Analysis of Nonlinear Fractal Optical Antenna

8/13/2019 Analysis of Nonlinear Fractal Optical Antenna

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Progress In Electromagnetics Research B, Vol. 56, 289308, 2013

ANALYSIS OF NONLINEAR FRACTAL OPTICALANTENNA ARRAYS A CONCEPTUAL APPROACH

Mounissamy Levy1, 2, *, Dhamodharan Sriram Kumar1, andAnh Dinh2

1National Institute of Technology, Tiruchirappalli 620015, India

2University of Saskatchewan, Saskatoon, SK S7N 5A9, Canada

AbstractFractal antennas have undergone dramatic changessince they were first considered for wireless systems. Numerousadvancements are developed both in the area of fractal shaped elementsand fractal antenna array technology for fractal electrodynamics. Thispaper makes an attempt of applying the concept of fractal antennaarray technology in the RF regime to optical antenna array technologyin the optical regime using nonlinear array concepts. The paper further

discusses on the enhancement of nonlinear array characteristics offractal optical antenna arrays using nonlinearities in coupled antennasand arrays in a conceptual manner.

1. INTRODUCTION

Nonlinear Antennas were extremely complicated in analysis, designand expensive [1]. In the 1980s and 1990s, however, came thedevelopment of antenna and antenna array concepts for commercial

systems, and increases in digital signal processing complexity thatpushed single antenna wireless systems close to their theoretical(Shannon) capacity. Nonlinear antennas are now seen as one ofthe key concept to further, many-fold, increase in both the linkcapacity, through pattern tailoring, spatial multiplexing, and systemcapacity, through interference suppression and beam forming, as well asincreasing coverage and robustness for antenna designs. Furthermore,decreases in integrated circuit cost and antenna advancementshave made these antennas attractive in terms of both cost andimplementation even on small devices. Therefore an exponential

growth have been seen in nonlinear antenna research, the inclusion

Received 26 September 2013, Accepted 6 November 2013, Scheduled 8 November 2013

* Corresponding author: Mounissamy Levy (levy [email protected]).


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290 Levy, Sriram Kumar, and Dinh

of nonlinear antenna technology into wireless standards, and thebeginnings of widespread deployment of nonlinear antennas in wireless

networks. The concept of nonlinear antennas can be extended to theoptical regime for Free Space Optics (FSO) also. Section 2 discusseson the fractal theory, Section 3 discusses on the fractal optical antennaarrays and its importance. Section 4 discusses on the Iterated functionsystems. Section 5 discusses on the nonlinear fractal optical antennaarray and finally Section 6 concludes the paper.

2. FRACTAL THEORY

Fractal theory is an emerging theory which revolutionized the way thescientists think about the nature of the world [210]. Derived from theLatin word meaning break apart the term fractal was originally coinedby Mandelbrot to describe a family of complex shapes that possess aninherent self-similarity or self-affinity in their geometrical structure. Afractal is a recursively generated object having a fractional dimension.These intricate iterative geometrical oddities first troubled the mindsof mathematicians around the turn of the twentieth century, wherefractals were used to visualize the concept of the limit in calculus.What particularly confounded the mathematicians is that when they

carry the limit to infinite, properties of these objects such as arclength would also go to infinity, yet the object would be boundedby a given area. However it was only during the mid-1970s that aclassification was assigned to these objects and the full significanceof fractal theory began to come to light. Fractal objects appear overand over throughout nature and are the product of simple stochasticmechanisms at work in the natural world. Fractal patterns oftenrepresent the most efficient solutions to achieving a goal whether it isdraining water from a basin or delivering blood throughout the human

body. These objects have also been used to describe the structuresof fern and trees, the erosion of mountains and coastlines, and theclustering of stars in a galaxy. For these reasons it is desirable toutilize the power of fractal geometry to describe the layout of antennaarrays in the RF regime and optical antennas in the optical regime.Fractal arrays are used to increase the bandwidth of the antenna and toreduce grating lobes. These arrays have fractional dimensions that arefound from generating sub array used to recursively create the fractalarray. Repetitive application of a generating sub array can form a richclass of fractal array. A generating sub array is a small array at scale

one (P = 1) where P is the scale factor, and is used to constructlarger arrays of higher scales (P > 1). The generating sub arrayelements are turned on and off in a particular pattern in many cases.


