Wave propagation inside a two-dimensional perfectly ... - IEEE Xplore

Wave Propagation Inside A Two-Dimensional Perfectly Conducting Parallel-Plate Waveguide: Hybrid Ray-Mode Techniques and their Visualizations Leopold B. Felsen‘, Funda Aklemad, and Levent Sevg? ’Dept. of Aerospace & Mechanical Engineering and Dept. of Electrical & Computer Engineering, Boston University 110 Cummington Street, Boston, MA 02215. USA E-mail: [email protected] ’Electronics and Communication Engineering Department, Istanbul Technical University 34469, Maslak. Istanbul, Turkey 3Electronicsand Communication Engineering Department, Doeug University Zeamet Sok. No. 21, AcibademIKadikdy, 34722 Istanbul. Turkev E-mail: [email protected]

Abstract

This article is intended as an educational aid, dealing with high-frequency (HF) electromagnetic wave propagation in guiding environments. It is aimed at advanced senior and first-year graduate students who are familiar with the usual engineering mathematics for wave equations, especially analytic functions, contour integrations in the complex plane, etc., and also with rudimentary saddle-point (HF) asymptotics. Afler an introductory overview of issues and physical interpretations pertaining to this broad subject area, detailed attention is given to the simplest canonical, thoroughly familiar, test environment: a (timeharmonic) line-source-excited two-dimensional infinite waveguide with perfectly conducting (PEC) plane-parallel boundaries. After formulating the Green’s function problem within the framework of Maxwell’s equations, alternative field representations are presented and interpreted in physical terms, highlighting two complementary phenomenologies: progressing (ray-type) and oscillatory (mode-type) phenomena, culminating in the self-consistent hybrid ray-mode scheme, which usually is not included in conventional treatments at this level. This provides the analytical background for two educational MATLAB packages, which explore the dynamics of ray fields, mode fields, and the ray-mode interplay. The first package, RAY-GUI, serves as a tool to compute and display eigenray trajectories between specified sourcelobserver locations, and to analyze their individual contributions to wave fields. The second package, HYBRID-GUI, may be used to comparatively display range andlor height variations of the wave fields, calculated via ray summation, mode-field summation, and hybrid ray-mode synthesis. Keywords: Electromagnetic propagation; parallel plate waveguides; spectral techniques; modes; ray tracing; hybrid ray-mode approach

w.

1. Introduction

hen high-frequency (HF) time-hannonic wave propagation IS spatially (transversely) constrained by physical impenetrable boundaries - or by “virtual” boundaries (ducts) established

through wave-trapping (graded-index) refraction in transversely inhomogeneous media - the resulting source-excited longitudinally guided (ducted) waves have traditionally been described in terms of two alternative phenomenologies: progressing and oscillatory

phenomena. The progressing formulation views the wave fields as continuous spectra o f waves that propagate (through the guiding environment) from the s o w e to the receiver via multiple transverse reflections and (or) refractions, which are localized around source-receiver-dependent “eigenray” trajectories. The oscillatory formulation views the wave fields as discrete (or discrete-continuous) superpositions o f frequency-dependent (guided-mode) “eigen-

IEEE Antennasandpropagation Magazine, Vol. 46, No. 6, December 2004

ISSN 1045-924312004B2002004 IEEE

69

spectra’,’-which are matched to the entire transversely confining cross section. These eigenspectra are individually independent of source and receiver locations, but have amplifudes of excitation and reception that do depend on these locations. In the “over-moded” HF regime, where the transverse dimensions span many wavelengths and many modes can propagate, the oscillatory formulation becomes unwieldy and is not well-matched to the wave physics for large source-receiver separations. This is because the fonvard-propagating lower mode eigenspectra (eigenangles) are found to form closely spaced clusters. Here, the eigenray progressing wave spectra can be more efficiently adapted to the wave phenomenology. On the other hand, at small sourcereceiver separations, the progressing formulation becomes unwieldy. This is because it now entails many closely spaced nearvertical multiple reflections. The oscillatory lower eigenmode spectra are widely spaced, and therefore more efficient. While excited in their totality very near the source, these modes decay exponentially (are evanescent) longitudinally away from the source when the number of oscillations in their transverse profile exceeds a frequency-dependent cutoff threshold. These circumstances have motivated efforts to combine the two complementary spectral methodologies - neither of which is convenient for all source-receiver locations - in a manner that seeks to exploit the best features of each. The outcome has been a comprehensive, rigorously based, self-consistent hybrid ray-mode algorithm. This has clarified the ray-mode interplay in the frequency and time domains through a series of spectral studies and wide-ranging applications to complex waveguiding environments [1-12]. The basic prototrpe problem by Felsen and Kamel [2] demonstrates alternative hybrid ray-mode partitioning of parallelplate-waveguide Green’s functions, with numerical examples that quantify all relevant dynamic constituents. Changes in the sourcereceiver locations give rise to intricate compensating wave mechanisms, which account for smooth transitions when a mode field or a ray field enters or leaves a specified angular-spectrum interval. These mechanisms can be systematically ‘examined in this idealized environment, thus furnishing cogent physical insight that is essential for treating more complicated guiding configurations. Although our presentation here is based on the (more advanced) treatment in 121, it is restructured in a tutorial fashion, so as to make it more easily accessible to our target student audience: advanced senior and first-year graduate students. (For a more formal mathematical treatment of waveguide Green’s functions, intended for advanced graduate students, see [13]; however, this does not contain the hybrid ray-mode decomposition). Two MATLAB packages have been prepared for tutorial visnalization of the hybrid ray-mode interplay. They involve MATLAB 6.5, and are compiled as standalone packages’.

sentation is given in Section 2. Alternative reductions in terms of global oscillatory wave objects (i.e., modes) and local progressing plane waves (i.e., geometrical-optic ray fields) are outlined in this section. The formulations used in the Waveguide simulator and in the MATLAB packages prepared for the reader are explained in Section 3. Numerical examples are presented in Section 4. Finally, conclusions are summarized in Section 5.

2. Propagation lnsidea Two-Dimensional PEC Parallel-Plate Waveguide The presentation in [2] deals with ray, mode, and hybrid field formulations for the two-dimensional parallel-plate waveguide with general impedance- (Cauchy-) type boundary conditions. There, rays between the transmitter and observer encounter lateral shifts at each reflection when they interact with the boundaries. When perfectly conducting (PEC) boundaries (i.e., Dirichlet-type for the TE, problem, and Neumam-type for the TM, problem) are assumed - as in our presentation here - the lateral shifts disappear, and rays are reflected from the boundary at the angle of incidence (Snell’s reflection law).

