Formulas of possible interest

Transcription

1 Name: PHYS 3410/6750: Modern Optics Final Exam Thursday 15 December 2011 Prof. Bolton No books, calculators, notes, etc. Formulas of possible interest I = ɛ 0 c E 2 T = 1 2 ɛ 0cE 2 0 E γ = hν γ n = c/v v = λν (v = vee ; ν = nu.) k =2π/λ ω =2πν τ =1/ν D = 1 ( 1 f =(n l 1) 1 ) R1 R2 1 s o + 1 s i = 1 f M T = y i y o = s i s o n 1 sin θ 1 = n 2 sin θ 2 θ r = θ i a(sin θ m sin θ i )=mλ P = σat 4 e iφ =cosφ + i sin φ

2 PHYS 3410/6750 Final Exam, Fall Problem 1 (20 points) An extremely simplified model for a compact digital camera is a single lens of diameter 1 cm and focal length 2.5 cm, together with a square pixelized digital detector of side length 0.5 cm. The detector is positioned 2.5 cm behind the lens so that real images of objects at infinity are focused onto it. The configuration is illustrated in the following diagram: lens detector 1 cm 0.5 cm 2.5 cm (a) What is the size of the angular field of view (in radians) of this camera? (b) Taking the diffraction limit as θ λ/d, what is the angular resolution of this camera in radians at a wavelength of 500 nm? (c) If the detector is composed of a total of 16 Megapixels (i.e., 16 million pixels) arranged in a square grid, what is the linear pixel scale of the system in radians per pixel? (d) The Nyquist criterion states that at least two samples per resolution element are necessary to record all information in a signal. Does this camera have enough megapixels by this standard? Too many? Too few?

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4 PHYS 3410/6750 Final Exam, Fall Problem 2 (15 points) The following ray diagram depicts a harmonic light wave incident on an interface from a medium with index of refraction n 1 into a medium with index n 2 (with n 2 >n 1 ), with its polarization entirely in the plane of incidence: θ i n 1 n 2 If the electric field vectors of the transmitted ray are parallel to the propagation direction of the reflected ray, the amplitude of the reflected ray will be zero. Showing all steps, derive the expression for the value of θ i (the Brewster angle ) at which this occurs, in terms of n 1 and n 2.

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6 PHYS 3410/6750 Final Exam, Fall Problem 3 (20 points) For light of vacuum wavelength λ 0 incident onto a thin film of refractive index n f and thickness d, immered in a medium with index of refraction n = 1, the phase shift between the front-surface and back-surface reflected rays as a function of incident angle θ i is given by δ = 4πd n 2 f λ θ i π 0 d n=1 n f n=1 θi (a) What is the explanation for the extra π term? (b) What is the condition on δ for there to be maximally constructive interference between the two reflected beams? (c) A 10 cm diameter soap film with index of refraction n =1.5 is oriented vertically and illuminated by white light at normal incidence (i.e., θ i = 0). Due to gravity, the thickness of the film varies smoothly from 500 nm at the top to 2100 nm at the bottom. How many bright interference fringes are seen in reflection over the surface of the film for violet light of wavelength 400 nm? How many for orange light of wavelength 600 nm? (d) Will the overall reflected fringe patterns have the appearance of well-ordered red-greenblue rainbow bands? Why or why not?

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8 PHYS 3410/6750 Final Exam, Fall Problem 4 (15 points) A star has a surface temperature T and a radius R. A planet of radius r orbits the star in a circular orbit at a distance d from the star. You may assume that the star radiates as a blackbody. d R T r Provide answers to the following two questions in terms of T, R, d, λ peak (defined below), and physical constants: (a) What is the total power (energy per unit time) from the star that hits the planet? (b) Make the approximation that all photons have the wavelength of the blackbody emission peak given by Wien s law : λ peak T = b, whereb is a constant. What is the number density of photons (number per unit volume) from the star at the orbital radius of the planet?

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10 PHYS 3410/6750 Final Exam, Fall Question 1: (6 points) Match each monochromatic aberration to its description: (a) Spherical aberration (b) Coma (c) Astigmatism (d) Distortion (e) Field curvature (i) Deviation of focal surface from a plane (ii) Off-axis variation of magnification over entrance pupil (iii) On-axis variation of focal length over entrance pupil (iv) Variation of magnification over image plane (v) Off-axis difference between tangential and saggital foci Question 2: (6 points) The following diagram represents a fiber optic. Draw two rays incident with different angles from the left-hand side throught the X, one of which is transmitted by the fiber (call it ray 1 ), and the other of which is not (call it ray 2 ). Follow the rays far enough to show the important differences, and be sure to clearly distinguish the two rays in each region of interest. Question 3: (6 points) Light of wavelength 600 nm is incident at an angle of 30 on a diffraction grating with a ruling density of 250 lines per millimeter. Into what angle is the (positive) second-order light diffracted? You may express your answer in terms of inverse trigonometric functions. Recall that sin(30 )=1/2.

11 PHYS 3410/6750 Final Exam, Fall Question 4: (6 points) Using the convention F (k) =F[f(x)] = f(x)e ikx dx, evaluate the Fourier transform of the function with the form f(x) =1for x <a/2and f(x) =0for x >a/2. Your final answer should not contain i. What diffraction irradiance pattern is described by the squared-modulus of this result? Question 5: (6 points) What is population inversion, why is it necessary for the operation of a laser, and how is it achieved in laser systems?

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