1. Waves and Particles 2. Interference of Waves 3. Wave Nature of Light

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1 1. Waves and Particles 2. Interference of Waves 3. Wave Nature of Light 1. Double-Slit Eperiment reading: Chapter Single-Slit Diffraction reading: Chapter Diffraction Grating reading: Chapter 22

2 Traveling Waves Chapter 20

3 Waves and Particles Wave periodic oscillations in space and in time of something t 0 It is moving as a whole with some velocity v vt t 0

4 Particle and Waves Sinusoidal Wave Particles

5 Sinusoidal Wave Plane wave y c - Speed of wave y changes only along one direction - wavelength Period of oscillation T (time to travel a distance of wavelength) Frequency of wave f c 1 c T maimum minimum

6 Sin-function Amplitude E( ) 0 E 0 Amplitude E( ) E0 sin 2 Phase (initial) 2 3

7 Sinusoidal (Basic) Wave E( t, ) E0sin2 2 ft E0sin 2 t Distribution of Electric Field in space at different time Source of the Wave E( ) E0 sin2 t c Distribution of some Field in space and in time with frequency f 1 c Time f T - usual sin-function with initial phase, depending on t as t t

8 Sinusoidal Wave E( t, ) E0sin2 2 ft E0sin 2 t Distribution of some Field in space and in time with frequency f At a given time t we have sin-function of with initial phase, depending on t E( ) E0 sin2 t t 2 ftt t At a given space point we have sin-function of t with initial phase, depending on E( ) E0 sint 2

9 Particle and Waves We can take the sum of many sinusoidal waves (with different wavelengths, amplitudes) = wave pack Sum = Any shape which is moving as a whole with constant velocity

10 Wave Pack Wave pack can be considered as a particle

11 Particle and Waves How can we distinguish between particles and waves? For waves we have interference, for particles not!

12 Interference of Waves Chapter 21

13 Sin-function E( ) Amplitude E( ) E sin 2 f t 0 Phase (initial) Amplitude 0 E 0 E( ) 1/ f 2/ f 3/ f t Amplitude 0 E 0 t Interference: THE SUM OF TWO SIGNALS (WAVES)

14 Sin-function: Constructive Interference Amplitude E( ) 0 E 0 Amplitude E( ) E sin 2 f t 0 Phase (initial) E( ) 1/ f 2/ f 3/ f t Amplitude 0 E 0 t Amplitude 0 2E 0 The phase difference between two waves should be 0 or integer number of 2 t

15 Sin-function: Destructive Interference Amplitude E( ) 0 E 0 Amplitude E( ) E sin 2 f t 0 Phase (initial) E( ) 1/ f 2/ f 3/ f t Amplitude o 180 ( ) E 0 t E( ) Amplitude (no signal) 0 The phase difference between two waves should be or plus integer number of 2 t

16 Waves and Particles Interference of waves: THE SUM OF TWO WAVES Analog of Interference for particles: Collision of two particles The difference between the interference of waves and collision of particles is the following: THE INTERFERENCE AFFECTS MUCH LARGER REGION OF SPACE THAN COLLISION DOES

17 Waves: Interference E1( t, ) E0sin2 2 ft Interference sum of two waves E1(, t) E0sin 2 ft 1 E2( t, ) E0sin2 2 ft E 2(, t) E0sin2 ft In constructive interference the amplitude of the resultant wave is greater than that of either individual wave In destructive interference the amplitude of the resultant wave is less than that of either individual wave

18 Waves: Interference Amplitude 1 2 E () t 1 1 E 0 E () t 2 Amplitude Et () Esin 2 ft 0 Phase (initial) 1/ f 2/ f 3/ f t Amplitude 2 2 E 0 2 t Constructive Interference: The phase difference between two waves should be 0 or integer number of 2 2 m 0, 1, 2... m 1 2 Destructive Interference: The phase difference between two waves should be or integer number of 2 2m m 0, 1,

19 Conditions for Interference To observe interference the following two conditions must be met: 1) The sources must be coherent - They must maintain a constant phase with respect to each other 2) The sources should be monochromatic - Monochromatic means they have a single (the same) wavelength

