# 3.2: In-lab Exercises

- Page ID
- 47360

**Diffraction **

### Introduction

The objectives of these laboratory exercises are to explore and investigate the phenomenon of diffraction through a series of simple and then more advanced experiments. The first set of diffraction experiments will be carried out with He-Ne laser light with a wavelength of 632.8 nm. These include:

- Fraunhofer diffraction from single slits, double slits, multiple slits, gratings, circular, rectangular, triangular and hexagonal apertures, and rectangular grids.
- Fresnel diffraction from circular apertures and obstacles.
- Investigation of the focusing and imaging properties of a Fresnel zone plate.

## A. Fraunhofer Diffraction

The setup we will use to demonstrate and study Fraunhofer diffraction is shown below. Note that it is important to illuminate the objects with a planar wavefront to observe Fraunhofer diffraction. In many cases, a reasonably accurate Fraunhofer pattern can be obtained with the raw laser beam (i.e., without the beam expander) and with the screen sufficiently far away from the diffracting object (see Figure 1).

### 3.1 Single Slit

- Set up and observe the Fraunhofer diffraction pattern owing to the variable single slit. Observe how the pattern changes as the slit-width is varied. Qualitatively sketch the typical pattern. Describe your observations. Explain why the pattern changes the way it does.
- What is the theoretical expression for the angular location of the bright fringes?
- Have the TA set the slit width to a specific value. Record the corresponding micrometer reading in your notebook. Measure the angular distribution of the bright fringes. Record the measurements in your notebook and determine the slit width from your measurements.

### 3.2 Double Slit

Set up and observe the Fraunhofer diffraction pattern owing to the given double slits. These are contained on the upper level of the plastic slide labeled **3.2/3.3**.

- Sketch the pattern, and measure the angular distribution of the bright fringes for the double slit with the widest separation. From your measurement calculate the slit width and the slit separation
- In addition, view the other double slits of varying separation on the top level of the same plastic slide. Qualitatively sketch your observations. Rank the double slits in order of increasing separation.
- Explain why and how varying the slit separation affects the diffraction pattern.
*Optional*: Using the CCD imager, as shown in Figure 1, capture an image of the two-slit diffraction pattern. It is important that you not move the imager from its current location. Using the captured image, employ Matlab to calculate the relative spacing of the slits. Provide a reconstruction of the slits, then compare their relative dimensions numerically.

### 3.3 Multiple Slits

Set up and observe the Fraunhofer diffraction pattern due to the N-slit gratings. These are contained on the bottom row of the plastic-mounted slide of **3.2**. The number of slits, N, in each grating is indicated beneath each set of the gratings.

- Sketch the pattern for various values of N and explain
**qualitatively**what happens as N increases from 2. What is the theoretically expected pattern as N → ∞. - Examine the Fraunhofer patterns from the three amplitude transmission diffraction gratings (labeled as 2,400, 7,500 and 15,000 lines/inch) and describe the patterns you see. Do the patterns from the gratings behave as predicted by the theory?
- For the grating labeled 15,000 lines per inch, measure the diffraction angles of the spots, and use this information to compute the actual periodicity of the grating. The grating material has aged over the past 10 years because of environmental effects. Did it shrink or expand?
- Note that only 5 diffracted spots are visible with the 15,000 lpi grating. Why do we not see an infinite number of spots? What determines the maximum number of diffraction spots? Derive a mathematical expression for the maximum number of spots.

### 3.4 Rectangular Grid

Set up and observe the Fraunhofer diffraction due to the rectangular grid. Make a qualitative sketch of the pattern and explain how it differs from that of the rectangular aperture (below).

### 3.5 Rectangular Aperture

Set up and observe the Fraunhofer diffraction pattern due to the rectangular apertures. There are five different rectangles on the plate. Refer to the diagram below to find their location on the plate. Illuminating the plate from behind with an incoherent light source (e.g., a flashlight) should prove helpful in locating and aligning the apertures in the plate.

- Choose any rectangular aperture and sketch its diffraction pattern on a separate sheet of paper (to be pasted into your lab notebook). Make angular fringe location measurements and label your sketch with the measurements.
- Use your measurements to compute the width (horizontal) and the height (vertical) of one of the apertures. If needed, have the TA or LA assist you in finding the diffraction objects on each plate provided. Size shown below is exaggerated.

### 3.6 Triangular and Hexagonal Apertures

Set up and observe the Fraunhofer diffraction pattern owing to the triangular and hexagonal apertures. These are located on the second row of the plate as illustrated in Figure 2.

- Roughly sketch the patterns next to each other in your lab notebook.
- What are the qualitative differences between the patterns?
- Using the Matlab simulation of the hexagon and triangle diffraction intensity patterns (in Problem set 3), describe the difference between the two simulated diffraction patterns. What are the observable differences between the computer simulation, and the actual observed pattern?

### 3.7 Circular Aperture

Set up and observe the Fraunhofer diffraction pattern due to the circular aperture.

- Qualitatively sketch the pattern, but quantitatively measure the diameter of the Airy disc, and label the sketch with your measurement. Make sure you measure the Airy disc correctly! Check with the T.A. or L.A. to be sure. In your lab writeup, be sure to include the sketched pattern, and indicate precisely the boundaries of the Airy disc.
- Use the results of (a) to calculate the size of the aperture.

### 3.8 2-D Grating Mirror

The thin substrate consists of a 2-D array of circular holes in a mirror substrate. Operate the system in reflection with an angle of incidence of 45 °. Set up and observe the Fraunhofer pattern due to this grating.

- Sketch the diffraction pattern.
- What is the packing symmetry of the holes in the mirror?
- Measure the dimensions of the pattern, and label the sketch with the dimensions.
- Compute the pitch of the grating.
- Is this structure a phase grating, or an amplitude grating? Why?

## B. Fresnel Diffraction

The setup for Fresnel diffraction is shown below. The wavefront incident on the apertures must be very nonplanar and the screen-to-object distance small (5 to 40 cm) to obtain dramatic results.

### 3.9 Circular Opening

Set up and observe the Fresnel diffraction pattern due to the small circular opening. Vary the distance between the source and the aperture. The circular aperture must be less than 1 cm away from the objective lens to see the Fresnel diffraction pattern.

- Sketch a few of the different patterns you get.
- Compare the Fresnel diffraction patterns with the Fraunhofer pattern from the circular aperture.
- Explain how and why the Fresnel patterns are different from the Fraunhofer patterns.

### 3.10 Circular Obstacle

Set up and observe the Fresnel diffraction patterns from the small spherical obstacle.

- Qualitatively sketch the pattern.
- Explain why there is always a bright spot in the center of the dark shadow.

### 3.11 Fresnel Zone Plate: Measurement of focal lengths

- Place the Fresnel zone plate in a well-collimated beam of He-Ne laser light and measure the positions of the farthest three or four focused spots you can see.
- From the data in (a) what are the primary, secondary and tertiary focal lengths (F1, F2, and F3, respectively) of the zone plate? What are your measured ratios: F2/F1, F3/F1 and F3/F2? Are these in agreement with the theoretical values? If not, why?
- From your data in (b) calculate the radius of the first zone of the zone plate. Measure the diameter of the first zone to the nearest half-millimeter with a ruler. Explain any significant difference between your measured and calculated values.