1

Field Synthesis with Azimuthally-Varying,Cascaded, Cylindrical Metasurfaces using a Wave

Matrix ApproachChun-Wen Lin, Student Member, IEEE, and Anthony Grbic, Fellow, IEEE

Abstract—In recent years, there has been extensive research onplanar metasurfaces capable of arbitrarily controlling scatteredfields. However, rigorous studies on conformal metasurfaces, suchas those that are cylindrical, have been few in number likelydue to their more complex geometry. Here, wave propagation incascaded cylindrical structures consisting of layers of dielectricspacers and azimuthally-varying metasurfaces (subwavelengthpatterned metallic claddings) is investigated. A wave matrixapproach, which incorporates the advantages of both ABCDmatrices and scattering matrices (S matrices), is adopted. Wavematrices are used to model the higher order coupling betweenmetasurface layers, overcoming fabrication difficulties associ-ated with previous works. The proposed framework providesan efficient approach to synthesize the inhomogeneous sheetadmittances that realize a desired cylindrical field transforma-tion. Design examples are reported to illustrate the power andpotential applications of the proposed method in antenna designand stealth technology.

Index Terms—Antenna radiation pattern synthesis, metasur-faces, curved metasurfaces, cylindrical scatterers, impedancesheets, wave matrix

I. INTRODUCTION

CYLINDRICAL metasurfaces, a popular category of con-formal metasurfaces, have been widely utilized in sce-

narios where planar metasurfaces are not applicable due tomechanical, aerodynamic or hydrodynamic reasons. Theirability to tailor both the amplitudes and phases of cylindricalwaves finds use in radiation pattern control [1-2], scatteringcontrol [3-5], cloaking [6-9], illusion [9-11], high gain antennadesign [9], beam steering [12] and angular momentum gen-eration [13]. Many electromagnetic design problems involvetransforming a known excitation field to a desired radiationfield.

In field transformations, conversion between azimuthalmodes is often necessary, and bianisotropic responses arereadily needed including electric, magnetic, and magneto-electric effects [14]. In order to achieve azimuthal mode

This work has been submitted to the IEEE for possible publication.Copyright may be transferred without notice, after which this version mayno longer be accessible.

This work was supported by the Office of Naval Research (ONR) un-der Grant N00014-18-1-2536, the Air Force Office of Scientific Research(AFOSR) Multidisciplinary University Research Initiative (MURI) programunder Grant FA9550-18-1-0379, and the Chia-Lun Lo Fellowship at theUniversity of Michigan. (Corresponding Author: Anthony Grbic.)

The authors are with the Radiation Laboratory, Department of ElectricalEngineering and Computer Science, University of Michigan, Ann Arbor, MI48109-2122 USA (email: chunwen@umich.edu; agrbic@umich.edu)

Fig. 1. Illustration of the cascaded, concentric cylindrical metasurfaces [3,19, 22]. Wave propagation along the radial direction is studied.

conversion, cylindrical metasurfaces are modeled with bian-isotropic polarizability tensors in [10]. The synthesis of a fieldtransformation simply involves finding the required surfacepolarization current densities that match the tangential com-ponents of the source field to the desired field. Nevertheless,complex polarizabilities, which lead to negative resistances,are unavoidable in this technique. By utilizing surface wavesin design, local power conservation can be satisfied, and fieldtransformation realized with passive and lossless cylindricalmetasurfaces [11]. Still, the reported method in [11] mandatesthat metasurfaces be impenetrable and of zero thickness si-multaneously. Such constraints become infeasible in practice.Due to the challenges associated with realizing magnetic andmagneto-electric effects, bianisotropic responses are typicallyimplemented by cascading layers of metasurfaces [15-16].In [9, 13-14], the bianisotropic responses required for fieldtransformation are first derived by a mode matching techniqueand then later replaced by cascaded structures. Althoughneat and elegant, directly replacing idealized bianisotropicboundaries with cascaded structures creates two significantissues for realization. First, to minimize the effects due tofinite thickness in cascaded structures, the separation betweenmetasurfaces has to be extremely small [9, 14]. This is becausesmall separation distances minimize transverse propagationcoupling within the cascaded metasurfaces. Alternatively, per-fect conducting baffles need to be inserted to prevent higherorder azimuthal modes from propagating between layers [9,13-14]. Both of these requirements greatly increase fabricationcomplexity as well as cost.

A more rigorous approach to analyzing cascaded cylindri-cal metasurfaces is to start from network parameters, wheremultiple azimuthal modes can be incorporated simultaneously.

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Hence, the higher-order coupling between layers can be intrin-sically modeled, and the aforementioned fabrication difficul-ties circumvented. An ABCD matrix formulation in cylindricalcoordinates can be adopted to accurately capture the radialwave propagation in cascaded structures [3]. However, ABCDmatrices involve total electric and magnetic fields and thusare not suitable for synthesizing scattering properties of themetasurfaces. Under this scenario, a wave matrix approach,relating the incident and scattered electric fields on one sideof the metasurfaces to those on the other side [17], is abetter choice. Not only do wave matrices provide scatteredfield information, but they also allow the simple analysis ofcascaded structures through matrix multiplication. In [18], thismethod was exploited to design cascaded, planar metasurfaceswith arbitrarily specified S matrices. The wave matrix theoryfor azimuthally invariant structures in cylindrical coordinateshas been derived in [19] to model the cascaded metasurfaces il-lustrated in Fig. 1. The power of this method was demonstratedby synthesizing interesting azimuthally invariant devices suchas polarization converter and polarization splitter analytically.

As noted, only azimuthally-invariant metasurfaces, which donot involve any conversion between azimuthal modes, wereinvestigated in this previous work [19]. Since fields alonga cylindrical geometry can be decomposed into elementarycylindrical waves with different azimuthal orders, arbitraryfield transformation is not possible with merely azimuthally-invariant structures. In order to address this issue, we combinethe concepts of multimodal matrix analysis [20] as well asthe mode matching technique [21]. By considering multipleazimuthal modes simultaneously, the proposed theory canbe generalized, and azimuthal mode converters successfullydesigned [22].

In this paper, the mathematical background for multimodalwave-matrix theory [22] is discussed in detail. We aim toaccomplish arbitrary field transformation through the design ofazimuthally-varying, cascaded, cylindrical metasurfaces. First,the definition of multimodal wave matrices, ABCD matrices, Smatrices in cylindrical coordinates, along with the conversionformulas between them are provided. Moreover, the wavematrix expressions of the building blocks that make up thecascaded structures are derived. Based on the proposed theory,an optimization process is employed to determine the metasur-face parameters required for a stipulated field transformation.Last but not least, several design examples, including az-imuthal mode converters, illusion devices, and multi-functionalmetasurfaces, are illustrated and verified through commercialelectromagnetic solvers. These intriguing results are free fromrealization difficulties such as close metasurface separations orperfect conducting baffles. The design examples showcase thecapability of the proposed approach in various applications.

