LIBRA FLY 
 
 OF THE 
 
 U N IVER.5ITY 
 
 Of ILLI NOIS 
 
 621.365 
 
 Ije655te 
 
 no. 40-49 
 
 cop. 2 
 
Digitized by the Internet Archive 
 in 2013 
 
 http://archive.org/details/studyofchromatic48loyt 
 
ANTENNA LABORATORY 
 
 Technical Report No. 48 
 
 A STUDY OF CHROMATIC ABERRATION OF A 
 COMA -CORRECTED ZONED MIRROR 
 
 by 
 Dr. Y. T. Lo 
 
 Contract AF33 (616) -6079 
 Project No. 9-(13-6278) Task 40572 
 
 Sponsored by. 
 
 WRIGHT AIR DEVELOPMENT DIVISION 
 
 Electrical Engineering Research Laboratory 
 
 Engineering Experiment Station 
 
 University of Illinois 
 
 Urbana, Illinois 
 
3fc5 
 
 f.ff ENGINEERING LIBRARY 
 
 ACKNOWLEDGMENT 
 
 It is a pleasure to acknowledge the comments on the manuscr]p T f 
 this report gnen by Professors G, A Deschamps and P E Mayes 
 
CONTENTS 
 
 Page 
 
 1„ Introduction 1 
 
 2. Results and Discussions 4 
 
 3. Simplified Analysis 18 
 Conclusion 28 
 Reference 29 
 
ILLUSTRATIONS 
 
 Figure 
 
 Number 
 
 Page 
 
 1. Cross-section of a zoned mirror 2 
 
 2„ Image pattern on the focal plane for a = 6 
 
 3„ Image pattern on the focal plane for a = 5° 7 
 
 4. Image pattern on the focal plane for a = 10° 8 
 
 5„ Image pattern on the focal plane for a = 15 9 
 
 6. Image paMprn on the focal plane for a = 20° 10 
 
 7 a) maximum field intensity of the image vs the angle of 
 
 incidence a, b) variation of directive gain vs scan angle 
 
 for the designed frequency f and 1.0625 f 12 
 
 o o 
 
 8„ Relation between the position of off-axis feed and beam 
 
 deflection angle a at 1„0625 f 13 
 
 o 
 
 9. Ratio of the two first secondary maxima vs scan angle a 
 
 for the designed frequency f and 1.0625 f 14 
 
 10. Magni + ude of secondary maximum vs its position in the 
 
 focal plane for the designed frequency f and 1.0625 f 16 
 
 o o 
 
 11, Geometry cf a typical zone of the mirror 19 
 
1 . INTRODUCTION 
 
 Briefly, a zoned mirror consists of sections of a set of confocal 
 
 parabolas with focus at c and axis VC, defined by P=2(f-nX./2)/(l+cos4 J ), 
 
 n = 0, 1, 2, ... where f is the focal length of the parabola with the 
 
 largest focal length as shown in Figure 1 „ This family of parabolas 
 
 intersect the circle 2 with center also at C and radius f at V , P , P„.c « 
 
 1 1 2 
 
 If a set of parallel planes perpendicular to the axis VC are drawn through 
 
 the vertices N , N„, N„, „ „ „ of the parabolas, they will intersect the 
 1' 2 3' 
 
 circle at Q , Q , Q , „ . , The sections of the parabolas cut out by the 
 lines parallel to VC and passing through Q , Q , Q , » . . form a zoned 
 
 mirror as shown by the solid line in Figure 1„ 
 
 1 
 In a previous report the properties of such a coma-corrected zoned 
 
 mirror are investigated by diffraction theory. The image pattern for various 
 incident angles of a plane electromagnetic wave and also the radiation 
 characteristics of the system are evaluated numerically by means of simple 
 fundamental functions and Fresnel integrals, It is concluded that for a 
 mirror with small F-number and nearly uniform illumination the zoned mirror 
 shows great effectiveness in coma correction. Since for such a two- 
 dimensional cylindric mirror there is no spherical aberration and astigmatism, 
 the only important defect of this system will be the chromatic aberration. 
 Unfortunately, such a system is inherently very frequency-sensitive; moreover, 
 to this author s knowledge there is yet no method available for chromatic 
 correction of this system. However, in contrast to many optical systems 
 a microwave device even with a limited bandwidth usually finds very wide 
 applications In this investigation the image pattern deterioration due to 
 chromatic aberration is studied. The purpose of this short report is to 
 
/ B 2^ 2 - 
 
 Figure i. Cross-section of a zoned mirror 
 
supplement some of the information on this point which is not found in 
 the previous report. 
 
