design and correction of optical systems - uni-jena.de...image construction of a lens by simple rays...
TRANSCRIPT
Overview
1. Basics 2012-04-18
2. Materials 2012-04-25
3. Components 2012-05-02
4. Paraxial optics 2012-05-09
5. Properties of optical systems 2012-05-16
6. Photometry 2012-05-23
7. Geometrical aberrations 2012-05-30
8. Wave optical aberrations 2012-06-06
9. Fourier optical image formation 2012-06-13
10. Performance criteria 1 2012-06-20
11. Performance criteria 2 2012-06-27
12. Measurement of system quality 2012-07-04
13. Correction of aberrations 1 2012-07-11
14. Optical system classification 2012-07-18
2012-04-18
4.1 Imaging - basic notations
- paraxial approximation
- linear collineation
- graphical image construction
- lens makers formula
4.2 Optical system properties - special aspects
- imaging
- multiple components
4.3 Matrix calculus - simple matrices
- relations
4.4 Phase space - basic idea
- invariant
Part 4: Paraxial Optics
� Optical Image formation:
All ray emerging from one object point meet in the perfect image point
� Region near axis:
gaussian imaging
ideal, paraxial
� Image field size:
Chief ray
� Aperture/size of
light cone:
marginal ray
defined by pupil stop
Optical Imaging
image
object
optical
system
O2field
point
axis
pupil
stop
marginal
ray
O1 O'1
O'2
chiefray
� Single surface between two media
Radius r, refractive indices n, n‘
� Imaging condition, paraxial
� Abbe invariant
alternative representation of the
imaging equation
'
1'
'
'
fr
nn
s
n
s
n=
−=−
−⋅=
−⋅=
'
11'
11
srn
srnQs
Single Surface
� Law of refraction
� Expansioin of the sine-function:
� Linearized approximation of the
law of refraction: I ----> i
� Relative error of the approximation
Paraxial Approximation
i0 5 10 15 20 25 30 35 40
0
0.01
0.02
0.03
0.04
0.05
((((i'- I') / I'
n' = 1.9
n' = 1.7
n' = 1.5
...!5!3
sin53
−+−=xx
xx
'sin'sin InIn ⋅=⋅
1
'
sinarcsin
'
'
''−
⋅
⋅
=−
=
n
inn
in
I
Iiε
'' inin ⋅=⋅
Linear Collineation
� General rational transformation
� Linear expression
� Describes linear collinear transform x,y,z ---> x',y',z'
� Analog in the image space
� Inserted in only 2 dimensions
� Focal lengths
� Principal planes
0
3
0
2
0
1 ',','F
Fz
F
Fy
F
Fx ===
3,2,1,0, =+++= jdzcybxaF jjjjj
0
3
0
2
0
1
'
',
'
',
'
'
F
Fz
F
Fy
F
Fx ===
3,2,1,0,'''''''' =+++= jdzcybxaF jjjjj
0
1
0
33 ','dzc
yay
dzc
dzcz
oo +=
+
+=
01
0303
0
1 ',ca
cddcf
c
af
−==
01
030313
'
0
01 ,ca
cddcacz
c
daz PP
+−=
−=
Linear Collineation
� Special choice of origin of coordinate systems: Newton imaging equations
� Finite angles: tan(u) must be taken:
Magnification:
Focal length:
Invariant:
O
O'
y'
y
F F'
P P'
N N'
f'
a'a
f
z'z
u'u
h
u
um
tan
'tan=
h
uu
f
tan'tan
'
1 −=
'tan''tan uynuny =
� Positive lens
Real image for s > f
� Negative lens
virtual image
F'F
y
f f
y'
s's
F'F
y
f f
y'
s'
s
Image Construction of a Lens by simple Rays
� Graphical image construction
according to Listing by
3 special rays:
1. First parallel through axis,
through focal point in image
space F‘
2. First through focal point F,
then parallel to optical axis
3. Through nodal points,
leaves the lens with the same
angle
� Procedure work for positive
and negative lenses
For negative lenses the F / F‘ sequence is
reversed
Graphical Image Construction after Listing
� First ray parallel to arbitrary ray through focal point, becomes parallel to optical axis
� Arbitrary ray:
- constant height in principal planes S � S‘
- meets the first ray in the back focal plane,
desired ray is S‘Q
General Graphical Ray Construction
� Principle setups of the image of a
positive lens
� Object and image can be real/virtual
Positive Lens
from infinity
real image
virtual image
F F'
F'F
F'F
virtual objectlocation
� Ranges of imaging
Location of the image for a single
lens system
� Change of object loaction
� Image could be:
1. real / virtual
2. enlarged/reduced
3. in finite/infinite distance
Imaging by a Lens
|s| < f'
image virtualmagnified F
Objekt
s
F object
s
Fobject
s
F
object
s
F'
F
object
image
s
|s| = f'
2f' > |s| > f'
|s| = 2f'
|s| > 2f'
F'
