1. The spectral energy density ρ(ν) in an electromagnetic field in
thermal equilibrium at temperature T is given by Eq. (26-6). Recall that
ρ(ν) is the energy per unit volume per unit frequency interval in the
field.a. Show that ρ(λ), which we define to be the energy per volume per
unit wavelength interval in the field, is ...b. Confirm that the
relation found in (a) is in agreement with the spectral exitance
associated with a blackbody given in Eq. (6-5) as ...Note that Mλ is the
the exitance per wavelength interval emitted by a blackbody source.
(Hint: The spectral energy density in (a) includes the energy in field
components moving in all directions while the spectral exitance accounts
for the power moving normally away from the blackbody surface.) Get solution
2. Consider a monochromatic electromagnetic field traveling with speed c in a given direction. Use a conservation of energy argument to show that the time-averaged energy density 〈u〉 associated with this field is related to the irradiance I of the field by 〈u〉 = I/c. Get solution
3. Show that Eqs. (26-17), (26-18), and (26-19) follow, in steady state, from Eq. (26-16) and Eq. (26-14). Get solution
4. One can define a saturation irradiance IS,abs for an absorptive medium as the irradiance for which the loss coefficient α is reduced by a factor of 2 from its small-signal value.a. Show that for the two-level absorptive medium considered in Section 26-2, ...where τ2 = 1/A21.b. Compare this relation to the saturation irradiance for the ideal four-level gain medium given in Eq. (26-39) and account, with a conceptual argument, for the factor of two difference between the two saturation irradiances. Get solution
5. Consider an amplifying medium composed of homogeneously broadened four-level atoms like the one depicted in Figure 26-5. Amplification is to occur on the 2-to-1 transition. The medium is pumped by a laser of intensity Ip, which is resonant with the 3-to-0 transition. The spontaneous decay processes are as indicated on the diagram. The total number density of gain atoms is NT = N0 + N1 + N2 + N3. The various parameters are ...a. Write down the rate equations for the population densities of the levels.b. Find and plot the steady-state small-signal population inversion N2 − N1 as a function of the pump irradiance. (Recall that “small signal” is code for “set I = 0.”)c. Find the pump irradiance Ip required to sustain a steady-state population inversion.d. Find the pump irradiance Ip required to sustain a small-signal gain coefficient of 0.01/cm.e. Find the pump irradiance Ip required to sustain a small-signal gain coefficient of 1/cm.f. Compare N0 to N1, N2, and N3 for the pump irradiances of parts (d) and (e). Is it reasonable to set N0≈ NT for either of these irradiances?g. Use the ideal four-level gain medium relation given as Eq. (26-38) together with the definition of the effective pump rate density given following Eq. (26-33) to estimate the pump irradiance required to sustain a small-signal gain coefficient of 0.01/cm and 1/cm. Compare these results to those obtained in parts (d) and (e). Get solution
6. Show that Eq. (26-34) follows from Eqs. (26-32) and (26-33). Get solution
7. Consider an amplifying medium composed of homogeneously broadened three-level atoms. Amplification is to occur on the 2-to-1 transition. The medium is pumped by a laser, of irradiance Ip, that is resonant with the 3-to-1 transition. Level 3 decays spontaneously only to level 2 and level 2 decays spontaneously to level 1, which is the ground state of the system. The total number density of gain atoms is NT = N1 + N2 + N3. The various parameters are ...a. Sketch a level diagram like Figure 26-5 appropriate for this case and indicate the various stimulated and decay processes with arrows on the level diagram.b. Write down the rate equations for the population densities of the levels. Include the presence of a field of irradiance I resonant with the 2-to-1 transition, the pump interaction, and the decay processes.c. Find and plot the steady-state small-signal population inversion N2 − N1 as a function of pump irradiance. (Recall that “small signal” is code for “set I = 0.”)d. Find the pump irradiance Ip required to sustain a steady-state population inversion. Compare this result to the answer obtained for part (c) of problem 1.e. Find the pump irradiance Ip required to sustain a small-signal gain coefficient of 0.01/cm. Compare