20 aberration theory

8/21/2019 20 Aberration Theory

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20. Aberration Theory

20. Aberration Theory

Wavefront aberrations (파면수차)

Chromatic Aberration (색수차)

Third-order (Seidel) aberration theory

Spherical aberrations

Coma

Astigmatism

Curvature of Field

Distortion


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Aberrations

Aberrations

ChromaticChromatic MonochromaticMonochromatic

UnclearUnclear

imageimage

DeformationDeformation

of imageof image

SphericalSpherical

ComaComa

astigmatismastigmatism

DistortionDistortion

Field CurvatureField Curvature

n (n (λλ))

Five third-order (Seidel) aberrations


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Aberrations: Chromatic

Aberrations: Chromatic

• Because the focal length of a lens depends on therefractive index (n), and this in turn depends on thewavelength, n = n(λ), light of different colors

emanating from an object will come to a focus atdifferent points.

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

n (n (λλ))


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Five monochromatic AberrationsFive monochromatic Aberrations

Unclear imageUnclear image Deformation of imageDeformation of image

SphericalSpherical

ComaComa

astigmatismastigmatism

DistortionDistortion

Field curvatureField curvature


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Spherical aberrationSpherical aberration

• This effect is related to rays which make large anglesrelative to the optical axis of the system

• Mathematically, can be shown to arise from the fact thata lens has a spherical surface and not a parabolic one

• Rays making significantly large angles with respect tothe optic axis are brought to different foci


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ComaComa

• An off-axis effect which appears when a bundle of incident raysall 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


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Astigmatism and curvature of fieldAstigmatism and curvature of field

Yields elliptically distorted imagesYields elliptically distorted images


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

• This effect results from the difference in lateralmagnification of the lens.

• If f differs for different parts of the lens,

o

i

o

iT

y

y

s

s M will differ also

objectobject

Pincushion imagePincushion image Barrel imageBarrel imagef f ii>0>0 f f ii


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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


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Now, let’s derivethe expression ofthe third-order aberrations

(Seidel aberrations, Ludwig von Seidel)

Now, let’s derivethe expression of

the third-order aberrations

(Seidel aberrations, Ludwig von Seidel)


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

LA

TALA : ray aberration - longitudinal

TA : ray aberration - transverse (lateral)

Actual wavefront

Wave aberration Ideal wavefront Paraxial Image plane

Paraxial Image point


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Longitudinal Ray AberrationLongitudinal Ray Aberration

Modern Optics, R.Guenther, p. 199.

n = 1.0nL = 2.0

R

sinsin sin sin

2

ii i t t t n n

R


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Longitudinal Aberration – cont’dLongitudinal Aberration – cont’d

n = 1.0nL = 2.0

Rsint

sinsin sin

t i i t

t i t

R R z

Z = Z ( i )

R

sin sini i t t

n n


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Modern Optics,R. Guenther, p. 199.

n = 1.0

nL = 2.0

( )

z/R

Longitudinal Aberration – cont’dLongitudinal Aberration – cont’d

Z

sin

sin t i t R

R z

sin sini i t t n n


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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 2opd a 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


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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 Rs R R R s R R

Substituting and rearranging we obtain

s R R R s R

l

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

Let’s describe l in terms of R, s, .


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R s’-Rℓ '


I

C

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’


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1/ 22 2 4

2

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 h

R R R

where we have used the binomial expansion

x x x

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 s

s R s

h R s h R ss

Rs R s

Use the same binomial ansion and neglecting terms of

order 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' 1 2 8 8h R s h R s h R ss

Rs R s R s


P

Q

I

O

h

Third-order angle effect


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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 hs

s Rs R s R s s Rs R s

h h h h h h hs

s Rs R s R s s Rs R s


1 2 2 1

'

n n n n

s 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 QWave aberration

2 24

41 21 1 1 1

8 ' '

n nha Q ch

s s R s s R


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20-3. Spherical Aberration20-3. Spherical Aberration

Optics, E. Hecht,p. 222.

