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20. Aberration Theory 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

(2)

Aberrations Aberrations

Chromatic

Chromatic MonochromaticMonochromatic

Unclear Unclear

image image

Deformation Deformation of image

of image

Spherical Spherical ComaComa

astigmatism astigmatism

Distortion Distortion

Field Curvature Field Curvature

n ( n ( λ λ ) )

 Five third-order (Seidel) aberrations

(3)

Aberrations: Chromatic Aberrations: 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))

(4)

Five monochromatic Aberrations Five monochromatic Aberrations

Unclear image

Unclear image Deformation of imageDeformation of image Spherical

Spherical ComaComa

astigmatism astigmatism

Distortion Distortion

Field curvature Field curvature

(5)

Spherical aberration Spherical 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

(6)

ComaComa

An off-axis effect which appears when a bundle of incident rays all make the same angle with respect to the optical axis (source at ∞)

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

(7)

Astigmatism and curvature of field Astigmatism and curvature of field

Yields elliptically distorted images Yields elliptically distorted images

(8)

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

i

T y

y s

M s will differ also

object object

Pincushion image

Pincushion 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)

(9)

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

(10)

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)

(11)

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

LA

TA LA : ray aberration - longitudinal

TA : ray aberration - transverse (lateral) Actual wavefront

Wave aberration Ideal wavefront

Paraxial Image plane Paraxial Image point

(12)

Longitudinal Ray Aberration Longitudinal Ray Aberration

Modern Optics, R.

Guenther, p. 199.

n = 1.0

nL = 2.0

R

sin sin sin sin

2

i

i i t t t

n n

R

(13)

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

n = 1.0

nL = 2.0

Rsint

sin

sin sin

t i i t

t

i t

R

R z

 

 

  Z = Z (i) R

sin sin

i i t t

n n

(14)

Modern Optics, R. Guenther, p. 199.

n = 1.0

nL = 2.0

( )

z/R

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

Z

sin

sin i t R t

R z

 

sin sin

i i t t

n n

(15)

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

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

   opd 1 2   1 2

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

(16)

R

P

Q

C O

s

2 2 2

2 2

2 2 2 2 2 2

2

2 2 2

2 2 2

cos sin

sin cos 1 2

2 cos

:

2 cos

R

s R s R

R s R R R

s R

s R R R s R R

Substituting 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, .

(17)

R ℓ ' s’-R

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

I

C O

2 2 2

2 2

2 2 2 2 2

2

2 2 2 2 2 2

2 2 2

cos sin

sin cos 1

2 2

cos 2 cos

R

s R s R

s R s R

R R R R s R

s R R s R R s R

 

 

 

l’

(18)

2 1/ 2 2 4

2

2 4

1/ 2 2

2

cos :

cos 1 sin 1 1

2 8

1 1

2 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 s

s R

     

 

4 1/ 2

2 3 2

2 4 1/ 2

2 3 2

6

2 4 4 2

2 3 2 2 4

4

' 1 4

exp

1 2 8 8

h R s

s R s

h R s h R s

s 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 s

s Rs R s R s

 

2

2 4 4

2 3 2 2 4

' 1 2 8 8

h R s h R s h R s

s Rs R s R s

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

P

Q

I O

h

Third-order angle effect

(19)

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

1 2 2 8 8 8 4 8

1 2 2 8 8 8 4 8

h h h h h h h

s s Rs R s R s s Rs R s

h h h h h h h

s s Rs R s R s s Rs R s

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

1 2 2 1

'

n n n n

s s R

Imaging formula (first-order approx.)

   1 2   1 2

a 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

  4 1 1 1 2 2 1 1 2 4

8 ' '

n n

a Q h ch

s s R s s R

 

(20)

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

Optics, E. Hecht, p. 222.

Transverse Spherical Aberration Longitudinal

Spherical Aberration

  4

a Q c h

b y

b z

3 2

4 '

y

b c s h

n 2 2

2

4 '

z

b c s h

n

h

n2

(21)

Spherical Aberration Spherical Aberration

Modern Optics, R. Guenther, p.

