Why study optics? CP2: Optics • History • Technology Jonathan Jones • Simplicity • Centrality Part 1: Geometric Optics • Passing CP2 The problem of teaching optics Optics around 1700 • Some feedback comments from 2011-12 • Lots of facts known 1. Too much A-Level content about light 2. This was not a basic intro to the course! I • Little understood hadn't studied optics before and found all about the underlying the work far too advanced for me to principles understand. • Newton making • Can’t keep everyone happy! This course is trouble as usual aimed squarely at beginners but does assume knowledge of the absolute basics • Waves or particles? 1 Waves or particles? Waves or particles? • Light travels in straight lines • Light bends (refracts) when moving – Waves travel in circles (chuck a rock in a between different media pond and watch the ripples spread out) n n – But particles in crossed beams would collide? 1 sin( θ1) = 2 sin( θ2 ) • Light reflects off mirrors and leaves at the – Newton had a semi-plausible explanation for same angle as it came in particles – Makes sense for particles (conservation of – Easy to explain for waves if they travel in momentum) straight lines! Waves or particles? Huygens’s Problem • Diffraction effects not really understood • For I do not find that any one has yet given – Newton’s rings provide excellent evidence for a probable explanation of … why it is not wave behaviour, but Newton was unhappy propagated except in straight lines, and with the wave model how visible rays … cross one another • Underlying basis of colour hardly without hindering one another in any way. understood at all • Christian Huygens “Treatise on Light” • Polarization only recently discovered translated by Silvanus P. Thompson (Iceland Spar) http://www.gutenberg.org/etext/14725 2 Huygens’s Principle Huygens’s Model • Huygens’s principle tells us to consider each • Light is made up of a series of pulsations point on a wavefront as a new source of in the ether, an otherwise undectable radiation and add the “radiation” from all of the substance filling all space new “sources” together. Physically this makes no sense at all . Light does not emit light; only • Each pulsation causes a chain of accelerating charges emit light. Thus we will secondary pulsations to spread out ahead begin by throwing out Huygens’s principle • In certain directions these pulsations completely; later we will see that it actually does reinforce one another, creating an intense give the right answer for the wrong reasons . (Melvin Schwartz, Principles of Electrodynamics) pulsation that appears as visible light Huygens’s Construction Straight lines • Every point on a wavefront may be • Straight wavefronts stay straight regarded as a source of secondary wavelets which spread out with the wave velocity. • Points on the wavefront all move forward • The new wavefront is the envelope of at the same speed in a direction normal to these secondary wavelets. the wavefront. All points on a wavefront correspond to the same point in time . • Light rays travel along these normals 3 Problems Reflection 1. Why do wavefronts travel forwards and • Wavefront propagates not backwards? in a straight line 2. What happens at the edges? • As it hits the surface it becomes a source of secondary wavelets • These questions can be answered with a • Wavelets all “grow” at more serious model but that is largely the same speed beyond the scope of this course. • Envelope of these forms new wavefront Reflection Image in a mirror • Reflected ray is at the same angle as Since light normally travels in incident ray straight lines, the light rays image appear to be coming from an • Reflected wavefront is “image” behind the mirror at the same angle as θ θ incident wavefront mirror • Occurs because the secondary wavelets object grow at the same rate in both wavefronts to eye screen blocks direct view 4 Refraction Refraction • Refraction is easily explained if wavelets • Refraction is easily explained if wavelets travel more slowly in glass than in air travel more slowly in glass than in air The two green lines are •Cut diagram down to both four wavelets long. essentials. The start points of each •Add construction lines line are points on a θ1 and rays wavefront and so the end points must also be •Note common angles corresponding points on the new wavefront θ2 Refraction Some materials • The light ray takes the same length of time to Air: n=1.0003 travel along the two green paths Water: n=1.33 • Travels at different speeds: v=c/n, where n is the refractive index sin( θ1)=d 1/D d1 Glass: n=1.5–1.7 d1/v 1 = d 2/v 2 Diamond: n=2.4 θ 1 D n sin( ) = n sin( ) θ2 1 θ1 2 θ2 d2 Snell’s law of refraction sin( θ2)=d 2/D 5 Fermat’s Principle Reflection (Fermat) • Fermat’s Principle of Least Time says that the A light ray takes • At constant speed least path adopted by a light ray between any two the shortest ( least time is equivalent to points is the path that takes the smallest time time ) path between shortest distance two points • Huygen’s model or ideas such as QED can be • Consistent with light used to show that the path must be a local moving in straight lines AB extremum (minimum, maximum, or inflection) • The green line is shorter • Basic ideas probably known by Hero of than the red and blue lines Alexandria and by Alhacen (Ibn al-Haytham) • Shortest path between A • Similar ideas will be seen in mechanics! and B via the mirror! Reflection (Fermat) Refraction (Fermat) AB • Need to minimise A light ray takes • At varying speed least time the shortest ( least y y total distance is not equivalent to time ) path between shortest distance s= yx22 ++ y 2 +−( ax ) 2 two points x • Light moves in straight a lines in one medium but − 1 − 1 ds 1222 1 2 2 2 A =2()yx + ×+2 xyax 2 () +− ( ) ×−−= 2( ax )(1)0 will bend at joins dx • The green line is the x= a − x • Or use geometrical insight to spot quickest path between A that the answer is obvious if you and B! x= a / 2 reflect point B in the mirror. B 6 Refraction (Fermat) Refraction (Fermat) • Minimise total time taken to travel along path d d nd nd t =+1 2 = 11 + 22 d1 y1 vv1 2 c c 22 22 x2 nx+ y + nx + y = 11 1 22 2 x1 c y2 d2 Solve dt/dx 1=0 Refraction (Fermat) Refraction (Fermat) − 1 dt n1 sin(θ 1 )− n 2 sin( θ 2 ) dt n 1 1 2 2 2 = =×2 ( x1 + y 1) × 2 x 1 dx1 c dx1 c 1 d1 n − θ1 d1 y 2 1 2 2 2 y 1 +×2 ()x2 + y 2 ×2 x 2 ×− ( 1) 1 c dt x2 x2 Solve= 0 to get dx1 x1 t nx nx x1 d 11 22 θ2 y2 = − y2 d2 d2 dx1 cd 1 cd 2 nsin(θ )= n sin( θ ) nsin(θ ) n sin( θ ) 1 1 2 2 =1 1 − 2 2 c c 7 Reversibility Critical angle • Optical paths are always reversible • A light ray travelling θ<θc from a material with •A light ray travelling high refractive index from glass into air will to one with low θ≈θ follow exactly the same c θ refractive index is 1 path as a light ray always bent away travelling from air into from the normal glass, just in the θ>θ opposite direction • Angle is limited to 90º c θ2 •Obvious from Fermat Beyond the critical angle light ray undergoes total internal reflection Critical angle Partial internal reflection θ<θ • For all angles less θ<θ n1sin( θ1) = n 2sin( θ2) c c that the critical angle there is both a At θ1 = θc, θ2 = 90º θ≈θ transmitted ray and a θ≈θ c reflected ray c sin( θc) = n 2/n 1 • Beyond critical angle For glass to air light ray undergoes −1 θ>θc θ>θc θc ≈ sin (1/n) total internal reflection Beyond the critical angle light ray undergoes The reflected ray is always reflected at the total internal reflection incident angle 8 Optic fibres (light pipes) Optic fibres (light pipes) • Light can travel along an optic fibre by a series • This process will also of total internal reflections work if the fibre is • If first reflection is beyond the critical angle then curved, as long as the all reflections will be; the limit of transmission is radius of curvature is set by the transparency of the glass not too small • Curves cause a small fraction of the light to leak out making the fibre visible • Real fibres are made from two sorts of glass Pane of glass Sign conventions • Light ray is refracted • Once we switch from pictures to at both surfaces calculations we need a sign convention • Ends up travelling in original direction but • Sign conventions give rise to more slightly offset confusion than any other topic, but • Weak reflections at fundamentally they are nothing more than each surface n a set of rules for choosing signs of • Both reflections travel distances in a consistent manner in same direction, but slightly offset • Similar problems occur in mechanics! 9 Sign conventions in mechanics Sign conventions in mechanics • The right way to do mechanics is to start • For simple problems where we know by defining an axis system and then stick roughly what the answer will be it is very rigorously to this through the calculation tempting to fudge the axes and equations • For example we might put the y-axis so that most things come out positive pointing up so that distances upwards are • This works nicely for simple problems but positive .