Geometrical Optics

1. kr kt ki n1 n2 Geometrical Optics • Represent EM radiation with lines in direction of propagation (parallel to k) • neglect interference/ diffraction effects • often gives useful picture of basic optical properties of a system • When λ << linear dimensions of an interface Light rays: PHY 530 -- Lecture 03

2. li lo R V P S n2 n1 q p Image Formation Imagine a spherical interface between two indices of refraction: C Optical axis PHY 530 -- Lecture 03

3. Definitions • S = point source of EM radiation • V = vertex of lens • P = image point (where rays/ spherical wavefronts converge) • p = “object distance” • q = “image distance” • R = radius of spherical interface • Assume n2 > n1 PHY 530 -- Lecture 03

4. Distance Conventions (Lenses) • For objects drawn on the left of the first vertex: • p: positive (object real) if to the left of V, negative if to the right (object virtual). • q: negative (image virtual) if to the left of V, positive if to the right (image real). • R: positive if C is to the right of V, negative if to the left. PHY 530 -- Lecture 03

5. Approximation To understand what is going on, let’s imagine that we Only care about rays traveling close to the optical axis. This means we will only deal with small angles. Try it! (q in radians) lp h q p PHY 530 -- Lecture 03

6. Image Formation f f1 f f2 lp h R lq f1 f V f2 x C I O n2 n1 q p PHY 530 -- Lecture 03

7. Now apply Snell’s Law Small angle approximation: Substitute for the angles: Small angle approximation in reverse: PHY 530 -- Lecture 03

8. Keep going… Looking at the picture: But x is a small number, so… PHY 530 -- Lecture 03

9. Finally… Rearranging, Or in other words, THIS DOES NOT DEPEND ON h, f1, or f2 !!! All rays close to the optical axis focus to the same point! PHY 530 -- Lecture 03

10. Implications... Choose P at infinity: “first focal length” = “object focal length” S V fo PHY 530 -- Lecture 03

11. More Implications... Choose S at infinity: “second focal length” = “image focal length” P V fi PHY 530 -- Lecture 03

12. li lo R1 R2 V2 V1 P S nl nm nm q p What is a Lens, Anyway? Two spherical surfaces: (assume nl>nm) PHY 530 -- Lecture 03

13. li lo R1 V2 V1 P P’ S nl nm nm q1 p Well, Consider the First Surface In the absence of the second surface, rays would converge at P’: PHY 530 -- Lecture 03

14. We Know the Solution! (*) Now, see figure on next slide. Can see that (Remember sign conventions! p2 is to the right of V2.) PHY 530 -- Lecture 03

15. Second Surface li lo R2 V2 V1 nm P P’ S nl nm q d p2 p q1 PHY 530 -- Lecture 03

16. We know the solution here too! Or, substituting for p2: (**) PHY 530 -- Lecture 03

17. Add two equations Adding (*) (slide 14) and (**) (slide 16), we have Now rearrange: PHY 530 -- Lecture 03

18. Thin Lens Approximation In the limit (lens thickness negligible compared with p, q) where PHY 530 -- Lecture 03

19. P S V1=V2 p q Thin lenses (2) In the thin lens approximation, V1, V2 coalesce into the same point. PHY 530 -- Lecture 03

20. Focal Lengths Note that, since the equation on slide 18) is symmetric in p and q, the image and object focal lengths are equal. S fo P fi Lensmaker’s Equation PHY 530 -- Lecture 03

21. P S V1=V2 p q Gaussian Lens Equation Gaussian form of the lens maker’s equation PHY 530 -- Lecture 03

22. L1 L2 P S Thin Lens Combinations d Similar to the two surface case, we will handle this in two pieces... PHY 530 -- Lecture 03

23. First Lens L1 L2 P P’ S q1 p In the absence of L2, image will form at P’, use Gaussian lens equation: PHY 530 -- Lecture 03

24. Second Lens L1 L2 P P’ S q d p2 q1 Real image Virtual image PHY 530 -- Lecture 03

25. Now the math... Okay, so lens 2 gives us Now, substitute for p2 Then substitute for q1. PHY 530 -- Lecture 03

26. Finally... Can show in the limit . (Lenses in contact.) In other words, the effective focal length of the combination is PHY 530 -- Lecture 03

27. N Thin Lenses in Contact One can keep going with the derivation to show that PHY 530 -- Lecture 03