1. After a while, a cube of rough steel (10 cm on a side) reaches
equilibrium inside a furnace al a temperature of 400°C. Knowing that its
total emissivity is 0.97, determine the rate at which the cube radiates
energy from each face. Get solution
2. A somewhat typical person has a total naked area of about 1.4 m2 and an average skin temperature of 33°C„Determine the net power radiated per unit area, the irradiance or more precisely the exitance, if the person's total emissivity is 97% and the environment is room temperature (20°C). How much energy does that body radiate per second? Get solution
3. Suppose that we measure the emitted exitance from a small hole in a furnace to be 22.8 W/cm2, using an optical pyrometer of some sort. Comptite the internal temperature of the furnace. Get solution
4. The temperature of an object resembling a blackbody is raised from 200 K to 2000 K. By how much does the amount of energy it radiates increase? Get solution
5. Your average skin temperature is about 33°C. Assuming you radiate as does a blackbody at that temperature, at what wavelength do you emit the most energy? Get solution
6. What is the wavelength that carries away the most energy when an object resembling a blackbody radiates energy into a room-temperature (20°C) environment? Get solution
7. The surface temperature of a class O blue-white star is around 40 × 103 K. At what frequency will it radiate most of its energy? Get solution
8. When the Sun's spectrum is photographed, using rockets to range above the Earth's atmosphere, it is found to have a peak in its spectral exitanee at roughly 465 nm. Compute the Sun's surface temperature, assuming it to be a blackbody. This approximation yields a value that is about 400 K too high. Get solution
9. An object resembling a blackbody emits a maximum amount of energy per unit wavelength in the red end of the visible spectrum p (λ = 680 nm). What's its surface temperature? Get solution
10. The energy per unit area per unit time per wavelength interval emitted by a blackbody at a temperature T' is given by...At a specific temperature, the total power radiated per unit area of the blackbody is equal to the area under the corresponding Iλ versus A curve. Use this to derive the Stefan-Boltzmann Law. [Hint: To clean up the exponential, change variables in the integral so that...Use the fact that ...where the gamma function is given by ... and the Riemann zeta function for n = 3 is ... Get solution
11. Get solution
12. In the atomic domain, energy is often measured in electron-volts. Arrive at the following expression for the energy of a light quantum in eV when the wavelength is in nanometers:...What is the energy of a quantum of 600-nm light? Get solution
13. Figure P.13.12 shows the spectral irradiance impinging on a horizontal surface, for a clear day, at sea level, with the Sun at the zenith. What is the most energetic photon we can expect to encounter (in eV and in J)?Figure P.13.12... Get solution
14. Suppose we have a 100-W yellow lightbulb (550 nm) 100 m away from a 3-cm diameter shuttered aperture. Assuming the bulb to have a 2.5% conversion to radiant power, how many photons will pass through the aperture if the shutter is opened for ... s? Get solution
15. The solar constant is the radiant flux density at a spherical surface centered on the Sun having a radius equal to that of the Earth's mean orbital radius; it has a value of 0.133-0.14 W/cm2. If we assume an average wavelength of about 700 nm, how many photons at most will arrive on each square meter per second of a solar cell panel just above the atmosphere? Get solution
16. A 50.0-cm3 chamber is filled with argon gas to a pressure of 20.3 Pa at a temperature of 0°C. All but a negligible number of these atoms are initially in their ground states. A flash tube surrounding the sample energizes 1.0% of the atoms into the same excited state having a mean life of 1.4 X 10-8as. What is the maximum rate at which photons are subsequently emitted by the gas, of course it falls off with time? Assume both that spontaneous emission is the only mechanism at work and that the medium is an ideal gas. Get solution
17. Show that for a system of atoms and photons in equilibrium at a temperature T the ratio of the transition "rates of stimulated to spontaneous emission is given by... Get solution
