/"••"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 ( 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 pupil z=z and the wave function l-'-OO in the image plane z=Zj can be expressed in terras of the unknown I! (x ). On the other hand U (x ) v o —o o —o can be calculated from the knowledge of P(\) and L'.(x),while with the knowledge of U (x ) the object might be reconstructed. The intensity distributions in the exit pupil and in the iraa^e plane are measurable quantities. Since the intensity distribution equals the squared modu- lus of the wave function, the phase of the wave function remain un- known in electron microscopy. We have to deal with two problems: Firstly, to try to calculate the unknown I'CO or l'.(x,), and from 1 I these subsequently the function U (x ), from the above mentioned in- tensity distributions. This problem is referred as the phase problem. The second problem is the reconstruction of the object from U (x ), which is referred to as the inverse scattering problem. For a better understanding of the last problem we have to discuss the scattering of the electrons by the object. The incident electrons, which have a high energy E(E>50keV) will 'see' the object, as if it were represen- ted by a three-dimensional (electrostatic) potential distribution. The amplitude attenuation (by absorption) anci phase shift of the in- cident wave describe mathematically the interaction of the incident electrons and this potential distrubution. The knowledge of only U (x ) is not sufficient to reconstruct the three dimensional object. This can be understood using a simple argument: A function U (x ) of two coordinates can, in general not contain enough information to recon- struct from it a function of three coordinates like the potential dis- tribution. In principle, however, it is possible to calculate the po- tential distribution from a set of different wave function I' (x ) . o —o obtained by choosing different illuminations (e.g. incident waves with different propagation directions | 1 |. In thir, thesis we do not discuss the inverse scattering problem | 1 | . We shall confine our considerious to the phase problem. Before we discuss the phase problem in electron microscopy in more detail, the question has to be answered how important it is (from a physical point of vieuw) to solve this phase problem. This means that we inquire, how much relevant information one can obtain from the micrographs without further (numerical) processing. Sometimes, con- trast can be (partially) assessed visually. Consider an object which is known to obtain heavy atoms, (we assume some a priori knowledge.'). The incident electrons will be scattered in various angles. These an- gles can be so large, that the electrons do not contribute, to the image, because they are intercepted by diaphragms. Consequently we do not observe these electrons. Their absence, give rise to scattering contrast in the image. From that, we still can obtain some l rough) information about the object structure, because those areas of the object, which contain the (heavy) atoms, from which the electrons are scattered over large angles, will be observed as spots on the micro- graph. We have to be very cautious, however, with a visual interpre- tation of the micrographs. It is not necessary that the contrast in the image is only due to scattering contrast. Hanssen I 2) has shown experimentally, that parasitic and spurious structures may be obser- ved. Of course, these structures depend not only strongly on the ob- ject, but they also depend on the choice of the illumination, on dia- phragms and on aberrations, including the defocusing. In order to overcome now the problems connected with the visual interpretation of the micrographs further (numerical) processing of the micrographs will be necessary for a reliable object structure determination. The solution of the phase problem is then the first obstackle to be over- come. The phase problem arises in all kinds of scattering physics, like nuclear scattering, coherence theory, x-ray and electron diffraction and, historically for the first time, in astronomy (Michelson's stel- lar interferometer) I 2]. The main complication when solving the phase problem is, that it is rather difficult to obtain different intensity distributions, from which the phase problem is uniquely solvable. (The phase retrieval problem, i.e. the problem of calculating uniquely the unknown argument of a complex function from its known modulus only cannot bo solved in general). Tn optics with visable light, it is possible to ,-ipply "pseudo" holographic procedures for solving the phase problem. Examples are interference microscopy and phase contrast microscopy. Unfortunately those techniques require beam splitters, |\-plates, which are difficult to vnanufarture for application in an electron microscope. Problems may also arise due to electric charging of the diaphragms by the reference beam. As we shall see in this the- sis it still is possible to solve the phase problem in electron mi- croscopy by using specially chosen illuminations, diaphragms and defocusing. In the first Chapter the significance of U (x ) in connection with o —o the object structure will be discussed. The mathematical formulation of the imaging process for coherent illumination will be given, in order to find the relations between the wave functions in the various planes of a microscope. Next we derive the relations between l'(") and the measured intensity distributions. In the last section of Chapter I a general mathematical discussion will be given of the phase retrieval problem. In Chapters II till IV several algorithms will be developed for solving the phase problem. After the evalua- tion of the different algorithms in Chapter V, we discuss in Chapter VI the applicability of those algorithms in transmission electron microscopy (TEM)• The partial coherence of the illumination beam will be taken into account. Bright-field and dark-field imaging will be compared. References I 1 J See for a survey of the inverse scattering problem: Hoenders, 'J.J., The inverse scattering problem in optics, (ed. H.P. Baltes, Springer), Chapter fll, (1978). I 2 ] For a survey of the phase retrieval problem in physics, see e.g. Kohler, P. and Mandel, L., J.O.S.A., 63, 126,(1973). I 3 | Hanssen, K.J., Image Proc. and comp. aided design in olfi-tr. optics, (ed. P.W. Hawkes, Academic Press), Manihestor, (197:2). See also Hanssen's comments conrorning dark-field ira.i);in>;: Hanssen, K.J., Optik, A6, 107, (1976). CHAPTER I THE PHASE PROBLEM IN ELECTRON MICROSCOPY 1.0 Introduction Plf) specimen, i i holder " ..lens system" • Z QXIS r ~R exit pupil image plane z=z. Fig. I.I: Scheme of imaging system. In the introduction of this thesis it has been mentioned that the phase problem in electron microscopy has to be solved in order to ob- tain the complex wave function from the measured intensity distribu- tion. The determination of this complex wave function is necessary in order to reconstruct the object I I I (see also Section I.I), in Section 1.3 it will be shown that the unknown phase of a wave function in a plane perpendicular to the symmetry z-axis of a microscope (sec fig. I.I) cannot be determined uniquely from its modulus. The unknown com- plex wave function in the exit pupil ?. = ?. , defined as the pupil wave function P(O, is related to the complex wave function in the plane z=z. (image wave function) and the wave function in the object plane z=z (object wave function). The aim is the determination of this ob- o ject wave function U (x ) or of tin.1 wave function l'('J. In Section ° o —o 1.2 relations will be derived between the unknown I1 (' J and the inten- sity distributions in the exit pupil and image plane. We shall develop in the following chapters algorithms for solving the phase problem in (electron) microscopy, based on these relations 1.1 The object wave function In electron microscopy, the coherence properties of illuminating beams can reach a level which is sufficient to meet the requirements for our calculations (see Chapter VI). We therefore consider perfectly coherent illuminations (not necessarily a plane wave only). It.is to be expected that due to the interaction of the incident electrons and the object, the complex wave function in a plane just in front of the object and the wave function in the plane z=z just behind the object will be different. The incident electrons will interact with the elec- tric field, caused by the charged particles of the object. We repre- sent the object by an electrostatic potential distribution V(x ,y ,?.) , o a where x and y are the lateral coordinates in the plane z=z . o o o For the Schrodinger equation, we obtain I 2 ) : -^ V2* I|I(X ,y ,z) = explik z + iu(x ,y ,z)| (1.1.2) Here, a(x ,y ,z) describes the disturbance of the incident plane wave by the object. Inserting (1.1.2) into (1.1.1) and using the relation (21 •> ') E = -~ '.1.1.3) !m we obtain: h2 2 -> ''^o » 2£Iiv~l(wz)-{Vni(wz)ri" ~"^ i(vyo-z)"eV(wz)- (1.1.4) If E is large enough, so that k is large, we have I 3 1: (1.1.5) In that case (1.1.4) reduces to: In the planv e z=zo ,' ue obtain for u(xo ,yo , zo ): V(x ,yo,z)dz (1.1.7) We have now calculated the phase shift (see (1.1.7) and (1.1.2)) caused by an object represented by an electrostatic potential distribution (see also I 4 ]). This result can be used for describing the object wave function U (x ,y ), being the wave function in the plant z=z . In ge- neral we can write: U (x ,y ) = U. (x ,y )T(x ,y ), (1.1.8) o o Jo me o"o o o where U. (x ,v ) is the undisturbed incident wave in the nl.ine z = z . inc o'-'o ' o T(x ,y ) is complex transmission function, describing the tlist\irbance of the incident wave: T(xo,yQ) = cxpl-£(xo,yo)+ i u(X(),y()) |, (1.1.9) in which fx ,y ) is the amplitude at Lenuat ion and "(x ,v ) tlie phase o J o o o ' shift by Llie object. Since the object is represented by an electrosta- tic potential distribution, the phase shift is related to the projec- tion of V(x ,y ,z) I 5 I on the plane Z=ZQ. According to (1.1.7) (f(x ,y )=0, we obtain: z o T(x ,y ) = expl-V" / V(x>y.;Odz (1.1.10) 00 li-k -«• We shall not discuss the inverse scattering problem, which is in our case the problem of obtaining the potential distribution from T(x ,y ) I 1 1 . We restrict the discussions to the problem of the re- construction of U (x ,y ) from one or more measured intensity distri- butions. In the next section, relations will be derived between P(O and the intensity distributions, so that P(O can be calculated from the measured quantities in electron microscopy. 2.1 The formulation of the imaye formation in electron mieroscovy A quantity of interest is the object wave function, i.e. the wave function in the plane z=z . Other wave functions of interest are the wave function in the exit pupil z=z and U.(x ), the wave function in the image plane z=z.. As both Maxwell's equations and Schrodinger's equation are linear as long as the materials involved are linear, we have in optical as well as in electron microscopy a linear relation between the image wave function U.(2C.) and the object wave function U (x ) i 6 | I) o o is the aperture area in the objoelplane /.=/. and l)(x ,x.) is the transmission function of ttie microscope I (•> |: D(x ,x ) = J expli k xU ,O |exp[-?'.if,.(x.-x )|d-., (1.2.2) •—o — l o ~-o — — — i —u — where o indicates the aperture in the exit pupil and ,. (x ,f) is the wave-aberration function. ^, C,,n), denotes the coordinate system in the exit pupil measured in units F (focal length of the imaging system. The position vector x is measured in units of wave length A (vacuum) (A=2ir/k o ) and —x tl, and therefore the coordinates (x,,y.II) in the image plane, are measured in units MA, where M is the lateral magnification of the optical imaging system. M equals to R/F, where R is the radius of the Gaussian reference sphere [ 6 ]. In the paraxial approximation (certainly valid in electron microscopy), R is approximately the dis- tance between the exit pupil and image plane (see Fig. I. 1). We assume that the imaging system is isoplanatic, which means that x(—xo ,£— ) depends on ~£~ only. In electron microscopy the main aberrations are the spherical aberration and defocusing (the latter introduced deliberately). Then x(.£) becomes [7] 2 2 2 2 X(i)=x(t,n) = " -fU ^ ) + ^(f, +n ), (1.2.3) where Sz is the defocusing and cg the coefficient of spherical aberra- tion. Substituting (1.2.2) in (1.2.1), we obtain the following expression for U£(x,) 18]: U.(x ) = J P(Oexp(-2irU.x,) P(O= Jdx U (x )|i k x('J lexp(2"ix . >.). (1.2.5) — —o o ™o o — —o ~ a o Using (1.2.4) and (1.2.5) we have found now the re hit ions between the pupil wave function P(p, the image wave function "-(^i) and t-he object wave function U (x ). If we know ]'(•',) or II. (x.) we ran calculate U (x ). o —o — l -1 o —o However, not the wave functions are measurable quantities, but the cur- rent density. The current density .1, |9|(tho intensity distribution) in the plane z=z or z=z. is related tu the wave functions in those p l planes (paraxial approximation). 2 2 Vxo,yo,Z)|z=z =cU(xo,yo,Z)| z=z (.J/m ) (..2.6) P P z=z. z=z. 1 1 where C is a constant from which J acquires a suitable dimension, 2 e.g. J/m . Measuring the intensity distribution in the exit pupil g(£) or the intensity distribution in the image plane Hx,), we obtain the following relations: g(O=P(OP*(p, (J/m2) (1.2.7) 2 I(x.)=U.(x.)U*(x]). (J/m ) (1.2.8) — i i — i l — i In Section 1.3 it will be shown that, in general, it is impossible to compute the phase of U.(x ) from a single intensity distribution. In the following Chapters, we shall discuss procedures for obtaining the unknown P(_O from one or more intensity distributions. Using the relations (1.2.4) - (1.2.8) we shall now derive equations which des- cribe the connection between P(O and the intensity distributions or quantities derivable from the latter. In the first instance we restrict our discussions to a single la- teral spatial dimension. We consider an exit pupil with the extent des- cribed by the interval: -g 10 / P(Oexp(-2TTix,Odf; - U. (>:,), (1.2.9) we obtain from (1.2.8): JBdC /Bdf;"P(S')P*(C11)exp|-2Trix.(f.1-f;1) I. (1.1.10) -3 -3 Defining now f(O as the Fouriertransform of l(x.), shifted by an amount 6 (half the extent of the pupil): Jdx. I(x )exp[2Tiix.(C+B)l» (1.2.11) i II 1 — CO we obtain by inserting (1.2.10) in (1.2.11) and carrying out the inte- gration over x., the following equation in the unknown P(t): f(O = / dc i dc"p(c')p*(c")6le+c-c'+!:") (1.2.12) -e -3 After integration over £", we find for -g<£;<0 J P(e')P*(el-5-3)d6t. (1.2.13) is the auto-correlation function of P(4) as well as the cross- correlation function of ?