2.2.1

Geometrical Optics

In the study of Geometrical Optics, we have used art works by René Magritte¹² and M. C. Escher¹³, see Fig. 4. We presented three objects (a glass ball and two Christmas tree balls) similarly to the work of M. C. Escher “Three Spheres II”, 1946, and we show an image of this work. We encourage a discussion about how light interacts with these bodies and what role materials and surfaces play. At the conclusion of this discussion we turn off the light of the room to observe the effects of a disco ball hanging from the ceiling and illuminated by the light of the projector. When students were encouraged to apply the conclusions of the discussion to the disco ball phenomena. We used several other works by Escher, as he has explored many optical phenomena, including when we did activities with kaleidoscopes and topology. We use several images of Magritte works to stimulate a discussion of image formation and other phenomena such as illumination. These discussions using the works of Escher and Magritte were especially stimulating for students, as it is a challenging approach to a subject that is usually very simplistically treated in physics courses.

Figure 4.

(a) Students were encouraged to discuss concepts of Geometrical Optics starting from questions about how light interacts with three spheres portrayed by M. C. Escher. (b) Then they had to apply their conclusions to explain what they observed with a disco ball hanging from the ceiling illuminated by the projector, see the small spots of light in the room. (c) La condition humanaine, (d) L’empire des lumières are examples of images of René Magritte’s works presented to stimulate a discussion involving image formation and illumination.

Practical activities with mirrors included flat and curved mirrors, kaleidoscopes, fractal images using kaleidoscopes of Christmas tree balls and stainless-steel spoons^3,14,15. We set up a Hall of Mirrors (Sala de espelhos), show in Fig. 5, which is a kaleidoscope with six large flat mirrors (0.9 m x 1.8 m each one) forming a hollow prism of regular hexagonal crosssection. This is a kaleidoscope in which people can get in, close the door and perceive the many different images of themselves. The students had this experience and showed great enthusiasm, taking photos and videos with their Smartphones. We discuss the Moebius transformations in the kaleidoscopes and topology aspects, which included the projections of the sphere (with application in cartography), where we also use images of Escher works. Concluding the discussion about topology the students constructed a tri-hexaflexagon using a sheet of paper, using the technique shown in our video: “Topologia: de Moebius ao Flexágono”¹⁶ (“Topology: from Moebius to Flexagon”).

Figure 5.

Students inside the big kaleidoscope called Hall of Mirrors no Espaço Ciência, Cultura e Educação (ECCE), which is a space for scientific dissemination in the EACH-USP.

The 7th lesson is about light refraction, when we discussed how light crosses transparent media with a microscopic model¹, how propagation speed varies with wavelength and what happens when light changes medium. We speak of homogeneous and non-homogeneous media¹⁷ and of total internal reflection. We use images, videos and demonstrations with prisms, glass sphere and liquids, showing the change of light direction in refraction, total internal reflection and how it can be frustrated. Before presenting rainbow images, we asked the students what details they could remember from their own observations and knowledge about this phenomenon. Most cited only colored bow with seven stripes, some quoted the possibility of two arcs but did not know about the inverted order of colors in the two arcs. Asked about the position of the image in relation to the Sun, few could answer correctly. Inspired by W. Lewin “Rainbows and Blue Sky”¹⁸ and in the work of H. M. Nussenzveig¹⁹, we present explanations of the details of the rainbow, describing this phenomenon as a colored caustic. Still on the rainbow, we made a video showing that it is polarized light²⁰, which in that lesson was shown only as motivation for the study of light polarization. This part of the lesson was summarized in the video with translated name as “Rainbow a colorful caustic”²¹.

On non-homogeneous media, where differential refraction occurs, we present videos showing the mirage effect, Fata Morgana effect¹⁷ and the distortion of the Sun near the horizon, as can be seen in Fig. 6. We also did a video showing and explaining the mirage effect happening in a hot road²². During the first time we taught this lesson, we used mirage-effect videos available on YouTube, where the mirage is seen on a road by a camera that has fixed position on the ground. At the end of the explanation and observation of the videos, a student asked, “There was water on the road, was not there?” Then we realized that it would be better to also show a video in which the camera travels along the road and shows that there is no water at the site of the mirage, as we show in this video.

Figure 6.

Example of differential refraction distorting the apparent format of the Sun during sunrise. Assembly made with images extracted from our video translated as “Sun rising (differential refraction)”²³.