18. A system of atoms in thermal equilibrium is emitting and absorbing 2.0-eV light photons. Determine the ratio of the transition rates of stimulated emission to spontaneous emission at a temperature of 300 K. Discuss the implications of your answer. [Hint. See the previous problem.] Get solution
19. Redo the previous problem for a temperature of 30.0 × 103K and compare the results of both calculations. Get solution
20. Get solution
21. Get solution
22. For a system of atoms (in equilibrium) having two energy levels, show that at high temperatures where kBT >> ...-...,-, the number densities of the two states tend to become equal. [Hint. Form the ratio of die transition rates for total emission to absorption.] Get solution
23. Radiation at 21 cm pours down on the Earth from outer space. Its origin is great clouds of hydrogen gas. Taking the background temperature of space to be 3.0 K, determine the ratio of the transition rates of stimulated emission to spontaneous emission and discuss the result. Get solution
24. Get solution
25. Get solution
26. Get solution
27. The beam (λ = 632.8 nm) from a He-Ne laser, which is initially 3.0 mm in diameter, shines on a perpendicular wall 100 m away. Given that the system is diffraction limited, how large is the circle of light on the wall? Get solution
28. Make a rough estimate of the amount of energy that can be delivered by a ruby laser whose crystal is 5.0 mm in diameter and 0.050 m long. Assume the pulse of light lasts 5.0 X 10-6 s. The density of aluminum oxide (A12O3) is 3.7 × 103 kg/m3. Use the data in the discussion of Fig. 13.6 and the fact that the chromium ions make a 1.79 eV lasing transition. How much power is available per pulse? Get solution
29. What is the transition rate for the neon atoms in a He-Ne laser if the energy drop for the 632.8 nm emission is 1.96 eV and the power output is 1.0 mW? Get solution
30. Get solution
31. Given that a ruby laser operating at 694.3 nm has a frequency bandwidth of 50 MHz, what is the corresponding linewidth? Get solution
32. Determine the frequency difference between adjacent axial resonant cavity modes for a typical gas laser 25 cm long (n ≈ 1). Get solution
33. Get solution
34. Get solution
35. A He-Ne c-w laser has a Doppler-broadened transition. bandwidth of about 1.4 GHz at 632.8 nm. Assuming n - 1.0, determine the maximum cavity length for single-axial mode operation. Make a sketch of the transition linewidth and the corresponding cavity modes. Get solution
36. Get solution
37. Show (hat the maximum electric-field intensity, Emax, that exists for a given irradiance l is...where n is the refractive index of the medium. Get solution
38. A He-Ne laser operating at 632.8 nm has an internal beam-waist diameter of 0.60 mm. Calculate the full-angular width, or divergence, of the beam. Get solution
39. What would the pattern look like for a laserbeam diffracted by the three crossed gratings of Fig. P. 13.29?Figure P.13.29... Get solution
40. Make a rough sketch of the Fraunhofer diffraction pattern that would arise if a transparency of Fig. P. 13.30a served as the object. How would you filter it to get Fig. P. 13.30b?Figure P.13.30(a) ...(b) ... Get solution
41. Repeat the previous problem using Fig. P.13.31 instead.Figure P.13.31 (Photos courtesy R. A. Phillips.)(a) ...(b) ... Get solution
42. Repeat the previous problem using Fig. P.13.32 this time.Figure P.13.32 (Photos courtesy R. A. Phillips.)(a) ...(b) ... Get solution
43. Returning to Fig. 13.32, what kind of spatial filter would produce each of the patterns shown in Fig. P.l 3.33?Figure P.13.33 (Photos courtesy D. Dutton, M. P. Givens, and R. E. Hopkins)(a) ...(b) ... Get solution
44. With Fig. 13.31 in mind, show that the transverse magnification of the system is given by -fi/fi and draw the appropriate ray diagram. Draw a ray up through the center of the first lens at an angle 6 with the axis. From the point where that ray intersects Σt draw a ray downward that passes through the center of the second lens at an angle .... Prove that.... Using the notion of spatial frequency, from Eq. (11.64), show that k1 at the object plane is related to k, at the image plane by...What docs this mean with respect to the size of the image which fi > f1? What can then be said about the spatial periods of the input data as compared with the image output? Get solution
45. A diffraction grating having a mere 50 grooves per cm is the object in the optical computer shown in Fig. 13.31. If it is coherently illuminated by plane waves of green light (543.5 nm) from a He-Ne laser and each lens has a 100,cm focal length, what will be the spacing of the diffraction spots on the' transform plane? Get solution
