Chapter 5 Geometrical Optics-Paraxial Theory

Transcription

1 Chapter 5 Gemetrcal Optcs-Paraxal Thery Hecht by YHLEE;100510; Intrductn An magng system Rays dverge frm S and cnverge t P P s a perfect mage f S An ptcal system n terms f wave: The bject space An ptcal system The mage space Cllectn Reshape f wavefrnt. f wavefrnt. The prncple f reversblty S s a perfect mage f P : Cnjugate pnts A real magng system: A pnt bject An ptcal system Blur spt (Stll an mage) Fnte extent. Dffractn Partal cllectn. The best pssble mage s called dffractn lmted λ 0 Rectlnear prpagatn f the lght : Gemetrcal ptcs N dffractn

2 5. Lenses A refractng devce that reshapes the wavefrnt. Hecht by YHLEE;100510; 5- A. Asphercal Surfaces Asphercs : Lenses and mrrrs wth nn-planar r nn-sphercal surfaces (Dffcult t make) Hyperbldal and ellpsdal refractng surfaces Perfectly parallel beam B. Refractn at Sphercal Surfaces Sphercal surfaces are easer t grnd than aspherc surfaces [p 153] A sphercal surface can have aberratns (Imagng errrs), but they can be crrected t make the system dffractn lmted. Refractn at a sphercal surface Optcal Path Length f a ray S A P : OPL = n l + n l 1 ( ) ( ) l = R + s + R R s + R csϕ ( ) ( ) l = R + s R + R s R csϕ Fermat s prncple dopl b g = 0 dϕ n l F HG n 1 ns ns + = l R l l 1 1 I KJ (1) Nte (1) Ths eq. s exact () Dfferent ϕ Dfferent A. Dfferent s, Dfferent pstn P Dfferent l, l. (NOT magng)

3 Paraxal rays : Rays nearly parallel t the ptcal axs Small ϕ and h csϕ 1, snϕ ϕ Hecht by YHLEE;100510; 5-3 Then (1) becmes n s n n + = s n 1 1 R : Independent f A (Perfect mage at P ) Ths s called Frst-rder ptcs, Paraxal ptcs, r Gaussan ptcs A perfect mage s pssble wth sphercal surfaces Devatns frm the paraxal apprxmatn (Ideal system) Aberratns The frst and secnd fcal lengths The frst fcal length r the bject fcal length n1 n n n1 + = s s R = f, = The secnd fcal length r the mage fcal length n1 n n n1 + = s s R =, = f A vrtual mage : Rays dverge frm t n the mage space A vrtual bject : Rays cnverge tward t n the bject space Nte s < 0, R < 0 Nte s < 0, R < 0 (Sgn cnvectn Table 5.1)

4 C. Thn Lenses Lens Smple lens Cmpund lens Thn lens Thck lens : It cnssts f tw r mre refractng surfaces. It has at least ne curved surface. : One element lens : Many element lens Hecht by YHLEE;100510; 5-4 Cnvex, cnvergng, r pstve lens Cncave, dvergng, r negatve lens Specal lenses Gradent-ndex lens (GRIN), Hlgraphc lens Thn-Lens Equatns At the frst refractng surface : n m nl nl nm + =, s 1 < 0, vrtual mage s1 s1 R1 nl nm nm nl At the secnd surface : + =, R < 0, s = s1 + d s s R Add tw eqs. : n m nm 1 1 nd l + = bnl nmg F I s1 s R1 R bs1 dgs1 A thn lens, d 0, n the ar, n m = F I + = n l s s R R b g : Thn-lens equatn r Lensmaker s frmula HG 1 KJ = : Gaussan lens frmula s s f The fcal lengths : 1 1 = = n l f f R R b g F HG HG 1 I KJ KJ +

5 Parallel rays at dfferent ncdent angles Dfferent fcal pnts n σ centered n C Hecht by YHLEE;100510; 5-5 Fr paraxal rays, σ becmes a plane near the ptcal axs Fcal plane In a lens, parallel paraxal rays fcus nt a secnd fcal plane, r back fcal plane. Smlarly the frst fcal plane r the frnt fcal plane s n the bject space. Fnte Imagery Imgng by a lens Three rays are used (1) A ray thrugh the ptcal center () A ray trugh the frst fcal pnt (3) A ray parallel t the ptcal axs

6 Hecht by YHLEE;100510; 5-6 A thn lens can be replaced by a plane Sgn cnventn y > 0, y < 0 Gaussan lens eq. Frm the trangles AOF and PPF 1 Frm the trangles SSO 1 and PPO 1 Frm (1) and () y y y y = f s f b g (1) s = () s = s s f Newtnan lens eq. Frm the trangles BOF and S SF 1 Frm (1) and (3) xx = f y y = b s f f g (3) x > 0 : The bject s t the left f F x > 0 : The mage s t the rght f F The mage by a thn lens

7 The ray parallel t the ptcal axs determnes the heght f the real mage Hecht by YHLEE;100510; 5-7 Nnlnear transfrmatn alng the axs The bject frm t F The mage frm F t F. Object mves alng the axs Axal change > Heght change Transverse magnfcatn : MT y / y Lngtudnal magnfcatn : M dx / dx = M L T Thn-Lens Cmbnatns [1] Ignre L fr a mment [] Rays 1 and 3 t fnd P 1 [3] Lne frm P 1 t O [4] Put lens L n O N change fr ray 4 But refractn f 3 by L [5] The mage by 3 and 4

