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06_optics

# At central rays ratio of collimated beam pupil sizes p

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Unformatted text preview: ratio of focal lengths: f1/f2 – magnification is just M = f· for small – so lens turns angle ( ) into displacement ( ) Winter 2008 Lecture 6 2/ 1 = f1/f2 • after all: magnification is how much bigger things look • displacement at focal plane, = f1 1 = f2 2 relation above • hint: look at central rays – ratio of collimated beam (pupil) sizes: P 1/P 2 = f1/f2 = M 27 Winter 2008 28 7 Geometrical Optics 01/31/2008 UCSD: Physics 121; 2008 UCSD: Physics 121; 2008 Reflector/Refractor Analogy Parabolic Example Take the parabola: y = x2 Slope is y ’ = 2x Curvature is y’’ = 2 So R = 1/y ’’ = 0.5 Slope is 1 (45°) at: x = 0.5; y = 0.25 • For the purposes of understanding a reflecting system, one may replace with lenses (which we know how to trace/analyze) So focus is at 0.25: f = R /2 – focal length and aperture the same; rays on other side – for a reflector, f = R/2 [compare to 1/f = (n 1)(1/R1 1/R2 ) for lens] • for n = 1.5, R2 = R1 (symmetric lens), f = R • so glass lens needs twice the curvature of a mirror Note that pathlength to focus is the same for depicted ray and one along x = 0 Winter 2008 29 Winter 2008 30 UCSD: Physics 121; 2008 UCSD: Physics 121; 2008 Cassegrain Telescope Cassegrain focus • Abstracting mirrors as lenses, then lenses as sticks: • A Cassegrain telescope can be modeled as as positive and negative lens – trace central ray with angle 1 – figure out 2 and then focal length given s’ and d 12 – eyepiece not shown: only up to focus • Final focus depends on placement of negative lens • • • • • • – if |s| = |f2|, light is collimated; if |s| > |f2 |, light will diverge • both s and f2 are negative • For the Apache Point 3.5 meter telescope, for example: – f1 = 6.12 m; f2 = 1 .60 m; d 12 = 4.8 m; s = f1 d12 = 1 .32 m – yields s’ = 7.5 m using 1/s + 1/s’ = 1/f2 Winter 2008 Lecture 6 31 Winter 2008 y2 = d 12 1 (adopt convention where 1 is negative as drawn) y1 = f2 1 (f2 is negative: negative lens) y2)/f 2 = 1(f2 d 12)/f2 2 = (y1 yf = y 2 + 2 s’ = 1 ( d12 + s ’( f2 d 12)/f 2 ) feff = d12 + s ’( f2 d 12)/f 2 = f 1 s’/s after lots of algebra for Apache Point 3.5 meter, this comes out to 35 meters 32 8 Geometrical Optics 01/31/2008 UCSD: Physics 121; 2008 UCSD: Physics 121; 2008 f-numbers f=D D f-numbers, compared f = 4D D f/4 beam: “slow” f/1 beam: “fast” • The f-number is a useful characteristic of a lens or system of lenses/mirrors • Simply = f/D – where f is the focal length, and D is the aperture (diameter) • “fast” converging beams (low f-number) are optically demanding fast” to make without aberrations • “slow” converging beams (large f-number) are easier to make slow” • aberrations are proportional to 1/ 2 • Lens curvature to scale for n = 1.5 1.5 – obviously slow lenses are easier to fabricate: less curvature – so pay the price for going “fast” Winter 2008 33 Winter 2008 34 UCSD: Physics 121; 2008 UCSD: Physics 121; 2008 Pupils Pupils within Pupils • Consider two “ field points” on the focal plane points” – e.g., two stars some angle apart • The rays obviously all overlap at the aperture • Looking at three stars (red, green, blue) through telescope, eye position is important • So is pupil size compared to eye pupil – called the entrance pupil • The rays are separate at the focus (completely distinct) • Then overlap again at exit pupil, behind eyepiece – want your pupil here –...
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