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 self-deflection-tuning property of the Airy SMPs depends on the direction of the external magnetic field applied, owing to the nonreciprocal effect.
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Airy wave packets, which were first proposed as nonspreading beams by Berry and Balaz , 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 . This kind of novel light beam has been widely studied in various materials, such as nonlinear mediums [4, 5] and uniaxial crystals , and broadly applied in particle cleaning  and light bullet generation  applications.
Compared with other diffraction-free wave packets, e.g. Bessel  and Mathieu  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) ) 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 ωp and the incident frequency ω, but also by the cyclotron frequency ωc, 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.
2. 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, as17, 19]:Eq. (2) . Considering the exponential decay of the SP waves in the metal/semiconductor material, we can express the electric field components as11], and consequently, it can be calculated by, where εV is the Voigt dielectric constant, defined by . ksmp is the propagation constant of the SMPs, calculated by a transcendental equation:Eqs. (2) and (3) into Eq. (1), and conduct the Fourier transform on the equations with respect to y, we obtainEq. (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 . To solve Eq. (5), we use and to express by Eq. (5c) and substitute it into Eqs. (5a) and (5b). The differential equations with respect to and are obtained asEq. (6) asEq. (7) up to the second and the zeroth order of ky /k0 and ky /ksmp, respectively. The simplified eigen values are calculated byEqs. (8a) and (8b) correspond to the TE (Ex = Ez = 0) and TM modes (Ey = 0) of a plane wave propagating in Voigt-magnetized plasma , respectively, except that in Ref , Λ2 = iksmp. Therefore, we choose Eq. (8b) as the eigen value in our calculations to ensure Ex is predominant in the electric field to excite plasmons on the surface. Consequently, the solution of Eq. (6) is1–6], we assume that the Ex component of the Airy SMPs in the metal/semiconductor layer at the input plane z = 0 takes a form of 3]. It can be inferred from the Fourier transform of Eq. (10) thatEq. (11) into Eq. (9), and conduct the inverse Fourier transform on the equation, we can obtain the electric field Ex of the Airy SMPs in the metal/semiconductor as11]. For simplicity, we only consider TM mode. Therefore, according to the Maxwell equations, other electric field components in regions (I) are derived as
In the dielectric, the electric field components can be calculated by the same procedure, by replacing εxx with εd, α1 with -α2, and εxx = 0, where . Together with the consideration of the boundary conditions, we obtain:
In Fig. 2 , the electric field intensity |E|2 distributions of Airy SMPs on both the x-y plane and y-z plane without the magnetic field are plotted. In order to ensure εV <0 (under this condition, SMPs can exists), the incident frequency and the magnetic field are set as ω = 0.8ωp and ωc = 0.1ωp, respectively. Without loss of generality, the characteristic width and the decay factor are chosen as w0 = 2λ and a = 0.1, respectively, where λ is the incident wavelength. We choose InSb as the semiconductor material, which is often used in the SMPs experiments [21–23]. The corresponding parameters of InSb are m* = 0.014m0, (m0 is the free electron mass in vacuum), ωp = 12.6THz, and ε∞ = 15.68 . It can be seen from Figs. 2(a) to 2(c) that the Airy plasmons are confined on the surface of the InSb material, keeping the diffraction-free characteristic even after propagation distance of 80λ. With the increase of the propagating length, the diffraction effect becomes obvious due to the decay factor a in the y-direction, which is the same as that of a free space Airy beam [2, 3]. When they propagate along the z-direction, the main lobe moves toward the + y-axis. Figure 2(d) shows the self-bending property of Airy SMPs.
3. Ballistic trajectory tuning by the external magnetic field
In this section, we will study the influence of the external magnetic field on the Airy SMPs. It is well known that the Airy beams and Airy plasmons perform ballistic dynamics similar to those of projectiles moving under gravity [11, 20]. From Eqs. (12) and (16), it is found that like Airy surface plasmons, the Airy SMPs also follow a ballistic trajectory in the y-z plane, which is described by
It can be clearly seen that if the transverse scale w0 is set, the ballistic dynamics of the Airy SMPs is mainly determined by ksmp. In Fig. 3 , we plot the dispersion curve of SMPs under different external magnetic field intensities with ω<ωp. It is found that when the magnetic field is applied along the + y-axis (denoted by B > 0 in the inset), the ksmp~ω curve nearly keeps unchanged. While, when the magnetic field is applied along the –y-axis (denoted by B<0), the dispersion curve moves toward lower frequencies side. This intriguing nonreciprocal effect can be understood by Eq. (4). When the magnetic field is applied in the opposite direction, εxz in the last term changes its sign. Thus the dispersion equations become different, having different cutoff frequencies .
From Fig. 3, it can be inferred that if the magnetic field increases in the region B<0, ksmp will increase for a monochromatic wave. Consequently, according to Eq. (20), the self-bending effect of the Airy SMPs will be alleviated. In Fig. 4 , the ballistic curve of Airy SMPs varying with respect to the magnetic field is plotted for an electromagnetic wave at ω = 0.85ωp. 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 ωc = 0.25ωp 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 . 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 FDTD-simulation 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 (ωc = 0.25ωp) 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.
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.
We would like to acknowledge financial support from Nanyang Technological University (NTU) (M58040017) and Ministry of Education, Singapore (MOE2011-T2-2-147). Support from the CNRS International-NTU-Thales Research Alliance (CINTRA) Laboratory, UMI 3288, Singapore 637553, is also acknowledged.
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