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www.iap.uni-jena.de
Imaging and Aberration Theory
Lecture 10: Sine condition, aplanatism and isplanatism
2015-01-15
Herbert Gross
Winter term 2014
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2
Preliminary time schedule
1 30.10. Paraxial imaging paraxial optics, fundamental laws of geometrical imaging, compound systems
2 06.11. Pupils, Fourier optics, Hamiltonian coordinates
pupil definition, basic Fourier relationship, phase space, analogy optics and mechanics, Hamiltonian coordinates
3 13.11. Eikonal Fermat principle, stationary phase, Eikonals, relation rays-waves, geometrical approximation, inhomogeneous media
4 20.11. Aberration expansions single surface, general Taylor expansion, representations, various orders, stop shift formulas
5 27.11. Representation of aberrations different types of representations, fields of application, limitations and pitfalls, measurement of aberrations
6 04.12. Spherical aberration phenomenology, sph-free surfaces, skew spherical, correction of sph, aspherical surfaces, higher orders
7 11.12. Distortion and coma phenomenology, relation to sine condition, aplanatic sytems, effect of stop position, various topics, correction options
8 18.12. Astigmatism and curvature phenomenology, Coddington equations, Petzval law, correction options
9 08.01. Chromatical aberrations Dispersion, axial chromatical aberration, transverse chromatical aberration, spherochromatism, secondary spoectrum
10 15.01. Sine condition, aplanatism and isoplanatism
Sine condition, isoplanatism, relation to coma and shift invariance, pupil aberrations, Herschel condition, relation to Fourier optics
11 22.01. Wave aberrations definition, various expansion forms, propagation of wave aberrations
12 29.01. Zernike polynomials special expansion for circular symmetry, problems, calculation, optimal balancing, influence of normalization, measurement
13 05.02. PSF and transfer function ideal psf, psf with aberrations, Strehl ratio, transfer function, resolution and contrast
14 12.02. Additional topics Vectorial aberrations, generalized surface contributions, Aldis theorem, intrinsic and induced aberrations, revertability
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1. Pupil aberrations
2. Sine condition
3. Isoplanatism
4. Herschel condition
5. Relation to Fourier optics and phase space
3
Contents
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Sine Condition
y y'
n n'
zU U'
y'
y
Lagrange invariante for paraxial angles U, U‘
sin-condition:
extension for finite aperture angle u
Corresponds to energy conservation in the system
Constant magnification for alle aperture zones
Pupil shape for finite aperture is a sphere
Definition of violation of the sine condition:
OSC (offense against sine condition)
OSC = 0 means correction of sagittal coma (aplanatic system)
'sin'
sin
'' un
un
Un
nUm
'sin''sin UynUny
'sin''sin uynuny
4
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Optical Sine Condition
Condition for finite angles
Condition for object at infinity
Condition for afocal system
In the formulation
the sagittal magnification
is used
Un
Un
un
num
sin
sin
U
h
u
hf
sin
x'p
x'
y'
x
y
z'
y'p
yp
xp
UU
U’
U’
y
y's
kk h
h
H
H 11
Un
Un
y
ym s
s
sin
sin
5
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If for example a small field area and a widespread ray bundle is considered, a perfect
imaging is possible
The eikonal with the expression
can be written for dL=0 as
In the special case of an angle 90°we get with cos(q)=sin(u) the Abbe sine condition
with the lateral magnification
6
Abbe Sine Condition
P
Qs
u dr'
P'
Q's'
u'drq q
rdsnrdsnL
'''d
'cos'cos
'cos''cos
'''
nn
drndrn
rdsnrdsn
rd
rdm
'
'sin'
sin
un
unm
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From geometry
Refraction
Division and
substitution
7
Derivation of the Sine Condition
y
Po
P
u
sR
S
Qi
i'
w
w
s'M
s'
u' M'o
M's
P'o
P'
C
y's
ideal
image
plane
sagittal
image
plane
surface
chief ray
marginal
ray
sR
i
R
u
)sin(sin
Rs
i
R
u
M
'
'sinsin
'sin'sin inin
'sin
sin
''sin
sin
sin
'sin'
u
u
n
n
u
u
i
i
sR
Rs M
CP
PP
CM
MM
o
o
o
so '
''
sin ' 'sin 'syn u y n u
sR
y
Rs
y
M
s
'
'
y
y
sR
Rs sM ''
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Vectorial Sine Condition
General vectorial sine condition:
spatial frequencies / direction cosines are linear related from entrance to exit pupil
Generalization can be applied for anamorphic systems
object
plane
system
image
plane
yo
zo
sy0
q0
yi
zi
