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Power tunneling and rejection fromfractal chiral–chiral interfaceArooj Hameeda, Muhammad Omara, Aqeel Abbas Syeda & QaisarAbbas Naqviaa Department of Electronics, Quaid-i-Azam University, Islamabad,Pakistan.Published online: 22 Aug 2014.

To cite this article: Arooj Hameed, Muhammad Omar, Aqeel Abbas Syed & Qaisar AbbasNaqvi (2014): Power tunneling and rejection from fractal chiral–chiral interface, Journal ofElectromagnetic Waves and Applications, DOI: 10.1080/09205071.2014.938448

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Journal of Electromagnetic Waves and Applications, 2014http://dx.doi.org/10.1080/09205071.2014.938448

Power tunneling and rejection from fractal chiral–chiral interface

Arooj Hameed, Muhammad Omar, Aqeel Abbas Syed and Qaisar Abbas Naqvi∗

Department of Electronics, Quaid-i-Azam University, Islamabad, Pakistan

(Received 24 December 2013; accepted 6 June 2014)

In this article, the reflection and transmission coefficients for a chiral–chiral interfaceof non-integer dimension has been determined theoretically. In order to realize theso-called fractal chiral–chiral interface, one of the two chiral media, forming theinterface, is assumed to be of fractal dimensions. The fields inside the fractal chiral mediaare expressed using a fractional parameter 1 < D ≤ 2, which for D = 2 correspondsto ordinary two-dimensional space and fields so obtained reduce to the original two-dimensional results. Having obtained the fields, the effect of dimension on tunnelingand rejection of power from the fractional chiral–chiral interface is investigated. Resultsare presented, for a circularly polarized incident field, depicting reflected power fromthe interface with parametric dependence on fractionality of the interface and chiralityof the media. As anticipated, the dimension of the interface, apart from the chirality ofthe media, is shown to have a strong effect on tunneling and rejection of power from theinterface. Therefore, fractionality of the interface, in addition to chirality of the media,can be used for controlling the reflected power from an interface.

Keywords: fractal; chiral; fractional space; transmission and reflection; tunneling;reflected power

1. Introduction

Fractal space formulation has proven useful in solving many real-world problems of com-plex geometry, in the area of physics. Mandelbrot identified with some of the complexstructures, which cannot be adequately described by Euclidean geometry, the property ofself-repetition on multiple scales. Such structures were termed as “Fractals” and were shownto have non-integer dimension in the Euclidean sense.[1] Many objects occurring in naturelike the roots of the trees, snow, dust particles, shapes of the clouds, or even galaxiesin space resemble fractals. The theory of fractals provides an excellent framework fordescribing such a highly complex shapes which cannot otherwise be described by Euclideangeometry. To model these complex structures, repeating themselves both at microscopic andmacroscopic level, the fractal framework requires a relatively small number of parameters,of which, the so-called fractal dimension D plays a central role.[2,3] Advances in fractionalcalculus, on the other hand, have provided excellent mathematical tools for efficientlyutilizing the concept of fractals to various disciplines including electromagnetic (EM)theory. For example, Zubair et al. proposed solutions of EM wave propagation in non-integer dimension space based on fractal analysis.[4–6] Asad et al. provided the expressionsfor Green’s function in a fractal dimension space.[7] In these works, it is shown that thedimension of media plays a vital role on the behavior of EM waves. Motivated by the idea,

∗Corresponding author. Email: qaisar@qau.edu.pk

© 2014 Taylor & Francis

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Omar et al. investigated the problem of reflection from a dielectric fractal–fractal inter-face.[8] This work forms the basis for studying various types of interfaces filled with fractalmedia. In the present article, the tunneling and reflection from a fractal chiral–chiral interfaceis investigated.