(O and P*(-£) [ 10]. Changing the defocusing by an amount (6z -6Z2), P(C) in (1.2.13) has to be replaced by (see (1.2.5)) (8J: P(£) -> P(5)exp(iAC2), (1.2.14) where: A = J ko(6Z)-6z2) . (1.2.15) 1 I Defining f.(0 as the shifted Fouriertransform of the intensity distribution I.(x.) according (1.2.10), where j labels the various exposures taken with different sLuges of defoc-using of the microscope, the following relation is valid: f.(f) = / df/P(f.')l'*(-.'--,-,-)exp|i/..;; •--(••.'-•-.•r'1'l d" r J Equation (2.1.16) describes the relation between the unknown !'(•') and f.(C), thu shifted Kouriertran.sform of the known I.(x.). Relations similar to (1.2.16) can be derived in two dimensions: We consider an exit pupil with a rectangular aperture (-•••_•'.•:•;-f'W t) . Defining f.(C,n) as: + fj(f,,n) = f dXj j"'dy] I (x|,y|)expl2T-i{x1CJ + :-)-'-y)(-i >)J| , j=l,2,.... (1.2.17) where I.(x.,y.) represents the known image intensity distributions for different stages of defocusing of the microscope indicated by the sub- script j, we can derive, analogously to (1.2.16), the following rela- tion between P(C,n) and the known function f.(f,,rl) MM: 1 2 2 f.(C,n) = / dS' J dn P(C'>n')P*('V-C-3,n'-Ti-a)exp|iA.{f,' -(t-,'-5-^) + J 5 n J 2 2 + n' -(n'-n-a) } 1, if -3 j = 1,2 (1.2.18) where: j = \ kQ &z.. (1.2.19) 12 Relations have now been derived between the unknown P(^) and y,(\) as well as the shifted Fourier transform of the intensity distribution I(x.). As we will see ?(.[/) or U.(x.) cannot be calculated from a single intensity distribution. The equations derived in the present section will be used in the following chapters to solve P(O from the intensity distributions in the exit pupil and image plane. I.."! flie yliac.c i'ft-t't'.L'i'di i'i\>bli'm As soon as l'(f.) is known the object wave function I) (x ) can be o o calculated by inverse Fourier transformation, after correction for possible aberrations. Unfortunately, we obtain only the moduli of ?(',) and U.(x ) by measuring the intensity distributions: = (gU))' (1.3.1) 2 |ui(x])| = (Hx,)) . (1.3.2) We may not expect that the phase of ?(.',) or U.(x ) can be calculated from its modulus only. We have therefore a phase retrieval problem, which is the problem of calculating the unknown argument of a complex function from its modulus, using additional knowledge concerning the function. In the present section we give a brief description of the theoretical background of the problem of obtaining the phase ^(x.) of U.(x.) if |u.(x.)| is known. For that purpose we use the relation be- tween P(O and Ui(x)): ILCx,) = / P(Uexp(-2Tiix]fJdfJ1 (1.3.3) U.(x.) belongs to a particular class of functions I 12], Due to the finite extend of the Fourier transformation of (1.3.3), U.(x.) is not only an entire, but even an band-limited function, i.e. an entire func- tion of the exponential type I 13] with a bandwidth 2£S. The question arises whether or not U.(Xj), and therefore P(O, can 13 be determined uniquely from the known value of jl!.(x,)j. If l'(;) is a solution of the problem, then the function ?*(-•'.) is also a solution, which we immediately deduce from the following n-l/it ion: Ai / P*(-fJ)oxp(-2"ix|'.)d'. = UttXj) (1.3.4) Hence we have at least a two-fold ambiguity. This ambiguity is related to the 'twin' image problem in holography I 14 |. However, it is not Ihe only possible ambiguity, as we shall sei1, The intensity distribution I(x.) can be continued analytically into Llie complex plane by: \*), (1-3.5) where U.U) = J P(f,)exp(-2~if.Odf. (1.3.6) 1 -B and U?(C*)=[U. U*)]*= / P*(-Oexp(-27,if.c)df; (1.3.7) 1 6 The asymptotic behaviour of U.(O if |r, \ *-" follows from (1.3.6) using partial integration I 14 1 (1.3.8) provided P(B) and P*(-B) are non-vanishing. In the complex plane, the zeros a of U.(c) for Id"*" are distributed as (see I 14 1,1 15) and [ 12): a ~ -~ {n + x~ £n|p(g)/p(-g) ] } ; j£|.*» (1.3.9) n ^g /Tti n being an integer. The zeros an according to (1.3.9) lie on a lin-; U parallel to the real axis. Suppose that these zeros a lie in the up- per half plane (u.h.p.), so that a finite number of zeros a (m=\,..,O are situated in the lower half plane (l.h.p). 1'ln-n we can ((instruct from U.(x.) a function U.(x ) without zeros in the l.h.p. by fiippin,, the zeros to their conjugate positions | Id I i.e. v 1 + ~ ' 1 m u.(x ) = i'.(x ) :: -—.— ; im .i • o . n.j.io) m= I '1 'in The product il (x,-a*) / (x ,-a ) in (I.J.10) is known as the Hl.ischke product, while the argument of this product is known as the lilasclike phase, lloenders I 14 | (see also Hof.stetter I ) r> I ) has shown that the following dispersion relation is valid: = ! * x^ ax; -2-i,x-2. arB + arg|!'(-;)| - '/I, (1.3.11) where $(x.) = arg|li(x )) . y denotes the Cauchy Principal Value. We can only determine the change in the phase as function of x.. The determination of 4>(x.) implies a constant phase ambiguity. The value of argjP(-f)] is such a constant overall phase. Thus arg|P(-h) ] has no physical meaning and we are free to choose this overall phase con- veniently. If we know the Blaschke phase, we can determine the phase of U.(x ) from the dispersion relation (1.3.11). Suppose we know that all zeros of U.(.'.) lie in the u.h.p. (inclu- ding the real axis). In that case no zero flipping is needed, so that the Blaschke phase vanishes I 17); the solution for 4(x.) is known as the Hilbert phase or minimal phase solution. A special case arises when all zeros lie on the real axis. Walther I 18 1 has shown that then 15 M are relevant, where M<«N. N is the Shannon number (see Chapter VI). The other zeros are the asymptotic zeros | 19 |. The zeros of L'f(',*) are the complex conjugates of those of I!. (•) . Tlieref ore i1. is impossible to decide whether a zero of !(',) is due LO a zero of L'. ( ') or L'f('*). In order to find the set of zeros a (m=!,...,•), determine the Blasc-h- ke phase in (1.3.11), we have to verify all Blasehke products which can be formed from tile N zeros of 1 (: ) in the l.li.p. This implies a possible ambiguity for y(x.) of 2". Unfortunately, .'ill the.se possible solutions of U.(x ) are band limited functions will] the same band width | 14 I ( 18 |, so that no discrimination by band widtli is possible. Therefore, we need another criterium if we want to solve the phase retrieval problem. In electron microscopy we are dealing with a finite extent of the object aperture, which implies that P(O is an entire function, boun- ded for 1^1""° along the real axis of ;,. Based on the principle of analytic continuation, Hoenders I 14 ] has shown that as long as , P(;,) N is an entire function, that only one of the possible 2 solutions for 4>(x ) will lead to an analytic solution for the wave function in the exit pupil, which is bounded for | T. | ••» along the real axis, viz. the solution P>.(,/ itself. All other solutions are not bounded for j •" j *•"* . (See (33) in i 14 ] ; see also [12 1 ). We have assumed that asymptoti- cally for |t I"**" the zeros of U.(.*) lie on a line parallel to the real axis in the u.h.p., so that |P(-S)]>|P(P)| (see (1.3.9); see also I 16] ). Using the additional a priori information that | P(r) 1 "• j P(-t) ] or |p(-8)|<|P(B)| even uniqueness of the solution can be established mathematically, because we can discriminate between P(O and its 'twin1 image P*(-£) I 14] (21 ]. The procedure for obtaining P(O implies the N verification of the asymptotic behaviour of atmost 2 possible solu- tions of the wave function in the exit pupil. The procedure is rather unpractical. A variant of this phase retrieval procedure has been proposed by Greenaway I 20 ] , who required the use of specially chosen spatial filters (selective stops). For a more realistic phase recovery, we have to resort to more than a single intensity distribution. This will be discussed in the follo- wing chapters. 16 I ! | Hoenders, B.J., The inverse source problem in Optics, (ed. H.I1. Baltes, Springer), Chap. Ill, (1978); other relevant references can be found in that paper. |2 1 Merzbacher, E., Quantum Mechanics, (Wiley), (1970). I 3 ] The first Born approximation is valid for incident electrons with an energy E exceeding 500-1000 eV. Mott, N.I', and Massey, H.S.W,, The theory of atomic collisions (Oxford I'niv. I'ress), Chap. XVI, (1971). Our approximation is therefore certainly valid for Es50 keV, the usual acceleration voltage in electron microscopy. 14 1 Cowley, J.M. and Moodie, A.F. , Acta Crystal logr. , J_2, 360, (1959), ibid., 20., 609, (1975). [ 5 1 From a single exposure we can only obtain one projection of the potential distribution. We need more than one projection on the plane z=z (exposures with different direction), if we want to calculate the three dimensional V(x ,y ,z); see also [ I | . [6 ] Born, M. and Wolf, E., Principles of Optics, (Pergamon), (1965). (7 1 See e.g. Hawkes, P.W., Image Proc. and comp.-aided Design in Elec. Optics (ed. Hawkes, P.W., Academic Press), Manch. Conf., p.2, (1972). [8 ] Drenth, A.J.J., Huiser, A.M.J. and Ferwerda, H.A. , Optica Acta, 2_2, 615, (1975). 19 ] Glaser, W., Grundlagen der Elektronen Optik, (Springer), p.551, (1953). (10| For -3g<£<-0 f(5) is the complex conjugate of f(O for -6<£<6(f(-(3+n)=f*(-3-n), 0;r,<2S). f(£) for -3P<£f-S does not contain additional information. The total bandwidth of f(O is Ill] It is not necessary to restrict the discussions to a rectangular aperture. If we take an arbitrarily shaped stop, which we can en- close within a rectangle, we can write for P(£,n): P(e,n)=S(C,n)P(C,n), where S(£,n)=l in the aperture and zero outside. [12] Ferwerda, H.A., The inverse source problem in Optics, (ed. H.P. Baltes, Springer), Chapter II, (1978). 17 ( 13 1 Paley, R.A.E.C. and Wiener, N., Fourier transforms in tho com- plex domain, (Am. Math. Soc), (1934). I 14] Hoenders, B.J., J. Math. Phys. , Jj>, 1719, (1975). Other relevant references may be found in this paper. (15] Hofstetter, E.M., I.E.E.E. trans, inf. theory, JJ), 119, (1964). [ 16 | If the asymptotic zeros lie in the l.h.p., we have to flip the zeros from the u.h.p. to the l.h.p. in order to obtain a zero- frce u.h.p. The mathematics changes s 1 i ght ly ,but not. fundamen- tally. I 17 1 It might happen that all zeros lie in the u.h.p. E.g. P(£) = exp(2iribO with Irob>0 gives rise to a function U.(c) with the zeros J-^ + b(Imb>0). i p [18] Walther, A., Optica Acta, _K), 41, (1963). [191 Burge, R.E., Fiddy, M.A., Greenaway, A.H. and Ross, G., Proc. R. Soc. London, Ser. A.350, 191, (1976). (20 1 Greenaway, A.H., Optics Letters, J_, 10, (1977). [21] If we have the additional information that |P(g)|=|p*(-g)j, then it is impossible to discriminate between P(£) and P*(-O, so that we are dealing with a two-fold ambiguity . [14]. 18 CHAPTER II ON THE SOLUTION OF THE PHASE PROBLEM US INC IMAGE INTENSITY DISTRIBUTIONS DETERMINED FOR DIFFERENT DEFOOUSINOS OF THE MICROSCOPE 2.0 Intpoduiitum In Section 1.3 we have seen that the phase of an image wave func- tion can, in general, not be calculated from its modulus only. We need additional information: such as a priori information concerning the object structure, from which the phase of the unknown wave func- tion in the image plane can be obtained | 1 1. Generally speaking, we need to know more than one intensity distri- bution, corresponding to different microscope settings. We can classi- fy the procedures for solving P(O roughly in three different cate- gories: 1) Image intensity distributions determined for different amounts of defocusing (defocused micrographs). 2) Intensity distributions in image plane and exit pupil. 3) Bright-field imaging with specially chosen stop, defocusing and il- lumination (in-line holography, half-plane apertures, off-axis holography). In the present chapter we discuss the first category of the solu- tion of the phase problem. 2.1 Theory As early as 1968, Schiske 1 2 ] mentioned the problem of solving P(O from defocused micrographs, while Misell | 3) in 1973 proposed a detailed algorithm. (This algorithm will be discussed in the last sec- tion of this chapter, where we will see that thp success of the algo- rithm is not guaranteed). Suppose we want to apply an algorithm for solving P(O (e.g. Misell's), then the question arises whether it is possible to 19 compute P(O uniquely from 1|U() and •.,(*]). where I j (x j ) and I.,(x];) are the intensity distributions of the defocused mi cro;;rapb;;. In Section 1.2 we derived equations, which describe the relation between the unknown P(O and I . Cx,) and 19(X|), viz., f CO = / P(r.')P*(C-t.-i')d:.', -''J-y> (2.1.1) I"' •> r I2(O = J PC',' >l'*( f.' - .-(-.)cxp I i/.{ ••,'-( • '-•>;• >" • Id", -•••J'J.. (2.1.2) where f.(t') and f,(O are the Fourier transforms of I.(x ) and I,,(x.) shifted by an amount p, and where .'. represents the change in del ocusing according to (1.2.15) I A 1. Hoenders I 5 1 has shown that P(O can be calculated up till a two- fold ambiguity, if the complex zeros of the analytic continuation of I (x.) and I7(x.) are known. In I 6 ) it has been proven that both the analytic functions P(O and the analytic function 1> (-?,) can be deter- mined uniquely from (2.1.1) and (2.1.2). P(f.) is treated as if it were mathematically independent of P (~O. Unfortunately we have no way to implement analyticity into an algo- rithm. Hence, we have to discuss the uniqueness problem even for non- analytic solutions. P(C) can be uniquely determined, if certain con- ditions are fulfilled | 7). •>£ (2.1.3a) b) P(O is a differentiable function for -iW,^. (2.1.3b) c) We have to restrict the change in defocusing between the micrographs: | A| <7r/8 . (2.1.3c) Developing now an algorithm for determing P(O from (2.1,1) and (2.1.2), we can easily fulfil condition c, while implementing the dif- ferentiability for P(O does not have to be a problem as we will see 20 in Section 2.2. We suppose that P(tf)l'*(-;-)#). (If p(r )!'*(-;• )=0 we have to choose another value for p (other aperture), such that condition a) is fulfilled.). (2.1.1) and (2.1.2) are two non-linear volti'rra integral equations of the first kind. If we want to solve a linear Volterra integral equation of the second kind, the usual method is the so-called Pi card- iteration procedure I 9 |. If we apply the Newlon-Kantorovich lineariza- tion procedure to the non-linear integral equations (2.1.1) and (2.1.2) I 10 | , we obtain such a system of linear Volierr.i integral equations of the second kind (see I II I, I 12 I for the solution by the 1'ic.ird method). Thus, we obtain correct .solutions of the problem, but the algorithm is rather slow I 12]. In the following sections we derive a faster algorithm to determine PU) from (2.1.1) and (2.1.2), which we shall call the "direct method". 2.2 The dii'b^t Jicihod foy th^ ,;>;. dirit'U;;:,»:;.' •;.:. In this section we develop an algorithm for the determination of P(f.) from the known f.