Lecturer: Ivan Kassamakov, Docent Assistants: Risto Montonen and Anton Nolvi, Doctoral

Transcription

1 Lecturer: Ivan Kassamakov, Docent Assistants: Risto Montonen and Anton Nolvi, Doctoral students Course webpage: Course webpage:

2 Personal information Ivan Kassamakov office: PHYSICUM - PHY C 318 (9:00-19:00) Risto Montonen risto.montonen@helsinki.fi office: PHYSICUM - PHY A 312 Anton Nolvi anton.nolvi@helsinki.finolvi@helsinki office: PHYSICUM - PHY C 317

3 Schedule: Lectures: Tuesdays: 10:15 12:00, , Lecture Room: PHYSICUM - PHY D116 SH; Exercises: Tuesdays: 12:15 14:00, , Lecture Room: PHYSICUM - PHY D116 SH; Demonstrations: Demonstrations: Electronics Laboratory: PHYSICUM - PHY C

4 Lectures Lecture # Week # Place - Lecture Room Date Starting time Ending time 01 OPTIIKKA : LUENTO 3 PHYSICUM - PHY D116 SH :15 12:00 02 OPTIIKKA : LUENTO 4 PHYSICUM - PHY D116 SH :15 12:00 03 OPTIIKKA : LUENTO 5 PHYSICUM - PHY D116 SH :15 12:00 04 OPTIIKKA : LUENTO 6 PHYSICUM - PHY D116 SH :15 12:00 05 OPTIIKKA : LUENTO 7 PHYSICUM - PHY D116 SH :15 12:00 06 OPTIIKKA : LUENTO 8 PHYSICUM - PHY D116 SH :15 12:00 07 OPTIIKKA : LUENTO 9 PHYSICUM - PHY D116 SH :15 12:00 08 OPTIIKKA : LUENTO 11 PHYSICUM - PHY D116 SH :15 12:00 09 OPTIIKKA : LUENTO 12 PHYSICUM - PHY D116 SH :15 12:00 10 OPTIIKKA : LUENTO 14 PHYSICUM - PHY D116 SH :15 12:00 11 OPTIIKKA : LUENTO 15 PHYSICUM - PHY D116 SH :15 12:00 12 OPTIIKKA : LUENTO 16 PHYSICUM - PHY D116 SH :15 12:00 13 OPTIIKKA : LUENTO 17 PHYSICUM - PHY D116 SH :15 12:00 14 OPTIIKKA : LUENTO 18 PHYSICUM - PHY D116 SH :15 12:00

5 Exercises Exercises # Week # Place - Lecture Room Date Starting time Ending time 01 OPTIIKKA : LUENTO 3 PHYSICUM - PHY D116 SH :15 16:00 02 OPTIIKKA : LUENTO 4 PHYSICUM - PHY D116 SH :15 16:00 03 OPTIIKKA : LUENTO 5 PHYSICUM - PHY D116 SH :15 16:00 04 OPTIIKKA : LUENTO 6 PHYSICUM - PHY D116 SH :15 16:00 05 OPTIIKKA : LUENTO 7 PHYSICUM - PHY D116 SH :15 16:00 06 OPTIIKKA : LUENTO 8 PHYSICUM - PHY D116 SH :15 16:00 07 OPTIIKKA : LUENTO 9 PHYSICUM - PHY D116 SH :15 16:00 08 OPTIIKKA : LUENTO 11 PHYSICUM - PHY D116 SH :15 16:00 09 OPTIIKKA : LUENTO 12 PHYSICUM - PHY D116 SH :15 16:00 10 OPTIIKKA : LUENTO 14 PHYSICUM - PHY D116 SH :15 16:00 11 OPTIIKKA : LUENTO 15 PHYSICUM - PHY D116 SH :15 16:00 12 OPTIIKKA : LUENTO 16 PHYSICUM - PHY D116 SH :15 16:00 13 OPTIIKKA : LUENTO 17 PHYSICUM - PHY D116 SH :15 16:00 14 OPTIIKKA : LUENTO 18 PHYSICUM - PHY D116 SH :15 16:00

6 Scattering When light encounters matter, matter not only reemits light in the forward direction (leading to absorption and refractive index), but it also re-emits light in all other directions. This is called scattering. A plane wave, incident from the left, sweeps across an atom and spherical wavelets are scattered. The process is continuous, and hundreds of millions of photons per second stream out of the scattering atom in all directions M. Smoluchowski (1908) and A. Einstein (1910) independently provided the basic ideas for the theory of scattering.

