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Phys 531 Lecture 24 15 November 2005 Polarizers Last time discussed basics of polarization Linear circular elliptical states Describe by polarization vectorj Today How to establish and manipulate polarization Outline o Polarizers Reflection scattering dichroism Calculations with polarizers o Birefringence Next time Retarders make circular and elliptical polarizations Jones calculus matrix method for calculations Polarizers Hecht 82 Most natural light sources are unpolarized Obtain polarized light with polarizer 2 filter passing only one polarization state Usually transmit linear polarization Plane of polarization given by transmission axis 9 y xamp Z Ideal polarizer Transmission for j axis 2 1 Transmission for it axis 2 O In general say axis 2 a and b L 21 Then in 1 2 basis j jHa l ij transmit amplitude j To get jcai 23 since aa 1 and baO Usuall a is real write 2 A 2 T lJHl lJ al If a real have linear polarizer Axis at angle 6 write a cos0 c l sin 637 If 5 linearly polarized j COSai l sin or Then it azjca COSOCOSa l sinesina cos0 oz Gives IlIalus s Law For linear polarization incident on polarizer out in COSQW 01 6 oz 2 angle between transmission axis and incident plane of polarization But T lj al2 is more general Example If jinc 373 XQEW what is transmission through linear polarizer at angle 0 A cos6 l z sin6 1 9 Have 3312 2 6Z 1 I l So T Eleml2 5 independent of 6 Question What would it mean if a were complex Other effect Light exitting ideal polarizer has jout a To see write jin jlla 4 ij a is transmitted b is blocked SO jout ja gt a amplitude j gives transmission Physically L component is absorbed or reflected only M component remains If two polarizers first at 61 second at 62 e1 92 Output of first polarized along 61 Transmission of second 2 cos262 61 Original version of Malus39s Law Or say three polarizers first at 00 second at 45quot third at 900 Without middle polarizer transmission is zero 1 1 1 With all three transmission is 7 x 7 7 2 2 4 Question This seems counterintuitive Where does the vertical component come from Real polarizers aren39t perfect 0 Transmission for j H a To lt 1 loss 0 Transmission for 31 a e gt O leakage 0 Output light not exactly polarized along a rarely specified Values depend on type of polarizer Discuss types of polarizers Constructing Polarizers Hecht 83 86 Already know one way to polarize light use Brewster39s angle When TM polarized light incident at angle 917 tan 1ntni Get 7quot 0 Two ways to make polarizer 0 Use reflected light get L component Then loss is very high glass get RL m 02 gt lose 80 Also leakage is fairly high hard to control angle accurately o Better use transmitted light and many surfaces Pile of plates polarizer Each surface transmits almost all of I and fraction TL of IL for glass TL m 08 For N plates total L transmission 2 TLtot TEN Say glass plates N 10 TLtot 001 Typically get total Tlltot 05 Pile of plates simple and robust Good extinction for large N But often awkward to use thick requires collimated light Rarely used now in optics Similar methods used for x rays other radiation Polarization by Scattering Brewster effect based on scattering properties Recall Brewster angle when kref Etrans Atoms in glass can39t radiate H E Charges radiate L acceleration Scattered light is generally polarized Example light from sky 9 b E y 008 6 q 39Q quot1 o 9 Q scattered light polarized Not typically useful as polarizer Dich roism Hecht 83 Dichroism 2 selective absorption of one linear polarization Clearly useful for polarizers Example microwave polarizer Array of parallel wires spacing ltlt A E aligned with wires drive current resistance gt power dissipation gt absorption E L wires little current no absorption Acts as a polarizer transmits only E L wires Watch out graphically want to picture vertical E squeezing through slots Actual effect is just the opposite Optical version wires gt long polymer chains Embed in clear plastic