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Unformatted text preview: 35. Using Eq. (3433) we have the following: (3.) Circular aperture of diameter d: A(;Â§:o)=s{m(?Â§)} using the similarity theorem for FourierBessel transforms (Eq. (234)) and the Fourier
Bessel transform pair of Eq. (2â€”35), fx=afl
ry=ï¬ï¬‚ ,3 (sham dJ(7rpd)
AGr)â€œ (â€œZL'lp ' 4 1:1 2
2
Finally, note that p = \/ fxz + f; = (ï¬r + (g) yielding (21;!) aâ€œ ( em (was) an =1â€ .._=
A( '0) (%)2+(f%)2 Xâ€˜ If 4 Â£22 5
(b) A circular opaque disk of diameter d can be modeled by the following amplitude transÂ»
mittance function: _ 2r
tA(x,y)â€”1â€”c1rc(g). Born the linearity theorem of Fourier analysis it follows that the angular spectrum of this
structure is pm}: km 4.4 OUWaMlottslwce Â£19 Frayed Prefagï¬llfï¬ Can A: dqeï¬Ã©q} by
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)8 < M. )SMC (7:3?) cog (Kiâ€”2r? 1(an considered 48. (a) The amplitude transmittance function is separable and each factor can be
separately; i.e. â€(531) = txï¬) ty (1')), where txlÃ©) = met (.5?) â€˜3 5(5) __= rect (%) _ 1 l a _n_
ty (:3) â€” [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) = Emï¬m â€” with), we have A
tym) = [teat (:37) Â® 2 Mn â€” 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 ï¬nd the Fâ€˜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 onedimensional transform on
each of the factors: Fï¬xfÃ©l} = Xï¬indxfx)
FHYWJ} = [Ysinc[fo) x Acomb(Afy)] c8 NASinCLVÃ©fy) i Ysinc (may) 6 (fy â€” Â£3) a Nasincwnfy) = NAY i sinc(mTY)smc[Na(f â€”Â§)]. m=â€”oo The full expression for the intensity distribution in the Fraunhofer diï¬'raction pattern
becomes: 2 co â€˜ 2
May) = (NAAZXY) Z sinC(1naâ€”Y)Sinc (if?) Sim [9â€˜sz (y _ mfg)â€  m=â€”oÂ¢ The sum appearing above can be viewed (considering only the y direction) as a sum of
terms of the form sinc (Â£3) sinc [LEâ€”3â€˜ (y â€” In?â€ , each with a weighting factor sinc (Hgâ€”Y).
We wish to ï¬nd 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 â€œzero order" remains at its maximum possible value. 49. The ampiitude transmittance of this aperture is given by a:
mung) = rent (â€”) rect (i) â€” rest (3) root (1) .
1â€.) we ILL'1'. w}; The Fourier transform of this transmittance function is
F{t,q{$, y)} = w: 51:14wa ) sinc{wofy) â€” w: sinciwzfx ) Sillc(wâ€œfy). It follows that the Fraunhofer diffraction pattern of this aperture is 411:3 2 , 1', Euros: . 2 21.901:
I(:i:,y) â€” (Az) smc ( Az )smc ( A: )
Await, 2 , 211mm _ 21991; _ 2min: _ Zwiy
â€” 2( Az )51nc( Az)smc( Az)smc(Mf)smo()m)
,2 2 . .
+ 4â€”1121. sine: 2101.2" 511162 2mm .
A2 2 Az ...
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