this result to the answer obtained for part (d) of problem 1.f. Find the pump irradiance Ip required to sustain a small-signal gain coefficient of 1/cm. Compare this result to the answer obtained for part (c) of problem 1.g. Summarize the important differences in the behavior of the three-level gain medium considered in this problem and the four-level gain medium considered in problem 1.Problem 1Consider an amplifying medium composed of homogeneously broadened four-level atoms like the one depicted in Figure 26-5. Amplification is to occur on the 2-to-1 transition. The medium is pumped by a laser of intensity Ip, which is resonant with the 3-to-0 transition. The spontaneous decay processes are as indicated on the diagram. The total number density of gain atoms is NT = N0 + N1 + N2 + N3. The various parameters are ...a. Write down the rate equations for the population densities of the levels.b. Find and plot the steady-state small-signal population inversion N2 − N1 as a function of the pump irradiance. (Recall that “small signal” is code for “set I = 0.”)c. Find the pump irradiance Ip required to sustain a steady-state population inversion.d. Find the pump irradiance Ip required to sustain a small-signal gain coefficient of 0.01/cm.e. Find the pump irradiance Ip required to sustain a small-signal gain coefficient of 1/cm.f. Compare N0 to N1, N2, and N3 for the pump irradiances of parts (d) and (e). Is it reasonable to set N0≈ NT for either of these irradiances?g. Use the ideal four-level gain medium relation given as Eq. (26-38) together with the definition of the effective pump rate density given following Eq. (26-33) to estimate the pump irradiance required to sustain a small-signal gain coefficient of 0.01/cm and 1/cm. Compare these results to those obtained in parts (d) and (e). Get solution
8. Show that if the irradiance throughout a gain cell described by Eq. (26-41) is much less than the saturation irradiance IS, the output irradiance IL is related to the input irradiance I0 by the simple relation ...That is, show that, in the small-signal regime, the irradiance exhibits exponential growth. Get solution
9. Show that if the irradiance throughout a gain cell described by Eq. (26-41) is much greater than the saturation irradiance IS, the output irradiance IL is related to the input irradiance I0 by the simple relation ...That is, show that for a very large input irradiance, the irradiance exhibits linear growth. [It may be somewhat simpler to implement the relation I ≫ IS in Eq. (26-40) and then integrate, than to use Eq. (26-41) directly.] Get solution
10. Consider the limit described in problem 1.a. Show that in this limit and for an ideal four-level gain medium, ...b. Argue that the relation in part (a) implies that for the large input-irradiance case of problem 1 every pump event leads to one photon added to the electromagnetic field being amplified.c. For the small input-irradiance of problem 2, even for an ideal four-level gain medium, it is not true that every pump event leads to one photon added to the electromagnetic field being amplified. Conceptually, account for the missing pump events.Problem 1Show that if the irradiance throughout a gain cell described by Eq. (26-41) is much greater than the saturation irradiance IS, the output irradiance IL is related to the input irradiance I0 by the simple relation ...That is, show that for a very large input irradiance, the irradiance exhibits linear growth. [It may be somewhat simpler to implement the relation I ≫ IS in Eq. (26-40) and then integrate, than to use Eq. (26-41) directly.]Problem 2Show that if the irradiance throughout a gain cell described by Eq. (26-41) is much less than the saturation irradiance IS, the output irradiance IL is related to the input irradiance I0 by the simple relation ...That is, show that, in the small-signal regime, the irradiance exhibits exponential growth. Get solution
11. A homogeneously broadened gain medium has a length of L = 2 cm, a small-signal gain coefficient at the transition linecenter of γ0(ν0) = 1/cm, and a saturation irradiance at the transition linecenter of IS(ν0) = 100 W/cm2. Assume that light of frequency ν′ − ν0 is input into the cell. Find the irradiance IL exiting the gain cell when the irradiance I0 input to the cell is (a) 1 W/cm2, (b) 10 W/cm2, (c) 100 W/cm2, (d) 1000 W/cm2, and (e) 10,000 W/cm2. Get solution
