6. Error in time reversing and a related theorem

Based on the equivalence of the TRT and the properties of the Green function, we can trust that any discussion about the interaction of metamaterials with electromagnetic field can be done through this function and simultaneously observe the effect of a time reversal. For this reason, we can now describe the error in terms of the Green function by the hypothesis that LHM can be put to test by forward and backward in time signals and read the results with two points of view: first, the direct effect of the loss of information because of the limited record time T or second, how the negative refraction index helps to preserve information. Now, we can review our previous results and generalize using the kernels, so we can characterize the capacity of a channel in many different circumstances. So, we have made use of the analogies [30] between the TRT and the employment of LHM to propose that we can express the capacity of any of these negative refraction index materials in the same terms or procedures as those of TRT. Also, we can propose an identical description for the channel capacity that is Eq. (24) and its generalization Eqs. (25) and (30). Then, the matrix formalism for discrete systems can be used to characterize the channel capacity of transmission of information in a process of time reversibility using the Fourier transforms of the Green functions (properly we use the kernels with the interaction matrix ) forward and backward. That is, by the first step, the signal transforms like (in the following equations I and F stand for initial and final places):

$$\mathbf{Y}\_F = [\mathbf{1} + \mathbf{R}(\rho)]\mathbf{X}\_I \tag{31}$$

In Eq. (37), the function has the form of the Fourier transform of

But with the time running backward, so, as we will show in a moment, if T is very short, the error will be very huge. On the contrary, if the time goes to infinity, the error will go to zero. Resuming, the new Eqs. (33)–(38), make possible a characterization of the lost information in left-hand materials not only for microwave range, but also for visible frequencies because we have extended recently the

and because the kernel of the Fourier transform of the generalized inhomoge-

While Eq. (41) exactly represents the problem with a finite recording time T, Eq. (42) represents a hypothetical problem in which the recording time is infinite.

Then, we can suppose that the two kernels in Eq. (40) represent the real and the

hypothetical problem described above. Of course, we see that if real conditions approximate the ideal ones, the error is clearly zero. But we can factorize the

But Eq. (44) says clearly that the error does not depend on the form of the interaction, only depends on the recording time T. Even we have supposed that the only source of error was the recording time, we do not suppose any particular behavior for the interaction. So, we have enunciated and proved a theorem:

In the time-reversal problem and for left-hand material conditions, the normal-

neous Fredholm's equation (GIFE) satisfies the following integral equations:

time-reversal techniques (see ref. [3, 12]).

Optimum Efficiency on Broadcasting Communications DOI: http://dx.doi.org/10.5772/intechopen.84954

So, we can write Eq. (37) like:

Substituting Eq. (41) into Eq. (40), we have:

interaction matrix in Eq. (43):

Theorem III.

ized error:

31

Now, we can define:

ð38Þ

ð39Þ

ð40Þ

ð41Þ

ð42Þ

ð43Þ

ð44Þ

the Green function but with the argument translated by an amount equal to the recording time that appears explicitly in Eq. (19) that is the Fourier transform of:

then in the second step, it returns to the initial place by means of the operation.

$$\mathbf{Z}\_I(\boldsymbol{\phi}) = [\mathbf{1} - \mathbf{K}^{(\circ)}(\boldsymbol{\phi})] \mathbf{Y}\_F(\boldsymbol{\phi}) \tag{32}$$

Then, the complete signal trip would be:

$$\mathbf{Z}\_I(\boldsymbol{\alpha}) = [\mathbf{1} - \mathbf{K}^{(\circ)}(\boldsymbol{\alpha})][\mathbf{1} + \mathbf{R}(\boldsymbol{\alpha})] \mathbf{X}\_I(\boldsymbol{\alpha}) \tag{33}$$

So that by defining the error in the time-reversing process by:

$$
\delta \mathbf{X}\_I = \mathbf{X}\_I - \mathbf{Z}\_I \tag{34}
$$

We can write this like:

$$\delta \mathbf{X}\_{\prime}(\boldsymbol{\alpha}) = \mathbf{X}\_{\prime}(\boldsymbol{\alpha}) - [\mathbf{1} - \mathbf{K}^{(\circ)}(\boldsymbol{\alpha})][\mathbf{1} + \mathbf{R}(\boldsymbol{\alpha})] \mathbf{X}\_{\prime} \tag{35}$$

or

$$\delta \mathbf{X}\_{\prime}(\boldsymbol{\phi}) = -[\mathbf{R}(\boldsymbol{\phi}) - \mathbf{K}^{(\circ)}(\boldsymbol{\phi}) - \mathbf{K}^{(\circ)}(\boldsymbol{\phi})\mathbf{R}(\boldsymbol{\phi})] \mathbf{X}\_{\prime}(\boldsymbol{\phi})\tag{36}$$

