circular waveguides with kronig–penney morphology as photonic band-gap filters

width, determined by 10-dB return loss, reaches 115 MHz (2400–2515 MHz), which covers the required bandwidth for WLANoperation in the 2.4-GHz band.

Figure 3 plots the measured radiation patterns at 2450 MHz.Radiation patterns for other operating frequencies across the 2.4-GHz band were also measured, and the obtained results are similarto those shown in Figure 3; that is, stable radiation patterns havebeen observed for the proposed antenna in the 2.4-GHz band.From the results shown in Figure 3, it is clearly seen that compa-rable E� and E� radiation fields are obtained. This characteristic ismainly due to the proposed inverted-L slot antenna with its twoequal horizontal and vertical slot arms. This also indicates that theproposed slot antenna has a dual-polarization radiation character-istic, which can enhance system performance for WLAN opera-tions, because their wave propagation environment is typicallycomplex. In addition, a good peak antenna gain of about 4.9–5.6dBi for operating frequencies across the 2.4-GHz band is obtainedfor the proposed slot antenna (see the measured peak antenna gainshown in Fig. 4).

4. CONCLUSION

A novel inverted-L slot antenna has been proposed, and a con-structed prototype placed at the corner of the display panel of anotebook computer for WLAN operation in the 2.4-GHz band hasbeen studied. The obtained impedance bandwidth (10-dB returnloss) covers the required bandwidth, and the measured radiationpatterns show comparable E� and E� radiation fields. In addition,good antenna gain for the constructed prototype has been ob-served. The obtained antenna performance makes the proposed

inverted-L slot antenna very suitable for WLAN applications in anotebook computer.

REFERENCES

1. E.B. Flint, B.P. Gaucher, and D. Liu, Integrated antenna for laptopapplications, U.S. Patent No. 6339400 B1, 2002.

2. S.N. Tsai, H.H. Shen, H.K. Dai, and K.T. Cheng, Arcuate slot antennaassembly, U.S. Patent No. 6373443 B1, 2002.

3. C.H. Huang, Slot antenna assembly having an adjustable tuning appa-ratus, U.S. Patent No. 6384794 B1, 2002.

4. C.M. Su, H.T. Chen, F.S. Chang, and K.L. Wong, Dual-band slotantenna for 2.4/5.2 GHz WLAN operation, Microwave Opt TechnolLett 35 (2002), 306–308.

© 2003 Wiley Periodicals, Inc.

CIRCULAR WAVEGUIDES WITHKRONIG–PENNEY MORPHOLOGY ASPHOTONIC BAND-GAP FILTERS

Alvaro Gomez,1,2 Miguel A. Solano,1 Akhlesh Lakhtakia,2 andAngel Vegas1

1 Departamento de Ingenierıa de ComunicacionesUniversity of CantabriaAvenida de los Castros s/n39005 Santander, Spain2 Department of Engineering Science and MechanicsPennsylvania State UniversityUniversity Park, PA 16802-6812

Received 27 November 2002

ABSTRACT: Guided wave propagation in a circular waveguide filledwith dielectric materials in the Kronig–Penney morphology is theoreti-cally examined. Allowed and forbidden frequency bands in these idealphotonic band-gap (PBG) structures are obtained for the TEz and TMz

propagation modes by invoking the Bloch theorem. Results show themany engineering possibilities for designing electromagnetic filters forideal structures. Filtering by real PBG structures with a finite numberof unit cells is also studied by using the scattering matrix technique. Asthe number of unit cells in a real filter increases, its transmission char-acteristics converge to those of its ideal analog in the forbidden fre-quency bands. © 2003 Wiley Periodicals, Inc. Microwave Opt TechnolLett 37: 316–321, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.10906

Key words: electromagnetic band gaps; Kronig–Penney morphology;circular waveguide; PBG filters

1. INTRODUCTION

PBG structures have received considerable attention due to theirtremendous potential for different applications [1–8]. As thesestructures are scalable, they are useful not only at optical frequen-cies but also at infrared, microwave, and millimeter wave frequen-cies. The benefits of PBG structures in electromagnetic technologycould turn out to be similar to that of semiconductors in electronictechnologies.

