Truncation Diffraction Phenomena of Floquet Waves Radiated From Semi-Infinite Phased Array Antenna...

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3592 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 62, NO. 7, JULY 2014

Truncation Diffraction Phenomena of Floquet Waves

Radiated From Semi-Infinite Phased Array Antenna

in a General Focus ProblemHsi-Tseng Chou , Fellow, IEEE

Abstract— This paper presents uniform formulations of radia-

tion mechanisms, analogous to the ray decompositions of diffrac-tion theory, from a semi-infinite and periodic array of antennas.

The antenna array is excited to radiate the electromagnetic (EM)

fields focused at a relatively arbitrarily selected location. The raydecomposition results from a closed-form formulation by asymp-totically evaluating the radiation integral of Floquetmodes thatare

obtained by applying the Poisson sum formula to the summationof elemental radiations. The analysis is relatively general and will

reduce to the case of a conventional phased array antenna that isexcited to radiate directive beams focused in the far zone, and ex-hibits consistent phenomena. Theoretical investigations as well asnumerical examinations are presented to demonstrate the radia-

tion mechanisms.

Index Terms— Electromagnetic radiation mechanism, Floquetmodes and general field focus, phased antenna array, Poisson sumformula.

I. I NTRODUCTION

THE phased array antenna has received intensive inves-

tigation in the literature because of its flexibility and potential to produce good radiation characteristics [1]. The con-

ventional investigations in the past focus on the examination of

far-field radiation applications [1]–[3] such as creating focused

and highly directive beams for point-to-point microwave trans-

missions and satellite communications [4]–[6]. As a result,

most explorations of previous wave phenomena and design

techniques have been limited to their realizations in these

applications [1]–[4], [7], [8] until recent interest in near-field

and short distance communications such as vital life-detection

systems, noncontact microwave detection systems and radio

frequency identification (RFID) [9]–[15]. Electrically large an-

tenna arrays, referred to as near-fi

eld focused antennas (NFAs)[11]–[15] in these applications, were found advantageous in

focusing the energy in a target area.

The NFA represents a generalization of antenna realization

relaxed from the far-field focused antenna (FFA) because its

concept is to focus the antenna radiation at a relatively arbitrary

Manuscript received October 03, 2013; revised March 31, 2014; accepted

April 03, 2014. Date of publication April 18, 2014; date of current version July02, 2014.

The author is with the Department of Communications Engineering, Yuan Ze

University, Chung-Li 320, Taiwan, R.O.C. (e-mail: [email protected]).

Digital Object Identifier 10.1109/TAP.2014.2318331

location including in the near- and far- zone. This generalization

reveals many shortages in the general understanding of radia-

tion and propagation phenomena and mechanisms, which were

not exhibited in the conventional investigations of FFAs and

needs to be explored. In this case, the conceptual formulations

in terms of the uniform geometrical theory of diffraction (UTD)

[8], [16]–[20] ray decompositions appear to be convenient in

the phenomenon interpretation, which has been successfully ap-

plied to analyze the radiation from conventional FFA arrays [8],[21].

The basic implementation of UTD ray decomposition starts

from a Floquet mode expansion using the Poisson Sum formula

[16], [17], [22], [23]. It transforms the superposition of discrete

radiation components from all antenna elements into an alter-

native superposition of integrals of Floquet modes. These inte-

grals can be asymptotically evaluated to decompose the fields

in terms of ray fields, in closed-form solutions, emerging from

corresponding radiation points on the array aperture. In the pre-

vious investigation of the FFA, it was found that each Floquet

mode of an infinite array appears only as a single radiation ray,

which is either propagating or evanescent for the observers inthe entire space regardless of their locations [8], [21]. However,

our recent investigations [16], [17] exhibit different characteris-

tics in the case of the NFA, where a selected Floquet mode may

have two rays that behave as either propagating or evanescent

waves depending on the field locations, except for the funda-

mental zero mode which has a single ray and is always propa-

gating. Many new wave propagation features have been discov-

ered in [16], [17] for the radiation of an infinite array of NFAs.

In particular, these two rays may converge into a single one and

form a curve of ray caustics in each Floquet mode when the two

radiation points on the array aperture coincide. This ray caustic

curve divides the space into two regions of evanescent and prop-

agating waves, respectively.

