Lens Conventions From Jenkins & White: Fundamentals of Optics, pg 50 Incident rays travel left to right Object distance s + if left to vertex, - if

Transcription

1 Len Convention From Jenkin & White: Fundamental o Optic, pg 50 Incident ray travel let to right Object ditance + i let to vertex, - i right to vertex Image ditance ' + i right to vertex, - i let to vertex Focal length meaured rom ocal point to vertex poitive or converging, negative or diverging r poitive or convex urace r negative or concave Object and Image dimenion + i up, - i down rom axi

2 Gauian Formula or a Spherical Surace The radiu o curvature r control the ocu Gauian Len ormula n n + n n r where n index on medium o light origin n index on medium entered r radiu o curvature o urace Clearly or ' ininite (parallel light output) then (primary ocal length) n n + n nr n n n n r

3 Thin Len Aume that thickne i very mall compared to, ' ditance Thi i oten true or large ocal length lene Primary ocu on let convex len, right concave Secondary ocu on right convex, let concave I ame medium on both ide then thin len approximation i

4 Baic Thin Len ormula Baic Thin Len ormula + Len Maker' ormula or calculating baed on len r, r & n ( ) r r n

5 Magniication and Thin Lene poitive or convex, negative or concave Magniication o a len i given by m M M ' Magniication i negative or convex, poitive or concave

6 Why i Light Focu by a Len Why doe the light get ocued by a len Conider a curved gla urace with index n on right ide Radiu o curvature r i centered at C Let parallel light ray P at height h rom axi hit the curvature at T Normal at T i through C orming angle φ to parallel beam Beam i reracted by Snell law to angle φ to the normal n in( ϕ ) n in( ϕ ) Auming mall angle then in(φ)~φ and n n in( ϕ ) in( ϕ ) or ϕ ϕ n n From geometry or mall angle h h in( φ ) or φ r r Angle θ the beam make to the axi i by geometry n n n h n n θ φ φ φ φ φ n n r n Thu the ocu point i located at h in h h r h nr ( θ ) θ n n n n Thu all light i ocued at ame point independent o h poition n

7 Simple Len Example Conider a gla (n.5) plano-convex len radiu r 0 cm By the Len Maker' ormula 0.5 ( n ) (.5 ) r r 0 0 cm Now conider a cm candle at 60 cm rom the vertex Where i the image ' cm M ' 30 Magniication m 0. 5 M 60 Image at 30 cm other ide o len inverted and hal object ize What i candle i at 40 cm (twice ) 0.05 ' 40 cm m Image i at 40 cm other ide o len inverted and ame ize ( cm)

8 Len with Object Cloer than Focu Now place candle at 0 cm ( < condition) m 0 ' cm Now image i on ame ide o len at 0 cm (ocal point) Image i virtual, erect and x object ize Virtual image mean light appear to come rom it

9 Graphic Method o Solving Len Optic Graphic method i why thi i called Geometric Optic Ue ome cale (graph paper good) Place len on axi line and mark radiu C & ocal F point Draw line rom object top Q to mirror parallel to axi (ray 4) Hit vertex line at T Then direct ray rom T through ocu point F and beyond Becaue parallel light rom object i ocued at Now direct ray rom object top Q through len center (ray 5) Thi interect ray 4 at image Q (point 7) Thi correctly how both poition and magniication o object Thi really how how the light ray are travelling Eg Ray through the ocal point F (ray 6) become parallel Interect ray 5 again at image Q

10 Thin Len Principal Point Object and image ditance are meaured rom the Principal Point Principal point H Location depend on the len hape H alo depend on a thin len orientation Note i you revere a len it oten doe not ocu at the ame point Need to look at len peciication or principal point Thick lene have eparate Principal point

11 Thick Len Formula A len get thicker optical urace may be not meet Len thickne t c (between vertex at the optical axi i.e. centre) Now len ormula much more complicated Ditance meaured relative to the principal point H or light coming rom the ront H or light coming rom the back o len ( n ) tc ( n ) + r r n r r Note imple len ormula aume t c 0 which i never true But i i large then r large and t c i mall o good approximation Note plano-convex r and thin thick but principal point change

