Chapter 3 Optical Systems with Annular Pupils

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Clifford Freeman
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1 Chapte 3 Optical Systems with Annula Pupils 3 INTRODUCTION In this chapte, we discuss the imaging popeties of a system with an annula pupil in a manne simila to those fo a system with a cicula pupil The two-mio astonomical telescopes discussed in Chapte 6 of Pat I ae a typical example of an imaging system with an annula pupil The linea obscuation atios of some of the wellknown telescopes ae 36 fo the -inch telescope at Mount Paloma, 37 fo the 84- inch telescope at the Kitt-Peak obsevatoy, 5 fo the telescope at the McDonald Obsevatoy, and 33 fo the Hubble Space Telescope Expessions fo the PSF, OTF, and encicled, ensquaed, and excluded powes ae given The Stehl atio of an abeated system is consideed and toleances fo pimay abeations ae discussed Symmety popeties of abeated PSFs ae discussed, and pictues of the PSFs fo pimay abeations ae given as examples The line of sight of an abeated system is discussed in tems of the centoid of its PSF Numeical esults ae given and compaed with the coesponding esults fo systems with cicula pupils wheeve possible and appopiate 3 ABERRATION-FREE SYSTEM We stat this chapte with a discussion of the PSF, encicled, ensquaed, and excluded powes, and the OTF of an abeation-fee system Equations ae developed in a way that the esults fo a cicula pupil can be obtained as a limiting case of the annula pupil It is shown that the obscuation in an annula pupil not only blocks the light incident on it, but it also educes the size of the cental disc and inceases the value of the seconday maxima of the PSF It also inceases the OTF value at high spatial fequencies while educing it at the low fequencies 3 Point-Spead Function Conside, as illustated in Figue 3-, an abeation-fee optical system imaging a point object with a unifomly illuminated annula exit pupil having oute and inne adii of a and a, espectively, whee is the linea obscuation atio of the pupil The iadiance at a point i in the image plane with espect to the Gaussian image point is given by Eq (-65), that is whee [ ] Û ı Ù - p i Ii( i; ) = Ii ( ; ) Sex ( ) exp Ê R p ˆ Á i d p, Ë l S ex p ( ) a = - (3-) (3-) 67

2 68 OPTICAL SYSTEMS WITH ANNULAR PUPILS ExP Defocused image plane a a Gaussian image plane z R Figue 3- Imaging by a system with an annula exit pupil of inne and oute adii a and a, espectively is the clea aea of the obscued exit pupil The quantity in Eq (3-) is sometimes efeed to as the aea obscuation atio The optical wavefont at the exit pupil is spheical with a adius of cuvatue R and cente of cuvatue at the Gaussian image point The cental iadiance is given by i = ex ex l (3-3a) I ; P S R = ppex - 4 l F (3-3b) whee P ex is the total powe in the exit pupil, and, theefoe, in the image Fo an object of intensity B o adiating at a wavelength l at a distance z fom the entance pupil of aea S en, the total powe is given by [ ] P h S z B, (3-4) = ex en o whee h is the tansmission facto of the system fo light popagation fom its entance to its exit pupil The quantity F in Eq (3-3b) is given by F = R D, (3-5) whee D = a is the oute diamete of the exit pupil It epesents the focal atio (f-numbe) of the image-foming light cone exiting fom the exit pupil The integation in Eq (3-) is caied ove the clea aea of the exit pupil such that the position vecto p of a point in its plane satisfies a a p As in Section, we expess the position vectos of points in the pupil and image planes in pola coodinates accoding to = ( cos q, sin q ), a a, q < p, (3-6) p p p p p p

