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Unformatted text preview: ' I '. ﬁeld in the planeat distancef 413 from the lens is given by 1 .ngm 22' ' [Hawa?!) “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 quadraticphase 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 quadraticphase 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 quadraticphase 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 52. 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 [jmmexprazwmg 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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 Spring '08
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