TFY4170 Fysikk 2 Justin Wells

Transkript

1 TFY4170 Fysikk 2 Justin Wells Forelesning 5: Wave Physics Interference, Diffraction, Young s double slit, many slits. Mansfield & O Sullivan: 12.6, 12.7, 19.4,19.5

2 Waves! Wave phenomena! Wave equation in one dimension! Energy, power and intensity of waves! Plane waves and dispersing waves. Huygens principle. Reflection and refraction (brytning).! Interference and diffraction. Young s double slit, many waves.! Diffraction in crystals, X- ray, neutron and electron diffraction.! Standing waves, resonance.! Doppler effect (classic and relativistic).! Lasers and coherent waves.! The wave equation. Mechanical waves and sound waves.! Electromagnetic waves, Maxwell s equations, polarisation.! Wave packets and envelopes, group velocity, dispersion.! Fourier analysis, bandwidth.

3 Repetition: Interference Example: 2-dimensional wave from two synchronised sources K 1 and K 2. We look at what is happening at point P, which is far from K1 and K2. K 1 difference in the distance from the sources: x 1 -x 2 x 1 P y= y 1 +y 2 = A[1+e i ϕ ] e i(ωt kx 1+y) K 2 x 2 ϕ=k (x 1 x 2 )

4 Repetition: Interference Constructive interference: Destructive interference: x 1 K 1 x 2 K 2

5 Repetition: Diffraction Incoming wave θ The weight of the wave spreads out over a range of angles Diffraction: We can think of this as interference between a number of point sources on the wavefront in the aperture/slit

6 Diffraction The resultant amplitude A θ : The sum of the N complex amplitudes (consider the phase difference) N 1 A θ = A' e (i ϕ n ) n=0 ϕ=k (x 1 x 2 )= ka N sin θ Im θ Maximum amplitude occurs when θ=0 the maximum amplitude is A 0 =NA. φ Re

7 Diffraction: Intensity The intensity of the diffracted wave is: I θ = A 2 θ I 0 A = sin2 α 2 α 0 α= π a λ sin θ a<<λ :the wave spreads evenly over a wide angle a>>λ :Most of the intensity is straight ahead, through the opening The maximum intensity occurs when: For larger angles, the intensity is smaller. The intensity varies with sin 2 α with a decreasing amplitude proportional to α 2. The intensity is minimum when: sin α=0, α=n π

8 Intensity variations Bi-maxima is observed when: α= 2n+1 2 They have a relative intensity: π This means that higher order bi-maxima are much weaker that the principle maxima The first bi-maxima is observed when: When the aperture is smaller than the wavelength, the diffraction variations are not seen.

9 Intensity variations A typical diffraction pattern when the opening is larger than the wavelength: Intensitet

10 Young s double slit experiment: We will now look what happens when we have two slits: Difference in distance: Phase difference: 2 γ= 2 π (a+b)sin θ=(a+b)k sin θ λ We add together the amplitudes from the two slits A A=A 1 [1+e i 2 γ ] A 2 = A 1 2 [1+e i 2 γ ][1+e i 2 γ ] =2 A 1 2 [1+cos(2 γ)] =4 A 1 2 cos 2 γ A 1 2γ A 1

11 Young s double slit experiment: For each single slit: derfor The total amplitude is: i A θ = sin A α α 0 A =2 A 1 cos γ=2a 0 sin α α cos γ The relative intensity at a given angle is therefore: I θ = A 2 I 0 (2A 0 ) = sin2 α 2 α 2 cos 2 γ The first term is the diffraction term, which is the same as for a single slit: sin 2 α α 2 = sin 2 ( π a λ sin θ) ( π a λ sin θ) 2

12 Young s double slit experiment: The second component is the two-wave interference component: The maximum (i.e.constructive interference) is found when: d=a+b b Typically, b>>a

13 Young s double slit experiment: Diffraction (first maximum): Young s double-slit: Total intensity: diffraction+interference

14 Diffraction and interference: Many slits In Young s two-slit experiment, we can measure the wavelength by looking at the interference+diffraction pattern. But, it is difficult because the maxima can be quite broad and badly defined. It is much easier (higher precision) to study the wave using many slits θ Interference between two waves There is a generalised expression to describe what happens when there are many slits...

15 Diffraction and interference: Many slits Generally, Many slits means N>1000 : i.e. a grid or lattice We now deal with interference of many waves The phase difference between waves from two neighbouring slits is: We need to add together N single slit amplitudes: 2γ A 1 A 1 2γ A 1

16 Diffraction and interference: Many slits Resultant amplitude: A N = A 1 [1+e i2 γ +e i2 γ 2 + +e i2 γ (N 1) ] N 1 = A 1 [e i2 γ ] n n=0 = A 1 1 (e i2 γ ) N 1 e i2 γ = A 1 e i γ N = A 1 e i γ N γ N e i γ N e i γ e i e i γ e i γ e sin(γ N ) i γ sin γ The relative difference in amplitude of N slits (compared to 1 slit) is therefore:

17 Amplitude for many waves: For a single slit: i A θ = sin A α α 0 derfor The angle dependent e amplitude amplituden for er many derfor slits is: A N = A 0 sin α α sin(γ N ) sin γ The resulting amplitude of waves which go straight through is: A 0T =NA 0 A N = A 0T N The angle dependent intensity is therefore: sin α α sin γ N sin γ where I 0 is the intensity at

18 Intensity for many waves: For small angles, we find: As before, the prefactor gives the 1-slit diffraction pattern: The second factor gives the many-slit interference:

19 Intensity for many waves: The interference pattern for many slits is a complicated function with many maxima and minima. It can be shown that the strongest maxima occur when: This is the same condition which we found for Young s double slit Example: To slits (N=2) Intensity θ

20 Intensity for many waves: What happens when we increase the number of slits? N=2 N=4 N=8 N=16

21 Intensity for many waves:! Maxima occur at the same angles.! The maxima get sharper when we use more slits.! The wavelength can therefore be found more accurately if we use many slits.! This is useful for spectroscopy.

22 Repetition forelesning 5 One slit Many slits