II. THE WAVE MATRIX THEORY

In this section, the basic assumptions and formulationemployed in this paper to derive the wave matrix theoryare introduced. The definition of network parameters andconversion formulas between them [19] are generalized toaccount for multiple azimuthal modes.

Fig. 2. The 2-D problem of concern, which is a cross-section of Fig. 1.Without loss of generality, it is assumed that the sources are located in thecentral region of the cascaded metasurfaces.

A. Problem Setup

Throughout this paper, a two-dimensional scenario is as-sumed in order to simplify the problems considered. All struc-tures, including the excitation, metasurfaces, and dielectricspacers are independent of z, and there is no propagationconstant along the z direction (kz = 0). Therefore, a cross-section of Fig. 1 along the z = 0 plane, shown in Fig. 2,is sufficient to describe all electromagnetic field distributions.Furthermore, as shown in Fig. 2, it is also assumed that thecascaded structure is excited by electric currents, resulting inTMz waves. The case of TEz waves can be derived directlyfrom duality.

As in [19, 22], the synthesis and analysis of the metasurfacesare conducted in the spectral domain. The time conventionof e+jωt is assumed, and is suppressed in all subsequentexpressions. For TMz waves with kz = 0, the only fieldcomponents tangential to the metasurfaces are Ez as well asHφ. The electric field Ez can be written as a summation overall azimuthal modes:

Ez(ρ, φ) =

+M∑m=−M

Em(ρ)e−jmφ

=

+M∑m=−M

E+m(ρ)e−jmφ +

+M∑m=−M

E−m(ρ)e−jmφ.

(1)

In the above equation, M is a sufficiently large number thatensures the series converges. Generally speaking, as we havemore layers of metasurfaces, or as the metasurfaces have morecomplicated spatial variations, M also needs to be larger.The quantity Em is the mth azimuthal mode of the totalelectric field, while E+m and E−m are its outward and inwardpropagating parts, respectively. Since H(2)

m and H(1)m represent

outward and inward propagating waves respectively undere+jωt convention, we can write:

E+m(ρ) = α+mH

(2)m (kρ)

E−m(ρ) = α−mH(1)m (kρ).

(2)

where α+m and α−m represent outward and inward traveling

wave amplitudes for azimuthal order m. These wave ampli-

3

Fig. 3. The annulus region enclosed by two cylindrical surfaces (denoted bytheir radii ρ1 and ρ2) is the region of interest. Network parameters relate theazimuthal modes of the fields at these two radii.

tudes are generally different in each dielectric layer. Similarly,the tangential magnetic field can be derived [23]:

Hφ(ρ, φ) =

+M∑m=−M

Hm(ρ)e−jmφ (3)

Hm(ρ) = −jωεk

[α+mH

(2)′

m (kρ) + α−mH(1)′

m (kρ)]. (4)

These azimuthal mode quantities will be used in the definitionof the network parameters, as discussed in the next part.

B. Definition of Network Parameters

Network parameters, commonly used to analyze and charac-terize microwave circuits [17, 24], have also proven very usefulfor solving field problems. Consider the region of interest inFig. 3 enclosed by cylindrical surfaces with an inner radius(inner port) ρ1 and an outer radius (outer port) ρ2. A wavematrix is defined to relate the incident and reflected waves onradius ρ1 to those at radius ρ2,[

E+(ρ1)E−(ρ1)

]=

[ ¯W++¯W+−

¯W−+¯W−−

]·[E+(ρ2)E−(ρ2)

](5)

where the column vectors E±(ρ) are composed of azimuthalmodes of the propagating electric fields at radius ρ:

E+(ρ) = [E+M (ρ), ..., E+0 (ρ), ..., E+−M (ρ)]T

E−(ρ) = [E−M (ρ), ..., E−0 (ρ), ..., E−−M (ρ)]T .(6)

Since E±(ρ) are both (2M+1)×1 vectors, the matrices ¯W++,¯W+−, ¯W−+, and ¯W−− are of dimension (2M+1)×(2M+1).

Accordingly, the wave matrix is a (4M+2)×(4M+2) matrix.In [19], only one azimuthal mode is considered, so the wavematrix in (5) reduces to a 2× 2 matrix for TMz waves.

Likewise, we define an ABCD matrix by relating theazimuthal modes of total electric and magnetic fields on thetwo cylindrical surfaces:[

E(ρ1)H(ρ1)

]=

[ ¯A ¯B¯C ¯D

]·[E(ρ2)H(ρ2)

](7)

where the column vectors

E(ρ) = [EM (ρ), ..., E0(ρ), ..., E−M (ρ)]T

H(ρ) = [HM (ρ), ...,H0(ρ), ...,H−M (ρ)]T .(8)

As in the case of a wave matrix, the dimension of the ABCDmatrix is also (4M + 2)× (4M + 2). The total field used to

define the ABCD matrices are directly related to the boundaryconditions. Hence, these matrices will be extensively studiedwhen complicated metasurface boundaries are involved.

On the other hand, a scattering matrix (S matrix) in cylin-drical coordinates bears a more complex form. For the casesof planar metasurfaces, an S matrix is just a rearrangementof its wave matrix counterpart, since the wave impedanceis identical everywhere. In cylindrical coordinates, the waveimpedance depends not only on the radius [19], but also theazimuthal mode of concern [23]. Normalization is required toensure that the S matrix is unitary when the system is lossless,and symmetric when the system is reciprocal. A reasonabledefinition of an S matrix is based on the ρ-directed TMz power[19, 25]. By generalizing the single mode S matrix derivationand expression in [19], we obtain:[

¯c−1 E−(ρ1)

¯c+2 E+(ρ2)

]=

[ ¯S11¯S12

¯S21¯S22

]·[

¯c+1 E+(ρ1)

¯c−2 E−(ρ2)

]. (9)

The matrices ¯c±1 and ¯c±2 are introduced for compactness andalgebraic purposes. For example, the diagonal matrix ¯c+i con-tains the normalization information for outward propagatingwaves at radius ρi:

¯c+i =

c+i,M . . . 0. . .

... c+i,0...

. . .0 . . . c+i,−M

(10)

in which the power normalization coefficients c+i,m is relatedto the radius ρi and the azimuthal mode m of concern [19]

c+i,m =

√√√√2πρiRe{jωεiki

H(2)′m (kiρi)

H(2)m (kiρi)

}. (11)

Finally, the diagonal matrix ¯c−i is similarly defined for inwardpropagating waves at ρi:

¯c−i =

c−i,M . . . 0. . .

... c−i,0...

