Low-cost optical home-lab experiments

Low-cost experiments have the advantage of being affordable to schools, universities, and families. In the particular case of optics, these experiments can be important to help students understand the properties of light and optical phenomena. In this article, we demonstrate optical experiments to be carried out at home by students learning remotely. These experiments were designed and performed at the 16th Summer School in Physics, which is a five-day activity for high school students organized by the University of Porto. This activity was organized remotely, with all lectures and activities being performed through video conferencing. The materials needed for the proposed experiments were sent by post to students before the beginning of school. The students were able to build optical experiments from scratch, perform measurements, analyse data and present their findings on the last day of the school in a public session dedicated to online presentations.


Introduction
Optics is a branch of physics concerned with the study of light, namely, its nature, properties, phenomena, and also its technological applications [1].However, optics goes beyond the physics departments around the world.Whether we are using a microscope in microbiology, a telescope Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. in astronomy, a spectroscopy technique in chemistry, or a basic camera, we are all using light as a tool to study other phenomena.Thus, learning the basics of optics is important for students no matter what academic path they will follow.And one way of teaching students is through handson experiments, which provide students with the opportunity to test their own ideas and build their own understanding [2].
In this paper, we present low-cost optical experiments to be performed at home by students under the supervision of a mentor via videoconferencing.For our definition of low-cost home lab experiments we used and adapted the definition of [3]: (i) the equipment is generally available The experiments described in this paper were performed by four students in the context of the Summer School in Physics at the University of Porto (a translation from Portuguese as 'Escola de Verão de Física da Universidade do Porto').This is a one-week hands-on summer school for highschool students, who are able to take advanced short courses (e.g.relativity, quantum mechanics, nanotechnologies and astrophysics) and engage in research activities with local researchers.Students are also able to choose research projects proposed by local researchers (henceforth known as 'supervisors') and are then divided into small groups of three or four members.In the final day of the school, each group presents their work to the public.Due to the restrictions related to the coronavirus pandemic, the 16th Summer School at Porto was completely online, which made the experimental part of this work challenging for both the students and the supervisors.Our proposed activities for students, in a project called 'An odyssey through the properties of light' (a translation from Portuguese 'Uma odisseia pelas propriedades da luz'), were to explore some of the properties of light through simple experiments that could be performed at home in a safe environment through videoconferencing.This would enable students to construct their own experimental setups, perform observations, data analysis and compare their results with the other members of the group.To this end, a 'home-lab kit' was developed in advance and sent to the students' addresses.Table 1 summarizes the material sent to the students.The laser sent was a custom-build diode-laser which has a potentiometer to regulate the intensity and was designed with a maximum power below 5 mW for safety reasons.Before starting the project, students were given instructions on laser safety, in particular, to have special attention to reflections, to turn on the laser only when the setup was completed, and to make sure that any reflections would happen parallel to the 'optical table '.This paper is structured as follows.In sections 2-5 we present the four optical experiments undertaken by the students at home, addressing the theoretical background, experimental methodology, and results.The first experiment presented in section 2 deals with imaging through a thick lens.The second and third experiments, described in sections 3 and 4, respectively, are both related to diffraction, where students generate and analyse diffraction patterns.Section 5 covers a simple polarization experiment.Finally, in section 6 we draw some conclusions regarding this event.

Determination of the focal length of a thick lens
The main goal of this experiment is to determine the focal length of a thick lens.To achieve this, students can use the two types of light sources sent: (i) the LEDs with different colours and sizes and (ii) the laser source.However, we encourage students to use LEDs for pedagogical reasons.The knowledge of the focal length will be important for the experiment described in section 3.

