Design and Correction of Optical Systemsand+Correction... · Optical System Classification...
Transcript of Design and Correction of Optical Systemsand+Correction... · Optical System Classification...
-
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Design and Correction of Optical
Systems
Lecture 3: Paraxial optics
2017-04-28
Herbert Gross
Summer term 2017
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2
Preliminary Schedule - DCS 2017
1 07.04. Basics Law of refraction, Fresnel formulas, optical system model, raytrace, calculation
approaches
2 21.04. Materials and Components Dispersion, anormal dispersion, glass map, liquids and plastics, lenses, mirrors,
aspheres, diffractive elements
3 28.04. Paraxial Optics Paraxial approximation, basic notations, imaging equation, multi-component
systems, matrix calculation, Lagrange invariant, phase space visualization
4 05.05. Optical Systems Pupil, ray sets and sampling, aperture and vignetting, telecentricity, symmetry,
photometry
5 12.05. Geometrical Aberrations Longitudinal and transverse aberrations, spot diagram, polynomial expansion,
primary aberrations, chromatical aberrations, Seidels surface contributions
6 19.05. Wave Aberrations Fermat principle and Eikonal, wave aberrations, expansion and higher orders,
Zernike polynomials, measurement of system quality
7 26.05. PSF and Transfer function Diffraction, point spread function, PSF with aberrations, optical transfer function,
Fourier imaging model
8 02.06. Further Performance Criteria Rayleigh and Marechal criteria, Strehl definition, 2-point resolution, MTF-based
criteria, further options
9 09.06. Optimization and Correction Principles of optimization, initial setups, constraints, sensitivity, optimization of
optical systems, global approaches
10 16.06. Correction Principles I Symmetry, lens bending, lens splitting, special options for spherical aberration,
astigmatism, coma and distortion, aspheres
11 23.06. Correction Principles II Field flattening and Petzval theorem, chromatical correction, achromate,
apochromate, sensitivity analysis, diffractive elements
12 30.06. Optical System Classification Overview, photographic lenses, microscopic objectives, lithographic systems,
eyepieces, scan systems, telescopes, endoscopes
13 07.07. Special System Examples Zoom systems, confocal systems
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1. Paraxial approximation
2. Ideal surfaces and lenses
3. Imaging equation
4. Matrix formalism
5. Lagrange invariant
6. Phase space considerations
7. Delano diagram
3
Contents
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Paraxiality is given for small angles
relative to the optical axis for all rays
Large numerical aperture angle u
violates the paraxiality,
spherical aberration occurs
Large field angles w violates the
paraxiality,
coma, astigmatism, distortion, field
curvature occurs
Paraxial Approximation
4
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Paraxial approximation:
Law of refraction for finite angles I, I‘
sin-expansion
Small incidence angles allows for a linearization of the law of refraction
Small angles of rays at every surface
All optical imaging conditions become linear (Gaussian optics),
calculation with ABCD matrix calculus is possible
No aberrations occur in optical systems
There are no truncation effects due to transverse finite sized components
Serves as a reference for ideal system conditions
Is the fundament for many system properties (focal length, principal plane, magnification,...)
