1. Carry out the “rearranging” called for in arriving at Eq. (20-18). Get solution
2. If image and object distance for a spherical refracting surface—in addition to satisfying Eq. (2-19)—also satisfy the relation 1/s′ = (1/s)+(1/R), show thata. s′=−(n1/n2)s, andb. a(Q) for spherical aberration in Eq. (20-18) vanishes.c. Show that a(Q) also vanishes for s′= R and for rays intersecting with the spherical surface vertex. Such image points arc called aplanatic points.d. Find the aplanatic points for a spherical surface of +8 cm separating two media of refractive indices 1.36 and 1.70, respectively. Get solution
3. A collimated light beam is incident on the plane side of a plano-convex lens of index 1.50, diameter 50 mm, and radius 40 mm. Find the spherical wave aberration and the longitudinal and transverse spherical ray aberrations. Get solution
4. Show that for a spherical concave mirror, a calculation like that done for a refracting surface gives a third-order aberration of ...where R is the magnitude of the radius of curvature. Get solution
5. Using the result of problem 20-4, determine the wave aberration, transverse aberration, and longitudinal aberration for a spherical mirror of 2-m focal length and 50-cm diameter, when it forms an image of a distant point object. Get solution
6. A reflecting telescope uses a spherical mirror with a 3-m focal length and an aperture given by f/3.75.a. Using the results of problem 20-4, determine the magnitude of the spherical wave aberration for the telescope.b. If a Schmidt-type correcting plane of refractive index 1.40 were installed to correct the spherical aberration, what would be the required difference in thickness between the center and edge of the plate? Get solution
7. In forming an image of an axial point object, a +4.0-diopter lens with a diameter of 6.0 cm gives a longitudinal spherical aberration of +1.0 cm. If the object is 50 cm from the lens, determine (a) the transverse spherical aberration and (b) the diameter of the blur circle in the paraxial focal plane. Get solution
8. Determine the longitudinal and lateral spherical ray aberration for a thin lens of n = 1.50, r1 = +10 cm, and r2 = −10 cm due to rays parallel to the axis and through a zone of radius h = 1 cm. Get solution
9. Using the equation for spherical aberration of a thin lens, see problem 20-8, find the longitudinal spherical ray aberration of a lens as a function of ray height h. Do this by plotting the longitudinal ray aberration as a function of ray height for h = 0, 1, 2, 3, 4, and 5 cm. The lens has a refractive index of 1.60 and radii r1 = 36 cm and r2 = −18 cm. The incident light rays arc parallel to the optical axis. Get solution
10. An equiconvex thin lens of index 1.50 and radius 15 cm forms an image of an axial object point 25 cm in front of the lens and for rays through a zone of radius h = 2 cm. Determine the longitudinal and lateral spherical ray aberration. (See problem 1.)Problem 1Determine the longitudinal and lateral spherical ray aberration for a thin lens of n = 1.50, r1 = +10 cm, and r2 = −10 cm due to rays parallel to the axis and through a zone of radius h = 1 cm. Get solution
11. Show that if L = (1/s′h) − (1/s′p), setting dL/dσ = 0 produces the condition for minimum spherical aberration: ... Get solution
12. A positive lens of index 1.50 and focal length 30 cm is “bent” to produce Coddington shape factors of 0.700 and 3.00. Determine the corresponding radii of curvature for the two lenses. Get solution
13. A positive thin lens of focal length 20 cm is designed to have minimal spherical aberration in its image plane, 30 cm from the lens. If the lens index is 1.60, determine its radii of curvature. Get solution
14. A thin, plano-convex lens with 1-m focal length and index 1.60 is to be used in an orientation that produces less spherical aberration while focusing a collimated light beam. Prove that the proper orientation is with light incident on the spherical side by comparing the Coddington shape factor for each orientation with the value giving minimum spherical aberration. Get solution
15. A positive lens is needed to focus a parallel beam of light with minimum spherical aberration. The required focal length is 30 cm. If the glass has a refractive index of 1.50, determine (a) the required Coddington factor and (b) the radii of curvature of the lens. (c) If the lens is to be used instead to produce a collimated beam, how do these answers change? Get solution
