1. Determine the Fourier series for the function of spatial period L given by ... Get solution
2. A half-wave rectifier removes the negative half-cycles of a sinusoidal waveform, given by E = E0 cos ωt. Find the Fourier series of the resulting wave.Figure 1... Get solution
3. Find the Fourier transform of the Gaussian function given by ...where h is the height and σ the “width.” (Hint: Remember how to complete a square? You will also need the definite integral ...in your calculations.) Does the transform, interpreted as the frequency spectrum, show the proper relationship to the original “pulse” width? Get solution
4. Using the Fourier transform, determine the power spectrum of a single square pulse of amplitude A and duration τ0. Sketch the power spectrum, locating its zeros, and show that the frequency bandwidth for the pulse is inversely proportional to its duration.Figure 1... Get solution
5. Two light filters are used to transmit yellow light centered around a wavelength of 590 nm. One filter has a “broad” transmission width of 100 nm, whereas the other has a “narrow” pass band of 10 nm. Which filter would be better to use for an interference experiment? Compare the coherence lengths of the light from each. Get solution
6. A continuous He-Ne laser beam (632.8 nm) is “chopped,” using a spinning aperture, into 1-μs pulses. Compute the resultant line width Δλ, bandwidth Δν, and coherence length. Get solution
7. The angular diameter of the sun viewed from the earth is approximately 0.5 degree. Determine the spatial coherence length for “good” coherence, neglecting any variations in brightness across the surface. Let us consider, somewhat arbitrarily, that “good” coherence will exist over an area that is 10% of the maximum area of coherence. Get solution
8. Michelson found that the cadmium red line (643.8 nm) was one of the most ideal monochromatic sources available, allowing fringes to be discerned up to a path difference of 30 cm in a beam-splitting interference experiment, such as with a Michelson interferometer. Calculate (a) the wavelength spread of the line and (b) the coherence time of the source. Get solution
9. A narrow band-pass filter transmits wavelengths in the range 5000 ± 0.5 Å. If this filter is placed in front of a source of white light, what is the coherence length of the transmitted light? Get solution
10. Let a collimatcd beam of white light fall on one refracting face of a prism and let the light emerging from the second face be focused by a lens onto a screen. Suppose that the linear dispersion at the screen is 20 Å/mm. By introducing a narrow “exit slit” in the screen, one has a type of monochromator that provides a nearly monochromatic beam of light. Sketch the setup. For an exit slit of 0.02 cm, what is the coherence time and coherence length of the light of mean wavelength 5000 Å? Get solution
11. A pinhole of diameter 0.5 mm is used in front of a sodium lamp (5890 Å) as a source in a Young interference experiment. The distance from pinhole to slits is 1 m. What is the maximum slit space insuring interference fringes that are just visible? Get solution
12. Determine the linewidth in angstroms and hertz for laser light whose coherence length is 10 km. The mean wavelength is 6328 Å. Get solution
13. a. A monochromator is used to obtain quasi-monochromatic light from a tungsten lamp. The linear dispersion of the instrument is 20 Å /mm and an exit slit of 200 μm is used. What is the coherence time and length of the light from the monochromator when set to give light of mean wavelength 500 nm?b. This light is used to form fringes in an interference experiment in which the light is first amplitude-split into two equal parts and then brought together again. If the optical path difference between the two paths is 0.400 mm, calculate the magnitude of the normalized correlation function and the visibility of the resulting fringes.c. If the maximum irradiance produced by the fringes is 100 on an arbitrary scale, what is the difference between maximum irradiance and background irradiance on this scale? Get solution
14. Determine the length and base area of the cylindrical volume within which light received from the sun is coherent. For this purpose, let us assume “good” spatial coherence occurs within a length that is 25% of the maximum value given by Eq. (9-38). The sun subtends an angle of 0.5° at the earth’s surface. The mean value of the visible spectrum may be taken at 550 nm. Express the coherence volume also in terms of number of wavelengths across cylindrical length and diameter.Equation 9-38... Get solution
16. Show that the visibility of double-slit fringes in the mth order is given by ...where λ is the average wavelength of the light and Δλ is its linewidth. Get solution
17. A filtered mercury lamp produces green light at 546.1 nm with a linewidth of 0.05 nm. The light illuminates a double slit of spacing 0.1 mm. Determine the visibility of the fringes on a screen 1 m away, in the vicinity of the fringe of order m = 20. (See problem 1.) If the discharge lamp is replaced with a white light source and a filter of bandwidth 10 nm at 546 nm, how does the visibility change?Figure 1...Problem 1Show that the visibility of double-slit fringes in the mth order is given by ...where λ is the average wavelength of the light and Δλ is its linewidth. Get solution
18. A Michelson interferometer forms fringes with cadmium red light of 643.847 nm and linewidth of 0.0013 nm. What is the visibility of the fringes when one mirror is moved 1 cm from the position of zero path difference between arms? How does this change when the distance moved is 5 cm? At what distance does the visibility go to zero? Get solution
19. a. Repeat problem 1 when the light is the green mercury line of 546.1 nm with a linewidth of 0.025 nm.b. How far can the mirror be moved from zero path difference so that fringe visibility is at least 0.85?Problem 1A Michelson interferometer forms fringes with cadmium red light of 643.847 nm and linewidth of 0.0013 nm. What is the visibility of the fringes when one mirror is moved 1 cm from the position of zero path difference between arms? How does this change when the distance moved is 5 cm? At what distance does the visibility go to zero? Get solution
