Sol_7 - I field in the plane-at distancef 413 from the...

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Unformatted text preview: ' I- '. field in the plane-at distancef 413 from the lens is given by 1 .ngm 22' ' [Haw-a?!) “4 if. f/PCILHyNfik' ; Ejki‘fr‘g eh'fi's‘f‘m «Midy- '- - _- - '-’Fhe._first_'glmrdratic phase factor in the integrand representfi the effect of the 1931s, while flhé: _ gemnd M1395 from the Fresnel. diffraction kernel. In Order for the diffraction pattefn to be: a};1:)r<',)'xi1n:-xtely Fraunhofer, we want the total quadratic-phase exponential factor to vanish. T1115 requires 'k r . 1 1 [9W *‘ '7” (7:3 * Q] i I 5—1 or, 933‘. 2.. ,zfiwéww '23 H’)f(f—~A) l radian. RIFLE In the worst case, 2:2 +1}? -+ (D2)? Also, assuming A is 51115111 compared tn f, f fl A «i f in the denm‘ninator. Thus, TTDJA NU? €121 C) 1' 4A;2 A (453 " 7NDa 541. Fourier planes win he found at the {allowing locatimns: s In the. plane where thaa illumirmtion beam mmes to firms; ixe. distance f to the right of the Objerzt. . u - n n 4' n a In the plane where the, abtwe Fnuner Diana 15' unaged. by the lens. Accordmg t0 the lens law, this will be at distance 2f to the: right of the lens. as: There will be only one image plane? namely the plane where the-{($118 law is satisfied for an object _3 f in front of the lens. We have ' ' v _ - -' _ .- 1 . 1 from whiéh it £0115va that, to. the right of the lens, 5m 13. . (a) 03) Expand the amplitude transmittance as follnws: ' 1 1.-,. . tfiflt(i+ffifl+ le‘jmfjra £1 . 3 and compare the snmnd and third term with the amplitude transmittance Of a lens with foca} length f: - - . . .,.2 no») a 5%? \Ne seen that; the second and third terms of this transmittannn functinn are of the same form as the transmittance function of a lens. Thus the structure lmhnvns simultaneously as two different 1011885, one positive and one negative, in addition to having a bias term that only nttnxmates the incident wavefront. If '15 1w5itive, the first qudratic~phase term in 331 can be interpreted as a negative lens with focal Iongth A: f:»; Ar while the muond quadratic-phase term can be interpreted an: a positive lens with fncaril length ' k: f3;- The focal: lengths given by the above twd equations are both functions of wavelength, since k := 27r/A. Therefme if the object has any significant spectral spread, the image wili e'xperienae severe degrmlation. 544. The circular bounding aperture will not; affect the problem, so we ignore it. From the (.lefinition' provided by Fig. P514, it is clear that the following is true: ' +“1,T( a T2 2 25%,11001‘37 ) m2) 2 Z [WWW/2)] exp (jaw-'2), M (7‘) H If 77” n:- - 00 where we have used the fact that the period X must he replacedby I -2? T. X. Noting that qtladmticqvhase structures can be. interpreted as being aquivalent to lenses, we 55% that the structure is equivalent to an infinite. number of positive and negative lenses of different focal lengths, pluS 2:1 bias term. Comparing; these terms with the amplitude transmittance of a _ _' m ~er . . . . lens, Mir) z: 6. M3“, the focal length of the nth term in the sewers ls glean to be h I i“. fn 2m] . a, where. the positive Sign is used for all terms having; a negative quadraticnpham factor and the negative. Sign is used for those with a positive quadratic-phase factor. The relative amount _ of optical power contributing to the nth term is the squared magnitude of the correspmxding '_ w Fourier coefiicié‘nt in the GKD‘dl'lSiQfl with respect to T2, Thus for the nth term the fraction of power contributing is - . 2 sm(¢m/2) an :2 WW . 7TH 5-2. The physical quantities to follow are amplitudes in the case of a coherent system and intensities in the case of an incoherent system. p(2:, :1) represents the (amplitude or intensity) pbintnspread fUI'iCfiOl’I. I (a) A line excitation lying along the .1: axis would he l'epmsentsxi by 002,31) 2 6hr)- The response to such an excitation would be “ii(m,y) 't pwaymrfiway} :mmmw) mama * mdédn = may) d5 : My) 1' E _' (b) C(hnsider a'onewdinwnSidnal Fourier transfnrm of the linedpread function: my» 2 [jmmexpra-zwmg 2 f / pmmxpiéjwaax + W] @542; 1W; x mm . Pym—#1 __OC, _ (c) The unit step function wilI be répregcnted by _ 0 y < 0 Therefore the response of the systemwill he' %-[.K£m%/_:1£5W53flfi f" m i2(m1 1 19(16: y) 3LT! _ " . ~00 rElms step response = /' Kayla. . “‘00 ' ...
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This note was uploaded on 06/04/2010 for the course EE 5621 taught by Professor Staff during the Spring '08 term at Minnesota.

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Sol_7 - I field in the plane-at distancef 413 from the...

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