Physics 116 Refraction and Lenses

Lecture 17 Refraction and lenses Oct 27, 2010

R. J. Wilkes Email: [email protected] Lecture Schedule (up to exam 2)

Today 2 Last time: Simple case: Refraction at a plane surface

•! Light bends at interface between refractive indices –! bends more the larger the difference in refractive index –! can be effectively viewed as a “least time” behavior •! get from A to B faster if you spend less time in the slow medium –! Object at B appears to be at location B’ •! Fish in tank appears to be displaced •! Put your feet in the lake and they seem bent A Exact formula: n sin = n sin !1 1 !1 2 !2 n1 = 1.0

n2 = 1.5

B B’ 3 Reflections, and refractive optical illusion

•! In a thick piece of glass (n = 1.5), the light paths are as shown –! About 4% of light energy is reflected from the surface (mirrorlike) •! Same thing happens at back surface (4% of that gets re-reflected from the inside front surface, and ends up going out the far side also…)

Apparent direction A to object n = 1.0 n1 = 1.5 n2 = 1.0 incoming ray 0 from object object appears displaced (100%) due to jog inside glass 96% 8% reflected in two reflections (front & back) 4% 92% transmitted ~ 4% 0.16% B 4 FYI: Fermat’s principle here too

Least-time principle applies to refraction also: Actual path is quickest path, taking into account changes in light speed! •! Fermat’s Least-Time principle –! Pierre de Fermat, French mathematician, 1601-1665 •! Best known for “Fermat’s Last Theorem” Has solutions (Pythagoras) But Never works, if N>2

•! Wrote a friend just before he died “I have discovered a truly remarkable proof which this margin is too small to contain” – but he never got to explain •! F’s Last Theorem was finally proven true in 1994 (by A. Wiles, Princeton U.)

5 Underwater refraction: Total Internal Reflection

•! Looking up toward air from under water, refraction bends light rays away from normal (going from higher to lower n) •! For large angle of incidence, angle of refraction becomes >90° ! –! thereafter, you get total internal reflection –! for glass, the critical internal angle is 42°, for water, it’s 49° –! a ray within the higher index medium cannot escape at greater incidence angles (look at sky from underwater…can’t see out!) Shallow-angle At critical angle, rays escape

refracted ray hugs surface n1 = 1.0

n2 = 1.5 At steeper angles, rays are 100% 42° reflected

–! Same principle is used to confine light within optical fibers 6 Polarization by reflection •! David Brewster (1781-1868): !I !R n1 Found that when !T - !R = 90deg reflected light is polarized parallel to surface n2 (perpendicular to plane of incidence) 90deg

By Snell's Law: !T

Brewster (polarizing) angle:

-1 e.g., air/glass has !B = tan (1.5)= 57deg Reason: –! incident light has E components parallel and perpendicular to plane of incidence

–! reflected light can only have component perpendicular to plane of incidence for !R = !T + 90deg •! Parallel component would have to be along propagation direction = longitudinal wave! Refraction at curved surfaces: Lenses

•! Lenses = medium with higher n than air, with curved surfaces

Lens with curved surfaces front and back (double-convex) bends light twice, each time refracting incoming ray towards normal line. Snell’s law of refraction applies at each surface. (One curved/one flat surface = plano-convex; inward cuving = concave lens) •! Convex (positive or converging ) lens: thicker at center than edges –! Positive lens (by convention, we say its focal length f is positive) focuses parallel incoming rays to a point at distance f behind it Focal length may be given in m or mm. Optometrists instead specify “refracting power” in diopters: higher power = shorter f F (in diopters) = 1/(f in meters) +1 diopter lens has f=1 m +2 diopter lens has f=1/2 m 8 Diverging lenses

•! Concave (negative or diverging ) lens: thinner at center than edges –! Negative lens (we say its f is negative) makes parallel incoming rays bend outwards, so it seems as if light were coming from a point in front of the lens (focal point) Converging (+) Diverging (-) Negative lens

f Meniscus lenses •! Meniscus lens: has both sided curved in the same direction, but one surface is more sharply curved than the other –! Commonly used for eyeglasses –! Can be either positive or negative (thicker at center, or thinner)

9 Ray tracing rules for lenses

Ray tracing rules for a (simple, thin) lens are: 1) Rays arriving parallel to the axis emerge to pass through the back focal point 2) Rays passing through the front focal point emerge parallel to the axis 3) Rays through center of the lens are undeviated

This lens setup could be used as a object 1 camera, or a slide projector 3 f o o o Focal pt 2 Focal pt image Object distance Image distance d O dI

Rays from object tip re-converge at a point, forming a real, inverted, magnified image there When object is close to converging lens

1) Rays arriving parallel to the axis emerge to pass through the back focal point 2) Rays through center of the lens are undeviated 3) Rays do not re-converge – image is virtual Gazing through the lens toward the object, we see rays object appearing to emerge from a 2 virtual, erect, magnified image image 1 f o o o Focal pt Focal pt Object distance

dO This lens setup could be used as a magnifying glass Image distance

dI Diverging lens, distant object

1) Rays arriving parallel to the axis emerge as if they came from the focal point 2) Rays through center of the lens are undeviated 3) The intersection of these rays shows image location, orientation, and size

image This lens could be used in object 1 eyeglasses for a nearsighted person 2 o o o Focal pt f Gazing through the lens toward Object distance the object, we see rays dO appearing to emerge from a virtual, erect, demagnified image Image distance that is closer than the object dI Diverging lens, nearby object

object image 1 2 o o o Focal pt f

Object distance Very little difference – image just dO moves a bit closer to the lens

Image distance

dI Ray tracing applet – try it yourself http://silver.neep.wisc.edu/~shock/tools/ray.html (original source http://webphysics.davidson.edu/Applets/Optics4/Intro.html)