T he phased array antenna (PAA) plays an important role in radar, broadcast, communication, etc.1,2. In

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1 OPEN SUBJECT AREAS: APPLIED PHYSICS MATERIALS FOR DEVICES Receive 7 August 014 Accepte 9 October 014 Publishe 30 October 014 Corresponence an requests for materials shoul be aresse to S.H. (sailing@kth.se) Extening the scanning angle of a phase array antenna by using a null-space meium Fei Sun 1, & Sailing He 1, 1 Centre for Optical an Electromagnetic Research, Zhejiang Provincial Key Laboratory for Sensing Technologies, JORCEP, East Builing #5, Zijingang Campus, Zhejiang University, Hangzhou 315, China, Department of Electromagnetic Engineering, School of Electrical Engineering, Royal Institute of Technology (KTH), S-1 44 Stockholm, Sween. By introucing a columnar null-space region as the reference space, we esign a raome that can exten the scanning angle of a phase array antenna (PAA) by a preetermine relationship (e.g. a linear relationship between the incient angle an steere output angle can be achieve). After some approximation, we only nee two homogeneous materials to construct the propose raome layer by layer. This kin of meium is calle a null-space meium, which has been stuie an fabricate for realizing hyper-lenses an some other evices. Numerical simulations verify the performance of our raome. T he phase array antenna (PAA) plays an important role in raar, broacast, communication, etc.1,. In practice, ue to various factors, the scanning range of a PAA is often within 60u to 160u 1,. It is still extremely challenging to achieve a scanning angle larger than 60u with an acceptable beam quality. One applicable metho is to set a raome behin the PAA to exten the angle of the output beam prouce by the PAA. Lam et al. has esigne an fabricate a raome that can exten the scanning angle of a PAA to almost 90u to 190u 3,4. However, the propose raome, accoring to heavy use of simulation optimization, cannot achieve a linear relationship between the input beam an the output beam, which is often esire in practice. Recently, we propose a general metho to esign the raome that can exten the scanning angle of a PAA by using transformation optics (TO) 5. The relationship between the input angle an the output angle of our raome can be manipulate as require (e.g. a linear relationship can be achieve). However, the material of the raome propose in reference [5] is very har to be acquire experimentally: it requires an inhomogeneous anisotropic meium (see Eq. () in [5]). In this paper, we will introuce a columnar null-space region as the reference space (but not free space as in [5]) to esign a novel raome that keeps the avantage of our previous esign (a linear relationship between input an output beams can be achieve) an only requires two homogeneous meia to realize it layer by layer. The main features of the propose raome in this paper inclue: 1) it can be use to exten the scanning range of a PAA (e.g. achieve a scanning angle larger than 60u); ) a preetermine linear relationship between the input an output beams can be achieve; an 3) the material requirement of the propose raome is much simpler than our previous esign 5. Only two layere homogeneous anisotropic meia can be use to construct the propose raome. Our methos in this paper will further avance the practical realization of this high quality raome with preetermine steering functionality. Methos Our esign is base on TO 6,7 an a null-space meium,9. TO is base on the form-invariant of Maxwell s equations uner coorinate transformation. A special meium (the so-calle transforme meium), which can be use to control the tracing of electromagnetic waves 6,7 or the istribution of DC electric or magnetic fiels in an unpreceente way, can be esigne by TO. This iea has been extene to control the tracing of acoustic waves 16 an thermal fiels 17,1. The null-space meium,9 is also esigne with transformation optics. We have previously use the iea of further transforming insie a null-space meium to reuce the material requirement of a DC magnetic lens 13.TO can help to establish a relationship between two spaces: one is the reference space an the other is the real space. By esigning the coorinate