Stretch plastic to align chains Material called polaroid Most common polarizer Great for demos Characteristics of polaroid 0 Somewhat lossy To m 07 0 Low leakage e m 10 3 Work best for visible light o Cheap 1 for 5 cm square Important restriction limited to low power plastic can melt Don39t use with high intensity beams max I s 1 Wcm2 2O Birefringence Hecht 84 Best bolarizers based on birefingence Property of certain crystals Generally different directions not equivalent Possible crystal lattice y T dots 2 atoms 90 and y axes different Note 90 and y determined by crystal 21 In asymmetric crystal index of refraction n depends on direction of E If E along a then have n mg E k R If E along y then n my 22 All crystals have three basic symmetry axes for now label 96 y and 239 Call no ny n principle indices of refraction Three different kinds of crystals 0 isotropic no no 2 n2 not birefringent O UniaXial mm my n z axis special called optic axis optical properties complicated Question Can a liquid be birefringent 23 Focus on uniaxial Symmetry like a cylinder 90 and y interchangable Terminology call nony no ordinary index Call n no extraordinary index Common optical materials Calcite no 2 1658 no 2 1486 Quartz no 2 1544 no 2 1553 Other examples ice mica sapphire LiNbO3 24 What happens if k is not along a crystal axis Exa m p l e E axis X k d9 Al i gt y Light propagates along 239 E along 90 optic axis at angle 7 in xz plane Question What is index if E along y 25 For E along 96 get effective index neff 1 cos27 sin27 2 2 2 quoteff no quot6 If y 00 neff no if y 900 neff 2 n6 Otherwise neff between no and ne Derivation a bit hard won39t go through See Klein and Furtak 94 Probably cover in Phys 532 26 Upshot In birefringent materials n depends on polarization Simple polarizer Calcite prism axis L to page L and H polarizations have different n39s Deflected by different amounts Separate outputs with lens or free propagation 27 Example of a polarizing beam splitter polarizer with two outputs one for each state But not a good design Deflection depends on A Significant reflection from surfaces Large common deflection inconvenient Improve by putting two prisms together 28 Wollaston prism K Typical angular separation 15 200 Good performance Loss m 10 or 1 if AR coated Leakage N 10 5 Works at high power Various other designs see optics catalogs 29 Another method Gian Thompson Uses total internal reflection Again calcite with optical axis L page Choose prism angle so that nesine lt 1 lt nosine 6 400 works Then 0 Iight is TIR e light is transmitted Gap is too big for frustrated TIR 30 Phys 531 Lecture 3 1 September 2005 Light in Matter Hecht Ch 3 Last time talked about light in vacuum Maxwell equations gt wave equation Light 2 EM wave Today What happens inside material typical example glass Important for understanding lenses prisms etc Consider o effect on Maxwell Eqns o index of refraction 0 atomic model for index Next time another perspective on same question What is matter Collection of atoms atom 2 positive nucleus l negative electron cloud The atom 1010 m l More detail use quantum mechanics plan to avoid here So matter contains charges can39t set p J O in Maxwell equations 8B 8E Vsz i VXB J 7 at 0 ouoat Do we really need to know p and J exactly don39t care about phenomena at atomic scale On macroscopic scale charge current gt 0 Is there any macroscopic effect Yes can have macroscopic dipole moment T Electrons move in applied E Nuclei fixed Ar T displace cloud by Ar gives atomic dipole moment p qAr q net charge displaced Expect lArl ltlt 10 10 m ql N electron charge 6 Many atoms add up give macroscopic polarization P net dipole moment per unit volume If density N atomsm3 then P Np units P Cmm3 Cm2 How does P come into Maxwell equations Must be related to p and J try to see how Suppose Pr test volume V lTT lllmTT T T Tl l TVTll T T T T TLT T TT T T T Claim net Charge enclosed Q P c dS From Gauss39s Theorem Q VCPdV But know QpdV Conclude Question