Zernike polynomials. Frederik Zernike, 1953 Nobel prize
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1 erike poloials Frederik erike, 95 Nobel prize
2 eider (,, ϕ Thierr Lépie - Optical desig... cos cos cos cos ϕ ϕ ϕ ϕ S
3 Advatages erike poloials are of great iterest i a fields : optical desig optical etrolog adaptive optics ophtalolog (coreal topograph, ocular aberroetr freefor optics For a circular pupil, erike poloials for a orthooral basis. Hece foralis is easier Set of basis shapes or topographies of a surface eal surface (wavefrot, irror is costructed fro liear cobiatio of basis shapes or odes Thierr Lépie - Optical desig
4 Disadvatages There are soe optical sstes with ocircular pupils : Telescopes : aular ad hexagoal pupils Lasers : squared pupils I these cases, there exist specific orthooral poloials other tha the erike oes Thierr Lépie - Optical desig
5 Defiitios ( ( ( if si ϕ Noralizatio costat adial part Aziuthal part arig : VEY CONFUSING - what axis is the referece axis? x, - [frige, stadard, Noll, go, at, Bor & olf ] orderig - oralizatio : zero-to-peak (-P, MS - uits : waves, icroeters ( ( ( ( ( ( ( ( ( ( eve et w,!!!! are defied such that :,,, The poloials if cos if si, ith k k k k N k k k < ϕ ϕ δ ϕ 5 Thierr Lépie - Optical desig
6 Cosequeces ( is a ( ( ( ( poloiali of degree ad is aught if > or ( - ueve. erike poloials are orthooral if π π (, ϕ (, ϕ ddϕ δ δ ad ol if : All wavefrots errors (,, ϕ c ( (, ϕ, ca the be expressed as a su of the More about field depedece of c, see for istace : obert. Gra, Christia Du, Kevi P. Thopso, ad Jaick P. ollad, "A aaltic expressio for the field depedece of erike poloials i rotatioall setric optical sstes," Opt. Express, ( Thierr Lépie - Optical desig : 6
7 This leads to ver iterestig properties: The coefficiet associated with the pisto is equal to the ea value of the aberratio fuctio : π ddϕ c π The quadratic su of the coefficiets, except that of the pisto, is equal to the variace of the aberratio fuctio : π π d d ϕ σ ( c (, (, Each coefficiet, except that of the pisto, is equal to the MS value of the associated aberratio The value of a coefficiet does ot deped o the uber of poloials used i the developeet : c π π ddϕ Thierr Lépie - Optical desig 7
8 The ost iportat! Each ter cotais the appropriate aout of lower order ters i order to iiize the MS wavefrot error of that ter. So erike poloials represet balaced classical aberratios Usig erike poloials, the evaluatio of the Strehl ratio is sipler (see further i this course Ma exaples will be studied i classwork Thierr Lépie - Optical desig 8
9 The first erike poloials poloial ae Notatio fro go, EOSC* pisto si ϕ cos ϕ ( 6 si ϕ 6 cos ϕ si cos tilt X - tilt Y defocus astigatis - astigatis ( ϕ ( ϕ coa X - coa Y ( spherical aberratio ( ( This lecture go,... * EOSC Thierr Lépie - Optical desig 9
10 The first erike poloials (fro Thierr Lépie - Optical desig
11 The first erike poloials (tilt X, c (defocus, c (astigatis, c (coa Y, c (spherical aberratio, c Thierr Lépie - Optical desig
12 rd order aberratios : best focus et toleraces e saw that : Ad we kow that : ( (,, ( c σ S σ π So : S σ (... (,, ( c c c c S π π Thierr Lépie - Optical desig
13 rd order aberratios : best focus et toleraces At the begiig, the referece sphere is cetered o the paraxial iage. The Strehl ratio depeds o the tilts et defocus coefficiets : c, c c, e ca chage the referece sphere, applig the opposite previous tilts et defocus to the ceter of the referece sphere ith respect to this ew referece sphere, tilts ad defocus have bee caceled : the Strehl ratio is axiized, ad the ceter of this ew sphere is the best focus! Thierr Lépie - Optical desig
14 rd order aberratios : best focus et toleraces Hece for the best focus : S π c (, (,,(,,(,,(, Thierr Lépie - Optical desig
15 Case of the rd order spherical aberratio eider : ith the Seidel foralis, we could deterie the positio of the best focus, usig rather tedious calculatios Thierr Lépie - Optical desig 5
16 eider : Seidel foralis defocus: e eed to itroduice a est iiized. the variace of axiized, ie. ratio is the Strehl At the best focus, b a d d b a P z θ π σ ε σ π focii! et argial halfwa betwee the paraxial The best focus is! so :, Hece:, ie. iiized, is for which e are lookig for the defocus (b LA a b b b ab a b ab a d d P z P z ε ε σ σ σ θ π π 6 Thierr Lépie - Optical desig
