An Arduino-based experiment designed to clarify the transition to total internal reflection

A quick scan of physics textbooks, old and new, and an internet search on the topic of total internal reflection (TIR), reveal that there is a widespread misunderstanding of what actually happens when light crosses the boundary of two transparent media of different refractive indices. Figure 1 encapsulates typical diagrams purporting to demonstrate the phenomenon.

Zoom In Zoom Out Reset image size

The figure claims to show the passage of light from an optically dense medium such as glass or Perspex (refractive index n  = 1.495) into air. The angle of incidence is θ. Diagram (a) suggests that there is no partial internal reflection (PIR) when the incident angle is less than the critical angle while in (b) at the critical angle θ_c there is a transmitted ray along the boundary with no internal reflection at all. Diagram (c) shows a boundary ray and an additional totally internally reflected ray! All three diagrams are commonly seen in the media, and all of them are wrong.

Such descriptions are very confusing for the student and, one suspects, for many teachers. What is the fact of the matter?

Figure 2 shows correct depictions.

Zoom In Zoom Out Reset image size

Notice that in (a) the incident light is split into a transmitted component which is refracted into the air at angle ϕ and a component which is internally reflected at an angle equal to the angle of incidence θ. In (b) the angle of incidence is equal the special value θ_c known as the critical angle. In this case, all of the light is internally reflected and we have the important case of total internal reflection (TIR). It is important to note that in this situation, there is no light along the boundary. In (c) the angle of incidence θʹ is greater than θ_c and all the light is internally reflected.

A common source of confusion is in consideration of the transition from partial internal reflection to TIR. It is often maintained that there is some kind of discontinuity at this point, but this not the case.

In Eugene Hecht's wonderful book Optics [1] we find

'It should be stressed that the transition......takes place without any discontinuities. As θ becomes larger, the reflected beam grows stronger and stronger while the transmitted beam grows weaker, until the latter vanishes and the former carries off all the energy......'

And this is the key. The intensity of the refracted and internally reflected beams is rarely mentioned in elementary texts. As we approach the critical angle, fewer and fewer photons are transmitted into the air, with more and more being turned back into the transparent medium. At the critical angle, none is transmitted and all are reflected. The confusion disappears along with any transmitted photons! We can think of the process in terms of probability. As θ increases, the probability of a photon emerging into the air becomes less and less, and finally becomes zero at the critical angle.

The origin of the confusion is probably that, within an educational context, it is relatively difficult to demonstrate these changes in intensity, particularly as changes in the latter are not appreciable for small angles of incidence. We need ideally to observe the reflected intensity and plot it as a function of θ. This may seem to require a laborious experimental procedure. However, I shall show how this can be achieved with relatively simple equipment engineered by a competent technician.

Figure 3 shows an arrangement I have used which facilitates the measurement of reflected and transmitted beams produced by a 15 mm thick semicircular Perspex block. The latter ensures light entry is normal and unchanging in reflectance for all angles of incidence. A small 1 mW red laser is used as a convenient light source.

Zoom In Zoom Out Reset image size

Measuring the transmitted beam directly would be problematical as the angle of refraction is not linearly related to θ (Snell's Law). Instead, the angle θ is measured along with the corresponding intensity of the reflected beam. Conservation of energy then permits the intensity of the transmitted beam to be deduced. Metal strips A, B, C and D, each of equal length, are linked in the form of a pantograph via pins P1, P2, P3 and P4 as shown. P1 is fixed, while P4 is free to move along a slot cut in the plastic base (of approximate dimensions 23 cm  ×  32 cm). Upward motion of P4 varies θ and ensures that the detector is always in the correct position to receive the reflected beam. The laser is mounted on strip A and the detector on strip B.

The Perspex block rests on three supports glued to the plastic base and which are arranged to allow free movement of strips A and B under the block. A small red plastic filter is placed in the output beam of the laser as it was found that without it, the detector would saturate.

The light detector consisted of the very cheap and convenient TAOS TSL 254R-LF¹ which incorporates a photodiode and op amp on a single chip. The device is robust and easy to use, requiring a supply in the range 2.7–5.5 V. It produces an output potential difference which is directly proportional to the light intensity. Figure 4 shows the pin connections for the sensor as viewed from the lens side of the device.

Zoom In Zoom Out Reset image size

Measurement of the angle θ is done indirectly. As P4 slides along the slot it is clear that θ will vary. P4 is connected to the slider of a linear potential divider glued to the underside of the base. Thus the electrical output of the divider will be a measure of θ (a rotary divider could be used but was found to be physically more difficult to attach to P1).

P4 is attached to the sliding tap of the potential divider under the base, and as P4 moves up the slot, the angle θ clearly changes. In figure 5 the relationship between the tap position and θ is shown. The length of the strips in the pantograph is S and y is the separation of P1 and P4.

