, it is difficult to obtain analytical derivations. While, in the paraxial approximation, a relatively simple expression on the electromagnetic field components of Airy SMPs can be derived. The analytical and simulation results show that the external magnetic field can manipulate the self-deflection property of the Airy SMPs by tuning the wave vector of SMPs. Furthermore, due to the nonreciprocal effect, the magnetic field applied in one direction can significantly change the tilting angle of the Airy SMPs, while has little effect when applied in the opposite direction.

Theory of Airy surface magneto plasmons
The schematic structure of Airy SMPs propagating at the interface of a metal (or semiconductor) (region I) and a dielectric (region II) is depicted in Fig. 1. The Airy SMP wave is excited at the plane z = 0, and propagates along the z-axis. An external static magnetic field B is applied uniformly on the whole structure along the y-axis, forming the so called Voigt configuration.  We first solve the wave equation of the electric field in the paraxial approximation in the region (I) (x<0). It can be written, according to the Maxwell equations, as where k 0 is the wave vector in free space. When B is applied, the permittivity of the metal/semiconductor m ε becomes a tensor caused by the Lorentz force on the free electrons, which is expressed by [17,19]: where, in the lossless case, , and ε yy = ε ∞ (1 -ω p 2 /ω 2 ), in which ω is the angular frequency of the incident wave, ω p is the plasma frequency of the metal/semiconductor, ε ∞ is the high-frequency permittivity, and ω c = eB/m* is the cyclotron frequency. e and m* are the charge and the effective mass of electrons, respectively. B is the applied external magnetic field. Here, it is noted that we use the Drude model to calculate the elements in Eq. (2) [19]. Considering the exponential decay of the SP waves in the metal/semiconductor material, we can express the electric field components as , where α 1 is the decay factor in x-direction. For a paraxial Airy SMP, α 1 does not change much with that of a plane SMP wave [11], and consequently, it can be calculated by 2 . k smp is the propagation constant of the SMPs, calculated by a transcendental equation: in which ε d is the permittivity of the dielectric. Substitute Eqs. (2) and (3) into Eq. (1), and conduct the Fourier transform on the equations with respect to y, we obtain where  , , ( , ) x y z y A k z are the Fourier transforms of , , ( , ) x y z A y z , respectively. From Eq. (5) it is observed that the electric field components are coupled due to the permittivity tensor, and cannot be solved by separated scalar wave equations like Airy SPs [11]. To solve Eq. (5) . D is the partial differential with respect to z. The eigen values can be calculated by nontrivial solutions of Eq. (6) as in which It can be found that Eqs. (8a) and (8b) correspond to the TE (E x = E z = 0) and TM modes (E y = 0) of a plane wave propagating in Voigt-magnetized plasma [18], respectively, except that in Ref [18], Λ 2 = ik smp . Therefore, we choose Eq. (8b) as the eigen value in our calculations to ensure E x is predominant in the electric field to excite plasmons on the surface. Consequently, the solution of Eq. (6) is where C 1 is a function of k y needing to be determined. Like other works of Airy beams [1][2][3][4][5][6], we assume that the E x component of the Airy SMPs in the metal/semiconductor layer at the input plane z = 0 takes a form of [11]  where w 0 is the characteristic width of the first Airy beam lobe, and a is the decay factor [3]. It can be inferred from the Fourier transform of Eq. (10) that Substitute Eq. (11) into Eq. (9), and conduct the inverse Fourier transform on the equation, we can obtain the electric field E x of the Airy SMPs in the metal/semiconductor as is the amplitude. because the electric field components are not invariant in the y-direction, the definition of TE and TM modes is not the same as that of an Airy beam in free space (for TE mode: H y = 0, TM mode: E y = 0). However, in the paraxial approximation, they can be divided by E z = 0 for TE mode and H z = 0 for TM mode [11]. For simplicity, we only consider TM mode. Therefore, according to the Maxwell equations, other electric field components in regions (I) are derived as varying with respect to the magnetic field is plotted for an electromagnetic wave at ω = 0.85ω p . It can be seen that with the increase of the magnetic field, the tilting angle of the Airy SMPs decreases. However, this phenomenon cannot be observed for B>0, which is clearly shown in the inset of Fig. 4. In this situation, the "gravity" in the Newtonian equation of Eq.
(21) changes very little (red dashed line), compared with that in the situation B<0 (blue line). In Fig. 5, the electric field distributions of Airy SMPs are plotted without any magnetic field, and with magnetic fields such that ω c = 0.25ω p along the + y-axis and -y-axis, respectively. It can be clearly seen that, when B < 0, the Airy SMPs can be tuned by the magnetic field. It should also be noted that the nonreciprocal effect can be observed by changing not only the direction of the external magnetic field, but also the propagation direction of the Airy SMPs. In order to verify our theoretical derivations, 3D finite-difference time-domain (FDTD) simulations are carried out to simulate Airy SMP waves propagating on a semiconductor surface. The parameters used in the simulation are the same as those in Fig. 5. The field distribution of the Airy SMP source in the semiconductor is calculated by Eq. (10), and that in the dielectric is calculated by Eq. (7) of Ref [11]. The source is launched along the + z direction at z = 0 plane. Due to the limitation on our PC memories, the FDTD simulation is conducted in the region (−2λ<x<5λ, −30λ<y<30λ, 0<z<50λ). The analytical and FDTDsimulation results are compared in Fig. 6. It shows that the theoretical model gives good predictions of the main lobe and the side lobes of the Airy SMPs except for a slight shift. We believe this shift is caused by the insufficient grid density. When a magnetic field is applied (ω c = 0.25ω p ) along the -y-axis, the main lobe moves about 1.5λ toward the -y direction, calculated by the theoretical model, while the FDTD simulation gives a shift of about 1.55λ.

Conclusions
In this paper, we study the Airy surface plasmon under an external magnetic field. When a magnetic field is applied perpendicular to the propagating direction of the Airy SMPs and perpendicular to the surface, the ballistic trajectory of the Airy SMPs can be tuned. When the applied magnetic field increases, the tilting angle of the Airy ballistic waves decreases. The FDTD full vectorial method demonstrates the accuracy of our analytical model. The proposed tuning mechanism, we believe, can be applied to design different types of Airy plasmonic devices.