1 Cardinal planes/points in paraxial optics Wednesday September 18, 2002

2 Thick Lens: Position of Cardinal Planes Consider as combination of two simple systems e.g. two refracting surfaces H’H H 1, H 1 ’ H 2, H 2 ’ Where are H, H’ for thick lens?

3 Cardinal planes of simple systems 1. Thin lens Principal planes, nodal planes, coincide at center V H, H’ V’ V’ and V coincide and is obeyed.

4 Cardinal planes of simple systems 1. Spherical refracting surface nn’ Gaussian imaging formula obeyed, with all distances measured from V V

5 Conjugate Planes – where y’=y H2H2H2H2 ƒ’ F2F2F2F2 PP 2 H1H1H1H1 ƒ F1F1F1F1 PP 1 ss’ nLnLnLnLnn’ y y’

6 Combination of two systems: e.g. two spherical interfaces, two thin lenses … n2n2n2n2nn’ H1’H1’H1’H1’ H1H1H1H1 H2H2H2H2 H2’H2’H2’H2’ H’ y Y d ƒ’ ƒ1’ƒ1’ƒ1’ƒ1’ F’ F1’F1’F1’F1’ 1. Consider F’ and F 1 ’ h’ Find h’

7 Combination of two systems: n2n2n2n2nn’ H1’H1’H1’H1’ H1H1H1H1 H2H2H2H2 H2’H2’H2’H2’ H y Ydƒ 1. Consider F and F 2 F2F2F2F2 ƒ2ƒ2ƒ2ƒ2 h F Find h

8 Combination of two systems: e.g. two spherical interfaces, two thin lenses … n2n2n2n2nn’ H1’H1’H1’H1’ H1H1H1H1 H2H2H2H2 H2’H2’H2’H2’ H’ y Y d ƒ’ ƒ2’ƒ2’ƒ2’ƒ2’ F’ F2’F2’F2’F2’ 1. Consider F’ and F 2 ’ F2F2F2F2 ƒ2ƒ2ƒ2ƒ2 h’ θθ y’ Find power of combined system

9 Summary H1’H1’H1’H1’ H1H1H1H1 H2H2H2H2 H2’H2’H2’H2’ HH’ ƒƒ’ hh’ F F’ d I II n2n2n2n2n’ n

10 Summary

11 Thick Lens n2n2n2n2 R1R1R1R1 R2R2R2R2 H 1,H 1 ’ H 2,H 2 ’ In air n = n’ =1 Lens, n 2 = 1.5 R 1 = - R 2 = 10 cm d = 3 cm Find ƒ 1,ƒ 2,ƒ, h and h’ Construct the principal planes, H, H’ of the entire system nn’

12 Principal planes for thick lens (n 2 =1.5) in air Equi-convex or equi-concave and moderately thick  P 1 = P 2 ≈ P/2 HH’HH’

13 Principal planes for thick lens (n 2 =1.5) in air Plano-convex or plano-concave lens with R 2 =   P 2 = 0 HH’HH’

14 Principal planes for thick lens (n=1.5) in air For meniscus lenses, the principal planes move outside the lens R 2 = 3R 1 (H’ reaches the first surface) R 2 = 3R 1 (H’ reaches the first surface) Same for all lenses HH’HH’HH’ HH’

15 Examples: Two thin lenses in air ƒ1ƒ1ƒ1ƒ1 ƒ2ƒ2ƒ2ƒ2 d H1’H1’H1’H1’ H1H1H1H1 H2H2H2H2 H2’H2’H2’H2’ n = n 2 = n’ = 1 Want to replace H i, H i ’ with H, H’ hh’ HH’

16 Examples: Two thin lenses in air ƒ1ƒ1ƒ1ƒ1 ƒ2ƒ2ƒ2ƒ2 d n = n 2 = n’ = 1 HH’ FF’ ƒ ƒ’ s’s

17 Huygen’s eyepiece In order for a combination of two lenses to be independent of the index of refraction (i.e. free of chromatic aberration) Example, Huygen’s Eyepiece ƒ 1 =2 ƒ 2 and d=1.5ƒ 2 Determine ƒ, h and h’

18 Huygen’s eyepiece H1H1H1H1 h=2ƒ 2 H2H2H2H2H d=1.5ƒ 2 h’ = -ƒ 2 H’