Chapter #13 Solutions - Optics - Eugene Hecht - 5th Edition

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

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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

Chapter #12 Solutions - Optics - Eugene Hecht - 5th Edition

1. Get solution

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8. Suppose we set up a fringe pattern using a Michelson Interferometer with a mercury vapor lamp as the source. Switch on the lamp in your mind's eye and discuss what will happen to the fringes as the mercury vapor pressure builds to its steady-slate value. Get solution

9. We wish to examine the irradiance produced on the plane of observation in Young's Experiment when the slits are illuminated simultaneously by two monochromatic plane waves of somewhat different frequency, E1, and E2, Sketch these against time, taking A! = 0.8A2. Now draw the product E1 E2 (at a point P) against time.What can you say about its average over a relatively long interval? What does (E1 + E2)2 look like? Compare it with Ef + E\. Over a time that is long compared with the periods of the waves, approximate ((E1 + E2)2)T Get solution

10. With the previous problem in mind, now consider things spread across space at a given moment in time. Each wave separately would result in an irradiance distribution I1 and I2. Plot both on the same space axis and then draw their sum I1 + I2. Discuss the meaning of your results. Compare your work with Fig. 7.16. What happens to the net irradiance as more waves of different frequency are added in? Explain in terms of the coherence length. Hypothetically, what would happen to the pattern as the frequency bandwidth approached infinity? Get solution

11. With the previous problem in mind, return to the autocorrelation of a sine function, shown in Fig. 11.37. Now suppose we have a signal composed of a great many sinusoidal components. Imagine that you take the autocorrelation of this complicated signal and plot the result (use three or four components to start with), as in part (e) of Fig. 11.37. What will the autocorrelation function look like when the number of waves is very large and the signal resembles random noise? What is the significance of the T - 0 value? How does this compare with the previous problem? Get solution

12. Imagine that we have the arrangement depicted in Fig. 12.3. If the separation between fringes (max. to max.) is 1 mm and if the projected width of the source slit on the screen is 0.5 mm, compute the visibility. Get solution

13. Referring to the slit source and pinhole screen arrangement of Fig. P. 12.6, show by integration over the source that...Figure P.12.6... Get solution

14. Carry out the details leading to the expression for the visibility given by Eq. (12.22). Get solution

15. Under what circumstances will the irradiance on ∑a in Fig. P.12.8 be equal to 4I0, where I0 is the irradiance due to either incoherent point source alone?Figure P.12.8... Get solution

16. Suppose we set up Young's Experiment with a small circular hole of diameter 0.1 mm in front of a sodium lamp ... as the source. If the distance from the source to the slits is 1 m, how far apart will the slits be when the fringe pattern disappears? Get solution

17. Get solution

18. Show that Eqs. (12.34) and (12.35) follow from Eqs. (12.32) and (12.33), Get solution

19. Return to Eq. (12.21) and separate it into two terms representing a coherent and an incoherent contribution, the first arising from the superposition of two coherent waves with irradiances of ... and ... having relative phase of α12(T), and the second from the superposition of incoherent waves of irradiance ... and .... Now derive expressions for Icoh/Iincoh and for Iincoh/Itotal, Discuss the physical significance of this alternative formulation and how we might view the visibility of fringes in terms of it. Get solution

20. Imagine that we have Young's Experiment, where one of the two pinholes is now covered by a neutral-density filter that cuts the irradiance by a factor of 10, and the other hole is covered by a transparent sheet of glass, so there is no relative phase shift introduced. Compute the visibility in the hypothetical case of completely coherent illumination. Get solution

21. Suppose that Young's double-slit apparatus is illuminated by sunlight with a mean wavelength of 550 nm. Determine the separation of the slits that would cause the fringes to vanish. Get solution

22. Get solution

23. Get solution

24. We wish to construct a double-pinhole setup illuminated by a uniform, quasimonuchromalic, incoherent slit source of mean wavelength 500 nm and width b, a distance of 1.5 m from the aperture screen. If the pinholes are 0.50 mm apart, how wide can the source be if the visibility of the fringes on the plane of observation is not to be less than 85%? Get solution

25. Suppose that we have an incoherent, quasimonochromatic, uniform slit source, such as a discharge lamp with a mask and fitter in front of it. We wish to illuminate a region on an aperture screen 10.0 m away, such that the modulus of the complex degree of coherence everywhere within a region 1.0 mm wide is equal to or greater than 90% when the wavelength is 500 nm. How wide can the slit be? Get solution

26. Figure P. 12.17 shows two incoherent quasimonochromatic point sources illuminating two pinholes in a mask. Show that the fringes formed on the plane of observation have minimum visibility when...where m = ±1, ±3, ±5....Figure P.12.17... Get solution

