1. Matrix methods, aberrations & optical systems
Friday September 27, 2002

2. System matrix

3. System matrix: Special Cases
(a) D = 0  f = Cyo (independent of o) f yo
Input plane is the first focal plane

4. System matrix: Special Cases
(b) A = 0  yf = Bo (independent of yo) o yf
Output plane is the second focal plane

5. System matrix: Special Cases
(c) B = 0  yf = Ayo yo yf
Input and output plane are conjugate – A = magnification

6. System matrix: Special Cases
(d) C = 0  f = Do (independent of yo) o f
Telescopic system – parallel rays in : parallel rays out

7. Examples: Thin lens Recall that for a thick lens For a thin lens, d=0

8. Examples: Thin lens Recall that for a thick lens For a thin lens, d=0
In air, n=n’=1

9. Imaging with thin lens in air’ o yo y’
Input
plane
Output plane s s’

10. Imaging with thin lens in air
For thin lens: A=1 B=0 D=1 C=-1/f
y’ = A’yo + B’o

11. Imaging with thin lens in air
For thin lens: A=1 B=0 D=1 C=-1/f
y’ = A’yo + B’o
For imaging, y’ must be independent of o
 B’ = 0
B’ = As + B + Css’ + Ds’ = 0
s (-1/f)ss’ + s’ = 0

12. Examples: Thick Lens H’ ’ yo y’ f’ n nf n’ x’ h’
h’ = - ( f’ - x’ )

13. Cardinal points of a thick lens

14. Cardinal points of a thick lens

15. Cardinal points of a thick lens
Recall that for a thick lens
As we have found before
h can be recovered in a similar manner, along with other cardinal points

16. Aberrations Monochromatic Chromatic Unclear image Deformation of image
Spherical
Coma
astigmatism
Distortion
Curvature
A mathematical treatment can be developed by expanding the sine and tangent terms used in the paraxial approximation

17. 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

18. Aberrations: Spherical
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

19. Aberrations: Coma
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 f

20. Aberrations: Astigmatism and curvature of field
Yields elliptically distorted images

21. Aberrations: Pincushion and Barrel Distortion
This effect results from the difference in lateral magnification of the lens.
If f differs for different parts of the lens,
will differ also
M on axis less than off axis (positive lens)
M on axis greater than off axis (negative lens)
fi>0
fi<0
object
Pincushion image
Barrel image

22. Stops in Optical Systems
In any optical system, one is concerned with a number of things including:
The brightness of the image
Two lenses of the same focal length (f), but diameter (D) differs
Image of S formed at the same place by both lenses S S’
Bundle of rays from S, imaged at S’ is larger for larger lens
More light collected from S by larger lens

23. Stops in Optical Systems
Brightness of the image is determined primarily by the size of the bundle of rays collected by the system (from each object point)
Stops can be used to reduce aberrations

24. Stops in Optical Systems
How much of the object we see is determined by:
(b) The field of View Q Q’
(not seen)
Rays from Q do not pass through system
We can only see object points closer to the axis of the system
Field of view is limited by the system

25. Theory of Stops Theory of Stops
We wish to develop an understanding of how and where the bundle of rays are limited by a given optical system
Theory of Stops

26. Aperture Stop
A stop is an opening (despite its name) in a series of lenses, mirrors, diaphragms, etc.
The stop itself is the boundary of the lens or diaphragm
Aperture stop: that element of the optical system that limits the cone of light from any particular object point on the axis of the system

27. Aperture Stop: ExampleAS