### Chapter #28 Solutions - Introduction to Optics - Leno M Pedrotti, Leno S Pedrotti, Frank L Pedrotti - 3rd Edition

1. a. Why are excimer lasers particularly useful in microsurgical applications?b. The absorption curves shown in Figure 28-1 begin at a wavelength of 0.4 μm, which is longer than the wavelengths of the excimer lasers. Do some research to find the absorption characteristics of water and hemoglobin at the wavelengths of the excimer lasers.c. Considering your findings in part (b), discuss the utility of the excimer laser as a scalpel and for photocoagulation. Get solution

2. After doing some research, summarize the current state of the art with regard to the development of fibers through which excimer lasers can propagate with low loss. Get solution

3. Consider the medical procedure called posterior capsulotomy, illustrated in Figure 28-2 and discussed in the text surrounding that figure. Using the analytical tools developed in Chapter 27, choose a lens focal length, a beam-waist spot size at the lens, and a lens-to-cornea distance for use in the procedure. Be sure to respect the 17° convergence angle requirement indicated in the caption to Figure 28-2. For the system you design, calculate the beam-waist spot size at the focused spot, and the Rayleigh range parameter (z0) for the focused beam. Comment on the importance of the Rayleigh range in this design problem. It may be useful to refer to the schematic eye shown in Figure 19-3. Get solution

4. For the system designed in problem 1, find the irradiance at the center of the focused beam waist for an Nd:YAG laser of pulse energy 4 mJ and pulse width 1 ns. Compare the answer obtained with the estimate given in Example 28-1.Problem 1Consider the medical procedure called posterior capsulotomy, illustrated in Figure 28-2 and discussed in the text surrounding that figure. Using the analytical tools developed in Chapter 27, choose a lens focal length, a beam-waist spot size at the lens, and a lens-to-cornea distance for use in the procedure. Be sure to respect the 17° convergence angle requirement indicated in the caption to Figure 28-2. For the system you design, calculate the beam-waist spot size at the focused spot, and the Rayleigh range parameter (z0) for the focused beam. Comment on the importance of the Rayleigh range in this design problem. It may be useful to refer to the schematic eye shown in Figure 19-3. Get solution

5. Estimate the electric field strength at the center of the focused spot of Example 28-1. Get solution

6. A laser system, fixed on a geosynchronous satellite, uses a Nd:YAG laser that emits 1.06 μm radiation in a highly collimated beam of 5-μrad beam divergence.a. If the laser is 36,205 km above Earth, what is the minimum diameter of the laser-beam “footprint” on the surface of Earth?b. If the laser emits pulses of 200-MW power, what is the average electric field per pulse in the laser beam at Earth’s surface? Get solution

7. Personnel who work with high-power lasers or with unshielded laser beams must take measures to protect themselves from laser damage to the skin and eyes. Toward that end, nominal hazard zones (NHZ) for given lasers can be defined. A NHZ outlines the conical space within which the level of direct, reflected, or scattered radiation—during normal operation—exceeds the assigned levels of maximum permissible exposure (MPE). Exposure levels outside the conical region are below the designated MPE level. The conical region, with lip emanating from the laser, extends to a range RNHZ, with sides spreading at the full-angle beam divergence ϕ. The range for a non focused laser beam is given by ...where P is the laser power, ϕ is the beam divergence, MPE is the maximum permissible exposure in power/area, and d is the aperture diameter of the laser exit port.a. Draw a sketch of the NHZ (conical shape), labeling the extent of the cone, RNHZ and the full-angle beam divergence ϕ.b. Determine RNHZ for a continuous wave Nd:YAG laser (λ = 1.06 μm) of 50-W power, 3.0-mrad beam divergence, MPE of 5.1 × 10−3 W/cm2, a 10-s exposure, and an exit aperture diameter of 3.0 mm. Get solution

