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4500HWD04

# 4500HWD04 - Diffraction by an Array of Apertures An...

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Unformatted text preview: Diffraction by an Array of Apertures An inﬁnite rectangular array of apertures is illuminated at normal incidence by a plane wave of freespace wavelength A. Each aperture has a transmittance of 100% and a diameter of d. The apertures are in a rectangular array with a center—to—center spacing of A35 in the a: direction and a center—to—center spacing of Ag in the y direction as shown in the drawing. At all locations outside of the apertures, the transmittance is zero. J_ Av ...dT 1 To 000 70000 ioooo oooo Calculate, showing all work, the radiant intensity of the far—ﬁeld diffraction pattern of the above array of apertures. Express your answer as a function of Am, Ay, d, A, 0m, By, and I0 only where 01 is the far-ﬁeld diffraction angle measured from the normal to the surface of the array in the a: direction, 6,, is the correSponding far—ﬁeld diffraction angle measured in the y direction, and I0 is the far-field radiant intensity at 0x = 6y 2 0. Sim— plify your answer as much as possible and put your final answer in the space provided. I(03,0y) = Diffraction by an Array of Apertures The transmittance is t 2 (2—D array of delta functions) * (circular aperture transmittance) Where “ * ” represents convolution. The 2—D array of delta functions itself can be repre- sented by a convolution. Thus The far—ﬁeld diffraction pattern (in amplitude) is the Fourier transform of the transmit- tance. Thus H +00 +00 f[ 2 6<m-mAx>* Z 5(y—nAy)]'ﬂtc(T/a)] m=—00 nz—oo F(jkx1jky) +00 +00 Puma) = [\$2 6<x—mAx>-IZ 6<y—nAy>]-ﬂtc<r/a)1 +00 +00 F(jkx,jky) : f: 6(m—mAx)-.7:Z 6(y—nAy)-F[tc(r/a)] FM .k) __ i *2” W _i)._1_ *2” W _i)_A0J1(7rdsm0/,\) J “J y ”‘ Amp“) x AI Ayizm y Ay (7rd sinQ/A) “there A0 is a constant and since kg + k3 2 kg and f3 + f; 2 f3 then sinzd = 527120,; + sinzﬁy. F(.k .k”) _ A_1_i +2” W __z'__)_ {5 6(f__i_).J1(7rdsin6//\) 7 ’6’] y ‘ OAx Ay 1.2m ”“ Am 1:00 y Ay (7rd sinH/A) 1 1 +°° sinO 2‘ +°° sinél i ‘ J1(7rd sinO/A) F.z. I __ at“, y___._____ 0'“ ’Jky) AOAxAy Z a A Am) 2 M A Ag) (wdsinH/A) i=——00 iz—oo The radiant intensity is Haney) oc |F(ﬂ<:g,.-,jky)l2 and so 2 ’ +00 . . +00 . . . 3m 6w 2 3m 9y 2 J1(7rd sm B/A) 12—00 l=—(X) ...
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4500HWD04 - Diffraction by an Array of Apertures An...

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