Spherical aberration(3rd order)

Aberration function

( ) 4

040

222

040 ρWyxWW PP =+=

•Independant of the field (y’) : we study it on the axis

•Independant of ϕ : we study it in the tangential plane (0z’y’)

•HP : the pupil is on the component (diopter, lens, mirror)

•We’ll see later that the spherical aberration does not depend on the position of the pupil

2Thierry Lépine - Optical design

X

Y

Z

Description

Paraxial image

Paraxial focus

Marginal focus

Axial caustic

Tangential caustic

Longitudinal spherical

aberration (LA)

Marginal ray

Least confusion

circle

Paraxial ray

3Thierry Lépine - Optical design

Description

X

Y

ZX

Y

Z

Under-correction Over-correction

4Thierry Lépine - Optical design

Formalism

( ) 4

040

222

040 ρWyxWW PP =+=( )

( ) 3

040)(

22

040

222

040

44 ρ

ε

WRn

RyyxW

Rn

R

y

yxW

Rn

R

y

W

Rn

R

PyzO

PPP

P

P

PP

PPP

y

××′

−=+××′

−=

∂+∂×

×′−=

∂∂×

×′−=

′′

A’0

A’’

I

Σ’0Σ’

JxP

yP

z’

Paraxial image

Exit pupil

LA

εy

5Thierry Lépine - Optical design040

1

3

040

8

8 : image theofdiameter theHence WRn

RW

Rn

R

PP′

=′

=Φ=ρ

ρ

Formalism

( ) 040

2

1

2

040

2

2

040

2

22

2

4

040)(

14 : Hence

14

140

: Hence

2

n

:naught is aberration e transversthefor which defocus theis

what know toneed we(LA), aberration allongitudin theevaluate To

WR

R

nLA

WR

R

nyW

R

R

ny

W

Rn

R

yRR

yWW

P

z

P

P

P

z

PP

y

PPz

PyzO

′−==

′−=

′−=⇒=

∂∂

′−=

××

′+=

=

′′

ρε

ρεε

ε

A’0

A’’

I

Σ’0Σ’

JxP

yP

z

Paraxial image

Exit pupil

LA=εz

εy

6Thierry Lépine - Optical design

Aberration function(the reference sphere is centered on the paraxial image)

λ3040 =W

7Thierry Lépine - Optical design

Viewing the PSF (star test)

Thierry Lépine - Optical design 8

λ3040 =W

-40

-35

-30

-25

-20

-15

-10

-5

0

-40

-35

-30

-25

-20

-15

-10

-5

0

Full field display

Thierry Lépine - Optical design 9

λ3040 =W

Image simulation

Thierry Lépine - Optical design 10

λ3040 =W

MTF

Thierry Lépine - Optical design 11

λ3040 =W

0 50 100 150 200 250 300 3500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

f (cyc/mm)

MT

F

MTF(u)

MTF(v)

diffraction

Transverse ray aberrations (paraxial image plane)

12Thierry Lépine - Optical design

-1 0 10

0.2

0.4

0.6

0.8

1

yp

Wy

y'=1

0 0.5 10

0.2

0.4

0.6

0.8

1

xp

Wx

y'=1

-1 0 1-10

-5

0

5

10

yp

epsy

y'=1

0 0.5 1-8

-6

-4

-2

0

xp

epsx

y'=1

-1 0 10

0.2

0.4

0.6

0.8

1

yp

Wy

y'=0.7

0 0.5 10

0.2

0.4

0.6

0.8

1

xp

Wx

y'=0.7

-1 0 1-10

-5

0

5

10

yp

epsy

y'=0.7

0 0.5 1-8

-6

-4

-2

0

xp

epsx

y'=0.7

-1 0 10

0.2

0.4

0.6

0.8

1

yp

Wy

y'=0

0 0.5 10

0.2

0.4

0.6

0.8

1

xp

Wx

y'=0

-1 0 1-10

-5

0

5

10

yp

epsy

y'=0

0 0.5 1-8

-6

-4

-2

0

xp

epsx

y'=0

Circle of least confusion

( )

