Voigt Airy surface magneto plasmons

doi:10.1364/OE.20.021187

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Voigt Airy surface magneto plasmons

Bin Hu, Qi Jie Wang, and Ying Zhang »View Author Affiliations

Optics Express, Vol. 20, Issue 19, pp. 21187-21195 (2012)
http://dx.doi.org/10.1364/OE.20.021187

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▸ Article Outline
– Abstract
1. Introduction
2. Theory of Airy surface magneto plasmons
3. Ballistic trajectory tuning by the external magnetic field
4. Conclusions

– References and links

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Abstract

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.

1. Introduction

Airy wave packets, which were first proposed as nonspreading beams by Berry and Balaz [1

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979). [CrossRef]

], have attracted a surge of interest since their experimental observation in free space [2

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007). [CrossRef] [PubMed]

, 3

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007). [CrossRef] [PubMed]

]. Airy beams are well known by the intriguing properties of nondiffracting, asymmetric field profile, self-bending, and self-healing [3

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007). [CrossRef] [PubMed]

]. This kind of novel light beam has been widely studied in various materials, such as nonlinear mediums [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). [CrossRef]

, 5

A. Rudnick and D. M. Marom, “Airy-soliton interactions in Kerr media,” Opt. Express 19(25), 25570–25582 (2011). [CrossRef] [PubMed]

] and uniaxial crystals [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). [CrossRef] [PubMed]

], and broadly applied in particle cleaning [7

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008). [CrossRef]

] and light bullet generation [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). [CrossRef] [PubMed]

] applications.

Compared with other diffraction-free wave packets, e.g. Bessel [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). [CrossRef] [PubMed]

] and Mathieu [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). [CrossRef] [PubMed]

] 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

A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett. 35(12), 2082–2084 (2010). [CrossRef] [PubMed]

, 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). [CrossRef] [PubMed]

] and experimental observations [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). [CrossRef] [PubMed]

, 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). [CrossRef] [PubMed]

] 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

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). [CrossRef]

]) 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

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). [CrossRef]

, 16

M. S. Kushwaha, “Plasmons and magnetoplasmons in semiconductor heterostructures,” Surf. Sci. Rep. 41(1-8), 1–416 (2001). [CrossRef]

]. 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

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). [CrossRef]

, 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). [CrossRef] [PubMed]

, 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). [CrossRef] [PubMed]

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.

Fig. 1 Schematic structure of Airy SMPs on the interface of a metal/semiconductor and a dielectric. The external magnetic field is applied along the y-axis.

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

\nabla \times (\nabla \times \vec{E}) - k_{0}^{2} {\tilde{ε}}_{m} \vec{E} = 0

(1)

where k₀ is the wave vector in free space. When B is applied, the permittivity of the metal/semiconductor

{\tilde{ε}}_{m}

becomes a tensor caused by the Lorentz force on the free electrons, which is expressed by [17

, 19

E. D. Palik and J. K. Furdyna, “Infrared and microwave magnetoplasma effects in semiconductors,” Rep. Prog. Phys. 33(3), 1193–1322 (1970). [CrossRef]

{\tilde{ε}}_{m} = [\begin{matrix} ε_{x x} & 0 & ε_{x z} \\ 0 & ε_{y y} & 0 \\ - ε_{x z} & 0 & ε_{x x} \end{matrix}]

(2)

where, in the lossless case, ε_xx = ε_∞ [1 – ω_p²/ (ω² – ω_c²)], ε_xz = –iε_∞ω_p²ω_c / [ω(ω² – ω_c²)], and ε_yy = ε_∞ (1 – ω_p²/ω²), 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

E. D. Palik and J. K. Furdyna, “Infrared and microwave magnetoplasma effects in semiconductors,” Rep. Prog. Phys. 33(3), 1193–1322 (1970). [CrossRef]

]. Considering the exponential decay of the SP waves in the metal/semiconductor material, we can express the electric field components as