7 Constructive & destructive interference Waves that combine in phase add up to relatively high irradiance. = Constructive interference (coherent) Waves that combine 180 out of phase cancel out and yield zero irradiance. = Destructive interference (coherent) Waves that combine with many different random phases nearly cancel out and yield very low irradiance. = Incoherent addition

8 Coherent & incoherent interference We ll check the interference one direction at a time, usually far away. This way we can approximate spherical waves by plane waves in that direction, vastly simplifying the math. Far away, spherical wave- fronts are almost flat Usually, coherent constructive interference will occur in one direction, and destructive interference will occur in all others. If incoherent interference occurs, it is usually omni-directional.

9 Coherent scattering occurs in one (or a few) directions, with coherent destructive scattering occurring in all others. A smooth surface scatters light coherently and constructively only in the direction whose angle of reflection equals the angle of incidence. Looking from any other direction you ll see no light at all due to Looking from any other direction, you ll see no light at all due to coherent destructive interference.

10 Incoherent scattering: reflection from a rough surface No matter which direction we look at it, each scattered wave from a rough surface has a different phase. So scattering is incoherent, and we ll see weak light in all directions. This is why rough surfaces look different from smooth surfaces and mirrors.

11 Scattering of light The scattering of light in the atmosphere depends on the size of the scattering particles, R, and on the wavelength,, of the scattered light. Geometric scattering: R>> Rain drops (R~ m) All wavelengths equally scattered Optical effects: white clouds Mie scattering: R~ Aerosols (R~ m) Red scattered better than blue Blue moon, blue sun Rayleigh scattering: R<< Air molecules (R~ m) Blues scattered better than red Blue sky, blue mountains, red sunsets R R R

12 Rayleigh Scattering Elastic ( does not change) Random direction of emission Little energy loss

13 Mie Scattering For particle sizes larger than a wavelength, Mie scattering predominates. This scattering produces a pattern like an antenna lobe, with a sharper and more intense forward lobe for larger particles. Mie scattering is not strongly wavelength dependent and produces the almost white glare around the sun when a lot of particulate material is present in the air. It also gives us the white light from mist and fog.

14 Scattering in tenuous media (d > ) Separation between the molecular scatterers is roughly a wavelength or more, as it is in a tenuous gas. The theory of Rayleigh Scattering has independent molecules randomly arrayed in space so that the phases of the secondary wavelets scattered off to the side have no particular relationship to one another and there is no sustained pattern of interference Lateral scattering: random phase Summation is a random walk (no interference).

15 Forward Propagation For a forward point P all the different paths taken by the light are about the same length; scattering alters the various path lengths by very little. Two molecules A and B, interacting with an incoming primary plane wave. In the forward direction the scattered wavelets arrive in-phase on planar wavefronts - trough with trough, peak with peak. Because of the asymmetry introduced by the beam itself, all the scattered wavelets add constructively with each other in the forward direction

16 The transmission of light through dense media (d<< ). At the wavelengths of light, the Earth's atmosphere at STP has (standard temperature and pressure, i.e., temperature of 273 K and pressure of 1 atm) about 3 million molecules in such a λ 3 - cube. The scattered wavelets (λ = 500 nm) radiated by sources so close together (=3 nm) cannot properly be assumed to arrive at some point P with random phases - interference will be important. Thus some molecule A radiates spherically out of the beam, but because of the ordered close arrangement, there will be a molecule B, a distance = /2 away, such that both wavelets cancel in the transverse direction. There is little or no light scattered laterally or backwards in a dense homogenous media. Forward scattering: in phase, constructive interference. Interference produces a redistribution of energy, out of the regions where it's destructive ti into the regions where it's constructive.

17 On-axis vs. off-axis light scattering Forward (on-axis) light Off-axis light scattering: scattered scattering: scattered wavelets have random relative phases wavelets have nonrandom in the direction of interest due to the (equal!) relative phases in the forward direction. often random placement of molecular scatterers. Randomly spaced scatterers in a plane Incident wave Incident wave Forward scattering is coherent even if the scatterers are randomly arranged in space. Path lengths are equal. Off-axis scattering is incoherent when the scatterers are randomly arranged in space. Path lengths are random.

18 Reflection When a beam of light impinges on the surface of a transparent material such as a sheet of glass, the wave "sees" a vast array of closely spaced atoms that will somehow scatter it. In the case of transmission through a dense medium, the scattered wavelets cancel each other in all but the forward direction and just the ongoing beam is sustained. - this can only happen if there are no discontinuities. This is not the case at an interface between two different transparent media (such as air and glass), which is a jolting discontinuity. When a beam of light strikes such an interface, some light is always scattered backward, and we call this phenomenon reflection. 4%

19 External & Internal Reflection Transition between two media is gradual - the change of n over a distance > λ Little reflection The change of n over a distance < λ /4 much like a totally t discontinuous change. Mechanism: In a uniform dense medium, the backward wavelets from different dipoles are paired and cancel each other. In an interface, this cancellation is not complete, which results in a net reflection from a thin layer about /2 deep on the surface. 4% 4% Beam I: Et External reflection: n incident < n transmitting Beam II: Internal reflection: n incident >n transmitting,180 o phase shift

20 The Law of Reflection As the wave-front descends, it energizes and re-energizes one scatterer after another, each of which radiates a stream of photons that can be regarded as a hemispherical wavelet in the incident medium. Because the wavelength is so much greater than the separation between the molecules, the wavelets emitted back into the incident medium advance together and add constructively in only one direction, and there is one well-defined reflected beam.

