ECE318 Uoft - Problem Set 1

Tutorial 1 Problems ECE 318 Jan 15 / 17, 2013 1) Briefly describe how the foci of an ellipse connect with Fermat’s Principle! 2) Use Fermat’s Principle to prove Snell’s law for ray propagation from point A to B, in the diagram on right. 3) A lamp is placed underwater in a pool, where water has a refractive index of 1.33. At what minimum angle of incidence with respect to the surface normal will the light be totally reflected by the water-air interface? Use this result to explain why a scuba diver may not see the lights from his boat during a night dive. 4) Pedrotti 2-5: A ray of light makes an angle of incidence of 45 o at the center of the top surface of a transparent cube of index 1.414. Trace the ray through the cube. 5) Hecht 5.5: Making use of Fig. P.55, Snell’s Law, and the fact that in the paraxial region, α = h/s o , φ = h/R, and β = h/s i , derive Eq. (5.8), for Gaussian Optics. Recall that this same formula was derived in class using Fermat’s Principle. 6) Pedrotti 2-13: a) At what position in front of a spherical refracting surface must an object be placed so that the refraction produces parallel rays of light? In other words, what is the focal length of a single refracting surface? b) Since real object distances are positive, what does your result imply for the cases n 2 > n 1 and n 2 < n 1 ? 7) Hecht 5.6: Show that in the paraxial domain, the magnification produced by a single spherical interface between two continuous media, as shown in Fig. P5.6, is given by Use the small-angle approximation for Snell’s Law and approximate the angles by their tangents. 8) Hecht 5.7: Imagine a hemispherical interface, with a radius of curvature of radius 5 cm, separating two media: air on the left, water on the right. A 3 cm-tall toad is in the air medium, facing the convex interface and located 30 cm from its vertex. Where in the water will it be imaged? How big will it appear to a fish in the water? Use the result of magnification of a single spherical interface from the previous problem, even though our frog is pushing the paraxial approximation limits: 9) Hecht 5.8: Locate the image of an object placed 1.2 m from the vertex of a gypsy’s crystal ball, which has a 20 cm diameter (n = 1.5). Make a sketch of the ray trajectories. 10) Pedrotti 2-16: A plano-convex lens having a focal length of 25.0 cm is to be made with glass of refractive index 1.52. Calculate the radius of curvature of the grinding and polishing tools to be used in making the lens. 1 2 i T o ns M ns = −

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Optics Problem Set

Transcript of ECE318 Uoft - Problem Set 1

Tutorial 1 Problems ECE 318 Jan 15 / 17, 2013

1) Briefly describe how the foci of an ellipse connect with

Fermat’s Principle! 2) Use Fermat’s Principle to prove Snell’s law for ray propagation

from point A to B, in the diagram on right. 3) A lamp is placed underwater in a pool, where water has a

refractive index of 1.33. At what minimum angle of incidence with respect to the surface normal will the light be totally reflected by the water-air interface? Use this result to explain why a scuba diver may not see the lights from his boat during a night dive.

4) Pedrotti 2-5: A ray of light makes an angle of incidence of 45o at the center of the top surface of a transparent cube of index 1.414. Trace the ray through the cube.

5) Hecht 5.5: Making use of Fig. P.55, Snell’s Law, and the fact that in the paraxial region, α = h/so, φ = h/R, and β = h/si, derive Eq. (5.8), for Gaussian Optics. Recall that this same formula was derived in class using Fermat’s Principle.

6) Pedrotti 2-13: a) At what position in front of a spherical

refracting surface must an object be placed so that the refraction produces parallel rays of light? In other words, what is the focal length of a single refracting surface?

b) Since real object distances are positive, what does your result imply for the cases n2 > n1 and n2 < n1 ?

7) Hecht 5.6: Show that in the paraxial domain, the magnification produced by a single spherical interface between two continuous media, as shown in Fig. P5.6, is given by

Use the small-angle approximation for Snell’s Law and approximate the angles by their tangents.

8) Hecht 5.7: Imagine a hemispherical interface, with a radius of curvature of radius 5 cm, separating two media: air on the left, water on the right. A 3 cm-tall toad is in the air medium, facing the convex interface and located 30 cm from its vertex. Where in the water will it be imaged? How big will it appear to a fish in the water? Use the result of magnification of a single spherical interface from the previous problem, even though our frog is pushing the paraxial approximation limits:

9) Hecht 5.8: Locate the image of an object placed 1.2 m from the vertex of a gypsy’s crystal ball, which has a 20 cm diameter (n = 1.5). Make a sketch of the ray trajectories.

10) Pedrotti 2-16: A plano-convex lens having a focal length of 25.0 cm is to be made with glass of refractive index 1.52. Calculate the radius of curvature of the grinding and polishing tools to be used in making the lens.

n sMn s

= −