12. For each case of problem 1, find the irradiance added by passage through the gain cell IL − I0 and describe how this added irradiance changes with increasing input irradiance.Problem 1A homogeneously broadened gain medium has a length of L = 2 cm, a small-signal gain coefficient at the transition linecenter of γ0(ν0) = 1/cm, and a saturation irradiance at the transition linecenter of IS(ν0) = 100 W/cm2. Assume that light of frequency ν′ − ν0 is input into the cell. Find the irradiance IL exiting the gain cell when the irradiance I0 input to the cell is (a) 1 W/cm2, (b) 10 W/cm2, (c) 100 W/cm2, (d) 1000 W/cm2, and (e) 10,000 W/cm2. Get solution
13. Repeat problems 1 and 2 for the case in which the field input into the cell has a frequency ν′ = ν0 + Δν/2, where Δν is the homogeneous linewidth of the gain medium.Problem 1A homogeneously broadened gain medium has a length of L = 2 cm, a small-signal gain coefficient at the transition linecenter of γ0(ν0) = 1/cm, and a saturation irradiance at the transition linecenter of IS(ν0) = 100 W/cm2. Assume that light of frequency ν′ − ν0 is input into the cell. Find the irradiance IL exiting the gain cell when the irradiance I0 input to the cell is (a) 1 W/cm2, (b) 10 W/cm2, (c) 100 W/cm2, (d) 1000 W/cm2, and (e) 10,000 W/cm2.Problem 2For each case of problem 1, find the irradiance added by passage through the gain cell IL − I0 and describe how this added irradiance changes with increasing input irradiance. Get solution
14. Reproduce the curves shown in Figure 26-7 but extend the maximum length of the gain cell on the plot to 10 cm. Get solution
15. Consider an ideal four-level gain medium in a ring cavity like the one of Figure 26-8 but with R1 = R2 = 1 and R3 = 1 − T3.a. Show that, for this case, Iout = Isat (γ0 − γth)Lb. Show that, for this ease and for γ0 ≫ γth, essentially every pump event leads to an output photon.c. Explain why, even when every pump event leads to an output photon, the efficiency of the laser system is less than 100%. Get solution
16. In this problem and the following two problems, consider a ring cavity like the one depicted in Figure 26-8. Let the cavity mirrors M1 and M2 have reflectances R1 = R2 and let mirror M3 have reflectance R3 = 1 − T3 − A3, where A3 characterizes the output mirror absorption. Let the gain medium be homogeneously broadened and have length L = 10 cm and a saturation irradiance (at the lasing frequency) of IS = 2000 W/cm2.a. Find the threshold gain coefficient if R1 = R2 = 1, R3 = 0.95, and T3 = 0.05.b. If the small-signal gain coefficient is twice the threshold value, find the irradiance of the laser output field. Get solution
17. Consider again the ring laser described in problem 1 but now take the small-signal gain coefficient to be 0.01/cm, R3 = 0.95, and T3 = 0.05. Plot the laser output irradiance as a function of the variable reflectance R = R1 = R2 of the other two cavity mirrors.Problem 1In this problem and the following two problems, consider a ring cavity like the one depicted in Figure 26-8. Let the cavity mirrors M1 and M2 have reflectances R1 = R2 and let mirror M3 have reflectance R3 = 1 − T3 − A3, where A3 characterizes the output mirror absorption. Let the gain medium be homogeneously broadened and have length L = 10 cm and a saturation irradiance (at the lasing frequency) of IS = 2000 W/cm2.a. Find the threshold gain coefficient if R1 = R2 =1, R3 = 0.95, and T3 = 0.05.b. If the small-signal gain coefficient is twice the threshold value, find the irradiance of the laser output field. Get solution
18. Consider again the ring laser described in problem 1. Let R1 = R2 = 0.99 and A3 = 0.01. Let the small-signal gain coefficient be 0.01/cm.a. Plot the laser output irradiance as a function of the variable transmittance T3 of the output mirror.b. Using the plot produced in part (a), determine the value of T3 that maximizes the laser output irradiance. Explain why, for this system in which there are unavoidable losses, the output irradiance is reduced from its maximum value if T3 is either too large or too small.Problem 1In this problem and the following two problems, consider a ring cavity like the one depicted in Figure 26-8. Let the cavity mirrors M1 and M2 have reflectances R1 = R2 and let mirror M3 have reflectance R3 = 1 − T3 − A3, where A3 characterizes the output mirror absorption. Let the gain medium be homogeneously broadened and have length L = 10 cm and a saturation irradiance (at the lasing frequency) of IS = 2000 W/cm2.a. Find the threshold gain coefficient if R1 = R2 =1, R3 = 0.95, and T3 = 0.05.b. If the small-signal gain coefficient is twice the threshold value, find the irradiance of the laser output field. Get solution
