Diffraction at a single slit

11.3.1 Sketch the variation with angle of diffraction of the relative intensity of light diffracted at a single slit.

11.3.2 Derive the formula = /b for the position of the first minimum of the diffraction pattern produced at a single slit.

11.3.3 Solve problems involving single-slit diffraction. Be able to apply the formula = /b.

Topic 11: Wave phenomena11.3 Diffraction

Diffraction at a single slit

Sketch the variation with angle of diffraction of the relative intensity of light diffracted at a single slit.

If a wave meets a hole in a wall that is of comparable size to its wavelength, the wave will be bent through a process called diffraction.

If the aperture (hole, opening, etc.) is much larger than the wavelength, diffraction will be minimal to nonexistent.

Topic 4: Oscillations and waves4.5 Wave properties

INCIDENT WAVE

DIFFRACTED WAVE

REFLECTED WAVE

Diffraction at a single slit

Sketch the variation with angle of diffraction of the relative intensity of light diffracted at a single slit.

Huygen’s principle states “Every point on a wavefront emits a spherical wavelet of the same velocity and wavelength as the original wave.”

Note that because of Huygen’s principle waves can turn corners.

Topic 4: Oscillations and waves4.5 Wave properties

Diffraction at a single slit

Sketch the variation with angle of diffraction of the relative intensity of light diffracted at a single slit.

The reason waves can turn corners is that the incoming wave transmits a disturbance by causing the medium to vibrate.

And wherever the medium vibrates it becomes the center of a new wave front as illustrated.

Note that the smaller the aperture b the more pronounced the effect.

Topic 4: Oscillations and waves4.5 Wave properties

b = 12b = 6

b = 2

b b b

Diffraction at a single slit

Sketch the variation with angle of diffraction of the relative intensity of light diffracted at a single slit.

Huygen’s wavelets not only allow the wave to turn corners, they also interfere with each other.

Topic 4: Oscillations and waves4.5 Wave properties

Constructive interference

RELATIVE INTENSITY

Destructive interference

EXAMPLE: If light is diffracted by a circular hole, a planar cross-section of the interference looks like the picture below. What will the light look like head-on?

Diffraction at a single slit

Sketch the variation with angle of diffraction of the relative intensity of light diffracted at a single slit.

Topic 4: Oscillations and waves4.5 Wave properties

Diffraction at a single slit

Derive the formula = /b for the position of the first minimum of the diffraction pattern produced at a single slit.

Consider a single slit of width b.From Huygen’s principle we know that every point within the slit acts as a wavelet.

At the central maximum we see that the distance traveled by all the wavelets is about equal, and thus has constructive interference.

Consider points x at one edge and y at the center of the slit:

At the 1st minimum, the difference in distance (dashed) must be /2. Why?

Topic 11: Wave phenomena11.3 Diffraction

b

1 st min x

y

Condition for destructive interference.

Diffraction at a single slit

Derive the formula = /b for the position of the first minimum of the diffraction pattern produced at a single slit.

We will choose the midpoint of the slit (y) as our reference.

And we will call the angle between the reference and the first minimum .We construct a right triangle as follows:

Why does the side opposite equal /2?

Topic 11: Wave phenomena11.3 Diffraction

b

1 st min x

yb2

y

x

2

Condition for destructive interference.

Diffraction at a single slit

Derive the formula = /b for the position of the first minimum of the diffraction pattern produced at a single slit.

From the right triangle we see that sin = (/2)/(b/2) sin = /b.Perhaps you recall that if is very small (and in radians) then

sin ( in rad).Finally…

Topic 11: Wave phenomena11.3 Diffraction

b2

y

x

2

location of first minimum in single slit diffraction

= /b ( in radians)

Topic 11: Wave phenomena11.3 Diffraction

INCIDENT WAVE

DIF

FR

AC

TE

D

WA

VE

DIF

FR

AC

TE

D

WA

VE

REFLECTED WAVE

Diffraction at a single slit

Solve problems involving single-slit diffraction. Be able to apply the formula = /b.PRACTICE: Sketch in the diffraction patterns in the double-slit breakwater with 5-m waves. Then map out MAX (10 m), MIN (-10 m) and 0 m points.

10

0

-10

Topic 11: Wave phenomena11.3 Diffraction

Diffraction at a single slit

Solve problems involving single-slit diffraction. Be able to apply the formula = /b.

Diffraction allows waves to turn corners.

All waves diffract-not just sound.

Topic 11: Wave phenomena11.3 Diffraction

Diffraction at a single slit

Solve problems involving single-slit diffraction. Be able to apply the formula = /b.

Huygen says the wavelets will be spherical.

Topic 11: Wave phenomena11.3 Diffraction

Diffraction at a single slit

Solve problems involving single-slit diffraction. Be able to apply the formula = /b.

Topic 11: Wave phenomena11.3 Diffraction

Diffraction at a single slit

Solve problems involving single-slit diffraction. Be able to apply the formula = /b.

Caused by path length difference (PLD) along b.

1st min PLD = /2. 2nd min PLD = 3/2.

1st 2nd

Central max PLD = 0. 2nd max PLD = .

Cent

2nd

Topic 11: Wave phenomena11.3 Diffraction

Diffraction at a single slit

Solve problems involving single-slit diffraction. Be able to apply the formula = /b.

= /b

d

tan = d/Dtan for small .

= d/D = /b. d = D/b.