drastic influence of the half-bowtie resonances on the focusing and collimating capabilities of 2-d...

1Hgclwftpapdu

cvimtetsmlaf

flacm(c

2442 J. Opt. Soc. Am. A/Vol. 27, No. 11 /November 2010 A. V. Boriskin and R. Sauleau

Drastic influence of the half-bowtie resonances onthe focusing and collimating capabilities of 2-D

extended hemielliptical and hemisphericaldielectric lenses

Artem V. Boriskin1,2,* and Ronan Sauleau2

1Institute of Radiophysics and Electronics NASU, Kharkiv, Ukraine2Institut d’Electronique et de Télécommunications de Rennes, Université de Rennes 1, Rennes, France

*Corresponding author: [email protected]

Received June 1, 2010; revised September 4, 2010; accepted September 18, 2010;posted September 21, 2010 (Doc. ID 129345); published October 20, 2010

The interplay between the optical focusing and wavelength-scale resonant features of extended hemielliptical(EHE) and extended hemispherical (EHS) lenses is studied in the two-dimensional (2-D) formulation usinghighly accurate in-house software based on the Muller boundary integral equations. The influence of the half-bowtie (HBT) resonances on the focusing and collimating capabilities of medium-size EHE and EHS lensesmade of silicon is characterized as a function of lens parameters and excitation conditions. As a result, factorsdetermining the parasitic impacts of the HBT resonances on the performance of integrated lens antennas arehighlighted. © 2010 Optical Society of America

OCIS codes: 220.0220, 220.3630, 040.2235, 040.1240.

ptst

f(tssssw

stpbivtapm

hpttNlt

. INTRODUCTIONemielliptical and hemispherical dielectric lenses elon-

ated with cylindrical extensions up to the true ellipse fo-al distance are essential building blocks of integratedens antennas (ILAs) operating at millimeter and shorteravelengths [1–8]. Integration of such lenses with planar

eeds enables the creation of ILAs with improved radia-ion and aperture efficiencies achieved thanks to im-roved matching of the feed, reduction of losses associ-ted with the surface waves intrinsic to on-substratelanar antennas, and correction of the primary feed ra-iation pattern. The latter is provided thanks to thenique focusing capabilities of the elliptical lenses.As it is known from geometrical optics (GO), an ellipti-

al dielectric lens, whose eccentricity is chosen as the in-erse of its material refractive index, is capable of collect-ng rays impinging the lens surface along the ellipse

ajor axis in its rear focus (Fig. 1) [1,9]. Reciprocally, inhe emitting mode (if fed by a point source located in thellipse focus) such a lens produces a locally plane wave inhe lens aperture. This focusing/collimating rule is wellatisfied for lenses of comparatively large optical size andade of low permittivity materials. For compact-size

enses and those made of high-index materials this is notlways true because of the finite curvature of the lens sur-ace and growing role of internal reflections.

To characterize and quantify the impacts of internal re-ections on the lens performance, one has to recall thatny dielectric lens is, in fact, an open dielectric resonatorapable of supporting the infinite number of resonantodes whose Q-factors depend of the lens parameters

size, shape, and material). Internal resonances are ex-ited each time the incident field frequency hits the real

1084-7529/10/112442-8/$15.00 © 2

art of the complex-valued frequency of a natural mode ofhe resonator. If excited, internal resonances maytrongly affect the near- and far-field distributions andhus spoil the ILA performance characteristics.

The resonances, which are of the primary importanceor the hemielliptic lenses, are the so-called half-bowtieHBT) resonances. The HBT modes constitute halves ofhe bowtie modes intrinsic to elliptical [10–12] andtadium-shaped [12–14] resonators. The latter corre-ponds to the stable four-bounce periodic orbits with thehape of a “bowtie.” These orbits have four equal (in ab-olute value) angles of incidence on the lens boundaryhich are below the critical angle [13].Unlike the other modes supported by elliptical and

tadium-shaped resonators (Fig. 2), the HBT modes arehe only ones that can be efficiently excited in and sup-orted by extended hemielliptical (EHE) lenses. This isecause (i) the bouncing rays may still experience the fullnternal reflection at the lens flat bottom boundary pro-ided the material permittivity is large enough, and (ii)he top of the resonant triangle leans on the lens bound-ry at the focal region where feed is located. The latterrovides the necessary condition for excitation of the HBTodes in both receiving and emitting modes.This is also true for the half-stadium [or extended

emispherical (EHS)] lenses that are often preferred inractice due to simplicity of fabrication, e.g., [2–5]. Notehat for high-index materials ��10� the ratio betweenhe ellipse semi-axes becomes negligible [see Fig. 1(b))].evertheless even a minor change in the lens front profile

eads to a significant shift of the geometrical focal pointhat is to be accounted for in design of ILAs.

