Adding more coherent light to the education

The laser technology and its usage are often neglected in physics courses at high schools. However, not only Generation Z but also the subsequent generations will be surrounded by technologies using (albeit covertly) lasers. It is desirable that students know about this technology and are able to use it. There are a plethora of STEM experiments that can be done with the help of laser light. We successfully demonstrate some of these experiments together with a short presentation about laser principles to high school students who visit our faculty as part of various events. These experiments can also be performed in high schools, each of them showing one or more of the three basic features of laser light - coherence, monochromaticity, and directionality.


Introduction
The 21 st century is sometimes called the century of the photon.This is caused mainly due to the huge boom in laser technology since its first lighting up in the year 1960 [1].Since then, laser light was slowly but steadily entering into all different kinds of technology.Lasers are all around us, we just do not realize it because we take them for granted.Many of us have a mobile phone equipped with a laser, we use laser printers, and the bar code scanners in the shops are based on lasers, too.We can see the more powerful units in the doctor's office, or when used for cutting and welding.Last but not least, one of the world's advanced laser facilities, ELI Beamlines, is located in the Czech Republic.The working principle of the laser, which is explained on a nice analogy e.g. in [2][3][4], ensures its three main properties -coherence, monochromaticity, and directionality [5].Today, the laser diode can be purchased for several tens of Euros, thus it is accessible even as the high school lab equipment, and it offers a nice STEM experimenting possibilities.

The Junior Academy project at Faculty of Science, University of South Bohemia
Under the title "Junior academy," interactive programs are taking place at the Faculty of Science, University of South Bohemia.The preparation and implementation of the programs involve not only experienced scientists from the Faculty of Science but also our students in teaching programs.The duration of each event is approximately one day, and high school students can choose topics from the fields of physics, chemistry, biology, mathematics, and geography.At the same time, students can explore the modern facilities of the scientific research institution informally accompanied by their peers, students in teaching programs.The high school students (usually from the last two years of studies) apply themselves to our coordinator.This is important step, since the student takes the whole responsibility on himself/herself.It also means that the group of visiting students can be a mix of students from the same school but different classes, or even a mix from different schools.This can challenge the students from the communication point of view.Depending on their high school, they usually have two-or four-years physics course.The overall number of students, who visited this program in the last year, is in dozens.More than twenty of them took part in the physics part.In the physics laboratory, a program primarily in the field of optics is prepared.However, we strive to design experiments that cover other fields as well, such as mathematics, biology, or chemistry.This approach has proven very successful because not only high school students but also students from other secondary schools, who may not have physics throughout their studies, come to us.First, an introductory session is held with the student.It includes a presentation, but the emphasis is on dialogue with students and their understanding of the basic principles and properties of lasers.This is followed by a tour of the ultrafast spectroscopy laboratory to see how lasers are used in scientific practice.Finally, the students themselves try their hand in the optical laboratory and assemble several experiments from scratch, as described in the following text.The combination of presentation with dialogue and manual work in the lab proved to be the right one, since some students appreciated more the first part, while others the second part.When the students arrive to the working place (optical lab), the empty optical table is there (figure 1), with all the optical elements, the students will need, disassembled on it.The students then split into three groups, each working on the separate optical path and separate experiment.The instructions for them are simple: First, clean the mirrors, you were given.Then guide the laser beam above the path, labelled on the optical table by the sticky tape with the equipment given to your group.The students have to figure out themselves, how to assemble the individual holders, which screwdrivers to use, which holder is best for certain position, how to guide the beam through a pair of irises, etc.They are enabled to have a look on the main optical table with the ultrafast spectroscopy setup, since the holders are assembled there.After the students succeed to guide the laser beam to the target position, each group is introduced with the final experiment (see below).We ask the students, how do they plan to assemble the setup and what result do they expect to observe.The expectations are then discussed thoroughly.