16. Answer problem 1 when the lens is designed to reduce coma.Problem 1A positive lens is needed to focus a parallel beam of light with minimum spherical aberration. The required focal length is 30 cm. If the glass has a refractive index of 1.50, determine (a) the required Coddington factor and (b) the radii of curvature of the lens. (c) If the lens is to be used instead to produce a collimated beam, how do these answers change? Get solution
17. A cm focal length positive lens is to be used as an inverting lens; that is, it simply inverts an image without altering its size. What radii of curvature lead to minimum spherical aberration in this application? The lens refractive index is 1.50. Get solution
18. Answer problem 1 when the lens is designed to reduce coma.Problem 1A cm focal length positive lens is to be used as an inverting lens; that is, it simply inverts an image without altering its size. What radii of curvature lead to minimum spherical aberration in this application? The lens refractive index is 1.50. Get solution
19. It is desired to reduce the curvature of field of a lens of cm focal length made of crown glass (n = 1.5230). For this purpose a second lens of flint glass (n = 1.7200) is added. What should be its focal length? Refractive indices are given for sodium light of 589.3 nm. Get solution
20. A doublet telescope objective is made of a cemented positive lens (n1 = 1.5736, f1 = 3.543 cm) and negative lens (n2 = 1.6039, f2= 5.391 cm).a. Determine the radius of their Petzval surface.b. What focal length for the negative lens gives a flat Petzval surface? Get solution
21. Design an achromatic doublet of 517/645 crown and 620/380 flint glasses that has an overall focal length of 20 cm. Assume the crown glass lens to be equiconvex. Determine the radii of curvature of the outer surfaces of the lens, as well as its resultant focal length for the D, C, and F Fraunhofer lines. Get solution
22. Design an achromatic doublet of 5-cm focal length using 638/555 crown and 805/255 flint glass. Determine (a) radii of curvature; (b) focal lengths for D, C, and F Fraunhofer lines; (c) powers and dispersive powers of the individual elements. (d) Is Eq. (20-44) satisfied? Get solution
23. Design an achromatic doublet of −10-cm focal length, using 573/574 and 689/312 glasses. Assume the crown glass lens to be equiconcave. Determine (a) radii of curvature of the lens surfaces; (b) individual focal lengths for the Fraunhofer D line; (c) the overall focal lengths of the lens for the Fraunhofer D, C, and F lines. Get solution
2. If image and object distance for a spherical refracting surface—in addition to satisfying Eq. (2-19)—also satisfy the relation 1/s′ = (1/s)+(1/R), show thata. s′=−(n1/n2)s, andb. a(Q) for spherical aberration in Eq. (20-18) vanishes.c. Show that a(Q) also vanishes for s′= R and for rays intersecting with the spherical surface vertex. Such image points arc called aplanatic points.d. Find the aplanatic points for a spherical surface of +8 cm separating two media of refractive indices 1.36 and 1.70, respectively. Get solution
3. A collimated light beam is incident on the plane side of a plano-convex lens of index 1.50, diameter 50 mm, and radius 40 mm. Find the spherical wave aberration and the longitudinal and transverse spherical ray aberrations. Get solution
4. Show that for a spherical concave mirror, a calculation like that done for a refracting surface gives a third-order aberration of ...where R is the magnitude of the radius of curvature. Get solution
5. Using the result of problem 20-4, determine the wave aberration, transverse aberration, and longitudinal aberration for a spherical mirror of 2-m focal length and 50-cm diameter, when it forms an image of a distant point object. Get solution
6. A reflecting telescope uses a spherical mirror with a 3-m focal length and an aperture given by f/3.75.a. Using the results of problem 20-4, determine the magnitude of the spherical wave aberration for the telescope.b. If a Schmidt-type correcting plane of refractive index 1.40 were installed to correct the spherical aberration, what would be the required difference in thickness between the center and edge of the plate? Get solution
7. In forming an image of an axial point object, a +4.0-diopter lens with a diameter of 6.0 cm gives a longitudinal spherical aberration of +1.0 cm. If the object is 50 cm from the lens, determine (a) the transverse spherical aberration and (b) the diameter of the blur circle in the paraxial focal plane. Get solution
8. Determine the longitudinal and lateral spherical ray aberration for a thin lens of n = 1.50, r1 = +10 cm, and r2 = −10 cm due to rays parallel to the axis and through a zone of radius h = 1 cm. Get solution