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Progress In Electromagnetics Research B, Vol. 56, 2013 291

A set formula for copying, scaling, and translation of the generatingsub array is then followed in order to produce the fractal array.

Hence, fractal arrays that are created in this manner will composeof a sequence of self-similar sub arrays. In other words, this may beconveniently considered as an array of arrays. Some recent researchon Sierpinski carpet arrays, multiband arrays, fractal ultra-widebandarrays, super wideband antennas gives promising and excellent resultsfor the application of fractal concept in antenna arrays [1116]. Thedifferent types of fractal antenna arrays include cantor recipe set forlinear fractal antenna array, planar Sierpinski carpet array, triangular,square and hexagonal antenna arrays.

3. FRACTAL OPTICAL ANTENNA ARRAYS

The absorption or emission of a photon by an electronic transition,e.g., in an atom, molecule, QD or color center is determined by thelight matter interaction in the optical regime [17]. Their absorptionand emission rate are limited by the weak and Omni directionalinteraction with light, since they are far less than the operating opticalwavelength of light. Radiation in the RF regime also faced similarproblems. However, these problems were encountered and addressed

long ago. Since electrical circuits are much smaller than the operatingwavelength usually around 50 to 60 Hz, they radiate very little. Thisproblem of antenna length for wireless communication was addressedby introduction of various modulation schemes for analog and digitalcommunication. To enable wireless communication they are connectedto antennas that have dimensions in the order of the wavelength.These antennas are designed to effectively convert electrical signals intoradiation and vice versa. Exactly the same concept can be applied inoptics. By near field coupling of light into the LSPR modes of a metal

NP, the interaction of quantum emitter with light can be improved toa great extent. A strong local field at the NP is formed by the LSPRsof a metal NP. If an emitter is placed in this field, its absorption andemission of radiation are enhanced. The function of the NP is thenanalogous to an optical antenna as in the case of an RF antenna in theRF regime. In this way excitation and emission rates can be increasedand the angular, polarization and spectral dependence controlled.

The fractal antenna array concept in the RF regime requiresan antenna array combined with fractal array processing technologysuch as Iterated Function Systems (IFS). Similarly for fractal optical

antenna arrays, for incorporating the fractal concept into the opticalantenna an Optical Signal Processor (OSP) combined with opticalantenna arrays needed. Antenna arrays are used in the design of


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apertures for modern radar and high performance communicationsystems in the RF regime. There are many advantages by using

arrays especially their high gain, directivity, and the possibility ofbeam steering which is called beam scanning. Arrays can be mainlyclassified into two types based on their dimension, single dimensionalarray and multidimensional array. The multidimensional array consistsof two dimensional arrays or planar array and three dimensional arrays.Depending on the Lattice structure arrays are classified as uniform andnon-uniform arrays. Non uniform linear arrays are called as nonlineararrays. Uniform arrays are called as periodic arrays. Depending uponthe distribution of array elements the arrays can be classified into densearrays and sparse arrays. Table 1 shows the trade-offs between denseand sparse antenna arrays for various properties [18]. Much of theexplanation about antenna array is available in the texts [1923].

Table 1. Trade-offs between dense and sparse antenna arrays.

Property Dense array Sparse array

Variance of the antenna match High Low

Average side lobe levels Low Moderate

Peak side lobe levels Moderate Low

Hardware density High Low

Power per element Low High

Aperture size needed Small Large

Number of elements for an aperture Many Few

Coupling Strong Weak

Achievable Bandwidth >10 : 1 10 : 1

4. ITERATED FUNCTION SYSTEMS (IFS)

For the construction of a broad spectrum of fractal geometries, IteratedFunction Systems (IFS) are powerful mathematical toolsets [18].These IFS are constructed from a finite set of contraction mappings,each based on an affine linear transformation performed in theEuclidean plane. The most general representation of an affine lineartransformation n consists of six real parameters (an, bn, cn, dn, en,fn) and is defined as

x

y

=n

xy

=

an bncn dn

xy

+

enfn

(1)

wn(x, y) = (anx+bny+en, cnx+dny+fn) (2)