19. Derive Eq. (26-47) by a procedure similar to that leading to Eq. (26-43). The linear cavity case is complicated by the fact that the field encounters the gain medium twice in each round-trip with the losses encountered at the mirrors interspersed between passes through the gain medium. Tt may be useful to research and then summarize the solution to this problem. Get solution
20. Show that Eq. (47) for a linear cavity reduces to ...for a cavity with R1 = 1. Here, S is the survival fraction in the linear cavity without gain (S − R2). Compare this result with the similar result given in Eq. (43) for the ring cavity and account for the differences between the two results. Get solution
21. Consider the CO2 transition described in Example 26-4. In addition to the information given in the example note that the spontaneous emission rate for the transition is A21= 0.34/s.a. What is the stimulated emission cross section σ for this transition?b. What must be the population inversion in the gain medium to produce a small-signal gain coefficient (at linecenter, ν ′ = ν0) of 0.03/cm?c. Treating this system as an ideal four-level system, estimate the saturation irradiance for this transition. Get solution
22. Find the Doppler-broadened gain bandwidth of the 633-nm He-Ne transition. Assume that the operating temperature is 400 K and recall that neon is the lasing species. Get solution
23. Use the method leading to Eq. (8-45) to show that the loss rate Γ for a ring cavity with round-trip survival factor S and perimeter P is ... Get solution
24. Reproduce the curves shown in Figure 26-13 using the parameters given in the figure caption. Note that the effective pump rate can be found from the listed condition γ0 = 2γth. Get solution
25. Produce curves like those shown in Figure 26-13 for the parameters given in the figure caption except let (a) κ = 10−8 s−1, (b) κ = 10−6 s−1, (c) γ0/γth = 1.1, and (d) γ0/γth= 4. In each case describe how changing the indicated parameter changes the curves. Get solution
27. The gain bandwidth (in nm) and the transition wavelength for three different laser systems are given below. Estimate the pulse width attainable with these laser systems if they are mode-locked. ... Get solution
28. In order to investigate the bandwidth theorem,plot the given function F as a function of time t for (a) N = 5, (b) N = 10, and (c) N = 50. In each case estimate the pulse width from the plot and compare the pulse width to the range of frequencies in the superposition. ... Get solution
29. Show that the sum E of the electric fields associated with N mode-locked cavity modes of equal amplitude and with frequencies νj = ν0 + jνfsr can be written as ...You may wish to review the mathematical procedure used to describe multislit diffraction in Section 11-6 as a guide for carrying out the indicated summation. Get solution
30. Use the relation in problem 1 to verify Eqs. (26-57) and (26-58).Problem 1Show that the sum E of the electric fields associated with N mode-locked cavity modes of equal amplitude and with frequencies νj = ν0 + jνfsr can be written as ...You may wish to review the mathematical procedure used to describe multislit diffraction in Section 11-6 as a guide for carrying out the indicated summation. Get solution
31. Estimate the peak power and pulse repetition rate in a mode-locked Nd:YAG laser pulse of pulse width 70 ps if the Nd:YAG laser cavity is 1.5 m long, and the CW output power of the Nd:YAG laser system is 10 W. Get solution
32. Estimate the diffraction-limited far-field divergence angles of a beam output from the heterojunction laser diode illustrated in Figure 26-19. Get solution
33. What is the band-gap energy of an AlGaAs semiconductor used in a laser diode device that emits light of wavelength 800 nm? Get solution
34. What must the reflectance of the cleaved ends of the laser diode illustrated in Figure 26-19 be if the small-signal gain coefficient of the medium is 40/cm? Get solution
