OII na Sperimentale

Di raction from Phase Steps (10 points) The Equipment Box gure 1: The equipment box gure 2: Inside the box, upview Points: 20 Time: 5.0 Hours IPhO 2024 Experiment, English (Official) Page 11 of 21

To release the setup, you need to unscrew the white plastic fastening rods in the direction indicated on the top. You can pull out the main platform from the box as shown in Figure 3. After this, remove the red plastic O-rings indicated in Figure 2 and remove the other components inside the box one by one. To remove the instruments hold them by their metallic parts or by their outer surface. Figure 3: Extracting the main platform from the box. Points: 20 Time: 5.0 Hours IPhO 2024 Experiment, English (Official) Page 12 of 21

Figure 4: The experimental setup and its components The setup (Details) Points: 20 Time: 5.0 Hours IPhO 2024 Experiment, English (Official) Page 13 of 21

1. The main platform of the experimental setup, which consists of: (a) A horizontal base. (b) A white rotating protractor: it can be rotated using the white plastic knob next to it. A reference mark on the metallic plate can be used to read the angle (see Figure 4-1). (c) A circular plate with 4 square holes to hold the container of an unknown liquid. (d) A red laser and (e) convex lenses for magnifying the di raction pattern, installed on the walls at the two sides of the platform: their height can be adjusted by turning the knobs at the top. (f) Four protrusions on the inner wall of the protractor to hold the glass pieces’ holders.

2. Holders S1 (2.1) and S2 (2.2): each Holder stands on the circular metallic plate concentric with protractor and the four protrusions (Figure 4-1f) keep it xed. The S1 holder includes a black piece which holds a thin microscope slide. The lower edge of the slide is completely free and laser light can be shone onto it. The S2 Holder is quite similar to S1, the only di erence being that it holds a thick microscope slide.

3. The observation screen: it can be placed at any distance from the setup.

4. The unknown liquid container: after removing the protective adhesive paper, it can be placed on the square holes in the middle of the protractor (Figure 4-1c). The e ect of container walls on the di raction pattern is negligible.

5. The pink liquid, inside the bottle on your desk, has an unknown refractive index.

6. The laser electronic board: it can be turned on by connecting the laser to the board (and the board to the power bank). Use the On/O switch on the board to turn the laser on or o . The intensity of the laser light can be adjusted by turning the current adjuster knob on the electronic board. Set the intensity of the laser to a level at which your eyes are comfortable.

7. Power bank and electrical cables. Please take note of the following:

8. Do not touch the glass lens and the microscope slides at all, because your ngerprints can a ect the results of your experiment, and the slides are rather thin and can easily break.

9. Do not drink the unknown liquid. Points: 20 Time: 5.0 Hours IPhO 2024 Experiment, English (Official) Page 14 of 21

10. Do not look directly into the laser. Theory When a laser beam is shone at the edge of a transparent slide, a phase di erence is introduced between the part that travels through the slide and the part that does not. This phase di erence results in a di raction pattern, the lines of which are parallel to the edge of the slide (see Figure 5). Figure 5: A theoretical di raction pattern (left), and the di raction pattern observed in the lab (right). Let us take the direction of the beam as the z-direction (see Figure 6), and at rst, we’ll assume that the slide is in the x-y plane and its horizontal edge coincides with the x-axis (i.e. the angle in Figure 6 is equal to zero). In this case the phase di erence between the two parts of the beam clearly is: (1) where is the thickness of the slide, is the wavelength of the laser beam, is the refractive index of the environment, and is the refractive index of the transparent slide. $\varphi 0$ = $2\text{pih}$ $\lambda$ (n $-N$) h $\lambda$ N n Points: 20 Time: 5.0 Hours IPhO 2024 Experiment, English (Official) Page 15 of 21

Figure 6: Schematic of the laser beam, the slide, and the screen. If we rotate the slide around the y-axis so that the normal to the surface of the slide makes an angle of with the incident beam, a simple calculation gives the following formula for the phase di erence (2) Hence the phase di erence is a function of . If we continuously change this angle, the phase di erence increases continuously and the shape of the pattern changes, but when the phase di erence reaches , the pattern reverts to its initial shape. We call this full cycle one fringe shift. Figure 7 displays the various stages of one fringe shift. $\theta$ $\varphi$ = $2\text{pih}$ $\lambda (\sqrt{}n2-N$ 2 sin2 $\theta -N$ cos $\theta$) $\theta$ $2\pi$ Points: 20 Time: 5.0 Hours IPhO 2024 Experiment, English (Official) Page 16 of 21

