(Solved): please calculation and conclusion Discussion Interference-Young's double-slit experiment Thomas Youn ...

Discussion Interference-Young's double-slit experiment Thomas Young's double-slit experiment is the first evidence for the wave nature of light. Consider plane waves of light of a single wavelength falling on a double slits, \( s_{1} \) and \( s_{2} \), as shown, in Figure 1. Because of diffraction, the waves leaving the two small slits spread out as shown in Figure 1. This is equivalent to the interference pattem produced when two rocks are thrown into a lake. If light consisted of particles, we would expect to see two bright lines on the screen behind the slits. Instead, Young observed many bright and dark lines. If the distance \( \mathrm{d} \) between the slits is very small compared to the distance to the screen, constructive interference occurs on the screen at an angle, when the path difference \( d \sin \theta \) equals a whole number of wavelength. \( d \sin \theta_{m}=m \lambda \quad m=0,1,2,3, \ldots \ldots \quad \) [constructive interference] The value of \( m \) is called the order of the interference fringe. For example, the first order ( \( m=1 \) ) bright fringe forms at angle \( \theta_{t} \) on each side of the center at \( \theta=0 \). Dark fringes of destructive interference occur when \( d \sin a_{n}=(m+1 / 2) \lambda \quad m=0,1,2,3, \ldots \ldots \quad \) [destructive interference] The intensity of the bright fringes is greatest at the center and decreases for higher orders. The interference fringes produced by the double-slit experiment can be detected by moving a detector across the fringes. In Figure 1 , the angle is equal to \( x / L \). Figure 1 Young's double-slit experiment. Diffraction- When plane waves of light of single wavelength illuminates a narrow slit, light can diffract around the edges, giving rise to bright and dark fringes. To see how a diffraction pattern arises, parallel rays fall on the slit of width \( D \) as shown in Figure 2. Figure 2 Single slit experiment. If the viewing screen is far away, the rays heading for any point on the screen are essentially parallel. Consider the waves emanating from the upper half and lower half of the slit. Destructive interference, a dark fringe, occurs if the path difference from any point in the upper half of slit and the corresponding points in the lower half of the slit is an integer multiple of \( \lambda / 2 \) so that the total electric field is zero. The angle at which this occurs can be seen from the diagram to be \[ D \sin \theta=\lambda \] [first minimum] (3) The intensity is a maximum at \( \theta=0 \) and decrease to zero at the angle given by (3). At a larger angle, there will be a bright line, but not nearly as bright as the central spot at \( \theta=0 \). As the path difference becomes an integer multiple of \( \lambda / 2 \), there will again be a minimum of zero intensity when \[ D \sin \theta_{m}=m \lambda \] Procedure Warning: A helium-neon laser is used in this experiment. Your eyes could be harmed if directly exposed to the intense laser beam. DIFFRACTION (1)Place the slit holder at \( 10 \mathrm{~cm} \) in front of the laser. Place the screen on the rail at about \( 60 \mathrm{~cm} \) in front of the laser. Turn on the helium-neon laser. Remove the filter cap if it is on. Place slit plate \#9165 A, containing a set of four single slits of various widths, on the holder. Adjust the position of the slit holder so that the laser beam illuminates one of the slits and a clear diffraction pattern appears on the screen. Observed the diffraction patterns for various slit width. The motion of the translation stage is controlled by a delicate high-precision micrometer with a travel range of \( 2.5 \mathrm{~cm} \). Do not force the micrometer to travel beyond the range of reading. (2) Remove the screen and place the detector at the same position. Adjust the height of the detector so that the diffraction pattern is at the same height as the marks on the detector. The photo detector is a solar cell which, when illuminated by light, generates electric current proportional to the intensity of light. In order to increase the spatial resolution the front center of the detector is covered by a 100 -micron-wide slit. Measure and record the distance between the slit and the detector. Screw the filter cap on the laser head. Set the reading of the micrometer on the translation stage to \( 1.5 \mathrm{~cm} \). Slide the slit holder laterally until the laser beam illuminates slit "D". Loosen the screw and slide the mounting plate of the detector laterally and find the position where the meter reading reaches maximum. (If the maximum reading is larger than \( 0.7 \mathrm{~mA} \), move the slit laterally until the maximum reading is within the scale.) Clamp the detector mounting plate on the translation stage by fastening the screw. Record the background current reading when the laser beam is blocked. This reading can be positive or negative. Record the current reading as a function of position for every half turn ( one tum \( =0.5 \) \( \mathrm{mm} \) ) of displacement. Plot the recorded photo-current as a function of position. The plot should include the central maximum and at least one secondary maxima on each side. Make sure to subtract the background current reading. (3) Place the viewing screen back on the rail at the end of the rail. Remove the filter cap from the laser head. Position plate \#9165 B, containing four pairs of double slits, so that the laser beam illuminates both slits. With proper illumination, the interference pattern appears clearly on the screen. The typical pattern is a superposition of the single-slit diffraction patterns which you have seen in PROCEDURES (1)-(2) and fine structures of the interference fringes of double slits. For this part of the experiment, we will concentrate on the fine structures within the central maximum of the diffraction pattern. Observe the interference fringes for slit pairs of various slit separation. Record the distance between the detector and the slits. (4) Remove the white screen. Place the filter cap on the laser head. Position the detector at the end of the rail and adjust the height so that the interference pattem is at the same height as the detector. Slide the slit holder so that the laser beam illuminates slit pair A. Clamp the slit holder. Record the photo-current as a function of position for every \( 1 / 4 \) turn of the micrometer. Take readings for at least four intensity maxima. Measure the distance between adjacent dark fringes by reading the distance of the positions of minimum photo-current. The accuracy can be improved by measuring the total distance of more than one fringe and take the average. (5) Measure the separation between the double slits using a microscope. The separation should be measured from center to center or from edge to edge. Calculations and Conclusions A. Plot the angular distribution of the data recorded in PROCEDURE (2) for the single slit on a graph paper. B. Calculate, from the given slit width and the laser wavelength \( =632.8 \mathrm{~nm} \), the angular separation from the central maximum to the first and second minimum of the diffraction pattern. Compare this calculated values with the measured values of PROCEDURE (A). C. Plot the angular distribution of the data recorded in PROCEDURE (4) for the double slit. D. From the measured fringe spacing in PROCEDURES (4) and the measured slit separation in PROCEDURE (5), calculate the wavelength of the He-Ne laser. Compare the measured values with the known value.