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The parameters of the IFS are often expressed using the compactnotation

ancn bn

dn enfn (3)

where coordinates x and y represent a point belonging to an initialobject and coordinates x and y represent a point belonging to thetransformed object. To scale rotate, shear, reflect and translateany arbitrary object, this general transformation can be used. Theparameters an, bn, cn and dn control rotation and scaling while enand fn control linear translation. Consider a set of N affine lineartransformations 1, 2, 3, 4, . . . , N. This set of transformations

forms an IFS that can be used to construct a fractal of stage +1 froma fractal of stage

F+1=W(F) =Nn=1wn(F) (4)where W is the Hutchinson operator and F is the Fractal of stage .The pattern produced by the Hutchinson operator is referred to as thegenerator of the fractal structure. If each transformation reduces thesize of the previous object, then the Hutchinson operator can be appliedan infinite number of times to generate the final fractal geometry, F.For example, if setF0represents the initial geometry, then this iterativeprocess would yield a sequence of Hutchinson operators that convergeupon the final fractal geometry F.

F1=W(F0) , F2 = W(F1) , . . . , F k+1 = W(Fk) , . . . , F = W(F) (5)

If the IFS is truncated at a finite number of stages L, then the objectgenerated is said to be a prefractal image, which is often described asa fractal of stage L.

The IFS approach is the most common method used to constructdeterministic fractal array geometries; however, deterministic fractals

may not resemble natural objects very closely because of their perfectsymmetry and order. But random fractals more closely resemblenatural objects because their objects are often created using purelystochastic means on the other hand. However these objects aredifficult to work with especially in the context of optimization,because their structures cannot be recreated with exact precision. Aspecialized type of fractal geometry called a fractal random tree wasdeveloped in an effort to bridge the gap between deterministic andrandom fractals. This new construct combines together propertiesof both deterministic and random fractal geometries. Therefore amore generalized expansion of deterministic fractal based geometryis introduced, called polyfractal geometry. In order to construct apoly fractal the IFS technique introduced must be expanded to handle


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multiple generators. Polyfratcal arrays are constructed from multiplegenerators, 1, 2, . . . , M , each of which is having a corresponding

Hutchinson operator W1, W2, W3, . . . , W M. Each Hutchinsonoperator Wm in turn contains Nm affine linear transformations,m,1, m,2, m,3, . . . , m,Nm. The IFS code for generating an invertedSierpinski gasket and the IFS code and associated connection factorsfor Sierpinski based polyfractal geometries are shown in Table 2 andTable 3 respectively. The same concept can be applied to opticalantenna arrays to get the novel smart optical antenna arrays usingsmart-fractal concepts. A fractal optical antenna can be designed usingthe application of fractal concept properly incorporated into an opticalantenna.

Table 2. IFS code for generating an inverted Sierpinski gasket.

w a b c d e f

1 1/2 0 0 1/2 0

3/4

2 1/2 0 0 1/2 1/2

3/4

3 1/2 0 0 1/2 1/4 0

Table 3. IFS code and associated connection factors for Sierpinski-based polyfractal geometries.

w a b c d e f :

Generator 11 1/2 0 0 1/2 0

3/4 : 1

2 1/2 0 0 1/2 1/2

3/4 : 1

3 1/2 0 0 1/2 1/4 0 : 2

Generator 2

1 1/2 0 0 1/2 0

3/4 : 2

2 1/2 0 0 1/2 1/2 3/4 : 13 1/2 0 0 1/2 1/4 0 : 24 1/2 0 0 1/2 1/4 3/2 : 1

5. NONLINEAR FRACTAL CONCEPT FOR OPTICALANTENNA ARRAY

The concept of Nano antennas has emerged in optics as an enabling

technology for controlling the spatial distribution of light on subdiffraction length scales. Analogously to classical antenna design, theobjective of optical antenna design is the optimization and control of


