Electrical Engineering 82 (2000) 353±361 Ó Springer-Verlag 2000

Line of periodically arranged passive dipole scatterers S. A. Tretyakov, A. J. Viitanen

Contents Electromagnetic properties of line-periodical arrangements of passive loaded dipole scatterers are studied. An analytical solution for eigenwaves propagating along in®nite lines of dipoles is presented. Re¯ective properties of arrays are studied, regimes of narrow-band and wide-band strong re¯ection are identi®ed and conditions for these operations found. Conditions of existence of guided-wave solutions are established. It is shown that in arrays of capacitively-loaded antennas very rapid phase variations along the line are possible, which can possibly be used to realize wide-band superdirective re¯ectors.

Periodisch linear angeordnete, passive DipolstreukoÈrper È bersicht Die elektromagnetischen Eigenschaften von U periodisch entlang einer Linie angeordneten, passiv belasteten DipolstreukoÈrpern werden untersucht. Die analytische LoÈsung von Eigenwellen, die sich entlang einer unendlichen Reihe linear angeordneter Dipole ausbreiten, wird angegeben. Die Re¯exionseigenschaften von linearen Anordnungen werden untersucht, die Funktionsweisen von Schmalband- und Breitbandre¯exionen werden identi®ziert und Vorraussetzungen fuÈr diese gezeigt. Vorraussetzungen fuÈr die Existenz von LoÈsungen fuÈr gefuÈhrten Wellen werden hergeleitet. Es wird gezeigt, dass in Anordnungen von kapazitiv belasteten Antennen sehr schnelle Phasenverschiebungen entlang der Antennenanordnung moÈglich sind, die moÈglicherweise fuÈr die Realisation von breitbandigen supergerichteten Re¯ektoren verwendet werden koÈnnen. 1 Introduction Electromagnetics of periodic structures is a very old and well developed ®eld of research. Various spatially-periodic arrangements are used in many practical devices, such as microwave and optical ®lters, array antennas, lasers. However, this topic remains of current interest. Very much attention in the literature has been recently payed to electromagnetic properties of periodical structures, especially in view of potential applications in light-wave

Received: 22 May 2000

S. A. Tretyakov (&), A. J. Viitanen Electromagnetics Laboratory, Helsinki University of Technology, P.O. 3000, FIN-02015 HUT, Finland

technology [1]. It is well known that when the wavelength of electromagnetic waves propagating in periodical structures is comparable to the period, strong spatial dispersion effects take place. In particular, in certain frequency ranges there can be no propagating modes, at least for some of the polarizations and propagation directions. These ranges are sometimes called band gaps or photonic band gaps, following an analogy with semiconductor physics. Also planar periodical arrangements of various inclusions are of interest, and they are used as frequency selective surfaces in microwave applications. Nowadays, new applications in microwave ®lters and other devices are discussed [2]. Linear periodical arrangements of small inhomogeneities (small disk patches or small holes in the ground plane) have been recently studied experimentally, with interesting resonance effects revealed [3, 4]. Waveguide channels in photonic crystals can be realized, for example, by removing one line of inclusions. This can be considered as a periodic perturbation of a regular crystal, and treated with similar techniques as other periodic arrangements. For review of recent results see [1]. Besides ``photonic crystals'', there are many other modern applications. One of them is in the design of optical ®lters using periodic gratings [5, 6]. Narrow-band ®lters can be realized if a layer with periodic spatial modulation of its properties (most often, the thickness is modulated) can support surface waves. In the antenna theory, electromagnetics of periodical structures is the key of understanding antenna arrays. Field coupling of antennas in an array is usually considered as a parasitic factor since this can cause scan blindness effect. Physically, this effect is connected to so called Wood anomalies of gratings, caused by possible excitation of higher-order Floquet modes or surface modes in the array [7]. This is in fact the same phenomenon which is utilized in guided-mode resonant optical ®lters [5, 6]. However, antenna ®eld coupling appears to offer interesting possibilities also in the array antenna techniques. Here, one of the fundamental problems is the practical design of superdirective antennas. They require current distributions which very quickly vary in space. In case of antenna arrays, to achieve a superdirective operation excitation of adjacent antenna elements should be out of phase or similar to that. It has been shown [8, 9] that such excitation is possible in arrays of passive resonant scatterers (conductive cavities with slot openings and resonant grooves have been considered). Recent advanced studies of periodic structures such as photonic crystals, waveguiding channels in these crystals

