diffraction around an edge and through an aperture huygens’ principle is consistent with...

$: Diffraction around an edge and through an aperture Huygens’ principle is consistent with diffraction: The larger the wavelength with respect to the aperture$
Diffraction around an edge and Diffraction around an edge and through an aperturethrough an aperture

Huygens’ principle is consistent with diffraction:

The larger the wavelength with respect to the aperture size, the more significant the diffraction

If light is a wave, interference effects will be seen, where one part of wavefront can interact with another part.

One way to study this is to do a double-slit experiment:

Young’s Double Slit Young’s Double Slit ExperimentExperiment

Two-Source Interference Two-Source Interference PatternPattern

Nodal points – blue dots

Anti-nodal points – red dots

Young’s Double Slit Young’s Double Slit ExperimentExperiment

If light is a wave, there should be an interference pattern

1) there is no difference

2) half a wavelength

3) one wavelength

4) three wavelengths

5) more than three wavelengths

In a double-slit experiment, what In a double-slit experiment, what

path differencepath difference have the waves have the waves

from each slit traveled to give a from each slit traveled to give a

minimum at the indicated minimum at the indicated

position?position?

Inte

ns

ity

Path DifferencePath Difference

Inte

ns

ity

7/

2

/2

3/

2

5/

2

For Destructive Interference

= 1/2 , 3/2 , 5/2 , 7/2 , …

= (m + 1/2)

23

1) there is no difference

2) half a wavelength

3) one wavelength

4) three wavelengths

5) more than three wavelengths

In a double-slit experiment, what In a double-slit experiment, what

path differencepath difference have the waves have the waves

from each slit traveled to give a from each slit traveled to give a

minimum at the indicated minimum at the indicated

position?position?

Path DifferencePath Difference

Diffraction by a Single Slit or Diffraction by a Single Slit or DiskDisk

Note: There are bright and dark fringes beyond the shadow. These resemble interference fringes of a double slit.

bright

dark

D

Path difference = m/2

D

Wave front going through the slit is made up of many wavelets.

Wavelets from the center and edge of the slit

Dark Band

(Interference of wavelets)


Contrary to expectation, an interference pattern is observed because the wave front approaching slit, D, is itself a source of wavelets. It is these individual wavelets that will interfere destructively at the part of the screen where the dark band is observed.

X

Y

If D is small we can use…..

similar triangles, for small angles

sin D = tan D = Y / X

so….. Y / X = m / D

or…. Y = m X / D

m = 1,2,3…….

D

Now from the diagram the angle to the first dark band is given by:

sinD = (/2) / (D/2) = / D

Therefore for any dark band

sin D = m / D

m = 1,2,3…….


If we look at the intensity of the bright bands we find that they are not the same. The brightest band is in the center and the bands seem to get dimmer as the angle increases. The diagram shows how the intensity changes with angle. You can see that as sin approaches / D the intensity falls to zero as expected from our theory. Also note that the second band is much less intense than the central band and is half as wide.Practice Problem

Light of wavelength 750 nm passes through a slit 1.0 x 10-3 mm wide. How wide is the central maximum a) in degrees and b) in centimeters on a screen 20 cm away?

= 750 x 10-9 m D = 1.0 x 10-6 m X = 0.2 m away m = 1

Width of central maximum is 2 or 2Y

a) Width = 2 = 2 sin-1 (m / D) = 2 sin-1 (1) (750 x 10-9m) / (1.0 x 10-6 m ) = 97.20

b) Width = 2Y = 2 X tan D = 2 (0.2) tan 48.590 = 0.45 m

Check Your UnderstandingCheck Your Understanding

The diffraction pattern below arises The diffraction pattern below arises

from a single slit. If we would like from a single slit. If we would like

to sharpen the pattern, i.e., make to sharpen the pattern, i.e., make

the central bright spot narrower, the central bright spot narrower,

what should we do to the slit width?what should we do to the slit width?

1) narrow the slit

2) widen the slit

3) enlarge the screen

4) close off the slit

The angle at which one finds the first minimum is:

The central bright spot can be narrowed by having a smaller angle. This in turn is accomplished by widening the slit.


dd

sin = d

The diffraction pattern below arises The diffraction pattern below arises

from a single slit. If we would like from a single slit. If we would like

to sharpen the pattern, i.e., make to sharpen the pattern, i.e., make

the central bright spot narrower, the central bright spot narrower,

what should we do to the slit width?what should we do to the slit width?

1) narrow the slit

2) widen the slit

3) enlarge the screen

4) close off the slit

Check Your UnderstandingCheck Your UnderstandingBlue lightBlue light of wavelength of wavelength passes passes

through a single slit of through a single slit of width width dd and and

forms a diffraction pattern on a screen. forms a diffraction pattern on a screen.

If the If the blue lightblue light is replaced by is replaced by red lightred light

of wavelength of wavelength 22, the original diffraction , the original diffraction

pattern can be reproduced if the slit pattern can be reproduced if the slit

width is changed to:width is changed to:

1) d/4

2) d/2

3) no change needed

4) 2 d

5) 4 d


dd

d sin = m (minima)

If 2 2 then we must have dd

2d2d for sin for sin to remain unchanged to remain unchanged

(and thus give the same diffraction

pattern).