8. The range RNHZ (refer to problem 1) for a laser that directs its exiting laser beam immediately onto a lens is given by ...where RNHZ defines the extent of the cone emanating from the point of beam focus, f is the focal length of the lens, b is the diameter of the beam exiling the laser, P is the power of the laser, and MPE is the maximum permissible exposure.a. Draw a sketch of the NHZ (conical space) for this focused laser beam.b. Determine RNHZ for a continuous wave Nd:YAG laser of 50-W power, exit beam diameter of 5.0 mm, and capped by a lens of focal length 7.5 cm, if the MPE of this laser for an 8-hr exposure is 1.6 × 10−3 W/cm2.Problem 1Personnel who work with high-power lasers or with unshielded laser beams must take measures to protect themselves from laser damage to the skin and eyes. Toward that end, nominal hazard zones (NHZ) for given lasers can be defined. A NHZ outlines the conical space within which the level of direct, reflected, or scattered radiation—during normal operation—exceeds the assigned levels of maximum permissible exposure (MPE). Exposure levels outside the conical region are below the designated MPE level. The conical region, with lip emanating from the laser, extends to a range RNHZ, with sides spreading at the full-angle beam divergence ϕ. The range for a non focused laser beam is given by ...where P is the laser power, ϕ is the beam divergence, MPE is the maximum permissible exposure in power/area, and d is the aperture diameter of the laser exit port.a. Draw a sketch of the NHZ (conical shape), labeling the extent of the cone, RNHZ and the full-angle beam divergence ϕ.b. Determine RNHZ for a continuous wave Nd:YAG laser (λ = 1.06 μm) of 50-W power, 3.0-mrad beam divergence, MPE of 5.1 × 10−3 W/cm2, a 10-s exposure, and an exit aperture diameter of 3.0 mm. Get solution

9. Investigate the tuning of the signal and idler frequencies of an OPO using a PPLN crystal. In particular, find a PPLN spatial frequency KC that will produce a signal of wavelength 2400 nm when the pump is (a) an Nd:YAG laser and (b) a frequency-doubled Nd:YAG laser. Assume that the operating temperature is room temperature. (To solve this problem you will have to do some research. The free software program called SNLO, available from the Sandia National Laboratory web site, would prove very useful.) Get solution

10. Consider an OPO system pumped with Nd:YAG laser radiation. For a certain configuration, the signal beam is found to have a wavelength of 1500 nm.a. Find the wavelength of the idler beam.b. Find a cavity length d, near 1 m, that will allow both signal and idler to resonate by modeling the cavity as having two flat mirrors and ignoring the influence of the PPLN crystal on the cavity mode frequencies. That is, take the cavity mode frequencies to be given by vq = qc/(2d).c. Refine the calculation of part (b) by finding a cavity length near 1 m that will allow both the signal and idler to resonate, in a TEM00 mode, in a cavity with one flat mirror and one mirror of radius of curvature 2 m. In this case, the cavity mode frequencies are given by Eq. (27-67). Get solution

11. Consider a two-mirror cavity and take the reflectance of each mirror to be R = 0.97. Let resonant light of power 100 mW be incident into the cavity. Find the (one-way) power in the intracavity beam. Get solution

12. Who won the Nobel prizes for physics in 1997 and 2001. Summarize the work that led to the awards. Get solution

13. Who won the Nobel Prize for chemistry in 1999. Summarize the work that led to the award. Get solution

14. Summarize the contents of a journal article describing a femtosecond dye laser experimental setup. Get solution

15. Find the aperture size and position used in a real Kerr-lens, mode-locked, Ti:sapphire laser system. Get solution

16. a. Calculate the root-mean-square room-temperature speed of an oxygen molecule.b. Find the root-mean-square speed of an atom in a gas sample of rubidium at a temperature of 2 mK. Get solution

17. Estimate the amount of Doppler detuning δ (see Figure 28-3) that should be used to cool a gas sample of rubidium atoms at a temperature of 3 mK. Take the transition wavelength to be 800 nm. (Equation (4-44) might prove useful.) Get solution

18. After doing research, answer the following questions.a. What is an atom-interferometer?b. Why is there an interest in atom-interferometers? Get solution

19. After doing research, describe the nature of the trapping force that exists in the focal region of an optical tweezers arrangement. Get solution

20. After doing research, describe some experiments designed to study DNA with optical tweezers. Get solution

21. After doing research, describe how the tip deflection in an atomic force microscope can be measured by piezoresistive means. Get solution

22. Summarize the contents of a journal article detailing the use of a scanning probe microscope to image a biological sample. Get solution

23. Summarize the contents of a journal article detailing the use of a NSOM to measure the optical properties of a surface. Get solution

### Chapter #27 Solutions - Introduction to Optics - Leno M Pedrotti, Leno S Pedrotti, Frank L Pedrotti - 3rd Edition

1. Describe the ways that the TEM00 Gaussian beam is similar to and different from (a) plane waves and (b) spherical waves. Get solution

2. Show that substitution of ... in the wave equation (27-3) leads to Eq. (27-7). Get solution

3. Show that Eq. (27-16) follows from Eq. (27-15). Get solution

4. A TEM00 He-Ne laser (λ = 632.8 nm) has a beam waist w0 (at z = 0) of 0.5 mm and a beam divergence of θFF = 0.4 mrad.a. Determine the value of the complex radius of curvature q at the beam waist.b. Determine a numerical expression for the complex radius of curvature q, at a distance of 50 m from the beam waist, using each of the following expressions: ...(Hint: Is the transverse plane, 50 m from the beam waist, far enough away to be in the far field? If so, what does this say about R and z?) Get solution

