Huygens and Fermat s s principles (Textbook 4.4, 4.5) Application to reflection & refraction at an interface

Transcription

1 Huygens and Fermat s s principles (Textbook 4.4, 4.5) Application to reflection & refraction at an interface 1

2 Propagation of light: Sources E.g. point source a fundamental source Light is emitted in all directions series of crests and troughs (stone dropped in water) Rays lines perpendicular to wave fronts λ Wave front - Surface of constant phase 2

3 Propagation of light: Sources Incandescent light bulb (blackbody radiation source) I ( s) ds lamp = B I A p A B 1. Glass bulb 2. Low pressure inert gas 3. Tungsten filament 4. Contact wire (goes out of stem) 5. Contact wire (goes into stem) 6. Support wires 7. Stem (Glass mount) 8. Contact wire (goes out of stem) 9. Cap (Sleeve) 10. Insulation (Vitrite) 11. Electrical contact 3

4 Two ideal light waves Spherical waves (from a point source) wave fronts are spherical Plane waves (from a point source at infinite) wave fronts are planes Rays lines perpendicular to wave fronts in the direction of propagation x Planes parallel to y-z plane 4

5 Huygens principle In the 17th Century, Christiaan Huygens ( ) proposed what we now know as Huygens Principle, one of the fundamental concepts of waves and wave optics. A typical statement of the principle is every point on a wavefront acts as a source of a new wavefront, propagating radially outward. 5

6 Huygen s principle Every point on a wave front is a source of secondary wavelets. i.e. particles in a medium excited by electric field (E) re-radiate in all directions i.e. in vacuum, E, B fields associated with wave act as sources of additional fields 6

7 Huygens wave front construction New wavefront Construct the wave front tangent to the wavelet r = c t λ Given wave-front at t Allow wavelets to evolve for time t What about r direction (focusing wave or beams)? 7

8 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 Note: no such thing as a perfect plane wave, or collimated beam 8

9 Geometric Optics As long as apertures are much larger than a wavelength of light (and thus wave fronts are much larger than λ) the light wave front propagates without distortion (or with a negligible amount) i.e. light travels in straight lines 9

10 Physical Optics If, however, apertures, obstacles etc have dimensions comparable to λ (e.g. < 10 3 λ) then wave front becomes distorted 10

11 Geometric Optics Let s reflect for a moment 11

12 Hero s principle Hero (150BC-250AD) asserted that the path taken by light in going from some point A to a point B via a reflecting surface is the shortest possible one. 12

13 Hero s principle and reflection A B R O O O A 13

14 Law of reflection θ i = θ r 14

15 Reflection by plane surfaces Reflecting through (x, z) plane y r 1 = (x,y,z) z r 2 = (-x,y,z) r 1 = (x,y,z) x r 3 =(-x,-y,z) x y r 4 =(-x-y,-z) r 2 = (x,-y,z) Law of Reflection r 1 = (x,y,z) r 2 = (x,-y,z) 15

16 Geometric Optics Let s refract for a moment 16

17 Speed of light in a medium v = c n n refractive index Light slows on entering a medium Huygens Also, if n ν = 0 i.e. light stops in its track!!!!! See: P. Ball, Nature, January 8, 2002 D. Philips et al. Nature 409, (2001) C. Liu et al. Physical Review Letters 88, (2002) 17

18 Index of refraction A property of a material that changes the speed of light. n α speed of light in vacuum = = speed of light in medium α c v n ε µ α = r r where ε r is the material's relative permittivity, and µ r is its relative permeability. For a non-magnetic material, µ r is very close to 1, therefore n is approximately ε r. 18

19 Speed of light in a medium v = c n Light slows on entering a medium Huygens Also, if n ν = 0 i.e. light stops in its track!!!!! See: P. Ball, Nature, January 8, 2002 D. Philips et al. Nature 409, (2001) C. Liu et al. Physical Review Letters 88, (2002) 19

20 Refraction: Bending light 20

21 Refraction: Bending light As light passes from one transparent medium to another, it changes speed, and bends. How much this happens depends on the refractive index of the mediums and the angle between the light ray and the line perpendicular (normal) to the surface separating the two mediums (medium/medium interface). Each medium has a different refractive index. The angle between the light ray and the normal as it leaves a medium is called the angle of incidence. The angle between the light ray and the normal as it enters a medium is called the angle of refraction. 21

22 Refraction: Bending light n 1 is the refractive index of the medium the light is leaving, θ 1 is the incident angle between the light ray and the normal to the medium to medium interface, n 2 is the refractive index of the medium the light is entering, θ 2 is the refractive angle between the light ray and the normal to the medium to medium interface. 22

