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801 over the range of space bounded by t,he maximum phase gradient, so that then t.he directivity D d of such an array would be width of real space 2 K -&Ti Dd = _ _ _ _ _~ --. (17)The dispersion is t,he ratio of the directivity of the dispelxed pattern to the directivity D of the in-phase pattern AT1 A ( d q / d t ) U ( d q / d x )To demonstrate the effectiveness of the phase perturbation met,hod for dispersion of the reflect.ion lobes, the appropriate calculations were performed for a linear array of 108 discrete elements. The element spacing mas chosen to be 0.2645 wavelengt.h, and the amplitude taper in the aperture was a modified sin x / x distribution with 30-dB sidelobes. A symmetrical phase diytribution was assumed with a maximum slope of = k f i (~/ A ) . The maximum reflection lobe was found to be 14.5 dB below the lobe that would occur if all the elements rvere added in phase. The t.heoretica1 dispersion for t.his array is R = -15.1 dB. (19)The difference bet-xeen the theoretical dispersion of 15.1 dB and t.he calculated value of 14.5 dB can be att.ributed to four effects: I) to the compntational errors in the approsimate integration of the ampliktde funct.ion due to the use of discrete intervals of the same size as the element spacing; 2) to the difference between the theoretically postulated continuous aperture dist.ribnt.ion and the discrete element array used in the calc.dation; 3) to the neglect of the higher order differen6ials in the derivation; and 4), most importantly, to the effect at the edge of t.he array where t.he integration or summation is t,rrmcated. The effect, of this truncation is an increase in the reflection-lobe level in the directions corresponding to the phase slope at the edge of t,he array. By means of calculations involving the Fresnel integral, it. can be shown that the t.runcation can lead to an increase in the reflection-lobe level of I .4 dB at. the edge of the dispersed radiation pattern. For the sake of comparison, the reflect.ion-lobe level =-as also calculated for a random phase distribution with a nlasimum phase slope of Y%;A.As aoald be expected, the sidelobe level within the dispersion region of u space varied both above and below (from -12 to -19 dB) the nearly uniform distribution generated by the optimum design. The calculations for the linear array can readily be extended to rectangular arrays with separable aperture distribut,ions. The optimum separable phase perturbation @(x,y), m-here W a ) = @Ar)@dd ("0) is given bswhere d , and A , are the separable components of t,he aperture dist.ribution.The theoretical limit on the dispersion R follows from (13) as [EEE TRdNS1CTIOXS ON ;\KTEhTAS AND PROPAGATION, SOVEXBER 1970 the dispersion t.hen can be nTit.ten as a 4a 477Eqtmtion (25) provides a simple means of estimat.ing the dispelsion that can be obt.ained in a planar array, and (21) and (22) provide a direct. method of calculating the desired aperture phase pert,urbat.ion. ACXKOWLEDGMEKTThe aut.hor would like to acknowledge the cont,ribution of A. E.Absfraci-...