5. a. In problem 1, use Eqs. (27-19), (27-21), and (27-22) to determine R(z) and w(z) at z = 50 m.b. Is it true that R(z) ≅ z in the far field? Is it true that tan θFF = w(z)/z, where θFF is the beam divergence angle, can be used as a good approximation to determine w(z) at z = 50 m?Problem 1A TEM00 He-Ne laser (λ = 632.8 nm) has a beam waist w0 (at z = 0) of 0.5 mm and a beam divergence of θFF = 0.4 mrad.a. Determine the value of the complex radius of curvature q at the beam waist.b. Determine a numerical expression for the complex radius of curvature q, at a distance of 50 m from the beam waist, using each of the following expressions: ...(Hint: Is the transverse plane, 50 m from the beam waist, far enough away to be in the far field? If so, what does this say about R and z?) Get solution

6. A TEM00 He-Ne laser (λ = 0.6328 μm) has a cavity that is 0.34 m long, a fully reflecting mirror of radius R = 10 m (concave inward), and an output mirror of radius R = 10 m (also concave inward).a. Determine the location of the beam waist in the cavity.b. Determine the spot size at the beam waist, wq.c. Determine the beam spot size w(z) at the left and right cavity mirrors.d. Determine the beam divergence angle θFF for this laser.e. Where is the far field for this laser if one uses the criterion zFF ≥ 50z0?f. If the laser emits a constant beam of power 5 mW, what is the on-axis irradiance at the position where zFF ≥ 50z0? Get solution

7. Refer to Figure 27-10, where the output element of the laser is a mirror-lens combination with thickness 0.004 m, mirror surface curvature of |R2| = 2 m, lens surface curvature of |R3| = 0.64 m, and lens refractive index of 1.50.a. Using the definitions given for the refraction and translation matrices in Chapter 18, set up the ABCD matrix for this element as follows: ...Pay particular attention to the changing meaning of n and n′ for the two refractions and to the sign conventions for R2 and R3 in the matrix formulations. Within rounding approximations, you should findb. Since L = 0.004 m is a very small dimension compared with |R2|= 2 m or |R3| = 0.64 m, repeat the ABCD calculation, replacing the translation matrix with the unit matrix ...What then is the result of the ABCD matrix for this “thin lens”? Get solution

8. a. Since the output element described in problem 1 is essentially a thin lens, compare the ABCD matrix obtained in problem 1(b) with the thin-lens matrix, namely,...and deduce the focal length of the output element.b. Use the expression for the focal length of a thin lens, ..., with careful attention to thin-lens sign conventions to obtain the focal length of the thin-lens output element. How do the results for parts (a) and (b) compare?Problem 1Refer to Figure 27-10, where the output element of the laser is a mirror-lens combination with thickness 0.004 m, mirror surface curvature of |R2| = 2 m, lens surface curvature of |R3| = 0.64 m, and lens refractive index of 1.50.a. Using the definitions given for the refraction and translation matrices in Chapter 18, set up the ABCD matrix for this element as follows: ...Pay particular attention to the changing meaning of n and n′ for the two refractions and to the sign conventions for R2 and R3 in the matrix formulations. Within rounding approximations, you should findb. Since L = 0.004 m is a very small dimension compared with |R2|= 2 m or |R3| = 0.64 m, repeat the ABCD calculation, replacing the translation matrix with the unit matrix ...What then is the result of the ABCD matrix for this “thin lens”? Get solution

9. Referring to Example 27-2 and Figure 27-10, (a) determine an expression for qx at the plane mirror; (b) solve Eq. 27-22 for the spot-size value w1; (c) obtain a numerical value for q1 ; (d) multiply q1 by the ABCD matrix to obtain q2; (e) use Eq. (27-17) and q2 from part (d) to obtain ℓ and w(ℓ). Get solution

10. a. Referring to Example 27-2 in which the external beam waist is focused at ℓ ≅ 0.06 m with a waist size w(ℓ) = 0.54 mm, use Eqs. (27-46) and (47) together with Figure 27-14 to obtain values for w02 and z2. How do these results compare with those for w0(ℓ) and ℓ obtained in the example?b. Explain why one cannot use the approximations ...in this instance. Get solution