27. Imagine that we have a wide quasimonochromatic source (λ - 500 nm) consisting of a series of vertical, incoherent, infinites-imally narrow line sources, each separated by 500 μm. This is used to illuminate a pair of exceedingly narrow vertical slits in an aperture screen 2.0 m away. How far apart should the apertures be to create a fringe system of maximum visibility Get solution

28. Get solution

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30. Get solution

Chapter #11 Solutions - Optics - Eugene Hecht - 5th Edition

1. Determine the Fourier transform of the function...Make a sketch of ...{E(x)}. Discuss its relationship to Fig. 11.11. Get solution

2. Determine the Fourier transform of...Make a sketch of it. Get solution

3. Determine the Fourier transform of...Make a sketch of F(to), then sketch its limiting form as T -> ±∞. Get solution

4. Show that = ... Get solution

5. Determine the Fourier transform of the function f(x) = A cosk0x. Get solution

6. Get solution

7. Get solution

8. Get solution

9. Given that ...{f(x)} = F(k) and ...{h(x}} = H(k) if a and b are constants, determine ...{af(x) + bh(x) Get solution

10. Figure P.l 1.7 shows two periodic functions, f(x) and h(x), which are to be added to produce g(x). Sketch g(x); then draw diagrams of the real and imaginary frequency spectra, as well as the amplitude spectra for each of the three functions.Figure P.11.7... Get solution

11. Compute the Fourier transform of the triangular pulse shown in ' Fig. P. 11.8. Make a sketch of your answer, labeling all the pertinent values on the curve.Figure P.11.8... Get solution

12. Given that ...{f(x)} = F(k), introduce a constant scaling factor 1/a and determine the Fourier transform of/f(x/a). Show that the transform of f(-x) is F(-k). Get solution

13. Show that the Fourier transform of the transform, ...{f(x)}, equals 2пf(-x), and that this is not the inverse transform of the transform, which equals f(x). This problem was suggested by Mr. D. Chapman while a student at the University of Ottawa. Get solution

14. The rectangular function is often defined as...where it is set equal to ½ at the discontinuities (Fig. P.11.11). Determine the Fourier transform of...Notice that this is just a rectangular pulse, like that in Fig. 11.16, shifted a distance x0 from the origin.Figure P.11.11... Get solution

15. With the last two problems in mind, show that ...{(l/2п)sinc (½x)} - rect(k). starting with the knowledge that ...{rect(x)} - sinc(½k),in other words, Eq.(7.58) with L = a, where a = 1. Get solution

16. Utilizing Eq. (11.38), show that ...-l{...{f(x)}} = f(x). Get solution

17. Given ...{f(x)}, show that ...{f(x - x0)} differs from it only by a linear phase factor. Get solution

18. Prove that f...h = h .../directly. Now do it using the convolution theorem. Get solution

19. Get solution

20. Get solution

21. Suppose we have two functions, f(x, y) and h(x, y), where both have a value of 1 over a square region in the xy-plane and are zero everywhere else (Fig. P.11.16). If g(X, Y) is their convolution, make a plot of g(X, 0).Figure P.11.16... Get solution

22. Referring to the previous problem, justify the fact that the convolution is zero for ... when h is viewed as a spread function. Get solution

23. Use the method illustrated in Fig. 11.23 to convolve the two functions depicted in Fig. P.11.18.Figure P.11.18... Get solution

24. Given that ... show that after shifting one of the functions an amount x0, we get ... Get solution

25. Get solution

26. Get solution

27. Prove analytically that the convolution of any function f(x) with a delta function, δ(x), generates the original function f(X). You might make use of the fact that δ(x) is even. Get solution

28. Prove that δ(x - x0) ... f(x) =f(X - x0) and discuss the meaning of this result. Make a sketch of two appropriate functions and convolve them. Be sure to use an asymmetrical f(x). Get solution

29. Show that ...{f(x) cos k0x} = [F(k - k0) + F(k + k0)]/2 and that ...{f(x) sin k0x] = [F(k-k0) - F(k + k0)]/2i. Get solution

30. Figure P.11.23 shows two functions. Convolve them graphically and draw a plot of the result. Get solution

31. Get solution

32. Given the function...determine its Fourier transform. (See Problem 11.11.) Get solution

33. Given the function f(x) = δ(x + 3) + δ (x - 2) + δ (x-5), convolve it with the arbitrary function h(x).Figure P.11.23... Get solution

34. Make a sketch of" the function arising from the convolution of the two functions depicted in Fig. P.l 1.26.Figure P.l 1.26... Get solution