During this lesson, the students did a quantitative practical activity by measuring the variation of the refractive index of the water, when it dissolves salt. An intense green laser diode beam is refracted by an acrylic cuvette filled with water, the incidence and refraction angles are measured, and the refractive index of the water is calculated. Without turning off the laser, they mixed salt into the water and watched the beam move. The visual effect is great because we use great distance, in this way, we are watching the refracted beam reach the wall on the other side of the room (about 5 meters away). The students are surprised by the large displacement of the light spot on the wall. Measuring the new angle of exit they were able to calculate the new refractive index and showed contentment by the understanding of the numbers, therefore, the measured quantities make some sense to them because they observed the phenomenon. We did not give instructions on how to measure the angles, an important part of the activity was that they planned and executed the measurements using simple tools available, such as rulers and protractor.

The next lesson was about applying refraction to lenses and devices with lenses. We start by putting the example of correction glasses and what is the relation between the focal distance and the diopter, called degree of glasses in Brazil, that is the magnitude of everyday life. Next, we discussed and demonstrated imaging effects with spherical and thin lenses, and students manipulated various devices with lenses. The practical activity was to set up a camera using a common magnifying glass, a cardboard box and a sheet of translucid paper as screen.

2.2.2

Introduction to Wave Optics

The 9th was the Second Test described in section 2.4 Evaluation. In the 10th and 11th lessons, we present Wave Optics or Physical Optics discussing the phenomena of diffraction, interference and structural colors. Practical activities were both qualitative and quantitative. We made demonstrations using cheap laser pointers and objects such as wires and hair of different thicknesses, razor blades borders, handmade slits, color observation in soap films, diffraction gratings and CDs. For the quantitative experiments the students used diffraction and interference kits from Pasco, to measure the width of a slit. And they measured the wavelength of a laser beam using the diffraction grating shown in Fig. 1, using the method very similar to that used in the Light Origin, as described in Fig. 7.

Figure 7.

Method for finding the wavelength of a laser beam using diffraction grating (grade de difração) and tape measure (trena). This method is very similar to the one described in Fig. 1. We can compare the distances between the first order spots (1^a ordem) and the zero order spot, to improve alignment and achieve a right triangle. We measure the legs (cateto oposto, cateto adjacente) with the tape measure, to calculate the hypotenuse and the sine of the angle corresponding to the first order of diffraction.

We also used peeled off CDs as a transmissive diffraction grating and DVDs as reflective diffraction grating, as shown in Fig. 8. The distances between the grooves are a = 1.6 x 10^-6 m for CD and a = 7.4 x 10^-7 m for DVD²⁴. In these experiments we used diode laser found in laser light projector for decorations, that has a good light intensity. Regarding the use of these discs as diffraction gratings, it should be noticed that grooves are curved in this case, which causes differences and asymmetries when compared to experiments using standard gratings. But students soon find the best positions the beam should reach the disc, to have more symmetry between the two-sided images and less distortion. The experimental results are as good as the ones reached by using standard gratings.

Figure 8.

To show that we can do Wave Optics experiments with everyday objects, students also used peeled off CDs (a) and DVDs (b) as transmissive and reflective diffraction gratings. The class of 2018 used the standard diffraction grating to find the wavelength of the laser. They exchanged the standard grid for the peeled off CD and found the CD constant.

2.2.3

Geometrical Theory of Diffraction

We started the 12th lesson showing the divisions of Optics (see Fig. 9) and the position of the Geometrical Theory of Diffraction (GTD) developed by Keller²⁵. We discuss the models used throughout the history of Optics and in every branches of that science: concepts such as particles, rays, waves, wave-particle duality.

Figure 9.

The divisions of Optics and the Geometrical Theory of Diffraction in the interface between Geometrical Optics and Wave Optics.

Keller’s theory is applied in Engineering, in systems involving acoustics and high-frequency radio waves. Keller reinterpreted the Huygens principle using Young’s principle, emphasizing the directionality of the wave and works well in the interface between Geometrical Optics and Wave Optics because he writes the electric field of light as a linear combination of wave and geometric models:

where α and β are adjusted as required²⁵. This theory was little used for the explanation of optical phenomena. By studying the laser beam scattering by soap foam²⁶, we came across with the light pattern shown in Fig. 10 and we noticed the need to apply the GTD to explain the cone of light diffracted by liquid wires present in the foam²⁷. By studying the optical phenomenon in detail, we notice that the half opening angle of the cone of diffracted light is equal to complementary angle of reflection, so, it is a wave phenomenon that occurs at an angle of reflection, as if it were a ray of light. This realization led us to GTD. We can change the radius of the circle by varying the angle of incident beam on the liquid thread, known as Plateau border. When the laser beam reaches the wire perpendicular to the direction defined by the thread, we have the Fraunhofer diffraction.

