Composite waveguide coupling aperture having a varying thickness dimension
This is a Continuation of application Ser. No. 06/819,041 filed Jan. 15, 1986 now abandoned. 
    
    
     CROSS-REFERENCE TO RELATED APPLICATION 
     This application is related to concurrently filed, commonly assigned application by the same inventor entitled SIDE-LOOKING AIRBORNE RADAR (SLAR) ANTENNA which is incorporated herein by reference. 
     FIELD OF THE INVENTION 
     The present invention relates to the coupling of electro-magnetic energy in general and in particular to coupling apertures or slots between waveguides. More particularly still, it relates to coupling apertures that have, in addition to dimensions in the slot-plane, a significant dimension (depth) perpendicular to the slot-plane. More particularly yet, the present invention provides a novel coupling aperture or slot which has a variable cross-sectional area along the coupling path. 
     BACKGROUND OF THE INVENTION 
     Co-pending, concurrently filed patent application Ser. No. 819,037, now U.S. Pat. No. 4,752,781 entitled &#34;Side-Looking Airborne Radar (SLAR) Antenna&#34; by the same inventor discloses a radar antenna array which, for mechanical reasons, required coupling between two waveguides separated by a 0.4 inch thick wall and a range of coupling of between -31 dB and -14 dB. But again for mechanical reasons, it was not possible to realize the degree of coupling by the prior art methods of displacing the coupling slot closer to or farther away from the centre line of the wall of the power feeding waveguide. A new composite coupling aperture (conduit actually) was devised to adjust the degree of coupling while accommodating the necessary mechanical constraints. 
     SUMMARY OF THE INVENTION 
     The present invention provides a composite coupling aperture or slot having at least three significant dimensions, instead of the two of conventional coupling slots. These dimensions are length l, width w and thickness t. 
     In its broader aspect, the composite coupling aperture is characterized by having non-uniform cross-sectional areas along its thickness dimension t. 
     In a narrower aspect, the cross-sectional area changes abruptly between the opposite, waveguide coupled ends of the aperture. 
     Due to the complexity of the composite coupling aperture, it is possible to synthesize such apertures only by a combination of calculation and experimentation. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The preferred embodiment of the present invention will now be described in conjunction with the annexed drawings, in which : 
     FIG. 1 depicts a thick-wall coupling aperture model helpful in understanding the theory of the present invention; 
     FIG. 2 depicts a test jig for experimental determination of design parameters of a composite coupling aperture according to the present invention; 
     FIg. 3 depicts a series of composite coupling apertures between the broad side of a feeder waveguide and the ends of a corresponding series of radiating waveguides in an antenna array as seen from inside the feeder waveguide. 
     FIG. 4 is a cross-sectional view of the antenna array shown in FIG. 3 taken along the line 4--4. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
     In order to provide a better understanding of the design procedure for the preferred embodiment, it is useful to begin with a theoretical treatment of a simple non-composite, thick-wall coupling aperture or slot. 
     With reference to FIG. 1 of the drawings, a thick-wall slot may be regarded as a conventional slot 10 terminating a rectangular coupling waveguide 11. The waveguide 11 as illustrated in the figure has a cross-section of (a×b) and a length l. The characteristic impedance of a rectangular waveguide may be defined in various ways. For the present purposes, it is convenient to imagine the slot as being driven by a voltage source connected across its centre (this configuration being compatible with the usual text book definition of the admittance of a slot radiating from a ground plane). It is then assumed that the central drive point current would be equal to the total current flowing through either of the broad faces of the rectangular coupling waveguide. From a different point of view, this hypotheses is equivalent to assuming that the shunt susceptance loading corresponding to the discontinuity at the slot end of the coupling waveguide may safely be neglected. 
     On the basis postulated above, the relevant definition of coupling waveguide inpedance may be described as `mid-section voltage÷total broad face current`. Thus, according to R. J. Collin&#39;s `Field of Guided Waves`, the coupling waveguide characteristic impedance is ##EQU1## and λ and λ g  are the wavelengths for free space, and in the coupling waveguide 11, respectively. 
