## Studio Session 10

### Diffraction and interference

When monochromatic light from a distant source passes through a narrow slit of width w in an opaque mask we observe a diffraction pattern on a distant screen.  The pattern is characterize by a central maximum and alternating dark and bright fringes, which appear symmetrically in both sides of the central maximum.  The central maximum is twice as wide, and much brighter than the other bright fringes.

The dark fringes in the diffraction pattern of a single slit are found at angles θ for which w sinθ = mλ, where λ is the wavelength of the light and m is an integer, m = 1, 2, 3, ... .

The intensity at the screen is proportional to the square of the electric field amplitude.

• If we block the slit completely with an opaque blocker, the electric field at a large distance is zero.
• If we remove mask with the slit, the electric field at a large distance is that of the non-diffracted beam

What if we remove the mask and only leave the blocker of width w?  Using Huygens' principle we have

Here Emask with slit is the field produced by sources at locations of the mask and Eblocker (no mask) is the field produced by source at locations of the blocker.
Therefore

For a laser beam the divergence angle θ0 is small, and for angles θ > θ0 we have

For angles θ > θ0 the average intensity, which is proportional to the square of the electric field, therefore is the same as that for the single slit.  Dark fringes in the diffraction pattern are found at angles θ for which w sinθ = mλ.

If light with wavelength λ passes through two or more slits separated by equal distances d, we will observe interference fringes inside the single slit diffraction pattern.  At certain angles we observe constructive interference.  These angles are found by applying the condition for constructive interference, which is

d sinθ = mλ, m = 0, 1, 2, ....

We will only see the bright interference fringes, if they do not appear at the angle θ of a diffraction minimum.  If d sinθ  = mλ = w sinθ, then the bright fringe of order m will be missing.

In this session you will use a helium-neon (HeNe) laser to determine the number of grooves per mm of a diffraction grating, to measure the distance d between adjacent wires and the width w of the gap between the wires of 2-dimensional wire mesh, and to measure the width of a human hair.

Open a Microsoft Word document to keep a log of your experimental procedures, results and discussions.  This log will become your lab report.  Address the points highlighted in blue.  Answer all questions.

Equipment needed:

• optical bench with optical holders
• helium neon (HeNe) laser (λ = 632.8 nm)
• transmission grating
• wire mesh
• slit and transverse ruler
• ruler, index card, tape

Experiment 1

Use the equipment provided to determine the number of grooves per mm of a diffraction grating.

• Mount the HeNe laser and the transverse ruler onto the optical bench.
• Mount the ruler 30 cm above the table.  That will allow you to move the other components into place.  Make sure the ruler is perpendicular to the track.
• Align the laser so that it shines through the hole in the center of the ruler and then tighten everything down securely.  Spend some time on this procedure, so the alignment is as good as possible.
• Move the grating into place.  Make sure the grooves are vertical.  Observe the first order interference maxima on the ruler.  Rotate the grating until they are symmetric about the middle, and move the grating along the track, so the distance from the center of the ruler to the maxima can be measured with a small uncertainty.
• Measure the distance from the grating to the ruler and then calculate the sine of the deflection angle θ.
• Determine the spacing d between the grooves of the grating from d sinθ = λ in units of mm and then determine the number of grooves/mm = 1/d.
• Make at least 2 independent measurements for two different grating-ruler distances.

Discuss and comment on your results.  Did anything surprise you?

Experiment 2

Use the equipment provided to measure the width of a human hair.

• Cut a hole into an index card and tape a hair across the hole.  Make sure the hair is stretched out and you can mount it vertically.
• Mount the index card into a holder close to the laser.
• Adjust its position until the laser shines onto the hair.
• Observe the diffraction pattern on the ruler.
• Measure the distance to the mth minimum, the one farthest away from the center that you can clearly identify.  Measure on both side of the center and average.
• Measure the distance from the hair to the ruler and then calculate the sine of the deflection angle θ.
• Determine the width of the hair w from w sinθ = mλ, where m is the order of the minimum you have chosen.
• Measure the width of a hair of at least two different students.  Choose a student who thinks his/her hair is thick and a student who thinks his/her hair is fine.

Discuss and comment on your results.  Did anything surprise you?

Experiment 3

Use the equipment provided to measure the distance d between adjacent wires and the width w of the gap between the wires of 2-dimensional wire mesh.

• Replace the ruler with the screen.  Replace the index card with the wire mesh.
• Observe the 2-dimensional diffraction/interference pattern on the screen.
• Measure the distance between the center of the pattern and the mth interference maximum and the mesh-screen distance to find sinθ.
• Calculate the distance d between adjacent wires from d sinθ = mλ.
• Measure the distance from the center of the pattern and the first diffraction minimum (where an interference maximum is missing) and find sinθ'.
• Calculate the width w of the gap between the wires from w sinθ' = λ.