35. Figure P. 11.27 depicts a rect function (as defined above) and a periodic comb function. Convolve the two to get g(x). Now sketch the transform of each of these functions against spatial frequency k/2π = 1/λ. Check your results with the convolution theorem. Label all the relevant points on the horizontal axes in terms of d—like the zeros of the transform of f(x).Figure P.11.27... Get solution

36. Figure P.l 1.28 shows, in one dimension, the electric field across an illuminated aperture consisting of several opaque bars forming a grating. Considering it to be created by taking the product of a periodic rectangular wave h(x) and a unit rectangular function f(x), sketch the resulting electric field in the Fraunhofer region.... Get solution

37. Show (for normally incident plane waves) that if an aperture has a center of symmetry (i.e., if the aperture function is even), then the diffracted field in the Fraunhofer case also possesses a center of 1 symmetry. Get solution

38. Suppose a given aperture produces a Fraunhofer field pattern E(Y, Z). Show that if the aperture's dimensions are altered such that the aperture function goes from ... (y, z) to ... (αy, βz), the newly diffracted field will be given by... Get solution

39. Show that when f(t) - A sin (ωt + ε), Cff(T) - (A2/2) cos ωt, which confirms the loss of phase information in the autocorrelation. Get solution

40. Suppose we have a single slit along the y-direction of width b where the aperture function is constant across it at a value of ... What is the diffracted field if we now apodize the slit with a cosine function amplitude mask? In other words, we cause the aperture function to go from ... at the center to 0 at ±b/2 via a cosinusoidal dropoff. Get solution

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45. Show, from the integral definitions, that ....... Get solution

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49. Figure P . l 1.34 shows a transparent ring on an otherwise opaque mask. Make a rough sketch of its autocorrelation function, taking l to be the center-to-center separation against which you plot that function.Figure P.11.34... Get solution

50. Consider the function in Fig. 11.35 as a cosine carrier multiplied by an exponential envelope. Use the frequency convolution theorem to evaluate its Fourier transform. Get solution

Chapter #10 Solutions - Optics - Eugene Hecht - 5th Edition

1. A point source S is a perpendicular distance R away from the center of a circular hole of radius a in an opaque screen. If the distance to the periphery is (R + ℓ), show that Fraunhofer diffraction will occur on a very distant screen whenλR» a2/2What is the smallest satisfactory value of R if the hole has a radius of 1 mm, ℓ≤λ/10, and λ - 500 nm? Get solution

2. Get solution

3. Referring back to the multiple antenna system on p.451, compute the angular separation between successive lobes or principal maxima and the width of the central maximum. Get solution

4. Examine the setup of Fig. 10.3 in order to determine what is happening in the image space of the lenses; in other words, locate the exit pupil and relate it to the diffraction process. Show that the configurations in Fig. P. 10.5 are equivalent to those of Fig. 10.3 and will therefore result in Fraunhofer diffraction. Design at least one more such arrangement.Figure P.10.5... Get solution

5. Get solution

6. The angular distance between the center and the first minimum of a single-slit Fraunhofer diffraction pattern is called the half-angular breadth; write an expression for it. Find the corresponding half-linear width (a) when no focusing lens is present and the slit-viewing screen distance is L, and (b) when a lens of focal length f2 is very close to the aperture, Notice that the half-linear width is also the distance between the successive minima. Get solution

7. A single slit in an opaque screen 0.10 mm wide is illuminated (in air) by plane waves from a krypton ion laser (λ0 — 461.9 nm). If the observing screen is 1.0 m away, determine whether or not the resulting diffraction pattern will be of the far-field variety and then compute the angular width of the central maximum. Get solution

8. A narrow single slit (in air) in an opaque screen is illuminated by infrared from a He-Ne laser at 1152.2 nm, and it is found that the center of the tenth dark band in the Fraunhofer pattern lies at an angle of 6.2° off the central axis. Please detennine the width of the slit. At what angle will the tenth minimum appear if the entire arrangement is immersed in water (nw, = 1.33) rather than air (na = 1.00029)? Get solution

9. A collimated beam of microwaves impinges on a metal screen that contains a long horizontal slit that is 20 cm wide. A detector moving parallel to the screen in the far-field region locates the first minimum of irradiance at an angle of 36.87° above the central axis. Determine the wavelength of the radiation. Get solution

10. Get solution

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14. Show that for a double-slit Fraunhofer pattern, if a = mb, the number of bright fringes (or parts thereof) within the central diffraction maximum will be equal to 2m. Get solution

15. Two long slits 0.10 mm wide, separated by 0.20 mm, in an opaque screen are illuminated by light with a wavelength of 500 nm. If the plane of observation is 2.5 m away, will the pattern correspond to Fraunhofer or Fresnel diffraction? How many Young's fringes will be seen within the central bright band? Get solution

