5. Aberration Theory5. Aberration Theory - Hanyangoptics.hanyang.ac.kr/~shsong/5-Aberration theory.pdf · 5. Aberration Theory5. Aberration Theory Last lecture •Matrix methods in

5. Aberration Theory5. Aberration TheoryLast lecture

•Matrix methods in paraxial optics •matrix for a two-lens system, principal planes

This lectureWavefront aberrationsChromatic AberrationThird-order (Seidel) aberration theory

Spherical aberrationsComaAstigmatismCurvature of FieldDistortion

AberrationsAberrations

• A mathematical treatment can bedeveloped by expanding the sine and tangent termsused in the paraxial approximation

ChromaticChromatic MonochromaticMonochromatic

Unclear Unclear imageimage

Deformation Deformation of imageof image

SphericalSpherical

ComaComa

astigmatismastigmatism

DistortionDistortion

CurvatureCurvature

n (n (λλ))

Third-order (Seidel) aberrationsThird-order (Seidel) aberrations

Paraxial approximation

Aberrations: ChromaticAberrations: Chromatic• Because the focal length of a lens depends on the

refractive index (n), and this in turn depends on the wavelength, n = n(λ), light of different colors emanating from an object will come to a focus at different points.

• A white object will therefore not give rise to a white image. It will be distorted and have rainbow edges

Aberrations: SphericalAberrations: Spherical• This effect is related to rays which make large angles

relative to the optical axis of the system• Mathematically, can be shown to arise from the fact that

a lens has a spherical surface and not a parabolic one• Rays making significantly large angles with respect to

the optic axis are brought to different foci

Aberrations: ComaAberrations: Coma• An off-axis effect which appears when a bundle of incident rays

all make the same angle with respect to the optical axis (sourceat ∞)

• Rays are brought to a focus at different points on the focal plane• Found in lenses with large spherical aberrations• An off-axis object produces a comet-shaped image

ff

Astigmatism and curvature of fieldAstigmatism and curvature of field

Yields elliptically distorted imagesYields elliptically distorted images

Aberrations: Pincushion and Barrel Distortions

Aberrations: Pincushion and Aberrations: Pincushion and Barrel DistortionsBarrel Distortions

• This effect results from the difference in lateral magnification of the lens.

• If f differs for different parts of the lens,

o

i

o

iT y

yssM =−= will differ also

objectobject

Pincushion imagePincushion image Barrel imageBarrel image

ffii>0>0 ffii<0<0

M on axis less than off M on axis less than off axis (positive lens)axis (positive lens)

M on axis greater than M on axis greater than off axis (negative lens)off axis (negative lens)

Ray and wave aberrationsRay and wave aberrations

LA

TALA : longitudinal ray aberrationTA : transverse (lateral) ray aberration

Actual wavefrontWave aberration

Ideal wavefrontParaxial Image plane

Paraxial Image point

Longitudinal Ray Aberration Longitudinal Ray Aberration

Modern Optics, R. Guenther, p. 199.

n = 1.0nL = 2.0

R

sinsin sin sin2

ii i t i tn n θθ θ θ= ⇒ =

Longitudinal Aberration – cont’d Longitudinal Aberration – cont’d


n = 1.0nL = 2.0

Rsinθt

( )

( ) sinsin sin

t i i t

ti t

RR z

γ π θ π θ θ θθγ θ θ

′′ = − − − = −

′′ = − =+ Δ

( ) ( ): , sin sin

sini i t t

i t i t

For paraxial rays Rρ θ θ θ θθ θ θ θ

≅ ≅

− ≅ −

ΔZ = ΔZ (θi )


n = 1.0

nL = 2.0

( ρ )

Δz/R

Longitudinal Aberration – cont’d Longitudinal Aberration – cont’d

ΔZ

Third-order (Seidel) aberrationsThird-order (Seidel) aberrations

Paraxial approximation

2 3( 1)(1 ) 1 ( )2!

n n nx nx x O x−+ = + + +

Third-order aberrations :On-axis imaging by a spherical interface

Third-order aberrations :On-axis imaging by a spherical interface

( ) ( ) ( ) ( )1 2 1 2opda Q PQI POI n n n s n s′ ′= − = + − +

To the paraxial (first-order) ray approximation, PQI = POI according to Fermat’s principle.

Beyond a first approximation, PQI ≠ POI , thus the aberration at Q is

Let’s start the aberration calculation for a simple case.