If a uniform electric field is applied to a glass cube as shown what is the resulting charge distribution Explain how it satisfies p V P E glass What happens if E is oscillating Also changing Pt gives current J Charges q moving at velocity v net current density qu So JquNq Ndip dt dt dt Example Understanding J An ionized gas has a density of 1010 moleculesm3 and car ries an average charge of 10 20 C per molecule The gas is flowing at a net speed of 100 ms How much charge passes through an area of 1 m2 in a time of 1 5 Solution Each 1 m3 of gas has charge 1010 molecules gtlt1O 20 Cmolecule 10 10 C One hundred cubes pass through test area in 1 s so net charge is 100 x 10 10 C 10 8 C Or J qu 1010 m 310 20 c1oo ms 2 108 Am2 11 Put in P Maxwell equations become 60VE VP VBO 8B 8P E V X E V X B at Oat60 08t Still need to specify P Expect Ar oc E therefore P oc E Write with X E electric susceptabiity dimensionless possibly complex Assumes steady state response Non linear optics P 60XE X2E2 X3E3 non linear function If E is small only linear term matters Characteristic scale electric field from nucleus E N 11 N 47r 0r2 N 10 Vm Corresponds to I N 1017 Wm2 We39ll assume I ltlt this Want more take Phys 532 next semester We have P 0XE Consider wave in infinite uniform medium Then X constant in space So eOVcEz VcP becomechEz XVcE Expect X gt 0 must have VcE 0 same as vacuum 8E 8P ButalsonBz 7 7 M0 at M060 at 8E becomes V x B quOX l DE Define electric permittivity e 60X l 1 Then Maxwell equations become B E Vsz a Vszeuoa 8t 6 Like vacuum but 60 gt 6 Still have waves but 1 0 speed c gt xwo v1 X Define xl l X n index of refraction then u cn ExpectXgtO s0ngt1 andvltc 0 Light is slower in medium than in vacuum Will see that39s not always the case Question Since the electrons are displaced and they have negative charge shouldn t we normally expect X lt O 16 Effect on plane waves Still need k 2 wv Frequency w doesn39t change SO k muc E nkro k0 vacuum wave number Wave number typically increases in medium Wave vector k nko Have A 27rk 2 mm A0 vacuum wavelength Wavelength typically decreases in medium Irradiance also changed Still 8 EO X B0 1A nA NOW BozikXEozikXEO U C So n A n A S lEol2k 2 lEol2k 770 21106 and TL 2 I E 277 O Model for index Hecht 35 o Index of refraction is important Could just measure for various materials but can we relate it to a microscopic model of atom Quantitative accuracy need quantum mechanics Get basic idea with classical approach Remember our atom model E Ar T Displaced cloud feels linear restoring force for small displacements Total force F qE nAr m 2 Ar dt2 n 2 spring constant m mass 20 Expect strong response 2 large n at w tag 2 Mrsm For simplicity take E polarized along 2 SO Ar gt 90 Differential equation i ng Et m Simple harmonic oscillator infinite response at too 21 Should also include damping force Differential equation becomes a 09b ng iEa m where a m Damped harmonic oscillator 22 Solve for plane wave Et Ema iWt look for solution xt woe m find 960 1 q 2 E0 wO in me 1 q 2 or generally Art 2 COO ZLUO39 w m Et Typical resonance response 23 Gives dipole moment pt qArt and macroscopic polarization Pt Npt N 2 1 7 1 21305 CU Pt 2 mwg l z39wa By definition P eOXE so Nq2 1 X 60m 418 iwa w2 o Predicts macroscopic quantity X in terms of mi croscopic quantities q m too a 24 Note X is complex for a 72 0 Really just our complex representation What does it mean n xl l X is complex write r7 as reminder So an l z m and kz ko Plane wave E Eoei Rm1k0 rWt1 Eoe njko re anO r wt Really E Edie quot1kmquot COS anO r cut I 5 25 Amplitude decays as wave propagates models absorption Comes from damping in atoms Usually write a gt n ii 2kg instead of Mg z39n1 Say 12 2 Then E EOe aZ26inkoZwtgt and irradiance is I LlEOlQe az 2770 So oz is absorption coefficient units m l I reduced by 16 in distance 101 26 2 Plot nw2 E 0 E u 15 8 E o a 8 1 10 E n 6 I D lt n l l l l l 1 2 3 0 1 2 3 who 2 0 mmresonancs resonance E X E g E E o g 5 10 n 539 V 39 