nodal aberration theory (nat) - institut optique

Nodal Aberration Theory(NAT)

Beyond Seidel theory…

Thierry Lépine - Optical design 2

Roland Shack Kevin Thompson

Jannick Rolland

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)


What is NAT ?


“…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 :


( )

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)


Y’

x’

xp

yp

zθ

ϕ

Exit pupil

Image planeSkew ray

Vector form of W

• If we consider the dot product :

• Then, we have :


( )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 :


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)


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


• 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 :


( ) ( )( ) ( ) ( )( ) ( )( )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


3rd order spherical aberration

• independent of field and

• so spherical aberration is unaffected by tilts and decenters


( )j

jW

2

040 .ρρ

jσ

3rd order coma

• with W131 = coma aberration coefficient of the centered system


( )( )( )

( )

( ) ( )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 :


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


( )( ) ( )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


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


( )

( )( ) ( )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 :


( )( ) ( ) ( )( ) ( )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


( )

( )

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


( ) 222222

2

222

2

222 0 biaHbaH ±==+−




nodes

Special case : W222=0



0222 =A

Case of a RC (Hubble space telescope)


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


nodal aberration theory (nat) - institut optique

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