Blue lightBlue light of wavelength of wavelength passes passes

through a single slit of through a single slit of width width dd and and

forms a diffraction pattern on a screen. forms a diffraction pattern on a screen.

If the If the blue lightblue light is replaced by is replaced by red lightred light

of wavelength of wavelength 22, the original diffraction , the original diffraction

pattern can be reproduced if the slit pattern can be reproduced if the slit

width is changed to:width is changed to:

1) d/4

2) d/2

3) no change needed

4) 2 d

5) 4 d

When violet light of wavelength 415 nm falls on a single slit, it creates a central diffraction peak that is 9.20 cm wide on a screen that is 2.55 m away. How wide is the slit?

= 415 x 10-9 m Y = 4.6 x 10-2 m X = 2.55 m away m = 1

D = ?

Because X is much bigger than Y, D is small, so we can use…..

Y = m X / D m = 1,2,3…….

D = m X / Y

= (1) (415 x 10-9 m) (2.55 m) / (4.66 x 10-2 m)

D = 2.30 x 10-5 m = 2.30 x 10-2 mm

Rayleigh Criterion for Resolution of Two Rayleigh Criterion for Resolution of Two Diffracted ImagesDiffracted Images

Two images are said to be just resolved when the central maximum of one image falls on the first minimum of the diffraction pattern of the other image.

From single slit diffraction

sinD = / D

If << d, sinD ≈ D

min = / D

Where min is in radians

The apertures of microscopes and telescopes are circular. Analysis shows…

min = 1.22 / D


An optical telescope has a 21 cm An optical telescope has a 21 cm

mirror diameter. To give the same mirror diameter. To give the same

angular resolution as an optical angular resolution as an optical

telescope the effective diameter of a telescope the effective diameter of a

radio telescope would have to be….radio telescope would have to be….

1) bigger

2) smaller

3) the same

4) doesn’t matter

The minimum angle of resolution is given by:

Radio waves have a much longer wavelength, so to give the same minimum angle of resolution the dish diameter has to be much larger


min= 1.22 d

An optical telescope has a 21 cm An optical telescope has a 21 cm

mirror diameter. To give the same mirror diameter. To give the same

angular resolution as an optical angular resolution as an optical

telescope the effective diameter of a telescope the effective diameter of a

radio telescope would have to be….radio telescope would have to be….

1) bigger

2) smaller

3) the same

4) doesn’t matter


A scientist wants to observe finer A scientist wants to observe finer

detail on a specimen under a detail on a specimen under a

microscope. They could….microscope. They could….

1) Make the objective bigger

2) Use smaller wavelengths

3) Use oil between the specimen and the objective

4) All of the above

The minimum angle of resolution is given by:

To make min smaller you would have

to make the objective lens bigger or use a smaller wavelength. Oil has a higher n than air so n = air / n.


min= 1.22 d

A scientist wants to observe finer A scientist wants to observe finer

detail on a specimen under a detail on a specimen under a

microscope. They could….microscope. They could….

1) Make the objective bigger

2) Use smaller wavelengths

3) Use oil between the specimen and the objective

4) All of the above

Spherical and chromatic aberration however limit the size of the objective, d. Also UV is absorbed by glass.

Electrons can be used because of their wave properties (E = h f)

Two of Jupiter’s largest moons (it has 16 or more) are separated by a distance of 3.0 x 106 km at a time when the planet is 3.3 x 108 km

from Earth. a) Would a person be able to resolve both moons with the unaided eye, assuming night-time pupil diameter of 7.5 mm? b) What

minimum diameter of mirror would be needed in a radio telescope?

Use: light = 550 nm, radio = 21 cm and s = r

= 550 x 10-9 m s = 3.0 x 109 m r = 3.3 x 1011 m

d = 7.5 x 10-3 m dmin = ?

= s / r = (3.0 x 109 m / (3.3 x 1011 m) = 0.009 rad

a) min = 1.22 / d = 1.22 (550 x 10-9 m / (7.5 x 10-3 m) = 0.000089 rad

Yes > min , so a person would be able to resolve both moons

b) dmin = 1.22 / = 1.22 (21 x 10-2 m) / 0.009 = 28 m

Now you can see why big dishes are needed for radio telescopes

A camera on a spy satellite orbiting at 200 km has a diameter of 35 cm. What is the smallest distance this camera can resolve on the

surface of the earth?

Use: light = 550 nm, and s = r

= 550 x 10-9 m s = ? r = 200 x 103 m d = 35 x 10-2 m

s = r min = (200 x 103 m)(1.92 x 10-6 ) = 0.38 m (38 cm)

min = 1.22 / d = 1.22 (550 x 10-9 m / (35 x 10-2 m) = 1.92 x 10-6 rad

This size of lens would not be good enough to read license plate numbers but good enough to resolve individual people.

diffraction around an edge and through an aperture huygens’ principle is consistent with...

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