A quantitative description of the interplay between the

010 Optical Society of America

oniwebeidb

tbsretlapl

2MWmcar=(

“lstvlt

8etc

b(tla

cbaCncadcqoff

3AImdflc[wen

Fsalmdi

Fp

Fommi

A. V. Boriskin and R. Sauleau Vol. 27, No. 11 /November 2010 /J. Opt. Soc. Am. A 2443

ptical-type focusing and wavelength-scale internal reso-ances in the compact-size dielectric lenses is a challeng-

ng task. This study requires utilization of adequate soft-are (capable of accurate characterization of the truelectromagnetic behavior) and careful consideration ofoth scattering and eigenvalue problems. Fortunately, anxhaustive description and classification of bowtie modess available in [11–14], whereas a trustable solution of theiffraction problem can be obtained using the Mulleroundary integral equation [15].In the paper, we first illustrate the interplay between

he optical (focusing) and modal (resonant) features in theehavior of medium-size EHE and EHS lenses made ofilicon. Then we describe the drastic influence of the HBTesonances on the performance characteristics of ILAs op-rating in the emitting and receiving modes and discusshe factors that help to minimize the HBT impacts on theens focusing and collimating capabilities. Finally, gener-lization of the obtained results is done based on the com-arison between focusing capabilities of EHE lenses ofarger size and made of different materials.

. PROBLEM FORMULATION ANDETHODS OF ANALYSISe consider the problem in the two-dimensional (2-D) for-ulation and model the lens by a homogeneous dielectric

ylinder whose contour is composed of two curves, namely,half-ellipse and a half-superellipse (rectangle with

ounded corners), smoothly joint at the points �x ,y��0, ±a�, where a is the minor semi-axis of the ellipse

Fig. 3). Hereafter, these points are referred to as the

(b)(a)

ig. 1. (Color online) EHE lens: (a) Geometry of the lens andchematic drawing of the ray-tracing focusing. The lens bound-ry is depicted by thick blue line. (b) Parameters of the EHEenses designed with respect to the GO focusing rule versus lens

aterial permittivity. Lens configurations marked by verticalotted lines correspond to materials typically used for ILA designn the millimeter and sub-millimeter wave ranges.

(a) Fabry-Perot (b) Triagle (c) Diamond

(d) WG-type (e) Bowtie

ig. 2. (Color online) Schematic drawing of different modes sup-orted by the stadium-shaped resonator [12,13].

edge points” because the focusing ability of the ellipticalens is determined by its front part, whereas the exten-ion is used to position the feed at the proper (focal) dis-ance. For the EHE lens the eccentricity equals the in-erse of the material refracting index e=1/�1/2 (that gives2= �� / ��−1��1/2, f= �1/ ��−1��1/2), whereas for the EHS lenshe extension parameter is defined as l2=1.0.

For numerical studies we select a medium-size lens of�0 in diameter (that is typical for practical applications,.g., [3,6]) and made of isotropic silicon ��=11.7�. The ma-erial is assumed to be lossless (unless different is indi-ated).

The lens is illuminated by either a plane wave or aeam radiated by the so-called complex source pointCSP) feed [16]. The two types of incident fields are usedo study the focusing and collimating properties of theenses associated with the ILA operating in the receivingnd emitting modes, respectively.The diffraction problem is solved using an in-house

omputer-aided design (CAD) tool based on the Mulleroundary integral equations (MBIEs) and the method ofnalytical regularization (MAR) [15,17,18]. Although theAD tool is only capable of dealing with 2-D models (thateglects the cross-polarization effects), it enables us to ac-urately study the resonant phenomena that often escapettention of antenna engineers due to limitations of stan-ard commercial software (namely, the lack of guaranteedonvergence of the numerical solutions and enormous re-uirements to computational time and resources). Detailsf the MBIE-algorithm and its validation against high-requency and finite-difference time-domain based solversor the ILA design can be found in [17,18].

