Paraxial raytracing: Refraction and Translation

1. Paraxial raytracing: Refraction and Translation

2. Sign Conventions: • angles • radii • distances r u y CC t

3. Sign Conventions Angles CCW from axis (+) CW from axis (-) u (+) u (-)

4. Sign Conventions Radii CC to right of surface: r (+) CC to left of surface: r (-) r (-) r (+) CC CC

5. Sign Conventions Distance Above axis: y (+) Above axis: y (+) Below axis: y (-) y (+) y (-)

6. Sign Conventions Distance Left to right: t (+) t (+)

7. n n' 1. Refraction at an interface θ’ θ

8. n n' 1. Refraction at an interface n sin(θ) = n’ sin(θ’) n sin(u - α) = n’ sin(u’ - α) For small angles: sin(θ) ~ θ n (u - α) = n’ (u’ - α) nu - nα = n’u’ – n’α nu + n’α- nα = n’u’ n’u’ = nu + α (n’ – n) θ’ = (u’ – α) θ = (u – α) (note: α is neg as drawn) θ’ θ u’ α α u u α u’

9. n n' 1. Refraction at an interface n’u’ = nu + α (n’ – n) r y α

10. n n' y r n’u’ = nu - (n’ – n) (n’ – n) r n’u’ = nu - y ] [ (n’ – n) r = optical power 1. Refraction at an interface n’u’ = nu + α (n’ – n) sin(α) = -y/r α = sin-1(-y/r) y r For small α: α = - r y α

11. (n’ – n) r = optical power n’ < n n’ > n r (+) r (-) (n’- n) > 0 r > 0 (n’- n) < 0 r < 0 (+) (+) (n’- n) < 0 r > 0 (n’- n) > 0 r < 0 (-) (-) n’ > n n’ < n r (+) r (-)

12. (n’ – n) r n’u’ = nu - y 1. Refraction at an interface (n’ – n) r nu n’u’ u’ u y u

13. 2. Translation between two surfaces n'

14. 2. Translation between two surfaces angle u’ is constant; height y  y’ n' u’ u’ y` y t

15. t n’ y’ = y + n’u’ t n’ y’=y + t •u’ = y + n’u’ 2. Translation between two surfaces y’ = y + t • u’ Since we’re carrying around index-angle product n’u’: n' u’ y` y t

16. t n’ y’=y + t •u’ = y + n’u’ n n' r (n’ – n) r n’u’ = nu - y Δy u’ y` y y u t 1. Refraction 2. Translation

17. Find: back focal length (BFL); distance from last surface to F’ n = 1.5 n = 1 n = 1 n = 1.4 r = 10 r = -6 r = -20 u = 0 y = 4 t = 7 t = 2