     The terminating impedance Z s  represented by the slot may be derived by applying considerations of Babinet&#39;s duality to the corresponding flat dipole (see C. Balanis, &#34;Antenna Theory, Analysis, and Design&#34;, pp. 497-8) and is given by 
     
         Z.sub.s =(377Ω).sup.2 /(4Z.sub.d),                   (2) 
    
     where Z d  is the dipole impedance. 
     We now define a dimensionless parameter δ according to ##EQU2## 
     The reflection coefficient ρ in the coupling waveguide is given by conventional transmission line theory (e.g. see Balanis, above) as ##EQU3## and the input impedance at the driven end of the coupling waveguide is similarly given by ##EQU4## 
     The ratio of voltage signals at either end of the waveguide is given (again by transmission line theory) as ##EQU5## 
     Since Z in  represents effectivrely a series connected element in the equivalent transmission line corresponding to a feeder waveguide, the overall transfer coefficient is given by ##EQU6## where Z eff  is an effective characteristic impedance of the feeder waveguide which takes into account the location of the slot aperture on the broad face. The overall voltage transfer coefficient is then represented by the equation ##EQU7## 
     As an example, the following Tables I and II show theoretically computed values of Z in  as a function of θ, and tZ eff  /Z o  as a function of 2a/λ, respectively. 
     
                       TABLE 1                                                     
______________________________________                                    
Calculated Impedance at Input of Coupling Guide                           
IMAGINARY PART                                                            
OF PROPAGATION                                                            
             INDEPENDANCE AT DRIVEN                                       
COEFFICIENT  END OF COUPLING                                              
FOR COUPLING GUIDE OHMS Z in                                              
GUIDE θ RADIANS                                                     
             REAL PART   IMAGINARY PART                                   
______________________________________                                    
0            1946         214                                             
0.05         1950         170                                             
0.1          1950          36                                             
0.15         1909        -178                                             
0.2          1783        -440                                             
0.3          1245        -830                                             
0.5           340        -590                                             
______________________________________                                    
 
    
     
                       TABLE II                                                    
______________________________________                                    
Voltage Transfer Coefficient                                              
for Thick-Wall Slot                                                       
                  OVERALL VOLTAGE                                         
                  TRANSFER                                                
                  COEFFICIENT,                                            
COUPLING WAVEGUIDE H                                                      
                  RELATIVE TO                                             
PLANE DIMENSION   RESONANT THIN                                           
PARAMETER 2α/λ                                               
                  WALLED SLOT CASE                                        
______________________________________                                    
1.0                1.0 &lt; -0°                                       
1.005             0.994 &lt; -11°                                     
1.01              0.955 &lt; -21°                                     
1.05              0.517 &lt; -62°                                     
1.1               0.312 &lt; -75°                                     
1.15              0.232 &lt; -80°                                     
1.3               0.149 &lt; -85°                                     
1.5               0.116 &lt; -87°                                     
______________________________________                                    
 
    
     Both Tables I and II correspond to a case specified by the geometrical parameters 
     b=0.16&#34; (coupling waveguide E-plane dimension) 
     l=0.4&#34; (coupling waveguide length) 
     f=9200 Mhz, and 
     a=0.65&#34; (coupling waveguide H-plane dimension) (assumed for Table I) 
     The tables also assume that the slot itself has an impedance of 1946 ohms when driven at its centre, corresponding to a resonant slot length. 
     From the data presented in Tables I and II, it may be seen that provided the `a` dimension of the feeder waveguide approaches the free space half-wave, the overall effective coupling level is very similar to that for a conventional thin wall slot. The bandwidth for the thick-wall design is, however, smaller. Also the phase angle of the voltage-transfer coefficient increases rapidly with `a`, when `a` is in the vicinity of λ/2 (maximum coupling case). 
     The basic theory as described above may be extended to the case of a composite coupling made up of two waveguides of different cross-sections, by applying transmission line principles. It is found that electrically such a composite structure behaves much as a single thick slot whose `a` and `b` dimensions are approximately given by the arithmetic mean of the `a` and `b` dimensions of the two parts of the composite aperture. 
