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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.

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

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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


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