Figure 7: Various stages of fringe shift as seen on the screen in the lab and as predicted theoretically (from left to right, each gure has phase di erence equals to $\varphi$, $\varphi +4\pi /9$, $\varphi +6\pi /9$, $\varphi +10\pi /9$, $\varphi +14\pi /9$, $\varphi +2\pi$). We can start from and gradually increase the angle. After such fringe shifts corresponding to a rotation by , we will have: (3) or: (4) Important note:

1. You only need to calculate the uncertainty in the nal results of each part (Errors need to be calculated and reported whenever the sign is present in the answer sheet).

2. You can use the provided calculator to nd the slope and the vertical axis intercept of the curves. $\theta$ = 0 m $\theta$ = $\text{thetam}$ $\varphi$ = $2\text{pih}$ $\lambda (\sqrt{}n2-N$ 2 sin2 $\text{thetam}-N$ cos $\text{thetam}$) = $2\text{pim}$ + $\varphi 0$ m = h $\lambda (\sqrt{}n2-N$ 2 sin2 $\text{thetam}-N$ cos $\text{thetam}$) $-\varphi 0$ $2\pi$ $\pm$ Points: 20 Time: 5.0 Hours IPhO 2024 Experiment, English (Official) Page 17 of 21

3. Note: Regression, r, is a number between 1 and -1 showing how much the data can be tted to a line. If |r|=1 it means data are completely on a line. In case we’re calculating slope (B) and intercept (A) using a calculator in linear mode, we can use these formulas below in order to calculate their uncertainty: Which is number of data points we’ve got, and is average of square of X. You must calculate uncertainty of the slope and the intercept only by the formulas above. Part A: Thickness of the thin slide (S1) (2.0 points) For the following tasks, take the refractive index of the glass components (S1, S2) to be and that of air to be . Take the wavelength of the red laser to be , and ignore any uncertainty in these values. Turn on the laser. Place the S1 Holder on the protractor, and adjust the height of the laser such that it shines on the bottom edge of the microscope slide. Then adjust the height of the lens until you can observe the di raction pattern on the screen (this height should almost be equal to the height of the laser beam). Note that the fringes in the di raction pattern are horizontal. gure 8 shows the experimental setup for part A. Now slowly turn the protractor and observe the fringe shift. $\text{DeltaB}$ = $B\sqrt{}$ 1 $(n-2$) ( 1 r2 $-1$) $\text{DeltaA}$ = $\text{DeltaB}\sqrt{}x2$ n x2

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5. 00 650 nm Points: 20 Time: 5.0 Hours IPhO 2024 Experiment, English (Official) Page 18 of 21

Figure 8: experimental setup in operation (part A) A-1 Starting with zero degrees, rotate the protractor and go up to 70 degrees. Watch the number of fringe shifts and write down the angle corresponding to each fringe shift number . Take at least 25 data points and ll out the table. 0.8 pt A-2 Draw appropriate graph. 0.3 pt A-3 Find the slope (B) and the vertical axis intercept (A). 0.1 pt A-4 Using the slope, nd the thickness of the thin slide. 0.8 pt $\text{thetam}$ m Points: 20 Time: 5.0 Hours IPhO 2024 Experiment, English (Official) Page 19 of 21

Part B: Thickness of the thick slide (S2) (1.6 points) Go back to the setup for Part A using the S2 Holder instead of the S1 Holder. B-1 Repeat the task A-1 for between 0 and 20 degrees and record at least 15 data points. 0.6 pt B-2 Assuming in Equation 4 is small enough, expand t

Topic: Wave Optics, Geometric Optics, Oscillations & Waves Metodi: Interference & Diffraction Analysis, Approximation & Series Expansion, Graph Linearization, Error Propagation, Experimental Data Analysis Competenze: Experimental Data Analysis, Graph Linearization, Error Propagation Objects: Lens, Screen Fonte: Testo (PDF) — p.1 Soluzione: Soluzioni (PDF)

