Samina Gulistan *, Farhat Majeed, and Aqeel A. Syed Department of Electronics, Quaid-i-Azam University, Islamabad 45320, Pakistan

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1 Progress In Electromagnetics Researc B, Vol. 5, 47 68, FIELDS IN FRACTIONAL PARALLEL PLATE D B, DB AND D B WAVEGUIDES Samina Gulistan *, Farat Majeed, and Aqeel A. Syed Department of Electronics, Quaid-i-Azam University, Islamabad 45, Pakistan Abstract D B, DB and D B boundary conditions are used to investigate te resulting field patterns inside a parallel plate waveguide. Te D B boundary conditions are accomodated by assigning te beavior of perfect magnetic conductor (PMC) for transverse electric mode (TE) and tat of perfect electric conductor (PEC) for transverse magnetic (TM) mode, to te boundary, respectively. Likewise, DB boundary conditions are incorporated by assuming te beavior of boundary as PMC for bot te TE mode and TM mode. Finally D B boundary conditions are realized by assigning PEC caracteristic to te boundary for bot TE and TM modes. A general wave propagating inside te parallel plate waveguide is assumed and decomposed into TE and TM modes for te purpose of analysis. Fractional curl operator as been used to study te fractional parallel plate D B, DB and D B waveguides for different values of fractional parameter. Beavior of te field patterns in te waveguides are studied wit respect to te fractional parameter describing te order of te fractionalization.. INTRODUCTION Fractional calculus is a branc of matematical analysis wic deals wit te differentiation and integration operators, of arbitrary real (non-integer) or complex order []. It as been demonstrated tat tese matematical operators are useful matematical tools in various disciplines of science and engineering including Electromagnetic teory [ 5]. Fractionalization of ordinary derivative and integral operators motivated te researcers in electromagnetics to explore te potential of fractionalization of oter operators in te field [6 5]. Engeta proposed a recipe to fractionalize te curl operator, describing Received Marc, Accepted April, Sceduled 8 April * Corresponding autor: Samina Gulistan (samina5@gmail.com).

2 48 Gulistan, Majeed, and Syed te differential form of Maxwell s equations [4]. He regarded te new solutions as intermediate between two dual solutions. In an isotropic, omogeneous, and source free medium described by wave number k and impedance η, te new set of solutions to te source-free Maxwell equations may be obtained by using te following relations [4] [ ] E fd = (ik) curl E [ ] () ηh fd = (ik) curl (ηh) From Eqs. () it can be seen tat for =, (E fd, ηh fd ) reduces to te original solutions wereas (E fd, ηh fd ) gives dual to te original solution to te Maxwell equations for =. Terefore for all values of between zero and unity, (E fd, ηh fd ) provides te new set of solutions wic can effectively be regarded as intermediate solutions. Tese solutions are also called te fractional dual fields as expressed wit te subscript fd. Naqvi and Rizvi extended Engeta s work on fractional curl operator by determining sources corresponding to te fractional dual solutions to te Maxwell equations. Results of teir work sow tat surface impedance of a planar reflector, an intermediate between PEC and PMC, is anisotropic in nature [6]. Naqvi et al. furter studied fractional dual solutions to te Maxwell equations for reciprocal, omogenous, and lossless ciral medium [7]. Laktakia pointed out tat any fractional operator tat commutes wit curl operator may yield fractional solutions []. Naqvi and Abbas studied te role of complex and iger order fractional curl operators in electromagnetic wave propagation [8]. Tey also studied te fractional dual solutions in double negative (DNG) medium [9]. Veliev et al. extended te work on te fractional curl operator by finding te reflection coefficients and surface impedance corresponding to fractional dual planar surfaces wit planar impedance surface as te original problem []. Te work on tis topic entered into new era wen concepts of fractional transmission lines, fractional waveguides, and fractional resonators were introduced [ 4]. Modelling of transmission of electromagnetic plane wave troug a ciral slab using fractional curl operator and fractional dual solutions in bi-isotropic medium are also available [4, 4]. After te introduction of niility concept by Laktakia [4], Tretyakov et al. incorporated te niility conditions to ciral medium and proposed anoter metamaterial termed as ciral niility metamaterial [44, 45]. Study of niility/ciral niility metamaterials is a topic of current researc by several researcers [46 57]. Naqvi contributed many researc articles on ciral niility and fractional dual solutions in ciral niility metamaterial [5

