1. Diffraction, Gratings, Resolving Power
Physics 102: Lecture 21
Diffraction, Gratings, Resolving Power
James Scholar Honors Credit article proposals due this Thursday, April 7!

2. Recall Interference (at least 2 coherent waves)
Constructive (full wavelength difference)
Destructive (half wavelength difference)
Light (1 source, but different paths)
Young’s double slit
Thin films
Multiple slit
X-ray diffraction from crystal
Diffraction/single slit
Last lecture
Today’s lecture

3. ACT: Double slit review
Which condition gives destructive interference? y θ θ d
dsin(q) L
1) dsin(q) = ml 2) dsin(q) = (m+1/2)l
m = 0, 1, 2

4. Multiple Slits: (Diffraction Grating – N slits with spacing d)
Assume screen is very far away (L>>d): L 1 2 3 4 θ d θ d d
Path length difference 1-2
= d sinq =l
Path length difference 1-3
= 2d sinq
=2l
Path length difference 1-4
= 3d sinq
=3l
Constructive interference for all paths when
dsin(q) = ml
m = 0, 1, 2

5. Multiple Slits: (Diffraction Grating – N slits with spacing d)
Assume screen is very far away (L>>d): L 1 2 3 4 θ d θ d d d
Constructive: dsin(q) = ml
m = 0, 1, 2
Same condition as Young’s double slit!
Holds for arbitrary N

6. Preflight 21.1 L 1 2 3 θ d θ d
All 3 rays are interfering constructively at the point shown. If the intensity from ray 1 is I0 , what is the combined intensity of all 3 rays? 1) I ) 3 I ) 9 I0
7% 63% 31%
Each slit contributes amplitude Eo at screen. Etot = 3 Eo. But I a E2. Itot = (3E0)2 = 9 E02 = 9 I0 = Imax

7. ACT/Preflight 21.2 L 1 2 3
these add to zero
this one is still there! θ d θ d
When rays 1 and 2 are interfering destructively, is the intensity from the three rays a minimum? 1) Yes ) No
46% %
Rays 1 and 2 completely cancel, but ray 3 is still there. Expect intensity Itot = I0 = 1/9 Imax

8. Three slit interference- - -

9. Multiple Slit Interference (Diffraction Grating)
Peak location depends on wavelength!
For many slits, maxima are still at
Region between maxima gets suppressed more and more as no. of slits increases – bright fringes become narrower and brighter.
2 slits (N=2)
intensity l 2l
d sin(θ)
10 slits (N=10)
intensity l 2l
d sin(θ)
Here do demo using diffraction grating for red and green laser

10. X-Ray Diffraction: A technique to study crystal structureθ
d sin(θ) d
Crystal solid such as sodium
Constructive interference: 2d sin(θ) = mλ
Mention that this is used by solid state physicists to determine lattice spacing in crystals
d ≈ 0.5nm in NaCl
1st maximum will be at 100
For =0.017nm
X-ray
Measure , determine d

11. Single slit interference?
Monochromatic light travels through a screen with opening
Bright spot
Shadow
This is not what is actually seen!

12. Diffraction/Huygens’ principle
Huygens: Every point on a wave front acts as a source of tiny wavelets that move forward. • • • • •
Light waves originating at different points within opening travel different distances to wall, and can interfere!
We will see maxima and minima on the wall!

13. Central maximum
1st minima

14. Single Slit Diffraction1 1 2 2 θ θ θ w
When rays 1 and 1
interfere destructively.
Rays 2 and 2 also start w/2 apart and have the same path length difference.
Under this condition, every ray originating in top half of slit interferes destructively with the corresponding ray originating in bottom half.
1st minimum at sin q = l/w

15. Single Slit Diffraction2 2 1 1 θ w
When rays 1 and 1
will interfere destructively.
Rays 2 and 2 also start w/4 apart and have the same path length difference.
Under this condition, every ray originating in top quarter of slit interferes destructively with the corresponding ray originating in second quarter.
2nd minimum at sin q = 2l/w

16. Single Slit Diffraction Summary
Condition for halves of slit to destructively interfere
Condition for quarters of slit to destructively interfere
Condition for sixths of slit to destructively interfere
(m = 1, 2, 3, …)
All together…
THIS FORMULA LOCATES MINIMA!!
Preflight 21.3
Narrower slit => broader pattern
Note: interference only occurs when w > λ

17. ACTS/Preflights 21.4, 21.5
A laser is shined onto a screen through a very small hole. If you make the hole even smaller, the spot on the screen will get:
(1) Larger (2) Smaller
Which drawing correctly depicts the pattern of light on the screen?
(1)
(2)
(3)
(4)
56%, 56%

18. Diffraction from Circular Aperture
Central maximum
1st diffraction minimum q
Diameter D
light
Maxima and minima will be a series of bright and dark rings on screen
First diffraction minimum is at

19. Intensity from Circular Aperture
First diffraction minimum
1.22λ/D

20. These objects are just resolved
Demo: Resolving Power
Not resolved
Just resolved
These objects are just resolved
Two objects are just resolved when the maximum of one diffraction pattern is at the minimum of the other.

21. Resolving Power To see two objects distinctly, need qobjects > qmin
qobjects is angle between objects and aperture:
qmin
qobjects ≈ tan-1(d/y)
qmin is minimum angular separation that aperture can resolve:
demo 752; 2 point sources and different size slits D
sin qmin ≈ qmin = 1.22 l/D y d
Improve resolution by increasing qobjects or decreasing qmin

22. ACT: Resolving Power
sin qmin ≈ qmin = 1.22 l/D
How does the maximum resolving power of your eye change when the brightness of the room is decreased?
1) Increases 2) Constant 3) Decreases
When the light is low, your pupil dilates (D can increase by factor of 10!) But actual limitation is due to density of rods and cones, so you don’t notice an effect!

23. Recap Interference: Coherent waves Multiple Slits Single Slit:
Full wavelength difference = Constructive
½ wavelength difference = Destructive
Multiple Slits
Constructive d sin(q) = m l (m=1,2,3…)
Destructive d sin(q) = (m + 1/2) l 2 slit only
More slits = brighter max, darker mins
Single Slit:
Destructive: w sin(q) = m l (m=1,2,3…)
Resolution: Max from 1 at Min from 2
opposite!

24. See you Monday!