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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 '
Refraction at a spherical interface – cont’dRefraction at a spherical interface – cont’d
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
Refraction at a spherical interface – cont’dRefraction at a spherical interface – cont’d
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
Refraction at a spherical interface – cont’dRefraction 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 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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Spherical AberrationSpherical Aberration
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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Spherical AberrationSpherical 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
p R R s s
22 12
L
L
n
pn
Spherical aberration is minimized when :
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Spherical AberrationSpherical Aberration
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
Optics, E. Hecht,p. 224.
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AstigmatismAstigmatism
Tangential plane(Meridional plane)
Sagittalplane
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AstigmatismAstigmatism
Coma
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AstigmatismAstigmatism
Modern Optics,R. Guenther, p.207.
Least
Astig.
Most
Astig.
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Field CurvatureField 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
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Field CurvatureField Curvature
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
Optics, E. Hecht,p. 232.
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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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Achromatic DoubletsAchromatic Doublets
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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Achromatic DoubletsAchromatic 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 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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Achromatic DoubletsAchromatic Doublets
Table 20-1