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
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
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