Eq. (36) is a corollary that shows explicitly the role of both the forward and backward Fourier transforms of the Green function (we have done on Eq. (8) for convenience and also for the complete kernels and ). Eq. (36) is very clear about the origin of the errors because we can see, for example, that in the case that the forward and backward Green functions are mathematically one the transpose conjugated of the other for a perfect time reversal (when acting the first on a column vector and on a row vector the other), we get that the error is zero and that the error increases as the differences of both functions also increase. In a very special case, we can then propose that and only differ by the factor or when the only source of error is the recording time , so that we obtain from Eq. (36) that:

$$\delta \mathbf{X}\_{\prime}(\boldsymbol{\alpha}) = -\left[e^{-2\pi i \frac{\boldsymbol{a}\cdot\boldsymbol{a}}{\boldsymbol{\alpha}\_{\prime}}} \mathbf{K}(\boldsymbol{\alpha}) - \mathbf{K}^{(\circ)}(\boldsymbol{\alpha}) - \mathbf{K}^{(\circ)}(\boldsymbol{\alpha}) e^{-2\pi i \frac{\boldsymbol{a}\cdot\boldsymbol{a}}{\boldsymbol{\alpha}\_{\prime}}} \mathbf{K}(\boldsymbol{\alpha})\right] \mathbf{X}\_{\prime}(\boldsymbol{\alpha})\qquad(37)$$

In Eq. (37), the function has the form of the Fourier transform of the Green function but with the argument translated by an amount equal to the recording time that appears explicitly in Eq. (19) that is the Fourier transform of:

$$\mathbf{K}(t - T) \tag{38}$$

But with the time running backward, so, as we will show in a moment, if T is very short, the error will be very huge. On the contrary, if the time goes to infinity, the error will go to zero. Resuming, the new Eqs. (33)–(38), make possible a characterization of the lost information in left-hand materials not only for microwave range, but also for visible frequencies because we have extended recently the time-reversal techniques (see ref. [3, 12]).

Now, we can define:

use of the analogies [30] between the TRT and the employment of LHM to propose that we can express the capacity of any of these negative refraction index materials in the same terms or procedures as those of TRT. Also, we can propose an identical description for the channel capacity that is Eq. (24) and its generalization Eqs. (25) and (30). Then, the matrix formalism for discrete systems can be used to characterize the channel capacity of transmission of information in a process of time reversibility using the Fourier transforms of the Green functions (properly we use the kernels with the interaction matrix ) forward and backward. That is, by the first step, the signal transforms like (in the following equations I and F stand for

Telecommunication Systems – Principles and Applications of Wireless-Optical Technologies

then in the second step, it returns to the initial place by means of the operation.

ð31Þ

ð32Þ

ð33Þ

ð34Þ

ð35Þ

ð36Þ

ð37Þ

initial and final places):

We can write this like:

obtain from Eq. (36) that:

or

30

Then, the complete signal trip would be:

So that by defining the error in the time-reversing process by:

Eq. (36) is a corollary that shows explicitly the role of both the forward and backward Fourier transforms of the Green function (we have done on Eq. (8) for convenience and also for the complete kernels and ). Eq. (36) is very clear about the origin of the errors because we can see, for example, that in the case that the forward and backward Green functions are mathematically one the transpose conjugated of the other for a perfect time reversal (when acting the first on a column vector and on a row vector the other), we get that the error is zero and that the error increases as the differences of both functions also increase. In a very special case, we can then propose that and only differ by the factor or when the only source of error is the recording time , so that we

$$\mathcal{K}\_- = e^{-2\pi i \frac{a}{\omega\_\eta}} \mathbf{K} \tag{39}$$

So, we can write Eq. (37) like:

$$\delta \mathbf{X}\_{\prime}(\boldsymbol{\omega}) = -[\mathcal{K}^{\prime}(\boldsymbol{\omega}) - \mathbf{K}^{(\circ)}(\boldsymbol{\omega}) - \mathbf{K}^{(\circ)}(\boldsymbol{\omega})\mathcal{K}^{\prime}(\boldsymbol{\omega})] \mathbf{X}\_{\prime}(\boldsymbol{\omega})\qquad(40)$$

and because the kernel of the Fourier transform of the generalized inhomogeneous Fredholm's equation (GIFE) satisfies the following integral equations:

$$\mathcal{K}\_- = \mathbf{K}^{(\circ)} + \mathbf{K}^{(\circ)} \mathcal{K} \tag{41}$$

$$\mathcal{K}\_- = \mathcal{K}\_-{(\circ)} + \mathcal{K}\_-{(\circ)}\mathcal{K}\_- \tag{42}$$

While Eq. (41) exactly represents the problem with a finite recording time T, Eq. (42) represents a hypothetical problem in which the recording time is infinite. Substituting Eq. (41) into Eq. (40), we have:

$$\delta \mathbf{X}\_{\prime}(\boldsymbol{\alpha}) = -[\mathcal{K}\_{\cdot}(\boldsymbol{\alpha}) - \mathcal{K}\_{\cdot}(\boldsymbol{\alpha})] \mathbf{X}\_{\prime}(\boldsymbol{\alpha}) \tag{43}$$

Then, we can suppose that the two kernels in Eq. (40) represent the real and the hypothetical problem described above. Of course, we see that if real conditions approximate the ideal ones, the error is clearly zero. But we can factorize the interaction matrix in Eq. (43):

$$\delta \mathbf{X}\_{\prime}(\boldsymbol{\omega}) = -\mathbf{V}[\mathcal{G}(\boldsymbol{\omega}) - \mathcal{G}(\boldsymbol{\omega})] \mathbf{X}\_{\prime}(\boldsymbol{\omega}) \tag{44}$$

But Eq. (44) says clearly that the error does not depend on the form of the interaction, only depends on the recording time T. Even we have supposed that the only source of error was the recording time, we do not suppose any particular behavior for the interaction. So, we have enunciated and proved a theorem:

Theorem III.

In the time-reversal problem and for left-hand material conditions, the normalized error:

$$\frac{\delta \mathbf{X}\_f(o)}{\|\mathbf{V}\|}\tag{45}$$

Then, each function has changed individually, but the ensemble as a whole is invariant under the transformation. Also, if we apply the operator T which gives for

It is possible to prove that if T is an invariant operator and the input ensemble Fαð Þt is stationary, the output ensemble Sαð Þt is also stationary. Now, for communication purposes, the operator T, which could be a modulation process, is not invariant because of the phase carrier that gives certain time structure, but if the translations are multiples of the periods of the carrier, then the modulation will be invariant. At this stage, it is important to remember that Wiener [6] has pointed out that if a device is linear as well as invariant (in the sense of the last definition), then the Fourier analysis is the appropriate mathematical tool for dealing with the problem. Now, suppose in addition that we are interested on functions that are limited to the band from 0 to Θ cycles per second, then we have the following theorem [10]:

> sin πð Þ 2Θt � n πð Þ 2Θt � n

sin ½ � πð Þ 2ωet � n πð Þ 2ωet � n

> n 2ω<sup>e</sup>

Every ω<sup>e</sup> allows us to build a decomposition like (54) but we expect that only a few terms are necessary for a well representation of Feð Þt . Next, we send separately each Feð Þt by its own device and it is all we need for broadcasting. To receive the

(53)

(55)

(54)

Let F tð Þ contain no frequencies over Θ. Then:

Optimum Efficiency on Broadcasting Communications DOI: http://dx.doi.org/10.5772/intechopen.84954

> F tðÞ¼ ∑ ∞ �∞ Xn

FeðÞ¼ t ∑

signal, we need a separate device for each ωe.

