Geometric aberrations : introduction, definitions ...
Transcript of Geometric aberrations : introduction, definitions ...
Geometric aberrations : introduction, definitions,
aberrations types, bibliography
• Geometric aberrations : sometimes called monochromatic aberrations
Thierry Lépine - Optical design 2
Transverse ray aberrations
x
y
z
x’
y’
z’
light
B’’B’
A’A
B
Transverse ray aberration :
Spot diagram = all the points B’’
′′′
y
xBB
εε
pupil
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Wavefront shape
x
y
z
x’
y’
z’
B’’B’
A’A
B
The wavefront is not spherical
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Wavefront shape
A’
A1’’
A2’’
A3’’
A4’’
Gaussian image plane
IJ
IΣ’0 Σ’
J
exit pupil
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Wavefront shape
• Consider the exit pupil
• Due to the aberration, the real and the reference wavefronts are different
• This difference ∆ is measured on a real ray and is equal to : IJ
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Aberration function
• Optical Path Difference (OPD)
• or WaveFront Error (WFE)
• or Wave aberration
• or Aberration function (W) :
IJnnW ×′=∆×′=
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Gouy’s theorem
W
W’
n n’
Perfect optical system
∆′×′=∆×⇔′= nnWW
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Gouy’s theorem(valid ONLY for 3rd order theory)
n1 n2 n3
1223312∆×+∆×=∆×⇔+= nnnWWW totalSStotal
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Relationships between transverse aberrations and aberration function
A’
B’’
I
Σ’
J
Exit pupil Plane of sensor
Y’
z’
B’
P
P1
Σ’0n’
( ) ( )
( ) ( )
∂∂×
′−≈
∂∂×
′−=
∂∂×
′−≈
∂∂×
′−=
′′′
P
PP
P
PPy
P
PP
P
PPx
Y
YXW
n
R
Y
YXW
n
R
X
YXW
n
R
X
YXW
n
R
BB,,
,,
1
1
ε
ε
( ) ( )( )BIR
BIBPR
′′=′=′==′Σ
1
0 radius sphere, reference
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XP
YP
Proof (for fanatics)from Principle of optics, Born and Wolf, and Aberrations of optical systems, Welford
[ ][ ] [ ][ ] [ ]BIBP
BIBJ
IJW
−=−=
=
z) y, x,,A( systeme coordinate space image in the
0
,0
0
, and
z), y, x,(A, system coordinate spaceobject in the
0
′
′
′
′
B
B
B
B
y
x
B
D
P
z
y
x
I
y
x
B
( ) ( )Hamilton offunction sticcharacteri V avec
,,;0,,,0,0;0,,
=−= zyxyxVDyxVW BBBB
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Proof
( ) ( ) ( )( ) ( )( ) ( )( )
x
z
z
V
x
V
x
W
yxzyxyxVcteyxzyxyxVDyxVW
DyxBPRRzyyxx
rel
BBBBBB
BBrel
BB
∂∂
∂∂−
∂∂−=
∂∂
−=−=
++=′==+−+− ′′′′
)2(
222222
)1(
222
,,,;0,,,,,;0,,,0,0;0,,
avec
R) (radius sphere reference on the is I
( ) ( ) ( )
z
xx
zyxBIR
R
znn
R
xnn
B′
′′′′
′′
′′
′′
−−=∂∂
+−+−=′′=
−×′=×′=∂∂−×′=×′=
∂∂
′′
x
z : rel(1) from Then,
yx :with
cosz
Vet
xcos
x
V
:such that ),,( cosinesdirection has 'IB'ray The
0
y
x
B note and
z
y
x
I recall We
22
B
2
B
22
1
11
B
B
B
γα
γβα
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Proof
( )
∂∂×
′−=
∂∂×
′−=
×′
−=∂
∂
×′
−=−×′
−=∂
∂′′′
y
W
n
Rx
W
n
R
R
n
y
W
R
nxx
R
n
x
W
y
x
y
xBB
1
1
1
11
:Then
:even And
:get we2,relation in sderivative partial theall putingThen
ε
ε
ε
ε
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Relationship between the longitudinal aberration and the aberration function
A’0
A’’
I
Σ’0 Σ’
JXP
YP
z
( )P
PP
P
zY
YXW
Yn
R
∂∂×
×′−≈ ,2
ε
′= 0, APdR
εz
Exit pupil
Plane of sensor
P
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Spherical wavefront
XP
YP
zExit pupil