11. Refer to the externally focused TEM00 laser beam shown in Figure 27-10, with beam waist w0(ℓ) = 0.54 mm, located at 0.06 m from the output element.a. Calculate the far-field distance zFF = 50z0 for the externally focused beam waist.b. Calculate the far-field beam divergence angle for the laser beam that emerges from the focused beam waist.c. Insert a 10 × beam expander in the beam at a distance z = 30 m past the focused beam waist. Calculate the beam spot size w at the entrance and exit faces of the beam expander.d. Now place a thin lens of focal length 10 cm and appropriate diameter at a distance of 20 cm from the output face of the beam expander. With reference to Figure 27-14 and Eqs. (27-46) and (27-47), calculate z2 and w02 for the newly focused beam. Could you have used the approximate formulas w02 ≅ fλ/πw01 and z2 = f in this instance? Why? How do the calculations for the exact formulas and approximate formulas compare? Get solution

12. a. Specialize Eqs. (27-46) and (27-47) for the case in which the lens is placed at the waist of the incident beam.b. For the case described in (a), show that the location of the beam waist can be written as ...where ....c. Investigate, for the case described in (a) and (b), whether Z2 ≈ f for a variety of reasonable choices for lens focal lengths and beam parameters. Get solution

13. Carry out the integration necessary to verily the claim that the total power carried in a TEM00 beam is ... Get solution

14. Carry out the integrations necessary to show that the fraction of the power in a TEM00 beam that is transmitted through a circular aperture of radius a is .... Get solution

15. Explain how you can use an adjustable circular aperture (iris) and a power meter to determine the spot size w of a TEM00 laser beam at any position along the beam. Get solution

16. Determine collimated beam lengths 2z0 for a TEM00 Nd:YAG laser beam (λ = 1.064 μm) focused by lenses of aperture diameters D = 1 cm, 2 cm, 3 cm, and 5 cm, respectively. Assume that the lens diameter D is related to the focused beam waist w0 by the equation .... Refer to Figure 27-13 for geometry and similar calculations made for He-Ne, HF, and CO2 TEM00 lasers. Get solution

17. Given the generating function, Eq. (27-58), for Hermite polynomials Hm(ξ), where ..., verify the particular cases for m = 0, 1, 2, … given in Eq. (27-59). Get solution

18. Fill in the steps to show how Eq. (27-65) follows from Eq. (27-64). Get solution

19. Refer to Figure 1. Extend the “table” to include the case m = 2, n = 0. Thus, in a third row, sketch in curves for column 1, Hm(xs), column 2 for the x-variation curves of the electric field, column 3 for the xs-variation curves of the irradiance, and column 4 for the expected burn pattern.Figure 1Laser-beam electric field and irradiance variations in the xs-direction for two values of the Hermite integer m. Corresponding burn patterns for m = 0, n = 0 and m = 1, n = 0 are shown.... Get solution

21. Plot each of the transmittance functions found in problem 1 as a function of a/w(z). Plot the four curves on the same set of axes.Problem 1Find expressions for the fraction of the total power in a beam of spot size w(z) that is transmitted through a circular aperture, centered on the beam, of radius a for a (a) TEM00, (b) TEM01, (c) TEM11, and (d) TEM02 beam. Get solution

22. Based on the plots obtained in problem 1, describe how an adjustable aperture can be used in a laser cavity to ensure that only the TEM00 cavity mode would be present in the laser output.Problem 1Plot each of the transmittance functions found in problem 2 as a function of a/w(z). Plot the four curves on the same set of axes.Problem 2Find expressions for the fraction of the total power in a beam of spot size w(z) that is transmitted through a circular aperture, centered on the beam, of radius a for a (a) TEM00, (b) TEM01, (c) TEM11, and (d) TEM02 beam. Get solution

23. The output from a single-mode TEM00Ar+ laser (λ = 488 nm) has a far-field divergence angle of 0.001 rad and an output power of 5 W.a. What is the spot size at the beam waist for this laser field?b. What is the irradiance at the center of the beam waist (ρ = 0, z = 0) for this field?c. What is the irradiance at the center of the beam 10 m from the beam waist? Get solution

24. Consider a laser cavity consisting of two spherical concave mirrors that are facing each other. Let the mirrors be separated by 20 cm and let each mirror have a radius of curvature of 100 cm. Find the mode-frequency separations: (a) v0,0,q+1 − v0,0,q, (b) vm,n,q+1 − vm,n,q, (c) v0,1,q − v0,0,q, (d) v1,0,q − v0,0,q and (e) v1,1,q − v0,0,q. Get solution

### Chapter #26 Solutions - Introduction to Optics - Leno M Pedrotti, Leno S Pedrotti, Frank L Pedrotti - 3rd Edition

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

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

### Chapter #25 Solutions - Introduction to Optics - Leno M Pedrotti, Leno S Pedrotti, Frank L Pedrotti - 3rd Edition

1. In general, the “electrical constant” K, the dielectric constant, is related to the refractive index by K = n2a. Show that if KR and KI are the real and imaginary parts of the dielectric constant, then ...and ...b. Calculate nR and nI for a dielectric, in terms of KI, at frequencies high enough such that KI = KR. Get solution

