2.1. Waves at an interface

To develop the TMM we must first know how the electromagnetic waves behave at the interface between two dielectrics. For plane waves at an interface, the electric and magnetic fields are given by

respectively. The two media separated by the interface will be characterized by permittivities ϵ1, ϵ2 and permeabilities μ1, μ2, and the geometry of the wave vectors at the interface is illustrated in Fig. (1). In this figure, we assume that the wave is propagating along the z-axis and the wavevector has two components, k⃗=(kx,0,kz).

Because the condition of continuity of the tangential components of both the electric and magnetic fields

must be satisfied for any point of the interface, the tangential components of k⃗ have to be equal

while the dispersion relation n2ω2/c2=k2 as written for the two media leads to

If k _x and k _z are real, we can define the angles of incidence and refraction as

and

respectively. From Eq. (4), we can see that

and if we apply the dispersion relationship, we obtain

If we now take the ratio of the last two equations we obtain Snell’s law

Considering the case of the TE polarization (s-type polarization), then E⃗ is parallel to the interface, i.e., we have

and

We also have the boundary condition that says that the component of the electric field which is parallel to the interface is the same on both sides of the interfaces

which holds for the magnetic field

as well as.

Using Maxwell’s equation

in Eq. (15), we obtain

Inserting (17) in (15) leads to

Now we can write Equations (19) and (14) in matrix form

or

where the transfer matrix of the interface

is introduced.

By a similar procedure, we can obtain the transfer matrix for the case of the TM polarization (p-type polarization)

For materials with only real dielectric constants we can express the wave vectors as

so the transfer matrices for the TE and TM cases are written as

where z21=μ2k1/μ1k2 for the s-polarization and z21=ϵ2k1/ϵ1k2 for the p-polarization.

2.2. The transfer matrix for a slab

For a set of interfaces, the systems behaves as a slab (sandwich of media). A simple example is given in Fig. (2).

For such systems, it is enough to apply the composition law for transfer matrices ³. In this way, the relationship of the coefficients of entry and exit is given by

where l is the length of the slab in the direction of propagation.

From the last equation, one can obtain the transfer matrix for the slab as

which can be used for the generalization to the multilayer case.

2.3. Wave propagation through a multilayered system

The next step is to consider a multilayered system as illustrated in Fig. (3).

For a system of this class, the transfer matrix is obtained by applying the composition law again. In this way, we have an interface matrix M ^i,i+1 for every interface and a propagation matrix of the form

for every slab in which the electromagnetic wave propagates. In this manner, for a ten multilayer slab system (5 A and 5 B layers) the transfer matrix is written as

where M _0X is the interface matrix between air and medium X, P _X is the propagation matrix of medium X, and M _XY is the interface matrix between medium X and medium Y. The M and M _XY matrices are transfer matrices for the s-polarization or the p-polarization, depending on the nature of the incident wave.

Once the transfer matrix for the multilayered system has been defined, one can proceed with the calculation of the transmission amplitude based on its definition for the case of electromagnetic waves given by ³

where det M stands for the determinant of M, and the transmittance as its square modulus

3.1. The effect of the number of layers

We begin by analyzing the effect of the number of layers upon the transmission spectrum. With this task in mind, we vary the number of periods “AB” crossed by the propagating wave using the values of the parameters given in Table I, where the parameters ϵ0 and μ0 correspond to the respective values for the relative permittivity and permittivity of the medium (usually air) in front and at the end of the “AB” multilayer structure. We call the “AB” period a block. In Fig. (7), the number of blocks is increased from two to five, (in terms of interfaces, from five to eleven), and each panel is labeled by the corresponding number of blocks. The regions of high transmittance show up as three symmetrical bubbles. Upon increasing the number of blocks in the system, one can see that the number of peaks inside each of the bubbles increases. The number of peaks, npeaks, is equal to twice the number of blocks, n _b , minus one, npeaks=2nb-1. This might be seen alike to the case of quantum systems if the number of quantum wells is put in correspondence with the number of peaks in the transmittance. One can also see that when the number of blocks increases the transmittance between the bubbles of high transmittance becomes smaller. This is a consequence of the change in the transmittance values that one can see in Fig. (4), where we can also notice that the transmittance bands get wider as the number of blocks increases. Moreover, in Fig. (7), we can see that we have a bubble of high transmittance inside the limits of the visible frequencies (black horizontal lines), which could be useful for camouflage purposes. In addition, as we increase the number of blocks, this high transmittance region entails a wider range of angular values. On the other hand, if we increase the number of blocks enough the simulation ends up in a completely opaque material with zero transmittance, as expected.