Theoretical background
With reference to figure 1, consider a thick lens, consisting of two spherical refractive surfaces separated by a thickness d between their vertices, V 1 and V 2 .Consider also that both surfaces have the same radius of curvature in magnitude, i.e. |R 1 | = |R 2 | = R. Assuming that the thick lens is immersed in air, one can use the following relation between the conjugated points Q and Q ′ [4]: where s o is the distance between the object and the primary principal plane, s i is the distance between the image and the secondary principal plane and f is the (effective) focal length.The latter is given by where n l is the refractive index of the thick lens.Note that we use the following sign convention: at its right.If we assume that the thick lens has a symmetric radius, i.e.R 1 = −R 2 = R, we have that Equation ( 3) is important from a manufacturer's perspective since to design a thick lens with a given focal length f, one needs to take into account the thickness d, the refractive index of the lens n l and radius R. The (transverse) magnification of an image is defined as [4] If an object is placed at a distance s o larger than the focal length of the lens (s o > f), a real image will be formed with a negative magnification (m t < 0) since both s i and s o are positive.If, on the other hand, the object is placed at a distance s o smaller than the focal length, a virtual image will be formed and s i will be negative, thus producing a positive magnification (m t > 0).In this situation the lens will act as a magnifying glass.Usually, when buying a magnifying glass, the manufacturer gives the magnifying power M, which is related to the focal length f through the following approximation where d o is the near point for the human eye, i.e. the closest distance that the human eye can focus (typically 250 mm [4]).

Experimental setup and procedure
The ultimate goal of this experiment is to experimentally estimate the focal length of the thick lens without having in advance any information about the lens.In this experiment we used the LEDs as object for two main reasons: (i) to give students the liberty to choose which and how many LEDs they want to use by building their own electrical circuit, and (ii) to easily explore the concept of image formation by varying the distance s o of equation ( 1) and the colour, the size and the position of LEDs with respect to each other.Note that they must be always parallel to the object plane.The first step in the experiment is to build a circuit containing LED sources.The LEDs can be inserted in series or in parallel if they have the same forward voltage (usually the same colour).In the former case, the maximum number of LEDs in series will be dependent on the voltage of the battery used since each LED will have a threshold voltage.For instance, if a given type of LED has a voltage threshold of 2.5 V and we use a 9 V battery with a resistor of 400Ω, we will only be able to put 3 LEDs in series.To place different LEDs in parallel, one must insert a dedicated resistor in series with the LED so that the voltage drop in each parallel path matches that of the battery.This way, the forward voltage for each LED is achieved with a specific dedicated resistor.It is also recommended to use the same procedure with LEDs of the same colour, since they may have slightly different forward voltage.Another advantage is that in this configuration the voltage drop across each parallel path remains the same even if one or more LEDs become faulty.For simplicity, we will demonstrate the placement of the LEDs in series.
Let A be the object plane at a distance s o from the primary principal plane B, as illustrated in figure 2. If all LEDs are placed in the object plane, then their image will be formed in the same plane, A ′ , at a distance s i .Thus, the second step of the experiment is to align all LEDs towards the lens so that all LEDs have the same distance s o .Note that if the distance is too small (s o < f), no real image is formed.After correctly placing the LEDs, one is left with placing a screen at a distance s i , where the student thinks the image is formed.One way to find s i is to find the position of the screen where the image has a relative minimum in terms of area.Note that one can do the same procedure but instead of fixing s o and try to find s i , one can do the inverse, i.e. choose a constant distance s i and vary s o .

Results and discussion
Students set up the experiment illustrated in figure 2 and varied the number of LEDs used, their size (3 mm and 5 mm), and their colour.Photographs sent by one of the students can be found in figure 3.After defining the object  and image planes, they measured the distances s o and s i and retrieved the focal length f by applying equation (1).The mean value obtained by the students was f avg = 25.6 mm.The thick lens sent to the students was a cheap magnifying glass with a magnifying power of M = 10, given by the manufacturer.Using equation (5) and assuming d 0 = 250 mm, the manufacturer's value for the focal length was considered to be 25 mm.Therefore, the value was very close to that given by the manufacturer.This value for the focal length will be important in the next experiment, where students will use the same lens to collimate light from LEDs.