2, RESULTS AND DISCISSIONS 
 
 Diffraction theory was used in the previous *ork c 1 nee coma aberrati >n has 
 
 a dominant effect on the minor lobes where the usual hirchhoff approximation 
 
 is generally susceptible to larger error However the results sho* that 
 
 the contribution to trie image field due to the coupling current? among 
 
 zones is negligible in comparison with that due to the geometric optics 
 
 current induced by an incident plane wave, Moreover. t he edge diffract. on 
 
 effect is also of higher order as compared to the latter (except at nulls 
 
 and small minor lobes), However, with considerable chromatic aoerration the 
 
 lower order contribution becomes appreciable (the nulls oi 'he pattern will 
 
 be filled up and the minor lobe level will be raised) therefore, tne 
 
 above mentioned high order effects become less significant when the mirror 
 
 is operated at a frequency other than that of proper design In such a 
 
 case the previously established theory'can be greatly simplified, since 
 
 it requires no near-field solution which is only essential in the evaluation 
 
 of coupling effed between zones, Such a simplified theory will be 
 
 established and discussed in the next section However., this theory is 
 
 not used here to evaluate the deterioration of the patterns =ince a 
 
 program for the ILLIAC (the electronic computer at the University of 
 
 UlinoJ ) based upon the more rigorous approach ha? already been established 
 
 th< previous investigation. With only a minor modification this 
 
 idapted to the evaluation of the performance of a zoned 
 
 dimensions at frequences equal to 1 0625 I 
 
 o 
 
 i ! 
 
(a) F-number = f/D = 0.556, 
 
 (b) the focal length = f 10\ where X. is the wavelength corresponding 
 to the design frequency f ; 
 
 (c) total number of zones = 11. 
 
 For the purpose of comparison, the performance of an equivalent smooth 
 parabola (one with same focal length and aperture) is also computed. As 
 expected the latter by comparison is a wide-band device since the only effect 
 of changing frequency is to vary the aperture dimension in wavelength accord- 
 ingly. 
 
 Figures 2 and 6 show the image pattern in the focal plane for an incoming 
 
 o o o o o 
 plane wave with incident angle equal to , 5 , 10 , 15 , and 20 with respect 
 
 ' 
 
 to the axis of the system. By comparing these patterns with those at the 
 
 design frequency f , it is seen that at a frequency deviated by 6.25 7c from 
 
 f the image patterns are changed considerably. First of all the pair 
 
 of minor lobes adjacent to the major beam are no longer distinct; instead they 
 
 are merged with the major lobe. This is evidently a result of overwhelming 
 
 chromatic aberration. However, it is interesting to see that other minor 
 
 lobes seem to be comparatively less affected by this aberration and also that 
 
 the symmetry of the image pattern is maintained up to an incident angle 
 
 equal to 10° „ For the smooth parabola, the image patterns remain very much 
 
 the same as those at the design frequency f Q , except that the beamwidths are 
 
 somewhat smaller and the gain is slightly higher up to a scan angle o. = 10 f 
 
 this being a consequence of an increment in aperture as referred to 
 
 wavelength. However, this is no longer true for large scan angle since a 
 
<0 
 
 *> ^ ro cvj 
 
 (AJLISN31NI 01313 3AllV13d) z 3 
 
W 
 II 
 d 
 
 u 
 
 o 
 
 0) 
 be 
 
 E 
 
 (A1ISN31NI Cl~l3ld 3AI1V1V13U) z 3 
 
(A1ISN31NI 01313 3AI1V13U) Z 3 
 
to 
 
 u 
 o 
 
 0) 
 
 c 
 ca 
 
 0) 
 
 U 
 
 a 
 be 
 
 in <fr ro cj 
 
 (A1ISN31NI 01313 3/VI±V13H) Z 3 
 
X 
 
 sr ro cj 
 
 (A1ISN31NI Q13IJ 3AllVH3d) z 3 
 
11 
 
 system of wider aperture has larger aberrations. These results can be seen 
 
 in Figure 6 which also shows the maximum field intensity (equivalent to the 
 
 gain except for a proportionality constant) and the normalized gain (referred 
 
 to the gain at a = 0) against the scan angle a for both mirrors at frequencies 
 
 f and 1.0625 f . 
 o o 
 
 It is seen from Figure 7(a) that the loss in gain of the zoned mirror 
 
 due to 6.25% increment in frequency is from 1 db at a = to about 2 db at 
 
 a = 20 . Figure 7(b) also shows that the gain at this frequency drops at 
 
 a faster rate than that at f as the scan angle a increases (somewhat close 
 
 o 
 
 to that of the smooth parabola) . 
 