20 Conditions for Interference: Coherence coherent E( ) E sin t 0 1 E( ) E sin t 0 2 The sources should be monochromatic (have the same frequency) E( ) E sin t 0 1 E( ) E sin t 0 2

21 Waves and Particles The difference between the interference of waves and collision of particles is the following: THE INTERFERENCE AFFECTS MUCH LARGER REGION OF SPACE THAN COLLISION AND FOR A MUCH LONGER TIME If we are looking at the region of space that is much larger than the wavelength of wave (or the size of the wave) than the wave can be considered as a particle

22 Chapter 22 Light as a Wave: Wave Optics

23 The Nature of Light Particle or Waves? WAVE? PARTICLES?

24 The Nature of Light Particle or Waves? Before the beginning of the nineteenth century, light was considered to be a stream of particles Newton was the chief architect of the particle theory of light He believed the particles left the object and stimulated the sense of sight upon entering the eyes But he was wrong. LIGHT IS A WAVE.

25 The Nature of Light Particle or Waves? How can we distinguish between particles and waves? For waves we have interference, for particles not!

26 The Nature of Light Wave Theory? Christian Huygens argued that light might be some sort of a wave motion Thomas Young (1801) provided the first clear demonstration of the wave nature of light He showed that light rays interfere with each other Such behavior could not be eplained by particles During the nineteenth century, other developments led to the general acceptance of the wave theory of light

27 Light as a Wave Plane wave y c - speed of light y changes only along one direction - wavelength Period of oscillation (time to travel distance of wavelength) Frequency of light f T c 1 c T maimum minimum

28 Light as a Wave Light is characterized by c its speed and wavelength (or frequency f ) Different frequency (wavelength) different color of light What is the speed of light?

29 Measurements of the Speed of Light Fizeau s Method (1849) d is the distance between the wheel and the mirror Δt is the time for one round trip Then c = 2d / Δt Fizeau found a value of c = m/s c = m/s - Speed in Vacuum!

30 Speed of Light What is the speed of light in a medium? The speed of light in a medium is smaller than the speed in vacuum. To understand this you can think about it in a following way: The medium consists of atoms (or molecules), which can absorb light and then emit it, so the propagation of light through the medium can be considered as a process of absorption and subsequent emission (AFTER SOME TIME t ) atoms light (free propagation) very schematic picture

31 Speed of Light v c n - The speed of light in the medium The properties of the medium is characterized by one dimensionless constant n, (it is called inde of refraction, we will see later why) which is equal to 1 for vacuum (and very close to 1 for air), greater then 1 for all other media

32 Light in the Media E at a given point T v c n - The speed of light in the medium t P 1 T t The same period (frequency) in all media, then c n T n air n

33 Light as a Wave E( t, ) E0sin2 2 ft E0sin 2 t Distribution of some Field inside the wave of frequency f At a given time t we have sin-function of with initial phase, depending on t E( ) E0 sin2 t t 2 ftt t At a given space point we have sin-function of t with initial phase, depending on E( ) E0 sint 2

34 Sin-function: Constructive Interference Amplitude E( ) 0 E 0 Amplitude E( ) E0 sin 2 Phase (initial) E( ) 2 3 Amplitude 0 Amplitude 0 E 0 2E 0 The phase difference between two waves should be 0 or integer number of 2

35 Sin-function: Destructive Interference Amplitude E( ) 0 E 0 Amplitude E( ) E0 sin 2 Phase (initial) E( ) 2 3 Amplitude o 180 ( ) E 0 E( ) Amplitude (no signal) 0 The phase difference between two waves should be or plus integer number of 2

36 Waves: Interference E1( t, ) E0sin2 2 ft Interference sum of two waves E1(, t) E0sin 2 ft 1 E2( t, ) E0sin2 2 ft E 2(, t) E0sin2 ft In constructive interference the amplitude of the resultant wave is greater than that of either individual wave In destructive interference the amplitude of the resultant wave is less than that of either individual wave