E_{x, y, z}^{(I)} (x, y, z) = A_{x, y, z} (y, z) e^{α_{1} x}

(3)

where α₁ is the decay factor in x-direction. For a paraxial Airy SMP, α₁ does not change much with that of a plane SMP wave [11

A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett. 35(12), 2082–2084 (2010). [CrossRef] [PubMed]

], and consequently, it can be calculated by

α_{1}^{2} = k_{s m p}^{2} - k_{0}^{2} ε_{V}

, where ε_V is the Voigt dielectric constant, defined by

ε_{V} = ε_{x x} + ε_{x z}^{2} / ε_{x x}

[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). [CrossRef]

]. k_smp is the propagation constant of the SMPs, calculated by a transcendental equation:

ε_{V} \sqrt{k_{s m p}^{2} - ε_{d} k_{0}^{2}} + ε_{d} \sqrt{k_{s m p}^{2} - ε_{V} k_{0}^{2}} + i (\frac{ε_{d} ε_{x z}}{ε_{x x}}) k_{s m p} = 0

(4)

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

\frac{\partial^{2} {\tilde{A}}_{x}}{\partial z^{2}} - α_{1} \frac{\partial {\tilde{A}}_{z}}{\partial z} + (ε_{x x} k_{0}^{2} - k_{y}^{2}) {\tilde{A}}_{x} - i k_{y} α_{1} {\tilde{A}}_{y} + ε_{x z} k_{0}^{2} {\tilde{A}}_{z} = 0

(5a)

\frac{\partial^{2} {\tilde{A}}_{y}}{\partial z^{2}} - i k_{y} \frac{\partial {\tilde{A}}_{z}}{\partial z} - i k_{y} α_{1} {\tilde{A}}_{x} + (α_{1}^{2} + ε_{y y} k_{0}^{2}) {\tilde{A}}_{y} = 0

(5b)

α_{1} \frac{\partial {\tilde{A}}_{x}}{\partial z} + i k_{y} \frac{\partial {\tilde{A}}_{y}}{\partial z} + k_{0}^{2} ε_{x z} {\tilde{A}}_{x} - (k_{0}^{2} ε_{x x} + α_{1}^{2} - k_{y}^{2}) {\tilde{A}}_{z} = 0

(5c)

where

{\tilde{A}}_{x, y, z} (k_{y}, z)

are the Fourier transforms of

A_{x, y, z} (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

A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett. 35(12), 2082–2084 (2010). [CrossRef] [PubMed]

]. To solve Eq. (5), we use

{\tilde{A}}_{x}

and

{\tilde{A}}_{y}

to express

{\tilde{A}}_{z}

by Eq. (5c) and substitute it into Eqs. (5a) and (5b). The differential equations with respect to

{\tilde{A}}_{x}

and

{\tilde{A}}_{y}

are obtained as

[\begin{matrix} \frac{k_{0}^{2} ε_{x x} - k_{y}^{2}}{κ_{1}^{2}} D^{2} + \frac{{(k_{0}^{2} ε_{x z})}^{2}}{κ_{1}^{2}} + k_{0}^{2} ε_{x x} - k_{y}^{2} & - \frac{i k_{y} α_{1}}{κ_{1}^{2}} D^{2} + \frac{i k_{y} k_{0}^{2} ε_{x z}}{κ_{1}^{2}} D - i k_{y} α_{1} \\ - \frac{i k_{y} α_{1}}{κ_{1}^{2}} D^{2} - \frac{i k_{y} k_{0}^{2} ε_{x z}}{κ_{1}^{2}} D - i k_{y} α_{1} & \frac{k_{0}^{2} ε_{x x} + α_{1}^{2}}{κ_{1}^{2}} D^{2} + α_{1}^{2} + k_{0}^{2} ε_{y y} \end{matrix}] (\begin{matrix} {\tilde{A}}_{x} \\ {\tilde{A}}_{y} \end{matrix}) = 0

