Superpositions of Laguerre Gaussian Beams in Strongly Nonlocal Left-handed Materials

Commun. Theor. Phys. (Beijing, China) 53 (2010) pp. 749 754 c Chinese Physical Society and IOP Publishing Ltd Vol. 53, No. 4, April 15, 2010 Superpositions of Laguerre Gaussian Beams in Strongly Nonlocal Left-handed Materials ZHONG Wei-Ping ( ), 1,2 Milivoj Belić, 2 HUANG Ting-Wen, 2 and WANG Li-Yang ( Û ) 1 1 Department of Electronic Engineering, Shunde Polytechnic, Guangdong Province, Shunde 528300, China 2 Texas A & M University at Qatar, 23874 Doha, Qatar (Received May 21, 2009; revised manuscript received July 27, 2009) Abstract We present beam solutions of the strongly nonlocal nonlinear Schrödinger equation in left-handed materials (LHMs). Different Laguerre Gaussian (LG) necklace beams, such as symmetric and asymmetric single layer and multilayer necklace beams are created by the superposition of two single beams with different topological charges. Such superpositions are then propagated through LHMs, displaying linear diffraction. It is found that the superposition of two LG nm beams with opposite topological charges does not show rotational behavior and that there exists rotation for other topological charge combinations. Our theory predicts that the accessible solitons cannot exist in LHMs. PACS numbers: 42.25.-p, 05.45.Yv Key words: Laguerre Gausssian beam, the strongly nonlocal nonlinear, left-handed material 1 Introduction Left-handed materials (LHMs) possess negative permittivity (ε < 0) and negative permeability (µ < 0). For an electromagnetic (EM) wave propagating in a LHM, the wave vector K, the electric field vector E, and the magnetic field vector H of the wave form the left-handed triplet, in contrast to the right-handed triplet in an ordinary medium. For this reason these materials are sometimes labeled as LHMs, in contrast to the ordinary righthanded media (RHM). They have been fabricated in experiments by Shelby, [1] Smith, [2] and Parini [3] et al., which has excited heated discussion on the topic. These novel metamaterials exhibit some intriguing phenomena, such as negative refraction, [1 3] reverse Goos Hanchen shift, [4] and negative Cerenkov radiation. [5] Recently Lazarides and Tsironis [6] presented a model for the propagation of EM waves in nonlinear (NL) metamaterials, having negative index of refraction. NL properties of LHMs have also been analyzed in Ref. [7]. Nonlocality plays an important role in the NL evolution of waves. It has been shown that there are some peculiar properties induced by the nonlocality, such as the suppression of wave collapse, [8] the support of vortex [9 10] and multipole solitons, [11] as well as the existence of accessible solitons (AS). [12] In NL optics especially, the nonlocality was found important in materials such as nematic liquid crystal [13 14] and lead glass. [15 16] In these materials, the characteristic length of the material response function can be much larger than the beam width; hence they are called the strongly nonlocal (NLO) nonlinear (SNN) media. In this paper, we introduce a class of spatial optical beams in SNN left-handed materials, the so-called Laguerre Gaussian (LG) beams, which are produced in the form of arbitrary symmetric and asymmetric singlelayer and multilayer necklace beams, formed by the superposition of two LG beams with different topological charges (TCs). The superposition of two beams with opposite TCs generates steady necklace beams, while the superposition of beams with different TCs gives rise to the more complex rotating necklace beams. We find that LHMs cannot support the propagation of AS beams. The paper is organized as follows. In Sec. 2 we introduce the SNN model in left-handed materials and find exact self-similar solutions. Different LG complex beams, produced by the superposition of two single beams, are presented in Sec. 3. Section 4 gives conclusions. 2 SNN Model in LHMs and Its Self-Similar Solution 2.1 Model We begin by considering the amplitude u( r, z) of a beam propagating in an NN LHMs along the z-axis of the coordinate system ( r, z), where r = (x, y) = (r, ϕ) denotes the transverse coordinates. The paraxial propagation of the beam is governed by the general NN Schrödinger equation (NNSE): [17 19] i u z d 2 2 u + k n 0 u R( r r ) u( r, z) 2 d r = 0, (1) where 2 is the transverse Laplacian, d = 1/k is the diffraction coefficient, k is the wave number, n 0 is the linear part of the refractive index, and R( r r ) is the nor- Supported by the Science Research Foundation of Shunde Polytechnic (2008-KJ06), China. Work at the Texas A&M University at Qatar is supported by the NPRP 25-6-7-2 project with the Qatar National Research Foundation

