Post on 18-Jan-2018
description
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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’
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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
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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)
)(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’
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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