Aberration theory.pptx

20. Aberration Theory
20. Aberration Theory
 Chromatic Aberration
 Third-order (Seidel) aberration theory
 Spherical aberrations
 Astigmatism
 Curvature of Field

Aberrations
Aberration Theory
A perfectly spherical wave will
converge to a point. Any deviation
from the ideal spherical shape is
said to be an aberration.
Optical systems convert
the shapes of wavefronts
Lens

Aberrations
Aberrations
Chromatic Monochromatic
Unclear
image
Deformation
of image
Spherical
Coma
astigmatism
Distortion
Field Curvature
n (λ)
 Five third-order (Seidel) aberrations

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 (λ)

Math of Chromatic Aberrations
 Consider a lens made of glass with n’(ω)
R1
R1 > 0
R2 < 0
R2
 Convergence
t
t
1 2
n
C  C1  C2  C C

n 1

1 n

(n 1)(1 n)t
R1 R2 nR1R2
(11)
Aberrations

Chromatic Aberrations
 Thin Lens: t = 0
f
 C 
1
1
 (n 1)
 1 
1 
 R R 
 1 2 
(12)
f
 If |R1|=|R2|
1

2(n 1)
f R
(13)
Aberrations

 The dispersion effect is very clear:
(14)
f ()  focal distance varies with color!
R
2[n() 1]
 Differentiate Eq. 14:
d
 1 
 d 2(n 1)
R
 
 f ()
 
(15)
 1 df

2 dn
f 2
d R d
(16)
df
 
2 f 2
dn
d R d
(17)
Aberrations

dn
 0  normal dispersion
 What is the negative sign telling?
d
df
 
2 f 2
dn
R d d
IR Red Green Blue UV
white light
marginal ray
longitudinal
chromatic aberration
OA
B Y R
Aberrations

Five monochromatic Aberrations
Correction of Chromatic Aberrations
• Chromatic aberration can be reduced by increasing the focal length of the
lens where possible
• Or by using an achromatic lens or achromat (in achromat lens, a compound
lens is formed by assembling together materials with different dispersion).
• The degree of correction can be further increased by combining more than
two lenses of different composition, as seen in an achoromatic lens.
Ref: https://www.quora.com/What-are-achromatic-lens-used-for

Correcting for Chromatic Aberration
• It is possible to have refraction without
chromatic aberration even when 𝑛 is a
function of 𝜆:
– Rays emerge displaced but parallel
– If the thickness is small, then there
is no distortion of an image
– Possible even for non-parallel surfaces:
– Aberration at one interface is compensated
by an opposite aberration at the other
surface.

Chromatic Aberration
• Focal length:
1
= 𝑛 − 1
ƒ
• Thin lens equation:
1
−
1
𝑅1 𝑅2
1
=
1
+
1
ƒ ƒ1 ƒ2
• Cancel chromatic aberration using a
combination of concave and convex lenses
with different index of refraction

Commercial Lens Assemblies
• Some lens components are made with ultra-
low dispersion glass, eg. calcium fluoride

Monochromatic Aberrations
Unclear image
Spherical
Astigmatism
Deformation of image
Field curvature

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

𝑠o i
𝑛1 𝑛2 𝑛2 − 𝑛1
𝑅
+
𝑠
= + ℎ2
o o
𝑛1 1 1
2𝑠 𝑠
+
𝑅
2
+
2𝑠i
𝑛2 1 1
𝑅
−
𝑠i
2
Paraxial approximation:
𝑛1
+
𝑛2
=
𝑛2 − 𝑛1
𝑠o 𝑠i 𝑅
Third order approximation:
Deviation from first-order theory

• The use of aspheric lenses, which have non-spherical surfaces, can reduce or
eliminate spherical aberration.
• Doublet or Triplet Lens Systems: Spherical aberration can be reduced by
using multiple lenses in a system, such as doublet or triplet lens
configurations.
Ref: https://www.quora.com/How-is-spherical-aberration-caused

20-2. Third-order treatment of refraction
at a spherical interface
Third-order treatment of refraction at a
spherical interface
opd
aQ PQI  POI   n1l  n2l  n1s  n2s
Let’s start the aberration calculation for a simple case.
P
Q
I
O
h
Figure 20-3
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

R
ℓ


P

Q
C
O
s
Substituting and rearranging we obtain :
 2
 2
cos 
  R
sin 

s  R s  R
sin2
  cos2
  1   s  R2
 2
 2R  R2
2
  R2
2
s  R2
s  R2
  s  Rcos  R
 s  R2
 R2
 2R s  Rcos
l2
 l2
 2R  R2
l2
l
Refraction at a spherical interface – cont’d
Let’s describe l in terms of R, s, .