syi
qi
so si
yyoyyi
xxoxxi
csms
csms
8
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Transfer of Energy in Optical Systems
Conservation of energy
Invariant local differential flux
Assumption: no absorption
Delivers the sine condition
'22 PdPd
ddudAuuLPd cossin2
T 1
y
dA dA's's
EnP ExP
n n'
F'F
y'
u u'
'sin''sin uynuyn
9
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Sine condition fulfilled: linear scaling from entrance to exit pupil
Pupil surface must be sperical
The pupil height scales with the sine of the angle
Pupil Sphere
object
entrance pupil
sphere
image
yo y'
u u'
hEnP=REnP
sin(u)
hExP=RExP
sin(u')
exit pupil
sphere
REnP
RExP
objectyo
equidistant
h =R sin(u)
angle not
equidistant
spherical pupil
surface
10
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Sine condition fulfilled: linear scaling from entrance to exit pupil
Offence against the sine condition (OSC):
Exit pupil grid is distorted
Consequences:
1. Photometric effect causes apodization
2. Wave aberration could be calculated wrong
3. Spatial filtering on warped grid
Pupil Distortion
xoxp
sphere
distorted entrance
pupil surface
object
exit pupil
optical
system
sx
u
xp
distorted
exit
pupil
grid
1sin
unf
xD
ap
p
11
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Afocal system:
Lagrange invariant
(classical Lagrange invariant for pupil imaging)
Magnification
Sine condition
Fulfillment of the sine condition: linear scaling of entrance to exit pupil
Pupil Distortion
entrance
pupil
xp exit
pupil
x'p
ww'
xp
x'p
optical
system
field
angle
z
''' wxnwxn pp
'w
wm
p
p
xn
xnm
''
12
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Typical high-NA system
Virtual pupil located inside
Typical grid distortion
Sine Condition in Microscopic Objective Lens
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1x 10
-3
chief
ray
exit
pupil
rear
stopobject
plane
pupil
13
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Photometric effect of pupil distortion:
illumination changes at pupil boundary
Effect induces apodization
Sign of distortion determines the effect:
outer zone of pupil brighter / darker
Additional effect: absolute diameter of
pupil changes
OSC and Apodization
focused +20 mm +50 mm-20 mm-50 mm
0 10 20 30 40 500
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
-0.05 - barrel
+0.05 - pincushion
no
distortion
rp
intensity
14
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Spherical aberration of the chief ray / pupil imaging
Exit pupil location depends on the field height
Pupil Aberrations
yobject
sP
chief rays
pupil position
pupil
location
15
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Pupil Aberration
Interlinked imaging of field and pupil
Distortion of object imaging corresponds to spherical aberration of the pupil
imaging
Corrected spherical pupil aberration: tangent condition
O O’
stop and
entrance pupil
optical system
exit pupil
objectimage
Object imaging Pupil imaging
Blue rays
Red rays
Marginal rays
Marginal raysChief rays
Chief rays
.tan
'tanconst
w
w
16
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Eyepiece with pupil aberration
Illumination for decentered pupil :
dark zones due to vignetting
Pupil Aberration
eyepiecelens and
pupil of
the eye
retina
caustic of the pupil
image enlarged
instrument
pupil
17
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Wavefront and Spot for Coma
Coma Seidel trans-
verse aberrations
Wavefront for coma
with
Relationship
Here
2
2
3 2 3
3 2
' ' ' (2 cos 2 )
'
' ' ' cos (2 ' ') ' ' cos ' '
' ' ' si ' 'n ' ' ' sinsin 2
p p p Pp
p p pp P
P
P
y S r A P y r D y
x S r P y
C y r
C r ry
q q
q
q
q q
' ',
p p
W y W x
y R x R
y x
W
3 2 3cosp p p p pW r x y yq
2 2
2 2 2
' 2 2 sin cos sin 2
' 3 2 cos 2
p p p p p p p
p
p p p p
p
Wx R R x y Rr Rr
x
Wy R R x y Rr
y
q q q
q
sin , cosp p p p p px r y rq q
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Wavefront and Spot for Coma
Schematic geometry:
Notice the doubled revolution in the image plane due to combined effect of azimuthal
rotation and tilt of wavefront
exit
pupiltangential
coma rays
sagittal
coma rays
image
plane
y'
x'
yp
xp
chief ray
coma
spot
90°
0°0°
180°
90°
45°
45° rays
projection
wavefront
with coma
tangential
coma rays
chief ray
wavefront
with coma image
plane
sagittal
coma rays
projection
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Tangential and Sagittal Coma
2 terms of tangential transverse aberration:
- Sagittal coma depends on xp, describes the asymmetry