35. a. Show that solving Eqs. (26-54) and (26-55) for the steady-state photon number density Np and population inversion N2 gives, ...b. Use the result for Np in (a) to form the following expression for the steady-slate output irradiance from a ring laser like the one discussed in connection with Figure 26-8: ...c. Show that the relation from part (b) agrees with Eq. (26-43) only if the survival fraction is close to 1. (Hint: ln(1 − x) ≈ x, for small x.)d. Which relation, the one from part (b) or the one given in Eq. (26-43), is correct when S is not close to 1? Explain. Get solution
2. Consider a monochromatic electromagnetic field traveling with speed c in a given direction. Use a conservation of energy argument to show that the time-averaged energy density 〈u〉 associated with this field is related to the irradiance I of the field by 〈u〉 = I/c. Get solution
3. Show that Eqs. (26-17), (26-18), and (26-19) follow, in steady state, from Eq. (26-16) and Eq. (26-14). Get solution
4. One can define a saturation irradiance IS,abs for an absorptive medium as the irradiance for which the loss coefficient α is reduced by a factor of 2 from its small-signal value.a. Show that for the two-level absorptive medium considered in Section 26-2, ...where τ2 = 1/A21.b. Compare this relation to the saturation irradiance for the ideal four-level gain medium given in Eq. (26-39) and account, with a conceptual argument, for the factor of two difference between the two saturation irradiances. Get solution
5. Consider an amplifying medium composed of homogeneously broadened four-level atoms like the one depicted in Figure 26-5. Amplification is to occur on the 2-to-1 transition. The medium is pumped by a laser of intensity Ip, which is resonant with the 3-to-0 transition. The spontaneous decay processes are as indicated on the diagram. The total number density of gain atoms is NT = N0 + N1 + N2 + N3. The various parameters are ...a. Write down the rate equations for the population densities of the levels.b. Find and plot the steady-state small-signal population inversion N2 − N1 as a function of the pump irradiance. (Recall that “small signal” is code for “set I = 0.”)c. Find the pump irradiance Ip required to sustain a steady-state population inversion.d. Find the pump irradiance Ip required to sustain a small-signal gain coefficient of 0.01/cm.e. Find the pump irradiance Ip required to sustain a small-signal gain coefficient of 1/cm.f. Compare N0 to N1, N2, and N3 for the pump irradiances of parts (d) and (e). Is it reasonable to set N0≈ NT for either of these irradiances?g. Use the ideal four-level gain medium relation given as Eq. (26-38) together with the definition of the effective pump rate density given following Eq. (26-33) to estimate the pump irradiance required to sustain a small-signal gain coefficient of 0.01/cm and 1/cm. Compare these results to those obtained in parts (d) and (e). Get solution
6. Show that Eq. (26-34) follows from Eqs. (26-32) and (26-33). Get solution
7. Consider an amplifying medium composed of homogeneously broadened three-level atoms. Amplification is to occur on the 2-to-1 transition. The medium is pumped by a laser, of irradiance Ip, that is resonant with the 3-to-1 transition. Level 3 decays spontaneously only to level 2 and level 2 decays spontaneously to level 1, which is the ground state of the system. The total number density of gain atoms is NT = N1 + N2 + N3. The various parameters are ...a. Sketch a level diagram like Figure 26-5 appropriate for this case and indicate the various stimulated and decay processes with arrows on the level diagram.b. Write down the rate equations for the population densities of the levels. Include the presence of a field of irradiance I resonant with the 2-to-1 transition, the pump interaction, and the decay processes.c. Find and plot the steady-state small-signal population inversion N2 − N1 as a function of pump irradiance. (Recall that “small signal” is code for “set I = 0.”)d. Find the pump irradiance Ip required to sustain a steady-state population inversion. Compare this result to the answer obtained for part (c) of problem 1.e. Find the pump irradiance Ip required to sustain a small-signal gain coefficient of 0.01/cm. Compare this result to the answer obtained for part (d) of problem 1.f. Find the pump irradiance Ip required to sustain a small-signal gain coefficient of 1/cm. Compare this result to the answer obtained for part (c) of problem 1.g. Summarize the important differences in the behavior of the three-level gain medium considered in this problem and the four-level gain medium considered in problem 1.Problem 1Consider an