The Commission shall adopt delegated acts in accordance with Article 21 of this Regulation. The Equipment Box Our first: The equipment box ♪ Inside the box, upview ♪ Score: 20 Time: 5.0 hours The following information shall be provided: Experiment, English (Official) Page 11 of 21

To release the setup, you need to unscrew the white plastic fastening rods in the direction indicated on the top. You can pull out the main platform from the box as shown in Figure 3. After This, remove the red plastic O-rings indicated in Figure 2 and remove the other components inside The box one by one. To remove the instruments hold them by their metallic parts or by their outer surface. Figure 3: Extracting the main platform from the box. Score: 20 Time: 5.0 hours The following information shall be provided: Experiment, English (Official) Page 12 of 21

Figure 4: The experimental setup and its components The setup (Details) Score: 20 Time: 5.0 hours The following information shall be provided: Experiment, English (Official) Page 13 of 21

1. The main platform of the experimental setup, which consists of: (a) A horizontal base. (b) A white rotating protractor: it can be rotated using the white plastic knob next to it. For reference mark on the metallic plate can be used to read the angle (see Figure 4-1). (c) A circular plate with 4 square holes to hold the container of an unknown liquid. (d) A red laser and (e) convex lenses for magnifying the dictionary pattern, installed on the walls at the two sides of the platform: their height can be adjusted by turning the knobs at the top. (f) Four protrusions on the inner wall of the protractor to hold the glass pieces holders.

2. Holders S1 (2.1) and S2 (2.2): each Holder stands on the circular metallic plate concentric with The two main types of protractor and the four protrusions (Figure 4-1f) keep it xed. The S1 holder includes a black piece which holds a thin microscope slide. The lower edge of the slide is completely free and laser light can be shone on it. The S2 Holder is quite similar to S1, the only d’erence being that it holds a It’s a thick microscope slide.

3. The observation screen: it can be placed at any distance from the setup.

4. The unknown liquid container: after removing the protective adhesive paper, it can be placed on the square holes in the middle of the protractor (Figure 4-1c). The ect of container walls on The ration pattern is negligible.

5. The pink liquid, inside the bottle on your desk, has an unknown refractive index.

6. The laser electronic board: it can be turned on by connecting the laser to the board (and the board to the power bank). Use the On/O switch on the board to turn the laser on or off . The The intensity of the laser light can be adjusted by turning the current adjuster knob on the electronic board. Set the intensity of the laser to a level at which your eyes are comfortable.

7. Power bank and electrical cables. Please note the following:

8. Do not touch the glass lens and the microscope slides at all, because your ngerprints can a ect The results of your experiment, and the slides are rather thin and can easily break.

9. Don’t drink the unknown liquid. Score: 20 Time: 5.0 hours The following information shall be provided: Experiment, English (Official) Page 14 of 21

10. Don’t look directly into the laser. Theory When a laser beam is shone at the edge of a transparent slide, a phase of erence is introduced between the part that travels through the slide and the part that doesn’t. This phase of erence results in a dictionary pattern, the lines of which are parallel to the edge of the slide (see Figure 5). Figure 5: A theoretical of the ration pattern (left), and the ration pattern observed in the lab (right). Let us take the direction of the beam as the z-direction (see Figure 6), and at rst, we’ll assume that the slide is in the x-y plane and its horizontal edge coincides with the x-axis (i.e. the angle in Figure 6 is equal to zero). In this case the phase of erence between the two parts of the beam clearly is: (1) where is the thickness of the slide, is the wavelength of the laser beam, is the refractive index of the environment, and is the refractive index of the transparent slide. $\varphi 0$ = $2\text{pih}$ $\lambda$ (n $-N$) h $\lambda$ N n Score: 20 Time: 5.0 hours The following information shall be provided: Experiment, English (Official) Page 15 of 21

Figure 6: Schematic of the laser beam, the slide, and the screen. If we rotate the slide around the y-axis so that the normal to the surface of the slide makes an angle of with the incident beam, a simple calculation gives the following formula for the phase of herring (2) Hence the phase of erence is a function of . If we continuously change this angle, the phase The pattern changes continuously and the shape of the pattern changes, but when the phase of erence reaches , the pattern reverts to its initial shape. We call this full cycle one fringe shift. Figure 7 displays the various stages of one fringe shift. $\theta$ $\varphi$ = $2\text{pih}$ $\lambda (\sqrt{}n2-N$ 2 sin2 $\theta -N$ cos $\theta$) $\theta$ $2\pi$ Score: 20 Time: 5.0 hours The following information shall be provided: Experiment, English (Official) Page 16 of 21