Diffraction by multiple slits
Using compact disks as diffraction gratings to perform diffraction experiments by multiple slits is very well-known and has been previously described by many authors (see, for example, [5][6][7][8][9][10][11][12][13]).It was also recently tested as a home-lab experiment during the lockdown due to COVID-19 pandemic [14].The ultimate goal of the experiments described in this section is to show students how diffraction works and how it can be useful, not only to study the diffracting object itself but also to learn more about the properties of the light source (e.g.study its spectrum through wavelength dispersion) and to understand how light interacts with matter.

Theoretical background
Consider a diffraction grating consisting of a large number N of grooves, separated by a distance d.The grating can work in transmission or reflection, as illustrated in figures 4(a) and (b), respectively.In both cases, the maximum constructive interference occurs at an angle θ m , given by [15] sin where θ i is the incident angle, θ m is the diffracted angle of order m and λ is the wavelength of light.The sign conventions of the angles θ i and θ m are as follows: if the incident and diffracted rays are on the same side of the grating normal ⃗ n, the diffraction angle θ m is considered positive (see figure 4(a)), otherwise, θ m is considered negative (see figure 4(b)).If the light beam direction is parallel to the surface's normal vector ⃗ n, i.e. θ i = 0, equation ( 6) reduces to In this particular case, the sign convention is irrel- Although this model give us information about where the diffraction orders will occur, it does not give us the relative intensity of each order.As will be observed in the experimental results, the diffraction orders for |m| > 1 will have lower intensity for increasing values of |m|.To explain this phenomenon one needs to use wave optics [16,17] or statistical optics in the case of partially coherent light [18], which is out of the scope of this work.

Optical media as diffraction gratings
Optical storage media (e.g.CDs, DVDs, and Blurays) are often used as examples of diffraction gratings.We will briefly describe why diffraction takes place, in particular, for recordable media (e.g.CD-R, DVD+R) since those are the samples used in the experiments.For the sake of simplicity, we will not go into many details on how optical media are created and how it works.For more details see, for example, [19][20][21].
In general, optical storage media have a single line of data in a spiral pattern, as depicted in figure 4(a).Depending on the type of optical media, the structure of the spiral pattern can differ.In recordable media such as CD-R, DVD+R, and DVD-R, a spiral pre-groove is already present in the disk and it is filled with a dye 'recording layer', as illustrated in figure 4(b).If a laser beam with a beam diameter much larger than the separation of the grooves d is incident on the optical disk, then the laser beam will be diffracted similarly to the situation of a multiple slit diffraction depicted in figure 4 and the diffraction angles will be given by equation (6).However, if we remove the reflector material, protector and label depicted in figure 5, the optical media will act as a transmission-type grating.We did this in the case of the CD-R sent to the students so they can use it as both reflective and transmissiontype grating.To save material, we actually 'cut' each optical disk (CD-R, DVD+R and Blu-ray) in two halves with appropriate tools in order to save material.
Optical disks act as diffraction gratings when light beams have diameters greater than the width of the grooves and, in the case of being much larger than the separations of grooves d, they act as reflective-type multiple-slit gratings.The width of the groove and the separation between adjacent  .a is the width of the pit and d is the pitch, i.e. the distance between sequential pits.On the bottom, we can see the read laser in two cases.Case 1: the laser is reflected back.Both these cases form the basis of how digital data is stored.Case 2: when the dye is 'burned', the laser is not reflected back.grooves (track spacing) depend on the type of recordable optical medium.Table 2 summarizes the three main optical media used and sent to the students.