(O and f,,(.rJ. The direct method is based on a numerical integration procedure of (2.1.1) and (2.1.2). We sample ', as follows: -P = -nh,. . . . ,nh=rS, where h is the sampling distance. We abbreviate P(kh) and f ,(kh)(k= -n,...,n) by P(k) and f (k) , respectively. Let us suppose for the moment that P(k) is known instead of f,(k) and f-(10. Applying the trapezium rule for computing f.(k) and f,(k) I 8 ], we guarantee that P(O represented by P(k), is dif ferentiable due to the fact that the trapezium rule is based on a linear inter- polation of P(k). Differentiability of P(f.) is a necessary condition (see (2.1.3b), I 7 |). Applying the Riemann sum for computing f.(k) and f2(k) from P(k), the differentiability of P(f,) , represented by P(k) , is, in general not guaranteed. However, P(f.) turns out to be a band- limited function: therefore P(£) can be written in terms of sine- functions according to the Whittaker-Shannon sampling procedure I 13] (see also Chapter IV). Consequently P(O is di f ferentiable in that case. Invoking now the Whit taker- Shannon sampling, f(O can be 21 approximated by a Riemann sum, while the differentiability of I'(O is guaranteed. For f.(O we obtain the Ricmann sum: n-l f (k) = h Z P(m)P*(m-k-n), k= -n,...,n-l. (2.2.1) m=k Furthermore, wo obtain as approximat ion for f,;(k): n-l f2(k) = li X P(m)P*(m-k-n)K(m,k), k= -n ,n-l. (2.2.2) where: K(m,k) = exp|iA{(mh)2 - ( (m-k-n)h)2}] . (2.2.3) We shall now discuss the direct method, assuming that l'(k) is the unknown quantity, while f,(k) and f.,(k) are known. The direct method is based on (2.2.1) and (2.2.2). From (2.2.1) we obtain for k=n-1: f,(n-l) = h P(n-l)P*(-n), (2.2.A) supposing f.(n-l)^0. (This condition corresponds to (2.1.3a), We choose for P(n-l) an arbitrary non-zero value. P*(-n) follows then from the known f (n-l). For k=n-2, (2.2.1.) and (2.2.2) lead to: f,(n-2) = h P(n-2)P*(-n) + h P(n-1)P*(-n+1), (2.2.5) f2(n-2) = h P(n-2)P*(-n)K(n-2,n-2) + h P(n-1)P*(-n+1)K(n-I,n-2). (2.2.6) The unknowns P(n-2) and P*(-n+l) can now be computed from (2.2.5) and (2.2.6) using the values of P(n-l) and P*(-n) just determined. In general we obtain for k=n-m(m>3): 22 m-1 f.(n-m) - h i: P(n-j)P*(m-n-j)=hP(n-m)P*(-n) j-2 hP(n-l)P*(m-n-l). and m-1 f,(n-m) - h Y. P(n-j )P*(m-n-j )Ktn-j ,n-m) = j 2 = hP(n-m)P*(-n)K(n-m,n-m)+hP(n-l)I'*(n-m-l)K(n-l,n-m). (2.2.8) The unknowns P(n-m) and P*(m-n-l) can now be computed successively, provided the determinant of the linear equations is non-vanishing, which is guaranteed if |A| <*/!-.2 (see (2.1.3c), I 14 I ). We have started from an arbitrary value for P(n-l). This implies that the solutions thus obtained, which we denote by P('") and P*(-'",), are almost certainly not each other's complex conjugate. This defect may be remedied by multiplying the solutions P(1',) and P*(-.r,) by a suitable complex factor such that in a certain point of the f, - inter- val, e.g. 5=0, the new functions P(O and P* (->',) are related to each other by complex conjugation. Hence: P(fj = \$(r,) (2.2.9a) P*(-O = iiP*(-fJ (2.2.9b) where Ap=l because of (2.1.1): f.(-fO= /f epp(e')p*(f,')df/ • p* v dv-= / 6 xpp(e- i)p*(c)dc'=>u• f (-B). (2.2.io) -B -e' The factors X and IJ can be calculated by choosing f,=0 (or for another value of C, if convenient), provided P(0)?t0. The constant phase 23 ambiguity (overall phase) is chosen, such time P(0) is real and posi- tive. In that case: \ = P(0)/f>(0); (2.2. lla) n = l'*(0)/i>(0) . (2.2.1 lb) Because Au=l, P(0) can now bv computed from:I IS | P(0) = P*(0) = '|P(O)P*(O) ' *. (2.2.12) 2.3 The extension of tlu dfi'ci n 'hod to ,i i" ••l.inj.uur atop We now extend the direct method to a two-dimensional case, viz. that of a rectangular aperture in the pupil plane. As in Eq. (1.2.18), we have: 3 a f.(£,n)= J dt1 J dn1P(r,',n1)P*(r,l-f;-e,nI-ri-.«)t .UV,,n',n), J C n J -P '.Un'.n) = 1 (2.3.2a) exp|iA{C'2-(C'-C-3)2+n'2-(ti'-n-a)2}] . (2.3.2b) We sample the functions P(f,,n), P*(-5,-n), f-C-^.n) along the 6 and n- directions in the points C=-P= -nh,,...,nh =3 and n=-a=-mh~,..., mh2=a, respectively. We abbreviate f.(kh.,Hh2) and P(kh.,Jlh2) by f.(k,Ji) and P(k,H), respectively. Applying the Riemann summation, we obtain for (2.3.1): n-I m-1 f.(k,e)=h h T. >: P(p,q)P*(p-k-n,q-?-m)K. (p.k.q.J) j = l,2 (2.3.3) J ' l p=k q=t J where: (2.3.4;i) and 2 2 2 2 K2(p,k,q,e)=exp[iA{(ph|) -{(p-k-n)h1l +(qh?) -{(q-."-m)h2' ;| (2.3.Ab) Following the direct method, we now observe that, for k=n-l and S.=m- 1 , f.(n-1,m-l)=h.h2P(n-l,m-l)P*(-n,-m)K.(n-1,n-I,ra-I,m-1). (2.3.5) Choosing an arbitrary non-vanishing value of P(n-I,m-I),P*(-n,-m) can be calculated from (2.3.5). For k=n-2 and l=m-l we get: f.(n-2,m-l)=h h2P(n-2,m-l)P*(-n,-m)K.(n-2,n-2,m-I,m-1) + +h|h2P(n-l,m-l)P*(-n+1,-m)K.(n-l,n-2,m-l,ra-l), (2.3.6) from which the unknowns P(n-2,m-l) and P*(-n+l,-m) can be found. For k=n-p and 8.=m-l we obtain (p>3): 25 p-1 f.(n-p,m-l)-h.h. £ P(n-r,m-l)P*(p-r-n,-m)K.(n-2,n-p,m-l J ' r=2 J =h h2P(n-p,n-l)P*(-n,-m)K.(n-p,n-p,m-l,m-l (2.3.7) from which P(n-p,m-l) and P*(p-n-l,-m) cm hi' computed successively. Thus for k n-1 f.(k,m-2)=h h, Z P(p,m-2)P*(p-k-n,-m)K.(p,k,m-2,m-2) J P=k J n-! t P(p,m-l)P*(p-k-n,-m+l)K.(p,k,m-l,m-2). (2.3.8) p=k J In this way P(k,m-2) and P*(k.-m+l) can be successively computed for k= -n,...,n-l, remembering that P(k,m-I) and P*(k,-m) are already known from (2.3.7). Generally we get for 8.=m-q(q>3): n-1 q-1 f.(k,m-q)-h.h9 I Z P(k,m-s)P*(p-k-n,-m+q-s) K.(p,k,m-s,m-q) = J ' l p=k s=2 J n-1 = h.h_ E P(p,m-q)P*(p-k-n,-m)K.(p,k,m-q,m-q) + p=k J n-1 +h h I P(p,m-l)P*(p-k-n,-Tn+q-l)K. (p,k,m-l ,m-q). p=k J (2.3.9) 26 Thus, we can calculate successively P(k,?.) and P*(k,t) for k=-n,...., n-I and £=-m, m-1. The quantities P(-n,-m) and P*(n-I,n-1), however, cannot be obtained from (2.3.9). After correction of the arbitrarily c'lfsen value of P(n-l,n-l), such that P(k,O and P*(k,f) have become complex conjugates (see (2,2.1 la,b) and (2.2.12)), P(-n,-m) and P*(n-l,n-l) can finally bo calculated from complex conju- gation of P*(-n,-m) and P(n-l,n-l). The direct method can be used analogously for an arbitrarily shaped stop, by enclosing the stop within a reel angle and putting l'(r.,'i)E0 outside the aperture. 2.4 Verification of the direct method using oinuuilad ob,'e>-lr, We will verify the direct method for a single spatial dimension (see Section 2.2) in the following way: firstly we derive objects in the form of a simulation of U (x ). Secondly, we assume (for con- venience) an optical system free of aberrations. In that case P(O is related to U (x ) by the finite Fourier transform: o o PU) = / U (x )exp(2Tiix f,)dx . (2.A.I) o o P(C) is computed using a Fast Fourier Transform (F.F.T.[16] ). By application of a numerical integration procedure, we compute f.(C) and f2(5) according to (2.1.1) and (2.1.2). Next we 'forget' how f.(C) and tAV) have been obtained and try to solve (2.1.1) and (2.1.2) for P(5). The reconstructed P(C is them compared with the true P(O- Since the direct method is based on the Riemann sum as an approxima- tion for the integral, we avoid the complications of truncation errors, provided that we compute f.(O and tA.V> with the Riemann summation, too. In order to test the numerical procedures we have made two dif- ferent choices for U (x ) (it is not pretended th.it these choices for o o U (x ) correspond to physical realizable objects). 27 -08 [ '. ' I •10 -08 -06 -04 -02 0 02 04 06 08 10 Fig. II.1: for Example I. Ke f^f) ( ); Im f (O (—-) -10 -08 -06 -04 -02 0 . 02 04 06 08 10 Fig. II.2: As Fig. II.1 for Example II. 28 -10 -OB -06 -04 -02 0 02 04 06 08 10 Fig. II. 3: ?(O for Example I. Re ?(!•/; (—-); Im P(f,) ( ). -10 -08 -06 -0.4 -02 0 02 04 06 08 10 Fig. II.4: As Fig. II.3 for Example II. 29 Example I: U (x )=exp(-x"/b) lsin(2:ix / I 50)+oos (2"x /500)| . (2.A.2) o o o o o Example II: U (x )=exp(-x~/b) lexpH |sin(2-x / 1 50|+cos(2'.x /500))>. (2.4.3; o o o o n with b=5.105 for x '0 ami b=IO6 if x '-0. o o~ For both examples the parameters of Llie transmission function are _2 4 the same, i.e. h=10 "", ^=4pm and A=1.18.IO , which corresponds to a change in defocusing of 15 am. P(f.) lias been sampled in a set of 489 equidistant points. In Figs. II. 1 and II.2, we present f.(t') for the two objects, re- spectively, in order to give an impression, what these functions look like. In Figs. II.3 and III.4, the simulated P('") for both examples are plotted. The reconstruction of P(O according to the direct method is almost identical to the values of the simulated P(O I 17]. The inaccuracy — ft —A (about 10 -10 %), caused by the roundingerrors in the computations increase for decreasing values of £,• Therefore Figs. II.3 and II.4 are also the plots of the reconstructed P(f) | 17|. 2.6 Discussion of the direct method We have derived a rather simple algorithm for the determination of P(O from (2.1.1) and (2.1.2). We have seen that this so-called direct method can be applied for obtaining values of P(£) from defocused micrographs. Up till now we neglected noise. In actual situations, however, the influence of noise cannot be neglected as we will see in Chapter V. These errors restrict drastically the applicability of the direct method (or any other algorithm, e.g. the Newton-Kantorovich algorithm) for determing P(5) from (2. I. I) and (2.1.2). 30 ".6 Miseli's algorithm Misell 1 3 1 proposed an algorithm for the solution of U.(x ) or P(") from two defocused micrographs. The wave functions l!.(x.) and U*(x.) (or P(O and P*(-O are not considered as independent quantities as in the case of the direct method). The algorithm starts with the computation of 1'C) from I.(x.) using a starting v.ilue for the phase of U. (x.) : ; ,(x.) (tlie subscript s stands for 'starting' value). 1 (x.) lias a band width of Ar . All quantities related to I.(x.) are considered to be quantities with a bandwidth 4P. P(O =J ll,(x.) |2exp|ii}' (x. ) lexp(2'i ix f )dx . (2.6.1) We now compute the image wave function corresponding to the second exposure: 22 U (x ) = / 0(OP(Oexp[iA;/|exp(-2iix O, (2.6.2) -26 where A represents the change in defocusing accordings to: A = j| kQ(*z2 " «z,), (2.6.3) and 0(O: 0(0=1 for [ C | 0(0=0 for 3 We introduce 0(0 to guarantee the bandwidth AB for |u.(x.)| as well as the correct sampling, using F.F.T. [16]. The obtained VAx^) will not have the correct absolute value. We change U (x.) by the replacement: i U2(x,) - {U2(x,)/|u2(x1)|}H2(x1)l . (2.6.5) 31 In this way the phase of lL(x ) is retained, while we enforce tlie correct modulus. Again, P(4) is calculated from: P(O = j U,,(x1)exp(2:iix1'.)dx] (2.6.6) and from this result, we can determine l'.(x.) using U,(x ) = / o(f,)P(Ooxp(-iAOoxp(-2':ix,Od-'. . (2.6.7) -2,-: For U.(x.), too, the absolute value is corrected (keeping the phase) by the replacement: lyx^-fiyx^/liyxplHijU,) p. (2.6.8) U.(x.) has now the correct modulus with a phase $ (x ), different from (\$ + j(x.)-<)> (x,) | < 10 ). If we have obtained this phase 4> . (,x ) and calculate P(O according to (2.6.1) and lL(x.) according to (2.6.2), then P(O has to vanish for Is^fl and the modulus of U (x.) has to be x equal to the square root of I?(x ), otherwise 4> |( i) cannot be ac- cepted as the solution of the phase problem. Our simulations have shown, that usually 1?(Ot0 for ]c|>|3 and that the modulus of U?(x ) is incorrect, so that usually Missell's algorithm fails. Only if the x is tne moduli are correct and P(£)=0 for |s|>8, the solution $ +i^ )) solution of the phase problem. References [ 1 ] An example of such a priori information is, when the object is an amplitude object 33 CHAPTER III THE SOLUTION OF THE PHASE PROBLEM USING THE INTENSITY DISTRIBUTIONS IN THE IMAGE PLANE AND EXIT PUPIL 3.0 Introduction In 1971 Gerchberg and Saxton I 1 I introduced a method for recovering the unknown phase of P(O, using the intensity distributions in the image plane I(x.) and the intensity distribution in the exit pupil (In the last section of the present chapter, we shall see that usually their algorithm fails). Applying their algorithm the question arises, whether P(O can be determined uniquely from the known inten- sity distributions. Huiser et.al, 121 have shown that the analytic functions P(£) and P*(-Q (P(C) and ?*(-£,) are treated as independent functions), can be determined to atmost a two-fold ambiguity. Unfor- tunately, the analyticity of P(O cannot be implemented in an algo- rithm. Schiske has shown [ 31 , only assuming integrability of the solution, that an infinity of solutions can be obtained. The continuity of the derivative of P(£) (see [ 4 ) ) can be used to reduce the number of solutions drastically. As we shall see, even uniqueness of the solution can be established under certain circumstances. In this chapter we develop another algorithm, the so called direct- method, for the solution of the unknown P(£) from the known g(S) and I(x.) and we verify the algorithm with the aid of simulated examples. In the last section of this present chapter we discuss the original algorithm according Gerchberg and Saxton [ 1] and compare the results with that from the direct method. 3.1 The direct method In this present section we shall develop an algorithm, which com- putes P(£) from the known intensity distributions g(S) and I(x.). The structure of the algorithm is analogous to the direct method for sol- 34 ving P(0 from two defocused images (See Chapter II). We have to solve P(£) from the following equations: g(O = P(C)P*(O, -\','J,y (3.1.1) f(O = J l f where f(C) is the shifted Fourier transform of tlu1 ima^e intensity distribution I(x.) (See Section 1.2). Following lines similar to I A I , we shall regard PC,), P*C), !'(-'',) and P*(-O as four independent [unctions. Consequently tlie following equations will be needed: g(-£) = P(-C)P*(-f,), -r^'st (3.1.3) and e f*(C) = / P*(C')P(C'-5-0)d^', -0£.r, The relations (3.1.1) - (3.1.4) form a set of four non-linear equations in the unknown functions PC), P*C), P(~O and P*(-O. The integrals in (3.1.2) and (3.1.4) will be replaced by sums, which is achieved by sampling the interval -3<£ -@=-nh, ,0, nh=B;h=B/n . (3.1.5) The integrals in (3.1.2) and (3.1.4) are computed using the trapezium rule [5] . We abbreviate f(3-kh) and P(8-kh) for k=0, ,2n as f(n-k) and P(n-k), respectively and obtain then the following relations be- tween P(n-k) and f(n-k): £f(n-k)=P(n-k)P*(-n)+P(n)P*(-n+k)+2(l-6 )Z P(n-j)P*(-n+k-j) k>l, n u j=, (3.1.6) 35 where Che sum in (3.1.6) is supposed to vanish if k=l and where ' is the Kronecker symbol. From (3.1.4), using Ihe notation according to (3.1.6), we obtain: 2 k' f-f*(n-k)=P*(n-k)P(-n)+P*(n)I'(-n+k)+2( I-' .. )l P(n-j)P*(-n+k-j) k'-l . 