This paper attempts to further develop a general formula-

tion for a semi-infinite NFA array with a purpose to examine

the diffraction mechanism due to the existence of a truncation

edge. Following a similar UTD-type formulation [8] by asymp-

totically evaluating the Floquet mode integrals with a semi-in-

finite integration interval, the radiation is decomposed into ra-

diation rays from the same array with infinite extent and edge

diffracted rays from the truncation edge. These edge diffracted

rays emerge from diffraction points on the edge in a fashion

similar to the radiation rays. This general formulation of NFA

field radiation/propagation mechanisms naturally reduces to the

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CHOU: TRUNCATION DIFFRACTION PHENOMENA OF FLOQUET WAVES 3593

Fig. 1. A one-dimensional, semi-infinite array configuration of line sources,

where the focus point and field observer are also shown.

case of FFA, previously explored in [8], and allows the char-

acteristics of wave propagation investigated in both near- and

far-zones. This UTD-type formulation remains valid when the

radiation point is close to the diffraction point, and results in a

half value of the radiation ray field at the boundary where these

two points coincide.

The insuf ficiency of the current formulation exists because

there are two radiation points and two diffraction points in the

general NFA problem. It becomes singular when any three or

more points are close to each other simultaneously. A typical

example is the field point on the ray caustic curve, in which

the two radiation points may coincide with a diffraction point.

These phenomena will be investigated in future phases.

This paper is organized in the following format. Section II

describes the formulation for the radiation from a one-dimen-

sional (1-D) semi-infinite NFA array. Section III presents the

case for a two-dimensional (2-D) semi-infinite NFA array. The

characteristics are investigated in Section III. In particular, the

study focuses on the truncation diffraction mechanism as a com-

pensation to the shortage of Floquet mode phenomena in [16],

[17], which examine only the radiation mechanisms of an in-

finite array of antennas. Numerical examples are presented inSection IV for demonstration and validation. Finally a short dis-

cussion is presented in Section V as a conclusion.

II. 2-D FINITE ARRAY R ADIATION PROBLEM

A. Problem Composition of 2-D Radiation From a 1-D Array

The semi-infinite, linear array of line sources under investi-

gation is illustrated in Fig. 1, whose elements are indexed by

and located at on the x-axis

with a period . The array is excited to radiate fields focused

at with the th element’s excitation, , given by

[16]

(1)

where is the wave number with being the wave-

length in free space, is used as a reference, and

. Thus the net potential at ra-

diated from this array can be expressed as

(2)

with .

The Floquet mode waves are obtained by using the following

Poisson sum formula [23]:

(3)

Applying (3) to (2), the th Floquet mode can be expressed as

(4)

where , ,

, and . The

evaluation of (4) can be performed asymptotically by em-

ploying the spectral representation of the Hankel functions

[25] to decompose the radiating fields into components of diffraction mechanisms [16].

B. Composition of Uniform Asymptotic Formulations

Within the UTD framework, the asymptotic evaluation of (4)

can be formulated into the following format [25]–[27]:

(5)

where is the Heaviside step function and [26]–[29]

is a UTD Fresnel transition function to assure a uniform field

distribution when the field point crosses the shadow boundary

of the direct field radiation. In (5), is the asymptotic so-lution when the size of the array is extended to infinity, while

accounts for the effects of truncation. These compo-

nents arise from a radiation point, , on the array aperture, and

a diffraction point, , on the edge, respectively. This formula-

tion remains valid as the radiation point approaches the diffrac-

tion point, i.e. . However, as mentioned in [16], [17],

there are two radiation points in a general NFA problem. The

solution in (5) becomes singular as these two radiation points

coincide near the diffraction point at the edge. The characteris-

tics of these terms are addressed in the following subsections.