V 39 V 39 V 39 l 10 a s 4 2 0 10 s e 4 2 o A resonance Resonance 27 Also see that n depends on w Resonance at w 2 too On resonance high absorption 0 bad for optics Good materials wo lies in UV o gives high frequency cutoff for transmis sron Below resonance n gt 1 and dndw gt 1 so 1 depends on w called dispersion More on this later 28 Quantum mechanics gives similar result 22 19 2 2 39 60m j wj w wag Xw Main difference Many resonant frequencies wj correspond to energy transitions Good optical materials no resonances in visible Weighting factors fj called oscillator strengths related to transition matrix elements 29 Index calculation has strange implications Since n nw wave velocity v vw o No longer have true wave equation o Non plane waves distorted in medium Artifact of steady state assumption Predict possible to have n lt 1 so u gt c o Meaning of v is tricky still can39t transmit info faster than c 0 But pretty strange Try to understand better next time 30 Phys 531 Lecture 18 25 October 2005 Fourier Optics Last time studied diffraction examples diffraction from an array diffraction from circular aperture Also gave important result get Fraunhofer pattern at focal plane of lens Today pursue this idea Fourier optics also discuss holograms Fanciest demos of whole course Outline o Lens as Fourier transformer 0 Optical filtering 0 Phase constrast imaging o Holography End of Fourier transform unit Next time start interference applications Optical Fourier Transform Hecht 132 Idea from end of previous class Suppose field Eacy0 Aay known Introduce thin lens at z so 0 H 506f z0 Then at back focal plane of lens 2 26 Exay380f with I 2 2s f C i zksofexp 2k Idea Lens maps 239 2 00 to back focal plane At 239 2 00 have Fraunhofer pattern regardless of so Allows observation of Fraunhofer pattern More important Gives access to Fourier transform A lLens Fourier transformerl Transform is not just a mathematical tool Lens sometimes called optical computer Useful if transform is what you really want Example Back scattering experiment sampIe beam splitter scattered light laser Shine laser on sample light scattered in all directions Angular distribution information about surface How to measure Use lens Focal plane all light at angle 6 focussed to 90 90 2 f6 Observe focal plane with camera gives 6 Note idea easy in ray optics Fourier connect 6 lt gt km So lens provides Fourier transform Limitations 0 bandwidth limited by lens aperture max km krDso o aberrations give errors worse for larger km o Hard to measure E in focal plane I only has lE not phase Still very useful Discuss some applications Optical Filtering Hecht 1323 Don39t just observe transform manipulate it System 1 1 1 Get Image of obJect at 7 7 7 52 f so gt inverts transform automatically Introduce filter in focal plane Characterize by transmission tay2 EOUtxa IX yEin33a sf l Example filter 2 circular hole radius a 1 x2 y2 lt a2 Then tay 0 else What is effect Point acy corresponds to kwky by Ol x y So aperture eliminates Fourier components with m 2 2 km kiy gt 7 Called low pass filter Generally low pass filter blurs image Need high k components to get sharp transitions Example 1D b Ia E Light with A 500 nm Object slit width b 40 um Image with lens f 100 mm Filter with slit width a 36 mm Incident field and transform 107 40 AX 20 057 0 00 200 100 0 100 2 0 3 2 1 3 X Hm k Hm391 X Filter cuts off ikxi gt ka2f 023 um l approx at first zero of A So after filter have 407 A lkx 1 o 11 kxium39 At image plane get inverse transform of A 2 image of A missing high km components A39x 107 Sharp edges smoothed out avoid oscillations with better filter function 15 Works just like low pass filter in electronics Applications Remove high frequency noise example dust on image Remove pixelation example digital image projector my Question What happens to the total power of the image when it is filtered Alternatively make filter 2 small opaque spot Disk radius a blocks components with k 13 k5 lt f Acts as high