Chiral media have long been known by researchers, in the field of electromagneticsand optics, for its peculiar properties of optical activity and circular dichroism.[9–17] Anobject is chiral, which has a non-superimpose able mirror image of itself; and the mediacomposed of chiral objects is termed as chiral media. The electrical properties of chiralmedia are described by the so-called chirality parameter κ introducing the cross-couplingbetween electric and magnetic field explaining the phenomena of optical activity and circulardichroism. The fields inside a chiral media are characterized by two eigen waves, namelyleft and right circularly polarized waves each having different refractive index and phasevelocity.

In this paper, the effect of two chiral media on energy flow through a fractal chiral–chiral interface is presented. Expressions for transmission and reflection coefficients forthe interface are derived in Section 2. The general solution for plane wave propagationin non-integer dimension is used to express the fields in fractal media. In Section 3,numerical results for reflected power for various parametric values of fractionality andchirality are presented. The proposed work gives necessary analytical approach to studychiral waveguides, multilayered structures, embedded in fractal space.

2. Fractal chiral–chiral interface

Consider a fractal interface of infinite extent located at z = d . The left half-space z < dis occupied by fractional chiral media with constitutive parameters (ε1, μ1, κ1, D1), where1 < D1 ≤ 2. While the right half-space z > d is filled with chiral media having constitutiveparameters (ε2, μ2, κ2) and D2 = 2. Geometry of the problem is shown in Figure 1.

For simplicity, it is assumed that the fractionality of media 1 exists only in z direction.Time dependency is taken as exp(iωt) and omitted throughout the paper for brevity. If a rightcircularly polarized (RCP) plane wave is made incident at the fractal chiral–chiral interface,the field expressions for left fractal half-space in term of eigen vectors are expressed as: [8]

Einc =(

x − ik−

z1

k−1

y + iky

k−1

z

)exp(−iky y)(k−

z1z)ne H2ne

(k−z1z) (1)

Ere f = AD

(x + i

k−z1

k−1

y + iky

k−1

z

)exp(−iky y)(k−

z1z)ne H1ne

(k−z1z)

+ BD

(x − i

k+z1

k+1

y − iky

k+1

z

)exp(−iky y)(k+

z1z)ne H1ne

(k+z1z) (2)

Hinc = − i

η1

(x − i

k−z1

k−1

y + iky

k−1

z

)exp(−iky y)(k−

z1z)nh H2nh

(k−z1z) (3)

Hre f = − i

η1[AD

(x + i

k−z1

k−1

y + iky

k−1

z

)exp(−iky y)(k−

z1z)nh H1nh

(k−z1z)

−BD

(x − i

k+z1

k+1

y − iky

k+1

z

)exp(−iky y)(k+

z1z)nh H1nh

(k+z1z)] (4)

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Journal of Electromagnetic Waves and Applications 3

Figure 1. Reflection and transmission in fractal chiral–chiral media: Incident RCP (solid lines),reflected LCP (dotted dashed line), reflected RCP (double doted dashed line), refracted LCP (dashedline), refracted RCP (long dashed line).

Figure 2. Co-component of reflected power versus angle of incidence for non-integer dimension,when μ1 = μ2 = 1, for impedance mismatch κ1 = 0.25, κ2 = 0.25.

Hankel function of first and second kind of order ne and nh are used for propagation ofwaves in z direction. Where ne = |3 − D1|/2, nh = |D1 − 1|/2 and D1 is the dimension ofmedia 1.[8] In the above expressions ηi = √

μiμ0/εiε0, for i = 1, 2 denotes the impedanceof respective media.

On the other hand, the field expressions for integer dimension right half-space can bewritten as:

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Figure 3. Cross-component of reflected power versus angle of incidence for non-integer dimension,when μ1 = μ2 = 1, for impedance mismatch κ1 = 0.25, κ2 = 0.25.

Figure 4. Co-component of reflected power versus angle of incidence for non-integer dimension,when μ1 = μ2 = 1, for impedance matching κ1 = 0.25, κ2 = 0.75.