C. The Asymptotic Solution of Direct Radiation From a

Non-Truncated Array,

In (5), is identical to the solution of radiation for an

infinite array, whose characteristics have been investigated in

[16]. The formulation is summarized in the f ollowing by

(6)

where with being the contributing saddle point in

[16]. In (6), and are and when . Also

is the distance to a ray caustic given by

(7)


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Fig. 2. Radiation mechanisms for different Floquet modes, where theand 1 cases are illustrated. The ray caustic curve for a nonzero mode is also

illustrated. Note that there are two radiation saddle points, and , in the

nonzero modes for the field points in the propagating region as shown in [16].(a) ; (b) .

with . The saddle point can be

found from the following equation:

(8a)

(8b)

The illustration of the saddle points is shown in Fig. 2(a) and (b)

for and , respectively, which are the points majorlycontributing to the radiation and referred to as the radiation

points. The basic phenomena have been investigated in [16] and

are omitted here. However, it is worth mentioning that there are

two radiation points from which emerge two radiation rays in

each Floquet mode, except in the fundamental zero mode where

only a single radiation ray exits. The illustration of different

saddle points is shown in Figs. 2(a) and (b). The ray arising

from the coincidence of these two radiation rays will form a

ray caustic curve. The ray tube diagram is shown in Fig. 3.

A uniform formulation has been developed in [16] to consider

the radiation field as these two radiation points come close and

become coincident. It is also noted that the existence of a raycaustic in (6) requires one to impose a phase change as the

field point crosses the caustic, as exhibited in the phenomena of

classic UTD solutions [26].

D. The Asymptotic Solution of Edge Point Contribution From

a Truncated Array

The asymptotic edge point contribution in (4) is developed in

Appendix A, and summarized in the following formulation:

(9)

Fig. 3. Ray tube diagrams for radiation and diffraction fields. The ray tube propagating through the ray caustic curve is als o shown.

where and as illustrated in

Figs. 2(a) and (b) for the and 1 modes, respec-

tively. In (9), the parameters associated with the diffraction

coef ficient (the term inside the last bracket) are defined by

(10)

It is noted that the point of diffraction is at the truncation edge

and remains fixed regardless of the Floquet modes under consid-

eration. Only the diffraction coef ficient, in addition to ,

is mode dependent. Thus for a non-uniform examination (i.e.,

in (5)), the summation of these truncation contribu-tions will form a Fourier series, which is combined with the first

term on the right-hand side of (3) to result in a net diffraction

contribution from the truncation given by

(11)

The detailed derivation of (11) is described in Appendix B.

E. The Characteristics of Special FunctionsThe diffraction coef ficient in (9) becomes singular as the ra-

diation point moves close to the edge diffraction point, i.e.,

, which makes . The standard

UTD Fresnel transition function [26]–[29] is given by

(12)

where the parameter, in [5], accounts for the phase

difference by


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(13a)

It is noted that can be approximated by considering

an expansion over (8a), which gives

(13b)

and makes (13a) become

(13c)

This expression allows one to find the argument in terms of the

parameters along the geometrical ray paths in Fig. 2 and sim-

plify the computation.

III. 3-D SEMI

-I NFINITE

ARRAY

R ADIATION

PROBLEM

A. Problem Composition of 3-D Radiation From a 2-D Array

The 2-D semi-infinite, linear array of point sources

is illustrated in Fig. 4. Its th element is located at

( , )

on the x-y plane with and being their inter-element

periods, respectively. The array is excited to focus its radiation

at by

(14)

where is the reference excitation weighting, and. The net radiation poten-

tial at is described by [17]

(15)

where . In (15), the

permittivity and permeability are omitted so that it can be used

to consider either electrical or magnetic current sources. The

polarization of the current sources is also omitted for simplifi-

cation. The Floquet modes are obtained by using the Poisson

sum formula by [23]:

(16)

Thus, the th Floquet mode of (15) can be expressed as

(17)

Fig. 4. A 2-D semi-infinite array with a truncation at .

where , and and are

and with and replaced by and , respectively. The

uniform asymptotic evaluation of (17) follows a form identical

to (5) in Section II-B, and will not repeated for brevity. The

components are described in the following subsections.

B. The Asymptotic Solution of Direct Radiation From a

Non-Truncated Array

A radiation point, , exists on the array aperture plane,

which satisfies the following conditions:

(18)

The field radiated directly from the same array with an infinite

extent has been investigated in [17] with significant phenomena

explored. The solution can be described as

(19)

where , and

(20)

with

(21)

and

(22)

In(19), and are and , respectivelycorresponding to

the distances between the field observer and focus point, and the


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Fig. 5. Ray tubes associated with the radiation and diffraction rays. After ra-diating from the array aperture, the radiation ray generally has two ray caustics

off the array aperture for a nonzero mode while the diffraction ray has only aray caustic off the truncation edge.