     With reference now to FIG. 2, the practical design steps are as follows for a composite coupling aperture: 
     (1) Calculate the H-plane feeder waveguide width required for the particular application. In the case of the preferred embodiment shown in FIG. 3 for a SLAR antenna as disclosed in the said copending, concurrently filed application, the feeder waveguide width would be that necessary to realize the desired beam angle in the azimuth-plane. Such calculation is a standard calculation and may follow Johnson and Jasik&#39;s &#34;Antenna Engineering Handbook&#34; (1984, McGraw-Hill). 
     (2) Derive the (conventional) slot coupling coefficients required as per the principles given in Johnson and Jasik, supra. Again for the preferred embodiment the coefficients would be those necessary to realize the azimuth array excitations. 
     (3) Using the theoretical treatment outlined hereinabove estimate the slot offset distance off the centre line of the feeder waveguide that is necessary to achieve the highest coupling coefficient required. This highest coupling coefficient is to be realized at approximately 6 dB below the peak of the slot resonance curve. This slot offset distance is to be used for all coupling apertures. 
     (4) Select suitable aperture widths for each of the two cross-sections of the composite aperture according to the mechanical constraints. In the case of the preferred embodiment, that means the widths chosen must 
     (a) be sufficiently wide to allow accurate milling, (i.e. such that the deflection of a milling cutter is insignificant); and 
     (b) constrain the narrow aperture to lie within the end wall of the coupled-to (radiating) waveguide. At the same time the narrow aperture must not conflict with other mechanical requirements such as fastening screws, e.g. of the back-plane cover of the SLAR array. This latter consideration did dictate the minimum thickness (depth) of the narrow aperture (a non-critical dimension). Of course, the thickness of the narrow and wide apertures add up to the total wall thickness between the two waveguides. 
     (5) No the test jig shown in FIG. 2 must be fabricated. It comprises a waveguide piece 20, with connecting flanges 21 and 22 at eigher end, which has the same dimensions as the feeder waveguide. A metal block 23, which has a composite aperture 24 as determined in step (4) above, closes an aperture in the feeder waveguide 20 with the wide end 25 of the composite aperture 24 opon onto the inside of the waveguide 20. When the composite aperture is being tested, flange 26 of a coupled-to waveguide 27 is connected to the upper surface of the block 23. Of course, at its other end the waveguide 27 must be properly loaded. Now using a microwave network analyzer (not shown) estimate the insertion loss and phase from the waveguide 20 to the waveguide 27. By constructing several such test jigs with different composite slot lengths l, coupling coefficient and insertion phase may be graphed as a function of the length l. 
     (6) Now the lengths l of the composite aperture corresponding to the requisite coupling coefficients determined in step (2) hereof may be read off the graph constructed under (5). The corresponding uncompensated insertion phases are also read off the graph. 
     (7) To compensate the insertion phases a longitudinal aperture displacement C (along the length of the feeder waveguide) is calculated as follows ##EQU8## where λ g  is the wavelength in the feeder waveguide. 
     Referring now to FIG. 3 the application of the above principles to design and construct 187 composite coupling apertures coupling a feeder waveguide 30 to 187 radiating waveguides is explained. In FIG. 3 only six coupling apertures 40, 41, 42, 43, 44 and 45 are shown, coupling the feeder waveguide 30 to associated radiating waveguides 50, 51, 52, 53, 54 and 55. As is apparent in the figure, the coupling apertures 40 to 45 are displaced with respect to the waveguides 50 to 55 along longitudinal axis 60, reflecting by way of illustration only, the insertion phase compensation displacement C referred to in step (7) hereinabove. The other composite aperture dimensions A and B are also shown at the aperture 41. The following pages give the dimensions A, B and C for the 187 composite coupling apertures designed within the context of the preferred embodiment of the said copending, concurrently filed application by the same inventor. Following the table a qualitative explanation of the design considerations is given. 