16. Get solution

17. What is the relative irradiance of the subsidiary maxima in a three-slit Fraunhofer diffraction pattern? Draw a graph of the irradiance distribution, when a = 2b, for two and then three slits. Get solution

18. Get solution

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21. Get solution

22. Starting with the irradiance expression for a finite slit, shrink, the slit down to a minuscule area element and show that it emits equally in all directions. Get solution

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25. Show that Fraunhofer diffraction patterns have a center of symmetry [i.e., I(Y, Z) = I( -Y, -Z)], regardless of the configuration of the aperture, as long as there are no phase variations in the field over the region of the hole. Begin with Eq. (10.41). We'll see later (Chapter 11) that this restriction is equivalent to saying that the aperture function is real. Get solution

26. With the results of Problem 10.14 in mind, discuss the symmetries that would be evident in the Fraunhofer diffraction pattern of an aperture that is itself symmetrical about a line (assuming normally incident quasimonochromatic plane waves). Get solution

27. From symmetry considerations, create a rough sketch of the Fraunhofer diffraction patterns of an equilateral triangular aperture and an aperture in the form of a plus sign. Get solution

28. Figure P. 10.17 is the irradiance distribution in the far field for a configuration of elongated rectangular apertures. Describe the arrangement of holes that would give rise to such a pattern and give your reasoning in detail.Figure P.10.17 (Photo courtesy R. G. Wilson, Illinois Wesleyan University.)... Get solution

29. In Fig. P.10.18a and b arc the electric field and irradiance distributions, respectively, in the far field for a configuration of elongated rectangular apertures. Describe the arrangement of holes that would give rise to such patterns and discuss your reasoning.Figure P.10.18 (Photo courtesy R. G. Wilson, Illinois Wesleyan University.)... Get solution

30. Figure P. 10.19 is a computer-generated Fraunhofer irradiance distribution. Describe the aperture that would give rise to such a pattern and give your reasoning in detail.Figure P.10.19 (Photo courtesy R. G. Wilson, Illinois Wesleyan University.)... Get solution

31. Figure P. 10.20 is the electric-field distribution in the far field for a hole of some sort in an opaque screen. Describe the aperture that would give rise to such a pattern and give your reasoning in detail.Figure P.10.20 (Photo courtesy R. G. Wilson, Illinois Wesleyan University.)... Get solution

32. In light of the five previous questions, identify Fig. P.10.21, explaining what it is and what aperture gave rise to it.Figure P.10.21 (Photo courtesy R. G. Wilson, Illinois Wesleyan University.)... Get solution

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36. Verify that the peak irradiance I1 of the first "ring" in the Airy pattern for far-field diffraction at a circular aperture is such that I1/I(0) = 0.0175. You might want to use the fact that... Get solution

38. No lens can focus light down to a perfect point because there will always be some diffraction. Estimate the size of the minimum spot of light that can be expected al the focus of a lens. Discuss the relationship among the focal length, the lens diameter, and the spot size. Take the f-number of the lens to be roughly 0.8 or 0.9, which is just about what you can expect for a fast lens. Get solution

39. Figure P. 10.24 shows several aperture configurations. Roughly sketch the Fraunhofer patterns for each. Note that the circular regions should generate Airy-like ring systems centered at the origin.Figure P.10.24... Get solution

40. Suppose that we have a laser emitting a diffraction-limited beam (λ0 = 632.84 nm) with a 2-mm diameter. How big a light spot would be produced on the surface of the Moon a distance of 376 × 103 Ion away from such a device? Neglect any effects of the Earth's atmosphere. Get solution

41. If you peered through a 0.75-mm hole at an eye chart, you would probably notice a decrease in visual acuity. Compute the angular limit of resolution, assuming that it's determined only by diffraction; take λ0 = 550 nm. Compare your results with the value of 1.7 × 10-4 rad, which corresponds to a 4.0-mm pupil. Get solution

42. Get solution

43. Get solution

44. Get solution

45. The neoimpressionist painter Georges Seurat was a member of the pointilhst school. His paintings consist of an enormous number of closely spaced small dots (...) of pure pigment. The illusion of color mixing is produced only in the eye of the observer. How far from such a painting should one stand in order to achieve the desired blending of color? Get solution

46. The Mount Palomar telescope has an objective mirror with a 508-em diameter. Determine its angular' limit of resolution at a wavelength of 550 nm, in radians, degrees, and seconds of arc. How far apart must two objects be on the surface of the Moon if they are to be resolvable by the Palomar telescope? The Earth-Moon distance is 3.844 × 3 08 m; take λ0 = 550 nm. How far apart must two objects be on the Moon if they are to be distinguished by the eye? Assume a pupil diameter of 4.00 mm. Get solution

