### Feb 19: 3rd Order Aberrations Lecture Notes

```Hope College, PHYS 352, Spring 2013
Lens Aberrations
Aberration: a departure from the paraxial limit.
Paraxial limit: (1) Rays are kept close to the optic axis.
(2) Small angle approximation (1st Order)
sin(  )   

3

3!
cos(  )  1 

5

5!
2

2!


4

4!
5th Order
3rd Order
1 Order
3rd Order aberration theory: Keep the 3rd order terms.
Hope College, PHYS 352, Spring 2013
Pages 153-159 of Hecht, 4th ed., treats refraction in the paraxial (1st Order) limit.
If the 3rd order terms are kept in ℓo and ℓi (p. 154) we find that extended images
don’t land on exactly the same focal surface (See Fig. 6.14, p, 254).
n1
n2
Q
ℓo
ℓi
Path 1
so
si
Paraxial Approximation: Path 1 and Path 2 have the same Optical Path Length
Quantify the aberration:
a(Q)=Path 2 – Path 1
a(Q)=(n1ℓo+n2ℓi)-(n1so+n2si)
a=0 means no aberration
Hope College, PHYS 352, Spring 2013
The Five Monochromatic Seidel Aberrations
n1
ℓo
Q
yi and yo are paraxial
image and object
heights.
n2
ℓi

r
yi
yo
a ( Q )  Ar  Br y i cos   Cr y i cos   Dr y i  Ery i cos 
4
3
2
2
2
2
2
3
Distortion
Curvature of field
Astigmatism
Coma
image point
Spherical aberration
(From Pedrotti, 3rd Ed., Section 20.2)
Hope College, PHYS 352, Spring 2013
1. Spherical aberration
The only aberration that exists even for objects on the optic axis (no yi dependence).
a spherical  r
4
X
Positive SA: The marginal rays converge left of the paraxial image (positive lens)
Negative SA: The marginal rays converge right of the paraxial image (negative lens)
Homework: Analyze this for a concave mirror.
Hope College, PHYS 352, Spring 2013
1. Spherical aberration
r2
r1
Minimizing spherical aberration
n
The Coddington shape factor:
 
r2  r1
r2  r1
Spherical aberration is minimized when   
(Shown in Jenkins & White, Sections 9.4-9.5)

2 n 1
2
n2
 s
i
 so
si  so
Homework: For what value of n does a planar convex lens produce a minimum
spherical aberration for an object located at infinity? Which side of the lens
should the light enter?
Hope College, PHYS 352, Spring 2013
2. Coma
a coma  r y i cos 
3
• Depends on yi. e.g. It’s an “off-axis” aberration.
• Not symmetrical, which is the origin of the name.
screen
off-axis object
www.telescope-optics.net
Negative coma: Marginal rays
focus closer to the optic axis.
Positive coma: …farther from..
The same lens designs minimize coma & spherical aberration.
www.ryokosha.com
Figure 6.22a (Hecht, page 260) shows the formation of a
comatic image from a series of comatic circles.
Hope College, PHYS 352, Spring 2013
3. Astigmatism
a  r y i cos 
2
2
2
Astigmatism: rays from off-axis source do not strike the lens symmetrically
(tangential rays versus saggital rays).
Tangential rays
Saggital rays
See Fig. 6.27 (page 263): tangential and saggital rays will fan out and
form line images of the point source at two different image surfaces.
Hope College, PHYS 352, Spring 2013
Circle of least confusion
http://www.olympus-ims.com/en/microscope/terms/classification/
Hope College, PHYS 352, Spring 2013
4. Curvature of Field
a  r yi
2
2
Curvature of field: Tangential and saggital rays do not form images on the same surface.
Very similar to astigmatism, but symmetric about the optic axis.
Tangential rays
Saggital rays
(T left of S: Positive astigmatism)
T S
Hope College, PHYS 352, Spring 2013
Elimination of astigmatism and curvature of field aberrations
Engineer the lens curvatures or spacing so that tangential and saggital surfaces coincide.
Two lenses will
have a flat P
surface if
n1f1=-n2f2
Ultimately, the
film must
conform to P.
A focal surface that
eliminates astigmatism to
3rd order is called a
Petzval surface.
Flat P surface – eliminates
both astigmatism and
curvature of field
aberrations.
T S P
P is always 3x farther from T than S
Hope College, PHYS 352, Spring 2013
5. Distortion
a  ry i cos 
3
http://www.olympus-ims.com/en/microscope/terms/classification/
Distortion aberration is caused by non-uniform lateral magnification
and is often minimized using aperture stops.
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