21 Angle of Incidence = Angle of Reflection The electric field wave-fronts are continuous at a boundary. The speed of light is the same in the incident and reflected media (because they re the same). Let r be the reflected-beam propagation angle. n i B vi t n i r E v i t r i A i D A r n i D AD = BD/sin( i ) AD = AE/sin( r ) BD/sin( i ) = AE/sin( r ) But: BD = v i t = (c 0 /n i ) t & AE = v t t = (c 0 /n i ) t So: (c 0 /n i ) t t /s sin( i ) = (c 0 /n i ) t t /s sin( r ) Or: sin( i ) = sin( r ) i = r

22 Coherent constructive scattering: Reflection from a smooth surface when angle of incidence equals angle of reflection A Abeam can only remain a plane wave ifth there s a direction for which coherent constructive interference occurs. Consider the different phase delays for different paths. i r Incident wave-front Potential outgoing wave-front Coherent constructive interference occurs for a reflected beam if the angle of incidence = the angle of reflection: i = r.

23 Coherent destructive scattering: Reflection from a smooth surface when the angle of incidence is not the angle of reflection Imagine that the reflection angle is too big. The symmetry is now gone, and the phases are now all different. = ka sin( too big ) i too big = ka sin( i ) a Potential wave front Coherent destructive interference occurs for a reflected beam direction if the angle of incidence the angle of reflection: i r.

24 The Law of Reflection The angle-of -incidence equals the angle-of-reflection. this is the first part of the Law of Reflection. Initially it appeared in the book Catoptrics, which was purported to have been written by Euclid. Normall incidence: id i 0 Glancing incidence: 90 i i = r Plane waves enter from the left and are reflected off to the right. The reflected wavefront CD is formed of waves scattered by the atoms on the surface from A to D. Just as the first wavelet arrives at C from A, the atom at D emits, and the wavefront along CD is completed.

25 Rays Spherical waves wave fronts are spherical Plane waves wave fronts are planes Rays lines perpendicular to wave fronts in the direction of propagation x Planes parallel to y-z plane

26 Reflection of light A ray of light bounces off a plane mirror. mirror This is an example of reflection of light.

27 Light reflection normal incident ray reflected ray mirror angle of incidence angle of reflection

28 Law of Reflection angle of incidence = angle of reflection normal incident ray mirror reflected ray

29 Law of Reflection, cont. The incident ray,the reflected ray and the normal all lie in the same plane. normal incident ray reflected ray mirror

30 Specular Reflection on a flat, smooth surface e.g. mirror parallel l incident id rays parallel l reflected rays

31 Multiple l Reflections The incident ray strikes the first mirror The reflected ray is directed toward the second mirror There is a second reflection from the second mirror Assume the angle between two mirrors is 90 o The reflected beam returns to the source parallel to its original path This phenomenon is called retroreflection

32 Diffuse reflection A surface behaves as a smooth surface as long as the surface variations are much smaller than the wavelength of the light on a rough, not perfectly smooth surface parallel l incident id rays reflected rays in different directions

33 Light scattering across an interface A plane wave sweeps in stimulating atoms across the interface. These radiate and reradiate, thereby giving rise to both the reflected and transmitted waves.

34 Refraction and Snell's Law The electric field (and its wave-fronts) are continuous at a boundary. But the speed AD = BD/sin( ( i ) of light will be different in the two media. AD = AE/sin( t ) n i B vi t So: BD/sin( i ) = AE/sin( t ) But: BD = v i t = (c 0 /n i ) t i A i D & AE = v t t =(c 0 /n t ) t v t t E t So: (c 0 /n i ) t / sin( i ) = (c 0 /n t ) t / sin( t ) n t t Or: n i sin( i ) = n t sin( t )

35 Refraction of Light Refraction is the bending of light when the light passes from one medium to another. air glass

36 Refraction of Light angle of incidence air glass normal angle of refraction

37 Refraction of Light, cont. From a less dense to a denser medium e.g. from air to glass air glass normal Light is bent towards the normal.

38 Refraction of Light From a denser to a less dense medium e.g. from water to air normal water air Light is bent away from the normal.