3 3 Abeation-Fee System 69 and = ( cos q, sin q ), q < p (3-7) i i i i i Substituting Eqs (3-6) and (3-7) into Eq (3-), we obtain a È i Ii i i Ii Sex R p (, q ; ) = ( ; ) ( ) exp Í i cos qp qi p d p d qp Î l a p [ ] Û ı Ù Û ı Ù - p ( - ) (3-8) Compaing Eqs (-7) and (3-8), we note that the significant diffeence between the two lies in the lowe limit of the integation ove p ; in Eq (-7), the lowe limit is, indicating an unobscued pupil; in Eq (3-8) it is, indicating an obscued pupil The values of S ex ae diffeent in the two equations by a facto of - The values of P ex would also be diffeent by this facto if the pupil iadiance wee the same in both cases Fo simplicity of equations as well as numeical analysis, we use nomalized quantities = a p = ( q q ) (3-9a) cos p, sin p, (3-9b) = l F (3-a) i = i i cos q, sin q, (3-b) and = [ ] I; I ; PS l R (3-) i i ex ex Note that in Eq (3-), we have nomalized the iadiance by the cental iadiance fo a system with a cicula pupil Using nomalized quantities, Eq (3-8) may be witten = p - - Û [ ] p I, qi; Ù Û exp i cos qp qi ddqp ı ı Ù -p - (3-) Integating ove q p by using Eq (-), we obtain [ ] È I ( ; ) = [ ( - )] Û Í J d ı Ù 4 Í ( p ) (3-3) ÎÍ Caying out the adial integation by using Eq (-4), we finally obtain

4 7 OPTICAL SYSTEMS WITH ANNULAR PUPILS ÈJ p J p I ( ; ) = - Í - Î p p (3-4) We note that the iadiance distibution is adially symmetic about the Gaussian image point =, as may be expected fo a adially symmetic (annula) pupil function It is not nomalized to unity at the cente Its cental value is given by - Except fo a nomalization facto, Eq (3-4) also gives the PSF of the system It follows fom Eq (- 6) that = (3-5) PSF ; I ; P i i i ex We note that as Æ, Eq (3-4) fo the annula pupil educes to Eq (-5) fo the cicula pupil In ode that the total powe be the same fo the two pupils, the iadiance acoss the annula pupil must be highe than that fo a cicula pupil by a facto of ( - ) - Fo a given total powe P ex in the exit pupil, the cental iadiance I( ;) is smalle by a facto of - compaed to that fo a cicula pupil Howeve, if the pupil iadiance is the same in both cases, as in astonomical obsevations, then P ex ( ) is also smalle than P ex ( ) by a facto of - Hence, I( ;) will be smalle than I( ; ) by a facto of - The pincipal maximum of the image iadiance distibution lies at the Gaussian image point =, since all the Huygens spheical wavelets oiginating at the spheical wavefont in the exit pupil aive in phase at this point and, accodingly, intefee constuctively Fom Eq (3-4), we note that the image iadiance is zeo at those values of fo which = ( p ) π J p J, (3-6) These values of locate the minima of the iadiance distibution Noting Eq (-9), we find that the seconday maxima lie at those values of that satisfy J p J, (3-7) = ( p ) π The iadiance distibution fo a system with a vey thin annulus pupil Æ may be obtained fom Eq (3-3) by noting that ~, and the vaiation of J ( p ) is negligibly small ove the vaiation of that it can be eplaced by J ( p ) Hence, Eq (3-3) educes to = I;Æ J p (3-8) when nomalized by the cental iadiance P S ex l R ex

5 3 Abeation-Fee System 7 3 Encicled Powe The amount of powe in the image plane contained in a cicle of adius c centeed at the Gaussian image point is given by c Û P i( c; ) = p Ù Ii( i; ) d i i (3-9) ı Substituting Eqs (3-a) and (3-) into Eq (3-9) and defining a nomalized o factional encicled powe = (3-) P; P ; P, we obtain c i c ex = p P ; I ; d c c l F Û ı Ù (3-) If we let c be in units of lf and substitute Eq (3-4) into Eq (3-), we obtain whee È P P P J u J u du ( c; ) = Í Û ( c ) + ( c ) - Ù ( p c ) ( p c ) 4, - Í (3-) ı u ÎÍ = - ( p ) - ( p ) P J J (3-3) c c c is the encicled powe fo a system with a cicula exit pupil 33 Ensquaed Powe The ensquaed powe in a squae egion of half-width s centeed on the Gaussian image point in the image plane is given by P ; I ; dd, i = Û ı ÙÛ ı Ù q (3-4) s s i i i i i whee the integation is caied ove the squae egion Following the same pocedue as in the case of cicula pupils (see Section 3), Eq (3-4) educes to Û ı Ù [ p ] 8 - P P J u J u u du s( s; ) = c( s; ) - ( s ) - ( p s ) cos, p - u (3-5) whee