3 Progress In Electromagnetics Researc B, Vol. 5, 49 57]. In computational electromagnetics, special attention as been paid to newly introduced DB and D B boundary conditions. A DB boundary requires tat te normal components of electric and magnetic flux densities vanis at a DB interface [58 65], i.e., ˆn D = () ˆn B = were ˆn is normal vector to te interface. A D B boundary is defined suc tat te derivatives of te normal components of te flux densities become zero, i.e., z D z = z B z = Tese conditions are in contrast wit traditional boundary conditions, like PEC (perfect electric conductor) or PMC (perfect magnetic conductor) boundary conditions, wic restrict te freedom of te tangential field components only. It as been noted tat PEC and PMC boundary conditions are special cases of DB or D B boundary conditions. Anoter pair of boundary conditions, namely, DB and D B can also be introduced along similar lines. All boundary conditions stated above are matematical concepts. From te practical point of view, tey can be realized in terms of psical structures, very precisely in many cases. In electromagnetics te PEC boundary corresponds to an interface of an ideal conducting material, wic can be approximated by metals. In [66], it was sown tat te DB boundary can be realized by an interface of uniaxial anisotropic medium, wose normal permittivity and permeability parameters become zero simultaneously. Suc a uniaxial medium was named as zero axial parameter (ZAP) medium in [64]. Realization of te D B boundary conditions is sown in [67], were it is suggested tat te planar D B boundary is realized by transforming a DB boundary, by means of a wave guiding quarter wave transformer. Suc a device is a quarter wave slab of uniaxial medium wit infinitely large axial parameter. It as been observed tat DB interface beaves like perfect reflector for te rigt anded circularly polarized (RHCP) and left anded circularly polarized (LHCP) incident fields [65]. Moreover, wen field is reflected from top and bottom of ciral niility coated DB interface, it keeps on rotating its plane of polarization and it appears as a circularly polarized field inside te core of te waveguide [68, 69]. Fractional dual solutions to te Maxwell equations for fields inside a parallel plate DB waveguide ave been discussed by Hussain et al. [7]. In te present work, to complete te study of set of boundary conditions requiring vanising of te normal components of te flux

4 5 Gulistan, Majeed, and Syed densities D and B (DB boundary) or teir normal derivatives (D B, DB, and D B boundary), we discuss fractional dual solutions to te Maxwell equations for fields inside a parallel plate D B, DB and D B waveguide. A variety of field configurations (electric and magnetic) can be obtained by applying eiter any of te D B, DB, D B boundary conditions or te fractionalization operator. Suc configurations may be required for some particular applications, e.g., couplers. So if any one desire to get some particular field pattern in any experiment or in some device, tis work can serve te purpose. In Section beavior of waves along a guiding structure is discussed. In Section fractional dual solutions of D B, DB, D B are derived. Section 4 deals wit results and discussions and paper as been concluded in Section 5.. GENERAL BEHAVIOUR OF WAVES ALONG A GUIDING STRUCTURE Consider a waveguide consisting of two parallel plates one located at y =, oter at y = b and separated by a dielectric medium aving constitutive parameters ɛ and µ. Te plates are assumed to be of infinite extent and te direction of propagation is taken along positive z-axis as sown in Figure. Electric and magnetic fields propagating in te source free dielectric region must satisfy te following omogeneous vector Helmoltz equations E(x, y, z) + k E(x, y, z) = H(x, y, z) + k H(x, y, z) = (a) (b) y y=b D'B' x z k k y= D'B' Figure. Plane wave representation of te fields inside te waveguide.

5 Progress In Electromagnetics Researc B, Vol. 5, 5 were = x + y + z is te Laplacian operator and k = ω µɛ is te wave number. By taking z dependance as exp(i), Eqs. (a) and (b) reduce to two dimensional vector Helmoltz equations as xye(x, y) + E(x, y) = xyh(x, y) + H(x, y) = (4a) (4b) were = k β, β is te propagation constant. Since propagation direction is along z-axis and te waveguide dimensions are taken to be infinite in xz-plane, so x-dependence can be ignored in Eqs. (4a) and (4b). Under tis condition, it will take te form of ordinary, second order differential equation as d E(y) dy + E(y) = (5a) d H(y) dy + H(y) = (5b) In general, for te waveguide problems, te Helmoltz equation is solved for te axial field components only. Te transverse field components can be obtained by using axial components of te fields and Maxwell equations. So scalar Helmoltz equations for te axial components can be written as d E z dy + E z = (5c) d H z dy + H z = (5d) General solution of te above equations is E z = a n cos() + b n sin() H z = c n cos() + d n sin() (6a) (6b) were a n, b n, c n and d n are constants and can be found from te boundary conditions. Using Maxwell curl equations, te transverse components can be expressed in terms of longitudinal components (E z, H z ), i.e., E x = ( iβ E z x + ik ηh ) z (7a) y E y = ( iβ E z y ik ηh ) z (7b) x