- Tangential coma depends on yp, corresponds to spherical aberration under skew conditions
larger by a factor of 3
Only asymmetry removed with sine condition: sagittal coma vanishes
exit
pupiltangential
coma rays
sagittal
coma rays
image
plane
y'
x'
yp
xp
chief ray
coma
spot0°
90°
45°
wavefront
with coma
ys'
2 2' 3 ' 'p p s ty R x y y y
yt'
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Linear coma (all orders)
Transverse aberrations
Sagittal coma yp = 0, xp = a
Sine condition fulfilled:
linear sagittal coma
vanishes
If in addition spherical
aberration is corrected
(aplanatic):
also tangential coma vanishes
Sine Condition and Coma
pupil
n n'
auxiliary axis
lower coma ray
upper coma ray
chief ray
sagittal rays
C
tangential
comasagittal
coma
astigmatic
difference between
coma rays
T
S
center of
curvature
surface R
chief ray
S
T
spot
S
lower
coma ray
upper coma ray
sagittal
rays
auxiliary axis
chief
ray
mppp
m
mppc yxyycyyxW 22')',,(
m
m
ppppm
p
cc yxymxcy
n
R
y
W
n
Ry
12222 )12('''
'
)',,0('
''
'
max,
2
,
yaWan
R
acyn
Ry
c
m
m
msagc
21
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Decomposition of coma:
1. part symmetrical around
chief ray: skew spherical
aberration
2. asymmetrical part:
tangential coma
Skew spherical aberration:
- higher order aberration
- caustic symmetric around
chief ray
Skew Spherical aberration
upper
coma ray
chief
ray
lower
coma ray exit
pupil
y'p
ideal image
location
S
sagittal image
point
tangential
image point
T
upper
coma ray
chief
ray
lower
coma ray
exit
pupil
y'p
ideal image
plane
common
intersection
point
2
lowcomupcom
tangcoma
yyy
2
lowcomupcom
skewsph
yyy
22
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Perfect imaging on axis due to conic section
- not aplanatic:
linear growth of coma with field size
Aplanatic:
- Perfect stigmatic imaging
on axis, spherical corrected
- linear coma vanishes:
good correction off-axis
but near to axis
- quadratic grows of spot size
due to astigmatism
- aplanatic and perfect
marginal ray quite different
Aplanatic and Perfect Imaging
0
100
50
Dspot
w in °0 1 2
mm]
real rays
ideal
lensideal
rays
real
lens
sin ureal' = 0.894
sin uideal' = 0.707
0
100
50
Dspot
w in °0 1 2
mm]
23
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Spherical Corrected Surface
Seidel contribution of spherical aberration
with
Result
Vanishing contribution:
1. first bracket: vertex ray
2. second bracket: concentric
3. bracket: aplanatic surface
Discussion with the Delano formula
2. concentric corresponds to i' = i
3. aplanatic condition corresponds to i' = u
24
jjjj
jjjsnsn
QS1
''
124
j
jh
h
1
jj
jjsR
nQ11
jjjjjj
j
j
jsnsnsR
nh
hS
1
''
1112
2
4
1
0jh
jjjj snsn ''
jj sR
j jjj
j
kkk
SPHSPHuU
uii
iih
n
n
uUn
uUnss
'sin'2
'sin2
2
'sin
''sin''
sin' 111
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Aplanatic Surfaces with Vanishing Spherical Aberration
s'
0 50 100 150 200 250 300-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
Saplanatic
concentricvertex
oblate ellipsoidoblate ellipsoid prolate ellipsoidhyperboloid+ power series + power series+ power series + power series
sp
he
re
sp
he
re
Aplanatic surfaces: zero spherical aberration:
1. Ray through vertex
2. concentric
3. Aplanatic
Condition for aplanatic
surface:
Virtual image location
Applications:
1. Microscopic objective lens
2. Interferometer objective lens
s s und u u' '
s s' 0
ns n s ' '
rns
n n
n s
n n
ss
s s
'
' '
'
'
'
25
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Aplanatic lenses
Combination of one concentric and
one aplanatic surface:
zero contribution of the whole lens to
spherical aberration
Not useful:
1. aplanatic-aplanatic
2. concentric-concentric
bended plane parallel plate,
nearly vanishing effect on rays
Aplanatic Lenses
A-A :
parallel offset
A-C :
convergence enhanced
C-C :
no effect
C-A :
convergence reduced
26
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General Aplanatic Surface
General approach
of Fermat principle:
aplanatic surface
Cartesian oval,
4th order
Special case OPD = 0:
Solution is spherical aplanatic surface
27
P
S
P'
oval
surface
r
ss'
z
nsnszsrnszrn '')'(')( 2222
2
2
2
2222222
22222222
2222
''
01'/'/'/21'/
'/2'/2'/
0)'/(')(
nn
snr
nn
snz
nnrnnnnzsnnz
nzsnsnnzrzsszrnn
zsnnrnszrn
nsns ''
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Isoplanatism
General definition of isoplanatism:
- Invariance of performance for small lateral shifts of the field position
- spherical aberration not necessarily corrected
Usual simple case: near to axis
Consequences:
- vanishing linear growing coma