2.1 Formulation of the TE- and TM-Type Problems The PEC parallel plate waveguide, with height a, has its propagation direction along the z coordinate of a Cartesian system, extending from -m to +m along the y and z directions (see Figure 1). Assuming that the medium between the plates is free space, one can use Maxwell’s curl equations in the form (a time dependence exp(jwt)is suppressed) [13, Section 5.51 VxE=-jopoH-M,

(la)

V x H = jw&,,E+J,

(Ih)

where boldface symbols denote vector quantities. The non-phased source currents, J and M I extend along the y direction, whence

throughout (i.e., the fields are functions of ( x , ~ ) ) .This reduces the vector equations in Equation (l), with Equation (2), to two sets of scalar equations - TM, (transverse magnetic with respect to z)

The layout of this paper is as follows. The two-dimensional (2D) Green’s function problem is formulated and its spectral repre-

+X

Line source

X-a,

‘The German phrase “eigen” (proper, characteristic) is used in the English mathematical literature to identify “eigenvalue problems,” the “eigensolutions” (eigenmodes) of which, parameterized by “eigenvalues” (spectral wavenumbers), describe source-free wavefields that individually satisfy the boundary conditions in wave-confining environments. ’The RAY-GUZ and HYBRID-GUI packages can he downloaded from the Web sites http://www3.dogus.edu.tr/lsevgi or

http://www.ehb.itu.edu.tr/funda. 70

,

/

Observer

/

PEC

/

/

i

.P

x=o

PEC

I

‘ -5

Figure 1. The physical configuration: A two-dimensional, homogeneously filled, parallel-plate waveguide with perfectly conducting boundaries (plate height = a). S is the source point and P is the observation point.

lEEE AntennasandPropagation Magazine, Vol. 46, No, 6 , December 2004

and TE, (transverse electric with respect to z) - which are given as follows:

outgoing wave (radiation) condition at z + A"

(9b)

The remaining two field components for this set are

SET#I: TM,

Since the TE, and TM, sets are decoupled, each can he excited independently of the other by appropriate selection of the sources, J and M . The line sources M y , J,. J , excite the TM, set, whereas the line sources

SET #2: TE,

M, , M, , J y excite the TE, set. Further

simplification can be obtained by setting the source components J , = 0 , J , = 0 for the TM, set, and MI = 0 , M, = 0 for the TE, set. These simplificationswill he assumed in what follows. aHx

'HZ - J y + j o x O E y ,

& a x

2.2 The Green's Function Problem

These equations can in turn be combined into second-order partial differential equations - the wave equations - as follows:

The Green's function problem, associated with both the TE, set (when M, = M , = 0 ) and the TM, set (when J , = J , =O), is defined by the equation

TM, wave equation

with boundary conditions g ( x , z ; x ' , z ' ) = O at x = O , a ,

with boundary conditions

a

-g

ax

outgoing wave (radiation) condition at z + t m ,

Here, k = 2lr/1= o &

is the free-space wavelength. The remaining two TM, field components can be derived from

(124

(TM,)

( 12b)

outgoing wave (radiation) condition at z + km.

(6b)

is the free-space wavenumher, and 1

( x , z; x',.') = 0 at x = 0 , a ,

(TE,)

(12c)

Here, ( x ' , ~ ' )and ( x , z ) specify source and Observation points, respectively, and S ( x ) , S ( z ) are the Dirac delta functions, with S(x)=O, x # o ; 6

lS(x)dx = 1, a < x < b . (1

The delta function is symmetric, i.e., J ( a - a ' ) = s ( a ' - a ) , TE, wave equation

whence g , ( x , z ; x ' , z ' )= g , ( x ' , z ' ; x , z ) , thereby exhibiting reciprocity under the interchange of source and observation points. The field, F ( x , z ) , excited by a physical distributed source, S ( x , z ) , can he synthesized in terms of the (symbolic) line-source Green's function as follows:

with boundary conditions E,=Oatx=O,a; IEEE Antennas andpropagation Magazine, Vol. 46, No. 6, December 2004

F ( x ,z) =

IJ dddz'S(x',z')g

( x , z; x',z').

(14)

SOWCe

regmn

71

one-dimensional (ID) eigenvalue problems based on Equations (19a) and (19b), respectively.

The solution of the coordinate-separable wave-equation and Green's-function problems is discussed next. In keeping with the tutorial focus in this paper, the spectral foundations of the solution strategy are summarized in some detail.

2.4.1 The x Domain

2.3 Accessing the Spectral Domain: Separation of Variables

For Equation (19a), the m-indexed eigensolutions (eigenhctions) Uxm(x) , corresponding to the eigenvalues k,, , are given by

To solve the wave equation

[$+,a22

%,,(I)=

+ k 2 ] U (x, z ) = 0 ,

A,sin(k,,x), (20)

k,,

ma

= -, m=1,2,3,._.,O S x < a ,

a

subject to the boundary conditions U ( x , z )= 0 at x = 0, a ; radiation condition at z

+ ?m,

where Am is an arbitrary constant. It is readily verified that for any two eigeufunctions with indexes m, and m2, respectively, the following relation - the orthogonality relation - applies:

(16)

assume that we can decompose

Y

U ( x , z )= U , ( x ) U z(21,

(17)

Am,Am2

pxm, ( 4 U X W( 2x ) A = 4 d m 2 S m ) % >

3

(21)

0

where U x satisfies the first boundary condition in Equation (16), and U2 satisfies the second boundruy condition in (16). Substituting Equation (17) into Equation (15), performing' the partial derivatives, and rearranging the result, one finds that

where the Kronecker delta S,,,m2

=0,

ml

;e m 2 , while

Sm,,m2= 1 ,

m, = m 2 . It is convenient to normalize the eigenfunctions with respect to Am so that

t

Y

f u ~ m ~ ( x ) ~ ~ m * ( x ) d x = ,i.e., S ~ , ,A, , * = -,

0

(22)

where a prime denotes the derivative with respect to the argument. Since the left-hand side in Equation (18) is a function of x only, while the right-hand side is a function of z only, and the equation is valid for arhitmy ( x , z ) in the domains of Equation (16), each

which renders the eigenset orthonormal. We shall use this normalized format throughout.

side must be a constant, which is denoted by k:. Thus, Equation (15) is reduced to two separate ordinary differential equations:

eigenexpansion of any "suitable" function, F, (x) (for which the

U:+k:U,=O, U: + k:Uz

OSxSa;

U , = O at x = O , a ;

The implied completeness of the eigenset {Uxn,)admits the series converges), m

(19a)

(23)

F,(x)= za,Uxm(x), O S x S a .

,=a = 0 , -m

< z < +=c; radiation condition at

z

+ *m To determine the amplitudes a m , multiply Equation (23) by any

(19h)

a

k Z = k,'

+ k,'.

other eigenfunction U,,

(194

( x ) , perform the integration f d x on both 0

The dispersion equation in Equation (19c) relates the x-domain and z-domain spectral wavenumbers, k, and k, , respectively, to the physical wavenumber, k.

2.4 Spectral Representations: Eigenvalue Problems'

sides, interchange the order of integration and summation on the right-hand side, and invoke Equation (22). This yields the result

Changing the arbitrary index mi to m, as in Equation (23), completes the expansion of F , ( x ) in Equation(23), with Equation (24), in terms ofthe { U x m }eigenhasis.

Spectral representations of fields are structured around discrete (or continuous) sums (or integrals), extending over complete (preferably orthogonal) sets of eigen (basis) functions that are matched to the problem conditions. The two-dimensional coordinate-separable problem in Equations (15) and (16) admits separate 72

The completeness of the eigenset can be formalized concisely by setting F, (x) = S(x-x'), which reduces the integral on the left-hand side of Equation (24) to U,,, (x') via the

IEEE Antennas andpropagation Magazine, Vol. 46, No. 6, December 2004

properties of the delta function. Accordingly, am,= Umb (x') , and

2.5 Spectral Representations: OneDimensional Characteristic Green's Function Problems

from Equation (23), m

S(x-x')

U,, (x)Uxm(x'), 0 I: (x,x') 5 a .