Experimental setup and procedure
Two goals were defined for this experiment with optical media: (i) determine the track spacing d of the optical media sent, and (ii) determine which LEDs are composed of only one 'pure' colour, i.e.LEDS that are not made of the merging of other colours.In the first experiment, one can use two types of light sources: a laser or an LED source.In the latter case, we will need to collimate light using the thick lens used in section 2. Since students already determined the focal length and are acquainted with imaging and light collimation, this procedure should not take too much time.Figure 6 illustrates the configuration that should be used to collimate three-point sources.Note that, in this case, the three source points belong to the same light source, since LEDs have a finite width.In this case, diffraction occurs since light impinging on the grating has a significant amount of spatial coherence when compared to the track spacing.The extent of spatial coherence, namely, the effective correlation length σ µ , can be calculated using van Citter-Zernike's theorem [16,22] where D is the diameter of the source.Thus, for 3 mm and 5 mm blue LEDs (λ 0 = 465 nm), the effective correlation lengths are 47.3 µm and 28.4 µm, respectively, which is larger than the track spacing of all the optical media sent, including the holographic diffraction grating.If one uses a laser source, this collimation is usually not needed, assuming that the divergence is neglected for the propagation distances of the experiment.Note that, if the incident angle θ i is nonzero, then according to equation ( 6), more diffraction orders can appear.
The second experiment uses the fact that the diffraction angle is wavelength dependent.Along with the traditional red, yellow, green and blue LEDS, we also send students white and magenta LEDs.These last two LEDs are actually made of more than one colour band.In this case, both the white and the pink LEDs are made by merging more than two colour bands.The idea is for students to find out by themselves the spectrum of these LEDs and compare them with the other traditional ones.

Results and discussion
Students started the diffraction experiments by using the laser (λ 0 = 650 nm) and the holographic diffraction grating since it was more easy to set up, as shown in figure 7(a).Since it has 1000 lines mm −1 , the track spacing is d = 1 µm.The main goal was for students to verify that the diffraction angle of order 1 was θ 1 ≈ 40.5 degrees (for θ i = 0) and that there was no second-order diffraction due to the limits of the sine function in equation ( 7).The angles were measured with paper rulers and using simple trigonometry.
Then, the students used the same procedure with a different grating, the CD-R.In this case, they started by using the CD-R as a transmission grating (this avoids insecure laser reflection) to measure the angle of diffraction, as depicted  7(c).Then, students tried using the DVD+R in reflection mode but had some difficulty extracting the angle θ m from the measurements.But the big surprise came when no diffraction was observed with the Blu-ray disk.Students were puzzled by this phenomenon and then, when given the track spacing (0.32 µm), they performed the calculation of the diffraction angle θ 1 for m = 1 and realized that for this wavelength of light, there was no solution for θ m in equation ( 7), concluding that in order to see diffraction they would need a light source with a shorter wavelength.The students were invited to discuss their results and think laterally.They could distinguish the CD from the DVD according to the differences in the diffraction pattern.They realised that this concept is also used in the identification and study of crystalline materials (e.g.metals, crystallised proteins) through a technique called x-ray diffraction [23,24].Moreover, they also understood why the electromagnetic radiation used for this technique is in the x-ray region.
Finally, the final goal was to use wavelengthdependent nature of diffraction to observe the spectral composition of two special LEDs sent to the students, namely, a white and a 'magenta' LED.Students used the experimental setup described in figure 8(a), which is the same as the one used for the results of figure 7(c).Some students used the transmission part of the CD-R, and others used the holographic grating.For the white LED, students could observe that the spectrum was composed of many colours.But the big surprise came with the magenta LED.Contrary to what happened with the blue LED in figure 7(c), the magenta LED did not have only one 'visible' colour, but two, red and blue, as depicted in the photograph of figure 8(c).Students were then shown the spectra of the white and magenta LED represented in figures 8(d) and (e), respectively, that were taken at the laboratory before sending them by post.

Diffraction by a single slit and Babinet principle
This experiment has the two following goals: (i) explore the concept of the wave nature of light with Babinet's principle and (ii) measure the width of the thin object using the same principle.