750 ZHONG Wei-Ping, Milivoj Belić, HUANG Ting-Wen, and WANG Li-Yang Vol. 53 malized radially symmetric spatial NLO response function of the medium. The response is assumed regular, real, and normalized, R( r )d 2 r = 1. For SNN media, the response function R( r r ) is much broader than the intensity distribution; this also means that the EM wavelength is much larger than the basic unit of the medium. The physical origin of phenomena can qualitatively be understood from the nonlocality of the medium. NL nonlocality means that the NL polarization of a medium in a small volume of radius r 0 (r 0 any characteristic wavelength involved) depends not only on the value of the electric field inside this volume, but also on the electric field outside. The stronger the nonlocality, the more the EM field is involved in contributing to the polarization. Assuming that the intensity distribution is peaked at the origin, the response function R( r r ) can be expanded using the Taylor series expansion with respect to r about r = r. In the limit of SNN response only the dominant term of the series is kept. In this case the strongly NNSE becomes the linear SE for harmonic oscillator [12,17] i u z d 2 2 u r 2 k 2n 0 γpu = 0, (2) where P = u 2 d r (> 0) is the beam power, γ (> 0) is a material constant, and n 0 is the linear part of the refractive index. 2.2 Self-Similar Solutions We rewrite Eq. (2) in the following form: i u z d 2 2 u sr 2 u = 0, (3) where s = kγp/2n 0. We note that LHMs possess negative index of refraction and negative wave number, [20] n 0 < 0 and k < 0; consequently d > 0 and s > 0. Note that the power P is constant, equal to the total input power P 0. It should also be noted that beam collapse cannot occur in Eq. (3) because it is a linear equation. The second term in Eq. (3) represents the diffraction and the third term originates from the optical NLO nonlinearity. We have treated Eq. (3) before, in polar coordinates in [21], by the self-similar method. We have found the exact LG solutions: u nm (z, r, ϕ) = A ( r ) m L ( m r 2 ) n w w w 2 e r2 /2w 2 +imϕ+id [a(z)+c(z)r 2], (4a) where w 2 = w0[ 2 1 + (1 + λ)sinh 2 ( 2dsz) ], (4b) 2ds(1 + λ)sinh(2 2dsz) c = 1 + (1 + λ)sinh 2 ( 2dsz), (4c) a = a 0 2n + m + 1 2dsλw 2 0 arctan [ λ tanh( 2dsz) ]. (4d) Here A = [n!/γ(n+m+1)] 1/2, λ = d/2sw0 4, w is the beam width, c is the chirp, and a is the propagation phase shift. It should be pointed out that the beam width is determined by s and d. We want to analyze the behavior of these solutions under propagation in LHMs; in particular to ascertain the existence or the lack of ASs in such materials. Fig. 1 Comparison of analytical solution (4) for different λ. (a) Beam width change with the propagation distance z; parameters are selected as d = 0.1, s = 0.05, andλ = 0, 0.8, 5 from bottom to top. (b) The optical field intensity distribution of the fundamental LG beam (n = m = 0) change with the propagation distance z and radial coordinate r; the parameters are d = 0.1, s = 0.05, w 0 = 1, and λ = 0,0.8, 5 from left to right.

No. 4 Superpositions of Laguerre Gaussian Beams in Strongly Nonlocal Left-handed Materials 751 It is known that an AS is a beam that maintains its width during propagation, or whose width oscillates periodically, and that its power remains unchanged during propagation. Even though ASs are solutions to a linear SE, they describe well the behavior of real soliton solutions of the NNSE in the strongly NLO regime. It can be seen from Eq. (4a) that w 2 /w0 2 is always larger than one, because λ > 0; the LG beam width expands as it travels in a straight path along the z axis. Consequently, there cannot exist ASs in LHMs. To illustrate this point, Fig. 1 shows how the beam width and the optical intensity distribution I = u 2 of the fundamental LG beam (n = m = 0) expand and diminish with the propagation distance z. As seen from Fig. 1, regardless of the value adopted for λ, the beam width spreads and the intensity reduces to zero. These characteristics can be explained easily. In SNN LHMs, the refractive index is negative, and the nonlocality leads to an increase in the refractive index in the overlap region. The effect of nonlinearity in LHMs is less than the diffraction effect and the accessible solitons cannot propagate in LHMs. 3 Discussion First, we discuss the optical EM field distribution of a single LG nm beam. We choose the initial conditions w 0 = 1, a 0 = 0, d = 0.1, s = 0.05, and perform numerical simulations. In Fig. 2 we present typical contour plots of the optical field u 2 distributions for the single propagating LG nm beam described by Eq. (4). As seen in Fig. 2, when m = 0 (Fig. 2(a)), the maximum optical intensity is located on the propagation axis; this is in contrast to the situation when m 0, when the optical intensity is zero at the central point (Fig. 2(b)). Fig. 2 Optical field intensity distributions of a typical LG nm beam, at different propagation distances z = 0, 5,10 from left to right. (a) LG 40 beam; (b) LG 41 beam. Second, we turn to the discussion of optical intensity distributions for the superposition of two LG nm beams. Because Eq. (3) is linear, the superposition of two or more solutions does not generate new exact solutions. However, new classes of beam shapes can be constructed by the superposition of single LG nm beams with equal beam widths w 0, equal layer numbers n (at the same propagation distance z), but with different TCs (positive or negative) m. Note that a fractional choice for m allows the possibility of having beams with fractional topological charge. Such a possibility has recently been discussed theoretically [22] and demonstrated experimentally. [23] 3.1 Opposite Topological Charges The superposition of linear waves is simply expressed as u = u n1m 1 + u n2m 2. First, we consider the simplest case the superposition of two LG nm beam with opposite topological charges m, namely m 1 = m 2. As said above, if u n1m 1 and u n2m 2 are the solutions to Eq. (3), then the superposition u = u n1m 1 +u n2m 2 is also a solution to Eq. (3). Typical examples of the patterns generated are displayed in Figs. 3 4. Even though such superpositions are an exercise in the presentation of special functions, we believe that they still carry useful information, especially the case with fractional TCs, which is not that well discussed the literature. When m ( 0) is an integer, we get the symmetric necklace beams. Figure 3 depicts some illustrative examples. We find that these beams posses adjacent alternating maximum and minimum spots along the azimuthal angle. For n = 0 the superposition of two beams forms a single layer necklace beam; for n > 0 we observe symmetric multilayer necklace beams. There are n + 1 layers, and the

