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IPhO 2019, Експеримент — Задача 1.1. Part A: The refractive index of a disk

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IPhO Israel 2019. Experiment Q1. Optical Measurements · 5,5 т.

Optical Measurements

In this experiment, we will measure the optical properties of samples at the highest achievable accuracy using the available equipment.

Note: under your table you can find 2 big water bottles that you need to use for the second experiment - do not drink them.

In part A, we will use two different methods to measure the refractive index of a transparent disk. The first method is traditional, while the second one is original and enables a higher level of accuracy.

In part B, we will measure the ratio between the laser wavelength $\lambda$ and the lattice constant of a diffraction grating $d$ , aiming at achieving the highest possible level of accuracy.

In part C, we will measure the refractive index of a triangular prism, again – attempting to reach the highest level of accuracy.

For the experiments, the exam hall will be darkened for 100 minutes, starting 20 minutes after the exam begins (you may use a table lamp when needed). It is more convenient to perform the measurements of part A in the dark, but it is still possible to perform most of them in light as well.

You may use the walls of your cubicle as a screen, and apply adhesive tapes to the cubicle walls.

In these experiments, you are using a diode laser as a light source.

Laser safety instructions:

NEVER look directly into the laser beam!
In all experiments, the laser beam is horizontal. When measuring the position of the laser beam on a surface make sure your head is ALWAYS above the level of the beam.
Do not aim the laser beam toward the opening of the experimental cubicle.
Turn the laser off using the designated switch whenever you are not performing measurements.

Equipment list

Pieces of equipment 1-9 are used in all parts of the question, and pieces of equipment 10-12 are used in separate parts of the questions. Note that you are given multiple pieces of optical equipment - make sure not to touch their vertical sides directly, to avoid fouling their surfaces.

Ruler, 60 cm long
Slider that can move along the ruler
Laser source, mounted on the slider. The laser can be set at two heights, or levels: low level 3A for part A and high level 3B for parts B and C. The on/off switch of the laser is shown in the figure as 3C
The tension of screws 4A and 4B controls the resistance to rotation and therefore the stability of the apparatus. Use the small metal bar 4C to change the direction of the laser. Rotate 4C by 180 degrees to change the height of the laser. Do not rotate the laser around the beam axis, since the beam polarisation is pre-tuned.
Screen: you may use the cubicle walls; you may assume the walls are perpendicular to each other
A roll of adhesive tape which may be used to affix equipment to the table
Flexible measuring tape
A variety of rulers
Table lamp
Round transparent disk with a diameter of 20.00 cm, fixed on a protractor, which is glued to a wooden base (for part A). You should remove the 4 small wooden cubes attached to the wooden base
Parchment paper that can be used as a transparent screen to be temporarily attached (by hand) to the side of the disk, allowing measuring the exit point without fouling the polished disk surface (for part A). It is more accurate to measure the beam exit point if you draw a line on the paper as shown in the figure.
A wooden piece (12A) and a cylinder holder (12B) capable of rotating along its vertical axis, for mounting the diffraction grating (12C) or the triangular prism (12D)

Photographs of the equipment are shown with Part A.

Условие

In this part, we will measure the refractive index of a transparent disk by observing the path taken by a beam of light as it is refracted and reflected inside the disk.

Schematic: a diode laser beam enters a yellow disk at incident angle α, refracts at angle β, reflects inside the disk (angle γ at the centre) and exits; the angle δ of the outgoing beam is marked with a dashed circle; the disk scale shows 0° and 180°. — A schematic view of the experiment

Definitions and symbols:

$\alpha$	The incident angle between the disk and the incoming beam
$2\Delta\alpha$	The angular spread of the incident angle, i.e. the size of the range of values of the incident angle $\alpha$
$\beta$	The refraction angle inside the disk
$\gamma$	$= 180^{\circ} - 2\beta$
$n$	The refractive index of the disk material
$N$	The number of times the beam hits the disk’s boundary before emerging from the disk into the air (in the sketch, $N = 3$ )
$\delta$	The angle between the direction opposite to that of the incoming beam and the direction of the outgoing beam, measured clockwise (the sketch shows the angle $\delta$ for $N = 3$ )
$2\Delta\delta$	The angular spread of $\delta$

It is possible to show that the angles $\alpha$ , $\beta$ and $\delta$ are related by:

\delta = 2\alpha + (N-1)(180^{\circ} - 2\beta)\ . \qquad (1)

You may use this equation without deriving it.

Affix the ruler to the table in order to control the incident angle of the laser beam, using the adhesive tape, and adjust the laser in a way that will allow you to easily measure the incident angle. Then, affix the disk to the table by attaching the adhesive tape to the corners of the exposed wooden base and the table. Adjust inclination with the metal bar 4C. The laser may be set to two different heights: the low level for part A and the high level for parts B and C. The laser was tuned in advance so that the incident beam will be in the S polarization (a polarization in which the reflection is higher). Do not change the polarization of the incident beam (do not rotate the laser around the beam axis)!

