20. Aberration Theory20. Aberration Theory - Hanyangoptics.hanyang.ac.kr/~shsong/20-Aberration theory.pdf · 20. Aberration Theory20. Aberration Theory Wavefront aberrations (파면수차)

20. Aberration Theory20. Aberration Theory

Wavefront aberrations (파면수차) Chromatic Aberration (색수차) Third-order (Seidel) aberration theory

Spherical aberrations Coma Astigmatism Curvature of Field Distortion

AberrationsAberrations

ChromaticChromatic MonochromaticMonochromatic

Unclear Unclear imageimage

Deformation Deformation of imageof image

SphericalSpherical

ComaComa

astigmatismastigmatism

DistortionDistortion

Field CurvatureField Curvature

n (n (λλ))

Five third-order (Seidel) aberrations

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

n (n (λλ))

Five monochromatic AberrationsFive monochromatic Aberrations

Unclear imageUnclear image Deformation of imageDeformation of image

SphericalSpherical

ComaComa

astigmatismastigmatism

DistortionDistortion

Field curvatureField curvature

Spherical aberrationSpherical aberration• 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

ComaComa• 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

Astigmatism and curvature of fieldAstigmatism and curvature of field

Yields elliptically distorted imagesYields elliptically distorted images

Distortion: Pincushion, BarrelDistortion: Pincushion, BarrelDistortion: Pincushion, Barrel

• 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)

A mathematical treatment of the monochromatic aberrations can be developed by expanding the binomial series

up to higher orders

Third-order aberration theory

Paraxial approximation

Now, let’s derive the expression of

the third-order aberrations

(Seidel aberrations, Ludwig von Seidel)

Now, let’s derive the expression of

the third-order aberrations

(Seidel aberrations, Ludwig von Seidel)

20-1. Ray and wave aberrations20-1. Ray and wave aberrations

LA

TALA : ray aberration - longitudinalTA : ray aberration - transverse (lateral)

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 t tn n

R

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

n = 1.0nL = 2.0

Rsint

sinsin sin

t i i t

ti t

RR z

Z = Z (i )

R

sin sini i t tn n


n = 1.0

nL = 2.0

( )

z/R

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

Z

sinsin t

i tRR z

sin sini i t tn n

20-2. Third-order treatment of refraction at a spherical interface

20-2. Third-order treatment of refraction at 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 (depends on the position of Q) ≠ POI.Thus we define the aberration at Q as

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

P

Q

I

O

h

Figure 20-3

Rℓ

P

Q

CO

s

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

l

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

Let’s describe l in terms of R, s, .

R s’-Rℓ '


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

R

s R s R

s Rs R

R R R R s R

s R R s R R s R

l’

1/ 22 2 42

2 4

21/ 2

2

cos :

cos 1 sin 1 12 8

1 12 8

exp

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 ressions for and and rearranging

h R ss

R

1/ 24

2 3 2

1/ 22 4

2 3 2

6

22 4 4

2 3 2 2 4

4

' 14

exp

12 8 8

h R ss R s

h R s h R ss

Rs R s

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

h R s h R s h R ss

Rs R s R s

22 4 4

2 3 2 2 4' 12 8 8

h R s h R s h R ss

Rs R s R s


P

Q

IO

h

Third-order angle effect

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 2a Q n n n s n s

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!

P

Q

O

h

a Q

Wave aberration

2 24

41 21 1 1 18 ' '

n nha Q chs s R s s R

20-3. Spherical Aberration 20-3. Spherical Aberration

Optics, E. Hecht, p. 222.

Transverse SphericalAberration

LongitudinalSphericalAberration

4a Q c h

yb

zb

3

2

4 'y

c sb hn

2

2

2

4 'z

c sb hn

h

n2

Spherical Aberration Spherical Aberration


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

32

2 23

2 1

2 1

21 1 4 1 3 2 18 1 1 1

where,

( hape factor),

L LL L L

h p L L L L

n nh n p n n ps s f n n n n

R R s ss pR R s s

22 12

L

L

np

n

Spherical aberration is minimized when :



= -1 = +1

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

worse better

2 1

2 1

, R R s spR R s s

22 12

L

L

np

n

Spherical aberration is minimized when :


2 1

2 1

R RR Rs sps s

~ 0.7

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

’r

h’

4 4

2 2 2

( ' )

' 2 cos , ; , , '

'

a Q a Q a O c b

r ba Q a Q r

r hh

b b

QQ

O

4 4

4 4

( ) '

( )opd

opd

a Q PQP PBP c BQ c

a O POP PBP c BO cb

On the lens surface

B

b

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

Spherical Aberration

Coma

Astigmatism

Curvature of Field

Distortion

Q

O

B

’r

On-axis imaging 에서의 a(Q) = ch4와일치

'h rC

20-4. Coma 20-4. Coma 3

1 31 ' cos ( ' 0, cos 0)a Q C h r h

Q

O

B

’r

Coma Coma


Coma Coma


LeastComa

MostComa

Coma Coma

20-5. Astigmatism 20-5. Astigmatism 2 2 2

2 22 ' cos ( ' 0, cos 0)a Q C h r h


Astigmatism Astigmatism

Tangential plane(Meridional plane)

Sagittalplane


Coma



Least Astig.

Most Astig.

Field Curvature Field Curvature

2 22 20 ' ( ' 0)a Q C h r h

Astigmatism when = 0.

A flat object normal to the optical axis cannot be brought into focus on a flat image plane.

This is less of a problem when the imaging surface is spherical, as in the human eye.

Q

O

B

’r

Field Curvature Field Curvature 2 2

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

If no astigmatism is present, the sagittal and tangential image surfaces coincide on the Petzval surface.

The best image plane, Petzval surface, is actually not planar, but spherical. This aberration is called field curvature.

20-6. Distortion 20-6. Distortion 3

3 11 ' cos ( ' 0, cos 0)a Q C h r h

20-7. Chromatic Aberration 20-7. Chromatic Aberration


Achromatic Doublet Achromatic Doublet

(1) (2)

Power of two lenses, P1D and P2D are different at the Fraunhofer wavelength, D = 587.6 nm.

20-13

* Note: What is the definition of the Fraunhofer wavelengths (lines)?

Figure 20-13

* Note: Fraunhofer wavelengths (lines)

Spectrum of a blue sky somewhat close to the horizon pointing east at around 3 or 4 pm on a clear day.

The dark lines in the solar spectrum were caused by absorption by those elements in the upper layers of the Sun.

D = 587.6 nm (yellow)

C = 656.3 nm (red)F = 486.1 nm (blue)

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

21 K

Total power is

1 21 2

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 derivat

:

., 486.1 656.3 .

F C

F C

F C

ive of refractive index withwavelength is approximated

n nn

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


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

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 V

Achromatic condition for doublets


1 2

1 22 1 1 2 1 2

2 1 2 1

1 1 2 2

1

:

0

where 1 1

D D D

D D D D D D

D D D

We can solve for the power of each lens in terms of the desired power of the doubletP P P

V VV P V P P P P PV V V V

P n K n K

PK

1 22

1 2

1 2

1211 21 12 22

2 12

,1 1

, 4

1

D D

D D

12

PKn n

From the values of K and K the radii of curvature for the two surfacesof the lenses can be determined. If lens1 is bi - convex with equal curvature for each surface :

rr r r r rK r

221 22

1 1where, Kr r


Table 20-1

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20. Aberration Theory20. Aberration Theory - Hanyangoptics.hanyang.ac.kr/~shsong/20-Aberration theory.pdf · 20. Aberration Theory20. Aberration Theory Wavefront aberrations (파면수차)