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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. C φ ’’’’ s s’OI y

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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 C ƒ ss’ ErectVirtualEnlarged e.g. shaving mirror What if s > f ?

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Ray diagrams: convex mirrors Cƒ ss’ ErectVirtualReduced What if s < |f| ? Calculate s’ for R=10 cm, s = 20 cm

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Thin lens First interfaceSecond interface

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Bi-convex thin lens: Ray diagram nn’ R1R1 R2R2 I f ‘f s s’ O ErectVirtualEnlargedErectVirtualEnlarged

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nn’ R1R1 R2R2 I f ‘f s s’ O InvertedRealEnlargedInvertedRealEnlarged Bi-convex thin lens: Ray diagram

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Bi-concave thin lens: Ray diagram n n’ R1R1 R2R2 I f ‘f s s’ O ErectVirtualReducedErectVirtualReduced

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

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Newtonian equation for thin lens nn’ R1R1 R2R2 I f ‘f s s’ O x x’

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Complex optical systems Thick lenses, combinations of lenses etc.. t nLnLnLnL n n’ Consider case where t is not negligible. We would like to maintain our Gaussian imaging relation 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. Focal (F) points & Principal planes (PP) and points nLnLnLnLnn’ Keep definition of focal pointƒ’ Keep definition of focal point ƒ’ H2H2H2H2 ƒ’ F2F2F2F2 PP 2

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Cardinal points and planes: 1. Focal (F) points & Principal planes (PP) and points nLnLnLnLnn’ Keep definition of focal pointƒ Keep definition of focal point ƒ H1H1H1H1 ƒ F1F1F1F1 PP 1

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Utility of principal planes H2H2H2H2 ƒ’ F2F2F2F2 PP 2 H1H1H1H1 ƒ F1F1F1F1 PP 1 ss’ nLnLnLnLnn’ h h’ Suppose s, s’, f, f’ all measured from H 1 and H 2 … Show that we recover the Gaussian Imaging relation…

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Cardinal points and planes: 1. Nodal (N) points and planes nn’ N2N2N2N2 NP 2 N1N1N1N1 NP 1 nLnLnLnL

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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.

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

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

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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’

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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

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Summary H1’H1’H1’H1’ H1H1H1H1 H2H2H2H2 H2’H2’H2’H2’ HH’ ƒƒ’ hh’ FF’ d

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