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EuPhO 2018, Експеримент — Задача 2. Interference

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Всички материали от EuPhO 2018

Porous membrane · 6 т.

General remarks

Provide detailed sketches of all the setups you use. Estimate errors only in part A. Depending on your method, you might be able to complete the tasks without using all the given equipment.

Porous membrane

You are going to study a membrane of anodic aluminum oxide. The membrane is transparent; it has thickness $h$ and cylindrical channels of diameter $d$ as shown in figure. Your goal is to determine parameters $h$ , $d$ and porosity $p$ (the volume fraction of channels in the membrane). Assume the membrane to be optically homogeneous and please do not touch the surface of the membrane!

Условие

Equipment Optical bench; membrane (identical to the one in part A) on a support; laser, $\lambda = 660\ \mathrm{nm}$ ; 2 polarizers (axis of polarizer is marked with a line and forms $45^{\circ}$ with the edge of the frame); photodiode (short-circuit current is proportional to the light intensity); multimeter; wires; clips; 2 rulers; glue pads; white paper.

The reflection intensity depends on the angle of incidence $\alpha$ due to interference of rays reflected from top and bottom surfaces of the membrane.

Cross-section of the membrane (grey slab): a red ray falls at angle α to the normal and splits into a ray reflected at angle α from the top surface and a ray refracted down to the bottom surface, reflected there and emerging parallel to the first reflected ray.

Task Determine the thickness $h$ of the membrane. Assume the refractive index of the membrane to be $n_o = 1.50$ . To avoid birefringence described in part C incident light should have polarization perpendicular to the plane of incidence (plane of the figure). If the contrast of interference is too weak, try the other surface of the membrane.

Отговори

Решение

Optical path difference between beams reflected from top and bottom sides of the membrane:

\delta = 2\delta_1 - 2\delta_2 + \frac{\lambda}{2}

$\frac{\lambda}{2}$ appears because of phase change in case of reflection from the surface with higher refractive index.

\frac{\sin\alpha}{\sin\beta} = n

Geometry of thin-film reflection: slab of thickness h between two horizontal lines, incoming rays at angle α to the dashed normals, refracted path δ1 going down at angle β to the bottom surface and back up (angles β, β), and the extra path δ2 of the ray reflected at the top surface, marked with right-angle marks.

\delta_1 = n\frac{h}{\cos\beta}

\delta_2 = \frac{h}{\cos\beta}\sin\beta\sin\alpha

\delta - \frac{\lambda}{2} = \frac{2h}{\cos\beta}(n - \sin\beta\sin\alpha) = \frac{2h}{\cos\beta}(n - n\sin^2\beta) = \frac{2nh}{\cos\beta}\cos^2\beta = 2nh\cos\beta = 2nh\sqrt{1 - \sin^2\beta}

= 2nh\sqrt{1 - \frac{\sin^2\alpha}{n^2}} = 2h\sqrt{n^2 - \sin^2\alpha}

\delta = 2h\sqrt{n^2 - \sin^2\alpha} + \frac{\lambda}{2}

(1)

Reflectance maximums can be observed if $\delta = m\lambda$ , where $m$ is integer. Reflectance maximums can be observed if $\delta = (m + 1/2)\lambda$ .

Rotating the sample, we can see dozens of minimums and maximums. $\sqrt{n^2 - \sin^2\alpha}$ depending on the number of minimum is linear according to the equation (1), as shown on the graph. Slope is equal $2h/\lambda$ .

Plot of the minimum number m' (0 to 70) against (n²− sin²α)^0.5 (1.2 to 1.5): red square data points labelled 16-16 on a black linear fit; inset table: Intercept −268.90752 (standard error 0.57503), Slope 226.05386 (standard error 0.42826).

Slope and its uncertainty were calculated with OLS.

h = 74.6 \pm 0.2\ \mu m

This data is shown for demonstration. During the competition, students were not supposed to measure positions of that much number of minimums and plot a graph, because this does not improve accuracy. The best solution is to rotate the sample and calculate number of minimums between two angles $\alpha$ . The more is number of minimums $N$ , the better is accuracy. In this case, uncertainty $\varepsilon_h$ may be estimated as $\varepsilon_h = 1/N$ .

The sample is not fixed on the rotating table. Before the measurement, one should set a zero using the reflected laser beam.

Colour photograph of the red laser beam reflected from the membrane onto a white paper screen, used to set the zero angle.

Оригинал в Архива: exp_pr.pdf · официални решения: exp_sol.pdf