LENSES s s’ h’ h f

Today . . . Lenses Summary of lenses and mirrors Lasers The Lens Equation The Lensmaker’s Formula Summary of lenses and mirrors Lasers Text Reference: Chapter 34.2 Examples: 34.3,4,6,7 and 8

Lenses A lens is a piece of transparent material shaped such that parallel light rays are refracted towards a point, a focus: Convergent Lens light moving from air into glass will move toward the normal light moving from glass back into air will move away from the normal real focus Positive f Negative f Divergent Lens light moving from air into glass will move toward the normal light moving from glass back into air will move away from the normal virtual focus

These are principal rays!!! The Lens Equation We now derive the lens equation which determines the image distance in terms of the object distance and the focal length. Convergent Lens: s h s’ f h’ Ray Trace: Ray through the center of the lens (light blue) passes through undeflected. These are principal rays!!! Ray parallel to axis (white) passes through focal point f. two sets of similar triangles: same as mirror eqn if we define s’ > 0 f > 0 eliminating h’/h: magnification: also same as mirror eqn!! M < 0 for inverted image.

Lens Equation summary Draw some ray diagrams: convergent lens example f h’ s h

Lecture 26, ACT 1 (pg. 9 of notes) object lens screen A lens is used to image an object on a screen. The right half of the lens is covered. What is the nature of the image on the screen? (a) left half of image disappears (b) right half of image disappears (c) entire image reduced in intensity

Lecture 26, ACT 1 (a) left half of image disappears object lens screen A lens is used to image an object on a screen. The right half of the lens is covered. What is the nature of the image on the screen? (a) left half of image disappears (b) right half of image disappears (c) entire image reduced in intensity All rays from the object are brought to a focus at the screen by the lens. The covering simply blocks half of the rays. Therefore the intensity is reduced but the image is of the entire object! ( one more thing… In our examples of image formation, we only needed the top half of the lens to form the image) s’ f h’ s h

Divergent lens example Because the lens is divergent, f is negative: -|f| s’ f h’ s h

The Lensmaker’s Formula So far, we have treated lenses in terms of their focal lengths. How do you make a lens with focal length f ? Start with Snell’s Law. Consider a plano-convex lens: q a R N air h q b a light ray (Nair=1.0) Snell’s Law at the convex surface: Assuming small angles, The bend-angle b is just given by: The bend-angle b also defines the focal length f : The angle q can be written in terms of R, the radius of curvature of the lens : Look, h and q drop out !! This holds for any incident parallel ray Putting these last equations together,

Lensmaker’s Formula The thin lens equation for a lens with two curved sides: R>0 if convex when light ray hits it See Text pages 1082-1085 for details and example

Lecture 26, ACT 2 (not in notes) A “meniscus” lens is one where boths sides are curved in the same direction. A meniscus lens always has (a) positive focal length (b) negative focal length (c) Can be either Depending on which radius is larger, the focal length can have either sign. In general, the action of the lens is determined by whether the center of the lens is thicker (converging lens) than the edge or thinner (diverging). Why ever use a meniscus lens? Because some of the aberations (described later) may be smaller (especially if used together with other lenses – see raytracing on page 1)

More generally…Lensmaker’s Formula Two curved surfaces… Two arbitrary indices of refraction The complete generalized case… Note: for one surface Planar, R > 0 if convex when light hits it R < 0 if concave when light hits it

Lecture 26, ACT 3 fair fwater (a) fwater < fair (b) fwater = fair What happens to the focal length of a lens when it is submerged in water? In air, the focal length of a glass lens (n=1.5) is determined to be fair. When the lens is submersed in water(n=1.33), its focal length is measured to be fwater. What is the relation between fair and fwater? air fair air water fwater water (a) fwater < fair (b) fwater = fair (c) fwater > fair

Lecture 26, ACT 3 fair fwater (a) fwater < fair (b) fwater = fair What happens to the focal length of a lens when it is submerged in water? In air, the focal length of a glass lens (n=1.5) is determined to be fair. When the lens is submersed in water(n=1.33), its focal length is measured to be fwater. What is the relation between fair and fwater? air fair air water fwater water (a) fwater < fair (b) fwater = fair (c) fwater > fair The refraction is determined by the difference in the two indices of refraction The smaller the difference, the less the bend the longer the focal length Think of the case where it is glass “in glass” ! f → infinity ! Quantitatively, look at the lensmaker’s formula

Summary of Lenses and Mirrors We have derived, in the paraxial (and thin lens) approximation, the same equations for mirrors and lenses: when the following sign conventions are used: Variable f > 0 f < 0 s > 0 s < 0 s’ > 0 s’ < 0 Mirror concave convex real (front) virtual (back) Lens converging diverging real (back) virtual (front) Principal rays “connect” object and image one goes through the center of the lens or mirror the other goes parallel to the optic axis and then is refracted or reflected through a focal point the third one is like the second one….

Optical Aberrations Why really good lenses cost a lot Chromatic aberration Due to dispersion (index of refraction depends on frequency), focal length can be different for different colors. Spherical aberration Outside the “paraxial limit”, optimal focusing occurs only for a parabolic lens. Spherical lenses look ~parabolic for narrow field of view. (But spherical lenses are much cheaper to grind & polish!) Astigmatism Curvature of the lens not symmetric in transverse directions → slightly cylindrical → different focal lengths

Incoherent versus Coherent Light (laser) In Physics 114 you will learn that electrons in atoms can only have certain particular energies, determined mostly by balancing kinetic energy (111) and electrostatic attraction to the nuclei (112), constrained by the “Pauli exclusion principle” (114). Electron promoted to higher energy state (i.e. farther from nucleus), e.g., by collision with another atom Within ~10ns electron falls from high to low energy state → emits 1 photon h = Planck’s constant = 6.6∙10-34 J-s In a normal light, the various atoms spontaneously emit their photons at random times and in random directions → light emitted in all directions, with no “coherence”, and maybe with lots of frequencies (“white light”). In a laser, the photon from one atom can stimulate the next atom to emit an identical photon in the same direction and in phase→ “peer pressure”. The atoms are placed in a high-Q cavity (highly reflective mirrors), which gives the photons more chances to stimulate more atoms → more photons → etc. Light Amplification by Stimulated Emission of Radiation

Lasers, cont. Characteristics Applications http://www.colorado.edu/physics/2000/index.pl Characteristics Well-defined direction (can bounce off the moon!) Very precise color ( ) Very “bright” (100 W light bulb vs. 10 W laser—cut steel!) “Coherent”: displays interference focusable to the smallest possible spot (~λ) Applications Checkout scanners Eye surgery “Star Wars” Laser sighting (construction) CD/DVD readers Fiber telecommunications Laser pointers Measuring the speed of light!

Summary Lens Equation Lensmaker’s Formula same as mirror equation Lensmaker’s Formula Laser – source of collimated, coherent light Next time: Multiple lenses, Optical instruments Reading assignment: Chapter 34.2, 4 Examples: 34.9,10,12,13,14 and 15