Fig. 6. Curved surface of radiation ray caustics for and modes,

where the two ray caustic distances are assumed to be the same for simplicity.

radiation point, respectively. The radiation ray tube is illustrated

in Fig. 5. As previously pointed out in [16], [17], there are two

ray caustics for each ray for each Floquet mode except at the

zeroth mode where all rays coincide at a point ray caustic (i.e.,

the focus point). Similar to the description of the 1-D case in

Section II-C, there are two radiation points for the propagating

fields for nonzero modes. The coincidence of these two points

will form a curved surface of ray caustics, where an example is

illustrated in Fig. 6 with and assumed. A uniform

formulation to account for the coincidence or close proximity of

these two radiation points is described in [17].

C. The Asymptotic Solution of Edge Point Contribution From

a Truncated Array

The diffraction contributions from the truncated edge diffrac-

tion point have been developed in Appendix C, and are summa-

rized in the following. Let and denote the distances

between focus and diffraction points, and between the field and

diffraction points, respectively, which are given by

(23)

with being a diffraction point located on the array trun-

cation edge. The propagation vector components are defined by

(24a)

(24b)

The diffractionpoint, , canbe found fromthe following

condition:

(25a)

(25b)

which fulfills the conditions in (A11) and (A12). This condition

will make the diffraction rays launched from the same diffrac-

tion point to form a cone in a fashion similar to the Keller cone

in the UTD wedge diffraction problem [26], which is illustrated

in Fig. 7. In particular, for the zeroth mode the focus point will

be located on the surface of this cone. For nonzero modes, the

span angles of the cones will become either larger or smaller in

comparison with that of the zeroth mode. The cone will disap-

pear and move into a complex space when the coordinate of the

diffraction point is complex. Thus the diffraction field can be

expressed as

(26)

where the distance toward the ray caustic is given by

(27)

The diffraction coef ficient is identified to be the term inside

the last bracket of (26). The diffraction ray tube is illustrated

in Fig. 5, where a ray caustic occurs at a position off the array

aperture. It is also noted that, similar to the arguments described

in Section II-D, the points of diffraction are at the same location

on the truncation edge for a selected value regardless of the

change of in the Floquet modes under consideration. In this

case, only the diffraction coef ficients, in addition to , are

mode dependent.Thus for a non-uniform case (i.e., in(5)), the summation of these truncation contributions will form


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Fig.7. Given a focus pointand diffraction point, thediffraction rays willformacone, which is similar to the Keller cone in the ordinary UTD wedge diffraction.

(a) mode; (b) mode.

a Fourier series, which is combined with the first term on the

right-hand side of (16) to result in a net contribution by

(28)

The detailed derivation is shown in Appendix D. However, the

diffraction phenomena are similar to the radiation from a 1-D

array as the condition of diffraction in (25) is in a form iden-

tical to (8). Thus, without the need of going through the de-

tailed derivations, one may easily conclude the existence of two

diffraction points. A uniform formulation with Airy functions as

the transition functions can be further performed to account for the effects of possibly close-by diffraction points as previously

examined in [16], [17]. This formulation is quite standard and

can be obtained by symbolic replacements in (28) of [16], which

will not be repeated for brevity. However, it is worth mentioning

that the coincidence of these two diffraction points occurs when

the field points are on the curved surface of diffracted ray caus-

tics. This curved surface of ray caustics is illustrated in Fig. 8.

At each cut of the curved surface along the truncation edge, the

curve will be identical to that in [16] for case of radiation from

a 1-D infinite array.

D. The Characteristics of Special Functions’ Argument

The diffraction coef ficient in (26) becomes singular as the

radiation point moves close to the edge point, i.e.,

Fig. 8. Curved surface of diffraction ray caustics, which is rotationally sym-

metric along the edge axis. Each cut of the surface intercepted by a plane con-

taining the truncation edge will show identical characteristics of the cau stic

curve as in the 1-D linear array described in [16] because of the rotational

symmetry.

