1. 1 Chapter 6 More on geometrical optics February 4 Thick lenses Review: Paraxial imaging from a single refracting spherical surface: Thin lens equation: Lens maker’s formula: Gaussian lens formula: Assumptions: 1) Paraxial rays. 2) Thin lenses. Additional assumption: 3) Monochromatic light. Question: What happens if these assumptions are not valid? Solution: Study the principles of thick lens and aberrations. Note the sign convention: everything has a sign.

2. 2 6.1 Thick lenses and lens systems Thick lenses: When d is not small, the lens maker’s formula and the Gaussian lens formula are not valid. Why thick lens? The image of a distant point source is not a point, but a diffraction pattern because of the limited size of the lenses.  Larger D produces clearer images. Some jargons in photography: Field of view (angle of view): The angle in the object space over which objects are recorded on the sensor of the camera. It depends on the focal length of the lens and the size of the sensor. Depth of field: The region in the object space over which the objects appear sharp on the sensor. f-number (f/#): The ratio of the focal length to the diameter of the entrance pupil: The f/# affects 1) the brightness of the image, 2) the sharpness of the image, and 3) the depth of field. ~ f f.o.v. D d.o.f. ~ f f D 

3. 3 Question: What are the formulas for f and s i for a thick lens? Terminology of thick lenses: Principal plane: The plane composed by the crossing points between the incident rays parallel to the optical axis and their emerged rays. Principal points: the intersects between the principal planes and the optical axis. H 1 and H 2. Note: 1) The principal planes are actually curved, while its paraxial regions forms a plane. 2) If one surface of the lens is planar, the tangent of the other surface should be a principal plane.  prove now, and soon again. 3) Generally (e.g., plane-convex lenses)  to be proved soon. FiFi b.f.l. V1V1 H1H1 V2V2 H2H2 Secondary principal plane FoFo f.f.l. V2V2 H2H2 V1V1 H1H1 Primary principal plane Note that we may later consider a three-segment ray as virtually two segments.

4. 4 N2N2 N1N1 O Nodal points: The crossing points between the optical axis and the rays passing through the optical center. Coincide with the principal points when both sides of the lens are in the same medium.  to be proved soon. Cardinal points of a lens: Two focal points + two principal points + two nodal points. When both sides of the lens are in the same medium: F i, F o, H 1, H 2 are the cardinal points. Optical center: All rays whose emerging directions are parallel to their incident directions pass through one common point. This point is called the optical center of the lens.  proved in chapter 5. R2R2 R1R1 C1C1 C2C2 A B O Points to remember for a lens: V 1, V 2, C 1, C 2, O, F i, F o, H 1, H 2, N 1, N 2. They are fixed to the lens on its optical axis. Plus S, P for object and image.

5. 5 Read: Ch6: 1 No homework

6. 6 February 6 Thick lens equations Locations of principal planes: Note that the principal planes may be external in some cases. However, people have developed a much simpler method which results in a set of thick lens equations that give exactly the same answer as the above method. Virtual rays: Fact: Under paraxial optics, it is proved that if we extend a ray’s two segments located in the air they then cross at the two principal planes at the same height. Solution: Virtual rays between lens surfaces and/or principal planes can be used to simplify the problem. The far-most rays are still real. Goal: Possible solution: Under paraxial optics we may use the formula for the imaging from a single refracting surface twice to locate the final image. H1H1 H2H2

7. 7 FiFi b.f.l. V1V1 H1H1 V2V2 H2H2 FoFo f.f.l. f f dldl soso sisi h2h2 h1h1 yoyo yiyi xixi xoxo Thick lens equations: When f, s o and s i are measured from the principal planes, we have h is positive when H is to the right of V. All are to be proved soon. I hate to believe anything that is not proved by my pencil. Newtonian form: Magnification: When  Light directed toward the first principal plane will emerge from the second principal plane at the same height. Eqs. 6.1-6.4 Locating the four cardinal points. Note the new three key rays. Here the rays inside the lens are virtual. The rays in the air are actual.

8. 8 Read: Ch6: 1 Homework: Ch6:4,6,8 Due: February 13

9. 9 February 9 Combination of thick lenses Thick lens equations: The procedure of locating an image from a thick lens: FiFi V1V1 H1H1 V2V2 H2H2 FoFo f f dldl soso sisi h2h2 h1h1 yoyo yiyi

10. 10 Example (P6.12): R 1 =4 cm, R 2 = -15 cm, d l =4 cm, n l =1.5, object =100 cm before lens. Everything has a sign!

11. 11 Combination of thick lenses: locating the overall cardinal points H1H1 FoFo Fo1Fo1 H 11 H 12 Fi1Fi1 Fo2Fo2 H 21 H 22 Fi2Fi2 FiFi H2H2 d f1f1 f1f1 f2f2 f2f2 ff Note the sign convention for

12. 12 Read: Ch6: 1 Homework: Ch6:12,13,14 Due: February 20

13. 13 February 11 Ray matrices 6. 2.1 Matrix method Ray tracing: Mathematically following the trace of a ray. Example: Ray tracing of a paraxial, meridional ray traversing a spherical lens. Meridional ray: A ray in a plane that contains the optical axis and the object point. (Opp: skew ray). I. Refraction (at P): Power of a refracting surface Refraction matrix Incident ray vector Refracted ray vector  P C y ntnt ii tt  R  nini

14. 14 II. Transfer (from P 1 to P 2 ): Transfer matrix P1P1 y1y1 y2y2 n 11 22 d P2P2 ii C P V ii rr R rr yiyi III. Reflection (Mirrors): Mirror matrix

15. 15 Note: 1)Different definitions for the ray vectors and matrixes may exist. 2)Merit of the current matrices: |R|=|T|=1, and their combinations. Examples of other definitions of ray vectors and matrices: System matrix A of a lens: Transforming an incident ray before the first surface to the emerging ray after the second surface: A→B→C→D Another popular form A C y2y2 nlnl 11 22 dldl y1y1 B D

16. 16 Read: Ch6: 2 Homework: Ch6: 16,20,22,23,24 Due: February 20

17. Eqs. 6.3-4 Eq. 6.2 17 Application I: Where are the cardinal points (F i, F o, H 1, H 2 )? Let the incident ray be parallel to the optical axis: February 13 Matrix analysis of lenses P1P1 FiFi V1V1 H1H1 V2V2 H2H2 P2P2 y1y1 y2y2  System matrix for a reversed lens

18. y2y2  18 Application II: Lens equation Eq. 6.1 If  = , then s i = s o = 0  Nodal points = Principal points y1y1 P V1V1 H1H1 V2V2 H2H2 P1P1 P2P2  S soso sisi

19. 19 Application III: Thin lens combination: where are the overall cardinal points? Thin lens system matrix Fi2Fi2 Fo1Fo1 Fi1Fi1 Fo2Fo2 O2O2 O1O1 d f1f1 f2f2

20. 20 Homework: Starting from Eq. 6.31, please prove Eqs. 6.1-4. Please include detailed drawings showing all the parameters. Due: February 20

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