Etra = CD

(x − i

k−z2

k−2

y + iky

k−2

z

)exp −i(ky y + k−

z2z)

+DD

(x + i

k+z2

k+2

y − iky

k+2

z

)exp −i(ky y + k+

z2z) (5)

Htra = − i

η2[CD

(x − i

k−z2

k−2

y + iky

k−2

z

)exp −i(ky y + k−

z2z)

−DD

(x + i

k+z2

k+2

y − iky

k+2

z

)exp −i(ky y + k+

z2z)] (6)

where,

k±(1,2) = ω

√μ0ε0(

√μ(1,2)ε(1,2) ± κ(1,2)) (7)

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Journal of Electromagnetic Waves and Applications 5

Figure 5. Cross-component of reflected power versus angle of incidence for non-integer dimension,when μ1 = μ2 = 1, for impedance matching κ1 = 0.25, κ2 = 0.75.

Figure 6. Co-component of reflected power versus chirality of the media 1 for non-integer dimension,when μ1 = μ2 = 1, for impedance mismatch κ2 = 0.25, θinc = 45◦.

κ(1,2) denotes chirality of media 1 and 2. AD, BD, CD, and DD are unknown reflectionand transmission coefficients to be determined. By imposing the boundary conditions, i.e.

Etinc(z = d) + Et

re f (z = d) = Ettra(z = d) (8)

Htinc(z = d) + Ht

re f (z = d) = Httra(z = d) (9)

the unknown coefficients AD, BD, CD, and DD are determined as follows:

AD = k−1

XD(a1a3η

21 + b1b3η

22)(k

−z1k+

1 − k−1 k+

z1)(k−z2k+

2 + k−2 k+

z2)

+ η1η2(a3b1 + a1b3)(−k+2 (k−

1 k−z2k+

z1 + k−z1(k

−z2k+

1 − 2k−2 k+

z1))

+ (−2k−1 k−

z2k+1 + k−

2 (k−z1k+

1 + k−1 k+

z1))k+z2) (10)

BD = −2k+1

XD(k−

1 k−z1(a1a2η

21 − b1b2η

22)(k

−z2k+

2 + k−2 k+

z2)

− η1η2(a2b1 − a1b2)(k−2 k−

z12k+

2 + k−1

2k−

z2k+z2)) (11)

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6 A. Hameed et al.

Figure 7. Cross-component of reflected power versus chirality of the media 1 for non-integerdimension, when μ1 = μ2 = 1, for impedance mismatch κ2 = 0.25, θinc = 45◦.

Figure 8. Co-component of reflected power versus chirality of the media 1 for non-integer dimension,when μ1 = μ2 = 1, for impedance matching κ2 = 0.25, θinc = 45◦.

CD = 1

XDη2k−

2 [(k−z1

2k+

2 k+1 + k−

12k+

z2k+z1)(a2b1 − a1b2)(a3η1 − b3η2)

+ (k−z1k+

2 k−1 k+

z1 + k−1 k+

z2k−z1k+

1 )(η1(a2a3b1 + a1a3b2 + 2a1a2b3)

+ η2(2a3b1b2 + a2b1b3 + a1b2b3))]DD = 1

XDη2k+

2 [(k−z1

2k−

2 k+1 − k−

12k−

z2k+z1)(a2b1 − a1b2)(a3η1 + b3η2)

+ (k−z1k−

2 k−1 k+

z1 − k−1 k−

z2k−z1k+

1 )(η1(a2a3b1 + a1a3b2 + 2a1a2b3)

− η2(2a3b1b2 + a2b1b3 + a1b2b3))] (12)

XD = k−1 ((a2a3η

21 + b2b3η

22)(k

−z1k+

1 + k−1 k+

z1)(k−z2k+

2 + k−2 k+

z2)

+ η1η2(a3b2 + a2b3)(k−z1(k

+2 (k−

z2k+1 + 2k−

2 k+z1) + k−

2 k+1 k+

z2)

+ k−1 (k−

z2k+2 k+

1 + (2k−z2k+

1 − k−2 k+

z1)k+z2))) (13)

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Journal of Electromagnetic Waves and Applications 7

Figure 9. Cross-component of reflected power versus chirality of the media I for non-integerdimension, when μ1 = μ2 = 1, for impedance matching κ2 = 0.25, θinc = 45◦.

