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Proposals for the Solution of the Phase Problem in Electron Microscopy

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Proposals for the Solution of the Phase Problem in Electron Microscopy /"••"I •"•"/ ' -> i- PROPOSALS FOR THE SOLUTION OF THE PHASE PROBLEM IN ELECTRON MICROSCOPY P. VAN TOORN PROPOSALS FOR THE SOLUTION OF THE PHASE PROBLEM IN ELECTRON MICROSCOPY PROHISCHRll T TI'R VERKRIJGING VAN HI T DOCTORAAT IN 1)1 WISKINDI I-N NATUURWITl-NSCMAPI'I'N AAN 1)1- RIJKSUN1V1 RS1TI:IT Tl; GRON1NG1-.N, 01' Gl'ZAG VAN Dl- RI.CTOR MAGNIFICUS DR. J. BORGMAN IN III T 0P1 NHAAR 11 VI.RDI.DIGl N OP VRIJDAG 2 MAART 1979 DIS NAMIDDAGS TI 2.45 UUR j l'RI CII S door Peter van Toorn geboren te Tiel •J •> J J RECEPTIE NA AFLOOP VAN DE PROMOTIE IN HET UNIVERSITEITSGEBOUW STKLI.INCKN. 1. Indien in STEM bij bekunde eleeironenstroom op het preparaat dr elastisch verstrooide. «leci. ronen gedetecteerd worden in het «e- hele detcctievlak, krijgen we een beeld dat geïnterpreteerd kan «or- den analoog aan het beeld verkregen in de licht microscopie bij in- coherente belichting. 2. Ue directe metbode mot behulp van de Gregory intergraal is de meest aangewezen methode voor het numeriek oplossen van Volterra integraal vorgel ijkingen. P. van Toorn en H.A. l'erwerda, Optica Acta, 2_J, 4f.9, (1976;. L.G, Kelly, Handbook, of numerical methods and applications (Addison- Wesley), pp 54, 252, (1969). 3. Het oplossen van het fase probleem met behulp van de algorithmen volgens Misell en Gerchberg/Saxton moet worden ontraden. Uit proefschrift, hoofstuk 2 en 3. 4. Een hoge waarde van de sferische aberratie constante, kan gunstig zijn voor de beeldvorming in de electronenmicrosconie. 5. Het verdient aanbeveling bij de publicatie van foto's gemaakt met een transmissie electronen microscoop, met name bij hoog scheidend ver- mogen, niet te volstaan met een gedeeltelijke vermelding van de be- lichtings omstandigheden en microscoop parameters. 6. Ondanks het hoge contrast in het beeld is de toepassing van donkerveld technieken zinloos in vergelijking met helderveld technieken. Dit proefschrift, hoofstuk 5 en 6. 7. De in veel leerboeken impliciet of expliciet gemaakte veronderstellingen dat dissipatieve krachten niet af te leiden zijn uit een gegeneraliseerde potentiaal is in ieder geval voor één dimensie onjuist, K. Courant en D. Hubert, Methoden der Mathematischen Physik, (Springer), p. 219,(1968). H. Goldstein, Classical Mechanics, (Addison Wesley), p. 21, (1977). L.D. Landau, E.M. Lifshitz, Mechanics (Pergamon) p. 76, (1960). L.A. Tars, Analytical Dynamics, (Heineman) p. 179, (1965) 8. Ui* veelvuldig gemaakte fout de bej-.rinpen st erii's.Mpi s< in- w.t.irnetui ni^ en drie dimensionale reconstruct it- mot elka.ir 11- verwisselen, leidt ten onrechte tot do opvatting, dat uit één huUn'.ram de niimulijke structuur van een voorwerp bepaald kan worden zonder n nriori infor- matie. 9, Daar Europese lijnvlucliten in du regel een IKIRV besett int'scm.id hebben, dient de prijs voor do vliegtickets verlaagd te worden in overeenstemming met de prijs voor de tickets op de t ransal tant isrht? route. 10. De uitspraak dat reform voedsel geen chemische bestanddelen bevat, is scheikundig onjuist. 11. Het optreden van filevorming op de rechter rijstrook van een twee- baans snelweg bij de passage van een Porsche van de rijkspolitie is uit overwegingen van verkeersvPiligheid ontoelaatbaar. Stellinpen behorende bij het proefschrift van P. van Toorn, Proposals for the solution of the phase problem in electron microscooy, Groningen, 2 mai,rt 1979. Voorwoord Hierbij wil ik allen bedanken, dio op enigerlei wijze hebben meegewerkt bij het tot stand komen van dit proefs-ciiri ft. Voor de kritische begeleiding bij liet srhri jven van dit proefschrift en do prettige samenwerking gedurende de afgelopen jaren ben ik mijn promotor Dr, 1). A. l'erwerda zeer erkentelijk. Tevens wil ik Prol". Dr. Ir. II. .1. Krankena bedanken voor x.ijn bereidheid om als coreferent te fungeren en voor het geven van opbouwende kritiek op het mamiskript. Speciaal wil ik liernhard Hoenders en André Iluiser bedanken voor onze samenwerking en voor de vrijwel dagelijkse discussies over ter zake doende fysische onderwerpen on allerlei discussies over andere onderwerpen. Elli Boswijk en Sietske Lutter ben ik erkentelijk voor het ontcijferen van het handschrift en het vele typewerk dat zij verricht hebben. Tenslotte > il ik Oomke Schutter en Henk Hron bedanken voor het verzorgen van de tekeningen. Contents Glossary. Introduction. I Chapter I: The phase problem in electron microscopy. 