Theoretical background
As we saw in section 3, light diffracts when an object is placed in its path.If the object is small enough, we will be able to see the diffraction pattern.Note that the fact that we do not see any diffraction pattern with our eyes, does not mean that diffraction has not occurred.In what follows, we will present a simplistic theoretical framework of Babinet's principle.Then, we will summarize the main results for the single-slit experiment, namely, the condition for finding a minimum in the diffraction pattern.For more details, we refer the reader to [4,16].
One of the most fascinating consequences of diffraction theory is what is called Babinet's principle.Suppose we have two situations as represented in figures 9(a) and (b).In the former case, light impinges on a screen (plane A) with a rectangular aperture of width a.A diffraction pattern is collected at a distance R. In the latter case, light hits an object with the same width as the previous aperture, a. Once again, the diffraction pattern is recorded at the same distance R. Babinet's principle states that in this case, where both screens are complementary, i.e. the opening of one screen corresponds to the opaque portion of the other [16], the field amplitudes will be such that that the sum of them will correspond the field produced by the 'sum' of both screens.In this case, the 'sum' of both screens will be a totally opaque screen and, therefore, Babinet's principle states that, in this case, both fields are symmetric and the sum is zero, as can be observed in figures 9(e) and (f).However, since our eyes detect the intensity and not the field, the squared of the field will be the same.Thus, Babinet's principle also states that the intensity (or diffraction pattern) of the two complementary fields are the same.In this particular case, the diffraction of an object with a width a will produce the same intensity pattern as that of the single-slit diffraction.We will next briefly describe a particular case to find the dark fringes (zero intensity) in the diffraction pattern, namely, through destructive interference conditions.For a detailed description using Fourier optics, we refer the reader elsewhere [17].
With reference to figure 10, consider a light beam with wavelength λ impinging on a slit with  width a and that a diffraction pattern is observed on a screen placed at a distance R from the slit.If the distance is large enough such that R ≫ π a 2 /λ, then we will be within the Fraunhofer approximation [4,16,17].In this case, the pattern is made up of bright and dark fringes, where a large central bright fringe is found at the centre of the screen.It can be shown that the dark fringes are observed at angles θ m given by [15,25] sin θ m = m λ a (m = ±1, ±2, ±3, . ..) .(9) Note that m is the order for destructive interference, contrary to equation (6), where the order n is for constructive interference.The angle θ m can be written as a function of y m and R, i.e.
For small angles of θ m , we can use the approximation sin θ m ≈ tan θ m .Then, if we substitute equation ( 10) into (9) and solve it for a, we obtain the following relation: Thus, by knowing the wavelength λ and by measuring the distances y m (for a given order m) and R, one can estimate the size a of the slit using equation (11).

Experimental setup and procedure
The main goal of this experiment was to measure the thickness of a thin wire sent to the students using Babinet's principle.The first step is to align the laser parallel to the screen at a large distance (R > 30 cm) to facilitate the observation of the diffraction pattern (see figure 9(b)).Then, a thin wire is placed immediately after the laser.The distance R, which is measured between the screen and the object, is annotated.Then, the laser is turned on and the brightness (i.e.optical power) can be adjusted using the potentiometer.When the diffraction pattern is visible, one uses a pen to mark in the paper screen the position of the central beam and of a given dark fringe, counting the order of the fringe (e.g.m = 1 for the dark fringe next to the central beam).After that, one uses equation (11) to determine the width of the object.

Results and discussion
Students started the experiment by setting up the configuration depicted in figure 9(b) using the available material, namely, the laser, a thin wire as the diffracting object, a ruller and a paper screen with ruled markings.An example of a setup made by the students is shown in the photograph of figure 11(a).By increasing the distance of the screen relative to the object, students were able to see the diffraction pattern (see figure 11(b)).
Then, using a pen, they chose a minimum and marked the position of that minimum and of the central lobe with a pen.Note that the contrast in this experiment may not be the best.Students took photographs of the pattern to figure out what was the order of the minimum they selected, as illustrated in figure 11(c).Then, they turned off the laser and measured the distance between the minimum and the central lobe with a ruler.At this point, students had enough data to estimate the width of the thin wire.The average value measured by the students was 0.076 mm and, therefore, the error was 9%.