Plane of sensor
A’
M
( ) ( )
R
YXz
RRzYXRAMd
PP
PP
×+=
<<<<
=−++⇔=′
2
: hence z, zet R Yet X general,In
,
:such that z) ,Y ,M(X points ofset theis R) (radius wavefrontspherical The
22
2
PP
2222
PP
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Defocus (longitudinal shift)
XP
YP
zExit pupil
Plane of sensor
A’
M
( )
2
221
21
1
22
: is sphere new theofequation the, defocusgiven aFor
2
22
z/
2
22
z
22
z
22
z
22
z
22
z
00
R
YX
n
W
R
YX
R
YX
RR
YX
R
R
YX
R
YXz
PP
PPPPPPPPPP
+−≈′
×+−
×+≈
−×+≈
+×+≈
+×+=
Σ′Σ ′′ ε
εεεε
ε
εz
n’
0Σ′0Σ ′′
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Change of the reference sphere, with defocus
XP
YP
zExit pupil
Plane of sensor
A’
2
22
z////
2
00000 R
YXnWWWW PP +′
+=+= Σ′Σ′Σ ′′Σ′Σ′Σ′Σ ′′Σ′ε
εz
n’
0Σ′0Σ ′′Σ′
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M
Depth of focus
2 2
max z z
2 2
max
2
z
2
z
We saw that :
1
2 2 4
It is assumed that n 1 (air), and that
the system is diffraction limited , ie. .4
Hence : 2
For the visible: 0,5µm, 2 (µm)
p pX YW
n R N
W
N
N
ε ε
λ
ε λλ ε
+= − = −
′′ =
= ±
= ±
≈ ≈
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( ) ( ) ( )
R
Y
n
W
R
Y
R
YXz
RRzYXRBMd
PyPyPP
yy
yPP
y
×−=
′
×−
×+=
<<<<<<
=−+−+⇔=′
Σ′Σ ′′ εε
εε
ε
ε
00 /22
22
PP
2222
PP
2
: hence ,et z, z R, Yet X general,In
,
:such that z) ,Y ,M(X points ofset theis
R) (radius wavefrontspherical The . isshift lateral The
Tilt (lateral shift)
XP
YP
zExit pupil
Plane of sensor
A’
M
εy
n’ B’
0Σ′0Σ ′′
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Change of the reference sphere, with tilt
00000 ////R
YnWWWW
Py ×′+=+= Σ′Σ′Σ ′′Σ′Σ′Σ′Σ ′′Σ′
ε
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Normalized coordinates
x’
y’
z’B’
A’
Exit pupil
z
yP
xP
xP
yP
1 Pϕ ρ
image
P B’’
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RpR
Relationship between transverse aberrations and aberration function
( )
PPP
P
P
y
PP
PP
P
PPy
y
W
Rn
R
Y
y
y
W
n
R
yR
RyY
Y
YXW
n
R
∂∂×
×′−=
∂∂×
∂∂×
′−=
≤≤×=
∂∂×
′−=
ε
ε
: Hence
.10 and pupil,exit theofdiameter -semi thewith
, : variableof Change
,
P
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Relationship between transverse aberrations and aberration function
( )
( )
pupil.exit theofdiameter -semi
sphere, reference theof radius
:with
10 ,10 ,,
,
==
≤≤≤≤
∂∂×
′−≈
∂∂×
′−≈
′′′
P
PP
P
PP
P
y
P
PP
P
x
R
R
yx
y
yxW
Rn
R
x
yxW
Rn
R
BB
ε
ε
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Change of the reference sphere
( ) PP
yPPP y
R
Rnyx
R
RnWW ′++
′+= Σ′Σ′Σ ′′Σ′
2
22
2
2
z// 00εε
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About the quantity Rp/R (infinite conjugate here)
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EP
H H’α’
F’
R
f’
RPDop/2
Nf
D
R
Rop
P
2
12 =′
=
Aberrations types• General case : with normalized variables
( )
• The optical system has rotational symmetry around its optical axis z
• Considering a ray BI, B on the object on the y axis (x’=0), I on the entrance pupil
• If we rotate this beam around the optical axis, the aberration function W is unchanged
• Hence, W has to be a combinaison of (x’), y’, xp et yp which is rotation invariant around z : –
–
–
( )PP yxyxWW ,,, ′′=
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22yx ′+′
22
pp yx +pp yyxx ′+′
1 and 0et 1 and 0 ≤′′≤≤≤ yxyx PP
• Validity :
• aperture : N > 8
• field : a few degres
• With normalized variables ( ) :
( )
( )
( ) ( ) ( )...