transformation between these two spaces, we can obtain the transforme meium in the real space with a pre-esigne function. We have previously provie a general metho to esign a raome of arbitrary shape that can exten the scanning angle of a PAA 5. However, as the reference space in that paper is free space, the material requirement of the esigne raome in [5] is very complex. In this paper, we will introuce the null-space meium as the reference space. In orer to escribe our iea analytically, we choose a columnar raome in our esign. First we will introuce the iea of a columnar null-space region. Assuming the reference space is free space, we perform the following transformation: SCIENTIFIC REPORTS 4 : 63 DOI: /srep063 1

2 R {D r,r[½0,r {DŠ < r ~ {D r{ D D R,h ~h,z ~z,,r[½r {D,R Š : r,r[½r,?) where 5 R,0, D,, R. The quantities in the reference or real space (before or after the transformation) are expresse without or with primes, respectively. If the reference space is fille with the free space, we can obtain the transforme meium in the real space after the coorinate transformation (1) 6,7 : R {D iag(1,1, ) D?0 R? iag(1,1, ), r [½0, Š < e cy ~m cy ~ iag(p, 1 P, D P) D?0, ðþ? iag(?,0,0), r [½,R Š : 1, r [½R,?) where P (/D 1) R /r9. The subscript cy means the quantity is expresse in the cylinrical coorinate system. If D approaches zero, the region R D, r, R in the reference space will approach a cylinrical surface an not a space. In this case, the region of space, r9,r in the real space also correspons to a surface an not a space. Therefore we refer to the meium in the region, r9,r in the real space as the null-space meium. It shoul be note that we have to introuce an anisotropic meium into the region r9, in orer to maintain a continuous transformation. The columnar null-space meium in Eq. () has been stuie in other literatures,9, e.g., a hyper-lens that can provie high-resolution imaging is a columnar null-space meium. In this paper, we will use the columnar null-space meium in Eq. () as the reference space, an then make a further transformation insie this null-space meium to achieve a raome that can exten the scanning angle of a PAA. We express the coorinates in the reference space an the real space as (x9, y9, z9) an (x0, y0, z0), respectively. The meium in the reference space is the columnar null-space meium escribe by Eq. (). We choose the following transformation: r ~r ; < < h, r [½0, Š h ~ f (h,r ), r [½,R Š ð3þ : h, r [½R,?) : z ~z ð1þ where f is a continuous function, which etermines the relationship between the input angle an output angle of the raome [5]. In this paper, we only consier the simplest relation between the input an output angles of the raome (an angle shifter in [5]), which requires that f satisfies the following bounary conitions: h ~f (h,r ~ )~h ð4þ h ~f (h,r ~R )~h zh 0 Eq. (4) inicates that we choose an ientical transformation at the input surface of our raome r0 5 r9 5 an a linear angle shift at the output surface of our raome r0 5 r9 5 R. Therefore, the raome esigne by Eq. (3) an (4) performs like a linear angle shifter: the angle of the output beam is the angle of the input beam shifte by a fixe angle h 0.ByusingTO 5,6, we can obtain the transforme meium of the transformation escribe in Eqs. (3) an (4) when the reference space is expresse by Eq. (): R {D iag(1,1, ), r [½0, Š 3 1 r f r 0 < e cy ~m cy ~ P r f r ðr f r Þ z 1 P ð f h Þ 0, r [½R f h 0 6 1,R Š 4 D : 1, r [½R,?) where f r ~Lf =Lr an f h ~Lf =Lh. Note that the region, r0,r in Eq. (5) is our raome, an the PAA is place in the region r0,. The region outsie the raome r0. R is free space, in accorance with the given situation. If we efine the null-space as the reference space, which means D approaches zero (P also approaches zero), the region, r0,r in Eq. (5) can be approximately expresse as (we rop the infinitesimal of higher orers): 3 1 r f r 0 e cy ~m cy < P r f r ðr f r Þ 0 6 f h 4 D 7 5,r [½,R Š ð6þ 0 0 The expression in Eq. (6) is still in the cylinrical coorinate system. We can make a iagonalization to express it uner the principal axis system: ð5þ Figure 1 (a) The basic structure of our raome, which can shift the incient beam by a fixe angle h 0. The raome is compose of two isotropic materials