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Design and Correction of optical Systems
Part 4: Paraxial optics
Summer term 2012Herbert Gross
1
Overview
1. Basics 2012-04-182. Materials 2012-04-253. Components 2012-05-024. Paraxial optics 2012-05-095. Properties of optical systems 2012-05-166. Photometry 2012-05-237. Geometrical aberrations 2012-05-308. Wave optical aberrations 2012-06-069. Fourier optical image formation 2012-06-1310. Performance criteria 1 2012-06-2011. Performance criteria 2 2012-06-2712. Measurement of system quality 2012-07-0413. Correction of aberrations 1 2012-07-1114. 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 imagingideal, paraxial
Image field size:Chief ray
Aperture/size of light cone:marginal raydefined by pupil stop
Optical Imaging
image
object
optical system
O2fieldpoint
axis
pupilstop
marginal ray
O1 O'1
O'2
chiefray
Single surface between two mediaRadius r, refractive indices n, n‘
Imaging condition, paraxial
Abbe invariantalternative representation of theimaging equation
'1'
''
frnn
sn
sn
'11'11sr
nsr
nQs
Single Surface
Law of refraction
Expansioin of the sine-function:
Linearized approximation of thelaw 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' = 1.9n' = 1.7n' = 1.5
...!5!3
sin53
xxxx
'sin'sin InIn
1
'sinarcsin
''
''
nin
nin
IIi
'' 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 ',','FFz
FFy
FFx
3,2,1,0, jdzcybxaF jjjjj
0
3
0
2
0
1
'',
'',
''
FFz
FFy
FFx
3,2,1,0,'''''''' jdzcybxaF jjjjj
0
1
0
33 ','dzc
yaydzcdzcz
oo
01
0303
0
1 ',ca
cddcfcaf
01
030313'
0
01 ,ca
cddcaczc
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
uum
tan'tan
huu
ftan'tan
'1
'tan''tan uynuny
Positive lensReal image for s > f
Negative lensvirtual image
F'Fy
f f
y'
s's
F'Fy
f f
y'
s'
s
Image Construction of a Lens by simple Rays
Graphical image constructionaccording to Listing by3 special rays:
1. First parallel through axis,through focal point in imagespace F‘
2. First through focal point F,then parallel to optical axis
3. Through nodal points,leaves the lens with the sameangle
Procedure work for positiveand negative lensesFor negative lenses the F / F‘ sequence isreversed
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 apositive 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 imagingLocation of the image for a singlelens system
Change of object loaction
Image could be:1. real / virtual2. enlarged/reduced3. in finite/infinite distance
Imaging by a Lens
|s| < f'image virtual
magnified 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'ufuf
yym
Magnification
z f f' z'
y
P P'
principal planes
objectimage
focal pointfocal point
s
s'
y'
F F'
Imaging on axis: circular / rotational symmetryOnly 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 bundlenot circular symmetric
- to distinguish:tangential and sagittalplane
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
hnnh
ww
Magnification
w
w'
'ff
Axial magnification
Approximation for small z and n = n‘
fzf
fzz
1
1'' 2
'tantan
2
22
uu
Axial Magnification
z z'
Distance object-image:(transfer length)
Two solution for a given L withdifferent magnifications
No real imaging for L < 4f
Transfer Length / Total Track L
mmfL 12'
''21
'2
2
fL
fL
fLm
| 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 setup2. M = -1: object in front focal plane 3. M = +1: object in infinity
The parameter M strongly influences the aberrations
1'
21211
''
sf
sf
UUUUM
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 infocalplane
lens acts asaperture stop
collimatedentrance bundle
image atinfinity
stop
image
eye lensfield 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 theprincipal planes P, P'
fss11
'1
ss'
Imaging Equation
s'
2f'
4f'
2f' 4f's
-2f'- 4f'
-2f'
- 4f'
real objectreal image
real objectvirtual objectvirtual image
virtual imagereal image
virtual image
Imaging equation in inverserepresentation with refractivepower and vergence 1/s:linear behavior
Imaging Equation
1/s'
2
4
2 41/s
-2- 4
-2
-4
real objectreal image
real objectvirtaul image
virtual objectvirtual image
virtual objectreal image
-
Imaging equation according to Newton:distances z, z' measured relative to the focal points
'' ffzz
Newton Formula
Two lenses with distance d
Focal lengthdistance of inner focal points e
Sequence of thin lenses closetogether
Sequence of surfaces with relativeray heights hj, paraxial
Magnification
nFFdFFF 21
21
eff
dfffff 21
21
21
k
kFF
k k
kkk
rnn
hhF 1'
1
kk
k
nn
ss
ss
ss
'''' 1
2
2
1
1
Multi-Surface Systems
Focal lengthe: tube length
Image location
Two-Lens System
eff
dfffff 21
21
21 '''''''
1
1
21
212 '
')'(''
')'('f
fdfdfffdfs
Matrix Calculus
Paraxial raytrace transfer
Matrix formulation
Paraxial raytrace refraction
Inserted
Matrix formulation
111 jjjj Udyy
1 jjjj Uyi inn
ijj
jj'
'
1' jj UU
1 jj yy
1'
''
j
j
jj
j
jjjj U
nn
yn
nnU
'' 1 jjjj iiUU
j
jj
j
j
Uyd
Uy
101
'' 1
j
j
j
j
j
jjj
j
j
Uy
nn
nnn
Uy
''
01
''
Linear Collineation
Matrix formalism for finite angles
j
j
j
j
uy
DCBA
uy
tan'tan'
Linear relation of ray transport
Simple case: free space propagation
General case:paraxial segment with matrixABCD-matrix :
ux
Mux
DCBA
ux''
z
x x'
ray
x'
u'
u
xB
Matrix Formulation of Paraxial Optics
A BC D
z
x x'
ray x'
u'u
x
Linear transfer of spation coordinate xand 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
uDxCuuBxAx
''
ux
Mux
DCBA
ux''
xxA /'
uuD /'
xuC /'
121 ... MMMMM kk
'det
nnCBDAM
Matrix Formulation of Paraxial Optics
System inversion
Transition over distance L
Thin lens with focal length f
Dielectric plane interface
Afocal telescope
ACBD
M 1
101 L
M
1101
fM
'0
01
nnM
0
1 LM
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
DsCBsAs
'
DsCBCAD
2'
DsCBCAD
dsds
'sCABCADDsC
CDBCADaH
CAaH
1'
CAaF ' C
DaF
Direct phase space representation of raytrace:spatial coordinate vs angle
Phase Space
Phase Space
z
x
I
x
I
x
u
x
u
x
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 1lens 2
free transfer
free transfer
free transfer
Direct phase space representation of raytrace:spatial coordinate vs angle
Grin lens with aberrations inphase 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 ' Chief ray: angle ---> 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 to1. Energy conservation2. Liouville theorem3. 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 changedat the transfer through an optical system
uDLGeo 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 beamsmall L corresponds to a good focussability
ooGB wL
GBL
12 nwL nnGB
Helmholtz-Lagrange Invariant
Outlook
Next lecture: Part 5 – Properties of optical systemsDate: 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