2. Show that in a nearly transparent medium, the absorption coefficient is related to the conductivity and refractive index by ... Get solution

3. Calculate and/or plot real and imaginary parts of the refractive index for a dielectric given the frictional parameter γ, the resonant frequency ω0, and the dipole density N. Check your calculations against Figure 25-2. Get solution

4. Assume that aluminum has one free electron per atom and a static conductivity given by 3.54 × 107/Ω-m. Determine (a) the frictional constant γ, (b) the plasma frequency ωp, (c) the real and imaginary parts of the refractive index at 550 nm. Get solution

5. Show that Eq. (25-58) for the skin depth at low frequency is an adequate approximation when ω ≪ γ and ω ≪ σ/ε0. Get solution

6. Calculate the skin depth in copper for radiation of (a) 60 Hz and (b) 3 m. First ensure that the approximations of problem 1 are satisfied. (Handbook data for copper: σ = 5.76 × 107/Ω-m.)Problem 1Show that Eq. (25-58) for the skin depth at low frequency is an adequate approximation when ω ≪ γ and ω ≪ σ/ε0. Get solution

7. Compare the skin depth of (a) aluminum, with conductivity of 3.54 × 107/Ω-m and (b) seawater, with conductivity of 4.3/Ω-m, for radio waves of 60 kHz. Get solution

8. Calculate the skin depth of a solid silver waveguide component for 10-cm microwaves. Silver has a conductivity of 3 × 107/Ω-m. Explain why a more economical silver-plated brass component will work as well. Get solution

9. The energy density of red light of wavelength 660 nm is reduced to one-quarter of its original value by passage through 342 cm of seawater.a. What is the absorption coefficient of seawater for red light of this wavelength?b. At what depth is red light reduced to 1 % of its original energy density? Get solution

10. Calculate and/or plot the real and imaginary parts of the refractive index for a metal, given the frictional parameter and the plasma frequency. Check your results against Figure 1.Figure 1Angular frequency dependence of the refractive index nR and the extinction coefficient nI for copper. Values assumed are ωp = 1.63 × 1016 s−1 and γ = 4.1 × 1013 s−1. The crossover point of the curves coincides with the plasma frequency.... Get solution

11. Determine the theoretical content of the constants A, B, and C used to express the Cauchy dispersion equation. Get solution

12. In writing Eq. (25-3) we neglected to include a contribution due to the magnetic force on the electron. Under what condition is the magnitude of the magnetic force exerted by a harmonic electromagnetic wave on an electron much less than the magnitude of the electric force exerted by the same harmonic electromagnetic wave acting on the electron? (Hint: see Eq. (4-30)). Get solution

### Chapter #24 Solutions - Introduction to Optics - Leno M Pedrotti, Leno S Pedrotti, Frank L Pedrotti - 3rd Edition

1. Write out the third-order terms of the polarization for a single beam described by a plane wave with amplitude E0 and frequency ω. What frequencies appear in the polarization wave? Get solution

2. Write out the third-order terms of the polarization for two- beam interaction, where the beams are plane waves having amplitudes E01 and E02 and frequencies ω1 and ω2, respectively. What frequencies are radiated by the polarization wave? Get solution

3. Write out the second-order terms of the polarization for three-beam interaction, where the beams are plane waves having amplitudes E01 and E02, and E03 frequencies ω1, and ω2, and ω3, respectively. What frequencies are radiated by the polarization wave? Get solution

4. Arguing from Eq. (24-7), show that the linear electro-optic effect is found only in crystals lacking inversion symmetry. Get solution

5. a. Determine the coherence length for second harmonic generation in KDP when subjected to pulsed ruby laser light at λ0 = 694 nm. Appropriate refractive indices are n(694 nm) = 1.505 and n(347 nm) = 1.534.b. The measured coherence length of barium titanate at λ0 = 1.06 μm is 5.8 μm. Calculate the expected change in refractive index at λ = 0.53 μm. Get solution

6. Determine the half-wave voltage for a longitudinal Pockels cell made of ADP (ammonium dihydrogen phosphate) at λ = 546 nm. What is its length? Get solution

7. A longitudinal Pockels cell is made from lithium niobate. Determine the change in refractive index and the phase difference produced by an applied voltage of 426 V when the light beam is from a He-Ne laser at 632.8 nm. The length of the crystal is 1 cm. Get solution

