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Cardinal planes and matrix methods

Monday September 23, 2002

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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' dhh

12

22

'ff

ndh

ff

ndh

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

32'

0

12

22

'ff

ndh

ff

ndh

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

'ff

ndh

ff

ndh

HH H’H’ HH H’H’ HH H’H’HH H’H’

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Examples: Two thin lenses in air

2

2

ffd

PPdh

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'ffd

PPdh

hh h’h’

HH H’H’

6

Examples: Two thin lenses in airƒƒ11 ƒƒ22

dd

n = nn = n2 2 = n’ = 1= n’ = 1

2121

2

2121

111,

ffd

fff

ornPPdPPP

HH H’H’

FF F’F’

ƒƒ ƒ’ƒ’ fss1

'11

s’s’ss

7

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)

)(21

21 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

fPPdh

fPPdh

2

2

2121

34

,

ff

ornPPdPPP

HH11

h=2ƒh=2ƒ22

HH22 HH

d=1.5ƒd=1.5ƒ22

h’ = -ƒh’ = -ƒ22

H’H’

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Two separated lenses in airff11’=2’=2ff22’’

d = 0.5 d = 0.5 ff22’’

HHH’H’

F’F’FF

f’f’

d = d = ff22’’

HHH’H’

F’F’FF

f’f’

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Two separated lenses in airff11’=2f’=2f22’’

d = 2d = 2ff22’’

HHH’H’

F’F’FF

f’f’

d = 3d = 3ff22’’

Principal points at Principal points at

e.g. Astronomical telescopee.g. Astronomical telescope

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Two separated lenses in airff11’=2f’=2f22’’

d = 5d = 5ff22’’

f’f’

e.g. Compound microscopee.g. Compound microscopeHH

F’F’FF

H’H’

12

Two separated lenses in airff11’=-2f’=-2f22’’

d = -d = -ff22’’

e.g. Galilean telescopee.g. Galilean telescope

Principal points at Principal points at

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Two separated lenses in airff11’=-2f’=-2f22’’

d = -1.5d = -1.5ff22’’e.g. Telephoto lense.g. Telephoto lens

HH H’H’

F’F’

f’f’

FF

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Matrices in paraxial OpticsTranslationTranslation

(in homogeneous medium)(in homogeneous medium)

00

LL

yyoo

yy

oo

oo

yLyy

101

101 L

T

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Matrix methods in paraxial optics

)'(')(''

nnnn

Refraction at a spherical interfaceRefraction at a spherical interface

yy

’’φφ

’’

nn n’n’

''''

''

''

nn

ny

Rnn

nnn

nn

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Matrix methods in paraxial optics

''

''nn

ny

Rnn

Refraction at a spherical interfaceRefraction at a spherical interface

yy

’’φφ

’’

nn n’n’

0' yy

''

01

nn

nP

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Matrix methods in paraxial optics1 2

Lens matrixLens matrix

nn nnLL n’n’

T

LL nn

nP11

01

''

0122

nn

nP L

101 d

TFor the complete systemFor the complete system

12 TL

Note order – matrices Note order – matrices do notdo not, in general, commute., in general, commute.

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Matrix methods in paraxial optics

22

11

'RnnP

RnnP

L

L

L

LL

nPd

nn

nP

nnd

nPd

TL2

1

12

1''

1

LnPPdPPPwhere 21

21,

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Matrix properties

DCBA

'det

nnBCAD

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Matrices: General Properties

1234 MMMMM

1234 detdetdetdetdet MMMMM

''det

12

1

3

23

nn

nn

nn

nn

nnM

For system in air, n=n’=1For system in air, n=n’=1

1det M

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System matrix

o

o

f

f yDCBAy

oof

oof

DCy

BAyy

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System matrix: Special Cases(a) D = 0 (a) D = 0 ff = Cy = Cyo o (independent of (independent of oo))

yyoo

ff

Input plane is the first focal planeInput plane is the first focal plane

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System matrix: Special Cases(b) A = 0 (b) A = 0 y yff = B = Boo (independent of y (independent of yoo))

oo

yyff

Output plane is the second focal planeOutput plane is the second focal plane

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System matrix: Special Cases(c) B = 0 (c) B = 0 y yff = Ay = Ayoo

yyff

Input and output plane are conjugate – A = magnificationInput and output plane are conjugate – A = magnification

yyoo

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System matrix: Special Cases(d) C = 0 (d) C = 0 ff = D = Doo (independent of y (independent of yoo))

Telescopic system – parallel rays in : parallel rays outTelescopic system – parallel rays in : parallel rays out

oo ff

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Examples: Thin lens

L

LL

nPd

nn

nP

nnd

nPd

TL2

1

12

1''

1

Recall that for a thick lensRecall that for a thick lens

For a thin lens, d=0For a thin lens, d=0

''

01

nn

nPL

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Examples: Thin lens

LnPPdPPP 21

21

'''

2121 f

nfn

Rnn

RnnPPP LL

Recall that for a thick lensRecall that for a thick lens

For a thin lens, d=0For a thin lens, d=0

In air, n=n’=1In air, n=n’=1

2121

11111'

11RR

nRn

Rn

ffP L

LL

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Imaging with thin lens in air

oo’’

ss s’s’

yyoo y’y’

Input Input

planeplaneOutput Output planeplane

11

01

fL