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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 19 — Sep. 10, 2012
  • pp: 21187–21195
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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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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.

© 2012 OSA

1. Introduction

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

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

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

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

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

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

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

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

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

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

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

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

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

, 12

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

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

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

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 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

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

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

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

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

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
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.
. 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
×(×E)k02ε˜mE=0
(1)
where k0 is the wave vector in free space. When B is applied, the permittivity of the metal/semiconductor ε˜m becomes a tensor caused by the Lorentz force on the free electrons, which is expressed by [17

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]

, 19

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

]:
ε˜m=[εxx0εxz0εyy0εxz0εxx]
(2)
where, in the lossless case, εxx = ε [1 – ωp2/ (ω2ωc2)], εxz = –ωp2ωc / [ω(ω2ωc2)], and εyy = ε (1 – ωp2/ω2), in which ω is the angular frequency of the incident wave, ωp is the plasma frequency of the metal/semiconductor, ε is the high-frequency permittivity, and ωc = 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

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
Ex,y,z(I)(x,y,z)=Ax,y,z(y,z)eα1x
(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

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 byα12=ksmp2k02εV, where εV is the Voigt dielectric constant, defined by εV=εxx+εxz2/εxx [15

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]

]. ksmp is the propagation constant of the SMPs, calculated by a transcendental equation:
εVksmp2εdk02+εdksmp2εVk02+i(εdεxzεxx)ksmp=0
(4)
in which εd 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
2A˜xz2α1A˜zz+(εxxk02ky2)A˜xikyα1A˜y+εxzk02A˜z=0
(5a)
2A˜yz2ikyA˜zzikyα1A˜x+(α12+εyyk02)A˜y=0
(5b)
α1A˜xz+ikyA˜yz+k02εxzA˜x(k02εxx+α12ky2)A˜z=0
(5c)
where A˜x,y,z(ky,z) are the Fourier transforms of Ax,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

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 A˜x and A˜y to express A˜zby Eq. (5c) and substitute it into Eqs. (5a) and (5b). The differential equations with respect to A˜x and A˜y are obtained as
[k02εxxky2κ12D2+(k02εxz)2κ12+k02εxxky2ikyα1κ12D2+ikyk02εxzκ12Dikyα1ikyα1κ12D2ikyk02εxzκ12Dikyα1k02εxx+α12κ12D2+α12+k02εyy](A˜xA˜y)=0
(6)
where κ12=k02εxx+α12ky2. D is the partial differential with respect to z. The eigen values can be calculated by nontrivial solutions of Eq. (6) as
Λ2=12[(κ22+κ32)±(κ22κ32)24(κ22κ12)εyyεxxky2]
(7)
in which κ22=k02εV+α12ky2and κ32=k02εyy+α12ky2εyy/εxx. Using the paraxial approximation expression ky<<k0 and ky<<ksmp, we expand Eq. (7) up to the second and the zeroth order of ky /k0 and ky /ksmp, respectively. The simplified eigen values are calculated by
Λ1=ik
(8a)
Λ2=i(ksmpεxx+εyy4εxxky2ksmp)
(8b)
where k2=k02εyy+α12. It can be found that Eqs. (8a) and (8b) correspond to the TE (Ex = Ez = 0) and TM modes (Ey = 0) of a plane wave propagating in Voigt-magnetized plasma [18

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]

], respectively, except that in Ref [18

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]

], 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) is
A˜x=C1exp[i(ksmpεxx+εyy4εxxky2ksmp)z]
(9)
where C1 is a function of ky needing to be determined. Like other works of Airy beams [1

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

6

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 Ex component of the Airy SMPs in the metal/semiconductor layer at the input plane z = 0 takes a form of [11

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

]
Ex(I)(x,y,0)=Ai(yw0)eay/w0eα1x
(10)
where w0 is the characteristic width of the first Airy beam lobe, and a is the decay factor [3

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
C1=w0exp[(aiw0ky)33]
(11)
Substitute Eq. (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 as
Ex(I)=E0(I)Ai[f1(y,p)]exp[f2(y,p)]eα1xeiksmpz
(12)
in which
f1(y,p)=yw0p2+i2ap
(13a)
f2(y,p)=ayw02ap2+i(23p3+a2p+yw0p)
(13b)
Ex0(I) is the amplitude. p=z(εxx+εyy)/(4εxxksmpw02). 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: Hy = 0, TM mode: Ey = 0). However, in the paraxial approximation, they can be divided by Ez = 0 for TE mode and Hz = 0 for TM mode [11

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

Ez(I)=ksmp2εxxk02εxzk02iksmpα1E0(I)Ai[f1(y,z)]exp[f2(y,z)]eα1xeiksmpz
(14)
Ey(I)=j1w0ksmpεxzk02iksmpα1ksmp2εxxk02ksmp2εyyk02E0(I)exp[f2(y,z)]eα1xeiksmpz×{f1(y,z)π3K2/3[23f13/2(y,z)]+(ip+a)Ai[f1(y,z)]}
(15)

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 α22=ksmp2k02εd. Together with the consideration of the boundary conditions, we obtain:
Ex(II)=E0(II)Ai[f1(y,z)]exp[f2(y,z)]eα2xeiksmpz
(16)
Ez(II)=ksmp2εdk02α2ksmpεd(εxxα12+εxz2k02)εxxα2(εxzksmpiεxxα1)E0(II)×Ai[f1(y,z)]exp[f2(y,z)]eα2xeiksmpz
(17)
Ey(II)=j1w0εdεxzksmpiεxxα1ksmp2εxxk02ksmp2εyyk02E0(II)exp[f2(y,z)]eα2xeiksmpz×{f1(y,z)π3K2/3[23f13/2(y,z)]+(ip+a)Ai[f1(y,z)]}
(18)
in which

E0(I)=εxzk02iksmpα1εxzksmp2iεxxksmpα1
(19a)
E0(II)=1εd
(19b)

In Fig. 2
Fig. 2 Electric field distributions of the Airy SMPs on the planes perpendicular and parallel to the surface when ωc = 0.1ωp and ω = 0.8ωp. (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.
, 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

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

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), ωp = 12.6THz, and ε = 15.68 [23

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

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

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.

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

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

, 20

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=14w03ksmp2z2
(20)
g=2yz2=14w03ksmp2
(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 ksmp. In Fig. 3
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.
, 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 [24

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]

].

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
Fig. 4 Ballistic trajectory curves of the Airy SMPs when the magnetic field is applied along the –y-axis. The cyclotron frequencies are ωc = 0, 0.1ωp, 0.2ωp, and 0.25ωp, respectively. The incident frequency is ω = 0.85ωp. 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.
, 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
Fig. 5 Electric field distributions of the Airy SMPs on the x = 0 plane with an incident frequency of ω = 0.85ωp. (a) No magnetic field is applied. (b) A magnetic field is applied along the + y-axis such that ωc = 0.25ωp. (c) A magnetic field is applied along the -y-axis such that ωc = 0.25ωp.
, 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 [11

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
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 = 0.85ωp. Solid lines: field distributions when ωc = 0ωp. Dashed lines: field distributions when ωc = 0.25ωp for B<0.
. 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.

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

  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. Photonics3(7), 395–398 (2009). [CrossRef]
  5. A. Rudnick and D. M. Marom, “Airy-soliton interactions in Kerr media,” Opt. Express19(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. Express20(3), 2196–2205 (2012). [CrossRef] [PubMed]
  7. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics2(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. Express19(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. B30(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. Express13(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. Matter36(11), 5960–5967 (1987). [CrossRef] [PubMed]

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