39 Refraction of Light The incident ray, the refracted ray, and the normalall lie in the same plane. air glass glass normal

40 Snel s law Willebrord Snel ( ) discovers the law of refraction Descartes ( ) publish the, now familiar, form of the law (viewed light as pressure transmitted by an elastic medium) n 1 sin 1 = n 2 sin 2 Willebrord Snel van Royen

41 Snell's Lawfor many parallel 1 n 1 layers n n 3 If the layers are parallel, then these angles are always equal. 5 n 4 n 5 n1sin 1 n2sin 2 n3sin 3... n m sin m So we can ignore the intermediate layers if we re only interested in the output angle!

42 Huygens Principle Every point on a propagating wavefront acts as the source of spherical secondary wavelets, the wavefront at a later time is the envelope of these wavelets. The secondary wavelets have the same frequency and speed as the propagating wave. proposed long before Maxwell first step towards scattering theory interference not included primary wave completely scattered still useful in inhomogeneous media Rays lines perpendicular to wave fronts Wave front - Surface of constant phase

43 Huygens wave front construction New wavefront Construct the wave front tangent to the wavelets r = c t λ Given wave-front at t Allow wavelets to evolve for time t

44 Plane wave propagation New wave front is still a plane as long as dimensions of wave front are >> λ If not, edge effects become important After all, if Einstein is right, there are only scattered photons; the wavelets themselves are a theoretical construct E. Hecht

45 Physical Optics If, however, apertures, obstacles etc have dimensions comparable to λ then wave front becomes distorted 45

46 The minimum path principle p Hero of Alexandria, who lived some time between 150 B.C.E. and A.D. 250, was the first to propose what has since become known as a variational principle. In his treatment of reflection, he asserted that the path taken by light in going from some point S to a point P via a reflecting surface was the shortest possible one. S Principle of least time - Fermat consequently Pierre de Fermat P ; was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to modern calculus. reformulated Hero's statement to read: the actual path between two points taken by a beam of light is the one that is traversed in the least time. Fermat s principle consequences: law of reflection, law of refraction

47 Optical path length (OPL) S n 1 n 2 n 3 n 4 n 5 P n m 47

48 Optical path length (OPL) Transit time from S to P t m P S 1 n s c i i i 1 OPL n( s) ds In an inhomogeneous medium m i 1 OPL n i s i OPL 1 P c S v ds 48

49 Fermat s principle t = OPL/c Light, in going from point S to P, traverses the route having the smallest optical path length t OPL c 49

50 Snell s Law causes thingsto to look bent in water.

51 Snell s Law of refraction: Bending light.

52 Snell's Law explains many everyday effects The refractive index increases with density (and hence decreases with temperature). Occasionally, astronomers speculate whether light can orbit a planet somewhere in the universe This effect may play a role in mirages.

53 Mirage Pictures

54 Snell s Law explains why the sun flattens as it sets. Light rays closer to the horizon bend more than rays further away.

55 Snell s Law explains why stars twinkle. The atmosphere has non-uniform temperature and hence nonuniform refractive index. And these regions move about in time. Star Cooler regions of air (with higher refractive index) As the air masses move about, the amount of light reaching our eyes from the star varies.

56 Dispersion is the tendency of optical properties to depend on frequency. Dispersion of the refractive index allows prisms to separate white light into its components and to measure the wavelength of light. Dispersed beam White light n( ) Dispersive element Dispersion can be good or bad, depending on what you d like to do.

57 Refraction in a Prism Since all the colors have different angles of deviation, white light will spread out into a spectrum Violet deviates the most Red deviates the least The remaining colors are in between

58 Rainbows result from dispersion in the refraction of sunlight in water droplets. Note that there can be two rainbows, and the top one is inverted. And the sky is much brighter below the bottom one.

59 The Rainbow At the back surface the light is reflected It is refracted again as it returns to the front surface and moves into the air The rays leave the drop at various angles The angle between the white light and the most intense violet ray is 40 The angle between the white light and the most intense red ray is 42

60 Primary Rainbow

61 Observing the Rainbow If a raindrop high in the sky is observed, the red ray is seen A drop lower in the sky would direct violet light to the observer The other colors of the spectra lie in between the red and the violet

62 Double Rainbow Rib The secondary rainbow is fainter than the primary The colors are reversed The secondary rainbow arises from light that makes two reflections from the interior surface before exiting the raindrop Higher-order rainbows are possible, but their intensity is low

63 Secondary Primary Secondary Rainbow The secondary rainbow is a rainbow of radius 51, occasionally visible outside the primary rainbow. It is produced when the light entering a cloud droplet is reflected twice internally and then exits the droplet. The color spectrum is reversed in respect to the primary rainbow, with red appearing on its inner edge.