An essential feature of PBG structures is the existence offrequency bands in which electromagnetic modes cannot propa-gate. Analogous to crystals, wherein periodic arrays of atoms mayproduce bandgaps where photon propagation is prohibited, a PBGstructure is a periodic array of macroscopic cells. These cells canbe implemented in many different forms and use different mate-rials (such as dielectrics, semiconductors, metals, or, even (re-cently) metallodielectrics [9]). The periodic nature creates electro-

Figure 3 Measured radiation patterns for 2450 MHz for the proposedantenna

Figure 4 Measured peak antenna gain for the proposed antenna

316 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 37, No. 5, June 5 2003

magnetic band gaps, within which waves are forbidden topropagate in certain directions. As a consequence, PBG structurescan exhibit filtering properties which make them very attractive formicrowave and millimeter-wave circuits [7, 10–13].

In the area of microwave engineering, among the most com-monly used devices for guided wave propagation are circularwaveguides. The goal of this paper is to analyze electromagneticwave propagation in a PBG device formed by inserting a sufficientnumber of cells composed of two different dielectrics inside acircular waveguide, as well as to determine the ability of thatdevice for filtering microwave signals. The PBG structure ana-lyzed in this paper is the . . . HLHLHL . . . multilayer, where Hstands for a high-permittivity dielectric material and L for alow-permittivity one [14]. The unit cell is constituted by the HL(or the LH) pair. The structure is periodic in the propagationdirection. This type of multilayer is typical in the optics area [15,16], and its permittivity profile is reminiscent of the Kronig–Penney model in the band theory of solids [17, 18].

The analysis of an ideal PBG structure with Kronig–Penneymorphology implemented in a circular waveguide is first presentedin section 2.1. An ideal PBG structure is assumed to be of infiniteextent in the propagation direction [16]. This idealization allowsthe imposition of the Bloch theorem, thereby greatly simplifyingmathematics. However, actual filters will contain a finite numberof unit cells, which would preclude application of the Blochtheorem; thus different analytical techniques must be employed.Real filters are treated in section 2.2, where a simple mode-matching technique gives the scattering matrix of each disconti-nuity between the different layers forming the device. Subse-quently, all the scattering matrices are joined to obtain thecorresponding generalized scattering matrix of the entire structure.Obviously, the response of a real filter will not be the same as thatof the ideal one; although the responses of an appropriate sequenceof real filters can be expected to converge to an ideal one’s.Finally, the transmission characteristics of ideal and real PBGfilters are presented and compared in section 3. An exp( j�t) timedependence is implicit in this work, with � as the angular fre-quency. A circular cylindrical coordinate system (r, �, z) isemployed.

2. THEORY

2.1. Ideal PBG FilterLet us consider the boundary value problem illustrated in Figure 1where the wall r � R is perfectly conducting. Slabs of twodifferent materials are placed in the region {0 � r � R and 0 �� 2�}, such that the permittivity is periodic along the z axis andhas a binary distribution in each unit cell. Mathematically,

��r, �, z � d� � ��r, �, z�,

0 � r � R, 0 � � � 2�, �z� �, (1)

with the reference unit cell characterized by

�� z� � ��1 � �0�r1 if 0 � z � a�2 � �0�r2 if �b � z � 0 ,

0 � r � R, 0 � � � 2�, (2)

where �r1,2are the relative permittivities of the two materials, and

�0 is the permittivity of free space (or vacuum). Thus, d � a � bis the spatial period of the chosen structure in the propagationdirection.

For propagation along the z axis, the wave equation

� 2

r2 �1

r

r�

1

r2

2

�2 �2

z2 � �2�� z��0��r, �, z� � 0 (3)

must be obeyed inside the circular waveguide. Here, we havedenoted �(r, �, z) � �( z), �0 is the permeability of free space,while az�(r, �, z) is the electric vector potential F for the TEz

modes or the magnetic vector potential A for the TMz modes [19].These two types of modes have to be analyzed separately.

