A four-layer slab waveguide including LHM is shown in 
), magnetic permeability (
) 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 (
,
and
), magnetic permeability (
,
and
) 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 
that multiplies all the field components is neglected from all equations. Where 
and 
denote angular frequency and longitudinal propagation constant. Usually, a slab waveguide can support TE and TM modes. In this paper,
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
, there exist two cases as follows:
Case 1 
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, … ,
Case 2 
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
. Let 
=
, the transfer matrix 
is rewritten as:
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.
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 
=
into Equation (4). The exact results can be obtained easily.
The total power flux (TPF) is defined as follows [
We know that power fluxes propagate forward along the conventional media and they are all positive, i.e. P0, P2 and P3 > 0. However, in the LHM medium, wave vector is opposite with Ponyting vector, thus, the corresponding power flux is negative, namely, P1 < 0. From a mathematical point of view, in terms of Equation (14), there should exist three cases: 1) P > 0, it means P0 + P2 + P3 > |P1| and is a case for the forward wave; 2) P < 0, it implies P0 + P2 +P3 < |P1| and is a case for the backward wave; 3) P = 0, it means P0 + P2 + P3 = |P1| and electromagnetic waves are stopped and all energy is stored in the waveguide.
4 GHz, 
10 GHz. As frequency 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
, 
, 
and permeability
, respectively. Using Equations (7) and (8), we plot some dispersive curves (the effective-refractive-index verse frequency) and discuss them as follows.
and 
modes) coexist and their dispersion curves are shown in 
mode, as h1 = h2 = 1 cm, its effectiverefractive-index decreases as frequency increases from 4.56 to 4.88 GHz. As h2 fixed and h1 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 h1 = h2 =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 h2 is fixed, and h1 changes, the curves almost overlap with each other. Besides, two types of fundamental modes have a common property, that is, their group velocity 
(
) are both negative. Negative group velocity implies energy propagates backward and reveals the special property in the LHM slab waveguide.
and 
modes coexist and their dispersion curves are plotted in 
mode, its effective-refractive-indexes increase as frequency changing from 4.33 GHz to 4.48 GHz. So, it has positive group velocity. 
mode exists as frequency from 4.14 GHz to 4.60 GHz. The bandwidth is 0.46 GHz. As frequency between 4.49 and 4.60 GHz, its effective-refractive-index has two different values
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.
and 
modes, their TPF curves are shown in 
mode, as h1 < h2, its TPF changes with frequency in a smaller range. However, as h1 = h2 and h1 > h2, 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 
modes, as frequency increases, TPF changes in a small region. For both h1 < h2 and h1 = h2, TPF is positive; for h1 > h2, TPF is negative, and zero TPF doesn’t occur.
and 
modes, their curves of TPF are plotted in 
mode, its TPF decreases with the frequency. But, for 
mode, its TPF increases with frequency, then, two different TPF values exist at the same frequency. It results from double-mode degeneracy.
and 
modes with m from 2 to 7, their TPF curves are plotted along the antihorizontal-axis in 
modes, their TPF decreases as frequency increases. But, they are all positive. For 
modes, at the same frequency, positive and negative TPF coexist. It means that two modes propagate along opposite directions. At maximum frequency, zero TPF can be found for each mode.