12 Very Thick Lene Now primary and econdary principal point very dierent A ront vertex (optical axi intercept o ront urace) H primary (ront) principal point A back vertex (optical axi intercept o back urace) H econdary (back) principal point t c centre thickne: eparation between vertex at optic axi Relative to the ront urace the primary principal point i n A H tc r Relative to the back urace the econdary principal point i n A H tc r el eective ocal length (EFL): uually dierent or ront and back

13 Numerical Aperture (NA) NA i the ine o the angle the larget ray a parallel beam make when ocued NA ( ) in θ where θ angle o the ocued beam φ diameter o the len NA < are common High NA lene are ater lene NA i related to the F# F# NA φ

14 Combining Lene Can combine lene to give Combination Eective Focal Length e I many thin lene in contact (ie no pace between them) then + + e Two lene and eparated by ditance d To completely replace two len or all calculation New image ditance or object at ininity (eg laer beam) e + d or 3 e L Ditance rom irt len primary principal point to combined len primary principal point d D e + Ditance rom econd len econdary principal point to combined len econdary principal point d D Combined "thick len" extend rom D to D' e d

15 Combining Two Len Element Combined object ditance e e D Combined image ditance ' e e D NOTE: Combined object/image ditance may change ign The thick len ollow the tandard ormula e + Combined magniication e me e Secondary ocu ditance relative to nd len vertex i: e + e Note ome device (e.g. telecope) cannot ue thee ormula D e

16 Human Eye Human eye i a imple ingle len ytem Cornea: outer urace protection Aqueou humor i water like liquid behind cornea Iri: control light Crytalline len provide ocu Retina: where image i ocued Note image are inverted Brain programming invert the image

17 Human Eye Ditance Crytalline len to retina ditance 4.4 mm Eye ocue object up to 5 cm rom it Called the near point or D v 5 cm Eye mucle to change ocal length o len over.<<.44 cm Near ighted: retina to len ditance too long, ocued in ront Ininity object ocued in ront o retina: out o ocu at it When bring object cloer ocu move to retina Near ighted people can ee object with D v < 5 cm Far ighted: eye i too hort, ocue behind retina, D v > 5 cm

18 Magniication o Len Lateral change in ditance equal change in image ize Meaure change in apparent image ize m M y y

19 Magniication with Index Change Many dierent way o meauring magniication With curved index o reraction urace meaure apparent change in ditance to image Called Lateral Magniication r m + r m i + i image virtual, - i real

20 Angular Magniication For the eye look at angular magniication m θ M θ Repreent the change in apparent angular ize

21 Simple Magniying Gla Human eye ocue near point or D v 5 cm Magniication o object: ratio o angle at eye between unaided and len Angle o Object with len y y tan( θ ) θ D 5 For maximum magniication place object at len (in cm) y θ Thu magniication i (where in cm) θ 5 m θ e.g. What i the magniication o a len inch.5 cm θ 5 5 m 0 θ. 5 v

22 Power o a Len or Surace Power: meaure the ability to create converging/diverging light by a len Meaured in Diopter (D) or /m For a imple curved urace n n P r For a thin len P Converging len have + D, diverging - D eg 50 cm, D + D -0 cm, D -5 D Recall that or multiple len touching + + e Hence power in Diopter i additive D D + D 3 L L

23 Human Eye: A two Len Sytem Eye i oten treated a ingle imple len Actually i a two len ytem Cornea with n.376 make main correction Aqueou humor i nearly water index Len n.406 relative to aqueou humor Δn caue change Eye mucle hape the len and adjut ocu Cornea give 44.8 D o correction Len give ~8.9 D o correction Cannot ee in water becaue water index.33 near cornea Thu cornea correction i not there.

24 Eyeglae (Hecht 5.7.) Ue Diopter in glae Farighted, Hypermetopia: ocu light behind retina Ue convex len, +D to correct Nearighted, Myopia: ocu in ront o retina ue concave len, -D to correct Normal human eye power i ~58.6 D Nearighted glae create a revere Galilean telecope Make object look maller.