2.1.1. TEz Modes. For a TEz mode of propagation, the electro-magnetic field can be obtained from the electric vector potential F� az�h, where [19]:

�h�r, �, z� � AmnJm�krmnr� � �U�cos�m�� V�sin�m��e�j 1mnz

� BmnJm�krmnr� � �U � cos�m�� V � sin�m��e j 1mnz, 0 � z � a,

(4a)

�h�r, �, z� � CmnJm�krmnr� � �U�cos�m�� V�sin�m��e�j 2mnz

� DmnJm�krmnr� � �U � cos�m�� V � sin�m��e j 2mnz, �b � z � 0.

(4b)

Here, the integers m, n � � are modal labels, with � the set of allnatural numbers; krmn

� �mn/R is the modal cut-off wavenumber,where �mn represents the nth zero (n � 1, 2, 3, . . .) of the firstderivative Jm of the Bessel function Jm of the first kind and orderm (m � 0, 1, 2, 3, . . .); 1,2mn

2 � �2�0�0�r1,2� krmn

2 ; whileAmn, Bmn, Cmn, and Dmn are constant coefficients. The first partsof the right sides of Eqs. (4a) and (4b) represent waves traveling inthe �z direction, and the second parts in the �z direction.

By ensuring the continuity of the tangential components of theelectromagnetic field [19] across the surface z � 0 inside thereference unit cell, the following relations are obtained:

1

�r1

� Amn � Bmn� �1

�r2

�Cmn � Dmn� � 0, � m, n � �, (5a)

1mn

�r1

� Amn � Bmn� � 2mn

�r2

�Cmn � Dmn� � 0, � m, n � �. (5b)

In addition, the periodicity of PBG structures requires the appli-cation of the Bloch theorem [20] to all of the tangential compo-nents; thus,

Er�r, �, z��z�a � Er�r, �, z��z��bej�d, (6a)

E��r, �, z��z�a � E��r, �, z��z��bej�d, (6b)

Hr�r, �, z��z�a � Hr�r, �, z��z��bej�d, (7a)

Figure 1 Schematic of the periodic variation of the relative permittivityin an ideal EBG structure realized as a circular waveguide with Kronig–Penney morphology

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 37, No. 5, June 5 2003 317

H��r, �, z��z�a � H��r, �, z��z��bej�d, (7b)

where � is the Bloch wavenumber and must be real-valued inallowed bands. Eqs. (6) and (7) lead to

1

�r1

�Amne�j� 1mm��a � Bmne

j� 1mn��a�

�1

�r2

�Cmnej� 2mn��b � Dmne

�j� 2mn��b� � 0, � m, n � �, (8a)

1mn

�r1

�Amne�j� 1mm��a � Bmne

j� 1mn��a�

� 2mn

�r2

�Cmnej� 2mn��b � Dmne

�j� 2mn��b� � 0, � m, n � �. (8b)

Eqs. (5) and (8) are conveniently written together in matrix nota-tion as

�1 1 �1 �1

1mn � 1mn � 2mn 2mn

e�j� 1mn��a e j� 1mn��a �e j� 2mn��b �e�j� 2mn��b

1mne�j� 1mn��a � 1mne

j� 1mn��a � 2mne�j� 2mn��b 2mne

�j� 2mn��b�

� �Amn

Bmn

Cmn

Dmn� � �0

000�, � m, n � �. (9)

This equation is the same as for the corresponding parallel-platewaveguide [21]. The 4 4 matrix must be singular for Eq. (9) tohave a nontrivial solution; thus, we obtain

cos��a � b�� cos� 1mna�cos� 2mnb�

� 1mn

2 � 2mn

2

2 1mn 2mn

sin� 1mna�sin� 2mnb�, � m, n � �, (10)

as the dispersion relation. From Eq. (10), it follows that

� �1

a � bcos�1�LTEmn

z �, � m, n � �, (11)

where

LTEmnz � cos� 1mna�cos� 2mnb� �

1mn

2 � 2mn

2

2 1mn 2mn

sin� 1mna�sin� 2mnb�,

� m, n � �. (12)