3.2. The effect of ϵ+ and ϵ-

Knowing that the number of blocks generates a series of transmittance bands, one naturally may ask if there is a way to control the width of these bands by means of the electric permittivities of the slabs. The values of the parameters that we use for this computation are given in Table II.

The results are presented in Fig. (8). We can see that the effect of increasing ϵ+ while keeping ϵ- fixed at -1.0, is to tighten the bubbles of high transmittance, and the magnitude of this effect is proportional to the increase in ϵ+. When ϵ+ is high enough, the bubbles lose the connection between them and become independent ovals. For this case, we see that the effect is similar for both the TE and TM modes, but for the TM mode the bubbles are more separated and there appears a series of smaller bubbles of high transmittance. Moreover, for both types of modes there is a shift to lower frequencies of these bubbles of high transmittance. This can be seen in the left panel of Fig. (5).

On the other hand, if we increase the value of |ϵ-| while keeping ϵ+ fixed at 2.0 we can see from Fig. (9) that the bubbles of high transmittance are absent and in their place a wide band with horizontal spikes occurs. Therefore, we conclude that the occurrence of bubbles depend on the chosen values of ϵ+. When ϵ- increases, we can see that the wide band is fragmented in three sections, and the separation between sections is proportional to the increment in |ϵ-|. In this case, there is no shift to lower frequencies. This fact can be observed in the right panel of Fig. (5). The effect is similar for both TE and TM modes, but in the TM case the decrease in transmittance is less but in a wider range of frequencies.

In general, we can say that the value of ϵ+ determines the form of the transmittance bands and the frequency range where they occur. On the other hand, the value of ϵ- determines the width of the horizontal transmittance spikes.

3.3. The effect of μ+ and μ-

We move now to the study of the effects of increasing the values of the permeabilities of the system. For this task, we use the values from Table III for the parameters under control.

From Fig. (6), we see that if we increase the magnitude of μ+ the bubbles of high transmittance have a shift to lower frequencies which is proportional to the magnitude of μ+. Unlike the case of varying ϵ+, we see that the bubbles remain connected, and as the value of μ+ increases the contact area between bubbles increases as well as. This is opposite to the case of varying ϵ+ and occurs for both the TE and TM modes.

From the left panel of Fig. (11), we can appreciate easily a shifting effect to lower frequencies. Also, and even more interesting, we see that if we increase the value of μ+ the valleys (regions of low transmittance) become less apparent. Thus, increasing μ+ has the effect of rising the valleys of transmittance diminishing the relative borders of the transmittance bands.

In the case of increasing the values of |μ-|, Fig. (11) shows that the main effect is to diminish the transmittance between the three regions of high transmittance. This effect is also visible on both the TE and TM modes. However, in this case, we notice that there is no shifting effect, which can be seen easily from the right panel of Fig. (6).

In addition, Figs. (7)-(11) show that the regions of high transmittance do not vary with the angle of incidence, which may be related to the independence of the polarization angle of the incident wave as also reported in ⁵. Considering that the reflectance and transmittance are complementary properties, our findings regarding the transmittance spectrum seem to be in agreement with those reported in ⁶ for an Ag-TiO₂ multilayer system, which is the standard multilayer candidate with high transmittance in the visible region.