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and of superdirective gratings have been performed using only numerical techniques. This makes it dif®cult to understand the governing physical phenomena and ®nd conditions necessary for realization of desired effects. Here we consider a conceptually simple periodic electromagnetic system in which all practically interesting effects can be realized. We solve the corresponding problem using analytical means and reveal various modes of operation as dependent on the properties of the array elements and the frequency. The chosen system is a periodical arrangement of small dipole antennas or polarizable particles when they are positioned along a line in isotropic space. This can be classi®ed as a one-dimensional periodical structure (although the associated electromagnetic problem is threedimensional). Of course, this is not a new object for study. Periodical inclusions in waveguides have been used for a long time in the design of microwave ®lters. Field coupling in an array of small dipole antennas has been studied for example in [10]. In the antenna theory, the main objective of these studies has been reducing the blidness effect, which requires calculations of the input impedances of array antennas as dependent on the scan angle. More recently, line-periodically perforated ground planes and substrates were considered [3, 4] and suggested to reduce parasitic resonances in microstrip ®lters. Our objections are different. We consider plane-wave excitation of straight lines with periodically positioned dipole inclusions and study eigenwaves which can propagate along such lines. In other words, we are interested in re¯ections from such structures and in waveguide modes propagating along such lines. In addition, we consider dipoles loaded by passive bulk loads, so that various excitation regimes can be realized. The solution can help to understand possible superdirectivity effects in arrays of loaded antennas. Another motivation for this study is the need for transmission lines with as small cross section as only possible. Here we show that waveguides can be manufactured as periodical arrangements of very small particles, and the distance between them can be quite comparable to the wavelength or even larger than that. As mentioned above, this object is also a model for waveguides in photonic crystals, since they can be formed by removing a line of inclusions, so the waveguide is formed by periodical arrangement of discrete elements. Furthermore, we show that periodical arrangements of geometrically very small inclusions can be very strong and wide-band re¯ectors, because of a certain resonance caused by electromagnetic interactions in the structure. In this theoretical study we assume that every inclusion can be modelled as an electric (or, using duality, magnetic) dipole. That is, the geometrical size of every separate inclusion is small compared to the wavelength. For example, these can be short pieces of conducting wires (lossless or lossy), small dielectric spheres or other similar objects. Loading short antennas by bulk loads (or using coated spheres or coated wire antennas for example), the polarizability can be changed. However, as the electrical size is assumed to be small, the radiation properties are still that of a short dipole antenna. Here we consider the case when

all the dipoles are directed along the axis of the structure. No assumption is made regarding the distance between inclusions and the full-wave interaction between all particles is taken into account.

2 Eigenvalue equation and the interaction field Geometry of the problem is shown in Fig. 1: small dipole particles are periodically arranged along a certain axis in space. Electromagnetic parameters of the surrounding isotropic space are denoted by 0 and l0 although the theory is not restricted to the free-space background. Polarizability of every inclusion we denote by a. All the induced dipole moments are directed along the axis z. 2.1 Eigenvalue equation For an arbitrarily chosen dipole on the line array (position z ˆ 0) p…0† ˆ aEloc

…1†

where Eloc is the local ®eld created by external sources and all the other particles. Assuming that the external ®eld Eext is a plane wave (or it is absent), we can make use of the Floquet theorem and write

p…nd† ˆ e

jqnd

p…0†

…2†

where q is the propagation factor which we will determine in this study. The local ®eld is created by external sources and by the other dipoles in the array: 1 X

Eloc ˆ Eext ‡ nˆ

1 1 jk 3‡ 2p …nd†2 0 …jnjd† 1; n6ˆ0 e

jkjnjd

e

jqnd

p…0†

!

