Cardinal planes and matrix methods

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 air H H’ ƒ1 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 air H H’ ƒ1 ƒ2 n = n2 = n’ = 1 F F’ d ƒ’ ƒ s s’

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

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

25. o f System matrix: Special Cases (d) C = 0  f = Do (independent of yo) 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’