46. Imagine that you have a cosine grating (i.e., a transparency whose amplitude transmission profile is cosinusoidal) with a spatial period of 0.01mm. The grating is illuminated by quasimonochromatic plane waves of A = 500 nm, and the setup is the same as that of Fig. 13.31, where the focal lengths of the transform and imaging lenses are 2.0 m and 1.0 m, respectively.a) Discuss the resulting pattern and design a filler that will pass only the first-order terms. Describe it in detail.b) What will the image look like on 2/ with that filter in place?c) How might you pass only the dc term, and what would the image look like then? Get solution
47. Suppose we insert a mask in the transform plane of the previous problem, which obscures everything but the m = +1 diffraction contribution. What will the reformed image look like on Σi? Explain your reasoning. Now suppose we remove only the m = +1 or the in m = -1 term. What will the re-formed image look like? Get solution
48. Reterring to the previous two problems with the cosine gratying oriented horizontally, make a sketch of the electric-field amplitude along y' with no filtering. Plot the corresponding image irradiance distribution. What will the electric field of the image look like if the dc term is filtered out? Plot it. Now plot the new irradiance distribution. What can you say about the spatial frequency of the image with and without the filter in place? Relate your answers to Fig. 11.13. Get solution
49. Replace the cosine grating in the previous problem with a "square" bar grating, that is, a series of many fine alternating opaque and transparent bands of equal width. We now filter out all terms in the transform plane but the zeroth and the two first-order diffraction spots. These we determine to have relative irradianccs of 1.00, 0.36, and 0.36: compare them with Figs. 7.32a and 7.33. Derive an expression for the general shape of the irradiance distribution on the image plane—make a sketch of it. What will the resulting fringe system look like? Get solution
50. A fine square wire mesh with 50 wires per cm is placed vertically in the object plane of the optical computer of Fig. 13.30. If the lenses each have 1,00-m focal lengths, what must be the illuminating wavelength, if the diffraction spots on the transform plane are to have a horizontal and vertical separation of 2.0 mm? What will be the mesh spacing as it appears on the image plane? Get solution
51. Imagine that we have an opaque mask into which arc punched an ordered array of circular holes, all of the same size, located as if at the corners of the boxes of a checkerboard. Now suppose our robot puncher goes mad and makes an additional batch of holes essentially randomly all across the mask. If this screen is now made the object in Problem 13.39, what will the diffraction pattern looklike? Given that the ordered holes are separated from their nearest neighbors on the object by 0.1 mm, what will be the spatial frequency of the corresponding dots in the image? Describe a filter that will remove the random holes from the final image. Get solution
52. Imagine that we have a large photographic transparency on which there is a picture of a student made up of a regular array of small circular dots, all of the same size, but each with its own density, so that it passes a spot of light with a particular field amplitude. Considering the transparency to be illuminated by a plane wave, discuss the idea of representing the electric-field amplitude just beyond it as the product (on average) of a regular two-dimensional array of top-hat functions (Fig. 11.4, p. 523) and the continuous two-dimensional picture function: the former like a dull bed of nails, the latter an ordinary photograph. Applying the frequency convolution theorem,, what does the distribution of light look like on the transform plane? How might it be filtered to produce a continuous output image? Get solution
53. The arrangement shown in Fig. P. 13.43 is used to convert a collimated laserbeam into a spherical wave. The pinhole cleans up the beam; that is, it eliminates diffraction effects due to dust and the like on the lens. How does it manage it?Figure P.13.43 (a) and (b) A high-power laserbeam before and after spatial filtering. (Photos courtesy Lawrence Livermore National Laboratory.)(a) ...(b) ...(c) ... Get solution