Figure 10. Co-component of reflected power versus chirality of the media 2 for non-integerdimension, when μ1 = μ2 = 1, for impedance mismatch κ1 = 0.25, θinc = 45◦.

where,

a1 = (k−z1 ∗ d)ne H2

ne(k−

z1 ∗ d) (14a)

a2 = (k−z1 ∗ d)ne H1

ne(k−

z1 ∗ d) (14b)

a3 = (k+z1 ∗ d)ne H2

ne(k+

z1 ∗ d) (14c)

b1 = (k−z1 ∗ d)nh H2

nh(k−

z1 ∗ d) (14d)

b2 = (k−z1 ∗ d)nh H1

nh(k−

z1 ∗ d) (14e)

b3 = (k+z1 ∗ d)nh H2

nh(k+

z1 ∗ d) (14f )

The classical results can be recovered by inserting integer values of dimension, i.e.D1 = 2. By setting D1 = 2, the order of Hankel function becomes ne = nh = 1/2. Forlarge arguments, the expression for Hankel function of the first kind becomes

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Figure 11. Cross-component of reflected power versus chirality of the media 2 for non-integerdimension, when μ1 = μ2 = 1, for impedance mismatch κ1 = 0.25, θinc = 45◦.

Figure 12. Co-component of reflected power versus chirality of the media 2 for non-integerdimension, when μ1 = μ2 = 1, for impedance matching κ1 = 0.25, θinc = 45◦.

H11/2(z) �

√2

π ze j (z) (15)

and the expression for the Hankel function of second kind is

H21/2(z) �

√2

π ze− j (z). (16)

Now by inserting (15) and (16) in ((10)–(13)), we get

AD=2 = e−2ik−z1d

XD=2[k+

2 (η1 − η2)2(k−

1 k−2 k+

1 ) − (−4η1η2(k−z2k−

z1

+ (η1 + η2)2k−

1 k−z2)k

+z1) + ((η1 − η2)

2k−2 k−

z1 − 4η1η2k−1 k−

z2)

+ 2η1η2(2(k+z1k−

z1k−2 k+

2 + k−z2k+

z2k+1 k−

1 ) + k+z1k−

z2k−1 k+

2 k+1

− (k−z2(η1 − η2)

2k−1 k−

2 k+z1)k

+z2)] (17)

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Journal of Electromagnetic Waves and Applications 9

Figure 13. Cross-component of reflected power versus chirality of the media 2 for non-integerdimension, when μ1 = μ2 = 1, for impedance matching κ1 = 0.25, θinc = 45◦.

BD=2 = −2k+1 k−

z1e−i(k−z1+k+

z1)d(η21 − η2

2)(k+2 k−

z2 + k−2 k+

z2)

XD=2(18)

CD=2 = 4η2k−2 k−

z2e−i(k−z1−k−

z2)d(η1 + η2)(k+2 k+

z1 + k+1 k+

z2)

XD=2(19)

DD=2 = −4η2k−z1k+

2 e−i(k−z1−k+

z2)d(η1 − η2)(k−z2k+

1 − k−2 k+

z1)

XD=2(20)

XD=2 = k+2 ((η1 − η2)

2k−z1k−

z2k+1 + (4η1η2k−

2 k−z1

+ (η1 + η2)2k−

1 k−z2)k

+z1) + (((η1 + η2)

2k−2 k−

z1

+ 4η1η2k−1 k−

z2)k+1 + (η1 − η2)

2k−1 k−

2 k+z1)k

+z2. (21)

These are the same results as obtained by Taj et al. [18], for LCP incident in integer-dimensional space of chiral–chiral interface at z = 0. Similarly for RCP incident wave,the results can easily be shown to fall in agreement with those for ordinary chiral–chiralinterface by Faiz et al. in [19] for D1 = 2.

3. Numerical results

In the simulations, reflected and transmitted power is calculated for both the RCP and LCPincident waves, separately. For each case, reflected power of both co- and cross-polarizedcomponents for non-integer dimension are presented. Additionally, two cases of impedancematching (η1 = η2) and mismatch (η1 �= η2) are also incorporated.