8. Using Eq. (24-9), show that the transmittance of a Pockels cell can also be written as I = Imax sin2(Φ/2).a. At what values of V and Φ (greater than zero) is the transmittance zero?b. If the Pockels cell is preceded by an ordinary half-wave plate, what is the irradiance when V = 0 and when V = VHW? Get solution

9. In what kinds of media are both longitudinal Pockels and Kerr effects present? To get some idea of their relative strengths, compare them by calculating the ratio of retardations produced by an appropriately applied 10 kV. Derive an expression for this ratio. Then do a numerical calculation by assuming a hypothetical medium with “typical” values of r = 10 pm/V, K = 1 pm/V2, L = 2 cm, d = 1 cm, and n0 = 2. Take λ = 550 nm. Get solution

10. Calculate the length of a Kerr cell using carbon disulfide required to produce half-wave retardation for an applied voltage of 30 kV. The electrodes of the cell have a separation of 1.5 cm. Is this cell practical? Get solution

11. Show that Eq. (24-19) is equivalent to the Doppler effect for light. Use the fact that the Doppler frequency shift Δν for light reflected from a moving object is twice that of light emanating from a moving object, or Δν = 2νup/ν, where ν is the light frequency, υ its velocity in the medium, and up is the component of the object velocity parallel to the light wave’s propagation direction. Use the geometry of Figure 24-11 and the Bragg condition. Get solution

12. The speed of sound in glass is 3 km/s. For a sound wave having a width of 1 cm, calculate the advance of the sound wave while it is traversed by a light wave. Take n = 1.50 for the glass. What is the significance of this result? Get solution

13. a. Show that a small change in angle Δθ around the direction of the diffracted beam in Figure 24-11 can be expressed approximately by Δθ = ΔkS/k.b. Show that this result can be expressed as Δθ = (λ/υS) ΔυSwhere λ is the wavelength in the medium.c. The factor by which Δθ exceeds the beam divergence is a practically useful number N called “number of resolvable spots.” This serves as a figure of merit, giving the number of resolvable positions that can be addressed by the beam deflector. If the beam divergence is expressed by the diffraction angle θD = λ/D, with D the beam diameter, show that ...where τ is the time for the sound to cross the optical beam diameter.d. As a numerical example, consider modulation of the sound frequency in the range 80–120 MHz in fused quartz, where υS = 5.95 × 105 cm/s. If the beam diameter is 1 cm, determine the number of resolvable spots. Get solution

14. What acoustic frequency is required of a plane acoustic wave, launched in an acousto-optic crystal, so that a He-Ne laser beam is deflected by 1o? The speed of sound in the crystal is 2500 m/s and its refractive index at 632.8 nm is 1.6. Get solution

15. In Bragg’s equation (24-18), the wavelength of the light and the angle arc those measured within the medium. Show that, if the medium is isotropic and its sides are parallel to the direction of a plane acoustic wave, the equation also holds for the wavelength and angle of diffraction measured outside the medium. Get solution

16. Determine the difference in deflection angle for a He-Ne laser beam that is Bragg-scattered by an acoustic plane wave when the frequencies are 50 MHz and 80 MHz. The acoustic crystal is sapphire, with n = 1.76 and a sound speed of 11 km/s. Get solution

17. Design an optical isolator, as in Figure 24-10, that uses ZnS as the active medium. Let the magnetic field be produced by winding a solenoid directly onto the ZnS crystal at a turn density of 60 turns/cm. Assume λ = 589 nm. Get solution

18. A sample of SF57 glass with polished, parallel sides and 2.73 cm in length is placed between the tapered poles of an electromagnet. A small, central hole is drilled through the pole pieces to allow passage of a linearly polarized He-Ne laser beam through the sample and parallel to the magnetic field direction. The magnetic field is set at 5.098 kG.a. When red He-Ne laser light (632.8 nm) is used, the measured rotation is 900 min. Determine the Verdet constant for the glass.b. When green He-Ne laser light (543.5 nm) is used, the measured rotation is 1330 min. Determine the Verdet constant for the glass. Get solution

19. A 5-cm-long liquid cell is situated in a magnetic field of 4 kG. The cell is filled with carbon disulfide and linearly polarized sodium light is transmitted through the cell, along the B-field direction. Determine both the net rotation of the light and the circular dispersion of CS2 at this wavelength. Get solution

20. Sketch the shape of a nonsymmetrical pulse before and after reflection from an ordinary mirror and before and alter reflection from a PCM. In the latter case, assume that the PCM is “turned on” by initiating the pump beams at the instant the entire pulse has moved inside the PC medium. Show how this effect might be used to correct dispersion broadening in an optical fiber. (If necessary, consult Vladimir V. Shkunov, and Boris Ya. Zel’dovich, “Optical Phase Conjugation,” Scientific American, Dec. 1985: 54.) Get solution

