Seidel sums and a p p l p icat a ion o s f or o r s imp m l p e e c as a es

Transcription

1 Seidel sums ad applicatios for simple cases

2 Aspheric surface Geerally : o spherical rotatioally symmetric surfaces but ca be off-axis coic sectios Greatly help to improve performace, ad reduce the umber of compoets (hece less weight)

3 Aspheric surface : coic I I h A A S I I sag h R d + h R d ( ε + ) ε type ε< hyperboloid, focii o the axis ε paraboloid <ε<0 ellipsoid, focii o the axis ε0 sphere ε>0 ellipsoid, focii ot o the axis

4 Aspheric surface : geeral equatio h Rd I I sag + a 4h + a6h + a8h + a0h + ( ε + ) h Rd coic base to correct rd order to correct higher orders Drawbacks : - higher order terms lead to «umerical oscillatios» durig optimizatio, - it is almost impossible to tolerace the a k Solutio : Q-type polyomials from Forbes. See these papers : - G.. Forbes, "Shape specificatio for axially symmetric optical surfaces Opt. Express 5, (007) -Kevi P. Thompso, Floria Fourier, Jaick P. Rollad, G.. Forbes The Forbes Polyomial : a more predictable surface for fabricators IODC, paper OtuA6, 00 4

5 Set up for a sigle surface P i y α θ j k h C p x 5

6 Defiitios ad otatios Curvature of the surface: C Lagrageivariat : H déf yα R d démo facile P C Petzval curvature ( kα hθ ) 6

7 The Seidel sums are geerally P 0S 8 S S S I II III S 4 4 S IV All surf All surf All surf 8 ( i) ( i)( j) ( j) Seidel sums determied for the maximum aperture ad field. α h + C α h + C α h + C ( S + S ) ( j) V III All surf Lateral chromatic aberratio IV All surf 4 ( S + S ) Logitudial chromatic aberratio δ j i III δ C ε h 4 ε h ε h k k ( j) More details:"aberratios of optical systems", elford. IV α h + C + C 00 II ε h k C I All surf ε h All surf k h ( i) H h C 7

8 Total aberratio fuctio 4 040ρ + y ρ cosϕ + y ρ cos ϕ + 0 y S ρ + y ρ cosϕ (ormalized variables) 8

9 Special cases Object o the surface (h 0) 040 0, o chromatism Applicatios : field les, field corrector (Smyth les ) Object at the ceter of curvature (i 0), spherical surface 040 0, o logitudial chromatic aberratio Pupil at the ceter of curvature (j 0), spherical surface 0, o lateral chromatic aberratio Applicatios : Schmidt telescope 9

10 Special cases eierstrass poits ( (α/) 0), spherical surface Applicatios : letille ½ boule des objectifs de microscope à immersio Aspheric surface At least oe aberratio ca be corrected : 040,,, If the pupil is o the surface (k 0) : oly 040 ca be corrected Applicatios : paraboloid mirror, Ritchey-Chrétie telescope 0

11 Special cases Pupil shift (keepig costat aperture) j et k are chaged : 040 is ivariat,,, ad the lateral chromatic aberratio are modified Symetric optical system ad ray paths All the aberratios depedig o a odd power of the field are aught (,, lateral chromatic aberratio) If the system is symetric ad ot the ray paths (ex : /F), all the aberratios depedig o a odd power of the field remai weak Applicatios : double Gauss

12 Pupil shift D après «Optical system desig», R. Fisher

13 Pupil shift e assume that the aperture is costat (ie. h cte). k The pupil shift is characterized by : E h Oe ca show that : 040 0P 0 ( δ00 ) ( δ ) 4 E E 0 E E ( + ) E δ P + E To get more details, see for istace:" Aberratios of + 4 E 040 optical systems", elford

14 Pupil shift No effect o Spherical Aberratio or Petzval terms Aberratio «trasfer» : Coma itroduced if SA is preset Astigmatism itroduced if SA or Coma are preset Distortio itroduced if SA, Coma or Astigmatism are preset For a aberratio free system, shiftig the stop does ot itroduce aberratios. If the spherical aberratio is corrected, the the coma does ot deped o the positio of the pupil If the optical system is aplaetic, the the astigmatism does ot deped o the positio of the pupil If the logitudial chromatic aberratio is corrected, the the lateral chromatic aberratio does ot deped o the positio of the pupil 4

15 Sigle les (pupil o the les, objet at ifiity) To get more details, see:" Aberratios of Notatios: R P radius of Φ power of b shape factor c déf α + α α α the pupil the les déf C C + C C f ( )( C C ) optical systems", elford 5

16 Seidel sums ( ) ( ) ( ) Φ Φ c b H R c c b R P P Φ Φ Φ R H H P P δ ν δ 6

17 40 Best shape les,5 ; c s b 7

18 Chage of the idex of refractio c- 40, s b 8

19 40 5 Variatios of 040 et,5 ; c- S S b 9

20 Spherical mirror (whatever the positios of the object ad the pupil) Notatio : C 040 0P i h 4 ijh C j j i h H C C R d C. e get : ( + ) 0P 0

21 Spherical mirror Cases of iterest: Pupil o the mirror (j θ) Pupil at the ceter of curvature (j 0) : coma, astigmatism ad distortio are corrected

22 Plae parallel plate (pupil o the plate, ay positio of the object) α d 040 0P δ δ 00 0 ( ) 8 ( ) 4 α d α θd ( ) d ( ) α θ αθ d α d αθd

23 Plae parallel plate These formulas are depedig o α et θ, ad ot h et k : the aberratios do ot deped o the positio of the plate, with respect to the beam The logitudial chromatic aberratio, the spherical aberratio ad the astigmatism are all over-corrected If the object is at ifiity (α 0), the all the aberratios are aught If the pupil is at ifiity (θ 0, telecetric system), the all the field aberratios are aught.