6 5 Gulistan, Majeed, and Syed were η = µ ɛ H x = ( iβ H z x ik η H y = ( iβ H z y + ik η ) E z y ) E z x is impedance of te medium inside te guide. (7c) (7d). FRACTIONAL DUAL WAVEGUIDES A wave of general polarization propagating in positive z-direction troug a parallel plate waveguide can be written as a linear sum of te transverse electric (T E z ) and transverse magnetic (T M z ) modes. A D B boundary can be simulated as te boundary wic beaves like perfect electric conductor (PEC) for (T M z ) and perfect magnetic conductor (PMC) for (T E z ) modes. Terefore fields inside a parallel plate D B waveguide may be obtained by linear superposition of two canonical solutions wic are transverse electric (T E z ) mode solution for PMC waveguide and transverse magnetic (T M z ) mode solution for PEC waveguide... D B Waveguide... Case : Transverse Electric (T E z ) Mode Propagation troug a PMC Waveguide Let us first consider tat (T E z ) mode is propagating troug a PMC waveguide. For tis mode, axial component of te electric field becomes zero and te corresponding transverse components can be found by using Eqs. (7a) (7d). ( ) ik E x = [ c n sin() + d n cos()] (8a) ( ) iβ H y = [ c n sin() + d n cos()] (8b) E y = (8c) H x = (8d) Using boundary conditions for PMC boundary, tat is, H x,z = y=,b, we get solutions as ( ) ik E x = [D n cos()] (9a) ηh y = ( iβ ) [D n cos()] (9b)

7 Progress In Electromagnetics Researc B, Vol. 5, 5 ηh z = [D n sin()] E y = H x = (9c) (9d) (9e) were D n = d n η = nπ b n =,,... By taking again te z-dependance exp(i) and writing Eqs. (9) in exponential form. Electric and magnetic fields inside te dielectric region will be obtained as sum of two plane waves given as E = E + E (a) ηh = ηh + ηh (b) were (E, H ) are te electric and magnetic fields associated wit one plane wave, and (E, H ) are te electric and magnetic fields associated wit te second plane wave. Tese fields are given as following ( ) ( ) Dn ik E = ˆx exp(i + i) (a) ( ) (ẑ Dn ηh = i + iβ ) ŷ exp(i + i) (b) ( ) ( ) Dn ik E = ˆx exp( i + i) (c) ( ) ( Dn ẑ ηh = + iβ ) i ŷ exp( i + i). (d) Tis situation is sown in Figure. Once we ave obtained electric and magnetic fields inside te dielectric region in terms of two plane waves, recipe for fractionalization [4, ] can be applied to get te fractional dual solutions as ( ) k π ) ( E TE PMCfd = D n i cos cos + π ) ˆx β ( π ) k sin sin( + π )ŷ i ( π ) ( k sin cos + π ) ] ẑ exp i π )] (a) ( ) k π ) ( ηh TE PMCfd = D n sin sin + π ) ˆx +i β ( π ) ( k cos cos + π ) ŷ + ( π ) ( k cos sin + π ) ] ẑ exp i π )] (b)

8 54 Gulistan, Majeed, and Syed... Case : Transverse Magnetic (T M z ) Mode Propagation troug a PEC Waveguide Similar to te treatment done in Case, using Eqs. (7a) (7d) results for transverse magnetic mode propagating troug a PEC waveguide can be written as, E TM PECfd = B n ηh TM PECfd = B n ( ) k [ sin +i β k cos ( π ( π ) ( sin + π ) ( cos + π ) ŷ ) ] ẑ exp i + ( π ) ( k cos sin + π ( ) k π ) ( i cos cos + π + β ( π ) ( k sin sin + π ) ŷ ( ] +i k sin ( π ) cos + π ) ŷ ) ˆx exp i ) ˆx π π )] (a) )] (b) Fractional dual solutions for te D B waveguide can be written by taking linear sum of te fractional dual fields of te above two cases as wic give E fd = ηh fd = ( ) k exp i E fd = E T PMCfd E + ET PECfd M ηh fd = ηh T PMCfd E + ηht PECfd M π )] [(B n S S y+ + id n C C y+ ) ˆx + β k (ib nc C y+ D n S S y+ ) ŷ + ] k (B nc S y+ id n S C y+ ) ẑ (4a) ( ) k exp i π )] [(D n S S y+ ib n C C y+ ) ˆx + β k (B ns S y+ + ic C y+ ) ŷ + ] k (ib ns C y+ + D n C S y+ ) ẑ (4b)

