Resonances on Discrete Electromagnetic Time Reversal Applications

J.M. Velázquez-Arcos1 C.A. Vargas2 J.L. Fernández-Chapou3 J. Granados-Samaniego4

1Departamento de Ciencias Básicas, Universidad Autónoma Metropolitana, Av. San Pablo 180. Col. Reynosa Tamaulipas, CP 02200, México D.F., México. e-mail: jmva@correo.azc.uam.mx, Tel: +52 5553189026, Fax: +52 5553189540. 2Departamento de Ciencias Básicas, Universidad Autónoma Metropolitana, Av. San Pablo 180. Col. Reynosa Tamaulipas, CP 02200, México D.F., México. e-mail: cvargas@correo.azc.uam.mx, Tel: +52 5553189020, Fax: +52 5553189540. 3Departamento de Ciencias Básicas, Universidad Autónoma Metropolitana, Av. San Pablo 180. Col. Reynosa Tamaulipas, CP 02200, México D.F., México. e-mail: jlfc@correo.azc.uam.mx, Tel: +52 5553189504, Fax: +52 5553189540. 4Departamento de Ciencias Básicas, Universidad Autónoma Metropolitana, Av. San Pablo 180. Col. Reynosa Tamaulipas, CP 02200, México D.F., México. e-mail: jgs3112@yahoo.com.mx , Tel: +52 5553182054, Fax: +52 5553189540.

Abstract – Electromagnetic Time Reversal Vector formalism [1, 2] can be employed not only to the purpose of achieve subwavelength focusing. We can go far away of this initial goal and take the formalism as the starting point for analyzing the behavior of discrete electromagnetic systems in a regime that represents special but interesting physical usefully situations. The advantage over other possible approximations is that we can guaranteeing the minimum loss of information because the discrete formalism was built by thinking that time reversal increase the scattering section and then the Green’s function carry the most complete information to recreate the source signals. In this work, we find that solutions of the corresponding homogeneous vector-matrix equation in Time Reversal formalism, have a strong similarity with the so called resonant states for nuclear reactions [3, 4] so they inherit some of their properties. The physical systems that we can model are reverberating or short living electromagnetic fields that can be localized inside real or imaginary cavities. Our ultimate purpose is the possible applications on communications related for example with MIMO microwave technology operating on imaginary cavities, that is, on some special regions of space.

Key Words – Time reversal, resonances, subwavelength focusing, communications.

1 INTRODUCTION

The concept of resonance is one of the most productive in the knowledge process of all physical phenomena. There is no any branch where this remarkable concept not be able exploited to bring a benefit. So we are confident that the resonances that we will obtain from the Electromagnetic Time Reversal Vector formalism (ETRVF) can give us a __________________________________________

powerful tool in many applications, particularly in the microwave technology of communications. In this work we make first reference to the vector matrix equation [1] for discrete electromagnetic time reversal and take them like the starting point, that is

mn

m

nff )()( AG1 (1)

Where )(G is the Fourier's transform of the complete Green’s function, A the complex

interaction, and )()( mf , )(nf are the source field and the arrived field respectively.

This equation clearly has a source term that is the key in the time reversal process to achieve a perfect subwavelength focusing. But now as we will see, we want to create conditions for emit radiation without the presence of an original source, that is for a resonance. In nuclear scattering we think that the physical meaning for a correspondent resonance is a decaying state, but in electromagnetism we can think in the transition since a vanishing field toward a traveler field like if media was a left-hand material [2]. It is very important also to distinguish between the resonance in a conventional electromagnetic circuit (even an antenna) and the resonances that beneath to the integral operator in ETRVF. For practical use, we will propose that each of the resonances can play the role of the centre of highways to information even when the media are not a left hand material and really we are not interested in recover information from evanescent fields [2] because we have guaranteed with the time reversal techniques that we have the best subwavelength focusing [1].