=

(25)

m=0

The spectral representation of the delta function in Equation (25) is referred to as the completeness relation, because any suitable function F , ( x ) can be expressed by performlng the (identity) a

operation F , ( x ) = IF,(x')S(x-x')dr'

The two-dimensional Green's function (GF) problem in the physical domain has been formulated in Equations (1 I ) and (12). It differs from the eigenvalue problem in Equation (15), due to the presence of the (delta-function) source terms on the right-hand side. The corresponding reduced one-dimensional spectral problems in the x and z domains are modified accordingly.

on the summand of the

0

right-hand side of Equation (25), which recovers Equation (23) with Equation (24). Equations (19a) and (22) form the hasis for an x-domain Fourier series representation of the total field (see Section 2.6.1).

2.5.1 The x Domain The spectral (also referred to as the "characteristic") Green's function g, is defined as follows:

{$

+ k : ] g x (x,x'; k,)

2.4.2 The z Domain

= -6 (x - x') , 0 5 (x,x') 5 (I ;

(28) g,=Oatx=O,a.

For Equation (19b) with its infinite domain, the eigenvalues coalesce into a continuum, ,k +k, , where k, is a continuous variable, io< k, < +m (this can be verified by starting with a finite domain, zI< z < z 2 , and letting q 2+ k m [14, Sec-

Now, k, is a specified spectral wavenumber. For the unique specification of g, in Equation(28), it is necessav that

tion 3.3~1. The corresponding trigonometric eigenfunctions,

k, t *km for any m, because the eigenfunctions satisfy the source-free form of Equation (28) (see Section 2.4.1), and can therefore be added to the source-driven solution without violating Equation (28). The solution for g, can he constructed explicitly in the following form ([14, Section 3 . 3 ~ 13, ; Section 3.21:

. U ( z , k , ) ,render the normalizing integral r U 2 ( z , k , ) d z diverm

gent. To assign a proper "weight" to the divergence, one can track the transition from the aforementioned initial finite z domain systematically to infinity to find for the orthonormality condition,

.-

L"

I

U(z,k,, )U' +

(&, )& = 6 (kz, - kz2), (26)

U ( z ,k ) = z

1

- m < (z,k,) < +m,

&

where the complex conjugate has been introduced in view of the convenient exponential form for the eigenfunctions. Thus, the for the discrete spectra in normalization expressed by S,,,, Equation (22) is replaced for continuous spectra by the delta function S( k,, - k+). By symmetry, interchanging z + k, , k, + z , leads to the completeness relation

+= [e-'kz(z-z')dkz= S ( z - z ' ) , 211 _m

-m x' , but x' for x < x' . Here, g, and g, are solutions of the homogeneous ( x f x ' ) form of Equation (28) that satisfy the boundary conditions at x = 0 and x = a , respectively. Thus, reciprocity is satisfied in Equation (29). The Wronskian,

is a function of k, only, independently o f x (or x'). The solution in Equation (29) is continuous at x = x' , but has a discontinuous x derivative there. This gives rise to the impulsive (delta-function) second derivative that accounts for the source term on the righthand side of Equation (28). Specifically, we can construct the solution in Equation (29) via

operation F , ( z ) = [ F z ( z ' ) S ( z - z ' ) d i ' . Equations(26) and (27) form the hasis for z-domain Fourier integral formulations of the total waveguide field (see Section 2.6.2).

IEEE Antennasandpropagation Magazine. Vol. 46, No. 6, December 2004

&(x,kx) = sin(k,x), (31)

Ex (x, k, ) = sin [k, ( a - x)] ,

73

2.6.1 Eigenfunction Expansion in the x Domain, One-Dimensional Green’s Function in the z Domain

whence

W, = -k, {sin(k,x) cos[ k, ( a - x ) ] + sin[k, ( a - x)]cos (k,x)} = -k,sin(k,a)

We write

from Equation (30). Accordingly, g, ( x , x ’ ; k , ) =

sin (k,x,)sin[k, ( a -I,)] k,sin(k,a)

(32) where Ur,( x ) is given in Equation (20), with Am ::

2.5.2 The z Domain The characteristic Green’s function ,g,, is defined as fol-

in view of Equations (19a) and 1 (19c). With this reduced x-domain formulation in mind, Equa-

lows:

obtains 8’ dx’ U,,

[$+ 41

( 1

g, (z,z’; k,) = -6 (z - z’), - m < (z,z’, k,) 5 +m

radiation condition at z + +m,

(see

Equation (22)). This form is suggestive in view of the delta-function representation in Equation(25), which is used to replace S ( x - x ’ ) on the right-hand side of Equation (11). Substituting Equation (37) into Equation (1 I), and performing the partial differentiations on the summand in Equation (37), for the .x domain one --f

-k:,U,,

tion ( l l ) becomes

(33)

with the solution (in the notation of Section 2.5.1) ,=I

(34) ‘Here, 8, and

g,

where k& are outgoing wave solutions in z < z‘ and z > z’ ,

respectively, in accord with the radiation condition (Re { k r ] > 0 ,

= k Z -k:m.

Equating the coefficients of the orthogonal

eigenfunction expansions on both sides, one identifies as a spectral Green’s function that satisfies

2,

(2-2’)

Im { k z ] < 0 for the assumed exp( jot) time dependence):

g = e +jk,z subject to the radiation condition at Izl + m . The solution ofEquation (39) is given in Equation (36), provided that k,

W, = -2 j k ,

+ kZm:

Thus, since ( z < - z , ) = - ~ z - z ‘ ~ ,

k,

2.6 The Two-Dimensional Green’s Function Problem: Alternative Representations

=

-(y)z.

Thus, the desired representation for g(x,z;x’,z’) is m

g(x,z;x’,z‘)=Cg,,(z,z’) “,=I

The two-dimensional Green’s function problem in physical space is defined in Equation (II), with Equation (12). Altemative field representations for the two-dimensional Green’s function can be generated by various combinations of the one-dimensional spectral-domain eigenfunction expansions and characteristic Green’s functions pertaining to the physical x domain and z domain, as treated in Sections 2.4 and 2.5 (for a more-general unified approach, see the Appendix).

74

which is convenient for evaluating the waveguide field away from the source plane z = z’ when the number of propagating modes

I 1 < 0,

with real k,, (i.e., k > m z l a ) is not too large. Since lm k,

the higher-order non-propagating modes with k < m z l a decay along z.

IEEE Antennas andPropagation Magazine, Vol. 46, No. 6, Deo?mber 2004

m

2.6.2.1 Eigenfunction Expansion in the z Domain, One-Dimensional Green’s Function in the x Domain Following the same procedure as in Section 2.6.1, and referring to the z-domain eigenexpansion in Section 2.4.2, we write CD

g ( x , z; x‘, 2’)=

-5

dk,

a(x,x’;k , ) U ( z,k,)U* (z’,k, ) , (42)

Replacing S ( z - 2 ‘ ) on the right-hand side of Equation(l1) by Equation (27), substituting Equation (42) into Equation (1l), and

(l-n)-’ = E R ”, n=O (45)

R = exp(-jlk,a), obtained by writing sin(k,a)

= [(l/2 j)(l -C2)exp(jkIa)].

