Prof. Jose Sasian

Overview of Aberrations

Lens Design OPTI 517

Prof. Jose Sasian

AberrationFrom the Latin, aberrare, to wander from; Latin, ab, away, errare, to wander.

Symmetry properties

Prof. Jose Sasian

Overview of Aberrations(Departures from ideal behavior)

• Basic reasoning• Wave aberration function• Aberration coefficients• Aspheric contributions• Stop shifting• Structural aberration coefficients

Prof. Jose Sasian

Wavefront

Prof. Jose Sasian

Basic reasoning

• Ideally wavefronts and rays converge to Gaussian image points. This implies that ideally wavefronts must be spherical and rays must be homocentric.

Prof. Jose Sasian

Basic reasoning• Actual image degradation by an optical system implies that the

collinear transformation can not model accurately imaging. In the wave picture for light propagation we notice that wavefronts must be deformed from the ideal spherical shape.

• Wavefront deformation is determined by the use of a reference sphere with center at the Gaussian image point and passing by the exit pupil on-axis point.

Exit pupil

Image plane

Reference sphereis centered at ideal image point

Referencesphere

Deformed wavefront

Prof. Jose Sasian

Basic reasoning• An axially symmetric system can only have an axially symmetric

wavefront deformation for an object point on-axis. In its simplest form this deformation can be quadratic or quartic with respect to the aperture. If the reference sphere is centered in the Gaussian image point then the quadratic deformation can not be present for the design wavelength.

Prof. Jose Sasian

Basic reasoning• For an object point that is off-axis the axial symmetry of the beam is

lost and is reduced to plane symmetry. Therefore for that off-axis beam the wavefront deformation can have axial, plane, or double plane symmetry.

Prof. Jose Sasian

Basic reasoning• The simplest plane symmetric wavefront deformation shapes

represent the primary aberrations. These are:

• Spherical aberration Axially symmetric• Coma Plane symmetric• Astigmatism Double plane symmetric• Field curvature Axially symmetric• Distortion Plane symmetric• Longitudinal Axially symmetric

chromatic• Lateral Plane symmetric

chromatic

Prof. Jose Sasian

Aberration forms:symmetry considerations

Focus

Spherical aberration

Distortion

Field curvatureFocus

Astigmatism

Coma

Sphericalaberration

On-axis

Off-axis

Prof. Jose Sasian

Wave aberration function

• The wave aberration function is a function of the field H and aperture ρ vectors. Because this function represents a scalar, which is the wavefront deformation at the exit pupil, it depends on the dot product of the field and aperture vectors. The assumed axial symmetry leads to a select set of terms.

( ) 2 2000 200 020 111

4 3 2 2 2040 131 222

2 2 3 4220 311 400

, , cos

cos coscos

...

W H W W H W W H

W W H W HW H W H W H

ρ θ ρ ρ θ

ρ ρ θ ρ θ

ρ ρ θ

= + + + +

+ + + +

+ + + ++

H

ρ{ } ∑=

nmj

mlkmlk HWHW

,,,, cos, θρρ

Prof. Jose Sasian

Wave aberration function

• The field vector has its foot at the center of the object plane and the aperture vector has its foot at the center of the exit pupil plane. Both are normalized and so their maximum magnitude is unity. For convenience we draw the first order image of the field vector.

Aperture and field vectors

Exit pupil

Aperture vector Field vector

Image plane

Optical axis

H

ρ

Prof. Jose Sasian

Wave aberration function

• Note that defocus W020 and the change of scale W111 terms are not needed because Gaussian optics accurately predict the location and size of the image. The piston terms W000, W200 and W400 represent a constant phase change that does not degrade the image. These piston terms do not depend on the aperture vector and so they do not produce transverse ray errors.

• Piston terms represent and advance or delay on the propagation of a wavefront.

Prof. Jose Sasian

Summary of primary aberrations

Prof. Jose Sasian

Aberration coefficients for a system of q surfaces

ISW81

040 =2

1

q

Ij

uS A yn=

= − ∆

∑

IISW21

131 = 1

q

IIj

uS AAyn=

= − ∆

∑

IIISW21

222 =2

1

q

IIIj

uS A yn=

= − ∆

∑

( )22014 IV IIIW S S= +

2

1

q

IVj

S Ж P=

= −∑

VSW21

311 =

2 2

1

q

Vj

A uS Ж P A yA n=

= − + ∆ ∑

( )22

1

1q

Vj

S A A y Ж Ay yPn=

= − ∆ − + ∑

Prof. Jose Sasian

Aberration coefficients for chromatic aberrations

LCW21

020 =δλ

1

q

Lj

nC Aynδ

=

= ∆

∑

TCW =111δλ

1

q

Tj

nC Aynδ

=

= ∆

∑

Prof. Jose Sasian

Aberration coefficient parameters

ninycnuA =+=

incynunA =+=

( )ncP /1∆=r

c 1=Ж nuy nuy= −

νδ

nn

nn 1−

=

∆

CF

d

nnn−−

=1

ν

Prof. Jose Sasian

Aberration coefficients

• c is the surface curvature, ν is the ν number or reciprocal dispersive power.

• All with the marginal and chief ray first-order ray traces !!!