layer by layer: the permittivity an permeability of the green material is extremely high, an the permittivity an permeability of the re material is nearly zero. The yellow region (r0, ) is the anisotropic meium. The white region is air. The materials istribution of the whole structure can be escribe by Eq. (9). (b) A multilevel raome that more closely resembles practical applications. This raome is ivie into three ifferent regions: region I with h 0 5 0u, which inicates that the incient beam will not be steere in this region; region III with a fixe shifting angle h 0, which is the same as (a) except that the lower part of region III is the mirror image of the upper part about the x0-axis; an region II, a transition region in which h 0 graually changes from 0u to the fixe angle in region III. SCIENTIFIC REPORTS 4 : 63 DOI: /srep063

3 Figure D TM wave FEM simulation results: the z-component of the magnetic fiel. (a) an (c): a beam with angle 10u incient onto the raome, which is compose of the two materials escribe by Eq. (11). The shift angle of their interface relative to the x0-axis is a a1 5 30u. The angles of the output beam are 39.9u an 39.5u, respectively. (b) an (): a beam with angle 45u incient onto the raome, which is compose of the two materials escribe by Eq. (11). The shift angle of their interface relative to the x0-axis is a a1 5 30u. The angle of the output beam is 74u an 70u, respectively. (a) an (b): the meium in the region [0, R1] is escribe by Eq. (9). (c) an (): the meium in the region [0, R1] is air. To conserve memory uring computation, we rop the lower part of the raome, which oesn t affect the results. ep ~mp ~iag( P D P 1zðr fr0 Þ,0, ) fh0 fh0 D?0? iag(?,0,0),r [½R1,R R P~1z {1 0 r D ð7þ The subscript p means that the quantity is expresse in its principal axis system. As we can see from Eq. (7), the raome can be approximately treate as a highly anisotropic homogeneous meium when D approaches zero. The irection of the local principal axis system can be obtaine by rotating an angle w from the cylinrical coorinate system11: tan w~ ðr fr Þ [ tan w~r fr 1{ðr fr Þ ðþ As we can see from Eq. (), w is relate with function f. Consiering that the function f can be any arbitrary continuous function that satisfies the bounary conition (4), w can thus also be arbitrary accoringly. To summarize the parameters of the whole structure: if D approaches zero (we use the null-space meium as the reference space), the whole structure escribe by Eq. (5) can be reuce to: R e ~m ~iag(1,1, ), r [½0,R1 cy < cy R1 ð9þ r [½R1,R e p ~m p ~iag(?,0,0), : e ~m ~1, r [½R,?) cy Figure 3 D TM wave FEM simulation results: the relation between the angle of the input beam an the angle of the output beam in the far fiel for the raome compose by the two materials escribe in Eq. (11). The shift angle of their interface with respect to the x0-axis is a a1 5 30u. SCIENTIFIC REPORTS 4 : 63 DOI: /srep063 cy We shoul note that although we can simplify the materials in the raome region R1, r0, R by introucing the null-space meium into the reference space, it also causes the region r0, R1 to be some anisotropic homogeneous meium rather than air. 3

4 Figure 4 D TE wave FEM simulation results: the z-component of the electric fiel. (a) an (c): a beam with angle 10u incient onto the raome (compose of the two materials escribe by Eq. (1)) with the shift angle of interface with respect to the x0-axis set to 30u. The angles of the output beam are 39.9u an 39.7u, respectively. (b) an (): a beam with angle 45u incient onto the raome (compose of the two materials escribe by Eq. (1)) with the shift angle of interface with respect to the x0-axis set to 30u. The angles of the output beam are 74.u an 70.u, respectively. (a) an (b): the meium in the region [0, R1] is escribe by Eq. (9). (c) an (): the meium in the region [0, R1] is air. To conserve memory uring computation, we rop the lower part of the raome, which oesn t affect the result. Since the material of our raome in its principle axis system is a highly anisotropic homogeneous one (see Eq. (7)), we can use two isotropic meia with extreme parameters to realize it layer by layer. Accoring to the effective meium theory, the effective permittivity (or permeability) of