 Figure 8 shows the relation between the position of the off-axis feed 
 
 and the beam deflection angle a at 1.0625f . If these curves are compared 
 
 o 
 
 with those at f , one finds practically no difference. This fact indicates 
 o 
 
 that the lateral chromatic aberration of the so-called "diffraction focus" 
 
 is not severe at all, at least in the range considered. As to the longitudinal 
 
 chromatic aberration no computation has been done. But as will be seen , later, 
 
 i 
 
 a very simple result can be obtained which states that as expressed in*" 
 
 percentage of the original focal length the longitudinal chromatic aberration 
 
 is equal to the percent change in wavelength as referred to \> . 
 
 Since the first pair of minor lobes are not distinct, in other words, 
 
 the coma aberration is overshadowed by a strong chromatic aberration at 
 
 this frequency, it becomes meaningless to evaluate the coma in terms of 
 
 their ratio as has been done previously. However, for the smooth parabola 
 
 this ratio is shown in Figure 9 at both frequencies together with that for 
 
 the zone mirror at f . Since a larger ratio indicates a larger coma aberration 
 
 o 
 
 this result is in agreement with the fact that a wider aperture (in wavelength) 
 
12 
 
 >- 
 
 (fi 
 
 UJ f- 
 
 Q X 
 
 Ld 
 
 '■0625 ft 
 
 x 
 
 < 
 
 
 
 10 15 20 
 
 SCAN ANGLE oc (DEGREE) 
 
 
 < 
 
 UJ 
 > 
 
 < 
 
 _l 
 UJ 
 
 or 
 
 10 15 20 
 
 SCAN ANGLE oC (DEGREE) 
 
 25 
 
 I 7. a) maximum field intensity of the image vs the angle of incidence Q, 
 
 b) variation of directive gain vs scan angle for the designed frequency 
 
 f and 1.0625 f 
 o o 
 
13 
 
 If) 
 CVJ 
 (0 
 
 o 
 
 ID 
 
 ro cvj 
 
 y/ k lN3IAI30V~ldSIQ Q333 
 
 s 
 
 CS 
 
 
 XI 
 
 ■a 
 c 
 
 03 
 
 T3 
 
 CD 
 
 <H 
 
 o o 
 
 C 
 
 o m 
 •h eg 
 
 +- 1 to 
 
 ■H O 
 
 O ,-i 
 
 a 
 
 0) a! 
 
 x: 
 
 i-H 
 
 bD 
 
 
 
 ce -a 
 

15 
 
 suffers more from aberrations. 
 
 Figure 10 shows the field intensity levels of the first pair of the 
 
 minor lobes (adjacent to the major beam) vs, their positions in the focal 
 
 plane of the smooth parabola for an incident wave at frequencies f and 
 
 o 
 
 1.0625f . Also shown in this figure are those of the zoned mirror at f 
 o o 
 
 only. It may be noted that the side lobe level of an image pattern is 
 
 not exactly the same thing as that of the corresponding radiation pattern. 
 
 The purpose of preparing Figure 10 is in an effort to relate these two 
 
 quantities,, 
 
 By reciprocity if a feed is at y/^ in the focal plane minor lobes of 
 
 corresponding intensity as shown in Figure 10 will be produced somewhere in 
 
 space. At the same time the major lobe field intensity corresponding to this 
 
 feed displacement can be found by first using Figure 8 to obtain the scan 
 
 angle # ( or the beam deflection angle in the terminology of transmitting) 
 
 for this value of y/^ , then referring Figure 7(a) for the intensity. The 
 
 ratio of these two intensities is the side lobe level of the radiation 
 
 pattern for the feed at y/^ . Incidentally Figure 10 also shows slightly 
 
 o 
 
 higher side lobe levels for the parabola operating at higher frequency, 
 especially at large scan angle. 
 
 In FiguVes 2 to 6 there are also shown the image patterns of the zoned 
 mirror at a frequency of 1 . 125f . Except the case of normal incidence all 
 images are so blurred" that the major and minor lobes are no longer 
 distinguishable A poor image in this case may be easily explained. At 
 this frequency the field contribution of the outer six zones (although 
 they are narrower in width) will have a component opposite to that due to 
 the central zone for the normal incidence case. On the other hand at a 
 
WniNIXVW AUVQN033S JO A1ISN31NI Q13IJ 
 
17 
 frequency equal to l,0625f , no such negative contribution arises except the 
 outer two narrow zones. If one uses this as a criterion to determine the 
 bandwidth such that no zone contributes a field at focus in opposite sense 
 then the maximum total bandwidth as referred to the frequency f is given 
 
 by — = .- — where N = total number of zones. For the present mirror 
 
 f N-l ^ 
 
 o 
 
 A f/f = O.lOo No computation for exactly this frequency has been done 
 since this criterion does not necessarily imply only a slight deterioration 
 of the pattern and a small minor lobe which are of primary interest for 
 .scanning purpose. It seems to be safe to infer that for the latter application 
 the bandwidth will be less than the value indicated above. 
 