. . .0 . . . c−i,−M

(12)

c−i,m =

√√√√2πρiRe{jωεiki

H(1)′m (kiρi)

H(1)m (kiρi)

}. (13)

C. Conversion Formulas between Network Parameters

After defining the network parameters in cylindrical coordi-nates, conversion formulas between them can also be derivedthrough algebraic manipulation. To convert from S matricesto wave matrices, we multiply out the first row of the blockmatrix equation (9),

¯c−1 E−(ρ1) = ¯S11¯c+1 E

+(ρ1) + ¯S12¯c−2 E−(ρ2). (14)

4

This equation (14) can be rearranged to the following

− ¯S11¯c+1 E+(ρ1) + ¯c−1 E

−(ρ1) = ¯OE+(ρ2) + ¯S12¯c−2 E−(ρ2).

(15)where ¯O represents a zero matrix of dimension (2M + 1) ×(2M + 1). Analogously, the second row of (9) becomes

− ¯S21¯c+1 E+(ρ1)+ ¯OE−(ρ1) = −¯c+2 E

+(ρ2)+ ¯S22¯c−2 E−(ρ2).

(16)Combining (15) and (16) yields a matrix equation[− ¯S11¯c+1 ¯c−1− ¯S21¯c+1

¯O

] [E+(ρ1)E−(ρ1)

]=

[ ¯O ¯S12¯c−2−¯c+2

¯S22¯c−2

] [E+(ρ2)E−(ρ2)

].

(17)By comparing (17) with the definition of a wave matrix (5),we arrive at the conversion formula from S matrices to wavematrices as[ ¯W++

¯W+−¯W−+

¯W−−

]=

[− ¯S11¯c+1 ¯c−1− ¯S21¯c+1

¯O

]−1 [ ¯O ¯S12¯c−2−¯c+2

¯S22¯c−2

].

(18)One can also express an S matrix in terms of a wave matrix

by expanding (5). For instance, the first row of this blockmatrix equation is equivalent to

¯OE−(ρ1)− ¯W++E+(ρ2) = − ¯IE+(ρ1)+ ¯W+−E

−(ρ2) (19)

where ¯I stands for an (2M + 1)× (2M + 1) identity matrix.In order to align (19) with the definition of S matrices (9), werewrite the above equation as

¯O(¯c−1 )−1¯c−1 E−(ρ1)− ¯W++(¯c+2 )−1¯c+2 E

+(ρ2)

= − ¯I(¯c+1 )−1¯c+1 E+(ρ1) + ¯W+−(¯c−2 )−1¯c−2 E

−(ρ2). (20)

Following the same procedure by which (18) is derived, weobtain the following conversion formula from wave matricesto S matrices,[ ¯S11

¯S12¯S21

¯S22

]=

[ ¯O − ¯W++(¯c+2 )−1

(¯c−1 )−1 − ¯W−+(¯c+2 )−1

]−1 [−(¯c+1 )−1 ¯W+−(¯c−2 )−1

¯O ¯W−−(¯c−2 )−1

].

(21)

On the other hand, the relation between wave matrices andABCD matrices can be easily obtained by generalizing theresult in [19]. In [19], a transformation matrix has been definedthat relates the incident and reflected electric fields to the totalelectric and magnetic fields on a cylindrical surface (denotedby a single radius ρ). For TMz waves, we have the singleazimuthal mode transformation matrix [19][

Em(ρ)Hm(ρ)

]=

[1 1−Y +

m Y −m

]·[E+m(ρ)E−m(ρ)

](22)

Y +m = +

j

η

H(2)′

m (kρ)

H(2)m (kρ)

, Y −m = − jη

H(1)′

m (kρ)

H(1)m (kρ)

. (23)

In fact, Y +m is exactly the outward wave admittance, while Y −m

is the inward wave admittance of azimuthal order m at radiusρ [23]. For the case of multiple modes, the network parameters

are defined by superposing each azimuthal mode. As a result,the multimodal transformation matrix can be expressed as:[

T(k,η)(ρ)]

=

[ ¯I ¯I

− ¯Y + ¯Y −

](24)

where the (2M + 1) × (2M + 1) diagonal matrices ¯Y + and¯Y − consist of entries Y +

m or Y −m for each mode

¯Y + =

Y+M . . . 0...

. . ....

0 . . . Y +−M

, ¯Y − =

Y−M . . . 0...

. . ....

0 . . . Y −−M

.(25)

The transformation matrix (24) satisfies the following relation[E(ρ)H(ρ)

]=[T(k,η)(ρ)

]·[E+(ρ)E−(ρ)

]. (26)

Therefore, the definition of an ABCD matrix (7) can berewritten by incorporating (26),[

T(k1,η1)(ρ1)] [E+(ρ1)E−(ρ1)

]=

[ ¯A ¯B¯C ¯D

] [T(k2,η2)(ρ2)

] [E+(ρ2)E−(ρ2)

].

(27)

The definition of a wave matrix (5) directly implies theconversion formula from ABCD matrices to wave matrices[ ¯W++

¯W+−¯W−+

¯W−−

]=[T(k1,η1)(ρ1)

]−1 [ ¯A ¯B¯C ¯D

] [T(k2,η2)(ρ2)

],

(28)as well as the one from wave matrices to ABCD matrices[ ¯A ¯B

¯C ¯D

]=[T(k1,η1)(ρ1)

] [ ¯W++¯W+−

¯W−+¯W−−

] [T(k2,η2)(ρ2)

]−1.

(29)

III. BUILDING BLOCKS OF THE CASCADED CYLINDRICALSTRUCTURE

Based on the aforementioned theory, the wave matricesdescribing the electromagnetic properties of dielectric spacers,dielectric interfaces and azimuthally-varying metasurfaces aredetermined in this section. The wave matrix of the cascadedsystem, which may be very complicated if derived by brute-force, can be easily obtained from the wave matrices ofbuilding blocks by multiplying them in sequential order.

A. Dielectric Spacers

Fig. 4 illustrates a dielectric spacer bounded by two cylin-drical surfaces with radii ρ1 and ρ2. We assume that thedielectric is isotropic as well as homogeneous. Accordingly,no azimuthal mode mixing occurs. Each mode propagatesindependently. This implies that every wave amplitude α±mremains the same throughout the dielectric spacer. We canrelate E±m(ρ1) and E±m(ρ2) by considering (2):

E+m(ρ1) = F+m · E+m(ρ2), E−m(ρ1) = F−m · E−m(ρ2) (30)

with the quantities

F+m =

H(2)m (kρ1)

H(2)m (kρ2)

, F−m =H

(1)m (kρ1)

H(1)m (kρ2)

. (31)

5

Fig. 4. A dielectric spacer with properties (k, η). The innermost radius ofthe dielectric layer is ρ1, while the outermost radius is denoted as ρ2.

Fig. 5. Example of an interface. The interface can either be a simple dielectricinterface or one with a metasurface.

By juxtaposing each azimuthal mode, the wave matrix of adielectric spacer can be written in the form of[

W]D

=

[ ¯W++¯W+−

¯W−+¯W−−

]=

[ ¯F+ ¯O¯O ¯F−

], (32)

where the (2M + 1) × (2M + 1) diagonal matrices ¯F+ and¯F− are defined as

¯F+ =

F+M . . . 0...