Figure 10.

(a) Light pattern obtained by projection on a screen of laser beam scattering in soap film structures called Plateau borders. (b) Names given to the pattern details: parlaseric circle (Círculo Parlasérico), diffraction (Difração), laser spot (Ponto do laser), laser dog (Cão de laser). We also call the diffraction straight lines as laser pillars. (c) The parlaseric circle is the projection of a conical shell of light with half angle of opening ϕ. The angle of incidence of the beam is θ, which is complementary to ϕ. The Plateau border that scatters the light has a triangular profile²⁷.

There are other details in the light pattern, such as Fraunhofer diffraction lines and crossing points between Fraunhoffer lines and the circle that we call laser dogs. Laser dogs are beam reflections on the walls of Plateau border. Plateau border is the meeting of three soap films, in other words, it is a liquid thread that has a triangular cross section. The laser spot is the undisturbed part of the beam. The laser dogs are reflections on the sides of the triangle and the Fraunhoffer lines are the diffraction in the films that are lying on that thread. It is worth mentioning that the circle has characteristic fringes of diffraction of light and we call it a parlaseric circle, by analogy to the parhelic circle observed in the sky that occurs due to the scattering of light by ice crystals. We call the reflections as laser dogs and Fraunhoffer lines as laser pillars, by analogy to the Sun dogs and Sun pillars, which are also light phenomena seen in the atmosphere. In a subsequent work we discuss more the apparent paradox of refracted rays, that is, light obeying simultaneously the law of the reflection of Geometrical Optics and diffracting like wave. This is not a paradox, because it is a phenomenon that lies in the limit between Geometrical Optics and Wave Optics which are models that the human beings elaborated to study the paths of light²⁸.

We discussed these concepts and demonstrated them to the students using the Soft Matter Laboratory equipment that we used in the research projects. Then we challenged them to reproduce the parlaseric circle using everyday objects such as the material shown in Fig. 11 and Fig. 12: an acrylic box (CD case), diode laser pointer, a pointer holder, aqueous solution of detergent, as shown in our video²⁹ “Faça um HALO com LASER” translated as “Make a HALO with LASER”. Thickness and smooth surfaces of soap films make them great candidates for Wave Optics experiments, as students had already learned in class about structural colors.

Figure 11.

The CD case has soap films that diffracts the laser beam (No estojo de CD tem filme de sabão que difrata o feixe laser). The laser beam must reach a Plateau Border. (O feixe laser deve incidir numa Borda de Plateau). The Plateau borders are formed in the encounter of liquid films of soap solutions. In these images we can see two Plateau borders that can be used in the experiment. To get these films, we plunged the opened acrylic box into the detergent solution, closed the box and took it out slowly, letting the solution run, more details in the video “Make a HALO with LASER”²⁹.

Figure 12.

Objects found in the house that can be used for observation of the parlaseric circle. CD case (narrow acrylic box), laser pointer, a tall and thin object used as a pointer holder and aqueous detergent solution.

We also present a demonstration of a parlaseric circle using a Ferrocell (Fig. 13), known as a magnetic lens³⁰. In that case, there is a superparamagnetic liquid called ferrofluid, which consists of ferromagnetic nanoscopic particles suspended in an oil. When there is no external magnetic field, the magnetic moment of the particles distributes randomly, so the liquid does not present macroscopic magnetization. When we apply a magnetic field, for example, by approximating a magnet, the magnetic moments of the particles align with the field and the attraction between the particles makes needle-like agglomerates as can be seen in the micrograph of Fig. 13. b. The needles are aligned with the magnetic field, so the needles are parallel to each other. When there is an incident laser beam, without the magnet field, we see only two laser spots, reflections on Ferrocell glass plates, Fig. 13. c. When the external magnetic field is applied, we see a parlaseric circle which is the sum of many parlaseric circles, one for each needle the beam strikes, Fig. 13. d. Here there are no fringes of diffraction because there is superposition of several circles. So, we have a fringe-free parlaseric circle when there is Keller diffraction on many aligned nanoscopic needles.

Figure 13.

(a) Ferrocell is composed of two flat glass plates, between them there is a thin layer of ferrofluid, as a sandwich. Ferrofluid is a suspension of ferromagnetic particles. (b) When an external magnetic field is applied, there is the alignment of the particles making needle-shaped agglomerates, which can be seen in the micrograph. (c) Without magnetic field, an incident laser beam is only reflected by the glass plates and a little bit scattered by the ferrofluid without macroscopic magnetization. (d) When we apply a magnetic field, the nanoscopic needles line up with the external field and we see a continuous parlaseric circle which is the overlap of diffraction in the various needles reached by the beam³⁰.