P

Q

I

O

h

Rℓ

α

β

φP

Q

CO

s+R

( )( )

( )

( ) ( )

( ) ( )

2 2 2

2 222 2 2 2 2

2

2 2 2

22 2

cos sin

sin cos 1 2

2 cos:

2 cos

Rs R s R

Rs R R R

s R

s R R R s R RSubstituting and rearranging we obtain

s R R R s R

β αφ φ α β

β αφ φ β β α

β β φ

φ

+= = = +

+ +

+ ++ = = ⇒ + = + + +

+

+ = + + = + −

= + + − +

l

Refraction at a spherical interface – cont’d Refraction at a spherical interface – cont’d

R s’-Rℓ '

αβ

φφ


IC

O

( )

( )( )

( )

( ) ( ) ( )

22 2

2 222 2 2 2

2

22 2 2 2 2

22 2

cos sin

sin cos 1

2 2

cos 2 cos

Rs R s R

s Rs R

R R R R s R

s R R s R R s R

α βφ φ α β

α βφ φ β α

α α β α

α φ φ

′= = = + +′ ′− −

+ ′+ = = ⇒ − = +′ −

′ ′= + + + = + + −

′ ′ ′ ′= − ⇒ = + − + −

l’

( )

( )

1/ 22 2 42

2 4

21/ 2

2

cos :

cos 1 sin 1 12 8

1 12 8

1

Writing the term in terms of h we obtain

h h hR R R

where we have used the binomial expansionx xx

Substituting into our expressions for and and rearranging

h R sRs

φ

φ φ⎡ ⎤⎛ ⎞= − = − ≅ − −⎢ ⎥⎜ ⎟

⎝ ⎠⎢ ⎥⎣ ⎦

+ ≅ + −

′

+= +

( )

( ) ( )

( ) ( ) ( )

1/ 24

2 3 2

1/ 22 4

2 3 2

6

22 4 4

2 3 2 2 4

4

14

12 8 8

1

h R sR s

h R s h R sRs R s

Use the same binomial expansion and neglecting terms oforder h and higher we obtain

h R s h R s h R sRs R s R s

⎧ ⎫⎡ ⎤+⎪ ⎪+⎨ ⎬⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

⎧ ⎫′ ′⎡ ⎤− −⎪ ⎪′ = + +⎨ ⎬⎢ ⎥′ ′⎪ ⎪⎣ ⎦⎩ ⎭

⎧ ⎫+ + +⎪ ⎪= + + −⎨ ⎬⎪ ⎪⎩ ⎭

′ = +( ) ( ) ( )22 4 4

2 3 2 2 42 8 8h R s h R s h R s

Rs R s R s

⎧ ⎫′ ′ ′− − −⎪ ⎪+ −⎨ ⎬′ ′ ′⎪ ⎪⎩ ⎭


2 2 4 4 4 4 4

2 2 2 3 4 3 2 2

2 2 4 4 4 4 4

2 2 2 3 4 3 2 2

12 2 8 8 8 4 8

12 2 8 8 8 4 8

h h h h h h hss Rs R s R s s Rs R s

h h h h h h hss Rs R s R s s Rs R s

⎧ ⎫= + + + + − − −⎨ ⎬

⎩ ⎭⎧ ⎫

′ ′= + − + − − + −⎨ ⎬′ ′ ′ ′ ′ ′ ′⎩ ⎭


1 2 2 1

'n n n ns s R

−+ =

Imaging formula(first-order approx.)

( ) ( ) ( )1 2 1 2

2 241 2

4

1 1 1 18 ' '

a Q n n n s n s

n nhs s R s s R

ch

′ ′= + − +

⎡ ⎤⎛ ⎞ ⎛ ⎞= − + + −⎢ ⎥⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

=Aberration for axial object points (on-axis imaging): This aberration will be referred to as spherical aberration.

The other aberrations will appear at off-axis imaging!

Third-Order Aberration :Off-axis imaging by a spherical interface

ThirdThird--Order Aberration :Order Aberration :OffOff--axis imaging by a spherical interface axis imaging by a spherical interface

Now, let’s calculate the third-order aberrations in a general case.

Q

O

B

ρ ’r

θ

h’

( ) ( ); , , 'a Q a Q r hθ=

QQ

O

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

opd

opd

a Q PQP PBP

a O POP PBP

a Q a Q a O

′ ′ ′= −

′ ′ ′= −

′ ′= −

Third-Order Aberration Theory Third-Order Aberration Theory After some very complicated analysis the third-order aberration equation is obtained:

( ) 40 40

31 31

2 2 22 22

2 22 20

33 11

cos

cos

cos

a Q C r

C h r

C h r

C h r

C h r

θ

θ

θ

=

′+

′+

′+

′+

Spherical Aberration

Coma

Astigmatism

Curvature of Field

Distortion

Q

O

B

ρ ’r

θ

On-axis imaging 에서의 a(Q) = ch4와일치

Spherical Aberration Spherical Aberration

Optics, E. Hecht, p. 222.

