1. When one mirror of a Michelson interferometer is translated by
0.0114 cm, 523 fringes are observed to pass the crosshairs of the
viewing telescope. Calculate the wavelength of the light. Get solution
2. When looking into a Michelson interferometer illuminated by the 546.1-nm light of mercury, one sees a series of straight-line fringes that number 12 per centimeter. Explain their occurrence. Get solution
3. A thin sheet of fluorite of index 1.434 is inserted normally into one beam of a Michelson interferometer. Using light of wavelength 589 nm, the fringe pattern is found to shift by 35 fringes. What is the thickness of the sheet? Get solution
4. Looking into a Michelson interferometer, one sees a dark central disk surrounded by concentric bright and dark rings. One arm of the device is 2 cm longer than the other, and the wavelength of the light is 500 nm. Determine (a) the order of the central disc and (b) the order of the sixth dark ring from the center. Get solution
5. A Michelson interferometer is used to measure the refractive index of a gas. The gas is allowed to flow into an evacuated glass cell of length L placed in one arm of the interferometer. The wavelength is λ.a. If N fringes are counted as the pressure in the cell changes from vacuum to atmospheric pressure, what is the index of refraction n in terms of N, λ, and L?b. How many fringes would be counted if the gas were carbon dioxide (n = 1.00045) for a 10-cm cell length, using sodium light at 589 nm? Get solution
6. A Michelson interferometer is used with red light of wavelength 632.8 nm and is adjusted for a path difference of 20 μm. Determine the angular radius of the (a) first (smallest- diameter) ring observed and (b) the tenth ring observed. Get solution
7. A polished surface is examined using a Michelson interferometer with the polished surface replacing one of the mirrors. A fringe pattern characterizing the surface contour is observed using He-Ne light of wavelength 632.8 nm. Fringe distortion over the surface is found to be less than one- fourth the fringe separation at any point. What is the maximum depth of polishing defects on the surface? Get solution
9. A Fabry-Perot interferometer is to be used to resolve the mode structure of a He-Ne laser operating at 632.8 nm. The frequency separation between the modes is 150 MHz. The plates are separated by an air gap and have a reflectance (r2) of 0.999.a. What is the coefficient of finesse of the instrument?b. What is the resolving power required?c. What plate spacing is required?d. What is the free spectral range of the instrument under these conditions?e. What is the minimum resolvable wavelength interval under these conditions? Get solution
10. A Fabry-Perot etalon is fashioned from a single slab of transparent material having a high refractive index (n = 4.5) and a thickness of 2 cm. The uncoated surfaces of the slab have a reflectance (r2) of 0.90. If the etalon is used in the vicinity of wavelength 546 nm, determine (a) the highest- order fringe in the interference pattern, (b) the ratio Tmax/Tmin, and (c) the resolving power.Figure 1... Get solution
12. White light is passed through a Fabry-Perot interferometer in the arrangement shown in Figure 1, where the detector is a spectroscope. A series of bright bands appear. When mercury light is simultaneously admitted into the spectroscope slit, 150 of the bright bands are seen to fall between the violet and green lines of mercury at 435.8 nm and 546.1 nm, respectively. What is the thickness of the etalon?Figure 1... Get solution
13. Apply the reasoning used to calculate the finesse of a Fabry- Perot interferometer to the Michelson interferometer. Using the irradiance of Michelson fringes as a function of phase, calculate (a) the fringe separation; (b) the fringe width at half-maximum; (c) their ratio, the finesse. Get solution
14. Assume that in a Mach-Zehnder interferometer (Figure 8-5), the beam splitter and mirror M3 each transmit 80% and reflect 20% of the incident light. Compare the visibility (Eq. (7-17)) when observing the interference of the two emerging beams (shown) with the visibility that results from the two beams emerging from M3 along a direction at 90° relative to the first (not shown). For the second case, beam (1) is reflected and beam (2) is transmitted at M3. Get solution
15. Consider the Fabry-Perot cavity shown in Figure 8-8.a. With the method used in Section 8-5 to derive the Fabry- Perot transmittance, find the reflectance, R = IR/It, of a Fabry-Perot cavity. (Note:The reflection coefficient for the external surface of the cavity mirror must be −r if that from the internal surface is r and the transmission coefficients t are real.)b. Using the result from (a) and Eq. (8-24) (or an equivalent form), show that the sum of the irradiances reflected by and transmitted through the Fabry-Perot cavity is equal to the irradiance in the field incident on the Fabry- Perot. That is, show that IR + IT = II. Get solution