17 et ( eider : Seidel foralis Let us tr to evaluate the axiu aberratio fuctio values for each case: paraxial BF paraxial ax ( ( MF paraxial ax he BF is axiized: ( BF ax For the best focus, the axiu wavefrot error is ties saller the it is for the paraxial focus, ad the wavefrot is retarded, with respect to the referece sphere. e ote also that the aberratio fuctio for the best focus is aught o the axis (obvious ad at the edge of the pupil :.8 paraxial best focus Thierr Lépie - Optical desig
18 e easil get : c Sae case with the erikes If the referece sphere is cetered o the paraxial iage, we kow that : e write as a su of the, c : At the best focus, we "cacel"the coefficiet c MS wavefrot error Applicatio : S π 6 Let us suppose that is equal to 5 c c at the edge of ( c, hece: c.so, the pupil (, ie. π ( c c,78 et S ( c, 99 is the copesated spherical aberratio (we itroduced the appropriate aout of defocus to iiize the paraxial focus best focus. Hece: Thierr Lépie - Optical desig 8
19 π - paraxial focus: π - best focus: Applicatio : ( c c Applicatio : e evaluate the tolerace at the focii, assuig that S,8. e get : Hece: ε,8 ( c,8,95 LA e evaluate the positio of the best focus. For this, we eed to itroduce a defocus ε z P z P, to copesate the ter c. Thierr Lépie - Optical desig 9
20 e write as a su of the e get : c At the best focus, we "cacel"the coefficiet c Applicatio :, c e evaluate the tolerace at the focii, assuig that : S π - paraxial focus: π - best focus: c Applicatio : : 8 ( c c,8, Coa If the referece sphere is cetered o the paraxial iage, we have:, hece:. The, (we itroduced the appropriate aout of tilt to iiize the MS wavefrot error So : ε ( e evaluate the positio of This tilt is equal to the third of For this, we eed to itroduce a tilt ε the greatest circle of coa ( c,8 eider : et are diesioless! P the best focus. c,6 ( c c,8. e get : P siϕ to copesate the ter c cosϕ the distace betwee the paraxial iage ad the ceter of Thierr Lépie - Optical desig is the copesated coa.
21 If the referece sphere is cetered o the paraxial iage, e write as a su of the The : c At the best focus, we "cacel" the ter c Applicatio :, c : e evaluate the tolerace at the focii, assuig that : S 6 c Astigatis, the :. Hece, (we itroduced the appropriate aout of defocus to iiize the MS wavefrot error π - paraxial focus : π - best focus : c Applicatio : ( e evaluate the positio of This defocus is equal to half ( c c,8 the best focus. P For this, we eed to itroduce a defocus ε z ε z P c,8, ( c c we have: the distace betwee the focii.,8,8. e get : to copesate the ter c cos ϕ is the copesated astigatis. Hece: Thierr Lépie - Optical desig
22 Field curvature For the referece sphere cetered o the paraxial iage, we have: e write as a su of the S e get : c The field curvature ISa defocus, depedig o the field! At the best focus, there is o aberratio! Applicatio : ( c π,8 Applicatio : here is the best focus? : alread deostrated previousl! c,5 P For this, we itroduce a defocus ε z ε z S ( c e evaluate the tolerace for the paraxial focus, P S assuig that : S to copesate the ter c Thierr Lépie - Optical desig S,8. e get :. Hece:
23 Distorsio For the referece sphere cetered o the paraxial iage, we kow that : e write as a su of the e easil get : c The distorsio IS a tilt, depedig o the field! At the best focus, there is o aberratio! Applicatio : ( c π,8 Applicatio : hat is the positio of To do this, we itroduce a tilt ε : alread deostrated previousl! c e evaluate the tolerace at the paraxial focus, assuig that : ε P, the best focus? P siϕ to copesate the ter c S Thierr Lépie - Optical desig,8. e get :. Hece: cosϕ
24 Toleraces : suar rd order aberratio Paraxial focus Best focus Spherical aberratio,,95 Coa,,6 Astigatis,8, Field curvature,5 Distorsio, Thierr Lépie - Optical desig
25 erike poloials for rd ad 5 th orders poloial ae pisto si ϕ tilt X - cos ϕ tilt Y ( defocus 6 si ϕ rd order astigatis - 6 cos ϕϕ rd order astigatis 5 ( ϕ 8 si rd order coa X ( ϕ - 8 cos rd order coa Y ( rd order spherical aberratio ( ϕ si 5 th order astigatis ( ϕ - cos 5 th order astigatis 5 ( ϕ ( ϕ 5 5 si 5 th order coa X 5-5 cos 5 th order coa Y ( th order spherical aberratio Thierr Lépie - Optical desig 5
26 To go further! Thierr Lépie - Optical desig 6
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