54. What would happen to the speckle pattern if a laserbeam were projected onto a suspension such as milk rather than onto a smooth wall? Get solution
2. A somewhat typical person has a total naked area of about 1.4 m2 and an average skin temperature of 33°C„Determine the net power radiated per unit area, the irradiance or more precisely the exitance, if the person's total emissivity is 97% and the environment is room temperature (20°C). How much energy does that body radiate per second? Get solution
3. Suppose that we measure the emitted exitance from a small hole in a furnace to be 22.8 W/cm2, using an optical pyrometer of some sort. Comptite the internal temperature of the furnace. Get solution
4. The temperature of an object resembling a blackbody is raised from 200 K to 2000 K. By how much does the amount of energy it radiates increase? Get solution
5. Your average skin temperature is about 33°C. Assuming you radiate as does a blackbody at that temperature, at what wavelength do you emit the most energy? Get solution
6. What is the wavelength that carries away the most energy when an object resembling a blackbody radiates energy into a room-temperature (20°C) environment? Get solution
7. The surface temperature of a class O blue-white star is around 40 × 103 K. At what frequency will it radiate most of its energy? Get solution
8. When the Sun's spectrum is photographed, using rockets to range above the Earth's atmosphere, it is found to have a peak in its spectral exitanee at roughly 465 nm. Compute the Sun's surface temperature, assuming it to be a blackbody. This approximation yields a value that is about 400 K too high. Get solution
9. An object resembling a blackbody emits a maximum amount of energy per unit wavelength in the red end of the visible spectrum p (λ = 680 nm). What's its surface temperature? Get solution
10. The energy per unit area per unit time per wavelength interval emitted by a blackbody at a temperature T' is given by...At a specific temperature, the total power radiated per unit area of the blackbody is equal to the area under the corresponding Iλ versus A curve. Use this to derive the Stefan-Boltzmann Law. [Hint: To clean up the exponential, change variables in the integral so that...Use the fact that ...where the gamma function is given by ... and the Riemann zeta function for n = 3 is ... Get solution
11. Get solution
12. In the atomic domain, energy is often measured in electron-volts. Arrive at the following expression for the energy of a light quantum in eV when the wavelength is in nanometers:...What is the energy of a quantum of 600-nm light? Get solution
13. Figure P.13.12 shows the spectral irradiance impinging on a horizontal surface, for a clear day, at sea level, with the Sun at the zenith. What is the most energetic photon we can expect to encounter (in eV and in J)?Figure P.13.12... Get solution
14. Suppose we have a 100-W yellow lightbulb (550 nm) 100 m away from a 3-cm diameter shuttered aperture. Assuming the bulb to have a 2.5% conversion to radiant power, how many photons will pass through the aperture if the shutter is opened for ... s? Get solution
15. The solar constant is the radiant flux density at a spherical surface centered on the Sun having a radius equal to that of the Earth's mean orbital radius; it has a value of 0.133-0.14 W/cm2. If we assume an average wavelength of about 700 nm, how many photons at most will arrive on each square meter per second of a solar cell panel just above the atmosphere? Get solution
16. A 50.0-cm3 chamber is filled with argon gas to a pressure of 20.3 Pa at a temperature of 0°C. All but a negligible number of these atoms are initially in their ground states. A flash tube surrounding the sample energizes 1.0% of the atoms into the same excited state having a mean life of 1.4 X 10-8as. What is the maximum rate at which photons are subsequently emitted by the gas, of course it falls off with time? Assume both that spontaneous emission is the only mechanism at work and that the medium is an ideal gas. Get solution
17. Show that for a system of atoms and photons in equilibrium at a temperature T the ratio of the transition "rates of stimulated to spontaneous emission is given by... Get solution
18. A system of atoms in thermal equilibrium is emitting and absorbing 2.0-eV light photons. Determine the ratio of the transition rates of stimulated emission to spontaneous emission at a temperature of 300 K. Discuss the implications of your answer. [Hint. See the previous problem.] Get solution