     
         ______________________________________                                    
SLOT NO.  &#34;A&#34; DIM     &#34;B&#34; DIM   &#34;C&#34; DIM                                   
______________________________________                                    
1         0.480       0.558     +0.083                                    
2         0.480       0.558     +0.083                                    
3         0.481       0.559     +0.083                                    
4         0.481       0.559     +0.083                                    
5         0.481       0.559     +0.083                                    
6         0.482       0.560     +0.083                                    
7         0.482       0.560     +0.083                                    
8         0.483       0.561     +0.083                                    
9         0.483       0.561     +0.083                                    
10        0.484       0.562     +0.083                                    
11        0.085       0.563     +0.083                                    
12        0.486       0.564     +0.083                                    
13        0.487       0.565     +0.083                                    
14        0.488       0.566     +0.083                                    
15        0.489       0.567     +0.083                                    
16        0.490       0.568     +0.083                                    
17        0.491       0.569     +0.083                                    
18        0.493       0.571     +0.083                                    
19        0.494       0.572     +0.083                                    
20        0.496       0.574     +0.082                                    
21        0.497       0.575     +0.082                                    
22        0.499       0.577     +0.082                                    
23        0.501       0.579     +0.082                                    
24        0.502       0.580     +0.082                                    
25        0.504       0.582     +0.082                                    
26        0.506       0.584     +0.082                                    
27        0.508       0.586     +0.082                                    
28        0.510       0.588     +0.081                                    
29        0.512       0.590     +0.081                                    
30        0.514       0.592     +0.081                                    
31        0.516       0.594     +0.081                                    
32        0.517       0.595     +0.080                                    
33        0.519       0.597     +0.080                                    
34        0.521       0.599     +0.080                                    
35        0.523       0.601     +0.080                                    
36        0.525       0.603     +0.079                                    
37        0.527       0.605     +0.079                                    
38        0.528       0.606     +0.079                                    
39        0.530       0.608     +0.078                                    
40        0.531       0.609     +0.078                                    
41        0.533       0.611     +0.078                                    
42        0.534       0.612     +0.077                                    
43        0.535       0.613     +0.077                                    
44        0.535       0.613     +0.076                                    
45        0.536       0.614     +0.076                                    
46        0.536       0.614     +0.075                                    
47        0.537       0.615     +0.075                                    
48        0.538       0.616     +0.074                                    
49        0.539       0.617     +0.074                                    
50        0.541       0.619     +0.073                                    
51        0.542       0.620     +0.073                                    
52        0.543       0.621     +0.072                                    
53        0.544       0.622     +0.072                                    
54        0.545       0.623     +0.071                                    
55        0.546       0.624     +0.071                                    
56        0.547       0.625     +0.070                                    
57        0.548       0.626     +0.069                                    
58        0.549       0.627     +0.069                                    
59        0.550       0.628     +0.068                                    
60        0.551       0.629     +0.067                                    
61        0.551       0.630     +0.067                                    
62        0.552       0.630     +0.066                                    
63        0.552       0.630     +0.066                                    
64        0.552       0.630     +0.065                                    
65        0.552       0.630     +0.064                                    
66        0.552       0.630     +0.063                                    
67        0.552       0.630     +0.063                                    
68        0.553       0.631     +0.062                                    
69        0.554       0.632     +0.061                                    
70        0.554       0.632     +0.060                                    
71        0.555       0.633     +0.059                                    
72        0.555       0.633     +0.058                                    
73        0.556       0.634     +0.057                                    
74        0.556       0.634     +0.056                                    
75        0.557       0.635     +0.055                                    
76        0.557       0.635     +0.053                                    
77        0.557       0.635     +0.052                                    
78        0.558       0.636     +0.051                                    
79        0.558       0.636     +0.050                                    
80        0.559       0.637     +0.048                                    
81        0.559       0.637     +0.046                                    
82        0.560       0.638     +0.044                                    
83        0.560       0.638     +0.042                                    
84        0.561       0.639     +0.040                                    
85        0.561       0.639     +0.038                                    
86        0.562       0.640     +0.036                                    
87        0.562       0.640     +0.033                                    
88        0.563       0.641     +0.031                                    
89        0.563       0.641     +0.028                                    
90        0.564       0.642     +0.025                                    
91        0.564       0.642     +0.022                                    
92        0.565       0.643     +0.019                                    
93        0.565       0.643     +0.016                                    
94        0.566       0.644     +0.013                                    