 In conclusion, it is seen that from the point of view of the image 
 sharpness, the gain, and side lobe level, the zoned mirror at 1.0625f or 
 higher is worse than a corresponding smooth parabola in scanning ability. 
 A similar conclusion may also be reached on the lower frequency side of f „ 
 
18 
 
 3. SIMPLIFIED ANALYSIS 
 
 Since for the zoned mirror at a frequency different from f , the design 
 frequency, the chromatic aberration becomes so overwhelming, a lower order 
 approximation will provide a fairly accurate solution. In this section 
 such a simplified theory will be given. 
 
 In Figure 11, a typical zone, (except the central one) is shown *ith a 
 plane wave incident at an angle a with the axis ox. Let the intersection of 
 the zone and the circle be Q (x ' , y' ), a typical point on the strip be 
 QCx', y') , the angle of the strip with respect to the y-axis be T and 
 
 0Q o = s > 0Q= s ^ 
 
 Then the current induced on the strip based upon the infinite plane 
 solution in the previous report . 
 
 ICQ) = I cos (« + x) e jk S Sin (a + T) Cl) 
 
 where *1 1 s the intrinsic wave impedance of free space , The f leld intensi ty 
 
 at observation point P (x,y) is given by 
 
 E(p) = -i--cos (a + T) f 2 H (2 Hr) e jkS Sin {a + T) ds (2) 
 2J 7 ! J o 
 
 S l 
 
 where s and s are distances of the edges of the strip function origin 
 1 ^ 
 
 0, and 
 
 f f ' N 2 / / N 2 
 
 r = | (x - x) + (y - y) 
 
 1/2 
 
 In fact this is generally true in optical systems c In systems with 
 large aberrations even geometrical optics gives a good description of 
 the image deterioration, See E Wolf, Rep Prog in Phvsics (London 
 Physical Society) 14, (1954), 95 Also~M BorrTand E„~Wolf , Principles 
 of Opt ics. Chapter IX, 1959, Pergamon Press 
 
19 
 
 9-X 
 
 Figure 11. Geometry of a typical zone of the mirror 
 
2'; 
 
 Let 
 
 x ! = x' + x , y' = y' + y 
 
 o o 
 
 2 2 V2 
 s" = (x- 2 + y" 2 ) 
 
 r = P Q , 
 
 o o ' 
 
 Then 
 
 5 = the angle between P Q and the axis x 0. 
 
 o 
 
 oo 2 
 
 r = r - 2r S" sin T cos 6 + 2r S" cos T sin 6 + S' (3) 
 o o o 
 
 = r 2 + S*' 2 + 2r S" sin (6 - t) 
 
 o o 
 
 For the observation point P in the neighborhood of focus F such that 
 
 r ^ f >> fS - S)/2, then 
 o 2 1 
 
 v m r [1 + S" sin (6 - T)/r ] (4) 
 
 o o 
 
 Since we are primarily interested in large aperture which implies that the 
 
 focal length f is considerably larger than wavelength X^ the Hankel function 
 
 in the integral can be replaced by its asymptotic expression 
 
 r~\ m — -, . „,. I „ ■ jkr [1 + s" sin (5 - r)/r 
 
 H^) ( k r) « PL e -^ + J */4 * /_£L- e J o L 
 o f kr WTTkr 
 
 Therefore 
 
 ti 
 
 o~ /-a T % "Jk[r - S sin (a + T )] f S 2 ., „ 
 
 w v c os (a + t ) o o J -Jkqs , „ 
 
 E(p) = ; 07r . , ~ e J „ e J M ds 
 ^ikr -^ s 
 
 cos fa + Tl -J fc t'o " S o Sin (a + T) ^ ^"^^ - ^^/J^ 
 t| /27Tjkr e 
 
 (5) 
 
 where q = sin f6 - t) - sin ( a + t). 
 
21 
 
 By assuming that s'' = - s" which is nearly true for most strips (or by 
 
 redefining the point Q as the center of the strip, not the interaction with 
 
 o > 
 
 the coma circle) then 
 
 ., , /Tcos (a + x) sln < k 1 S 2> -JK[r o - S o sin(a + x)] 
 
 E<P) = n^ST — e 
 
 This is equivalent to the field due to an inhomogeneous line source located 
 
 at Q with a current 
 o 
 
 o "H kq 
 
 since q is a function of 6, depending on the coordinates of observation point P. 
 