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the energy transfer between a localized source acting as a receiver ortransmitter, and the free radiation field. Most of the implemented

optical antenna designs operate in the linear regime that is theradiation field and the polarization fields are linearly dependent oneach other. When this linear dependence breaks down, however newinteresting phenomena arise, such as frequency conversion, switchingand modulation. Beyond the ability of mediating between localized andpropagating fields, a nonlinear optical antenna provides the additionalability to control the interaction between the two. The nonlinearantenna converts the frequency of the incident radiation centered atby a predefined amount into a new frequency band centered at. Itis vital to know the basic properties of nonlinear antenna and thenfocus on the nonlinearities achievable either in single NP systems ormore complex-coupled NP systems. In practice the use of nonlinearmaterials either metals or dielectrics in the design of optical antennasis a promising route towards the generation and control of opticalinformation. Optical antennas link objects to light [17]. The mainidea is shown in the block diagrams in Figure 1 and Figure 2. Inthe first method as shown in Figure 1, the linear optical antennaarray is considered first and then the nonlinear fractal antenna conceptusing fractal theory is applied which results in a nonlinear fractal

optical antenna array. The optical signal processor controls the opticalantenna array and the nonlinear fractal part. In the Second methodshown in Figure 2, the nonlinear optical antenna array is consideredfirst and the linear fractal concept is applied using fractal theory whichresults in the nonlinear array. The optical signal processor controlsboth the part, for tailoring the beam shape and for producing the beamin the desired direction. Figures 3 to 6 show the array factor patternand directivity pattern [7] of the fractal antenna array in the RF regimewhich is essentially the same for optical antenna array when operated

Linear

Optical

antenna array

Nonlinear

Fractalantennaconcept and

Nonlinear

fractaloptical

antenna

Optical

Signal

Processor

(OSP)

fractal theory array

Figure 1. Block diagram ofnonlinear fractal optical antennaarray system, First method.

Nonlinear

Optical

antenna array

Linear

Fractalantennaconcept and

Nonlinear

fractalopticalantenna

Optical

SignalProcessor(OSP)

fractal theory array

Figure 2. Block diagram ofnonlinear fractal optical antennaarray system, Second method.


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ArrayFactor

-2 -1.5-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1 -0.5

Sweep (Radiaan) for P= 4 ; = 5

0 0.5 1 1.5 2

Figure 3. Fractal optical an-tenna array pattern for P = 4;= 3 in the optical regime is thesame as in RF regime when oper-ated in fundamental mode.

P = 1P = 2P = 3P

= 4

ArrayFactor

-2 -1.5

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-1 -0.5

Sweep (Radiaan) for P= varying; = 7

0 0.5 11 1.5 2-1

Figure 4. Fractal optical an-tenna array pattern for P= 1 to4; = 7 in the optical regime isthe same as in RF regime in fun-damental mode.

Figure 5. Fractal optical an-tenna array directivity pattern forP = 4;= 3 in the optical regimeis the same as in RF regime in fun-damental mode.

Figure 6. Fractal optical an-tenna array directivity pattern forP = 1 to 4; = 3 in the opti-cal regime is the same as in RFregime in fundamental mode.

in the fundamental mode of operation [17]. Consider a quantum object,such as a molecule, QD or atom. The typical timescale for an electricdipole transition to emit a photon is on the order of nanoseconds andthe photon is emitted in dipole angular pattern. This slow undirected

interaction places several limits on their absorption and emission oflight. First, the long radiative lifetime limits the maximum amount ofphotons that can be emitted per second, i.e., the maximum brightness.Second, if faster competing loss channels and/or depahsing are present


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as is often the case for condensed matter at room temperature; a slowinteraction becomes a weak interaction. The emitter then absorbs

only a small fraction of the incident light, and radiates its energy witha low efficiency. Third, the undirected nature of the interaction makesit challenging to efficiently collect the emission and further reduces theprobability of absorption under illumination.

