Physics 272 Laboratory Experiments: Interference and Diffraction

4. Wave Properties of Light: Interference and Diffraction

The goal of this experiment is to measure the wavelength ($\lambda$) of a laser using two different methods that exploit the wave nature of light. The research questions are these:

With what experimental precision may the wavelength be measured using multislit interference patterns for N (# of slits) = 2,3,5? What's the least significant figure or digit that you are sure of with this sort of measurement? Does it improve with N? How well does it agree with the manufacturers estimate of the wavelength?
Measuring the wavelength with a ruler (hahaha, like that's possible!....it sort of is!) actually utilizes many more slits than is possible with the Pasco Multislit Apparatus. Again, how many significant digits does the method permit? Is it better or worse than using the Pasco Multislit Apparatus?

Safety Precaution: Never look directly into the laser beam or its specular reflection.

4.1 Task #1: Determining the Wavelength ($\lambda$) using Multi-Slit Gratings

Theory and Background

When coherent light passes through multiple slits, the resulting pattern is a combination of interference (from $N$ slits) and diffraction (from the finite width of a single slit). The intensity peaks are 'modulated' by the single-slit diffraction envelope. For interference, the simplest physical model, where the small-angle approximation prevails ($\sin{\theta} \approx \tan{\theta} \approx \theta$), is given by:

\begin{equation} d \sin{\theta_n} \approx d \frac{y_n}{D} = \left\{ \begin{array}{rl} n \lambda & \mbox{ (Constructive Interference, CI) }\\ (n+\frac{1}{2})\lambda & \mbox{ (Destructive Interference, DI)} \end{array} \right. \end{equation}

Here, $d$ is the slit separation, $D$ is the distance from the slits to the screen, $y_n$ is the distance from the central maximum to the $n^{\text{th}}$ order maximum, and $\lambda$ is the wavelength. For single-slit diffraction minima (the overall envelope), the condition is:

\begin{equation} a \sin{\theta_m} \approx a \frac{y_m}{D} = m \lambda \quad (m=1, 2, 3, \dots) \end{equation}

Collage of images depicting the setup for double-slit diffraction. — Figure 1. Setup for multi-slit diffraction, showing the geometric variables ($\theta$, $D$, $y_n$) and the resulting pattern.

Procedure: Measurements and Modeling

Setup and Record Parameters:
1. Set up the laser, the multi-slit wheel, and the screen (or paper-covered board).
2. Measure and record the distance from the slit to the screen ($\mathbf{D}$) and estimate its uncertainty ($\mathbf{\Delta D}$).
3. Record the manufacturer's specified slit separation ($\mathbf{d}$) and slit width ($\mathbf{a}$) for two patterns: the 2-slit pattern and one other (e.g., $N=3, 4,$ or $5$). Record the laser's manufacturer wavelength ($\lambda_{\text{man}}$) and any serial numbers.
4. Sketch a cartoon of the setup in your notebook, visually defining the variables $d, D,$ and $y_n$, including where and how each measurement is taken.
Observe and Qualitatively Analyze:
1. Turn on the laser and select the 2-slit pattern ($N=2$).
2. Sketch the observed pattern. Note the brightness variation (modulation) caused by the single-slit diffraction envelope.
3. Preliminary Calculation Check: Select and measure the position ($y_n$) of one principal maximum (e.g., $n=3$). Use this single measurement with the known $d$ and $D$ in Equation (1) to calculate a preliminary value for $\lambda$.
4. Also calculate the predicted wavelength using the $\mathbf{m=1}$ single-slit diffraction null ($y_{\min, 1}$) in Equation (2). Compare the two preliminary values. This too an intermediate check on progress (addresses the question, `is my data any good, do my results make sense').
Collect Data for All Principal Maxima:
1. Measure and record the position ($\mathbf{y_n}$) of all visible principal maxima from the central maximum ($y_0=0$).
2. Create a data table with columns for: order ($\mathbf{n}$), position ($\mathbf{y_n}$), and uncertainty ($\mathbf{\Delta y_n}$).
Modeling the Wavelength (FITTEIA):
1. Using Equation (1), rearrange the model to a linear or near-linear form (e.g., plot $y_n$ vs. $n$).
2. Use FITTEIA to plot your data and fit a modeling curve where the wavelength ($\lambda$) is the fitting parameter.
3. Determine the best estimate of $\lambda$ and its associated uncertainty ($\Delta \lambda$) (e.g., using the Goldilocks plot).
4. Discuss: Why is the uncertainty of the fit superior a superior estimate of the uncertainty of a single measurement (Step 2)?
5. Write a clear HOW-TO statement that clarifies the relationship of the Physical Model (PM) Computational Model (CM), and all that goes with that (units, Goldilocks analysis, etc. etc.)
Intensity Pattern Modeling (Desmos or Fooplot):
The full intensity profile for $N$ slits is given by the product of the diffraction and interference terms:
\begin{equation} I (\theta) \propto \left(\frac{ \sin{ \left(k\frac{a}{2}\sin{\theta}\right) } }{ \left( k\frac{a}{2}\sin{\theta}\right) }\right)^2 \left[ \frac{ \sin{\left( N k\frac{d}{2}\sin{\theta} \right) } }{ \sin{ \left( k\frac{d}{2}\sin{\theta} \right) }} \right]^2 \end{equation}
To simplify, define the phase angle variable $x \equiv \frac{\pi d \sin{\theta}}{\lambda}$. The intensity can be written as:
\begin{equation} I (x) = \left( \frac{ \sin{ \left( (a/d) \cdot x \right) } }{ \left( (a/d) \cdot x \right) } \right)^2 \left[ \frac{ \sin{\left( N \cdot x \right)} }{ \sin{ \left( x \right) }} \right]^2 \end{equation}
1. Use Desmos or Fooplot to plot $I(x)$ for your chosen $N$ (e.g., $N=2$). Use your best fit $\lambda$ and the known $a$ and $d$.
2. Compare the plot with your qualitative sketch (Step 2). Count the number of $N$-slit principal maxima predicted to fall underneath the central diffraction envelope. Is this count consistent with your observation?
Repeat Steps 3 and 4 for your second and third chosen multi-slit pattern (e.g., two of $N=3,4,5$).