2

3

2

1

040

2

2

3

3

040

2

0402

2

3

0401n

3

1 : caustic theofequation cartesian then theand

8et 12

:branch)(upper caustic theof equations parametric thededucecan weSo

0 :point contact for the hence caustic, theo tangent tisray This

4y

: b and a constants theof sexpression thededucecan weSo

0et

0 points he through tgoesray This

: pupil (exit) thefrom emergingray afor Equation

zWR

Ry

WR

RyW

R

Rz

dρ

yd

zR

RW

R

RzLA

LA

LA

bazy

P

PP

P

P

−

=′

−=′

=′−=

=′

−−=−=′

+=′

ρρ

ρρε

ε

ρρ

ρ

ρ

ρ

13Thierry Lépine - Optical design

2 1 3

2 2040 0403(lower branch)

2

rd

At the circle of least confusion (CLC), the marginal ray ( 1) intersects the negative part

of the caustic :

14

3

This is a 3 degree equation of unk

P P

P

R RRW z W z

R R R

ρ

−

=

− − = − −

1

nown z.

3We just check that LA is a solution.

4

The image position for the circle of least confusion occurs three-quarters of the way from

the paraxial focus to the marginal focus.

The diameter of th

z ρ ==

1

1

3 040LA

4

24 2 4

040 0402 3LA

4

e CLC is : 2 2 . It is the quarter of the diameter

of the paraxial image.

If the center of the reference sphere is at the center of the CLC :

1,52

z

zP

Pz

Ry W

R

RW W W

R

ρ

ρερ ε ρ ρ ρ

=

=

=

=

′ =

= + = −( )2

Circle of least confusion

14Thierry Lépine - Optical design

Best focus

! foci marginal and paraxial ebetween thhalfway is focusbest The

! 2

2

2 :or , : Hence

,0 ie. minimum, is for which (b) defocus for the looking are We

12645

4

325

1W

23

1

W

2 then defocus, a introduce weIf

minimum. is ie. maximum, is ratio Strehl thefocus,best At the

0402

2

0402

2

22

222

222

0

1

0

22

2

0

1

0

222

242

2

24

0401

2

LAW

R

RW

R

Rab

b

baba

babaddW

baddWW

W

baR

RWW

P

zP

z

WW

W

W

Pz

n

W

=−=⇒−=−=

=∂

∂

++=⇒

++==

+==

−=

+=+=

∫ ∫

∫ ∫

=′

εε

σσ

σ

ϕρρπ

ϕρρπ

σ

ρρρερ

σ

π

π

15Thierry Lépine - Optical design

Best focus

( )( )

( )

pupil. theof edges at the and

(obvious) axis on thenaught is focusbest for thefunction aberration that thenote alsocan We

sphere. reference therespect to with retarded, is wavefront the0, Wif and,

image, paraxial at the valueitsan smaller th times4 is WFEmaximum thefocus,best At the

42

10 :maximum is for which for looking are We

: casesboth for functions aberration theof valuemaximum thedetermine usLet

040

040

max

24

040

0401max

4

040

>

=⇒=⇒=∂

∂−=

=⇒

=

=

WW

WW

WW

WW

WW

BFBF

BF

BF

paraxial

paraxial

ρρ

ρ

ρρ

ρ

ρ

16Thierry Lépine - Optical design-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

y

W paraxial

W best focus

Aberration function(reference sphere centered at best focus)

λλ 3,3 020040 −== WW

17Thierry Lépine - Optical design

Aberration correction

• Compensation : we use 2 components for which aberrations are compensated– Examples : doublet (Fraunhofer, Gauss,

Littrow…), Schmidt telescope, …

• Aspheric surfaces :– Example : parabolic mirror…

Thierry Lépine - Optical design 18