- caustic symmetrical around chief ray
28
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Isoplanatism Condition of Staeble-Lihotzky
Sagittal coma aberration:
from the geometry of the figure and Lagrange invariant
Condition of Staeble-Lihotzky
Problems:
- no quantitative measure
- only tangential rays are considered
- integral criterion
m
un
un
m
sSss
p
p'sin'
sin''''
m
ssS
sS
un
un
m
yy
sphp
p
s'''
''
'sin'
sin''
exit pupil
real
tangential
image plane
ideal
gaussian
image plane
Q' P'
Q'chief ray
optical axis
marginal ray
s
Q't
projection of
sagittal coma ray y'
s'
S'
s'
last
surface
sp'
u'
ys'
ys'
P't
29
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Isoplanatism from Wave Aberrations
Lateral shift of object point
Change in image
Change of wave aberration
Isoplanatism: change is equal dW' = dW
30
R
udy
W
P
chief ray dy'' dy'
R'
s'
undydW sin
'
''
'
'''''
R
dsRdym
R
dsRdydy
'sin''
'''sin'''' un
R
dsRdymundydW
01'
''
'sin'
sin1
'sin''
''sin
R
dsR
un
un
m
unR
dsRmun
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Isoplanatism
Berek's condition of proportionality
Berek's coincidence condition
Isoplanatism in case of defocussing:
can only be fulfilled in one plane or for telecentricity
31
mun
un
sS
m
sS p
'sin'
sin
''''
.''sin'
sin''' consts
un
un
m
sSs p
p
zs
s
zs
su
zs
s
zs
su
p
p
p
p
p
p
p
p1cos1
''
'1
''
''cos1d
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Piecewise Isoplanatism
Invariance of PSF: to be defined
Possible options:
1. relative change of Strehl
2. correlation of PSF's
Examples for microscopic lenses
with and without flattening
correction
In medium field size:
small isoplanatic patches
On axis:
large isoplanatic area
Criteria not useful at the
edge for low performance
Plane MO 100x1.25
0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Strehl correlation
Strehl correlation
no plane MO
40x0.85normalized
field position
System
MO plane 100x1.25 isoplanatic patch size in
mm
MO not plane 40x0.85 isoplanatic patch size in
mm
Strehl 1%
Psf correlation
0.5%
Strehl 1%
Psf correlation
0.5%
on axis 70 72 81 100
half field 3.8 3.8 27 3.1
field zone 2.5 2.5 29 39
full field 45 3.8 117 62
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exit pupil
realtangential
image plane
idealgaussian
image plane
Q' P'
Q'chief ray
optical axis
marginal ray
s
Q'P'
t t
y's
y't
sagittalcoma ray
yo
s'
S's'
Offence Against the Sine Condition
Conradys OSC (offense against sine condition):
- measurement of deviation of sagittal coma
- quantitative validation of the sine condition
Only sagittal coma considered
in case of OSC=0 the Staeble-Lihotzky-
condition is automatically fulfilled
OSC allows for the definition of surface
contribution
''
''
'sin'
sin1
'
''
p
p
t
stOSC
ss
sS
unm
un
y
yy
OSCtppcoma yrryW )0,,(
ExP
yp
z
CR
Q'1
Q's
Q'
P'1
P'
ideal
y's
y'y's
y't
k kkk
CR
kkkkOSC
unh
inQQ
u
w
'''
)'(
sin
sin )(
1
1
'sin'
sin3'
un
unmyyt
33
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OSC
Coma and isoplanatism are strongly connected
Vectorial OSC:
linear scaling of spatial frequencies:
perturbation of the linearity
exit
pupiltangential
coma rays
sagittal
coma rays
image
plane
y'
x'
yp
xp
chief ray
coma
spot
apl
p
apl vm
v
1
'
Wvvv yxapl ,
1''
34
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Rotation around axis for small angle
calculation of change in wave aberration
Welfords condition
All other conditions can be obtained as special cases:
1. sine condition
2. off axis isoplanatism
3. Herrschel condition
4. Smith cos-invariant
35
General Invariant of Welford
d d dW n p D e n p D e ' ' ' , ' , ' , ,
Surface
n n'
ray
p
axis of
rotation
p'
s
s'
D
d
q
q'
d'
D'
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From Eikonal theory: General condition of Smith: Invariance of the scalar product
Special case: P on axis, q = 90°: Abbe sine condition, invariant transverse magnification
Special case: P on axis, q = 0°: Herschel condition, invariant axial magnification
'cos''cos qq drndrn
''' esnesn
dd
Cos-Condition of Smith
P
q
Q
dr
Q'
P'
dr'q'
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Herschel condition:
Invariance of the depth magnification
In principle not compatible with the sine condition
Therefore a perfect imaging of a volume is impossible
2
'sin''
2sin 22 u
nzu
nz
Herschel Condition
P Qdz Q'P' dz'
u'u
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Overview on conditions for aberrations and aplanatism-isoplanatism
38
Overview Aplanatism-Isoplanatism
Nr Sine
cond.