Polarization of light
The last experiment aims to experimentally demonstrate to students the electromagnetic nature of light, namely, the ability of the electric field of light to oscillate in several directions.

Introduction
In classical electrodynamics, light is described as an electromagnetic wave, composed of both electric and magnetic fields, which are mutually orthogonal.The direction of propagation of light (defined by the so-called Poynting vector) is also orthogonal to both the electric and magnetic fields.The polarization of light is commonly associated with the direction of the oscillation of the electric field, which is a transversal polarization since it is in the transverse plane of the direction of propagation.This polarization is usually classified into three categories: (i) linear, (ii) circular, and (iii) elliptical.The latter can be viewed as the most general case, where linear and circular polarization are particular cases of elliptical polarization.Here we will only deal with the case of linear polarization.For more details on the polarization of light, we refer the reader to [26].Natural light sources, such as the one generated by blackbody radiation, are unpolarized, meaning that the direction of oscillation of the electric field is random.Other light sources, such as lasers, have some degree of polarization, where a great part of the electric field is oscillating in a given direction.Some materials show different absorption spectra when light has different polarization states.When these materials are used as optical filters, i.e. they are used to filter or to transmit a given polarization state, they are called polarizers. Figure 12 illustrates how these materials can be used to block a given polarization state.

Experimental setup and procedure
This experiment had two main goals: (i) to determine the polarization state of the laser diode of each student, and (ii) to check if the LEDs have visible polarization state.
As mentioned in section 1, one custom-built laser diodes were sent to each of the four students.Two lasers were horizontally polarized and the other two were vertical polarized.To verify the polarization direction, students have a polarizer with marking stating the transmitted polarization direction.The experimental setup is to simply place the polarizer after the light source and then place a screen after the polarizer.If one observes a change in intensity by rotating the polarizer, then that means that there is a visible polarization direction that is more predominant.Once again, if the light source used is an LED, it is important to perform a collimation as represented in figure 6.

Results and discussion
Students performed the experiments using both the collimated LEDs and the laser.For the LEDs, students did not observe any noticeable change in intensity by rotating the polarizer.Figures 13(a) and (b) show examples where the polarizer is rotated by 90 degrees using a blue LED as the light source.However, when students used the laser diode as a light source, a visible difference in the intensity of light was observed (see  figure 13(c)) when using the polarizer in 0 or 90 degrees.Students were thus able to correctly determine the polarization direction of their laser according to the specifications of our custom-built laser diode.After that, students used the polarizers to polarize light from the LEDs and verified it by adding another polarizer (the analiser) rotated by 90 degrees, as illustrated in figures 13(d) and (e).They were able to track differences in the intensity of the LED light on the screen by changing the angle of the analyser with respect to the polarizer.If they were able to measure the intensity of light they would verify the Malu's law [27,28].

Conclusion
The 16th edition of the summer school in physics organized by the University of Porto for high-school students was for the first time realised remotely using online free videoconferencing software.The main part of this school was dedicated to hands-on physics projects, which had also to be done remotely.Experiments traditionally taught in the laboratory had to be redefined and designed to be done at home with a low-cost kit.Here we presented a project where students, under remote supervision, were invited to explore optical phenomena and learn about the properties of light using geometric optics and wave and electromagnetic theories.In the end, students performed a video conferencing public presentation, explaining their project and showing the results.The feedback given by the students was positive, although they would certainly had advantages in working together in the same environment at the laboratory, sharing ideas more easily, and having access to more sophisticated lab equipment.On the other hand, the kit is reusable and with it, the students can reproduce the experiments on their own and try new ones.

Figure 1 .
Figure 1.Image formation by a thick lens.H 1 and H 2 are the first and second principal points, respectively, where the primary and secondary principal planes intersect the optical axis.F 1 and F 2 are the first and second focal points, respectively, d is the thickness of the lens, Q and Q ′ are the object and image points, respectively.

