Minimizing spherical aberrations Exploiting the existence of conjugate points in spherical lenses

Transcription

1 Mmzg sphecal abeatos Explotg the exstece of cojugate pots sphecal leses Let s ecall that whe usg asphecal leses, abeato fee magg occus oly fo a couple of, so called, cojugate pots ( ad the fgue below) Oval Fg. 1 s a pefect mage of (ad vce vesa) whe a pope efactve oval suface s used We have sad befoe that, geeal, sphecal efactve suface (ot beg a oval) wll suffe fom abeatos. Howeve, t tus out, Huyges may have bee the fst to dscove that a gve object pot could be pefectly mage also by a sphecal suface!! That s, all the magg ays efactg at the sphecal suface covege at the same (vtual ths case) mage pot. (smla to the case whe usg a oval.) We wll cosde two cases. I the fst (secod) case the object ad vtual mage pots ae sde the medum of hghe (lowe) dex of efacto. ase 1: Imagg fom glass to a ( ad mage mmesed sde the glass.) Let s cosde a efactg sphecal suface of adus. ck up a abtay object pot. The ay BB mages pot at pot (the mage s vtual.)

2 ASE: > B N B Fg. maged though a efactg sphecal suface of adus. But otce that, geeal, ot ecessaly all the ays leavg fom wll be maged at the same pot. Ou objectve s to show that, whe the locato of the pot s popely chose, deed all the ays leavg fom ad efacted at ay pot of the sphecal suface wll poduce a (vtual) mage at the commo mage pot. Fo coveece, let s expess the posto of ad, wth espect to, uts of p b q a B N a B b B B Fg. 3 Scalg the postos of ad tems of the adus.

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4 Expessos (7) ad (8) gve mplctly the values of p ad. Let s obta a moe explct expesso fo p, whch wll allow locatg the mage pot (sce = p) Fom (4) ad (8) S S p 1 pcos( ) q 1 qcos( ) Usg ths esult (7) oe obtas,

5 p q p q 1 pcos( ) 1 q cos( ) (9) Gve q ad (that s, gve the posto of ad B), the value of p (the posto of pot ) s mplct ths fomula. Summay: Fo a gve pot, whose dstace fom ( tems of, s q;.e. q s gve), we wat to fd the locato of ts mage. Fo that pupose, we sed a geec ay B. The agula posto of B, wth espect to the cete, s detemed by the agle. Gve q ad, we calculate usg expesso (4) expesso (9) gves mplctly the value of p, calculate p usg (9); the s obtaed usg expesso (8) Notce, fo the same : dffeet values of (.e. a dffeet pot B) wll gve dffeet values fo p (.e a dffeet pot ). Lookg fo abeato-fee magg We wode f the posto of ca be wsely chose, such that fo ay pot B, the coespodg posto of s the same? That s, ca we choose the value of q wsely eough, such that p ( expesso (9)) becomes depedet of?

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7 Thus, fo a gve efactg sphecal suface of adus, we have foud a patcula posto fo a object pot that wll be pefectly maged at pot. ASE: > N B Implemetato usg a covegg mescus Istead of usg a bulky (ad costly) sphee, we ca use a mescus stead. The stategy s to gd aothe sphecal suface wth cete at ; the adus of ths ew e sphecal teface s ot ctcal.

8 Ie sphecal suface wth cete at Image The e sphecal suface does ot affect the tajectoes of the ays leavg the object ASE: >

9 Assgmet

10 ase : Imagg fom a to glass ( ad mage mmesed sde the a medum) ASE: < B N B We ealze the pocedue s gog to be detcal to the pevous case-1. The choce of q = ( / ) -1 (whch gve the soluto p=( / ) wll ema the same, but ths tme t wll mply that s fathe away fom the cete tha. B N B q p

11 Implemetato usg a dvegg mescus Istead of usg a bulky (ad costly) sphecal les, we ca use a mescus stead. The stategy s to gd aothe sphecal suface wth cete at ; the adus of ths ew oute sphecal teface s ot ctcal. ASE: < N B B q p Sphee wth cete at Ths sphee does ot affect the outgog ays efacted fom the suface of adus

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