5 1.0 Introduction 5 1.1 The object wave function 6 1.2 The formulation of the image formation in electron 8 microscopy 1.3 The phase retrieval problem 13 Chapter II: On the solution of the phase problem using image 19 intensity distributions determined for different defocusings of the microscope. 2.0 Introduction 19 2.1 Theory 19 2.2 The direct method for the one dimensional case 21 2.3 The extension of the direct method to a rectangular 24 aperture 2.4 Verification of the direct method using simulated objects 27 2.5 Discussion of the direct method 30 2.6 Misell's algorithm 31 Chapter III: The solution of the phase problem using the inten- 34 sity distributions in the image plane and exit pupil. 3.0 Introduction 34 3.1 The direct method 34 3.2 The selection procedure 38 3.3 Reconstruction of the pupil wave function using the direct 41 method 3.4 Discussion of the direct method 45 3.5 Gerchberg-Saxton algorithm 46 Chapter VI: The phase problem in the case uf weakly scattering 49 objects. 4.0 Introduction 49 4.1 Theory 49 4.2 Some comments on the weak scattering approximation 51 4.3 Iterative algorithms for the reconstruction ol" the pupil 53 wave function 4.3.0 Introduction 55 4.3.1 Sampling procedure 55 4.3.2 The sampling procedure for semi-wc.ik objects 57 4.3.3 Reconstruction of the pupil wave function from two 58 defocused images 4.3.4 The weak phase approximation 60 4.3.5 Iterative algorithm for the reconstruction of semi- 61 weak objects from two or more defocused images and the intensity distribution in the exit pupil Chapter V: The possibilities of applying phase retrieval algo- 66 rithins in electron microscopy. 5.0 Introduction 66 1.I Possibilities for applying direct methods in electron 67 microscopy 5.2 Possibilities for applying the iterative algorithms to 69 semi-weak objects in electron ricroscopy 5.2.0 Introduction 69 5.2.1 Simulations of noisy intensity distributions and 69 the reconstruction using the iterative algorithms for semi-weak objects 5.2.2 Conclusion 78 Chapter IV: The phase problem for different operation modes of 81 a transmission electron microscope. 6.0 Introduction 81 6.1 The phase problem in CTEM and STEM 8! 6.1.1 The imaging in CTEM 81 6.1.2 The imaging in STEM 83 6.1.3 Some special operation modes of CTKM and STEM 85 6. I.A Inelastic scattering ')- 6.1.5 STEM or CTEM 9 3 6.2 Dark-field imaging and selective diaphragms 96 Summary 101 Samenvatting 10 3 ^ : tlu' w.'ivc I i'ii>;t II. k : the wave number, o x : tin- coordinate in tin1 object p I .un- /-•/. , O I' Treasured in units É. X : lin1 coordi nat o in tin' im.icr plane •"=•'., measured in units M'fM is the m.inii t i c.it ion). : the ('onrilin.it L1 in tin' exit pupil y = z , measured in units (îor.i I length). x : tin1 riHirdi nato in tin' Muiri'c p I .mi1, IIK-.ISII n-J in units • (d i s t ance between source pi.me .itul object plane), x : tbe coordinate in the detect ion piane,measured in units • (distance between source plant' and detection plane). I' (x ) : the wave function in the object plane z = z . o o o PC") : the wave function in the exit pupil 7 = 7. U. (x ) : tlu' wave function in the image plane 7.-7... 'Mx ), !>('.), 'f:(x ) : the phnsc of the wave functions in the planes z=z , z = z and z = 7.., respect i ve 1 v . op i D(x ,x.) : the transmission function ot tbe microsccpe. 01 X(^) : tbe wave aberration function. 6z : the di'focUHing. C : the constant of spherical aberration. s C . : the constant of chromatic aberration, c A : i,k A ='k (<Sz,-(Sz,). " o 7. o 2 1 O : the extent of the object plane, a : the extent of the exit pupil. In one lateral dimension: -Ivf,•'(">. P(O : the scattered wave function, s : the scattering percentage. I.(x ) : the intensity distribution in the image plane. (j labels the different .stages of defocusing). g(O : the intensity distribution in the exit pupil. f.(O : the shifted Fourier transform of l.(x.). J J 1 I (x ) : the intensity distribution in the .sourft- plane, s s V(x ,y ,z) : the object potential distribution. T(x ) : tlie complex transmission function of the object. T (O, T(f.) : the Fourier transform of T(x ) with: p o T (r.) = cl»(') ^ T<;). P F (x ,x!) : the mutual intensity distribution in the object o o o piane. r(x.,x!) : the mulu.il intensity distribution in the ima^e plane. C , (C+B) : the phase contrast function, ph C (£+_) : tlie amplitude contrast function, am — r D : the dose. The aim in electron microscopy is In obtain 1 rLMn electron micro- graphs information about the objectstruclure. 'I he question arises then how those micrographs are related to the obje<t structure. An incident wave will be modified, when n,is:,in^ through ,1 three-di- mensional object. The modified wave function in a plane /. = / behind the object (see l'ig. I. I of Chapter 1) will be called the object wave function I) (x ).
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