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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 farﬁeld 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 farfield 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<mmAx>* 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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 Spring '08
 Gaylord

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