47. Get solution

48. Get solution

49. Get solution

50. Get solution

51. A transmission grating whose lines are separated by 3.0 × 10-6 m is illuininated by a narrow beam of red light (λ0 = 694.3 nm) from a ruby laser. Spots of diffracted light, on both sides of the undefeated beam, appear on a screen 2.0 m away. How far from the central axis is each of the two nearest spots? Get solution

52. A diffraction grating with slits 0.60 × 103 cm apart is illuminated by light with a wavelength of 500 nm. At what angle will the third-order maximum appear? Get solution

53. A diffraction grating produces a second-order spectrum of yellow light (λ0 = 550 nm) at 25°. Determine the spacing between the lines on the grating. Get solution

54. Get solution

55. White light falls normally on a transmission grating that contains 1000 lines per centimeter. At what angle will red light (λ0 = 650 nm) emerge in the first-order spectrum? Get solution

56. Light from a laboratory sodium lamp has two strong yellow (components at 589.5923 nm and 588.9953 nm. How far apart in the first-order spectrum will these two lines be on a screen 1.00 m from a grating having 10 000 lines per centimeter? Get solution

57. Get solution

58. Get solution

59. Sunlight impinges on a transmission grating that is formed with 5000 lines per centimeter. Does the third-order spectrum overlap the second-order spectrum? Take red to be 780 nm and violet to be 390 nm. Get solution

60. Get solution

61. Light having a frequency of 4.0 × 1014 Hz is incident on a grating formed with 10 000 lines per centimeter. What is the highest-order spectrum that can be seen with this device? Explain. Get solution

62. Suppose that a grating spectrometer while in vacuum on Earth sends 500-nm light off at an angle of 20.0° in the first-order spectrum. By comparison, after landing on the planet Mongo, the same light is diffracted through 18.0°. Determine the index of refraction of the Mongoian atmosphere. Get solution

63. Prove that the equationa(sin θm - sin θ1) = mλ [10.61]when applied to a transmission grating, is independent of the refractive index. Get solution

64. Get solution

65. A high-resolution grating 260 nun wide, with 300 lines per millimeter, at about 75° in autocollimation has a resolving power of just about 106 for λ = 500 nm. Find its free spectral range. How do these values of R and (∆λ)fsr compare with those of a Fabry-Perot etalon having a 1-cm air gap and a finesse of 25? Get solution

66. What is the total number of lines a grating must have in order just to separate the sodium doublet (λ1 = 5895.9 Å, λ2 = 5890.0 A) in the third order? Get solution

67. Imagine an opaque screen containing 30 randomly located circular holes. The light source is such that every aperture is coherently illuminated by its own plane wave. Each wave in turn is completely incoherent with respect to all the others. Describe the resulting far-field diffraction pattern. Get solution

68. Imagine that you are looking through a piece of square woven cloth at a point source (λ0 = 600 nm) 20m away. If you see a square arrangement of bright spots located about the point source (Fig. P. 10.41), each separated by an apparent nearest-neighbor distance of 12 cm, how close together are the strands of cloth?Figure P.10.41 (Photo by E. H.)... Get solution

69. Perform the necessary mathematical operations needed to arrive at Eq. (10.76). Get solution

70. Referring to Fig. 10.38, integrate the expression dS = 2πρ2 sin (φ dφ over the lth zone to get the area of that zone,...Show that the mean distance to the lth zone is...so that the ratio Al/rl is constant Get solution

71. 'Derive Eq. (10.84). Get solution

72. Get solution

73. Collimated light from a krypton ion laser at 568.19 nm impinges normally on a circular aperture. When viewed axially from a distance of 1.00 m, the hole uncovers the first half-period Fresnel zone. Determine its diameter Get solution

74. Plane waves impinge perpendicularly on a screen with a small circular hole in it. It is found that when viewed from some axial point P the hole uncovers P of the first half-period zone. What is the irradiance at P in terms of the irradiance there when the screen is removed? Get solution

75. Get solution

76. Get solution

77. Get solution

78. Get solution

79. Get solution

80. Get solution

81. Get solution

82. Get solution

83. A collimated beam from a ruby laser (694.3 nm) having an irradiance of 10 W/m2 is incident perpendicularly on an opaque screen containing a square hole 5.0 mm on a side. Compute the irradiance at a point on the central axis 250 cm from the aperture. Get solution

84. Use the Comu spiral to make a rough sketch of ... (w1+w2)/2 for ∆w = 5.5. Compare your results with those of Fig. 10.57. Get solution

85. The Fresnel integrals have the asymptotic forms (corresponding to large values of w) given by......Using this fact, show that the irradiance in the shadow of a semi-infinite opaque screen decreases in proportion to the inverse square of the distance to the edge, as z1 and therefore v1 become large. Get solution

