Physics 11b Lecture #19

Transcription

1 Physics b Lecture #9 Geometrical Optics S&J Chapter 34, 35

2 What We Did Last Time Itesity (power/area) of EM waves is give by the Poytig vector See slide #5 of Lecture #8 for a summary EM waves are produced by acceleratig charges Atea produces varyig E ad B field EM radiatio Started discussig optics Ray approximatio works as log as λ is small Light travels straight Light paths are reversible S = E B µ 0

3 Today s Goals Geeral rules of Geometrical Optics Huyges priciple Fermat s priciple Reflectio of light o a smooth surface This is easy Refractio of light as it eters differet medium Speed of light chages Idex of refractio Sell s law Dispersio = differet speed for differet wavelegth Chromatic aberratio

4 Law of Reflectio Whe light reflects, icidet agle = reflected agle θ = θ The surface must be smooth or specular If the surface is rough, light is scattered i radom directios by diffuse reflectio Plai white paper reflects all visible light by diffuse reflectio θ θ The law of reflectio seems atural if you cosider light as particles How do we explai it with EM waves?

5 Huyges Priciple Christiaa Huyges ( ) Draw circles to costruct successive wavefrots Each circle has r = λ ad cetered o the previous wavefrot Draw a commo taget of all circles New wavefrot Easy examples: Plae waves Circular waves

6 Huyges ad Reflectio Huyges costructio shows that waves reflect with θ = θ Laws of reflectio ca be satisfied by particles or waves We ll use Huyges priciple to study wave propagatio i this lecture

7 Light i Matter Light ca travel iside certai material (water, glass, etc.) The speed of light decreases iside material vacuum c = m/s air v = c glass v = 0.66 c Defie the idex of refractio: c = > v.0003 for air,.33 for water,.52 for ordiary glass, 2.4 for diamod, etc. See Table 35. i the textbook The slowdow is caused by the electros i the material They move as light passes through, ad emit EM radiatio

8 Wavelegth i Matter Light eters from oe material to aother Speed of light chages v c = v2 c = 2 2 v λ v 2 λ 2 Frequecy is the same Wavelegth chages λ = v c f = v c λ 2 2 = = λ f f f = λ22 2 Wavelegth shriks whe light eters material with larger

9 Refractio Light chages directio (or refracts) as it eters differet Air material You probably kow which directio Refractio is caused by the chage of the speed (or wavelegth) Water? All you eed to kow are idices of refractio ad 2 Huyges priciple ca tell us the exact agle of refractio

10 Sell s Law Apply Huyges priciple to the refractio problem Wavelegth λ chages at the boudary λ = λ 2 2 Cosider wavelegth alog the surface λ λ2 λs = = siθ siθ siθ siθ 2 λ = = λ siθ = siθ 2 2 λ θ λ s λ θ 2 2 Law of Refractio, or Sell s Law

11 Sell s Law siθ = siθ 2 2 Directio of refractio bed depeds o / 2 Fast slow < 2 θ > θ2 bed dow Slow fast < θ > θ bed up 2 2 θ < 2 θ 2 θ 2 bed toward ormal < 2 θ 2 2 bed away from ormal

12 Total Iteral Reflectio For slow fast trasitio, there is a upper limit i icomig agle θ siθ = siθ2 < < θ < arcsi Example: glass ( =.5) air θ critical = arcsi(/.5) = 42 At θ = θ critical the outgoig ray is parallel to the surface What happes whe θ > θ critical? Critical agle θ critical Total iteral reflectio No light gets through Perfect reflectio

13 Optical Fibers Total iteral reflectio makes glass rod a light pipe Light gets trapped iside ad bouce alog Basic idea of optical fibers Real-world optical fibers made of two types of glass core: large claddig: small Light loss due to impurity i glass must be very small Typical glass is ot so trasparet beyod a few meters Trasoceaic fiber liks exted several 000 km

14 Time Cosider (agai) light eterig water from air Light goes from poit A to poit B How log does it take? Defie the x-y coordiates: A = (0, a) B= (, b a) C = ( x,0) Total time is 2 T = AC+ CB= a + x + 2 c c c a + ( b x) What is the fastest path? { } dt x 2( b x) = = 0 dx c a + x a + ( b x) a a A Where are we goig? b C B

15 Fermat s Priciple dt x ( b x) dx c a + x a + ( b x) 2 = = Agles θ ad θ 2 are x siθ = siθ = a + x ( b x) 0 a + ( b x) 2 2 a a A b C B Above equatio becomes siθ = siθ Sell s law! 2 2 Fermat s Priciple of Least Time: The actual path betwee two poits take by a beam of light is the oe which is traversed i the least time

16 Fermat s Priciple Light chooses the fastest path to get from A to B Souds good, but how does it kow which path to take? Aswer: it does t EM waves go all over the place All possible paths are i fact take Each path gives a certai wave amplitude at poit B Whe itegrated, cotributios from all but the fastest path cacel out Similar priciples gover broader rage of physics Feyma s path itegral i Quatum Mechaics Hamilto s priciple i Classical Mechaics

17 Dispersio Idex of refractio is ot really a costat It depeds o the wavelegth λ This is called dispersio That meas the agle of refractio depeds o the wavelegth, too air glass Purple light has larger i glass beds more Red light has smaller i glass beds less This is how a prism works

18 Chromatic Aberratio Leses use refractio to create optical images Will discuss them i the ext lecture Dispersio makes rays of differet λ bed differetly Differet colors arrive at a differet poits o the image Example: a cocave les focuses blue light at a shorter distace tha red light This is bad ews for telescopes ad cameras It s called chromatic aberratio

19 Achromatic Leses How ca we get rid of chromatic aberratio? Idea: use two kids of glass with differet dispersio flit glass crow glass Blue beds more tha red Combie covex ad cocave leses so that the dispersio cacels out Achromatic les Every (respectable) optical equipmet uses this

20 Achromatic Leses Flit glass was created i early 8 th cetury Achromatic les was iveted by C.M. Hall i the 730s He kept it secret, orderig the two leses to differet les compaies G. Bass, a subcotractor of the two les compaies realized the secret, but also kept it to himself Idea leaked to J. Dollad, who re-iveted achromatic les i 759, got patet, ad made big moey

21 Summary Law of reflectio: Huyges priciple Idex of refractio θ = θ Wavelegth is affected Sell s law of refractio: = c v> λ = λ 2 2 siθ = siθ 2 2 θ θ θ < 2 Total iteral reflectio if 2 θ < arcsi 2 θ 2 Fermat s priciple: The actual path betwee two poits take by a beam of light is the oe which is traversed i the least time Dispersio ad chromatic aberratio < θ 2 2 θ 2