pass filter Not as directly useful as low pass filter but related to phase contrast imaging In slit example say filter 2 stripe Use width a 10 mm Cutoff km 2 063 um1 Blocks five central maxima Gives filtered fields 1n Iv 05 Ak x A39X 0 00 V V 10 i i i i i 05 i 3 2 1 o 11 2 3 200 100 o 100 20 k um X X Image bright at edges of object In general many types of filter possible If known artifact or noise in image design filter to remove Hecht figure 1334 1335 has nice examples Question Suppose one used a edge filter that blocked all kw lt O and passed all kw gt 0 What would be the effect on the image Demo Set up as in examples fll Ff l Object 1 2 wire grid array of square holes Object 2 circular hole Filter 2 circular aperture low pass or sphere mounted on glass slide high pass Cameras view transform plane and image plane 20 Phase Contrast Imaging Hecht 1324 Filters can let you see things otherwise invisible Suppose object is phase object Example 1D 1496 z39EO lxl ltb2 E0 lwlgtb2 1 Then 32 ilEOlQ constant IO 2770 Object not visible with conventional detector 21 Phase objects 2 variations in index of refraction w little absorption or reflection Simple example glass plate with 250 nm deep slot since 250 nm of glass gt 7r2 phase shift Real life examples bacteria in water density variations in water or air stresses in glass or plastic All hard to see normally 22 Phase objects have nontrivial transform Easy way to calculate Write Ax 2 E0 Ba Then 190039 1 lxl lt 52 Bac 1403 E0 O 33 gt b2 Like normal slit function amplitude Eoz39 1 Transform 3kx E0z39 1bsinc 23 Also transform of constant E0 is 27rE06lltx So Aac 2 E0 l Bac has transform Awm 2 E0 27r6kx l i Ubsmc Akx 24 In focal plane of lens 6 function gt Airy pattern spike width 122 AfD Width of sinc function 2 2Afb If lens diameter D gtgt slit width b central spike small compared to sinc pattern Suppose high pass filter size a m AfD Blocks most of spike negligible effect on sinc 25 kxb Then A km s E0z 1bsrnc transform of ordinary slit function So image looks like ordinary slit Am Eoltz 1 lxl lt W O 90 gt b2 Irradiance O lacl gt b2 Observe bright slit on dark background 26 Filter converts phase variation of object to irradi ance variation in image Called phase contrast imaging Understand idea from scattered waves scattered wave incident wave object Think of object as small transparent lens 27 Several types of phase contrast imaging High pass filter 2 dark field method also called dark ground background is dark Demo same setup as before Object 2 scotch tape pattern Similar idea use edge filter block 6 function l half of scattered field Called schlieren method Easier to align don39t need small dot Demo movie 28 Third method bright field Like dark field but different filter Instead of opaque spot use phase plate 2 glass plate with small phase spot z39 x2 y2 lt a2 1 ac l y gt a Suppose object field is Away EOelWay where 5 is phase shift due to object 29 Typically lt15 ltlt 1 so Away m Eoll i xyl Like in dark field example get transform Alltx kg 27rE06kx6ky iE0lt1gtkx kry where CD is transform of 5 3O If phase plate radius a ltlt width of ltlgt then filtered transform is viikm kry 27rz39E06kx6ky iEolt1gtkx kry Image gives inverse transform Axa with irradiance mm Ioil km s Ioil 2qgtacy 31 Looks like regular absorptive object Signal depends linearly on 5 Main advantage of bright field Dark field schlieren give signal oc 52 gt Bright field signal stronger for small 5 Disadvantage bright background Source of noise Not as pretty Bright field common in biology 32 Holograms Another neat diffraction trick Question Why is it so easy to tell the difference between an object and a photograph of the object Some points We percieve scattered light in either case Get same irradiance from each 33 To get 3D effect recreate complete scattered field Real object scattered object wave eye illumination 34 In some plane 239 0 have Eacy0 Aacy Could recreate field with filter tay oc Aay plane wave igt filter Since