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Условие Experimental Apparatus Experiment Requirement Explanation and Suggestion Решение

IPhO 1994, Експеримент — Задача 1. Determination of light reflectivity of a transparent dielectric surface

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Всички материали от IPhO 1994

XXV International Physics Olympiad — Beijing, People’s Republic of China — Practical competition, July15,1994 · 15 юли 1994 г. · 10 т.

Внимание: Бележка към темата

The complete archived booklet contains two whole experiments, including Black Box, which the archive task listing omits. Experiments are shown here using original pages22–36. The round total20 is the sum of the native10-point marking schemes. Archived worked solutions have no established official authorship. Native differences/errors are retained: polarizerA/P1 labels, the linearity prose ratio versus the displayed formula, differing Brewster-angle uncertainties, low-frequency impedance/inductance wording, high-frequency Z_L approximately0 statement and original uncertainty values. Referenced separate oscilloscope/frequency-generator panel illustrations and plotting papers are absent from the obtained booklet; no missing illustrations or measurements are invented.

Practical competition — July15,1994 — Time available:2.5hours

Use only the ball pen provided.
Your graphs should be drawn on the answer sheets attached to the problem.
Write your solution on the marked side of the paper only.
The draft papers are provided for doing numerical calculations and draft drawings.
Write at the top of every page:
- The number of the problem
- The number of the page of your report in each problem
- The total number of pages in your report to the problem
- Your name and code number

Условие

Experimental Apparatus

He-Ne Laser ( $\sim1.5\ \mathrm{mW}$ ). The light from this laser is not linearly polarized.
Two polarizers ( $P_1$ , $P_2$ ) with degree scale disk (Fig. 1), one ( $P_1$ ) has been mounted in front of the laser output window as a polarizer, and another one can be fixed in a proper place of the drawing board by push-pins when it is necessary.
Two light intensity detectors ( $D_1$ , $D_2$ ) which consisted of a photocell and a microammeter (Fig. 2).
Glass beam splitter (B).
Transparent dielectric plate, whose reflectivity and refractive index are to be determined.
Sample table mounted on a semicircular degree scale plate with a coaxial swivel arm (Fig. 3).
Several push-pins for fixing the sample table on the drawing board and as its rotation axis.
Slit aperture and viewing screen for adjusting the laser beam in the horizontal direction and for alignment of optical elements.
Lute for adhere of optical elements in a fixed place.
Wooden drawing board.
Plotting papers.

Original Fig1 polarizer with both push-pins and complete degree-scale disk.

Original complete Fig2 photocell and microammeter controls, multiplier ranges and cable labels.

Original complete Fig3 sample table, both semicircular degree scales, coaxial swivel arm and all labels.

Experiment Requirement

Determine the reflectivity of the $p$ -component as a function of the incident angle (the electric field component, parallel to the plane of incidence is called the $p$ -component).

1(a) Specify the transmission axis of the polarizer (A) by the position of the marked line on the degree scale disk in the $p$ -component measurement (the transmission axis is the direction of vibration of the electric field vector of the transmitted light). [1 т.]

1(b) Choose any one of the light intensity detector and set its micro-ammeter at the range of “×5”. Verify the linear relation ship between the light intensity and the micro-ammeter reading. Draw the optical schematic diagram. Show your measured data and calculated results (including the calculation formula) in the form of a table. Plot the linear relationship curve. [2 т.]

1(c) Determine the reflectivity of the $p$ -component as a function of the incident angle. Draw the optical schematic diagram. Show your measured data and calculated reflectivity (including the calculation formula) in the form of a table. Plot the reflectivity as a function of the incident angle. [4 т.]

2) Determine the refractive index of the sample as accurate as possible. [3 т.]

Explanation and Suggestion

Laser radiation avoid direct eye exposure.
Since the output power of the laser beam may fluctuate from time to time, the fluctuation of light output has to be monitored during the performance of the experiment and a correction of the experimental results has to be made.
The laser should be lighting all the time, even when you finish your experiment and leave the examination hall, the laser should be keeping in work.
The reflected light is totally plane polarized at an incident angle $\theta_B$ while

\operatorname{tg}\theta_B=n\quad\text{(refractive index).}

Insert the plug of photocell into the “INPUT” socket of microammeter.
Switching on the microammeter.
Blocd off the light entrance hole in front of the photocell and adjust the scale reading of micro ammeter to “0”.
Set the “Multiple” knob to a proper range.