two isotropic homogeneous meia with e1 an e (or m1 an m) can be given as: {1 e\ ~e1 {1 f1 ze {1 f < e\\ ~e1 f1 ze f m\ ~m1 {1 f1 zm {1 f : m\\ ~m1 f1 zm f ð10þ Figure 5 D TM wave FEM simulation results: the z-component of the magnetic fiel. The multilevel raome use here is shown in Fig. 1(b). (a), (b), an (c): a beam with angle 0u, 0u, an 45u incient onto the multilevel raome, respectively. The raome is compose of two materials escribe by Eq. (11), an the angle of their interface with respect to the x0-axis graually changes from 0u to 30u. The lower part of the raome is the mirror image of the upper part about the x0-axis. SCIENTIFIC REPORTS 4 : 63 DOI: /srep063 4

5 The subscript \\ inicates the irection along the interface between two meia an the subscript H inicates the irection perpenicular to the interface between two meia. f 1 an f correspon to the filling factors of two meia (f 1 1 f 5 1). We can choose e 1 an m 1 to be very large; e an m become nearly zero to mimic a highly anisotropic meium, accoring to Eq. (9). In this case, the irection (angle w) of the principal axis in Eq. () is along the irection of the interface between two meia. As f can be an arbitrary continuous function, w can be arbitrarily an continuously changing insie the raome. The only thing we nee to note is that it shoul satisfy bounary conition (4), which requires the angle shift of angle a 1 (the rotation angle of intersection between the input surface of the raome an the interface of the two meia with respect to the x0- axis) an a (the rotation angle of intersection between the output surface of the raome an the interface of the two meia with respect to the x0-axis) to be exactly h 0. Results The schematic iagram of our raome can be shown in Fig. 1(a): the whole raome is compose of two isotropic meia layer by layer (the materials requirement is greatly simplifie compare with the former material requirement in [5]). One meium requires high permittivity an permeability, an the other meium requires both the permittivity an permeability to be nearly zero. High permittivity can be simply achieve by using metal (metal performs as a perfect conuctor in the microwave ban). High permeability can be achieve through the use of artificial perfect magnetic conuctors 19. The meium with nearly zero permittivity an permeability can be achieve with impeance-matche zero-inex metamaterials 0. Some literatures have stuie how to realize such a null-space meium,1,. Next we will utilize finite element metho (FEM) to verify the performance of our raome. The FEM we use is base on the commercial software COMSOL Multiphysics. We consier a D TM wave case, an the two homogenous materials are chosen as: e 1 ~10 {3 ; m 1 ~10 {3 ; e ~10 3 ; m ~10 {3 : ð11þ Fig. shows the performance of the propose raome. As we can see from Fig. (a) an (b), this null-space meium (compose of two homogeneous meia in Eq. (11)) performs like a special waveguie: it confines the incient beam insie it, an bens the light beam as itself is bening. At the output surface of the raome, this bening waveguie rotates exactly an angle h 0, which leas to an angle shift h 0 between the input beam an the output beam. Strictly speaking, in orer to make the layere composite material perform as an effective meium for electromagnetic waves, the perio of these two materials shoul be much smaller than the wavelength of the electromagnetic wave (e.g. less than l/10). However, the computational omain in our simulation is very large (larger than 10 l): if the perio of these two materials is too small (e.g. less than l/4), the computation will excee the memory limit of the computer. Therefore we have to set the perio to be about l/3 in our simulation, which leas to some reflections at the input surface of our raome. We also simulate the case where we replace the anisotropic meium in the region r0, (see Eq. (9)) with air. As shown in Fig. (c) an (), the raome still works in this case (note that the output angle is slightly affecte: see Fig. 3). For a D TE wave case, the two homogenous materials are chosen as: e 1 ~10 {3 ; m 1 ~10 {3 ; e ~10 {3 ; m ~10 3 : ð1þ Simulation results verify the performance of raome compose by the two materials escribe by Eq. (1) (see Fig. 