2.1.2. TMz Modes. For a TMz mode of propagation, the electro-magnetic field can be obtained from the magnetic vector potentialA � az�e, where [19]:

�e�r, �, z� � FmnJm�krmnr� � �U�cos�m�� V�sin�m��e�j 1mnz

� GmnJm�krmnr� � �U�cos�m�� V�sin�m��e j 1mnz, 0 � z � a,

(13a)

�e�r, �, z� � KmnJm�krmnr� � �U�cos�m�� V�sin�m��e�j 1mnz

� LmnJm�krmnr� � �U�cos�m�� V�sin�m��e j 1mnz, �b � z � 0.

(13b)

Here, krmn� �mn/R is the modal cut-off wavenumber, where �mn

represents the nth zero (n � 1, 2, 3, . . .) of the Bessel functionJm. The meanings of the other symbols are similar to thoseprevious section. Following exactly the same procedure as for TEz

modes, we find that eq. (11) remains valid upon replacing LTEmnz by

LTMmnz � cos� 1mna�cos� 2mnb�

� 1mn

2 �r2

2 � 2mn

2 �r1

2

2 1mn 2mn�r1�r2

sin� 1mna�sin� 2mnb�, � m, n � �. (14)

2.2. Real PBG FilterA real PBG filter (see Fig. 2) does not have longitudinal periodicitybecause it has a finite number Ncell of unit cells, characterized byEq. (2), inside the circular waveguide. The input and the outputwaveguide sections are also similar, and are usually unfilledwaveguides. The solution of this boundary value problem is bestformulated in terms of a generalized scattering matrix (GSM).Each transverse discontinuity is treated as the junction of twosemi-infinite waveguides filled homogeneously with different ma-terials, in a form such that the usual TEz and TMz modes arematched at the junction to produce a GSM. With no internaldiscontinuity in any transverse direction, each individual TEz orTMz mode satisfies the appropriate continuity condition at everylongitudinal discontinuity plane. Then, the GSM is reduced to theusual 2 2 circuital scattering matrix for a two-port circuit. Thus,the scattering matrix for any of the discontinuities between thedifferent dielectrics shown in Figure 2 could be written as

�S� � �S11 S12

S21 S22� , (15)

where the diagonal terms are reflection coefficients and off-diag-onal terms are transmission coefficients. The voltage-normalizedvalues of these coefficients are [22]:

S11 � �S22 � �Zright � Zleft�/�Zright � Zleft�, (16a)

S21 � �1 � S11�Zleft/Zright, (16b)

S12 � �1 � S22�Zright/Zleft, (16c)

with Zleft � ��0/ left and Zright � ��0/ right the wave impedancesfor the TEz mode chosen in the discontinuity, and Zleft � left/(��left) and Zright � right/(��right) for the TMz mode. The sub-

Figure 2 Schematic of the variation of the relative permittivity in a realEBG filter implemented in a circular waveguide. The first and the lastsections are unfilled and extended semi-infinitely


script left applies to the waveguide section on the left of thediscontinuity and subscript right to the waveguide section on theright of the discontinuity; also, the subscript mn has been removedfrom in every instance because this formulation is applicable toany single mode.

The overall scattering matrix of the cascade of discontinuities isobtained by consecutively linking the scattering matrix of eachdiscontinuity with that of the closest following one, as described indetail by Van Blaricum and Mittra [23]. The complex-valuedtransmission coefficient �T� across the Ncell unit cells of a real PBGfilter, can then be easily computed from the overall scatteringmatrix.

3. RESULTSThe functions LTEmn

z and LTMmnz were computed for a circular

waveguide endowed with the Kronig–Penney morphology. Bothdispersion and dissipation were ignored; thus �r1

� 1 was keptfixed, whereas �r2

� �r1could vary. We focused our attention on

the lowest-order modes—TE11z and TM01

z —of which the former isregarded as the fundamental mode.