,,
3
311
222
220
22
222
22
131
222
040
111
22
020
4
400
2
200000
PPPSPPPPPP
P
PP
PP
yyWyxyWyyWyxyyWyxW
yyW
yxW
yWyWW
yxyWW
′++′+′++′+++
′+++
′+′+=
′=
Aberration types : 3rd order
Seidel aberrations (1859)(4th order for W,
3rd order for transverse aberrations)
10 and 1et 0 ≤′≤≤≤ yyx PP
defocus
tilt
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piston
Aberration types : 3rd order• Using polar coordinates in the exit pupil, with
normalized variables ρ and y’ :
( )
...
coscoscos
cos
,,
3
311
22
220
222
222
3
131
4
040
111
2
020
4
400
2
200000
ϕρρϕρϕρρϕρ
ρ
ϕρ
yWyWyWyWW
yW
W
yWyWW
yWW
S′+′+′+′++
′++
′+′+=
′=
( )4 3 2 2 2 2 2 31 1 1 1 1cos cos cos
8 2 2 4 2I II III III IV VS S y S y S S y S yρ ρ ϕ ρ ϕ ρ ρ ϕ′ ′ ′ ′+ + + + +
10 and 10 ≤≤≤′≤ ρy
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( )kji
ijk yW ϕρ cos′
Old formalism from Seidel
Wijk = amount of wavefront error associatedwith this aberration term at the edge of thepupil (ρ = 1) and the edge of the field (y’ = 1)
HH. Hopkins, The wave theory of aberrations, Oxford at the clarendon Press (1950)
Aberration types : 3rd order
Coefficient Expression Name
W040 Spherical aberration
W131 Coma
W222 Astigmatism
W220S Field curvature
W311 Distortion
( ) 4222 ρ=+ PP yx
( ) ϕρ cos322yyxyy PPP′=+′
ϕρ 22222 cosyyy P′=′
( ) 22222 ρyyxy PP′=+′
ϕρ cos33yyy P′=′
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Classification : 5th order• Validity (3rd + 5th) : aperture N > 2, field < 25°
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Coefficient Expression Name
W060 Spherical aberration
W151 Field-linear Coma
W422 Astigmatism
W420S Field curvature
W511 Distortion
W240S, W242 W240 : a component of oblique sphericalW242 : oblique spherical
W331S, W333 W331 : a component of elliptical coma (trefoil), also knownas field-cubed comaW333 : elliptical coma (trefoil)
6ρ
ϕρ cos5y′
ϕρ 224 cosy′
24ρy′
ϕρ cos5y′
ϕρρ 242
242
42
240 cosyWyW ′+′
ϕρϕρ 333
333
33
331 coscos yWyW ′+′
Important to know
• The geometric aberrations depend on : – the position of the objet,
– the position of the pupil,
– the geometry of the system : rotational symetry or not (off-axis systems [TMA…])
– The quality of the opto-mechanical design : aberrations due to decentered and tilted elements
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Kevin Thompson : Nodal Aberration Theory (NAT), Best starting point : « Description of the third-order optical Aberrations of near-circular pupil optical systems without Symmetry », vol 22, n° 7, july 2005, JOSA A
References• All my lectures are here : http://paristech.institutoptique.fr/index.php?domaine=168
Click on « Optical design (Saint-Etienne) », then on « ressources pédagogiques »
• Handbook of optical systems– Edited by Herbert Gross, Wiley-VCH (vol 1 to 5)
• Field guide to lens design– J.Bentley, C. Olson, SPIE Press
• Aberrations of optical systems– W. T. Welford, Adam Hilger (1991)
• Modern optical engineering– W. J. Smith, Mac Graw-Hill
• Optical system design– R. E. Fisher, B. Tadic-Galeb, Mac Graw-Hill
• Handbook of optical design– D. Malacara, Z. Malacara, Marcel Dekker
• Lens design– M. Laikin, Marcel Dekker
• Optical shop testing– D. Malacara, Wiley – Interscience
• Principles of optics– M. Born and E. Wolf, Cambridge University Press
• Optics– E. Hecht, Addison Wesley
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