9 Progress In Electromagnetics Researc B, Vol. 5, 55 wit ( π ) ( S =sin S y+ = sin + π ) ( π ) ( C =cos C y+ = cos + π ) B n, D n are te constants to be determined from te initial conditions... DB and D B waveguides In DB waveguide, te DB boundary beaves like PMC boundary for te bot modes, i.e., (T E z ) and (T M z ). After solving on similar lines as for D B waveguide, fractional dual solutions for te DB waveguide can be written as ( ) k [( E fd = id n C C y+ exp i π )] A n S C y+ exp i + π )]) ˆx ( ) β ( D n S S y+ exp i π )] k +ia n C S y+ exp i + π )]) ŷ ( ) ( + id n S C y+ exp i π )] k +A n C C y+ exp i + π )]) ] ẑ (5a) ( ) k [( ηh fd = D n S S y+ exp i π )] +ia n C S y+ exp i + π )]) ˆx ( ) β ( + id n C C y+ exp i π )] k A n S C y+ exp i + π )]) ŷ ( ) ( + D n C S y+ exp i π )] k +ia n S S y+ exp i + π )]) ] ẑ (5b) In D B waveguide, te D B boundary beaves like PEC boundary for te bot modes, i.e., (T E z ) and (T M z ). After solving along

10 56 Gulistan, Majeed, and Syed similar lines as for D B waveguide, fractional dual solutions for te D B waveguide can be written as ( ) k [( E fd = B n S S y+ exp i π )] ic n C S y+ exp i + π )]) ˆx ( ) β ( + ib n C C y+ exp i π )] k +C n S C y+ exp i + π )]) ŷ ( ) ( + B n C S y+ exp i π )] k id n S S y+ exp i + π )]) ] ẑ (6a) ( ) k [( ηh fd = ib n C C y+ exp i π )] C n S C y+ exp i + π )]) ˆx ( ) β ( + B n S S y+ exp i π )] k ic n C S y+ exp i + π )]) ŷ ( ) ( + ib n S C y+ exp i π )] k +C n C C y+ exp i + π )]) ] ẑ (6b) Te fields given in Eqs. (4a) (6b) are plotted in Figures, and 4 by varying values of between [, ] at an observation point (, ) = (π/4, π/4). From Figures, and 4 it can be seen tat principle of duality is being satisfied by fractional dual fields, i.e., for = E fdx = E x, ηh fdx = ηh x E fdy = E y, ηh fdy = ηh y E fdz = E z, ηh fdz = ηh z and for = E fdx = ηh x, ηh fdx = E x E fdy = ηh y, ηh fdy = E y E fdz = ηh z, ηh fdz = E z

11 Progress In Electromagnetics Researc B, Vol. 5, E X H X real of (E x,h x ) - E X H X image of (E x,h ) x real of (E y,h y ) E Y H Y image of (E y,h y ) E Y H Y E z Hz.5 E X HX real of (E z,h z ) image of (E x,h x ) (a) (b) Figure. Plots of fractional dual fields for D B waveguide, (a) real parts, (b) imaginary parts. 4. RESULTS AND DISCUSSION In order to analyze te beavior of fractional fields inside te waveguides, plots of electric and magnetic field lines in te yz-plane are presented and are sown in Figures 5, 6 and 7. We ave taken yz-

12 58 Gulistan, Majeed, and Syed.5.5 E X H X E X H X real of (E x,h x ) real of (E y,h y ) real of (E z,h z ) E z Hz E Y H Y image of (E x,h x ) image of (E y,h y ) image of (E x,h ) x E z Hz E Y H Y (a) (b) Figure. Plots of fractional dual fields for DB waveguide, (a) real parts, (b) imaginary parts.