37 Waves: Interference Amplitude 1 2 E () t 1 1 E 0 E () t 2 Amplitude Et () Esin 2 ft 0 Phase (initial) 1/ f 2/ f 3/ f t Amplitude 2 2 E 0 2 t Constructive Interference: The phase difference between two waves should be 0 or integer number of 2 2 m 0, 1, 2... m 1 2 Destructive Interference: The phase difference between two waves should be or integer number of 2 2m m 0, 1,

38 Conditions of Interference coherent E( ) E sin t 0 1 E( ) E sin t 0 2 The sources should be monochromatic (have the same frequency) E( ) E sin t 0 1 E( ) E sin t 0 2

39 1. Double-Slit Eperiment (interference) d d 2. Single-Slit Diffraction 3. Diffraction Grating

40 Young s Double-Slit Eperiment Thomas Young first demonstrated interference in light waves from two sources in 1801 The narrow slits S 1 and S 2 act as sources of waves The waves emerging from the slits originate from the same wave front and therefore are always in phase

41 Double-Slit Eperiment: Interference E( ) E0 sin t 2 The phase of wave 1: 1, The phase of wave 2: 2,2 2 Constructive Interference: (bright fringe) Destructive Interference: (dark fringe) where n is integer,2,1 2 n n 2 1 n where n is integer,2,1 2 n 2 1 n 2

42 Double-Slit Eperiment: Interference Constructive Interference: (bright fringe) n 2 1 Destructive Interference: (dark fringe) 2 1 n

43 Double-Slit Eperiment: Interference The path difference, δ, is found from the tan triangle δ = 2 1 = d sin θ This assumes the paths are parallel Not eactly true, but a very good approimation if L is much greater than d 1 2

44 Double-Slit Eperiment: Interference δ = 2 1 = d sin θ 1 2 Bright fringes (constructive interference): δ = d sin θ = nλ n = 0, ±1, ±2, n is called the order number - when n = 0, it is the zeroth-order maimum - when n = ±1, it is called the first-order maimum Dark fringes (destructive interference): δ = d sin θ = (n + ½)λ n = 0, ±1, ±2,

45 Double-Slit Eperiment: Interference δ = 2 1 = d sin θ The positions of the fringes can be measured vertically from the zeroth-order maimum θ is small and therefore the small angle approimation tan θ ~ sin θ can be used y = L tan θ L sin θ For bright fringes For dark fringes y λl bright n (n 0, 1, 2 ) d λl 1 ydark n ( n0, 1, 2 ) d 2

46 1 2 Constructive Interference: (bright fringe) where m is,2,1 2 m integer m 0, 1, 2,... m 2 1 m y λl bright m (m 0, 1, 2 ) d Destructive Interference: (dark fringe),2,1 2m where m is integer m 0, 1, 2, m 2 λl 1 ydark m ( m0, 1, 2 ) d 2

47 Double-Slit Eperiment: Eample The two slits are separated by mm, and the incident light includes light of wavelengths 1 540nm and 2 450nm. At what minimal distance from the center of the screen the bright line of the 1 light coincides with a bright line of the light 2 y y Bright lines: λl 1 m (m 0, 1, 2 ) d λ2l m (m 0, 1, 2 ) d bright,1 1 1 bright, mm ybright,1 m,, 3 1(m) 5 m1(mm) (m ) m ybright,2 m,, 3 2(m) 4 m2(mm) (m ) y bright,1 0, 5, 10, 15, 20, 25...(mm) y bright,1 0, 4, 8, 12, 16, 20...(mm)

48 Double-Slit Eperiment: Eample Light with a wavelength of 442 nm passes through a double-slip system that has a slip separation d=0.4 mm. Determine L so that the first dark fringe appears directly opposite both slits. Dark lines: λl 1 ydark,n m ( m0, 1, 2 ) d 2 d ydark,1 2 y dark,1 1 2 λ L d d λ L d 0.4mm d m L 036. m36cm 9 λ m 1 2

49 Chapter 22 Diffraction Pattern and Interference

50 Diffraction Diffraction: Light spreads beyond the narrow path defined by the slit into regions that would be in shadow if light traveled in straight lines Wrong picture if d Diffraction d Diffraction Pattern d d Geometric Optics - if d Diffraction and Interference are closely related; Diffraction Patterns are due to Interference