95        0.566       0.644     +0.009                                    
96        0.567       0.645     +0.006                                    
97        0.567       0.645     +0.002                                    
98        0.568       0.646     -0.001                                    
99        0.568       0.646     -0.005                                    
100       0.569       0.647     -0.009                                    
101       0.569       0.647     -0.012                                    
102       0.570       0.648     -0.013                                    
103       0.570       0.648     -0.015                                    
104       0.571       0.649     -0.017                                    
105       0.572       0.650     -0.019                                    
106       0.572       0.650     -0.020                                    
107       0.573       0.651     -0.022                                    
108       0.573       0.651     -0.023                                    
109       0.574       0.652     -0.024                                    
110       0.574       0.652     -0.026                                    
111       0.575       0.653     -0.027                                    
112       0.575       0.653     -0.028                                    
113       0.576       0.654     -0.029                                    
114       0.576       0.654     -0.030                                    
115       0.577       0.655     -0.031                                    
116       0.577       0.655     -0.031                                    
117       0.578       0.656     -0.032                                    
118       0.058       0.656     -0.032                                    
119       0.579       0.657     -0.033                                    
120       0.579       0.657     -0.033                                    
121       0.580       0.658     -0.034                                    
122       0.580       0.658     -0.934                                    
123       0.581       0.659     -0.034                                    
124       0.580       0.659     -0.035                                    
125       0.581       0.659     -0.035                                    
126       0.582       0.660     -0.035                                    
127       0.582       0.660     -0.035                                    
128       0.582       0.660     -0.035                                    
129       0.582       0.660     -0.036                                    
130       0.583       0.661     -0.036                                    
131       0.583       0.661     -0.036                                    
132       0.583       0.661     -0.037                                    
133       0.583       0.661     -0.037                                    
134       0.584       0.662     -0.037                                    
135       0.584       0.662     -0.037                                    
136       0.584       0.662     -0.037                                    
137       0.584       0.662     -0.937                                    
138       0.584       0.662     -0.037                                    
139       0.584       0.662     -0.037                                    
140       0.584       0.662     -0.037                                    
141       0.584       0.662     -0.037                                    
142       0.584       0.662     -0.038                                    
143       0.584       0.662     -0.038                                    
144       0.584       0.662     -0.038                                    
145       0.584       0.662     -0.037                                    
146       0.584       0.662     -0.037                                    
147       0.584       0.662     -0.037                                    
148       0.584       0.662     -0.037                                    
149       0.584       0.662     -0.037                                    
150       0.584       0.662     -0.037                                    
151       0.583       0.661     -0.037                                    
152       0.583       0.661     -0.036                                    
153       0.583       0.661     -0.036                                    
154       0.583       0.661     -0.036                                    
155       0.583       0.661     -0.036                                    
156       0.582       0.660     -0.035                                    
157       0.582       0.660     -0.035                                    
158       0.582       0.660     -0.035                                    
159       0.582       0.660     -0.035                                    
160       0.581       0.659     -0.035                                    
161       0.581       0.659     -0.035                                    
162       0.581       0.659     -0.035                                    
163       0.580       0.658     -0.034                                    
164       0.580       0.658     -0.034                                    
165       0.580       0.658     -0.034                                    
166       0.580       0.658     -0.034 - 167 0.579 0.657 -0.034           
168       0.579       0.657     -0.034                                    
169       0.579       0.657     -0.033                                    
170       0.579       0.657     -0.033                                    
171       0.579       0.657     -0.033                                    
172       0.579       0.657     -0.033                                    
173       0.579       0.657     -0.033                                    
174       0.579       0.657     -0.033                                    
175       0.579       0.657     -0.033                                    
176       0.579       0.657     -0.034                                    
177       0.580       0.658     -0.034                                    
178       0.580       0.658     -0.034                                    
179       0.581       0.659     -0.035                                    
180       0.581       0.659     -0.035                                    
181       0.582       0.660     -0.035                                    
182       0.583       0.661     -0.036                                    
183       0.584       0.662     -0.037                                    
184       0.585       0.663     -0.038                                    
185       0.586       0.664     -0.039                                    
186       0.587       0.665     -0.040                                    
187       0.588       0.666     -0.040                                    
______________________________________                                    
 
    
     The SLAR antenna subject of the copending application comprises 187 waveguides, each containing radiating slots. These radiating waveguides are all excited from a single feeder or &#34;manifold&#34; waveguide, which is 17 feet long. Excitation of each radiating guide is via a coupling apterture in the broad wall of the manifold guide. 