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2 THE HOMOGENEOUS EQUATION

Now, we are going to build a homogeneous equation from equation (1). We must remember that our vector matrix equation is indeed an algebraic form of an integral Fredholm's equation, so we will follow the procedure based in the correspondent theory to obtain the resonances, that is, we eliminate the source term in the inhomogeneous equation. By using the Fredholm's alternative [5], the resonances are the solutions to the resultant homogeneous equation and these solutions exist only for a discrete set of values of the eigenvalue )( . Recalling that

kjifrrGA

kjif

jinm

nmij

mnkj ),;(

0)(

,,,,K (2)

Then, equation (1) can be written as

nm

n

nff

)()( K1 (3)

But if we suppose that n

f)(

vanish, the equation must be written as

0)( nm

n fK1 (4)Where

kjifrrGAkjif

jin

mn

mij

mnkj ),;(

0)( )(

,,

)(,,K

(5)Now the Fredholm's alternative demand to introduce an eigenvalue because the equation (3) does not have solutions for any condition, so we must write the proper equation:

0)()()( )( ne

m

ne wK1 (6)

In this equation )(e is the eigenvalue, )(new

the resonances and subindex e denotes the specific resonance.

3 THE ORTHOGONALITY RELATION FOR RESONANCES

In this section we will obtain a very interesting relation for the resonant solutions, that is the orthogonality between them. To this end, we assume that the theory of homogeneous Fredholm's equations [3, 4, 5] is applied and written explicitly the kernel

)()(K we can then put

0)( )( ne

m

ne wAG1 (7)

Also we recall that Fredholm's determinant [5]

))()(1())((

ee

(8)must observe that

0))(( (9)So we have that for each eigenvalue )(e we have

an eigenfunction (or eigenvector) )(pew By

developing expression (7) we obtain, with AA )(ee (10) that

)()()( )( se

p

se

pe ww AG (11)

So that also we have

s

pe

se

pe ww AG )()()( )(††

(12) Making the same procedure but for another eigenfunction

)(quw (13)

We have

)()()( )( ru

q

ru

qu ww AG (14)

and

r

qu

ru

qu ww AG )()()( )(††

(15)

By multiplying (86) with A)(†p

ew and then

equation (84) by A)(†q

uw we obtain

)()()()(†

)()(† r

uwqru

pew

quw

pew AGAA

(16)

and

)()()()(†

)()(† s

ewpse

quw

pew

quw AGAA

(17) Now we can explicitly write ee

1)(A and

uu1)(A to obtain the equations

)()()()(†1)()(

† ruw

qr

pewu

quw

pew AGAA

(18)

and

168

Figure 1:Sanayei a

)(†q

uw A

Now we mto obtain t

(†p

ewu And

)(†q

uwe

if we takobservingequations

pew

This is tresonanceresonant classical interactionbelongs torthogonasense of th

4 MIMO

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multiply (90) the following

)()pew

quwA

)()quw

pewA

ke the differeng that the t are equal we

()(† q

upe wAthe rule of es and is sstates in quan

electroman potential, thto an eigenval to other wihe last equatio

O AND RES

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nce between (wo right mhave

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ith different eon.

ONANT SO

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)()( sew

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qrA

(

)()sew

psA

((93) and (94) embers in b

0 (2y for the vehe correspondnics (but now nce A is sonant signal

1e must

eigenvalue in

OLUTIONS

ations, orthogose do not interl ones [6, 7]. ve to find a sebe used to assnna as the cen

es in order t multiple-outp

sign the signa

)(se

(19) )(

(20)

(21) and

both

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et of sign ntre

to put)

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5

Lbctaebis

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orthogonal) stadevice (may beoriginal signalhave expectedphenomena likhe signals w

frequencies aralso provides make simultanbecause of the

)()(G andadvantage fromantennas and manner in figuand A. Nosraorthogonal bloappropriate wa

5 FINDING

Let’s find thebut very simpconvenient kerake into acc

electromagnetibut we have tws:

)()(K

different comp

ur proposal isa set of reso

ates, next, assoe an antenna) l around the cd that becauke superoscilla

without the usround these p

the best perfneously a time knowledge d )(G them the usual tim

other deviceure 1 we showatinia [6] whiock codes p

ay for direct th

RESONANT

e resonant freple example. rnel )()(Kcount the thic field and suwo emitting an

di

d

)0(

)0cos()0(

)0sin(

ponents as pro

s, as we havenant (and alociate to each and finally, d

central resonanuse of the eation [8], we se of a large

principals. Theformance if y

me reversal prof the Greenen we haveme reversal teces [9]. As a

w a proposal byich alternativeproperties to he signals.