. NUMERICAL RESULTS. Emitting Mode

n the 2-D formulation, an ILA operating in the emittingode can be modeled as a homogeneous dielectric cylin-

er excited by a directive source located close to the lensat bottom. We simulate the source as a CSP that is a lineurrent located in a point with a complex coordinate16,19]. In the real space such a feed radiates a beamhose monolobe pattern is controlled by a single param-ter kb, where b is the imaginary part of the CSP coordi-ate, and k is the free space wavenumber (Fig. 4). In

ig. 3. (Color online) Geometry and notations of the 2-D modelf an EHE lens antenna excited by a plane wave (in the receivingode) or by an aperture feed simulated by CSP (in the emittingode). In the latter case, the curvy line indicates the branch cut

n the real space due to CSP [16].

sflladHP

EHimrFlFdbattfmptto

EtstF

s“hpsld

rcfnmtf

rwlm

1Aswecgiwbnt

FEL

FCt

FCp

FEsr


imulations the distance between the feed and the lensat bottom is �=�0 /10, where �0 is the free space wave-

ength. The radiation characteristic that is considered asmeasure of the lens collimation ability is the broadside

irectivity defined as D=2��U��bs��2 /Ptot, where U=Ez orz for E- or H-polarizations, respectively; �bs=180°; andtot=�0

2��U��2d� is the total radiated power.As already discussed, the strongest influence on the

HE lens performance characteristics comes from theBT resonances. Their impact on the collimating capabil-

ty of the lens is demonstrated using the simplest antennaodel that is an EHE lens excited by a cylindrical wave

adiated by the CSP feed with kb=0. As one can see inig. 5, the highest directivity is achieved for the extension

ength parameter close to the value suggested by GO.urther increase in the lens extension leads to directivityegradation and rise of periodic resonances. This is trueoth for E- and H-polarizations although the resonancesre more pronounced in the E-case. The difference be-ween polarizations in terms of the directivity value andhe impact of internal resonances can be explained by dif-erent levels of backreflection of the incident wave (that isuch higher for the H-case) and the difference in trans-

arency of the lens boundaries [20]. As we focus our at-ention on the resonance properties of the lenses, our fur-her discussion will be limited to the E-polarization casenly.

As it was shown in [21], the resonances in l1 for theHE lenses correspond to HBT modes that differ between

hemselves by the number of field variations along theide of the resonance triangle. The near-field maps plot-ed for the lenses, whose configurations are marked inig. 5 as (A) cut-through-focus and (B) cut in a way to

ig. 4. (Color online) Free space radiation patterns of the 2-DSP feed with different values of the kb parameter. The values of

he CSP broadside directivity are given in parentheses.

. . . . .

ig. 5. (Color online) Broadside directivity (linear scale) of theSP feeds �kb=0� assisted by the EHE lens versus lens extensionarameter ��=11.7, a=4� �.
0
upport the nearest HBT resonance, clearly evidence theoptical focusing” and “resonant” regimes in the lens be-avior (Fig. 6). The resonant distortion of the near-fieldattern is reflected on the far-field pattern in the form ofplit main beam and significant increase in the sidelobeevel (Fig. 7) that explains degradation of the broadsideirectivity observed in Fig. 5.Finally, as it is seen in Fig. 8, the lens with a non-

esonant extension (at the feed central frequency) is stillapable of supporting multiple HBT resonances at otherrequencies within a given frequency band. Thus a limitedumber of HBT modes are always involved in the perfor-ance of hemielliptic ILAs. This is one of the reasons for

he often reported “higher-than-expected” sidelobe levelor wideband ILAs, e.g., [22].

The tight relation between the optical focusing andesonant features of the EHE lenses provides no simpleay to avoid excitation of the HBT resonances. Neverthe-

ess, there are some factors (discussed below) that help toinimize their impacts.