Photograph of the 60 cm ruler taped to the table with the red slider carrying the laser; labels 1 (ruler) and 2 (slider). — Equipment 1, 2

Two photographs of the laser mounted on the slider: left, the laser at the low level with labels 3A and 3C (on/off switch); right, the laser at the high level with labels 3B and 3C. — Equipment 3

Photograph of the laser holder with screws 4A and 4B and the metal bar 4C; photograph of a roll of adhesive tape labelled 6. — Equipment 4, 6

Photographs of the flexible measuring tape (7), a set of rulers and a protractor in a bag (8), and a table lamp (9). — Equipment 7, 8, 9

Photograph of the round transparent disk fixed on a protractor on a wooden base (10); photograph of parchment paper with a drawn vertical line and a red laser spot, held against a scale (11). — Equipment 10, 11

Photographs of the wooden piece 12A with the cylinder holder 12B holding the diffraction grating 12C (left) and the triangular prism 12D (right). — Equipment 12

A.1 Draw a sketch of the system showing the ruler with the slider, the disk, and the path taken by the laser beam. Indicate the incident angle $\alpha$ .
Perform a series of measurements between the angles $15^{\circ} \le \alpha \le 75^{\circ}$ , and write down $\alpha, \Delta\alpha, \delta, \Delta\delta$ in Table 1.
Note: for measuring $\delta$ , it is more convenient to measure $\delta/2$ directly on the disk. [1 т.]

A.2 Using the measurement from the previous step, draw the appropriate graph, from which you can extract the refractive index $n$ and the error $\Delta n$ . Should you need to calculate additional quantities, fill the calculated numbers in the empty columns of Table 1. Find $n$ and $\Delta n$ . [1 т.]

A.3 For the measurements performed in A1, draw a graph of $\delta$ as a function of $\alpha$ . Mark on each measured point the values of $\Delta\delta$ and $\Delta\alpha$ using bars. Make an additional measurement to accurately find the minimal $\delta$ , and the corresponding $\alpha$ . Denote them by $\delta_{\mathrm{min}}$ and $\alpha_{\mathrm{min}}$ .
In order to identify the minimum point most accurately, you may use the walls of your cubicle as a screen for the outgoing beam. [0,5 т.]

A second method to measure the refractive index

In this section, you will develop an alternative method that will allow you to get very accurate results. Even though you are required to measure at the highest achievable accuracy, you are not required to perform error calculations. Nevertheless, you are required to detail the equations you use to obtain your results. Write them in the answer sheets.

A.4 Based on the behavior of the graph you got in A3, make a choice of the optimal angle to perform the measurements to find the refractive index. Write down the equation which one could use to obtain the refractive index using the developed method. [0,7 т.]

A.5 For $N = 3$ , perform the necessary measurements to calculate the refractive index at high accuracy, using the method developed in A4

Draw a diagram of the disk and the beam path, and indicate on the diagram the quantities you have measured.
Document the measurements you have performed.
Perform an analysis of the measurements and calculate the refractive index $n$ of the disk, at the highest achievable accuracy. You may use the additional supplied graph pages if needed. [0,8 т.]

A.6 Repeat the process you performed in the previous task, for $N = 4$ and $N = 5$ (no need to draw the system and beam path).

Document the measurements you have performed at $N = 4$ .
Perform an analysis of the measurements at $N = 4$ , and calculate the refractive index $n$ using these measurements at the highest achievable accuracy.
Document the measurements you have performed at $N = 5$ .
Perform an analysis of the measurements at $N = 5$ , and calculate the refractive index $n$ using these measurements at the highest achievable accuracy.
From the results you obtained of the value of the refractive index using the measurements at $N = 3$ , $N = 4$ and $N = 5$ , calculate the average value $\langle n \rangle$ of the refractive index. [1,5 т.]

Отговори

Решение

Optical Measurements – Solution

Part A: The refractive index of a disk

A.1: A sketch of the experimental setup for $N = 3$

Sketch: the ruler with the laser on its slider set at an angle; the beam enters the yellow disk at angle α, the angles δ/2 and β are marked at the centre, the beam reflects and exits; the outgoing angle δ is marked; 0° reference line. — A.1: A sketch of the experimental setup for N = 3

A.1: table of measured and calculated values

$\alpha(\circ)$	$\Delta\alpha(\circ)$	$\delta/2(\circ)$	$\Delta\delta/2(\circ)$	$\delta(\circ)$	$\Delta\delta(\circ)$	$\beta(\circ)$	$\sin\alpha$	$\sin\beta$
15	0.25	174.5	0.25	349	0.5	10.25	0.259	0.178
20	0.25	173	0.25	346	0.5	13.5	0.342	0.233
25	0.25	172	0.25	344	0.5	16.5	0.423	0.284
30	0.25	171	0.25	342	0.5	19.5	0.500	0.334
35	0.5	170	0.25	340	0.5	22.5	0.574	0.383
40	1	169.5	0.25	339	0.5	25.25	0.643	0.427
45	1	169	0.25	338	0.5	28	0.707	0.469
50	1	169	0.25	338	0.5	30.5	0.766	0.508
55	1	169	0.25	338	0.5	33	0.819	0.545
60	1.5	170	0.25	340	0.5	35	0.866	0.574
65	1.5	171	0.5	342	1	37	0.906	0.602
70	1.5	173.5	1	347	2	38.25	0.940	0.619
75	2	176.5	1.5	353	3	39.25	0.966	0.633

A.2:

A.3:

Handwritten graph on blue millimetre paper of δ (vertical, 336°–354°) against α (horizontal, 15°–75°) with error bars on each point and a curve with a minimum near α = 50°, δ = 338°. — A.3

By observing the remote screen, it is possible to identify the point in which $\delta$ is minimal at the highest accuracy.

