Physics 4BL Lab 6

Laboratory 6

Diffraction and Interference

Max Woo
12/11/14
Lab 11, Michael Ip
Partners: Zoe Zhu

Introduction

In this laboratory, we investigate the wave characteristics of light by observing the diffraction of a laser beam. In the first part, we analyze the diffraction pattern through a double slit to determine the width and spacing of the slits. In the second part, we pass the laser through a diffraction grating and measure the angles of the outgoing light to verify the spacing of the diffraction grating. In the third part, we pass white light through the diffraction grating and use the angles of each color of light to measure the wavelength of each. In the last part, we measure the width of a human hair by measuring the diffraction pattern.

Experimental Data

Part 1: Double-slit Diffraction/Interference Patterns

First, we set up the laser, photometer, and linear translator along the bench according to the diagram below (note that the diffracting element will be added later):

Fig. 1: Diagram of the setup

We connected the myDAQ so that channel 0 was connected to the potentiometer in the linear translator, and channel 1 was attached to photometer. We measured the potentiometer-to-distance conversion rate by moving the linear translator from the 1 cm mark to the 4 cm mark, using 0.5 cm increments and measuring the potentiometer voltage at each stop.

Next, to optimally align the laser to give the best photometer readings, we first moved the linear translator to the 2.5 cm mark. We then adjusted the photometer to the lower sensitivity, and adjusted the laser until it shined directly on the fiber optic. We changed the photometer sensitivity until the reading was between 4 and 8, and then made small adjustments to the laser alignment to maximize the photometer reading. Then we placed the aperture slide in front of the fiber optic, and positioned it to maximize the photometer reading. Finally, we readjusted the sensitivity of the photometer so that the new maximum reading is between 4 and 8.

Next, we observed the graph of the position (potentiometer voltage) versus the intensity (photometer voltage) as the linear translator moved from the 1cm mark to the 4cm mark. It looked like this:

Graph 1: Position-intensity graph with no refraction

We then added a double slit element a measured distance $D$ from the aperture slide, starting with the double slit with width $b = 0.04 mm$ and spacing $d = 0.125 mm$. We recorded the position voltage and photometer voltage as the linear translator moved from the 1cm mark to the 4cm mark, and then repeated the same with 3 more double slits, this time with unknown widths and spacings. The graphs are shown below

Graph 2: Graph for known-width double slit

Graph 3: Graph for unknown double slit 1

Graph 4: Graph for unknown double slit 2

Graph 5: Graph for unknown double slit 3

Part 2: Diffraction Grating

For this part, we first positioned a beam expander horizontally in the path of the laser, between the laser and the diffraction element, turning the laser into a sheet beam perpendicular to the table. In addition, by adding two vertical metal sheets between the laser and beam expander to create an adjustable vertical iris, we reduced the width of the laser beam before it reaches the beam expander, and increased the resolution of the sheet beam.

We replaced the double slit with a diffraction grating, positioned so that the lines are vertical. The diffraction grating split the sheet beam into multiple beams, whose angles we measured relative to the center beam using a protractor. We got angles $16° \pm 0.5°$ on both sides of the center beam.

Part 3: Dispersion of White Light by a Grating

We removed the vertical metal sheets and the beam expander. Then we replaced the laser with a ray box, adjusting the ray box to output a single ray of light perpendicular to the table and through the diffraction grating. The white beam of light split into blue, green, and red beams of light, with the blue beam closest to the center and the red beam furthest from the center. We measured the angles relative to the center beam and got $blue = 18.0° \pm 0.5°$, $green = 20.0° \pm 0.5°$, and $red = 22.5° \pm 0.5°$.

Part 4: Human Hair

For the last part, we replaced the ray box with the laser, and replaced the diffraction grating with a single human hair clamped vertically to the table and in the path of the laser. We then placed a white sheet of paper 90cm away from the hair, and measured the distance between the maxima of the diffraction pattern projected onto the paper. We got an average separation distance of $0.67 \pm 0.05 cm$.

