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Transcript of 1 Chapter 6 More on geometrical optics February 4 Thick lenses Review: Paraxial imaging from a...
1
Chapter 6 More on geometrical opticsFebruary 4 Thick lenses
Review:
Paraxial imaging from a single refracting spherical surface:
Thin lens equation:
Lens maker’s formula:
Gaussian lens formula:
Assumptions: 1) Paraxial rays. 2) Thin lenses.Additional assumption: 3) Monochromatic light.Question: What happens if these assumptions are not valid?Solution: Study the principles of thick lens and aberrations.
21
11)1(
1
RRn
f l
fss io
111
Note the sign convention: everything has a sign.
R
nn
s
n
s
n
io
1221
21
11)1(
11
RRn
ss lio
2
6.1 Thick lenses and lens systemsThick lenses: When d is not small, the lens maker’s formula and the Gaussian lens formula are not valid.Why thick lens?The image of a distant point source is not a point, but a diffraction pattern because of the limited size of the lenses. Larger D produces clearer images.D/22.1
Some jargons in photography:Field of view (angle of view): The angle in the object space over which objects are recorded on the sensor of the camera. It depends on the focal length of the lens and the size of the sensor.
Depth of field: The region in the object space over which the objects appear sharp on the sensor.
f-number (f/#): The ratio of the focal length to the diameter of the entrance pupil:The f/# affects 1) the brightness of the image, 2) the sharpness of the image, and 3) the depth of field.
Dff //#
~ f
f.o.v.
D
d.o.f.
~ f
f
D
3
Question: What are the formulas for f and si for a thick lens?
Terminology of thick lenses:Principal plane: The plane composed by the crossing points between the incident rays parallel to the optical axis and their emerged rays.Principal points: the intersects between the principal planes and the optical axis. H1 and H2.
Note:1) The principal planes are actually curved, while its paraxial regions forms a plane.2) If one surface of the lens is planar, the tangent of the other surface should be a principal plane. prove now, and soon again. 3) Generally
(e.g., plane-convex lenses) to be proved soon.
|).| and 1.5(for )3/1( 212121 -RRdnVVHH l
Fi
b.f.l.
V1 H1V2H2
Secondaryprincipal plane
Fo
f.f.l.
V2H2V1 H1
Primaryprincipal plane
Note that we may later consider a three-segment ray as virtually two segments.
4
N2
N1 O
Nodal points: The crossing points between the optical axis and the rays passing through the optical center.Coincide with the principal points when both sides of the lens are in the same medium. to be proved soon.
Cardinal points of a lens:Two focal points + two principal points + two nodal points.When both sides of the lens are in the same medium:Fi, Fo, H1, H2 are the cardinal points.
Optical center: All rays whose emerging directions are parallel to their incident directions pass through one common point. This point is called the optical center of the lens. proved in chapter 5.
R2R1
C1C2
A
BO
Points to remember for a lens:V1, V2, C1, C2, O, Fi, Fo, H1, H2 , N1, N2. They are fixed tothe lens on its optical axis.Plus S, P for object and image.
5
Read: Ch6: 1No homework
6
February 6 Thick lens equations
Locations of principal planes: Note that the principal planes may be external in some cases.
However, people have developed a much simpler method which results in a set of thick lens equations that give exactly the same answer as the above method.
Virtual rays:Fact: Under paraxial optics, it is proved that if we extend a ray’s two segments located in the air they then cross at the two principal planes at the same height.Solution: Virtual rays between lens surfaces and/or principal planes can be used to simplify the problem. The far-most rays are still real.
Goal:
?
?,,, 21
i
i
o
o
ll
y
s
y
s
dRRn
Possible solution: Under paraxial optics we may use the formula for the imaging from a single refracting surface twice to locate the final image.
H1H2
7
Fi
b.f.l.
V1
H1
V2
H2Fo
f.f.l.
f fdl
sosi
h2h1
yo
yi
xixo
Thick lens equations:When f, so and si are measured from the principal planes, we have
fss
Rn
fdnh
Rn
fdnh
RRn
dn
RRn
f
io
l
ll
l
ll
l
lll
111
)1(
)1(
)1(11)1(
1
12
21
2121
h is positive when H is to the right of V.
All are to be proved soon. I hate to believe anything that is not proved by my pencil.
Newtonian form: 2fxx io
Magnification:
o
i
o
i
o
iT s
s
x
f
f
x
y
yM
When Light directed toward the first principal plane will emerge from the second principal plane at the same height.
.1,,,0 have we,0 Mfxfxss ioio
Eqs. 6.1-6.4
io
oill
FHHFf
FFHHhhfdRRn
21
212121 ,,,,,,,
Locating the four cardinal points.
Note the new three key rays. Here the rays inside the lens are virtual. The rays in the air are actual.
8
Read: Ch6: 1Homework: Ch6:4,6,8Due: February 13
9
February 9 Combination of thick lenses
fss
Rn
fdnh
Rn
fdnh
RRn
dn
RRn
f
io
l
ll
l
ll
l
lll
111
)1(
)1(
)1(11)1(
1
12
21
2121
Thick lens equations:
The procedure of locating an image from a thick lens:
). from distance (image '
) from distance(object '
.,,, points cardinal,,,,,
2
2
1
1
212121
o
iT
i
o
io
o
oill
s
sM
Vs
sh
s
f
sVs
h
FFHHhhfdRRn
FiV1
H1
V2
H2Fo
f fdl
sosi
h2h1
yo
yi
10
Example (P6.12):R1=4 cm, R2= -15 cm, dl =4 cm, nl =1.5, object =100 cm before lens.
minified. is image The1
inverted. is image The0
real. is image The0
.072.028.7
6.100
.02.5surface)back thefrom distance (image'
26.2
28.7
79.6
6.100100)surfacefront thefrom distance(object '
60.0
.26.2,60.0,79.6,,,
2
1
2121
T
T
i
o
iT
i
o
i
io
o
ll
M
M
s
s
sM
cms
cms
cms
cmh
cms
cmf
cmscms
cmh
cmhcmhcmfdRRn
Everything has a sign!
