Beam Splitting by the Use of Waveguide Airy Beams
 Author: Kim KyoungYoum, Kim Saehwa
 Organization: Kim KyoungYoum; Kim Saehwa
 Publish: Journal of the Optical Society of Korea Volume 15, Issue4, p403~406, 25 Dec 2011

ABSTRACT
Here we report Airy beams coupled with waveguide modes. These waveguide Airy (WAiry) beams propagate through layered planar structures inheriting the characteristics of waveguide modes as well as those of Airy beams, such as diffractionfree and accelerating features. In particular, we focus on the WAiry beams associated with
backward waveguide modes, showing that the backward feature can alter the trajectories of the WAiry beams significantly. Based on this, we propose a new scheme of waveguidetype polarization/power beam splitters.

KEYWORD
Airy beam , Waveguides , Metamaterials

I. INTRODUCTION
Airy beams, optical versions of the original Airy wavepackets in the context of quantum mechanics [1], are one of several nondiffracting beam solutions whose intensity profiles remain invariant as they propagate through an optical medium. Compared with other types of diffractionfree solutions such as Bessel [2] and Mathieu [3] beams, Airy beams have several unique properties: they are the only nondiffracting solution in the most simple (1+ 1)D configuration and are
accelerated during the propagation [4,5]. This acceleration or, to put it more correctly, selfbending is a phenomenal one caused by initial field configurations lacking spatial symmetry. Not only the diffractionfree but also the selfbending features have their origin in the multiple interference between various field components comprising the Airy beam, which maintains its profile during propagation, only changing the peak positions. These attractive characteristics have already been exploited in various applications such as optical trapping/particle manipulation [6,7], plasma waveguiding [8], implementation of Airy plasmons [9], optical bullets [10], dual Airy beams [11], slow nondispersing wavepackets [12], and the generation of Bessellike beams [13].In this paper, we propose waveguide Airy (WAiry) beams: Airy beams propagating through inhomogeneous media with the help of waveguide modes. They not only retain the diffractionfree and selfbending properties of the Airy beam but also inherit the unique features of the waveguide mode such as polarization dependency and slowlight characteristics. Here, we will focus on the WAiry beam associated with the socalled
backward waveguide mode [14] and show that the backward feature can alter the trajectories of the WAiry beams significantly, which makes them applicable to the implementation of polarization or power beam splitters.II. WAVEGUIDE AIRY BEAMS
Let us consider a layered or stacked planar waveguide shown in Fig. 1. We have
where
φ denotes eitherE_{x} (TEpolarized case) orH_{x} (TM case) field.k _{0} is 2π /λ whereλ is the wavelength of lightin vacuum,
ε andμ are the relative permittivity and permeability of the material. We can easily show thatwhere
ψ (y ,z ) and g(y ) denote a waveguide mode and its transverse profile, respectively, satisfies Eq. (1).A (x ,z ) is the conventional (1 1)D Airy beam profile given bywhere
x _{0} is an arbitrary scaling factor and α = α ' jα " , α' and α " being related to the initial field profilewhere we can find that α' is the exponential apodization parameter for the finite energy carried by the beam and α" is the phase modulation factor along the scaled transverse direction [15].
The above results imply that an Airy beam can propagate in any layered planar waveguides by forming a hybrid beam with waveguide modes. These Airylike diffractionfree beams, hereafter called waveguide Airy beams or WAiry beams, inherit the characteristics of both the waveguide mode and the Airy beam. We note that WAiry beams are nondiffracting along all spatial directions: diffraction along the y direction is prohibited by the waveguide structure. Along the two other directions diffraction is prevented by the Airy beam profile.
III. BACKWARD MODE AND THE SPLITTING OF WAIRY BEAMS
The trajectory of the WAiry beam can be derived by setting the real argument of the Airy function in Eq. (3) to be zero:
where