. . ....

0 . . . F+−M

, ¯F− =

F−M . . . 0...

. . ....

0 . . . F−−M

.(33)

It is worth mentioning that in the limiting case where theradii are very large, the Hankel functions behave similarly tothe exponential functions. The entries of the wave matrix (31)become just phase delays between the input and output ports,which is identical to the case of a planar dielectric slab [18].

B. Dielectric Interfaces

Consider the interface (a cylindrical surface with radius ρ)between two dielectric materials as shown in Fig. 5. The valueρ− is taken to be infinitely close to but smaller than ρ, so that itfalls in the inner dielectric. Contrarily, the value ρ+ is definedto be infinitely close to but larger than ρ, so that it belongs tothe outer dielectric. Since no surface current density exists onthe interface, there is continuity of the tangential electric andmagnetic fields,

Ez(ρ−, φ) = Ez(ρ

+, φ), Hφ(ρ−, φ) = Hφ(ρ+, φ). (34)

These field quantities in (34) can be written as summationsover azimuthal modes, as in (1) and (3). Mode matchingimplies:

Em(ρ−) = Em(ρ+), Hm(ρ−) = Hm(ρ+) (35)

for all azimuthal orders m. Hence, in vector form (8), we write[E(ρ−)H(ρ−)

]=

[E(ρ+)H(ρ+)

]. (36)

Applying transformation matrices (24) to (36), we arrive at[T(kA,ηA)(ρ

−)] [E+(ρ−)E−(ρ−)

]=[T(kB ,ηB)(ρ

+)] [E+(ρ+)E−(ρ+)

](37)

which results in a compact form for the wave matrix of adielectric interface:[

W]I

=[T(kA,ηA)(ρ

−)]−1 · [T(kB ,ηB)(ρ

+)]. (38)

C. Metasurfaces

Now, let us assume that an azimuthally-varying metasurfaceexists at the cylindrical boundary shown in Fig. 5. In this paper,metasurfaces with only electric responses are considered sincethese responses can be easily realized by metallic patterning.Mathematically, an electric response is often characterized byan electric admittance which is the ratio of surface currentdensity to averaged electric field [26]. Since the metasurfaceis periodic in φ (with period 2π), we can express its admittanceprofile YMS(φ) as a Fourier series,

YMS(φ) = p0 +

K∑k=1

qk cos(kφ) +

K∑k=1

rk sin(kφ)

= p0 +

K∑k=1

(qk2

+rk2j

)e+jkφ +

K∑k=1

(qk2− rk

2j)e−jkφ.

(39)

In (39), the highest azimuthal order K, along with the param-eters p0, qk, and rk define the metasurface. There is a trade-offbetween higher K values and the computational complexity.Larger K values of a single-layer metasurface generate morecomplicated azimuthal mode mixing. However, more com-putational resources are required to design the metasurfaceadmittance, as well as the metallic patterning.

The tangential fields across a metasurface at a radius ρ obeythe following boundary conditions

Ez(ρ−, φ) = Ez(ρ

+, φ) (40)

Hφ(ρ−, φ) = Hφ(ρ+, φ)− YMS(φ)Ez(ρ+, φ). (41)

In order to derive the wave matrix of this metasurface, we startwith its ABCD matrix representation,[ ¯A ¯B

¯C ¯D

]MS

=

[ ¯AMS¯BMS

¯CMS¯DMS

](42)

which is defined based on the matrix equations (7),

E(ρ−) = ¯AMSE(ρ+) + ¯BMSH(ρ+) (43)

H(ρ−) = ¯CMSE(ρ+) + ¯DMSH(ρ+). (44)

Our target is to decompose the boundary conditions (40) and(41) into their azimuthal modes Em(ρ±) and Hm(ρ±). Next,we cast the azimuthal modes into vector form E(ρ±), H(ρ±)as defined by (8), and substitute these vectors into the matrixequations (43), (44) so that ¯AMS , ¯BMS , ¯CMS and ¯DMS canbe found.

6

First, the electric field boundary condition (40) can bedecomposed as∑

m

Em(ρ−)e−jmφ =∑m

Em(ρ+)e−jmφ, (45)

indicating that Em(ρ−) = Em(ρ+) for all azimuthal orders m.By casting each mode into vector form (8), we arrive at

E(ρ−) = E(ρ+) = ¯IE(ρ+) + ¯OH(ρ+), (46)

which combines with the definition (43) to yield¯AMS = ¯I, ¯BMS = ¯O. (47)

Decomposing the fields in (41) into azimuthal modes gives us∑m

Hm(ρ−)e−jmφ

=∑m

Hm(ρ+)e−jmφ − YMS(φ)∑m′

Em′(ρ+)e−jm′φ.

(48)

The first summation on the right hand side of (48) togetherwith (44) implies

¯DMS = ¯I. (49)

On the contrary, the expression for ¯CMS is more complicatedsince the admittance profile YMS(φ) also contains a φ de-pendence. Substituting (39) into the second summation on theright hand side of (48) yields

−∑m′

p0 · Em′(ρ+)e−jm′φ

−∑m′

∑k

(qk2

+rk2j

) · Em′(ρ+)e−j(m′−k)φ

−∑m′

∑k

(qk2− rk

2j) · Em′(ρ+)e−j(m

′+k)φ.

(50)

To perform mode matching, the summation variables arechanged. In (50), we let m′ = m in the first summation,m′ = m + k in the second summation, and m′ = m − kin the last one. Accordingly, (50) becomes

−∑m

p0 · Em(ρ+)e−jmφ

−∑m

∑k

(qk2

+rk2j

) · Em+k(ρ+)e−jmφ

−∑m

∑k

(qk2− rk

2j) · Em−k(ρ+)e−jmφ.

(51)

In matrix form, the first summation in (51) can be repre-sented by an (2M + 1) × (2M + 1) identity matrix ¯I . Toexpress the second summation in (51) in matrix form, wedefine a set of new matrices ¯L(k), all with the same dimension(2M + 1) × (2M + 1). The (i, j)-th element of ¯L(k) matrixsatisfies

¯L(k)(i, j) =

{1, if i > k, and i = j + k.

0, otherwise.(52)

The matrix ¯L(k) can be viewed as shifting all the ones in anidentity matrix ¯I to the left by k positions. For example, ifM = 1, 2M + 1 = 3, then

¯L(1) =

0 0 01 0 00 1 0

, ¯L(2) =

0 0 00 0 01 0 0

.

Similarly, to express the third summation in (51) in matrixform, another set of (2M + 1)× (2M + 1) matrices ¯R(k) aredefined by shifting all the ones in an identity matrix to theright by k positions:

¯R(k)(i, j) =

{1, if i ≤ (2M + 1)− k, and i = j − k.0, otherwise.