This

series can be ordered into four i-indexed infinite periodic “image arrays” in the x domain. Interchanging the orders of summation and integration, each of the integrals represents an exact twodimensional “image” line-source Green’s function, in free space (see Equation (46b)). This can be identified asymptotically with a ray-optical field reaching the observer along a direct or multiply reflected ray path, and is frequently referred to as an “eigenray” from a specified source point to a specified observation point. Following this procedure, one obtains the Green’s function fields as

it),

~)+ we obtain recalling that ( ~ ? ~ / & ~ ) U ( z , k-k:U(z,k,),

-

where

tm

=-

dk,U(z,k,)U’ (z‘,k,)S(x

~

1’).

(43)

Equating the (unique Fourier) integrands on both sides identifies A(x,x’;k,) as the x-domain Green’s function in Equation (32), so that

This representation is useful for field evaluations in the near zone of the source, (z(e L’ ,because the large number of closely spaced,

is the “image” line-source Green’s function in free space, the closed-form and plane-wave spectral representations of which are given by the first and second equalities in Equation (46h), respectively. When the 2;) integral is approximated asymptotically in the high-frequency range, one obtains, by the stutionary-phase method [14, Chapter 41,

weakly decaying evanescent modes, which cannot be neglected there, render the series in Equation (41) poorly convergent. It is also useful at high frequencies when Equation (41) encompasses manypropagating modes. Note that the integrand in Equation (44) contains pole singularities at sin(k,,a)=O, m =+l,i2,..., i.e.,

k,, = ( m x / a ) , which must be circumnavigated by analytic extension of the real-axis integrand into the complex k, plane . The residues at these poles generate precisely the x-domain eigenfunction expansion in Equation (41). We shall elaborate on this observation in Section 2.6.3 (also, see the Appendix).

(46~) where a prime denotes the derivative with respect to the argument. In Equation (46b), H i z ) is the Hankel function of the second kind

]

112

with argument kr = k [ (x - $1)’

+ ( z - 2’)’

. The four different

ray species are defined by the following phase accumulations:

2.6.2.2 Alternative Representation: Eigenray Solution

q;” = ( x , - x , ) c o s ~+ zsin w + 2 u n c o s ~ ,

The integral representation for eigenfunctions in the z domain and propagation along x in Equation(44) can be restructured in a manner that expresses the x-domain Green’s function behavior in terms of multiply reflected traveling plane waves between the boundaries at x=O,a, rather than the sinusoidal standing waves as shown. We expand the resonant denominator in the integrand into an infinite power series,

IEEE Antennasandpropagation Magazine, Vol. 46, No. 6, December 2004

75

qi4) = - ( x , -x,)cosW

i = 1 (- -)

+ rsinW + 2 a ( n + I)cosW. (47d)

i = 2 (- +)

i = 3 (+ -)

i = 4 l+ +I

Here, the angular spectrum variable, W , denotes the angle between the ray (i.e., the local plane wave) and the normal to the boundaries, as shown in Figure 2, whereas n is the number of reflections from both boundaries. The eigenrays emanating from the source and reaching the observer are determined from the eigenray (stationary-phase)condition:

w 1= o ~ w = w , ( ' ) ,

-4;)

('(

dW

i=l,2,3,4.

(48)

The four eigenray species in Equation (48) are pictured in Figure 3, and their trajectories are organized according to their departure directions (up and down) at the source, and their arrival directions (up and down) at the observer ( x , = source, x p = observer):

-

i = 1 : n reflections at both boundaries (down, down for x, = x s , x, = x p ,and up, up for x, = x p , x with k x = Rk . ,The first two terms on the right-hand side of Equation (50) represent traveling (i.e., ray-type) fields having undergone n = 0 to N reflections ( n = O,N being the limiting rays). Conceming Equation (SI), if, in the integrand, the term in brackets is ignored for the moment, the remaining integral has the form of &)(x,z;x,$),z'), i.e., is of ray type. Assuming that the bracketed term is slowly vruying, then the entire integral behaves asymptotically (via the saddle-point method) like a ray field with N reflections, with the addition of residue contributions due to the pole singularities at k, = ( m z / a ), m = 1,2,3,... , (i.e., k,

mz/a) ) m ' the bracketed term, which lie between

= -/,

k, = 0 and k, = k:gs , where kSs is the saddle point for the limit: ing ray, N (see Figure 20; the pole singularities are intercepted during the deformation of the original integration path along the real axis into the local steepest-descent path through the saddle point, k 3 $ , and their residues fumish the partial mode fields). This leads to the following asymptotic approximation:

defined byk!A, thereby implying that SJk! < k . The second term represents a weighting of the limiting eigenray, n = N. The third term represents the mode contributions that compensate for the remaining eigenrays with kgi < k$Jx in the eigenriiy expansion (see Figure 20). It should be noted that the decomposi1,ion of g, in Equation (44) into traveling-wave 'constituents generates within each such constituent branch-point singularities at k, = ?kx in the complex k, plane (see Figure20), which are not encountered when all of these constituents are recombined into complete mode fields. In fact, g, in Equation(44) is an even function of k, (replacing k, by -k, does not change the result), so that its behavior for small k, is proportional to k,'= k2-kz, which behaves regularly near k, = + k (see the Appendix for a unified approach). Some fundamental and remarkably simple insights emerge from the mechanism that explains the proper mix of ray and modal fields: representing each mode in terms of two ray congruencies that propagate transversely at the modal (characteristic) angles, the required modes have those values that fill the void left by a buncated angular spectrum of rays. Conversely, the required rays have orientation angles that till the void left by an incomplete series of modes. The hybrid formulation is, in general, numerically more efficient than either the mode formulation alone or the ray formulation alone, when the number of contributing modes is very large. Selection of the best combination of rays and modes depends on the problem at hand.

3. Two-Dimensional PEC Parallel-Plate Waveguide Simulator A MATLAB simulation package bas been developed for the calculation of wave fields inside the two-dimensional PEC parallel-plate waveguide subjected to time-harmonic line-source excitation.

(52) where M ( N ) is the N-dependent last intercepted pole in Figure 20, kt;,

iz)

$1

Ikz+:; ,

k$i8

= k,

( k $ $ ) , and the saddle point,

is defined by (d/dk,)q!)(x,z;x:),I)=O

at k, =k$%.

Moreover,

2.1 Representations Used for Mode, Ray, and Hybrid Solutions The Green's function, g(x,z;x',z'), of the TE, problem given in Equation (41) with Equation (40) is used for the mode solution, The line-source-excited electric field, Ey , is then given by Ey = -jo&g (see Equation (Z)), so that

112

with

=[(x-x!))?+(z-d)']

(cf. Equation(46b)). Expres-

(55)

sion (52) is valid provided that cot(k, a ) is slowly varying, i.e., 78

IEEE Antennasandfropagation Magazine, Vol. 46, No. 6, December 2004

propagating) is required to reconstruct the line-source field in the near zone of the source, higher-order non-propagating modes are damped as the wave propagates longitudinally inside the waveguide, and their contribution becomes negligible after a few tens of wavelengths in range. Therefore, it is adequate to retain only the propagating modes in Equation ( 5 5 ) at longer ranges. The modal sum in Equation ( 5 5 ) tends to diverge for a parameter set (frequency - waveguide width) such that k, + 0 (modal resonance condition). To obtain an acceptable result in this case, the mode at or near cutoff should be omitted.

