Curved mirrors, thin & thick lenses and cardinal points in paraxial optics

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. • Drop by in person • Slip an anonymous note under my door • …

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

Sign convention: Mirrors • Object distance • S >0 for real object (to the left of V) • S<0 for virtual object • Image distance • S’ > 0 for real image (to left of V) • S’ < 0 for virtual image (to right of V) • 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 C What if s < |f| ? s s’

Thin lens First interface Second interface

I O f f ‘ s s’ Bi-convex thin lens: Ray diagram Erect Virtual Enlarged n n’ R1 R2

O f f ‘ I s s’ Bi-convex thin lens: Ray diagram R1 R2 Inverted Real Enlarged n n’

O I f f ‘ s s’ Bi-concave thin lens: Ray diagram n’ n R1 R2 Erect Virtual Reduced

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

O x f f ‘ x’ I s s’ Newtonian equation for thin lens R1 R2 n n’

Complex optical systems Thick lenses, combinations of lenses etc.. Consider case where t is not negligible. We would like to maintain our Gaussian imaging relation n n’ t nL 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 nL n’ F2 H2 ƒ’ PP2 Keep definition of focal pointƒ’

Cardinal points and planes:1. Focal (F) points & Principal planes (PP) and points n nL n’ F1 H1 ƒ PP1 Keep definition of focal pointƒ

Utility of principal planes Suppose s, s’, f, f’ all measured from H1 and H2 … n nL n’ h F1 F2 H1 H2 h’ ƒ’ ƒ s s’ PP1 PP2 Show that we recover the Gaussian Imaging relation…

Cardinal points and planes:1. Nodal (N) points and planes n n’ N1 N2 nL NP1 NP2

Cardinal planes of simple systems1. Thin lens V’ and V coincide and V’ V H, H’ is obeyed. Principal planes, nodal planes, coincide at center

Cardinal planes of simple systems1. Spherical refracting surface n n’ Gaussian imaging formula obeyed, with all distances measured from V V

Conjugate Planes – where y’=y n nL n’ y F1 F2 H1 H2 y’ ƒ’ ƒ s s’ PP1 PP2

Combination of two systems: e.g. two spherical interfaces, two thin lenses … n H1 H1’ n2 H’ h’ n’ H2 H2’ 1. Consider F’ and F1’ Find h’ y Y F’ F1’ d ƒ’ ƒ1’

d Combination of two systems: H2 H2’ h H H1’ Find h H1 y Y F2 F ƒ ƒ2 1. Consider F and F2 n n2 n’

Summary H H’ H1’ H1 H2 H2’ F F’ d h h’ ƒ ƒ’

Summary

Curved mirrors, thin & thick lenses and cardinal points in paraxial optics