1. A biconvex lens of 5 cm thickness and index 1.60 has surfaces of
radius 40 cm. If this lens is used for objects in water, with air on its
opposite side, determine its effective focal length and sketch its
focal and principal points. Get solution
2. A double concave lens of glass with n = 1.53 has surfaces of 5 D (diopters) and 8 D, respectively. The lens is used in air and has an axial thickness of 3 cm.a. Determine the position of its focal and principal planes.b. Also find the position of the image, relative to the lens center, corresponding to an object at 30 cm in front of the first lens vertex.c. Calculate the paraxial image distance assuming the thin-lens approximation. What is the percent error involved? Get solution
3. A biconcave lens has radii of curvature of 20 cm and 10 cm. Its refractive index is 1.50 and its central thickness is 5 cm. Describe the image of a 1-in.-tall object, situated 8 cm from the first vertex. Get solution
4. An equiconvex lens having spherical surfaces of radius 10 cm, a central thickness of 2 cm, and a refractive index of 1.61 is situated between air and water (n = 1.33). An object 5 cm high is placed 60 cm in front of the lens surface. Find the cardinal points for the lens and the position and size of the image formed. Get solution
6. Light rays enter the plane surface of a glass hemisphere of radius 5 cm and refractive index 1.5.a. Using the system matrix representing the hemisphere, determine the exit elevation and angle of a ray that enters parallel to the optical axis and at an elevation of 1 cm.b. Enlarge the system to a distance x beyond the hemisphere and find the new system matrix as a function of x.c. Using the new system matrix, determine where the ray described above crosses the optical axis. Get solution
7. Using Figure 18-12b and c, verify the expressions given in Table 18-2 for the distances q, f2, s, and w. Get solution
8. A lens has the following specifications:R1 = 1.5 cm = R2, d(thickness) = 2.0 cm,n1 = 1.00, n2 = 1.60, n3 = 1.30.Find the principal points using the matrix method. Include a sketch, roughly to scale, and do a ray diagram for a finite object of your choice. Get solution
9. A positive thin lens of focal length 10 cm is separated by 5 cm from a thin negative lens of focal length −10 cm. Find the equivalent focal length of the combination and the position of the foci and principal planes using the matrix approach. Show them in a sketch of the optical system, roughly to scale, and use them to find the image of an arbitrary object placed in front of the system. Get solution
10. A glass lens 3 cm thick along the axis has one convex face of radius 5 cm and the other, also convex, of radius 2 cm. The former face is on the left in contact with air and the other in contact with a liquid of index 1.4. The refractive index of the glass is 1.50. Find the positions of the foci, principal planes, and focal lengths of the system. Use the matrix approach. Get solution
11. a. Find the matrix for the simple “system” of a thin lens of focal length 10 cm, with input plane at 30 cm in front of the lens and output plane at 15 cm beyond the lens.b. Show that the matrix elements predict the locations of the six cardinal points as they would be expected for a thin lens.c. Why is B = 0 in this case? What is the special meaning of A in this case? Get solution
12. A gypsy’s crystal ball has a refractive index of 1.50 and a diameter of 8 in.a. By the matrix approach, determine the location of its principal points.b. Where will sunlight be focused by the crystal ball? Get solution
13. A thick lens presents two concave surfaces, each of radius 5 cm, to incident light. The lens is 1 cm thick and has a refractive index of 1.50. Find (a) the system matrix for the lens when used in air and (b) its cardinal points. Do a ray diagram for some object. Get solution
14. An achromatic doublet consists of a crown glass positive lens of index 1.52 and of thickness 1 cm, cemented to a flint glass negative lens of index 1.62 and of thickness 0.5 cm. All surfaces have a radius of curvature of magnitude 20 cm. If the doublet is to be used in air, determine (a) the system matrix elements for input and output planes adjacent to the lens surfaces; (b) the cardinal points; (c) the focal length of the combination, using the lensmaker’s equation and the equivalent focal length of two lenses in contact. Compare this calculation of f, which assumes thin lenses, with the previous value. Get solution
15. Enlarge the optical system of Figure 18-15 to include an object space to the left and an image space to the right of the lens. Let the new input plane be located at distance 5 in object space and the new output plane at distance s′ in image space.a. Recalculate the system matrix for the enlarged system.b. Examine element B to determine the general relationship between object and image distances for the lens. Also determine the general relationship for the lateral magnification.c. From the results of (b), calculate the image distance and lateral magnification for an object 20 cm to the left of the lens.d. What information can you find for the system by setting matrix elements A and D equal to zero? (See Figure 18-9.) Get solution
