Guided Modes in a Four-Layer Slab Waveguide with Dispersive Left-Handed Material

A four-layer slab waveguide including left-handed material is investigated numerically in this paper. Considering left-handed material dispersion, we find eight TE guided modes as frequency from 4 GHz to 6 GHz. The fundamental mode can exist, and its dispersion curves are insensitive to the waveguide thickness. Besides, the total power fluxes of TE guided modes are analyzed and corresponding new properties are found, such as: positive and negative total power fluxes coexist; at maximum value of frequency, we find zero total power flux, etc. Our results may be of benefit to the optical waveguide technology.


Introduction
Since Smith et al. [1] made firstly the left-handed material (LHM) with negative permittivity and negative permeability in microwaves, it has attracted much attention due to their novel electromagnetic properties.Now, negative refraction has been successfully realized in THz waves, and optical waves [2,3].Many scholars [4][5][6] have analyzed symmetric slab waveguide containing LHM.Typical properties of these waveguides including the absence of the fundamental mode, backward propagating waves with negative power flux have been found.The LHM asymmetric slab waveguides and the slab waveguides with LHM cover or substrate have also been investigated [7][8][9].Besides, the five-layer slab waveguides with LHM have been investigated and several new dispersion properties have been discovered [10][11][12].J. Zhang etc. [13] have studied a four-layer slab waveguide with LHM core by using a graphical method.We know that the graphical method can only determine whether or not the mode exists.Furthermore, most above researches are neglecting LHM dispersion.This is not the practical case.
In this paper, the four-layer slab waveguide with LHM in one layer and right-handed materials (RHMs) in the other layers is investigated.The material dispersion of LHM has been considered.Through Maxwell's equations, by using a transfer matrix method, two dispersion equa-tions for the TE guided modes are obtained.Solving these equations, we plot some dispersion curves.Compared these curves, some dispersion properties of TE guided modes are obtained.Besides, power fluxes of TE guided modes are calculated in the waveguide and the corresponding curves are plotted, respectively.From these curves we find some new power flux properties.

Dispersion Equations
A four-layer slab waveguide including LHM is shown in Figure 1.Medium 1 is the LHM, i.e. its dielectric permittivity ( 1  ), magnetic permeability ( 1  ) and refractive index ( ) are all negative.However, the cover (medium 0) and the substrates (media 2 and 3) are different conventional materials, thus, their dielectric permittivity (  ), magnetic permeability ( 0  , 2  and 3  ) and refractive index ( , and ) are all positive.The thicknesses of media 1, 2 is and , respectively.Besides, we assume that media 0 and 3 extend to infinity.For simplicity, the time-and z-factor we study TE guided modes.For TM modes, they will be investigated in other papers.By using Maxwell's equations, the only electric field for TE modes satisfies the following equation: where ,  is the wavelength in vacuum, denotes refractive indexes in media i with = 0, 1, 2 and 3, respectively.For different i n i  , there exist two cases as follows: Case In this case, guided mode fields decay in media 0 and 3, and oscillate in media 1 and 2. We call these modes as the first guided modes and note them .From Equation (1), their electric fields in the slab waveguide are as follows: where A is an undetermined constant, and With continuous conditions of the transverse electromagnetic fields and by using the transfer matrix method, a dispersion equation for mode is obtained as follows: where After some algebraic manipulation, Equation ( 6) can be rewritten as: where m = 0, 1, 2, 3, … , Under this condition, mode fields are oscillating in medium 1 while decay in the other media.We define these modes as the second guided modes and note them

Substituting
' into Equation ( 6), we obtain a dispersion equation for modes where , and m = 0, 1, 2, 3, … Although the forms of two dispersion Equations ( 7) and ( 8) are similar, they have different physical properties.For TM modes, their dispersive equations are similar with that of the corresponding TE modes.But, their magnetic permeability in the equations is replaced by dielectric permittivity.

The Total Power Flux (TPF)
Power fluxes inside the slab waveguide are calculated by an integral of Poynting vector.For TE guided modes, their power flux ( ) in each layer can be obtained through a following equation.
Substituting Equations ( 2)-( 5) into Equation ( 9), after some algebraic manipulation, we have the power fluxes inside the waveguide as follows: where denote power fluxes of the first TE guided modes in media 0, 1, 2 and 3. Similarly, for the second TE guided modes, their power fluxes are obtained by substituting We know that power fluxes propagate forward along the conventional media and they are all positive, i.e.P 0 , P 2 and P 3 > 0. However, in the LHM medium, wave vector is opposite with Ponyting vector, thus, the corresponding power flux is negative, namely, P 1 < 0. From a mathematical point of view, in terms of Equation ( 14), there should exist three cases: 1) P > 0, it means P 0 + P 2 + P 3 > |P 1 | and is a case for the forward wave; 2) P < 0, it implies P 0 + P 2 +P 3 < |P 1 | and is a case for the backward wave; 3) P = 0, it means P 0 + P 2 + P 3 = |P 1 | and electromagnetic waves are stopped and all energy is stored in the waveguide.

The Dispersive Properties of the TE Guided Modes
Material dispersion should be considered because it is one of essential properties of LHM [9].In this paper, we employ an experimental model [8] with dielectric per-mittivity and magnetic permeability being dependent on frequency as: quency increases from 4 GHz to 6 GHz, its dielectric permittivity and magnetic permeability become negative simultaneously.For simplicity, we assume that waveguide thickness of media 2 is fixed and equals to 1 cm.For other media, their permittivity is , respectively.Using Equations ( 7) and ( 8), we plot some dispersive curves (the effective-refractive-index verse frequency) and discuss them as follows.