, which makes and

. The parameter, , accounts for the phase difference by

(29)

One performs a 2-D Taylor expansion over (29) with the condi-

tion of (25) in mind to give

(30)

where

or (31)

Also one approximates and by considering

an expansion over (18), which gives

(32)

Substituting (32) into (30) gives

(33)

where the parameters can be found using a similar

formula in (20)–(22) with the related parameter changed by

and the propagation vectors replaced by (24).


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Fig. 9. Diagram of approach to build up a valid solution formulation when twocritical points become close together. The two radiation points are indicated

by A and B while the two diffraction points are indicated by C and D. Thecoincident points are indicated by with being A or B, and being C or

D, respectively. is the coincident point of all four critical points.

E. Contribution From the Edge 1-D Array

The contribution of Floquet modes from the edge linear 1-D

array, i.e., the first term on the right-hand side of (16), is con-

sidered. The th mode is defined by

(34)

which can be evaluated by following the procedure in

Appendix C. As a result, (34) can be expressed as

(35)

where the ray parameters are identical to those defined in

Section II-C.

F. Validity of Formulation for Near-by Radiation and Diffraction Points

It was shown that there are two radiation points and two

diffraction points for the radiation from a semi-infinite array of

antennas in this general focus problem. It is thus required to

consider the situations when the radiation and diffraction points

coincide. Fig. 9 illustrates the strategy to build up a uniform

formulation that provides smoothly convergent solutions when

the radiation and diffraction points come close-by. In particular,

points A and B stand for the two separated radiation points while

C and D stand for the diffraction points on the truncation edge.

Also ( and are either A, B, C or D) represent the co-

incided point of any two points. represents the coincided

point of the four points.

In the case of two close-by radiation points (i.e., points A and

B), the uniform formulation for the field radiation from an in-

finite array has been developed in [16], [17] via the utilization

of Airy functions. The argument to assure the uniform conver-

gence of fields has been discussed in [16]. Similarly, the uni-

form formulations contributing to the fields of truncation ef-

fects, which are radiated from the two diffraction points C andD, can be formatted according to the discussion of diffraction

phenomena in Section III-C, whose form was developed in [16].

This case will occur when the observer is on the curved surface

of ray caustics in Fig. 6.

On the other hand, the formulation in (5) provides a uni-

form solution for the fields radiated from a pair of radiation

and diffraction points (for example, A and C). This uniform for-

mulation utilizes the Fresnel integral to cancel the discontinuity

caused by the diffraction coef ficient. In particular, at the coinci-

dent point, such as at , the net field is equal to a half of the

radiation field resulting from an infinite array with a radiation

point at . A similar formulation can be built for the case of

radiation and diffraction points B and D. This case will occur

when the observer is on the curved surface of caustics in Fig. 8.

The situation when all four points coincide can be formulated

by considering the coincidence of and at . As

discussed in the previous paragraph, the radiation fields at the

observer contributed from and are equal to half of

their individual radiation from an infinite array with radiation

points at and , respectively. Thus the uniform solu-

tion can be formulated by utilizing the Airy functions as demon-

strated in the case of two close-by radiation points in [16], [17].

This case occurs at the observer located on the overlap curve of ray caustic surfaces in Fig. 6 and 8. However, the uniform for-

mulation in the case with any three of radiation and diffraction

points close-by and coincident is not validated in this work. It

will be investigated in a future phase.

IV. NUMERICAL EXAMPLES

The numerical examples first consider the radiation in the

near-zone from a 1-D semi-infinite array of

line current sources, where the inter-element period is ,

and thus the edge element is at . In this

case, the focus point is at , the frequencyis 10 GHz. The near-field is observed at , and is com-

pared with the reference results obtained by numerical integra-

tion. Fig. 10(a) shows the pattern of the mode, where

the shadow boundary (SB) of edge diffraction is also shown.