. Impact of the Excitation Conditionss was noted, one corner of the HBT resonant triangletands on the center of the lens flat bottom [see Fig. 6(b)]here feed is located. Thus, for ILAs operating in themitting mode, one of the necessary conditions for effi-ient excitation of HBT modes is always satisfied. The de-ree of freedom that still remains for HBT suppression isn adjustment of the primary feed radiation pattern in aay to minimize the illumination of the arcs of the lensoundary supporting the two other corners of the reso-ant triangle. Proper mutual adjustment of the feed pat-ern and lens geometry can provide a twofold advantage.

ig. 6. (Color online) Normalized near-field maps ��Ez�� for theHE lenses illuminated by the E-polarized CSP feeds �kb=0�.ens configurations correspond to A and B marks in Fig. 5.

ig. 7. (Color online) Normalized radiation patterns of the-polarized CSP feed �kb=0� assisted by two lenses whose de-

igns correspond to marks A and B in Fig. 5. Solid line: non-esonant lens; dashed line: resonant lens.

Fchlos(aesf

feratosvtl

2Atb

fglinbptphamm

3TatbpnvE1

ais

FE�qs

FCms

FE4e


irst, it can improve broadside directivity thanks to effi-ient exploitation of the lens aperture. Second, it mayelp to minimize the amount of power trapped within the

ens in the HBT modes and thus minimize their impactsn the radiation characteristics of the antenna. As it iseen in Fig. 9, the optimal performance of the silicon lenshighest directivity and minimal impact of resonances) ischieved when the lens is excited by a low directive feed,.g., with kb=0.5 (this value will be used in further analy-is, although it may require adjustment for feeds with dif-erent radiation patterns).

The quantitative description of the ratio between theeed and lens parameters can be given in terms of thedge illumination level defined similarly as it is done foreflector antennas: as the ratio between the powers radi-ted in the broadside and edge point directions. Exhaus-ive research on this topic was reported in [23], so here wenly summarize the conclusions relevant to the currenttudy: the optimal edge illumination does not fully pre-ent excitation of HBT resonances, although it reducesheir impact by inducing a decay of field intensity at theens aperture edges.

. Optimal Size of the Lens Extensionnalysis of the curves in Fig. 9 enables one to conclude

hat silicon EHE lens of the selected size is less affectedy internal resonances when its extension equals the GO

. . . . .

ig. 8. (Color online) Broadside directivity (linear scale) of theSP feed �kb=0� assisted by the cut-through-focus EHE lens (seeark A in Fig. 5) versus frequency parameter that is lens bottom

ize normalized by the incident field wavelength.

. . . . .

ig. 9. (Color online) Broadside directivity (linear scale) of the-polarized CSP feeds with different radiation patterns (see Fig.) assisted by EHE lens ��=11.7, a=4�0 , l2=1.046� versus lensxtension parameter.

ocal distance or shorter �l1�0.3�. To check if it is true ineneral, we have studied the performance of the EHEenses with different extensions (Fig. 10). As one can see,ndeed shorter extension �l1=0.26� results in less pro-ounced resonances but at the same time causes lowerroadside directivity. The longer the extension, the moreronounced the resonances. There is an exception fromhis general rule (i.e., l1=0.32) that shows the potentialossibility for designing EHE lenses that will not supportigh-Q resonances in the given frequency band via properdjustment of the lens extension size. This possibilityay open doors for design of ILAs with improved perfor-ance characteristics.

. Impact of Shape: EHE Lens versus EHS Onehe simplicity of fabrication makes EHS lenses a favor-ble choice for many practical applications, e.g., [2–5]. Al-hough performance characteristics of the EHS lens-ased ILAs have been studied in a number ofublications, the resonant properties of EHS lenses haveever been described in the literature. Our analysis re-eals the following differences in the behavior of EHS andHE lenses of the selected size and material (Figs. 11 and2):The highest broadside directivity for the EHS lens is

chieved for extensions larger than the GO focus location,.e., l1=0.35–0.38 (Fig. 11) that is in line with earliertudies [24].

. . . . . .

ig. 10. (Color online) Broadside directivity (linear scale) of the-polarized CSP feed �kb=0.5� assisted by EHE lenses

�=11.7, a=4�0 , l2=1.046� with different extensions versus fre-uency parameter (i.e., lens bottom size normalized by the freepace wavelength).