…3†

where we have substituted the electric dipole ®elds (we use p the usual notation k ˆ x 0 l0 ). Let us denote by b the interaction constant

Fig. 1. Periodical arrangement of longitudinally directed dipole particles along a straight line. The system is in®nite in the z direction. Each inclusion is an electric dipole (wire antenna) which can be loaded by bulk passive loads. Examples of capacitive and inductive loadings are shown

S. A. Tretyakov and A. J. Viitanen: Line of periodically arranged passive dipole scatterers

! 1 1 jk ‡ e 2p0 …jnjd†3 …nd†2 1; n6ˆ0

1 X

bˆ nˆ

jkjnjd

1 qkq x2 jp…0†j2 2 Sz ˆ g d2 2 16 pkq q

jqnd

e

…12†

This component of the Poynting vector does not contribute to the power balance, because every dipole receives With this notation, at the position of the reference dipole and re-radiates this amount of power in the longitudinal p…0† the local ®eld is direction. The power radiated by each dipole in the q-direction Eloc ˆ Eext ‡ bp…0† …5† reads If there is no incident ®eld, then p…0† ˆ abp…0†. Thus, we 2 1 gk3q x2 jp…0†j2 4pqd gk2q jp…0†j have the eigenvalue equation

…4†

Prad ˆ Sq 2pqd ˆ

ab ˆ 1 or

" # 1 X 1 1 1 jk ˆ e ‡ a nˆ 1; n6ˆ0 2p0 …jnjd†3 …nd†2

jkjnjd

e

1

ab

Eext

pkq q

ˆ

k

80 l0 d

In the absence of dissipation loss in the scatterers, this power is equal to the power received by each dipole:

2.2 Energy conservation condition If the distance between the dipoles and the frequency satisfy kd < 2p (for real wavenumbers k), only one cylindrical wave excited by the averaged current line exists in the far zone (one fundamental Floquet mode). This fact can be used to establish a relation between the imaginary part of the interaction constant and the imaginary part of the particle polarizability, similarly to the theory developed in [11] for planar two-dimensional arrays. Let us assume that the background medium is lossless (k is real) and  the transverse propagation factor p kq ˆ k2 q2 is real. This means that the line radiates a cylindrical wave in the far zone, and the q-component of the Poynting vector is not zero. In this case the imaginary part of the interaction constant can be found exactly from the power balance condition. Let us suppose that the array is excited by a plane wave and the z-component of the external ®eld at z ˆ 0 is Eext . Then, the dipole moment of the particle located at z ˆ 0 is p…0† ˆ

d2

…13†

jqnd

…6†

a

2 16k

…7†

Prec ˆ

  x a jEext j2 Im 2 1 ab

…14†

This leads to the following relation between the imaginary part of the interaction constant and the particle polarizability:

  1 Imfbg ˆ Im a

  gxk2q 1 ˆ Im 4kd a

  k2 kq 2 40 d k

…15† This result can be checked by direct summation of the dipole ®elds. Indeed,

" 1 1 X k cos knd Imfbg ˆ p0 nˆ1 …nd†2

# sin knd cos…nqd† …nd†3

…16† Using the known summations valid for 0 < a < 2p 1 X cos na p2 pa a2 ‡ ; ˆ 6 4 n2 2 nˆ1 1 X sin na p2 a pa2 a3 ‡ ˆ 4 12 n3 6 nˆ1

…17†

The averaged (along the z direction) ®eld components are we ®nd [12]

k2q jxp…0† jqz …2† e H0 …kq q† …8† 4k d qkq jxp…0† jqz …2† e H1 …kq q† Eq ˆ jg …9† 4k d kq jxp…0† jqz …2† e H1 …kq q† Hu ˆ j …10† 4 d p where g ˆ l0 =0 is the wave impedance. The amplitude of the normal to the line direction (which means the direction along q) component of the Poynting vector in the far zone (kq q ! 1) is ( ) gk3q x2 jp…0†j2 2 1 …11† Sq ˆ Re 16k d2 2 pkq q Ez ˆ

g

Also, there is longitudinal (along z) power ¯ow:

8 3 …kd† 2 p > > …kd†2 Š; 0 < qd < kd > 6 ‡ 4 ‰…qd† > 3 1 < …kd† ; kd < qd < 2p kd 6 Imfbg ˆ …kd†3 p0 d3 > > ‡ p4 ‰…2p qd†2 …kd†2 Š; > > : 6 2p kd < qd < 2p ‡ kd …18† 1

(this function is 2p-periodic with respect to qd† In the region q < k, where the line of dipoles radiates a cylindrical wave, (18) gives the same result as (15), because for dipole scatterers [11]

  1 k3 ˆ Im 6p0 a

…19†

1 This summation was done in [10], but not all the terms in the dipole ®eld were included.

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Electrical Engineering 82 (2000)

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The mathematical fact that the series expressing the imaginary part of the interaction constant reduces to a simple closed form although the real part can only be calculated numerically is related to the physics of the problem. The imaginary part is determined by the energy conservation requirement which relates the imaginary part of the interaction constant with far-zone radiated ®eld. The real part of the interaction constant describes the nonradiating near ®eld created by the array. Consider the case of the normal incidence on the array (q ˆ 0). Of course, (15) is valid only if no higher-order Floquet modes are excited, that is, for kd < 2p. This is illustrated by Fig. 2, where the imaginary part of the normalized interaction constant is plotted as a function of kd. For kd < 2p, we have

  1 Imfbg ˆ Im a

  x 1 ˆ Im l0 4d a 2

2

k 40 d

…20†

For larger kd, the imaginary part of the interaction constant is much smaller than the right-hand side of (20), because some part of the radiated power is scattered into higher-order Floquet modes. Thus, although the power radiated by every dipole sharply increases with the frequency, the imaginary part of the sum of these ®elds (the imaginary part is responsible for the energy transport) at the position of every inclusion does not follow this trend. If kq is purely imaginary (for real k this means that q > k), the averaged ®elds become

2 jkq j2 jxp…0† jqz …21† e K0 …jkq jq† p 4k d 2 qjkq j jxp…0† jqz e K1 …jkq jq† Eq ˆ g …22† p 4k d 2 jkq j jxp…0† jqz e K1 …jkq jq† Hu ˆ …23† p 4 d In this case there is no power ¯ow in the q direction because RefEz Hu g ˆ 0 and the ®elds exponentially decay with increasing q. The general expression for the imaginary part of the interaction constant (18) is still valid. Ez ˆ jg

3 Reflection at the normal incidence For any dipole particle we can write       1 1 1 ‡ Im ˆ Im Im a arad aloss   3 k 1 ˆ ‡ Im 6p0 aloss

…24†

where the two parts in the right-hand side refer to the radiation (``rad'') and dissipation loss, respectively. If a line of dipoles is excited by a plane wave propagating normally to the z axis, we ®nd, using (7), that for kd < 2p

p…0† ˆ

Re

1

a

1 h n o i Eext 2 1 ‡ 4k0 d b ‡ j Im aloss

…25†

Relations (18) and (24) have been taken into account. The re¯ected ®elds are given by (8)±(10). For larger distances between the inclusions the scattering loss is higher because relation (15) is not valid and the dipole radiation term, proportional to …kd†3 , does not cancel from (25) (see illustration in Fig. 2). The real part of the interaction constant in (25) is

" # 1 1 X cos knd k sin knd Refbg ˆ ‡ p0 nˆ1 …nd†3 …nd†2

…26†

Typical behaviour of this function is shown in Fig. 3. The series converges rather quickly, and the dominate term is kd sin …kd†, that is why the maxima of the function grow for large kd. Resonances in re¯ection can appear when this quantity equals Ref1=ag. For short metal inclusions of length 2l  k made of round conducting wires of radius r0

Refag ˆ

p0 l3 log r2l0

…27†

Fig. 3. Real part of the normalized interaction constant for the Fig. 2. Imaginary part of the interaction constant for the normal normal incidence of plane waves. Here, the inclusions are thin round conducting wires with l=r0 ˆ 5 and d ˆ 3l incidence of plane waves

