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IFEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MIT-34, NO. 3, MARCH 1986 321
Excitation of Waveguide by Stripline- and
Microstrip-Line-Fed Slots
B. N. DAS, K. V. S.V. R. PRASAD, AND K. V. SESHAGIRI RAO, MEMBER, IEEE
Ab.vtract—This paper presents investigations on coupling between strip- tion required for the analysis of both stripline-fed and
fhre/microstrip fine and arectangular ‘wavegnidecoupled through aslot in
microstrip-fed slots is determined from one common gen-
the ground plane which is fixed in the cross-sectionaf plane of the wave-
eral formulation. The impedance seen by the line exciting
guide. A closed-form expression for the impedance loading on stripliue/
microstip line is evahrated from knowledge of the complex power flowing the waveguide is determined from the complex power
down the rectangular waveguide supporting the dominant modeanddiseoq- flowing down the waveguide and the discontinuity in modal
tinuity in the modal voltage in stripline/microstrip line. The reactance voltage in the stripline or rnicrosttip line. The analysis is
cancellation is obtained by terminating the stripline/microstrip tine excit-
then extended to the case of a waveguide supporting TEIO,
ing the slot in a short-circuited stub. The structure under this condition
TE20, TE30 modes for slot locations corresponding to eigen
forms atransition between stripline/microstrip fine and awavegnide. The
design curves on slot length versusfrequency are presented for different excitations of the form (1/2,1,1/2), (fi/2, O,– &/2)
values of dielectric constants. The variation of coupling asa fnnction of and (1, – 1,1), when used in the construction of waveguide
frequency and also the location of the slot is evaluated. Numerical results simulators [7]. Numerical results on the normalized resis-
for slot coupling useful for the design of wavegnide simulators are also
tance and the reactance are presented. Further, the effect
presented.
of a reactive stub in ‘the feedline terminating the slot is
L INTRODUCTION investigated. The results on coupling between the
s stripline/microstrip line and waveguide have also been
OME INVESTIGATIONS on aperture coupling be-
evaluated. Finally, the design curves on slot length L for
tween stripline/microstrip line and waveguide through
which the normalized resistance becomes unity are
an aperture in the common wall have been reported [1]–[4].
presented as a function of frequency for various values of
Design of a multi-aperture directional coupler has also
dielectric constants.
been carried out by Kumar and Das [5]. It is found that the
coupling between the stripline and the waveguide attained II. GENERAL ANALYSIS
in this process is limited to low values. In order to obtain
The method of exciting a rectangular waveguide by a
higher coupling, the length of the coupling region becomes
slot, the center of which is located at (xO, yO) on the
very large. The method of exciting a multimode guide
ground plane of a, stripline, is shown in Fig. l(a). The
through a stripline-fed slot in the transverse section of
length of the slot is parallel to the x-axis. The analysis is
waveguide has recently been discussed in the literature [6].
carried out for the general case of a stripline having its
The electric-field distribution required for the analysis has
center strip arbitrarily located parallel to the ground planes
been obtained from conformal transformation in the case
and embedded at the interface between the two different
of a symmetric stripline and from the equivalent parallel-
dielectric media.
plate configuration in the case of a rnicrostrip line. When
The electric field in the cross section of the stripline/
an equivalent parallel-plate configuration is used in the
microstrip line can be expressed as [9]
case of microstrip line, the maximum dimension of the
aperture is restricted. Evaluation of the electric-field distri- E=d” (1)
bution by the method of conformal transformation is quite where 2 is the modal vector and V is the modal voltage
involved in the cases of asymmetric stripline, symmetric
satisfying the orthonormality condition
and asymmetric striplines with partial dielectric filling, and
microstrip line. e2ds=l
/
In the present work, excitation of a waveguide by strip-
line- and microstrip-line-fed slots in the transverse cross where ds represents the area of an infinitesimally small
section of the waveguide is investigated. The field distribu- element in the cross section of the waveguide.
In Fig. l(a), the coupling of the slot is transverse to the
axis of the stripline. Using the formulas by Marcuvitz and
Manuscript received May 29, 1985; revised October 28, 1985.
Schwinger [13], it can be shown easily that such a slot
B. N. Das and K.V.S.V.R. Prasad are with the Department of Elec-
tronics and Electrical Communication Engineering, Indian Institute of produces discontinuity in the modal voltage, and the equiv-
Technology, Klmragpur 721302, India. alent network of the slot appears as a series element in the
K. V. Seshagiri Rao is with the Radar and Communication Centre,
transmission-line representation of Fig. l(c). From a
Indian Institute of Technology, Kbaragpur 721302, India.
IEEE Log Number 8406848. knowledge of this discontinuity in modal voltage AV, the
0018-9480/86/0300-0321 $01.00 01986 IEEE
322 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-34, NO. 3, MARCH 1986
The electric field in the coupling aperture given by (3)
excites the TE to x mode in the waveguide, which can be
expressed as the superposition of TE and TM modes [9].
The modal amplitudes of the TE~.-mode fields excited
in the rectangular waveguide of Fig. l(a) can be obtained
from the formula [9]
‘z where
&fl=l “for rt=O
(a)
. 2 for n >0.
The complex power in the waveguide is given by [9]
(5)
where Z= is the unit vector along the axis of the waveguide.
Substituting (3) in (4) and using (5), an expression for the
complex power in the waveguide is obtained as
(b)
r
mrd 12
--+=+&l
v
)$( STUB I nTyO mtrxO
.cos2— .sin2 —
e I b a
(c)
“KL[ ‘mrL 12
Fig, 1. Configuration of the excitation of a waveguide by a slot in the
ground plane of the stnpline. (b) Configuration of the stripline used for CosT – Cos 2a
(6)
the determination of modal vector function. (c) Equivalent circuit of the mn 2
stub-terminated feed arrangement. – K2
(-)a
with
normalized admittance loading on the stripline resulting
from the power coupled to the waveguide is obtained from
the relation
P
F= (2)
YO(AV)2
where YOis the characteristic admittance of the stripline, where d is the width of the slot, V. is the maximum voltage
AV is the discontinuity in modal voltage in the stripline, across the slot, EOd, a and b are the broad and narrow
and P is the complex power flowing down the waveguide. dimensions of the waveguide, and p and c are, respectively,
The electric-field distribution in the aperture plane of a the permeabi~liy and permittivity of the medium inside the
slot exciting the guide is assumed to be of the form waveguide. Slrice the waveguide is airfilled, the values of p
and c are assumed to be the same as those of free space.
(L
EY=EOsin K ~–lx–xO\ (3) The expression for the modal vector function required for
) the evaluation of discontinuity in the modal voltage in the
where E. is the maximum value of the electric field in the structure of Fig. l(b) having the slot in its ground plane is
derived from the potential function. Assuming that the
aperture plane of the slot, L is the length of the slot,
K = (27T/A o)~, and the expression for c: is given by [8] charge distribution in the center strip is of the form
p(x) =
*’
where c; is the effective dielectric constant and c, is the
relative dielectric constant of the medium filling the region (=)<’<($+:)=0
‘thertise
between the ground planes of the stripline.
DAS @f(d.: STRIPLINE- AND MICROSTIUP-LINE-FED SLOTS 323
Following the method suggested by Yamashita and Atsuki
[10], the expression for the potential function O(X, z) for
the TEM-mode field is obtained as
@(x, z)
=“=~,,? .JO(%)sin(~)
.sirih(~)sin(~) sinh(~)
O<.z<hl 02~—L_—L_
2.6 3 3,4 3.8 4
FREQUENCY [GHZ)
(a)
where JOindicates the Bessel function of zeroth order, and
08~_L_L_
‘=’lcosh(%)sinh(%) 10.8 11.6 12
FREQUENCY [I; Hz]
(b)
“,cosh(%”sFiig.n2.(Th%e variation of resonant slot length as a function of frequency
with the dielectric constant of the substrate asaparameter.
Following the method suggested by Barrington, the modal where
vector function Z appearing in (1) can be found for the l“(=)[co’h(+)+coth
TEM mode from the relation
R2= f –
n n~hl n~h2 1, “
r?=l,3,5 c1 coth — +E2coth —
where Vt k the derivative operator in the cross section [ A A
perpendicular to the axis of the line, @ in the potential After obtaining the relation for the normalized modal
function for the TEM mode. vector function Z, the discontinuity in modal voltage in the
The expression for the normalized modal vector function stritiine due to the slot cut on its mound Plane can be
Z for the electric field is obtained as determined from [8]
AV= ]/iix~s. ~cosfiyds (8)
slot
where /3 is the phase constant in the y direction, ~s is the
.si~(~)sinh(%)
electric-field distribution in the aperture plane of the slot,
and ~ is the normalized transverse modal vector function
“[cos(%sinh(w for the magnetic field of the dominant TEM mode in the
stripline, which is related to the normalized modal vector
function E for tlie electric field by the relation [9]
O<z<hl (7a) Z=iiyxz. (9)
+s+Hcost34
Using (7a), (8), and (9), and taking the effect of asymmetry
into account, an expression for the discontinuity in the
modal voltage for a slot in the lower ground plane of the
stripline (z = O)is found to be of the form
sin(~)sinh(~) JG (n)77W
AV=VO ~ — —
nTX ,,=1,3,5 ARAnJO 2A
- Cos~sinh ;(B–z) iiX
[ ( )
)1
hl<z<B sinh(~)K~:~~~K?l ’10)
-Sin(%”cosh(%z) ‘z ~
(7b)
324 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-34, NO. 3. MARCH 1986
‘r 55
a
5“o
IL
. I
wuz6 Luz<IJ co c\
1(-II \
ciui!c 3.0 I \\\
8 10 12 12”4
FREQUENCY [GHz)
2“0 3,0 tio
FREQUENCY (GHz)
1.5
~--
‘ t _- 1,0
lx
F
lx /“
w. 0 20 ,. I / coI FREQUENCY (GHz)
u //
<. /
~ -,.0
w /
e
/
0
~ -2.0
i
a
.%
0 -Yo
z
d
he’’
-co
Fig. 3. Variation of normalized resistance and reactance loading on the -C5 b
stripline/microstrip line with frequency (X-band).
Fig. 4. Variation of normalized resistance and reactance loading on the
Curve Description stnpline\microstrip line with frequency (~-band).
u, b Impedance seen by amicrostrip line without Curve Description
astub in the feedline.
a, b Impedance seenby amicrostrip line without
C,d Impedance seen by astripline without a
astub in the feedline.
stub in the feedline.
c, d Impedance seen by astripline without a
e Reactance seen by amicrostrip line, with
stub in the feedline.
astub in the feedIine.
e Reactance seen by amicrostrip line, with a
f Reactance seen by astripline with astub stub in the feedline,
in the feedline.
f Reactance seenby astnpline with astub
in the feedline.
III. EVALUATION OF NORMALIZED SLOT
design curves on this slot length L versus frequency are
ADMITTANCE
presented in Fig. 2.
The normalized admittance presented to the stripline by After selecting the slot length, corresponding to a par-
the slot located at a position (xO, yo) in the cross section of ticular frequency fd, the variation of both 7 and ~ as a
the waveguide and radiating into the guide is given by function of frequency for YO= 0.02 mhos is obtained using
1 P,+ jP, (6), (10), and (11) for the following cases.
F=g+jz= :=-= (11)
Case 1. Impedance loading due to the slot in the ground
z ~+jx YO(AV)2”
plane of symmetric stripline at X-band with A/l?= 10,
P. is evaluated for n = Oand m = 1 and P, is calculated for (1 = ~z= 2.56, fd=9.375 GHz, a = 2.286 cm, b = 1.016
the other values of m and n in a single-mode guide. For cm, XO=1.143 cm, -yO= 0.508 cm, d = 0.1 cm, and L =
the multimode guide supporting TEIO, TEZO, and TE30 2.0516 cm. The results for this case are presented in Fig. 3.
modes, P, is evaluated for n = Oand m =1, 2, 3, and P, is Case 2. Impedance loading on the microstrip line excit-
calculated for the other values of m and n. ing a single-mode guide through a slot in the ground plane
Using (6), (10), and (11), the slot length L, which offers at X-band with A=1O cm, Cl= 2.56, Cz=1, hz=m, d =
a resistive part of the impedance equal to 50 Q (; =1), has 0.1 cm, ~d=11.5 GHz, a = 2.286 cm, b =1.016 cm, XO=
been calculated for different frequency bands (S and X) 1.143 cm, yO= 0.508 cm, and L = 2.0908 cm. The results
for the structure having a slot in the ground plane of a on 7 and Z for this case are also presented in Fig. 3.
stripline exciting the rectangular waveguide when the di- Case 3. Impedance loading due to the slot in the ground
electric constant ~, filling the stripline is 2.56 and 10. The plane of symmetries stripline at S-band with A/B= 25,
DAS d d.: STRIPLINE- AND MICROSTIUP-LINE-FED SLOTS 325
fd=
Cl= ez= 2.56, 3 GHz, d = 0.1 cm, a = 7.214 cm, b = X- BAND
10
3.404 cm, XO= 3.607 cm, yO= 1.702 cm, and L = 5.14065 J — MICROSTRIP
cm. The results on normalized impedance are given in --– 5YM sTRIP
L L=3mm,6tfi=rH , d=o’lcm /
Fig. 4.
! /
I /
Case 4. Impedance loading on the microstrip line excit-
ing a single-mode guide through a slot in the ground plane 3 b /
: /
at S-band with A=1O cm, C1=2.56, C2=1, h2=m, d= /
0.1 cm, fd=3 GHz, a = 7.214 cm, b = 3.404cm, XO= 3.607 1: ~ ,,
/
cm, y. =1.702 cm, and L = 6.8184 cm. The plots on /
/
impedance of slot for this case are in Fig. 4. 1 /
. ya
\
\ /
IV. THE EFFECT OF A REACTIVE STUB IN THE FEEDLINE ‘,11, .
0
TERMINATING THE SLOT ON THE IMPEDANCE , 8’2 9.0 9“8 10’6 11”.4 12”2 12,L
SEEN BY THE FEEDLINE FREQUENCY (GHz)
(a)
It is found from the results presented in Figs. 3 and 4
10 –
that the imaginary parts of the impedance seen by the S–BAND
J
stripline and microstrip line do not become zero in the
frequency range of interest. It k possible to cancel L [ /
b
the reactive parts of these impedances by terminating the
It
striplke and microstrip line feeding the slot k a stub. The 3 / — MICROSTRIP
2
length of the stub becomes shorter if an exponentially ‘0 .\ I --- SYM. STRIP
tapered line k used as a stub. The expression for the 1: \ I L=lOmm, 61fi=0’5, d= Olcm
2 \ I
reactance Xl due to the stub is given by [12]
\ I
\
1
\
\
o I
2.0 3.0 co
where KO1 is the impedance level of the line at y = yO, 1is FREQUENCY (GHz)
the length of the line, ~ is the propagation constant, and 8 (b)
is the rate of the taper. i3/~ is positive when the impedance
level changes from a lower value to ahigher value from the
input end to the short-circuited end of the stub. In the 5-
;.o.5—
other case, 8//3 is negative. The modified reactance in
= 0.1
presence of the stub are shown in Fig. 3 as curyes (f )and 54-
(e) for 1= 3 mm, and 8/~= 0.1 for an X-band waveguide. 1: = E:=
o
The modified reactance in presence of the stub are shown 23 -
i 7
in Fig. 4 as curves(f) and (e) for 1=10 mm and ii/fl = 0.5 a
//’
for an S-band waveguide. $2 - /’
x’
/’
V. ESTIMATION OF COUPLING IN THE PRESENCE OF 1:-... ,/”
-.. ,/
----
A STUB TERMINATION .~ ----- _____
o
The reflection coefficient at the slot location because of 8 9 10 11 12
FREQUENCY (GHZ)
the impedance seen by the slot into the waveguide when
(c)
the slot is terminated by the reactive stub is given by
Fig. 5. (a) Variation of coupling in decibels as a function of frequency
ZL – Z. (X-band).
~: (12)
ZL + Z. curve Description
a Stripline-fed waveguide slot with astub
where Z~ is the effective impedance offered by the slot in
in the feedline.
the presence of a stub and ZO is the characteristic imped- b Microstrip-fed waveguide slot with astub
ance of the feedline. in the feedline.
Assuming the losses are negligible, the coupling in de- (b) Variation of coupling in decibels asafunction of frequency (S-band).
cibels between the feedline and waveguide is calculated
Curve Description
from
a Stripline-fed waveguide slot with astub
Z=lologlo(l– 1712). (13) in the feedline.
b Microstrip-fed waveguide slot with astub
Using (12) and (13) and the data of curves (c) and (f) of in the feedline.
Fig. 3, the variation of coupling between a symmetric
(c) Variation of coupling asafunction of frequency for different values of
stripline and an X-band waveguide, as a function of slot offset positions.
326 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTF34, NO. 3, MARCH 1986
a\ F 4.29 cm, and -yO= 0.76 cm for each XO.The locations of the
‘i
slots correspond to eigen excitations of the form
(1/2, 1,1/2), (fi/2,0, – ~/2), and (1, – 1,1) in the
waveguide simulator, where these arrangements are used.
For the multimode guide supporting TEIO, TEZO, TE~O
modes, the waveguides dimensions are a = 5.145 cm, b =
1.52 cm. The normalized impedances are evaluated using
(6), (10), and (11) for the above values of X. and yO and
c1= C2= 2.56, A/B = 25.0, frequency = 9.375 GHz, d =
0.1 cm, y.= 0.02 mho, as a function of length of the slot.
The results are presented in Fig. 6. The numerical results
obtained from conformal transformation [6] are also pre-
0 O“L 0“8 1’2 1’6 2“0 24 2’8 sented in the same figure for the sake of comparison.
sLOT LENGTH [cm)
VII. DISCUSSIONS
SLOT LENGTH (cm)
The investigations reported in this paper have thrown a
o lYL 0.8 1.2 1.6 20 2.4 2.8
new light regarding the realization of a transition between
a stripline/microstrip line and a rectangular waveguide.
The reactive stub is used to tune out the reactance at the
position of the slot. Such a configuration can provide very
strong coupling. It is found from the results presented in
nw -10 L // ll l CCINFCIRTMRANASFOLRMATION Figs. 5 and 6 that the values of coupling of the order of O
dB can be achieved using the structure shown in Fig. 1. It
is worthwhile to point out that, in the case of a symmetric
T junction, the highest value of coupling is limited to 3 dB
only.
_18 L
It is found from the results presented in Fig. 5(a) that for
Fig. 6. Variation of normalized resistance and reactance seen by a a stub-terminated slot of length 2.055 cm, the coupling
stripline feeding a multimode guide as a function of slot length.
between a symmetric stripline and a single-mode X-band
cuNe Description waveguide is nearly OdB over a frequency band of 9.0 to
a, b X. = 0.86 cm, X. = 4.29 cm, and y. = O 9.6 GHz, centered at 9.3 GHz. For the same arrangement,
c’,d xo = 2.57 cm and y. = 0,0, the coupling is within 1dB over a frequency range of about
2.5 GHz. For a microstrip line coupled to a single-mode
frequency, is computed and the results are presented as X-band guide, the coupling is nearly OdB over a frequency
curve (a) in Fig. 5(a). The corresponding variation of range 0.54 GHz, centered at 11.5 GHz for a slot length
coupling between a microstrip line and a rectangular wave- equal to 2.09 cm. The coupling is found to be within 1 dB
guide is presented ascurve (b) of Fig. 5(a). over a frequency range of about 2.5 GHz,
Using (12) and (13) and the data of curves (c) and (j’) In the case of an S-band guide and a symmetric strip-
of Fig. 4, the variation of coupling between a stripline and line, the coupling is nearly OdB over a frequency band of
an S-band guide, as a function of frequency, is evaluated 2.9–3.2 GHz, and it is within 1 dB over a frequency range
and the results are presented as curve (a) of Fig. 5(b). The 0.74 GHz for a slot length equal to 5.14 cm. For coupling
corresponding results for coupling between a microstrip between a microstrip line and an S-band guide, the cou-
line and an S-band guide as a function of frequency is pling is nearly O dB over a frequency band of 0.1 GHz,
computed and the results on coupling are presented as centered at 3.0 GHz, and it is within 1 dB over a frequency
curve (b) of Fig. 5(b). range of 0.46 GHz for a slot length equal to 6.818 cm.
The effect of the slot location in the y-direction on In order to study the effect of slot location in the
coupling has also been estimated at X-band for the case of y-direction, (13) is evaluated for the values of yO/b = 0.1,
coupling between stripline and the waveguide. The results 0.2,0.3,0.4, and 0.5, and the results are plotted in Fig. 5(c).
on coupling as a function of frequency (X-band) for vari- From the computed results, it is found that the coupling
ous locations of the slot in the y-direction have been for yO/b = 0.1 and yO= 0.4 are identical. Further, coupling
presented in Fig. 5(c). for yO/b = 0.2 and 0.3 are also found to be identical. As
the slot is displaced from the symmetric location, it is
VI. EXCITATION OF MULTIMODE GUIDE BY A SLOT observed that for frequencies higher than the resonant
IN THE GROUND PLANE OF SYMMETRIC STRIPLINE frequency, the coupling first increases with a change in
The normalized impedances of three slots a’, b’, c’ displacement from the symmetric location and then de-
arranged in the cross section of the multimode guide, creases. For frequencies lower than the resonant point, the
shown as the onset in Fig. 6, are evaluated. The locations coupling decreases with a change in position of slot from a
of the centers of the slot are XO= 0.86 cm, 2.57 cm, and symmetric location and again increases. Results of Fig. 5(c)
DAS et u[.: STRIPLINE- AND MICROSTRIP-LINE-FED SLOTS 327
reveal that coupling varies between O and 0.5 dB over a far published more, than ninety papers in journaJs in the U.S.A., U.K.,
frequency range of 8.375 GHz to 10.5 GHz for yO/b = 0.5, U.S.S.R., and India. His current research interests are electromagnetic,
microwave networks, antenna pattern synthesis, printed circuit antennas,
8.75 GHz to 11.125 GHz for yO/b = 0.1,0.4, and 9 GHz to
and EMI/EMC studies.
11.25 GHz for yO/b = 0.2 and 0.3. Prof. Das is aFellow of the Institution of Engineers (India) and also of
the Indian National Science Academy.
REFERENCES
[1] H. Perini and P. Sferrazza, “Rectangular waveguide to strip trans-
mission line directional couplers,” in IRE WESCON Conu. Rec.,
vol. 1, pt. 1, 1957, pp. 16–21.
[2] M. Kumar and B. N. Das, “Coupled transmission lines,” IEEE
Trans. Microwave Theory Tech., vol. MTT-25, pp. 7-10, Jan. 1977.
[3] S. N. Prasad and S. Mahapatra, “Waveguide to microstrip multi-
aperature directional coupler, “ in Proc 9th Eur. Microwave Conf.
(Brighton, England), Sept. 17-21, 1979, pp. 425-429.
[4] J. S. Rae, K. K. Joshi, and B. N. Das, “Analysis of smafl aperture
coupling between rectangular wavegpide and microstripline,” IEEE
Trans. Micr-owaue Thecvy Tech., vol. MTT-29, pp. 150-154, Feb.
1981.
[5] M. Kumar and B. N. Das, “Muftiaperture directional coupler using
dissimilar lines,” Proc. Inst. Elec, Eng., vol. 123, no. 12, pp.
1299–1301, Dec. 1976.
[6] B. N. Das and A. Biswas, “Excitation of multimode guide by
stnpline-fed slots,” Electron. Lelt., vol. 18, no. 18, pp. 801-802, K. V. S. V. R. Prasad was born in Srikakularn, India, on March 27, 1945.
Sept. 1982; Correction in Electron. Lett., vol. 19, no. 8, pp. 309, He received the B.E. degree in telecommunication engineering in 1967.
Apr. 1983. Since 1981, he has been working toward the Ph.D. degree at the Indian
[7] J. J. Gustincic, “The determination of active array impedance with Institute of Technology, Kharagpnr, under the Quality Improvement
multielemcnt waveguide simulator,” IEEE Trans. Antennas Propa- Program. He is a member of the faculty at the University Collegi of
gut., vol. AP-20, pp. 589-595, Sept. 1972. Engineering Osmania University, Hyderabad.
[8] J. S. Rao and B. N. Das, “Impedance characteristics of transverse
slots in the ground plane of astripline,” Proc. Inst. Elec. Eng., vol.
125, no. 1, pp. 29-31, JarL 1978.
[9] R, F, Barrington, Time Harmonic Electromagnetic Fields. New
York: McGraw-Hill, 1961.
[10] E. Yamashita and K. H. Atsuki, “Stripline with rectangular outer
conductor and three dielectric layers,” IEEE Trans. Microwave
Theoiy Tech., vol. MTT-18, pp. 238-244, May 1970.
[11] B. N. Das and K. K. Joshi, “Impedance of a radiating slot in the
ground plane of amicrostripline~’ IEEE Trans. Antennas Propagat.,
vol. AP-30, pp. 922–926, Sept. 1982.
[12] Charles P. Womack, “The use of exponential transmissions lines in
microwave components,” IRE Trans. Microwave Theory Tech., vol.
MTT-10, pp. 124-132, Mar. 1962.
[13] N. Marcuvitz and J. Schwinger, “On the representation of electric
and magnetic fields produced by currents and dicontinuities in
waveguides,” J. Appl. Phys., vol. 22, pp. 806-819, June 1951.
K. V. Seshagiri Rao (S’78-M84) received the B.
Tech. degree in electronics and communication
engineering from the College of Engineering
B. N. Dasreceived the M.SC.(Tech.) degree from Kakinada, Andhra Pradesh, in 1976, the M.E.
the Institute of Radio Physics and Electronics, degree from Jadavpur University, Calcutta, in
University of Calcutta in 1956. He received Ph.D. 1978, and the Ph.D. degree from Indian Institute
degree in electronics and electrical communica- of Technology, Kharagpur, India, in 1984.
tion engineering from I.I.T. Kharagpur in 1967. He joined the Radar and Communication
He joined the faculty of the Department of Elec- Centre, I.I.T. Kharagpur, in 1979 as a Senior
tronics and Electrical Communication, I.I.T., Research Assistant, and subsequently joined the
Kharagpur in 1958, and at present he is aProfes- Faculty as a Lecturer in 1984, He is engaged in
sor of the same department. He has been actively different sponsored research projects in the areas of microw~ve; and
guiding research in the fields of slot arrays, phased phased Array sub-systems, His current interests are microwave and milli-
arrays, striplines and microstrip lines. He has so meter-wave circuits and slotted array antennas.
328 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-34, NO. 3, MARCH 1986
Calculation of Propagation Constants
and Cutoff Frequencies of Radially
Inhomogeneous Optical Fibers
CHING-CHUAN SU AND CHUN HSIIUNG CHEN
.-tbstract—Based on the finite-difference technique, anefficient numeri- In this investigation, from the rigorous vectorial wave
cal method that can treat both the propagation constants and cutoff equations, we present an efficient finite-difference method
frequencies of optical fibers with arbitrary permittivity profiles isdeveloped
in the H@– H, formulation, which does not suffer from
in the rigorousvector form. Such a propagation problem is formulated in
spurious modes and can handle both the propagation
transverse fields so that the proposed method does not suffer from spurious
modes. The associated boundary conditions including those at cntoff are constants and cutoff frequencies of optical fibers with
derived in anovel way. ‘f’hereafter, numerical results of the cutoff frequency arbitrary permittivity profiles. In Section II, two coupled
and propagation constant of afiber with the parabolic profile are presented. second-order differential equations in H* and H, are
formulated. Thereafter, the associated boundary conditions
I. INTRODUCTION are derived, including those at cutoff. The calculated re-
sults for the parabolic profile are presented in Section IV.
T O SOLVE THE propagation constants of guided
modes of optical fibers with arbitrary permittivity II. FORMULATION
profiles in the rigorous vector form, several numerical
Consider an isotropic, rotationally symmetric optical
methods have been developed. They include the stair-case
fiber of which the relative permittivity profile, in general,
approximation [1], [2], direct numerical integration of four
can be expressed as
coupled first-order differential equations [3]–[6], and the
6( R)=61+(62–6JP(R), R<l
finite-element method [7]. In the stair-case approximation,
the radially inhomogeneous fiber is divided into and is thus =el, R>l. (1)
approximated by a series of homogeneous regions. In each
Here, R = r/a, r is the radial variable in cylindrical
such region, linear combinations of analytic field distribu-
coordinates, a is the core radius of the fiber, and P(R) is
tions (the Bessel or modified Bessel functions) are related
an arbitrary function whose value never exceeds unity such
to those in the neighboring regions. In the second method,
that the relative permittivity never exceeds c~ (~~> (l).
the four tangential components of the electromagnetic fields
Along such a fiber, a time-harmonic electromagnetic wave
are related by four coupled first-order differential equa-
of anular frequency Q propagates with fixed variations
tions. The propagation problem is solved by matching at
with respect to the axial (z) and the azimuthal (O) direc-
the boundary two independent sets of solutions, which in
tions as exp (– jm+ – j~z), where the azimuthal mode
turn are obtained numerically by direct integrations of the
number m is an integer. From Maxwell’s equations, one “
four first-order differential equations. The finite-element
obtaines
method in [7] is formulated in the axial fields, which may
suffer from spurious modes when the node points are not VC(R)
chosen carefully [8]. k;f(R)fi+v2H+— Xvx%=o. (2)
c(R)
Now let us turn to the methods of determining cutoff
From such a relation, the propagation problem of the
frequency. As early as 1973, Dil and Blok [3] proposed
radially inhomogeneous fiber can be described in terms of
associated equations, from the four coupled first-order
differential equations, for treating the cutoff frequencies of the transverse magnetic fields, M@ and H,, as in the
following two coupled second-order differential equations:
guided modes, but no serious data was presented at that
time. Perhaps, the first detailed evaluation of ‘cutoff fre-
quencies in the vector form is due to Bianciardi and Rizzoli H(’(R)+;H:(R)
[1], who have developed a method based on the stair-case
approximation for dealing with the cutoff frequency aswell 1 mz 1
+ a2(k;c(R)–~2)–z–~ H+(R)
as propagation constant.
[
E’(R)
Manuscript received July 16, 1985; revised October 28, 1985,
–#Hr(R)– —
C-C. Su is with the Department of Electrical Engineenng, National c(R)
Tsing Hua University, Hsinchu, Taiwan,
-%@;)K+(&R@) )=+0
C. H. Chen is with the Department of Electrical Engineering, National 1
Taiwan University, Tarpei, Taiwan. (3a)
IEEE Log Number 8406850. [
0018-9480/86/0300-0328$01.00 01986 IEEE