The sag of optical surfaces (difference in z between vertex plane and real surface
intersection point) can be neglected
All waves are plane of spherical (parabolic)
The phase factor of spherical waves is quadratic
Paraxial approximation
'' inin
R
xi
eExE 2
0)(
...36288050401206
sin9753
iiii
ii
'sin'sin InIn
5
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Taylor expansion of the
sin-function
Definition of allowed error
10-4
Deviation of the various
approximations:
- linear: 5° - cubic: 24° - 5th order: 542°
6
Paraxial approximation
0 10 20 30 40 50 60 70 80 900
0.2
0.4
0.6
0.8
1
sin(x)
x [°]
exact sin(x)
linear
cubic
5th order
x = 5° x = 24° x = 52° deviation 10-4
-
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
Ii
'' inin
7
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8
Generalized Paraxiality
Pitfalls in the classical definition of paraxiality:
1. Central obscurartion and ring-shaped pupil:
- Paraxial marginal ray of no relevance
- Reference on centroid ray
2. General 3D system without straight axis:
Central ray as reference, calculated finite
parabasal rays in the neighborhood of the
real chief ray
Distortion information is lost
General quasi-parabasal rays:
- macroscopic astigmatism
- aberrations reference definition more complicated
- separated view on cheif ray / marginal ray
M
F
M1circular
symmetric
asphere
M2pupil
freeform M3freeform
image
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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
image
object
optical
system
O2field
point
axis
pupil
stop
marginal
ray
O1 O'1
O'2
chief
ray
Optical imaging
9
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Single surface
imaging equation
Thin lens in air
focal length
Thin lens in air with one plane
surface, focal length
Thin symmetrical bi-lens
Thick lens in air
focal length
'
1'
'
'
fr
nn
s
n
s
n
21
111
'
1
rrn
f
1'
n
rf
12'
n
rf
21
2
21
1111
'
1
rrn
dn
rrn
f
Formulas for surface and lens imaging
10
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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
y
n'ny'
r
C
ray through center of curvature C principal
plane
vertex S
s
s'
object
surface
image
arbitrary ray
'
11'
11
srn
srnQs
Single Surface
11
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Imaging with a lens
Location of the image:
lens equation
Size of the image:
Magnification
Imaging by a Lens
object image
-sf
+s'
system
lens
y
y'
fss
11
'
1
s
s
y
ym
''
12
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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 virtual
magnified FObjekt
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 at
infinity
image real
magnified
image real
1 : 1
image real
reduced
13
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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'
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
s
sm
'
14
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Distance object-image:
(transfer length)
L = |s| + |s‘|
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
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Lateral magnification for finite imaging
Scaling of image size
'tan'
tan'
uf
uf
y
ym
z f f' z'
y
P P'
principal planes
object
imagefocal pointfocal point
s
s'
y'
F F'
Magnification
16
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Afocal systems with object/image in infinity
Definition with field angle w
angular magnification
Relation with finite-distance magnification
''tan
'tan
hn
nh
w
w
'f
fm
Angle Magnification
h
h'w'
w
17
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Imaging equation according to Newton:
distances z, z' measured relative to the
focal points
'' ffzz
Newton Formula
-z -f f' z'
y
P P'
principal planes
image
focal
point F'focal
point F
-s s'
object
18
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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
F
F'
y
y'
P'P
1
2
3
F'
F
y
y'
P'P
1
2
3
19
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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
F'F
P'P
Qarbitrary ray
desired output ray
parallel ray
through F
f'
S'S
20
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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 2121
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
sm
'
''' 1
2
2
1
1
Multi-Surface Systems
21
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System of two separated thin lenses
Variation of the back principal plane
as a function of the distribution of
refractive power
22
Principal Planes
P' L1 L2P F
plate
plate
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Imaging with tilted object plane
If principal plane, object and image plane meet in a common point:
Scheimpflug condition,
sharp imaging possible
Scheimpflug equation
tan'tan
tantan
'
s
s
Scheimpflug Imaging
'
y
s
s'
h
y'
h'
tilted
object
tilted
image
system
optical axis
principal
plane
23
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General :
1. Image plane is tilted
2. Magnification is anamorphic
Example :
Scheimpflug-Imaging
