MODERN OPTICS -1. Tutorial Solutions

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1 MODERN OPTICS - Tutorial Solutions Reision and General Optics All questions ere are general background optics wic will be assumed in tis course. Problem 3 requires tecnical details o te microscope wic most o you will require to look up in a standard textbook.. Simple Lens Use geometric optics and Snell s law to sow tat or a tin lens o ocal lengt te object and image plane distances are related by u + = I te lens is made o glass wit reractie index n sow tat =(n ) R + R 2 were R and R 2 are te radii o curature o te lens suraces. Solution Part A For a lens o ocal lengt, rays parallel to te optical axis will be ocused a distance beind te lens. Taking rays rom an object we get, u Object β α β α Image Tus so tat tanα = = Lens = u & tanβ = u = ) = u Tis is te magniication o te system. We ten ae tat tat = u ) = u( ) Department o Pysics and Astronomy Reised: August 2

2 MODERN OPTICS -2 Wic can ten be rearranged to gie u + = te expected ormula. Note: Tis is te real-is-positie expression or a lens, see text-books or alternaties. Part B Consider a lens wit radii R and R 2, made o glass wit reractie index n, R R 2 n Take a ray parallel to te optical axis at eigt incident on te irst surace. θ R θ2 I θ is small ten: From Snell s Law, we ae tat θ ß R n sin θ = n 2 sinθ 2 were, in tis case n = (air) and n 2 = n (glass). Now, again i bot θ and θ 2 are small ten θ 2 = n θ So te ray is deiated, downwards, by angle, α = θ θ 2 = θ = n R n Tis ray ten alls on te second surace, α R θ 3 2 θ4 Department o Pysics and Astronomy Reised: August 2

3 MODERN OPTICS -3 ten, i te lens is tin, ten ß,sotat θ 3 ß R 2 + α and rom Snell s Law (noting tat tis is a glass!air interace), θ 4 = nθ 3 So, on reraction, tis ray is deiated by a urter angle β = θ 3 θ 4 =( n)θ 3 = ( n) R 2 + α so wile passing troug te lens, te ray is deiated by te total angle γ = α+β = (n )» + R R 2 A ray parallel to te optical axis is deiated by and angle γ, tis will intersect te optical axis is distance beyond te lens, were γ ß were, rom Part A, is te ocal lengt o te lens. So we ae tat: so tat te ocal lengt is gien by: =(n )» + R R 2 =(n )» R + R 2 wic is known as te Lens Maker s Formula wic we will derie again rom scalar diraction teory in Lecture 4..2 Binocular, and Telescopes You are asked by a orse racing punter to supply tem wit a pair o binoculars tat will allow tem to read te starting prices rom te ar side o Musselburg race course, wic is approximately 4 m across. Te starting prices are written in calk in approximately cm ig letters. Speciy a suitable magniication, and sketc a possible optical system or te binoculars. Hint: Most people can easily obtain te minumum standard o ision required to drie a car (read a numberplate at 25 yards). Te optical system in a binocular is similar to a telescope, but it is olded wit a prism(s) tat also re-inerts te image. Department o Pysics and Astronomy Reised: August 2

4 MODERN OPTICS -4 Solution I you ae a driing licence you must be able to read standard car number plate at 25 yards (22.75 m). Tese letters are 75 mm ig so a reasonable estimate o te required angular size o readable letters is: θ = 3:3 3 Rad We will compare tis wit te diraction limited resolution o te eye later in te course. In tis case, te letters are cm ig at a distance o 4 m, so teir angular extent is δθ = 2:5 4 Rad to in order to see te letters easily, we need an optical system wit a magniication o θ = 3:2 δθ Te optical system in a binocular is just a olded telescope, so take te system o: d d 2 2 For a telescope, te angular magniication M = 2 so by similar triangles, te lens sizes are also gien by: M = d 2 d We need to matc te d to te size o te pupil o te eye. In normal dayligt te pupil is 2! 4 mm in diameter and to make te binoculars easy to use we require tat d > pupil diameter. d = 4 mm would be typical. Want to keep te optics simple (and tus ceap), so we must not be too sort. For example i ß 4d F No = 4 ten we would be able to use a simple doublet lens or te eyepiece. Aside: i < 6 mm ten a more complex optical system would be required to preent lens aberrations making te system unusable. Tis is discussed in more detail in Lecture 8. Sooramagniicationo3:2 we require 2 = 2mm and d 2 = 53mm Te binocular system is olded by use o a pair o silered prisms. Tis as two eects, a) re-inerts te image so it is te correct way-up wen iewed, b) reduces te lengt o te system to make it easier to old. Department o Pysics and Astronomy Reised: August 2