Iso-planat cond.
Isoplanatism
condition
Spherical
aberration
Sagittal
coma
Tangential
coma
Imaging system
1 # # # # # general
2a # OSC=0, Conrady # 0 # isoplanatic-I
2b # Staeble-Lihotzky / Berek
# 0 0 isoplanatic-II
3a 0 0 axial aplanatic
3b 0 (skew) 0 0 off-axis aplanatic
Tangential coma
Isoplanatism
Staeble-
Lihotzky
Sagittal coma
Spherical aberration
Isoplanatism
Conrady
OSC
sine condition
off-axis
Aplanatism
0
0 00 0
0
Skew Spherical aberration
sine condition
axial
Aplanatism
0
0 0
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Overview on invariants and conditions
39
Overview
only
translation
general invariance
(Welford)
off axis isoplanatismcos-law
(Smith)
translation
along z
change
object
position dy
off axis z-invariance
(Herrschel
axis isoplanatism
(Staeble-Lihotzki /
Berek)
special on
axis y'=0
sine condition
(Abbe)
special for
sph=0
on axis z-invariance
(classical Herrschel)
special on
axis x'=y'=0
only
transverse
translation
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Phase Space: 90°-Rotation
Transition pupil-image plane: 90° rotation in phase space
Planes Fourier inverse
Marginal ray: space coordinate x ---> angle q'
Chief ray: angle q ---> space coordinate x'
f
xx'
q
q'
Fourier plane
pupil
image
location
marginal ray
chief
ray
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Nearfield - Farfield
2f-setup:
Fourier-conjugated
planes
Angle and spatial
frequency are equivalent
Angle- and spatial coor-
dinate are interchanged:
x ---> q'
q ---> x'
Corresponds to nearfield <---> farfield
Relationship:
q v
qq fxf
x','
x'
pupil
Fourier domain
angle/frequency
object space
spatial domain
coordinate x
ff
xuu
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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
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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'CR
yMR
yCR
B A
O'
Q
S
P
ExPMRExPMR
CR swuwynssws
ynyunL ''''''''
'''''
)(')('' zyuzywnL CRMR
43
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Photometry in Phase Space
Radiation transport in optical systems
Phase space area changes its shape
Finite chief ray angle:
parallelogram geometry y
p
2y
2y'
2sinu
2sinu'
sinw'
y
y'
s'
Uw' U'
y
y'
s
lens stop
44
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Aberrations in Phase Space
• Angle diviations due to aberrations in the pupil
• Increased spatial extention in the focus region
x
upupil
u
x
ufocus
x
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Ray Caustic
caustic
• Special case of vanishing determinante of Jacobian matrix: ray caustic
• Singulare solution of the wave equation
• Two ray directions in one point
• Special characterization with Morse- and
Maslov index
x
u
xc
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1. Slit diffraction
Diffraction angle inverse to slit
width D
2. Gaussian beam
Constant product of waist size wo
and divergence angle qo
q 00w
D
q
q
D D
q
qo
wo
x
z
Uncertainty Relation in Optics
47
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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
Light Sources in Phase Space
plane wave
(laser)
x
u
ray
spherical
wave
gaussian
beam
LED
point
source
x
u
discrete
mode
points
quasi continuum
48