(6)

where

κ_{1}^{2} = k_{0}^{2} ε_{x x} + α_{1}^{2} - k_{y}^{2}

. D is the partial differential with respect to z. The eigen values can be calculated by nontrivial solutions of Eq. (6) as

Λ^{2} = \frac{1}{2} [- (κ_{2}^{2} + κ_{3}^{2}) \pm \sqrt{{(κ_{2}^{2} - κ_{3}^{2})}^{2} - 4 (κ_{2}^{2} - κ_{1}^{2}) \frac{ε_{y y}}{ε_{x x}} k_{y}^{2}}]

(7)

in which

κ_{2}^{2} = k_{0}^{2} ε_{V} + α_{1}^{2} - k_{y}^{2}

and

κ_{3}^{2} = k_{0}^{2} ε_{y y} + α_{1}^{2} - k_{y}^{2} ε_{y y} / ε_{x x}

. Using the paraxial approximation expression k_y<<k₀ and k_y<<k_smp, we expand Eq. (7) up to the second and the zeroth order of k_y /k₀ and k_y /k_smp, respectively. The simplified eigen values are calculated by

Λ_{1} = i k_{∥}

(8a)

Λ_{2} = i (k_{s m p} - \frac{ε_{x x} + ε_{y y}}{4 ε_{x x}} \frac{k_{y}^{2}}{k_{s m p}})

(8b)

where

k_{∥}^{2} = k_{0}^{2} ε_{y y} + α_{1}^{2}

. 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

], Λ₂ = 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

{\tilde{A}}_{x} = C_{1} \exp [i (k_{s m p} - \frac{ε_{x x} + ε_{y y}}{4 ε_{x x}} \frac{k_{y}^{2}}{k_{s m p}}) z]

(9)

where C₁ is a function of k_y needing to be determined. Like other works of Airy beams [1

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979). [CrossRef]

–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). [CrossRef] [PubMed]

], 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

A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett. 35(12), 2082–2084 (2010). [CrossRef] [PubMed]

]

E_{x}^{(I)} (x, y, 0) = A i (\frac{y}{w_{0}}) e^{a y / w_{0}} e^{α_{1} x}

(10)

where w₀ is the characteristic width of the first Airy beam lobe, and a is the decay factor [3

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007). [CrossRef] [PubMed]

]. It can be inferred from the Fourier transform of Eq. (10) that

C_{1} = w_{0} \exp [\frac{{(a - i w_{0} k_{y})}^{3}}{3}]

(11)

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

E_{x}^{(I)} = E_{0}^{(I)} A i [f_{1} (y, p)] \cdot \exp [f_{2} (y, p)] \cdot e^{α_{1} x} \cdot e^{i k_{s m p} z}

(12)

in which

f_{1} (y, p) = \frac{y}{w_{0}} - p^{2} + i 2 a p

(13a)

f_{2} (y, p) = a \frac{y}{w_{0}} - 2 a p^{2} + i (- \frac{2}{3} p^{3} + a^{2} p + \frac{y}{w_{0}} p)

(13b)

E_{x 0}^{(I)}

is the amplitude.

p = z (ε_{x x} + ε_{y y}) / (4 ε_{x x} k_{s m p} w_{0}^{2})

. For an Airy surface plasmon, 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

A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett. 35(12), 2082–2084 (2010). [CrossRef] [PubMed]

]. For simplicity, we only consider TM mode. Therefore, according to the Maxwell equations, other electric field components in regions (I) are derived as

E_{z}^{(I)} = \frac{k_{s m p}^{2} - ε_{x x} k_{0}^{2}}{ε_{x z} k_{0}^{2} - i k_{s m p} α_{1}} E_{0}^{(I)} A i [f_{1} (y, z)] \exp [f_{2} (y, z)] e^{α_{1} x} e^{i k_{s m p} z}

