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 smallerRaaRf-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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