TE
As m ＝ 0, two guided modes ( and modes) coexist and their dispersion curves are shown in Figure 2. It is a unique property of the waveguides considering left-handed material dispersion.If neglecting material dispersion, we find the absence of the fundamental mode [6].For mode, as h 1 = h 2 = 1 cm, its effectiverefractive-index decreases as frequency increases from 4.56 to 4.88 GHz.As h 2 fixed and h 1 modified (from 0.1 cm to 10 cm), the curves coexist in two frequency regions from 4.735 to 4.88 GHz and 4.835 to 4.88 GHz, respectively.Especially, as frequency is between 4.843 to 4.88 GHz, their dispersion curves are almost overlap.For mode, as h 1 = h 2 =1 cm, its effective-refractiveindex decreases with frequency increasing from 4.14 GHz to 4.735 GHz.The bandwidth is 0.595 GHz.On the contrary, if h 2 is fixed, and h 1 changes, the curves almost overlap with each other.Besides, two types of fundamental modes have a common property, that is, their group velocity ( 0 ) are both negative.Negative group velocity implies energy propagates backward and reveals the special property in the LHM slab waveguide.corresponding to the same frequency i.e. double-mode degeneracy.This is because the dispersion equation has two different solutions at the same frequency.This property can be found in other LHM slab waveguides [4,6].Besides, its positive and negative group velocities coexist.

The Higher Order TE
2) As m increases from 2 to 7, there exist six TE guided modes and their dispersion curves are plotted in Figure 3.For the same m, two types of TE guided modes exist and their curves keep continuous.As m increases, their curves shift to left and their cutoff frequencies be-come less.This is different from that of omitting materials dispersion [6].For the first type modes, their group velocities are positive.However, for the second type of modes, their double-mode degeneracy appears and their positive and negative group velocities coexist.

The Total Power Flux (TPF) of TE Guided Modes
Employing Equations ( 10)-( 14) and dispersion Equations ( 7) and ( 8), we choose the same parameters as Subsection 2.1.The curves of the TPF versus frequency for TE guided modes are plotted in Figures 4 and 5, respectively.The results are as follows: ) curves represent h 1 = 0.1 cm, 1 cm and 10 cm, respectively.Clearly, they have a common property that their TPF becomes small with h 1 increased.This is because their power fluxes in the LHM medium increases with h 1 , and they are negative.This makes TPF small and even negative with the increase of h 1 .For 0 TE Ⅰ mode, as h 1 < h 2 , its TPF changes with frequency in a smaller range.However, as h 1 = h 2 and h 1 > h 2 , its TPF changes with frequencies in a bigger range.Furthermore, the TPF is positive, negative, and zero at different frequencies.Zero TPF implies that electromagnetic waves are stopped in the waveguide.This property may have some potential applications in the optical waveguide technology.For Ⅱ 0 TE modes, as frequency increases, TPF changes in a small region.For both h 1 < h 2 and h 1 = h 2 , TPF is positive; for h 1 > h 2 , TPF is negative, and zero TPF doesn't occur.total power flux for TE guided modes is calculated and its corresponding curves are plotted.Through these curves, we find some new properties, such as: positive and negative total power fluxes coexist.At maximum frequency, we find zero total power flux.This property may find some potential applications in the optical waveguide technology.

2 Figure 1 .
Figure 1.The geometry for a four-layer slab waveguide including left-handed material . The exact results can be obtained easily.The total power flux (TPF) is defined as follows[8

3 Figure 4 .Figure 5 .Figure 4 ,
Figure 4.The total power flux of the fundamental TE mode for different slab thicknesses.The parameters are the same as Figure 2. The dashed curves stand for modes, the solid curves correspond to modes modes, their curves of a plotted in Figure5TPF re .these curves, we find that the TPF of the mer is er than that of the latter and they are both positive.For 1 TE Ⅰ mode, its TPF decreases with the frequency.But, f Ⅱ 1 TE mode, its TPF increases with frequency, then, two diffe ent TPF values exist at the same frequency.It result m double-mode degeneracy.2) For m TE Ⅰ and m TE Ⅱ modes with m from 2 to 7, their TPF cu From bigg for or rv gu es are d along the anti-horizonl-axis in Fi re 5. T rmer is always bigger than the latter.For m TE Ⅰ modes, their TPF decreases as frequency increases.But, they are all positive.For m TE Ⅱ modes, at the same frequency, positive and negative TPF coexist.It means that two modes propagate along osite directions.At maximum frequency, zero TPF can be found for each mode.
layer slab w RHMs in other lay dispersion equations of two types of the TE guided modes are obtained and dispersion curves are plotted.Compare these curves, we find some dispersion properties of TE modes, such as: two types of the fundamental modes exist, moreover, in some frequency regions, they are insensitive to the waveguide thickness.Besides, the key project of the Nation ina (60937003) and STCSM REFERENCES [1] D. R. Smith, W et al., "Composite Medium with eability and elengths in a Two-Dimensional at Optical Wavelengths," Science, Vol. e Index Wavee Num-.Y. Luo, "Surface Modes .Xiao and Z. H. Wang, "Dispersion Characteristics of Shen, "Guided Optical Modes in nces with Zhang and C. F. Li, "Guided Modes in a Symnd Z. H. Wang, S. P. Li, "Propagation Propoupling Characteristics d F. M. Zhang, "Guided