Introductory Optomechancal Engneerng 2) Frst order optcs Moton of optcal elements affects the optcal performance? 1. by movng the mage 2. hgher order thngs (aberratons) The frst order effects are most mportant Snell s law for refracton ncdent n 2 θ 1 θ 2 n 1 refracted n θ = n 1 sn 1 2 sn θ 2 In ar n = 1.000 so sn n θ 1 = 2 snθ 2 Recprocty: Works the same from left to rght as rght to left, same comng and gong. Small angle approxmaton: n 1 θ 1 = n 2 θ 2 Expanson 3 5 θ θ sn θ = θ +... 3! 5! J. Burge Unversty of Arzona 1
Small angle prsm n ar devaton δ = α(n-1) α Defne Lne of Sght (LOS): Where the optcal system s lookng apparent mage LOS real object One easy way to determne the lne of sght s to magne that your optcal system s projectng lke a laser projector. Lght travels the same path n ether drecton. Your Lne of Sght wll be defned by ths magnary projected beam J. Burge Unversty of Arzona 2
A rgd body always has 6 degrees of freedom: Translaton n x, y, and z Rotaton about x-axs, y-axs, z-axs y θ y z θ x (x axs out of page) θ z Moton of thn prsm: The only moton that affects the lne of sght lght s θ z, rotaton about the optcal axs. J. Burge Unversty of Arzona 3
Rsley prsms Steer the lne of sght by usng rotaton of prsms Rotaton of one prsm moves LOS n a crcle Separate rotaton of a second prsm allows two-axs control of LOS (www.optra.com) φ 2 φ 1 δ = δ + δ 1 2 δx = δ cosφ + δ cosφ 1 1 2 2 δ y = δ snφ + δ snφ 1 1 2 2 J. Burge Unversty of Arzona 4
Plane Parallel Plate No angular change of lne of sght However, tlted plate causes a lnear devaton Approxmately: θ t θ(n-1) n y t t For glass, 45 tlt, y 3 J. Burge Unversty of Arzona 5
Plane Parallel Plate Focus shft n a convergng or dvergng beam Approxmately: t (n-1) t n t For glass, z 3 Does not depend on poston or orentaton J. Burge Unversty of Arzona 6
Plane Parallel Plate n a convergng or dvergng beam : causes aberratons W SA 2 t( n 1) = 4 ( f /#) 128n 3 W COMA W ASTIG 2 tθ ( n 1) = 3 ( f /#) 16n 2 tθ ( n = ( f /#) 2 2 1) 3 8n 3 transverse color longtudnal color t n = θ ( 1) n v W x λ 2 t( n 1) = n v W z λ 2 J. Burge Unversty of Arzona 7
Reflecton from a Plane mrror normal θ θr Law of reflecton θ = θ r Reflected ray n plane wth ncdent ray and surface normal In vector form: kˆ r = kˆ 2(ˆ k nˆ) nˆ J. Burge Unversty of Arzona 8
P Plane mrror creates mrror mage θ θ' P' J. Burge Unversty of Arzona 9
Image orentaton J. Burge Unversty of Arzona 10
R R J. Burge Unversty of Arzona 11
Moton of a plane mrror Tlt: LOS s rotated 2 tmes the mrror moton J. Burge Unversty of Arzona 12
Moton of a plane mrror Z translaton : mage moves two tmes mrror moton Z 2 Z J. Burge Unversty of Arzona 13
Moton of a plane mrror rotaton Three degrees of freedom do not matter translaton or J. Burge Unversty of Arzona 14
Postve thn lens, creates real mage Imagng systems f f = focal length object mage Choosng +o, + n drectons shown: o 1 f = 1 o 1 + magnfcaton y o y mage s rotated 180, mantans handedness m = = lateral magnfcaton y o o Object at nfnty, m = 0 Object at focal pont, m = -nfnty: Object at o = 2f, mage at = 2f, m = -1 y y shown as negatve here J. Burge Unversty of Arzona 15
If the object moves, how much does the mage move? For lateral moton, smply scales by magnfcaton Moton of y 0 n object space appears as y n mage space y y = o m What about moton along the axs: do d For axal moton, dfferentate: d do + = 2 2 o 0 d do = o 2 2 = m 2 Ths s often called the axal magnfcaton (Object and mage always move n the same drecton) J. Burge Unversty of Arzona 16
Focal rato Smple case stop at lens, object at nfnty f D θ f/number : F # = f D 100 mm focal length, 10 mm dameter lens -- f/10 Numercal aperture NA (n medum wth refractve ndex n) u=snθ NA = nsnθ 1 2F# Dffracton lmt: Wdth of Ary functon = 2.44 λ F# (FWHM = λ F#) Depth of focus : z = ±2 λ (F#) 2 MTF cutoff : f c = 1/(λ F#) J. Burge Unversty of Arzona 17
Postve lens m > 1 J. Burge Unversty of Arzona 18
Negatve Lens Image at Infnty Vrtual object at focal pont J. Burge Unversty of Arzona 19
Unfoldng systems wth mrrors convex J. Burge Unversty of Arzona 20