4.2 Task #2: Measuring Wavelength ($\lambda$) with a Steel Rule Reflection Grating

Theory and Background

This method, based on the work of Nobel laureate A. L. Schawlow, uses the fine markings on a steel ruler as a reflection grating (where $N \gg 1$). The laser beam diffracts upon reflection, producing bright spots (principal maxima) on the wall. The grating spacing $d$ is the distance between the markings (e.g., $1/32^{\prime\prime}$ or $1/64^{\prime\prime}$).

When the laser is leveled and the ruler is slightly tilted (as shown in Figure 3, right), the diffraction equation simplifies. Assuming the small-angle approximation is justified for the vertical displacements ($\left(y_m/R\right)^2 \ll 1$), the relationship between the order $m$ and the position $y_m$ is:

\begin{equation} m \lambda \approx \frac{d}{2 R^2} \left( y^2_m - y_m y_o \right) \end{equation}

Here, $R$ is the horizontal distance from the ruler to the wall, $y_m$ is the vertical displacement of the $m^{\text{th}}$ order maximum, and $y_o$ is the position of the mirror mode (where incidence and reflection angles are equal).

Figure from Schawlow's paper showing the reflection grating geometry. — Figure 3. On the left, Schawlow's original setup. On the right, the setup used in this lab: a level laser, slightly tilted ruler, and a screen at distance $R$ (labeled $x_o$ in the figure).

Horizontal laser setup with a tilted ruler acting as a reflection grating. — Figure 3. On the left, Schawlow's original setup. On the right, the setup used in this lab: a level laser, slightly tilted ruler, and a screen at distance $R$ (labeled $x_o$ in the figure).

Procedure: Measurements and Modeling

Setup and Define Variables:
1. Set up the laser and steel ruler as shown in Figure 3, right, with the laser leveled and the ruler slightly tilted to reflect the beam onto the wall. This will likely be done for us (thank you Mr. Matt Searle, our Lab Tech!)
2. Measure and record the horizontal distance from the ruler to the wall ($\mathbf{R}$) and its uncertainty.
3. Determine and record the grating spacing ($\mathbf{d}$) of the ruler scale being illuminated.
4. Estimate the number of `slits' illuminated on the ruler by the laser beam at grazing incidence.
5. Sketch the apparatus in your notebook, clearly defining $R$, $y_m$, $y_o$, and the angles of incidence ($\alpha$) and diffraction ($\beta$) relative to the surface normal.
Check the Small-Angle Approximation:
1. Measure the largest observed vertical displacement ($y_{\text{max}}$).
2. Calculate the ratio $\mathbf{(y_{\text{max}}/R)^2}$. Exhibit the calculation to justify whether the Taylor expansion leading to Equation (6) is valid.
Collect Data for Maxima:
1. Contrive to measure the position ($y_m$) of the mirror mode ($m=0$, which is $y_o$) and all visible principal maxima ($m=\pm 1, \pm 2, \dots$). Use tape or paper on the wall to mark the spots.
2. Create a data table with columns for: order ($\mathbf{m}$), position ($\mathbf{y_m}$), and uncertainty ($\mathbf{\Delta y_m}$).
3. Preliminary Calculation Check: Reduce one of your measurements to a single $\lambda$ estimate using the full Grating Equation (see Schawlow's paper for the full equation). Quantify its uncertainty.
Modeling the Wavelength (FITTEIA):
1. Equation (6) is quadratic in $y_m$. Rearrange it to solve for $y_m$: $$y_m = \frac{y_o}{2} \pm \sqrt{\left(\frac{y_o}{2}\right)^2 + m \lambda \frac{2 R^2}{d}}$$ You must choose the physical solution ($+$ or $-$) based on your coordinate system.
2. Use FITTEIA to fit your data, using the selected solution as the C-function fitting model. Ensure $\lambda$ is a fitting parameter.
3. Write a clear HOW-TO statement that clarifies the relationship of the Physical Model (PM) Computational Model (CM), and all that goes with that (units, Goldilocks analysis, etc. etc.)
4. Determine the best estimate of $\lambda$ and its uncertainty ($\Delta \lambda$). Save you modeling output with an appropriate file name, and put the name, file path, and file contents in the record for this experiment. Include the plots in page-images upload.

4.3 Conclusion and Summary

Compare the two measurements of $\lambda$ (from Task #1 and Task #2) with the accepted manufacturer's value. Write an abstract that:

Briefly reviews the measurement methodology and approach, that is understandable for someone who has not read the materials or has just completed the experiment. This is meant to be brief.
Quotes your final results in the form $\lambda \pm \Delta \lambda$, using appropriate significant figures.
Compares the experimental uncertainties and the discrepancy relative to the manufacturer's value.
Interprets which method you found to be the most reliable and explains why, above all
Explicitly addresses the two research questions, making clear assertions about their answers, and how those answers are supported by the data and the modeling.

4.4 References

A L Schawlow, "Measuring the Wavelength of Light with a Meter Stick," American Journal of Physics 33 922 (1965). (Available through Copley Library databases.)
Pasco Precision Diffraction Slits Manual