(14)

\begin{array}{l} E_{y}^{(I)} = - j \frac{1}{w_{0}} \frac{k_{s m p}}{ε_{x z} k_{0}^{2} - i k_{s m p} α_{1}} \frac{k_{s m p}^{2} - ε_{x x} k_{0}^{2}}{k_{s m p}^{2} - ε_{y y} k_{0}^{2}} E_{0}^{(I)} \exp [f_{2} (y, z)] e^{α_{1} x} e^{i k_{s m p} z} \\ \times {- \frac{f_{1} (y, z)}{π \sqrt{3}} K_{2 / 3} [\frac{2}{3} f_{1}^{3 / 2} (y, z)] + (i p + a) A i [f_{1} (y, z)]} \end{array}

(15)

In the dielectric, the electric field components can be calculated by the same procedure, by replacing ε_xx with ε_d, α₁ with -α₂, and ε_xx = 0, where

α_{2}^{2} = k_{s m p}^{2} - k_{0}^{2} ε_{d}

. Together with the consideration of the boundary conditions, we obtain:

E_{x}^{(I I)} = E_{0}^{(I I)} A i [f_{1} (y, z)] \exp [f_{2} (y, z)] e^{- α_{2} x} e^{i k_{s m p} z}

(16)

\begin{array}{l} E_{z}^{(I I)} = \frac{k_{s m p}^{2} - ε_{d} k_{0}^{2}}{α_{2} k_{s m p}} \frac{ε_{d} (ε_{x x} α_{1}^{2} + ε_{x z}^{2} k_{0}^{2})}{ε_{x x} α_{2} (ε_{x z} k_{s m p} - i ε_{x x} α_{1})} E_{0}^{(I I)} \\ \times A i [f_{1} (y, z)] \exp [f_{2} (y, z)] e^{- α_{2} x} e^{i k_{s m p} z} \end{array}

(17)

\begin{array}{l} E_{y}^{(I I)} = - j \frac{1}{w_{0}} \frac{ε_{d}}{ε_{x z} k_{s m p} - i ε_{x x} α_{1}} \frac{k_{s m p}^{2} - ε_{x x} k_{0}^{2}}{k_{s m p}^{2} - ε_{y y} k_{0}^{2}} E_{0}^{(I I)} \exp [f_{2} (y, z)] e^{- α_{2} x} e^{i k_{s m p} z} \\ \times {- \frac{f_{1} (y, z)}{π \sqrt{3}} K_{2 / 3} [\frac{2}{3} f_{1}^{3 / 2} (y, z)] + (i p + a) A i [f_{1} (y, z)]} \end{array}

(18)

in which

E_{0}^{(I)} = \frac{ε_{x z} k_{0}^{2} - i k_{s m p} α_{1}}{ε_{x z} k_{s m p}^{2} - i ε_{x x} k_{s m p} α_{1}}

(19a)

E_{0}^{(I I)} = \frac{1}{ε_{d}}

(19b)

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 ε_V <0 (under this condition, SMPs can exists), the incident frequency and the magnetic field are set as ω = 0.8ω_p and ω_c = 0.1ω_p, 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

I. L. Tyler, B. Fischer, and R. J. Bell, “On the observation of surface magnetoplasmons,” Opt. Commun. 8(2), 145–146 (1973). [CrossRef]

–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). [CrossRef] [PubMed]

]. The corresponding parameters of InSb are m* = 0.014m₀, (m₀ is the free electron mass in vacuum), ω_p = 12.6THz, and ε_∞ = 15.68 [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). [CrossRef] [PubMed]

]. 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

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007). [CrossRef] [PubMed]

, 3

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007). [CrossRef] [PubMed]

]. 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.