(53)Truncating the summation range of (51) from m = −M tom = +M and applying the ¯L(k) and ¯R(k) matrices, (51) canbe cast into a matrix form as required in (44):[

(−p0) ¯I −K∑k=1

(qk2

+rk2j

) ¯L(k) −K∑k=1

(qk2− rk

2j) ¯R(k)

]E(ρ+),

(54)which indicates that

¯CMS = (−p0) ¯I −K∑k=1

(qk2

+rk2j

) ¯L(k) −K∑k=1

(qk2− rk

2j) ¯R(k).

(55)Finally, the wave matrix of a metasurface described by anadmittance profile (39) can be derived by converting from theABCD matrix using (28),[W]MS

=[T(kA,ηA)(ρ

−)]−1 [ ¯I ¯O

¯CMS¯I

] [T(kB ,ηB)(ρ

+)].

(56)Note that if the metasurface is absent, p0, qk and rk are allzero, so ¯CMS = ¯O and (56) reduces to the wave matrix of asimple dielectric interface (38).

IV. SYNTHESIS OF FIELD TRANSFORMING DEVICES

In this section, the design procedure for cascaded metasur-faces that realize a specified field transformation is outlined.Strictly speaking, when a general field distribution is expressedas a summation of its azimuthal modes, an infinite number oforders should be taken into account. In reality, computationalresources are limited so the summation must be truncated.Hence, (1) is an approximation, and synthesizing the fieldtransformation analytically is arduous.

Instead, an optimization process is adopted in this paper.As shown in Fig. 2, it is assumed that the sources are locatedin the central region of the metasurfaces. The first step of thedesign is to stipulate the field transformation that transformsthe excitation field to some desired field in the outer region(outside the metasurfaces). Next, we choose the number ofcylindrical metasurface layers (denoted by N ), as well as thehighest order of variation K for each admittance profile. Ifthe wanted field transformation is complicated, larger N andK values should be chosen in order to provide more degreesof freedom in the structure. Furthermore, we also need toassign the radius of each cylindrical metasurface ρ1, ρ2, ..., ρNand select the dielectric substrates between the layers. Allthese parameters are regarded as fixed throughout the synthesisprocess. As a result, the targeted structure is illustrated in Fig.6.

In the synthesis process, the Fourier cosine and sine co-efficients p0, qk and rk of the admittance profile for every

7

Fig. 6. The cascaded structure, composed of N layers of azimuthally-varyingmetasurfaces, is considered in the synthesis problem.

metasurface layer are optimized in order to realize the desiredfield outside the metasurfaces. The coefficients of each layerdo not have to be the same, so there are (2K+1)×N variablesin total. To initiate the first iteration of the optimization, p0,qk and rk for each layer are selected randomly. The wavematrix of the entire cascaded structure is obtained from matrixmultiplication as[

W]

all =[W]MS1

[W]D1

[W]MS2

[W]D2. . .

. . .[W]MSN−1

[W]DN−1

[W]MSN

.(57)

The wave matrices of the building blocks are found bysubstituting the physical parameters and Fourier coefficientsinto (32), (38), and (56). We then convert this wave matrix[W ]all to its corresponding S matrix [S]all by employing (21).

Now that the scattering matrix [S]all of the entire cascadedstructure is known, the field distribution in both the central andouter regions can be calculated explicitly. In Fig. 6, nothingexists at infinity, so the electric field at ρN simply propagatesoutward without any reflection. This indicates that E−(ρN ) isa zero vector. Multiplying out the block matrix equation of Smatrix definition (9) yields

¯c−1 E−(ρ1) = ¯S11, all · ¯c+1 E+(ρ1) (58)

¯c+2 E+(ρN ) = ¯S21, all · ¯c+1 E+(ρ1). (59)

The first equation (58) can be solved by separating the incidentfield generated by the excitation from the field scattered bymetasurfaces. Since it is assumed that all the sources are inthe central region of the cascaded metasurfaces, the incidentfield generated by the sources must be outward propagating atρ1. Therefore,

E+(ρ1) = E+inc(ρ1) + E+

sca(ρ1)

E−(ρ1) = E−sca(ρ1).(60)

In (60), E+inc(ρ1) is already known because it is simply the

field generated by the excitation when the metasurfaces areabsent. On the other hand, the fields E+

sca(ρ1) and E−sca(ρ1)together form standing waves since they are caused by scat-tering, and not by real sources. We can write

E+sca(ρ1) =

[βMH

(2)M (kρ1), ..., β−MH

(2)−M (kρ1)

]TE−sca(ρ1) =

[βMH

(1)M (kρ1), ..., β−MH

(1)−M (kρ1)

]T,

(61)

where the 2M + 1 coefficients βM , ..., β−M are the onlyunknowns in the 2M+1 equations (58). Accordingly, E+(ρ1)and E−(ρ1) can be solved explicitly from (58). Substitutingthe calculated E+(ρ1) into (59), the vector E+(ρN ) and thusthe field in the outer region can be determined as well.

The calculated E+(ρN ) for this current optimization it-eration is then compared with the desired field outside thecascaded metasurfaces. A cost function C is defined to evaluatethe error between the calculated field and the targeted field.It is minimized through the built-in fmincon optimizationalgorithm in MATLAB. Useful example forms of the costfunction will be shown in the next section. At the end ofevery iteration, the Fourier coefficients p0, qk and rk of eachmetasurface layer are updated, brought into the wave matrix(57), and a new E+(ρN ) is calculated until the optimizationgoal is achieved.

V. DESIGN EXAMPLES

In order to verify the proposed wave matrix theory anddemonstrate the field transforming ability of cylindrical meta-surfaces, three metasurface designs are synthesized, simulatedand discussed in this section. The full-wave simulations ofthe targeted structure illustrated in Fig. 6 are conducted usingthe COMSOL Multiphysics 2D finite element electromagneticsolver. The metasurface admittance (39) is modeled withsurface current densities that are dependent on electric fields.The central region (where ρ < ρ1) and the outer region(ρ > ρN ) are taken to be free-space. The outermost edge ofthe computational domain (at some ρ� ρN ) is terminated bya Perfect Matched Layer (PML).

A. Case 1: Azimuthal Mode Converter

The first device we are interested in is an azimuthal modeconverter that transform a m = 0 (0th azimuthal mode)excitation field to a m = 1 (1st azimuthal mode) field outsidethe cascaded metasurfaces. Three metasurfaces (N = 3) withradii ρ1 = 1.6λ, ρ2 = 2.2λ, and ρ3 = 2.8λ are assumed in theazimuthal mode converter design (λ stands for the operatingwavelength). The dielectric spacers between the metasurfacesare chosen to be air. In addition, the order of spatial variationon each metasurface is chosen to be K = 5.