EOUtxa tEO OC 143 propagates as original E 35 How to make filter Easiest way is with interference quotobject wavequot photographic film reference wave 36 At film get interference pattern Etot Eobj Eref Record irradiance lEtotl2 lEobjl2 lErefl2 EobjEFef E bjEref Make t oc lEtotl2 Film contains information about both waves 37 To observe illuminate film with same Eref Then 1433 tayEref 1403 0 lEobleEref lEreleEref EgbjEref2 Eoblerefl2 Since Eref is plane wave Erele is constant So boxed term oc Eobj desired field Other terms waves propagating in other directions 38 film extraneous waves virtual image object wave reference wave Percieve 3D representation of object 39 Of course have demo Simple holograbh method requires laser Possible to avoid holograbh on credit card uses ambient light Laser method easiest to understand also works best 40 Phys 531 Lecture 11 7 October 2004 Survey of Optical Systems Last time Developed tools to analyze optical systems ray matrix technique thick lens picture Today look at several common systems Won39t use matrix methods explicitly but many lenses thick implicitly use thick lens picture Outline 0 the eye 0 eye glasses 0 magnifying glass 0 microscope These and more examples Hecht 57 Next time advice for designing systems of your own The Eye Hecht 571 lost basic optical system Components 7 cornea n 1376 itreous humor m water 17 133 7 lens 17 139 7 41 index varies high in center low at edges 7 iris variable aperture stop diameter 278 mm 7 retina detector Again use thick lens picture lens system has well defined focal length 3 Sclera Vitreous humor Optic Nerve Retina Note detector is in medium m 133 Lens equation becomes 1 ni1ni amp fo object focal length fi 2 image focal length 50 Irrelevant for ray matrix Modifies thick lens picture f0 applies to front focal point fi applies to back focal point System focal length variable max object distance 2 oo relaxed min object distance x 25 cm varies What focal lengths Distance from lens to retina m 24 mm m si Relaxed eye so 2 00 gt fi si 1 i7izgtfi22 mm 239 30 TL39 n For so 25 cm 1 8 2 So 1222 24 mm Called accommodation Most focusing power from cornea air interface n f2 n 1 Rs9 mm gt fim33 mm Remaining surfaces 1 1 1 N N ftotal fcornea frest So 1 1 1 1 E E E 88 mm for relaxed eye frest 66 mm for accommodated eye frest adjusted by squeezing lens muscles relaxed f long muscles tense f short Minimum achievable so 2 near point depends on flexibility of lens varies with age Question Why can t you see well under water Capabilities of the eye Resolution angular resolution A6 m 00170 03 mrad Just adequate to resolve crescent of Venus Correpsonds to about 5 um on retina At so 25 cm spatial resolution 2 SCAG 75 um Also wide field of view corresponds to 100 Mpixels Resolution best in center Sensitivity Fully expanded pupil can see I g 10 10 Wm2 from point source Power A Area 7r4 mm2 gt P s 10 14 W Maximum irradiance sunlight Is 250 Wm2 pupil area 7rl mm2 Max power 2 10 3 W But sun is not point source power spread out on retina Sun subtends angle 10 mrad m 30 x A0 Same intensity from point source illuminate area 302x smaller on retina m 1000gtlt higher image irradiance Max power from point source s 10 6 W m damage threshold for laser Dynamic range of eye 10 14 to 10 6 W eight orders of magnitude Instantaneous range lower five orders of magnitude Best artificial detectors photographic film high end CCDs dynamic range m four orders of magnitude 10x worse than eye Upshot Can39t build a detector nearly as good as the eye Eyeglasses Hecht 572 Common problem focal length of eye isn39t right Too strong 2 near sighted myopic relaxed eye has fi lt 24 mm 1 Somax 239 5239 so can39t focus at 00 Maximum distance of focus 2 far point Easy to measure For me far point z 25 cm Also moves near point closer for me 50min m 7 cm What is my range of f assuming si 2 24 mm 1 1 1 fmm 13370 mm 24 mm gt fmm 19 mm and 1 1 1 fmax 133 O 250 mm 24 mm gt fmax 22 mm Fix with eye glasses Relaxed eye 9 25 cm 