Fig. 2 Electric field distributions of the Airy SMPs on the planes perpendicular and parallel to the surface when ω_c = 0.1ω_p and ω = 0.8ω_p. (a)-(c) Electric field distributions in the x-y plane when z = 0, 50λ, and 100λ, respectively. (d) Electric field distribution in the y-z plane when x = 0.

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

A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett. 35(12), 2082–2084 (2010). [CrossRef] [PubMed]

, 20

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33(3), 207–209 (2008). [CrossRef] [PubMed]

]. 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

y = \frac{1}{4 w_{0}^{3} k_{s m p}^{2}} z^{2}

(20)

g = \frac{\partial^{2} y}{\partial z^{2}} = \frac{1}{4 w_{0}^{3} k_{s m p}^{2}}

(21)

It can be clearly seen that if the transverse scale w₀ is set, the ballistic dynamics of the Airy SMPs is mainly determined by k_smp. In Fig. 3 , we plot the dispersion curve of SMPs under different external magnetic field intensities with ω<ω_p. It is found that when the magnetic field is applied along the + y-axis (denoted by B > 0 in the inset), the k_smp~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, ε_xz in the last term changes its sign. Thus the dispersion equations become different, having different cutoff frequencies [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). [CrossRef] [PubMed]

Fig. 3 Dispersion relation of SMPs under different external magnetic field intensities. If the external magnetic field is along the -y-axis (denoted by B<0), the dispersion curve is red-shifted with the increase of the magnetic field. However, if the external magnetic field is along the + y-axis (denoted by B>0 in the inset), the dispersion curve is nearly unchanged.

From Fig. 3, it can be inferred that if the magnetic field increases in the region B<0, k_smp 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 = 0.85ω_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.

Fig. 4 Ballistic trajectory curves of the Airy SMPs when the magnetic field is applied along the –y-axis. The cyclotron frequencies are ω_c = 0, 0.1ω_p, 0.2ω_p, and 0.25ω_p, respectively. The incident frequency is ω = 0.85ω_p. The inset shows the “gravity” g as a function of the external magnetic field when the magnetic field is applied in the + y-axis and –y-axis, respectively.

Fig. 5 Electric field distributions of the Airy SMPs on the x = 0 plane with an incident frequency of ω = 0.85ω_p. (a) No magnetic field is applied. (b) A magnetic field is applied along the + y-axis such that ω_c = 0.25ω_p. (c) A magnetic field is applied along the -y-axis such that ω_c = 0.25ω_p.

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

A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett. 35(12), 2082–2084 (2010). [CrossRef] [PubMed]

]. 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 (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.

Fig. 6 Comparison of theory (red lines) and FDTD simulation (blue lines) results of normalized electric field distributions of the Airy SMPs on the x = 0 plane at z = 50λ. The incident frequency is ω = 0.85ω_p. Solid lines: field distributions when ω_c = 0ω_p. Dashed lines: field distributions when ω_c = 0.25ω_p for B<0.

4. 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.

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.

Acknowledgements

References and links

1.	M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979). [CrossRef]
2.	G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007). [CrossRef] [PubMed]
3.	G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007). [CrossRef] [PubMed]
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). [CrossRef]
5.	A. Rudnick and D. M. Marom, “Airy-soliton interactions in Kerr media,” Opt. Express 19(25), 25570–25582 (2011). [CrossRef] [PubMed]
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). [CrossRef] [PubMed]
7.	J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008). [CrossRef]
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). [CrossRef] [PubMed]
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). [CrossRef] [PubMed]
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). [CrossRef] [PubMed]
11.	A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett. 35(12), 2082–2084 (2010). [CrossRef] [PubMed]
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). [CrossRef] [PubMed]
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). [CrossRef] [PubMed]
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). [CrossRef] [PubMed]
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). [CrossRef]
16.	M. S. Kushwaha, “Plasmons and magnetoplasmons in semiconductor heterostructures,” Surf. Sci. Rep. 41(1-8), 1–416 (2001). [CrossRef]
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). [CrossRef] [PubMed]
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). [CrossRef] [PubMed]
19.	E. D. Palik and J. K. Furdyna, “Infrared and microwave magnetoplasma effects in semiconductors,” Rep. Prog. Phys. 33(3), 1193–1322 (1970). [CrossRef]
20.	G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33(3), 207–209 (2008). [CrossRef] [PubMed]
21.	I. L. Tyler, B. Fischer, and R. J. Bell, “On the observation of surface magnetoplasmons,” Opt. Commun. 8(2), 145–146 (1973). [CrossRef]
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). [CrossRef]
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). [CrossRef] [PubMed]
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). [CrossRef] [PubMed]