F'
F'
F'
image
image
image
image
image atinfinity
image realmagnified
image real1 : 1
image realreduced
� Lateral magnification for finite imaging
� Scaling of image size'tan'
tan'
uf
uf
y
ym
⋅
⋅−==
Magnification
z f f' z'
y
P P'
principal planes
object
imagefocal pointfocal point
s
s'
y'
F F'
� Imaging on axis: circular / rotational symmetry
Only spherical aberration and chromatical aberrations
� Finite field size, object point off-axis:
- chief ray as reference
- skew ray bundels:
coma and distortion
- Vignetting, cone of ray bundle
not circular symmetric
- to distinguish:
tangential and sagittal
plane
Definition of Field of View and Aperture
� Afocal systems with object/image in infinity
� Definition with field angle w
angular amgnification
� Relation with finite-distance magnification
''tan
'tan
hn
nh
w
w==γ
Magnification
w
w'
'f
f−=⋅γβ
� Axial magnification
� Approximation for small ∆z and n = n‘
f
zf
f
z
z
∆⋅−
⋅⋅−=∆
∆=
ββα
1
1'' 2
'tan
tan2
2
2
u
u−=−= βα
Axial Magnification
∆∆∆∆z ∆∆∆∆z'
� Distance object-image:
(transfer length)
� Two solution for a given L with
different magnifications
� No real imaging for L < 4f
Transfer Length / Total Track L
++⋅=
mmfL
12'
''21
'2
2
f
L
f
L
f
Lm −
±−=
| m |
0 1 2 3 4 5 6 7 80
1
2
3
4
5
6
L / f'
magnified
reduced
Lmin
= 4f' mmax
= L / f' - 2
4f-imaging
� Magnification parameter M:
defines ray path through the lens
� Special cases:
1. M = 0 : symmetrical 4f-imaging setup
2. M = -1: object in front focal plane
3. M = +1: object in infinity
� The parameter M strongly influences the aberrations
1'
21
2
1
1
'
'−=+=
−
+=
−
+=
s
f
s
f
UU
UUM
β
β
Magnification Parameter
M=0
M=-1
M<-1
M=+1
M>+1
� Image in infinity:
- collimated exit ray bundle
- realized in binoculars
� Object in infinity
- input ray bundle collimated
- realized in telescopes
- aperture defined by diameter
not by angle
Special Setups of Object / Image
object atinfinity
image in
focalplane
lens acts as
aperture stop
collimatedentrance bundle
image atinfinity
stop
image
eye lens
field lens
� Imaging by a lens in air:
lens makers formula
� Magnification
� Real imaging:
s < 0 , s' > 0
� Intersection lengths s, s'
measured with respective to the
principal planes P, P'
fss
11
'
1=−
s
s'=β
Imaging Equation
s'
2f'
4f'
2f' 4f'
s-2f'- 4f'
-2f'
- 4f'
real object
real image
real object
virtual object
virtual image
virtual image
real image
virtual image
� Imaging equation in inverse
representation with refractive
power and vergence 1/s:
linear behavior
Imaging Equation
1/s'
2ΦΦΦΦ
4ΦΦΦΦ
2ΦΦΦΦ 4ΦΦΦΦ
1/s-2ΦΦΦΦ- 4ΦΦΦΦ
-2ΦΦΦΦ
-4ΦΦΦΦ
real objectreal image
real objectvirtaul image
virtual objectvirtual image
virtual objectreal image
ΦΦΦΦ
ΦΦΦΦ
- ΦΦΦΦ
� Imaging equation according to Newton:
distances z, z' measured relative to the
focal points
'' ffzz ⋅=⋅
Newton Formula
� Two lenses with distance d
� Focal length
distance of inner focal points e
� Sequence of thin lenses close
together
� Sequence of surfaces with relative
ray heights hj, paraxial
� Magnification
n
FFdFFF 21
21
⋅⋅−+=
e
ff
dff
fff 21
21
21⋅
=−+
⋅=
∑=k
kFF
( )∑ ⋅−⋅=k k
kkk
rnn
h
hF
1'
1
kk
k
n
n
s
s
s
s
s
s
'
''' 1
2
2
1
1 ⋅⋅⋅⋅⋅=β
Multi-Surface Systems
� Focal length
e: tube length
� Image location
Two-Lens System
e
ff
dff
fff 21
21
21 ''
''
'''
⋅=
−+
⋅=
1
1
21
21
2'
')'(
''
')'('
f
fdf
dff
fdfs
⋅−=
−+
⋅−=
Matrix Calculus
� Paraxial raytrace transfer
� Matrix formulation
� Paraxial raytrace refraction
� Inserted
� Matrix formulation
111 −−− ⋅+= jjjj Udyy
1−+⋅= jjjj Uyi ρ in
nij
j
j
j''
=
1' −= jj UU
1−= jj yy
( )1
'
'' −+
−⋅−= j
j
j
j
j
jjj
j Un
ny
n
nnU
ρ
'' 1 jjjj iiUU +−= −
⋅
=
−
j
jj
j
j
U
yd
U
y
10
1
'
'1
( )
⋅
−⋅
−=
j
j
j
j
j
jjj
j
j
U
y
n
n
n
nnU
y
'
'01
'
' ρ
� Linear relation of ray transport
� Simple case: free space propagation
� General case:
paraxial segment with matrix
ABCD-matrix :
=
=
u
xM
u
x
DC
BA
u
x
'
'
z
x x'
ray
x'
u'
u
x
B
Matrix Formulation of Paraxial Optics
A B
C D
z
x x'
ray x'
u'u
x
� Linear transfer of spation coordinate x
and angle u
� Matrix representation
� Lateral magnification for u=0