Visualization of the diffusion process
Diffusion is a spontaneous process in which particles (atoms, molecules) disperse from areas of higher concentration to areas of lower concentration.This phenomenon occurs in various fields, such as technology (desalination, gas dilution), pharmacology (pharmacokinetics), and medicine (drug and toxin transport).High school students encounter diffusion in physics, biology, and chemistry.In biology, it is important for understanding cellular and tissue processes, while in chemistry, it is associated with osmosis.Despite its significance, it receives relatively little time in teaching.Research among first-year university students has shown that many students have limited knowledge of diffusion and lack an understanding of basic quantities and concepts, such as Fick's law or gradient [6].Experimentally, diffusion is typically demonstrated in schools by dropping ink into a glass of water.However, this example not only shows diffusion but also other physical processes.The water on the surface evaporates, cooling the water and creating downward flow.This causes the ink to mix more quickly than if the entire process relied solely on diffusion.In the next part, we will focus on what to consider when conducting diffusion experiments and how to perform a simple laser experiment that provides a unique view of diffusion.A light beam refracts at the interface of media with different refractive indices.This also applies to less distinct interfaces, such as air above a hot road (heat shimmer).Stable layers of air with different temperatures and thus different refractive indices form in such conditions.A similar gradient can be created at the interface of two liquids with different refractive indices.Thanks to diffusion and the method used, we can observe its development over time.However, to achieve pure diffusion without additional parasitic phenomena, we must pay attention to a few things.Firstly, we need to limit evaporation to prevent liquid turbulence.This can be achieved with two simple steps: (i) Reduce the evaporation area by choosing a container with the smallest possible surface area for the liquid's surface.(ii) Limit the air flow above the surface by covering the container.Also, avoid moving the container after filling it to avoid disturbing the sample.The experiment's schematic is shown in figure 2. The experiment involves spreading a laser beam into a plane using a cylindrical lens with a 6 mm diameter.This spread beam passes through a transparent container measuring 5x5x1 cm.The container is first filled with water to half, and the second liquid, a salt solution (0.2 g/l), is gently introduced to the bottom of the cuvette using a pipette or syringe.The salt solution, having a higher density than water, remains at the bottom of the container until the two liquids mix through diffusion.It's worth noting that the transition between the liquids can be observed with the naked eye.After passing through the sample, the laser beam hits a screen where the pattern can be observed, as shown in figure 3.Alternatively, you can place a camera behind the screen to capture the pattern at different times, discussing the changes (decreasing of the peak and its widening).

Diffraction on bird's feather
When electromagnetic radiation passes through a material with distances between planes comparable to the wavelength of the incident radiation, diffraction occurs.This phenomenon is commonly used in X-ray crystallography to determine internal structures.Since the typical distances between crystal planes are between 1 and 100 Å, this method requires the use of X-ray radiation, leading to the name "X-ray crystallography."The necessity of using X-ray radiation was realized by Paul Peter Ewald and Max von Laue in 1912, 17 years after the discovery of X-rays.Von Laue, along with two colleagues, invented the first device that could record X-ray diffraction and successfully tested it on a crystal of blue vitriol [7,8].Von Laue later expanded the theory of X-ray crystallography, and two years later, he received the Nobel Prize for it.This method was used thirty years ago to decipher the double-helix structure of DNA (see photo 51 [9]).With a spring from a pen, this method can be demonstrated using a laser [10].Although this experiment can be conducted with simple equipment, it requires a relatively complex optical system because the spring has large dimensions compared to the wavelength of laser radiation.However, the spring can be replaced with a biological material, such as bird feathers.This material has been used as a simple optical grating since the time of Isaac Newton [11].The experiment's schematic is shown in figure 4a.The obtained diffraction pattern on the screen (figure 4b) can then be used to calculate the angles between individual branches, rays, and hooks of the feathers and their widths.The results can be verified through direct observation under a microscope or by comparing them to literature.

Fresnel-Arago-Poisson Spot with a Twist
The history associated with observing the Fresnel-Arago-Poisson spot (FAP spot) is a beautiful example of the need to experimentally verify scientific theory, even when it predicts highly counterintuitive results.According to the wave theory of light presented by Augustin-Jean Fresnel, this theory predicted that there would be a bright point in the middle of the shadow of a circular obstacle.This result was calculated by a critic of the wave nature of light, Siméon Denis Poisson.To experimentally confirm this fact, Dominique-François-Jean Arago was assigned the task and, to the delight of A.-J. Fresnel, he indeed observed a bright point in the middle of the shadow of a circular obstacle [12].Arago created coherent light necessary for this experiment using a flame and a set of filters.Nowadays, we have a much simpler task, as we can use coherent light from a laser (figure 5a).However, what happens if we use an elliptical obstacle instead of a classic circular one to demonstrate the FAP spot?An elliptical obstacle can be achieved in two ways.If we use a flat circle as the obstacle, such as a coin, we slightly tilt it to demonstrate the FAP spot.The second option is to use an elliptical obstacle directly, which can be 3D-printed.Whichever method you choose, it is recommended to attach the obstacle to a microscopic cover glass, making it easy to place it in the experiment and avoiding the need to address parasitic effects that would occur if, for example, a thin wire were used to position the obstacle.The experiment is conducted as follows: Allow a light beam from a laser with an appropriate coherence length (a HeNe gas laser is suitable) to pass through a telescope (or lens) to enlarge its diameter.Place the obstacle into the laser beam so that it nearly completely obstructs it.Then, let the remaining beam hit a distant wall, and observe the FAP spot in the center of the shadow.When using an elliptical obstacle, you won't observe a bright point in the middle of the shadow but a pattern called an evolute (figure 5b) [13].The evolute is determined by the envelope of its normals (figure 5c), and students can create it during a mathematics or computer science class using a computer.

