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MIT 2 710 - Lecture notes

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2. (40%)Calculate and sketch the Fourier transformF(u) of the functionf(x) = sincxbcos2x 1+cos2x 2:Assume that the following condition holds:1b1 1;1 2;1 1;1 2:3. (30%)Avery large observation screen (e.g., a blank piece of paper) is placed inthe path of a monochromatic lightbeam(wavelength). Asinusoidal interferencepattern of the formI(x)=I01 + cos2x is observed on the screen, whereI0is a constant with units of optical intensity, is a constant with units of distance, andxis a distance co ordinate measuredon the observation screen.3.a)Describ e quantitatively two alternative optical elds that could have led tothe same measurement on the observation screen.3.b)Describ e an exp erimental pro cedure by whichwe can determine whichoneof the two alternative elds is illuminating the observation screen.GOOD LUCK2Goodman, problem 5-9 solutionMASSACHUSETTS INSTITUTE OF TECHNOLOGY2.710 Optics Fall '01Final Examination Wednesday, December 19, 2001: 9am{No onInstructionsa) Use the paraxial approximation in all problems.b) Use one-dimensional derivations unless it is clearly necessary to do otherwise.c) If you need to make assumptions b eyond those given in the problems, state themclearly.d) Lab el your sketches thoroughly and consistently with your derivations.e) Keep Occam's razor in mind: \All else being equal, the simplest solution mustbe the correct one."11. (45%)We intend to use a spherical ball lens of radiusRand refractive indexnas magnier in an imaging system, as shown in Figure A. The refractive indexsatises the relationship 1<n<4=3, and the medium surrounding the ball lensis air (refractive index = 1).1.a)Calculate the eective fo cal length (EFL) of the ball lens. Use the thicklens mo del with appropriate parameters.1.b)Lo cate the back fo cal length (BFL), the front fo cal length (FFL) and theprincipal planes of the ball lens.1.c)An ob ject located at distancedto the left of the back surface of the balllens, as shown in Figure A, whered=R4;3n4(n;1):Show that the ob ject is one half (EFL) b ehind the principal plane, and usethis fact to nd the lo cation of the image plane.1.d)Is the image real or virtual? Is it erect or inverted? What is the magni-cation?1.e)Lo cate the aperture stop and calculate the numerical ap erture (NA) of theoptical system of Figure A.1.f )Sketchhowahuman observer using the optical system of Figure A as inputto her eye would form the nal image of the ob ject on her retina.dRobject ball-lensFigure A22. (20%)Consider the 4F optical system shown in Figure B, where lenses L1, L2are identical with focal lengthf. A thin transparency with arbitrary transmissionfunctiont(x) is placed at the input plane of the system, and illuminated with amono chromatic, coherent plane wave at wavelength, incident on-axis. At theFourier plane of the system we place the amplitude lter shown in Figure C.The lter is opaque everywhere except over two thin strip es of widtha, lo catedsymmetrically around they00axis. The distance between the strip e centers isx0>a.2.a)Which range of spatial frequencies mustt(x) contain for the system totransmit any light to its image plane?2.b)Write an expression for the eld at the image plane as convolution oft(x)with the coherent impulse resp onse of this system.f f f finputplaneopticalaxisFourierplaneimageplaneL1 L2t(x)x’"xxFigure Bx0aa"yx"Figure C33. (35%)The p oint-spread function (PSF) of a one-dimensional (1D) optical systemin resp onse to mono chromatic, spatially incoherent illumination is~h(x)=sinc4ucx0;(1)wherex0is the output plane co ordinate.3.a)Sketch the modulation transfer function (MTF) of this optical system withas much detail as you can (but do not derive an explicit expression).3.b)Sketch the MTF and PSF of the corresponding diraction-limited opticalsystem,i.e.a diraction-limited optical system which has the same spatialbandwidth as the system describ ed by (1).3.c)An amplitude grating with transmission function as shown in Figure D isilluminated with mono chromatic, uniform, spatially incoherent light. Usingthe mo dulated light as input, derive the resp onse of the diraction-limitedoptical system.3.d)Using the results of the ab ove questions, discuss why the diraction-limitedsystem is sup erior to the system of (1) for the purp ose of imaging.xt(x)01......uc1c2u1Figure D (xis the input plane co ordinate)GOOD LUCK42.710 Final exam fall ’01SolutionsG. Barbastathis & D. GilThere are((iii)2.c) What is the intensity measured at the observation plane?2.d)Comparing your answers (a) and (c), how is T2 helpful in imaging the phaseobject T1?(Note: the symbols a, b are defined in Figure B.)3. Figure 3 below shows the schematic diagram of a simple grating spectrometer. Itconsists of a sinusoidal amplitude grating of period Λ and lateral size (aperture)a followed by a lens of focal length f and sufficiently large aperture. To analyzethis spectrometer, we will assume that it is illuminated from the left in spatiallycoherent fashion by two plane waves on–axis. One of the plane waves is atwavelength λ and the other is at wavelength λ + ∆λ, where |∆λ| ¿ λ. (Thetwo plane waves at different wavelengths are mutually incoherent.) Since the twocolors are diffracted by the grating to slightly different angles, the goal of thissystem is to produce two adjacent but sufficiently well separated bright spots atthe output plane, one for each color.’xoutputplanexf faΛgrating lens                                               Figure 33.a) Estimate the minimum aperture size of the lens so that it does not impairthe operation of the spectrometer.3.b)What is the maximum power efficiency that this spectrometer can achieve?3.c)Show that a condition for the two color spots to be “sufficiently well” sep-arated isλ|∆λ|<aΛ.This result is often stated in spectroscopy books as follows: The resolvingpower of a grating spectrometer, defined as the ratio of the mean wavelengthλ to the spectral resolution |∆λ|, equals the number of periods in the grating.32.c) What is the intensity measured at the observation plane?2.d)Comparing your answers (a) and


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MIT 2 710 - Lecture notes

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