� Angle magnification of conjugated planes
� Refractive power for u=0
� Composition of systems
� Determinant, only 3 variables
uDxCu
uBxAx
+=
+=
'
'
⋅=
⋅
=
u
xM
u
x
DC
BA
u
x
'
'
β== xxA /'
γ== uuD /'
xuC /'=
121... MMMMM kk ⋅⋅⋅⋅= −
'det
n
nCBDAM =−=
Matrix Formulation of Paraxial Optics
� System inversion
� Transition over distance L
� Thin lens with focal length f
� Dielectric plane interface
� Afocal telescope
−
−=
−
AC
BDM
1
=
10
1 LM
−=
11
01
f
M
=
'0
01
n
nM
ΓΓ=0
1L
M
Matrix Formulation of Paraxial Optics
� Calculation of intersection length
� Magnifications:
1. lateral
2. angle
3. axial, depth
� Principal planes
� Focal points
Matrix Formulation of Paraxial Optics
DsC
BsAs
+⋅
+⋅='
DsC
BCAD
+⋅
−=β
( )2
'
DsC
BCAD
ds
ds
+⋅
−==α
'sCA
BCADDsC
⋅−
−=+⋅=γ
C
DBCADaH
−−=
C
AaH
1'
−=
C
AaF ='
C
DaF −=
Phase Space
z
x
x
u
1
1
2 2'
2 2'
3 3' 4
33'
5
4
5
6
6
free
transfer
lens 1
lens 2
grin lens
lens 1
lens 2
free
transfer
free transfer
free transfer
Direct phase space
representation of raytrace:
spatial coordinate vs angle
� Grin lens with aberrations in
phase space:
- continuous bended curves
- aberrations seen as nonlinear
angle or spatial deviations
Phase Space
z
x
x
u u
x
u
x
Phase Space: 90°-Rotation
� Transition pupil-image plane: 90°rotation in phase space
� Planes Fourier inverse
� Marginal ray: space coordinate x ---> angle θ'
� Chief ray: angle θ ---> space coordinate x'
� Product of field size y and numercial aperture is invariant in a paraxial system
� The invariant L describes to the phase space volume (area)
� The invariance corresponds to
1. Energy conservation
2. Liouville theorem
3. Constant transfer of information
y
y'
u u'
marginal ray
chief ray
object
image
system
and stop
''' uynuynL ⋅⋅=⋅⋅=
Helmholtz-Lagrange Invariant
� Geometrical optic:
Etendue, light gathering capacity
� Paraxial optic: invariant of Lagrange / Helmholtz
� Invariance corresponds to conservation of energy
� Interpretation in phase space:
constant area, only shape is changed
at the transfer through an optical
system
uD
LGeo sin2
⋅=
''' uynuynL ⋅⋅=⋅⋅=
Helmholtz-Lagrange Invariant
� Laser optics: beam parameter product
waist radius times far field divergence angle
� Minimum value of L:
TEMoo - fundamental mode
� Elementary area of phase space:
Uncertainty relation in optics
� Laser modes: discrete structure of phase space
� Geometrical optics: quasi continuum
� L is a measure of quality of a beam
small L corresponds to a good focussability
ooGB wL θ⋅=
π
λ=GBL
( )12 +⋅=⋅= nwL nnGBπ
λθ
Helmholtz-Lagrange Invariant
Summary of Important Topics
� Nonlinearity of the law of refraction defines the paraxial approximation
� Linera collineation: general approach of linear mapping, in case of larger angles with tan(u)
� Graphical image construction in paraxial optics:
3 rays determine location and size of the image: nodal ray, rays through focal points F, F‘
are parallel to axis
� Imaging condition of a single lens: real image for object distances s>2f , virtual images for
closer object points
� Definition of lateral magnification, angle and depth magnification
� Lens makers formula -1/s+1/s‘=1/f allows for paraxial imaging calculations
� Combination of several lenses, cascaded systems
For thin components near together: focal power F=1/f is additive
� Matrix calculus: practical calculation scheme with 2x2 ABCD matrices, connect ray
coordinate and angle between two planes
� Phase space in optics: spatial coordinate y and angle u
� Illustration of optical systems and ray paths by a line in the phase space
� Lagrange invariant: constant area of a ray bundle in phase space, corresponds to the
conservation of energy
40
Outlook
Next lecture: Part 5 – Properties of optical systems
Date: Wednesday, 2012-05-16
Content: 5.1 Pupil - basic notations
- pupil
- special rays
- vignetting
- ray sets
5.2 Special imaging setups - telecentricity
- anamorphotic imaging
- Scheimpflug condition
5.3 Canonical coordinates - normalized properties
- pupil sphere
5.4 Delano diagram - basic idea
- examples