86. What would you expect to see on the plane of observation if the half-plane 2 in Fig. 10.58 were semi-transparent? Get solution

87. Plane waves from a collimated He-Ne laserbeam (λ0 = 632.8 nm) impinge on a steel rod with a 2.5-mm diameter. Draw a rough graphic representation of the diffraction pattern that would be seen on a screen 3.16 m from the rod. Get solution

88. Make a rough sketch of the irradiance function for a Fresnel diffraction pattern arising from a double slit. What would the Cornu spiral picture look like at point P0? Get solution

89. Make a rough sketch of a possible Fresnel diffraction pattern arising from each of the indicated apertures (Fig. P. 10.50).Figure P.10.50... Get solution

90. Suppose the slit in Fig. 10.54 is made very wide. What will the Fresnel diffraction pattern look like? Get solution

91. A long narrow slit 0.10 mm wide is illuminated by light of wavelength 500 nm coming from a point source 0.90 m away. Determine me irradiance at a point 2.0 m beyond the screen when the slit is centered on, and perpendicular to, the line from the source to the point of observation. Write your answer in terms of the unobstructed irradiance. Get solution

92. Get solution

93. Get solution

Chapter #9 Solutions - Optics - Eugene Hecht - 5th Edition

1. Returning to Section 9.1, let...And ...where the wavefront shapes are not explicitly specified, and ... and ... are complex vectors depending on space and initial phase angle. Show that the interference term is then given by...You will have to evaluate terms of the form...for T>> T (take another look at Problem 3.10). Show that Eq. (9.109) leads to Eq. (9.11) for plane waves. Get solution

2. In Section 9.1 we considered the spatial distribution of energy for two point sources. We mentioned that for the case in which the separation a >> λ, I12 spatially averages to zero. Why is this true? What happens when a is much less than λ? Get solution

3. Return to Fig. 2.22 and prove that if two electromagnetic plane waves making an angle θ have the same amplitude, Eθ, the resulting interference pattern on the yx-plane is a cosine-squared irradiance.. distribution given by...Locate the zeros of irradiance. What is the value of the fringe separation? What happens to the separation as θ increases? Compare your analysis with that leading to Eq. (9.17). [Hint: Begin with the wave expressions given in Section 2.7, which have the proper phases already worked out, and write them as exponentials.] Get solution

4. Will we get an interference pattern in Young's Experiment (Fig. 9.8) if we replace the source slit S by a single long-filament light-bulb? What would occur if we replaced the slits S1, and S2 by these same bulbs? Get solution

5. Figure P.9.5 shows an output pattern that was measured by a • tiny microphone when two small piezo-loudspeakers separated by 15 cm were pointed toward the microphone at a distance of 1.5 m away. Given that the speed of sound at 20°C is 343 m/s, determine the approximate frequency at which the speakers were driven. Discuss the nature of the pattern and explain why it has a central minimum.Figure P.9.5 (Data courtesy of CENCO.)... Get solution

6. Two 1.0-MHz radio antennas emitting in-phase are separated by 600 m along a north-south line. A radio receiver placed 2.0 km east is equidistant from both transmitting antennas and picks up a fairly strong signal. How far north should that receiver be moved if it is again to detect a signal nearly as strong? Get solution

7. Get solution

8. Get solution

9. An expanded beam of red light from a He-Ne laser (λ0 = 632.8nm) is incident on a screen containing two very narrow horizontal slits separated by 0.200 mm. A fringe pattern appears on a white screen held 1.00 m away.(a) How far (in radians and millimeters) above and below the central axis ate the first zeros of irradiance?(b) How far (in mm) from the axis is die fifth bright band?(c) Compare these two results. Get solution

10. Get solution

11. Red plane waves from a ruby laser (λ0 = 694.3nm) in air impinge on two parallel slits in an opaque screen. A fringe pattern forms on a distant wall, and we see the fourth bright band 1.0° above the central axis. Kindly calculate the separation between the slits. Get solution

12. A 3 × 5 card containing two pinholes, 0.08 mm in diameter and separated center to center by 0.10 mm, is illuminated by parallel rays of blue light from an argon ion laser (λ0 = 487.99 nm). If the fringes on an observing screen are to be 10mm apart, how far away should the screen be? Get solution

13. White light falling on two long narrow slits emerges and is observed on a distant screen. If red light (λ0 = 780 nm) in the first-order fringe overlaps violet in the second-order fringe, what is the latter's wavelength? Get solution

14. Get solution

15. Get solution

16. Considering the double-slit experiment, derive an equation for the distance ym from the central axis to the m'th irradiance minimum, such that the first dark bands on either side of the central maximum correspond to m' = ±1. Identify and justify all your approximations. Get solution