19. Redo the previous problem for a temperature of 30.0 × 103K and compare the results of both calculations. Get solution
20. Get solution
21. Get solution
22. For a system of atoms (in equilibrium) having two energy levels, show that at high temperatures where kBT >> ...-...,-, the number densities of the two states tend to become equal. [Hint. Form the ratio of die transition rates for total emission to absorption.] Get solution
23. Radiation at 21 cm pours down on the Earth from outer space. Its origin is great clouds of hydrogen gas. Taking the background temperature of space to be 3.0 K, determine the ratio of the transition rates of stimulated emission to spontaneous emission and discuss the result. Get solution
24. Get solution
25. Get solution
26. Get solution
27. The beam (λ = 632.8 nm) from a He-Ne laser, which is initially 3.0 mm in diameter, shines on a perpendicular wall 100 m away. Given that the system is diffraction limited, how large is the circle of light on the wall? Get solution
28. Make a rough estimate of the amount of energy that can be delivered by a ruby laser whose crystal is 5.0 mm in diameter and 0.050 m long. Assume the pulse of light lasts 5.0 X 10-6 s. The density of aluminum oxide (A12O3) is 3.7 × 103 kg/m3. Use the data in the discussion of Fig. 13.6 and the fact that the chromium ions make a 1.79 eV lasing transition. How much power is available per pulse? Get solution
29. What is the transition rate for the neon atoms in a He-Ne laser if the energy drop for the 632.8 nm emission is 1.96 eV and the power output is 1.0 mW? Get solution
30. Get solution
31. Given that a ruby laser operating at 694.3 nm has a frequency bandwidth of 50 MHz, what is the corresponding linewidth? Get solution
32. Determine the frequency difference between adjacent axial resonant cavity modes for a typical gas laser 25 cm long (n ≈ 1). Get solution
33. Get solution
34. Get solution
35. A He-Ne c-w laser has a Doppler-broadened transition. bandwidth of about 1.4 GHz at 632.8 nm. Assuming n - 1.0, determine the maximum cavity length for single-axial mode operation. Make a sketch of the transition linewidth and the corresponding cavity modes. Get solution
36. Get solution
37. Show (hat the maximum electric-field intensity, Emax, that exists for a given irradiance l is...where n is the refractive index of the medium. Get solution
38. A He-Ne laser operating at 632.8 nm has an internal beam-waist diameter of 0.60 mm. Calculate the full-angular width, or divergence, of the beam. Get solution
39. What would the pattern look like for a laserbeam diffracted by the three crossed gratings of Fig. P. 13.29?Figure P.13.29... Get solution
40. Make a rough sketch of the Fraunhofer diffraction pattern that would arise if a transparency of Fig. P. 13.30a served as the object. How would you filter it to get Fig. P. 13.30b?Figure P.13.30(a) ...(b) ... Get solution
41. Repeat the previous problem using Fig. P.13.31 instead.Figure P.13.31 (Photos courtesy R. A. Phillips.)(a) ...(b) ... Get solution
42. Repeat the previous problem using Fig. P.13.32 this time.Figure P.13.32 (Photos courtesy R. A. Phillips.)(a) ...(b) ... Get solution
43. Returning to Fig. 13.32, what kind of spatial filter would produce each of the patterns shown in Fig. P.l 3.33?Figure P.13.33 (Photos courtesy D. Dutton, M. P. Givens, and R. E. Hopkins)(a) ...(b) ... Get solution
44. With Fig. 13.31 in mind, show that the transverse magnification of the system is given by -fi/fi and draw the appropriate ray diagram. Draw a ray up through the center of the first lens at an angle 6 with the axis. From the point where that ray intersects Σt draw a ray downward that passes through the center of the second lens at an angle .... Prove that.... Using the notion of spatial frequency, from Eq. (11.64), show that k1 at the object plane is related to k, at the image plane by...What docs this mean with respect to the size of the image which fi > f1? What can then be said about the spatial periods of the input data as compared with the image output? Get solution
45. A diffraction grating having a mere 50 grooves per cm is the object in the optical computer shown in Fig. 13.31. If it is coherently illuminated by plane waves of green light (543.5 nm) from a He-Ne laser and each lens has a 100,cm focal length, what will be the spacing of the diffraction spots on the' transform plane? Get solution