∞ �∞ Xn,e

Xn,e ¼ Fe

(52) where,

Xn <sup>¼</sup> <sup>F</sup> <sup>n</sup>

In this expansion, F tð Þ is represented as a sum of orthogonal (basis) functions. The coefficients Xn of the various terms can be considered as coordinates in an infinite dimensional "functions space." We will take the last theorem (Eqs. (52) and (53)) as a very suggestive rule to consider the recently obtained resonant frequencies. If we use physical arguments about the reasons of the presence of a resonance, we can be sure that channels available for broadcasting are also limited in number. Indeed, in a recent paper, we have generalized the procedure for electromagnetic scalar and vector potentials [30] and we have established that we can use either the electromagnetic field or the potentials for obtaining the resonances and also for the use of the recording time as a resource to optimize communications. And now, we can build information packs (IP) that are functions, which represent a part of the signal we want to send with the minimum loss of information. The resultant

2W

SαðÞ¼ t TFαð Þt (50)

Sαð Þ¼ t þ t<sup>1</sup> TFαð Þ t þ t<sup>1</sup> (51)

each member

expression is:

where,

33

It implies that

is independent of the explicit form of the interaction provided the last is isotropic.

$$\left< \mathbf{V}^{-1} = \mathbf{V}^{\prime \ast} \right>$$

Returning to the time representation, for the time-dependent retarded isotropic (remember that in the following expression, the indices m and n indicates components of the field and can be omitted), free Green function related to , we can write explicitly.

$$G^{mm(\circ)+}({\bf r},t;{\bf r}',t') \equiv G^{(+)}({\bf r},t;{\bf r}',t') = \frac{\delta[t'-(t-\frac{|{\bf r}-{\bf r}'|}{c})]}{\left|{\bf r}-{\bf r}'\right|}\tag{46}$$

and for the advanced time-dependent free Green function related to :

$$\mathcal{G}^{\text{un}(\circ)-}(\mathbf{r},t;\mathbf{r}',t') \equiv \mathcal{G}^{(-)}(\mathbf{r},t;\mathbf{r}',t') = \frac{\delta[t'-(t+\frac{|\mathbf{r}-\mathbf{r}|}{c}-T)]}{\left|\mathbf{r}-\mathbf{r}'\right|}\tag{47}$$

That is the recording time appears explicitly in the advanced Green function and we can show that its value makes possible to blend many signals on the same channel without interference. It is important to note that for resonances, the relevant Green functions are precisely the free ones and not the complete ones as we can see in Eqs. (5) and (6).

### 7. Information packs

In this section, we present the support and the definition of the information packs that are required for the adequate performance of the device shown in Section 6 and that by him constitute a method to improve the broadcasting efficiency. To this end, we must remember that on communication theory [9, 10] are defined the so-called ensembles of functions dependent on time. One of their properties is really a group one from the mathematical point of view and lies in that any ensemble transforms into another member of the same ensemble when we change the function at any certain amount of time. To illustrate this property, we shift by an amount t<sup>1</sup> the argument of all the members of the ensemble defined as follows:

$$F\_{\theta}(t) = \sin\left(t + \theta\right) \tag{48}$$

where θ is distributed uniformly from 0 to 2π. Then, we have:

$$F\_{\theta}(t+t\_{\text{l}}) = \sin(t+t\_{\text{l}}+\theta) \ = \sin(t+\varphi) \tag{49}$$

where φ is distributed uniformly from 0 to 2π.

Optimum Efficiency on Broadcasting Communications DOI: http://dx.doi.org/10.5772/intechopen.84954

Then, each function has changed individually, but the ensemble as a whole is invariant under the transformation. Also, if we apply the operator T which gives for each member

$$S\_a(t) = TF\_a(t) \tag{50}$$

It implies that

ð45Þ

ð46Þ

ð47Þ

ð49Þ

is independent of the explicit form of the interaction provided the last is isotropic.

Telecommunication Systems – Principles and Applications of Wireless-Optical Technologies

Returning to the time representation, for the time-dependent retarded isotropic (remember that in the following expression, the indices m and n indicates components of the field and can be omitted), free Green function related to , we

and for the advanced time-dependent free Green function related to :

That is the recording time appears explicitly in the advanced Green function and

In this section, we present the support and the definition of the information packs that are required for the adequate performance of the device shown in Section 6 and that by him constitute a method to improve the broadcasting efficiency. To this end, we must remember that on communication theory [9, 10] are defined the so-called ensembles of functions dependent on time. One of their properties is really a group one from the mathematical point of view and lies in that any ensemble transforms into another member of the same ensemble when we change the function at any certain amount of time. To illustrate this property, we shift by an amount t<sup>1</sup> the argument of all the members of the ensemble defined as follows:

where θ is distributed uniformly from 0 to 2π.

where φ is distributed uniformly from 0 to 2π.