Figure 2 .
Figure 2. Illustration for the experimental setup for imaging LEDs.A and A ′ are the object and image planes, respectively.B and B ′ are the primary and secondary principal planes, respectively.

Figure 3 .
Figure 3. Imaging of two LEDs (3 mm yellow and blue).(a), (b) Photographs of the experiment taken by a student from different perspectives.(c) Illustration of the inverted magnification caused by the positive thick lens.

Figure 4 .
Figure 4. Notation used for the diffraction of light by a transmission and reflection grating.(a) Transmission grating.(b) Reflection grating.

Figure 5 .
Figure 5. CD-R typical characteristics.(a) Illustration of the spiral pre-groove imprinted in polycarbonate (not to scale).(b) Different layers of a CD-R (not to scale).a is the width of the pit and d is the pitch, i.e. the distance between sequential pits.On the bottom, we can see the read laser in two cases.Case 1: the laser is reflected back.Both these cases form the basis of how digital data is stored.Case 2: when the dye is 'burned', the laser is not reflected back.

Figure 6 .
Figure 6.Collimation of light of three source points.Here, A and C are the front and back focal plane of the lens, respectively.

Figure 7 .
Figure 7. (a) Students' results of diffraction in transmission.(a) Diffraction by a grating with a known track spacing (1000 lines mm −1 ) using a laser diode.(b) Diffraction by a CD-R using the laser diode.(c) Diffraction using collimated light from a LED source.

Figure 8 .
Figure 8. Diffraction of collimated light from white and pink LEDs.(a) Illustration of the experimental setup used.Note that we are simplifying by illustrating an LED as a point source, but one should expect a small divergence as shown in figure 6.(b) Screen with the diffraction pattern of the white LED.(c) Screen with the diffraction pattern of the magenta LED.(d) White LED spectrum.(e) Orange LED spectrum.

Figure 9 .
Figure 9. Babinet's principle.(a), (b) Complementary screens, where in (a) we have a rectangular aperture with width a and in (b) we have a rectangular opaque object with width a. (c), (d) Intensity I(x) simulated for both cases (a) and (b), respectively, where the following parameter values were used: λ = 650 nm, R = 0.5 m and a = 0.07 mm.(e), (f) Field amplitude simulated for both cases (a) and (b), respectively.Notice that, although the intensity pattern is the same, the sum of the field amplitudes give zero, as expected from Babinet's principle.

Figure 10 .
Figure 10.Single-slit experiment parameters for determining the intensity minima (Fraunhofer diffraction).a: width of the slit.A and B: source and observation planes, respectively.R: Distance between A and B. ym: distance between the centre of the screen and the minimum of order m (m = 3 represented).θm: angle between the line segments OO ′ and ODm, where Dm is the point representing the mth dark fringe (D 3 represented).

Figure 11 .
Figure 11.Results for the measurement of the thin wire width using Babinet's principle.(a) Photo of experimental setup sent by the students.(b) Photo sent by the students of the diffraction pattern obtained.(c) Image of (b) with gamma correction used to observe the first-order diffraction.

Figure 12 .
Figure 12.Polarizer filtering light.(a) A polarizer that blocks horizontally polarized light.(b) A polarizer that blocks vertically polarized light.

Figure 13 .
Figure 13.Polarization experimental results.(a), (b) Collimated light from a blue LED passes through a polarizer that transmits horizontal (a) and vertical (b) polarized light.(c) The laser beam passes through a polarizer that transmits vertically polarized light.(d), (e) Light from an LED (no collimation) passing through a horizontal polarizer then by another horizontal polarizer (d) or by a vertical polarizer (e).The white double arrows indicate the transmitting axis of the polarizer.

Table 1 .
List of materials sent to each student by post.
and low-cost; (ii) the experimental setup is simple to construct; (iii) the presentation explaining the activity is short; (iv) the experiments are motivated by real-world technological applications; and (v) the observed experimental results should cause surprise and discussion among the participants.

Table 2 .
Characteristics of optical media used in the experiments.