The interaction of the emitter with light can be improved by nearfield coupling to a second larger but still small object: an antenna.The emitter now mainly absorbs and emits light through the modes ofthe antenna. By suitably designing the antenna, the absorption andemission rates can be enhanced. Furthermore the angular, polarizationand spectral dependence of both the emission and absorption canbe controlled. Consider a number of such objects are placed nearthe light source in an angular split. Let there be n such objects.Considering uniform distribution in space of a two dimensional mediumthe angular separation between one object and the other object forlinear spacing in the theta plane is given by

=2

n (6)

Let us consider a simple nonlinear spacing of the objects placed at0, nX, n2X, n3X, . . .. A suitable mechanism should be devisedto select a particular antenna. Optical Signal Processor (OSP)is usedto select a particular object depending upon the requirement and theoptical antenna will radiate in that particular direction. Dependingupon the element selected the excitation ratio on the selected elementto the reference element is given by [17]

exc,nexc,0

= |p.Eloc(r,,,)|2|p Eloc,0(r,,,)|2

(7)

whereexc,n is the excitation rate on the nth element andexc,0 is the

excitation rate on the reference element, p is the unit vector in thedirection of the orientation of the dipole moment, Eloc(r, , , ) isthe resulting local electric field at the emitter position on the emitterdipole moment, Eloc,0(r, , , ) is the resulting local electric field atthe reference position on the reference dipole moment.

The power from each emitter is given by

P(r,,,) =|E(r,,,)|2

2Z0(8)

where E(r, , , ) is the electric field on each emitter and P(r) isgiven by

P(r) =

2

P(r,,,)sin dd (9)


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The ratio of excitation to the nth element with respect to the referenceelement is given by

r.nr,0

= Pn(r)P0(r0)

(10)

The coupling efficiency is given by

c(r) =

2P(r,,,)sin dd

P(r) (11)

The ratio of power dissipated by the nth element to the referenceelement power emitted is given by

n,dissr,0= Pn,dissP0(r0)

(12)

q= r

r+nr(13)

nr=diss+int (14)

= r+nr (15)

The loss in power is proportional to the square of the current and thetotal loss in the antenna array follows the equation

Ptot=n (Pdir+Pfeed+Pref) =n

I2dir+I2feed+I2ref

R (16)

withnbeing the number of antenna elements, Idir,Ifeed, andIref beingthe current inside the director, feed element and reflector respectivelyandRbeing the resistance of the nano rod.

A measure of the total loss in the array is the absorption A, whichcan be deduced from the reflectance R, transmittanceT and scatteredintensity Sby the simple expression

A= 1 T R S (17)where scattering is negligible in the array structure. Since the reflectedintensity of the antenna array depends on the direction of illumination,and transmittance is equal for the different illumination angles, theabsorption depends on the angle of incidence.

For any antenna, both at RF and optical frequencies, it is veryimportant to quantify how directed the emission is. The antennadirectivity D is defined as the power emitted into desired directioncompared with the power averaged over all directions. The angulardirectivityD describes how effectively the power is concentrated into

a particular direction, a very small solid angle or approximately a planewave. D= 1 for a hypothetical isotropic emitter, while the maximumof the angular directivity for an EHD in an isotropic homogeneousenvironment is 1.5. An even more important figure of merit for


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in order to de-embed the intrinsic impedance of the optical antennawhen evaluated with full-wave simulations. The input impedance of

fractal antenna is given byzin(FRACT) =F zin (24)

whereFaccounts for the fractal array impedance factor. If a NP loadwith arbitrary permittivity L in the gap is included, then the inputimpedance is changed to

Rin = g2Ra

g2 2LgSXa+22LS2(R2a+X2a)) (25)

Xin = g(gX

aS(R2

a+X2

a)

Lg2 2LgSXa+22LS2 (R2a+X2a)) (26)

For simplicity, complete filing of the gap by the NP is assumed.However, more complex configurations with parallel or seriescombination of NPs with different fractal configurations can beconsidered. Some fractal configurations can include the Cantor recipeset for the linear array or the Sierpinski carpet for planar array. Theopen-circuit resonance of interest for matching and radiation purpose isobtained whenRinis a maximum andXin= 0 at the radian frequency

given by0=

gXaS(R2a+X

2a)L

(27)

which is the most celebrated parallel resonance equation betweenthe intrinsic impedance and the load impedance. Enabling tuningof the resonance frequency by increasing the load permittivity, theoperational bandwidth of the antenna also called the sensitivity s tothe load permittivity is defined as

s=

0L (28)

