PHYS 408 HOMEWORK 9 SOLUTIONS

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±raction .............................................. 5 3.1 ²eld, 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 . 5 m and R 1 = 2 m. This Fxes a numerical value for the minimum waist: w 0 = ° λz 0 π = 525 μ m (1) 1.3 (c) waist at mirror 2 w (mirror 2) = w (1 . 5m)=(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 ±rom 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 re²ector 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 re²ector surfaces, where the curvature of the wavefronts must match the curvature of the re²ector surfaces. This value is 2 m. 1.5 (e) resonance spectral width ±ollow page 254 and 255 in Saleh and Teich. The spectral width is a³ected by the cavity Fnesse, for Frstly calculate the Fnesse. Stated re²ectivities (on page 374 Saleh, referred to as re²ectances) are R j = | r j | 2 . 9, thus the Fnesse-invoking parameter r = r 1 r 2 . 9. The Fnesse is F = π ± | r | 1 −| r | = 29 . 8 (3) ±or 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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PHYS 408 HOMEWORK 9 SOLUTIONS - solution set 9 December 4...

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