25 Anamorphic Lene Lene & Mirror do not need to be cylindrically ymmetric Anamorphic Lene have dierent characteritic in each axi Sphero-cylinderical mot common One axi (eg vertical): cylindrical curve jut like regular len Other axi (e.g. horizontal): ha no curve Reult light i ocued in horizontal axi but not vertical Oten ued to create a line o light

26 Atigmatim Atigmatim mean light i ocued in on axi not other Cylinderical len caue a Atigmatim: ocu in one plan In eye atigmatim caued by hape o eye (& len) Image i compreed in one axi and out o ocu Typically meaure D in both axi Rotation o atigmatim axi i meaured Then make len lightly cylindrical i.e. perpendicular to axi may have higher D in one than other eg. eyegla atigmatim precription give +D and axi angle +D i dierence between the two axi.

27 Ray Tacing (Hecht 6.) For more complicated ytem ue CAD tool Both are baed on Ray Tracing concept Solve the optical ytem by tracing many optical ray Ue exact urace poition & urace Do not make parallex aumption ue Snell law Eg.o program Z max, Code 5

28 Matrix Method in Optic(Hecht 6..) Alternative Matrix method Both matrix & CAD are baed on Ray Tracing concept Solve the optical ytem by tracing many optical ray In ree pace a ray ha poition and angle o direction y i radial ditance rom optical axi V i the angle (in radian) o the ray Now aume you want to a Tranlation: ind the poition at a ditance t urther on Then the baic Ray equation are in ree pace making the parallex aumption y y + Vt V V

29 Matrix Method: Tranlation Matrix Can deine a matrix method to obtain the reult or any optical proce Conider a imple tranlation ditance t Then the Tranlation Matrix (or T matrix) V y 0 t V y D C B A V y The revere direction ue the invere matrix 0 V y t V y A C B D V y D C B A V y

30 General Matrix or Optical Device Optical urace however will change angle or location Example a len will keep ame location but dierent angle Reerence or more len matrice & operation A. Gerrard & J.M. Burch, Introduction to Matrix Method in Optic, Dover 994 Matrix method equal Ray Trace Program or imple calculation

31 General Optical Matrix Operation Place Matrix on the let or operation on the right Can olve or calculate a ingle matrix or the ytem [ ][ ][ ] V y M M M V y object len image V y V y

32 Solving or image with Optical Matrix Operation For any len ytem can create an equivalent matrix Combine the len (mirror) and pacing between them Create a ingle matrix [ ] [ ][ ] [ ] D C B A M M M M ytem n L Now add the object and image ditance tranlation matrice [ ][ ][ ] object len image M M M D C B A 0 D C B A 0 D C B A ( ) D C C D C B A C A D C B A Image ditance i ound by olving or B 0 Image magniication i D m

33 Example Solving or the Optical Matrix Two len ytem: olve or image poition and ize Biconvex len 8 cm located 4 cm rom 3 cm tall object Second len biconcave - cm located d6 cm rom irt len Then the matrix olution i X D C B A X D C B A X D C B A X D C B A Solving or the image poition uing the matrix & X matrix: cm D B X or XD B B 0 + Then the magniication i D D m Thu the object i at cm rom nd len, -3 cm high

34 Matrix Method and Spread Sheet Eay to ue matrix method in Excel or matlab or maple Ue mmult array unction in excel Select array output cell (eg. matrix) and enter mmult( Select pace cell then comma Select len cell (eg mmult(g5:h6,i5:j6) ) Then do control+hit+enter (very important) Here i example rom previou page

35 Optical Matrix Equivalent Len For any len ytem can create an equivalent matrix & len Combine all the matrice or the len and pace The or the combined matrix A B [ M n] L [ M ][ M] [ M ytem] C D Table how how to calculate ocal, nodal and principal point where RP irt len let vertex RP lat len right mot vertex n index o reraction beore t len n index o reraction ater lat len

36 Example Combined Optical Matrix Uing Two len ytem rom beore Biconvex len 8 cm Second len biconcave - cm located 6 cm rom Then the ytem matrix i A C B D 0 0 Second ocal length (relative to H ) i cm C Second ocal point, relative to RP (econd vertex) A 0. 5 rp. 400 cm C Second principal point, relative to RP (econd vertex) A 0. 5 H cm C