21. Sketch an arrangement using a PCM to project a sharp, high-intensity image of a mask onto the photo-resist layer on a semiconducting chip without using lenses. This provides a means of doing photolithography without placing a mask in direct contact with the chip. (If necessary, consult Vladimir V. Shkunov, and Boris Ya. Zel’dovich, “Optical Phase Conjugation,” Scientific American, Dec. 1985: 54.) Get solution

### Chapter #23 Solutions - Introduction to Optics - Leno M Pedrotti, Leno S Pedrotti, Frank L Pedrotti - 3rd Edition

1. Show that the vanishing of the reflection coefficient in the TM mode, Eq. (23-28), occurs at Brewster’s angle, θp = tan−1(n). Get solution

2. The critical angle for a certain oil is found to be 33°33′. What are its Brewster’s angles for both external and internal reflections? Get solution

3. Determine the critical angle and polarizing angles for (a) external and (b) internal reflections from dense flint glass of index n = 1.84. Get solution

4. For what refractive index are the critical angle and (external) Brewster angle equal when the first medium is air? Get solution

5. Show that the Fresnel equations, Eqs. (23-27) to (23-30), may also be expressed by ... Get solution

6. Show that Eq. (23-37) follows from Eq. (23-35). Get solution

7. Using Eqs. (23-27) through (23-30) and a computer program or a computer algebra system, reproduce Figures 23-3 and 23-5. Also, change the value of n to produce graphs for the case of external and internal reflection from diamond (n = 2.42). Get solution

8. Use a computer to calculate and plot the reflectance curves of Figure 23-4. Also plot the corresponding transmittance. Get solution

9. Use a computer to calculate and plot the phase shifts on reflection as a function of angle of incidence for θ > θc. Take n = 1/1.5 to reproduce Figure 23-6 and then make similar plots for n = 1/1.3 and n = 1/2.42. Get solution

10. A film of magnesium fluoride is deposited onto a glass substrate with optical thickness equal to one-fourth the wavelength of the light to be reflected from it. Refractive indices for the film and substrate are 1.38 and 1.52, respectively. Assume that the film is nonabsorbing. For monochromatic light incident normally on the film, determine (a) reflectance from the air–film surface; (b) reflectance from the film–glass surface; (c) reflectance from an air–glass surface without the film; (d) net reflectance from the combination. See Eq. (22-43). Get solution

11. Calculate the reflectance of water (n = 1.33) for both (a) TE and (b) TM polarizations when the angles of incidence are 0°, 10°, 45°, and 90°. Get solution

12. Light is incident upon an air–diamond interface. If the index of diamond is 2.42, calculate the Brewster and critical angles for both (a) external and (b) internal reflections. In each case distinguish between polarization modes. Get solution

13. Calculate the percent reflectance and transmittance for both (a) TE and (b) TM modes of light incident at 50° on a glass surface of index 1.60. Get solution

14. Derive Eqs. (23-29) and (23-30) for the transmission coefficients both by (a) eliminating Er from Eqs. (23-20) to (23-23) and by (b) using the corresponding equations for the reflection coefficients, together with the relationships between reflection and transmission coefficients implied by Eqs. (23-20) and (23-22). Get solution

15. Unpolarized light is reflected from a plane surface of fused silica glass of index 1.458.a. Determine the critical and polarizing angles.b. Determine the reflectance and transmittance for the TE mode at normal incidence and at 45°.c. Repeat (b) for the TM mode.d. Calculate the phase difference between TM and TE modes for internally reflected rays at angles of incidence of 0°, 20°, 40°, 50°, 70°, and 90°. Get solution

17. Determine the reflectance for metallic reflection of sodium light (589.3 nm) from steel, for which nR = 2.485 and nI = 1.381. Calculate reflectance for (a) TE and (b) TM modes at angles of incidence of 0°, 30°, 50°, 70°, and 90°. Get solution

18. Determine the reflectance from tin at angles of incidence of 0°, 30°, and 60°. Do this for the (a) TE and (b) TM modes of polarization. Real and imaginary parts of the complex refractive index are 1.5 and 5.3, respectively, for light of 589.3 nm. Get solution

19. a. What is the absorption coefficient for tin, with an imaginary part of the refractive index equal to 5.3 for 589.3-nm light?b. At what depth is 99% of normally incident sodium light absorbed in tin? Get solution

20. a. From the power conservation requirement, as expressed by Eq. (23-47), show that for an external reflection the transmission coefficient t must be less than 1, but for an internal reflection t′ may be greater than 1.b. Show further, using the Fresnel Eqs. (23-29) and (23-30), that as the angle of incidence approaches the critical angle, t′ must approach a value of 2 in the TE mode and 2/n in the TM mode.c. Plot the transmission coefficient t′ for an interface between glass (n = 1.5) and air. Get solution