 Let, t 5 , q , s , r , and s" (half width of the strip) be those 
 ' n' n' n' on' on' n 
 
 values for strip number n. The total field at P(x^y) is 
 
 N /2 cos (a + t ) sin(kq s" ) -jk[ r - s sin (a + t )] 
 *, \ *, , \ ^ n nn on on n J 
 
 E(p) = E (p) + 2 — e 
 
 n=-N T] /j77kr on n (g) 
 
 nj6l 
 
 where E (p) is the field at P due to the central zone which has been obtained 
 previously. To conform with our present notations, it is rewritten as follows 
 0, 
 
 ■* , x /jf~~ f 1 cos(4j/2 + a) -jk[r-f tan iJj/2 (tan i|i/2 cos a + 2 sin a)] 
 
 ^1 (P) =t\ \ 2 , /Q e 
 
 v J _ cos 4V2 
 
 where 
 
 = f sec ty/2 + P - 2fp sec \\>/2 cos (9 - i|0 
 
 (P,Q) is the polar coordinate of P(x, y) with respect to the focus F 
 and axis FO. 
 
22 
 
 20 is the angular aperture of the central zone (parabola) referred to F. 
 For P in the neighborhood of F, f > > P, then 
 
 2 
 r *» f sec 4V2 - p cos (9-4) 
 
 Let z = tan 4V 2, and the coordinates of P with respect to F be (Ax, y) ; then 
 
 2 
 
 dz = dLJJ/2 cos i|j/2 
 
 cos 4j = (1 - z ) / (1 + z 2 ) 
 
 2 2 
 
 sin (^ = 2z / (1 + z ) 
 
 2 2 
 
 cos 4*/2 = 1/(1 + z ) 
 
 2 2 2 
 
 sin i^/ 2 = z /(I -*- z ) 
 
 Substituting these quantities into (9), one obtains 
 
 z 
 
 v ( \ n h f cos a - z sin a. -jk(p(z) . . . 
 
 E 1 (p)=2 ^ - / ^ — eJr dz (10) 
 
 1 J -Z V 1 4- z 
 
 where 
 
 z 2 = tan Q (10) 
 
 <p(z) = i x-2(f sin a. + y ) z + [f(2-cos a) + A x]z -2f sin a z + f(l-cos a.) z 4 ] 
 
 „ 2.-1 (11) 
 
 X (1 + z ) 
 
 and 
 
 z 2 = I (X/f - X 2 /4f 2 ) (12) 
 
 Since this mirror is primarily for large aperture application and also 
 
 2 
 since the smallest F-number is 1/2, then z is a small number given by 
 
 2 
 
 z < X/4f < V2D << 1 '13) 
 
23 
 
 2 
 For D = 20\, which is about the case computed before, z < 0.025. 
 
 By introducing this approximation, the integration in (9) can be carried 
 
 out in terms of basic functions and Fresnel integrals as shown in the 
 
 following. Now 
 
 2 3 4 
 
 <P(z) = x-2 (f sin a +y) z + (f + 2 A x-2 cos a )z +2yz -2Axz 
 
 5 6 7 
 
 -2yz +2Axz +0(z). (14) 
 
 Since P is in the neighborhood of F, Ax «f or x, and y « f . 
 
 Further using the condition indicated by (13), one obtains the following 
 
 approximations: 
 
 2 
 (p(z) » x-2 (f sin a- + y) z + (f + 2Ax-2 cos a) z 
 
 = A(z + B/2) 2 + (C-B 2 /4) = A U + D 
 where A = f + 2Ax - 2 cos <*■; 
 
 AB = -2(f sin a. + y) ; (15) 
 
 AC = x 
 U = z + B/2; 
 D = C - B /4 
 Similarly the other factor in the integrand of (10) can be approximated 
 
 as follows, 
 
 2 " 1/2 2 
 
 (cos a ~ z S in a) (l + z ) -cos a - z sin a - z (cos a)/2 
 
 + z 3 (sin a)/2 = P(z) (16) 
 
 Substituting (U-B/2) for z in P(z), one obtains a polynomial of third 
 
 degree in U which may be written as 
 
 2 3 
 g(U) =a +bU+ c U+dU (17) 
 
 when a, b, c, and d can be obtained by comparing (16) with (17). 
 
Therefore 
 
 24 
 
 E l( p) = 2 Jg. e jk ° I 
 
 <18J 
 
 where 
 
 g(U) e JkAU dU, 
 
 (18a; 
 
 U = -z + B/2 
 
 U = z, + B/2 
 2 1 
 
 (18b; 
 
 The first term of I is given by 
 
 .2 
 
 U 
 
 r 2 
 
 J l = a J 
 1 J 
 
 e JkAU dU = 
 
 J F( v/kA U ) - F( VkA U ) 
 
 (19) 
 
 oo . 2 
 
 I IX 
 
 where the Fresnel integral F(x) = e dx is available in tabulated form, 
 
 x 
 
 The second tern of I is given by 
 
 U 
 
 I = b 
 2 J 
 
 jkAU* jkAU* 
 
 Ue JkAU dU = £ S Lf e 
 
 2 jkA 
 
 b sin kAz B jkA(z 2 + B 2 /4) 
 ______ e 
 
 (20) 
 
 The third term of I is given by 
 
 I, = C ^ U 2 e^ 1 ' 2 dB = ° 
 
 3 , 2jkA 
 
 1 
 
 ^ UdeJ"" 2 
 
 u. 
 