16. The reflectance R (see Problem 1) of a Fabry-Perot etalon is 0.6. Determine the ratio of transmittance of the étalon at maximum to the transmittance at halfway between maxima.Problem1Consider the Fabry-Perot cavity shown in Figure 8-8.a. With the method used in Section 8-5 to derive the Fabry- Perot transmittance, find the reflectance, R = IR/It, of a Fabry-Perot cavity. (Note:The reflection coefficient for the external surface of the cavity mirror must be −r if that from the internal surface is r and the transmission coefficients t are real.)b. Using the result from (a) and Eq. (8-24) (or an equivalent form), show that the sum of the irradiances reflected by and transmitted through the Fabry-Perot cavity is equal to the irradiance in the field incident on the Fabry- Perot. That is, show that IR + IT = II. Get solution
17. Find the transmittance, T = IT/II, and the reflectance, R = IR/II, of a Fabry-Perot cavity with mirrors of (internal) reflection coefficients r1 and r2 ≠ r1]. Take the mirror separation to be d and see the note given in part (a) of Problem 1.Problem 1Consider the Fabry-Perot cavity shown in Figure 8-8.a. With the method used in Section 8-5 to derive the Fabry- Perot transmittance, find the reflectance, R = IR/It, of a Fabry-Perot cavity. (Note:The reflection coefficient for the external surface of the cavity mirror must be −r if that from the internal surface is r and the transmission coefficients t are real.)b. Using the result from (a) and Eq. (8-24) (or an equivalent form), show that the sum of the irradiances reflected by and transmitted through the Fabry-Perot cavity is equal to the irradiance in the field incident on the Fabry- Perot. That is, show that IR + IT = II. Get solution
18. Consider the transmittance of the variable-input-frequency Fabry-Perot cavity shown in Figure 8-15. Assume that the Fabry-Perot cavity used has a length of 10 cm and that the nominal frequency of the laser input is 4.53 × 1014 Hz. Finda. The finesse, ..., of the cavity.b. The free spectral range, vfsr, of the transmittance.c. The FWHM, 2Δv1/2, of a transmittance peak.d. The quality factor, Q, of the cavity.e. The photon lifetime, τp, of the cavity. Get solution
19. Plot the transmittance, T, as a function of cavity length, d, for a scanning Fabry-Perot interferometer with a monochromatic input of wavelength 632.8 nm if the finesse, ..., of the cavity is 15. In the plot let d range from 5 cm to 5.000001 cm. Get solution
20. Find the values of all the quantities listed in the first column of Table 8-2 for a mirror reflection coefficient of 0.999. Get solution
21. Consider a light source consisting of two components with different wavelength λ1 and λ2. Let light from this source be incident on a scanning Fabry-Perot interferometer of nominal length d = 5 cm. Let the scaled transmittance through the Fabry-Perot as a function of the change in the cavity length be as shown in Figure 1a and 1b. Figure 1b shows the first set of dual peaks of Figure 1a over a smaller length scale in order to allow a closer examination of the structure of the overlapping peaks.a. What is the nominal wavelength of the light source?b. Estimate the difference λ2 − λ1 in wavelength of the two components presuming that the overlapping transmittance peaks have the same mode number, m2 = m1 = m.c. Estimate the difference λ 2 − λ1 in wavelength of the two components presuming that the overlapping transmittance peaks have mode numbers that differ by 1, so that m2 = m1+ 1.Figure 1... Get solution
22. In this problem we examine experimental absorption spectroscopy data. The output of a variable-frequency diode laser is divided at a beam splitter so that part of the laser beam is incident on a Fabry-Perot cavity of fixed length and part of the laser beam passes through a sample cell containing atmospheric oxygen, as shown in Figure 1a. An overlay of the scaled transmittance through the Fabry- Perot cavity (solid curve) and the scaled transmittance through the oxygen cell (curve made with + symbols) as functions of the laser frequency change is shown in Figure 1b. The dips in the transmittance through the oxygen cell indicate that the oxygen molecule strongly absorbs these frequencies. The free spectral range of the Fabry-Perot cavity used in the experiment was known to be 11.6 GHz. The free spectral range can be taken to be the distance between the tall transmittance peaks, indicated by the arrows in Figure 1b. (A spherical-mirror Fabry-Perot cavity was used in the experiment and so the transmittance includes peaks corresponding to both longitudinal and transverse modes.)a. Estimate the difference in the frequencies of the two absorption dips shown in Figure 1b.b. Estimate the “full-width-at-half-depth” of each absorption dip.Figure 1 (a) Experimental arrangement, (b) Overlay of transmit- tancc curves. (Courtesy of R. J. Brecha, Physics Department, University of Dayton.)... Get solution
23. Consider the transmittance through a Fabry-Perot interferometer as a function of the variable wavelength λ of its input field. Show that the FWHM of the transmittance peaks is ... and the separation between transmittance peaks is λfsr = λ/m. (Here m = 2d/λ, where d is the length of the Fabry-Perot interferometer.) Get solution