S. A. Tretyakov and A. J. Viitanen: Line of periodically arranged passive dipole scatterers

Thus, the equation for re¯ection resonances becomes  3 d 2l log ˆ p0 d3 Refbg …28† l r0 Normally, the left-hand side is a large number, see example in Fig. 3 for l=r0 ˆ 5 and d ˆ 3l. For larger kd there are solutions but there the inclusions are not any more small compared to the wavelength. The diverging behaviour of the interaction constant at high frequencies is restricted by lossess in space. If the wavenumber k ˆ k0 jk00 is a complex number, (26) becomes 1 1 X 00 Refbg ˆ e k nd p0 nˆ1 " # cos k0 nd k0 sin k0 nd ‡ k00 cos k0 nd  ‡ …nd†3 …nd†2

where l is the semilength of the dipole and

Cˆ

pl0 log…2l=r0 †

…31†

is the equivalent capacitance. The real part of (30) can be changed by loading the dipole by reactive bulk loads. If the dipoles are loaded by inductances L, we get

 1 k3 1 ‡ 1 ˆj 6p0 l2 C a

k2 k20



…32†

where k20 ˆ 0 l0 =…LC†. Moreover, capacitance C can also be changed by bulk capacitive loads, so in general the following is realizable (with geometrical size of an inclusion much smaller than the wavelength):

 1 k3 ˆj ‡A 1 6p0 a

k2 k20



…33†

where A is a frequency-independent constant. Consider the electric ®eld of the cylindrical wave in the far zone, which we write in the form This function tends to zero at d ! 1. However, if losses are small, quite high values of the interaction constant are e jkq E ˆ E …34† possible, see an example in Fig. 4 for sc 0 p kq Imfkg=Refkg ˆ 0:05. If the dipole particles are loaded by bulk reactances, the In the following numerical examples the normalized re¯ected ®eld is calculated, which is de®ned as polarizability can be very large even for small inclusion

…29†

E0 R ˆ ˆ  Eext p0 d3 A 1

k2 k20



p 2p…kd†2  1
P …kd†3 cos knd kd sin knd ‡ ‡ j n3 n2 6

1 P

nˆ1

nˆ1

kd cos knd n2

sin knd n3




…35†

length, re¯ection coef®cient can be very large, either as a For comparison, re¯ection coef®cient from a continuous resonance effect or even in wide frequency bands. For thin conducting wire of the same radius r0 as the dipole's short ideally conducting pieces of thin wire wire is also shown. The last quantity is

1 k3 1 ˆj ‡ 2 6p0 l C a

…30†

Rwire

r 2 ˆ p1

1 j p2

log kr20


‡ 0:5772

…36†

Here, the wire has been modelled as a line of current. First, we consider an array of dipoles with only inductive loads, such that at the resonance k0 d ˆ p. The other parameters are the following: d=l ˆ 30, l=r0 ˆ 50. At the resonance frequency of the dipoles we observe a very narrow resonant re¯ection. Re¯ection from the line of dipoles can be compared also with re¯ection from a single dipole in free space. Radar cross section of a dipole particle with the polarizability a is

1 …xkg†2 jaj2 4p Thus, we can compare formula (35) with p 1 RCS ˆ p k2 jaj 2 p0 p d p …kd†2   ˆ 2 p d3 A 1 k22 ‡ j…kd†3 =6 0 k RCS ˆ

Fig. 4. Real part of the interaction constant for the normal incidence of plane waves. Lossy background medium (Imfkg=Refkg ˆ 0:05)

…37†

…38†

0

The comparison shows that the resonance seen in Fig. 5 is mainly due to the individual inclusion resonance. The line

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Electrical Engineering 82 (2000)

is that due to the resonance conditions the currents induced in the inclusions have very high amplitudes. The resonance here means that Ref1=ag becomes close to Refbg. This effect is much more broadband as compared to the single particle resonance effect (compare with Fig. 5). Finally, if the load contains both capacitances and inductances, so that p0 d3 A ˆ 50 and k0 d ˆ p, the result is shown in Fig. 7. Here, a broad and strong resonant re¯ection peak exists at the resonance frequency, as the single inclusion resonance is enhanced due to strong interaction with other particles.