17. Get solution

18. With regard to Young's Experiment, derive a general expression for the shift in the vertical position of the mth maximum as a result of placing a thin parallel sheet of glass of index n and thickness d directly over one of the slits. Identify your assumptions. Get solution

19. Plane waves of monochromatic light impinge at an angle θi on a screen containing two narrow slits separated by a distance a. Derive" an equation for the angle measured from the central axis which locates the mth maximum. Get solution

20. Sunlight incident on a screen containing two long narrow slits 0.20 mm apart casts a pattern on a white sheet of paper 2.0 m beyond. What is the distance separating the violet (λ0 = 400 nm) in the first-order band from the red (λ0 = 600 nm) in the second-order band? Get solution

21. To examine the conditions under which the approximations of H3q. (9.23) are valid:(a) Apply the law of cosines to triangle S1S2P in Fig. 9.8c to get...(b) Expand this in a Maclaurin series yielding...(c) In light of Eq. (9.17), show that if (r1, - r2) is to equal a sin θ, it is required that r1, >> a2/λ. Get solution

22. A stream of electrons, each having an energy of 0.5 eV, impinges on a pair of extremely thin slits separated by 10-2mm. What is the distance between adjacent minima on a screen 20 m behind the slits? (me, = 9.108 × 10-31kg, 1 eV = 1.602 × 10-19J.) Get solution

23. It is our intention to produce interference fringes by illuminating some sort of arrangement (Young's Experiment, a thin film, the Michelson Interferometer, etc.) with light at a mean wavelength of 500 nm, having a linewidth of 2.5 X 10-3 nm. At approximately what optical path length difference can you expect the fringes to vanish? [Hint: Think about the coherence length and revisit Problem 7.39.] Get solution

24. Imagine that you have an opaque screen with three horizontal 1 very narrow parallel slits in it. The second slit is a center-to-center distance a beneath the first, and the third is a distance 5a/2 beneath the first, (a) Write a complex exponential expression in terms of δ for the amplitude of the electric field at some point P at an elevation θ on a distant screen where δ = ka sin θ. Prove that...Verify that at θ =0,I(θ) = I (0). Get solution

25. Get solution

26. In the Fresnel double mirror s = 2 m, λ0 = 589 nm, and the separation of the fringes was found to be 0.5 mm. What is the angle of inclination of the mirrors, if the perpendicular distance of the actual point source to the intersection of the two mirrors is I m? Get solution

27. Show that a for the Fresnel biprism of Fig. 9.13 is given by a = 2d(n - l)α. Get solution

28. The Fresnel biprism is used to obtain fringes from a point source that is placed 2 m from the screen, and the prism is midway between the source and the screen. Let the wavelength of the light be λ0 = 500 nm and the index of refraction of the glass be n = 1.5. What is the prism angle, if the separation of the fringes is 0.5 mm? Get solution

29. What is the general expression for the separation of the fringes of a Fresnel biprism of index n immersed in a medium having an index of refraction n'? Get solution

30. Get solution

31. Using Lloyd's mirror, X-ray fringes were observed, the spacing of which was found to be 0.ÖÖ2 5 cm. The wavelength used was 8.33 Ǻ. If the source-screen distance was 3 m, how high above the min or plane was the point source of X-rays placed? Get solution

32. Imagine that we have an antenna at the edge of a lake picking up a signal from a distant radio star (Fig. P.9.24), which is just coming up above the horizon. Write expressions for δ and for the angular position of the star when the antenna detects its first maximum.Figure P.9.24... Get solution

33. If the plate in Fig. 9.17 is glass in air, show that the amplitudes of Elr, E2r, and E3r are respectively 0.2 E0i, 0.192 E0i. and 0.008E0i where E0i is the incident amplitude. Make use of the Fresnel coefficients at normal incidence, assuming no absorption. You might repeal the calculation for a water film in air. Get solution

34. A soap film surrounded by air has an index of refraction of 1.34. If a region of the film appears bright red (λ0 = 633 nm) in normally reflected light, what is its minimum thickness there? Get solution

35. A thin film of ethyl alcohol (n = 1.36) spread on a flat glass plate and illuminated with white light shows a color pattern in reflection. If a region of the film reflects only green light (500 nm) strongly, how thick is it? Get solution

36. A soap film of index 1.34 has a region where it is 550.0 nm thick. Detennine the vacuum wavelengths of the radiation that is not reflected when the film is illuminated from above with sunlight. Get solution

37. Get solution

38. Consider the circular pattern of Haidinger's fringes resulting from a film with a thickness of 2 mm and an index of refraction of 1.5. For monochromatic illumination of λ0 = 600 nm, find the value of m for the central fringe (θt, = 0). Will it be bright or dark? Get solution