FθðÞ¼ t sin ð Þ t þ θ (48)

we can show that its value makes possible to blend many signals on the same channel without interference. It is important to note that for resonances, the relevant Green functions are precisely the free ones and not the complete ones as we

can write explicitly.

can see in Eqs. (5) and (6).

7. Information packs

Then, we have:

32

$$S\_a(t+t\_1) = TF\_a(t+t\_1) \tag{51}$$

It is possible to prove that if T is an invariant operator and the input ensemble Fαð Þt is stationary, the output ensemble Sαð Þt is also stationary. Now, for communication purposes, the operator T, which could be a modulation process, is not invariant because of the phase carrier that gives certain time structure, but if the translations are multiples of the periods of the carrier, then the modulation will be invariant. At this stage, it is important to remember that Wiener [6] has pointed out that if a device is linear as well as invariant (in the sense of the last definition), then the Fourier analysis is the appropriate mathematical tool for dealing with the problem. Now, suppose in addition that we are interested on functions that are limited to the band from 0 to Θ cycles per second, then we have the following theorem [10]:

Let F tð Þ contain no frequencies over Θ. Then:

$$\text{where,}\tag{5}\\
\text{where,}\\
\qquad F(t) = \sum\_{-\infty}^{\infty} X\_n \frac{\sin \pi (2\Theta t - n)}{\pi (2\Theta t - n)}\tag{52}$$

$$X\_n = F\left(\frac{n}{2W}\right) \tag{53}$$

In this expansion, F tð Þ is represented as a sum of orthogonal (basis) functions. The coefficients Xn of the various terms can be considered as coordinates in an infinite dimensional "functions space." We will take the last theorem (Eqs. (52) and (53)) as a very suggestive rule to consider the recently obtained resonant frequencies. If we use physical arguments about the reasons of the presence of a resonance, we can be sure that channels available for broadcasting are also limited in number. Indeed, in a recent paper, we have generalized the procedure for electromagnetic scalar and vector potentials [30] and we have established that we can use either the electromagnetic field or the potentials for obtaining the resonances and also for the use of the recording time as a resource to optimize communications. And now, we can build information packs (IP) that are functions, which represent a part of the signal we want to send with the minimum loss of information. The resultant expression is:

$$F\_{\epsilon}(t) = \sum\_{-\infty}^{\infty} X\_{n,\epsilon} \frac{\sin\left[\pi(2\alpha\_{\epsilon}t - n)\right]}{\pi(2\alpha\_{\epsilon}t - n)}\tag{54}$$

where,

$$X\_{n,\epsilon} = F\_{\epsilon} \left(\frac{n}{2a\_{\epsilon}}\right) \tag{55}$$

Every ω<sup>e</sup> allows us to build a decomposition like (54) but we expect that only a few terms are necessary for a well representation of Feð Þt . Next, we send separately each Feð Þt by its own device and it is all we need for broadcasting. To receive the signal, we need a separate device for each ωe.

A very important feature is that because of the properties of the modulation process stated in Eqs. (50) and (51), we can recover, for any arbitrary signal, the behavior under spectral representation and under separated pack representation. So we can either talk about Feð Þt in Eq. (54) as the representation of some element of the basis function for the spectral representation or directly as the e component of an arbitrary signal S tðÞ¼ TF tð Þ. Now, we recall the two resonances founded in another work [3]:

$$
\rho\_1 = \frac{\pi}{4d} + \rho\_0 \tag{56}
$$

and

$$
\alpha\_2 = \frac{3\pi}{4d} + a\_0 \tag{57}
$$

It implies that

Figure 3.

cies will be obtained by the same procedure.