11 lk j l (3.1.7) Sampling in the same set of points, we find for (3.1.1) and (3.1.3): g(n-k)=P(n-k)P*(n-k) kM) (3.1.8) g(-n+k)=P(-n+k)P*(-n+k) k^O (3.1.9) Suppose we know P(n-m), P*(n-m), p(-n+m) and P*(-n+m) for m=0,1,2,....,k-J. Then, the pquations (3.1.ft) - (3.1.9) can be consi- dered as a system of four equations in the four unknowns P(n-k), P*(n-k), P(-n+k) and P*(-n+k). By eliminating, we derive from (3.1.6) - (3.1.9) a quadratic equation in P(n-k): 7*(n-k)P*(-n)P2(n-k)+{-g(-n)g(n-k)+g(n)g(-n+k) + - 7(n-k)}2 P(n-k)+?(n-k)g(n-k)P(-n)=O (3.1.10) where: 2 k"' f(n-k) = f f(n-k)-2(l-.S k)>: P(n-j)P*(-n+k-j) (3.1.11) jo, and where f*(n-k) has a similar meaning. In general, the quadratic equation (3.1.10) has two solutions, and we have to design a method in order to discriminate the roots of (3.I. 10). This method will be dis- cussed in the following section. If we have selected one of the two possible solutions of P(n-k), the values of P*(n-k), P(-n+k) and P*(-n+k) can be calculated using 36 the relations (3.1.b), (3.1.7) and (3.1.9). This enables us to proceed to the next step in the calculation of P({,). It remains to be shown that the procedure can be started, i.e. that we can determine I'fn), P*(n), P(-n) and P*(-n). From (3.1.8) we conclude that g(n)=l'(n)I'* (n). Since P(f,) can only be determined apart from a constant phase factor, one may choose P(n) real and positive, i.e. I'(n) = l'*(n) = [g(n)l ' . Differentiating (3.1.2) with respect to '. we find for '=r! > I jr f (Ol f_,, = -l'(")I *(-l-) (3.1.12) which determines P*(-n) if the left hand side of (3.1.12) does not vanish. From g(-n) we compute P(n). Hence the procedure can be started. We discuss now (3.1.10) in more detail. Ignoring the sampling of P(£), the discriminant equals: 7 2 ,^. ,9 ^ ••-•-•-•-••• (3.1.13) According to (3.1.11) and (3.1.6) we can write f(f.) in terms of the unknowns P(£) and P*(-£): ?(£) = P(C)P*H) + P(3)P*(-r.) (3.1.14) Inserting (3.1.14) in (3.1.12), we obtain after a straightforward calculation the following relation between Dis(f,) and the unknown P(O and P(-£): Dis(C) = {2 Im P*(-P)P*(B)P(5)P(-C)}2. (3.1.15) If Dis(C)=0, the two roots of (3.1.10) coincide. This is an interesting result, especially if we take into account the results derived in I 4|. There has been proven that if P(O is defined on |-B,|-!] and is dif- ferentiable with a continuous derivative, P(O can be uniquely deter- mined from (3.1.1) - (3.1.4) provided that: 37 a) P(g)P*(-g)y!O (see also 3. 1. 12) (3.1.Ida) b) |P(6)|2?«|P(-[3)|2 (3.1.16b) c) A(a )= U Sl_Hi '±- - 2 (3.1.16c) (-(<) h~ del Il(',)l . where a denote the zeros of det »(•'.) and del li(') I 6 1 is defined as: n det B(C)=2Im P*(fi)P*(-f)P(f,)I>(-O. (3.1.17) If condition b) is not satisfied a two fold ambiguity may occur [ 4 | . (see also (3.2. 14). From (3.1.15) and (3.1.17) we see: Dis(O ={det B(<:) }2 (3. 1. 18) When ambiguities occur, if det B(r,) vanishes for v"=a and A(a )>2, we might be inclined to believe that the reason for these ambiguities lies in the fact that the selection procedure for the dis- crimination between the two roots of (3.1.10) fails for C=a . However, because of relations (3.1.18) the roots are equal if det B(O=0. Therefore, if the algorithm fails, it will not be a failure of the selection procedure. 3.2 The selection procedure We have assumed the existence of a criterium according to which one of the possible values of P(n-k) may be selected as the 'correct1 one. The absence of such a criterium would imply, that the total number of N solutions is in the order of 2 I 7 ]| 8 )| 12], since we have two alterna- tives for each sampling point, apart from £,= S and the points where 38 det B(O vanishes. A similar result has been found by Schiske I 3] and Dallas 19]. Now we formulate the selection criterium. Let us suppostc that P(n), P(n-1),.,..,P(n-m) have been calculated. in order to select the correct value of l'(n-m-l). We estimate this value by extrapolation from P(n,m). As long as wo know the value of P (n-m-1), we select now that root of (3.1.10), which is closest to Pextr(n~m~')' iie< if |P (n-m-l) - P. (n-m-1) | = . . j = l ,2 , (3.2.1) we choose the value of P.(n-m-1) for which i. i as the smallest value J J (j=l,2). The value of P (n-m-1) is still nor known, and therefore we estimate P (n-m-1) by extrapolation from P(n-m). Hence: GX t r P (n-m-l) = P(n-m) - Pj(n-m) (3.2.2) where Pj(n—m) stands for the left-hand side derivative: P'tt) - Lim *"+•>-'<" . (3.2.3) x. s sto The simplest method is to use P'(n-m+l) as an approximation of P!(n-m): = |P(n-m+l) - P(n-m)l/h (3.2.A) Hence: Pextr(n-m-I) = P(n-m)-h P^(n-m)«P(n-m)-h P^( = 2P(n-tn) - P(n-m+l). (3.2.5) A second, more elaborate, method proceeds as follows: 39 Introducing the integration variable -f,"=(. '-?-?• in (3.1.2) and dif- ferentiating the resulting equation with respect tn ', yields I 7 | : f (C)= -P*(-O!'(^) + / P*(-<",")T^rP('>'."+f)d>:" (3.2.6) where f'(C) denotes the derivative of f('.). Approximating the integral with the trapezium rule, (3.2.6) gives: f (n-m)= -P*(-n+m)P(n)+|h P*(-n+m)l" (n) + ih I'* (-n)P'(n-m) + m-1 + h(l-A ) >: P*(n-j)P'(n-m+j), m=l,2, ,2n-l (3.2.7) jl Equation (3.2.7) yields P'(n-m) if all other quantities are known. Both methods, however, fail to give an estimate for P'(|J). From (3.1.2) we derive: ,2 [~ f(O],=R = -P*(-B)P'(P) + P(P)P*'(-:O, (3.2.8) ,2 -^y f*(£)l - ,, = -P(H)P*'(e) + P*(fi)P'(-i1.). (3.2.9) Using (3.I.1) we find: .2 S =P = P(P)P*'(6) + P*(6)P'(H) . (3.2.10) df/ ^ |J The fourth equation we need, can also be derived from (3.1.2): .2 fi 1-^ f(C)I, _„= -P*(-6)P'(-B) + P(B)P*'(-ti) + / PW)P"((,')dC . AC K -\i (3.2.11) Remarking that: 40 PU)P*"(f.) - P*(rJP"(f.') = jr I P(;jp*'('-) - P*(-'.)P'(-.) I , 0.2.12) it can be seen that: 2 = P(P)P*'(I<) - P+d-OP'd*) + P(-.-)P*'(-i-O-P*(-.-)P'(-.-). (3.2. IS) If the determinant of this system of equations for the unknowns P'(e), P*'(ti), P'(-H) and P*'(-i-.) is non-vanishing, !"(;•) can be cal- culated. This determinant " is equal to: 0 = 2P(;-.)P*(-!-')[g(-,-.)-g(.') I. (3.2.14) We notice that 0=0 if g(B) = g(~l0, which means a two-fold ambiguity for the solution of P(O, if the intensity distribution in the exit pupil for £>B and C=-B are equal to each other | 4 | I 7 |. The determi- nation of P'(B), etc., will be difficult in experimental circumstances, because higher derivatives of f('") and g(",) are needed. P'(S) can be approximated by the derivative from the left: P'(B) = [ P(fl)-P(S-h)|/h. This implies, however, a two fold ambiguity, because we are dealing witli two possible solutions for P(B-h) and so with two solutions for P'(3). 3.3 Hcoonstruation of the pupil uaViifunction usbij the dir&jL method We now verify the direct method for simulated examples. Problems with the uniqueness of the solution can be expected if det B(O = 0 for £=a , while A(a )>2 (See (3.1.16c) and (3.1.17)). n n - Therefore it is useful to discuss examples for the following cases | 8J. A) det B(O#) for -0<£ C) det B(C)=O for C=o , while lA(an)|"'=0. (Multiple zero of det B(£)). Uniqueness cannot be guaranteed mathematically. We present two examples: 1) the direct method fails. 2) the direct method succeeds. D) det B(O=0 for t=cx , while A(a )>2 ( lA(f< )l~Vo). Uniqueness is not expected mathematically. In all examples, sampling is done for 101 points, while r> I 1 10]. This corresponds to a sampling distance h=0.02. We take for P(f,) a simulated function and compute the corresponding f(r.) and g(''). Then we 'forget' how f(O and g(?) have been computed and try to solve PC1") according to the direct method. Here,we present examples for the fol- lowing cases: A,C,,C_, and D. The functions P(O below do not neces- sarily correspond to physically realizable objects!. Example I (case A) ) + HnU2+2) (3.3.1) det B(£)j*O for -1<5<1, thus we may expect that P(4) can be computed without complications in the whole interval. We notice that in Figs. Ill la and III lb the reconstructed P(£) is identical to the simulated one. Example II (case C.): P(S) = sin(27r£) + exp(2nH2) + i. (3.3.2) Both det B(£) and its derivative vanish for 5=0, so we may not expect a unique solution. In Fig. III. 2 we have, plotted the re- sults for P(C). Applying the direct method wt obtain a solution for P(£) and for the function P(-O. In Fig. Ill 3 we have plotted the real part of the reconstructed P(O and the real part of Q(-£), where Q(5) = P(~£). P(5) and Q(-O are not identical, so the solution for P(O (as well as that for Q(-£) has to be discarded on mathematical grounds. However, det B(f)j*O for 0-•<,< 1, so that the solution for P(') and P(-s) is unique in that interval. Hence, P(O can be determined from the results obtained from P(",) and P(-O. Kxample III (case C_). P(O = sin|n(r-l)] (3.3.3) Both det B(O and its derivative vanish for ''.=0. For C-0, the results for this example have an inaccuracy of about 10 % and for f.'O, tins is increasing until 10 % for f.=-1 (Figs. Ill 4a and III 4b). Example IV (case D): ^nMO + 1; M=l ,3,5,7,9,... (3.3.A) det B(C) vanishes for several values of C, depending on the value of M. If Mf3 the reconstruction succeeds (inaccuracy less than 10 %), but for M>5 the direct method fails. As A(a )~-2 for the zeros a of - n - n det B(C), theory I A] has shown that uniqueness of the solution is not guaranteed. (Figs. Ill 5a and III 5b for M=7; Fig. Ill 6 for the real part of the reconstructed P(O and Q(-£.) when M=t3). Pill 26 26 18 IB REAL PART la) IMAGINARY PARTlbl 10 -10 02 02 -06 -06 -U -14 •22 -22 -30 -30L -10 -Q5 OS -1.0 -Q5 0.5 Fig. III. 1: Simulated P(O ( ) and reconstructed P(Q ( ) for Example I. (On this scale the dotted line cannot be distinguished from the solid line). 43 viy^ \\\,2: Real part of l'( ' ) for Example- 1 I . -10 .0.1 -01 -01 .01 0 01 04 Ot 111 10 Fit;. 111. 3 : Reconstructed Re l'(r) (-•-•) and recon- structed Re Q(-'") ( ) for Kxample II witli Q(r.) = P(-".). IMAGINARY PAR1(b) Fig. III.4: As Fig. III.1 for Example III, Pig Pig * t 1 2.6 26 REALPARTIa) IMAGINARY PARTI b) 18 10 10 "iv 02 iii 02 -06 -06 •14 •14 •22 •22 •30 - -30 -1 0 -05 0 05 -1 -QS 05 Fig. III.5: As Fig. III. I Cor Example IV with M=7. 26 o • 5 Fig. III.6: As Fig. III.3 for example IV with M=9. 3.4 Discussion of the direct method We have seen in the preceding section, that the solution obtained for the Examples I, II, III and IV are completely in agreement with the experimental expectations |4] . The direct method, which selects a continuously differentiable, not necessarily analytic, solution leads to a unique solution, provided that for the zeros a of det we have A(a )<2. Even in the case that A(a )>2, the algorithm can pro duce the correct solution (see Example III), but this is not guaran- 45 teed (see Example II). If the direct method fails, it can be explained from the quantity dot B(O and its derivative, why this failure occurs. In experimental cases we do not know anything about !'(•') and so we cannot predict the behaviour of det B(",) and A(\). This implies, that we do not know in advance, whether the direct method will succeed or fail. This problem is the first limitation tor tin- application of tIn- direct method. A second complication is, that we ii.ive to compute the higher derivatives of f(\) and ;;(',), in order to determine !'(,-) and Cerchberg and Saxton proposed an iterative algorithm ( 1 I for com- puting the unknown phase from 'l'0".)| and ;l'.(x );. The following iteration scheme was proposed. : Pn(O = J |u.(x])]exp(i*n(x])]exp(2;iix. .)dx] (3.5.1) (3.5.2) 2P / C(fJ)Pn(C)exp(-27iix]f,)dfJ I 13), (3.5.3) -2e> (3.5.4) where U.(x.) has to be read as u"(x,) = lu.Cxplexpli^U,)) , (3.5.5) and where: 0(5) = 1 for |c| 46 u(O = 0 for U<\r.\<2i-.. (3.5.6) We take a starting value for References I 1) Gerchberg, R.W. and Saxton, W.O. , Optik, _34, 275, 35_, 237, (1971) [2] Huiser, A.M.J., Drenth, A.J.J. and Perwerda, H.A., Optik, 4j>, 303, (1970). [3] Schiske, P., Optik, 4£, 261, (1974). 141 Huiser, A.M.J. and Ferwerda, H.A., Optik, 46, (1976). (5J We apply the trapezium rule in stead of the Riemann sum for two reasons: 1) det B(£) and its dertivative (see (3.1.16) and (3.1.17)) re- main unknown, because P(3) and P'(f>) cannot be computed. 2) Applying the Riemann sum to e.g. (3. 1.2) we guarantee the integrability of P(£), but the determination of P1 (f.) is not guaranteed, which is necessary in order to compute P(O 1 4 1. (see also Section 3.2). ( 6 | The theoretical study of the problem of the uniqueness leads to 47 a matrix integral equation | 4], where the determinant of the matrix B(C) is denoted as dot B(fJ. 1 7] Huiser, A.M.J., van Toorn, P. and Ferwerda, H.A., Optik, 47, I, (1977). |8] van Toorn, P. and Ferwerda, H.A. , Optik, 4_7, 123, (1977). [9] Dallas, W.J., Optik, kk_, 45, (1975). - 3 -2 [10] In experimental cases h=10 -10 "~ for an electron microscope objective, but using a scaling factor of •',, it is always pos- sible to normalize to fs=l. (II) Misell, D.L., J. Phys. D., 6, L6, 2200, 2217, (1973); see also Section 2.6. 112 | N is the Shannon number; see I 8]; see also Chapter IV. |13] The image intensity distribution in the image plane has a band- width 48. We introduce 0(f.) to guarantee the bandwidth of |U.(x.) . (See also the discussion concerning Misell's algorithm I 10 ] in Section 2.6). 48 CHAPTER IV THE PHASE PROBLEM IN THE CASK OV WKAK! Y SCATTER INC OBJECTS '!.-.' l>itt>od:ui !L». In previous chapters we derived algorithms for solving !'(•') fruin known intensity distributions. 1H'), the pupil wave function, is the finite Fourier transform of the object wave function (see Eq. (1.2.5)). In the algorithms we did not assume any restrictions concerning the behaviour of P(C) and U (x ), except for the differentiability of l'(t,)> However, due to the structure of many objects (thin/lightly stained), the disturbance of the incident wave will be rather small. These kind of objects will be referred as "weakly scattering" objects. In this chapter we discuss the phase problem for such weakly scattering objects. In particular we discuss the phase problem for the case that the inci- dent wave is a plane wave. 4.1 Theory We consider the unaberrated wave function P(S) in the interval -&<£,<&. Tbe basic equations for describing the phase problem are (see Chapter II): 6 + J S gCS) = P(S)P*CS), (4.1.2) where X-CS) represents the aberrations of the microscope, including the various settings of the defocusing of the microscope, labeled by the index j (j = 1,2). can be written as: PCS) = PRCS) + PCS), for -0<£ ;- K J + / p(t";l)p*(.;:I-f"-3)tj(',','",)d;/ + + / >(f;')P*(f/-C-3)t.(5',C)d-.'. (4.1.4) with c.(x:,y) = exp[ik Cx)-Xj(x-y-B)}]. (4.1.5) Suppose now that B f At J ® (4.1.6) ;|pR( -B -B then, it is possible that the quadratic term in P(O of (4.1.4) can be neglected with respect to the term linear in P(Q and P (-?), provided the linear term does not vanish. By imaging in focus (e.