2. When looking into a Michelson interferometer illuminated by the 546.1-nm light of mercury, one sees a series of straight-line fringes that number 12 per centimeter. Explain their occurrence. Get solution
3. A thin sheet of fluorite of index 1.434 is inserted normally into one beam of a Michelson interferometer. Using light of wavelength 589 nm, the fringe pattern is found to shift by 35 fringes. What is the thickness of the sheet? Get solution
4. Looking into a Michelson interferometer, one sees a dark central disk surrounded by concentric bright and dark rings. One arm of the device is 2 cm longer than the other, and the wavelength of the light is 500 nm. Determine (a) the order of the central disc and (b) the order of the sixth dark ring from the center. Get solution
5. A Michelson interferometer is used to measure the refractive index of a gas. The gas is allowed to flow into an evacuated glass cell of length L placed in one arm of the interferometer. The wavelength is λ.a. If N fringes are counted as the pressure in the cell changes from vacuum to atmospheric pressure, what is the index of refraction n in terms of N, λ, and L?b. How many fringes would be counted if the gas were carbon dioxide (n = 1.00045) for a 10-cm cell length, using sodium light at 589 nm? Get solution
6. A Michelson interferometer is used with red light of wavelength 632.8 nm and is adjusted for a path difference of 20 μm. Determine the angular radius of the (a) first (smallest- diameter) ring observed and (b) the tenth ring observed. Get solution
7. A polished surface is examined using a Michelson interferometer with the polished surface replacing one of the mirrors. A fringe pattern characterizing the surface contour is observed using He-Ne light of wavelength 632.8 nm. Fringe distortion over the surface is found to be less than one- fourth the fringe separation at any point. What is the maximum depth of polishing defects on the surface? Get solution
9. A Fabry-Perot interferometer is to be used to resolve the mode structure of a He-Ne laser operating at 632.8 nm. The frequency separation between the modes is 150 MHz. The plates are separated by an air gap and have a reflectance (r2) of 0.999.a. What is the coefficient of finesse of the instrument?b. What is the resolving power required?c. What plate spacing is required?d. What is the free spectral range of the instrument under these conditions?e. What is the minimum resolvable wavelength interval under these conditions? Get solution
10. A Fabry-Perot etalon is fashioned from a single slab of transparent material having a high refractive index (n = 4.5) and a thickness of 2 cm. The uncoated surfaces of the slab have a reflectance (r2) of 0.90. If the etalon is used in the vicinity of wavelength 546 nm, determine (a) the highest- order fringe in the interference pattern, (b) the ratio Tmax/Tmin, and (c) the resolving power.Figure 1... Get solution
12. White light is passed through a Fabry-Perot interferometer in the arrangement shown in Figure 1, where the detector is a spectroscope. A series of bright bands appear. When mercury light is simultaneously admitted into the spectroscope slit, 150 of the bright bands are seen to fall between the violet and green lines of mercury at 435.8 nm and 546.1 nm, respectively. What is the thickness of the etalon?Figure 1... Get solution
13. Apply the reasoning used to calculate the finesse of a Fabry- Perot interferometer to the Michelson interferometer. Using the irradiance of Michelson fringes as a function of phase, calculate (a) the fringe separation; (b) the fringe width at half-maximum; (c) their ratio, the finesse. Get solution
14. Assume that in a Mach-Zehnder interferometer (Figure 8-5), the beam splitter and mirror M3 each transmit 80% and reflect 20% of the incident light. Compare the visibility (Eq. (7-17)) when observing the interference of the two emerging beams (shown) with the visibility that results from the two beams emerging from M3 along a direction at 90° relative to the first (not shown). For the second case, beam (1) is reflected and beam (2) is transmitted at M3. Get solution
15. Consider the Fabry-Perot cavity shown in Figure 8-8.a. With the method used in Section 8-5 to derive the Fabry- Perot transmittance, find the reflectance, R = IR/It, of a Fabry-Perot cavity. (Note:The reflection coefficient for the external surface of the cavity mirror must be −r if that from the internal surface is r and the transmission coefficients t are real.)b. Using the result from (a) and Eq. (8-24) (or an equivalent form), show that the sum of the irradiances reflected by and transmitted through the Fabry-Perot cavity is equal to the irradiance in the field incident on the Fabry- Perot. That is, show that IR + IT = II. Get solution