5 MODERN OPTICS -5 Prism (wit silered aces) 4mm 53mm Tis would be a pysically big and airly expensie pair pair o binoculars (see below). (Can also get roo-prism style binoculars tat ae te eyepiece and ront objectie in-line. Tese use a double doe prism and are more compact and rugged, oweer te optics are more complex and tey are ery muc more expensie to make.) Binoculars are speciied by Typical systems are Magniication Objectie size in mm 6 2 Compact pocket binocular 8 3 Standard Field Glasses 7 5 Low ligt leel binoculars 4 Bird watcers binocular 2 5 Horse racing / iewing sport 2 72 Bridge o sip or obseration post A binocular is te igest magniication tat you can easily and old. For larger magniications you need a support to keep ten still. Note: small moements in te binocular are also magniied. Optics.3 Microscopes A microscope objectie is marked 2; :35NA. Explain wat tese terms mean. Wat is te ocal lengt o tis objectie. Sketc an optical system to gie a 3 isual magniication, and calculate te required ocal lengt o te eye piece. Hint: Assume a tube lengt o 6 mm and a isual near point o 254 mm. You will need to look-up tese terms in a standard optics book. Te easiest explanation is in Hect s Optics page 9(is). Solution Tis solution contains a lot o background inormation about microscopes tat it not part o te course but people inoled in optics sould really know about! I an objectie is marked 2.35 NA tis gies te magniication (and tus ocal lengt), and eectie diameter o te lens as ollows: Numerical Aperture, (NA): I we ae a lens o ocal lengt and diameter d Department o Pysics and Astronomy Reised: August 2

6 MODERN OPTICS -6 θ d ten te Numerical Aperture is deined as NA = sin θ So tat NA = :35 means tat te al-angle θ = 2:5 i, or te ull exceptance angle o te objectie 2θ = 5 i. For most good microscope objecties θ is NOT small, and good quality, ig magniication objectie can ae NA ß :85 wic gies θ ß 3 i. For ery ig magniication work te eectie NA is increased by illing te space between te lens and te specimen wit oil o reractie index n. Te Numerical Aperture is ten gien by NA = nsinθ so wit typical alues o nß:5 it is possible to ae NA >. (.25 NA oil objecties are aailable at a price.) Te alternatie measure is F-Number F No, wic is deined by i θ is small ten we ae te relation tat F No = d NA ß 2F No Magniication: deines te ocal lengt wen used in a Standard Microscope wit a tube lengt o 6 mm. u Object Back Focal Plane t = 6mm Image Te tube lengt is te distance rom te back-ocal-plane o te objectie to te image plane. Magniication is M = u but we ae tat t =, so rom te lens ormula ) = u( ) u + = Department o Pysics and Astronomy Reised: August 2

7 MODERN OPTICS -7 so we ae tat = ut ) u = t = M so or a 2 objectie and a 6 mm tube-lengt te ocal lengt = 8mm. Te use o a standard tube-lengt allows microscope objecties to be moed between microscopes. Most microscope objecties are multi-element systems, but tey are all designed wit a standard location or te back ocal plane as sown: Back Focal Plane Working Distance Tis allows te objectie to be canged and te image to stay in ocus. Te oter important parameter is Working Distance wic is te pysical distance rom te object to te ront element. Tis can be as small as. mm. Microscope objecties come in a range o standard magniications, tese being: Magniication Typical NA Working Distance Focal Lengt 5 2 mm 32 mm 5mm 6 mm 2 2mm 8mm 4 :45! :65.5 mm 4mm Objecties wit iger magniication and longer working distance or a gien magniication are aailable but inole complex optics and tend to be ery expensie. Aside: Te standard tube-lengt was a standard tat as lasted or about years, and is still used on most low and medium quality instruments. Howeer te igest quality instruments ae ininity corrected microscope objecties were te image plane is at ininity. Tis as been introduced to allow easy insertion o polarisers, beam-splitter, intererence prisms etc., between te objectie and te image witout te need or re-ocusing. Howeer, just to make matters conusing, tese objecties are still marked wit te Mag notation, wic is te equialent magniication wen combined wit a suitable eyepiece. Objecties o tis type will ae te symbol beside te and NA. Simple Microscope: In addition to te objectie, an eyepiece is added wit acts like a magniier on te image plane. u e Object t = 6mm Image eyepiece Department o Pysics and Astronomy Reised: August 2