Optimum Efficiency on Broadcasting Communications DOI: http://dx.doi.org/10.5772/intechopen.84954

in which the coefficients are given by:

Se0ðÞ¼ t ∑

∞ �∞ Xn,e<sup>0</sup>

Xn,e<sup>0</sup> ¼ Se<sup>0</sup>

0

and δ are preconceived constants but otherwise arbitrary. With these preliminaries, we can build the first IP:

H1ðÞ¼ t ∑

∞ �∞ Xn, <sup>1</sup>

first for a non modulated beam:

the new basis functions or as the e

35

Hβð Þ¼ t þ t<sup>1</sup> ΩSβð Þ t þ t<sup>1</sup> (64)

Now, the operator Ω is a generic operator like T but acting over the ensemble Sβð Þt . Some care must be taken when reading the WS information, because the translations stated in Eqs. (63) and (64) were multiples of the periods of the carrier, and then as we said above, the modulation will be invariant. The resonant frequen-

The former radio broadcasting procedure: modulated amplitude. Image given by Pérez-Martinez [31].

In order to complete the methodology, we recall the concept of group velocity cgð Þt and construct this inherent quotient between them and the enveloping frequency ω<sup>g</sup> which results in the wave number κg, so we associate them with the resonance frequencies in a similar form as we styled with microwaves, but now these last signals come from the measured properties of the Green's function associated with the modulated signal. In this way, in Eq. (54), we put directly the WS

> sin π 2ω<sup>e</sup> ½ � ð Þ <sup>0</sup>t � n π 2ω<sup>e</sup> ð Þ <sup>0</sup>t � n

(66)

component of an arbitrary amplitude-modulated

H tðÞ¼ a cosð Þ ΘAt þ δ (67)

sin ½ � πð Þ 2ω1t � n πð Þ 2ω1t � n

n 2ω<sup>e</sup><sup>0</sup>

The signal Se0ð Þt in (65) can be viewed as the representation of some element of

signal He0ð Þt . Now, we can give an example where we use the same values for the resonances on Eqs. (56) and (57) and where we propose an arbitrary amplitude modulated or WS (for a modulated visible light beam) signal given as follows:

In Eq. (67), Θ<sup>A</sup> ¼ Θ<sup>p</sup> � Θ<sup>m</sup> is an arbitrary frequency, and in a same manner, a

(65)

(68)

Suppose that S tð Þ is the signal

$$S(t) = \frac{\sin\left[\pi(2\Theta t)\right]}{\pi(2\Theta t)}\tag{58}$$

Then, we have the first pack:

$$S\_1(t) = \sum\_{-\infty}^{\infty} X\_{n,1} \frac{\sin\left[\pi(2\omega\_1 t - n)\right]}{\pi(2\omega\_1 t - n)}\tag{59}$$

$$X\_{n,1} = \mathcal{S}\left(\frac{n}{2a\_1}\right) \tag{60}$$

And, we have the second pack

$$S\_2(t) = \sum\_{-\infty}^{\infty} X\_{n,2} \frac{\sin\left[\pi(2\alpha\_2 t - n)\right]}{\pi(2\alpha\_2 t - n)}\tag{61}$$

with

$$X\_{n,2} = \mathcal{S}\left(\frac{n}{2a\_2}\right) \tag{62}$$

We can see that if Θ ¼ ω1, the only coordinate distinct to zero is X0, <sup>1</sup> ¼ 1 and if Θ ¼ ω2, only survives the term X0, <sup>2</sup> ¼ 1. So, we remark self-consistency of the method.

Even VMF has a broad application on the microwave range, maybe it would be more useful to apply for larger frequencies. But even the great technological boom, there is not any device that could manipulate visible light at length as happens with microwaves. Whatever we can recall some of the basic early ideas on radio broadcasting when the option was sending information by means of modulating the wave's amplitude as appears in Figure 3. However, we can take our definition of information packs and put it in a modulated visible-light signal taking the enveloping of the signal we name the wrapping signal (WS) as the information that can be injected inside Eq. (54). Technically, we rewrite Eqs. (50) and (51) in the form:

$$H\_{\beta}(t) = \mathfrak{Q} \mathbb{S}\_{\beta}(t) \tag{63}$$

Optimum Efficiency on Broadcasting Communications DOI: http://dx.doi.org/10.5772/intechopen.84954

#### Figure 3.

A very important feature is that because of the properties of the modulation process stated in Eqs. (50) and (51), we can recover, for any arbitrary signal, the behavior under spectral representation and under separated pack representation. So we can either talk about Feð Þt in Eq. (54) as the representation of some element of the basis function for the spectral representation or directly as the e component of an arbitrary signal S tðÞ¼ TF tð Þ. Now, we recall the two resonances founded in