 2jkA 
 
 2 U 2 
 [ (Ue JkAU ) 
 
 U 
 
 r 2 2 
 
 e JkAU dU 
 
 °1 " u l 
 
 j jkAU 
 
 Ff /kAl I 
 2 J 
 
 (21) 
 
25 
 
 The last term of I is given by 
 
 U 
 
 . (" 2 TT 3 jkAU 2 dU = — — 
 
 I = d J D e J 2jkA . 
 
 U, 
 
 U 2 d e JkAU 
 
 2jkA 
 
 U 
 
 U 2 e jkAU 
 
 jkAU 2 
 e J dU 
 
 2jkA 
 
 jkAU 2 jk AU 2 
 (U 2 e - U x e 
 
 ± jkAU 2 jkAU^ 
 
 ) r-r (e - e 
 
 jkA 
 
 (22) 
 
 To sum up the field at P near F is given by Equation (8) in which the 
 field due to the first zone 
 
 V P> - > /4 * ikD = ' p 
 
 V P=l P 
 
 where 
 
 z i> l 2> V X 4 are given b y (19) to (22) 
 
 A, B, C } D, are given by (15) 
 
 U U are given by (18b) and (10a); 
 
 a, b, c, d are given by (16) and (17). 
 
 In Equation (8), the focal length of the first zone is practically inde- 
 pendent of frequency; however that of the rest zones for normal incidence can be 
 easily found. In this case by symmetry the focus must lie on the axis. Let P 
 be such a point on the axis with coordinates (Ax, o) . Then by definition 
 
 2 2 2 
 r = f + (Ax) + 2 f (Ax) cos 2t and 
 o 
 
 Ax 1 Ax 2 A Ax 
 
 r = f Q + =- cos 2T + - — 7t sin 2T + 9 — - 
 
 o f 2 * \ f 
 
 (23) 
 
1" Vt 
 
 Now the phase function of the field due to the n zone with n^l 
 in Equation 8) becomes for a - 
 
 CD = k S sin t - k r «. kx' - k f (1 + ^ cos 2*0 . (24) 
 T n on n on 
 
 However x' = (n-1) *. 
 on o 
 
 where p = an integer Then 
 
 kx 7 
 
 - k f (1 + . 
 
 Ax 
 
 on 
 
 f 
 
 = P X 
 o 
 
 
 
 <P = 2 TT (n-1) rfi— -27TP -^ (1 + -^£_ cos 2T) 
 
 n a. /v j 
 
 A. \r% Ax 
 
 = - 2ir(p-n+l) — jr^_ - 2ir p — £— — ~" cos 2T * (25) 
 
 Let ^ = ^ + d\ then 
 o 
 
 <P = - 2 tt (p-n+1) (l-d\ X ) - 2 it p *■ /M cos 2t (Ax/f) 
 n o 
 
 = 2ir(p-n+l) + 2ir(p-n+l) d^\ -2irp C^o/M cos 2T (Ax/f) (26) 
 Putting 2TT(p-n-^l) d\A ■ 2l7p (\yA) COS 2T (Ax/f), 
 then Ax/f = [l - (n-1) / p] sec 2T d\^ 
 
 o 
 
 Since x - (n-1) ^ = f(l-cos 2'«") , 
 on o 
 
 cos 2T _ i _ ( n _D X. /f = l - (n-1) / p, 
 
 o' 
 
 Thus the chromatic aberration expressed in fraction of the focal length is 
 
 glVen by Ax/f = dV* (27) 
 
 o 
 
 which is independent of n to the approximations assumed in (24). Therefore 
 
 the best focal point for all the zones except the central one is at 
 
 f(l + d^A ) d i a wavelength ^ 4- dV 
 o o 
 
27 
 
 It is obvious that the relation (27) also holds for high mode operation 
 
 when ^ = nA + d^ where m = integer. 
 
28 
 CONCLUSION 
 
 A coma-corrected cylindric mirror which has been studied previously for 
 its coma aberration is re-examined for its chromatic aberration. Although 
 such a device can be operated at multiple modes, at each mode it has a 
 narrow frequency bandwidth. At a frequency 6.25% higher than the designed 
 one, there is a slight loss in gain but the side lobes are raised to a 
 very high level. At such a frequency the scanning performance becomes 
 inferior to a corresponding smooth parabolic reflector. At a frequency 
 12.5% higher than the designed one, the image formation becomes so poor 
 that no focus is generally recognizable. It may be inferred that such a 
 device has a total bandwidth of perhaps only five or six percent, depending 
 on what deterioration can be tolerated on the side lobes. 
 