Scheimpflug System
object plane
lens
image plane
'
s
s'
'sin
sin2
oy mm
tan
'tan'
s
smm ox
24
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Linear relation of ray transport
Simple case: free space
propagation
Advantages of matrix calculus:
1. simple calculation of component
combinations
2. Automatic correct signs of
properties
3. Easy to implement
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
25
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Matrix Calculus
Paraxial raytrace transfer
Matrix formulation
Matrix formalism for finite angles
Paraxial raytrace refraction
Inserted
Matrix formulation
111 jjjj Udyy
1 jjjj Uyi in
nij
j
j
j''
1' jj UU
1 jj yy
1
'
''
j
j
j
j
j
jjj
j Un
ny
n
nnU
'' 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
'
'
j
j
j
j
u
y
DC
BA
u
y
tan'tan
'
26
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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
'
'
mxxA /'
uuD /'
xuC /'
121 ... MMMMM kk
'det
n
nCBDAM
Matrix Formulation of Paraxial Optics
27
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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
0
1L
M
Matrix Formulation of Paraxial Optics
28
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Calculation of intersection length
Magnifications:
1. lateral
2. angle
3. axial, depth
Principal planes
Focal points
Matrix Formulation of Paraxial Optics
DsC
BsAs
'
DsC
BCADm
2'
DsC
BCAD
ds
ds
'sCA
BCADDsC
C
DBCADaH
C
AaH
1'
C
AaF '
C
DaF
29
-
Decomposition of ABCD-Matrix
2x2 ABCD-matrix of a system in air: 3 arbitrary parameters
Every arbitrary ABCD-setup can be decomposed into a simple system
Decomposition in 3 elementary partitions is alway possible
Case 1: C # 0 one lens, 2 transitions
System data
MA B
C D
L
f
L
1
0 1
1 01
1
1
0 1
1 2
LD
C1
1
LA
C2
1
fC
1
Output
xo
Input
xi
Lens
f
L2
L1
-
Decomposition of ABCD-Matrix
Case 2: B # 0 two lenses, one transition
System data:
MA B
C D f
L
f
1 01
11
0 1
1 01
12 1
fB
A1
1
L B
fB
D2
1
OutputInput
Lens 1
f1
L
Lens 2
f2
-
Product of field size y and numercial aperture is
invariant in a paraxial system
Derivation at a single refracting surface:
1. Common height h:
2. Triangles
3. Refraction:
4. Elimination of s, s',w,w'
The invariance corresponds to:
1. Energy conservation
2. Liouville theorem
3. Invariant phase space volume (area)
4. Constant transfer of information
''' uynuynL
Helmholtz-Lagrange Invariant
y
u u'
chief ray
marginal ray
w
h
ss'
w'
n n'
surface
''usush
'
'',
s
yw
s
yw
''wnnw
32
-
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
33
-
Basic formulation of the Lagrange
invariant:
Uses image heigth,
only valid in field planes
General expression:
1. Triangle SPB
2. Triangle ABO'
3. Triangle SQA
4. Gives
5. Final result for arbitrary z:
Helmholtz-Lagrange Invariant
ExPCR sswy ''''
ExP
CR
s
yw
''
''
s
yu MR
z
y'y yp
pupil imagearbitrary z
chief
ray
marginal
ray
s's'Exp
y'CRyMR
yCRB A
O'
Q
S
P
ExPMRExPMR
CR swuwynssws
ynyunL ''''''''
'''''
)(')('' zyuzywnL CRMR
34
-
Simple example:
- A microscope is a 4f-system with
objective lens (fobj = 3 mm) and tube lens (fobj = 180 mm)
- the numerical aperture is NA = 0.9 and the intermediate Image size D = 2yima = 25 mm
- magnification
- image sided aperture
- pupil size
- object field
60/ objTL ffm
Helmholtz-Lagrange Invariant
yobjyimauima
image diameter
25 mm
tube lens:
focal length 180 mm
uobj
objective lens
focal length 3 mm
NA = 0.9
pupil
015.0/ muu objima
mmNAfD objpup 7.2
mmuuyy objimaimaobj 42.0/
35
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Geometrical optic:
Etendue, light gathering capacity
Paraxial optic: invariant of Lagrange / Helmholtz
General case: 2D
Invariance corresponds to
conservation of energy
Interpretation in phase space:
constant area, only shape is changed
at the transfer through an optical
system
unD
Lfield
Geo sin2
''' uynuynL
Helmholtz-Lagrange Invariant
space y
angle u
large
aperture
aperture
small
1
23
medium
aperture
y'1
y'2
y'3u1 u2
u3
36
-
Direct phase space
representation of raytrace:
spatial coordinate vs angle
Phase Space
z
x
x
u u
x
x1
x'1
x2
x'2
x1 x'1
x2
x'2
u1 u2 u1 u2
11'
2
2'
1
1'
2
2'
space
domain
phase
space
37
-
Phase Space
z
x
I
x
I
x
u
x
u
x
38
-
Phase Space
z
x
x
u
1
1
2 2'
2 2'
3 3'4
3
3'
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
39
-
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
40
-
1. Slit diffraction
Diffraction angle inverse to slit
width D
2. Gaussian beam
Constant product of waist size wo
and divergence angle o
00w
D
D D
o
wo
x
z
Uncertainty Relation in Optics
41
-
• Angle u is limited
• Typical shapes:
Ray : point (delta function)
Coherent plane wave: horizonthal line
Extended source : area
Isotropic point source: vertical line
Gaussian beam: elliptical area with
minimal size
• Range of small etendues: modes, discrete
structure
• Range of large etendues: quasi continuum
Phase Space in Optics
plane wave
(laser)
x
u
ray
spherical
wave
gaussian
beam
LED
point
source
x
u
discrete
mode
points
quasi continuum
42
-
Simplified pictures to the changes of the phase space density.
