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Physics 11bLecture #20Lenses, Mirrors, and ImagesS&J Chapter 36What We Did Last Time Law of reflection: Huygens’ principle Index of refraction Wavelength is affected Snell’s law of refraction: Total internal reflection if Fermat’s principle: The actual path betweentwo points taken by a beam of light is the onewhich is traversed in the least time Dispersion and chromatic aberration11θθ′=11 2 2nnλλ=112 2sin sinnnθθ=1ncv=>1θ′1θn1n2θ1θ212nn<n2n1θ2θ121nn<211arcsinnnθ<Today’s Goals Discuss simple optical devices Lenses and mirrors – Building blocks of optical devices Use Geometrical Optics approximation Convex and concave lenses Focal points, focal lengths Spherical aberration Concave mirrors Lens formula for ideal lenses Analyze a magnifying glassGeometrical Optics Optical devices are combination of lenses and mirrors Telescopes, microscopes, cameras, binoculars … Assume all elements (lenses/mirrors) are much larger than the wavelength in aperture and thickness We can treat light as if it’s a particle Trajectory in each medium is a straight line At boundaries, it either reflects or refracts Refraction angle given by Snell’s law Everything is determined by the elements’shapes, indices of refraction, and theirgeometrical arrangement Geometrical Optics = Analysis of optical devices using this approximation1221sinsinnnθθ=1θ2θOptical Elements There are 4 major types: Concave and convex lenses Concave and convex mirrors Lenses may have differentradii on two surfaces Consider them as a combinationof two lenses with one side flatFocal Points Lenses (mirrors) turn plane waves into spherical waves “Origin” of the spherical wavesis the focal point Light may or may not actuallygo through the focal point If yes Æ real focal point If not Æ virtual focal point Distance between the lens and itsfocal point = focal length f If real Æ f > 0 If virtual Æ f < 0ff−Real FPVirtual FPconventionDiopter Your optometrist prescribes your glasses using diopter It’s just 1/f – called the optical power Larger number Æ shorter focal length Æ stronger bending of light rays e.g. a –1.5 diopter lens hasf = –0.7m Æ concave lens A nearsighted (myopic) eyeneeds a concave lens forcorrection Now you know what thosemysterious numbers on yourprescription areConvex Lens Let’s start with a flat-convex lens Convex side is spherical Snell’s law For small angles Incoming light converges at the focal pointR1θ1θ2θ12sin sinnθθ=index ny21θθ−f2sinnyRθ=21 21tan( ) 1ny yRRyyyRfnθθ θθ=≈≈=−−−−Focal lengthBut there are approximationsConcave Lens Now a flat-concave lens Snell’s law For small angles Same formula, just a negative signR1θ1θ2θ12sin sinnθθ=yf−2sinnyRθ=21tan( ) 1yRfnθθ−= ≈−−Yeah, what about those approximations?index n21θθ−sign!Aberration Two approximations were made Angles θ1and θ2are small Index n is a constant Both are incorrect for real lenses Angles may get large if the aperture is largeÆ Rays at different y do not converge at the same f Spherical Aberration Index varies with wavelength λdue to dispersionÆ Rays with different λdo not converge at the same f Chromatic AberrationR1θ1θ2θy21θθ−fSpherical Aberration For lenses with large aperture a, rays passing near the perimeter over-refract Negligible if Camera lenses withsmall “f-stop” sufferfrom spherical aberration Smaller f-stop = larger aperture = brighter (faster) lens Good 50mm lenses have f/1.4 or smallerRaaRf-stop(1)fRaan==−Large Aperture Lens Canon EL 50mm f/1.0(!) lens It reduces spherical aberration usingaspherical lenses and glass withhigh index of refraction Latter is simple Larger n makes R larger for the same fÆ smaller aberration for the same a Flint glass has n = 1.575–1.89Æ n –1larger than normal glass by max 78%1Rfn=−Spherical Aberration Spherical aberration can also be reduced by Aspherical (hyperbolic) lens shape Difficult to make with traditionalpolishing technique Combining multiple lenses so thataberrations cancel Mathematical technique known since 1830 Designing good lens remains on borderline between art and science Photographers still believe 60-year-old Zeiss lenses are better than modern computer-designed ones…AsphericallensesMirrors Mirrors are simpler than lenses For small angle θ No chromatic aberration To avoid spherical aberration,you need a parabolic mirror Concave mirrors are used in place ofconvex lenses in telescopes Easier to make a large mirror than a large lens Can make the overall length shorterRθyfθ2fR=Hubble Space Telescope Hubble Space Telescope launched in April 1990with a spherical primary mirror Spherical aberration made it nearly useless Corrective optics (COSTAR) added in December 1993 “Eyeglasses for Hubble”Ideal Lens An ideal lens would use very-high-index, non-dispersive material Since such a lens have very large R It can be made very thin, with no spherical aberration In the limit, we would have an infinitely thin film Light entering the film magicallybends by that satisfies1Rfn=−We can dream…n →∞fFrtanrfθ=θ()rθθ=Ideal Lens What about a concave lens? Easy: Negative signs on θand fcancel each other An ideal lens (concave or convex) bendsthe light that passes at radius r by θaccording to Let’s this idealized formula to analyze a very simple optical device – a magnifying glassf−Frθ−tanrfθ=tanrfθ=Lens Formula First, we trace rays of light from a point through a lens We assume ideal lens with no aberration o = distance from the object i = distance to the image Assuming small angles This is more useful that it looks…oiθtanrfθθ≈=12rroiθθ θ=+ ≈+rgeometryideal lens11 1oi f+=General “lens formula”1θ2θLens Formula The formula works in all combinations of o, i, and f Negative i Æ Image is virtual, i.e. light does not actually focus in a pointoif foi−f f11 1oi f+=of>0ofiof=>−of<0ofiof=<−Lens Formula It works with concave lenses as well Since f is negative, i is always negativeÆ Image is virtual What do we mean by “images”? So far our “object” is a point How does a real object


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HARVARD PHYS 11b - Lecture 20

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