Transverse SphericalAberration

LongitudinalSphericalAberration

( ) 4 40 4 0a Q C r c h= =



Least SA

Most SA


For a thin lens with surfaces with radii of curvature R1 and R2, refractive index nL, object distance s, image distance s', the difference between the paraxial image distance s'p and image distance s'h is given by

( ) ( ) ( ) ( )

2 1

2 1

2 32 2

3

1 1 2 4 1 3 2 18 1 1 1

L LL L L

h p L L L L

R R s spR R s s

h n nn p n n ps s f n n n n

σ

σ σ

′+ −= =

′− +

⎡ ⎤+− = + + + + − +⎢ ⎥′ ′ − − −⎣ ⎦

( )22 12

L

L

np

nσ

−= −

+Spherical aberration is minimized when :



σ = -1 σ = +1

For an object at infinite ( p = -1, nL = 1.50 ), σ ~ 0.7

worse better

2 1

2 1

R RR Rs sps s

σ +=

−′ −

=′ +

( )22 12

L

L

np

nσ

−= −

+Spherical aberration is minimized when :

σ


σ

2 1

2 1

R RR Rs sps s

σ +=

−′ −

=′ +

Coma Coma ( ) 3

1 31 ' cos ( ' 0, cos 0)a Q C h r hθ θ= ≠ ≠

Coma Coma


Coma Coma


LeastComa

MostComa

Astigmatism Astigmatism ( ) 2 2 2

2 22 ' cos ( ' 0, cos 0)a Q C h r hθ θ= ≠ ≠


Astigmatism Astigmatism

Tangential plane(Meridional plane)

Sagittalplane

Astigmatism Astigmatism


Least Astig.

Most Astig.

Distortion Distortion ( ) 3

3 11 ' cos ( ' 0, cos 0)a Q C h r hθ θ= ≠ ≠

Field Curvature Field Curvature


( ) 2 22 20 ' ( ' 0)a Q C h r h= ≠ Astigmatism when θ = 0.

Chromatic Aberration Chromatic Aberration


Achromatic Doublet Achromatic Doublet


Achromatic Doublets Achromatic Doublets ( ) ( )

( ) ( )

( )

1 1 1 11 11 12

2 2 2 22 21 22

1 2 1 21 2 1 2

1 2 1 1 2

1 1 11 1

1 1 11 1

587.6

:

1 1 1

0 :

1

D D DD

D D DD

D

P n n Kf r r

P n n Kf r r

nm center of visible spectrum

For a lens separation of L

L P P P L P Pf f f f f

For a cemented doublet L

P P P n K n

λ

⎛ ⎞= = − − = −⎜ ⎟

⎝ ⎠⎛ ⎞

= = − − = −⎜ ⎟⎝ ⎠

= =

= + − ⇒ = + −

=

= + = − + ( ) 21 K−

1 11 2

1

1 2

:

0

, " " :

0

, .

Chromatic aberration is eliminated when

n nP K K

In general for materials with normal dispersionn

That means that to eliminate chromatic aberrationK and K must have opposite signsThe partial deriva

λ λ λ

λ

∂ ∂∂= + =

∂ ∂ ∂

∂<

∂

:

., 486.1 656.3 .

F C

F C

F C

tive of refractive index withwavelength is approximated

n nn

for an achromat in the visible region of the spectrumIn the above equation nm and nm

λ λ λ

λ λ

−∂≅

∂ −

= =

Achromatic Doublets Achromatic Doublets

( )

( )

1 11 1 11 1

1 1

2 22 2 22 2

2 2

1 2

1:

11

11

D

F C

F CD D D

F C D F C

F CD D D

F C D F C

nDefining the dispersive constant V Vn n

we can write

n nn n PK Kn V

n nn n PK Kn V

V and V are functions only of the material

λ λ λ λ λ

λ λ λ λ λ

−=

−

⎛ ⎞⎛ ⎞−∂ −= =⎜ ⎟⎜ ⎟∂ − − −⎝ ⎠⎝ ⎠

⎛ ⎞⎛ ⎞−∂ −= =⎜ ⎟⎜ ⎟∂ − − −⎝ ⎠⎝ ⎠

( ) ( )1 2

2 1 1 21 2

.

0 0D DD D

F C F C

properties of the two lenses

P P V P V PV Vλ λ λ λ

+ = ⇒ + =− −

Achromatic condition for doublets


( ) ( )

1 21 2 1 2

2 1 2 1

1 21 2

1 2

1 2

:

1 1

D D D D D D D

D D

D D

We can solve for the power of each lens in terms of the desiredpower of the doublet

V VP P P P P P PV V V V

P PK Kn n

From the values of K and K the radii of curvature for the two surfacesof the lenses can

= + ⇒ = − =− −

= =− −

1211 21 12 22

2 12112

be determined. If lens1 is bi - convex with equal curvaturefor each surface :

rr r r r rK r

= − = =−


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5. Aberration Theory5. Aberration Theory - Hanyangoptics.hanyang.ac.kr/~shsong/5-Aberration theory.pdf · 5. Aberration Theory5. Aberration Theory Last lecture •Matrix methods in