Measuring the wavelength of light

Activity 2 | Week 4 | Experiments 1 and 2

1 What to expect

In this session, you will work on two simple experiments to measure the wavelength of light from a laser.¹

In the first experiment, you will use a diffraction grating. In the second, you will use the markings on a ruler as a reflective diffraction grating. The second method is based on the article by¹, distributed in class.

Even though the apparatus is simple, you need to exercise care to obtain a precise result. I hope you bring the same patience and precision that you developed during the pendulum experiment.

A central question I would like you to focus on in this session is:

Which individual measurements have the greatest influence on the final result, and therefore need to be made most carefully?

Do not worry if you cannot complete both experiments this week. You can continue your measurements during next week’s IS session.

2 Safety first

2.1 Ground rules

You will be working with lasers. A laser produces a narrow, concentrated beam of light that can damage the eye if used incorrectly or carelessly.

Use only the provided laser.
Do not substitute any other laser for this experiment.
Never look directly into the laser beam or its reflections.
Even a low-power Class 2 laser can cause eye damage.

Additionally, follow these rules throughout the session:

Never point a laser at another person.
Keep the beam below eye level, whether people are sitting or standing.
Switch the laser on only after the apparatus has been arranged.
Identify where the beam will travel and where it will terminate before switching it on.
Keep watches, jewellery, polished metal, glassware, and other unnecessary reflective objects out of the beam path: they may reflect the beam into someone else’s eyes.
Do not view the beam through a magnifying glass, camera, telescope, or other optical instrument.
Do not bend down so that your eyes are at the height of the beam.
Switch off the laser before adjusting or moving the apparatus.

2.2 Things you should know about lasers

Do not assume that a laser is safe merely because its beam looks dim. Apparent brightness is not a reliable measure of its power or its potential to damage the eye. For instance, the eye is particularly sensitive to green light, so a green laser can appear much brighter than a red laser of the same power.
Laser classes indicate the potential hazard a laser poses. Table 1 is a simplified guide for common visible lasers:

Class	Approximate power \(P\)	General hazard
2	\(P \leq 1\ \mathrm{mW}\)	Prolonged deliberate viewing of the beam is hazardous
3R or 3A	\(1\ \mathrm{mW} < P \leq 5\ \mathrm{mW}\)	Direct viewing of the beam can be hazardous
3B	\(5\ \mathrm{mW} < P \leq 500\ \mathrm{mW}\)	Direct and mirror-like reflected beams are hazardous to the eye
4	\(P > 500\ \mathrm{mW}\)	Direct and reflected beams can be hazardous to eyes and skin

Table 1: Simplified laser hazard classes.

Power alone does not determine the danger a laser can pose. Classification also depends on wavelength, exposure duration, beam properties, and how the radiation is accessed. Treat the simplified guide above with caution, and always err on the side of safety.
Use of Class 3B and Class 4 lasers usually require a license from NEA.
Be careful with laser pointers bought over the internet: they may emit more power than their labels claim, and some may also emit invisible radiation. See NEA’s laser safety FAQ for more information.

3 Planning ahead

When possible, it is good to plan ahead and estimate which measurements will be most useful and which need the greatest care.

Let me take measuring the wavelength with a diffraction grating as an example. We discussed in class that, for a diffraction grating at normal incidence,

\[ d\sin\theta_m=m\lambda, \tag{1}\]

where:

\(d\) is the spacing between adjacent grating lines;
\(\theta_m\) is the angle of the \(m\)th-order maximum;
\(m=0,\pm1,\pm2,\ldots\) is the diffraction order; and
\(\lambda\) is the wavelength of the light.

The central maximum has \(m=0\). The first bright spots on either side normally correspond to \(m=+1\) and \(m=-1\).

If a screen is a perpendicular distance \(L\) from the grating and a maximum has a displacement \(y_m\) from the centre, along the screen, then

\[ \sin\theta_m = \frac{y_m}{\sqrt{L^2+y_m^2}}. \tag{2}\]

Combining Equation 1 and Equation 2 gives,

\[ \lambda = \frac{d}{m} \frac{y_m}{\sqrt{L^2+y_m^2}}. \tag{3}\]

3.1 Making estimates

Before collecting data, use an approximate calculation to decide:

how far apart the diffraction spots are likely to be;
whether they will fit on the available screen;
which distances need to be measured most carefully; and
whether a proposed arrangement is practical.

Let’s plan by assuming a typical wavelength for the laser. For a red laser pointer, we can use roughly \(650\ \mathrm{nm}\). This is only a ballpark value to help us plan the experiment.