=l), while e.g. P (C) =P£(-£) and P(^) = -P (-£) (phase objects), the linear part in P(^) and P (-£) vanishes. Condition (4.1.6) is necessary but not suffi- cient. If the quadratic term can be neglected, and knowing the function Pn(O> (4.1.4) is a system of Volterra integral equations which can be solved by direct methods as introduced in Chapter II. In the case of (4.1.1) as well as in the case of (4.1.4) we have to solve Volterra integral equations. Thus from a mathematical point of view we cannot expect any particular improvement if we try to solve (4,1.4) instead of (4.1.1) for and arbitrary function PR(£). However, imaging in ab- sence of an object can lead to a strongly peaked pupil wave function (e.g. illumination with a plane wave leads to a sine-function for P(£)). This implies that it is possible to achieve that both P(£) and Pn(£) are R 50 strongly peaked functions. This occurs when the disturbance of the in- cident plane wave is small (see (4.1,6). Kor sake of simplicity we re- present the strongly peaked P ('",) by a P(O = c6(>> a) + I'(-J. (4.1.7) Substituting (4.1.7) in (4.1.1), we obtain for -r"_a: i £.(£) = c P*(a-f:-p)L.(a,-,)+c*] (ii + ->,-0' .(a+-+.-,•) + + / 'hi:')V* (,''-•-ii), .(".1,;,)d--,t. (4.1.8) f, J The linear part in P(",) vanishes for •" a (a-0). For c, =-(3 we obtain: f.(-B) = J |P('')|2d.-'. (4.1.9) J -I3. The quadratic term in P(O of (4.1.8) equals the last term on the right hand side of (4.1.4). Neglecting these quadratic terms in both equat- ions, we notice a great difference between (4.1.4) and (4.1.3). In (4.1.4), P(Q has to be solved from a convolution with P,,('") while, because P (5) in (4.1.8) has the character of a -- function, there nK has to be dealt with a simple algebraic sum of the unknowns P (a-"-B) and P(a+C+S). The mathematical structure of (4.1.8) suggests iterative procedures for the solution of phase problem. In the last section of the present chapter we shall derive iterative algorithms for solving P(O and in Chapter V we shall see, that those iterative algorithms have great advantages as compared with the direct method for solving (4.1.1) or (4.1.4). 4.2 Some aormenis on the weak sixitlcriiij a;>i roxfnat >\'>i For the object wave function in the plane z = z we can write the o relation: U (x ) = U (x ) exp |-f(x ) + i |-f(x ) + U(x )j:''- I, (4 we can approximate (4.2.1) by: U (x ) - I!. I l-r(x ) + i:(x ) I. (4.2. i) a o inc o o If we take a plane wave along the u-axis as i 1 lumiinu ion, we tiave U. (x ) = 1. Then: inc o U (x ) = l-f(x ) + i.'(x ). (4.2.4) o o o o The Fourier transform P(rJ of (4.2.4) can be written as: P(O = M-)tQ(-); Q(:J = !*('".). (4.2.5) For a = 0 (illumination along the z-axis), we obtain a relation be- tween P(O according to (4.2,5) and P(.\) according to (4.1.7) (|c|=l) | 17 1. The approximation according to (4.2.2) and (4.2.4) is the so-called weak phase/amplitude approximation | 2 |. Now, we shall give some critical comments concerning the definition of weak phase/ amplitude approximation and shall show that the condition implied by inequality (4.2.2) is sometimes too severe. Suppose | 52 represent as sine-functions. We take for U (x ) the following function: U (x ) = exp [ i{a sin(2"vBx ) + b sin(2:i4;-;x )} | (4.2.6) o o 4 o o with a << 1 and b >N 1. The inequality (4.2.2) is certainly not valid in the case thai b >> 1, but sin(2:i4Dx ) in the exponent of (4.2.b) will lead to dif- fraction peaks outside the interval -['>• •'.• r, so that they do not contri- bute to the image, because they are filtered away by the stop. The point is that the approximation (4.2.4) and (4.2.2) does not take into account the influence of the transmission function of the microscope. For this reason we formulate more appropriate definitions for the weak approximation. Definition 1 A weakly scattering object is, by definition, an object for which the so-called "scattering percentage within the objective aperture" s, is small (s « 1). s is defined by: 8 „ cs s = / |P(O|2d£/ / |p(O|?d,'. (4.2.7) -8 -3 Definition 2 A semi-weak object is defined as a weakly scattering object, for which the terms linear in P(a+£+|3) and P*(a-t"-S) on the right-hand side of (4.1.8) are, in the interval -(3 Definition 3 A weak object is defined as a weakly scattering object, for which the quadratic term in P(^) of (4.1.8) can be neglected. (The weak phase/amplitude approximation). The condition for s, mentioned in Definition I , has to be valid if 53 we deal with a (setni)-weak object. This follows from (A.2.7), because if s » I, we may not expect that the terms linear in i' and P* in (4,1.8) are large compared to the quadratic term. The condition is necessary, but not sufficient. Now, we shall discuss relation (4.1.8) in further detail for the case a=0. (illumination along the K-axis). If a^O, the case is not completely different, but for the sake of simplicity we restrict the considerations to a=0. Then we obtain from (4,1.8): ; : i : f.(f,) = c tP(^>i-;)+!>(-.'-j-,)} c|m( .+.-.)-u- • \H +.-)-\ {-;-•)-c^h{ +.•) + i : 1 1 + J P(f/)'i *( ,'-'>,- .).t.j(-:',';)d-- , -.••••,•_ 0 (4.2.10) where the overall phase is chosen, such that c it, real and positive. The "amplitude" contrast function|18| CJ (•">.-.) is equal to C (C+6) = Li zl cj(Ojf;)+cj(-:+B,-.) 1 = cos Uo,,j(-,+^.)] , (4.2.11) while i^e obtain for the "phase" contrast function | 18]: C (f +P) = ph - IT ' ^(o.fj-Cja+e,--;) ] = sin i k0Xj(-;+«) 1 , (4.2.12) Defining now: e U?(x.) = / P(U exp I ik x-(Ol exp (-2n ix,5)dC, (4.2.13) 1 B °J we obtain for the image intensity distributions I.(x.): ? 2 2 UCx,) = |c+u{(X))| = |c| +2Re {c u|(X|)} + |W(Xl)| . (4.2.14) According to Definitions 2 and 3 we have to guarantee, that linear terms in P(£) and P (-£) are large with respect to the quadratic term. This implies that for any particular object (and hence for any particular P(£)) we have to choose the defocusing such that the linear part in P(£) and P (-£) is in the interval -g<£<0 as large as 54 possible, which because of (4.2.14) implies that lie 'c I' . ( x ) is as large as possible, while as a consequent:!.' In 'c l.(x )• is small. In the previous Sections o!" this i-h.ipLi-I" Wi' have principles of the imaging process for semi-weak objects. In the (allow- ing sections of (.his chapter we start with the derivation ol the sampling procedure, liased on this sampling procedure we derive itera- tive algorithms, with which the phase problem for semi-weak objects from (defocused) image intensity distributions, either in cor.ihin, t.. n with the intensity distribution in the exit pupil or not, can be skived. •I.S.I We consider the unaberrated pupil wave function l'() in the inter- val ~M^'">':" (•' is the (.angular) extent of the pupil plane (see Section 1.2)). Due to the finite extent o (measured in units •) we can trans- o form the integral equation (4.1.1) into an algebraic equation by ex- panding P(.") according to Lhe '..'hi ttaker - Shannon theorem I 4 |: !'(•;) = >: V sine I ('.- nh)/h] , (4.3.1.1) n= — " where the sarapling distance equals li = •' , while 1' = P(nh) and o n sinc(x) = sin(flx)/;tx. Substituting (4.3.1.1) in (4.1.1), we obtain for f(rJ in the sampling points •'.= kh, Lhe following expression: fk= * PnCCk (m'n) (4-3'K2) n,m = - " with f, = f(kh)/h. (4.3.1.3) The symbol C, (m,n) stand for: R jNh c (m,n) = f / sinc I (••'-iil')/kl sine | (• '-kh-jNh-mh) /hi d: ' k h kh 6 : " n,n-k-!X _' sim- (x)dx = •,1)>n_k_,., ('..).1.4, if kMijN. We have used thai N = ~r/h, from which !•'. = ;Nh. N is the su-c al ] od Shannon-number. WL> have put !'(•') = 0 tor , ' . . I.ft tin1 .sampling points inside -,v-.:i I1- be -,••= -iNli, ljN~l)li, WIRTC tor ronvi'iiiciuf we assurau that N is evun, Wo then have: JN-1 f. = J: I1 P* , ,., ($.}. 1.5) k , n n-k-.N; n=k f, can be computed from the image intensity distributions ](x.). R i In doing this, we have to notice that I(:O IKIS a bandwidth 4.-. Sampl- ing I(x ) accordins to the l.'h it taker-Shannon theorem, we generate the following Riemann sum for the integral: N f, = — Z K—r) exphi(k+j\-WNl 14.3.1.6) k 2Nhr n = -N A" for k = -JN, ,JN-I, where we have taken into account tlie relation (4.3.1.3). Relation (4.3.1.6) permits us to compute f using a Fast Fourier TransOmi I 5 1 [6! . We reserve the symbol P for the unaberrated sampled pupil wave function. Then we obtain for the various defocused images: £k=\ Vn-k-lN Kj (n'n"k) j=1'2 (4-3-U7) where: K.(m,n) = exp | ik {x-(mh) - x•(nh-ANh)r | . (4.3.1.8). In the next section we shall apply the sampling procedure to the case of semi-weak objects. 56 Let the object be illuminated by ;i plane wave U. (x ) = e>;]>(-2 ' ia:-.o) , whore a describes the direction of the incident wave in tlie plane '&-''•• Let us now assume that the point •'=a in the exit pupil, which corre- sponds to this direction, coincides with a sampling point .h: a=.h. Furthermore, we assume that most of the electrons pa.ssinj; this plane are concentrated around "=;i. For that reason we rewrite (4.3.1,1) as follows: P(O = I1, sine | (',- rh)/k| + :. p(nj sine I (:-nli)/l>] , ('i.i.2.1) n = - •• where p(n) = P for n ? •, p(>) = 0 and ]P ' • p . Substituting (A.3.2.I) in (4,1.1), we obtain after a straight- forward calculation a system of equations, (see the way in which (4.3.1.7) has been derived). We consider three different cases (we choose ?.>0). f^ = P P*('--klN)K. k nA j 1N-1 '.'I p(n)p (n-k-jN)K.(n.n-k). (4.3.2.2) n=k J In this case the linear part contains p (i-k-jN) and p(i'+k+jN). ii) -JN<-V J f = P[)p*(Z-k-JN)K.(n,n-k) + JN-I + Y. p(n)p*(n-k-jN)K.(n,n-k). (4.3.2.3) n=k J In this case the linear part contains p (5,-k-jN) only. iii) k>Z, 57 k-- iS-1 , f.J = :: p(n)p*(n-k-(N)K.(n,n-lO (4.3.2.4) n=k Notice that the linear part is absent. I'or k = -(N we obtain: fJ,.=» >.' ;p(n),'K.(ii,n+\Nj+ I1 "K.I. ,• t-iN). O.5.2.J) ~-'N n=-5N J ' •' The .structure of the liquations (A.S.2..!) and ('.,!._'.)) Mii;i;i'su ,m iterative algor i Lhra. In the fo) Lowing sections w>- derive two algo- rithms for solving p from f alonj; tiiese lines. We choosy the overall phase such that I1 is real and positive. A first approximation of I' from (4.3.2.5) is: p (1> 1 r = ! ^JJJ {Kj(.,.+J.\)}" |- (4.J.J.I) Tlie equations (4.3.2.2) - (4.3.2.J) witliout the quadratic term in p in botli equations allow the approximate determination oC p*(..-k-5N) and p(.+k+jN), provided that the determinant of p (.-k-;,N) and p(.'+k+,>:) is non-vanisliing. This certainly is true if the change in delncusiii" [i\z\-~}/fi'' (see also Cliapter 11). In this way the approximation p (£+k+;,N) and p \--k-\S,) is obtained. ( I'J' of Ptj is computed from (4.3.2.5) by substituting the value p (n) of p(n) in the quadratic term. A second approximation p (n) of p(n) is computed from (4.3.2.2) and (4.3.2.J) by giving p(n) in the quadra- (i ) tic term the values p (n). Continuing the iteration scheme, we arrive at: JN-1 (v-1).. f^, = {p/vV K.(;.,£+£N) + :•: |p(n)| K.(';,;+4N>. (4.3.3.2) 5N " J n=-JN J I'o r - J N!+ 1 < k< •' we ob Lain: p*(w(,_k_,N)K (i),_kJ J l) + " E p^'^Ui) p*^"'~ (n-k-J.\)K. (n,»-k) (..J.i.i) n=k ' Equation (4,3,3.3) follows frum (,.'•.3.2.J.) and (•'<.!. 2.J) by put tint; p(n) = 0 for nSN-l. The expression;. (•'<. J. i.2) .ind (A.i.J.i) represent an iterative algorithm for solving 1'(' ) from twi> det ocused image in- tensity distributions. The weak phase/amp Li Hide approximation (see Definition 3 in Section 4.2) corresponds U> the very lirst stej) ol iteration scheme. Tlie approach of previous and present sections ol' this chapter strongly reminds of holography: The various types of holography are connected with the value for •. If •=0 (illumination directed along the £-axis; bright-field imaging), we have the well-known case of in-line (Gabor) holography I 7| . In general we need two different defocused images (j=l,2) in order to compute p (-k-jN) and p(k+jN"). If O.J.M-l, we also need two intensity distributions; in the interval -> -k^. , however, p H,-k-^N) can be computed from a single distribution (PU+k+^N) = 0 for -V. Suppose tliat we are dealing with objects for wliich the weak phase approximation according to Definition .1 (Section 4.2) applies. Illu- minating by a plane wave along the x-auLs, we may expect that P(:,) = - -P*(- f"). In this case it is possible to recover l'(<:) from a single image intensity distribution, provided that the aberrations are ade- quately chosen (see Section A.2). A very specially chosen "aberration" is the phase plate in the phase contrast microscope. If we put |X-plate in the exit pupil just at the position of the spot of the unscattered beam, then P.(in our case 1=0) is transformed into P exp(- Jni) = iP . Considering now a weak phase object, we may put £*(•',) = - P (-r,) . Hence (see(4.3.2.2)): f = -iP p*(-k-jN) + iP p(k+lN) = = 2iP p(k+iN) with -iN+Kk^O, (4.3.4.1) while PQ = P* = |f_iNl^ • (4.3.4.2) In this way p(n) can be computed. Phase plates are very difficult to produce | 12 1 and its adjustment requires a great effort for the elec- tron microscopist. Therefore, in order to achieve phase contrast in electronmicroscopy, we have to resort to the phase shift produced by the aberrations. From (4.3.2.2) we obtain: fk = "Po P where P and P follow from (4.3.4.2). Using symmetry relations we find after a straightforward calculation (see also (4.2.1.2)): 60 f = -2ilJ p(k+jN) lmiK(o,-k) i, (4.3.A.4) K O where we used K(o,-k) = K (k+iN,;N). If 1mlK(o,-kj; does not v.mibh for certain values of k 4 -!N,p(k+jN) can be computed l rum f , and thus Erom a single image intensity distribution. Using the relation (4.3.4.4) and also using the available knowledge concerning the intensity distribution in the exit pupil, L'nwin .mil Henderson | 13 | have shown, that the phase problem in experimental situations can he solved for a particular class ol objects (e.g. thin unstained biological objects like calalase I 15| ). I hough, in principle, the intensity distribution in the exit pupil does not have to be known in this special case, the knowledge of |p(n)| can lead to a more stable algorithm (we discuss this further in Chapter V). In the next section we shall derive an algorithm for the case of (semi)-weak objects, using the intensity distribution in the exit pupil and at least two defocused ima^e intensity distributions. 4.3.i> Iterative aijoi'i.iiitn -\'i' the lwu'.^ti'iwiSo': L>/" ncrr'-:.; .;-• L'^V.';; j'vom two oi' more U('j'^\-;n'c\; inujt a ant to '> ...'.."•'.';« ii;'nt ril-utSx* U2 cte exit [lUi'H. We rewrite (4.3.2.5) for the case k = -5N as: JN-1 f^ - S |p(n)|?K.(n,n+iN) = l'^K. (f , i+^N) . (4.3.5.1) For k=-jN+l ,...,)!. we obtain from (4.3.2.2) and (4.3.5.2) (we put p(£+k+iN) = 0 for k>-Jt): F^ = P. |p*(H-k-|N)K.(«,,H-k)+p(?-+k+^N)K. (k+jN,t+iN) ], (4.3.5.2) where Fu = fu " l p(n)p*(n-k-jN)K.(n,n-k). (4.3.5.3) As long as the intensity distribution in the exit pupil is known, we know |p(n)| and we have only to determine the phase if>(n) of p(p). 