196.

Least SA

Most SA

(22)

Spherical Aberration Spherical Aberration

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 2  2 3

3

2 1

2 1

2

1 1

4 1 3 2 1

8 1 1 1

where,

( hape factor),

L L

L L L

h p L L L L

n n

h n p n n p

s s f n n n n

R R s s

s p

R R s s

2

2 1

2

L L

n p

  n Spherical aberration is minimized when :

(23)

Spherical Aberration Spherical Aberration

Optics, E. Hecht, p. 222.

= -1 = +1

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

worse better

2 1

2 1

,

R R s s

R R p s s

2

2 1

2

L L

n p

  n

Spherical aberration is minimized when :

(24)

Spherical Aberration Spherical Aberration



2 1

2 1

R R

R R

s s p s s

 

 

~ 0.7

(25)

Third-Order Aberration :

Off-axis imaging by a spherical interface Third

Third--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 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

(26)

Third-Order Aberration Theory Third-Order Aberration Theory

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

  0 40 4 3 1 31

2 2 2

2 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 imaging 에서의 a(Q) = ch4 와 일치

'

h

C

r

(27)

20-4. Coma 20-4. Coma

  1 31 ' 3 cos ( ' 0, cos 0)

a Q C h r h

Q O

B

 r

(28)

Coma Coma

Optics, E.

Hecht, p.

224.

(29)

Coma Coma

Modern Optics, R. Guenther, p.

205.

Least Coma

Most Coma

(30)

Coma Coma

(31)

20-5. Astigmatism 20-5. Astigmatism

  2 22 '2 2 cos2 ( ' 0, cos 0)

a Q C h r h

Optics, E. Hecht, p. 224.

(32)

Astigmatism Astigmatism

Tangential plane (Meridional plane) Sagittal

plane

(33)

Astigmatism Astigmatism

Coma

(34)

Astigmatism Astigmatism

Modern Optics, R. Guenther, p.

Least 207.

Astig.

Most Astig.

(35)

Field Curvature Field Curvature

  2 20 '2 2 ( ' 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

(36)

Field Curvature Field Curvature

  2 20 '2 2 ( ' 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.

(37)

20-6. Distortion 20-6. Distortion

  3 11 '3 cos ( ' 0, cos 0)

a Q C h r h

(38)

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

Optics, E. Hecht, p. 232.

(39)

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

(40)

* 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

absorptionby those elements in the upper layers of the Sun.

D = 587.6 nm (yellow)

C = 656.3 nm (red)

F = 486.1 nm (blue)

(41)

Achromatic Doublets Achromatic 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 1

1 1

1 1 1

1 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

L P P P L P P

f f f f f

For a cemented doublet L

P P P n K n

1 K 2

Total power is

1 2

1 2

1 2

:

0

, " " :

0

, .

Chromatic aberration is eliminated when

n n

P K K

In general for materials with normal dispersion n

That means that to eliminate chromatic aberration K 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 n

n

for an achromat in the visible region of the spectrum

In the above equation nm and nm

(42)

Achromatic Doublets Achromatic Doublets

1 1

1 1 1

1 1

1 1

2 2

2 2 2

2 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

Defining the dispersive constant V V n

n n

we can write

n n

n n P

K K

n V

n n

n n P

K K

n V

V and V are functions only of the material









1 1 2 2 2 1 1 2

.

0 0

D D

D D

F C F C

properties of the two lenses

P P

V P V P

V V

Achromatic condition for doublets

(43)

Achromatic Doublets Achromatic Doublets

1 2

1 2

2 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

K P

 

1 1 2 2 2

1 2

12

11 21 12 22

2 12

1 , 1

, 4

1

D D

D D

12

K P

n n

From the values of K and K the radii of curvature for the two surfaces of the lenses can be determined.

If lens 1 is bi - convex with equal curvature for each surface :

r r r r r r

K r

 

2 21 22

1 1

where, K

r r

(44)

Achromatic Doublets Achromatic Doublets

Table 20-1

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