amplifying medium composed of homogeneously broadened four-level atoms like the one depicted in Figure 26-5. Amplification is to occur on the 2-to-1 transition. The medium is pumped by a laser of intensity Ip, which is resonant with the 3-to-0 transition. The spontaneous decay processes are as indicated on the diagram. The total number density of gain atoms is NT = N0 + N1 + N2 + N3. The various parameters are ...a. Write down the rate equations for the population densities of the levels.b. Find and plot the steady-state small-signal population inversion N2 − N1 as a function of the pump irradiance. (Recall that “small signal” is code for “set I = 0.”)c. Find the pump irradiance Ip required to sustain a steady-state population inversion.d. Find the pump irradiance Ip required to sustain a small-signal gain coefficient of 0.01/cm.e. Find the pump irradiance Ip required to sustain a small-signal gain coefficient of 1/cm.f. Compare N0 to N1, N2, and N3 for the pump irradiances of parts (d) and (e). Is it reasonable to set N0≈ NT for either of these irradiances?g. Use the ideal four-level gain medium relation given as Eq. (26-38) together with the definition of the effective pump rate density given following Eq. (26-33) to estimate the pump irradiance required to sustain a small-signal gain coefficient of 0.01/cm and 1/cm. Compare these results to those obtained in parts (d) and (e). Get solution
8. Show that if the irradiance throughout a gain cell described by Eq. (26-41) is much less than the saturation irradiance IS, the output irradiance IL is related to the input irradiance I0 by the simple relation ...That is, show that, in the small-signal regime, the irradiance exhibits exponential growth. Get solution
9. Show that if the irradiance throughout a gain cell described by Eq. (26-41) is much greater than the saturation irradiance IS, the output irradiance IL is related to the input irradiance I0 by the simple relation ...That is, show that for a very large input irradiance, the irradiance exhibits linear growth. [It may be somewhat simpler to implement the relation I ≫ IS in Eq. (26-40) and then integrate, than to use Eq. (26-41) directly.] Get solution
10. Consider the limit described in problem 1.a. Show that in this limit and for an ideal four-level gain medium, ...b. Argue that the relation in part (a) implies that for the large input-irradiance case of problem 1 every pump event leads to one photon added to the electromagnetic field being amplified.c. For the small input-irradiance of problem 2, even for an ideal four-level gain medium, it is not true that every pump event leads to one photon added to the electromagnetic field being amplified. Conceptually, account for the missing pump events.Problem 1Show that if the irradiance throughout a gain cell described by Eq. (26-41) is much greater than the saturation irradiance IS, the output irradiance IL is related to the input irradiance I0 by the simple relation ...That is, show that for a very large input irradiance, the irradiance exhibits linear growth. [It may be somewhat simpler to implement the relation I ≫ IS in Eq. (26-40) and then integrate, than to use Eq. (26-41) directly.]Problem 2Show that if the irradiance throughout a gain cell described by Eq. (26-41) is much less than the saturation irradiance IS, the output irradiance IL is related to the input irradiance I0 by the simple relation ...That is, show that, in the small-signal regime, the irradiance exhibits exponential growth. Get solution
11. A homogeneously broadened gain medium has a length of L = 2 cm, a small-signal gain coefficient at the transition linecenter of γ0(ν0) = 1/cm, and a saturation irradiance at the transition linecenter of IS(ν0) = 100 W/cm2. Assume that light of frequency ν′ − ν0 is input into the cell. Find the irradiance IL exiting the gain cell when the irradiance I0 input to the cell is (a) 1 W/cm2, (b) 10 W/cm2, (c) 100 W/cm2, (d) 1000 W/cm2, and (e) 10,000 W/cm2. Get solution
12. For each case of problem 1, find the irradiance added by passage through the gain cell IL − I0 and describe how this added irradiance changes with increasing input irradiance.Problem 1A homogeneously broadened gain medium has a length of L = 2 cm, a small-signal gain coefficient at the transition linecenter of γ0(ν0) = 1/cm, and a saturation irradiance at the transition linecenter of IS(ν0) = 100 W/cm2. Assume that light of frequency ν′ − ν0 is input into the cell. Find the irradiance IL exiting the gain cell when the irradiance I0 input to the cell is (a) 1 W/cm2, (b) 10 W/cm2, (c) 100 W/cm2, (d) 1000 W/cm2, and (e) 10,000 W/cm2. Get solution