358

4 Propagation of waves along lines of dipoles 4.1 General relations To study travelling or exponentially decaying waves along the z axis we shall study the interaction constant (4) and Fig. 5. Normalized re¯ected ®eld for an inductive load. The resonance frequency corresponds to k0 d ˆ p the eigenvalue equation (6) in more detail, for the case when q 6ˆ 0. If there is some radiation loss of energy and width is only slightly wider due to the interaction effects, there is energy dissipation in the inclusions, the eigenvalue and the resonance frequency shift described by the real equation reads (assuming that part of the interaction constant is rather small. Imf1=ag ˆ Imf1=arad g ‡ Imf1=aloss g)     Next, for particles with capacitive loads (the polariz3 ability is given by (30), and parameter p0 d3 =…l2 C† ˆ 50 in Re 1 ‡ j k ‡ jIm 1 ˆb …39† 6p0 a aloss this example) we observe the behaviour shown in Fig. 6. In this case, there is no particle resonance in this frequency This is a complex equation in which the real and imagirange. We observe that the re¯ection grows in the region nary parts should vanish. If the particles have no dissifrom kd ˆ 0 to kd ˆ 2p, and then there are sharp knees at pation, the propagation factor q can be real. For real kd ˆ 2p and kd ˆ 4p. This is because new higher-order propagation factors, the last equation takes the form modes start to radiate at these frequencies, and part of the     1 k3 1 power is scattered in other directions. As a result, the re‡j Re ‡ jIm ¯ection in the main cylindrical mode gets weaker. This 6p0 a aloss " # effect is sometimes called Wood anomaly, or classical 1 1 X cos knd k sin knd Wood anomaly to distinguish from anomalies connected ˆ jImfbg ‡ cos qnd ‡ with excitation of surface modes. p0 nˆ1 …nd†3 …nd†2 Note that the inclusion electric size here is the same as …40† in the previous example, that is, inclusions radiate as small dipoles. However, re¯ection is much stronger. The reason

Fig. 6. Normalized re¯ected ®eld for a capacitive load. No particle resonance. Classical Wood anomalies can be seen

Fig. 7. Normalized re¯ected ®eld for a combined resonance load. The resonance frequency of a single inclusion corresponds to k0 d ˆ p

S. A. Tretyakov and A. J. Viitanen: Line of periodically arranged passive dipole scatterers

where Imfbg is given by (18). Because in the region2 kd < qd < 2p kd the dipole radiation term k3 =…6p0 † in equation (40) cancels out [see (18)], propagating guided waves can exist under this condition. Physically, this cancellation comes about because of the energy conservation requirement: the power radiated by each particle equals the power received by the same particle, so that the line does not radiate any power in the far zone. The transverse wavenumber kq is imaginary in this case. If the line is lossy, the propagation factor obviously must be a complex number. Guided waves (with some loss of energy) are possible as in other slow-wave structures. The ®eld is con®ned to the line if kq has non-zero imaginary part. If q is real but smaller than k, no guided waves can exist in lines of passive scatterers. It is obvious [see (18)] that in this case the imaginary part of the right-hand side of (40) is smaller that of the left-hand side. The power received by every particle (right-hand side) is smaller than that radiated by the same particle (left-hand side) Fig. 8. Normalized imaginary part of the interaction constant for the guided-mode regime because some part of the power is radiated into a cylindrical wave. < 2p kd where the guided solutions are possible (the imaginary part of the dispersion equation is satis®ed). On 4.2 the ¯oor of the graph, curves of constant levels of the Guided waves in lossless structures For the guided-wave solutions with real propagation fac- function are shown. These curves show dispersion curves tors we have q > k and imaginary transverse wave number for the guided modes in case if the left-hand side of the real part of the dispersion equation (that is, Ref1=ag) is kq . In this situation electromagnetic ®elds exponentially decay in the transverse direction. Existence of guided-wave frequency-independent. The family of dispersion curves solutions and the dispersion relation of these waves can be for this case is also shown in Fig. 10. Here, both guidedwave and leaky wave solutions are shown. The two regimes determined from the corresponding complex eigenvalue are separated by the line q ˆ k. For q < k the solutions to equation discussed above. the dispersion For lossless particles, the scattering loss of every p equation are leaky modes, because kq ˆ k2 q2 is real and the line of dipoles radiates inclusion should be balanced by the interaction ®eld power. For larger normalized frequencies kd > p there are created by other inclusions. To determine if this is no guided-wave solutions, since the energy conservation possible, we consider the imaginary part of the requirement cannot be satis®ed. This is so because for eigenvalue equation:   larger kd there exist higher-order radiating Floquet modes 3 1 k which take power away from the guide. ˆ Im 6p0 a Thus, depending on the value of the real part of the " # inverse polarizability, guided waves can exist even at low 1 1 X d cos nkd sin nkd ˆ cos nqd …41† frequencies, having zero cut-off value. Let us consider an

