Sol_3 - 3-5 Using Eq(3-433 we have the following(3 Circular...

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Unformatted text preview: 3-5. Using Eq. (3-433) we have the following: (3.) Circular aperture of diameter d: A(;§:o)=s{m(?§)} using the similarity theorem for Fourier-Bessel transforms (Eq. (2-34)) and the Fourier- Bessel transform pair of Eq. (2—35), fx=afl ry=fifl ,3 (sham dJ(7rpd) AG-r)“ (“ZL'lp ' 4 1:1 2 2 Finally, note that p = \/ fxz + f; = (fir + (g) yielding (21;!) a“ ( em (was) an =1” .._= A( '0) (%)2+(-f%)2 X‘ If 4 £22 5 (b) A circular opaque disk of diameter d can be modeled by the following amplitude trans» mittance function: _ 2r tA(x,y)—-1—c1rc(g). Born the linearity theorem of Fourier analysis it follows that the angular spectrum of this structure is pm}: km 4.4 OUWaM-lottslwce £19 Frayed Prefagfillffi Can A: dqefiéq} by (the Wsfir WCUncM-fl H Gd». £4: EKP(/Jh2‘gje~np (—1" m gmfiuflzfl Pronaa’rm Over SW‘TR‘ ORA-WW Enéy “:31. Can be PepTE’sCch/ 513 muHiPltcfir‘é’M on: @e Successwe “Hm-Fer ”Eng—{7,415 “(My Nil—W Wan) 1]" WM,- an ) Pagofwmj (The PTMIUC’I) H (£1,413; £.+2L+ ~-+ an) ; :2fo [j k<a1+£L+--'+.2HX¥KL1_§EL/] Clearly; SM“ '2: £fi21+‘”+ ‘2“; PTG?O~3A}I.W over (KG—Ema? E” is Qj Uni/01W? ”('0 Pfifdtamfllw over The SUV” 0+ WedlSWCfl El) %L) 23’ Ma ‘2’] . a] 77L: aperjwrc can lo: 'cfrxcribca’ as 0: Fwd ¥un<+réw {:4 7749 if; péme gamed upward Md a‘awmward {:9 “(We Wonf 4/2. $1M "(74¢ 0513:27. A single +r‘msmfilm6 fume-Um 'inmL 03mm“ ’Ws c/Mév/e-g’nL aper'éwe {is : - * f(xtr I 2 Q .3.“— «. at .- - ‘ 0] and <33: led (TI—J} *édG-‘J'é/EJ‘LJ/Eh'é)? The. 1%ch firm a warm”?! WKMPM'!‘ {Marne wave (M215 :ZMP/r'H/e) {is .' J 80% "2% {umxumflmsmc (Efifim 1% {7 WM Eeéwaflg-mtasgfl file-wee 33(- l ; ' ‘ I n! ldidném‘cm (5 +510” : (z. _ 2- i 6 min) ~ (an! — Hi: YL . ' "—75% 17.1% wyo ‘1 6 )8 < M.- )SMC (7:3?) cog (Ki—2r? 1(an considered 4-8. (a) The amplitude transmittance function is separable and each factor can be separately; i.e. ”(531) = txfi) ty (1')), where txlé) = met (.5?) ‘3 5(5) __= rect (%) _ 1 l a _n_ ty (:3) — [tact (Y) :8 Acomb (13)] met (NA) The behavior of tag (5) is quite clear. The behavior of ty (1}) requires more thought. Since leomb (35) = Emfim —- with), we have A tym) = [teat (:37) ® 2 Mn —- min] rect (Wig) . aced that the width of the rectangles. function subtends a symmetrical Since A > Y, the delta functions are more widely sp The fact that N is odd means that the outer rectangle To find the F‘raunhofer diffraction pattern, we must Fourier transform the amplitude transmittance and evaluate the scaled transform at properly scaled frequencies. Since the amplitude transmittanCe is separable. we can perform one-dimensional transform on each of the factors: Ffixfél} = Xfiindxfx) FHYWJ} = [Ysinc[fo) x Acomb(Afy)] c8 NASinCL-Véfy) i Ysinc (may) 6 (fy — £3) a Nasincwnfy) = NAY i sinc(mTY)smc[Na(f —§)]. m=—oo The full expression for the intensity distribution in the Fraunhofer difi'raction pattern becomes: 2 co ‘ 2 May) = (NAAZXY) Z sinC(1na—Y)Sinc (if?) Sim [9‘sz (y _ mfg)” - m=—o¢ The sum appearing above can be viewed (considering only the y direction) as a sum of terms of the form sinc (£3) sinc [LE—3‘- (y — In?” , each with a weighting factor sinc (Hg—Y). We wish to find conditions under which the weighting factors of the terms for even values of m will be as small as pOSsible. Since the sine function has zero value at integer arguments, this requires QkY/A = integer for each integer k. This will be the case if Y/A is any integer multiple of 1/2. For example, if the slit spacing is twice the slit width. this will be the case. Note that the weighting factor for the m = 0 term is independent of ratio of Y to A. so the strength of the “zero order" remains at its maximum possible value. 4-9. The ampiitude transmittance of this aperture is given by a: mung) = rent (—) rect (i) — rest (3-) root (1) . 1”.) we ILL-'1'. w}; The Fourier transform of this transmittance function is F{t,q{$, y)} = w: 51:14wa ) sinc{wofy) — w: sinci-wzfx ) Sillc(w“fy). It follows that the Fraunhofer diffraction pattern of this aperture is 411:3 2 , 1', Euros: . 2 21.901: I(:i:,y) — (Az) smc ( Az )smc ( A: ) Await-,- 2 , 211mm _ 21991; _ 2min: _ Zwiy — 2( Az )51nc( Az)smc( Az)smc(Mf)smo()m) ,2 2 . . + 4—1121. sine: 2101.2" 511162 2mm . A2 2 Az ...
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