At the operating optical frequency, the intrinsic impedance of thenano antenna is determined by the sensitivity. The sensitivity of theoptical antenna is optimized by the proper choice of the geometricalantenna parameters and the operational bandwidth for the applicationof interest. The fractal geometry plays a major role in this part.Linear array, planar array, triangular array, hexagonal array, concentriccircular ring sub-array generators plays a vital role in determining thegeometrical parameters of the optical antenna array using smart fractalconcepts. This is particularly important for smart antenna, biosensingor SERS applications as the proper tailoring of the antenna intrinsicimpedance is directly related to its sensitivity and bandwidth.


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Nonlinear optics exploits the nonlinear relationship between theexciting electric field Eand the resulting polarization P in general.

The relationship between Eand P is linear for weak excitation fields,and most optical antennas operate in this regime. However, for strongexcitations the response Pdepends on higher powers of E, which givesrise to interesting and technologically important phenomena, such asfrequency mixing, rectification or self-phase modulation. The nonlinearrelation between P and Ecan be expressed as a series given by

P =0

(1) E+(2) EE+(3) EEE+(4) EEEE+. . .

(29)

where the susceptibilities are tensors of rankn +1. The polarizationP constitutes a secondary source current that, when inserted intoMaxwell equations, gives rise to a set of nonlinear differential equations.Since the light matter interaction is inherently weak, however it isvery usual to approximate the response of a charged NP by a drivenharmonic oscillator that is Lorentz atom model. The charges oscillatingat the new frequencies induce polarization currentsP(t), that give riseto electromagnetic radiation at those new frequencies. For the drivingfieldE(t) =Ecos(t), the induced SHG nonlinear polarization is givenby

P2 (t) =0(2)E2 (t) = 20

(2)E2 + 20(2)E2 cos(2t) (30)

A constant dispersion free nonlinear susceptibility is assumed forsimplicity. The THG components are given by

P3 (t) =0(3)E3 (t) =

1

40

(3)E3 cos(3t) +3

40

(3)E3 cos(t) (31)

where the first term describes the THG and the second term is theKerr nonlinearity which is the change of the refractive index of the

material at the fundamental frequency. The driving field contains morethan one discrete frequency, nonlinear response at mixing frequencies1 +2, difference 1 2, or four wave mixing (4WM) given by1 2 3. Another class of nonlinear process involves a sequenceof distinct optical interactions, such as the absorption of two photonsfollowed by the emission of one photon, a process referred as TPLwhich is a third order nonlinear process. It is a very powerful tool tostudy local fields near metal nano antennas because of its quadraticintensity dependence. The Poynting theorem for the average rate ofenergy dissipation in a polarizable metal is given by

Pabs =V

j(r, t) E(r, t)

dV (32)


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where. . .denotes the time average and j =d P/dtis the polarizationcurrent density. For a time harmonic field given by E(r,t) =

Re[ E(r,)exp(it), the Equation (32) changes toPabs =1

2

V

Rej(r,) E(r,)

dV (33)

The lowest order term in Equation (29) contributes to the Pabs thatis associated with(1) and is responsible for linear absorption is givenby

P(1)

abs =

2 V Im (1) ()

EEdV (34)

Similarly the next highest contributing term associated with (3) andresponsible for two-photon absorption is given by

P(3)

abs =

2

V

Im

(3) (, , )

EE EEdV (35)

For an extended isotropic material irradiated by a plane wave, theabove equation reduces to

P(3)

absIm (3) E4 (36)

The assemblies of metal NPs play a vital role in many applications.Optical plasmonic properties depend on the interplay between itsconstituting NPs. However, the near field interaction of an NP with itsadjacent NP results in couple LSPRs. Even though the absolute LSPRshift depends on the size, shape, type, surrounding medium, a universalrule of the fractional shift of the wave length /0 can be observedwhen the distance S between NPs is scaled by their characteristicdimension D given by