The pattern on the left-hand side of the SB is purely caused by

the edge diffraction, which varies monotonically because the

array has only a truncation. On the other hand, the pattern on

the right-hand side is caused by the superposition of direct fields

and diffracted fields, and exhibits an oscillating curve, where a

single saddle point exists for the direct radiation field. Fig. 10(b)

shows the patternsof the mode, where two saddlepoints

exist for the direct fields. The two curves associated with these

two saddle points almost overlap on each other. The boundary


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Fig. 10. Validation of the radiation pattern in the near-zone of a 1-D array in

comparison with a reference solutionobtained by numerical integration. (a)mode; (b) mode.

between the propagation and evanescent regions are also shown,

where the two saddle points tend to coincide. The fields dif-fracted from the array truncation are also shown in Fig. 10(b),

which exist in both propagation and evanescent regions. In the

deep evanescent region, the diffracted fields will dominate since

the evanescent waves decay significantly. Thus in the propa-

gating field region, the pattern is formed by the superposition

of the contributions from these two saddle points and the edge

diffraction field. On the other hand, in the evanescent field re-

gion, the pattern is formed by the superposition of the contribu-

tions from evanescent waves and the edge diffracted field. Thus

an oscillating curve is observed in both regions, where the am-

plitude of oscillation is larger in the propagating field region. As

expected, the agreements with the results of numerical integra-

tion are excellent in both cases.

One next considers the radiation from a 2-D semi-infinite

array ( and ) of point current

sources, where and are examined.

One first examines the radiation of the mode

at the observers along the and line, where

is considered for the array. As shown in Fig. 11(a), the

results exhibit phenomena similar to the case of the mode

radiation of a 1-D array with a faster oscillation. The agreement

with the reference result obtained by numerical integration is ex-

cellent. One next considers the mode radiation.In this case the observation line is moved outward to

Fig. 11. Validation of the radiation pattern in the near-zone of a 2-D array in

comparison with a reference solution obtained by numerical integration. (a)mode; (b) mode.

with for the array dimension. The observers are all in

the propagation region, so the field decays in the deep shadow

region of direct radiations as can be observed. In the lit region,

there is only one saddle point found so the field variation ex-

hibits oscillation because of the interaction with the truncation

diffraction field. The agreement with the reference result ob-

tained by numerical integration is again excellent.

V. CONCLUSION

A uniform asymptotic formulation for the EM radiation from

a semi-infinite, periodic array is presented in this paper. It de-

composes the radiation mechanisms with respect to the funda-

mental concepts of UTD, and can be applied to interpret the

wave propagation phenomena in a general radiation problem of

a focused field. In particular, the diffraction mechanisms from

the array truncation are investigated with numerical example

demonstrations. This formulation remains valid as the radiation

point approaches the diffraction point on the truncation edge

of the array. As discussed in [16], [17], there are two radiation

points on the array aperture when the radiation focuses in the

near-zone. The coincidence of these two radiation point occurs

in the case that the field points are on the caustic curves of each

Floquet mode. The presented formulation becomes singular as

the diffraction point is near the coincident point of these two ra-

diation points. These phenomena will be investigated in a future phase of this work.


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APPENDIX

A. Truncation Diffraction of Floquet Waves for a 1-D

Semi-In finite Array

One considers (4) and substitutes the spectral representations

of the Hankel functions, and , into (4)

which gives

(A1)

The integral inside the bracket is first found in closed form. Re-

arranging each term makes (A1) become

(A2)

The evaluation of (A2) can be performed asymptotically in a

sequential manner. The integral inside the bracket of (A2) is

first evaluated by the stationary phase method to consider the

contributions from the saddle points and poles via the modified

Pauli-Clemmow method [28], [29]. Afterward, the integration

over is subsequently performed to consider the saddle point

contribution. The procedure resulting in the formulation in (5) is

quite standard and omitted. In this case, the contribution arising

from the pole at (poles and saddle point

contributions in and integrations, respectively) gives the

direct radiation from the same array configure with an infinite

extent, whose results are summarized in Section II-C. On the

other hand, the double saddle point contributions in and

integrations give the diffraction effects arising from the edge

truncations, and are summarized in Section II-D.

B. Net Truncation Diffraction From a 1-D Semi-In finite Array

One considers (2) and substitutes the spectral representations

of Hankel functions, and , into (2) which

gives

(A3a)

with

(A3b)

The summation inside the bracket can be found in closed-form.

As a result, (A4) becomes

(A4)

The evaluation of (A4) can be performed in an identical fashion

as (A1) by considering the contributions from the poles and

saddle points. It is straightforward to show that there exist an

infinite number of poles with each resulting in identical formula-tions shown in Section II-C. On the other hand, the saddle point

contribution results in a single formulation, which corresponds

to the net diffraction effect and is equal to superposition of all

diffraction terms in part A of this appendix and the first term in

(3). Without going through the details, the saddle point contri-

bution gives (11).

