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 …

Reflection at a curved mirror interface in paraxial approx.

''

CC φφ ’’

ss

s’s’

OO II

yy

fRss

12

'

11

Sign convention: Mirrors

Object distanceS >0 for real object (to the left of V)S<0 for virtual object

Image distanceS’ > 0 for real image (to left of V)S’ < 0 for virtual image (to right of V)

RadiusR > 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,In previous example,

0

0',

R

ss

So we can write more generally,So we can write more generally,

Rss

2

'

11

'

121

fRf

2'

Rff

's

sm

Ray diagrams: concave mirrors

CC ƒ

ss s’s’

ErectErect

VirtualVirtual

EnlargedEnlarged

e.g. shaving mirror

What if s > f ?What if s > f ?

Ray diagrams: convex mirrors

CCƒ

ss s’s’

ErectErect

VirtualVirtual

ReducedReduced

What if s < |f| ?What if s < |f| ?

Calculate s’ for R=10 cm, s = 20 cmCalculate s’ for R=10 cm, s = 20 cm

Thin lens

1" R

nn

s

n

s

n LL

2

'

'

'

" R

nn

s

n

s

n LL

21

'

'

'

R

nn

R

nn

s

n

s

n LL

First interface Second interface

Bi-convex thin lens: Ray diagram

nn n’n’R1 R2

II

f ‘f

'

''

'

'

21 s

n

s

n

R

nn

R

nn

f

n

f

nP LL

s

s’

OO

ErectErect

VirtualVirtual

EnlargedEnlarged

ErectErect

VirtualVirtual

EnlargedEnlarged

nn n’n’

R1 R2

IIf ‘f

'

''

'

'

21 s

n

s

n

R

nn

R

nn

f

n

f

nP LL

s

s’

OO

InvertedInverted

RealReal

EnlargedEnlarged

InvertedInverted

RealReal

EnlargedEnlarged

Bi-convex thin lens: Ray diagram

Bi-concave thin lens: Ray diagram

nnn’n’

R1 R2

IIf ‘f

'

''

'

'

21 s

n

s

n

R

nn

R

nn

f

n

f

nP LL

ss’

OO

ErectErectVirtualVirtualReducedReduced

ErectErectVirtualVirtualReducedReduced

Converging and diverging lenses

Why are the following lenses Why are the following lenses convergingconverging or or divergingdiverging??

Converging lenses Diverging lenses

Newtonian equation for thin lens

nn n’n’

R1 R2

IIf ‘f

s

s’

OOx x’

f

x

x

fm

fxx

'

' 2

Complex optical systems

Thick lenses, combinations of lenses etc..Thick lenses, combinations of lenses etc..

tt

nnLL

nn n’n’

Consider case where t is not Consider case where t is not negligible. negligible.

We would like to maintain our We would like to maintain our Gaussian imaging relationGaussian imaging relation

Ps

n

s

n

'

'

But where do we measure s, s’ ; f, f’ But where do we measure s, s’ ; f, f’ from? How do we determine P?from? How do we determine P?

We try to develop a formalism that We try to develop a formalism that can be used with any system!!can be used with any system!!

Cardinal points and planes:1. Focal (F) points & Principal planes (PP) and points

nnLLnn n’n’

Keep definition of focal pointKeep definition of focal point ƒ’ƒ’

HH22

ƒ’ƒ’

FF22

PPPP22

Cardinal points and planes:1. Focal (F) points & Principal planes (PP) and points

nnLLnn n’n’

Keep definition of focal pointKeep definition of focal point ƒƒ

HH11

ƒƒ

FF11

PPPP11

Utility of principal planes

HH22

ƒ’ƒ’

FF22

PPPP22

HH11

ƒƒ

FF11

PPPP11

s s’

nnLLnn n’n’

hh

h’h’

Suppose s, s’, f, f’ all measured from HSuppose s, s’, f, f’ all measured from H11 and H and H22 … …

Show that we recover the Gaussian Imaging relation…Show that we recover the Gaussian Imaging relation…

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

nn n’n’

NN22

NPNP22

NN11

NPNP11

nnLL

Cardinal planes of simple systems1. Thin lens

Ps

n

s

n

'

'

Principal planes, nodal planes, Principal planes, nodal planes,

coincide at centercoincide at center

VV

H, H’H, H’

V’V’

V’ and V coincide andV’ and V coincide and

is obeyed.is obeyed.

Cardinal planes of simple systems1. Spherical refracting surface

nn n’n’

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

VV

Ps

n

s

n

'

'

Conjugate Planes – where y’=y

HH22

ƒ’ƒ’

FF22

PPPP22

HH11

ƒƒ

FF11

PPPP11

s s’

nnLLnn n’n’

yy

y’y’

Combination of two systems: e.g. two spherical interfaces, two thin lenses …

nn22nn n’n’HH11’’HH11

HH22 HH22’’

H’H’

yyYY

dd

ƒ’ƒ’

ƒƒ11’’

F’F’ FF11’’

1. Consider F’ and F1. Consider F’ and F11’’

h’h’

Find h’Find h’

Combination of two systems:

nn22nn n’n’

HH11’’HH11

HH22 HH22’’HH

yyYY

ddƒƒ

1. Consider F and F1. Consider F and F22

FF22

ƒƒ22

hh

FF

Find hFind h

Summary

HH11’’HH11 HH22 HH22’’

HH H’H’

ƒƒ ƒ’ƒ’hh h’h’

FF F’F’

dd

Summary

2

2121

211

2

2

2

12

1

2

21

2

,

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'

'

'

'

''

''

''

n

PPdPPP

or

f

n

ff

dn

f

n

f

n

f

n

hn

n

P

PdHH

f

fdh

hn

n

P

PdHH

f

fdh