wave propagation inside a two-dimensional perfectly conducting parailel-plate waveguide: hybrid...
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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
3Electronics and 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 (time- harmonic) 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
1. Introduction
hen high-frequency (HF) time-hannonic wave propagation IS spatially (transversely) constrained by physical impene-
trable 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
IEEE Antennasandpropagation Magazine, Vol. 46, No. 6, December 2004
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 trans- verse reflections and (or) refractions, which are localized around source-receiver-dependent “eigenray” trajectories. The oscillatory formulation views the wave fields as discrete (or discrete-continu- ous) superpositions o f frequency-dependent (guided-mode) “eigen-
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ISSN 1045-924312004B20 02004 IEEE
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 dimen- sions 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 source- receiver separations, the progressing formulation becomes unwieldy. This is because it now entails many closely spaced near- vertical 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 parallel- plate-waveguide Green’s functions, with numerical examples that quantify all relevant dynamic constituents. Changes in the source- receiver locations give rise to intricate compensating wave mecha- nisms, 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’.
The layout of this paper is as follows. The two-dimensional (2D) Green’s function problem is formulated and its spectral repre-
‘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.
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 disap- pear, and rays are reflected from the boundary at the angle of inci- dence (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 Fig- ure 1). Assuming that the medium between the plates is free space, one can use Maxwell’s curl equations in the form (a time depend- ence exp(jwt)is suppressed) [13, Section 5.51
V x E = - j o p o H - 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)
+X Line source Observer X - a , / PEC / , / /
i . P I ‘ -5
x = o PEC
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.
70 lEEE AntennasandPropagation Magazine, Vol. 46, No, 6 , December 2004
and TE, (transverse electric with respect to z) - which are given as follows:
SET#I: TM,
SET #2: TE,
aHx 'HZ - J y + joxOEy, & a x
These equations can in turn be combined into second-order partial differential equations - the wave equations - as follows:
TM, wave equation
with boundary conditions
outgoing wave (radiation) condition at z + t m , (6b)
Here, k = 2lr/1= o& is the free-space wavenumher, and 1 is the free-space wavelength. The remaining two TM, field components can be derived from
TE, wave equation
with boundary conditions
E , = O a t x = O , a ;
outgoing wave (radiation) condition at z + A" (9b)
The remaining two field components for this set are
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 My, J,. J , excite the TM, set,
whereas the line sources 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 simplifications will he assumed in what follows.
2.2 The Green's Function Problem
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
with boundary conditions
g(x , z ;x ' , z ' )=O at x = O , a , (TE,) (124
(TM,) ( 12b) a
(12c)
- g ( x , z; x',.') = 0 at x = 0 ,a , ax
outgoing wave (radiation) condition at z + km.
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 ) , 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:
F ( x , z ) = IJ dddz'S(x',z')g ( x , z; x',z'). (14) SOWCe regmn
IEEE Antennas andpropagation Magazine, Vol. 46, No. 6, December 2004 71
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.
2.3 Accessing the Spectral Domain: Separation of Variables
To solve the wave equation
[ $ + ,2 a2 + k 2 ] U (x, z ) = 0 ,
subject to the boundary conditions
U ( x , z ) = 0 at x = 0, a ; radiation condition at z + ?m, (16)
assume that we can decompose
U ( x , z ) = U, ( x ) U z (21, (17)
where Ux satisfies the first boundary condition in Equation (16), and U2 satisfies the second boundruy condition in (16). Substitut- ing Equation (17) into Equation (15), performing' the partial derivatives, and rearranging the result, one finds that
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
side must be a constant, which is denoted by k:. Thus, Equa- tion (15) is reduced to two separate ordinary differential equations:
U:+k:U,=O, O S x S a ; U , = O at x = O , a ; (19a)
U: + k:Uz = 0 , -m < z < +=c ; radiation condition at z + *m
(19h)
kZ = k,' + k,'. (194
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'
Spectral representations of fields are structured around dis- crete (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 coordi- nate-separable problem in Equations (15) and (16) admits separate
one-dimensional (ID) eigenvalue problems based on Equations (19a) and (19b), respectively.
2.4.1 The x Domain
For Equation (19a), the m-indexed eigensolutions (eigenhctions) Uxm (x) , corresponding to the eigenvalues k,, ,
are given by
%,,(I)= A,sin(k,,x), (20)
m a k,, = - , m=1,2,3 ,._., O S x < a , a
where Am is an arbitrary constant. It is readily verified that for any two eigeufunctions with indexes m, and m2, respectively, the fol- lowing relation - the orthogonality relation - applies:
Y
Am,Am2 pxm, ( 4 U X W 2 ( x ) A = 4 d m 2 S m ) % > 3 (21) 0
where the Kronecker delta S,,,m2 = 0 , ml ;e m2, while Sm,,m2 = 1,
m, =m2. It is convenient to normalize the eigenfunctions with respect to Am so that
Y
f u ~ m ~ ( x ) ~ ~ m * ( x ) d x = S ~ , , , * ,i.e., A, = - , (22) 0 t
which renders the eigenset orthonormal. We shall use this normal- ized format throughout.
The implied completeness of the eigenset {Uxn,) admits the eigenexpansion of any "suitable" function, F, (x) (for which the series converges),
m
F , ( x ) = z a , U x m ( x ) , O S x S a . (23) ,=a
To determine the amplitudes am, multiply Equation (23) by any
other eigenfunction U,,, (x) , perform the integration f d x on both
sides, interchange the order of integration and summation on the right-hand side, and invoke Equation (22). This yields the result
a
0
Changing the arbitrary index mi to m, as in Equation (23), com- pletes the expansion of F , ( x ) in Equation(23), with Equa-
tion (24), in terms ofthe {Uxm} eigenhasis.
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
72 IEEE Antennas andpropagation Magazine, Vol. 46, No. 6, December 2004
properties of the delta function. Accordingly, am, = Umb (x') , and from Equation (23),
m
S(x-x ' ) = U,, (x)Uxm (x'), 0 I: (x,x') 5 a . ( 2 5 ) m=0
The spectral representation of the delta function in Equation (25) is referred to as the completeness relation, because any suitable func- tion F,(x) can be expressed by performlng the (identity)
operation F, (x )= IF,(x')S(x-x')dr' on the summand of the
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).
a
0
2.4.2 The z Domain
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 +km [14, Sec- tion 3.3~1. The corresponding trigonometric eigenfunctions,
. U ( z , k , ) , render the normalizing integral r U 2 ( z , k , ) d z diver-
gent. To assign a proper "weight" to the divergence, one can track the transition from the aforementioned initial finite z domain sys- tematically to infinity to find for the orthonormality condition,
m
L" .- I U(z,k,, )U' (&, )& = 6 (kz, - kz2 ), +
(26) 1
U ( z , k ) = - m < (z,k,) < +m, z &
where the complex conjugate has been introduced in view of the convenient exponential form for the eigenfunctions. Thus, the normalization expressed by S,,,,,, for the discrete spectra in
Equation (22) is replaced for continuous spectra by the delta func- tion S( k,, - k+). By symmetry, interchanging z + k, , k, + z , leads to the completeness relation
+= [e-'kz(z-z')dkz = S ( z - z ' ) , -m<(z,z ' ,k,)<+m. (27)
211 _m
2.5 Spectral Representations: One- Dimensional Characteristic Green's Function Problems
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 prob- lems in the x and z domains are modified accordingly.
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,) = -6 (x - x') , 0 5 (x,x') 5 (I ;
(28) g , = O a t x = O , a .
Now, k, is a specified spectral wavenumber. For the unique specification of g, in Equation(28), it is necessav that 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:
where xc = x for x < x' , but x' for x > x' ; similarly, x, = x for x > 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, recip- rocity 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 right- hand side of Equation (28).
Specifically, we can construct the solution in Equation (29) Any suitable function F,(z) can thereby be represented via the via
operation F , ( z ) = [Fz (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).
- &(x,kx) = sin(k,x) ,
Ex (x, k, ) = sin [ k, ( a - x)] , (31)
IEEE Antennasandpropagation Magazine. Vol. 46, No. 6, December 2004 73
whence
W, = -k, {sin(k,x) cos[ k, ( a - x ) ] + sin[k, ( a - x)]cos (k ,x)}
= -k,sin(k,a)
from Equation (30). Accordingly,
sin (k,x,)sin[k, ( a -I,)] g, (x ,x’ ;k , ) = (32) k,sin(k,a)
2.5.2 The z Domain
The characteristic Green’s function ,g,, is defined as fol- lows:
[ $ + 41 g, (z,z’; k,) = -6 (z - z’), - m < (z,z’, k,) 5 +m
radiation condition at z + +m, (33)
with the solution (in the notation of Section 2.5.1)
(34)
‘Here, 8, and g, are outgoing wave solutions in z < z‘ and z > z’ , respectively, in accord with the radiation condition (Re { k r ] > 0 ,
Im { k z ] < 0 for the assumed exp( jot) time dependence):
+ jk,z g = e
W, = -2 jk ,
Thus, since ( z < - z , ) = - ~ z - z ‘ ~ ,
2.6 The Two-Dimensional Green’s Function Problem:
Alternative Representations
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 uni- fied approach, see the Appendix).
74 IEEE Antennas andPropagation Magazine, Vol. 46, No. 6, Deo?mber 2004
2.6.1 Eigenfunction Expansion in the x Domain, One-Dimensional Green’s
Function in the z Domain
We write
where Ur, ( x ) is given in Equation (20), with Am :: (see Equation (22)). This form is suggestive in view of the delta-func- tion 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 differ- entiations on the summand in Equation (37), for the .x domain one obtains 8’ dx’ U,, --f -k:,U,, in view of Equations (19a) and
(19c). With this reduced x-domain formulation in mind, Equa- tion ( l l ) becomes
( 1 1
,=I
where k& = k Z -k:m. Equating the coefficients of the orthogonal
eigenfunction expansions on both sides, one identifies 2, (2-2’)
as a spectral Green’s function that satisfies
subject to the radiation condition at Izl + m . The solution of Equa- tion (39) is given in Equation (36), provided that k, + kZm:
k, = -(y)z .
Thus, the desired representation for g(x,z;x’,z’) is
m
g(x,z ;x ’ ,z ‘ )=Cg, , (z ,z ’ ) “,=I
which is convenient for evaluating the waveguide field away from the source plane z = z’ when the number of propagating modes with real k,, (i.e., k > m z l a ) is not too large. Since lm k, < 0,
the higher-order non-propagating modes with k < mzla decay along z.
I 1
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 refer- ring 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 (1 l), and recalling that ( ~ ? ~ / & ~ ) U ( z , k ~ ) + -k:U(z ,k , ) , we obtain
t m
= - 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, 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 exten- sion of the real-axis integrand into the complex k, plane . The residues at these poles generate precisely the x-domain eigenfunc- tion expansion in Equation (41). We shall elaborate on this obser- vation in Section 2.6.3 (also, see the Appendix).
2.6.2.2 Alternative Representation: Eigenray Solution
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,
m
(l-n)-’ = E R ” , n=O
R = exp(-jlk,a), (45)
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 two- dimensional “image” line-source Green’s function, i t), 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
where
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), respec- tively. When the 2;) integral is approximated asymptotically in the high-frequency range, one obtains, by the stutionary-phase method [14, Chapter 41,
( 4 6 ~ )
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
with argument kr = k [ ( x - $1)’ + ( z - 2’)’ ] . The four different
ray species are defined by the following phase accumulations:
112
q;” = ( x , - x , ) c o s ~ + zsin w + 2 u n c o s ~ ,
IEEE Antennasandpropagation Magazine, Vol. 46, No. 6, December 2004 75
i = 1 (- -) i = 2 (- +) i = 3 (+ -) i = 4 l+ +I qi4) = - ( x , -x,)cosW + rsinW + 2 a ( n + I)cosW. (47d)
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:
-4;) ( ' ( w 1 = o ~ w = w , ( ' ) , i=l ,2,3,4. (48) dW
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< =Is 1;
* i = 1 : n reflections at upper and ( n + I) reflections at lower boundary (down, up),
- i = 3 : (n + I ) reflections at upper and n reflections at lower boundary (up, down),
i = 4: ( E + ' ) reflections at both boundaries (up, up for x, = 4, x< = x p , and down, down for x , = x p , x , = x , )
2.6.3 The Relation Between the Eigenmode Representation in Equation (41) and the Eigenray
Representation in Equations (46)-(48)
The generating function for either the eigenmode or the eigenray representation is the denominator shown in
z
Figure 2. The spatial and spectral coordinates (transverse and longitudinal wave numbers); local plane-wave (i.e., ray) trajec- tories showing a generic ray and an n-indexed eigenray; and the spectral angles Wand W, (measured with respect to the x direction).
Figure 3a. A phenomenological map showing the trajectories for eigenrays, where n and i tag the number of reflections and the eigenray species, respectively. This figure shows the trajec- tories for x, = x s , x< = x p . The relationships between n and i are as follows: i = l : n reflections at both boundaries (--); i = 2 : n reflections at upper and n + l reflections at lower boundaries (- +); i = 3 : n + 1 reflections at upper and n reflec- tions at lower boundaries (+ -); i = 4 : n + 1 reflections at both boundaries (+ +). This i-ordering corresponds to the upward (+) or downward (-) departure direction at the source (the first symbol inside the parenthesis), and the corresponding arrival direction at the observer (the second symbol inside the paren- thesis).
i = 1 [+ +) i = 2 ( - + ) i = d 1. .I
Figure 3h. A phenomenological map as in Figure 3a, showing the trajectories for eigenrays for x, = x p , x.: = x s . The
relationships between n and i are as follows: i = 1 : 11 reflections at both boundaries (+ +); i = 2 : n reflections at upper and n + 1 reflections at lower boundaries (- +); i = 3 : n + 1 reflections at upper and n reflections at lower boundaries (+ -); i = 4 : n + 1 reflections at both boundaries (--). The i-ordering is as in Figure 3a.
Equation (49, pertaining to the- spectral integral formulation in Equation (44). The eigenmodes are parameterized by the mode eigenvalues W, , obtained from the resonance equation sin(k,,a) = sin(kasinW,) = 0 , m = +l,+2,..., which keeps the denominator intact. The eigenrays are parameterized via the series expansion in Equation (45) by the eigenangles W,?, which satisfy
the stationary-phase condition (d/dW)qt)(W,z,z') = 0 at
W = W,? in Equation (48). These alternative parameterizations are summarized in Figure 4, which displays the eigenmode and eigenray "densities," respectively. The complementan behavior of the mode and ray eigenangles plays an important role in the hybrid ray-mode representations in Section 2.7 (see Figure 5).
76 IEEE Antennasandpropagation Magazine, Vol. 46, No. 6, December 2004
2.7 Most Versatile Representation of Two- Dimensional Green’s Function:
Hybrid Ray-Mode Decomposition
The computationally most useful and phenomenologically most intriguing representation of the two-dimensional Green’s function is the hybrid ray-mode decomposition. This splits the propagating angular wavenumber spectral domain into sectors, which can he “filled” sequentially either by ray fields or mode fields, so as to take advantage of the “best” properties of each. This algorithm bas been described conceptually in Section 1, and isdis- cussed in detail in Section3, below. By an entirely different approach, the hybrid ray-mode scheme can be derived directly through application of finite Poisson summation to convert a finite series of ray fields into a finite series of mode fields (plus remain- ders), and vice versa (see [6] ) .
Eigenray expansion (n= ray index)
( i ) Small Wn (many near-vertical reflections): Many tightly packed eigenrays
Hybrid ray-mode decomposition
/ -x ._ ’-.” , Ray-weighted modes ’, ....... .~ .... i y Retained rays
t modes eigenrays
z
Figure 5a. As in Figure 4, but for hybrid ray-mode partitioning in the physical domain.
Hybrid ray-mode complementarity
Many eigenrays c, few eigenmodes (see Fig. 4)
+ X
n=2 n=1 n=o k, jF,-:; : k,, = k sin W,, s w
.x
0 z
z k2 m=3 m=2 m=l - Figure 4a. A schematic diagram of eigenray congruencies, with inclusion of the analytic generating function for eigenray
I k,, = ksin W,
expansions.
Eiaenmode exDansion Im= mode index)
Figure 5b. As in Figure 4, but for hybrid ray-mode schematiza- tion.
mr 1 - R(k,,) = 0, k , = - a
ro
2‘’’ = i:), Eigenmode 2;) : mth residue of (1 - a)-’ “$4
Small W$ (near-vertical congruences): Few (widely spaced) elgenmodes
r X
.P m = 3 m = 2 m = l
S . . z
Figure 4b. A schematic diagram of eigenmode ray congruen- cies, with inclusion of the analytic generating function for eigenmode expansions.
The hybrid ray-mode formulation can be justified by retum- ing to the two-dimensional Green’s-function representation in Equation (44), and expanding the resonant denominator sin(k,a) in g, not into an infinite power series, as in Equation (4% but into an exact decomposition with a finite power series plus a resonant remainder, according to
(49)
Q=exp(-jZk,n)
This can be shown (by algebraic manipulations in Equation (44)) to lead, instead of Equation (46), to the following exact expansion:
IEEE AntennasandPropagation Magazine, Vol. 46, No. 6, December 2004 77
where
with k x = R . , k 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:
(52)
where M ( N ) is the N-dependent last intercepted pole in Fig-
ure 20, iz) $1 I k z + : ; ; , k$i8 = k, ( k $ $ ) , and the saddle point,
kt;, is defined by (d/dk,)q!)(x,z;x:),I)=O at k, =k$%.
Moreover,
112 with = [ ( x - x ! ) ) ? + ( z - d ) ' ] (cf. Equation(46b)). Expres-
sion (52) is valid provided that cot(k, a ) is slowly varying, i.e.,
that k,, does not lie near a saddle point, k!Js ; when this is not the case, uniform asymptotics are required (see ([14, Chapter 41).
The result in Equation (52) constitutes the desired hybrid ray-mode algorithm. The first term on the right-hand side represents the first N eigenrays (i.e., those with large k,, small k,)
defined byk!A, thereby implying that k!JS <!f!a < k . The second term represents a weighting of the limiting eigenray, n = N. The third term represents the mode contributions that compensate for the remaining eigenrays with kgi < k$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 bun- cated 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 formu- lation 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.
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
(55 )
78 IEEE Antennasandfropagation Magazine, Vol. 46, No. 6, December 2004
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 con- dition 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-
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 (55 ) at longer ranges. The modal sum in Equation (55 ) tends to diverge for a parameter set (frequency - waveguide width) such that k, + 0 (modal reso- nance condition). To obtain an acceptable result in this case, the mode at or near cutoff should be omitted.
, . 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, beta- kO=2*pi*fr/300; % wavenumber mp=O; for m=l:mm
% number of propagating mode counter
beta(m)=complex(O,O); if (kOA2-(m*pi/a)"2) > 0
beta(m)=sqrt(kOA2-(m*pi/a)"2); mp=mp+l;
beta(m)=i*sqrt((m*pi/a)"2-kOA2); else
end end yo ....__......._.__ ~ ........................................................ ~ ____....._ ~ _ _ _ _ nx=100; delx=a/(nx- 1); %Height increments
% Number of height points
for n=l:nx xx(n)=(n- l)*delx; yr(n)=complex(O,O); for m=l :mm
end yr(n)=yr(n)*(Z/a)+( l/a)*exp(i*kO*z)i(2*i*kO);
y r ( n ) = y r ( n ) + c o s ( m * p i * ~ a ) * c o s ( m * p i * x g / a ) ;
end
plot(abs(yr),xx); __...__ ~ _...._____...______... Plot mode fields ys. height __..____...._____.....--
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(Wumber of Reflections = ? ')
yo ..................................... solution ........................................ gl=complex(O,O); g2=complex(O,O); g3=complex(O,O); g4=complex(O,O);
yo ..__.............---____ Calculate +enray angles ................................... for n=l:nray+l
nr=n-I; wn 1 (n)=fsolve(@myfun 1 ,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);
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));
yo _.___________........ Calculate four my contributions ._...._____________...-.....----.
end green=g 1 +g2+g3+g4
yo .............................. 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(wn 1); 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) 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); yo ................................... End 0fRay.m ..........................................
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 con- stants are calculated in the first loop. Mode fieid as a function of height calculations are performed in the second loop, and field pro- files 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 calcu- late 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 con- stant 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 con- struct 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 per- formed 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, Ni
tU=O G N , g(x ,z ;x ' ,z ' )= G,(LOM)+ 1 G,,(rays)
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 num- ber of reflections (NI < n 2 N 2 ) , and higher-order modes (HOM) ( M 2 < m < M 3 ) , respectively. Here, Mi is the highest mode index of the lower-order modes, M , is the lowest mode index of the higher-order modes, and M3 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 MI being 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 ray- mode 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
- EZZ - Eigenrays s$j Higher 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 or 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
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) solu- tion, and searches for the eigenrays for the specified geometri- cal parameters. The user may trace the eigenray paths indi- vidually in the right-hand graphics window, and compare their contributions on the left with the modal solution.
H
L
02 IEEE AntennasandPropagation Magazine, Vol. 46, No, 6, December 2004
Figure 9. The front window of the HYBRID-CUI package, where field as a function of height and/or range can he calcu- lated 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 calcu- . , I
lates the modal solution, and searches for the eigenrays for the specified geometrical parameters.
excited wave- fields in terms of ray and mode solutions. The run- time 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
Observation heights. There are three operational buttons. When the first huttou is clicked, the package calculates the numher of propa- gating 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.
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 plot- ted 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 MY- GUI package. The user must supply three parameters for the range calculations: initial range [m], final range [m], and the range incre- ment [m]. The last two parameters are required for the hybrid ray- mode analysis. The minimum and ,maximum numher of reflected rays are stored to provide N , and Nz in Equation (57), respec- tively.
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
Figure 10. The eigenray paths calculated via the MY-GUI 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 ) .
/Green’s function1
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 dif- ferent numbers of eigenrays are plotted in Figure 11. The summa- tion of 15 propagating modes may be considered to be the refer-
ence solution, since all higher-order non-propagating modes con- tributed 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 solu- tion. The difference between field plots for n = 6 and n > 6 was almost indistinguishable for the set of chosen parameters.
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.
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 the- same as in Figure 13, except that the range was more than doubled. The number of reflections was n = 10.
84 IEEE Antennasandpropagation Magazine, Vol. 46, No. 6, December 2004
. .
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 higher- order 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 cal- culated 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].
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 horizon- tal axis). Therefore, different numbers of lower-order and higher- on$ 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 dou- bled (range = 12 m), and eigenray contributions with I O 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 func- tion representations by spectral wavenumber techniques. The com- plementary roles of progressing (ray-type) and oscillatory (mode- type) wave dynamics were emphasized, culminating in the self- consistent hybrid ray-mode approach.
level have
The intent has been tutorial, targeting graduate and upper- undergraduate students. MATLAB visualization packages 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 bound- ary 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.
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 func- tion via analytic extension and contour deformations in the com- plex (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 Equa- tions (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 (Sturm- Liouville (SL) theory [13, 14]), the corresponding separation con- stants are 5 = k,’, ( = k,‘, which we shall use from now on.
6.1 One-Dimensional Green’s Functions
6.1 .I The 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:
where 5, = k:, = (mrr/a)’. By substituting Equation (58 ) into Equation (28), interchanging the order of summation and differen- tiation, utilizing Equation (19a), and invoking the orthogonality condition in Equation(22), one may show that
1 a, ( x ’ , 4 ) = U,, ( x ‘ ) ( 5 - 5,)- , Thus,
(59)
Now, integrate Equation (59) along an infinite contour Cs, enclos-
ing 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 ( ~ - x ‘ ) . (60)
m=l
~
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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 com- pleteness 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 ( l l ) , (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 connect-
ing 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
2 via 5 = k,’ + d5 = 2k,dkx , with the breakup 5 - 5, = k, - 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 con- tour 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 com- plex 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,,) 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 (=-CO to ( = +m (see Equation (19c) and Section 2.4.2), the x- domain 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 3.3~1 and [13, Section 3]), pro- vided 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
Riemann sheet with Re&> 0, I*& < 0 . Thus, the generalized completeness relation becomes (c.f. Equations (61) and (36)),
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 Two- Dimensional Green’s Function:
Characteristic Green’s Functions in the x and z Domains
We shall now perform the spectral synthesis of ‘the two- dimensional 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
Figure 16a. The integration paths and pole singularities per- taining to g, (x ,x ’ ;5 ) in the complex 5 plane (see Equa- tion (61)).
~~ . Figure 16b. The integration paths and pole singularines per- taining to g,(x.x’;{) in the complex k, plane (see Equa- tion(61); the contour can be deformed into the lower half plane, h ( k , } < O ) :
1 - - Figure 17. The integration path, Cs, and the branch-
pointbranch-cut singularities pertaining to g, (=,;<<) on the
“proper” Riemann sheet (Re{&}>O, lm{&}~;O) in the
complex < plane (see Equation (62)). Mapping inlo the com-
plex k z ( = f i ) plane is analogous to Figure 16h (the proper
sheet in the plane maps into the lower half of the k, plane, and the branch point is eliminated: see also Figure 20).
Singularitles I Deformed I contour
Figure 18. The integration paths, C, and Ci , pertaining to the two-dimensional Green’s function g,(x,z;x’,z’) on the
proper Riemann sheet ( R e ( 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) ofg, (z ,z ’ ;< (c) (= k Z - 5 ) ) ; the integration con-
tour, Cs, enclosing all singularities of g, ( x , x ‘ ; { ) , but exclud-
ing the (branch-cut) singularities of g, ( z , z ’ ; c ( c ) ) ; and the deformed contour, C’,, enclosing the singularities of
gz(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
where Cs is a contour that tightly encircles the positive real 5
86 lEEE Antennasandpropagation Magazine, Vol. 46, No. 6 , December 2004
2 imag 5 <(= k, ) - plane
t Singuiarities of g,
axis with the pole singularities at 5, = ( m z / a ) 2 , m =1,2,3, ..., m
(see Figure 16a). The singularities of g , (z,z’,C(c)) (see Equation(36)) lie outside the contour C,, with
k , + & --f (see Figure 18). Evaluation of Equation (63)
in terms of the residues enclosed by Cs recovers Equation (41).
Now, invoke analytic continuation to extend g, into the complex 5 plane, and deform C, away from the real axis into the complex
Singularlties of g,
Figure 19. As in Figure 18, but in the complex 6 plane. The contour Cs corresponds to Ci in Figure IS.
h a g k, k-plane
\ / Real k.
Figure 20. The integration paths and singularities pertaining to the hyhrid ray-mode decomposition of the two-dimensional Green’s function g ( x , z ; x ’ , z ’ ) in the complex k,(=&) plane
(see Equations (50) and (51)). The contour Ck, corresponds to
C, in Figure 19, eliminating the <-plane branch cut; the pole set <, in Figure 19 maps into the two sets of poles
* 112 k , =i kZ ( m d a ) ] in the upper (improper) and lower [ - (proper) k, (half planes). The branch cuts emanating from k, = i-k arise from the integrand in Equation (51), which is associated with one of the four i-indexed modal ray congruen- cies in Equation (50). There is no corresponding branch cut in Figure 16h that embodies the “full-mode’’ representation of g , in Equation (44). As noted in the text, th ibranch cuts disap- pear in the i = 1 to 4 full-mode superposition. The contour Ck,
can he deformed into the lower (proper) half plane without contributions from path segments at Ik,l+m, Im{k,} C O . The
deformed contour passes through the saddle point k$!s in the asymptotic (local-steepest-descent path) evaluation of Equation (51), due account being taken of residues contributed by intercepted poles 9. *.” denotes propagating-mode poles, “v” denotes evanescent-mode poles.
5 plane. During this deformation, there is no contribution from contour segments at 151 + m because the characteristic Green’s functions decay exponentially there (on the proper Riemann sheet) [14].’ This analytically extended form of Equation (63), with its arbitrary contour Cs (separating all singularities of g , from those of g , ) is the most general form of the physical two-dimensional
~
IEEE Antennasandfropagation Magazine, Vol. 46, No. 6, December 2004 87
Green’s function. When C, is deformed tightly into Ci around the branch-point and branch-cut singularities of g , as in Figure 18, and the branch-cut integral is reduced, one obtains the altemative z-domain eigenfunctiodx-domain Green’s function representation in Equation (44). Thus, either of the two alternative spectral representations in Sections 2.6.1 and 2.6.2 can be derived from Equation (63), with its contour Cg extended along an infinite path in the complex 5 plane (see Figure 18) or 6 plane (see
Figure 19), which are related via c + < = k 2 . Note that in Figures 16, 18, and 19, the spectral-pole and/or branch-point singularities actually lie on the real 5 - or 5 -plane axes. Their displacement away from the axes serves only to clarify intermingling of integration paths and singularities when all are arranged along the real axes.
The analytic continuations into the complex (c,C) planes remain valid also for the eigenray expansions in Section 2.6.2.2, the relation between the eigenray and eigenmode expansions in Section 2.6.3, and the hyhrid ray-mode decompositions in Sec- tion 2.7, leading to the results in Equations (50)-(54). The general- ized formulation has, in fact, been used to implement the contour deformations in Figure 20.
6. References
1. L. B. Felsen and T. Ishihara, “Hybrid Ray-Mode Formulation of Ducted Propagation,” J. Acoust. Soc. Am., 65,1979, pp. 595-607.
2. L. B. Felsen and A. H. Kamel, “Hybrid Ray-Mode Formulation of Parallel Plate Waveguide Green’s Functions,” IEEE Transac- tions on Antennas and Propagalion, AP-29, 4, July 1981, pp. 637- 649.
3. A. Kamel and L. B. Felsen, “On the Ray Equivalent of a Group ofModes,”l Acoust. Soc. Am., 71,1982, pp. 1445-1452.
4. L. B. Felsen, “Progressing and Oscillatory Waves for Hybrid Synthesis of Source-Excited Propagation in Layered Media,” Geo- phys. J. Roy. Astron. Soc., 79, 1984, pp. 11-33.
5. L. B. Felsen, “Progressing and Oscillatory Waves for Hybrid Synthesis of Source-Excited Propagation and Diffraction,” IEEE Transactions on Anlennas and Propagation, AP-32, 1984, pp. 7 7 5 - 7 9 6.
6. L. B. Felsen (ed.), Hybrid Formulation of Wave Propagation and Scattering, Nijhoff Publishing Co., 1984.
7. L. B. Felsen and E. Heyman, “Hybrid Ray-Mode Analysis of Transient Scattering,” in V. K. Varadan and V. V. Varadan (eds.), Low and High Frequencj, Asympfotics, New York, Elseiver Sci- ence, 1986.
8. E. Heyman, G. Friedlander, and L. B. Felsen, “Ray-Mode Analysis of Complex Resonance of an Open Cavity,” IEEE Pro- ceedings, 77, 1989, pp. 780-787.
9. L. Sevgi and L. B. Felsen, “A New Algorithm for Ground Wave Propagation Based on a Hybrid Ray-Mode Approach,” Int. J . of NumericaIModeling, 11,2, March 1998, pp. 87-103.
10. L. Sevgi, F. Akleman, and L. B. Felsen, “Ground Wave Propa- gation Modeling: Problem-Matched Analytical Formulations and Direct Numerical Techniques,” IEEE Antennas and Propagation Magazine, 44, 1, Februiuy 2002, pp. 55-75.
I I . L. B. Felsen, “A Review of Frequency and Time Domain Hybrid Ray-Mode Representations,” 2003 IEEE Infernational Symposium on Antennas and Propagation and USNC/URSI North America Radio Science Meeting, June 22-27, Columbus, Ohio.
12. L. Sevgi, F. Akleman, and L. B. Felsen, “Electromagnetic Wave Propagation in Waveguiding Environments: Modeling and Simulation,” 2003 IEEE International Symposium on Antennas and Propagalion and USNC/URSI North America Radio Science Meeting, June 22-27, Columbus, Ohio.
13. D. G. Dudley, Mathemafical Foundationsfor Electromagnetic Theory, New York, IEEE Press, New York, 1994.
14. L. B. Felsen and N. Marcuvitz, Radiation and Scatfering of Waves, Englewood Cliffs, New Jersey, Prentice-Hall, 1973; New York, IEEE Press, 1994.
15. T. Ishihara and L. B. Felsen, “Hybrid Ray-Mode Parameteriza- tion of High Frequency Propagation in an Open Waveguide with Inhomogeneous Transverse Refractive Index: Numerical Results and Quality Assessment,” IEEE Transacfions on Antennas and Propagation, AP-39,6, 1991, pp. 789-797.
16. L. B. Felsen and L. Sevgi, “Adiabatic and Intrinsic Modes for Wave Propagation in Guiding Environments with Longitudinal and Transverse Variations: Continuously Refracting Media,” IEEE Transactions on Antennas and Propagation, AP-39, 8, August 1991, pp. 1137-1 143.
Introducing the Feature Article Authors
Leopold B r 7, 1924. He
iich, Germany, DEE degrees fi
on .om
the Polytechnic Institute of Brooklyn, Brooklyn, NY, in 1948, 1950, and 1952, respectively. He emigrated to the Unil.ed States in 1939 and served in the US Army from 1943 to 1946. After 1952, he remained with the Polytechnic (now Polytechnic University), gaining the position of University Professor in 1978. From 1974 to 1978, be was Dean of Engineering. In 1994, he resigned from the full-time Polytechnic faculty and was granted the status of Univer- sity Professor Emeritus. He is now Professor of Aerospace and Mechanical Engineering and Professor of Electrical and Computer Engineering at Boston University, Boston, MA (part-time).
Dr. Felsen is the author or co-author of over 350 papers and of several hooks, including Radiation and Scattering of Waves (IEEE Press, 1994). He is an Associate Editor of several profes- sional journals, and was an Editor of the Wave Phenomena Series (Springer-Verlag). His research interests encompass wave propa- gation and diffraction in complex environments and in various dis- ciplines, high-frequency asymptotic and short-pulse techniques, and phase-space methods with an emphasis on wave-oriented data processing and imaging.
Dr. Felsen is a member of Sigma Xi, a Life Fellow of the IEEE, and a Fellow of the Optical Society of America and of the Acoustical Society of America. He has held named Visiting Pro- fessorships and Fellowships at universities in the United States and abroad, including the Guggenheim in 1973 and tho Humboldt Foundation Senior Scientist Award in 1981. In 1974, he was an IEEEIAP-S Distinguished Lecturer. His Poet’s Corner appears sporadically in the IEEE Antennas and Propagation Magazine.
Dr. Felsen received the AP-S Best Paper Award for 1969, and was best-paper co-author for 1974 and 1981; he was contrib- uting author to papers selected for the R.W.P. King Award for 1984, 1986, and 2000. He received the Balthasar van der Pol Gold Medal from URSI in 1975; an honorary doctorate from the Techni- cal University of Denmark in 1979; the IEEE Heinrich Hertz Gold Medal for 1991; the AP-S Distinguished Achievement Award for 1998; the IEEE Third Millennium Medal in 2000; an honorary Laurea degree from the University of Sannio in Benevento, Italy, in 2003; the IEEE Electramagnetics Award for 2003; an honorary doctorate from the Technical University of Munich, Germany, in 2004; three Distinguished Faculty Alumnus Awards from Poly- technic University; and an IEEE Centennial Medal in 1984. In 1977, he was elected to the US National Academy of Engineering.
He has served on the AP-S Administrative Committee from 1963-1966, and was Vice Chair and Chair for both the USNC (1966-1973) and the international (1978-1984) URSI Commission B.
*, ’ *>* 1~
Funda Akleman was hom in Canakkale, Turkey, on March 17, 1973. She received her BS, MS, and PhD degrees in Electron- ics Engineering from the Electrical Engineering Department of Istanbul Technical University, Turkey, in 1995, 1998, and 2002, respectively. She worked as a test engineer in NETAS AS.
88 /E€€ AntennasandPropagation Magazine, Vol. 46, No. 6. December 2004
(Northern Telecom) in Istanbul between 1995 and 1996 (in the^ GSM, DECT project). In 1997, she joined the Electrical Engineer- ing Department of Istanbul Technical University as a research assistant, and in 2003 she became an Assistant Professor in the same department.
She was awarded a Scholarship of the TUBITAK, Turkish National Scientific and Technical Research Centre, which allowed her to work as a visiting scientist for three months in the Radio Communications Research Unit of Rutherford Appleton Lahora- tory, Chilton, Didcot, Oxfordshire, and three months at Boston University (with Prof Felsen) for her PhD studies. She also received a Young Scientist Award at the XXVIth General Assem- bly of URSl in 1999, and an Army Grant for Young Investigators Award at the IEEE A P - S Conference on Antennas and Propagation for Wireless Communications, Boston, USA, in 1998. Recently, she was awarded a grant from the NATO Science Fellowship Pro- gram by TUBITAK for a period of six months, to study in the Electrical Engineering Department of Pennsylvania State Univer- sity.
Her interests and research involve electromagnetic problems, such as analytical solutions and numerical modeling in EM propa- gation, RCS analysis, numerical modeling of antennas, and EMCiEMI engineering. She is a member of Prof Levent Sevgi’s Numerical Electromagnetics Group in Turkey, and is experienced with numerical techniques including the Finite-Difference Time- Domain, Parabolic Equation, and Transmission Line Matrix Meth- ods and their application to various electromagnetic problems. She is a member of the IEEE Antennas and Propagation and Micro- . . wave Theory and Techniques Societies.
Levent Sevgi was horn in AkhisariManisa on January 1, 1958. He received his BSEE, MSEE, and PhD degrees in Elec-
tronic Engineering from Istanbul Technical University (ITU) in 1982, 1984, and 1990, respectively. In 1982, he joined the faculty as a research assistant. In 1987, while working on his PhD, he was awarded a fellowship that allowed him to work with Prof L. B. Felsen at New York Polytechnic University for two years. His work at the Polytechnic concerned the propagation phenomena in non-homogeneous open and closed waveguides.
In 1990, he returned to the Electrical and Electronics Engi- neering Faculty of ITU. He became an Assistant, Associate, and full Professor in 1991, 1996, and 2002, respectively. He was with the Center for Defense Studies, ITW-SAM, during 1993-1997, for the Long Horizon (UZUN UFUK) Project studies for the Turkish Navy. By invitation he spent September, 1998, to June, 1999 with the Scientific Research Group of Raytheon Systems Canada for the Integrated Maritime Surveillance System trials, where surface-wave HE radars were of interest. By invitation he was with the TUBITAK-MRC Information Technologies Research Institute as the Chair of the Electronic Systems Department during lune, 1999, to August, 2000, where he led the project for develop- ing hand-held and vehicle-mounted multi-sensor mine detection systems. He was also with the Center for Defense Studies, ITUV- SAM, from 2000 to 2002, for installation of the Vessel Traffic System for the Turkish Straits. Since February, 2002, he has been with the Electronics and Communication Engineering Department of the Engineering Faculty of the Do@$ University in Kadikoy, Istanbul.
His research studies have focused on propagation in complex environments, analytical and numerical methods in electromagnet- ics, and radar systems. His recent interests include EMCIEMI modeling and measurement; surface-wave HF radars; FDTD, TLM, SSPE, and MOM techniques and their applications; radar cross section, and bioelectromagnetics. He has also been involved in discussions and studies of novel challenges in engineering edu- cation in today’s complex computer and technology-driven world. He is the author or co-author of more than 30 journal and 60 international conference papers. He authored the book Complex Elecfromagnefic Problems and Numerical Simulation Approaches (IEEE PressIJohn Wiley, 2003). He has also published two books in Turkish (1999 and 2001), and a book chapter. He is a Senior Member of the IEEE, and a member of the Turkish Chamber of Electrical Engineers - EMO. @
IEEE AntennasandPropagation Magazine, Vol. 46, No. 6, December 2004 89