Laser Lunar ranging
As part of the presentation, we also discuss one interesting experiment -Laser Lunar ranging.The method is very simple in theory.We send a laser pulse to the object of interest.The light reflect from the object (or the so-called retroreflector on its surface to get more reflected light), and returns to the measuring device, which detects the returning pulse.The time t between releasing and obtaining the laser pulse is recorded.Since the speed of light c is known, it is easy to calculate the distance as We need to realize that the light travels to the object and back, thus we have to divide the recorded time by two.The experiments with the laser distance measure are nicely presented for example in the Interactive Physics Lab at the Faculty of Mathematics and Physics at the Charles University in Prag [14].The students here measure with the help of laser tool the same distance in two different mediums -air and water.For the air, the measured distance is as expected.But since the speed of light in water is slower than in the air, the laser measure shows higher value for this case.By comparing these two measured values, students can calculate the index of refraction of water.Nonetheless, we will now focus on the larger scale distance measurement.
To introduce this so-called Lunar laser ranging experiment to students, the scene from the TV show The Big Bang Theory (season 3, episode 23, named The Lunar Excitation) available on YouTube can be used.Here, the group of scientist uses the rather small equipment, which they brought on the roof of their apartment, to measure the distance to the Moon.The principle of the experiment is described here pretty well.The light pulse is sent to the retroreflector on the Moon, bounces back, and it is then detected by a photomultiplier.But would the experiment be so easy, so one can conduct it by himself on the roof?The astronaut carrying the retroreflectors.Images taken from [17,18].most used retroreflector is the one from the last Apollo 15 mission, since it provides the largest reflecting surface.We can notice one more nice detail in the figure 6b, which shows the astronaut carrying the retroreflector for the Lunar laser ranging experiment.The retroreflector weight was about 20 kg [16].Nonetheless the lower gravitational force on the Moon compared to the Earth enables to this equipment quite comfortable.
As it was said in the show, our naked eye is not able to see the returning photons.Neither a common detector would be good enough to "see" the light coming back.The major problem is that no light source can be totally directional, meaning there is always some divergence [5].This causes there are less and less photons, which can make all the way from the laser source back to our detector, as it is shown in the scheme of the Lunar laser ranging experiment in figure 7. On the contrary the divergence of the beam makes the pointing to the retroreflector placed on Moon a bit easier, since the irradiated area is larger.To be able to calculate, how many photons we will detect, we need to know the basic input information, such as the number of photons in one pulse, approximate distance to the Moon, which can be obtained by other methods, and the divergence angle.Let's begin with calculating the number of photon in one pulse.The typical parameters of the laser used for the Lunar laser ranging experiment are: wavelength λ = 532 nm, pulse duration ca 100 ps, and the pulse energy   = 100 mJ [19].The pulse duration only affects the precision of the measurement, not the number of the photons.
To calculate the number of photons N, we divide the energy of the pulse Epulse by energy of the individual photon (Ephoton) of given wavelength, where h is the Planck constant, equal to 6.626 × 10 -34 m 2 kg/s and c is the speed of light, 3 × 10 8 m/s.In the beginning of the calculation, we should check if the students understand the notation of the numbers, e.g., nm, ps, or mJ.If so, we can proceed to the calculation itself, preferably without any calculators, or software.The resulting number of photons per pulse is approximately 4×10 17 .On this number, we can demonstrate the different notations, which can be found for example in the literature.
The other way to write this number is to use the so-called grouping.We would call this number 400 billiard then.The problem with this notation is that it is not unified and differs in US and Europe.The notation, we used, is the European one, while the same number would be called 400 quadrillion in US.Another, scientific way to write this number is 4e17, which translated to words is 4 multiplied by 10 to the 17th power (the 'e' meaning 'exponent').At this point, you can ask the students, how many photons do they think will come back?The next step is taking into account the divergence of the laser beam, which is expressed by means of the divergence angle θ1 (figure 7), and is a measure for how fast a laser beam expands.The common values for the Lunar laser ranging experiment lasers are around 1 arcsec (1/3600°) [19].We can try to imagine this as if we would try to see a small coin (2 euro cent or a dime) from a distance 4 km.Nonetheless, this does not help much with the idea of this angle.Lot easier is to visualize 1°, which is ca a little finger measure at arm's length, but then we have to divide it in our minds by 3600.The main purpose of these visualizations is not the precise picture of the angle, but rather the awareness that it is small.Despite the tiny divergence angle, we have to keep in mind that the light travels 400 000 km to the Moon.The students then may be surprised by the diameter of the laser beam on the Moon surface, which can be easily calculated as   = 400 000 × ( 0.5 3600 ).
(3) Note, we took only half of the angle, in order to calculate the radius.We also neglected the initial beam diameter (3.5 m) in our calculations.This is caused by the fact that the initial laser beam diameter is spread by the telescope.Nonetheless, we can make one simple assumption that the initial beam diameter is so small compared the Earth-Moon distance that we can take it as a point source.It will make the calculation easier, and it will also teach the students to deal with the approximations.
The value of the beam radius on the moon (  ) then equals to ca 1 km, and thus, the area on the Moon enlightened by the laser beam is ca 3 km 2 .The retroreflector placed by Apollo 15 is composed of 300 individual small retroreflectors, each with diameter 3.8 cm, resulting in an area of 1.36 m 2 .To simplify further, we will round this number to 1 m 2 .By making a ratio of these areas, we see that only 1 one in 3×10 6 will hit the retroreflector.The divergence angle θ2 of the returning beam is even larger, due to the optics of the retroreflector, and it is ca 8 arcsec.This results in the radius of the beam on the Earth surface being equal to  ℎ = 400 000 × ( ), Which is approximately 8 km.Thus, the area enlightened on the Earth is 200 km 2 .The area of the detector is 10 m 2 .Again, by making the ratio, we get that only 1 one in 2×10 7 will hit the detector.This means, from the 4×10 17 photons at the beginning, only approximately 7 000 photons make it back to the detector.And this would happen only when the conditions were ideal.When conducting the Lunar laser ranging experiment in real, the scientists have to deal with additional problems, such as weather conditions, or the rotation of Earth and Moon.Thus, the number of returned photons further decreases, sometimes to the individual photons, and this makes the measurement even more complicated.Nonetheless, they are able to routinely conduct this experiment.This is done by employing several steps.First, the detectors are tuned to the wavelength of the laser.Thus, we observe only a spectrally narrow region.The detectors are cooled down to decrease the electronic noise, too.We also know quite accurately when the pulse should come back, and we detect the incoming beam only in the times around the supposed return of the photons.Last, but not least, the series of pulses is sent in a short time and the statistics helps us to distinguish the real signal from the background noise.