T FREQUE

equencies in aTo this end w, for simplicit

hree componeuppose we havntennas. A pos

d

d

id

(

sin()(

cos(

oposed by S.

e mentioned, lso mutually resonance a

distribute the nt ones. We

existence of can localize e spread of e formalism you want to rocedure [1] n’s functions e additional chniques for an example y S. Sanayei ely think in choose the

ENCIES

an academic we choose a ty we do not ents of the ve only one, ssible kernel

d

d

d

d

)0

)0

)0

)0

169

(23) The conditions for resonances are:

0

)0(

)0sin(

)0(

)0cos()0(

)0cos(

)0(

)0sin(

d

d

d

di

d

di

d

d

(24) And also that

1 (25)These two conditions give us the two resonant frequencies:

01 4d (26)And

02 43

d (27)

6 CONCLUDING REMARKS

As suggested by N. Wiener [10,11], an appropriate language for communications is the use of Fourier's transforms of the functions involved, so our formalism have adopted this point of view. Even when we have omitting many steps on justifying our results, we can guaranty that all of them are original ones (for example equations 6, 7 and 22). Besides that is the first time that an approximation based on resonant solutions of a vector matrix equation is outlined, we think that we can use some of the properties like orthogonality to perform efficiency on communications particularly in the use of simultaneity display of a signal toward a set of devices as occur in MIMO technology [6,7]. In summary the method here presented is as follows: we must find the appropriate kernel )()(K , then

make 01K ))()(( )( and finally set1)( the results obtained are the resonant

frequencies q . Additionally, we propose that resonant frequencies can be used to select which ranges of frequencies must be assigned to each device (it may be a set of antennas).

References

[1] J. M. Velázquez-Arcos, J. Granados-Samaniego, J. L. Fernández-Chapou, C. A. Vargas, Vector generalization of the discrete Time Reversal

formalism brings an electromagnetic application on overcoming the diffraction limit. Electromagnetics in Advanced Applications (ICEAA), 2010 International Conference on. pp.264-267, 20-24 Sept. 2010 doi: 10.1109/ICEAA.2010.5653059. [2] J. M. Velázquez-Arcos, J. Granados-Samaniego, J. L. Fernández-Chapou and A. L. Rodríguez-Soria. The Equivalence Between Time Reversed Means and Employment of Left Hand Materials to Overcome the Diffraction Limit. PIERS Proceedings (ISSN 1559-9450), Vol. PIERS 2009 in Moscow Proceedings (ISBN: 978-1-934142-10-3, doi: 10.2529/PIERSOnline), 520-528, published by The Electromagnetics Academy, Cambridge, MA. USA. [3] J. M. Velázquez-Arcos, C. A. Vargas, J. L. Fernández-Chapou and A. L. Salas-Brito. On computing the trace of the kernel of the homogeneous Fredholm's equation. J. Math. Phys. Vol. 49, 103508 (2008) doi: 10.1063/1.3003062. [4] A. Mondragón, E. Hernández and J. M. Velázquez-Arcos. Resonances and Gamow States in Non-Local Potentials. Ann. Phys. (Leipzig) Vol. 48, 503-616 (1991) doi: 10.1002/andp.19915030802. [5] N. von Der Heydt. Schrödinger equation with non-local potential. I. The resolvent, Ann. Phys. (Leipzig) Vol. 29, 309-324 (1973). [6] S. Sanayei and A. Nosratinia, Antenna Selection in MIMO Systems, Communications Magazine, IEEE, Vol. 42, No. 10, pp. 68-73 October, 2004. [7] I. Hen, MIMO Architecture for Wireless Communication, Mobility Group, Intel Corporation Index Vol. 10 Issue 02 Published, May 15, 2006 ISSN 1535-864X doi: 10.1535/itj.1002. [8] N. I. Zheludev, What diffraction limit?, Nature Materials Vol. 7, 420-422 (2008). [9] J. M. F. Moura and Y. Jin, Detection by Time Reversal: Single Antenna, IEEE Transactions on Signal Processing, Vol. 55, No. 1, January, 2007. [10] C. E. Shannon, A Mathematical Theory of Communication, The Bell System Technical Journal, Vol. 27, pp. 379-423, 623-656, July, October, 1948. [11] N. Wiener, The ergodic theorem, Duke Mathematical Journal, Vol. 5, pp. 1-20, 1939.

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