Nodal Aberration Theory (NAT) - Institut Optique
Transcript of Nodal Aberration Theory (NAT) - Institut Optique
Bibliography
Kevin Thompson
Aberration fields in tilted and decentered optical systems
The University of Arizona, PhD, 1980
Supervisor : Pr Roland Shack
Kevin Thompson
Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry
J. Opt. Soc. Am. A 22, 1389-1401 (2005)
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What is NAT ?
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“…the figure demonstrates the underly-ing concept of nodal aberration theory: that tilted, but oth-erwise rotationally symmetric subsystems contribute aber-ration patterns that are shifted in the image field. That is,element 1 contributes coma that increases linearly from apoint, and astigmatism that increases quadratically fromthat same point. Similarly, element 2 also contributes comaand astigmatism, but centered on a different point in thefield. Nodal aberration theory deals with how the twoshifted ~ but overlapping ! coma patterns combine to form aresultant coma field for the entire system, and how theoverlapping astigmatism contributions combine to form anastigmatism field for the system…”
Techniques and tools for obtaining symmetricalperformance from tilted-component systemsJohn R. RogersOpt. Eng. 39(7) 1776–1787 (July 2000)
Rotationally symmetric systems• We already know that :
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( )
surface j for the j
)y (from pupilexit in theazimuth
pupil,exit in the coordinate radial normalized
field, normalized y'et
th
p
m2nl m,2pkwith
cos
=
=
=
+=+=
′=∞ ∞ ∞
ϕ
ρ
ϕρj p n m
mlk
jklm yWW
Non rotationally symmetric systems• For perturbed optical systems, one must consider not
only rays travelling in the tangential plane
• So we introduce 2 angles : – θ is the azimuth angle in the image plane (from y’)
– ϕ is the azimuth angle in the exit pupil plane (from yp)
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Y’
x’
xp
yp
zθ
ϕ
Exit pupil
Image planeSkew ray
Vector form of W
• If we consider the dot product :
• Then, we have :
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( )ϕθρρ −= cos. HH
( ) ( ) ( ) ( )
field image in theposition
normalizedH and m2nl m,2pkwith
...
=+=+=
=∞ ∞ ∞ mn
j p n m
p
jklm HHHWW ρρρ
Example : 3rd order aberrations
• For instance :
• is now written as :
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ϕρρϕρϕρρ coscoscos 3
311
22
220
222
222
3
131
4
040 yWyWyWyWW S′+′+′+′+
( ) ( )( ) ( ) ( )( ) ( )( )ρρρρρρρρρ ........ 311220
2
222131
2
040 HHHWHHWHWHWW S ++++
Perturbed systems(decenters, tilts)
• Case of just one surface (tilt for instance)
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y
P
CH
jσAjH
Center of the gaussian image plane
Center of aberration fieldfor the tilted surface
Ajj HH += σ
Optical Axis Ray
Perturbed systems
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• Now, W is written as :
( ) ( ) ( ) ( )( ) ( )( )( ) ( ) ( )( )
m2nl m,2pkwith
...
...
+=+=
−−−=
=
∞ ∞ ∞
∞ ∞ ∞
m
j
n
j p n m
p
jjjklm
m
Aj
n
j p n m
p
AjAjjklm
HHHW
HHHWW
ρσρρσσ
ρρρ
Example : 3rd order• For instance :
• is now written as :
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( ) ( )( ) ( ) ( )( ) ( )( )ρρρρρρρρρ ........ 311220
2
222131
2
040 HHHWHHWHWHWW ++++
( )( )( )( )( )( )( )( )( )( )( )( )( )( )( )
−−−
+−−
+−
+−
+
j
jjj
j
jj
j
j
j
j
j
HHHW
HHW
HW
HW
W
j
j
j
j
j
ρσσσ
ρρσσ
ρσ
ρρρσ
ρρ
..
..
.
..
.
311
220
2
222
131
2
040
Very important
• (Wklm)j are not affected by tilts and decenters as there are determined onlywith paraxial quantities
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3rd order spherical aberration
• independent of field and
• so spherical aberration is unaffected by tilts and decenters
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( )j
jW
2
040 .ρρ
jσ
3rd order coma
• with W131 = coma aberration coefficient of the centered system
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( )( )( )
( )
( ) ( )ρρρσ
ρρρσ
ρρρσ
..
..
..
131131
131131
131
−=
−
=
−=
j
j
j
j
j
j
j
j
jj
j
WHW
WHW
HW
3rd order coma
• Let us define :
• And then the same normalized vector :
• This gives for the coma :
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131131 AWj
jj=
σ
131
131131
W
Aa =
( )( ) ( )131 131 . .W H a ρ ρ ρ−��� ���� �� �� ��
3rd order coma
• This expression show that now the nodeof the coma field of the perturbedsystem is shifted to a point in the image plane defined by the vector
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( )( ) ( )131 131 . .W H a ρ ρ ρ−��� ���� �� �� ��
131a
3rd order coma
• We define
• Then the magnitude of the coma isproportional to
• And it is oriented along
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131131 aHH −=
131131 HW
131H
X’
y’
From K. Thompson thesis
Special case• W131 = 0 : optical system corrected for coma
• This new coma is constant, ie. uniform in the field
• The magnitude of the coma is proportionalto and it is oriented along
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( )
( )( ) ( )131
131
. .
. .
j j
j
W W
A
σ ρ ρ ρ
ρ ρ ρ
= −
= −
��� �� �� ��
���� �� �� ��
131A131A
3rd order astigmatism and fieldcurvature
• We consider the best images, associatedwith the coefficient :
• Then we have :
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( )( ) ( ) ( )( ) ( )2
222 220. . .j jj S j j
j j
W W H W H Hσ ρ σ σ ρ ρ= − + − − ��� ��� �� ��� ��� ��� ��� �� ��
2222202202
1WWW SM +=
( ) ( )( ) ( ) ( )2 2
220 222
1. . .
2Mj j j j j
j
W W H H W Hσ σ ρ ρ σ ρ = − − + −
��� ��� ��� ��� �� �� ��� ��� ��
Shack vector product : ( )
i
AB A B eα β+=
���� �� ��
3rd order astigmatism
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( )
( )
2
222
222
2
2222
222
222
2
2222
222
222
222
222
222222
22
222
2
222222
22
222222
2
222
22
222
et
:avec
.2
1
.22
1
.2
1
aW
W
aW
Bb
W
W
W
Aa
baHW
WWHHW
HW
j
jj
j
jj
j j j
jjjjj
j
jj
−=−===
+−=
+−=
−
σσ
ρ
ρσσ
ρσ
3rd order astigmatism
• We are looking where in the image plane the astigmatism is zero :
• The astigmatism is zero at two points in the field.
• It’s called binodal astigmatism
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( ) 222222
2
222
2
222 0 biaHbaH ±==+−
From K. Thompson thesis
Case of a RC (Hubble space telescope)
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Kevin ThompsonDescription of the third-order optical aberrations of near-circular pupil optical systems without symmetryJ. Opt. Soc. Am. A 22, 1389-1401 (2005)
Conclusion
• For a rotationally symmetric optical system,with tilts and decenters, there are no new3rd order aberrations. It is the fielddependance of these aberrations that ischanged, with the development of node(s) inthe field
• This theory has been extended by Thompsonand Rolland for higher orders, for off-axissystems, and for systems including freefomssurfaces
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