1. A collimated beam of mercury green light at 546.1 nm is normally
incident on a slit 0.015 cm wide. A lens of focal length 60 cm is placed
behind the slit. A diffraction pattern is formed on a screen placed in
the focal plane of the lens. Determine the distance between (a) the
central maximum and first minimum and (b) the first and second
minima.Figure 1. ... Get solution
2. Call the irradiance at the center of the central Fraunhofer diffraction maximum of a single slit I0 and the irradiance at some other point in the pattern I. Obtain the ratio I/I0 for a point on the screen that is 3/4 of a wavelength farther from one edge of the slit than the other. Get solution
3. The width of a rectangular slit is measured in the laboratory by means of its diffraction pattern at a distance of 2 m from the slit. When illuminated normally with a parallel beam of laser light (632.8 nm), the distance between the third minima on either side of the principal maximum is measured. An average of several tries gives 5.625 cm.a. Assuming Fraunhofer diffraction, what is the slit width?b. Is the assumption of far-field diffraction justified in this case? What is the ratio L/Lmin?Figure 1. ... Get solution
4. In viewing the far-field diffraction pattern of a single slit illuminated by a discrete-spectrum source with the help of absorption filters, one finds that the fifth minimum of one wavelength component coincides exactly with the fourth minimum of the pattern due to a wavelength of 620 nm. What is the other wavelength? Get solution
5. Calculate the rectangular slit width that will produce a central maximum in its far-field diffraction pattern having an angular breadth of 30°, 45°, 90°, and 180°. Assume a wavelength of 550 nm. Get solution
6. Consider the far-field diffraction pattern of a single slit of width 2.125 μm when illuminated normally by a collimated beam of 550-nm light. Determine (a) the angular radius of its central peak and (b) the ratio I/I0 at points making an angle of θ = 5°, 10°, 15°, and 22.5° with the axis. Get solution
7. a. Find the values of β for which the fourth and fifth secondary maxima of the single-slit diffraction pattern occur. (See the discussion surrounding Figure 11-3.)b. Find the ratio of the irradiance of the maxima of part (a) to the irradiance at the central maximum of the single-slit diffraction pattern. Get solution
8. Compare the relative irradiances of the first two secondary maxima of a circular diffraction pattern to those of a single-slit diffraction pattern. Get solution
9. The Lick Observatory has one of the largest refracting telescopes, with an aperture diameter of 36 in. and a focal length of 56 ft. Determine the radii of the first and second bright rings surrounding the Airy disc in the diffraction pattern formed by a star on the focal plane of the objective. See Figure 11-8b.Figure 1. ... Get solution
10. A telescope objective is 12 cm in diameter and has a focal length of 150 cm. Light of mean wavelength 550 nm from a distant star enters the scope as a nearly collimated beam. Compute the radius of the central disk of light forming the image of the star on the focal plane of the lens. Get solution
11. Suppose that a CO2 gas laser emits a diffraction-limited beam at wavelength 10.6 μm, power 2 kW, and diameter 1 mm. Assume that, by multimoding, the laser beam has an essentially uniform irradiance over its cross section. Approximately how large a spot would be produced on the surface of the moon, a distance of 376,000 km away from such a device, neglecting any scattering by the earth’s atmosphere? What will be the irradiance at the lunar surface? Get solution
12. Assume that a 2-mm-diameter laser beam (632.8 nm) is diffraction limited and has a constant irradiance over its cross section. On the basis of spreading due to diffraction alone, how far must it travel to double its diameter? Get solution
13. Two headlights on an automobile are 45 in. apart. How far away will the lights appear to be if they are just resolvable to a person whose nocturnal pupils are just 5 mm in diameter? Assume an average wavelength of 550 nm. Get solution
14. Assume that the pupil diameter of a normal eye typically can vary from 2 to 7 mm in response to ambient light variations.a. What is the corresponding range of distances over which such an eye can detect the separation of objects 1 mm apart?b. Experiment to find the range of distances over which you can detect the separation of lines placed 1 mm. apart. Use the results of your experiment to estimate the diameter range of your own pupils. Get solution