9. Using the equation for spherical aberration of a thin lens, see problem 20-8, find the longitudinal spherical ray aberration of a lens as a function of ray height h. Do this by plotting the longitudinal ray aberration as a function of ray height for h = 0, 1, 2, 3, 4, and 5 cm. The lens has a refractive index of 1.60 and radii r1 = 36 cm and r2 = −18 cm. The incident light rays arc parallel to the optical axis. Get solution
10. An equiconvex thin lens of index 1.50 and radius 15 cm forms an image of an axial object point 25 cm in front of the lens and for rays through a zone of radius h = 2 cm. Determine the longitudinal and lateral spherical ray aberration. (See problem 1.)Problem 1Determine the longitudinal and lateral spherical ray aberration for a thin lens of n = 1.50, r1 = +10 cm, and r2 = −10 cm due to rays parallel to the axis and through a zone of radius h = 1 cm. Get solution
11. Show that if L = (1/s′h) − (1/s′p), setting dL/dσ = 0 produces the condition for minimum spherical aberration: ... Get solution
12. A positive lens of index 1.50 and focal length 30 cm is “bent” to produce Coddington shape factors of 0.700 and 3.00. Determine the corresponding radii of curvature for the two lenses. Get solution
13. A positive thin lens of focal length 20 cm is designed to have minimal spherical aberration in its image plane, 30 cm from the lens. If the lens index is 1.60, determine its radii of curvature. Get solution
14. A thin, plano-convex lens with 1-m focal length and index 1.60 is to be used in an orientation that produces less spherical aberration while focusing a collimated light beam. Prove that the proper orientation is with light incident on the spherical side by comparing the Coddington shape factor for each orientation with the value giving minimum spherical aberration. Get solution
15. A positive lens is needed to focus a parallel beam of light with minimum spherical aberration. The required focal length is 30 cm. If the glass has a refractive index of 1.50, determine (a) the required Coddington factor and (b) the radii of curvature of the lens. (c) If the lens is to be used instead to produce a collimated beam, how do these answers change? Get solution
16. Answer problem 1 when the lens is designed to reduce coma.Problem 1A positive lens is needed to focus a parallel beam of light with minimum spherical aberration. The required focal length is 30 cm. If the glass has a refractive index of 1.50, determine (a) the required Coddington factor and (b) the radii of curvature of the lens. (c) If the lens is to be used instead to produce a collimated beam, how do these answers change? Get solution
17. A cm focal length positive lens is to be used as an inverting lens; that is, it simply inverts an image without altering its size. What radii of curvature lead to minimum spherical aberration in this application? The lens refractive index is 1.50. Get solution
18. Answer problem 1 when the lens is designed to reduce coma.Problem 1A cm focal length positive lens is to be used as an inverting lens; that is, it simply inverts an image without altering its size. What radii of curvature lead to minimum spherical aberration in this application? The lens refractive index is 1.50. Get solution
19. It is desired to reduce the curvature of field of a lens of cm focal length made of crown glass (n = 1.5230). For this purpose a second lens of flint glass (n = 1.7200) is added. What should be its focal length? Refractive indices are given for sodium light of 589.3 nm. Get solution
20. A doublet telescope objective is made of a cemented positive lens (n1 = 1.5736, f1 = 3.543 cm) and negative lens (n2 = 1.6039, f2= 5.391 cm).a. Determine the radius of their Petzval surface.b. What focal length for the negative lens gives a flat Petzval surface? Get solution
21. Design an achromatic doublet of 517/645 crown and 620/380 flint glasses that has an overall focal length of 20 cm. Assume the crown glass lens to be equiconvex. Determine the radii of curvature of the outer surfaces of the lens, as well as its resultant focal length for the D, C, and F Fraunhofer lines. Get solution
22. Design an achromatic doublet of 5-cm focal length using 638/555 crown and 805/255 flint glass. Determine (a) radii of curvature; (b) focal lengths for D, C, and F Fraunhofer lines; (c) powers and dispersive powers of the individual elements. (d) Is Eq. (20-44) satisfied? Get solution
23. Design an achromatic doublet of −10-cm focal length, using 573/574 and 689/312 glasses. Assume the crown glass lens to be equiconcave. Determine (a) radii of curvature of the lens surfaces; (b) individual focal lengths for the Fraunhofer D line; (c) the overall focal lengths of the lens for the Fraunhofer D, C, and F lines. Get solution
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