In Figure 3 the computed values of LTE11z and LTM01

z are shownwith �r2

� 2, R � 300 �m, d � 600 �m and a � 0.85 d.Because � must not be real-valued in a forbidden band, satisfactionof the condition �1 � cos(�d) � 1 identifies allowed bands forboth TE11

z and TM01z modes. Therefore, the lines LTE11

z � �1 andLTM01

z � �1 are also shown in Figure 3. Clearly, several forbiddenbands exist. Significantly, in addition to �LTEmn

z � 1 or �LTMmnz �

1 in the first allowed band, the frequency must be greater thanthe cut-off frequency because for guided-wave propagation, 1mn

2

� �2�0�1 � krmn

2 and 2mn

2 � �2�0�2 � krmn

2 must be realpositive.

Figure 4 shows the dependence of the band gaps (that is, theforbidden bands) against frequency for the fundamental nodeTE11, for four different values of the permittivity contrast �r2

/�r1

and for a fixed-cell geometry. Inspecting these figures, we inferthat the number of band gaps increases for a fixed value of the unitcell thickness (in this case 600 �m), as the permittivity contrast isincreased; that is, there is a compression of the allowed andforbidden bands, so that the widths of both allowed and forbiddenbands decrease. It is possible that, in some cases, the number of

allowed bands decreases when the permittivity increases. This isbecause allowed and forbidden bands migrate to lower frequencieswhich lie in the cut-off regime, and this regime is always aforbidden band. In Figure 5 analogous results can be seen for afixed permittivity contrast �r2

/�r1and variable cell thickness. Same

conclusions are inferred: as the cell thickness is increased, thenumber of band gaps also increases; the allowed as well as theforbidden bands are compressed, and thus their bandwidths arereduced.

In Figures 6–8 the locations of the photonic band gaps for theTE11

z mode are identified with respect to geometric and constitu-tive parameters. These figures are called gap maps. The gap mapsare very useful to demonstrate the engineering potentially of thePBG structures. Figure 6 shows the allowed (white zones) andforbidden (black zones) bands with respect to the frequency (hor-izontal axis) and radius R of the waveguide (vertical axis) for fourdifferent values of cell thickness d while the radio a/d is fixed.Obviously, the first allowed band appears when the frequency isgreater than the cut-off frequency for the considered mode. Oninspecting these four plots, two important consequences are de-

Figure 3 Variation of LTE11z and LTM01

z with frequency f � �/ 2� for �r1�

1, �r2� 2, R � 300 �m, d � 600 �m, and a � 0.85 d in a circular

waveguide. Propagation is forbidden if �LTE11z � � 1 or �LTM01

z � � 1, asapplicable

Figure 4 Variation of LTE11z with f in a circular waveguide for different

values of �r2when �r1

� 1, R � 300 �m, d � 600 �m, and a � 0.85 d

Figure 5 Varation of LTE11z with f in a circular waveguide for different

values of the unit cell thickness, where �r2� 2, �r1

� 1, R � 300 �m,and a � 0.85 d


duced. First, the band gaps do not critically depend on the radiusof the waveguide as it is increased beyond a threshold value. Thisis because, after the threshold value is reached, the propagationconstant in any waveguide section is significantly greater than thecut-off wavenumber. Hence the propagation constant on the TE11

z

mode in any section approaches the wavenumber in that sectionand is not critically dependent on the radius. In this situation, theguide is practically nondispersive. Second, as the cell thickness isincreased the band gaps are compressed, which agrees with Fig-ures 3–5.