Discussion and conclusion
The example of a STEM exercises described above shows a couple of interesting facts to the students.First, they should learn the main properties of the laser light and notice their practical usage.Also, they should realize the limitations of these properties in the real world.They learn important manual skills used in the lab, too.The overall evaluation from the students is positive.It is based on the discussion with the course leader, university students participating on the project, and the coordinator.
The students appreciate the equality between them and the tutors, the opportunity to learn new stuff in an interesting way, the freedom in the lab, and the combination of presentation and manual work.We also observe the shift in the ability to organize the work.We do not provide instruction manual with the step-by-step instructions.We only tell the students a brief introduction in the lab and their goal.Thus the students have to decide what and when to do.First, they are usually not self-confident, since they are not used to organize the work in the lab without precise instructions.Nonetheless, in a while they can adapt for the new situation and they appreciate this approach at the end.

Figure 1 .
Figure 1.The optical table prepared for the students.

Figure 2 .
Figure 2. Scheme of the experiment.

Figure 3 .
Figure 3.An example of the resulting pattern observed on the screen.(b) An image composed of three frames at different times (overlaid).

Figure 5 .
Figure 5. Image of the FAP pattern behind a circular aperture.(b) Image of the FAP pattern behind an elliptical aperture.(c) Evolute created in Python as the envelope of normals of an ellipse (the ellipse is depicted with a bold line).

Figure 7 .
Figure 7. Scheme of Lunar laser ranging experiment showing the divergence-related challenges.