Midterm Exam ECE 181 Spring 2013 Calculators are allowed, and will be useful. One page of notes are allowed; otherwise the exam is closed book. Name ______________________________________________ Problem Score 1 2 3 4 Total Ques%on 1: Waveguides A long parallel slab waveguide with a glass core (ncore = 1.5) is covered top and boOom with a cladding (nclad = 1.4) and held in air. You can assume the cladding of the waveguide is painted black – so light that enters the cladding will be absorbed. (1) What is the maximum angle of incident light which will be captured and guided without internal loss to the opposite end? (2) Assuming the input end is illuminated with all angles of light, what is the maximum angle of emission of light from the end of the waveguide? input light input end waveguide core output end θout, max θin.max Ques%on 1 con%nued Now the input end of the waveguide is immersed in water, with index n=1.3333. (3) What is the maximum angle of incident light which will be captured and guided without internal loss to the opposite end? (4) Assuming the input end is illuminated with all angles of light, what is the maximum angle of emission of light from the end of the waveguide? input light θ'in.max water θ'out, max Ques%on 2: thin lens imaging (1) An ideal thin lens of focal length f = +1m is used to create a real image of an object with lateral magniﬁcaYon M = -‐3. Calculate the posiYon of the object and image relaYve to the lens, and draw the system, showing the object, lens and image posiYons (to approximate scale), and labeling the chief, focal, and axial rays, and the total length between the object and image. (2) Assuming you have only ideal thin lenses with +1 m focal length, describe how to make an imaging system with the same magniﬁcaYon but a total length of less than 1 meter, and calculate the new length. 1m Ques%on 2 con%nued: (3) Using as many as you need of the same ideal thin lenses with focal length f = +1 m, describe and draw roughly to scale an imaging system with lateral magniﬁcaYon M = +3. (No need for the length to be limited) (4) What is the axial magniﬁcaYon for this system? 1m Ques%on 3: Wave Op%cs x An inﬁnite unit amplitude plane wave of wavelength λ, = 500 nm is propagaYng in an inﬁnite material with index of refracYon ng = 1.5. The k-‐vector of the plane wave is in the x-‐z plane, and pointed downward at an angle of θ = 60˚ relaYve to the z axis. (1) What is the frequency of this wavefront, in Hz? (2) Write the k-‐vector for this wavefront, with units of microns-‐1. (3) Express the complex wavefuncYon Up(x,y,z,t) in terms of the variables given above. (4) Express the real valued wavefuncYon up(x,y,z) corresponding to the complex wavefuncYon Up(x,y,z,t = 0). Don't worry about parts 5-‐7; not covered in 2015 midterm (5) Express the paraxial approximaYon to the complex wavefuncYon Up(x,y,z,t = 0). (6) Is the paraxial approximaYon to this wavefront of part 4 reasonably accurate? Why or why not? (7) Does the funcYon described in part 3 obey the Helmholtz equaYon? Explain why or why not. θ k z Ques%on 4: Ray tracing aCer dark The air bubbles in a glass of champagne each act as a Yny lens. Assuming that champagne has an index of refracYon similar to water (n = 1.333) and the gas an index of 1, answer the following quesYons. (1) If a ray was incident at the center of the bubble, what fracYon of the incident light energy would reﬂect from the ﬁrst champagne-‐air interface? (2) If a ray was incident near the top of the bubble, what fracYon of the incident light energy would reﬂect from the ﬁrst champagne-‐air interface? (3) If the ray was incident above the center of the bubble, as drawn, would it bend upwards or downwards? (4) Assuming that champagne obeys normal dispersion, which would deﬂect more, a red ray or a blue ray? (5) Assuming that the rays illuminaYng the bubble were limited to the central 10% of the overall diameter of the bubble, would paraxial raytracing (as we did with 2x2 matrices) provide a good esYmate of the image formed? (explain why or why not). (6) Could a thin lens approximaYon accurately model the bubble lens? (explain why or why not) Q1 Q2 Q3-‐6