Diffraction at Multiple Slits and Diffraction Gratings. The role of Interference

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Diffraction at Multiple Slits and Diffraction Gratings. The role of Interference. For multiple apertures, the effects of interference and diffraction cannot be readily separated . Diffraction at Multiple Slits and Diffraction Gratings. The role of Interference.

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### Diffraction at Multiple Slits and Diffraction Gratings. The role of Interference

For multiple apertures, the effects of interference and diffraction cannot be readily separated

### Diffraction at Multiple Slits and Diffraction Gratings. The role of Interference

In Young’s Double Slit experiment, diffraction must have been occurring as well as interference

Narrow slits minimize any visible role of diffraction in the screen intensity profile

### Diffraction at Multiple Slits and Diffraction Gratings. The role of Interference

The same mathematical constructions as for diffraction are used to define path length differences leading to interference minima and maxima

Waves traveling

to point P

on screen

A

E

q

q

d

D

C

B

Double Slit Interference: Path Difference

Double

Slit

to central

maximum

Fig 32 Page 43

d

_

BC =

l

Waves traveling to point Q on screen corresponding to . Waves in phase  constructive interference

DOUBLE

SLIT

INTERFERENCE

A

to central

maximum

q

q

B

C

When BC =  , waves traveling through the two slits are in phase because their path difference is equal to

Fig 33 Page 45

Slit

to P

A

0

d

D

B

_

C

BC =

l

Diffraction - First Order Minimum

SINGLE

SLIT

DIFFRACTION

to P

Each wave through upper half cancels corresponding wave through lower half of slit destructive interference

Fig 34b Page 46

I

interference

plus diffraction

diffraction only

q

I

q

Intensity Profile - Interference vs. Diffraction

Double slit

interference

narrow slits

Double slit

interference

wider slits

Fig 35 Page 43

Interference Maxima: Young’s Double Slit

Source Slit

Double Slit

(m = 0 · · · · · · · 6)

Practice Problem 1

Yellow light ( = 595 nm) is used to produce interferencefringes with Young’s double-slit set-up. Find angular location of the third order interference maximum for a slit separation of 0.10 mm:

• 0.349O
• 1.023O
• 1.875O
• 3.550O

Source Slit

Double Slit

Practice Problem 1

Yellow light ( = 595 nm) is used to produce interferencefringes with Young’s double-slit set-up. Find angular location of the third order interference maximum for a slit separation of 0.10 mm:

• 0.349O
• 1.023O
• 1.875O
• 3.550O

Practice Problem 2

For the same interference set-up (slit separation 0.10 mm; yellow light,  = 595 nm), find linear position on the screen of the third order interference maximum if the screen is 7.5 meters from the double slit

• 4.95 cm
• 7.03 cm
• 11.95 cm
• 13.39 cm

 = 7.5 meters

Source Slit

Source Slit

x

 = 7.5 meters

Practice Problem 2

For the same interference set-up (slit separation 0.10 mm; yellow light,  = 595 nm), find linear position on the screen of the third order interference maximum if the screen is 7.5 meters from the double slit

• 4.95 cm
• 7.03 cm
• 11.95 cm
• 13.39 cm

Interference Pattern for Relatively Narrow Slits

Source Slit

Double Slit

Source Slit

Double Slit

Widen the slits in the Double Slit

Source Slit

Double Slit

Farther widen the slits in the double slit

Source Slit

Double Slit

Slits wider again: almost an entire diffraction max

Source Slit

Double Slit

Farther widen the slits in the double slit

As the slits become wider, the first order diffraction minimum moves closer to the center of the screen profile

### Double Slit Interference: Diffraction versus Slit Width

For narrow slits, the angle of the first order diffraction minimum is >> than that of the first order interference maximum

### Double Slit Interference: Diffraction versus Slit Width

As the slits become wider, the angle of diffraction decreases, and diffraction has a more visible effect on the screen intensity profile

A

B

Practice Problem 3

The two figures below show screen intensity profiles for a double slit. Slit separation is the same in both cases, but slit width differs. In which figure are the slits wider?

1st order Diffraction minimum

Interference maxima

Practice Problem 3

The two figures below show screen intensity profiles for a double slit. Slit separation is the same in both cases, but slit width differs. In which figure are the slits wider?

1st order Diffraction minimum

A

B

Practice Problem 3

The two figures below show screen intensity profiles for a double slit. Slit separation is the same in both cases, but slit width differs. In which figure are the slits wider?

A

B

Practice Problem 4

All four figures show screen intensity profiles for a double slit. In which figure is the slit separation greatest?

Practice Problem 4

Greatest slit separation?

Interference

Greatest separation  smallest angle (maxima closest together)

Practice Problem 4

Greatest slit separation?

Interference

Greatest separation  smallest angle (maxima closest together)

Smallest slit separation?

Practice Problem 5

Practice Problem 6

All four figures show screen intensity profiles for a double slit.

In which figure is the slit width greatest?

Practice Problem 6

Greatest slit width?

Diffraction

Greatest width  smallest angle of diffraction (1st order minimum)

 Same?

Practice Problem 6

Greatest slit width?

Diffraction

Greatest slit separation

Smallest slit separation

Greatest slit separation

Widest slit?

Widest slit

Slit Width 0.5 m

Slit Sepn 10 m

Slit Width0.25 m

Slit Sepn2.5 m

Slit Width2.5 m

Slit Sepn10 m

Slit Width2.5 m

Slit Sepn 5 m

### Double Slit Interference: Diffraction versus Slit Width

Take a double slit and systematically vary slit width, while keeping slit separation constant

diffraction only

interference plus diffraction

P

0

l

slit separation = 20

slit width =

l

30

15

15

30

q

Fig S6a Page S6

Screen Intensity Profile

Slit Width  e.g. 5  104 mm

Slit Separation 20  102 mm

diffraction only

interference plus diffraction

P

0

l

slit separation = 20

slit width = 2

l

30

15

15

30

q

Fig S6b Page S6

Screen Intensity Profile

Slit Width 2 103 mm

Slit Separation 20  102 mm

diffraction only

interference plus diffraction

P

0

l

slit separation = 20

slit width = 5

l

30

15

15

30

q

Fig 7a Page S7

Screen Intensity Profile

Slit Width 5 2.5  103 mm

Slit Separation 20  102 mm

diffraction only

interference plus diffraction

P

0

l

slit separation = 20

slit width = 10

l

30

15

15

30

q

Fig S7b Page S7

Screen Intensity Profile

Slit Width 10 5  103 mm

Slit Separation 20  102 mm

Slit Width  e.g. 5  104 mm

Slit Separation 20  102 mm

Slit Width 2 103 mm

Slit Separation 20  102 mm

Slit Width 5 2.5  103 mm

Slit Separation 20  102 mm

Slit Width 10 5  103 mm

Slit Separation 20  102 mm

30

15

15

30

q

P

0

Slit Width 5 2.5  103 mm

Slit Separation 20  102 mm

Add slits at same separation as first two slits

Intensity Profile: 2, 3 and 4 Slits

I

As # slits increases: maxima becoming narrower  minima becoming broader

q

Fig S8 Page S9

m = 2

m = 1

_

_

Many Slits: “Diffraction” Grating

Showing all wavelengths

I

Green

Yellow

Blue

Orange

Violet

Red

m = 1

m = 2

m = 0

q

Fig S9 Page S10