Paraxial into real surfaces

Transcription

1 Paraxial into real surfaces Curvature, Radius Power lens and mirrors lens maker equation mirror and lens in contact Principle planes

2 Real Surfaces Refractive via Fermat s Principle Calculate optical path length. Set equal to our definition of OPL for thin lens Paraxial power of curved refractive surface.

3 Real Surfaces refractive via Snell s Law Paraxial Snell s Law Replace refraction with ray angles Paraxial approximations to angle, obeying sign conv. Cancel y, rearrange.

4 F1 2 lens separated by d F2 d y i [ ] [ ] M = R 3 T 2 R 1 system matrix u i = T 4 R 3 T 2 R 1 T 0 y o u o M = 1/ f d d / / f 1 f f 2 2 1/ f 2 d / d f M 21 = 1 f eff = 1 f f 2 d f f 1 2

5 Lens Maker s Equation Not that anyone would make a lens with it! This is why it is not good to make lens with

6 Real Surfaces reflective Derive power of mirror by placing object at center of curvature. Since ray strikes surface normally, it must return to the same point. Remember that by our sign convention, R<0 in the geometry shown, yielding a lens power which is positive.

7 Lenses in Contact Example: Magnin Mirror = a symmetric concave lens with curvatures c 1 + a convex mirror of curvature c 2

8 If real lenses have significant thickness, why have we wasted all this time with thin lens? It turns out that a thick lens (or system of lens) can be replaced with 2 surfaces called principle planes. This assumes axially symmetric optics. All of our paraxial lens equations work with distances measured from the principal planes. However, for high NA optics the planes are actually spherical surfaces.

9 Cardinal Points Guass showed that these quantities completely characterize a thick lens system. Front and back focal points/planes Principle points/planes First and second nodal points

10 Principle Surfaces and Focal points In the paraxial limit, all rays exiting a compound lens will appear to have encountered a single plane P with power. Ditto for reverse travelling rays, but in general P is not at P.

11 Understanding Principle Planes Overlay previous two drawings and reverse the direction of the ray in the bottom picture (if this bothers you, hold on, it will make sense in a moment). Note that incident rays 1 and 2 appear to converge towards a virtual object at P. Note also that exiting rays 1 and 2 emanate from a virtual image at P. And finally note that the object and image height are equal.

12 Principle Planes: What are they good for? Since the principal planes are the conjugates of unit magnification, we can replace any compound lens element with just P and P. Rays that hit P must be imaged with unit transverse magnification and unit angular magnification to P. In other words, they teleport from P to P. Ditto on the reverse trip. All the element power F is applied on the second plane (P if going forward, P if going backwards).

13 Principle Planes P is roughly 1/3 glass thickness Rule of thumb Note: focal points for negative are reversed Warren Smith, Modern Optical Engineering, SPIE

14 Nodal Points A ray crossing the axis at the first nodal point emerges from the second nodal points parallel to itself. Obviously, for lenses with the same index in object and image space, the nodal points are at the intersection of the principal planes and the axis. This is not true in general (e.g. the eye).

15 Ray Tracing Thick Lenses F F P1 P2 Same rules as before just rays are extended to principle planes and then they skip the distance in between the principle planes

16 Vertex and BFL/FFL

17 Summary of a 2 Surface System N - Conjugate Matrix

18 Summary of a 2 Surface System Remember ϕ 1 = (n-1)/r 1

19 Lens Example R1 = 40mm R2 =20mm n =1.5 d = 5mm ϕ 1 = (n-1)/r 1 = ϕ 2 = ffl bfl F 1/F = ϕ 1 + ϕ 2 -d ϕ 1 ϕ 2 F = 27.8mm compared to 26.67mm for d=0 bfl = F(1-d ϕ 1 ) = 26.1mm ffl = F(d ϕ 2-1) = -24.3mm H2= = 1.7mm H1= =3.5mm

20 We know how to solve problems Now we can calculate F, bfl, ffl from lens parameters With bfl, ffl and F you can find location of principle planes (h 1 and h 2 on figure) With measuring distances from principle planes All other thin lens equations still work for image position magnification etc WOW!!!!

21 Compound lens We will do lots of problems next lecture to see how this works

22 Design Problem 1 If you are making a system with multiple elements that have to be hard mounted (i.e. no adjustments), how can you use the result of HW problem 1) to help align the system (assuming that you can shim the mounting plate of the plate). Can use you plates to change the location of the focus? Are their issues with using a plate in a converging beam? Is there anyway to minimize these issues? The problem of not having mechanical adjustments is typical in optical engineering as the system is being converted from a research testbed into the final product.

23 Design Problem 1 Solution 1: Tip/tilt plate as fine position adjustment of a nearly collimated beam. Δθ = 1 = 17mrad, t=1mm, n=1.5. The beam displacement will be ~5 microns. Can slim in both directions to get X and Y offset. Solution 2: Put 2 plates tilted at fixed angles in rotating barrels spin barrels to reach and XY. Similar to Risley prism set. Comment: can change focus position but with consequences.

24 Design Problem 1

25 Design Problem 1 VERY LIMITED NA, much worse if plate is tilted

26 Design Problem 2 You are given an elliptical beam that has a 2:1 aspect ratio. Use prism(s) to circularize the beam. What are the tolerances on angles, index, etc to get a circular beam to within 5% of perfectly circular?

27 Nichia Laser Diode Spec

28 Nichia Laser Diode Spec

29 Nichia Laser Diode Spec

30 Nichia Multimode LD Spec

31 Design Problem 2 x 2x Solutions Single prism 2 prisms Toric/astigmatic lens

32 Sample Spec for prism in this use

33 Homework #2 Available at the website under homework Due in 2 weeks W. Smith Modern Optical Engineering Chapter 4 (again) and Chapter 13 (will help for next week) Reading Chapter 17 cover camera lenses we discussed last time.