6 7 OPTICAL SYSTEMS WITH ANNULAR PUPILS and = (3-6) P ; P P s s i s ex P ( ) = P( ) (3-7) c c c ae the factional ensquaed and encicled powes, espectively, and s and c ae in units of lf The fist tem on the ight-hand side of Eq (3-5) epesents the image powe contained in a cicle of adius s The second tem gives the image powe contained in a egion lying between a cicle of adius s and a squae of full-width s Both of these tems equie numeical integation 34 Excluded Powe The excluded image powe X i, ie, the powe contained outside a cetain aea in the image plane can be calculated quite accuately in closed fom if the included aea is lage enough so that Xi Pex Fo lage aguments we can use the asymptotic expession fo Bessel functions; namely, Eq (-33) Thus, fo lage values of, Eq (3-4) can be witten [ ] 8 I ( ; ) = sin p ( -4) - sin p ( -4) p (3-8) Noting that the aveage of a sine squae is half and the aveage of the poduct of two sines with diffeent aguments is zeo, the aveage iadiance (indicated by a ba) fo lage values of may be witten 4 I ( ; ) ~ 4 3 (3-9) p - Hence, the excluded encicled powe is given by Xc( c; ) ~ ( p ) Û ı Ù I( ; ) d c = p c -, (3-3) and the excluded ensquaed powe is given by Xss; ~ Û Û ( ) Ù dx dyi ; ı ı x > s y > s Ù 4 = p 3 s -, (3-3)

7 3 Abeation-Fee System 73 whee = x + y and the subscipts c and s on X indicate a lage cicula o a squae egion of exclusion centeed on the Gaussian image point Fom Eqs (3-3) and (3-3), we note that = X ; 9 X ; (3-3) s c c c The facto of 9 between X s and X c is independent of We also note that excluded powe fo an annula pupil is lage by a facto of ( - ) - compaed to that fo a coesponding cicula pupil The appoximate esult of Eq (3-9) and those that follow fom it, although obtained fo the abeation-fee case, ae valid even when abeations ae pesent in the system This may be seen by substituting Eq (3-4) given late into Eq (-54) and consideing the nomalizations used in Eq (3-9) It should be noted, howeve, that the value of fo which Eq (3-9) is valid inceases as abeations ae intoduced into the system 35 Numeical Results Figue 3- shows the iadiance distibution and encicled powe fo seveal values of including zeo The iadiance is nomalized to unity at the cente in Figue 3-b The values of fo the fist seveal minima ae given in Table 3- fo = 9 ( ) We note that the adius of the cental bight disc (fist dak ing) deceases monotonically as inceases As Æ, this adius appoaches a value of 76 [fist zeo of J ( p )] compaed to a value of [fist zeo of J ( p )] when = Moeove, the seconday maxima become highe as inceases Fo example, when = 5, the fist seconday maximum has a value of 963% of the pincipal maximum compaed to a value of 75% fo a cicula pupil The adius of the second dak ing fist inceases with, achieves its maximum value fo = 4, and then deceases The adius of the thid dak ing fist deceases, then inceases, achieving its maximum value fo = 5, and then deceases again The adius of the fouth dak ing fist inceases, then deceases, inceases again, and finally deceases as inceases Figue 3-3 shows the iadiance distibution given by Eq (3-4) fo =, 5, 5, and 75 nomalized to unity at the cente This figue and Table 3- also show how Pc, Ps and Ps - Pc vay with c in gaphical and tabulated foms The value of P c in a given dak ing deceases o inceases with in a manne simila to how its adius vaies, although a peak o a valley in its vaiation is not achieved fo the same value of Fo small values of, the fist maximum of Ps - Pc is the highest Howeve, fo lage values of one of the seconday maxima is the highest As inceases, the seconday maxima of iadiance become inceasingly moe significant, and theefoe, Eqs (3-3) and (3-3) give accuate esults fo inceasingly lage values of c and s, espectively Fo example, if the diffeence between actual esults (given in Table 3-) and those obtained fom Figue 3- is to be less than 6% of the total powe, c must be lage than 7, 8, and 38 when is equal to 5, 5, and 75, espectively A geneal ule of thumb is

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