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PHYS 408 HOMEWORK 5 SOLUTIONS - solution set 5 Contents 1 2...

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solution set 5 November 10, 2009 Contents 1 beam focused into aquarium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1 spot size in air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 spot size and location in aquarium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 gaussian aperture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1 divergence angle and minimum waist location . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 equivalent ABCD matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.3 creation of film . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 lens of finite thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.1 lens of negligible thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3.2 lens of finite thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1
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1 beam focused into aquarium 1.1 spot size in air θ 0 w 0 z 0 w 0 ( λ z 0 / π ) 1 / 2 w 0 = λ πθ 0 (1) Minimum spot size: 2 w 0 = 2 λ πθ 0 = 4 × 10 4 m (2) Spot size at 12 cm to left of minimum spot size location, noting z 0 = w 0 / θ 0 = 0 . 2 m: 2 w ( z = 12 cm) = 4 × 10 4 × 1 + (0 . 12 / 0 . 2) 2 1 / 2 m = 4 . 66 × 10 4 m (3) 1.2 spot size and location in aquarium Just before the aquarium, noting z is negative b/c it is to the left of the minimum waist: q a = 0 . 12 0 . 2 i (4) The e ff ective ABCD matrix for entering into to the glass, then propagating through the glass, then entering into the water is: A B C D = 1 0 0 n glass /n water 1 d glass 0 1 1 0 0 n air /n glass = 1 0 . 02 1 0 . 75 (5) The q at just to the right of the glass (inside the aquarium) is: q b = (1)( 0 . 12 0 . 2 i ) + (0 . 02) (1)( 0 . 12 0 . 2 i ) + 0 . 75 = 0 . 1333 0 . 2667 i (6) So the “new” minimum waist is located 0.133 m to the right of point b, equivalently 0.163 to the right of the outside edge of the aquarium. The new minimum spot size is 2 w 0 = 2((2 π · 10 7 / 1 . 3333)0 . 2667 / π ) 1 / 2 = 4 × 10 4 m.
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