Contents. Diffraction by 1-D Obstacles. Narrow Slit. Wide Slit. N Slits. 5 Infinite Number of Slits

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Eugene Stone
5 years ago
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1 Diffraction

2 Contents 1 2 Narrow Slit 3 Wide Slit 4 N Slits 5 Infinite Number of Slits

3 - geometric arrangement diffraction pattern amplitude Fk ( ) ik r F( k)= f( r) dr all r f( r) : amplitude function xz plane e obstacle- along x-axis r = ( x,,) f() r f() x k = ( kx,, kz) k r = k x = kx sin θ θ : scattering angle x

4 - geometric arrangement ik r ikxsin F( k)= f ( r) e d r = θ f ( x) e dx - all r ikxsinθ (sin ) ( ) ( : constant) F θ = f x e dx k - - in general, F(sin θ ) is complex intensity of diffraction pattern F(sin θ ) 2

5 - one narrow slit an infinite opaque sheet along the x-axis containing narrow slit at the origin - narrowness- compared to the wavelength - amplitude function- δ function

6 - f( x) = δ ( x) ikxsinθ ikxsinθ - F(sin θ) = f ( x) e dx = δ( x) e dx - - ikxsinθ = e = x= - intensity ~ F(sin θ ) = intensity is uniform at all angles - a single narrow slit an active point source 1

7 - two narrow slit a pair of δ functions, one at + x,and the other at x - f( x) = δ( x+ x ) + δ( x x ) ikxsinθ ikxsinθ = + = ikxsinθ F θ = f x e dx - (sin ) ( ) - ikxsinθ ikxsinθ δ( x x) e dx δ( x x) e dx - - = + + e - e 2 2 2cos( kx sin θ ) ( f( x) δ ( x x) dx= f( x) ) - F(sin θ) = 4cos ( kx sin θ)

8 Interference

9 Young s Experiment maxima dsinθ = mλ

10 - three narrow slits f( x) = δ( x+ x ) + δ( x) + δ( x x ) ikxsinθ F f x e dx - (sin θ ) = ( ) - = 1+ 2cos( kx sin θ ) 2 2 = + kx θ - F(sin θ) [1 2cos( sin )] nine times more intense

11 - N narrow slits equally spaced by a distance N = 2 p+ 1 (odd number) x N δ function n= p - f ( x) = δ ( x nx ) n= p

12 ikxsinθ F θ = f x e dx - (sin ) ( ) - = e + e e ikpx sin θ ik ( p 1) x sinθ ikpx sinθ ikpxsinθ ikxsinθ ik 2 pxsinθ = e (1 + e + + e ) ik (2 p+ 1) xsinθ iknxsinθ ikpxsinθ 1 e ikpxsinθ 1 e = e ( ) = e ( ) ikx sinθ ikx sinθ 1 e 1 e Nkx sinθ sin = 2 kx sinθ sin *main peak 2 2 Nkx sinθ angular width- 4 π / Nkx sin 2 - F(sin θ ) = 2 separated by- 2 π / kx 2 kx sinθ N - 2 subsidiary peaks sin 2

13 - infinite number of narrow slits separated by a distance x infinite array of δ function n= - f ( x) = δ ( x nx ) n= n= 2nπ - F(sin θ) = δ(sin θ ) kx n= *infinitely sharp peak (sin θ ) = 2π kx

14 Grating Spectroscope dsinθ = mλ visible emission line of cadmium visible emission line from hydrogen

15 - one wide slit an opaque screen containing a slit which is wide ~1λ - f ( x) = if - < x < -X - X 1 if - X < x <+ X - F(sin θ )= f ( x) e = X - f ( x) e if + X < x < ikxsinθ ikxsinθ dx dx ikxsinθ e = ik sinθ X - X

16 - F(sin θ ) = 2X = ik sinθ = ik sinθ ikxsinθ ikx sinθ ikx e e e - X sin( kx sin θ ) kx sinθ - intensity~ F(sin θ ) 4-2X (sin θ) = 2 π / kx = X 2 ( kx sin θ ) - the wider slit, the narrow the diffraction pattern - secondary peak-very rapidly weak X sinθ sin ( kx sin θ )

$Single-Slit Diffraction δ dsin$

17 Single-Slit Diffraction δ dsin θ, for minima sinθ = m d λ

18 Single-Slit Diffraction - δ d sinθ λ for minima sin θ = m, m=1,2,3, d

19 Single-Slit Diffraction

21 Convolution - c() u = f()* r g() r = f() r g( u r) dr = f( u r) g() r dr all r all r - integrand is a function of u and r integration is taken over r function of u - gr ( ) vs. gu ( r) or gx ( ) vs. gu ( x) for 1-D reflection+displacement - example f( x): δ function, gx ( ): arbitrary function c(u)= f ( x) g( u - x) dx = δ ( x + x ) g( u - x) dx δ ( x x ) g( u - x) dx - = gu ( + x) + gu ( - x) o

22 inversion in 3-D

27 - two wide slit slits, each of width 2 X, centered at x= - x and x= x - f ( x) = if - < x < -( x + X ) 1 if - ( x + X ) < x < -( x X ) if -( x X ) < x< ( x X ) 1 if ( x X ) < x < ( x + X) if ( x + X ) < x < - f( x) is convolution of one wide slit and two narrow slit f ( two wide slit) = f ( one wide slit)* f ( two narrow slit)