R s’-R
ℓ '



I

C
O
 
2 2 2
2
2
 R 2
 2
cos 
 sin 

s R s R
 2

sin2
  cos2
  1   s  R   

s  R
 R2
 2R 2
 2
 R2
 2R  s R2
  s Rcos   R2
 s R2
 2Rs  Rcos
l2
l2
l2
l’

h2
h4
2R2
8R4
1/ 2
Writing the cos term in terms of h we obtain :
 1  
where we have used the binomial expansion
x x2
2 8
1 x  1  
Substituting into our expressions for l and l and rearranging

1/ 2
 h 
2

cos  1 sin2
  1   

  R  

 
 
1/ 2
4
Rs2
4R3
s2
1/ 2
4
Use the same binomial exp ansion and neglecting terms of
order h6
and higher we obtain
2Rs2
8R3
s2
8R2
s4


h2
R  s h R  s

 

l  s 1 



 

 
h2
R  s

 

Rs2

h R  s
4R3
s2


 
l  s ' 1 

 h2
R  s h4
R  s h4
R  s2


  


l  s 1



 h2
R  s h4
R  s h4
R  s2


 
2Rs2
8R3
s2
8R2
s4



l  s '1


P
Q
I
O
h
Third-order angle effect

2 3 4 2 2
2
h2
h2
h4
h4
h4
h4
h4
 
     
2s 2Rs 8R2
s2
8R s 8s 4Rs3
8R s
 
 
 



h2
h2
h4
h4
h4
h4
h4
2s
     
2Rs 8R2
s2
8R3
s 8s4
4Rs3
8R2
s2

l  s 1
l  s 1
n1

n2

n2  n1
s s' R
Imaging formula
(first-order approx.)
aQ n1l  n2l  n1s  n2s
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
Wave aberration
aQ
h4 n  1
aQ 
n  1
1 
2
1 
2

 2
     ch4
s'  s' R  


 1
 
8 s  s R 

20-3. Spherical Aberration
20-3. Spherical Aberration
Optics, E. Hecht,
p. 222.
Transverse
Spherical
Aberration
Longitudinal
Spherical
Aberration
b y
b z
3
2
4 cs '
b y  h
n
4 cs '2
2
2
b z  h
n
a Q   c h 4
h
n2

Spherical Aberration
Modern Optics,
R. Guenther, p.
196.
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

 
n3
h2
2 2
3
 
R2  R1
(shape factor), p 
s s
R2  R1 s s
L
n  2
1

1

sh
 4 nL 1 p  3nL  2 nL  
where,
1 p 

L

nL 1

 
sp 8 f nL nL 1 n 1
 L
 2
L
p
2 n 1
nL  2
Spherical aberration is minimized when :   

Optics, E. Hecht,
p. 222.
 = -1  = +1
For an object at infinite ( p = -1, nL = 1.50 ),  ~ 0.7
worse better
 
R2  R1
, p 
s s
s s
 2
L
p
2 n 1
  
nL  2 R2  R1
Spherical aberration is minimized when :

Third-Order Aberration :
Off-axis imaging by a spherical interface
Third-Order Aberration :
Off-axis imaging by a spherical interface
Now, let’s calculate the third-order aberrations in a general case.
Q
O  ’

r
h’
aQ aQaOc( '4
b4
)
 '2
 r2
b2
 2rbcos, b  h'
 aQ  aQ; r,,h'
Q
Q
aQ  PQP  PBP  c(BQ)4
 c '4
opd
aO  POP  PBP  c(BO)4
 cb4
opd
On the lens surface
O
B
b
B

Third-Order Aberration Theory
After some very complicated analysis the third-order aberration equation is obtained:
1 31
2 20
3 11
4
0C40 r
aQ 
 C hr3
cos
 2C22 h r cos 
2 2 2
h2
r2
 C
 C h3
rcos
Coma
Astigmatism
Curvature of Field
Distortion
Q
B
 ’
r
O

On-axis imaging a(Q) = ch4
h'Cr

Astigmatism
• Parallel rays from an off-axis object arrive in the plane of the lens in one direction,
but not in a perpendicular direction:
https://www.physics.purdue.edu/~jones105/phys42200_Spring2013/notes/Phys42200_Lecture33.pdf

20-5. Astigmatism
20-5. Astigmatism
aQ  2C22 h' r cos  (h'  0, cos  0)
2 2 2
Optics, E. Hecht,
p. 224.

Astigmatism
Astigmatism
Tangential plane
(Meridional plane)
Sagittal
plane

Astigmatism
Astigmatism
Modern Optics,
R. Guenther, p.
207.
Least
Astig.
Most
Astig.

Field of Curvature
• It is the area in the image space where the optical system
brings light into sharp focus.
• The field of curvature becomes important because astigmatic
optical systems can have multiple fields of curvature.
• This means that different sets of light rays focus at different
points within the image space, creating multiple fields of
curvature. The result is a distorted image with variations in
sharpness and clarity across the field of view.
• To mitigate astigmatism and improve image quality, optical
systems may use complex lens designs or aspheric surfaces
that aim to minimize the differences in curvature and provide a
larger, more uniform field of curvature.
https://www.physics.purdue.edu/~jones105/phys42200_Spring2013/notes/Phys42200_Lecture33.pdf

Field Curvature
Field Curvature
aQ  C h'2
r2
(h'  0)
2 20
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
B
 ’
r
O


Field Curvature
Field Curvature
aQ  2C20 h' r
2 2 (h'  0) 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.

Field Curvature
Field Curvature Extra Math
https://www.erbion.com/index_files/Modern_Optics/Ch20.pdf

Aberration theory.pptx

Recommended

Recommended

More Related Content

Similar to Aberration theory.pptx

Similar to Aberration theory.pptx (20)

Recently uploaded

Recently uploaded (20)

Aberration theory.pptx