16. Find the system matrix for a Cooke triplet camera lens (see Figure 3-23a). Light entering from the left encounters six spherical surfaces whose radii of curvature are, in turn, r1 to r6. The thickness of the three lenses are, in turn, t1 to t3, and the refractive indices are n1 to n3. The first and second air separations between lens surfaces are d1 and d2. Sketch the lens system with its cardinal points. How far behind the last surface must the film plane occur to focus paraxial rays? Data:r1= 19.4 mmt1 = 4.29 mmn1 = 1.6110 r2 = −128.3 mmt2 = 0.93 mmn2 = 1.5744 r3 = −57.8 mmt3 = 3.03 mmn3 = 1.6110 r4 = 18.9 mm r5 = 311.3 mmd1 = 1.63 mm r6 = −66.4 mmd2 = 12.90 mm Get solution
17. Process the product of matrices for a thick lens, as in Eq. (18-15), without assuming the special conditions, n = n′ and t = 0. Thus find the general matrix elements A, B, C, and D for a thick lens. Get solution
18. Using the cardinal point locations (Table 18-2) in terms of the matrix elements for a general thick lens (problem 1), verify that f1, and f2 are given by Eqs. (18-1) and (18-2).Problem 1Process the product of matrices for a thick lens, as in Eq. (18-15), without assuming the special conditions, n = n′ and t = 0. Thus find the general matrix elements A, B, C, and D for a thick lens. Get solution
19. Using the cardinal point locations (Table 18-2) in terms of the matrix elements for a general thick lens (problem 1), verify that the distances r, s, v, and w are given by Eqs. (18-3) and (18-4).Problem 1Process the product of matrices for a thick lens, as in Eq. (18-15), without assuming the special conditions, n = n′ and t = 0. Thus find the general matrix elements A, B, C, and D for a thick lens. Get solution
20. Write a computer program that incorporates Eqs. (18-34) to (18-41) for ray tracing through an arbitrary number of refracting, spherical surfaces. The program should allow for the special cases of rays from far-distant objects and for plane surfaces of refraction.... Get solution
21. Trace two rays through the hemispherical lens of Figure 18-15. The rays originate from the same object point, 2 cm above the optical axis and an axial distance of 10 cm from the first surface. One ray is parallel to the axis and the other makes an angle of −20° with the axis. Get solution
23. Trace two rays, both from far-distant objects, through a Protor photographic lens, one at altitude of 1 mm and the other at 5 mm. Determine where and at what angle the rays cross the optical axis. The specifications of this four-element lens, including an intermediate air space of 3 mm, are as follows, with distances in mm: R1 = 17.5t1 = 2.9n1 = 1.6489R2 = 5.8t2 = 1.3n2 = 1.6031R3 = 18.6t3 = 3.0n3 = 1R4 = −12.8t4 = 1.1n4 = 1.5154R5 = 18.6t5 = 1.8n5 = 1.6112R6 = −14.3 Get solution
2. A double concave lens of glass with n = 1.53 has surfaces of 5 D (diopters) and 8 D, respectively. The lens is used in air and has an axial thickness of 3 cm.a. Determine the position of its focal and principal planes.b. Also find the position of the image, relative to the lens center, corresponding to an object at 30 cm in front of the first lens vertex.c. Calculate the paraxial image distance assuming the thin-lens approximation. What is the percent error involved? Get solution
3. A biconcave lens has radii of curvature of 20 cm and 10 cm. Its refractive index is 1.50 and its central thickness is 5 cm. Describe the image of a 1-in.-tall object, situated 8 cm from the first vertex. Get solution
4. An equiconvex lens having spherical surfaces of radius 10 cm, a central thickness of 2 cm, and a refractive index of 1.61 is situated between air and water (n = 1.33). An object 5 cm high is placed 60 cm in front of the lens surface. Find the cardinal points for the lens and the position and size of the image formed. Get solution
6. Light rays enter the plane surface of a glass hemisphere of radius 5 cm and refractive index 1.5.a. Using the system matrix representing the hemisphere, determine the exit elevation and angle of a ray that enters parallel to the optical axis and at an elevation of 1 cm.b. Enlarge the system to a distance x beyond the hemisphere and find the new system matrix as a function of x.c. Using the new system matrix, determine where the ray described above crosses the optical axis. Get solution
7. Using Figure 18-12b and c, verify the expressions given in Table 18-2 for the distances q, f2, s, and w. Get solution
8. A lens has the following specifications:R1 = 1.5 cm = R2, d(thickness) = 2.0 cm,n1 = 1.00, n2 = 1.60, n3 = 1.30.Find the principal points using the matrix method. Include a sketch, roughly to scale, and do a ray diagram for a finite object of your choice. Get solution
9. A positive thin lens of focal length 10 cm is separated by 5 cm from a thin negative lens of focal length −10 cm. Find the equivalent focal length of the combination and the position of the foci and principal planes using the matrix approach. Show them in a sketch of the optical system, roughly to scale, and use them to find the image of an arbitrary object placed in front of the system. Get solution
10. A glass lens 3 cm thick along the axis has one convex face of radius 5 cm and the other, also convex, of radius 2 cm. The former face is on the left in contact with air and the other in contact with a liquid of index 1.4. The refractive index of the glass is 1.50. Find the positions of the foci, principal planes, and focal lengths of the system. Use the matrix approach. Get solution