Lateral moton of lens We treat the case where the lens moves, yet the object and the mage plane do not. To calculate the amount of mage moton, smply sketch ths out. You can solve ths usng smlar trangles o = m New Axs, angle Image moves X X = X = X L L α = For object at nfnty, X L o α ( o + ) = X o + o ( 1 m) X = X L (Mrrors behave the same way) J. Burge Unversty of Arzona 21
Axal moton of lens We treat the case where the lens moves axally, yet the object and the mage plane do not. To calculate the amount of mage defocus, you need to be careful. Make a good sketch! Absolute mage moton = Lens moton + (Image moton relatve to lens) - z f Object at nfnty, m = 0, 1:1 conjugate, m = -1, (statonary pont) Be careful wth mrrors! J. Burge Unversty of Arzona 22
Tlt of optcal element Tlt an element about ts center, what happens to the mage? For thn lens- No sgnfcant effects (Large tlt cause aberratons) For Mrrors Mrror Tlt α Follow the chef ray!! J. Burge Unversty of Arzona 23
Moton of detector The detector could be flm, CCD, fber end, What we care about s moton of the mage wth respect to the detector. Ths moton would cause a blurred mage, trackng error, or degraded couplng effcency. If the mage and detector move together, the system performs perfectly. Moton of the detector has the same (but opposte sgn) as moton of the mage. Although pontng performance s defned by mage moton on the detector, t s usually not specfed n mage space where problem occurs, but t s referred back to object space. You must be able to go effcently back and forth between these two spaces: For object at nfnty, m = 0 x = m x o x = EFL α o Where α 0 gves the angle n object space. J. Burge Unversty of Arzona 24
Defnton of cardnal ponts project rays from object and mage space Image space PP 2 Object space RFP EFL BFD PP 1 FFP PP1: Front prncpal pont PP2: Rear prncpal pont FFP: Front focal pont RFP: Rear focal pont (mage of object at nfnty) EFL: Effectve focal length BFD: Back focal dstance J. Burge Unversty of Arzona 25
Nodal pont at rear prncpal plane α EFL * α EFL In ar, object at nfnty, nodal pont s concdent wth rear prncpal pont Rotaton of lens system about nodal pont does not move mage Smple proof (for mages n ar): Object at feld angle α has mage heght of EFL x α relatve to axs Lens rotaton α about PP 2 moves system axs at focal plane by EFL x α Lens rotaton α causes a fxed object to shft by angle -α relatve to axs The absolute mage moton s mage moton relatve to lens axs = EFL x -α + moton of lens axs + EFL x α 0, no moton Only for the case where the system s rotated about the rear prncpal pont. J. Burge Unversty of Arzona 26
Rgd body rotaton Rotaton about one pont on an object s equvalent to rotaton about any other pont plus a translaton. α rotaton about ths corner s equvalent to plus a translaton α rotaton about ths corner (Calculate the magntude of the translaton usng trgonometry) You can choose any pont you want to rotate about as long as you keep track of the translaton To calculate effect of rotatng an optcal system: 1. Decompose rotaton to a. translaton of the nodal pont b. rotaton about that pont 2. Image moton wll be caused only by translaton of nodal pont J. Burge Unversty of Arzona 27
Aperture stop Actual hole that defnes whch rays get through the system Margnal ray on axs ray that goes through edge of stop Chef ray off axs ray that goes through center of stop Defnton of pupls Entrance pupl Image of the stop n object space Located where chef ray cross the axs n object space Szed by margnal ray heght of pupl mage n object space Ext pupl Image of the stop n mage space Located where chef ray cross the axs n mage space Szed by margnal ray heght of pupl mage n mage space J. Burge Unversty of Arzona 28
Afocal systems Do not create a real mage -- object at nfnty, mage at nfnty D 1 D 2 α 1 α 2 D 1 = Entrance Pupl D 2 = Ext pupl It makes stuff appear larger magnfyng power MP = α α 2 1 LaGrange Invarant requres D 1α 1 = D2α 2 Examples: Gallean, Kepleran telescope, laser beam projector Bnoculars J. Burge Unversty of Arzona 29