. . . . .

Fig. 11. (Color online) Same as Fig. 9 for EHS lens �l =1.0�.
2

nftq

one

4TdmtssmdodGicasi

BIsne

mbs2EtcsllMfiffdrfibdtlps[asFtseot

sHsple

FCrstd

Frmo


For EHS lenses internal resonances in l1 are more pro-ounce when fed by a directive feed (Fig. 11). Moreover, asar as it can be stated based on the solution of the scat-ering problem, they have denser spectrum and higheruality.In the given frequency range (Fig. 12), the performance

f the EHS lens is strongly affected by internal reso-ances irrespective to its extension size. The longer thextension, the more pronounced the resonances.

. Impact of Dissipation Losseshere is a natural remedy from high-Q resonances that isissipation losses. To assess their influence on the colli-ating characteristics of the resonant lens, we have plot-

ed radiation patterns of the CSP feed assisted by an EHEilicon lens with different loss-factors (Fig. 13). As one canee, for the lens of the selected size and made of low-lossaterial �tan ��10−2�, the resonance behavior remains

ominant. The “positive” impact of losses becomes evidentnly for a material with tan �=10−2 that is roughly 1 or-er more lossy than standard silicon material at 100–400Hz [25]. Thus we can conclude that dissipation losses,

ntrinsic to standard dielectric materials, do not signifi-antly affect the interplay between the optical focusingnd resonant mechanisms in the performance of medium-ize silicon lenses, although for larger-size lenses theirmpacts may become more pronounced.

. . . . . .

Fig. 12. (Color online) Same as Fig. 10 for EHS lens �l2=1.0�.

ig. 13. (Color online) Normalized radiation patterns of theSP feed �kb=0� assisted by EHE silicon lenses whose configu-

ations are marked in Fig. 5. The family of four curves corre-ponding to the “in-resonance” case differ by the value of the ma-erial loss-factor (see legend). The value of the broadsideirectivity for each case is shown in parentheses.

. Receiving Moden most cases, receiving and emitting modes can be con-idered as equivalent ones. For the current study this isot true because the efficiency of the internal resonancexcitation depends on the excitation conditions.

In 2-D formulation, an ILA operating in the receivingode can be modeled as a dielectric cylinder illuminated

y a plane wave. In simulations, the lens shape is definedimilarly as it is done in the emitting mode (see Section). The lens is illuminated by a unit-amplitude-polarized plane wave, and its focusing ability is charac-

erized by the peak field value in the focal spot. Note thatontrary to the GO prediction, the focal spot of compact-ize dielectric lenses has a finite size, and its shape andocation depend on the lens parameters as well as on po-arization and angle of the plane wave incidence [20].

oreover, multiple internal reflections may result in theormation of several hot spots with nearly the same fieldntensity (Fig. 14). This phenomenon may significantly af-ect the performance of ILAs especially those excited byeed arrays. This is because the focal spot behavior is veryifferent for lenses operating in the “non-resonant” andesonant regimes. To illustrate this we have plotted theeld amplitude distribution inside the lens close to its flatottom (=�e /10, where �e is the wavelength in theielectric—see Fig. 15). As one can see in Fig. 15(a), forhe non-resonant lens the focal spot migrates along theens flat bottom versus the plane wave angle of incidencereserving its shape and peak field amplitude that isimilar to what is observed for low-permittivity lenses20]. This phenomenon enables one to design multi-beamntennas excited by feed arrays where each feed is re-ponsible for a given direction, e.g., [24,26]. As it is seen inig. 15(b), excitation of a HBT mode can completely spoil

he performance of such antennas because of the multi-pot focal pattern that simultaneously covers several feedlements. The overlap of the hot spots for different anglesf the plane wave incidence prevents unique definition ofhe incoming wave direction.

Finally, it is important to note that the resonant focalpot pattern is determined by the order of the excitedBT mode, and thus it strongly depends on the lens

hape and size. In particular, as it was shown in [21], ex-loitation of the low-order modes in small-size resonantenses, whose near-field patterns are less sensitive to thexcitation conditions, may potentially lead to the develop-

(b)(a)

ig. 14. (Color online) Normalized near-field maps (focal spotegion) for the silicon EHE lenses whose configurations arearked in Fig. 5 illuminated by the E-polarized plane waves: (a)

ff-resonance case, (b) in-resonance case.

ms

1VIattEhtFHctc(dottltf

fs

dlst

2FbcfaF

lalF�

FE�E

FElwl


ent of narrowband integrated receivers with improvedensitivity.