Analysis

Part 1

First, we used linear regression on our potentiometer callibration data, and found that for any difference in voltage $v$, the corresponding distance $x$ is given by

$x = (0.8769 \pm 0.00975) v$ (Eq. 1)

Now we know that, for the diffraction pattern of a light wave with wavelength $\lambda$ through a double slit with separation $d$, the maxima number $m$ is related to the angle $\theta$ by the equation:

$d\sin(\theta) = m\lambda$ (Eq. 2)

with error from propagation of uncertainty

$\delta_d = d \cot(\theta) \delta_\theta$ (Eq. 3)

We know that for small angles, $\sin(\theta) \approx \tan(\theta)$, which gives us

$d\tan(\theta) = d\frac{x}{D} = m\lambda$ (Eq. 4)

where $x$ is the distance from the center to the maxima, and $D$ is the distance from the center to the double slit. This means that, for two maxima separated by distance $M$:

$d\frac{M}{D} = \lambda$ (Eq. 5)

with an error formula derived using propagation of uncertainty:

$\delta_d = d \sqrt{(\frac{\delta_M}{M})^2 + (\frac{\delta_D}{D})^2}$ (Eq. 6)

In addition, we know that the intensity of a double slit diffraction pattern is the product of the double slit interference pattern and the single slit intensity envelope pattern. We know that the minima of the single slit intensity pattern for slit width $b$ are given by:

$b \frac{x}{D} = m\lambda$ (Eq. 7)

which, by the same logic as above, means that for two minima separated by a distance $N$,

$b\frac{N}{D} = \lambda$ (Eq. 8)

$\delta_b = b \sqrt{(\frac{\delta_N}{N})^2 + (\frac{\delta_D}{D})^2}$ (Eq. 9)

Thus, we can find the slit width $b$ by measuring the space between minima in the intensity envelope.

Starting with the diffraction pattern data for the known-width double slit, we found the average separation between maxima to be $(0.219 \pm 0.014) V$. Converting this to distance using Eq. 1 and then using Eq. 5 and 6 to calculate separation gives us $d = (0.137 \pm 0.0122) mm$. The double slit was known to have a separation of $0.125 mm$, which falls within the error bounds of our result, verifying our method.

With our process confirmed to be working, we calculate the separation for the unknown double slits to be:

Results 1: Table of calculated separations for unknown double slits

We noticed from comparing the graphs and the calculated results that slit separation and the distance between maxima seemed to be inversely related, meaning the smaller the distance between maxima the larger the slit separation, and vice versa.

Lastly, the intensity envelope of unknown double slit #2 was distinct enough to allow us to measure the average distance between minima and use Eq. 8 and 9 to find the width $b = (0.124 \pm 0.00002) mm$.

Part 2

The equations for the extrema in a diffraction grating are the same as the equations for double slits. Thus, we can use Eq. 3 and 4 on the angles we measured to find the separation between each line in the diffraction grating: $d = (0.00243 \pm 0.00007) mm$. However, the actual separation distance, $1/600 = 0.00167 mm$ does not fall within the error bounds of the calculated result. One possible reason could be that when we measured the angle, we had to tilt the paper upwards so that the light beam made a visible projection on the paper. However, tilting the paper upwards makes the projections more parallel to the center line, decreasing the angle and in turn increasing the calculated separation, explaining why our result was so much larger than expected.

Part 3

We rearrange Eq. 3 to get it in terms of wavelength:

$\lambda = d\sin(\theta) /m$ (Eq. 10)

with error from propagation of uncertainty

$\delta_\lambda = d\cos(\theta) \delta_\theta /m$ (Eq. 11)

Then we use Eq. 10 and 11 to find the wavelengths of the 3 colors:

Results 2: Calculated wavelengths for the three colors

The expected values of the wavelengths just from the observed colors (blue ≈ 495 nm, green ≈ 550 nm, red ≈ 650 nm) fell within the error bounds of our results, confirming our calculations.

Part 4

For this part, we already found the equations for calculating the width of a single slit from the distance between the minima of the intensity graph, given in Eq. 8 and 9. A laser shined through a strand of hair acts like a “reverse” single slit, with the diffraction pattern the reverse of the diffraction pattern of a single slit. This is why we measured the distance between maxima and not minima. Thus, we can still use Eq. 8 and 9 to calculate the width of the slit, or in this case, the strand of hair: $d = (0.0704 \pm 0.0053) mm$.

Conclusion

To conclude, we were generally able to achieve the purpose of this experiment, which was investigate the wave characteristics of light, and to use the diffraction of light to calculate various characteristics of the diffraction element. In the first part, we successfully verified the spacing of a double slit, and then calculated the spacing of 3 unknown double slits. In the second part, our calculation for the spacing of the diffraction grating was a little off, most likely due to the inaccuracy of measuring the angle of the light using a tilted piece of paper. For the third part, our measurements for the wavelengths of colored diffracted beams were within expected ranges, and for the last part, we were successfully able to measure the width of a strand of hair using single slit diffraction principles.