6U Add lens to put image of 00 at 25 cm What focal length required want so oo 8239 25 cm So f 25 cm This is my prescription 1 D f 4 diopters How close is near point with glasses on so such that si 2 7 cm for f 25 cm 1 1 1 1 s 0E 10cm Other VlSlOn problems Far sighted hyperopia eye39s lens too weak Correct With positive lens Astigmatism asymmetry in lens f39s different along x y Correct With cylindrical lens Question If you want to start a fire With your glasses should you be neareslghted or fareslghted7 17 Magnifying glass Hecht 5 7 3 At 25 cm typical eye can resolve 75 pm Use a lens to see something smaller what kind7 Want erect magnified image of real object Real object so gt 0 Erect m 75150 gt 0 so 51 lt 0 Magnified m gt 1 so lstl gt le Have 1 1 1 5 0 5 f Want 150 positive and large lsi negative and small Means f should be positive Recall get virtual image with positive lens When 80 lt f Picture See image is magnified But also further away Resolution improved if image on retina is bigger 2O Note size of image on retina proportional to angular size of object W U Don39t really know f just consider oz Size on retina ozf 21 Define magnifying power of system MP 2 angular magnification oz with lens 2 oz without lens Write as MP 2 5x etc Could make oz without lens very big hold object right up to eye But can39t focus if so lt near point 22 For magnifying glass microscope etc not telescope Define oz for standard distance so 25 cm Example IfI take off my glasses near point is 7 cm Object at 7 cm subtends oz 2 y7 cm Object at standard 25 cm subtends a0 y25 cm 25 Magnifying power 2 36 010 7 My bare eyes have MP 2 36x 23 What is MP of glass iii Am a L Express in terms of practical parameters d 2 distance from eye to glass L 2 distance from eye to image f focal length of glass Have object size yo image size yi 24 39 m Angular size of Image oz 2 i L magnification m sisoi 80 L yo Without glass a0 do So MP 2 3 do so L 25 Eliminate so 5239 Have si 2 d L 1 1 1 and 50 5239 f So 1d7L1 so f f L Gives MPlt1 fdgtdfo 26 Two reasonable ways to use o Make L gt oo Achieve by making so gt f so si gt 00 View image with relaxed eye d doesn39t matter Get MP gtd70 Large MP for small f 27 Other method 0 Lens Close to eye d gt O MP gtd0 To get arge MP waht L sma mihimum L 2 near point 2 do Then MP gt1CO Large if f is small 28 Where do we need to put object 111 so f 8239 l i f d0 d0 f SOSOZdio MP Recall eye example MP 2 dOso same 29 Object looks as it would if you could focus at so Lens makes eye stronger like being near sighted Works well if you can hold object up to eye 30 Either method works up to about 4x f down to 6 cm For higher MP not paraxial lens aberrations important requires more complex lens Still works Method 2 Jeweler39s loupe Get MP up to 30x Impractical if object position fixed or MP so high you can39t hold steady Already a problem at 10x 31 Method 1 put lens very close to object since 50 f and f is small Problem exit pupil is small and far away can39t see very much Question Where is the exit pupil in this case Solution compound microscope 32 Microscope Hecht 575 Use two lenses objective short f close to object eyepiece collects light match to eye pupil MM lVVW e e iece objective y p 33 Typically don39t care about inversion object creates real inverted image Objective collects rays at steep angles important to control aberrations Eyepiece o puts final image at 00 view with relaxed eye o provides additional magnification 0 matches exit pupil to eye Aberrations less important than in objective 34 What is magnifying power Objective magnification siso Angular magnification not appropriate intermediate image not viewed by eye Even so there is a standard length scale Want objectives interchangeable standard position for object image Set by tube length 2 distance from back focal point to image 2 160 mm 35 Then si160 mm l f 11 1 50 f 5239 Mpz z l 80 f 160 mm l f 116O mm f f This is magnification written on objective So 20x objective has f 8 mm 36