     The very large number of radiating guides needed to obtain a sufficiently narrow antenna azimuth beam for a SLAR, were manufactured by milling from a single block of metal. The slot coupling ratios are chosed to couple out the majority (say 90% or more) of the power in the manifold guide, whilst maintaining an excitation of the radiating guides corresponding to a smoothly tapering function towards edges of the antenna. 
     On the basis of established design principles as outlined above and taking the parameters of the SLAR antenna as an example, this would imply slot coupling coefficients of up to about -14 dB. The maximum slot offset (or displacement of slot centre line from the centre line of the broad face of the manifold guide) would then be about 0.06&#34;. 
     In common with most conventional shunt displaced series feed slot devices, the signs of the slot offsets alternate along the feeder guide to permit proper phasing. 
     For practical reasons associated with limiting the deflection of a milling cutter when machining through a 0.4&#34; thickness of material, the slot needs to be about 3/16&#34; wide. It is found that there is not sufficient room for such a slot to break through within the cross-section of the radiating guide without (for one sign of offset) interfering with the attachment screw for the cover plate. 
     In the present coupling aperture design, a composite slot is formed, comprising two slots of differing widths in a staggered geometry. The positions of the aperture cross-section centre line relative to the centre line of the broad-face of the feeder waveguide determines the coupling. With suitable choice of parameters, the composite apertures are sufficiently close together where they break through the end-wall of the radiating guides to achieve a viable mechanical design. For example, for the SLAR antenna, the slot apertures span a total width of 0.416&#34; at the broad face of the feeder guide, but only 0.26&#34; at the radiating guides interface. 
     As cited earlier, a maximum coupling of -14 dB is needed. However the required coupling varies from aperture to aperture, being only about -31 dB at the input end of the feeder guide. In a conventional slotted waveguide series feed device, the smaller coupling ratios are realized by reducing the offset of the slots, as measured from the centre line of the broad face of the feeder waveguide. This method is satisfactory for small arrays. However, in the case of a SLAR array, if the conventional approach were adopted the slots with small coupling ratios would be displaced only about 0.008&#34; from the centre line, which was considered to be impractical to realize, given the 17 feet length of two separately machined pieces. 
     A further disincentive for 0.008&#34; offsets is that if the offset of the wide slot is made equal to this amount, the offset of the narrow slot will of course be much larger. Nominally it is the offset of the wide slot which matters. However, to the extent that asymmetrical higher order modes can penetrate the wide slot, the narrow slot is important. With the CAL Antenna geometry, the relevant order mode (TE11) has a calculated attenuation of 22 dB through the 0.15&#34; thickness of the wide slot, which is hardly enough to permit the five times larger offset for the thin slot in the 0.008&#34; case. 
     In the present coupling slot configuration, the range of slot couplings required is satisfied by varying aperture-length rather than aperture-offset. A potential problem then arises in that not only coupling but also insertion phase tends to vary. This in turn would result in a phase error associated with the excitations of the radiating guides, degrading the azimuth beam shape and increasing the level of the side lobes. By using a larger aperture offset (0.114&#34; rather than 0.06&#34;), even the slot with maximum coupling has a length which is shorter than the resonant length. From the theoretical treatment presented herein it can be seen that this approach reduces the insertion phase variation (maximum coupling phase--minimum coupling phase) from 80° (0.06&#34; offset) to 30° (0.114&#34; offset). The residual variation may now be compensated by spacing the apertures in a slightly irregular fashion along the 17 ft. length. Those near the driven end are miscentred relative to their radiating guides in such a fashion as to be further away from the source, whereas those at the load end are miscentred so as to be nearer to the source.