The eigenangle, W, , of the mth mode is given by

k,

=e

For a distributed excitation, the

Ey field in Equation (8) that satis-

fies the boundary conditions in Equation (9) may be obtained from the Green's function via Equation (14). Three of the lower-order-mode profiles (m = 1, m

= 3,

and

m = 6 ) are pictured in Figure 6. Any transverse source profile can

be synthesized with a complete set of these modes, the excitation amplitudes of which are determined from the orthonormality condition in Equation (22). The propagating and non-propagating (evanescent cut-om modes have positive real and purely negative imaginary propagation constants k,, , respectively. By equating k,, to zero, the number of propagating modes can be determined, once the frequency, m=kf&, of the source is specified. Although an infinite number of modes (both propagating and non-

, ,,

,/"

I,

.

.- ....

,

/

Mode profiles

Figure 6. The mode profiles of the first, third, and sixth modes.

Table 1. A MATLAB module for the calculation of wave field as a function of.height at a given range in terms of mode summation. ..

%Program : M0DE.m %Mode summation inside a ZD-PED parallel plate waveguide % Supply : Waveguide height (a), frequency (fr) in MHz, % source height (xk), range (2). # ofmodes (mm) % --Calculate propagating modes and propagation constants, betakO=2*pi*fr/300; % wavenumber % number of propagating mode counter mp=O; for m=l:mm beta(m)=complex(O,O); if (kOA2-(m*pi/a)"2) > 0 beta(m)=sqrt(kOA2-(m*pi/a)"2); mp=mp+l; else beta(m)=i*sqrt((m*pi/a)"2-kOA2); end end

yo....__......._.__ ........................................................ nx=100; % Number of height points ~

delx=a/(nx-1);

~

____

____....._ ~

%Height increments

for n=l:nx xx(n)=(n- l)*delx; yr(n)=complex(O,O); for m=l :mm yr(n)=yr(n)+cos(m*pi*~a)*cos(m*pi*xg/a); end yr(n)=yr(n)*(Z/a)+(l/a)*exp(i*kO*z)i(2*i*kO); end ___...._____...______... ...__ Plot mode fields ys. height __..____...._____.....-plot(abs(yr),xx); yo..____..._____..__ ___...____ -... End ofmodule ._____..__ __....______....____... ~

~

~

IEEE Antennasandpropagation Magazine, Vol. 46, No. 6.December 2004

~

79

~

Table 2. A MATLAB module for the calculation of eigenrays and their field contributions for given sourcdreceiver locations. function varargout = ray(varargin) % Program :R.4Y.m

%Ray summation inside a 2D-PEC parallel plate waveguide

a = input('Plate Height (a) = ? ')

fr = input('Frequency [MHz] = ? ') k0=2*pi*fr/300; xk=input('Source Height [m] = ? ') xg=input('Observation Height [m] = ? ') z=input('Ohservation Range [m] = ? ') nray=input(Wumberof Reflections = ? ') solution ........................................ gl=complex(O,O); g2=complex(O,O); g3=complex(O,O); g4=complex(O,O);

yo.....................................

yo..__.......... Calculate ...---_ +enray ___ angles ...................................

for n=l:nray+l nr=n-I; wn 1(n)=fsolve(@myfun1,pi/4,[],a,kO,xg,xk,z,nr); wn2(n)=fsolvc(@myfun2,pi/4,[],a,kO,xg,xk,z,m); wn3(n)=fsolve(@myfun3,pi/4,[],a,kO,xg,xk,z,ur); wn4(n)=fsolve(@myfun4,pi/4,[],a,kO,xg,xk,z,nr); yo_.___________........ Calculate four my contributions ._...._____________...-.....----. gl=g l+i/(4*pi)*sqrt((2*pi*i)/(kO*abs(ql d2(wn 1(n),a,kO,xg,xk,z,nr))))*exp(i*kO*ql (wnl(n),a,kO,xg,xk,z,nr)); g2=g2+i/(4*pi)*sqrt((2*pi*i)/(kO*abs(q2d2(wn2(n),a,kO,xg,~,z,nr))))*exp(i*kO*q2(wn2(n),a,kO,xg,~,z,nr)); g3=g3+i/(4*pi)*sqrt((2*pi*i)/(kO*abs(q3d2(wn3(n),a,kO,xg,xk,z,nr))))*exp(i~kO*q3(wn3(n),a,kO,xg,xk,z,m)); g4=g4+i/(4*pi)*sqrt((2*pi*i)/(kO*abs(q4d2(wn4(n),a,kO,xg,~,z,m))))*exp(i*kO*q4(wn4(n),a,kO,xg,~,z,nr)); end green=g 1+g2+g3+g4 Functions for ray equations __.......... _ ... _ ._ _ _ ._.... function F=myfunl(wnl ,a,k,xg,xl,z,n) F=-(xg-xl)*sin(wnl)+z*cos(wn 1)-2"a*n*sin(wnl); function F=myfun2(wn2,a,k,xg,xl,z,n) F=-(xg+xl)*sin(wn2)+z*cos(wn2)-2*a*n*sin(wn2); function F=myfun3(wn3,a,k,xg,xl,z,n) F=(xg+xl)*sin(wn3)+z*cos(wn3)-2*a*(n+ l)*sin(wn3); function F=myfun4(wn4,a,k,xg,xl,z,n) F=-(xl-xg)*sin(wn4)+z*cos(wn4)-2*a*(n+l)*sin(wn4); function ql=ql(wnl ,a,k,xg,xl,z,n) ql=(xg-xl)*cos(wnl)+z*sin(wn 1)+2*a*n*cos(wn1); Gnction ql d2=ql d2(wnl ,a,k,xg,xl,z,n) qld2=-(xg-~l)*cos(wnI)-z*sin(wnl)-2*a*n*cos(wnl); function q2=q2(wn2,a,k,xg,xl,z,n) q2=(xg+xl)*cos(wn2)+z*sin(wn2)+2*a*n*cos(wn2); function q2d2=q2dZ(wn2,a,k,xg,xl,z,n) yo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

~

q2d2=-(xg+xl)*cos(wn2)-z*sin(wn2)-2*a*n*cos(wn2); function q3=q3(wn3,a,k,xg,xl,z,n) q3=-(xg+xl)*cos(wn3)+z*sin(wn3)+2*a*(n+l)*cos(~3); function q3d2=q3d2(wn3,a,k,xg,xl,z,n) q3d2=(xg+xl)*cos(wn3)-z*sin(wn3)-2*a*(n+ l)*cos(wn3); function q4=q4(wn4,a,k,xg,xl,z,n) q4=(xl-xg)*cos(wn4)+z*sin(wn4)+2*a*(n+ l)*cos(wn4); function q4d2=q4d2(wn4,a,k,xg,xl,z,n)

q4d2=-(xl-xg)*cos(wn4)-z*sin(wn4)-2*a*(n+l)*cos(~4); End 0fRay.m ..........................................

