## Sweet Mirage

#### Abstract:

Fata Morgana is the name given to a particular form of mirage. A similar effect can be replicated in the laboratory by shining a laser through a fluid with a refractive index gradient. This effect will be investigated by using a green point laser. A mathematical model will be developed and its predictions will be compared to the results of the experiment.#### Bibliography/Citations:

E. Fabri, G. Fiorio, F. Lazzeri and P. Violino, "Mirage in the Laboratory", Am, J. Phys. 50 517 (1982)

R. Barker, P.R.M Crofts, and M. Gal, "A superior 'superior' mirage", J. Am. Phys. 57 953 (1989)

K.C. Manola, W.F. Mueller, and B. Regittko, "Light rays in gradient index media: a laboatory exercise", Am, J. Phys. 60 527 (1992)

R.G Greenler, "Laboratory simulation of inferior and superior mirage", J. Opt. Soc. Am. A 4 589 (1987)

J.G. Speight, "Properties of Inorganic Compounds", Environmental Inorganic Chemistry. Elsevier (2017)

T. Lopez, G. Calza, L.M. Gratton, and S. Oss, "Mirages in a bottle", IOP (2009)

https://www.sciencedirect.com/topics/earth-and-planetary-sciences/refractivity

https://tutorial.math.lamar.edu/Classes/DE/HeatEqnNonZero.aspx

https://en.wikipedia.org/wiki/Solubility#Rate_of_dissolution

https://dharmapaudelphy.files.wordpress.com/2016/07/akur-project.pdf

#### Additional Project Information

#### Research Plan:

Is this project, I will replicate the Fata Morgana effect in the Laboratory using a sugar-water solution with a refractive index gradient. The different steps of the experiment are outlined below:

1- Fill the container progressively with warm water

2- Pour sugar in the container until in falls to the bottom and form a layer

3- wait about two days and then measure the sugar concentration gradient as a function of height

4- Outside of the container, on a given side and at the same height as the sugar level, place un object

5- On one side of the container, you shine a laser such that the light enters the container at the same height as the height of the object and bends until it hits the top corner of the object placed on the other side.

6- You record the angle at which the light bends to hit the top of the object and you take a picture of the trajectory of the laser

7- Take a picture of the object through the container

8- Remove the container, then take again a picture of the object

7- Use mathematics to do a fit of the trajectory of light

8- Compute the angle at which the trajectory hit the object

9- Use mathematics to develop a model to predict the trajectory of light ray and compare it to the recorded trajectory.

10- Develop a mathematical model to determine how the image would be distorted

The accuracy of the model will be determined by comparing the experimental results and the theoretical predictions.

At this point, I have all the experimental results and I am in the process of developing the theoretical model.

#### Questions and Answers

1. What was the major objective of your project and what was your plan to achieve it?

I started working on this project for the IYPT (international Young Physicist Tournament) in 2020, unfortunately the competition was canceled because of Covid-19. I was supposed to present this project for the Swiss Young Physicist tournament, but again due to Covid-19, the competition did not take place. So I kept working on the project during the summer and again this year.

a. Was that goal the result of any specific situation, experience, or problem you encountered?

Since I love mathematical modeling, when I saw the problem posted by the IPYT, I immediately got interested in the topic and wanted to contribute to it.

b. Were you trying to solve a problem, answer a question, or test a hypothesis?

My project was to check whether it will be possible to replicate a natural effect in the lab and develop a mathematical model for the effect. So yes I am trying to answer a question and test a hypothesis.

2. What were the major tasks you had to perform in order to complete your project?

1- Learn higher level of mathematics

2- Because of Covid-19 confinement, I had to do the experiment at home in a creative way with the minimum of material

that I had access to.

a. For teams, describe what each member worked on.

3. What is new or novel about your project?

a. Is there some aspect of your project's objective, or how you achieved it that you haven't done before?

- Learned more advanced MATH

- Learned how lasers function

- Learned how to mathematically model certain chemical reactions

- Learned more about optics and how light propagate in matter

- Learned more advanced computing

b. Is your project's objective, or the way you implemented it, different from anything you have seen?

I believe that this project is higher in complexity than what one would expect from a high school project.

c. If you believe your work to be unique in some way, what research have you done to confirm that it is?

I believe similar experiment were done before, but I believe that the mathematical work I did is specific to this experiment.

4. What was the most challenging part of completing your project?

a. What problems did you encounter, and how did you overcome them?

- Doing the experiment at home with the minimum of resources I had access to during the pandemic

- I had to take a new math class in order to understand the math I found in the literature

b. What did you learn from overcoming these problems?

- I learned more academically in terms of Math

- I learned how to be more creative with the minimum resources I had in order to complete my project

5. If you were going to do this project again, are there any things you would you do differently the next time?

- Definitely, I will use more accurate tools in order to perform the experiment and minimize the experimental errors

- I would also work on other aspects of the project, such as the color of the image.

- I would use different types of lasers with different wavelengths

- I would try to replicate the experiment with a temperature generated refractive index gradient, instead of a concentration induced one.

6. Did working on this project give you any ideas for other projects?

- Develop a more advanced mathematical model for the trajectory of light using curvatures instead of Snell's Law.

- Find a way to determine the temperature gradient at the periphery of earth surface

- Find a way the calculate the size and color of a rainbow

7. How did COVID-19 affect the completion of your project?

Covid-19 made the resources hard to acquire. Also staying at home all the time did not help.