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).

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).