α_{x} denotes the acceleration along the transverse (+x ) direction andθ _{z} is the initial launch angle [see Fig. 2(a)]. As was clearly discussed in [15], Eq. (5) is equivalent to that describing the constant acceleration motion of particles. When we consider a WAiry beam associated with thebackward waveguide mode (in the sense that the overall power flow direction is opposite to that of the phase velocity [14]),β becomes negative for forward power coupling. This negativeβ flips the initial launch angleθ _{z} with respect toz since it retains the sign ofβ . This flipping, although the acceleration direction does not change at all, enables us to separate WAiry beams associated with forward and backward modes.We would like to propose two such cases in which the relevant two modes are forward and backward ones. In all cases, slab waveguides composed of
ε negative (withε _{E} < 0 and μ_{E} > 0) and μnegative (withε _{M} > 0 and μ_{M} <0) metamaterials are adopted. For the first caseε negative and μnegative metamaterials are dual singly negative ones (i.e.,ε _{E} / μ_{E} =ε _{M} / μ_{M}) As was shown in [16], TE and TM modes in slab waveguides made of dual singly negative metamaterials are degenerate, i.e., βTE= βTM, but one of them is a backward mode, resulting in βTE = βTM. Therefore, when the polarization state of light changes, the forward waveguide mode also changes into a backward mode (or vice versa). This implies that the WAiry beams associated with TE and TM waveguide modes can be separated due to the difference in their initial launch angles. Figure 2 shows the splitting of TE and TM WAiry beams in slab waveguides made of dual singly negative metamaterials. We assumed ε_{E} = 1, μ_{E} = 1, ε_{M} = 2, μ_{M} = 2, a = 0.05 2j, and x0 =1.5λ, and a μ negative / ε negative / μnegative slab structure was considered. The results were obtained using a commercial software (COMSOL) based on the finite element method with the effective index method to reduce the 3D structure into a 2D one. Most optical power flows through the core layer made of anε negative medium. The timeaveraged Poynting vector <S >, which is inversely proportional to μ_{E} andε E for TE and TMpolarized light, respectively, becomes forward (TE: // <β S > ) or backward (TM: // <β S >), depending on the polarization state of light. Therefore, we haveθ _{z} ^{(TE)} >0 andθ _{z} ^{(TM)}<0, resulting in the separation of TE and TM WAiry beams (i.e., polarization beam splitting).For the second case the waveguide supports dual modes. Dual modes have the same parity and the same number of zerocrossings, but their directions of overall power flow
are opposite to each other [14], meaning that dual modes include forward and backward modes. Therefore, the corresponding WAiry beam associated with each of them can also be separated. This characteristic, i.e., power beam splitting, is shown in Fig. 3 where we assumed TMpolarized light waves incident to a μ negative/εnegative/
μ negative slab withε _{E} = 2, μ_{E} = 3.5,ε _{M} = 3, and μ_{M} = 1. The fact that dual modes have the same parity as well as the same number of zerocrossings, as can be found in the mode profiles shown in the inset of Fig. 3, entails that the coupling efficiencies to two respective WAiry beams associated with dual modes can be made nearly the same, which is very advantageous in the actual implementation of power beam splitters.IV. CONCLUSION
We proposed WAiry beams: Airylike beams associated with waveguide modes and propagating through planar waveguides. They not only retain the nondiffracting and accelerating properties of the Airy beam but also inherit the unique features of the waveguide mode such as backward powerflow characteristics. We showed that this additional feature of the backward power flow can be utilized to develop a new kind of polarization/power beam splitter.

[FIG. 1.] Geometry of the layered planar structure. ψ (y, z) andg(y) denote a waveguide mode and its profile along thetransverse (y) direction, respectively. A(x, z) describes theconventional (1 1)D Airy beam profile.

[FIG. 2.] (a) Polarizationdependent trajectories of WAiry beams. (b), (c) Power flows (<S> ) through TE and TM WAiry beams, calculated numerically using a commercial software (COMSOL) based on the finite element method. We assumed the operation wavelength (λ) of 1550 nm and a μnegative/ εnegative/μnegative slab waveguide with the core thickness of 300 nm. εE = 1, μE = 1, εM = 2, μM = 2, α= 0.05 2j, and x0 =1.5λwere used. β and <S>denote the propagation wavevector and the timeaveraged Poynting vector, respectively.

[FIG. 3.] Power beam splitting of TM WAiry beams associated with backward and forward dual modes. We also assumed the operation wavelength of 1550 nm and a μnegative / εnegative/μnegative slab waveguide with the core thickness of 89.5 nm. εE = 2, μE = 3.5, εM = 3, μM = 1, α= 0.03 2j, and x0 =1.5λwere used. Inset: actual profiles of dual modes.