Telecommunication Systems – Principles and Applications of Wireless-Optical Technologies

<sup>ω</sup><sup>1</sup> <sup>¼</sup> <sup>π</sup>

<sup>ω</sup><sup>2</sup> <sup>¼</sup> <sup>3</sup><sup>π</sup>

S1ðÞ¼ t ∑ ∞ �∞ Xn, <sup>1</sup>

S2ðÞ¼ t ∑ ∞ �∞ Xn, <sup>2</sup>

S tðÞ¼ sin ½ � <sup>π</sup>ð Þ <sup>2</sup>Θ<sup>t</sup>

(59) with

Xn, <sup>1</sup> <sup>¼</sup> <sup>S</sup> <sup>n</sup>

Xn, <sup>2</sup> <sup>¼</sup> <sup>S</sup> <sup>n</sup>

Θ ¼ ω2, only survives the term X0, <sup>2</sup> ¼ 1. So, we remark self-consistency of the

sin ½ � πð Þ 2ω1t � n πð Þ 2ω1t � n

sin ½ � πð Þ 2ω2t � n πð Þ 2ω2t � n

2ω<sup>1</sup> 

2ω<sup>2</sup> 

We can see that if Θ ¼ ω1, the only coordinate distinct to zero is X0, <sup>1</sup> ¼ 1 and if

Even VMF has a broad application on the microwave range, maybe it would be more useful to apply for larger frequencies. But even the great technological boom, there is not any device that could manipulate visible light at length as happens with microwaves. Whatever we can recall some of the basic early ideas on radio broadcasting when the option was sending information by means of modulating the wave's amplitude as appears in Figure 3. However, we can take our definition of information packs and put it in a modulated visible-light signal taking the

enveloping of the signal we name the wrapping signal (WS) as the information that can be injected inside Eq. (54). Technically, we rewrite Eqs. (50) and (51) in the

HβðÞ¼ t ΩSβð Þt (63)

<sup>4</sup><sup>d</sup> <sup>þ</sup> <sup>ω</sup><sup>0</sup> (56)

<sup>4</sup><sup>d</sup> <sup>þ</sup> <sup>ω</sup><sup>0</sup> (57)

<sup>π</sup>ð Þ <sup>2</sup>Θ<sup>t</sup> (58)

(60)

(61)

(62)

another work [3]:

Suppose that S tð Þ is the signal

Then, we have the first pack:

And, we have the second pack

and

with

method.

form:

34

The former radio broadcasting procedure: modulated amplitude. Image given by Pérez-Martinez [31].

It implies that

$$H\_{\beta}(t + t\_1) = \mathfrak{Q} \mathbb{S}\_{\beta}(t + t\_1) \tag{64}$$

Now, the operator Ω is a generic operator like T but acting over the ensemble Sβð Þt . Some care must be taken when reading the WS information, because the translations stated in Eqs. (63) and (64) were multiples of the periods of the carrier, and then as we said above, the modulation will be invariant. The resonant frequencies will be obtained by the same procedure.

In order to complete the methodology, we recall the concept of group velocity cgð Þt and construct this inherent quotient between them and the enveloping frequency ω<sup>g</sup> which results in the wave number κg, so we associate them with the resonance frequencies in a similar form as we styled with microwaves, but now these last signals come from the measured properties of the Green's function associated with the modulated signal. In this way, in Eq. (54), we put directly the WS first for a non modulated beam:

$$S\_{\epsilon'}(t) = \sum\_{-\infty}^{\infty} X\_{n,\epsilon'} \frac{\sin\left[\pi(2\alpha\_{\epsilon'}t - n)\right]}{\pi(2\alpha\_{\epsilon'}t - n)}\tag{65}$$

in which the coefficients are given by:

$$X\_{n, \epsilon'} = \mathbb{S}\_{\epsilon'} \left( \frac{n}{2a\_{\epsilon'}} \right) \tag{66}$$

The signal Se0ð Þt in (65) can be viewed as the representation of some element of the new basis functions or as the e 0 component of an arbitrary amplitude-modulated signal He0ð Þt . Now, we can give an example where we use the same values for the resonances on Eqs. (56) and (57) and where we propose an arbitrary amplitude modulated or WS (for a modulated visible light beam) signal given as follows:

$$H(t) = a\cos\left(\Theta\_A t + \delta\right) \tag{67}$$

In Eq. (67), Θ<sup>A</sup> ¼ Θ<sup>p</sup> � Θ<sup>m</sup> is an arbitrary frequency, and in a same manner, a and δ are preconceived constants but otherwise arbitrary.