Now the analogy with the light patterns seen in the sky makes more sense, if we think of the ice crystals in suspension in the atmosphere as needles aligned with each other. Then there is the possibility that for certain atmospheric conditions, where ice crystals are microscopic needles or long prisms, that their alignment occurs by effects such as aerodynamics or electric field, since the ice crystals may be electrically charged, or polarized, as they are made of water. We use Figure 14, to discuss this model in which the atmosphere is seen as a complex fluid with properties similar to ferrofluid, liquid crystal, and layers of micelles. Sunlight is a plane wave like the laser beam, but with all wavelengths visible. This model may be one of the possibilities for explaining the occurrence of the parhelic circle in the atmosphere. The most common explanation is also presented to students, in which sunlight is scattered by ice crystals in the form of horizontal hexagonal plates in all directions, following the laws of Geometrical Optics³¹. Other atmospheric phenomena were presented with our images or others available online, such as halos, glory effect, sun miracle, iridescent clouds. We also present results from our work with jumping laser dogs in which we make analogy with the jumping Sun dogs, using the model of suspended crystals in the atmosphere as a complex fluid.

Figure 14.

Diagram used to make the analogy between the parlaseric circle and the parhelic circle, modeling the suspended ice crystals in the atmosphere as a complex fluid.

2.2.4

Light Polarization and Birefringence

The 13th lesson is about light polarization and birefringence. We discuss theoretical concepts and applications by making various demonstrations with polarizing filters, LCD displays, polarizing (linear) sunglasses, 3D cinema glasses (circular polarizers), birefringent objects (calcite and plastics), examples in Figure 15, several are in our video about light polarization and birefringence³². The videos are good for showing blue sky and rainbow polarization effects and online images like F. Snik’s “Reflecting Beetles”³³ showing circular polarization effect of reflected light on beetles are examples of application of concepts in natural phenomena that we could not reproduce in the didactic laboratory.

Figure 15.

Examples of demonstrations of polarization and birefringence. (a) Overlap of polarizers on the LCD screen of a notebook. (b) Plastic object superimposed on the LCD screen, observed without and with polarizer, showing a birefringence colored effect.

The good conceptual discussion of this lesson was achieved with the seemingly simple demonstration using only three polarizing filters, which can be explained by the vector transverse-wave model. First the students see the light from a ceiling lamp of the room cross a polarizer, then add a second polarizer and rotate one of them observing the variation of the luminous intensity that crosses the pair of filters. With the axes of the double crossed, in such a way that the luminous intensity is minimal, the third polarizer is added between them with axis at an angle of approximately 45° with respect to the axes of the pair. The visual effect is wow type. The students discussed and experimented until they found the transversal field vector explanation.

Another important activity was students wearing 3D movie glasses, looking at the mirrors and closing one eye. For example, when we close the left eye and only looking through the right eye, we cannot see the right eye in the mirror. The light could not pass through the same polarized lens twice, because one lens is a left circular polarizer, the other lens is right circular polarizer, and the reflection in the mirror inverts the circular polarization. We heard some wow moments. We discussed the differences between sunglasses that have two equally linearly polarizing lenses and the 3D cinema glasses with different circular polarization orientations lens. We also discussed the stereoscopic effect of vision to explain how the 3D cinema works.

We present a demonstration using a Ferrocell under the influence of the magnetic field of a super magnet, in which the ferrofluid acts as a non-homogeneous polarizer, because nanoscopic needles are formed, which align with the magnetic field³⁴. In Figure 16, we have the Ferrocell between two polarizers with crossed axes, as a sandwich. The white illumination is of a panel-shaped luminaire, the dark circle in the middle is the shadow of the cylindrical magnet that is leaning against one of the polarizers. In Ferrocell plane, the magnetic field lines are radial, so the nanoscopic needles are radially aligned. In the vertical and horizontal directions of the image, the needles are aligned with the axis of one of the polarizers, cutting much light, we have dark regions. In other directions the needles aligned at intermediate angles allow more light intensity to cross the assembly. Ferrocell is functioning as a non-homogenous polarizer with the polarizing axes in the radial directions.

Figure 16.

Ferrocell between two polarizers with crossed axes, under the influence of a cylindrical super magnet with a pole leaning on the assembly. The lighting is white and diffused, the middle circular shadow is of the magnet. The magnetic field in the Ferrocell plane is radial, which makes it function as a non-homogeneous polarizer with polarizing axes in the radial direction³⁴.