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

3Thierry Lépine - Optical design

Wavefront shape

x

y

z

x’

y’

z’

B’’B’

A’A

B

The wavefront is not spherical


Wavefront shape

A’

A1’’

A2’’

A3’’

A4’’

Gaussian image plane

IJ

IΣ’0 Σ’

J

exit pupil


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


Aberration function

• Optical Path Difference (OPD)

• or WaveFront Error (WFE)

• or Wave aberration

• or Aberration function (W) :

IJnnW ×′=∆×′=


Gouy’s theorem

W

W’

n n’

Perfect optical system

∆′×′=∆×⇔′= nnWW


Gouy’s theorem(valid ONLY for 3rd order theory)

n1 n2 n3

1223312∆×+∆×=∆×⇔+= nnnWWW totalSStotal


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


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


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


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


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


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


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 ′′


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′


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

+= − = −

′′ =

= ±

= ±

≈ ≈


( ) ( ) ( )

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 ′′


Change of the reference sphere, with tilt

00000 ////R

YnWWWW

Py ×′+=+= Σ′Σ′Σ ′′Σ′Σ′Σ′Σ ′′Σ′

ε


Normalized coordinates

x’

y’

z’B’

A’

Exit pupil

z

yP

xP

xP

yP

1 Pϕ ρ

image

P B’’


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


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


Change of the reference sphere

( ) PP

yPPP y

R

Rnyx

R

RnWW ′++

′+= Σ′Σ′Σ ′′Σ′

2

22

2

2

z// 00εε


About the quantity Rp/R (infinite conjugate here)


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 ,,, ′′=


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


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


( )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′=′


Classification : 5th order• Validity (3rd + 5th) : aperture N > 2, field < 25°


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


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


geometric aberrations : introduction, definitions

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