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 fnumbers
f=D
D fnumbers, compared
f = 4D D
f/4 beam: “slow” f/1 beam: “fast” • The fnumber 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 fnumber) are optically demanding
fast”
to make without aberrations
• “slow” converging beams (large fnumber) 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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 Winter '08
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 Physics, Light, Geometrical optics

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