4). For practical applications, we can make a multilevel raome base on our above results. A multilevel raome is shown in Fig. 1(b), which is also compose of two isotropic meia with extreme parameters layer by layer an ivie into three ifferent regions. In region I, h 0 5 0u, which means that the electromagnetic beam will not be shifte when it is incient to this region; in region III, h 0 is a fixe value (e.g. 30u), which is the same as our previous analysis an simulations; in region II, h 0 graually changes from 0u to 30u, which forms a transition region between I an III. We also make some preliminary simulations on this multilevel raome (see Fig. 5). The performance of this multilevel raome is relate to many factors: 1) the zoning of these ifferent regions; ) how to etermine the graual change in the transition region; 3) the thickness of each layer; 4) the with of the incient beam; 5) the size of the whole raome, etc. This is beyon the stuy of the present paper. To realize our raome, we nee to realize the null-space meium that is a kin of extremely anisotropic meium whose parameters are nearly zero in one main principle irection an extremely large in the other main principle irection. Some experimental results on how to achieve such kin of meium with some metamaterials in practice have been reporte,. We shoul also mention that recent stuy shows such a null-space meium can be realize by metallic structures base on Fabry-Pérot (FP) resonances, which may enable our raome operative at multiple frequencies an meanwhile maintain its functionality. Note that such kin of null-space has also appeare in other evices esigne by TO (e.g. the inner bounary of the invisibility cloak 7,3 an the hyper-lens 9,4 ). Another feasible metho to realize such an extremely anisotropic meium is to use layere positive-negative meium 3. The losses of meta-materials (e.g. loss in zero-inex meta-materials or negative-inex materials) will certainly egrae the performance of our raome to some exten. As the loss is mainly relate to the specific unit of meta-material use to realize such a null-space meium, the stuy of novel low-loss meta-materials will be an effective way to reuce the loss effect on the performance of our raome. Another way to reuce the loss effect on our raome is to use metallic structures base on Fabry-Pérot (FP) resonances (which oes not contain any zero-inex or negative-inex meium) to equivalently realize the null-space meium. Recent experiment result shows that the null-space meium realize by metallic structures base on Fabry-Pérot (FP) resonances can give a very goo performance (i.e. loss is very small in this kin of materials). Discussion Base on transformation within a null-space meium, we propose a raome esign that can exten the scanning angle of a PAA. It only requires two homogeneous materials (null-space meia) to construct the propose raome. Simulation results inicate goo performance of our raome: these two materials form a special waveguie for the incient electromagnetic wave (like a rotating hyper-lens). The electromagnetic wave is confine insie this waveguie, steere as the waveguie bens, an exits with the esire shifte angle at the output surface of the waveguie. Our analysis is mainly focuse on a D structure. Extening our calculation to the 3D case (a spherical structure) is fairly straightforwar: the null-space meium is still highly anisotropic with permittivity an permeability extremely high along the raial irection, an near zero along the angle irection. We can also rotate a D raome (see Fig. 1b) along the x0-axis to form a quasi-3d raome. Our work in this paper follows our previous work on how to esign a highperformance raome that can exten the scanning angle of a PAA with a preetermine way, simplifies the material requirement of the propose raome, an avances closer to practical realization. 1. Mailloux, R. J. Phase Array Antenna Hanbook, n e. Boston, MA: Artech House (5).. Amitay, N., Galino, V. & Wu, C. P. Theory an analysis of phase array antennas New York: Wiley-Interscience (197). 3. Lam, T. A., Parazzoli, C. G. & Tanielian, M. H. Negative inex metamaterial lens for the scanning angle enhancement of phase array antennas. 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