51 Diffraction Pattern Diffraction d Diffraction Pattern secondary maima d central maimum Diffraction d Diffraction Pattern d Diffraction Pattern is similar to Interference Pattern

52 Huygens s Principle

53 Huygens s Principle Huygens s Principle is a geometric construction for determining the position of a new wave at some point based on the knowledge of the wave front that preceded it All points on a given wave front are taken as point sources for the production of spherical secondary waves, called wavelets, which propagate outward through a medium with speeds characteristic of waves in that medium After some time has passed, the new position of the wave front is the surface tangent to the wavelets

54 Single-Slip Diffraction

55 Single Slit Diffraction Each portion of the slit acts as a source of light waves Therefore, light from one portion of the slit can interfere with light from another portion

56 Intensity of Single-Slit Diffraction Pattern I( ) I sin( / 2) /2 ma 2 2 a sin I( ) I sin( a sin / ) a sin / ma 2 The first minimum occurs at sin( / 2) 0 or 2 or sin / a dark /2

57 Diffraction I( ) I sin( a sin / ) a sin / ma 2 The size of the spot a sin / a L / a dark L a sin / a dark L / a The size of the spot (image) L

58 Diffraction: Eample The source of the light emits the light with wavelength 540nm. The diffraction pattern is observed in the water, n L = 10m, a=0.5 mm What is the size of the spot, D? wavelength in the water sin / dark water a L na D L / a water 9 D 3 m 810 m 8mm 3 water / n n 1.33 a sin / dark water a L water The size of the spot (image) / a L

59 Diffraction Grating Chapter 22

60 Diffraction Grating The diffraction grating consists of a large number of equally spaced parallel slits A typical grating contains several thousand lines per centimeter The intensity of the pattern on the screen is the result of the combined effects of interference and diffraction Each slit produces diffraction, and the diffracted beams interfere with one another to form the final pattern

61 N-Slit Interference: Intensity Graph Two Slits 2 4E 0 Three Slits 2 E 0 2 4E 0 For N slits, the primary maima is N 2 times greater than that due to a single slit

62 Diffraction Grating The condition for maima is 2 m, m 0, 1, 2,... d sin bright 2 2 then d sin bright m The integer m is the order number of the diffraction pattern 2 m

63 Diffraction Grating All the wavelengths are seen at m = 0 This is called the zerothorder maimum The first-order maimum corresponds to m = 1 Note the sharpness of the principle maima and the broad range of the dark areas d sin bright m

64 Diffraction Grating Spectrometer The collimated beam is incident on the grating sharp peaks The diffracted light leaves the gratings and the telescope is used to view the image The wavelength can be determined by measuring the precise angles at which the images of the slit appear for the various orders

65 Diffraction Grating: Eample Three discrete spectral lines occur at angles , , and in the first order spectrum of a grading spectrometer. If the grading has N=3600 slits per centimeter, what are the wavelength of the light? d sin bright m First order means that m=1, then 1 d sin1 2 d sin2 3 d sin3 d 1cm N Then 1 sin10.09 sin sin1 cm 480nm N 3600 cm nm sin14.77 cm 708nm 3600

66 Michelson Interferometer Chapter 22

67 Michelson Interferometer A ray of light is split into two rays by the mirror M o The mirror is at 45 o to the incident beam The mirror is called a beam splitter It transmits half the light and reflects the rest The two rays travel separate paths L 1 and L 2 2( L L ) Maimum (constructive interference): 2( L L ) 2( L L ) m m 2 1 m 0, 1, 2,...

68 Michelson Interferometer The maimum with and without glass film. What is the value of d? Without glass film: maimum (constructive interference): Glass film d 2( L L ) m m1 0, 1, 2,... With glass film: maimum (constructive interference): 2L2 2( L1 d) 2d 2 2 2n 2 m 2 2[ L2 L1 d( n1)] m2 2 dn ( 1) ( mm) m m2 0, 1, 2,... m3 0, 1, 2,...

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