1. Cardinal planes and matrix methods
Monday September 23, 2002

2. Principal planes for thick lens (n2=1.5) in air
Equi-convex or equi-concave and moderately thick  P1 = P2 ≈ P/2 H H’ H H’

3. Principal planes for thick lens (n2=1.5) in air
Plano-convex or plano-concave lens with R2 = 
 P2 = 0 H H’ H H’

4. Principal planes for thick lens (n=1.5) in air
For meniscus lenses, the principal planes move outside the lens
R2 = 3R1 (H’ reaches the first surface) H H’ H H’ H H’ H H’
Same for all lenses

5. Examples: Two thin lenses in airƒ1 ƒ2
n = n2 = n’ = 1
Want to replace Hi, Hi’ with H, H’ d h h’ H1
H1’ H2
H2’

6. Examples: Two thin lenses in airƒ1 ƒ2
n = n2 = n’ = 1 F F’ d ƒ’ ƒ s s’

7. Huygen’s eyepiece ƒ1=2ƒ2 and d=1.5ƒ2
In order for a combination of two lenses to be independent of the index of refraction (i.e. free of chromatic aberration)
Example, Huygen’s Eyepiece
ƒ1=2ƒ2 and d=1.5ƒ2
Determine ƒ, h and h’

8. Huygen’s eyepiece H1 H’ H2 H
h’ = -ƒ2
h=2ƒ2
d=1.5ƒ2

9. Two separated lenses in air
f1’=2f2’ H’ H H’ H F’ F’ F F f’ f’
d = f2’
d = 0.5 f2’

10. Two separated lenses in air
f1’=2f2’
Principal points at  H’ H F’ F f’
d = 3f2’
d = 2f2’
e.g. Astronomical telescope

11. Two separated lenses in air
f1’=2f2’
e.g. Compound microscope H H’ F’ F f’
d = 5f2’

12. Two separated lenses in air
f1’=-2f2’
e.g. Galilean telescope
d = -f2’
Principal points at 

13. Two separated lenses in air
f1’=-2f2’ H H’ F F’ f’
e.g. Telephoto lens
d = -1.5f2’

14. Matrices in paraxial Optics
Translation
(in homogeneous medium)  0 y yo L

15. Matrix methods in paraxial optics
Refraction at a spherical interface  y ’  φ ’ n n’

16. Matrix methods in paraxial optics
Refraction at a spherical interface  y ’  φ ’ n n’

17. Matrix methods in paraxial optics
Lens matrix n nL n’
For the complete system
Note order – matrices do not, in general, commute.

18. Matrix methods in paraxial optics

19. Matrix properties

20. Matrices: General Properties
For system in air, n=n’=1

21. System matrix

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

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

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

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

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

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

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