15. A double-slit diffraction pattern is formed using mercury green light at 546.1 nm. Each slit has a width of 0.100 mm. The pattern reveals that the fourth-order interference maxima are missing from the pattern.a. What is the slit separation?b. What is the irradiance of the first three orders of interference fringes, relative to the zeroth-order maximum? Get solution
16. a. Show that the number of bright fringes seen under the central diffraction peak in a Fraunhofer double-slit pattern is given by 2(a/b) − 1, where a/b is the ratio of slit separation to slit width.b. If 13 bright fringes are seen in the central diffraction peak when the slit width is 0.30 mm, determine the slit separation. Get solution
17. a. Show that in a double-slit Fraunhofer diffraction pattern, the ratio of widths of the central diffraction peak to the central interference fringe is 2(a/b), where a/b is the ratio of slit separation to slit width. Notice that the result is independent of wavelength.b. Determine the peak-to-fringe ratio, in particular when a = 10b. Get solution
18. Calculate by integration the irradiance of the diffraction pattern produced by a three-slit aperture, where the slit separation a is three times the slit width b. Make a careful sketch of I versus sin θ and describe properties of the pattern. Also show that your results are consistent with the general result for N slits, given by Eq. (11-32). Get solution
19. Make a rough sketch for the irradiance pattern from seven equally spaced slits having a separation-to-width ratio of 4. Label points on the x-axis with corresponding values of α and β. Get solution
20. A 10-slit aperture, with slit spacing five times the slit width of 1 × 10−4 cm, is used to produce a Fraunhofer diffraction pattern with light of 435.8 nm. Determine the irradiance of the principal interference maxima of orders 1,2,3,4, and 5 relative to the central fringe of zeroth order. Get solution
21. Show that one can arrive at Eq. (11-32) by taking the origin of coordinates at the midpoint of the central slit in an array where N is odd. Get solution
22. A rectangular aperture of dimensions 0.100 mm along the x-axis and 0.200 mm along the y-axis is illuminated by coherent light of wavelength 546 nm. A 1-m focal length lens intercepts the light diffracted by the aperture and projects the diffraction pattern on a screen in its focal plane. See Figure 11-21.a. What is the distribution of irradiance on the screen near the pattern center as a function of x and y (in mm) and I0, the irradiance at the pattern center?b. How far from the pattern center are the first minima along the x and y directions?c. What fraction of the I0 irradiance occurs at 1 mm from the pattern center along the x- and y-directions?d. What is the irradiance at the point (x = 2, y = 3) mm?Figure 1.... Get solution
23. What is the angular half-width (from central maximum to first minimum) of a diffracted beam for a slit width of (a) λ; (b) 5λ; (c) 10λ? Get solution
24. A property of the Bessel function J1(x) is that, for large x, a closed form exists, given by ...Find the angular separation of diffraction minima far from the axis of a circular aperture. Get solution
25. We have shown that the secondary maxima in a single-slit diffraction pattern do not fall exactly halfway between minima, but are quite close. Assuming they are halfway:a. Show that the irradiance of the mth secondary peak is given approximately by ...b. Calculate the percent error involved in this approximation for the first three secondary maxima. Get solution
26. Three antennas broadcast in phase at a wavelength of 1 km. The antennas arc separated by a distance of ...km and each antenna radiates equally in all horizontal directions. Because of interference, a broadcast “beam” is limited by interference minima. How many well-defined beams are broadcast and what are their angular half-widths? Get solution
27. A collimated light beam is incident normally on three very narrow, identical slits. At the center of the pattern projected on a screen, the irradiance is Imax.a. If the irradiance IP at some point P on the screen is zero, what is the phase difference between light arriving at P from neighboring slits?b. If the phase difference between light waves arriving at P from neighboring slits is π, determine the ratio IP/Imax.c. What is IP/Imax at the first principal maximum?d. If the average irradiance on the entire screen is Iav, what is the ratio IP/Iav at the central maximum? Get solution