TransverseSpherical Aberration

LongitudinalSpherical

Aberration

4a Q c h

yb

zb

3

2

4 ' y

c sb h

n

2

2

2

4 ' z

c sb h

n

h

n 2


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Spherical AberrationSpherical Aberration

Modern Optics,R. Guenther, p.

196.

Least SA

Most SA


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For a thin lens with surfaces with radii of curvature R1 andR2, 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

322 2

3

2 1

2 1

21 14 1 3 2 1

8 1 1 1

where,

( hape factor),

L L L L L

h p L L L L

n nhn p n n p

s s f n n n n

R R s ss p

R R s s

22 12

L

L

n p

n

Spherical aberration is minimized when :


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= -1

= +1

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

worse better

2 1

2 1 ,

R R s s

p R R s s

22 12

L

L

n

pn

Spherical aberration is minimized when :


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2 1

2 1

R R R R

s s p

s s

~ 0.7


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Third-Order Aberration :

Off-axis imaging by a spherical interface

ThirdThird--Order Aberration :Order Aberration :

Off Off --axis imaging by a spherical interfaceaxis imaging by a spherical interfaceNow, 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 b

a Q a Q r

r h

h

b b

Q

Q

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


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Third-Order Aberration TheoryThird-Order Aberration Theory

After some very complicated analysis the third-order aberration equation is obtained:

40 40

3

1 31

2 2 22 22

2 2

2 20

3

3 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 imaging에서의 a(Q) = ch 4와일치

'h r

C


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20-4. Coma20-4. Coma

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

Q

O

B

’r


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ComaComa

Optics, E.Hecht, p.224.


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ComaComa

Modern Optics,R. Guenther, p.205.

LeastComa

MostComa


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ComaComa


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20-5. Astigmatism20-5. Astigmatism

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



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AstigmatismAstigmatism

Tangential plane(Meridional plane)

Sagittalplane


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Coma


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Modern Optics,R. Guenther, p.207.

Least

Astig.

Most

Astig.


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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


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2 22 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.


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20-6. Distortion20-6. Distortion

33 11 ' cos ( ' 0, cos 0)a Q C h r h


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20-7. Chromatic Aberration20-7. Chromatic Aberration



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Achromatic DoubletAchromatic Doublet

(1) (2)

Power of two lenses, P1D and P2D are differentat 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)


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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 byabsorption 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)


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Achromatic DoubletsAchromatic Doublets

1 1 1 1

1 11 12

2 2 2 2

2 21 22

1 2 1 2

1 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 D

D

D D D

D

D

P n n K f r r

P n n K f r r

nm center of visible spectrum

For a lens separation of L

LP P P L P P

f 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 nPK K

In general for materials with normal dispersion

n

That means that to eliminate chromatic aberrationK and K must have opposite signs

The partial derivat

:

.

, 486.1 656.3 .

F C

F C

F C

ive of refractive index with

wavelength is approximated

n nn

for an achromat in the visible region of the spectrum

In the above equation nm and nm


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1 11 1 11 1

1 1

2 22 2 22 2

2 2

1 2

1:

1

1

1

1

D

F C

F C D D D

F C D F C

F C D D D

F C D F C

n Defining the dispersive constant V V

n n

we can writen nn n P

K K n V

n nn n PK K

n V

V and V are functions only of the material

1 2

2 1 1 2

1 2

.

0 0 D D D DF C F C

properties of the two lenses

P PV P V P

V V

Achromatic condition for doublets

A h ti D bl t


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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 doublet

P P P

V V V P V P P P P P

V 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

PK

n n

From the values of K and K the radii of curvature for the two surfaces

of the lenses can be determined.

If lens1 is bi - convex with equal curvature for each surface :

r r r r r r

K r

2

21 22

1 1where, K

r r

A h ti D bl t


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Table 20-1

20 aberration theory

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