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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
Modern Optics, R. Guenther, p. 199.
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 '
Refraction at a spherical interface – cont’d Refraction at a spherical interface – cont’d
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
Refraction at a spherical interface – cont’d Refraction at a spherical interface – cont’d
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
Refraction at a spherical interface – cont’d Refraction at a spherical interface – cont’d
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
Modern Optics, R. Guenther, p. 196.
Least SA
Most SA
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
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 :
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 spR R s s
22 12
L
L
np
n
Spherical aberration is minimized when :
Spherical Aberration Spherical Aberration
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
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
Optics, E. Hecht, p. 224.
Coma Coma
Modern Optics, R. Guenther, p. 205.
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
Optics, E. Hecht, p. 224.
Astigmatism Astigmatism
Tangential plane(Meridional plane)
Sagittalplane
Astigmatism Astigmatism
Coma
Astigmatism Astigmatism
Modern Optics, R. Guenther, p. 207.
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
Optics, E. Hecht, p. 232.
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
Achromatic Doublets Achromatic Doublets
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
Achromatic Doublets Achromatic 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
Achromatic Doublets Achromatic Doublets
Table 20-1