p0 nˆ1

…nd†2

or, in the normalized form, 1  …kd†3 X kd cos nkd ˆ 6 n2 nˆ1

…nd†3

 sin nkd cos nqd n3

…42†

and use the summation (18). Obviously, the guided-mode solutions are permitted by the energy conservation in the region kd < qd < 2p kd (this is also true in the regions kd ‡ 2pm < qd < 2p…m ‡ 1†, m ˆ 1; 2; . . .). This is illustrated by Fig. 8, where the regions of the validity of the energy conservation are clearly seen. Next, let us consider the real part of the eigenvalue equation. The real part of the interaction constant has to be evaluated numerically. The result is shown in Fig. 9. Calculations are made only for the region kd < qd 2 Note here that the imaginary part of the interaction constant is Fig. 9. Normalized real part of the interaction constant for the guided-mode solutions 2p-periodic with respect to qd.

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Electrical Engineering 82 (2000)

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Fig. 10. Dispersion curves for frequency-independent values of Ref1=ag. The corresponding normalized values of p0 d3 Ref1=ag are shown near each curve. Curves for p < qd < 2p kd can be obtained by re¯ecting the picture around the line kd ˆ p. Solid lines correspond to guided-wave solutions, and dash lines show the curves for leaky modes

Balance between the power scattered by every inclusion and that received by the same inclusion from other inclusions in the array has been discussed and used to determine the region of existence of eigenwaves propagating along the line. It has been shown that both narrow-band and wide-band strong re¯ection is possible depending on the load impedance of the inclusions. Here, Wood anomalies can be identi®ed from the analysis of the power balance. In particular, guided eigenmodes have been studied in detail. Guided-wave modes cannot be excited by plane waves in the in®nite line, since these modes have propagation constants larger than the wave number in the background medium. That is why in re¯ection curves3 only the classical Wood anomalies related to excitation of higher-order Floquet modes can be seen. Waveguide modes can be excited in arrays of a ®nite number of inclusions or by ®nite sources. These modes exist if the particles are loaded by reactive elements (either capacitive or inductive) where electromagnetic energy can be stored. It is well known that superdirective properties of array antennas (or passive arrays, such as an array of grooves in conducting plane [9]) realize if the adjacent elements are located at a distance smaller than k=2 and excited out of phase. The same is true for arrays of resonant re¯ectors, see e.g. [8]. Our results show that waveguide solutions for a line of loaded dipole antennas indeed exist in the range of propagation factors which include this case. Thus, we expect that similar phenomena can be realized in simpler systems with bulk reactive loads. Interesting enough that the required energy storage can be provided by non-resonant capacitive loads, which means that guided modes with very rapid variations of the ®elds along the line of antennas can exist in wide frequency bands, so that no high-quality resonators are necessary. Naturally, if a narrow-band operation is required, resonance loads can be used.