8 MODERN OPTICS -8 Te total magniication o te system is now M = M obj M eye Te M eye is deined wit respect to te Visual Near Point wic is te closest most people can iew an object wit te naked eye. Tis is conentionally taken to be (254 mm). I we iew a small object o size rom a distance D ten te angular size o tis object is θ n = D I tis same object is iewed wit an lens o ocal lengt e e θ e eyepiece ten te angular extent o te object is θ e = e Te angular magniication, seen by te eye, is now M eye = θ e θ n = D e were, by conention, we take D = 254 mm, and e is te ocal lengt o te magniier (or eyepiece). So now i we ae a objectie and we want a 3 oerall magniication we need M eye = 5 ) e = 7mm Usually eyepieces are multi-element systems to allow a wide ield o iew witout te eye being too close to te lens, but te principle is te same..4 Lens Power Somebody wo is myopic (sort sigted), tells you tat teir prescription is 3:75D in te rigt eye and 4:5D in te let. Wat are te ocal lengts o te teir lenses? Suggest a simple sceme to ceck tat tese ocal lengts are rigt. (You may assume tat you ae a single positie lens o ocal lengt mm tat you may wis to use). Solution Lens power is measured is Diopters, (D) and is deined as D = Department o Pysics and Astronomy Reised: August 2

9 MODERN OPTICS -9 were is te ocal lengt in meters. Sotat 3:75D 4:5D ) = 26:67cm ) = 22:2cm by conention, optalmic lenses are quoted in Diopters speciied to te nearest.25 D. Tis is because.25 D is te smallest cange tat can be perceied by most people. Aside: many people require a more complex lens tat includes a cylinderical component to correct aberrations in te eye. Bot o tese lenses are concae, (negatie). Te typical sape o suc a lens is: Sallow Conex Steep Concae Tis sape allows te lens to be closer to te eye, so extending te ield o iew. Also it looks better since te tick edge o te lens is idden beind te rame. A quick test o a positie lens is to orm te image o a distant object, wic will be located at te ocal lengt o te lens. Distant object eg. te sun Sarp image but tis will not work wit negatie lens since you will get a irtual image. I you ae a positie lens o know ocal lengt, in tis case = mm, so power is D. I two lenses are placed togeter ten te powers-add D D so te eectie power o tese two lenses is 6.25 D, so te ocal lengt = 6cm. Similarly wit te 4:5D lens, te eect power is 5:5D so te ocal lengt is = 8:8cm. You can ten measure te ocal lengt o te combinations using a distant object as described aboe. Tis is a rater crude metod. Te power (or ocal lengt), o a lens is usually measured by and imaging instrument called a posometer. Department o Pysics and Astronomy Reised: August 2

10 MODERN OPTICS -.5 A Tricky Puzzle A metal plate wit a ole on unknown diameter is placed at one end o an optical benc and a screen at te oter. Te metal plate is illuminated rom beind wit a ordinary ligtbulb and a lens is used to orm a sarp image on te screen. Te diameter o te image is 4.5 cm. Te lens is ten moed towards te screen until a scond sarp image is ormed, tis time o 2. cm in diameter. Wat is te size o te ole in te metal plate. Hint: Draw out te two conditions or a sarp image and remember tat object and image are intercangable. Yes tere is enoug inormation to sole tis. Solution Assume tat te radius o te ole in te plate is, so te irst condition or imaging is, u wereweaetat In tis coniguration te te object size is: u + = = u were = 2:25cm, being te radius o te irst sarp image. We now moe te lens towards te screen until we get te second sarp image, wic will be at, u were now we ae tat In tis coniguration te object size is, u = = 2 u 2 2 were 2 = :cm, being te radius o te second sarp image. Department o Pysics and Astronomy Reised: August 2

11 MODERN OPTICS - Now i we intercange object and image in te second coniguration (all rays pats are reersible), ten we see tat, 2 = u and u = 2 I we now substitute or u 2 and 2 is te aboe expression or we get te relation tat, so giing tat, u 2 = 2 2 = u = 2 u and tus We now take te square root o bot sides and inert to get, u = r 2 2 = 2 u 2 wic is now just te inerse o te magniication o te irst system! Substitute tis into te irst expression or to get = r 2 = p 2 Simply put in te alues o 9/4 and or and 2 being te radius o te two images, to get = :5cm wic is te radius o te object, so te diameter is 3 cm. Department o Pysics and Astronomy Reised: August 2

OPTI-502 Optical Design and Instrumentation I John E. Greivenkamp Homework Set 5 Fall, 2018

OPTI-502 Optical Design and Instrumentation I John E. Greivenkamp Homework Set 5 Fall, 2018 Homework Set 5 all, 2018 Assigned: 9/26/18 Lecture 11 Due: 10/3/18 Lecture 13 Midterm Exam: Wednesday October 24 (Lecture 19) 5-1) Te following combination of tin lenses in air is in a telepoto configuration: