Theory of Slots in Rectangular Waveguides J.appl. PH 1948
Theory of Slots in Rectangular Waveguides J.appl. PH 1948
Theory of Slots in Rectangular Waveguides J.appl. PH 1948
Guides
Cite as: Journal of Applied Physics 19, 24 (1948); https://doi.org/10.1063/1.1697868
Submitted: 03 March 1947 • Published Online: 28 April 2004
A. F. Stevenson
A basic theory of slots in rectangular wave-guides is given. The analogy with a transmission
line is developed and established, and detailed formulae for the reflection and transmission
mefficients and for t he "voltage amplit ude" in t he slot generated by a given incident wave
are given. While the complete expressions for these quantities are quite complicated and
involve the summation of inlinite series, certain parts of the expressions are comparatively
simple. I II particular, the "resistance" or "conductance" of slots which are equivalent to
series or shunt elements in a transmission line are given by fairly simple closed expressions.
Guide-to-guide coupling by slots and slot arrays are also considered. A more detailed summary
of the main results of the paper is given in Section 1.
1. INTRODUCTION. FUNDAMENTAL ASSUMPTIONS cates that, in some cases, the penetration of the
AND SUMMARY OF MAIN RESULTS field behind the face is by no means negligible.
In the case of an array of slots, however, the
T HE use of slots in wave-guides has proved a
promising means of launching high fre- assumption is probably a good one. The fourth
assumption is not in any way essential to the
quency radiation. The subject has been exten-
sively investigated by Watson. I In this paper, a development of the theory, but is made because
fundamental theory of slots in wave-guides, it is probably satisfied in cases of practical im-
based on the field equations, is attempted. portance. Much of the work is independent of
Attention is confined, for the most part, to this assumption.
rectangular guides, but the principle of the In Section 2, as a preliminary, the problem of
method is general, and other shapes of guide determining the field generated in a guide of ar-
could be considered if necessary. bitrary section by an assigned tangential electric
The fundamental assumptions on which the field in the wall of the guide is solved. In Sec-
theory is based are the following: tion 3, some general considerations relating to
the slot problem are given, and the analogy of a
(1) The walls of the guide are perfectly conducting and
of negligible thickness.
slot to an antenna is pointed out. The contents
(2) The slot is narrow; to be more precise, we assume of this section are probably not essentially new,
that but are given here for subsequent convenience.
2 log (length of slot/width of slot)>> 1. In Section 4 the problem of a slot in a rectan-
(3) In considering the field outside the guide, the gular guide which transmits only the HOI-wave
penetration of the field into the region behind the face is considered. A quantity called the "voltage"
containing the slot is neglected. In other words, we treat at any point in the slot is introduced, which is
the problem as if the guide-face containing the slot had an the analog of the current in an antenna. This
infinite perfectly conducting flange on it.
(4) The guide transmits only the HOI-wave, and the voltage varies approximately sinusoidally along
length of the slot is near that of the first "resonance" the slot, as does the current in a half-wave
(i.e., near X/2). antenna. Expressions for the amplitudes of the
HOI-waves scattered in either direction, in terms
Of these assumptions, the third is probably the
one most seriously at fault.Ia Experiment indi- of the voltage amplitude, are given, with par-
ticular cases, in Eqs. (25-31). It is then shown
* This paper is based on Radio Reports Nos. 12 and 13 that, in a certain sense, the slot is equivalent to
of the Special Committee on Applied Mathematics of the
National Research Council of Canada. a network in a transmission line and can be
1 W. H. Watson, The Physical Principles of Wave Guide characterized by the reflection and transmission
Transmission (Oxford University Press, London, 1947).
The writer's work is there referred to, but the notation is
different. of the walls might lead to large errors. A rough calculation
Ia The referee has suggested that the finite conductivity made in the appendix indicates that this is not so, however.
We may conveniently refer to 1/1, 'l', as the where the Green's function G2 satisfies conditions
generating functions for E- and H-waves, re- which are the same as for GIt except that condi-
spectively. tion (2) is replaced by aG2/ a" = 0 on S, and that,
It is assumed that all complex field quantities in condition (3), 1/41rr is to be replaced by
vary with the time according to the factor e- iwt , -1/41rr.
this factor being omitted, and that the actual By repeated integration by parts with respect
physical field quantities are the real parts of to z' or s', and by using the conditions that E" Es
the corresponding complex expressions. vanish at z= ± CIJ, we can replace (4) by
Let S denote the surface of the guide. Then . a
we regard E z , E. as being assigned functions of
position on S, it being assumed in the first
H.(P) =~
k s
f [(-+k
2
az'2
2)G2(P, P') ·Es(P')
i I/; n 2dxdy = 1. j
E z, lIz at any point in the guide. If we now use
(2), and the third and sixth of (1), we obtain
(no additive functions evidently being neces-
sary) :
Substituting (8) in (6), using (9) and the ortho-
gonality relations of the ,pn, we find 1
,pCP) = - L --,pn(X, y)
(d 2a n /dz 2 ) +u n 2a n n 2iu n !J.n 2
where 1
(11) 'l'(P) = L --'l'n(X, y)
n 2kU"
the positive square root being taken in (11) if Un
is real and the positive-imaginary root if Un is
pure imaginary. We now solve (10) by the
x f s
'l' ,,(x', y')eiUnlz-z'IE8(P')dS'
-,/4
(26)
cients a', f3' by
where J(~)/a denotes the component of H along
a'=B'/A', f3'=1+C'/A'. (23) the slot at the point ~ in an Horwave of unit
If the slot is in the narrow face of the guide, amplitude traveling in the positive z-direction. 6
and we keep the axes of Fig. 2, we must write y • This voltage does not vanish at the ends unless the
in place of x in (19) and (21); otherwise the same length of the slot is exactly "/2. It is permissible to suppose
this, however, as far as this part of the calculation is con-
definitions (22), (23) hold for the reflection and cerned (see Section 6).
6 That r must be expressible in this manner becomes
transmission coefficients.
clear when the problem is treated by Bethe's method
According to the previous section (and as (reference 2).
and for the slot in the narrow face, PIA =r/Ka, P'/A'=s*/Ka, (32)
where r denotes the same quantity as defined in
(26) above, and K is another (complex) dimen-
We thus find, for the slot in the broad face: sionless constant whose complete expression is
given in Section 6. These formulae hold whether
r = -11'2 cosO[1I'Xl
cos-I (0) -
1I'XI]
i sin-J (0) the slot is in the broad or the narrow face.
ka a . a From Eqs. (22), (23), (25), and (32), we now
have for the reflection and transmission cocfn-
1I'U [1I'XI 1I'Xl] cients:
+--- sin8 cos-J(8) -i sin-I(8) , (27)
k a a ex=yrz/K, ex'='Yr*2/K,
{j=/3'=1+'Y!r!2/K, (33)
where
where
cos(p1l' /2) cos(q1l' /2) (33')
1(8) =
I-p2
+ l-q2
,
\Ve thus see that the four coefficients ex, {j, a', fJ'
cos(p1l' /2) cos(q1l' /2) are not independent. They are subject to the two
J(O)=-- relations:
I-p2 l-q2
/3={3', aex'=(1-fJ)2. (34)
P = (U/k) cos8- (1I'/ka) sin8,
The first of the relations (34) shows that, as
q= (U/k) cosO+(1I'/ka) sin8. far as HOI-waves are concerned, the slot is com-
pletely analogous to a network connecting two
As particular cases of this,7 we have for the
portions of an infinite transmission line, if we
centered inclined slot (Xl =a/2):
define the reflection and transmission coefficients
r = (i1l' /k)[ U sin8I(8) + (11' fa) cos8J(8) J; (28) by means (say) of the voltage waves. The
equality of the transmission coefficients for waves
for the longitudinal slot (0==0):
incident from either direction in a transmission
r=2ka cos(1I'U/2k) cos(1I'xI/a); (29) line is, in fact, easily shown to hold under all
circumstances by means of ordinary line theory. 8
and for the transverse slot (8=11'/2):
The second of the relations (34) shows, however,
S = - (211'ik/ U) cos(1I'2/2ka) sin(1I'xI/a). (30) that the equivalent network for a slot is not of
the most general type: there is an iden tical
For the slot in the narrow face we find relation connecting the three quantities which
211' 2k characterize the network.
i:= -- cos8cos(UlcosO), (31) It is now dear that thl' two coefficients ex, jj
a(k 2 - U2 cos 20)
8 This was pointed out to the writer by J. R. Pounder.
the result being in this case independent of Xl. That some identical relation must exist between the four
coefficients a, a', {J, f3' for a transmission line folIows from
7 The terminology here used is that of Watson (refer- the fact that a network can be characterized by only three
ence 1). independent constants.
Equating (37) to the difference between (35) and G. eries = Rshunt = 73/12011"1' / r 12. (44)
(36), and dividing by / C/ 2 , we obtain, on From (28)-(31), we therefore have:
Transverse slot in broad face (series):
9 We could, of course, obtain the real part from the
complete expression for K to be given in Section 6; but 1 48011"3 1'k2 11"2 1I"XI
the present method is much simpler and forms a useful -=-_·-·cos 2-·sin 2- . (45)
check on the work. G 73 U2 2ka a
2
'T- cose sine L: '1'"
2k 11
this, since the imaginary part of K, and hence the sin 20 Un a'l'"
reactance or susceptance, varies rapidly with t. + - L : 2- -
2k n Mn ax
6. CALCULATION OF VOLTAGE AMPLITUDE
We proceed now to the more difficult problem
xfs eiU"lz-z" _a'¥ax_n', E/ded 71', (50)
of calculating the voltage amplitude when the
incident Horwave is given. where S now denotes the area of the slot. In (SO)
Referring to Fig. 3, our problem is, in ac- we have written for brevity
cordance with what was said in Section 3, to find
1/In=1/In(X, y), 'l'n = 'l'n(X, y), Vtn'=1/In(X', y'),
the field component E~ in the slot (or the voltage
V) in order that H t may be continuous as we go '¥n' ='I'nex', y'), E/ =E~(e, 11'),
from inside to outside the slot. We also impose the
conditions10 and we are to put, after differentiation, y=y' =0.
The functions Vtn, '1'" and constants Jl.n, M n are
the normalized eigenfunctions and eigenvalues
i.e.,
~=±l'} (49) for the E- and H-waves in the rectangular guide,
Vw=O when ~=±l. as defined by (9) and (14). All these are, of
course, damped waves except the HOi-wave.
From 0), (16), (17), we have for the com- Now we have (Fig. 3),
ponent H~ in the slot of the field generated inside
a a a
the guide by the field E~: - = cose-+ sin 0-,
a~ az ax
10These are special cases of the condition that in an
aperture in a perfectly conducting screen the component a2 a 2
a2
a2
of E parallel to the edge of the aperture tends to zero as - = cos 6--+sin 26--+ 2 cosO sin 0--,
2
X JG2(~'
R
1}; ~/, 'f/)Ev'd~/dr/
+ f -I
I
F(~, ~/) V(nd~', (52)
at the beginning of this section, the difference
between (55) and (52) must be equal to the com-
ponent ll~ of the incident field. We denote this
latter component by Af(l;)/a, where A is the
11 Each of the three tenns actually consists of the sum
of two parts, the infinities of which cancel. The considera- amplitude of the incident wave at the center of
tions of Section 3 show that the only singularity which the slot, so thatfW is identical with the function
should occur in H~ as t-O is correctly given by the first
term of (51), introduced in (26). The continuity condition then
=- III
- (1/471') JI
-I
de Jr.
integral equation occurring in an antenna prob- We now see that (with our approximations) the
lem. We rewrite the integral on the left-hand side integral in (60) must be independent of 11, which
shows that E~ must vary across the slot in the
12 E. Hallen, Nova Acta Reg. Soc. Upsaliensis 11, No.4
same way that the electrostatic charge density
(1938). varies across an infinite conducting strip of the
-I
[G 2 (1, n+G 2 ( -I, t')J cosk~'de
E= E/2. Collecting our results, we now see that the
i
integral equation (57) can be replaced by
i 41 A J~
+-J7r
1
-I l - f
eik(l-eJ
--coskedf
--log- VW = - cosk~ sinkU(~)d~
27rk E k -I I
A
--- sil1k~
It coskUWd~
-JJF(~, n cosk~ l'oskfd~d~'. (64)
k -I
-I
+--
i
27rk.- l
f --[I eikl t-n
lt-eJ
Vee) - VW]de
.
tude pI is given by
PljA'=s*/Ka. (65)
i t The results (63) and (65) are those which have
+- cosk~J
k -1-1
dt j I
(68) (69)
>
=!,
unlesR n or m=O,
if n or m=O.
-i II
-I
[G 2(l)(l, n+G 2 (l)( -I, nJcosk(de
7. GUIDE-TO-GUIDE COUPLING
-f I F(2)(~, () cosk~cosk~'d~d(, (70)
-/
Consider now the case where the slot couples
two infinite guides-guide 1, where we have an the superscripts 1, 2 in the functions G 2 , Fin (70)
incident (exciting) wave, and guide 2, where referring to guides 1, 2, respectively. If the inci-
waves are generated, Suppose that Fig. 3 gives dent wave is traveling in the negative z-direction,
the orientation of the slot relative to the standard we must write tl* in place of in (69). rl
axes for guide 1. Then the orientation of the slot Energy considerations similar to those used ill
13 The term proportional to cose sine in (53) has been
Section 5 show that
omitted from F(~, ~'), since it can be seen that this vanishes
on integration.
termination. In particular, the voltage amplitude where l' is a length of the order of the length of the slot,
in any slot will be and 2E is, as before, the width of the slot.
Comparing (A3) with the expression (37) for the mean
flux of energy out of the slot, we see that the error involved
in assuming perfect conductivity of the waIls is of the
where PI, P 2 , are the voltage amplitudes for the order
two solutions for the infinite guide.
.!(L)l~.
~ 4<T E