11
Combination of thick lenses: locating the overall cardinal points
2111
12222
2121
1
21
12111
2
2
1
1
1
12
1
222
111
, , ,
1111
f
fdHH
f
fdfsHH
ff
d
ffff
fd
sf
fsfdsfsss
s
s
s
s
s
sM
fdfsfs
i
i
ioioo
o
i
o
i
o
iT
oi
H1 Fo Fo1 H11 H12Fi1 Fo2 H21 H22 Fi2 Fi H2
df1
f1 f2 f2f f
21221121 ,,,,,, HHfHHdff
Note the sign convention for . and 222111 HHHH
12
Read: Ch6: 1Homework: Ch6:12,13,14Due: February 20
yR
nnnn
R
yn
R
yn
nnnnnn
itttii
ttii
ttiittiittii
)()(sinsin
13
February 11 Ray matrices
6. 2.1 Matrix methodRay tracing: Mathematically following the trace of a ray.Example: Ray tracing of a paraxial, meridional ray traversing a spherical lens.Meridional ray: A ray in a plane that contains the optical axis and the object point.(Opp: skew ray).
I. Refraction (at P):
Power of arefracting surface R
nnD it
i
ii
t
tt
it
iitt
y
nD
y
n
yyy
Dynn 10
1
i
ii
t
tt
y
nD
y
n 10
1
Refraction matrix Incident ray vectorRefracted ray vectorit Rrr
P
Cy
nt
i
t
R
ni
14
II. Transfer (from P1 to P2):
1
1
2
2
112
12
1/
01
y
n
ndy
n
ydy
nn
1
1
2
2
1/
01
y
n
ndy
n
12 Trr Transfer matrix
P1
y1
y2
n
1
2
d
P2
i
C PV
ir
R
r
yi
III. Reflection (Mirrors):
i
i
r
r
ir
iir
rii
iii
y
nR
n
y
n
yy
R
yR
y
10
21
2
2
10
21
R
n
Mirror matrix
15
Note:1) Different definitions for the ray vectors and matrixes may exist.2) Merit of the current matrices: |R|=|T|=1, and their combinations.
Examples of other definitions of ray vectors and matrices:
i
i
titt
t
i
ii
t
tt y
nnnD
y
y
nD
y
n
//
01
10
1
System matrix A of a lens:Transforming an incident ray before the first surface to the emerging ray after the second surface: A→B→C→D
2221
1211
1
2121
2
12121221
1
1
10
1
1/
01
10
1
aa
aa
n
dD
n
dn
dDDDD
n
dD
D
nd
D
l
l
l
l
l
l
l
l
ll
RTRA
Another popular form
A Cy2
nl
1
2
dl
y1
BD
16
Read: Ch6: 2Homework: Ch6: 16,20,22,23,24Due: February 20
211
12
11
12
22111
122
12
2221222
2121
212112
12
1222
112
12221
1211
2
)1()1(
1
'
'1 Similarly,
)1()1(
1
)1(11)1(
11
0
Rn
fdnaf
a
a
a
aHVh
Rn
fdnaf
a
ayyHVh
RRn
dn
RRn
n
dDDDDa
yFHf
yay
ya
yaa
aa
y
l
ll
l
ll
l
lll
l
l
i
Eqs. 6.3-4
Eq. 6.2
17
Application I: Where are the cardinal points (Fi, Fo, H1, H2 )?Let the incident ray be parallel to the optical axis:
February 13 Matrix analysis of lenses
l
l
l
l
l
l
l
l
n
dD
n
dn
dDDDD
n
dD
aa
aa
1
2121
2
2221
1211
1
1A
.,,,,,,,, 212121 HHFFhhfdRRn oill
P1
Fi
V1 H1V2H2
P2
y1 y2
System matrix for a reversed lens
fss
fa
ss
ssa
aaaa
sa
aa
saa
a
as
a
asaa
a
as
sa
a
asaa
aa
a
as
a
ashsy
a
ashsy
yaa
aa
y
ioio
io
o
o
i
oi
o
oi
oo
ii
111
/1
1
1
1
1
11
)1(
11
1)(
1))((
12
12
12212211
12
12
112221
12
22
12
112221
12
22
12
12
112221
1211
12
22
12
1111
12
2222
12221
1211
2
y2
18
Application II: Lens equation
Eq. 6.1
If =, then si = so= 0 Nodal points = Principal points
y1 P
V1 H1V2H2
P1P2
S
so si
19
Application III: Thin lens combination: where are the overall cardinal points?
Thin lens system matrix
10
/1121
fA
112
2222
212
1111
212112
1
21212
12
1
1
111
1
111
10
/11
1
01
10
/11
f
fd
a
aHO
f
fd
a
aHO
ff
d
ffa
f
f
dd
ff
d
fff
d
f
d
f
Fi2
Fo1
Fi1
Fo2
O2O1
d
f1
f2
20
Homework:Starting from Eq. 6.31, please prove Eqs. 6.1-4. Please include detailed drawings showing all the parameters.
Due: February 20
21