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Cardinal planes/points in paraxial opticsWednesday September 18, 2002

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Thick Lens: Position of Cardinal PlanesConsider as combination of two simple systemsConsider as combination of two simple systems

e.g. two refracting surfacese.g. two refracting surfaces

H’H’HH

HH11, H, H11’’ HH22, H, H22’’

Where are H, H’ Where are H, H’ for thick lens?for thick lens?

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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.

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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

'

'

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Conjugate Planes – where y’=y

HH22

ƒ’ƒ’

FF22

PPPP22

HH11

ƒƒ

FF11

PPPP11

s s’

nnLLnn n’n’

yy

y’y’

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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’

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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

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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’

ƒƒ22’’

F’F’ FF22’’

1. Consider F’ and F1. Consider F’ and F22’’

FF22

ƒƒ22

h’h’

θθθθ

y’y’

Find power of combined systemFind power of combined system

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Summary

HH11’’HH11 HH22 HH22’’

HH H’H’

ƒƒ ƒ’ƒ’hh h’h’

FF F’F’

dd

I II

nn22 n’n’nn

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Summary

2

2121

211

2

2

2

12

1

2

21

2

,

''

'

'

'

'

'

''

''

''

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

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Thick Lens

nn22

RR11 RR22

HH11,H,H11’’ HH22,H,H22’’

In air n = n’ =1In air n = n’ =1

Lens, nLens, n22 = 1.5 = 1.5

RR11 = - R = - R22 = 10 cm = 10 cm

d = 3 cmd = 3 cm

Find Find ƒƒ11,ƒ,ƒ22,ƒ, h and h’,ƒ, h and h’

Construct the Construct the principal planes, H, principal planes, H, H’ of the entire H’ of the entire systemsystem

nn n’n’

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Principal planes for thick lens (n2=1.5) in air

Equi-convex or equi-concave and moderately thick Equi-convex or equi-concave and moderately thick PP11 = P = P22 ≈ P/2≈ P/2

3'd

hh

12

22

'f

f

n

dh

f

f

n

dh

HH H’H’ HH H’H’

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Principal planes for thick lens (n2=1.5) in air

Plano-convex or plano-concave lens with RPlano-convex or plano-concave lens with R22 = =

PP22 = 0= 0

dh

h

3

2'

0

12

22

'f

f

n

dh

f

f

n

dh

HH H’H’ HH H’H’

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Principal planes for thick lens (n=1.5) in air

For meniscus lenses, the principal planes move For meniscus lenses, the principal planes move outside the lensoutside the lens

RR22 = 3R = 3R11 (H’ reaches the first surface) (H’ reaches the first surface)

P Same for all lensesSame for all lenses

12

22

'f

f

n

dh

f

f

n

dh

HH H’H’ HH H’H’ HH H’H’HH H’H’

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Examples: Two thin lenses in air

2

2

f

fd

P

Pdh

ƒƒ11 ƒƒ22

dd

HH11’’HH11 HH22 HH22’’

n = nn = n2 2 = n’ = 1= n’ = 1

Want to replace HWant to replace Hii, H, Hii’ with H, H’’ with H, H’

1

1'f

fd

P

Pdh

hh h’h’

HH H’H’

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Examples: Two thin lenses in air

ƒƒ11 ƒƒ22

dd

n = nn = n2 2 = n’ = 1= n’ = 1

2121

2

2121

111

,

ff

d

fff

or

n

PPdPPP

HH H’H’

FF F’F’

ƒƒ ƒ’ƒ’ fss

1

'

11

s’s’ss

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Huygen’s eyepieceIn order for a combination of two lenses to be independent of In order for a combination of two lenses to be independent of the index of refraction (i.e. free of chromatic aberration)the index of refraction (i.e. free of chromatic aberration)

)(2

121 ffd

Example, Huygen’s EyepieceExample, Huygen’s Eyepiece

ƒƒ11=2=2ƒƒ22 and d=1.5 and d=1.5ƒƒ22

Determine ƒ, h and h’Determine ƒ, h and h’

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Huygen’s eyepiece

21

22

'

2

fP

Pdh

fP

Pdh

2

2

2121

3

4

,

ff

or

n

PPdPPP

HH11

h=2ƒh=2ƒ22

HH22 HH

d=1.5ƒd=1.5ƒ22

h’ = -ƒh’ = -ƒ22

H’H’