. HBT Resonances in the EHE and EHS Lenses witharying Extension Sizen the ray-tracing approximation, the best focusability ofn EHE lens is achieved when the lens extension equalshe ellipse focal distance. Our analysis shows that excita-ion of internal resonances within compact-size siliconHE and EHS lenses may result in the near-field en-ancement twice higher than that achieved due to the op-ical focusing (Fig. 16). Comparison between Fig. 16 andigs. 9 and 11 reveals that these resonances correspond toBT modes. It is interesting to note that some resonances

learly defined in the emitting mode escape detection inhe receiving mode. This can be explained by differentlasses of symmetry of the corresponding HBT modes [14]in calculations, the peak field value in the focal spot isetermined along the x-axis �y=0�; thus only modes withdd numbers of field variations along the y-axis are de-ected). Nevertheless complementarity of the problems inhe receiving and emitting modes enables one to designenses with favorable resonant (or non-resonant) proper-ies by solving a simpler problem of the plane wave dif-raction.

In particular, comparison of the peak field values in theocal spot of the EHE and EHS silicon lenses plotted ver-us frequency parameter (Fig. 17) confirms conclusions

3. . . . . . . .

. . . . . . .. .

. . . . . . . .

. . . . . . . .(b)

(a)

ig. 15. (Color online) Field amplitude distribution along theens flat bottom for different angles of incidence of the unit-mplitude E-polarized plane wave illuminating the EHE siliconenses ��=11.7, a=4�0� whose configurations are marked inig. 5: (a) non-resonant lens �l1=0.3�, (b) resonant lens

l1=0.343�.

erived in the emitting mode: the performance of the EHSens is more affected by the HBT resonances, and ahorter lens extension is the only partial remedy forhem.

. Impact of the Lens Size and Materialinally, a quantitative characterization of the interplayetween the optical and resonant mechanisms is given viaomparison of the focusability of EHE lenses made of dif-erent materials (Fig. 18). Here the lens configurationsre those marked in Fig. 1(b), and the corresponding val-

. . . . .

ig. 16. (Color online) Field amplitude in the hot spot of theHE and EHS lenses versus length extension. The lenses

�=11.7, a=4�0� are illuminated by the unit-amplitude-polarized plane waves under normal incidence �=0°�.

......

......

(b)

(a)

ig. 17. (Color online) Field amplitude in the hot spot of the (a)HE and (b) EHS silicon lenses versus frequency parameter. The

enses are illuminated by the unit-amplitude E-polarized planeaves under normal incidence and differ by the extension

engths.

ucoroqnfiacsrmgo

3TiwToaf

4Tt

mbsstabtsamitlstmthwropl

ATrMdtR

R

FEnel


es of the extension parameter suggested by the GO fo-using rule for each material are marked by triangles. Asne can see in Fig. 18(a), internal resonances (observed asipples over the curves with single maxima near the GO-ptimal extension size) become detectible already for theuartz lens ��=3.8�. For the Macor lens ��=5.6�, the reso-ant near-field enhancement (evidenced by the clearly de-ned resonant spikes) becomes comparable with thatchieved due to the optical focusing, whereas for the sili-on lens the resonant behavior becomes dominant. As it iseen in Fig. 18(b), the relevant impact of the optical andesonant mechanisms in the near-field enhancement re-ains similar for lenses of larger size. This enables us to

eneralize the conclusion derived throughout the papern the lenses of different sizes.

. Additional Remark as to the ILA Performancehere is one more factor that may attenuate the impact of

nternal resonances on the ILA performance and is thusorth to be mentioned in the scope of the current paper.his is the presence of a matched feed that sucks powerut of the lens. Unfortunately, the model used for thenalysis does not enable us to assess the impact of such aeed.

. CONCLUSIONShe optical and modal features of EHE and EHS lenses,ypically used as building blocks of ILAs operating at the

. . . .