16. The reflectance R (see Problem 1) of a Fabry-Perot etalon is 0.6. Determine the ratio of transmittance of the étalon at maximum to the transmittance at halfway between maxima.Problem1Consider the Fabry-Perot cavity shown in Figure 8-8.a. With the method used in Section 8-5 to derive the Fabry- Perot transmittance, find the reflectance, R = IR/It, of a Fabry-Perot cavity. (Note:The reflection coefficient for the external surface of the cavity mirror must be −r if that from the internal surface is r and the transmission coefficients t are real.)b. Using the result from (a) and Eq. (8-24) (or an equivalent form), show that the sum of the irradiances reflected by and transmitted through the Fabry-Perot cavity is equal to the irradiance in the field incident on the Fabry- Perot. That is, show that IR + IT = II. Get solution
17. Find the transmittance, T = IT/II, and the reflectance, R = IR/II, of a Fabry-Perot cavity with mirrors of (internal) reflection coefficients r1 and r2 ≠ r1]. Take the mirror separation to be d and see the note given in part (a) of Problem 1.Problem 1Consider the Fabry-Perot cavity shown in Figure 8-8.a. With the method used in Section 8-5 to derive the Fabry- Perot transmittance, find the reflectance, R = IR/It, of a Fabry-Perot cavity. (Note:The reflection coefficient for the external surface of the cavity mirror must be −r if that from the internal surface is r and the transmission coefficients t are real.)b. Using the result from (a) and Eq. (8-24) (or an equivalent form), show that the sum of the irradiances reflected by and transmitted through the Fabry-Perot cavity is equal to the irradiance in the field incident on the Fabry- Perot. That is, show that IR + IT = II. Get solution
18. Consider the transmittance of the variable-input-frequency Fabry-Perot cavity shown in Figure 8-15. Assume that the Fabry-Perot cavity used has a length of 10 cm and that the nominal frequency of the laser input is 4.53 × 1014 Hz. Finda. The finesse, ..., of the cavity.b. The free spectral range, vfsr, of the transmittance.c. The FWHM, 2Δv1/2, of a transmittance peak.d. The quality factor, Q, of the cavity.e. The photon lifetime, τp, of the cavity. Get solution
19. Plot the transmittance, T, as a function of cavity length, d, for a scanning Fabry-Perot interferometer with a monochromatic input of wavelength 632.8 nm if the finesse, ..., of the cavity is 15. In the plot let d range from 5 cm to 5.000001 cm. Get solution
20. Find the values of all the quantities listed in the first column of Table 8-2 for a mirror reflection coefficient of 0.999. Get solution
21. Consider a light source consisting of two components with different wavelength λ1 and λ2. Let light from this source be incident on a scanning Fabry-Perot interferometer of nominal length d = 5 cm. Let the scaled transmittance through the Fabry-Perot as a function of the change in the cavity length be as shown in Figure 1a and 1b. Figure 1b shows the first set of dual peaks of Figure 1a over a smaller length scale in order to allow a closer examination of the structure of the overlapping peaks.a. What is the nominal wavelength of the light source?b. Estimate the difference λ2 − λ1 in wavelength of the two components presuming that the overlapping transmittance peaks have the same mode number, m2 = m1 = m.c. Estimate the difference λ 2 − λ1 in wavelength of the two components presuming that the overlapping transmittance peaks have mode numbers that differ by 1, so that m2 = m1+ 1.Figure 1... Get solution
22. In this problem we examine experimental absorption spectroscopy data. The output of a variable-frequency diode laser is divided at a beam splitter so that part of the laser beam is incident on a Fabry-Perot cavity of fixed length and part of the laser beam passes through a sample cell containing atmospheric oxygen, as shown in Figure 1a. An overlay of the scaled transmittance through the Fabry- Perot cavity (solid curve) and the scaled transmittance through the oxygen cell (curve made with + symbols) as functions of the laser frequency change is shown in Figure 1b. The dips in the transmittance through the oxygen cell indicate that the oxygen molecule strongly absorbs these frequencies. The free spectral range of the Fabry-Perot cavity used in the experiment was known to be 11.6 GHz. The free spectral range can be taken to be the distance between the tall transmittance peaks, indicated by the arrows in Figure 1b. (A spherical-mirror Fabry-Perot cavity was used in the experiment and so the transmittance includes peaks corresponding to both longitudinal and transverse modes.)a. Estimate the difference in the frequencies of the two absorption dips shown in Figure 1b.b. Estimate the “full-width-at-half-depth” of each absorption dip.Figure 1 (a) Experimental arrangement, (b) Overlay of transmit- tancc curves. (Courtesy of R. J. Brecha, Physics Department, University of Dayton.)... Get solution
23. Consider the transmittance through a Fabry-Perot interferometer as a function of the variable wavelength λ of its input field. Show that the FWHM of the transmittance peaks is ... and the separation between transmittance peaks is λfsr = λ/m. (Here m = 2d/λ, where d is the length of the Fabry-Perot interferometer.) Get solution
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