Edward S. Rogers Sr. Department of Electrical and Computer Engineering ECE426F Optical Engineering Final Exam Dec. 17, 2003 Exam Type: D (Close-book + one 2-sided aid sheet + a non-programmable calculator) Exam Duration: 2 ½ hours Marks: Q1 Q2 Q3 Q4 Q5 Total 20 20 20 20 20 + 5 100 + 5 (Note: You may, but do not have to, supply explanations to your answers. If you choose to do so, partial marks will be awarded for wrong answers accompanied by correct and relevant explanations. However, marks for correct answers accompanied by wrong explanations will be deducted accordingly.)
ECE426F Optical Engineering Final Exam Page 2 of 15 Q1 Answer the following short questions: (a) Identify the locations (A, B, C or D) of aperture stop, field stop, entrance pupil, and exit pupil of a compound microscope shown in the following diagram: D A B (b) Aperture Stop: Field Stop: Entrance Pupil: Exit Pupil: The following diagram illustrates the operation of a Wollaston prism. Is the crystal used for this prism a positive or a negative uniaxial crystal? C (2 marks) e-ray o-ray (The lines and dots illustrate the direction of the optical axis.) positive (n e >n o ) negative (n e <n o )
ECE426F Optical Engineering Final Exam Page 3 of 15 (c) Please match the aperture shapes with their far-field diffraction patterns by linking each pair with a solid line. (d) What are the two necessary conditions for achieving the state of lasing? (2 marks)
ECE426F Optical Engineering Final Exam Page 4 of 15 (e) Briefly explain why polarized sunglasses help reduce glare from some reflective surfaces, such as puddles on the road or the back window of a car ahead of you. Which polarization (horizontal or vertical) should be blocked by the sunglasses? (5 marks) (f) Briefly explain why more densely packed materials (such as solids) usually exhibit higher refractive index than less densely packed materials (such as liquids) in the transparent region. (3 marks)
ECE426F Optical Engineering Final Exam Page 5 of 15 Q2. A beam having a vacuum wavelength of 500 nm is incident on a glass-air interface from the glass side (n g = 1.5) at 45 o. The beam is linearly polarized with equal, in-phase TE and TM component before reflection. (a) At what distance from the surface is the amplitude of the evanescent wave 1/e of its value at the surface? (8 marks) (b) What is the phase difference between the reflected TE and TM waves? Based on this result, determine the polarization state of the reflected beam. (8 marks) (c) Draw the time evolution of the electrical field vector E of the reflected wave on a 2D graph, with the TE direction as x and the TM direction as y.
ECE426F Optical Engineering Final Exam Page 6 of 15
ECE426F Optical Engineering Final Exam Page 7 of 15
ECE426F Optical Engineering Final Exam Page 8 of 15 Q3. Consider a diffraction grating spectrometer as shown. N θ d 2.0 cm f=20 cm (Only the first-order diffracted rays are drawn.) (a) What is the minimum number of lines N the grating must have in order to resolve the sodium D lines (λ 1 = 589.00 nm and λ 2 =589.59 nm) in the first order? (b) For the number of lines calculated above, and the total grating width of 2.0 cm, determine the angle θ between the two first-order diffracted rays corresponding to the two wavelengths. (c) If the focal length of the convex lens is 20 cm, what is the linear separation d between the two D lines at the focal plane? (d) What would be the separation d if third-order diffracted rays were used instead? Assume the same total number of lines N. (6 marks) (6 marks) (Use the small angle approximation in your calculations.)
ECE426F Optical Engineering Final Exam Page 9 of 15
ECE426F Optical Engineering Final Exam Page 10 of 15
ECE426F Optical Engineering Final Exam Page 11 of 15 Q4. Assume a Fabry-Pérot interferometer is adjusted such that a zero-radius bright fringe just starts to emerge at the centre of the interference pattern. (a) What is the relationship between the mirror spacing d, refractive index n, the wavelength λ, and the order m of this central fringe? (b) Show that the radii of the other interference fringes are approximately proportional to 1, 2, N, where N is an integer, and inversely 2 proportional to m. [Hint: use cos θ 1 ( θ / 2) ]. (12 marks) (c) Using the above results, comment on how you could make the fringe spacing larger in order to count the fringes more easily in the Fabry-Pérot interferometer experiment.
ECE426F Optical Engineering Final Exam Page 12 of 15
ECE426F Optical Engineering Final Exam Page 13 of 15 Q5. An Ar + -ion laser emits at 515 nm. Its gain medium has a transition cross-section σ 0 of 3.1 10-16 m 2. The gain of this medium is primarily Doppler-broadened with a bandwidth ν G of 3.5 GHz: 2 4ln 2( ν ν 0 ) g( ν ) = g0 exp 2. ν G The gain medium has a refractive index of 1, and fills up a 1-m long resonator cavity with mirror reflectances of 100% and 98%. Other loss mechanisms are negligible. (a) Calculate the threshold gain coefficient g th. (b) At threshold, that is, when g 0 = g th, calculate the population inversion density. (c) Calculate the longitudinal mode spacing of the resonator. Please express your answer both in frequency and in wavelength. (d) If a peak gain coefficient of 0.03 m -1 is measured without feedback, how many longitudinal modes will be lasing with feedback from the resonator? (e) If the resonator length (as well as the gain medium length) is reduced by half, how many longitudinal modes will be lasing? (6 marks) (6 marks) (Bonus 5 marks)
ECE426F Optical Engineering Final Exam Page 14 of 15
ECE426F Optical Engineering Final Exam Page 15 of 15