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PHYS 408 HOMEWORK 9 SOLUTIONS - solution set 9 December 4...

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solution set 9 December 4, 2009 Contents 1 Resonator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1 (a) minimum waist location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 (b) minimum waist . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 (c) waist at mirror 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.4 (d) upper and lower bounds, radius of curvature, inside resonator . . . . . . . . . . . . . . . . 2 1.5 (e) resonance spectral width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1 (a) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 (b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 (c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.4 (d) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.5 (e) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Fraunhofer di ff raction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.1 field, interpreted contour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 intensity, interpreted contour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3 intensity, 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.4 scripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.4.1 p3.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.4.2 g.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 List of Figures 1 p3a.jpg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2 p3b.jpg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 p3c.jpg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1
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1 Resonator 1.1 (a) minimum waist location The curvature of the wavefronts will match the curvature of the mirrors at the location of the mirror, and this requires the location of the minimum waist to be to the left of mirror (1), in accordance with the shape of a gaussian beam, regardless of the magnitude of the radius of curvature. We can argue this same physics but with these alternative words: the location of minimum waist cannot be to the right of mirror (1) because the corresponding wavefronts would have a curvature of sign opposite that of mirror (1) at the location of mirror (1). 1.2 (b) minimum waist Matching wavefronts from each mirror requires R 1 = z 1 + z 2 0 /z 1 (see page 383, Saleh and Teich), which yields z 0 = 0 . 866 m for the stated z 1 = 0 . 5 m and R 1 = 2 m. This fixes a numerical value for the minimum waist: w 0 = λ z 0 π = 525 μ m (1) 1.3 (c) waist at mirror 2 w (mirror 2) = w (1 . 5 m) = (0 . 866 m) 1 + (1 . 5) 2 / (0 . 866) 2 = 1 . 05 mm (2) 1.4 (d) upper and lower bounds, radius of curvature, inside resonator From R = z + z 2 0 /z we see that extremal values of the radius of curvature occur at z = ± z 0 . The value z = z 0 is inside the resonator, and noting R ( z 0 ) > 0, we know this value of the curvature is a minimum. There are no mechanisms in this simple two reflector system to decrease this radius of curvature further, so we know it’s a global minimum, and certainly a minimum inside the cavity. The curvature at this point is R ( z 0 ) = 1 . 73 m. The maximum radius of curvature occurs at each of the reflector surfaces, where the curvature of the wavefronts must match the curvature of the reflector surfaces. This value is 2 m. 1.5 (e) resonance spectral width Follow page 254 and 255 in Saleh and Teich. The spectral width is a ff ected by the cavity finesse, for firstly calculate the finesse. Stated reflectivities (on page 374 Saleh, referred to as reflectances) are R j = | r j | 2 = 0 . 9, thus the finesse-invoking parameter r = r 1 r 2 = 0 . 9. The finesse is F = π | r | 1 | r | = 29 . 8 (3) For F 1, which is the case here, the approximate spectral width is δν ν F F = c 2 F d = 5 . 03 MHz (4) 2
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2 Polarization Use the convention e ikz corresponds to a wave propagating in the + z direction. Also, the top component of the Jones vector
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