28 - Fourier transform Tf ( two wide slit) = T[ f ( one wide slit)* f ( two narrow slit)] - Fourier transform of a convolution is the product of the individual Fourier transforms Tf ( two wide slit) = Tf ( one wide slit) Tf ( two narrow slit)] - one wide slit F(sin θ ) = 2X - two narrow slit sin( kx sin θ ) kx sinθ F(sin θ) = 2cos( kx sin θ)

29 - two wide slit sin( kx sin θ ) F(sin θ) = 4X cos( kx sin θ) -intensity kx sinθ sin ( kx sin θ ) F(sin θ) 16X cos ( kx sin θ) = 2 ( kx sin θ )

30 - two wide slit 2 form of diffraction pattern is a series of cos fringes modulated by (sin α / α) 2 - cos function- first zero kxsin θ1 = π / 2 sin θ = π /(2 kx ) (sin αα / ) function- first zero kx sinθ2 - x sin θ = π /( kx ) 2 X θ > θ = π

31 - three wide slit - amplitude function f ( three wide slit) = f ( one wide slit)* f ( three narrow slits) Tf ( threeo wide slit) = T[ f ( one wide slit)* f ( three narrow slits)] Tf ( threeo wide slit) = Tf ( one wide slit) Tf ( three narrow slits) sin( kx sin θ ) - F(sin θ) = 2 X [1 + 2cos( kx sin θ)] kx sinθ

32 - three wide slits - intensity sin ( kx sin θ ) F(sin θ) 4 X [1 2cos( kx sin )] = + 2 θ ( kx sin θ )

33 - N wide slits width of 2X, centered on a δ function - amplitude function f ( N wide slit) = f ( one wide slit)* f ( N narrow slits) Tf ( N wide slit) = T[ f ( one wide slit)* f ( N narrow slits)] Tf ( N wide slit) = Tf ( one wide slit) Tf ( N narrow slits)

34 - N wide slits - intensity F(sin θ ) 4 sin ( kx sin θ ) = X 2 ( kx sin θ ) sin sin Nkx 2 kx 2 sinθ 2 sinθ 2

35 Experiments

36 (a) s = 12 µm (b) s = 3 µm Slit spacing s and slit width w

$Diffraction$ from

37 Diffraction patterns from gratings (a) and (b).

40 The zebra

41 - infinite wide slits - amplitude function f ( wide slit) = f ( one wide slit)* f ( narrow slits) Tf ( wide slit) = T[ f ( one wide slit)* f ( narrow slits)] Tf ( wide slit) = Tf ( one wide slit) Tf ( narrow slits)

42 - infinite number of wide slits - intensity sin ( kx sin θ ) 2nπ F(sin θ) 4 δ(sin θ ) 2 n= 2 2 = X 2 ( kx sin θ ) n= kx 2

43 - significance of the diffraction pattern

44 - narrow slits 1. As the number of slits increases, the main peaks become sharper and narrower, whilest the subsidiary peaks become rapidly less intense 2. The position and separation of the main peaks is constant independent of the number of peaks. The man peaks are separated by an angular deflection given by (sin θ ) = 2π kx The distance is determined soly by λ and separation x of any two neighboring narrow slits.

45 - narrow slits - The position of the main peak in a diffraction pattern is determined solely by the lattice spacing in an obstacle. - The shape of the main peak is determined by the overall shape of the obstacle.

47 - wide slits - The effect of the motif (one wide slit) is to alter the intensity of each main peak, but the position of the main peaks are unchanged - intensity envolope structure of motif

48 - another way of looking at N wide slits - shape function is zero everywhere outside an obstacle corresponds to the macroscopic shape of the obstacle within the obstacle

49 - finite lattice f ( finite lattice) = f ( infinite lattice) f ( shape function) f ( obstacle) = f ( motif )* f ( finite lattice) f ( obstacle) = f ( motif )*[ f ( infinite lattice) f ( shape function)] - diffraction pattern F(sin θ ) = Tf ( obstacle) = T{ f ( motif )*[ f ( infinite lattice) f ( shape function)]} = Tf ( motif ) T [ f ( infinite lattice) f ( shape function)] = Tf ( motif ) [ Tf ( infinite lattice)* Tf ( shape function)]

50 - N wide slits f ( N wide slits) = f ( one wide slit)* f ( N narrow slits) f ( N wide slits) = f ( one wide slit)* [ f ( narrow slits) f ( shape function)]

51 - diffraction pattern Tf ( N wide slits) = T{ f ( one wide slit)* [ f ( narrow slits) f ( shape function)]} = Tf ( one wide slit) T [ f ( narrow slits) f ( shape function)] = Tf ( one wide slit) Tf ( narrow slits)* Tf ( shape function) - three types of structural information that concerning the lattice that concerning the motif that concerning the shape of the entire crystal

54 Summary - the position of the main peaks gives information on the lattice - the shape of each main peak gives information on the overall object shape - the set of intensities of all the main on the structure of the motif peaks gives information

PH 222-3A Spring 2010

PH 222-3A Spring 2010 PH -3A Spring 010 Interference Lecture 6-7 Chapter 35 (Halliday/Resnick/Walker, Fundamentals of Physics 8 th edition) 1 Chapter 35 Interference The concept of optical interference is critical to understanding