The source is assumed to be an electric line current withunit amplitude located at the origin, which produces an m = 0excitation. In this case, the incident field at ρ1 can be foundfrom applying Ampere’s law around the line source,

Ez,inc(ρ1, φ) = − k204ωε0

H(2)0 (k0ρ1). (62)

The incident field vector E+inc(ρ1) in (60) can be derived by

decomposing (62) into azimuthal modes. Specifically, only them = 0 component E+0,inc(ρ1) is nonzero in E+

inc(ρ1):

E+0,inc(ρ1) = − k204ωε0

H(2)0 (k0ρ1). (63)

With the expression of the vector E+inc(ρ1), the optimization

process proposed in section IV can be performed. Outside thedevice an m = 1 field is desired, so Ez(ρ3, φ) should be of

8

(a)

(b)

Fig. 7. The designed azimuthal mode converter which converts the m = 0line current excitation to m = 1 output field. The cross symbols in the fieldplot (from COMSOL simulation) indicates the location of the excitation linecurrent. Admittance profiles of the metasurfaces are also shown. (a) The caseutilizing reflection waves within the central region. (b) The reflectionless case.

the form H(2)1 (k0ρ3)e−jφ. Consequently, the following cost

function can be defined [22],

C =[Power of H(2)

1 mode in the outer regionTotal power in the outer region

− 1]2. (64)

In (64), a restriction is placed only on the field outside thecascaded metasurfaces, which means that reflections within thecentral region are allowed. A total of 2M + 1 = 31 azimuthalorders are considered in all regions during the optimizationprocess. The admittance profiles obtained through optimizationare simulated in COMSOL, resulting in the field plot shownin Fig. 7 (a). The specified field transformation is clearlyaccomplished.

The azimuthal mode converter can also be designed tominimize reflections in the central region. Since the field trans-formation has the additional constraint of being reflectionless,we choose N = 4 (four metasurfaces) and K = 5. The radiiof the metasurfaces are ρ1 = 2.5λ, ρ2 = 2.8λ, ρ3 = 3.1λ,ρ4 = 3.4λ, and air is used for the dielectric spacers in between.The cost function is now defined as [22],

C = a[Power of H(2)

1 mode in the outer regionTotal power in the outer region

− 1]2

+ b[Power of H(2)

0 mode in the central regionTotal power in the central region

− 1]2.

(65)

In this example 2M + 1 = 41, and the ratio of a/b is takento be 10. The optimized admittance profiles, together withthe full-wave simulation result, are displayed in Fig. 7 (b).Although the field outside the device is not as perfect as inthe previous case, reflections are significantly reduced in the

central region. This example helps demonstrate the versatilityof the proposed wave matrix framework.

Recently, light beams with orbital angular momenta (OAM)have been used in particle trapping and manipulation, aswell as high-speed communication [13, 27-29]. The azimuthalmode converters reported here demonstrate the possibilityof generating OAM beams using relatively simple structuresinstead of bulky antenna arrays [30].

B. Case 2: Illusion DeviceThe next example proposed in this paper is an illusion

device. We again excite the device using a line current source,but design the metasurfaces so that the source appears as if itwas displaced in space. Four metasurfaces (N = 4) with radiiρ1 = 2.0λ, ρ2 = 2.3λ, ρ3 = 2.6λ, and ρ4 = 2.9λ are usedto synthesize the device. The order of spatial variation K oneach metasurface is set to be 4. Again, the dielectric spacersare all chosen to be air.

The electric field at (ρ,φ) generated by an off-centered linecurrent located at (ρ′, φ′) is given by [23]

Ez,inc(ρ, φ) = − k204ωε

∑m

H(2)m (kρ′)Jm(kρ)e−jm(φ−φ′)

(66)when ρ ≤ ρ′, and

Ez,inc(ρ, φ) = − k204ωε

∑m

Jm(kρ′)H(2)m (kρ)e−jm(φ−φ′)

(67)when ρ ≥ ρ′. Here the device is excited by a line source at0.5λ to the right of the origin (ρ′ = 0.5λ, φ′ = 0). Therefore,equation (67) is used to determine the incident field at ρ1. Theazimuthal components of the incident field at ρ1 are

E+m,inc(ρ1) = − k204ωε0

Jm(π)H(2)m (k0ρ1)

= α+m,incH

(2)m (k0ρ1).

(68)

On the other hand, the illusion is set to be at 0.5λ to the leftof the origin (ρ′ = 0.5λ, φ′ = π). The targeted output fieldat ρ4 can also be expressed using (67). Ideally, the azimuthalcomponents are

E+m,ideal(ρ4) = − k204ωε0

(−1)mJm(π)H(2)m (k0ρ4)

= α+m,idealH

(2)m (k0ρ4).

(69)

In order to define the cost function, we denote the calculatedazimuthal components at each iteration as

E+m,cal(ρ4) = α+m,calH

(2)m (k0ρ4). (70)

In this case, a possible cost function to be minimized can be

C =∑m

∣∣α+m,cal − α

+m,ideal

∣∣2, (71)

where the quantities α+m,cal and α+

m,ideal are normalized waveamplitudes:

α+m,cal =

α+m,cal√∑

m |αm,cal|2, α+

m,ideal =α+m,ideal√∑

m |αm,ideal|2.

(72)

9

Fig. 8. The designed illusion device that tailors electromagnetic fields so thatthe source appears displaced in space. The real source is marked as the whitecross symbol, and the illusion of the source is represented by the green one.

The line current is an impressed source, and the cascadedmetasurface structure appears as a load to the source. Foreach optimization iteration, the load is changed, so the powertransmitted to the outer region also changes. Therefore, therecould be an overall magnitude difference between the calcu-lated wave amplitudes α+

m,cal and the ideal ones α+m,ideal.

Hence, the wave amplitudes are normalized to remove thismagnitude difference. In essence, this cost function demandsthat the optimized azimuthal components be as close to theideal ones as possible.

Employing the cost function (71), the optimized admittanceprofiles of the four metasurfaces are plotted in Fig. 8. The totalnumber of azimuthal modes considered in the optimization is2M + 1 = 49. These admittance profiles are again simulatedin COMSOL, yielding the field plot in Fig. 8. From outsidethe cascaded metasurfaces it seems that the field is originatingfrom (ρ′ = 0.5λ, φ′ = π), while the actual source is locatedat (ρ′ = 0.5λ, φ′ = 0). This example illustrates possibleapplications to stealth technologies, where illusions or radarcross-section reductions are of concern.

C. Case 3: Multi-functional Metasurface

The final device presented in this paper is a multi-functionalmetasurface. The multi-functional metasurface performs dif-ferently when it is excited by different sources. For instance,when the device is excited by a line current source locatedat (ρ′ = 1.5λ, φ′ = 0), it generates an m = 0 outputfield. However, when the same device is fed by a source at(ρ′ = 1.5λ, φ′ = π) , it produces an m = 2 output field. Forconvenience, the line current sources at (ρ′ = 1.5λ, φ′ = 0)and (ρ′ = 1.5λ, φ′ = π) are named source 1 and source 2,respectively.