752 ZHONG Wei-Ping, Milivoj Belić, HUANG Ting-Wen, and WANG Li-Yang Vol. 53 maximum optical intensity is evenly distributed at the farthest outside spots along the radial direction. The number of spots in each layer is determined by m, and the number of layers by n. The total number of spots in a necklace beam equals 2m(n + 1), one-half of these being maximum spots and the other half minimum spots. Fig. 3 Symmetric single-layer (a) and multilayer (b) necklace beams for m 1 = m 2 ( 0) being integer, at different propagation distances z = 0, 5, 10 from left to right. (a) n = 0, m 1 = m 2 = 9; (b) n = 2, m 1 = m 2 = 5. Fig. 4 Asymmetric single-layer and multilayer necklace beams for m 1 = m 2 being a fraction, at different propagation distances z = 0,5, 10 from left to right. (a) n = 0, m 1 = m 2 = 7/2; (b) n = 1, m 1 = m 2 = 5/2. When m is a fraction, we find asymmetric single-layer and multilayer necklace beams. In Fig. 4 we present some illustrative examples of the asymmetric necklace beams. We note that when m becomes sufficiently large, the asymmetric beams become symmetric beams. These beams also form adjacent alternating minimum and maximum spots along the radial and the azimuthal direction. There exist n + 1 layers, and the maximum optical intensity is evenly distributed at the most outside spots along the radial direction. The number of spots in each layer is decided by m, and the layer number is decided by n; there are (2m 1)(n + 1) spots overall, half of them minimum and the other half maximum. As one can see in Figs. 3 4, with the increase in the

No. 4 Superpositions of Laguerre Gaussian Beams in Strongly Nonlocal Left-handed Materials 753 propagation distance the beams spread, their amplitudes reduce, but the total power remains unchanged. This is an expression of the energy conservation. Further, we find that the distributions change regularly with azimuthal angle for all the spots. When m is large enough, the spots form an alternating min/max beam ring. We find that the beams do not rotate. 3.2 Arbitrary Topological Charges Next, we explore more complex field propagation in LHMs. We consider the combination of two LG nm beams with arbitrary TCs; then in general the superposition beam displays rotation. Possible examples include: both ms are integer but not zero; one is zero and the other integer, one is integer and the other fraction; one is zero and the other fraction; both are fractions. A simulation of the propagation of the superposition of two optical fields, produced by the special choices of LG beams is shown in Fig. 5. When one of the TCs is a fraction, the symmetric shape is lost. The central part develops a large spot; it is noted that the optical intensity is maximum there. As shown in Fig. 5, the distributions rotate counterclockwise around the center, as the propagation distance increases. In addition to rotation, the shape of the distributions with fractional TC is more deformed. Fig. 5 Optical intensity distributions for the superposition of two LG beams with different topological charges, at different propagation distances z = 0, 5,10 from left to right. (a) n = 0, m 1 = 3, m 2 = 1/2; (b) n = 0, m 1 = 0, m 2 = 3/2. It should be pointed out that of the NL polarization of the wave combination in LHMs has the symmetry of the electric field, due to strong nonlocality. We also note that the fractional TC leads to an asymmetric polarization in LHMs; it destroys the nonlocality characteristics of LHMs. Under proper conditions, the nonlocality leads to a change of the refractive index in the overlap region, giving rise to the formation of symmetric and asymmetric beams. 4 Conclusions In conclusion, we have considered the strongly NN model of beam propagation in LHMs and obtained its selfsimilar solution. We considered superpositions of two LG beams in propagation. Such superpositions can create complex patterns. The combination of two LG nm beams with opposite TCs produces steadily expanding symmetric and asymmetric single-layer and multilayer necklace beams; for other TC combinations there exists rotation of the intensity distributions. Our theoretical analysis predicts that ASs cannot exist in LHMs. References [1] R.A. Shelby, D.R. Smith, and S. Schultz, Science 77 (2001) 292. [2] D.R. Smith and N. Kroll, Phys. Rev. Lett. 85 (2000) 2933; D.R. Smith, et al., Phys. Rev. Lett. 84 (2000) 4184. [3] P. Parini, et al., Nature (London) 426 (2003) 404. [4] P.R. Berman, Phys. Rev. E 66 (2002) 067603. [5] C. Luo, M. Ibanescu, S.G. Johnson, and J.D. Joannopoulos, Science 299 (2003) 368.

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