28. Draw phasor diagrams illustrating the principal maxima and zero irradiance points for a four-slit aperture. Get solution
2. Call the irradiance at the center of the central Fraunhofer diffraction maximum of a single slit I0 and the irradiance at some other point in the pattern I. Obtain the ratio I/I0 for a point on the screen that is 3/4 of a wavelength farther from one edge of the slit than the other. Get solution
3. The width of a rectangular slit is measured in the laboratory by means of its diffraction pattern at a distance of 2 m from the slit. When illuminated normally with a parallel beam of laser light (632.8 nm), the distance between the third minima on either side of the principal maximum is measured. An average of several tries gives 5.625 cm.a. Assuming Fraunhofer diffraction, what is the slit width?b. Is the assumption of far-field diffraction justified in this case? What is the ratio L/Lmin?Figure 1. ... Get solution
4. In viewing the far-field diffraction pattern of a single slit illuminated by a discrete-spectrum source with the help of absorption filters, one finds that the fifth minimum of one wavelength component coincides exactly with the fourth minimum of the pattern due to a wavelength of 620 nm. What is the other wavelength? Get solution
5. Calculate the rectangular slit width that will produce a central maximum in its far-field diffraction pattern having an angular breadth of 30°, 45°, 90°, and 180°. Assume a wavelength of 550 nm. Get solution
6. Consider the far-field diffraction pattern of a single slit of width 2.125 μm when illuminated normally by a collimated beam of 550-nm light. Determine (a) the angular radius of its central peak and (b) the ratio I/I0 at points making an angle of θ = 5°, 10°, 15°, and 22.5° with the axis. Get solution
7. a. Find the values of β for which the fourth and fifth secondary maxima of the single-slit diffraction pattern occur. (See the discussion surrounding Figure 11-3.)b. Find the ratio of the irradiance of the maxima of part (a) to the irradiance at the central maximum of the single-slit diffraction pattern. Get solution
8. Compare the relative irradiances of the first two secondary maxima of a circular diffraction pattern to those of a single-slit diffraction pattern. Get solution
9. The Lick Observatory has one of the largest refracting telescopes, with an aperture diameter of 36 in. and a focal length of 56 ft. Determine the radii of the first and second bright rings surrounding the Airy disc in the diffraction pattern formed by a star on the focal plane of the objective. See Figure 11-8b.Figure 1. ... Get solution
10. A telescope objective is 12 cm in diameter and has a focal length of 150 cm. Light of mean wavelength 550 nm from a distant star enters the scope as a nearly collimated beam. Compute the radius of the central disk of light forming the image of the star on the focal plane of the lens. Get solution
11. Suppose that a CO2 gas laser emits a diffraction-limited beam at wavelength 10.6 μm, power 2 kW, and diameter 1 mm. Assume that, by multimoding, the laser beam has an essentially uniform irradiance over its cross section. Approximately how large a spot would be produced on the surface of the moon, a distance of 376,000 km away from such a device, neglecting any scattering by the earth’s atmosphere? What will be the irradiance at the lunar surface? Get solution
12. Assume that a 2-mm-diameter laser beam (632.8 nm) is diffraction limited and has a constant irradiance over its cross section. On the basis of spreading due to diffraction alone, how far must it travel to double its diameter? Get solution
13. Two headlights on an automobile are 45 in. apart. How far away will the lights appear to be if they are just resolvable to a person whose nocturnal pupils are just 5 mm in diameter? Assume an average wavelength of 550 nm. Get solution
14. Assume that the pupil diameter of a normal eye typically can vary from 2 to 7 mm in response to ambient light variations.a. What is the corresponding range of distances over which such an eye can detect the separation of objects 1 mm apart?b. Experiment to find the range of distances over which you can detect the separation of lines placed 1 mm. apart. Use the results of your experiment to estimate the diameter range of your own pupils. Get solution