To synthesize this device, five metasurfaces are used (N =5), with order of variation K = 5. The radii are chosen tobe ρ1 = 4.0λ, ρ2 = 4.3λ, ρ3 = 4.6λ, ρ4 = 4.9λ, and ρ5 =5.2λ. Finally, all the dielectric spacers are set to be air. Beforeintroducing the cost function, we define two power ratios:

R1 =Power of H(2)

0 mode in the outer regionTotal power in the outer region

(73)

(a)

(b)

Fig. 9. An example of multi-functional metasurfaces. (a) The figure on theleft displays the field plot when the device is fed by source 1. The figure onthe right shows the field plot when source 2 is used as the excitation. Anm = 2 dependence in the output field is clear. (b) The optimized admittanceprofiles for all five metasurface layers.

in which both the numerator and denominator are evaluatedwhen only source 1 is excited. Similarly,

R2 =Power of H(2)

2 mode in the outer regionTotal power in the outer region

(74)

in which both the numerator and denominator are evaluatedwhen only source 2 is excited. The cost function takes theform:

C = a(R1 − 1)2 + b(R2 − 1)2. (75)

In this example, both a and b are set to be 1 so that the twocases have equal weightings.

Based on the proposed cost function (75), optimization inMATLAB and simulation in COMSOL are carried out. Atotal of 2M + 1 = 61 azimuthal modes are included in theoptimization process. The electric field plots, along with therequired admittance profiles, are shown in Fig. 9 (a) and (b)respectively. The synthesized multi-functional device success-fully transforms source 1 to an m = 0 field. It also transformssource 2 to an m = 2 field. This device may be a promisingantenna feed for multi-channel orbital angular momentum-based links, and Multi-Input Multi-Output (MIMO) antennasin general. More importantly, it demonstrates the power andthe feasibility of the proposed framework to realize arbitraryfield transformation.

VI. CONCLUSION

Field synthesis and transformation with cascaded cylindricalmetasurfaces have been investigated in literature using ide-alized bianisotropic boundary conditions. However, physical

10

realizations continue to be a challenge due to the requirementsimposed by these idealized boundary conditions, such asextremely close metasurface separations as well as the need forperfect conducting baffles. These intricate and costly structuresare needed to prevent higher order azimuthal mode couplingfrom propagating between metasurface layers. In this paper,these realization issues are resolved since the complicatedwave propagation phenomena between metasurface layerscan be accurately captured by the wave matrix approach.Generalizing our earlier theory that described only a singleazimuthal order, the wave matrices, ABCD matrices, and Smatrices for cylindrical structures presented here account formultiple azimuthal orders, and are defined in a multimodalsense. Conversion formulas between different network pa-rameters needed for analysis and synthesis convenience arealso provided. Additionally, wave matrix expressions for thebuilding blocks of the cascaded cylindrical metasurfaces arederived and discussed in detail. The azimuthal variation of thecylindrical metasurfaces is accounted for by a Fourier seriesexpansion. Using this comprehensive wave matrix theory, anoptimization technique is utilized to synthesize specified fieldtransformations. Finally, the design and full-wave simulationof three interesting and powerful devices (azimuthal mode con-verters, illusion devices, and multi-functional metasurfaces)are conducted to verify the synthesis method. These designexamples demonstrate the ability to perform arbitrary fieldtransformation using the proposed framework. Future workincludes the integration of real-world feeding structures suchas coaxial [31,32] or waveguide excitations, and the fabricationand measurement of prototypes.

REFERENCES

[1] B. O. Raeker, S. M. Rudolph, “Arbitrary transformation of antennaradiation using a cylindrical impedance metasurface,” IEEE AntennasWirel. Propag. Lett., pp. 1101–1104, vol. 15, Oct. 2015.

[2] B. O. Raeker, S. M. Rudolph, “Verification of arbitrary radiation patterncontrol using a cylindrical impedance metasurface,” IEEE AntennasWirel. Propag. Lett., pp. 995–998, vol. 16, Oct. 2016.

[3] Z. Sipus, M. Bosiljevac, and A. Grbic, “Modelling cascaded cylindricalmetasurfaces using sheet impedances and a transmission matrix formu-lation,” IET Microw. Antennas Propag., Vol. 12, no. 7, pp. 1041-1047,Jan. 2018.

[4] X. -Y. Luo, W. -L. Guo, K. Chen, J. Zhao, T. Jiang, Y. Liu, and Y.Feng, ”Active Cylindrical Metasurface With Spatial Reconfigurabilityfor Tunable Backward Scattering Reduction,” IEEE Trans. AntennasPropag., vol. 69, no. 6, pp. 3332-3340, June 2021.

[5] S. Vellucci, A. Monti, A. Toscano, and F. Bilotti, “Scattering manipula-tion and camouflage of electrically small objects through metasurfaces,”Phys. Rev. Appl., vol. 7, 034032, Mar. 2017.

[6] J. C. Soric, A. Monti, A. Toscano, F. Bilotti and A. Alu, “Multibandand Wideband Bilayer Mantle Cloaks,” IEEE Trans. Antennas Propag.,vol. 63, no. 7, pp. 3235-3240, Jul. 2015.

[7] S. Vellucci, A. Monti, M. Barbuto, A. Toscano and F. Bilotti,“Waveform-Selective Mantle Cloaks for Intelligent Antennas,” IEEETrans. Antennas Propag., vol. 68, no. 3, pp. 1717-1725, Mar. 2020.

[8] P. -Y. Chen and A. Alu, “Mantle cloaking using thin patterned metasur-faces,” Phys. Rev. B, Condens. Matter, vol. 84, 205110, Nov. 2011.

[9] G. Xu, G. Eleftheriades, and S. Hum, “Discrete-Fourier-Transform-Based Framework for Analysis and Synthesis of Cylindrical Omega-Bianisotropic Metasurfaces,” Phys. Rev. Appl., vol. 14, 064055, Dec.2020.

[10] M. Safari, H. Kazemi, A. Abdolali, M. Albooyeh, and F. Capolino,“Illusion mechanisms with cylindrical metasurfaces: A general synthesisapproach,” Phys. Rev. B, Condens. Matter, vol. 100, 165418, Oct. 2019.

[11] D. -H. Kwon, “Illusion electromagnetics for free-standing objects usingpassive lossless metasurfaces,” Phys. Rev. B, Condens. Matter, vol. 101,235135, Jun. 2020.

[12] H. Li, C. Ma, F. Shen, K. Xu, D. Ye, J. Huangfu, C. Li, L. Ran, and T.A. Denidni, ”Wide-Angle Beam Steering Based on an Active ConformalMetasurface Lens,” IEEE Access, vol. 7, pp. 185264-185272, 2019.