46. Imagine that you have a cosine grating (i.e., a transparency whose amplitude transmission profile is cosinusoidal) with a spatial period of 0.01mm. The grating is illuminated by quasimonochromatic plane waves of A = 500 nm, and the setup is the same as that of Fig. 13.31, where the focal lengths of the transform and imaging lenses are 2.0 m and 1.0 m, respectively.a) Discuss the resulting pattern and design a filler that will pass only the first-order terms. Describe it in detail.b) What will the image look like on 2/ with that filter in place?c) How might you pass only the dc term, and what would the image look like then? Get solution
47. Suppose we insert a mask in the transform plane of the previous problem, which obscures everything but the m = +1 diffraction contribution. What will the reformed image look like on Σi? Explain your reasoning. Now suppose we remove only the m = +1 or the in m = -1 term. What will the re-formed image look like? Get solution
48. Reterring to the previous two problems with the cosine gratying oriented horizontally, make a sketch of the electric-field amplitude along y' with no filtering. Plot the corresponding image irradiance distribution. What will the electric field of the image look like if the dc term is filtered out? Plot it. Now plot the new irradiance distribution. What can you say about the spatial frequency of the image with and without the filter in place? Relate your answers to Fig. 11.13. Get solution
49. Replace the cosine grating in the previous problem with a "square" bar grating, that is, a series of many fine alternating opaque and transparent bands of equal width. We now filter out all terms in the transform plane but the zeroth and the two first-order diffraction spots. These we determine to have relative irradianccs of 1.00, 0.36, and 0.36: compare them with Figs. 7.32a and 7.33. Derive an expression for the general shape of the irradiance distribution on the image plane—make a sketch of it. What will the resulting fringe system look like? Get solution
50. A fine square wire mesh with 50 wires per cm is placed vertically in the object plane of the optical computer of Fig. 13.30. If the lenses each have 1,00-m focal lengths, what must be the illuminating wavelength, if the diffraction spots on the transform plane are to have a horizontal and vertical separation of 2.0 mm? What will be the mesh spacing as it appears on the image plane? Get solution
51. Imagine that we have an opaque mask into which arc punched an ordered array of circular holes, all of the same size, located as if at the corners of the boxes of a checkerboard. Now suppose our robot puncher goes mad and makes an additional batch of holes essentially randomly all across the mask. If this screen is now made the object in Problem 13.39, what will the diffraction pattern looklike? Given that the ordered holes are separated from their nearest neighbors on the object by 0.1 mm, what will be the spatial frequency of the corresponding dots in the image? Describe a filter that will remove the random holes from the final image. Get solution
52. Imagine that we have a large photographic transparency on which there is a picture of a student made up of a regular array of small circular dots, all of the same size, but each with its own density, so that it passes a spot of light with a particular field amplitude. Considering the transparency to be illuminated by a plane wave, discuss the idea of representing the electric-field amplitude just beyond it as the product (on average) of a regular two-dimensional array of top-hat functions (Fig. 11.4, p. 523) and the continuous two-dimensional picture function: the former like a dull bed of nails, the latter an ordinary photograph. Applying the frequency convolution theorem,, what does the distribution of light look like on the transform plane? How might it be filtered to produce a continuous output image? Get solution
53. The arrangement shown in Fig. P. 13.43 is used to convert a collimated laserbeam into a spherical wave. The pinhole cleans up the beam; that is, it eliminates diffraction effects due to dust and the like on the lens. How does it manage it?Figure P.13.43 (a) and (b) A high-power laserbeam before and after spatial filtering. (Photos courtesy Lawrence Livermore National Laboratory.)(a) ...(b) ...(c) ... Get solution
54. What would happen to the speckle pattern if a laserbeam were projected onto a suspension such as milk rather than onto a smooth wall? Get solution
