We permeability tensors of the indefinite metamaterial

We  consider
a  1D  PC
with  the  periodic
structure   embedded  in
air,  as  shown
in  Fig.  1.
Here,    represents an isotropic
dielectric layer with the permittivity, permeability, and thickness, and  is a uniaxial indefinite
metamaterial with thickness. N is the period number, and a plane wave is incident at an angle  upon the 1DPC from air. The
interfaces of the layers are parallel to the
plane, and the  axis is normal to the structure.
We assume that the optical axis of the indefinite medium lies in the  plane and makes angle  with the  axis. In this case, the
permittivity and permeability tensors of the indefinite metamaterial medium are
given by 24, 33,

Fig. 1. Schematic of proposed of
the 1DPC consisting of alternate layers of isotropic material (B) and uniaxial
indefinite metamaterial (A), and  is the
number of periods.

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,
(1)

Where

,
(2)

Here,  ,    ,   and   are the principle elements of
the permittivity and permeability tensors of the layer  along the optical axis and
perpendicular to the optical axis, respectively, and   is the angle between the optical
axis and the -axis. The permittivity and permeability of layer A are complex given by
34, 35:

(3)

Where   is the angular frequency of the
incident wave, and is measured in units of (109 rad ? s).Consider an
electromagnetic wave with frequency of, electric and magnetic fields of  and, respectively, incident to the structure with angle  with respect to the -axis. The fundamental equations for an electromagnetic wave are given
by the following Maxwell equations:

(4)

where  and  is the relative permittivity and permeability tensors, which, for anisotropic metamaterial with
arbitrary optical axis  is described  Eq. (1). At first, we focus only on the TE
waves. According to the Maxwell equations, the electric field  inside the indefinite layer
satisfies the wave equation:

,
(5)

where  is the vacuum wave vector. By
imposing the continuity condition on   and at the interfaces and introducing a wave function as,

,
(6)

The following relation is derived between
the electric and magnetic fields at any two positions  and  of the same medium:

(7)

here,  is the transfer matrix of the
indefinite medium,

(8)

where,     and    .Similar results can be
obtained for the isotropic layer :

,
(9)

where  is the   component of the wave vector in
the medium B , and c is the light speed in vacuum, and .For the waves, the wave equation in the metamaterial layer  can be obtained similarly
as:

,
(10)

here,  is the transfer matrix of the
indefinite medium for TM polarization:

,
(11)

Where,     and    . By means of the transfer
matrix method 29, we obtain the transmission of the structure as,

,
(12)

where    are the elements of the total
matrix  , and  for the surrounding medium (air).

We  consider
a  1D  PC
with  the  periodic
structure   embedded  in
air,  as  shown
in  Fig.  1.
Here,    represents an isotropic
dielectric layer with the permittivity, permeability, and thickness, and  is a uniaxial indefinite
metamaterial with thickness. N is the period number, and a plane wave is incident at an angle  upon the 1DPC from air. The
interfaces of the layers are parallel to the
plane, and the  axis is normal to the structure.
We assume that the optical axis of the indefinite medium lies in the  plane and makes angle  with the  axis. In this case, the
permittivity and permeability tensors of the indefinite metamaterial medium are
given by 24, 33,

Fig. 1. Schematic of proposed of
the 1DPC consisting of alternate layers of isotropic material (B) and uniaxial
indefinite metamaterial (A), and  is the
number of periods.

,
(1)

Where

,
(2)

Here,  ,    ,   and   are the principle elements of
the permittivity and permeability tensors of the layer  along the optical axis and
perpendicular to the optical axis, respectively, and   is the angle between the optical
axis and the -axis. The permittivity and permeability of layer A are complex given by
34, 35:

(3)

Where   is the angular frequency of the
incident wave, and is measured in units of (109 rad ? s).Consider an
electromagnetic wave with frequency of, electric and magnetic fields of  and, respectively, incident to the structure with angle  with respect to the -axis. The fundamental equations for an electromagnetic wave are given
by the following Maxwell equations:

(4)

where  and  is the relative permittivity and permeability tensors, which, for anisotropic metamaterial with
arbitrary optical axis  is described  Eq. (1). At first, we focus only on the TE
waves. According to the Maxwell equations, the electric field  inside the indefinite layer
satisfies the wave equation:

,
(5)

where  is the vacuum wave vector. By
imposing the continuity condition on   and at the interfaces and introducing a wave function as,

,
(6)

The following relation is derived between
the electric and magnetic fields at any two positions  and  of the same medium:

(7)

here,  is the transfer matrix of the
indefinite medium,

(8)

where,     and    .Similar results can be
obtained for the isotropic layer :

,
(9)

where  is the   component of the wave vector in
the medium B , and c is the light speed in vacuum, and .For the waves, the wave equation in the metamaterial layer  can be obtained similarly
as:

,
(10)

here,  is the transfer matrix of the
indefinite medium for TM polarization:

,
(11)

Where,     and    . By means of the transfer
matrix method 29, we obtain the transmission of the structure as,

,
(12)

where    are the elements of the total
matrix  , and  for the surrounding medium (air).

x

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