THz Bandpass Filter Design Using Metamaterial-Based Defected 1D …
3
defected photonic crystal up to which the value of defect density can be accepted so
that it will not affect the proper filtering action.
3 Mathematical Formulation
Computation of optical bandwidth for photonic filter is a combination of multiple
measures, where in the initial step, transmission profile is calculated in given spectra,
and then, bandwidth can be estimated considering the wavelength difference between
two subsequent notches. In this section, we have shown a detailed computation
procedure.
At first, we consider the normal incidence of e.m wave propagation. Furthermore,
the oblique incidence is considered. The well-known transfer matrix approach is
applicable for both type of polarized incident waves, namely TE polarization, and
TM polarization of light is shown in details in Sect. 3.1, whereas for the unit section,
it is simplified and presented in Sect. 3.2. From the definition, TE-polarized wave
is also called as s-polarized wave; where electric field vector of the propagating
wave is parallel to the stack layers of the structure. TM-polarized light is popularly
known as p-polarized wave; where magnetic field vector is exactly parallel to the
same stack layers, i.e. the electric field is now perpendicular. Therefore, the problem
can now be reformulated as the investigation of characteristics and performance of
electromagnetic field when we consider its propagation through the perpendicular
direction of a multiple stack layer where all the layers are made by different (two in
this case in alternative mode) dielectric media with different refractive indices. The
solution is simple and can easily be derived from Maxwell’s equations by decoupling
it in two independent formats in the two media under consideration and therefore
use of suitable boundary conditions in the terms of electric and magnetic fields at
the interface. Here, we may consider both tangential as well as transverse solutions,
though tangential components are extremely crucial and their continuity property
will determine the final solution.
3.1 Transmission and Reflection Coefficients for Polarized
Wave Incidence
We consider propagation of transverse wave along the z-direction in a liner medium
where ‘n’ is the refractive index. Here for generalization, we can presume that ‘n’
is homogeneous in X–Y plane, but its property should have a dependence along the
field direction, i.e. in this case, it is dependent on Z. Therefore, the wave equation
may be formulated in the following form:
∂
2 E
∂z 2 = −κ
2
0 n
2 E
(3.1)
3
defected photonic crystal up to which the value of defect density can be accepted so
that it will not affect the proper filtering action.
3 Mathematical Formulation
Computation of optical bandwidth for photonic filter is a combination of multiple
measures, where in the initial step, transmission profile is calculated in given spectra,
and then, bandwidth can be estimated considering the wavelength difference between
two subsequent notches. In this section, we have shown a detailed computation
procedure.
At first, we consider the normal incidence of e.m wave propagation. Furthermore,
the oblique incidence is considered. The well-known transfer matrix approach is
applicable for both type of polarized incident waves, namely TE polarization, and
TM polarization of light is shown in details in Sect. 3.1, whereas for the unit section,
it is simplified and presented in Sect. 3.2. From the definition, TE-polarized wave
is also called as s-polarized wave; where electric field vector of the propagating
wave is parallel to the stack layers of the structure. TM-polarized light is popularly
known as p-polarized wave; where magnetic field vector is exactly parallel to the
same stack layers, i.e. the electric field is now perpendicular. Therefore, the problem
can now be reformulated as the investigation of characteristics and performance of
electromagnetic field when we consider its propagation through the perpendicular
direction of a multiple stack layer where all the layers are made by different (two in
this case in alternative mode) dielectric media with different refractive indices. The
solution is simple and can easily be derived from Maxwell’s equations by decoupling
it in two independent formats in the two media under consideration and therefore
use of suitable boundary conditions in the terms of electric and magnetic fields at
the interface. Here, we may consider both tangential as well as transverse solutions,
though tangential components are extremely crucial and their continuity property
will determine the final solution.
3.1 Transmission and Reflection Coefficients for Polarized
Wave Incidence
We consider propagation of transverse wave along the z-direction in a liner medium
where ‘n’ is the refractive index. Here for generalization, we can presume that ‘n’
is homogeneous in X–Y plane, but its property should have a dependence along the
field direction, i.e. in this case, it is dependent on Z. Therefore, the wave equation
may be formulated in the following form:
∂
2 E
∂z 2 = −κ
2
0 n
2 E
(3.1)