Angular dependence of reflected power is shown through Figures 2–5. Figure 2 showsthe co-component of reflected power for (η1 �= η2) in fractional space. The values of theconstitutive parameters are arbitrarily taken as ε1 = 4, ε2 = 1, for impedance mismatchand ε1 = 1, ε2 = 1 for impedance matching. For both impedance matching and mismatch,the maximum power rejection (|RC O | + |RC R | = 1) occurs at θinc = 90◦. Additionally,the reflected power is plotted for various values of fractal dimensions. Initially, there seemsnot much effect of dimension on reflected power for θinc = 0◦ − 19◦; however, fromθinc = 19◦ −90◦, the magnitude of reflected power is shown to decrease with dimension ofthe media. Finally, at θinc = 90◦, the magnitude of reflected power approaches to unity, for

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all parametric values. Cross-component of reflected power is shown in Figure 3. In this case,the magnitude of reflected power increases with the decrease in the dimension of mediafor all values of incident angles θinc and eventually reduces to 0 at θinc = 90◦. Angulardependence of reflected powers for impedance matching case, i.e. (η1 = η2), is shown inFigures 4 and 5 corresponding to co- and cross-components, respectively. For impedancematching situation, the maximum power rejection occurs after θinc = 19◦ for all the valuesof dimension.

The variation in magnitude of co- and cross-components for both impedance matchingand mismatch situation in fractional space versus chirality of media 1 are shown by Figures6–9. Rejection and tunneling of power with different dimensions for a wide range of chiralityparameter has been observed for fixed value of θinc = 45◦. Power rejection is observed forwhole range of chirality except for κ1 = 1.2 − 2.7, where power tunneling is observed,instead, as depicted in Figure 6. Dimension of the media is seen to have a strong effecton tunneling, that is, with the decrease in dimension the tunneling also decreases. Figure 7shows that the magnitude of cross-component of reflected power increases with decreasein dimension for all range of chirality of media 1. The results for impedance matchingare presented in Figures 8 and 9. For all range of chirality of media 1, magnitude of co-component of reflected power decreases as the dimension decreases except for κ1 = 0.75−1.25, but power rejection holds for all values of dimension shown in Figure 8. Figure 9shows the magnitude of cross-component of reflected power increases with decrease in thedimension of the space.

Figures 10–13 show change in magnitude of co- and cross-components of reflectedpower versus chirality parameter of fractional space media 2 keeping κ1 and θinc constant.The co- and cross-components of reflected power for impedance mismatching are depictedby Figures 10 and 11, respectively. For the co-component, the strong power rejection isobserved for κ2 = 0.5 − 1.5, whereas, for values of κ2 ≥ 2.25, the magnitude of reflectedpower decreases sharply, as shown by Figure 10. For all values of κ2, the reflected powerhas a decreasing trend with the dimension parameter. For the cross-component, on the otherhand, the reflected power increases with decrease in dimension for all range of chiralityparameter of media 2, as shown in Figure 11. A similar behavior of reflected powers vs.chirality of media 2 and different values of dimension, for the impedance matching case,depicted in Figures 12 and 13, are also observed.

4. Conclusions

A fractal chiral–chiral interface is realized by assigning a fractal dimension to one of thetwo interfacing chiral mediums. The reflection and transmission of electromagnetic wavesfor so-formed fractal chiral–chiral interface are analyzed for various parametric values offractionality of the interface and chirality of the media. It was found that dimension ofmedia, in addition to chirality, has a strong effect on tunneling and rejection of power. Thefields in the fractional media are expressed using a fractional parameter limiting values ofwhich yield the results corresponding to usual integer dimension space, thus validating ourapproach. Our results show that fractionality of the interface, in addition to chirality, canbe used to control the power reflection and rejection from an interface. These results mayalso be the basis for future study of multilayered fractal structures and wave guides filledwith fractal-chiral media.

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