. . . .

(b)

(a)

ig. 18. (Color online) Field amplitude in the hot spot of theHE lenses made of standard dielectric materials and illumi-ated by the unit-amplitude E-polarized plane waves versus lensxtension parameter: (a) medium-size lens �a=4�0�, (b) large-sizeens �a=12�0�.

illimeter and sub-millimeter wavelength ranges, haveeen carefully studied using highly accurate in-houseoftware based on the MBIE/MAR approach. It washown that HBT resonances are efficiently excited withinhe EHE and EHS lenses illuminated by plane waves andperture feeds located close to the center of the lens flatottom. The drastic influence of the HBT resonances onhe collimating and focusing capabilities of the medium-ize EHE and EHS lenses have been described by the ex-mple of a silicon lens of 8�0 in diameter. The lens perfor-ance characteristics have been studied as a function of

ts optical size and extension length. It was demonstratedhat (i) resonant features may become dominant forenses made of materials with ��5, (ii) losses intrinsic totandard dielectric materials do not significantly affecthe ratio between focusing and resonant properties ofedium-size lenses, and (iii) proper mutual adjustment of

he primary feed radiation pattern and lens geometryelps to suppress excitation of the HBT modes. Finally, itas shown that EHE lenses are less affected by the HBT

esonances than EHS ones. Furthermore, proper selectionf the EHE lens extension size may help to minimize thearasitic influence of the HBT resonances on the lens col-imating capabilities.

CKNOWLEDGMENTShis work was supported in part by the Université Eu-opéenne de Bretagne, Rennes, France; by the Fondationichel Métivier, France; by the European Science Foun-

ation in the framework of the RNP-NEWFOCUS; and byhe North Atlantic Treaty Organization under grantIG983313.

EFERENCES1. D. F. Filipovic, S. S. Gearhart, and G. M. Rebeiz, “Double

slot antennas on extended hemispherical and elliptical sili-con dielectric lenses,” IEEE Trans. Microwave Theory Tech.41, 1738–1749 (1993).

2. T. H. Buttgenbach, “An improved solution for integrated ar-ray optics in quasioptical mm and submm receivers: Thehybrid antenna,” IEEE Trans. Microwave Theory Tech. 41,1750–1761 (1993).

3. S. Raman and G. M. Rebeiz, “Single- and dual-polarizedmillimeter-wave slot-ring antennas,” IEEE Trans. Anten-nas Propag. 44, 1438–1444 (1996).

4. P. Otero, G. V. Eleftheriades, and J. R. Mosig, “Integratedmodified rectangular loop slot antenna on substrate lensesfor millimeter- and submillimeter-wave frequencies mixerapplications,” IEEE Trans. Antennas Propag. 46, 1489–1497 (1998).

5. J. van Rudd and D. Mittleman, “Influence of substrate-lensdesign in terahertz time-domain spectroscopy,” J. Opt. Soc.Am. B 19, 319–329 (2002).

6. G. Godi, R. Sauleau, and D. Thouroude, “Performance of re-duced size substrate lens antennas for mm-wave communi-cations,” IEEE Trans. Antennas Propag. 53, 1278–1286(2005).

7. S. Cherednichenko, V. Drakinskiy, T. Berg, P. Khosropanah,and E. Kollberg, “Hot-electron bolometer terahertz mixersfor the Herschel Space Observatory,” Rev. Sci. Instrum. 79,034501 (2008).

8. A. Neto, “UWB, non dispersive radiation from the planarlyfed leaky lens antenna. Part 1: theory and design,” IEEETrans. Antennas Propag. 58, 2238–2247 (2010).

1

1

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2


9. P. Varga, “Focusing of electromagnetic radiation by hyper-boloidal and ellipsoidal lenses,” J. Opt. Soc. Am. A 19,1658–1667 (2002).

0. S. V. Boriskina, T. M. Benson, P. D. Sewell, and A. I. Nosich,“Directional emission, increased free spectral range, andmode Q-factors in 2-D wavelength-scale optical microcavitystructures,” IEEE J. Sel. Top. Quantum Electron. 12, 1175–1182 (2006).