 A simplified formula has been obtained for the image field in the 
 neighborhood of the focus. Such a formula will be found particularly useful 
 when the chromatic aberration is large. It may also be used to evaluate the 
 field witn coma aberration at the design frequency if some errors at the minor 
 lobes and nulls are tolerable. 
 
 Although no computation has hitherto been done to show how close a 
 solution this theory provides, yet it is clear that the major approximation 
 involved is that only the geometric optics current is considered. Even in the 
 absence of chromatic aberration, this has been previously shown to be a close 
 approximat ion. 
 
 A simple formula of the longitudinal chromatic aberration for the case 
 of normal incidence is obtained. The lateral chromatic aberration for incident 
 angle up to 20 and for a frequency range of + 6.257o is very small as indicated 
 by the numerical results. 
 
29 
 
 REFERENCE 
 
 1. S. Dasgupta and Y. T. Lo, "A Study of the Coma-corrected Zoned Mirror 
 by Diffraction Theory." Technical Report No. 40, Ant. Lab., University 
 of Illinois, July, 1959. 
 
 For other references, see pp. 87 and 88 of the above report. 
 
ANTENNA LABORATORY 
 TECHNICAL REPORTS AND MEMORANDA ISSUED 
 
 Contract AF33 (616 ) -310 
 
 "Synthesis of Aperture Antennas/' Techn ical Report No, 1, C.T.A. Johnk 
 October, 1954.* 
 
 "A Synthesis Method for Broad-band Antenna Impedance Matching Networks " 
 Technical Repo rt No. 2, Nicholas Yaru, 1 February 1955,* 
 
 "The Asymmetrically Excited Spherical Antenna/' Techn ical Re port No. 3, 
 Robert C Hansen, 30 April 1955.* 
 
 "Analysis of an Airborne Homing System/' Technical Report No. 4, Paul E. 
 Mayes, 1 June 1955 (CONFIDENTIAL). 
 
 "Coupling of Antenna Elements to a Circular Surface Waveguide/' Technical 
 Repor t No 5, H. E. King and R. H. DuHamel, 30 June 1955.* 
 
 "Axial ly Excited Surface Wave Antennas," Technical Report No. 7, D. E. Royal, 
 10 October 1955„* 
 
 "Homing Antennas for the F-86F Aircraft (450-2500mc)/' Technical Report No. 8, 
 P.E, Mayes. R.F. Hyneman, and R.C. Becker, 20 February 1957, (CONFIDENTIAL). 
 
 "Ground Screen Pattern Range," Technical Memorandum No. 1, Roger R. Trapp, 
 10 July 1955 " 
 
 Contract AF33 (616 ) -3220 
 
 "Effective Permeability of Spheroidal Shells," Technical Report No. 9, E. J. 
 Scott and R H DuHamel, 16 April 1956. 
 
 "An Analytical Study of Spaced Loop ADF Antenna Systems," Technical Report 
 No. 10 , D G Berry and J. B. Kreer, 10 May 1956. 
 
 "A Technique for Controlling the Radiation from Dielectric Rod Waveguides," 
 Technical Report No. 11, J, W. Duncan and R. H. DuHamel, 15 July 1956. 
 
 "Directional Characteristics of a U-Shaped Slot Antenna," Techn ical Report 
 No. 12, Richard C, Becker, 30 September 1956.** 
 
 "impedance of Ferrite Loop Antennas," Technical Report No. 13, V. H. Rumsey 
 and W. L Weeks, 15 October 1956, 
 
 "Closely Spaced Transverse Slots in Rectangular Waveguide," Technical Report 
 No, 14, Richard F. Hyneman, 20 December 1956, 
 
"Distributed Coupling to Surface Wave Antennas," Technical Report No, 15, 
 Ralph Richard Hodges, Jr., 5 January 1957. 
 
 "The Characteristic Impedance of the Fin Antenna of Infinite Length," Technical 
 Report No 16, Robert L Carrel, 15 January 1957. 
 
 "On the Estimation of Ferrite Loop Antenna Impedance," Technical Report No. 17, 
 Walter L. Weeks. 10 April 1957.* 
 
 "A Note Concerning a Mechanical Scanning System for a Flush Mounted Line Source 
 Antenna,' Technical Report No. 18, Walter L. Weeks, 20 April 1957. 
 
 "Broadband Logarithmically Periodic Antenna Structures,' Technical Report No^ 1, 
 R. H, DuHamel and D E. Isbell, 1 May 1957. 
 