2. A half-wave rectifier removes the negative half-cycles of a sinusoidal waveform, given by E = E0 cos ωt. Find the Fourier series of the resulting wave.Figure 1... Get solution
3. Find the Fourier transform of the Gaussian function given by ...where h is the height and σ the “width.” (Hint: Remember how to complete a square? You will also need the definite integral ...in your calculations.) Does the transform, interpreted as the frequency spectrum, show the proper relationship to the original “pulse” width? Get solution
4. Using the Fourier transform, determine the power spectrum of a single square pulse of amplitude A and duration τ0. Sketch the power spectrum, locating its zeros, and show that the frequency bandwidth for the pulse is inversely proportional to its duration.Figure 1... Get solution
5. Two light filters are used to transmit yellow light centered around a wavelength of 590 nm. One filter has a “broad” transmission width of 100 nm, whereas the other has a “narrow” pass band of 10 nm. Which filter would be better to use for an interference experiment? Compare the coherence lengths of the light from each. Get solution
6. A continuous He-Ne laser beam (632.8 nm) is “chopped,” using a spinning aperture, into 1-μs pulses. Compute the resultant line width Δλ, bandwidth Δν, and coherence length. Get solution
7. The angular diameter of the sun viewed from the earth is approximately 0.5 degree. Determine the spatial coherence length for “good” coherence, neglecting any variations in brightness across the surface. Let us consider, somewhat arbitrarily, that “good” coherence will exist over an area that is 10% of the maximum area of coherence. Get solution
8. Michelson found that the cadmium red line (643.8 nm) was one of the most ideal monochromatic sources available, allowing fringes to be discerned up to a path difference of 30 cm in a beam-splitting interference experiment, such as with a Michelson interferometer. Calculate (a) the wavelength spread of the line and (b) the coherence time of the source. Get solution
9. A narrow band-pass filter transmits wavelengths in the range 5000 ± 0.5 Å. If this filter is placed in front of a source of white light, what is the coherence length of the transmitted light? Get solution
10. Let a collimatcd beam of white light fall on one refracting face of a prism and let the light emerging from the second face be focused by a lens onto a screen. Suppose that the linear dispersion at the screen is 20 Å/mm. By introducing a narrow “exit slit” in the screen, one has a type of monochromator that provides a nearly monochromatic beam of light. Sketch the setup. For an exit slit of 0.02 cm, what is the coherence time and coherence length of the light of mean wavelength 5000 Å? Get solution
11. A pinhole of diameter 0.5 mm is used in front of a sodium lamp (5890 Å) as a source in a Young interference experiment. The distance from pinhole to slits is 1 m. What is the maximum slit space insuring interference fringes that are just visible? Get solution
12. Determine the linewidth in angstroms and hertz for laser light whose coherence length is 10 km. The mean wavelength is 6328 Å. Get solution
13. a. A monochromator is used to obtain quasi-monochromatic light from a tungsten lamp. The linear dispersion of the instrument is 20 Å /mm and an exit slit of 200 μm is used. What is the coherence time and length of the light from the monochromator when set to give light of mean wavelength 500 nm?b. This light is used to form fringes in an interference experiment in which the light is first amplitude-split into two equal parts and then brought together again. If the optical path difference between the two paths is 0.400 mm, calculate the magnitude of the normalized correlation function and the visibility of the resulting fringes.c. If the maximum irradiance produced by the fringes is 100 on an arbitrary scale, what is the difference between maximum irradiance and background irradiance on this scale? Get solution
14. Determine the length and base area of the cylindrical volume within which light received from the sun is coherent. For this purpose, let us assume “good” spatial coherence occurs within a length that is 25% of the maximum value given by Eq. (9-38). The sun subtends an angle of 0.5° at the earth’s surface. The mean value of the visible spectrum may be taken at 550 nm. Express the coherence volume also in terms of number of wavelengths across cylindrical length and diameter.Equation 9-38... Get solution
16. Show that the visibility of double-slit fringes in the mth order is given by ...where λ is the average wavelength of the light and Δλ is its linewidth. Get solution
17. A filtered mercury lamp produces green light at 546.1 nm with a linewidth of 0.05 nm. The light illuminates a double slit of spacing 0.1 mm. Determine the visibility of the fringes on a screen 1 m away, in the vicinity of the fringe of order m = 20. (See problem 1.) If the discharge lamp is replaced with a white light source and a filter of bandwidth 10 nm at 546 nm, how does the visibility change?Figure 1...Problem 1Show that the visibility of double-slit fringes in the mth order is given by ...where λ is the average wavelength of the light and Δλ is its linewidth. Get solution
18. A Michelson interferometer forms fringes with cadmium red light of 643.847 nm and linewidth of 0.0013 nm. What is the visibility of the fringes when one mirror is moved 1 cm from the position of zero path difference between arms? How does this change when the distance moved is 5 cm? At what distance does the visibility go to zero? Get solution
19. a. Repeat problem 1 when the light is the green mercury line of 546.1 nm with a linewidth of 0.025 nm.b. How far can the mirror be moved from zero path difference so that fringe visibility is at least 0.85?Problem 1A Michelson interferometer forms fringes with cadmium red light of 643.847 nm and linewidth of 0.0013 nm. What is the visibility of the fringes when one mirror is moved 1 cm from the position of zero path difference between arms? How does this change when the distance moved is 5 cm? At what distance does the visibility go to zero? Get solution
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