Zoom In Zoom Out Reset image size

We have

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle y={{y}_{{\rm s}}}-L\nonumber \end{align} \tag{ 1 }$$

where y_s is the maximum separation of P1 and P4.

Also

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle y= 2S\cdot \cos \theta .\nonumber \end{align} \tag{ 2 }$$

For the potential divider

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{V}_{{\rm out}}}=\left\{\frac{L}{{{y}_{{\rm s}}}-{{y}_{{\rm e}}}} \right\}{{V}_{{\rm in}}}\nonumber \end{align} \tag{ 3 }$$

where y_e is the minimum separation of P1 and P4.

Eliminating y between (1) and (2) and using (3), we obtain

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \theta ={\rm arcos}\left[ \frac{1}{2S}\cdot \left\{{{y}_{{\rm s}}}-\frac{{{V}_{{\rm out}}}}{{{V}_{{\rm in}}}}\cdot ({{y}_{{\rm s}}}-{{y}_{{\rm e}}} \right\} \right].\nonumber \end{align} \tag{ 4 }$$

The output voltage, V_D, from the light sensor and V_out from the potential divider are both sent to the analog input ports of the Arduino microcontroller. The system is shown in figure 6 and the Arduino code in figure 7.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The 'float' statements at the head of the program contain values for y_s, y_e and S. In my system these were 184 mm, 138 mm and 100 mm respectively. A voltmeter indicated that the maximum from the detector was 3.9 V when the laser was switched on. This corresponds to the incident intensity of the light. The θ offset value of 0.25° was included to allow for some latitude in the pantograph movement.

The reflectance R is defined as the relative intensity of the light internally reflected and is the ratio of the detector voltage to its maximum value.

The transmittance T is the relative intensity of light refracted out of the Perspex block. By conservation of energy T  +  R  =  1.

The connections shown in figure 6 are made and the Arduino is connected to a PC or laptop already loaded with the code in figure 7. Any voltage in the range 0–5 V presented at an analog port (such as A0 and A1) generates an integer in the range 0–1023 which explains the presence of the conversion factor G  =  0.004 8876  =  5/1023 in the program. After the program has been uploaded, it is also necessary to load a suitable graph-plotting package such as MakerPlot² which I have used on numerous occasions with some success. The MakerPlot XY plotting interface should be opened and suitable scales chosen. In this case, the X-axis can be used for θ and the Y-axis for both the reflectance R and transmittance T. A convenient range for θ is 30°–50° (the critical angle for Perspex is 42°).

With both programs running it is then necessary to move P4 along the slot in a smooth fashion. (I have found that varying θ from its maximum to its minimum value gives the best results.) The curves for T and R will then appear automatically in the graph area. A sample output is shown in figure 8.

Zoom In Zoom Out Reset image size

Figure 9 shows the pantograph board on the bench and figure 10 shows the system with a simple black cowl fitted. The latter largely eliminates problems arising from ambient light entering the system. A black-painted piece of plastic gutter with glued end-pieces makes an effective cowl.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

The experimental curves in figure 8 clearly show that the reflectance (green curve) smoothly increases with the angle of incidence, reaching a maximum at 42°—the critical angle for Perspex. At the same time the transmittance (red curve) falls smoothly, reaching zero at the critical angle.

The electromagnetic wave theory of light produces the well known Fresnel equations, one of which gives the reflectance for light which is linearly polarised normal to the plane of incidence:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{R}_{}}=~{{\left\{\frac{\sin (\theta -~\varphi)}{\sin (\theta +~\varphi } \right\}}^{2}}\nonumber \end{align} \tag{ 5 }$$

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {\rm where}\,\varphi ={\rm arcsin}(n.\sin \theta).\nonumber \end{align} \tag{ 6 }$$

A plot of this function and corresponding transmittance for Perspex is shown in figure 11.

Zoom In Zoom Out Reset image size

It can be seen that, although not exact, there is general agreement with the experimental curves. The solid-state laser used in these experiments was not characterised in terms of the type and degree of polarization of the light it emitted and this fact doubtless accounted for the small discrepancies between the experimental and theoretical curves.

Using the apparatus described above, it should be possible, as a lecture demonstration, to show changes in reflectance and transmittance occurring in real time, thus serving to dispel the all-too-common misrepresentations of the transition to TIR seen in the media and text books on optics. Alternatively, the construction and assembly of such a system would make an ideal project for a small student group to work on. The pantograph concept would also be applicable to other optical investigations such as polarization by reflection and the Brewster angle.

It is hoped that interested readers will reproduce and hopefully improve these experiments. My other Arduino-based experiments [2–4] may also be of interest.

I should like to thank Byron Blackadder for his skilful construction of the pantograph mechanism.

My appreciations also go to Dr Tim Searle and Simon Gray for taking the time to read this paper and for their expert, critical, and constructive comments.