13. Repeat problems 1 and 2 for the case in which the field input into the cell has a frequency ν′ = ν0 + Δν/2, where Δν is the homogeneous linewidth of the gain medium.Problem 1A homogeneously broadened gain medium has a length of L = 2 cm, a small-signal gain coefficient at the transition linecenter of γ0(ν0) = 1/cm, and a saturation irradiance at the transition linecenter of IS(ν0) = 100 W/cm2. Assume that light of frequency ν′ − ν0 is input into the cell. Find the irradiance IL exiting the gain cell when the irradiance I0 input to the cell is (a) 1 W/cm2, (b) 10 W/cm2, (c) 100 W/cm2, (d) 1000 W/cm2, and (e) 10,000 W/cm2.Problem 2For each case of problem 1, find the irradiance added by passage through the gain cell IL − I0 and describe how this added irradiance changes with increasing input irradiance. Get solution
14. Reproduce the curves shown in Figure 26-7 but extend the maximum length of the gain cell on the plot to 10 cm. Get solution
15. Consider an ideal four-level gain medium in a ring cavity like the one of Figure 26-8 but with R1 = R2 = 1 and R3 = 1 − T3.a. Show that, for this case, Iout = Isat (γ0 − γth)Lb. Show that, for this ease and for γ0 ≫ γth, essentially every pump event leads to an output photon.c. Explain why, even when every pump event leads to an output photon, the efficiency of the laser system is less than 100%. Get solution
16. In this problem and the following two problems, consider a ring cavity like the one depicted in Figure 26-8. Let the cavity mirrors M1 and M2 have reflectances R1 = R2 and let mirror M3 have reflectance R3 = 1 − T3 − A3, where A3 characterizes the output mirror absorption. Let the gain medium be homogeneously broadened and have length L = 10 cm and a saturation irradiance (at the lasing frequency) of IS = 2000 W/cm2.a. Find the threshold gain coefficient if R1 = R2 = 1, R3 = 0.95, and T3 = 0.05.b. If the small-signal gain coefficient is twice the threshold value, find the irradiance of the laser output field. Get solution
17. Consider again the ring laser described in problem 1 but now take the small-signal gain coefficient to be 0.01/cm, R3 = 0.95, and T3 = 0.05. Plot the laser output irradiance as a function of the variable reflectance R = R1 = R2 of the other two cavity mirrors.Problem 1In this problem and the following two problems, consider a ring cavity like the one depicted in Figure 26-8. Let the cavity mirrors M1 and M2 have reflectances R1 = R2 and let mirror M3 have reflectance R3 = 1 − T3 − A3, where A3 characterizes the output mirror absorption. Let the gain medium be homogeneously broadened and have length L = 10 cm and a saturation irradiance (at the lasing frequency) of IS = 2000 W/cm2.a. Find the threshold gain coefficient if R1 = R2 =1, R3 = 0.95, and T3 = 0.05.b. If the small-signal gain coefficient is twice the threshold value, find the irradiance of the laser output field. Get solution
18. Consider again the ring laser described in problem 1. Let R1 = R2 = 0.99 and A3 = 0.01. Let the small-signal gain coefficient be 0.01/cm.a. Plot the laser output irradiance as a function of the variable transmittance T3 of the output mirror.b. Using the plot produced in part (a), determine the value of T3 that maximizes the laser output irradiance. Explain why, for this system in which there are unavoidable losses, the output irradiance is reduced from its maximum value if T3 is either too large or too small.Problem 1In this problem and the following two problems, consider a ring cavity like the one depicted in Figure 26-8. Let the cavity mirrors M1 and M2 have reflectances R1 = R2 and let mirror M3 have reflectance R3 = 1 − T3 − A3, where A3 characterizes the output mirror absorption. Let the gain medium be homogeneously broadened and have length L = 10 cm and a saturation irradiance (at the lasing frequency) of IS = 2000 W/cm2.a. Find the threshold gain coefficient if R1 = R2 =1, R3 = 0.95, and T3 = 0.05.b. If the small-signal gain coefficient is twice the threshold value, find the irradiance of the laser output field. Get solution