39. Illuminate a microscope slide (or even belter, a thin cover-glass slide). Colored fringes can easily be seen with an ordinary fluorescent lamp (although some of the newer versions don't work well at all) serving as a broad source or a mercury street light as a point source. Describe the fringes. Now rotate the glass. Does the pattern change? Duplicate the conditions shown in Figs. 9.18 and 9.19. Try it again with a sheet of plastic food wrap stretched across the lop of a cup. Get solution

40. Fringes are observed when a parallel beam of light of wavelength 500 nm is incident perpendicularly onto a wedge-shaped film with an index of refraction of 1.5. What is the angle of the wedge if the fringe separation is ... cm?Figure P.9.31... Get solution

41. Suppose a wedge-shaped air film is made between two sheets of glass, with a piece of paper 7.618 X 10-5 m thick used as the spacer at their very ends. If light of wavelength 500 nm comes down from directly above, determine the number of bright fringes that will be seen across the wedge. Get solution

42. Get solution

43. Figure P.9.31 illustrates a setup used for testing lenses. Show thatd = x2(R1 – R1,)/2R1R2when d1 and d2 are negligible in comparison with 2R1, and 2R2, respectively. (Recall the theorem from plane geometry that relates the products of the segments of intersecting chords.) Prove that the radius of the mth dark fringe is then...How does this relate to Eq. (9.43)? Get solution

44. Newton rings are observed on a film with quasimonochromatic that has a wavelength of 500 nm. If the 20th bright ring has a radius of 1 cm, what is the radius of curvature of the lens forming one part of the interfering system? Get solution

45. Get solution

46. Get solution

47. Get solution

48. One of the mirrors of a Michelson Interferometer is moved, and 1000 fringe-pairs shift past the hairline in a viewing telescope during the process. If the device is illuminated wilh 500-nm light, how far was the mirror moved? Get solution

49. Get solution

50. Suppose we place a chamber 10.0 cm long with flat parallel windows in one arm of a Michelson Interferometer that is being illuminated by 600-nm light. If the refractive index of air is 1.000 29 and all the air is pumped out of the cell, how many fringe-pairs will shift by in the process? Get solution

51. Get solution

52. A form of the Jamin Interferometer is illustrated in Fig. P.9.38. How does it work? To what use might it be put? Get solution

53. Starting with Eq. (9.53) for the transmitted wave, compute the flux density, thai is, Eq. (9.54). Get solution

54. Given that the mirrors of a Fabry-Perol Interferometer have an...amplitude reflection coefficient of r = 0.8944, find(a) the coefficient of finesse,(b) the half-width,(c) the finesse, and,(d) the contrast factor defined by... Get solution

55. To fill in some of the details in the derivation of the smallest phase increment separating two resolvable Fabry-Perot fringes, that is,...satisfy yourself that...Show that Eq. (9.72) can be rewritten as...When F is large γ is small, and sin (∆δ) = ∆δ. Prove that Eq. (9.73) then follows. Get solution

56. Consider the interference pattern of the Michelson Interferometer as arising from two beams of equal flux density. Using Eq.(9.17), compute the half-width. What is the separation, in S, between adjacent maxima? What then is the finesse? Get solution

57. Satisfy yourself of the fact that a film of thickness λf/4 and index n1, will always reduce the reflectance of the substrate on which it is deposited, as long as ns, > n1, >n0. Consider the simplest case of normal incidence and n0 = 1. Show that this is equivalent to saying that the waves reflected back from the two interfaces cancel one another. Get solution

58. Verify that the reflectance of a substrate can be increased by coating it with a λf/4, high-index layer, that is, n1 > ns* Show that the reflected waves interfere constructively. The quarter-wave stack g(HL)"'Ha can be thought of as a series of such structures. Get solution

59. Determine the refractive index and thickness of a film to be deposited on a glass surface (ng = 1.54) such that no normally incident light of wavelength 540 nm is reflected. Get solution

60. A glass microscope lens having an index of 1.55 is to be coated with a magnesium fluoride film to increase the transmission of normally incident yellow light (λ0 = 550 nm). What minimum thickness should be deposited on the lens? Get solution

61. A glass camera lens with an index of 1.55 is to be coated with a cryolite film (n ≈ 1.30) to decrease the reflection of normally incident green light (λ0 = 500 nm). What thickness should be deposited on the lens? Get solution

62. Using Fig. 9.60, which depicts the geometry of the Shuttle -radar interferometer, show thatz(x) = h – r1, cos θThen use (he Law of Cosines to establish that Eq. (9.108) is correct. Get solution