Etendue is enlarged, but no complete filling.
x
u
x
u
x
u1) before array
x
u
x
u
x
u
2) separation of
subapertures
3) lens effect
of subapertures
4) in focal plane 5) in 2f-plane 6) far away
1 2 3 4 5 6
Example Phase Space of an Array
43
-
Analogy Optics - Mechanics
44
Mechanics Optics
spatial variable x x
Impuls variable p=mv angle u, direction cosine p,
spatial frequency sx, kx
equation of motion spatial
domain
Lagrange Eikonal
equation of motion phase space Hamilton Wigner transport
potential mass m index of refraction n
minimal principle Hamilton principle Fermat principle
Liouville theorem constant number of
particles
constant energy
system function impuls response point spread function
wave equation Schrödinger Helmholtz
propagation variable time t space z
uncertainty h/2p l/2p
continuum approximation classical mechanics geometrical optics
exact description quantum mechanics wave optics
-
Delano Diagram
Special representation of ray bundles in
optical systems:
marginal ray height
vs.
chief ray height
Delano digram gives useful insight into
system layout
Every z-position in the system corresponds
to a point on the line of the diagram
Interpretation needs experience
CRyy
lens
y
field lens collimatormarginal ray
chief ray
y
y
y
lens at
pupil
position
field lens
in the focal
plane
collimator
lens
MRyy
-
46
Delano Diagram
Delano’s
skew ray
Image
d2 d1
yC yM
(yC,yM) yM
a a
b
c c
d d
yC Delano ray (blue)=
Chief ray (red) in x +
Marginal ray (green) in y
Delano Diagram =
Delano ray projected
into the xy-Plane
Substitution
x -->
y = Pupil coordinate
= yc Field coordinate
Stop
Lens
a b
c
d
y (or yM)
y
y
y
Ref.: M. Schwab / M. Geiser
chief ray
marginal ray
Delano diagram:
projection
along z
b
x
y
marginal
ray
chief
ray
skew ray
y
y
y
y
diagram
image
object
-
Pupil locations:
intersection points with y-axis
Field planes/object/image:
intersectioin points with y-bar axis
Construction of focal points by
parallel lines to initial and final line
through origin
y
y
object
plane
lens
image
plane
stop and
entrance pupil
exit pupil
y
y
object
space
image
space
front focal
point Frear focal
point F'
Delano Diagram
-
Delano Diagram
Influence of lenses:
diagram line bended
Location of principal planes
y
y
strong positive
refractive power
weak positive
refractive power
weak negative
refractive power
y
y
object space image space
principal
plane
yP
yP
-
Delano Diagram
Afocal Kepler-type telescope
Effect of a field lens
y
y
lens 1
objective
lens 2
eyepiece
intermediate
focal point
y
y
lens 1
objective
lens 2
eyepiece
field lensintermediate
focal point
-
Delano Diagram
Microscopic system
y
y
eyepiece
microscope
objective tube lens
object
image at infinity
aperture
stop
intermediate
image
exit pupil
telecentric
-
Conjugated point are located on a
straight line through the origin
Distance of a system point from
origin gives the systems half
diameter
Delano Diagram
y
y
object
space image
space
conjugate
line
conjugate line
with m = 1
principal point
conjugate
points
y
y
maximum height of
the coma ray at
lens 2
curve of
the system
lens 1lens 2
lens 3
D/2
51
-
Location of principal planes in the Delano diagram
Triplet Effect of stop shift
Delano Diagram
y
y
object
plane
lens L1
lens L2
lens L3
image
plane
stop shift
y
y
object
spaceimage
space
principal plane
yP
yP
-
Vignetting :
ray heigth from axis
Marginal and chief ray considered
Line parallel to -45° maximum diameter
yya
Delano Diagram
object
pupil
chief ray
marginal ray
coma ray
yyy + y
y
y
maximum height at
lens 2
system polygon line
lens 1
lens 2
lens 3
D/2