13 Progress In Electromagnetics Researc B, Vol. 5, 59 real of (E x,h x ) real of (E y,h y ) real of (E z,h z ) (a) E X H X E Y H Y E z Hz image of (E x,h x ) image of (E y,h y ) image of (E x,h ) x (b) E X H X E z Hz E Y H Y Figure 4. Plots of fractional dual fields for D B waveguide, (a) real parts, (b) imaginary parts. plane as an observation plane. Te instantaneous field expressions are obtained by multiplying te pasor vector expressions wit exp(jωt) and taking te real part of te product. Equation tat describe te beaviour of fractional fields at a given time t can be found from te

14 6 Gulistan, Majeed, and Syed following relation. dy E fdy = dz E fdz (7) Field lines beavior is obtained by integrating above equation. Tese plots are for te mode propagating troug te guide at an angle π/6 so tat β/k = cos(π/6), /k = sin(π/6). Initial conditions for bot te modes are taken same. Electric as well as magnetic field plots for waveguides are sown by solid lines. From Figure 5 we see tat tere is no tangential component of te electric or te magnetic field for =. =.5.5 E fd.5.5 H fd = = = = Figure 5. Field lines in yz-plane at different values of ; for D B waveguide.

15 Progress In Electromagnetics Researc B, Vol. 5, 6 = =. = E fd H fd = = Figure 6. Field lines in yz-plane at different values of ; for DB waveguide. Tis is because for D B waveguide, te plates of te guide beave as perfect magnetic conductors for transverse electric components wile tey beave as perfect electric conductor for transverse magnetic mode. For te DB waveguide at =, tere is no normal component of te electric filed at te guide surface wile magnetic field as no tangential component. Tis is because for DB waveguide, te plates of te guide beave as perfect magnetic conductors for bot, te transverse electric mode and transverse magnetic mode. For te D B waveguide

16 6 Gulistan, Majeed, and Syed =.5.5 E fd H fd = = = = Figure 7. waveguide. Field lines in yz-plane at different values of ; for D B at =, tere is no tangential component of te electric filed at te guide surface wile magnetic field as no normal component. Tis is because for D B waveguide, te plates of te guide beave as perfect electric conductors for bot, te transverse electric mode and transverse magnetic mode. For all tree cases at =, we can see clearly tat electric field lines ave attain te sape of magnetic field lines of =, and magnetic field lines ave attain te sape of electric field lines of = wit opposite direction of arrows, i.e., solutions corresponds to dual waveguides. Wile for < <, te electric and magnetic field distributions corresponds to fractional dual waveguides.

17 Progress In Electromagnetics Researc B, Vol. 5, 6 5. CONCLUSIONS Fractional dual solutions to te Maxwell equations for te fields inside a parallel plate D B, DB and D B waveguides are derived using fractional curl operator. Te purpose of tis work was to complete te study of set of boundary conditions requiring vanising of te normal components of te flux densities D and B (DB boundary) or teir normal derivatives (D B, DB and D B boundary). Electric and magnetic field distributions for limiting value of corresponds to D B, DB or D B waveguide and dual waveguide, wile for < < distributions of fractional dual fields are obtained. Tis work can serve te purpose to get a variety of field distributions. REFERENCES. Oldam, K. B. and J. Spanier, Te Fractional Calculus, Academic Press, New York, Hilfer, R., Applications of Fractional Calculus in Psics, World Scientific,.. Podlubny, I., Fractional Differential Equations, Matematics in Science and Engineering, Vol. 98, XV XXIV, Academic Press, Das, S., Functional Fractional Calculus for System Identification and Controls, Springer-Verlag, Debnat, L., Recent applications of fractional calculus to science and engineering, International Journal of Matematics and Matematical Sciences, Vol. 54, 4 44,. 6. Engeta, N., Note on fractional calculus and te image metod for dielectric speres, Journal of Electromagnetic Waves and Applications, Vol. 9, No. 9, 79 88, Engeta, N., Use of fractional calculus to propose some fractional solution for te scalar Helmoltzs equation, Progress In Electromagnetics Researc, Vol., 7, Engeta, N., Electrostatic fractional image metods for perfectly conducting wedges and cones, IEEE Transactions on Antennas and Propagation, Vol. 44, , Engeta, N., On te role of fractional calculus in electromagnetic teory, IEEE Antennas and Propagation Magazine, Vol. 9, 5 46, Engeta, N., Pase and amplitude of fractional-order intermediate wave, Microwave and Optical Tecnology Letters, Vol., 8 4, 999.

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