• Lens optimization started!!!

is the Lagrange invariant.Ж

Prof. Jose Sasian

Comments on aberrations• Third-order or fourth-order ?• A well corrected system has its third-order aberrations

almost zero• Aberration cancellation is the main mechanism for

image correction• Presenting to the optimization routine a system with its

third-order aberrations corrected is a good starting point

• Some simple systems are designed by formulas that relate third-order aberrations

• Note symmetry in third-order aberration coefficients• Wave aberrations seem to be simpler to understand

than transverse ray aberrations

Prof. Jose Sasian

Example

Prof. Jose Sasian

Example

Prof. Jose Sasian

Summary of aberrations

Aspheric surfaces(non-spherical)

• Conic (or conicoids)• Cartesian ovals• Polynomials on x/y, r-theta, Zernikes• Bernstein polynomials, Bezier curves• Splines• NURBS• Freeform surfaces• User definedProf. Jose Sasian

Prof. Jose Sasian

Conic plus polynomial(much used in lens design)

( ) [ ] ...)1(11

1010

88

66

4422

2

++++++−+

= SASASASAScK

cSSZ

S x y2 2 2= +

C is 1/r where r is the radius of curvature; K is the conic constant; A’s are aspheric coefficients

Aspheric contribution can be thought of as a cap to the spherical part

Sag= sphere + aspheric cap

Prof. Jose Sasian

Aspheric contributions to the Seidel sums

IS aδ =

0IVSδ =

0=LCδ

IIyS ay

δ =

3

VyS ay

δ

=

0=TCδ

2

IIIyS ay

δ

= nyca ∆−= 432ε

nyAa ∆= 448

ISW81

040 =

IISW21

131 =

IIISW21

222 =

( )22014 III IVW S S= +

VSW21

311 =

Prof. Jose Sasian

Aspheric contributions explanation

Prof. Jose Sasian

Aspheric contributions depend on chief ray height at the surface

•When the stop is at the aspheric surface only spherical aberration is contributed given that all the beams see the same portion of the surface .•When the stop is away from the surface, different field beams pass through different parts of the aspheric surface and other aberrations are contributed

Prof. Jose Sasian

Stop Shifting

Exit pupil Image plane

Optical axis

•Stop shifting is a change in the location of the aperture stop along the optical axis•Stop shifting does not change the f/#•Stop shifting does not change the optical throughput•Stop shifting selects a different portion of the wavefront for off-axis beams

Prof. Jose Sasian

Stop shifting may produce light vignetting

Prof. Jose Sasian

Stop shifting

Prof. Jose Sasian

Change of Seidel sums with stop shifting

0=ISδ

III SyyS δδ =

IIIIII SyyS

yyS

2

2

+

∂=

δδ

0=IVSδ

{ } IIIIIIIVV SyyS

yySS

yyS

32

33

+

++=

δδδδ

0=LCδ

LT CyyC ∂

=δ

ISW81

040 =

IISW21

131 =

IIISW21

222 =

( )22014 III IVW S S= +

VSW21

311 =

Prof. Jose Sasian

Wave coefficients in terms of Seidel sums

ISW81

040 =

IISW21

131 =

IIISW21

222 =

( )22014 III IVW S S= +

VSW21

311 =

Prof. Jose Sasian

The ratio yyδ

Can be calculated at any plane in the optical system

new old new old new oldu u y y A ASu y A− − −

= = =

is the stop shifting parameterS

Prof. Jose Sasian

Structural coefficients σ

IpI yS σφ 34

41

=

3

2

2V V

p

ЖSy

σ=

2 212II p IIS Жy φ σ=

LpL yC φσ2=

2III IIIS Ж φσ=

2T TC Жσ=

2IV IVS Ж φσ=

Marginal ray height atthe principal planes

py

Prof. Jose Sasian

Structural coefficients:Thin lens (stop at lens)

[ ]DCYBXYAXySI ++−= 2234

41 φ

[ ]2 212IIS Жy EX FYφ= −

2IIIS Ж φ=

2 1IVS Ж

nφ=

0=VS

2 1LC y φ

ν=

0=TC

( )212

−+

=nnnA

( )2

2

1−=

nnD

( )( )1

14−+

=nnnB

( )11−+

=nn

nE

nnC 23 +

=

nnF 12 +

=

12

12

21

21

rrrr

ccccX

−+

=−+

=

uuuu

mmY

−+

=−+

=''

11

))(1( 1 xccncn −−=∆∆=φ

Surface optical power φ

Prof. Jose Sasian

Bending of a lens• Maintains the

optical power• Changes the

optical shape• Meniscus, plano-

convex, double-convex, etc.

• Shape factor 12

12

21

21

rrrr

ccccX

−+

=−+

=

Prof. Jose Sasian

Using a lens design program• Must be able to interpret correctly the information

displayed by the program• In some instances the program is right but we think it is

wrong. So we must carefully review our assumptions• When there is disagreement between you an the

program, there is an opportunity to learn• Verify that the program is modeling what you want• Check and double check• You must feel comfortable when using a program. • Read the manual• Play with the program to verify that it does what you

think it does• Must reach the point when it is actually fun to use the

program

Prof. Jose Sasian

Summary

• Review of aberrations• Aspheric surfaces• Stop shifting• Aberration coefficients• Structural aberration coefficients

• Next class derivation of coefficients

Prof. Jose Sasian

Mode matching concept

• Same mode diameter• Same amplitude distribution• Same phase distribution• Same polarization• Same x,y,z position• Same angular position

Prof. Jose Sasian

Fiber coupling efficiency

( )2 2

20 0

0 0 0 0

4

z z

εω ω λω ω πω ω

=

+ + +

( ) ( )m mx x dx+∞

∗

−∞

Ψ Ψ∫( ) ( ) 1m mx x dx+∞

∗

−∞

Ψ Ψ =∫

“overlap integral”

z

Herwig Kogelnik