## 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 isotropicdielectric layer with the permittivity, permeability, and thickness, and  is a uniaxial indefinitemetamaterial with thickness. N is the period number, and a plane wave is incident at an angle  upon the 1DPC from air.

Theinterfaces 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, thepermittivity and permeability tensors of the indefinite metamaterial medium aregiven by 24, 33, Fig.

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1. Schematic of proposed ofthe 1DPC consisting of alternate layers of isotropic material (B) and uniaxialindefinite metamaterial (A), and  is thenumber of periods.             ,                                                                       (1)Where,                                                     (2) Here,  ,    ,   and   are the principle elements ofthe permittivity and permeability tensors of the layer  along the optical axis andperpendicular to the optical axis, respectively, and   is the angle between the opticalaxis and the -axis. The permittivity and permeability of layer A are complex given by34, 35:                                                                                                                         (3) Where   is the angular frequency of theincident wave, and is measured in units of (109 rad ? s).Consider anelectromagnetic 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 givenby the following Maxwell equations:                                                                                                                       (4)where  and  is the relative permittivity and permeability tensors, which, for anisotropic metamaterial witharbitrary optical axis  is described  Eq. (1).

At first, we focus only on the TEwaves. According to the Maxwell equations, the electric field  inside the indefinite layersatisfies the wave equation:,                                                                                   (5) where  is the vacuum wave vector. Byimposing the continuity condition on   and at the interfaces and introducing a wave function as, ,                                                                                                                                     (6)    The following relation is derived betweenthe electric and magnetic fields at any two positions  and  of the same medium:                                                                                                                     (7) here,  is the transfer matrix of theindefinite medium,                                                                         (8)                             where,     and    .Similar results can beobtained for the isotropic layer :          ,                                                                          (9)                                                                          where  is the   component of the wave vector inthe medium B , and c is the light speed in vacuum, and .For the waves, the wave equation in the metamaterial layer  can be obtained similarlyas:                                                    ,                                                                              (10)                                                               here,  is the transfer matrix of theindefinite medium for TM polarization:   ,                                                                      (11)                                                             Where,     and    . By means of the transfermatrix method 29, we obtain the transmission of the structure as,   ,                                                                                          (12)                                                                               where    are the elements of the totalmatrix  , 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 isotropicdielectric layer with the permittivity, permeability, and thickness, and  is a uniaxial indefinitemetamaterial with thickness. N is the period number, and a plane wave is incident at an angle  upon the 1DPC from air. Theinterfaces 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, thepermittivity and permeability tensors of the indefinite metamaterial medium aregiven by 24, 33, Fig. 1. Schematic of proposed ofthe 1DPC consisting of alternate layers of isotropic material (B) and uniaxialindefinite metamaterial (A), and  is thenumber of periods.             ,                                                                       (1)Where,                                                     (2) Here,  ,    ,   and   are the principle elements ofthe permittivity and permeability tensors of the layer  along the optical axis andperpendicular to the optical axis, respectively, and   is the angle between the opticalaxis and the -axis. The permittivity and permeability of layer A are complex given by34, 35:                                                                                                                         (3) Where   is the angular frequency of theincident wave, and is measured in units of (109 rad ? s).

Consider anelectromagnetic 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 givenby the following Maxwell equations:                                                                                                                       (4)where  and  is the relative permittivity and permeability tensors, which, for anisotropic metamaterial witharbitrary optical axis  is described  Eq. (1). At first, we focus only on the TEwaves. According to the Maxwell equations, the electric field  inside the indefinite layersatisfies the wave equation:,                                                                                   (5) where  is the vacuum wave vector.

Byimposing the continuity condition on   and at the interfaces and introducing a wave function as, ,                                                                                                                                     (6)    The following relation is derived betweenthe electric and magnetic fields at any two positions  and  of the same medium:                                                                                                                     (7) here,  is the transfer matrix of theindefinite medium,                                                                         (8)                             where,     and    .Similar results can beobtained for the isotropic layer :          ,                                                                          (9)                                                                          where  is the   component of the wave vector inthe medium B , and c is the light speed in vacuum, and .For the waves, the wave equation in the metamaterial layer  can be obtained similarlyas:                                                    ,                                                                              (10)                                                               here,  is the transfer matrix of theindefinite medium for TM polarization:   ,                                                                      (11)                                                             Where,     and    . By means of the transfermatrix method 29, we obtain the transmission of the structure as,   ,                                                                                          (12)                                                                               where    are the elements of the totalmatrix  , and  for the surrounding medium (air).

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