The gap maps in Figure 7 have variations with respect todimension a for four different values of the unit cell thickness dwith �r2

/�r1� 2 and R � 300 �m. For a � 0 and a � d, the

entire waveguide is filled with only one medium so that theforbidden bands are naturally absent. For a fixed cell of thicknessd, new bandgaps emerge on the low-frequency side as the ratio a/dis increased. This is most clearly shown in the gap map for d �600 �m in Figure 7. Furthermore, the number of band gaps

increases as d increases, but the bandwidths are reduced. Finally,gap maps are shown in Figure 8 for four different values of thepermittivity contrast �r2

/�r1. These gap maps support the conclu-

sions drawn from Figure 7.All the results presented in Figures 3–8 hold for ideal PBG

filters, whose allowed and forbidden bands are sharply differen-tiable from each other. Notionally, the transmission coefficient �T�� 1 in allowed bands and �T� � 0 in forbidden bands. But, asmentioned before, ideal PBG structures cannot be implementedbecause they must contain infinite number of unit cells. In contrast,a real PBG filter must have a finite number of unit cells. Thetransmission coefficient magnitudes of a sequence of real PBGfilters (with Ncell � 10, 25, 50, and 75) are shown in Figure 9.Clearly, as Ncell increases, the spectrum of the transmission coef-

Figure 6 Allowed and forbidden bands for TE11z mode with respect to f

and R for different values of d when �r1� 1, �r2

� 2, and a � 0.85 d.Allowed and forbidden bands are shown in white and in black, respectively


and a for different values of d when �r1� 1, �r2

� 2, and R � 300 �m


and a for different values of �r2where �r1

� 1, d � 150 �m, and R �300 �m

Figure 9 Transmission coefficient magnitudes in dB of a sequence ofreal EBG filters (Ncell � 10, 25, 50, and 75) when �r1

� 1, �r2� 2, a �

50 �m, d � 300 �m, and R � 300 �m. For comparison, the notionaltransmission coefficient of the analogous ideal EBG structure has also beenplotted


ficient proceeds closer to that of the notional transmission coeffi-cient of the analogous ideal PBG filter.

Although the ideal PBG filter cannot be practically imple-mented, its analysis is very important to quickly obtain initialvalues of the relative permittivities �r1

and �r2, the radius R of the

waveguide, and practically the unit-cell dimensions a and b. Oncethese values have been ascertained, the real filter with a finitenumber of unit cells can be designed with the optimum number ofunit cells for the desired response.

4. CONCLUSION

In this paper, we analyzed guided wave propagation in a circularwaveguide filled nonhomogeneously in the longitudinal directionin order to emulate the Kronig–Penney model in the band theory ofsolids. First, the ideal case was studied considering an infinitenumber of unit cells; consequently the Bloch theorem could beinvoked. The theory confirmed the existence of forbidden andallowed frequency bands, so that these periodic structures can beuseful for designing electromagnetic filters. The results for thetransmission characteristics of ideal PBG filters were comparedwith those computed for a sequence of real PBG filters by usingthe scattering matrix technique. Numerical results clearly showedthat the response of a real filter converges to that of the idealfilter’s as the number of unit cells increases.

ACKNOWLEDGMENT

This work and the stay of A. Gomez at the Pennsylvania StateUniversity were partially supported by the Direccion General deInvestigacion, MCyT, Spain, under project no. TIC2000-1612-C03-01.

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© 2003 Wiley Periodicals, Inc.

AUSTRALIA AND NEW ZEALANDSATELLITE COVERAGE USING AMICROSTRIP PATCH REFLECTARRAY

J. A. Zornoza1 and M. E. Bialkowski21 Department of Electromagnetism and Circuit TheoryPolytechnic University of MadridAv Complutense s/n 28040 Madrid, Spain2 School of Information Technology & Electrical EngineeringSt. Lucia, University of QueenslandQueensland 4072, Australia

Received 15 November 2002

ABSTRACT: This paper presents the design of Ku-band (12.25–12.75GHz) dual-polarized reflectarrays for Optus B1 satellites to obtain acontoured beam for Australia and New Zealand. The specified radiationpattern is synthesized using a phase-only synthesis method based on theconcept of intersection approach. Having determined the phasing data,single- and double-layer reflectarrays are designed using variable-sizerectangular patches. The performances of the two reflectarrays are as-sessed by comparing their radiation patterns with the assumed pattern.© 2003 Wiley Periodicals, Inc. Microwave Opt Technol Lett 37: 321–325,2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.10907

Key words: reflectarray; multi-layer microstrip patch array; radiationpattern synthesis; antennas for satellite broadcasting


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