15. A double-slit diffraction pattern is formed using mercury green light at 546.1 nm. Each slit has a width of 0.100 mm. The pattern reveals that the fourth-order interference maxima are missing from the pattern.a. What is the slit separation?b. What is the irradiance of the first three orders of interference fringes, relative to the zeroth-order maximum? Get solution
16. a. Show that the number of bright fringes seen under the central diffraction peak in a Fraunhofer double-slit pattern is given by 2(a/b) − 1, where a/b is the ratio of slit separation to slit width.b. If 13 bright fringes are seen in the central diffraction peak when the slit width is 0.30 mm, determine the slit separation. Get solution
17. a. Show that in a double-slit Fraunhofer diffraction pattern, the ratio of widths of the central diffraction peak to the central interference fringe is 2(a/b), where a/b is the ratio of slit separation to slit width. Notice that the result is independent of wavelength.b. Determine the peak-to-fringe ratio, in particular when a = 10b. Get solution
18. Calculate by integration the irradiance of the diffraction pattern produced by a three-slit aperture, where the slit separation a is three times the slit width b. Make a careful sketch of I versus sin θ and describe properties of the pattern. Also show that your results are consistent with the general result for N slits, given by Eq. (11-32). Get solution
19. Make a rough sketch for the irradiance pattern from seven equally spaced slits having a separation-to-width ratio of 4. Label points on the x-axis with corresponding values of α and β. Get solution
20. A 10-slit aperture, with slit spacing five times the slit width of 1 × 10−4 cm, is used to produce a Fraunhofer diffraction pattern with light of 435.8 nm. Determine the irradiance of the principal interference maxima of orders 1,2,3,4, and 5 relative to the central fringe of zeroth order. Get solution
21. Show that one can arrive at Eq. (11-32) by taking the origin of coordinates at the midpoint of the central slit in an array where N is odd. Get solution
22. A rectangular aperture of dimensions 0.100 mm along the x-axis and 0.200 mm along the y-axis is illuminated by coherent light of wavelength 546 nm. A 1-m focal length lens intercepts the light diffracted by the aperture and projects the diffraction pattern on a screen in its focal plane. See Figure 11-21.a. What is the distribution of irradiance on the screen near the pattern center as a function of x and y (in mm) and I0, the irradiance at the pattern center?b. How far from the pattern center are the first minima along the x and y directions?c. What fraction of the I0 irradiance occurs at 1 mm from the pattern center along the x- and y-directions?d. What is the irradiance at the point (x = 2, y = 3) mm?Figure 1.... Get solution
23. What is the angular half-width (from central maximum to first minimum) of a diffracted beam for a slit width of (a) λ; (b) 5λ; (c) 10λ? Get solution
24. A property of the Bessel function J1(x) is that, for large x, a closed form exists, given by ...Find the angular separation of diffraction minima far from the axis of a circular aperture. Get solution
25. We have shown that the secondary maxima in a single-slit diffraction pattern do not fall exactly halfway between minima, but are quite close. Assuming they are halfway:a. Show that the irradiance of the mth secondary peak is given approximately by ...b. Calculate the percent error involved in this approximation for the first three secondary maxima. Get solution
26. Three antennas broadcast in phase at a wavelength of 1 km. The antennas arc separated by a distance of ...km and each antenna radiates equally in all horizontal directions. Because of interference, a broadcast “beam” is limited by interference minima. How many well-defined beams are broadcast and what are their angular half-widths? Get solution
27. A collimated light beam is incident normally on three very narrow, identical slits. At the center of the pattern projected on a screen, the irradiance is Imax.a. If the irradiance IP at some point P on the screen is zero, what is the phase difference between light arriving at P from neighboring slits?b. If the phase difference between light waves arriving at P from neighboring slits is π, determine the ratio IP/Imax.c. What is IP/Imax at the first principal maximum?d. If the average irradiance on the entire screen is Iav, what is the ratio IP/Iav at the central maximum? Get solution
28. Draw phasor diagrams illustrating the principal maxima and zero irradiance points for a four-slit aperture. Get solution
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