curved mirrors, thin & thick lenses and cardinal points in paraxial optics hecht 5.2, 6.1 monday...
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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 …
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
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 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
'
'
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