61 To this aim we rewrite (4.J.5.J) .is: ?l = ? {|p(''-k-]N)| exp | -i ; (.. -k-J. NJ1 K . t- ,--k) + + |p('+k+lN)| exp | i ; (• +k+;,Xj| K. (.+k+;.N, . +.'N) •. (4.i.rj.4j Let us assurai' that we «.-an détermine the two im.i;;e i nt en.-, j t v distribu- tions witli different stages ot del 'netisi iij; .nul that F fan he iunsidtred an a known quantity in the unknown.'; exp | -i ; (•-k-J\') | .ind exp | ij'(*' + k+l.N) |. In general, we shall I i ml .1 eoiiiplex solution for '. , because there is no guarantee tluit the unknowns have .1 unit modulus. For this reason, we prefer a diflerenL approach. We deline the lol low- ing functional: m G. = I |F.J - ]',,jp(.'-k-!,N)i exp I -i;(.-k-lN)|K.l. ,.-k)- k j= ] k ^ .1 1 i -1' |p(..+k + 2 N)| exp I i:.(.+k+aNJ|K.(.+k+i.N, .+;.N);- . (4.3.5.5) m is tlie number of defoeused images (m '2) | 14 ] . The solution for the phases is identified witli the set of phases ,* (n) which makes G minimal. The search of this minimum can be done iteratively by adopt- ing the following iteration scheme. (V) (V U (v) G = I! |FJ - -P |p(.:-k-iN>! exp |-i,;, (.-k-iX)]K.(.:,;-k) - k j=l k >. j (v/ JpU+k+jN) exp I i1f) ('.:+k+^')| k = -|N+I e, (4.3.5.6) where v numbers the iteration step. We start the procedure by assum- ing ij> (n) = 0 for n = -JN, ....jN-I ( 16 | . For a certain step in v in the iteration scheme, the phases k = n = 2.-k-|N, 3(|)(v)(n) - k = +iN+l, ...,l. (4.3.5.7) Equation (4.3.5.7) has to be satisfied in order to minimize G, with is. 62 respect to ; ("), and gives rise to a relation between ; (.-k-J.N) and ;• (."'+k+jN) bavin}1, tin1 following .structure: (v) (v) ,'' (.'-k-5x) = S|;. (.+k+].\H + ni _,._.A ' (4.3.5.8) with - ;i|~ ' (•-k-lX) ', wliilt- m _ ,,. = 0,1 or -I. U'e obtain two values Cor m , , because (4.S.5.7) leads to ,1 ini n inium .is well as 1~k~ j N to a maximum. Uy inspection we select the v.ilue ut r.i , wl > i <.' 11 minimizes C . The function S |; '(^k+jN)| in ('i.l.').Hi is equal to: K Slv (•+k+1,N) | = [in {[ ): {In | I-:1 .p(.-k-'M) I" K*t • , • -k ) - j=l ' ' 1 - I ;' jp(.-k-jN) I p C • +k+IN) ,K. C.+k+|.N, .+iN)K!v ,.-^] ;]• • cxp \^V^'} (•+k+lN)] •. (4.3.5.9) The minimum o[ G according to (4.3.5.6) is now found by vary- ing the value of ij> (•'+k+;X) between -•: and - and computing the corresponding values of ;• (.-k-jX) according to (4.J.5.8). Now varying .+k+\K) we find a set of minima for G ' . We select from these minima the combination of <; (?+k+iN) and j * (.'-k-jN') which gives the absolute minimum of G. . In this way .' ' (n) is calculated for n = -5N, ..., JN-I. The procedure is terminated when We expect that this procedure will give better results than the algorithm of Section 4.J.3, because of the "taming" effect from the knowledge of |p(n)|. Unfortunately, the search for the absolute mini- mum of G needs a lot of computation time. Therefore, the second K. algorithm can still be a factor 8 slower than the first one. In the next chapter we shall discuss the resuJts using these algorithms applied to simulated intensity distributions including noise. References [ 1 ] Born, M. and Wolf, E., Principles of Optics, (Pergamon), 143, (1965). 63 [2| see e.g.: Schiske, 1'., Eur. Reg. Genf. E I. Micr., Rome, Vo] 1, K(5, (1968). Lenz, F.A., t'l. Micr. in MaL. Science (ed, Valdré, U: Erice Lectures), 541-569, (1971). Hanssen, K.J. Optik, _35_> 431> (ly7-)> Hoenders, B.J., Thesis, State University at CroninHi-n, ( 1 9 7 2 J . Hoenders, B.J., Optik, 35_, lib, (197.2). Miacll, D.L., Adv. in Electronics and VA. Physics, i2_, f>4, (iy7J>. [3] The exit pupil is always finite, so CliaL the elvcIrons scattered in angles larger than the extent of the exit pupil, will not be observed. Consequently a small image contrast will always be observed. Ideal imaging ( 1 1 is not physically realizable. [4] Goodman, J.W., Introduction to Fourier optics (Mc-Graw Hill), p. 21, (1968). [5] Van Toorn, P., Huiser, A.M.J., Ferwerda, H.A., Optik, _5J_, 309, (1978). [6] Singleton, R.C., I.E.E.E. trans, audio electro, acoust., 77, 93, (1969). [7] See 4 , p 205, 206. [8] Misell, D.L., Bürge, R.E., Greenaway, A.H., J. Phys. D, ]_, L27, (1973), Misell, D.L., Greenaway, ibid 832. 19] Leith, E.N. and Upatnieks, J., J. Opt. Soc. Am., 5_4_, 1295, (1964) Bates, R.H.T. and Lewitt, R.M. , Optik, 44_, I, (1975). Lohmann, A.W., Optik, ^J_, 1, (1974). Greenaway, A.H. and Huiser, A.M.J., Optik, 4_5, 295, (1976). see also: Saxton, W.O., J. Phys. D., ]_ L63, (1973) concerning the connection between off-axis holography and Hubert method (8l. [10] Van Toorn, P. and Ferwerda, H.A., Optica Acta, _23, 457, ibid, £3, 469, (1976). [11] Huiser, A.M.J. and Ferwerda, H.A., Optica Acta, 23_, 445, (1976). [ 12] see e.g.: Möllenstedt, G., Speidel, R., Hoppe, W., Langer, R., Katerbau, K.H., Thon, F., 4. Eur. Reg. Conf. El. Micr., Rome Vol. I, 125, (1968). 6A I 13] Unwin, P.N.T. and Henderson, R., J. Mol. Hiol., <^_, 425, (l'J73; I 14 | If m = 1 we can expect problems, because it lias been shown thai a single image intensity distribution with the intensity distri- bution in the exit pupil does not guarantee a unique solution ot the phase problem. See Chapter 11; see also | r) | . I 15 | a gives the position of the peak. We cm obtain .such a peak e.g. by illumination with a plane wave. Then a depends on llie direc- tion of the incident wave; .see Section •'»,), I I6| In the next chapter we verify the a I norithms. In order to justify the validity of the weak phase/amplitude approximation, we start the algorithm with the starting values ; p (n)|= (). [17 | (4.2.5) is an approximation of (4,1.7). [18) C (C+B) is referred to as the amplitude contrast (unction I 2) , am because in the case of weak amplitude objects according to (4.2.4) P(Q - P(-••") vanishes, while in the case of the weak phase approximation according to (4.2.4): !'('.) = ~P(~")i so that C . (C+3) is referred to as the phase contrast function I 2) . (In the case of the weak phase/amplitude approximation according to (4.2.4) the quadratic term in !'(•") of (4.2.10) is neglected 121 ). 65 CHAPTKK V THE POSSIBILITIES FOR APPLYING PHASK KKTRIHVAI. ALGORITHMS IN' KUiCTRDN MICROSCOPY 5. 0 I>it.)'oJ:<.'i ;'..»; In previous chapters, wo .derived several algori thins for solving the phase problem using more than one intensity distribution dutermi' ed from different adjustments of the microscope. It was supposed, that these intensity distributions can be measured in actual situations within any wanted accuracy. However, several problems appear to arise, when measuring them. One of the major problems in electron microscopy is how to cope with the radiation damage of the object, caused by the incident electrons I U. In order to reduce this radiation damage, contamination, etc. ot a particular object we have to lower the dosage on the object. This can result in a very poor signal to noise ratio, especially for (unstained) biological objects. This (quantum) noise is in electron microscopy one of the major limitations for applying phase retrieval algorithms. The algorithms are based on the use of various measured intensity distributions, taken with different adjustments of the microscope (e.g. defocusing). We cannot be sure that after changing the defocusing, no errors like misalignment will arise. This problem of misalignment can to a large extent be overcome by cross-correlation of the different intensity distributions ( 2 ]. Error sources like quantum noise, misalignment I 2) and systematic errors of the detection system have to be taken into account, if we want to verify the practical use of the various phase recovery algo- rithms in electron microscopy. In the present chapter we shall discuss the possibilities of apply- ing the earlier developed algorithms in electron microscopy (see Chap- ters II, III, IV). The conclusions of this chapter will, in the next 66 chapter, lead to a discussion on the usefulness of the different modes of operation of a transmission electron microscope. 5.7 Possibilities for applying dh'CL lit !.'n'i,: 'u ••'< •':'> >. n:''-r',-.;oopy In Chapters II and III we developed algorithms, the so rolled di- rect methods, for solving the phase problem. The first algorithm was developed for solving the phase problem if two inlonsity distributions in the image plane for different defocusings of the microscope ;jre de- termined. This implies, that we have to solve the following system of Volterra integral equations (see (2.1.1) and (2.1.2): e l 1 r t r : f.(O = J dC Pa )P*a - ,-r0expli.'J.U *'-( .'- ,-:-rn j = l,2 (5.1.1) The direct method yields the unique solution F(r.) of (5.1.1). From the examples studied in Chapter II we have seen that the reconstructed P(5) can hardly be distinguished from the input function, because we have avoided all inaccuracies in the calculation of f •(•",) from the chosen input function P(C) as far as possible. Defining the inaccuracy (due to both measuring and calculating er- rors) in f-(C) as Af.(£), the conjugation relation between P(O and P*(-£) only can be satisfied if Af.((;)=0 I 3] (uniqueness of the pro- blem). In our simulations (see Chapter II), Af.(O was very small -12 J (Af.(5)<10 ). (only rounding errors from the computer handling oc- curred) and the conjugation relation was practically satisfied. As soon as Af.(£,);*O, the conjugagion relation is not satisfied and whether or not P(O can be accepted as a solution of the phase problem, depends on which degree of violation of the conjugation relation we are willing to accept. Our computer simulations have shown that even small values of Af.(O (when | Af. (£)|/|f • (O j is a few percent) can lead to a recon- U structed ?(O and P*(-C) with |[ P(OI ^=_p- [P*(-fJ]f =fi | > 1(T . For this reason the direct method (or any other algorithm which computes P(O and P*(-£), e.g. the Newton-Kantorovich algorithm I4||5l) is hardly recommendable using experimental data, where Af.(fJ cannot be neglec- ted. 67 However, if we know the actual values of \'(:) from the intensity distribution in the exip pupil, we ran improve the results of the di- rect method. For each value of •', we obtain a system of two linear al- gebraic equations in the unknowns !'(•".) and \'*(-:.) (see direct method in Chapter II). lioth |P(',)| and |l'*(-:.)! are known. The arguments of P(O and P*(-O are the only unknown quantities. The unknown arguments of P(O and P*(-t") can now be determined by a least squares fit, simi- lar to the algorithm of Section 4.T.5 for the solution of !'(•') in the case of a semi-weak object. Applying such a least squares fit proce- dure, we have obtained a large improvement over the results obtained from the direct method of Chapter II. Hut also in this algorithm we have to calculate P(O for •',=:•', i—li ,....,-:• and we are still faced with an accumulation of errors in P(;,), which may lead to a large inaccu- racy in the argument of P(\) (between the actual phase and the calcu- lated phase, a big difference somewhere between " and 2"1 may occur for 5 near -6). These problems for solving P(f,) from (5.1.1) have great consequen- ces when special operation modes of an eletron microscope are studied. In the next chapter we shall explain that they make phase recovery from only dark-field images a senseless enterprise. The second direct method was developed for solving the phase pro- blem from the image and diffraction patterns. In that case (see Chap- ter III) we have to solve the following system of equations: 6 f(O = / P(e')P*(C'-C-P)dCl. (5.1.2) gU) = P(OP*(O. (5.1.3) As we have seen.this direct method (described in Chapter 111) [ 6 ]| 7] succeeds or fails depending on the behaviour of det B(O. ( 8|(see (3.1.16) and (3.1.17)). This is because the behaviour of det B(f,) de- pends on the functions to be solved (P(O and P*(-r,)). We cannot, how- ever,predict whether the algorithm wil1 succeed or fail. Hence, sol- ving the phase problem from (5.1.2) and (5.1.3) is not recommendable for the solution of actual problems. The application of the algorithm 68 may occasionally lead to the solution of the problem, but success is not guaranteed. In the next section of the present chapter the possibilities of ap- plying the iterative algorithms of Chapter IV will be discussed. These algorithms do not suffer from an accumulation of errors. Moreover, the presence of the linear term of "?(') (see (4.1.8) improves the results of the algorithms drastically, as we shall see. 6.2 Possibilities for a[>]>Ujiiuj the il<:i'*:'..'\v .;,';/'')'.''' limit I- i;.>t'i'-\><' 5.2.0 Introduction In the preceding section of this chapter we mentioned that the di- rect method for solving P(O from two defocused images cannot always be applied, because the result is very sensitive to noise in the mea- sured intensity distributions. It only plays a role in imaging of non- weakly scattering objects ("strong") and in dark-field imaging. In Chapter IV we developed two algorithms for the case of semi-weak objects. These algorithms are based on the presence of a single strong peak in the exit pupil. This peak plays a role similar to that of the reference beam in holographic reconstruction procedures. In the next section the algorithms of Chapter IV will be examined on their sensi- tivety to noise. We assume, that we already have corrected for the misalignment [ 2 ] . We restrict our considerations to the influence of quantum noise. The latter is the most important source of inaccuracy in electron microscopy. 5.Z.I Simulations of noisy intensity distributions and the reconstruc- tion using the iterative algorithms for semi-weak objects In this section we describe the reconstruction of P(O for semi- weak objects from noisy intensity distributions. We use the algorithms described in Section 4.3.3 and 4.3.5, which we briefly denote by Al-1 and Al-2, respectively. We assume that all exposures are taken with a 69 plane wave incident along the axis of the optical system and thai the transmission function of the microscope is known. We consider the else in which we have the fol lowing parameters of the mi i roscupc: r- = 2.10 (measured in units • ). \ = 4 pm (corresponding to an acceleration voltage of rt7 kV). ii = 5000 (measured in units •). o C =1.6 mm. r>.2. I . 1) s With those values we find for the Shannon number N:N~2' .-=J(). The o sampling distance h in the exit pupil is 2./N = 2.ll) . I he sampling distance in the object plane is \x =" /N=(J,) =250, corresponding to a sampling interval of 250' = 1 mil for V (x ). Applying the algorithms Al-1 and Al-2 is only allowed, if the semi- weakness of the object is guaranteed. This can be achieved by choosing the circumstances such that the linear part of Kquations (4.3.2.2) and (4. 3.2. 3) is transmitted as strong as possible (see Section 4.2). i-'or the case of phase objects, the transmission of the terms linear in P(O is described by the phase contrast transmission function: see Section 4.2, especially (4.2.10) and (4.2.12): C . (t>.