1. 1 Chapter 5 Geometrical optics January 21,23 Lenses 5.1 Introductory remarks Image: If a cone of rays emitted from a point source S arrives at a certain point P, then P is called the image of S. Diffraction-limited image: The size of the image for a point source is not zero. The limited size of an optical system causes the blur of the image point due to diffraction effect: Geometrical optics: When  <<D, diffraction effects can be neglected, and light propagates on a straight line in homogeneous media. Physical optics: When  ~ D or >D, the wave nature of light must be considered.

2. 2 5.2 Lenses Lens: A refracting device that causes each diverging wavelet from an object to converge or diverge and to form the image of the object. Lens terminology: Convex lens, converging lens, positive lens Concave lens, diverging lens, negative lens Focal points Real image: Rays converge to the image point Virtual image: Rays diverge from the image point Real object: Rays diverge from the object point Virtual object: Rays converge to the object. S P S P P S

3. 3 5.2.1 Aspherical surfaces Determining the shape of the surface of a lens: The optical path length (OPL) from the source to the output wavefront should be a constant. Example: Collimating a point source (image at infinity) The surface is a hyperboloid when n ti >1, and is an ellipsoid when n ti <1. F A(x,y) D d x y nini ntnt Aspherical lens can form perfect image, but is hard to manufacture. Spherical lens cannot form perfect image (aberration), but is easy to manufacture. Example: Imaging a point source. The surface is a Cartesian oval.

4. 4 5.2.2 Refraction at spherical surfaces Terminology: vertex, object distance s o, image distance s i, optical axis. S A P ii tt  sisi soso lili lolo C V n1n1 n2n2 R Gaussian (paraxial, first-order) optics: When  is small, cos  ≈1, sin  ≈   Paraxial imaging from a single spherical surface:

5. 5 Paraxial imaging from a single spherical surface: Note: 1) This is the grandfather equation of many other equations in geometrical optics. 2) For a planar surface (fish in water): (A bear needs to know this.) 3) Magnification: (P5.6).

6. 6 Object (first) focal length: when s i = , Image (second) focal length: when s o = , fofo FoFo FiFi fifi EVERYTHING HAS A SIGN! Sign convention for lenses (light comes from the left): s o, f o + left of vertex s i, f i + right of vertex x o + left of F o x i + right of F i R + curved toward left y o, y i + above axis  Virtual image (s i < 0) and virtual object (s o < 0): sisi V FiFi C soso V C FoFo

7. 7 Read: Ch5: 1-2 Homework: Ch5: 1,5,6 Note: In P5.1 the expression should be (s o +s i -x) 2. Due: January 30

8. 8 January 26, 28 Thin lenses 5.2.3 Thin lenses Thin lens: The lens thickness is negligible compared to object distance and image distance. Thin lens equations: Forming an image with two spherical surfaces: SP'P 1 st surface 2 nd surface (R 1, n m, n l ) (R 2, n l, n m ) C1C1 V1V1 V2V2 P'P' P si1si1 so2so2 nmnm d si2si2 R2R2 R1R1 C2C2 nlnl S so1so1

9. 9 If the lens is thin enough, d → 0. Assuming n m =1, we have the thin lens equation: Remember them together with the sign convention. Gaussian lens formula: Lens maker’s equation: Question: what if the lens is in water?

10. 10 Optical center: All rays whose emerging directions are parallel to their incident directions pass through one special common point inside the lens. This point is called the optical center of the lens. Proof: R2R2 R1R1 C1C1 C2C2 A B O Conversely, rays passing through O refract parallelly.  For a thin lens, rays passing through the optical center are straight rays. Corollary: For a thin lens, with respect to the optical center, the angle subtended by the image equals the angle subtended by the object.

11. 11 Focal plane: A plane that contains the focal point and is perpendicular to the optical axis. In paraxial optics, a lens focuses any bundle of parallel rays entering in a narrow cone onto a point on the focal plane. Proof: 1 st surface, 2 nd surface C C’ Focal plane Focal plane FiFi Image plane: In paraxial optics, the image formed by a lens of a small planar object normal to the optical axis will also be a small plane normal to that axis. C S P Image plane

12. 12 Read: Ch5: 2 Homework: Ch5: 7,10,11,15 Due: February 6

13. 13 January 30 Ray diagrams Finding an image using ray diagrams: Three key rays in locating an image point: 1)Ray through the optical center: a straight line. 2)Ray parallel to the optical axis: emerging passing through the focal point. 3)Ray passing through the focal point: emerging parallel to the optical axis. Newtonian lens equation: Transverse magnification: yoyo 1 2 3 FoFo FiFi S P S'S' P'P' O soso sisi f f xixi yiyi A B xoxo Longitudinal magnification: Meanings of the signs: + – s o Real objectVirtual object s i Real imageVirtual image fConverging Diverging lens y o Erect objectInverted object y i Erect imageInverted image M T Erect imageInverted image

14. 14 Thin lens combinations I. Locating the final image of L 1 +L 2 using ray diagrams: 1)Constructing the image formed by L 1 as if there was no L 2. 2)Using the image by L 1 as an object (may be virtual), locating the final image. The ray through O 2 (Ray 4, may be backward) is needed. II. Analytical calculation of the image position: Fi2Fi2 Fo1Fo1 si1si1 so2so2 d Fi1Fi1 Fo2Fo2 d<f 1, d<f 2 O2O2 O1O1 4 Total transverse magnification: s i2 is a function of (s o1, f 1, f 2, d)

15. 15 Back focal length (b.f.l.): Distance from the last surface to the 2 nd focal point of the system. Front focal length (f.f.l.): Distance from the first surface to the 1 st focal point of the system. Special cases: 1) d = f 1 +f 2 : Both f.f.l. and b.f.l. are infinity.  Plane wave in, plane wave out (telescope). 2) d → 0: effective focal length f: 3) N lenses in contact:

16. 16 Read: Ch5: 2 Homework: Ch5: 20,25,26,32,33 Due: February 6

17. 17 5.4 Mirrors 5.4.1 Planar mirrors 1)|s o |=|s i |. 2)Sign convention for mirrors: s o and s i are positive when they lie to the left of the vertex. 3)Image inversion (left hand  right hand). 5.4.3 Spherical mirrors The paraxial region (y<<R): x y February 2 Mirrors and prisms

18. 18 The mirror formula: S C P F V ii ff A f sisi soso R Four key rays in finding an image point: 1)Ray through the center of curvature. 2)Ray parallel to the optical axis. 3)Ray through the focal point. 4)Ray pointing to the vertex. Transverse magnification: S P V C F 1 2 3 4 Ray diagrams of mirrors:

19. 19 5.5 Prisms Functions of prisms: 1)Dispersion devices. 2)Changing the direction of a light beam. 3)Changing the orientation of an image. 5.5.1 Dispersion prisms Apex angle, angular deviation  i1i1 t1t1 i2i2 t2t2 

20. 20 Minimum deviation: The minimum deviation ray traverses the prism symmetrically. At minimum deviation, This is an accurate method for measuring the refractive indexes of substances.

21. 21 Read: Ch5: 3-5 Homework: Ch5: 54,60,61,64,65,67,68 Due: February 13