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**Hecht 5.2, 6.1 Monday September 16, 2002**

Curved mirrors, thin & thick lenses and cardinal points in paraxial optics Hecht 5.2, 6.1 Monday September 16, 2002

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**General comments Welcome comments on structure of the course.**

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

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**Reflection at a curved mirror interface in paraxial approx.**

y φ ’ O C I s’ s

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**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)

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**Paraxial ray equation for reflection by curved mirrors**

In previous example, So we can write more generally,

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**Ray diagrams: concave mirrors**

Erect Virtual Enlarged C ƒ e.g. shaving mirror What if s > f ? s s’

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**Ray diagrams: convex mirrors**

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

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Thin lens First interface Second interface

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**Bi-convex thin lens: Ray diagram**

Erect Virtual Enlarged O f f ‘ n n’ R1 R2 s s’

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**Bi-convex thin lens: Ray diagram**

Inverted Real Enlarged O I f f ‘ n n’ s s’

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**Bi-concave thin lens: Ray diagram**

f f ‘ n’ n R1 R2 s’ s Erect Virtual Reduced

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**Converging and diverging lenses**

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

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**Newtonian equation for thin lens**

x I f f ‘ x’ n n’ s s’

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**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!!

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**Cardinal points and planes: 1**

Cardinal points and planes: 1. Focal (F) points & Principal planes (PP) and points n nL n’ F2 H2 ƒ’ PP2 Keep definition of focal point ƒ’

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**Cardinal points and planes: 1**

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

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**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…

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**Cardinal points and planes: 1. Nodal (N) points and planes**

nL NP1 NP2

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**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

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**Cardinal planes of simple systems 1. Spherical refracting surface**

Gaussian imaging formula obeyed, with all distances measured from V V

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**Conjugate Planes – where y’=y**

nL n’ y F1 F2 H1 H2 y’ ƒ’ ƒ s s’ PP1 PP2

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**Combination of two systems: e. g**

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’

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**Combination of two systems:**

H2 H2’ h H Find h H1’ H1 y Y F2 F ƒ d ƒ2 1. Consider F and F2 n n2 n’

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

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Summary

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Images formed by lenses. Convex (converging) lenses, f>0.

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