8 Hecht by YHLEE;100510; 5-8 Analytc calculatns Analytc calculatns Fr lens L 1 : Fr lens L : = s s f =, where s1 + s = d s s f Cmbne the three equatns f d f s f / bs f s = d f s f / s f s 1, s, s s 1 b g g ( ) ( ) f d f1 = d f + f b b 1 f1 d f d f + f 1 g g : back fcal length : frnt fcal length When d=0, f. f. l = b. f. l. = f1f f + f = + : Effectve Fcal Length f f1 f The ttal transverse magnfcatn M = M M T T1 T

9 5.3 Stps A. Aperture and Feld Stps Aperture stp : It lmts the amunt f lght reachng the mage (The rm f a lens) Hecht by YHLEE;100510; 5-9 Feld stp : It lmts the sze f the mage (The edge f a flm) It determnes the angular extent f the bject, Feld f Vew. B. Entrance and Ext Pupls Entrance pupl : The mage f A.S. frmed tward the bject. Ext pupl : The mage f A.S. frmed tward the mage. The cne f EnP The amunt f lght enterng the system. The cne f ExP The amunt f lght leavng the system.

10 Chef ray Margnal ray : A ray frm an ff-axs bject pnt crssng the center f A.S It passes thrugh the centers f EnP and ExP. : A ray frm the axal bject pnt tuchng the edge f A.S Hecht by YHLEE;100510; 5-10 Vgnettng : Cne f rays becmes narrwer fr ff axs bject pnts. Image fade ut frm n-axs t ff-axs. C. Relatve Aperture and f-number The rradance at the mage ~ D / f f f-number s defned as f /# D : speed f camera lens f/1.4 at 1/500 sec = f/ at 1/50 sec :The same amunt f lght at the flm

11 5.4 Mrrrs Fr hgh reflectn Fr ultrahgh reflectn : Plshed substrate + Al catng + Prtectve catng. : Multlayered delectrc catngs. Hecht by YHLEE;100510; 5-11 A. Planar Mrrrs Same dstance frm mrrr t bject and t mage Inversn va reflectn Twce tlt f the reflected beam B. Asphercal Mrrrs C. Sphercal Mrrrs The same lens maker s frmula = = : f > 0 fr cncave mrrrs, f < 0 fr cnvex mrrrs s s f R

12 5.5 Prsms Applcatns : Beam spltters, Plarzng devces, Interfermeters, Measurng n( ω ). Hecht by YHLEE;100510; 5-1 Man characterstcs : 5.6 Fber Optcs Dspersn. Change n the mage rentatn. Change n the beam prpagatn. 5.7 Optcal Systems A. Eyes (1) A sngle centered lens : Vertebrates. A real mage n retna () A multfaceted cmpund lens : Insects. N real mage n retna Electrcal synthess n the nervus system (3) A smple lensless hle Structure f the Human Eye The eye s almst sphercal (4mm lng, mm acrss) Crnea : The strngest cnvex element n c = (water n w = 133. ). Crystallne lens : A layered fbrus mass Crnea + Crystallne lens (Duble-lens system) Fsh mves the lens tself. Shell fsh cntract r expand the whle eye. Brds f prey change the crnea curvature. Irs Aqueus humr Retna : Aperture stp (Eye clr, mm ~ 8mm) : Thck gel : electrchemcal reactns. Tw knds f phtreceptr cells n retna Rds : Hgh speed, peratn at dm lght, black and whte, sharp mage, Cnes : Lw-speed, peratn at brght lght, clr mage. Operatn wavelength 390 nm ~ 780 nm (310 nm ~ 1050 nm, fr sme peple) Macula Fvea centrals : Center f the retna. Twce as many cnes as rds. : 0.3 mm n da. at the center f macula. Rd-free regn. (The mage f full mn ~0.mm) The perceptn f an mage A cntnuus analyss f the tme-varyng retnal mage by the eye-bran system Fr detaled vew f mage The cntnuus mvement f eye ball Cntnuus shft f the mage acrss the phtreceptrs. N fadng ut f mage.

13 Hecht by YHLEE;100510; 5-13 Accmmdatn (The fne fcusng) The lens Lgaments Clary muscles. Near pnt Far pnt : The clsest pnt the eye can fcus. 7cm fr teens 1 cm fr yung adults 8~40 cm fr mddle-aged 100 cm fr 60 years f age : The mst dstant pnt the eye can fcus. Infnty fr the nrmal eye. B. Eyeglasses Dpter = 1 f : f n m = + a = a1 + a f f1 f The nrmal eye = 58.6a : 19=a frm the crystallne lens, 43=a frm the crnea) The secnd fcal pnt nt n the retna due t crnea, lens r eye ball length Farsghtedness, nearsghtedness, astgmatsm Cmmn cause

14 Nearsghtedness Parallel rays are fcused n frnt f the retna. The far pnt dstance s nt nfnty. Hecht by YHLEE;100510; 5-14 The fcal length f the crrectn lens = The far pnt dstance Mvng the far pnt t nfnty. Example An eye wth a far pnt f m = + f s s f = m (a=-0.5d) =, -, Brng bjects frm nfnty t wthn m

15 Farsghtedness Parallel rays are fcused behnd the retna. The near pnt dstance s lnger than the nrmal (5cm). Hecht by YHLEE;100510; 5-15 Example An eye wth the near pnt f 15cm = + f = 31cm (a=+3.d) f s s 5cm -15cm Astgmatsm Due t an uneven curvature f the crnea.

16 C. The Magnfyng Glass D. Eyepeces r Ocular E. The Cmpund Mcrscpe Hecht by YHLEE;100510; 5-16 F. The Camera The pnhle camera Well defned, undstrted mage ver a wde angular feld and a large range f dstances Great depth f fcus Great depth f feld [Fg ] [p. 17] G. The Telescpe 5.8 Wavefrnt Shapng A. Adaptve Optcs B. Phase Cnjugatn