Physics 11b Lecture #20

Transcription

1 Physics 11b Lecture #20 Lenses, Mirrrs, and Images S&J Chapter 36

2 What We Did Last Time Law relectin: Huygens principle Index reractin θ = θ 1 1 Wavelength is aected Snell s law reractin: n = c v> λ n n 1 = λ n sinθ = n sinθ n 1 θ 1 θ 1 θ 1 n < n 1 2 Ttal internal relectin i 2 θ 1 < arcsin n n 1 n 2 θ 2 Fermat s principle: The actual path between tw pints taken by a beam light is the ne which is traversed in the least time Dispersin and chrmatic aberratin n < θ 1 n 2 1 n 1 n 2 θ 2

3 Tday s Gals Discuss simple ptical devices Lenses and mirrrs Building blcks ptical devices Use Gemetrical Optics apprximatin Cnvex and cncave lenses Fcal pints, cal lengths Spherical aberratin Cncave mirrrs Lens rmula r ideal lenses Analyze a magniying glass

4 Gemetrical Optics Optical devices are cmbinatin lenses and mirrrs Telescpes, micrscpes, cameras, binculars Assume all elements (lenses/mirrrs) are much larger than the wavelength in aperture and thickness We can treat light as i it s a particle Trajectry in each medium is a straight line At bundaries, it either relects r reracts eractin angle given by Snell s law Everything is determined by the elements sinθ1 n2 shapes, indices reractin, and their sinθ = gemetrical arrangement 2 n1 Gemetrical Optics = Analysis ptical devices using this apprximatin θ 1 θ 2

5 Optical Elements There are 4 majr types: Cncave and cnvex lenses Cncave and cnvex mirrrs Lenses may have dierent radii n tw suraces Cnsider them as a cmbinatin tw lenses with ne side lat

6 Fcal Pints Lenses (mirrrs) turn plane waves int spherical waves Origin the spherical waves is the cal pint Light may r may nt actually g thrugh the cal pint I yes real cal pint I nt virtual cal pint Distance between the lens and its cal pint = cal length I real > 0 I virtual < 0 cnventin Virtual FP eal FP

7 Dipter Yur ptmetrist prescribes yur glasses using dipter It s just 1/ called the ptical pwer Larger number shrter cal length strnger bending light rays e.g. a 1.5 dipter lens has = 0.7m cncave lens A nearsighted (mypic) eye needs a cncave lens r crrectin Nw yu knw what thse mysterius numbers n yur prescriptin are

8 Cnvex Lens Let s start with a lat-cnvex lens Cnvex side is spherical Snell s law nsinθ = sinθ 1 2 sinθ = 2 ny Fr small angles y y y = = ny y tan( θ θ ) θ θ n θ θ 2 y 1 θ2 1 θ 1 Incming light cnverges at the cal pint But there are apprximatins index n Fcal length θ

9 Cncave Lens Nw a lat-cncave lens Snell s law nsinθ = sinθ 1 2 sinθ = Fr small angles sign! 2 ny y = tan( θ θ ) n θ 1 Same rmula, just a negative sign y θ θ 2 1 index n θ 2 θ 1 Yeah, what abut thse apprximatins?

10 Aberratin Tw apprximatins were made Angles θ 1 and θ 2 are small Index n is a cnstant Bth are incrrect r real lenses Angles may get large i the aperture is large ays at dierent y d nt cnverge at the same Spherical Aberratin Index varies with wavelength λ due t dispersin ays with dierent λ d nt cnverge at the same Chrmatic Aberratin θ θ 2 y 1 θ2 θ1 θ 1

11 Spherical Aberratin Fr lenses with large aperture a, rays passing near the perimeter ver-reract Negligible i a Camera lenses with small -stp suer rm spherical aberratin -stp = = a a( n 1) Smaller -stp = larger aperture = brighter (aster) lens Gd 50mm lenses have /1.4 r smaller a

12 Large Aperture Lens Cann EL 50mm /1.0(!) lens It reduces spherical aberratin using aspherical lenses and glass with high index reractin Latter is simple = n 1 Larger n makes larger r the same smaller aberratin r the same a Flint glass has n = n 1larger than nrmal glass by max 78%

13 Spherical Aberratin Spherical aberratin can als be reduced by Aspherical (hyperblic) lens shape Diicult t make with traditinal plishing technique Cmbining multiple lenses s that aberratins cancel Mathematical technique knwn since 1830 Aspherical lenses Designing gd lens remains n brderline between art and science Phtgraphers still believe 60-year-ld Zeiss lenses are better than mdern cmputer-designed nes

14 Mirrrs Mirrrs are simpler than lenses Fr small angle θ = 2 N chrmatic aberratin T avid spherical aberratin, yu need a parablic mirrr Cncave mirrrs are used in place cnvex lenses in telescpes Easier t make a large mirrr than a large lens Can make the verall length shrter y θ θ

15 Hubble Space Telescpe Hubble Space Telescpe launched in April 1990 with a spherical primary mirrr Spherical aberratin made it nearly useless Crrective ptics (COSTA) added in December 1993 Eyeglasses r Hubble

16 Ideal Lens An ideal lens wuld use very-high-index, nn-dispersive material We can dream Since = such a lens have very large n 1 It can be made very thin, with n spherical aberratin In the n limit, we wuld have an ininitely thin ilm Light entering the ilm magically r bends by θ = θ () r that satisies θ r F tanθ =

17 Ideal Lens What abut a cncave lens? r Easy: tanθ = Negative signs n θ and cancel each ther An ideal lens (cncave r cnvex) bends the light that passes at radius r by θ accrding t r tanθ = Let s this idealized rmula t analyze a very simple ptical device a magniying glass F r θ

18 Lens Frmula First, we trace rays light rm a pint thrugh a lens We assume ideal lens with n aberratin = distance rm the bject i = distance t the image r θ θ1 2 θ Assuming small angles i r r θ = θ1+ θ2 + i gemetry This is mre useul that it lks θ tanθ = = General lens rmula i r ideal lens

19 Lens Frmula = i The rmula wrks in all cmbinatins, i, and i > i = 0 > i Negative i Image is virtual, i.e. light des nt actually cus in a pint < i = 0 <

20 Lens Frmula = i It wrks with cncave lenses as well i Since is negative, i is always negative Image is virtual What d we mean by images? S ar ur bject is a pint < 0 i = 0 < Hw des a real bject (with size) lk thrugh lenses?

21 Magniying Glass Simplest ptical device: a magniying glass = i We knw image distance rm the lens rmula ays passing the middle the lens dn t bend i We can trace rays rm varius pints the bject

22 Magniicatin Magniicatin pwer is i m = Using the lens rmula = i I bject is utside the cal pint Image is inverted m = Magniicatin is again i m = = Negative = inverted i Object shuld be placed slightly inside the cal pint i

23 Practical Issues Have yu ever seen a 100x magniying glass? What s wrng with it? Try t make small i = Image is bigger, but it s ar away Desn t help seeing mre details Usually i = 25 cm r easy viewing Hw abut making small? Object must it between and the lens! i i m = m = 25 cm

24 Summary Fcal lengths lenses and mirrrs > 0 r real, < 0 r virtual cal pints Fr lenses: =± + r cncave, r cnvex n 1 Fr a cncave mirrr: = 2 Lens (r mirrr) with large aperture Spherical aberratin Lens rmula: = i Applied t magniying glass magniicatin: m = i i