OCIS Codes
(230.3810) Optical devices : Magneto-optic systems
(240.6680) Optics at surfaces : Surface plasmons

ToC Category:
Optics at Surfaces

History
Original Manuscript: July 9, 2012
Revised Manuscript: August 19, 2012
Manuscript Accepted: August 19, 2012
Published: August 31, 2012

Citation
Bin Hu, Qi Jie Wang, and Ying Zhang, "Voigt Airy surface magneto plasmons," Opt. Express 20, 21187-21195 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-19-21187

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References

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys.47(3), 264–267 (1979). [CrossRef]
G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett.99(21), 213901 (2007). [CrossRef] [PubMed]
G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett.32(8), 979–981 (2007). [CrossRef] [PubMed]
T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics3(7), 395–398 (2009). [CrossRef]
A. Rudnick and D. M. Marom, “Airy-soliton interactions in Kerr media,” Opt. Express19(25), 25570–25582 (2011). [CrossRef] [PubMed]
G. Zhou, R. Chen, and X. Chu, “Propagation of Airy beams in uniaxial crystals orthogonal to the optical axis,” Opt. Express20(3), 2196–2205 (2012). [CrossRef] [PubMed]
J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics2(11), 675–678 (2008). [CrossRef]
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). [CrossRef] [PubMed]
C. J. Zapata-Rodríguez, S. Vuković, M. R. Belić, D. Pastor, and J. J. Miret, “Nondiffracting Bessel plasmons,” Opt. Express19(20), 19572–19581 (2011). [CrossRef] [PubMed]
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). [CrossRef] [PubMed]
A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett.35(12), 2082–2084 (2010). [CrossRef] [PubMed]
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). [CrossRef] [PubMed]
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). [CrossRef] [PubMed]
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). [CrossRef] [PubMed]
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). [CrossRef]
M. S. Kushwaha, “Plasmons and magnetoplasmons in semiconductor heterostructures,” Surf. Sci. Rep.41(1-8), 1–416 (2001). [CrossRef]
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). [CrossRef] [PubMed]
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). [CrossRef] [PubMed]
E. D. Palik and J. K. Furdyna, “Infrared and microwave magnetoplasma effects in semiconductors,” Rep. Prog. Phys.33(3), 1193–1322 (1970). [CrossRef]
G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett.33(3), 207–209 (2008). [CrossRef] [PubMed]
I. L. Tyler, B. Fischer, and R. J. Bell, “On the observation of surface magnetoplasmons,” Opt. Commun.8(2), 145–146 (1973). [CrossRef]
L. Remer, E. Mohler, W. Grill, and B. Lüthi, “Nonreciprocity in the optical reflection of magnetoplasmas,” Phys. Rev. B30(6), 3277–3282 (1984). [CrossRef]
J. Gómez Rivas, C. Janke, P. H. Bolivar, and H. Kurz, “Transmission of THz radiation through InSb gratings of subwavelength apertures,” Opt. Express13(3), 847–859 (2005). [CrossRef] [PubMed]
M. S. Kushwaha and P. Halevi, “Magnetoplasmons in thin films in the Voigt configuration,” Phys. Rev. B Condens. Matter36(11), 5960–5967 (1987). [CrossRef] [PubMed]

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