The values we find are

$\alpha = 49^{\circ} \pm 0.25^{\circ}$ and $\delta = 338^{\circ} \pm 0.5^{\circ}$

A.4:

When $\delta$ is minimal, $\frac{d\delta}{d\alpha} = 0$ .
Differentiating the relation $\delta = 2\alpha + (N-1)(180^{\circ} - 2\beta)$ by $\alpha$ we get:

$2 - 2(N-1)\frac{d\beta}{d\alpha} = 0$ and therefore $\frac{d\beta}{d\alpha} = \frac{1}{N-1}$ .

By differentiating Snell’s law $\sin\alpha = n\sin\beta$ we get $\cos\alpha = n\cos\beta\cdot\frac{d\beta}{d\alpha} = \frac{n\cos\beta}{N-1}$

Squaring this result, as well as Snell’s law and summing the expressions we get:

1 = \sin^2\alpha + \cos^2\alpha = n^2\sin^2\beta + \frac{n^2\cos^2\beta}{(N-1)^2}

Hence: $\frac{1}{n^2} = \sin^2\beta + \frac{\cos^2\beta}{(N-1)^2}$

We got an explicit relation between the refraction angle $\beta$ and the refraction index of the material. Due to the multiple reflections inside the disk it is possible, by following all the point in which the beam hits the disk-air interface, to measure the angle $\beta$ at very high accuracy.

A.5: a sketch showing all the measured quantities:

Define the angle $\gamma = 180^{\circ} - 2\beta$ , as shown in the sketch. In fact, after two reflections inside the disk the beam exits at a point very close to the entering point. We will measure the angular location of the points where the beam hits the interface after $k$ reflection, for as many values of $k$ as we can:

$k$	$\alpha + k\gamma$
0	49
1	168.5
2	288.5
3	409

Note: for the case of $N = 3$ it is not possible to measure for $k > 3$ as in this case, starting from $k = 3$ the impact points co-inside with previous points.

Next we draw a graph of $y = \alpha + k\gamma$ vs. $k$ and find the linear regression slope, $\gamma$ :

Sketch of the yellow disk with the laser beam entering at angle α, refraction angle β, and the successive impact points labelled α+γ, α+2γ and α+3γ, with the central angles γ marked; 0° reference line. — A.5: a sketch showing all the measured quantities

Chart of α + kγ (0–900) against k (0–7) with four points and a dotted linear fit labelled y = 120x + 48.75, R² = 1. — Graph of y = α + kγ vs. k for N = 3

From $\gamma = 120^{\circ}$ we get $\beta = 30^{\circ}$ , and using the equation we derived in A.4 we get:

n = \frac{1}{\sqrt{(\sin\beta)^2 + (\cos\beta)^2/(N-1)^2}} = 1.512

A.6: We will identify the beam exiting the disk after 4 refractions/reflections $(N = 4)$ and we will change the incident angle until we get $\delta_{min}$ for $N = 4$ . We will measure $\alpha + k\gamma$ as a function of the number of times the beams hits the disk-air interface, $k$ :

$k$	$\alpha + k\gamma$
0	67
1	172
2	278
3	383
4	488
5	593
6	698.5

Chart of α + kγ (0–800) against k (0–7) with seven points and a dotted linear fit labelled y = 105.23x + 67.089, R² = 1. — Graph of α + kγ vs. k for N = 4

n = \frac{1}{\sqrt{(\sin\beta)^2 + (\cos\beta)^2/(N-1)^2}} = 1.511

We’ll repeat this process for $N = 5$ :

We will identify the beam exiting the disk after hitting the disk-air interface 5 times $(N = 5)$ and measure $\alpha + k\gamma$ as a function of the number of hits, $k$ :

$k$	$\alpha + k\gamma$
0	72.5
1	174.5
2	276.5
3	379.5
4	480.5
5	582.5
6	685

Chart of α + kγ (0–800) against k (0–7) with seven points and a dotted linear fit labelled y = 102.05x + 72.554, R² = 1. — Graph of α + kγ vs. k for N = 5

n = \frac{1}{\sqrt{(\sin\beta)^2 + (\cos\beta)^2/(N-1)^2}} = 1.519

Averaging the three results we get: $n = \frac{1.519 + 1.511 + 1.512}{3} = 1.514 \pm 0.004$

Handwritten graph on red millimetre paper of sin α (vertical, 0.1–1) against sin β (horizontal, 0.1–0.7) with circled data points on a straight line; written: n = Δy/Δx = 23.2/15.2 = 1.526, n = 1.53 ± 0.02. — A.2

Оригинал в Архива: exp1.pdf · официални решения: exp1_sol.pdf