Page 455 - House of Maxell

MORE COMPLICATED ELEMENTS OF FEED LINES 435

1 +1 1 1 +1

� � = � / +1 � 11 � � � = � 11 � � � (8.17)

2 1 +1 1+ 11 1 +1
�1−|
11 | 11 11
The next step is to take into consideration that the output incident +1 and reflected +1 waves
should be defined on (i+1)-interface, i.e. shifting the reference plane for them as +1 →
+1 and +1 → +1 − where = . Therefore, the scattering transfer matrix

relating waves on i- and (i+1)-interface can be written in the form

1 −
11
= � � (8.18)

1+ 11 −
11
+1 = − . If so,

On the next (i+1)-boundary where +2 > +1 we evidently have 11 11

1 − −
11
= � � (8.19)
+1
1− 11 − −
11
According to (7.23), the product of these two matrices is the transfer matrix of single bilayer,
i.e. = . As soon as the dielectric stack is formed by N identical bilayers, its transfer
+1
matrix yields = ( ) where ( ) is the matrix product of N copies of . Completing

all multiplications analytically or numerically, we could obtain the transparency characteristics
⁄
of multilayer thin-film stack analyzing the matrix component 22 = 1 in the frequency
21
domain (look back at expression (7.19)). However, more urgent task is not such simple analysis.
The manufacturing team is able to produce the filter but only if you tell them how many layers
is required and what the thickness and refractive index of each of them are. Furthermore, these
refractive indexes must belong to the real materials in Table 8.4 or some extended one that
makes the optical filter synthesis very challenging not only analytically but numerically too.
According to (8.18) and (8.19), we have only three independent parameters for
optimization: , , and the problem of synthesis does not look difficult. However, the

11
number of layers might reach hundreds and that is a real problem. We can simplify the byline
synthesis assuming that the electrical thickness of each layer is close to a quarter-wavelength,
i.e. ≅ ≅ /2, and thus the byline is the half-wavelength. Then each byline can play a

role of in-line resonator we have analyzed at the beginning of the current chapter. If so, such
optical filter is the cascade of resonators with equal loaded Q-factor. That explains the fact that
the transfer matrix of such filter is the transfer matrix of single byline aka resonator raised to
power N. Recall the far-reaching analogy between the formation of the filter transfer function
in the frequency domain and the development of radiation pattern of the linear antenna in space.
At the heart of both phenomena is the EM wave interference. In Chapter 5 we have
demonstrated that a linear antenna uniformly excited has relatively high sidelobe level. The
similar effect can be expected in filters with the same Q-factor, i.e. the excessive level of ripples
in the stopband. Figure 8.4.16d on the next page confirms this fact perfectly well. The central
wavelength of 10 bylines was chosen 1490 nm (green arrow). The passband characteristic is
almost flat, but the peak oscillation in the stopband reaches -2dB. Nevertheless, such filters are
simple in production and thus low-cost, as their fabrication requires a minimum set of dielectric
materials (only two).