1. J. Wiersig, “Formation of long-lived, scarlike modes nearavoided resonance crossings in optical microcavities,” Phys.Rev. Lett. 97, 253901 (2006).

2. H. G. L. Schwefel, N. B. Rex, H. E. Tureci, R. K. Chang, A.D. Stone, T. Ben-Messaoud, and J. Zyss, “Dramatic shapesensitivity of directional emission patterns from similarlydeformed cylindrical polymer lasers,” J. Opt. Soc. Am. B 21,923–934 (2004).

3. C. Gmachl, F. Capasso, E. E. Narimanov, J. U. Nockel, A. D.Stone, J. Faist, D. L. Sivco, and A. Y. Cho, “High-power di-rectional emission from microlasers with chaotic resona-tors,” Science 280, 1556–1564 (1998).

4. S. Y. Lee, M. S. Kurdoglyan, S. Rim, and C. M. Kim, “Reso-nance patterns in a stadium-shaped microcavity,” Phys.Rev. A 70, 023809 (2004).

5. S. V. Boriskina, T. M. Benson, P. Sewell, and A. I. Nosich,“Accurate simulation of 2D optical microcavities withuniquely solvable boundary integral equations andtrigonometric-Galerkin discretization,” J. Opt. Soc. Am. A21, 393–402 (2004).

6. E. Heyman and L. B. Felson, “Gaussian beam and pulsed-beam dynamics: complex-source and complex-spectrum for-mulations within and beyond paraxial asymptotics,” J. Opt.Soc. Am. A 18, 1588–1611 (2001).

7. A. V. Boriskin, G. Godi, R. Sauleau, and A. I. Nosich, “Smallhemielliptic dielectric lens antenna analysis in 2-D: bound-ary integral equations versus geometrical and physical op-tics,” IEEE Trans. Antennas Propag. 56, 485–492 (2008).

8. A. V. Boriskin, A. Rolland, R. Sauleau, and A. I. Nosich,

“Assessment of FDTD accuracy in the compact hemiellipticdielectric lens antenna analysis,” IEEE Trans. AntennasPropag. 56, 758–764 (2008).

9. A. V. Boriskin and A. I. Nosich, “Whispering-gallery andLuneburg-lens effects in a beam-fed circularly-layered di-electric cylinder,” IEEE Trans. Antennas Propag. 50, 1245–1249 (2002).

0. A. V. Boriskin, R. Sauleau, and A. I. Nosich, “Exact off-resonance near fields of small-size extended hemielliptic2–D lenses illuminated by plane waves,” J. Opt. Soc. Am. A26, 259–264 (2009).

1. A. V. Boriskin, S. V. Boriskina, A. I. Nosich, T. M. Benson, P.Sewell, and A. Altintas, “Lens or resonator? electromag-netic behavior of an extended hemielliptical lens for asub-mm wave receiver,” Microwave Opt. Technol. Lett. 43,515–518 (2004).

2. A. D. Semenov, H. Richter, H.-W. Hübers, B. Günther, A.Smirnov, K. S. Il’in, M. Siegel, and J. P. Karamarkovic,“Terahertz performance of integrated lens antennas with ahot-electron bolometer,” IEEE Trans. Microwave TheoryTech. 55, 239–247 (2007).

3. A. V. Boriskin, R. Sauleau, and A. I. Nosich, “Performanceof hemielliptic dielectric lens antennas with optimal edge il-lumination,” IEEE Trans. Antennas Propag. 57, 2193–2198(2009).

4. D. F. Filipovic, G. P. Gauthier, S. Raman, and G. M. Rebeiz,“Off-axis properties of silicon and quartz dielectric lens an-tennas,” IEEE Trans. Antennas Propag. 45, 760–766(1997).

5. J. W. Lamb, “Miscellaneous data on materials for millime-ter and submillimeter optics,” Int. J. Infrared Millim.Waves 17, 1997–2034 (1996).

6. X. Wu, G. V. Eleftheriades, and T. E. van Deventer-Perkins,“Design and characterization of single and multiple-beammm-wave circularly polarized substrate lens antennas forwireless communications,” IEEE Trans. Microwave Theory
Tech. 49, 431–441 (2001).

drastic influence of the half-bowtie resonances on the focusing and collimating capabilities of 2-d...

Documents