example of dipole particles made of short metal sections of conducting wires. The corresponding values of Ref1=ag [see (27) and (28)] are frequency-independent but large, as discussed above, and there are no guided-wave solutions. To make the guided-wave solutions possible, the wire dipole particles must be loaded. There are two possibilities: loads which increase capacitance, for example bulk capacitances or high-permittivity coverings of wires, and resonance loads. In the ®rst case, which corresponds to weak frequency dependence of the polarizability (frequency independent in the quasi-static approximation), dispersion curves have the form shown in Fig. 10. (in case of capacitive loads the real part of the polarizability is of course positive, so not all the curves can be realized in this References way). In case of the resonant loads (inductive loading of 1. Scherer A, Doll T, Yablonovitch E, Everitt H, Higgins A (eds.) short dipole antennas), the left-hand side of the real part of (1999) Mini-special issue on Electromagnetic Crystal the dispersion equation (40) quickly varies with the freStructures, Design, Synthesis, and Applications. IEEE Trans. quency. Thus, guided-wave solutions exist only in a very Microw. Theory Techn. 47: no. 11; Scherer A, Doll T, Yablonovitch E, Everitt H, and Higgins A (eds.) (1999) Special narrow frequency band. This situation can be also visusection on Electromagnetic Crystal Structures, Design, alized using Fig. 10, where one should assume that with Synthesis, and Applications. J. Lightwave Techn. 17: no. 11 changing frequency (that is, varying kd), the constant level 2. Sievenpiper D, Zhang L, Broas RFJ, Alexopoulos NG, Yablshown at the curves quickly changes. In this regime, very onovich E (1999) High-impedance electromagnetic surfaces sharp resonances in the electromagnetic response of the with a forbidden frequency band. IEEE Trans. Microwave array should be expected. Theory Techniques 47: 2059±2074 These results show that very quick spatial variations of 3. Laso MAG, Erro MJ, Benito D, Garde MJ, Lopetegi T, Falcone F, Sorolla M (1999) Analysis and design of 1-D photonic the currents induced on the inclusions (large qd) are bandgap microstrip structures using a ®ber grating model. possible even at low frequencies (small kd), if the antennas Microw. Optical Technol. Lett. 22: 223±226 are capacitively loaded. We also observe that the propa4. Falcone F, Lopetegi T, Sorolla M (1999) 1-D and 2-D photonic gation factor q in this case slowly depends on the frebandgap structures. Microw. Optical Technol. Lett. 22: quency. This feature can be possibly used to generate 411±412 current distributions needed to realize superdirective 5. Tibuleac S, Magnusson R (1997) Re¯ection and transmission antenna patterns. guided-mode resonance ®lters. J. Opt. Soc. Am. A 14: 1617±1626

5 Conclusion Using analytical means, properties of periodical arrangements of small passive scatterers have been explored.

6. Wang SS, Magnusson R (1993) Theory and applications of guided-mode resonance ®lters. Appl. Optics 32: 2606±2613

3 In this paper, only curves for the normal incidence are shown, but the conclusion is general.

S. A. Tretyakov and A. J. Viitanen: Line of periodically arranged passive dipole scatterers

7. Knittel GH, Hessel A, Oliner AA (1968) Element pattern nulls 11. Tretyakov SA, Viitanen AJ, Maslovski SI, Saarela IE (1999) Impedance boundary conditions for regular arrays of in phased arrays and their relation to guided waves. Proc. of dipole scatterers, Electromagnetics Laboratory Report the IEEE 56: 1822±1836 Series, Helsinki University of Technology, Report 304, 8. Veremey V (1995) Superdirective antennas with passive reAugust 1999 (submitted to IEEE Transactions Antennas ¯ectors. IEEE Antennas and Propag. Magazine 37: 16±27 Propagation) 9. Skidin DC, Veremey VV, Mittra R (1999) Superdirective ra12. Felsen LB, Marcuvitz N (1994) Radiation and scattering of diation from ®nite gratings of rectangular grooves. IEEE waves. IEEE Press (originally published: Englewoods Cliffs, Trans. Antennas Propag. 47: 376±383 N.J.: Prentice-Hall, 1972) 10. Wasylkiwskyj W, Kahn WK (1968) Mutual coupling and element ef®ciency for in®nite linear arrays. Proc. of the IEEE 56: 1901±1907

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