0

= 1 0

0 exp

S

D

(37)

For the coherent control of nano optical excitations, the relationbetween local electric field and the external incident electric field isgiven by

E(r,t) =

space

dr t

dtS

r, r, t tEextr, t (38)For frequency domain description, the convolution leads to a simplemultiplication

E(r,) =S (r,) Eext() (39)


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where S stands for the tensorial linear response function. The aboveequation can be written in component notation as a sum over two

terms

E(r,) =

Ex(r,)Ey(r,)Ez(r,)

=

2j=1

Sjx(r,)Sjy(r,)

Sjz(r,)

Ejext() (40)

Both the external transverse frequency domain field and the local fieldare complex valued quantity given by

Ejext() =Ajext() e

ijext

() (41)

and the Fourier inverse givesE(r,t) =F1

E(r,)

(42)

and the interaction with the wave function (r, t) of a quantummechanical system gives

V (r,t) = E(r,t) (43)with the dipole operator in electric dipole approximation. For localpolarization mode interference, the far field polarization componentsare orthogonal to each other and the total far field intensity is givenby

Iext() Eext()2 = A1ext()2 + A2ext()2 (44)

For the near field case the total local intensity is obtained from

I(r,)

E(r,)

2=

n=x,y,z

2

j=1Sjn(r,) A

jext() e

ijext

()

2

(45)

For local pulse compression Equations (40) and (41) can be written as

E(r,) =

S1x (r,)

S1y(r,)

S1z(r,)

A1ext()+

S2x (r,)

S2y(r,)

S2z(r,)

A2ext() eiext()

ei1ext() (46)

where the external phase difference is defined in the frequency domainby

ext() =1ext() 2ext() (47)

For fixed phase difference ext(), the local spectral phase can bechanged to a modified phase

modx (r,) =x(r,) +1ext() (48)


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For a short pulse, modx (r,) = 0, therefore

1

ext

() =

x(r,) (49)

For optimal control, we have to identify a suitable observable whoseexpectation value shall approach a given target. For controlling localfields on a spatial scale, let us start with the local spectral densitydefined in Equation (45) and integrate over frequency to obtain thelocal linear flux

F(r) =

0

d

n=x,y,z

2j=1

Sjn (r,)A

jext

() eijext

()2

0

dI(r, ) (50)

The objective is to enhance the local flux at an arbitrary position r1

and to suppress it at a different position r2. The fitness function is

f

Eext()

= F(r1) F(r2) (51)A necessary condition for global optimal control is given by

[ext()]f

Eext()

= 0 (52)

The solution for Equation (52) for a linear response is given by

ext()=arctan

Smix(r2, ) sin[mix(r2, )] Smix(r1, ) sin[mix(r1, )]

Smix(r1, )cos[mix(r1, )] Smix(r2, )cos[mix(r2, )]

+k (53)

where k is an integer. The response-function mixing amplitude withthe conditionSmix(r,)0 and mixing angle mix(r,) is given by

Smix(r,) eimix(r,) =

n=x,y,z

S1n(r,)

S2n(r,)

(54)

The optimum external field amplitude is given by

Ajext() =j ()

Iin() (55)

For optimum transmission coefficients, the solution pairs are given by1 () , 2 ()

[0, 0] , 1,Cmix()C2()

,

Cmix()

C1() , 1

, [1, 1]

(56)

with the coefficients given by

Cj() =

n=x,y,z Sjn(r1,)

2

Sjn(r,)

2

(57)

andCmix() = Smix(r1, )cos[mix(r1, ) + ext()]

Smix(r2, )cos[mix(r2, ) + ext()] (58)