11. a. Find the matrix for the simple “system” of a thin lens of focal length 10 cm, with input plane at 30 cm in front of the lens and output plane at 15 cm beyond the lens.b. Show that the matrix elements predict the locations of the six cardinal points as they would be expected for a thin lens.c. Why is B = 0 in this case? What is the special meaning of A in this case? Get solution
12. A gypsy’s crystal ball has a refractive index of 1.50 and a diameter of 8 in.a. By the matrix approach, determine the location of its principal points.b. Where will sunlight be focused by the crystal ball? Get solution
13. A thick lens presents two concave surfaces, each of radius 5 cm, to incident light. The lens is 1 cm thick and has a refractive index of 1.50. Find (a) the system matrix for the lens when used in air and (b) its cardinal points. Do a ray diagram for some object. Get solution
14. An achromatic doublet consists of a crown glass positive lens of index 1.52 and of thickness 1 cm, cemented to a flint glass negative lens of index 1.62 and of thickness 0.5 cm. All surfaces have a radius of curvature of magnitude 20 cm. If the doublet is to be used in air, determine (a) the system matrix elements for input and output planes adjacent to the lens surfaces; (b) the cardinal points; (c) the focal length of the combination, using the lensmaker’s equation and the equivalent focal length of two lenses in contact. Compare this calculation of f, which assumes thin lenses, with the previous value. Get solution
15. Enlarge the optical system of Figure 18-15 to include an object space to the left and an image space to the right of the lens. Let the new input plane be located at distance 5 in object space and the new output plane at distance s′ in image space.a. Recalculate the system matrix for the enlarged system.b. Examine element B to determine the general relationship between object and image distances for the lens. Also determine the general relationship for the lateral magnification.c. From the results of (b), calculate the image distance and lateral magnification for an object 20 cm to the left of the lens.d. What information can you find for the system by setting matrix elements A and D equal to zero? (See Figure 18-9.) Get solution
16. Find the system matrix for a Cooke triplet camera lens (see Figure 3-23a). Light entering from the left encounters six spherical surfaces whose radii of curvature are, in turn, r1 to r6. The thickness of the three lenses are, in turn, t1 to t3, and the refractive indices are n1 to n3. The first and second air separations between lens surfaces are d1 and d2. Sketch the lens system with its cardinal points. How far behind the last surface must the film plane occur to focus paraxial rays? Data:r1= 19.4 mmt1 = 4.29 mmn1 = 1.6110 r2 = −128.3 mmt2 = 0.93 mmn2 = 1.5744 r3 = −57.8 mmt3 = 3.03 mmn3 = 1.6110 r4 = 18.9 mm r5 = 311.3 mmd1 = 1.63 mm r6 = −66.4 mmd2 = 12.90 mm Get solution
17. Process the product of matrices for a thick lens, as in Eq. (18-15), without assuming the special conditions, n = n′ and t = 0. Thus find the general matrix elements A, B, C, and D for a thick lens. Get solution
18. Using the cardinal point locations (Table 18-2) in terms of the matrix elements for a general thick lens (problem 1), verify that f1, and f2 are given by Eqs. (18-1) and (18-2).Problem 1Process the product of matrices for a thick lens, as in Eq. (18-15), without assuming the special conditions, n = n′ and t = 0. Thus find the general matrix elements A, B, C, and D for a thick lens. Get solution
19. Using the cardinal point locations (Table 18-2) in terms of the matrix elements for a general thick lens (problem 1), verify that the distances r, s, v, and w are given by Eqs. (18-3) and (18-4).Problem 1Process the product of matrices for a thick lens, as in Eq. (18-15), without assuming the special conditions, n = n′ and t = 0. Thus find the general matrix elements A, B, C, and D for a thick lens. Get solution
20. Write a computer program that incorporates Eqs. (18-34) to (18-41) for ray tracing through an arbitrary number of refracting, spherical surfaces. The program should allow for the special cases of rays from far-distant objects and for plane surfaces of refraction.... Get solution
21. Trace two rays through the hemispherical lens of Figure 18-15. The rays originate from the same object point, 2 cm above the optical axis and an axial distance of 10 cm from the first surface. One ray is parallel to the axis and the other makes an angle of −20° with the axis. Get solution
23. Trace two rays, both from far-distant objects, through a Protor photographic lens, one at altitude of 1 mm and the other at 5 mm. Determine where and at what angle the rays cross the optical axis. The specifications of this four-element lens, including an intermediate air space of 3 mm, are as follows, with distances in mm: R1 = 17.5t1 = 2.9n1 = 1.6489R2 = 5.8t2 = 1.3n2 = 1.6031R3 = 18.6t3 = 3.0n3 = 1R4 = −12.8t4 = 1.1n4 = 1.5154R5 = 18.6t5 = 1.8n5 = 1.6112R6 = −14.3 Get solution
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