C. Truncation Diffraction From a 2-D Semi-In finite Array

One considers (17) and uses the spectral representations of

the free space Green’s function by

(A5)and

(A6)

where and .

Afterward, one uses the delta function identity

(A7)

and (17) becomes

(A8)


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where . The integral inside the

first bracket can be exactly found while the integral inside the

second bracket can be evaluated asymptotically to account for

the saddle point contribution. These two integrals, denoted by

and , respectively, are given by

(A9)

and

(A10)

with and

. Here satisfies the following

equation:

(A11)

It is noted that has a pole singularity at .

This pole contribution in (A8), in conjunction with the saddle

point contribution from the integral of the variable,

will result in the direct radiation of an infinite array as sum-

marized in Section III-B. The double saddle point contribution

from and will result in the truncation diffraction. These

double saddle points occur at

(A12)

The result of (A8) will reduce to the summary in Section III-C.

D. Net Truncation Diffraction From a 2-D Semi-In finite Array

One considers (15) and employs the Poisson’s sum formula

over the infinite summation, which gives

(A13a)

with

(A13b)

Using identities similar to (A5)–(A7) and (A3b), (A13) becomes

(where (A6) is used explicitly in the following)

(A14)

where and are the sameas in(A6) and (A8), respectively.

It is noted that (A14) has a form almost identical to (A8). Thus

without repeating the detailed procedure, the result of (A8) can

be employed here with the following replacement:

(A15)

which gives the results in (28).

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Hsi-Tseng Chou (S’96–M’97–SM’01–F’12) was born in Taiwan, in 1966. He received the B.S. degree

from National Taiwan University, in 1988, and the

M.S. and Ph.D. degrees from Ohio State University(OSU), Columbus, OH, USA, in 1993 and 1996,

respectively, all in electrical engineering.In August 1998, he joined Yuan-Ze University

(YZU), Taiwan, where he is currently a Professor in the Department of Communications Engineering.

After completing his military obligation, he workedfor one year as an R&D Engineer at China Raydon

Corp., during which time he was sent to Mitsubishi Electronic, Japan, for

three months’ technical training. From 1991 to 1996, he was a GraduateResearch Associate with the ElectroScience Laboratory (ESL), OSU, and

from 1996 to 1998, a Postdoctoral Researcher. After joining YZU in 1998,he was simultaneously a Technical Consultant to several industries including

Wistron NeWeb, Zinwell, Jonsa and Skyworks. His research interests includewireless communication network, antenna design, antenna measurement, elec-

tromagnetic scattering, asymptotic high frequency techniques such as uniform

geometrical theory of diffraction (UTD), novel Gaussian beam techniques, andUTD type solution for periodic structures. He has published more than 400

journal and conference papers.Dr. Chou is an IEEE Fellow, an IET Fellow, and an elected member of URSI

International Radio Science US Commission B. He has received many nationalawards to recognize his distinguished contributions in technological develop-

ments. Some important ones include a Young Scientist Research Award fromAcademia Sinica of Taiwan, a Distinguished Contribution Award in promotinginter-academic and industrial cooperation from the Ministry of Education, a

Distinguished Engineering Professor Award from the Chinese Institute of En-gineers, a Distinguished Electrical Engineering Professor Award from the Chi-

nese Institute of Electrical Engineering, and University’s Industrial EconomicsContribution Award and National Award for Industrial Innovation—Key Tech-

nology Elite Award both from Ministry of Economics. He was elected in 2004

as one of the nation’s ten outstanding young persons by the Junior Chamber International, in 2005 he was awarded a National Young Person Medal from

China Youth Corps of Taiwan, and was named one of the Top 10 Rising Starsin Taiwan by the Central News Agency of Taiwan. He has served as the Chair

of IEEE AP-S Taipei Chapter and received the Best Chapter Award in 2012. Healso received Outstanding Branch Counselor Awards from the IEEE including

IEEE Headquarter, R-10 and the Taipei Section, respectively. In addition, hereceived the IEEE Technical Field Undergraduate Teaching Award in 2014.

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