-)=sin{k |- ~ O(<)4 + — (•.+:•)" |:; -.••••() (5.2.1.2) pn o 4 1 - - The defocusing Az should be chosen as to make 0 . ('.+.) as large as possible in the interval -l-^V'0. Using the values given in (5.2.1.1) this requires a minimal defocusing in the range of about 500-900 nm. I 9 |. The measurements should be conducted such that at least one ex- posure is taken with a defocusing in the range 500-900 nm. The change in defocusing, A =ic/Z«-Az of the second exposure with respect to the first, has to be chosen, such that the determinant of the linear equa- Az 7 tions, which is proportional to sin k \-~- Or)"i , is non-vanishing at the interval -ft^C^Q. Considering the values given in (5.2.1.1), this l is certainly true if A '1000 nm. Again, the defocusing z9 should be chosen such that the transmission of the linear part is as strong as possible, with AZ^^Z.. All these conditions are satisfied in our si- 70 ! mulations when we take Sz =800 nrn, 6z.,= 1000 nm or z9=1300 nm. This choice implies two changes in defocusing: ,\ =200 nm and .". =500 nm. In the discussion of the influence of noise we restrict the calcu- lations to quantum noise. Let L denote the number of detected elec- trons per exposure. We consider the fol lowing choices for 1.: L. = <" (el) (no noise) 1.. = 1012"J (el) : j = 2,...,8. (5.2. 1.3) Since the number of electrons which reach a certain area in tlie image plane has a Poisson distribution, ilic .standard deviation of this num- ber of electrons is equal to the square root of the number of detected electrons. This leads to the following stochastic model of a noisy image. The noisy image intensity distribution over the sampling points k I(k),is the sum of the noiseless distribution l(k) and a stochastic quantity, which we take to be proportional to the standard deviation I 10]: T(k) = I(k) + Ak|I(k)L/Z I(n)]' I I(n)/L (5.2.1.4) ; The proportionality constant ^k(~l ''•' k' ') is a random number genera- ted by the computer. The object wave function U (x ) is assumed to be of the form: U (x ) = explx G.(x ) + ix G.(x )| , x >o and x ^o (5.2.1.5) oo alo pzo a p where x and x are parameters which can be chosen freely. G,(x ) is a p i o generated by the computer through selection, in a random way, of num- bers in the interval (-TT,O). (X G,(X ) describes the amplitude atte- nuation). In this way we avoid any unintended regularity in the con- struction of G (x ). G_(x ) is constructed similarly by selection from 1 o 2 o J the interval (-TT,TT). The scattering percentage s within the aperture is defined by (see also Definition 1 of Chapter IV). 71 JN-1 | p(m) , s = Tv-Y ()..'.1.6) h p(i;O m= -tS We have studied niiu1 combinations of x ami x . Hie corresponding va- lues of s are computed according L<> ( r>. _'. 1 . b) . ]hi- results havi- been 1 is ted in Table 1. TAHI.K I : Scattering percentawes s. X X s -. a P 0.01 0.01 0.05 0.00 0.03 0.35 0.00 0.05 0.95 0.00 0.07 1 .85 0.00 0. 10 3.75 0.00 0.12 5.40 0.00 0.15 8. 30 0.00 0. 18 1 1 .80 0.01 0.20 14.40 Next we shall introduce the definition of the cm struction. Let P (Q be the simulated ("correct") wa' c let P (C) be the reconstructed wave function according to Al-1 or Al-2. Because the wave functions can only be determined apart from a constant phase factor (overall phase), we eliminate a constant phase difference between P (O and P (O by introducing the quantity T: 4N-I , T = I |P - P exp(iv)r (5.2.1.7) m=__,N c,m r,m 72 Wo now compute the value for y, which minimizes T (if the reconstruc- ted wave function equals the correct wave function the minimum is zero). With this computed y, P exp(ii) will he used .is the new value of P . We define the error between I' and I' : r,m c r rJA'-l , JN-I U t = • P -P "/ • I1 ('>..'. 1.8) ,,, c ,in r,m ' ,., ' c,m (the value f=o corresponds to the direction of the incident plane wave). In both cases, Al-I and A1 - 2, we aiV interested in the results of the algorithms after a single interaction .is well ,>s .-irter i(ijl) iterations, if i is the smallest number for which the results of the algorithms do not change any more, when the number of iterations is increased. We are particularly interested in the first iteration, be- cause this corresponds to the result of the weak phase/amplitude ap- proximation (see Definition 3 in Section 4.2). In Table 2 we present the result for one of the combinations of x and x (x =0.0), x =0.2), corresponding to a scattering percentage s=14.A%. c is the error after the first iteration and i . after the th 1 i iteration. We consider various values for the number I. of the de- tected electrons per exposure. Until now the simulations have been per- formed for one-dimensional images. In order to discuss the reconstruc- tion of two-dimensional micrographs, we suppose that the physical cha- racteristics of the object are independent of the y -coordinate: T(xo,yo)*T(xo). We assume a square diafragm in the exit pupil, centred around the z-axis. The length of its sides are taken to be 2t->. This implies that the sampling distance in the x and y directions are the same and oo equal to d=jVl3. In making the transition to two dimensional objects, we assume the object to be divided into strips of width x in the x wo direction and d in the y direction (x =a \= 20 nm and d = 1 nm). We o wo assume that the results of the one dimensional theory apply to such strips. This seems to be a reasonable guess, if there is no signifi- cant change in structural detail in the y -direction within the sam- 73 TABLE 2: Calculations of the errors in the result of the algorithms Al - 1 and Al - 2. x = 0.01, .x = 0.2 and s = 14.4%. a p Al - 1 Al - 2 A = 200 nm A = 500 nm = 200 nm = 500 nm z z £ L 1 c.Z i E,Z c.Z i L I r r 00 49 '1 21 48 -1 18 30 9 27 '1 7 10 .o 47 2 21 47 <] 21 30 -1 9 27 -\ 6 9 .o 47 5 20 47 <\ 18 30 9 27 •-] 7 8 •o 47 8 21 48 2 18 30 10 27 - 1 7 7 ,o 47 32 18 47 7 20 30 10 28 1 7 iO6 - - - 48 17 17 32 1 1 28 - 1 10 105 - - - 54 27 15 47 1 1 r-> 3 1 1 10* 60 10 bO 59 6 pling distance d. Let here L denote the total number of detected elec- trons per strip per exposure (corresponding to the in the case of one-dimensional imaging). The dose 1) per exposure is then given by: '' = dxT t"'/(nm)':' (•''•-'• K9) ' w where 1' is tile detection oft'i c ii-ncy (I_l). In the else 1 = 1 with I he values quoted in (5.2.1.1), we obtain I) = 0.051. e. /(inn)" (S.J.I. 10) It we do not accept errors > larger than 10." we inter Iron consulting Table 2 that L'-I.o. (see (5.2.13)). From (5.2.1.10) we then calculate that D'-~-500 e./(nm)*~. In general, the maximal dose 1) , which can be tolerated in view of the radiation damage is I) RS 30-1000 v/(nm)~. ma x Then it seems to be hardly possibl" to find a compromise between the dose, necessary for obtaining sufficient accuracy,and the limitation of the dose because of radiation damage. A method to overcome this problem lias been proposed by I'nwin and Henderson | 12 1. They used crystalline specimens consisting of a large number of unit cells. The required number of unit cells can he esti- mated as follows: accepting an accuracy of maximal >%, we need a mini- mal dose D per exposure. If we take m exposures (m=2 for Al-I and at least m=3 for Al-2), the total dose equals ml) . If 1' is the maxi- i max mal dose that can be allowed before radiation damage sets in I 11 ], we find that the required number of unit cells, n, has to satisfy: n>n . =mD /I) (5.2. I . 1 I) - nun E max We shall now study the dependence of n . on the scattering percen- tage s. This is done for a numerical example. The simulated object is described by the wave function U (x ,y )=U (x ) specified in (5.2.1.5). This has the advantage that we can use some results of the one-dinen- sionai computations. In Fig. V.I we plot D n . as a function of the 75 scattering percentage s, where the maximal tolerated error < equals 10%. The results of both algorithms Al-1 and AI — 2 have been prest.-nt.ed. The change in defocusing is 500 nm. In Fig. V._'a and V._'b we show for both algorithms • . -> as a function ot s lor two d i I ! erences in defo- cusing (viz. A =500 and .'. =200 nm, corresponding to / =800 nm and Sz =100 or 1300 nm). The dose I) is the name for each micrograph and for each value of s. Kacli simulated intensity di si r i but i on has been generated with the same dose. The dose has been chosen so large that c. This result also holds for the other iombinations (x ,x ). Ihus we con- .1 p elude tentatively, that great improvements mav be exp'ei ted by perfor- ming more than a single iteration, in particular lor s • 1 ''. Further we see that the weak-objeel approximation is only reliable for s().")-!•'', when we tolerate an error of about 10/7. 60 90 120 ^ '50 Fig. V.I: D n . as function of x . 0 results according to max nun s ° Al-1 and + results according to Al-2. A =500 ran. 76 • . 50 • 40 • 30 • 20 • o 10 >• o + - SV. Fig. V.2a: (E -c.) as function of x ('-. 12). 0 results accor- ding to Al-1 and + according to Al-2. .', =500 nm. 1 e, - £,)•/. 1 (b) 40 30 20 IS —- s1/. Fig. V.2b: The same as Fig. V.2a. A =200 nm. 77 Since tin1 rt'.sulls obtained in the prrviuus set ion are based mi nu- merical studies of simulated objotts I Ml, mir • <>m lusinn cannot be mure tlKin tentative. The results show, however, some s. I 1 i < • n t I e.it ur< ::.' 1) The weak pli;ise/ampl i t ude .ipprohim.it i nn, whiih i <>rrespi>nds to the first step ol the iteration already breaks down .it low Mattering pri- centages. Increasing the number ot iterations, whiih means iiulusi.m of the non-linear terms in Kq.s. Hi. 3..!..') .iml (•'<. 1..'. i), give'- tl (•.real improvement as shown in V'\\\. V. i ami especially in lie.. V.J. 2) In connection with the preceding remark, it should be stressed that tlie inlaying should be arranged such as to ensure a strong trans- mission of the part linear in p(k) and >*(k) in liquation (•<. '•.Z.I) and (4.3.2.3) (see also Section 4.2 and the remarks lo]lowing ( T . 2 . I. _') 1 , by choosing the do focus ins in the proper range. 3) The non-linear terra contains useful information about the object which can be retrieved by the algorithms. Hence, tor image processing we do not have to restrict to weak object s. >'ore~<>ver, by increasing s, e.g. by staining of the object or lowering the accelaration Voltage, which leads to a larger phase shift, we can restrict our calculations to a lower number of units cells as can be seen Irom Fig. V.I. I I I By a simple calculation we can estimate the amount of energy loss caused by the incident electrons. The mean free path length of inclas- tically scattered electrons (predominantly due to plasmons) is about 50-150 nm (see e.g. Riemer, L. , Elektronenmikroscopische 1'ntersuchungs- und Praparations-methoden, (Springer),56.U, (1967)). Assuming that every inelastically scattered electron looses 22 eV (22 eV is the va- lue for the inelastic peak in the energy loss-spectrum of carbon) and 2 that the dose in low (f»100 eS./nm ), we are dealing with extremely high 8 9 ^ radiation of 10 till 10 rad. (1 rad.=IOO erg/cm ). This radiation can be compared with the radiation at a distance of 10-100 m from the centre of a small nuclear explosion. This amount of radiation is also 78 found by Lippert, who has shown that 1 C/cm gives a radiation of 10 rad.O00e!/nnT is about IO~ C/cm"). Lippert.. W. , Optik, J_5, 293, (1958). |2| Our computer simulations have shown that the correction for mis- alignment (on the assumption that only translation occurs) can be done for a change of defocusing of almost —T till ——r . See also: Frank,.!., 21" " A ;• ~ Proc. 5th. Eur. Cong. El. Micr., Manchester, 622, (1972); Frank, .J. , Advanced Tech.: in liiol. El. Micr. (ed. J.K. Koeliler; Springer), 2)5, (1973); Al-Ali, I..S., 1'roc. KMA« 75, d-d. J.A. VVnaMcs; Academic Press), 225, (1976). [31 Huiser, A.M.J. and Ferwerda, II.A., Optica Ada, 2/3, 445, (1976). [4) van Toorn, P. and Ferwerda, H.A., Optica Acta, 2_3, 457, (1976). I 5] van Toorn, P. and Ferwerda, H.A. , Optica Acta, 2_3, 469, (1976). 161 Huiser, A.M.J. , van Toorn, P. and Ferwerda, H.A., Optik, 4_7, 1, (1977). |7] van Toorn, P. and Ferwerda, H.A., Optik, V7, 123, (1977). [8] Huiser, A.M.J. and Ferwerda, H.A., Optik, 46, 407, (1976). [9] The influence of C =1.6 mm (>=4pm) can be neglected for a resolu- tion Ax >0.5 nm, so that C (r, + (J) depends only on the defocusing. If B=2.10 a defocusing of 500-900 nm leads to the best approximation of the result, which can be obtained with a -y >-plate. Decreasing r to 6. (B 80 CHAPTER VI THE PHASE PROBLEM TOR DIFFERENT OPERATION MODES 01" A TRANSMISSION ELECTRON MICROSCOPE It.! J In the preceding chapters, where wo developed si'Vcral algorithms lur solving the phase problem, we have assumed lh.it the i 1 lumi 11.1L ion is perfectly coherent. Perfectly coherent as well a.s completely incoherent illuminations are not realizable. In this chapter we discuss the phase problem in the case of partially coherent illumination. In this con- text, we further will compare two operation modes of a transmission electron microscope (TEM), viz. the conventional transmission electron microscope (CTEM) and the scannings transmission electron microscope (STEM). In the Introduction of Chapter V we already mentioned the problem of radiation damage. Radiation damage is caused mainly by heating of tbe specimens which finds its origin in energy loss by incident elec- trons. We briefly discuss the influence of this inelastic scattering on the imaging process. In the last section we will discuss dark-field imaging and the possibilities of applying selective aperture. 6.1 The rhucc ypoblem in CTKM ut:u tTi'/-/ 6.1.1 The Imaging in CTb'M In the case of optics with visible light the imaging process in terms of the mutual intensity distribution for a partially coherent illumination has been described [ 1 ) . Expressions of the same kind can be found for the imaging process in electron microscopy (see e.g. Hoenders [21 ). We restrict our consideration to quasi-monochromatic illumination (|£ ~ io~5, for E = 80 - 100 kV). For the sake of sim- plicity we also restrict the discussions to a single lateral 81 dimension, but requires a more elaborate notation, (If necessary the extension to two dimensions will bo made; see e.g. Tig. VI.2). From [ 1 ] , we take the formula: r(x1>xj) = J dx0(/ dx' ro(xo,^)l)(xo,x|)I)*(x^,xp, (6.1.1.1) o o where F(x.,x ) and V (x ,x') are the mutual intensity distributions in image and object plane, respectively. I)(x ,x.) in the transmission function of the microscope Ml. Considering a "spatially" stationary i 1 luinina'. io-i I 18 | , we obtain the following expression for 1' (x ,x.): i*(x^), (6.1.1.2) where T(x ) is tlie complex transmission function of the object (see Section 1.1) and where ( 3 I : y F(x -x1) = / d I (x ) exp I -2iiix (x -x') ]. (6.1.1.3) oo xss soo a s a and y are the angular limits of the source. As seen from the object plane x is meausured in units i. , w'nere t is the distance between r s s s object plane and source plane. We now shall express the (known) intensity distributions in the image plane 1. (x ) in the (unknown) T(x ), in order to find relations, so that eventually T(x ) (or quantities related to T(x ) can be calcu- lated from the known 1. (x ). (j labels the various stages of defocusing). We suppose that I (x ) is known. Inserting (6.1.1.3) into (6.1.1.1) and using (6.1.1.2) and the rela- tion for D(x ,x') (see Chapter I): 3 D(xo,x'|) = / explik^S) ] exp [ 2-niax^x,) ] , (6.1.1.4) (where x(£) i-s the wave aberration function and 2$ the extent of the pupil), we obtain: 82 I (x ) = Jdx Jdx^Jdx Jd- ' Jd-" ls(xs)T(x )T (x;; 0o oo u -;-, -[•; exp |ik {,<('",' )-•>.