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In the case of linear optical antenna array, the emission pattern steadilyrotates from the dipole pattern of the horizontal emitter towards

the dipole pattern of the vertical antenna dipole moment, when theresonant length of the antenna is reached. For nonlinear distributionof the array elements, the rotation pattern is nonlinear and dependson the assembly of the adjacent NPs. The horizontal dipole momentof the emitter plus the transverse response of the antenna and thevertical dipole moment of the antenna mode effectively determines theangular emission. As the antenna is tuned into resonance with theemission wavelength, the balance progressively shifts from the emitterdipole towards the perpendicular oriented antenna dipole, until theantenna mode dominates and fully determines the angular emission.Hence, the presence of an antenna at resonance has a major effect.It can enhance the rate with almost three orders of magnitude whilethe emission pattern can be fully redirected. This concept can bemade use for redirecting the beam in different directions. By varyingthe antenna length,tuning the antenna resonance to the excitationwavelength can be done. For studying the antenna resonance patterns,monopole antennas are fabricated with different lengths. The antennaresponse as a function of antenna length is studied. The observednarrow molecular fluorescence spots confirm the role of the sharp tip.

In principle any sharp metallic tip can confine the field through thelightening rod effect. Thus it is important to verify the role of theantenna resonance. To this end the antenna length was controllablyvaried between approximately 30 and 140 nm.

Optical antennas work both in excitation and emission. Toenhance the excitation level, spatial localization and enhancementare used in an antenna. Let us consider the emission control of asingle quantum system by the antenna. The spectral, polarization andangular response of the antenna-emitter can be studied, independent ofthe details of the measurement procedure and the absolute fluorescenceintensity by carefully observing a single emitter. Unlike in the case ofensemble emitters, the polarization and angular emission observed areindependent of how the system was illuminated, as emitter-antennaposition and orientation are well defined, in a single emitter. Thispolarization and angular emission of a single emitter are very crucial,when it is coupled to an optical antenna. For monopole and dipoleantennas, the polarization of the emission provides information aboutthe orientation of the effective dipole moment dominating that emissionwhich in turn determines the optical beam focusing direction. If

fractal ensemble pattern is used, the fractal array concept increasesthe computation speed rapidly as the number of elements increase forrapid beam form computation as given in the Table 4. Prefers to the


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stages of growth andrefers to the number of elements in each stage.For example for P= 3 and = 7, the fractal array factor is 300 times

faster to calculate than the conventional array for rapid beam formingcomputation.

Table 4. Illustrating the number of times faster for rapid beamforming computations.

P= 1 P= 2 P= 3 P= 4 P= 5

= 3 1 2 4 10 24

= 5 2 6 20 78 312

= 7 3 12 57 300 1680 = 9 4 20 121 820 5904

The size, shape, material and enclosing medium determine theresonance of a single plasmonic nanostructure. The inter particledistance in an ensemble of NPs influences the spectral position,extinction amplitude as well as the decay dynamics (line width ofthe LSPR). The entire property of the array is determined by thecharacteristics of each particle in the array. Dense arraying or sparse

arraying can be used to fulfill the requirement. A transition from near-field coupling to far field coupling, which extends to only a few tens ofnanometers, takes place with increasing distance between the NPs. Atthe same time, there are some limitations in increasing the distance tobeyond some extent. Also in the far field regime, for NPs sufficientlysmall compared to the operating wavelength (electrostatic limit), theNPs in the array can be seen as individual scatterers of the incomingradiation, with the scattered field being of dipolar character. However,for sufficiently large distances, the interference of the scattered fielddetermines the optical properties of the array configuration.

6. CONCLUSION

A proper nonlinear fractal array configuration is required where noradiation is scattered into diffraction orders, for efficient receiving andfocusing of optical radiation to the sub wavelength region withoutlosing energy by scattering. By reciprocity, in order to achievehighly directive radiation into one direction, the opening of diffractionchannels has to be carefully suppressed by carefully choosing the non-

linearity determined by the fractal array configuration using dense orsparse array with respect to the resonant wavelength of the individualantennas in the array. The optical antenna array along with the


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nonlinearities in coupled antennas and arrays concept increases therapid beam form computation and pattern tailoring for smart beam

forming using the iterated function system (IFS) concept for thenonlinear fractal optical antenna array.

ACKNOWLEDGMENT

This work was supported in part by the Canadian Government underGraduate Student Exchange Program (GSEP).

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Analysis of Nonlinear Fractal Optical Antenna

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