 "Frequency Independent Antennas," Technical Report No. 20, V. H, Rumsey, 25 
 October 1957 
 
 "The Equiangular Spiral Antenna," Techn ical Report No. 21, J. D. Dyson, 15 
 September 1957 
 
 "Experimental Investigation of the Conical Spiral Antenna," Technical Report 
 No. 22, R L. Carrel, 25 May 1957.** 
 
 "Coupling between a Parallel Plate Waveguide and a Surface Waveguide," Technical 
 Rgport _No ^_2_3. E, J. Scott, 10 August 1957. 
 
 "Launching Efficiency of Wires and Slots for a Dielectric Rod Waveguide " 
 Tech nical Report No. 24, J. W. Duncan and R. H t DuHamel, August 1957. 
 
 "The Characteristic Impedance of an Infinite Biconical Antenna of Arbitrary 
 Cross Section, Technical Report No. 25, Robert L Carrel, August 1957. 
 
 "Cavity-Backed Slot Antennas," Techn ical Report No. 26, R. J. Tector, 30 
 October 1957 
 
 "Coupled Waveguide Excitation of Traveling Wave Slot Antennas," Technical 
 Report No, 27, W, L Weeks, 1 December 1957. 
 
 "Phase Velocities in Rectangular Waveguide Partially Filled with Dielectric," 
 Tec hnical Report NJ o 1 _28 J W. L. Weeks, 20 December 1957. 
 
 "Measuring the Capacitance per Unit Length of Biconical Structures of Arbitrary 
 Cross Secti I'chni cal Report No. 29, J„ D. Dyson, 10 January 1958. 
 
 "Non-Planar Logan I hrr.ically Periodic Antenna Structure," Tech nical Rep ort N^__3J 
 D W Isbell. 20 February 1958 
 
 "Electromagnetic Fields in Rectangular Slots," Technical Report No. 31 , N. J. 
 Kuhn and P E Masl 10 March 1958 
 
 The Efficiency of Excitation of a Surface Wave on a Dielectric Cylinder," 
 Tech no . ■ Reporl J W. Duncan, 25 May 1958, 
 
"A Unidirectional Equiangular Spiral Antenna/' Technical Report No. 33, J. D. 
 Dyson, 10 July 1958 
 
 "Dielectric Coated Spheroidal Radiators," Technical Report No. 34 W. L Weeks 
 12 September 1958. 
 
 "A Theoretical Study of the Equiangular Spiral Antenna/' Technical Report 
 No. 35, P E. Mast, 12 September 1958. ' ~~ 
 
 Contract AF33 (616) -6079 
 
 "Use of Coupled Waveguides in a Traveling Wave Scanning Antenna/' Technical 
 Report No 36, R H. MacPhie, 30 April 1959. 
 
 "On the Solution of a Class of Wiener-Hopf Integral Equations in Finite and 
 Infinite Ranges," Technical Report No. 37, Raj Mittra, 15 May 1959. 
 
 "Prolate Spheroidal Wave Functions for Electromagnetic Theory/' Technical 
 Report No. 38, W. L, Weeks, 5 June 1959. 
 
 "Log Periodic Dipole Arrays," Technical Rep ort No. 39, D,E„ Isbell, 1 June 1959, 
 
 "A Study of the Coma-Corrected Zoned Mirror by Diffraction Theory " Technical 
 Report No. 40, S. Dasgupta and Y. T. Lo, 17 July 1959, 
 
 "The Radiation Pattern of a Dipole on a Finite Dielectric Sheet," Technical 
 Report No, 41 , K, G, Balmain, 1 August 1959, 
 
 "The Finite Range Wiener-Hopf Integral Equation and a Boundary Value Problem 
 in a Waveguide," T echnical Report No. 42 , Raj Mittra, 1 October 1959. 
 
 "impedance Properties of Complementary Mul titerminal Planar Structures," 
 Technical Report No, 43, G. A, Deschamps, 11 November 1959. 
 
 "On the Synthesis of Strip Sources," Technical Repor t No, 44, Raj Mittra, 
 4 December 1959. 
 
 "Numerical Analysis of the Eigenvalue Problem of Waves in Cylindrical Waveguides,' 
 Tech nical Report Jjo ^_45 , C. H. Tang and Y. T. Lo, 11 March 1960. 
 
 "New Circularly Polarized Frequency Independent Antennas With Conical Beam or 
 Omnidirectional Patterns,'' Technical Report No. 46, J.D c Dyson and P.E, Mayes, 
 20 June 1960. 
 
 "Logarithmically Periodic Resonant-V Arrays," Technical Report No. 47, P,E = 
 Mayes and R, L, Carrel, 15 July 1960. 
 
 ~ Copies" available for a three week loan period. 
 ** Copies no longer available, 
 
AF 33(616) -6079 
 
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