Решение

1(a)

Determine the transmission axis of the polarizer and the Brewster angle $\theta_B$ of the sample by using the fact that the reflectivity of the $p$ -component $R_p=0$ at the Brewster angle.

Change the orientation of the transmission axis of $P_1$ , specified by the position of the marked line on the degree scale disk $(\psi)$ and the incident angle $(\theta_i)$ successively until the related intensity $I_r=0$ .

Original polarizer orientation/Brewster-angle optical arrangement, all incident/reflected beam arrows and angular labels.

Now the incident light consists of $p$ -component only and the incident angle is $\theta_B$ , the corresponding values $\psi_1$ and $\theta_B$ are shown below:

$\psi_1$	$140.0^\circ$	$322.0^\circ$	$141.0^\circ$	$322.5^\circ$
$\theta$	$56.4^\circ$	$56.4^\circ$	$56.2^\circ$	$56.2^\circ$

\psi_1=140.5^\circ\pm0.5^\circ\quad\text{or}\quad322.3^\circ\pm0.1^\circ.

The Brewster angle $\theta_B$ is $56.3^\circ\pm0.1^\circ$ .

1(b) — Verification of the linear relationship between the light intensity and the microammeter reading

Original two-polarizer/beam-splitter arrangement with both detectors and all intensity labels.

The intensity the transmitted light passing through two polarized $P_1$ and $P_2$ obeys Malus’ law

I(\theta)=I_0\cos^2\theta,

where $I_0$ is the intensity of the light polarized by $p_1$ and incident, $I$ is the intensity of the transmitted light, and $\theta$ is the angle between the transmission axes of $P_1$ and $p_2$ . Thus we can obtain light with various intensities for the verification by using two polarizers.

The experimental arrangement is shown in the figure.

The light intensity detector $D_1$ serves to monitor the intensity fluctuation of the incident beam (the ratio of $I_1$ to $I_2$ remain unchanged), and $D_2$ measures $I_2$ . Let $i_1(\theta)$ and $i_2(\theta)$ be the readings of $D_1$ and $D_2$ respectively, and $\psi_2(\theta)$ be the reading of the marked line position. $i_2=0$ when $\theta=90^\circ$ , the corresponding $\psi_2$ is $\psi_2(90^\circ)$ , and the value of $\theta$ corresponding to $\psi_2$ is

\theta=\left\lvert\psi_2-\psi_2(90^\circ)\pm90^\circ\right\rvert.

Data and results;

\psi_2(90^\circ)=4^\circ.

$\psi_2$	$94.0^\circ$	$64.0^\circ$	$49.0^\circ$	$34.0^\circ$	$4.0^\circ$
$\theta$	$0.0^\circ$	$30.0^\circ$	$45.0^\circ$	$60.0^\circ$	$90.0^\circ$
$i_1(\theta)$ , $\mu\mathrm{A}$	$6.3\times1$	$5.7\times1$	$5.7\times1$	$5.7\times1$	$5.7\times1$
$i_2(\theta)$ , $\mu\mathrm{A}$	$18.7\times5$	$12.7\times5$	$8.2\times5$	$4.0\times5$	$0.0\times5$

From the above data we can obtain the values of $I(\theta)/I_2(\theta)$ from the formula

\frac{I(\theta)}{I_0}=\frac{i_2(\theta)}{i_1(\theta)}\cdot\frac{i_1(0)}{i_2(0)}

and compare them with $\cos^2\theta$ for examining the linear relationship. The results obtained are:

$\theta$	$0.0^\circ$	$30.0^\circ$	$45.0^\circ$	$60.0^\circ$	$90.0^\circ$
$\cos^2\theta$	$1.00$	$0.75$	$0.50$	$0.25$	$0.00$
$I(\theta)/I_0$	$1.00$	$0.75$	$0.49$	$0.24$	$0.00$

Original measured normalized-intensity versus cos-squared graph with full axis formula and numeric scales.