Suppose the grating has \(500\) lines per millimetre. Its line spacing is

\[ d = \frac{1\ \mathrm{mm}}{500} = 2.00\times10^{-6}\ \mathrm{m}. \]

For the first-order maximum, \(m=1\),

\[ \sin\theta_1 \approx \frac{650\times10^{-9}}{2.00\times10^{-6}} = 0.325 \approx19^\circ. \]

Let’s quickly estimate the displacement \(y_1\) for a chosen screen distance \(L\) using Python. For this, note that

\[ \tan\theta_m = \frac{y_m}{L} \quad\Rightarrow\quad y_m=L\tan\theta_m. \tag{4}\]

For now, we will consider the first-order maximum, \(m=1\).

wavelength = 650e-9
d = 1e-3 / 500
m = 1

theta_m = np.arcsin(m * wavelength / d)

print("L (m)    y (cm)")
for L in np.array([0.5, 1, 2, 5, 10]):
    y_m = L * np.tan(theta_m)
    print(f"{L:5.2f}    {y_m*100:6.1f}")

L (m)    y (cm)
 0.50      17.2
 1.00      34.4
 2.00      68.7
 5.00     171.8
10.00     343.7

As \(L\) increases, the spots move farther apart. This may make their positions easier to measure, but it also requires a wider screen.

3.2 A Monte Carlo method to model uncertainty

A quick way to investigate measurement sensitivity is to simulate the experiment many times. In each simulated trial, values of \(L\) and \(y\) are drawn randomly from distributions representing their measurement uncertainties.

rng = np.random.seed(3176)

# Just to keep the units in control!
meter = 1.0
cm = 1e-2 * meter
mm = 1e-3 * meter
nm = 1e-9 * meter

# Grating and diffraction order
lines_per_mm = 500
d = 1*mm / lines_per_mm
m = 1

# Typical values I estimated earlier
L_mean, y_mean = 1.00 * meter, 34.4 * cm
# L_mean, y_mean = 2.00 * meter, 68.7 * cm

# A function to run the Monte Carlo estimates


def simulate_wavelength(u_L, u_y):
    L = np.random.normal(L_mean, u_L, 100_000)
    y = np.random.normal(y_mean, u_y, 100_000)
    wavelength = (d / m) * y / np.sqrt(L**2 + y**2)
    return wavelength.mean(), wavelength.std(ddof=1)

Now let’s check for various situations.

print(f'For L={L_mean} m and y_m={y_mean/cm} cm')

u = 0.5
print(f"L uncertain by {u:0.3f} cm:", end="\t")
wavelength_mean, wavelength_std = simulate_wavelength(u_L=u*cm, u_y=0)
print(f"{wavelength_mean / nm:.1f} ± {wavelength_std / nm:.1f} nm")


print(f"y uncertain by {u:0.3f} cm:", end="\t")
wavelength_mean, wavelength_std = simulate_wavelength(u_L=0, u_y=u*cm)
print(f"{wavelength_mean / nm:.1f} ± {wavelength_std / nm:.1f} nm")

u = .1
print(f"y uncertain by {u:0.3f} cm:", end="\t")
wavelength_mean, wavelength_std = simulate_wavelength(u_L=0, u_y=u*cm)
print(f"{wavelength_mean / nm:.1f} ± {wavelength_std / nm:.1f} nm")

For L=1.0 m and y_m=34.4 cm
L uncertain by 0.500 cm:    650.6 ± 2.9 nm
y uncertain by 0.500 cm:    650.6 ± 8.5 nm
y uncertain by 0.100 cm:    650.6 ± 1.7 nm

It is clear that this experiment is more sensitive to the measurement of \(y_m\) than \(L\)! So, you need to be more careful with this.

Remember that, this simulation includes only random uncertainty in \(L\) and \(y\). It does not automatically account for systematic effects such as:

measuring from the wrong reference point,
a screen that is not perpendicular to the central beam,
an asymmetric diffraction pattern, or
a laser striking the grating at an unintended angle.

4 Experiment 1: Transmission diffraction grating

4.1 Apparatus

supplied laser,
diffraction grating,
screen or large sheet of paper,
metre rule or measuring tape,
stands and clamps

4.2 A procedure

Arrange the laser, grating, and screen so that the beam is below eye level.
Place the screen approximately perpendicular to the undeviated beam.
Position the diffraction grating with its ruled lines vertical. This should produce a horizontal diffraction pattern.
Ensure that the screen intercepts the beam and the diffraction maxima you intend to measure.
Mark the position of the central maximum, \(m=0\).
Identify and mark the maxima on both sides of the centre.
Measure the perpendicular grating-to-screen distance \(L\).
For each order, measure the positions of both the \(+m\) and \(-m\) maxima.
If possible, repeat the experiment using a different value of \(L\).

Think carefully about smart ways to measure these distances to minimise measurement errors.

5 Experiment 2: Using a ruler as a reflective grating

Conduct the experiment described in the article by^{Schawlow 1}. Draw on your experience and the estimation techniques you learned in the previous experiment to guide your planning and measurements.

References

Schawlow, A. L. Measuring the Wavelength of Light with a Ruler. Am. J. Phys. 33, 922–923 (1965).

Footnotes

LASER is an acronym for Light Amplification by Stimulated Emission of Radiation.↩︎