23 Snell s law Willebrord Snell ( ) 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 sinθ 1 = n 2 sinθ 2 23

24 Snell s law n 1 sinθ 1 = n 2 sinθ 2 24

25 Pierre de Fermat s principle 1657 Fermat ( ) proposed a Principle of Least Time encompassing both reflection and refraction The actual path between two points taken by a beam of light is the one that is traversed in the least time 25

26 Pierre de Fermat s principle "Light, in going between two points, traverses the route having the smallest optical path length." v = α c n α t d 1 1 = = v c c α α ( n d ) = OPL OPL = n α d = index of refraction distance traveled 26

27 Fermat s principle The actual path between two points taken by a beam of light is the one that is traversed in the least time Light, in going from point S to P, traverses the route having the smallest optical path length t = OPL c 27

28 Application of Fermat s principle o 28

29 Application of Fermat s principle o ( v = c n ) 29

30 Optical path length S n 1 n 2 n 3 n 4 n 5 P n m 30

31 Optical path length Transit time from S to P 1 t = c m i= 1 n i s i OPL n( s) ds = P S m OPL = i= 1 n i s i OPL P = S c v Same for all rays ds 31

32 Optical path length In an inhomogeneous medium the refractive index n(r) is a function of the position. The optical path length along a given path between two points A and B is therefore where ds is the differential element of the length along the path (Could be more than one path). r(x,y,z) A ds B n(r) 32

33 Nature of light Law of Reflection: θ = θ i r Law of Refraction Snell s Law: n sinθ = n sinθ i i r r 33

34 Refraction by plane interface & Total internal reflection n 2 θ 2 θ 2 n 1 > n 2 θ 1 θ 1 θ C n 1 P Snell s law n 1 sinθ 1 =n 2 sinθ 2 34

35 Total internal reflection (TIR) 1611 Discovered by Kepler θ C n 1 n 2 θ c Critical angle n θ 1 c sinθ = n = c sin 1 2 n n 2 1 sin 90 n 1 > n 2 35

36 TIR Critical Angle Light bends toward the normal when the light enters a medium of greater refractive index, and away from the normal when entering a medium of lesser refractive index. As you approach the critical angle the refracted light approaches 90 and, at the critical angle, the angle of refraction becomes 90 and the light is no longer transmitted across the medium/medium interface. For angles greater in absolute value than the critical angle, all the light is reflected. This is called total internal reflection. 36

37 Refraction by plane interface & Total internal reflection n 2 θ c = sin 1 n n 2 1 n 1 > n 2 θ C θ 1 θ 1 n 1 P Snell s law n 1 sinθ 1 =n 2 sinθ 2 37

38 Refraction by plane interface & Total internal reflection n 2 θ 2 θ 2 n 1 > n 2 θ 1 θ C θ 1 θ 1 θ 1 n 1 P Snell s law n 1 sinθ 1 =n 2 sinθ 2 38

39 Examples of prisms and total internal reflection 45 o 45 o 45 o Totally reflecting prism 45 o Porro Prism 39

40 Porro prism A Porro prism, named for its inventor Ignazio Porro, is a type of reflection prism used in optical instruments to alter the orientation of an image. An image travelling through a Porro prism is rotated by 180 and exits in the opposite direction offset from its entrance point. Since the image is reflected twice, the handedness of the image is unchanged. 40

41 Porro prism An image travelling through a Porro prism is rotated by 180 and exits in the opposite direction offset from its entrance point. Since the image is reflected twice, the handedness of the image is unchanged. 41

42 Porro prism An image travelling through a Porro prism is rotated by 180 and exits in the opposite direction offset from its entrance point. Since the image is reflected twice, the handedness of the image is unchanged. Porro prism binoculars 42

43 Image inversion and reversion 43

44 Optical waveguide Many devices take advantage of the total internal reflection, including optical waveguides (like optical fiber). A waveguide is a length of transparent material that is surrounded by material that has a lower index of refraction. Rays that intersect the interface between the waveguide material and the surrounding material at angles equal to or larger than the critical angle are trapped in the waveguide and travel losslessly along it. Rays Can be bound (trapped) in a waveguide through total internal reflection. 44

45 Optical waveguide 45

46 Optical waveguide Many devices take advantage of the total internal reflection, including optical waveguides (like optical fiber). A waveguide is a length of transparent material that is surrounded by material that has a lower index of refraction. Rays that intersect the interface between the waveguide material and the surrounding material at angles equal to or larger than the critical angle are trapped in the waveguide and travel losslessly along it. Rays Can be bound (trapped) in a waveguide through total internal reflection. 46

47 Natural phenomenon 47