Curved mirrors thin thick lenses and cardinal points

Curved mirrors, thin & thick lenses and cardinal points in paraxial optics Hecht 5. 2, 6. 1 Monday September 16, 2002

General comments Welcome comments on structure of the course. n Drop by in person n Slip an anonymous note under my door n… n

Reflection at a curved mirror interface in paraxial approx. y φ C O ’ I s’ s

Sign convention: Mirrors n Object distance ¨S >0 for real object (to the left of V) ¨ S<0 for virtual object n Image distance ¨ S’ > 0 for real image (to left of V) ¨ S’ < 0 for virtual image (to right of V) n Radius ¨R > 0 (C to the right of V) ¨ R < 0 (C to the left of V)

Paraxial ray equation for reflection by curved mirrors In previous example, So we can write more generally,

Ray diagrams: concave mirrors Erect Virtual Enlarged C ƒ e. g. shaving mirror What if s > f ? s s’

Ray diagrams: convex mirrors Calculate s’ for R=10 cm, s = 20 cm Erect Virtual Reduced ƒ s s’ C What if s < |f| ?

Thin lens First interface Second interface

Bi-convex thin lens: Ray diagram Erect Virtual Enlarged I O f f‘ n n’ R 1 s s’ R 2

Bi-convex thin lens: Ray diagram R 1 R 2 Inverted Real Enlarged O n I f f‘ n’ s s’

Bi-concave thin lens: Ray diagram O I f f‘ n’ n s Erect Virtual Reduced R 1 s’ R 2

Converging and diverging lenses Why are the following lenses converging or diverging? Converging lenses Diverging lenses

Newtonian equation for thin lens R 1 R 2 O f n f‘ x’ n’ s s’ I x

Complex optical systems Thick lenses, combinations of lenses etc. . Consider case where t is not negligible. n We would like to maintain our Gaussian imaging relation n’ t n. L But where do we measure s, s’ ; f, f’ from? How do we determine P? We try to develop a formalism that can be used with any system!!

Cardinal points and planes: 1. Focal (F) points & Principal planes (PP) and points n n. L n’ F 2 H 2 ƒ’ PP 2 Keep definition of focal point ƒ’

Cardinal points and planes: 1. Focal (F) points & Principal planes (PP) and points n n. L F 1 H 1 ƒ PP 1 Keep definition of focal point ƒ n’

Utility of principal planes Suppose s, s’, f, f’ all measured from H 1 and H 2 … n h n. L n’ F 1 F 2 H 1 H 2 h’ ƒ’ ƒ s s’ PP 1 PP 2 Show that we recover the Gaussian Imaging relation…

Cardinal points and planes: 1. Nodal (N) points and planes n n’ N 1 N 2 n. L NP 1 NP 2

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

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

Conjugate Planes – where y’=y n y n. L n’ F 1 F 2 H 1 H 2 y’ ƒ’ ƒ s s’ PP 1 PP 2

Combination of two systems: e. g. two spherical interfaces, two thin lenses … n H 1 ’ n 2 H’ n’ h’ H 2 ’ 1. Consider F’ and F 1’ y Find h’ Y F’ d ƒ’ ƒ 1’ F 1 ’

Combination of two systems: h Find h H 1 H 2 ’ H H 1 ’ Y F 2 y F ƒ d ƒ 2 n 1. Consider F and F 2 n’

Summary H H’ H 1 ’ H 2 ’ F F’ d ƒ h h’ ƒ’

Summary