1(c) — Reflectivity measurement

The experimental arrangement shown below is used to determine the ratio of $I_0$ to $I_1$ , which is proportional to the ratio of the reading $(i_{20})$ of $D_2$ to the corresponding reading $(i_{10})$ of $D_1$ .

Original reference beam-splitter optical circuit with both intensity detectors.

Then used the experimental arrangement shown below to measure the relativity $R_p$ of the sample at various incident angle $(\theta)$ while the incident light consists of $p$ -component only. Let $i_1(\theta)$ and $i_2(\theta)$ be the readings of $D_1$ and $D_2$ respectively.

Original reflectivity optical circuit including sample, both detector paths and2theta marker.

Then the reflectivity is

R_p(\theta)=\frac{I(\theta)}{I_0}=\frac{i_2(\theta)}{i_1(\theta)}\cdot\frac{i_{10}}{i_{20}}.

Data and results:

\psi_1=140.5^\circ,\qquad i_{20}=19.8\times5\ \mu\mathrm{A},\qquad i_{10}=13.3\ \mu\mathrm{A}.

$\theta$ ( $^\circ$ )	$i_2(\theta)$	$i_1$ ( $\mu\mathrm{A}$ )	$R_p(\theta)$
5	$15.1\times0.2$	11.1	0.037
10	$14.9\times0.2$	11.2	0.036
20	$13.3\times0.2$	11.1	0.032
30	$11.4\times0.2$	12.2	0.025
40	$7.8\times0.2$	14.7	0.014
50	$2.3\times0.2$	16.9	0.0037
53	$0.7\times0.2$	11.3	0.0017
55	$0.3\times0.2$	11.3	0.00059
56.3 (dark)	$\sim0$	11.5	$\sim0$
58	$0.3\times0.2$	11.5	0.0007
60	$1.1\times0.2$	13.5	0.0024
64	$6.5\times0.2$	16.7	0.011
66	$7.8\times0.2$	11.8	0.018
68	$16.3\times0.2$	15.0	0.029
72	$5.3\times0.1$	11.7	0.061
76	$13.1\times1$	14.0	0.13
80	$4.4\times5$	11.7	0.25
84	$9.1\times5$	14.5	0.42

The curve of reflectivity of $p$ -component as a function of incident in plexiglass

Original reflectivity graph, complete angle/reflectivity scales and Brewster-angle minimum.

2)

The Brewster angle $\theta_B$ can be found from the above date as

\theta_B=56.3^\circ\pm0.2^\circ.

The index of refraction can be calculated as

n=\tan\theta_B=1.50\pm0.01.

The sources of errors are:

Detector sensitivity is low.
The incident light does not consist of $p$ -component only.
The degree scales are not uniform.

Grading Scheme (10 points)

Part 1. Reflectivity of the $p$ -component. 7 points, distributed as follows.

a. Determination of the transmission axis of the polarizer (A) in $p$ -component measurement, 1 point.

Error	Points
less than $\pm2^\circ$	1.0
less than $\pm3^\circ$	0.7
less than $\pm4^\circ$	0.3
less than $\pm5^\circ$	0.1

b. Verification of the linearity of the light intensity detector (2 points). Draws the optical schematic diagram correctly, 1.0 point; (Without the correction of the fluctuation of the light intensity, 0.4 point only);

Uses $I/I_0\sim\cos^2\theta$ figure to show the “linearity”, 0.5 point;

Tabulate the measured data (with 5 points at least) correctly, 0.5 point.

c. Determination of the reflectivity of the $p$ -component of the light as a function of incident angle, 4 points, distributed as follows.

Draws the optical schematic diagram correctly and tabulate the measured data perfectly, 2.0 points;

Plot the reflectivity as the function of incident angle with indication of errors, 2 points.

Part 2. Determination of the refractive index of sample, 3 point.

Brewster angle of sample, 1 point;

Error	Points
less than $\pm1^\circ$	1.0
less than $\pm2^\circ$	0.5
less than $\pm3^\circ$	0.2
larger than $\pm3^\circ$	0

The refractive index of sample, 0.5 point.

Discussion and determination of errors, 1.5 points.

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