N. Mahmudi 1 , S. Rendevski 2 , F. Ajredini 1 and R. Popeski-Dimovski 3

1. A STUDENT’S EXPERIMENT WITH MULTIPLE REFLECTIONS AND REFRACTIONS ON A GLASS PLATE AND VALIDATION OF THE FRESNEL’S EQUATIONS N. Mahmudi1, S. Rendevski2, F. Ajredini1 and R. Popeski-Dimovski3 1Faculty of Natural Sciences and Mathematics, State University of Tetovo, bul. Ilinden, b.b., 1200 Tetovo, Republic of Macedonia 2 Faculty of Electrical Engineering, University “Goce Delcev”, ul. Krste Misirkov, b.b., 2000 Stip, Republic of Macedonia 3Faculty of Natural Sciences and Mathematics, University “Ss. Cyril and Methodius”, ul. Gazi Baba, b.b., 1000 Skopje, Republic of Macedonia 1

2. Introduction • -Very important practical interest of experimenting with multiple reflections and • refractions on a layer in the last decade, • -the examination of multiple reflections and refractions in a single partially transmitting • layer with thickness D much larger than the wavelength of the incident light, . • -We choose the ray tracing method for calculating the transmission T, reflection R • and absorption coefficients A in such a layer . • -The other method that gives same equations for calculation of A, R and T • is the net radiation method described in the literature. • mediums at the front and back side of the thick layer are decided to be air (index of • refraction n1) and the thick layer to be a glass with index of refraction n2. • We assumed that the thickness of the layer is thick enough that there is no interference • effect. Hence, the transmission and reflection of the interface 1 are the same as reflection • and transmission of the interface 2. The same is valid for interface 3 and interface 4. Surface reflectance of a layer if front and back mediums are different

3. Experimental ,Results and Discussion -we define  as reflection power, which is the ratio of energy reflected from a surface to energy of the wave from the incident direction to that surface. -If the front and the back environments of the layer are the same then the surface reflectance and transmission on the top are the same as the bottom surface, 1 = 2 = . -The reflection power for unpolarized incident light is defined according to the Fresnel’s equation, where  is derived from the Snell’s law sin = (1/n)·sin:

4. On the upper surface (between interface 1 and interface 2) an incident light beam with unity intensity is considered (Fig.1). On the interface 1, an amount 1 is reflected and 1- 1 enter the non-attenuating material. Fig.1 Multiple reflections in a non-attenuating material

5. The fraction of incident energy reflected by the layer (material) is the sum of the terms leaving the top surface: (1) The fraction of incident energy transmitted by the layer (material) is the sum of the terms leaving the bottom surface: (2) Because there is no absorption the fraction of incident energy absorbed by the layer is A = 0. The equations (1) and (2) are known as Fresnel’s equations for multiple internal reflections for non-attenuating material.

6. -The reflection power R and transmission power T depend only on the surface reflectance. Also, the thickness of the non-attenuating material has no effect on reflection and transmission power of the material. -In a special case where 1 = 2= , we have: (3) (4)

7. For attenuating materials with transmittance power , (5) (6) According to the energy conservation principle, the fraction of incident energy absorbed in the layer A is: A = 1 – R – T. (7)

8. Experimental setup glass with thickness of 5.0 mm and index of refraction n of 1.50 for monochromatic laser light with wavelength 690 nm was used. -Detection of reflected and transmitted laser beams from the glass plate was made with PASCO ultrasensitive light sensor PS-2176. Fig.2. Experimental setup

9. -We worked with seven different angles of incident light beam: 350, 400, 450, 500, 550, 600 and 650. -Three light beams of multiple reflections going out of the top surface of the glass plate had enough intensity to be measured. -From the bottom surface of the glass plate, three light beams of multiple refractions were detected in transmission from the bottom surface of the plate. -In the experimental conditions limited to low power of the laser pointer and light absorption properties of the glass plate, it was hard to obtain fourth or higher order of reflected and transmitted light.

10. Fig.3. Dependences of R on the incident angle for the first reflected light beam. Fig.4. Dependences of R on the incident angle for the second reflected light beam. Fig.5. Dependences of R on the incident angle for the third reflected light beam.

11. Fig.6. Dependences of T on the incident angle for the first transmitted light beam. Fig.7. Dependences of T on the incident angle for the second transmitted light beam. Fig.8. Dependences of T on the incident angle for the third transmitted light beam.

12. Conclusion -Experimenting with the multiple reflections and refractions on a glass plate is suitable for students learning physics at the university level. -A practical validation of such experiment is in connection to ray tracing analysis in single or multilayer materials for cooling and heating processes. -For undergraduate students, this experiment could contribute to their broader understanding of optics, in this case, of Fresnel’s equations and its practical usage.