Figure 7: Various stages of fringe shift as seen on screen in the lab and as predicted theoretically (from left to right, each gure has phase di erence equals to $\varphi$, $\varphi +4\pi /9$, $\varphi +6\pi /9$, $\varphi +10\pi /9$, $\varphi +14\pi /9$, $\varphi +2\pi$). We can start from And gradually increase the angle. After such fringe shifts corresponding to a rotation by , we will have: (3) or: (4) Important note:

1. You only need to calculate the uncertainty in the final results of each part (Errors need to be calculated and reported whenever the sign is present in the answer sheet).

2. You can use the provided calculator to nd the slope and the vertical axis intercept of The curves. $\theta$ = 0 m $\theta$ = $\text{thetam}$ $\varphi$ = $2\text{pih}$ $\lambda (\sqrt{}n2-N$ 2 sin2 $\text{thetam}-N$ cos $\text{thetam}$) = $2\text{pim}$ + $\varphi 0$ m = h $\lambda (\sqrt{}n2-N$ 2 sin2 $\text{thetam}-N$ cos $\text{thetam}$) $-\varphi 0$ $2\pi$ $\pm$ Score: 20 Time: 5.0 hours The following information shall be provided: Experiment, English (Official) Page 17 of 21

3. Notes: Regression, r, is a number between 1 and -1 showing how much the data can be tted I’m going to the line. Ifr is equal to 1 it means data is completely on line. In cases we’re calculating slope (B) and intercept (A) using a linear mode calculator, we can use these formulas below in order to calculate their uncertainty: Which is number of data points we’ve got, and is average of square of X. You must calculate uncertainty of the slope and the intercept only by the formulas

• I’m going to go upstairs. Part A: Thickness of the thin slide (S1) (2.0 points) For the following tasks, take the refractive index of the glass components (S1, S2) to be and That of air to be . Take the wavelength of the red laser to be , and ignore any Uncertainty in these values. Turn on the laser. Place the S1 Holder on the protractor, and adjust the height of the laser such that It shines on the bottom edge of the microscope slide. Then adjust the height of the lens until you can observe the diction pattern on the screen (this height should almost be equal to the height of the laser beam). Note that the fringes in the di-raction pattern are horizontal. Our 8 shows The experimental setup for part A. Now slowly turn the protractor and observe the fringe shift. $\text{DeltaB}$ = $B\sqrt{}$ 1 $(n-2$) ( 1 r2 $-1$) $\text{DeltaA}$ = $\text{DeltaB}\sqrt{}x2$ n x2

1. 51
2. 00 650 nm Score: 20 Time: 5.0 hours The following information shall be provided: Experiment, English (Official) Page 18 of 21

Figure 8: experimental setup in operation (part A) A-1 Starting with zero degrees, rotate the protractor and go up to 70 degrees. Watch the number of fringe shifts and write down the angle corresponding to each fringe shift number . Take at least 25 data points And I’ll get out of the table. 0.8 pt A-2 Draw appropriate graphs. 0.3 pt A-3 Find the slope (B) and the vertical axis intercept (A). 0.1 pt A-4 Using the slope, nd the thickness of the thin slide. 0.8 pt $\text{thetam}$ m Score: 20 Time: 5.0 hours The following information shall be provided: Experiment, English (Official) Page 19 of 21

Part B: Thickness of the thick slide (S2) (1.6 points) Go back to the setup for Part A using the S2 Holder instead of the S1 Holder. B-1 Repeat the task A-1 for between 0 and 20 degrees and record at least 15 data points. 0.6 pt B-2 Assuming In Equation 4 is small enough, expand t

Topic: Wave Optics, Geometric Optics, Oscillations & Waves Metodi: Interference & Diffraction Analysis, Approximation & Series Expansion, Graph Linearization, Error Propagation, Experimental Data Analysis Competenze: Experimental Data Analysis, Graph Linearization, Error Propagation Objects: Lens, Screen Fonte: Testo (PDF) — p.1 Soluzione: Soluzioni (PDF)