21. A narrow beam of light (λ = 546 nm) is rotated through 90° by TIR from the hypotenuse face of a 45°–90°–45° prism made of glass with n = 1.60.a. What is the penetration depth at which the amplitude of the evanescent wave is reduced to 1/e of its value at the surface?b. What is the ratio of irradiance of the evanescent wave at 1 μm beyond the surface to that at the surface? Get solution

### Chapter #22 Solutions - Introduction to Optics - Leno M Pedrotti, Leno S Pedrotti, Frank L Pedrotti - 3rd Edition

1. Show that when the incident ...-field is parallel to the plane of incidence, γ1 has the form given in Eq. (22-37). Get solution

2. A transparent film is deposited on glass of refractive index 1.50.a. Determine values of film thickness and (hypothetical) refractive index that will produce a nonreflecting film for normally incident light of 500 nm.b. What reflectance does the structure have for incident light of 550 nm? Get solution

3. Show from Eq. (22-42) that the normal reflectance of a single half-wave thick layer deposited on a substrate is the same as the reflectance from the uncoated substrate. ... Get solution

4. A single layer of SiO2 (n = 1.46) is deposited to a thickness of 137 nm on a glass substrate (n = 1.52). Determine the normal reflectance for light of wavelength (a) 800 nm; (b) 600 nm; (c) 400 nm. Verify the reasonableness of your results by comparison with Figure 22-2. Get solution

5. A 596-Å-thick layer of ZnS (n = 2.35) is deposited on glass (n = 1.52). Calculate the normal reflectance of 560 nm light. Get solution

6. Determine the theoretical refractive index and thickness of a single film layer deposited on germanium (n = 4.0) such that normal reflectance is zero at a wavelength of 2 μm. What actual material could be used? Get solution

7. A double layer of quarter-wave layers of Al2O3 (n = 1.60) and cryolite (n = 1.30) are deposited in turn on a glass substrate (n = 1.52).a. Determine the thickness of the layers and the normal reflectance for light of 550 nm.b. What is the reflectance if the layers are reversed? Get solution

8. Quarter-wave thin films of ZnS (n = 2.2) and MgF2 (n = 1.35) are deposited in turn on a substrate of silicon (n = 3.3) to produce minimum reflectance at 2 μm.a. Determine the actual thickness of the layers.b. By what percentage difference does the ratio of the film indices differ from the ideal?c. What is the normal reflectance produced? Get solution

9. By working with the appropriate transfer matrix, show that a quarter-wave/half-wave double layer, as in Figure 22-5, produces the same reflectance as the quarter-wave layer alone. Get solution

10. Write a computer program that will calculate and/or plot reflectance values for a double layer under normal incidence. Let input parameters include thickness and indices of the layers and the index of the substrate. Check results against Figure 22-4. Get solution

11. Prove the condition given by Eq. (22-47) for zero reflectance of three-layer, quarter-quarter-quarter-wave films when used with normal incidence. Do this by determining the composite transfer matrix for the three quarter layers and using the matrix elements in the calculation of the reflection coefficient in Eq. (22-36). Get solution

12. Using the materials given in Table 22-1, design a three- layer multifilm of quarter-wave thicknesses on a substrate of germanium that will give nearly zero reflectance for normal incidence of 2 μm radiation. Get solution

13. Determine the maximum reflectance in the center of the visible spectrum for a high-reflectance stack of high-low index double layers formed using nL = 1.38 and nH = 2.6 on a substrate of index 1.52. The layers are of equal optical thickness, corresponding to a quarter-wavelength for light of average wavelength 550 nm. The high-index material is encountered first by the incident light, as in Figure 22-8. Assume normal incidence and stacks of (a) 2; (b) 4; (c) 8 double layers. Get solution

14. A high-reflectance stack of alternating high-low index layers is produced to operate at 2 μm in the near infrared. A stack of four double layers is made of layers of germanium (n = 4.0) and MgF2 (n = 1.35), each of 0.5-μm optical thickness. Assume a substrate index of 1.50 and normal incidence. What reflectance is produced at 2 μm? Get solution

15. What theoretical ratio of high-to-low refractive indices is needed to give at least 90% reflectance in a high-reflectance stack of two double layers of quarter-wave layers at normal incidence? Assume a substrate of index 1.52. Get solution

16. Show that Rmax in Eq. (22-53) approaches 1 when either N approaches infinity or when the ratio nL/nH approaches zero. ... Get solution