[13] J. Li, A. Dıaz-Rubio, C. Shen, Z. Jia, S. Tretyakov, and S. Cummer,“Highly Efficient Generation of Angular Momentum with CylindricalBianisotropic Metasurfaces,” Phys. Rev. Appl., vol. 11, 024016, Feb.2019.

[14] G. Xu, G. Eleftheriades, and S. Hum, “Analysis and Synthesis of Cylin-drical Omega-Bianisotropic Metasurfaces with Mode Expansion,” IEEEInternational Symposium on Antennas and Propagation and USNC-URSI Radio Science Meeting, pp. 1-2, Montreal Canada, Jul. 5-10, 2020.

[15] C. Pfeiffer and A. Grbic, “Cascaded metasurfaces for complete phaseand polarization control,” Appl. Phys. Lett., vol. 102, 231116, Jun. 2013.

[16] C. Pfeiffer and A. Grbic “Bianisotropic metasurfaces for optimal polar-ization control: Analysis and synthesis,” Phys. Rev. Appl., vol. 2, 044011,Oct. 2014.

[17] R. Collin, Field Theory of Guided Waves. New York, NY, USA:McGraw-Hill, 1960.

[18] A. Ranjbar and A. Grbic, “Analysis and synthesis of cascaded metasur-faces using wave matrices”, Phys. Rev. B, vol. 95, 205114, May 2017.

[19] C. -W. Lin and A. Grbic, “Analysis and Synthesis of Cascaded Cylindri-cal Metasurfaces using a Wave Matrix Approach,” IEEE Trans. AntennasPropag., vol. 69, no. 10, pp. 6546-6559, Oct. 2021.

[20] F. Alsolamy and A. Grbic, ”Modal Network Formulation for theAnalysis and Design of Mode-Converting Metasurfaces in CylindricalWaveguides,” IEEE Trans. Antennas Propag., vol. 69, no. 8, pp. 4598-4611, Aug. 2021.

[21] Z. Sipus, Z. Eres, D. Barbaric, “Modeling Cascaded Cylindrical Meta-surfaces with Spatially-Varying Impedance Distribution,” Radioengi-neering, Vol. 28, No. 3, pp. 505-511, Sep. 2019.

[22] C. -W. Lin and A. Grbic, “A Wave Matrix Approach to Design-ing Azimuthally-Varying Cylindrical Metasurfaces,” IEEE InternationalSymposium on Antennas and Propagation and USNC-URSI RadioScience Meeting, Marina Bay Sands, Singapore, Dec. 4-10, 2021.

[23] R. F. Harrington, Time-Harmonic Electromagnetic Fields. IEEE-Press,2001.

[24] D. M. Pozar, Microwave Engineering, 4th ed. New York, NY, USA:Wiley, 2012.

[25] K. Kurokawa, “Power Waves and the Scattering Matrix,” IEEE Trans.Microw. Theory Tech., vol. 13, no. 2, pp. 194-202, Mar. 1965.

[26] S. Tretyakov, Analytical Modeling in Applied Electromagnetics. Nor-wood, MA, USA: Artech House, 2003.

[27] H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop,”Direct Observation of Transfer of Angular Momentum to AbsorptiveParticles from a Laser Beam with a Phase Singularity,” Phys. Rev. Lett.,vol. 75, pp. 826-829, Jul. 1995.

[28] V. Bobkova, J. Stegemann, R. Droop, E. Otte, and C. Denz, ”Opticalgrinder: sorting of trapped particles by orbital angular momentum,” Opt.Express, vol. 29, pp. 12967-12975, Apr. 2021.

[29] J. Wang, J. -Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren,Y. Yue, S.l Dolinar, M. Tur, and A. E. Willner, ”Terabit free-space datatransmission employing orbital angular momentum multiplexing,” Nat.Photonics., vol. 6, pp. 488-496, Jun. 2012.

[30] X. Song, Z. Su, J. Ma, Y. Yao, Y. Bai, Z. Zheng, X. Gao, and S. Huang,”Generation of rotational orbital angular momentum beams in the radiofrequency based on an optical-controlled circular antenna array,” Opt.Express, vol. 29, pp. 23717-23728, Jul. 2021.

[31] J. D. Heebl, M. Ettorre, and A. Grbic, ”Wireless Links in the RadiativeNear Field via Bessel Beams,” Phys. Rev. Appl., vol. 6, 034018, Sep.2016.

[32] Z. Shen, J. L. Volakis and R. H. MacPhie, ”A coaxial-radial line junctionwith a top loading disk for broadband matchings”, Microw. Opt. Technol.Lett., vol. 22, no. 2, pp. 87-90, Jul. 1999.

11

Chun-Wen Lin received the B.S. degree in electricalengineering and M.S. degree in communication en-gineering from National Taiwan University, Taipei,Taiwan, in 2016 and 2018, respectively.

In 2019, he joined Prof. Anthony Grbic’s groupwith the University of Michigan, Ann Arbor, MI,USA, where he is currently pursuing the Ph.D. de-gree in electrical and computer engineering. His re-search interests include metamaterials, curved meta-surfaces, antenna designs, and study of electromag-netic theory.

Anthony Grbic received the B.A.Sc., M.A.Sc., andPh.D. degrees in electrical engineering from theUniversity of Toronto, Canada, in 1998, 2000, and2005, respectively. In 2006, he joined the Depart-ment of Electrical Engineering and Computer Sci-ence, University of Michigan, Ann Arbor, MI, USA,where he is currently a Professor. His research in-terests include engineered electromagnetic structures(metamaterials, metasurfaces, electromagnetic band-gap materials, frequency-selective surfaces), anten-nas, microwave circuits, plasmonics, wireless power

transmission, and analytical electromagnetics/optics.Dr. Grbic served as Technical Program Co-Chair in 2012 and Topic Co-

Chair in 2016 and 2017 for the IEEE International Symposium on Antennasand Propagation and USNC-URSI National Radio Science Meeting. He was anAssociate Editor for IEEE Antennas and Wireless Propagation Letters from2010 to 2015. Dr. Grbic was the recipient of AFOSR Young InvestigatorAward as well as NSF Faculty Early Career Development Award in 2008,the Presidential Early Career Award for Scientists and Engineers in January2010. He also received an Outstanding Young Engineer Award from theIEEE Microwave Theory and Techniques Society, a Henry Russel Awardfrom the University of Michigan, and a Booker Fellowship from the UnitedStates National Committee of the International Union of Radio Sciencein 2011. He was the inaugural recipient of the Ernest and Bettine KuhDistinguished Faculty Scholar Award in the Department of Electrical andComputer Science, University of Michigan in 2012. In 2018, Prof. AnthonyGrbic received a University of Michigan Faculty Recognition Award foroutstanding achievement in scholarly research, excellence as a teacher, advisorand mentor, and distinguished service to the institution and profession.