19. Derive Eq. (26-47) by a procedure similar to that leading to Eq. (26-43). The linear cavity case is complicated by the fact that the field encounters the gain medium twice in each round-trip with the losses encountered at the mirrors interspersed between passes through the gain medium. Tt may be useful to research and then summarize the solution to this problem. Get solution
20. Show that Eq. (47) for a linear cavity reduces to ...for a cavity with R1 = 1. Here, S is the survival fraction in the linear cavity without gain (S − R2). Compare this result with the similar result given in Eq. (43) for the ring cavity and account for the differences between the two results. Get solution
21. Consider the CO2 transition described in Example 26-4. In addition to the information given in the example note that the spontaneous emission rate for the transition is A21= 0.34/s.a. What is the stimulated emission cross section σ for this transition?b. What must be the population inversion in the gain medium to produce a small-signal gain coefficient (at linecenter, ν ′ = ν0) of 0.03/cm?c. Treating this system as an ideal four-level system, estimate the saturation irradiance for this transition. Get solution
22. Find the Doppler-broadened gain bandwidth of the 633-nm He-Ne transition. Assume that the operating temperature is 400 K and recall that neon is the lasing species. Get solution
23. Use the method leading to Eq. (8-45) to show that the loss rate Γ for a ring cavity with round-trip survival factor S and perimeter P is ... Get solution
24. Reproduce the curves shown in Figure 26-13 using the parameters given in the figure caption. Note that the effective pump rate can be found from the listed condition γ0 = 2γth. Get solution
25. Produce curves like those shown in Figure 26-13 for the parameters given in the figure caption except let (a) κ = 10−8 s−1, (b) κ = 10−6 s−1, (c) γ0/γth = 1.1, and (d) γ0/γth= 4. In each case describe how changing the indicated parameter changes the curves. Get solution
27. The gain bandwidth (in nm) and the transition wavelength for three different laser systems are given below. Estimate the pulse width attainable with these laser systems if they are mode-locked. ... Get solution
28. In order to investigate the bandwidth theorem,plot the given function F as a function of time t for (a) N = 5, (b) N = 10, and (c) N = 50. In each case estimate the pulse width from the plot and compare the pulse width to the range of frequencies in the superposition. ... Get solution
29. Show that the sum E of the electric fields associated with N mode-locked cavity modes of equal amplitude and with frequencies νj = ν0 + jνfsr can be written as ...You may wish to review the mathematical procedure used to describe multislit diffraction in Section 11-6 as a guide for carrying out the indicated summation. Get solution
30. Use the relation in problem 1 to verify Eqs. (26-57) and (26-58).Problem 1Show that the sum E of the electric fields associated with N mode-locked cavity modes of equal amplitude and with frequencies νj = ν0 + jνfsr can be written as ...You may wish to review the mathematical procedure used to describe multislit diffraction in Section 11-6 as a guide for carrying out the indicated summation. Get solution
31. Estimate the peak power and pulse repetition rate in a mode-locked Nd:YAG laser pulse of pulse width 70 ps if the Nd:YAG laser cavity is 1.5 m long, and the CW output power of the Nd:YAG laser system is 10 W. Get solution
32. Estimate the diffraction-limited far-field divergence angles of a beam output from the heterojunction laser diode illustrated in Figure 26-19. Get solution
33. What is the band-gap energy of an AlGaAs semiconductor used in a laser diode device that emits light of wavelength 800 nm? Get solution
34. What must the reflectance of the cleaved ends of the laser diode illustrated in Figure 26-19 be if the small-signal gain coefficient of the medium is 40/cm? Get solution
35. a. Show that solving Eqs. (26-54) and (26-55) for the steady-state photon number density Np and population inversion N2 gives, ...b. Use the result for Np in (a) to form the following expression for the steady-slate output irradiance from a ring laser like the one discussed in connection with Figure 26-8: ...c. Show that the relation from part (b) agrees with Eq. (26-43) only if the survival fraction is close to 1. (Hint: ln(1 − x) ≈ x, for small x.)d. Which relation, the one from part (b) or the one given in Eq. (26-43), is correct when S is not close to 1? Explain. Get solution
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