H.O. Meyer 12/19/2004
Lab #1: Index of Refraction of Objects with Flat Surfaces
Physics:
For all following experiments, Snell’s Law is the only theoretical concept used. We also
assume that the index of air is 1. Thus,
sin ϑ = n ⋅ sin ϑn
(1)
Goal:
In the first two labs we will explore many ways to measure the index of objects with
plane surfaces. We will practice to experimentally map out functional dependences, and
compare the results with theoretical expectations. We will also learn to deal with errors,
to make proper graphs, to keep a good lab book, etc.
Equipment:
All following experiments require a laser (color does not matter), and a gradated rotating
stage that allows one to rotate a prism with respect to the beam direction. Care should be
taken to aim the beam at the center of the rotating stage.
Preparation:
Before coming to the lab, you should read these instructions and work out the
assignments shown in italics (green) in the text below.
1 parallel-plane prism, transmission
As a transparent slab is tilted with respect to
the beam, the transmitted beam is shifted
parallel to the incident beam.
θ
1.1 Theory
Derive an expression for dT in terms of the
index, n, the thickness t, and sinθ.
dT
θn
t
1.2 Setup
Setup the laser, rotating stage and an acrylic
block. To measure the displacement dT,
setup the translator stage with a fiber connection to the photometer. Put a diffuser in front
of the fiber, and about 2 cm in front of that a 0.5 mm slit. The beam position is located by
searching for maximum intensity. The reading of the translator must be converted to dT,
so you should also locate the translator position without the prism.
1.3 Measurement
Measure dT as a function of the plate angle θ over the full visible range. To identify a
possible angle offset, measure on both sides of the incident beam. Estimate the reading
uncertainty of both, θ and dT.
1.4 Analysis
See sect. 2.4.
1
H.O. Meyer 12/19/2004
2 parallel-plane prism, reflection
dR
The two beams reflected from the two surfaces
are shifted in a manner similar to exp.1
2.1 Theory
Derive an expression for dR in terms of the index,
n, the thickness t, and sinθ.
θ
2.2 Setup
t
To measure the displacement dR, setup the
rotating arm with a projection screen with mm
scale. The screen is rotated to be perpendicular to
the exiting beams, and the beam separation is read off the screen directly.
θn
2.3 Measurement
Measure dR as a function of the plate angle θ over the full visible range. To identify a
possible angle offset, measure on both sides of the incident beam. Estimate the reading
uncertainty of both, θ and dR.
2.4 Analysis
Experiments 1 and 2 are analyzed together. Enter your data in a spreadsheet, convert the
translator reading of Exp.1 to dT, and graph dT and dR versus θ. Program the two
expressions for dT and dR with variable index n. Plot the calculations as curves on the
graphs and adjust n for best fit. Estimate the uncertainty of the resulting n. Discuss
possible source for systematic errors.
3 semi-circular prism
3.1 Theory
The incident beam is not deflected. Snell’s law
applies to the exit surface directly.
θn
3.2 Setup
Care must be taken to aim the beam at the center
of the flat surface.
θ
3.3 Measurement
Measure θ as a function of θn (5o steps) on both sides of the beam.
3.4 Analysis
Each data point i yields a measurement of n. Calculate the error of each ni, assuming δθ =
δθn = ±0.5o. Plot ni ± δni versus θ, and calculate the weighted average <ni> and its error
δ<ni>.
2
H.O. Meyer 12/19/2004
4 minimum beam deflection by a prism
4.1 Theory
The smallest deflection angle βmin that can be
obtained by rotating the prism around a
vertical axis is related to the index by
sin({β min + γ } / 2)
n=
(2)
sin(γ / 2)
(see Hecht, eq. 5.54).
γ
β
4.2 Measurement and analysis
Determine βmin for a number of prisms that are
available in the lab.
There is also a hollow prism that can be filled with a liquid (e.g., distilled water, n =1.33).
Prove that the walls of the prism have no effect on the deflection and that one determines
the index of refraction of the liquid.
5 beam deflection by a 45o prism
5.1 Theory
Derive an expression for the deflection angle β in terms of θ, the angle of incidence, and
n. What is the incident angle for which the beam undergoes total internal reflection on
the long side of the prism?
5.2 Setup
Place the prism on the rotating stage. Think of a
way to make sure that θ = 0 corresponds to normal
incidence. Measure the outgoing direction as in
Exps. 3 and 4.
5.3 Measurement
Measure β as a function of θ (about a dozen points).
γ=45o
θ
β
5.4 Analysis
Enter the data in a spreadsheet and make a plot. Also plot the theoretical expectation,
from 5.1, calculated for a variable n. Adjust n for best fit and discuss the uncertainty.
Locate the place in the graph that corresponds to total internal reflection.
In the course of this measurement you have also determined the minimum deflection
angle. Apply the method of Exp.4 to determine the n, and compare with the fit value.
3