yo. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

80

IEEE Antennasandhopagation Magazine, Vol. 46,No. 6, December 2004

A simple MATLAB program file mode. m is given in Table I . It computes mode fields as a function of height at a specified range when the waveguide width and the frequency are supplied. In the program, modal eigenvalues and longitudinal propagation constants are calculated in the first loop. Mode fieid as a function of height calculations are performed in the second loop, and field profiles as well as height values are stored in arrays y r ( I and xx ( ) , respectively. Finally, the amplitude of the mode field as a function of height is plotted. The module may easily he modified to calculate only propagating or non-propagating mode contributions. To calculate the eigenangles from the eigenray equation, Equation (48), with Equation (47), for specified sourcelobserver locations, some sort of root-search algorithm is required. Since the eigenrays are sourcelobserver dependent, the root search must be repeated whenever one or both sourceiobserver location(s) change. For example, for plots of wave field as a function of range at constant height, or wave field as a function of height at constant range, roots at every range or every height must he~. calculated and stored. More specifically, suppose that 100 height points are used to construct a wave field profile as a function of height at a given range, with the number of ray reflections chosen to be 10 (i.e., n = IO): then, the number of eigenray angles that should be calculated by a root-search algorithm will be I00 x 10 x 4 = 4000. ~

Equations (46c)-(48) are used for the ray field calculations and eigenray visualizations. A simple MATLAB program file ray. m is given in Table 2. It computes eigenray paths and ray fields for given sourcelreceiver locations, once the waveguide width and the frequency are specified. First, the eigenray angles m e calculated via the root-search function f solve in MATLAB; the ray field is calculated thereafter. The program can easily be modified to calculate ray fields as a function of range andor height for the set of supplied parameters. Wave-field calculations in large waveguides have been performed hy direct summation of either the modal or the ray-field series. In both instances, it is customary to truncate the series when the number of contributing terms is exceedingly large. Inefficient calculations involving a finite number of modes plus a remainder term, or a finite number of eigenrays plus a remainder term, may be used for most of the sourcelobserver locations, with truncations that render contributions from the remainder terms negligible. The alternative self-consistent hybrid ray-mode approach (see Equation (52)) utilizes a group of rays together with a group of lower- and/or higher-order modes plus modified remainder terms, tuned so as to render the wave-field computation more efficient. The physical interpretation of the process is pictured in Figure 7. The propagating angular-spectrum intervals that are inefficient for modes are filled by rays, and vice versa. As the observer moves either longitudinally or transversely, the ray-mode combination must he readjusted whenever the departure angle of a limiting eigenray coincides with the constituent plane-wave propagation angle of a limiting mode. The adjustment can be made by either retaining the same number of rays and altering the number of modes, or by retaining the same number of modes and altering the number of rays. The former approach is used in the numerical packages that are explained in Section 3. The hybrid ray-mode package comprises the following constituents:

M,

G,(LOM)+

g(x,z;x',z')=

Ni 1 G,,(rays) GN,

tU=O

M.

+

2

G, ( H O M ) + Remainder.

~

(57)

I=M2

The first, second, and third terms on the right-hand side correspond to lower-order modes (LOM) ( 0 5 m < M I ) ,rays with a finite number of reflections (NI < n 2 N 2 ) , and higher-order modes (HOM) ( M 2 < m < M 3 ) ,respectively. Here, M iis the highest mode index of the lower-order modes, M , is the lowest mode index of the higher-order modes, and M 3 is the highest mode (truncation) index of the higher-order modes. Also, NI and N 2 are the included rays with minimum and maximum number of reflections, respectively. Setting N I equal to zero results in M Ibeing equal to zero, which means that the ray-mode mix involves a number of rays plus a few higher-order modes. If N2 is large, then M 2 will be equal to M 3 , which means that the ray-mode mix involves a few lower-order modes plus a number of rays. The package may be used in one of two ways: (i) one can specify the number of eigenrays (i.e., NI and N z ) , and use the algorithm to add lower-order modes (LOM) and higher-order modes (HOM) automatically for a given set of parameters (fix the number of rays and float the number of LOM and HOM); (ii) one can specify a sector of angular spectrum (i.e., lower-order modes and higher-order modes) and use the algorithm to add eigenrays automatically (fix the number of LOM and HOM, float the number of eigenrays). Here, the latter option is used for the hybrid raymode calculations.

3.2 MATLAB Packages: RAY-GUI and HYBRID-GUI Two MATLAB packages have been prepared. The first is the RAY-GUI package, which is designed to investigate line-source-

tx

E ZZ Eigenrays

s$jHigher order Modes

l l l i / Lower order Modes

~

Figure 7. Multi-sector partitioning as in Equation (57). A generic geometrical schematization of the angular-spectrum distribution for arbitrarily selected groupings of eigenrays and modes. These intervals (horizontal or vertical shadings) can be filled with either rays o r modes, depending on the problem conditions. For example, a few lower- and higher-order modes plus a few eigenrays may be used in an efficient hybrid form to account for the field distrihution.

IEEE AntennasandPropagation Magazine, Vol. 46, No. 6,December 2004

81

'

Observation heights. There are three operational buttons. When the first huttou is clicked, the package calculates the numher of propagating modes (i.e., the modes with positive real modal propagation constants) and displays it. When the second button - the “Calculate Rays” button - is clicked, the package calculates eigenray angles by using a built-in root-search algorithm, fsalve, of MATLAB. This may take time, especially when the numher of required rays is large. The last operational button - the “Add Ray” button - is reserved to plot eigenrays and wave-field variations. It should be noted that the “Add Ray” button must be used after the calculations.

Figure S. The front window of the RAY-Curpackage. The user supplies the plate height, sourcelobserver heights, horizontal distance, frequency, and nnmher of reflections (n) for the ray solution. The package calculates the modal (reference) solution, and searches for the eigenrays for the specified geometrical parameters. The user may trace the eigenray paths individually in the right-hand graphics window, and compare their contributions on the left with the modal solution.

H

L

Figure 9. The front window of the HYBRID-CUI package, where field as a function of height and/or range can he calculated via (i) mode summation, (ii) ray summation, and (iii) a hybrid ray-mode formulation. The user supplies the plate height, sourcelobserver heights, first and last observation ranges, range increments, frequency, and rays with minimum and maximum numbers of reflections (n). . , The Dackace calculates the modal solution, and searches for the eigenrays for the specified geometrical parameters. I

excited wave- fields in terms of ray and mode solutions. The runtime window is shown in Figure 8, depicting the waveguide, the eigenray plot, and the plot of wave field as a function of height. The geometrical parameters that should he supplied by the user are the waveguide (plate) height [m] and the distance From the source [m]; the operational parameters are the frequency in MHz and the number of reflections (n) for the ray solution. The sliders at the lef? and right end of the waveguide are used to specify the source and 02

Two graphics are reserved for the output. Eigenrays between source and observer can he plotted on the bottom right, whereas wave fieid as a function of height at a specified range can he plotted in terms of both mode solutions and ray solutions on the bottom left. The package is designed in such a way that eigenrays with different numbers of reflections, and their contributions, can be plotted separately, so that the user may analyze the effects of the group of eigenray species. Each time the user changes any of the parameters, the calculations must be repeated by clicking the operation “Calculate Rays” button. This initiates both the mode and ray calculations. The second package is the HYBRID-GUI package, which is designed to calculate wave field as a function of range at a constant height and/or wave field as a function OP height at a specified range via the mode representation, the ray representation, and the hybrid form. The runtime window is shown in Figure 9. It looks similar to the RAY-GUI package. The waveguide schematic, input windows for waveguide height and frequency, and the sliders for the sourcelohserver heights are exactly the same as in the MYGUI package. The user must supply three parameters for the range calculations: initial range [m], final range [m], and the range increment [m]. The last two parameters are required for the hybrid raymode analysis. The minimum and ,maximum numher of reflected rays are stored to provide N , and N z in Equation (57), respectively. Two graphics are used in the HYBRID-GUIpackage,plotting either the range profile or the height profile of the line-source wave field. The two “Calculate” buttons are used to perform the calculations first, and then to plot the profiles. The range profile is plotted from the given initial range to the final range with the range increment supplied. The height profile is calculated for the last range only and is plotted on the right graphics. The number of lower-order modes and higher-order modes at each range for calculations of wave field as a function of range, and at each height for calculations of wave field as a function of height, are stored in the output files range.dat and height. dat, respectively. The angular sector at each point is also recorded in the output files.

4. Numerical Examples The two MATLAB packages have been run with characteristic parameter sets and the outputs are presented in this section. Tests were performed with the MY-GUI package; results are given in Figures 10 and 11 for the same set of parameters. The waveguide width was chosen to be 1 m. The heights of the source and observer were 0.3 m and 0.7 m, respectively. The range was 5.6 m, and the frequency was 2387 MHz (Le., ka = 50). Rays with up to the tenth reflection were used in the ray representation. The number of propagating modes for these parameters was 15. Figure IO shows eigenray photos with different numbers of

IEEE AntennasandPropagation Magazine, Vol. 46, No, 6,December 2004

ence solution, since all higher-order non-propagating modes contributed negligibly at this range (which corresponds to a distance of a few hundred wavelengths). Whenever the “Add Ray” button is clicked, a group of eigenrays with one higher reflection number is added. The figure again displays results for three different n values ( n = 0, n = I , n = 6 ). When the number of eigenrays increased to 24 or higher, the ray summation results approached the mode solution. The difference between field plots for n = 6 and n > 6 was almost indistinguishable for the set of chosen parameters.

Figure 10. The eigenray paths calculated via the M Y - G U I package for the following parameters: U = 1 m, distance = 5.6 m, source height = 0.3 m, observer height = 0.7 m, f = 2387 MHz (ku = 50), n = 6 . The number of propagating modes for these parameters was 15 (top: n = 0, middle: n = 1, bottom: n = 6 ) .

0

Normalized range [e] Figure 12. The field as a function of range, calculated via the HYBRID-GUZ package. The parameters were: a = 1 m, source height = 0.2 m, observer height = 0.5 m, first range = 5 m, last range = 6.5 m, range increment = 0.01 m, frequency = 2500 MHz. Rays with n = 2 , n = 3 , and n = 4 were included.

/Green’sfunction1 Figure 11. The field as a function of height, calculated via both mode summation and ray summation in the RAY-CUlpackage. The parameters were n = 1 m, distance = 5.6 m, source height = 0.3 m, f = 2387 MHz (ku = 50). The number of propagating modes for these parameters was 15 (left: n = 0 , middle: n = 1 , right: n = 6 ) .

reflections ( n = O , n = l , n = 6 ) . As observed from the figure, eigenrays with n = 0 and n = 1 were far from filling the entire angular spectrum. On the other hand, eigenrays with up to six reflections (i.e., n = 6 ) covered the angular spectrum almost adequately. Eigenrays with n = 0 included the direct ray, the ray with a single reflection from the bottom boundary, the ray with a single reflection from the top boundary, and the ray with a single reflection from both boundaries (see Figure 3). Height profiles of the wave field at a range of 5.6 m with different numbers of eigenrays are plotted in Figure 11. The summation of 15 propagating modes may be considered to be the refer-

Figure 13. The field as a function of height, calculated via the HYBRID-GUI package. The parameters are: were: a = 1 m, source height = 0.2 m, observer height = 0.5 m, range = 6.5 m, height increment = 0.01 m, frequency = 2500 MHz. Rays with n = 2 , n = 3 , and n = 4 were Included.

IEEE AntennasandPropagatbn Magazine, Vol. 46, No. 6, December 2004

83

I

-

... .

Figure 14. Eigenrays and wave field as a function of height, obtained via the RAY-CUI package. All parameters were thesame as in Figure 13, except that the range was more than doubled. The number of reflections was n = 10.

The hybrid ray-mode results presented here were based on filling the angular spectrum partially with modes and rays in a few different ways. The modal eigenangles were determined from the waveguide width ( a ) and the frequency 0,and were constant once they were specified. On the other hand, eigenray spectral angles are sourceiohserver dependent. The longer the range, the larger is the ray eigenangle, for a fixed height and frequency (recall that the eigenray angle is measured from the vertical axis, not the horizontal axis). Therefore, different numbers of lower-order and higheron$ modes may be required as the range changes, in order to fill the angular spectrum let? empty by the eigenrays. To illustrate this, the range in the first example (range = 5.6 m) was more than doubled (range = 12 m), and eigenray contributions with IO reflections and 19 reflections were plotted in Figures 14 and 15. Although eigenrays with up to I O reflections gave quite good results at the range of 6 m for the given parameters, they were incapable of reproducing the reference solution calculated from the first 15 propagating modes. Even increasing the number of reflections of the eigenrays to 19 (i.e., a total of 72 eigenrays) did not meet the requirement, It was obvious that more than 20 eigenrays were required to construct acceptable results.

5. Conclusions A simple line-source-excited two-dimensional parallel-plate, homogeneously filled waveguide has been used here to introduce the analytical background for developing alternative Green's function representations by spectral wavenumber techniques. The complementary roles of progressing (ray-type) and oscillatory (modetype) wave dynamics were emphasized, culminating in the selfconsistent hybrid ray-mode approach.

.. Figure IS. The same as in Figure 14, hut with n = 19n = 19.

The HYBRID-GUI package is designed to test the sensitivity of various hybrid ray-mode combinations to frequency and sourcelobserver locations. Figures 12 and 13 display two typical results for the following parameters: waveguide width = 1 m, source height = 0.2 m, observer height = 0.5 m, frequency = 2400 MHz, first range = 5 m, last range = 6.5 m, range increment = 0.01m. Eigenrays with n = 2 , 3, and 4 ( N I = 2 , N 2 = 4 ) were chosen for inclusion in the hybrid ray-mode formulation. In this case, the number of propagating modes was 15, and all higherorder non-propagating mode contributions were negligible in the range of interest. The range profile was calculated first, and is plotted in Figure 12; the height profile at the final range was calculated next; and both profiles are plotted in Figure 13. Reasonably good agreement was obtained.behveen the mode sum (the reference solution) and the hybrid solution. Ranges and heights where discrepancies occurred require the addition of the remainder term [2].

84

The intent has been tutorial, targeting graduate and upperlevel undergraduate students. MATLAB visualization packages have been prepared, documented, and used to illustrate the ray, mode, and hybrid ray-mode wave dynamics through parametric numerical experiments. For the hybrid formulation, it has been shown that there exists a variety of well-defined combinations of mode and ray fields that expresses the total field inside the waveguide with good accuracy. For each combination, the modal fields account for the omitted rays or vice versa; for certain parameter ranges, the addition of a remainder (in easily computable form) may be required. The modes @Ius remainder term, when necessary) quantify the truncation error of an incomplete ray series, whereas the rays (plus remainder term, when necessary) quantify the truncation error of an incomplete mode series. The two MATLAB packages, designed to investigate ray, mode, and hybrid options for the canonical parallel-plate waveguide problem, can serve as effective educational tools toward understanding the physics-based, observable wave objects in this environment. The packages may easily be extended to include the remainder terms and to handle impedance-type boundary conditions. Moreover, based on the experience and physical insight gained here with these packages, the user may extend the algorithms to waveguiding structures with virtual boundaries (i.e., trapping inhomogeneous refractivities) where modal as well as ray solutions undergo critical transitions, such as modal cutoff, ray caustics, etc. [IS, 161.

IEEE Antennasandpropagation Magazine, Vol. 46, No. 6, December 2004

~

6. Appendix: The Relationship Between Characteristic Green’s Functions and Eigenfunction Expansions In this Appendix, we summarize a generalized method, based on the spectral theory of operators, which unifies the derivation of alternative representations for the two-dimensional Green’s function via analytic extension and contour deformations in the complex (k,, k, ) wavenumber planes. As noted in the text, this method is presented separately because it requires more complex-plane familiarity than the procedure in Section 2. We begin with some clarifying remarks, which are relevant for overall perspective. In reducing Equation (18) to Equations (19a)-(19c), the separation “constant” has been chosen as the square of another constant, k, (or k , , subject to Equation (19~)). This anticipates the utility of k, as the x-domain spectral wavenumher within the context of the wave equation. However, in the general theory of coordinate-separable configurations (SturmLiouville (SL) theory [13, 14]), the corresponding separation constants are 5 = k,’, ( = k,‘, which we shall use from now on.

6.1 One-Dimensional Green’s Functions 6.1.IThe x Domain Using k,

+fi,we expand g,(x,x’;{)

as in Section2.5.1

in terms of the (assumed complete) orthogonal eigenset {Uxm}in Section 2.4.1:

In Equation (60), Csm(in the bracketed term) is a closed contour around each pole location 5 = (, yielding unity by the residue theorem (see Figure 16a). The last equality follows from Equation (25). This yields the desired construction of the completeness relation (delta-function representation) in terms of a contour integral over the closed-form spectral characteristic Green’s function,

which plays a critical role in generating altemative representations for the two-dimensional field defined in Equations (ll), (12), and (14). The characteristic Green’s function has the property that it decays exponentially at 161--t m , Im{{} # 0 (see [13, Section 31 and [14, Section 3.41). Therefore, the open contour, Cs , can he terminated anywhere at 161+ m , Im(e} # 0, because the connecting path segments at 15)=m do not contribute when invoking Cauchy’s theorem. The same is true for the z-domain formulation in the complex ( plane in Equation (62). It should be noted that the 5 - and ( -plane formulations in Equations (61) and (62) have their exact counterpart in the spectral k, and k, planes, respectively. Accessing the spectral k, domain via 5 = k,’ + d5 = 2k,dkx , with the breakup 5 - 5, = k,2 - k:”, = (k, + kIm)(kx- kx,,,),unfolds the integration contour Cs and the pole set along the positive real 5 axis in Figure 16a into the contour Ck., extending from -m to +m above the pole set along the entire real axis in the complex k, plane (Figure 16b). The entire spectral 5 plane is thereby mapped into the lower half of the complex k, plane (see Figure 16b). Closing the integration contour Ck* by a semicircle of infinite radius in the lower half plane, and evaluating the integral in terms of the residues at the poles k,, = (mn/a), yields exactly the same results as in Equations (60) and (61); the 2k, term in the numerator cancels the (k, +k,,)

where 5, = k:, = (mrr/a)’. By substituting Equation ( 5 8 ) into Equation (28), interchanging the order of summation and differentiation, utilizing Equation (19a), and invoking the orthogonality condition in Equation(22), one may show that a, ( x ’ , 4 ) = U,, ( x ‘ ) ( 5 - 5,)-

1

, Thus,

term in the denominator as k, approaches k , (see Figure 16b).

6.1.2 The z Domain Since the z-domain eigenvalues form a continuum from to ( = +m (see Equation (19c) and Section 2.4.2), the xdomain procedure in Section 6.1.1 is not directly applicable here. It can be shown that the formal structure of EquFtion (61) does remain applicable (see [14, Section 3.3~1and [13, Section 3]), provided that the contour Cc in the complex 5 plane (see Figure 17) now surrounds the branch-point singularity at ( = 0 (and the associated “spectral” branch cut) of g,(r,z’;() on the top (=-CO

(59)

Now, integrate Equation (59) along an infinite contour Cs,enclosing all of the pole singularities of g, , at 5 = 5, = (mrr/a)’ in the complex 5 plane (see Figure 16a), perform the integration on the summand on the right-hand side, and invoke the residue theorem, to obtain

=

-

C Uxm(x)UIm

(1‘) =d

Riemann sheet with Re&> 0, I*& < 0 . Thus, the generalized completeness relation becomes (c.f. Equations (61) and (36)),

( - x~‘ ) . (60)

m=l

IEEE Anfennasandf‘ropagation Magazine, Vol. 46, No. 6 , December 2004

85

When the contour C: closely hugs the branch cut along the positive real axis, and the resulting real portions on the upper and lower sides are combined, followed by the mapping’& = k,, one recovers the formulation in Equation (27).

6.1.3 Most General Representation of TwoDimensional Green’s Function: Characteristic Green’s Functions in the x and z Domains We shall now perform the spectral synthesis of ‘the twodimensional Green’s function in terms of an integration of the product of the two individual one-dimensional spectral Green’s functions over a properly chosen infinite contour in the complex 5 (or

< = kZ- 5 ) plane. Starting with Equations (37) and (39), and

1- Figure 17. The integration path, Cs, and the branchpointbranch-cut singularities pertaining to g, (=,; O , I m { & ] ~ O , < = k 2 - 5 ) in the complex 5 plane (see Eqnation(63)). Shown are singularities (simple poles) of g, ( X , X ’ ; :~ ) singularities (branch-cuts) o f g , ( z , z ’ ; < ( c ) ( = k Z - 5 ) ) ; the integration contour, Cs,enclosing all singularities of g, ( x , x ‘ ; { ) , but excluding the (branch-cut) singularities of g, ( z , z ’ ; c ( c ) ) ;and the deformed

contour,

C’,,

enclosing the singularities

of

g z ( z , z ‘ ; < ( < ) ) , but excluding the singularities of g,, ( x , x ‘ ; C ) .

refemng to Equation (61), replace the delta-function kernel in Equation (37) by the contour-integral operator in Equation (61) to obtain .

~~

Figure 16b. The integration paths and pole singularines pertaining to g,(x.x’;{) in the complex k, plane (see Equation(61); the contour can be deformed into the lower half plane, h ( k , } < O ) : 86

where Cs is a contour that tightly encircles the positive real

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lEEE Antennasandpropagation Magazine, Vol. 46, No. 6 , December 2004

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t

Singuiarities of g,

imag 5

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