PHYS 408 HOMEWORK 9 SOLUTIONS

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Unformatted text preview: ................................ 7 8 9 List of Figures 1 2 3 p3a.jpg p3b.jpg p3c.jpg 1 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 2 Matching wavefronts from each mirror requires R1 = z1 + z0 /z1 (see page 383, Saleh and Teich), which yields z0 = 0.866 m for the stated z1 = 0.5 m and R1 = 2 m. This ﬁxes a numerical value for the minimum waist: w0 = 1.3 ￿ λ z0 = 525 µm π (c) waist at mirror 2 w(mirror 2) = w(1.5 m) = (0.866 m) 1.4 (1) ￿ 1 + (1.5)2 /(0.866)2 = 1.05 mm (2) (d) upper and lower bounds, radius of curvature, inside resonator 2 From R = z + z0 /z we see that extremal values of the radius of curvature occur at z = ±z0 . The value z = z0 is inside the resonator, and noting R￿￿ (z0 ) &gt; 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(z0 ) = 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 aﬀected by the cavity ﬁnesse, for ﬁrstly calculate the ﬁnesse. Stated reﬂectivities (on page 374 Saleh, referred to as reﬂectances) are Rj =...
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