With these preliminaries, we can build the first IP:

$$H\_1(t) = \sum\_{-\infty}^{\infty} X\_{n,1} \frac{\sin\left[\pi(2\alpha\_1 t - n)\right]}{\pi(2\alpha\_1 t - n)}\tag{68}$$

where explicitly the coefficients are:

$$X\_{n,1} = H\left(\frac{n}{2\alpha\_1}\right) \tag{69}$$

8. Conclusions

Optimum Efficiency on Broadcasting Communications DOI: http://dx.doi.org/10.5772/intechopen.84954

efficiency.

Author details

37

Juan Manuel Velazquez Arcos\*, Ricardo Teodoro Paez Hernandez, Tomas David Navarrete Gonzalez and Jaime Granados Samaniego Universidad Autónoma Metropolitana, Mexico City, Mexico

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*Address all correspondence to: jmva@correo.azc.uam.mx

provided the original work is properly cited.

In Eqs. (25), (29), (30), (34)–(40), we have shown that it is possible to use an operator language and the properties of the Green function to define the capacity of a channel, the loss of information, and finally, the error in the time-reversal process. Therefore, we can use our results to describe the behavior of LHM interacting with electromagnetic field whether forward or backward in time. Thanks to our interpretation of a resonance in the broadcasting problem with the left-hand material conditions, and the application of the model PSM, we make up a broadcasting system that has the power for distinguishes between signals according to their recording time, and allows to superpose signals in the same frequency range having different recording times with the minor loss because of resonance technology; to this end, we have presented a detailed support and definition of the information packs (IP) and the possibility of application for visible light. In addition, we have enunciated and proved a theorem (theorem III) that establishes: for the TRT and LHM, the normalized error is independent of the particular behavior of the interaction. Summarizing, we give a complete recipe for optimizing communications

And taking expression (67)

$$X\_{n,1} = a \cos \left[ \Theta\_A \left( \frac{n}{2\omega\_1} \right) + \delta \right] \tag{70}$$

In a similar manner, the second IP will be:

$$H\_2(t) = \sum\_{-\infty}^{\infty} X\_{n,2} \frac{\sin\left[\pi(2\alpha\_2 t - n)\right]}{\pi(2\alpha\_2 t - n)}\tag{71}$$

in which

$$X\_{n,2} = H\left(\frac{n}{2a\_2}\right) \tag{72}$$

Also, by taking Eq. (67):

$$X\_{n,2} = a \cos\left[\Theta\_A \left(\frac{n}{2a\_2}\right)\delta\right] \tag{73}$$

As we said above, the resonances must come also for the WS. By this procedure, we have enlarged the scope of the formalism we named vector-matrix or VMF [1–3].

In order to complete our example, we put explicit values of the resonances for the two visible light IP:

$$H\_1(t) = \sum\_{-\infty}^{\infty} X\_{n,1} \frac{\sin\left[\pi \left(2\left[\frac{\pi}{4d} + o\_0\right]t - n\right)\right]}{\pi \left(2\left[\frac{\pi}{4d} + o\_0\right]t - n\right)}\tag{74}$$

And explicitly

$$X\_{n,1} = a \cos\left[\Theta\_A \left(\frac{n}{2\left[\frac{\pi}{4d} + o\_0\right]}\right) + \delta\right] \tag{75}$$

For the second IP

$$H\_2(t) = \sum\_{-\infty}^{\infty} X\_{n,2} \frac{\sin\left[\pi \left(2\left[\frac{3\pi}{4d} + o\_0\right]t - n\right)\right]}{\pi \left(2\left[\frac{3\pi}{4d} + o\_0\right]t - n\right)}\tag{76}$$

in which

$$X\_{n,2} = a \cos\left[\Theta\_A \left(\frac{n}{2\left[\frac{3\pi}{4d} + o\_0\right]}\right) + \delta\right] \tag{77}$$