(•'") H exp | 2 ' i1. x (x'-x ) + ' ' (x -x, ; + ' "(x ,-x ): o "j sou o 1 1 o with: (6.1.1.5; (6.1.1.6; In this way wu have found a relation beLwoon T(x ; and the known l.(x.). In the next section we derive an equation similar to (,0,1.1.5; for the case of STKM, from wliich we invest igate the correspondence between CTKM and STKM. C.I.!: Trie imuih:; .'/. ,'l:'-l -X, Xj. CTEM STEM Figure VI.1: Schemes of STEM and CTEM. Using geometrical optics, Cowley |4) obtained already that, in principle, there will be no difference between the imaging properties 83 and STEM (see Fig. VI.1). We now shall derive equations similar to (6.1.1.5), form which we hope to understand the correspondence between CTEM and STEM. Consider a plane wave illumination propagating along the z-axis with an amplitude A. Then the wave just in front of the object I'. (x. ) is given by | 12) : r ; Uinc(x ) ~ A j ti/ uxp | iko/.( )| uxp(-j'ii x)) (6.1.2.1) where again j labels the particular liefoeun ing. The wave function behind the object I! (x ;x ) reads: U (x ;x ) = U. (x )T(x ) (6.2.2.2) o p o me p o where x stands for x-x ; x, is the translation by the deflection p 1 o 1 coils. These deflection coils lead to a change in the propagation di- rection of the incident plane wave. \ is measured in units of the focal length; x is measured in units ^ and x is measured in units MX (M is the magnification). The important approximation in STEM is 16.1.2.3). We ciesune that 7(x ) :\; ;V:\;. >:d,. >.: ,•;' ".i.e.). Xotice that according (6.1,2.1) U. (x ) (with x = x-x ) corresponds to D(x ,x ) me p p I o o 1 from (6.1.1.4) in CTEM. The distance i between object plane and detec- tion plane is chosen large, so that the wave function in the detection plane, U.(x jx ), can be derived using the 1'raunbofer approximation: d p a U,(x ,x.) = J dx U (x ;x ) exp(-2nix,x ), (6.1.2.3) dpd Joopo do o where x, has been measured in units I,. The intensity distribution in d o the direction plane can now be written as follows: 0 3 * 2 I;J(x ;x.) = A / dS'/dFJ'/dx /dx'T(x )T (x') exp I ik {X. (5' )-\ •(?"))] a o I -6 -3 o a ° ° ° J J o o {C'(xo-x])+C"(x]-x^)-xd(xo-x^)}l (6.1.2.4) The detector has a finite extent described by the angular range 84 Y Y , Y 3 3 x lj(x ;x )h(x )a Jdx Jdx Jdx lj(xd;x )h(xd)a JdxQ Jdx' /dxd d JdM /df,"h(xd)T(x )T*(x^) a a a a -8 -8 o o (6.1.2.5) We now see that (6.1.1,5) and (6.1.2,5) are similar. The angular limits of the detector and the position of dutuctor corresponds to tlu> angular limits and the position of the source in CTEM (sec also Fig. VI. 1), respectively. The intensity distribution of the source, l.(x_), corre- sponds to the detector sensitivity function h(x.) in STEM. 6.1.S Some special operation modes of CTEM and S'l'EM In this section we shall discuss briefly some special cases of (6.1.1.5), expecially for the solution of the phase problem in CTEM. Due to the analogy of (6.1.1.5) and (6.1.2.5) we can restrict the dis- cussions only to CTEM. If we draw conclusions from (6.1.1.5), similar results apply mutatis mutandis to STEM. Firstly, we derive equations similar to (4.1.8), from which we can solve the phase problem in the case of semi-weak objects for partially coherent illumination. We de- fine f.(£) as the shifted Fourier transform of the known I.(x.): oo £.(£) = /dx.I.(x) exp [ 2nix (£+8)1 • (6.1.3.1) Defining: T (£) = JT(X ) exp(2nix £)dx . (.6.1.3.2) P 0 o o o From (6.1.1.5), we obtain after a straightforward calculation f.(£) = jdx I (x ) Jd£'T (£'-x )T*(£'-£-3-x ) J ctss£p sp s exp I ik {x-CO-X-tC'-C-B)} ] , -3<£<3. (6.1.3.3) 85 We take a central symmetric source, so that I (x ) = l_(-x ) in the interval a T (C) = c In the absence of an object T (?,) is a sine-function, which is approxi- mately a 5-function, because (-S is much larger than Che width of the sine-function. T(O represents the disturbance caused by the object. Inserting (6.1.3.4) into (6.1.3.3) we obtain, choosin« c to be real and positive: f,(Q = c + ic {T(f,+3)T(C3)}C, (f, ph V 3 + /dx I (x ) /dC' T(C'-x )T*(f;l-t>B-x ) a s s s ? s s exp [ikolx.(f,)-x(C'-C-6)} I j=l ,2,..., -6 Here we introduced tue function a J C (C+3) = /dx I (xc) exp lik (x,(x )-x.(x S S S 015 IS -a J J s |xs/Pi, (6.1.3.6) and defined d (C+3) and c\ (C+S) C^ (C+3) = Re CJ (C+3), (6.1.3.7a) j Cjjh (C+6) = Im C (5+3). (6.1.3.7b) 0(x) is defined by: 0(x) = 1 for \x\ < 1 (6.1.3.8a) 0(x) = 0 for |x| > 1. (6.1.3.8b) 36 For the derivation of (6.1.3.5) we have used the symmetry relation 0(x) = 0(-x) and x-(C) = X •(-£)• If the linear part in T and T in (6.1.3.5) is large with respect to the quadratic term, we may apply an iterative algorithm to solve c and T(C), such as the algorithm Al-1 discussed in Section 4.3.3. Figure VI.2a: C (C+3) with 3 = 2.IO"3, a= 10 aim 6z=8OO nm(n=0). Ph "~ "~ ( ) coherent; ( ) partial coherent. Figure VI.2b: As fig. VI.2a with a=5.10"" and 6z=800 nm. 87 Figure VI.2c: As Fig. VI.2a with ct=10~'and t5z=l3O0 nm. 10: Figure VI.2d: As Fig. VI.2a with n=5.10""and Sz=l300 nm. For imaging of the (simulated) objects of Chapter V, we introduce the following parameters of the microscope: C =1.6 mm, Sz. = 800 nm S I 3 and &zo = 1000 or 1300 nm, X = 4 pm and 3= 2.10~ . We introduce the 88 coordinates x , (x ,y ), with x2 + y? In Fig. VI,2 we have plotted C (f>0) witli n=0 and 1 (x )=l. We con- ph — -~ s "*s sider two values of u, viza= 10~'*anda= 5.I0"11, while 6z = 800 or 1300 run. From Fig. VI.2 we notice that, if <' dark.field bright-field dark.field Figure VI.3: Dark-field and bright-field in STEM. For STEM this implies (see Eq. (6.1.2.5)), that the angular limits of the detector have to be much smaller than the extent of the bright- 89 field image in STEM (ct<<3), (see Kij>. 6.3). A second proposal for the solution of the phase problem is to use two exposures with different sources. Let us consider a microscope without aberrations (including defocusing), with which we take two exposures using different sources: one source occupying the interval o f'(C) = c 1 dx 0|(x -.f.-J -Jfi h b IB + c / dx 0|(x +C -IB S S 4 J3 6 -* 3 + / dx / df/ TCC'-x -}P)T (/.'-/.—B-x ) (6.3.2.9) -is s c s After performing the integration with respect to x we obtain: f'(£) = e q,(OT(£+B)+c q (fJT (-ZS) + 1 2 IB 3 . 3 tj dx / dC'T(C'-x -i6)T (£;'-f,~B-x ), ~3<^ where: q,(O = - C for -6<£ b q2(f,) = 3-C for o Analogously, we obtain for the source in the interval -3 / dx / d£' T(C'-x -J3)T (?'-C-jP-x ), -13 s C s (6.3,2.12) 90 Defining the function h~(C) by = f'(Q + f2(C) (6.1.3.13) the following system of equations is obtained: }8 H J dx / dr,'(T (f.'-x -l -{6 s f; s (6.I.3.J4) fgnoring the quadratic part in (6.1.3.14) we can directly calcu- late T(£) in the interval -23's<2B. Applying the iterative procedures as proposed in Section 4.3.3., we can also take into account the quadratic part in T(£), provided that the part linear inTand T is large with respect to the quadratic part. Because of the analogy be- tween STEM and CTEM this procedure for computing T (C) can also be applied in STEM, by using two detectors: one on the interval c'.x 91 as evidenced e.g. by (6.1.1.2) | 31 . In the next section we will discuss briefly the influence of inelastically scattered electrons on the imagine process. G.I.4 Inelasli-v c For STEM it has been shown experimentally that the smaller details of the object can be better observed by applying oneryy filtering |8| . This can be explained if we realize, that the energy loss of the in- cident electrons is "absorbed"' by the object, which predominantly leads to heating of the object (this ran result in radiation damage). Inelastic scattering is mainly due to plasimms | 6 | . If we can neylect changes in the object structure during the exposure, we may represent the inelastic scattering with an energy-loss AE by an appropriate broadening of the source (or detector) in STEM I 14] |8) (effective sources). In the case of plasmons the broadening leads to a Lorentz peak of which the tails extend along a finite portion of the fre- quency axis (6| I 7) | 14 ] . In the case of an energy-loss spectrum we have to convolute the effective source for energy-loss AE with the energy-loss spectrum. This leads to a broadening of the transfer function | 14 J which means a decrease of the degreee of coherence of the illumination. The inelastic scattering occurs in C'l'EM before the electrons are "imaged" by the objective lens (In STEM behind the objective lens). Therefore, we have to take into accaount in CTEM the influence of, chromatic aberrations {13 1 - (6.1.4.!) whsre C is the chromatic aberration coefficient (a typical value is C «s2nun). The presence of C in combination with a realistic value for E in (6,1.4.1) leads (including the broadening of the effective source) to a drastic change in the behaviour of CJ (£+(3) and CJ (£+3) ptn am especially for large values of £» we can obtain small values of CJ(C+3) in (6.1.3.6). Thus, it nuy happen that small details (corre- 92 s pond ing to large values of £) will not be observed in the imago.. Energy filtering now explains why smaller details of the object can be still observed |8| . Energy filtering in STEM (in order to improve the coherence) is rather simple. In CTEM we can separate the elastic and inelastic scattering by introducing filter lenses. Filter lenses however, present severe problems |9) and they are not yet available for commer- cial microscopes. In electron microscopy we can neglect the absorption and back- scattering of the incident electrons by the object. Therefore, the object is a phase object. However, application of energy filtering, such that the inelastically scattered electrons will not be observed, will lead effectively to an object which has to be considered as an amplitude/phase object. This has to be taken into account when choosing the aberrations (defoccusing) in order to obtain a large linear term in T and T in E. (6.1.3.5). 6.1.5 STEM OR CTEM In this section we shall compare CTEM and STEM in connection with the phase problem in the case of semi-weak objects. We assume that the transmission function T(x ) of the object is independent of the (quasi-monochromatic) illumination. STEM CTEM 1) delealion The detector signal and sometimes We use a photographic plate in even the number of counted elec- combination with a high-resolution trons can be stored and pro- densitometer with a digital output, cessed by an on-line computer. suitable for analyzing with a di- gital computer. 2) object and vadialion damage The same as in CTEM, but the A very low dose can be neces- radiation damage can be larger sary, in order to prevent compli- 93 than in CTEM. If the dose in cations with the radiation damage. STEM is the same as in CTEM, A rather poor S/N ratio can be the the dose per second is much result. larger in STEM (we take the exposure time the same in STEM and CTEM). Therefore the ra- diation damage is at least as large as in CTEM. 3) S/N ratio and a ai-iiijlo u'w.i We detect only a fraction »/ The S/N r.-uio depends on the of the electrons ( I I | in STKM, allowable dose I) . Small sources m>i x where 2n is the angular extend are necessary (i • V) in order to of the detector and 23 the obtain a high degree of coherence, angular extend of the bright- causing the part linear in T and f field image ( »< •JJ S/N ratio and more lhan one aourae or de tea tor In STEM we can use an array The phase problem can be solved of small detectors (a<<8) in using various sources (different order to detect all electrons tilted illuminations); see Eqn. forming the bright-field image. (6.1.3.9) et seq. Each detector can give the solu- tion of the phase problem with a very poor S/N ratio. Averaging these solutions for the wave function obtained from each detector separately, (using defocused images), we obtain the solution of the phase pro- blem with the same accuracy as in CTEM from a single small source. For special detection systems with more than one bright-field detector, see (6.1.3.9) et seq. ; I 131 . £.' Illumination Extremely good coherence A rather good coherence is ne- is necessary, e.g. field emission cessary (small sources for the gun (or conventional gun of small solution of the phase problem, extent, obtained by using dia- phragms, etc). The same bright- ness of the source is necessary as in CTEM, (if the source was brighter the radiation damage would increase. The dose in STEM cannot be larger as in CTEM). 6) Energy selection Energy selection is necessary Energy selection is necessary, to improve coherence. STEM with but very difficult. Filterlenses more than one detector needs an are not available for commercial energy selector for each detector microscopes. Summing up our conclusions: In STEM it will be necessary to apply special detection systems with more than one bright-field detector (e.g. for differential phase contrast I 13 ] ). Except for these special kinds of detection systems in STEM, CTEM in combination with filter lenses is more recommendable than STEM. 95 6.2 Dark—field unagitig and oetectlo, ui In the present section we shall discuss dark-field imaging in connection with the phase problem and compare dark-field and bright- field imaging. In bright-field imaging (see 12q. (4.1.8)) we obtain the following equation for the shifted Fourier transform of the intensity distri- bution in the image plane: f(O = c P (a-£-8)r(n,O + cfya+f.+i-iH (n+f.+IV. ) + + / ^(OpV'-C-P.k-.aVJcK.1, (6.2.1) with |P(C)|2= 0 for C>S. The function c is defined by: e(y,x) = exp I iko fx(y) - x(y-x-S)}l. (6.2.2) Dark-field imaging is achieved by intercepting the unscattered beam. This can be done by inserting a stop at £ = a. Consequently we obtain for dark-field imaging: n fd(?) - / P(OP*(V<-8)t(V,OdV (6.2.3) where the superscript d denotes that we consider dark-field imaging. Let U.(x.) be the wave function in the case of dark-field imaging: U.(x ) = / £(O exp(-2nUx )d£. (6.2.4) -B Hence, in the case of dark-field imaging we obtain for the image intensity distribution: d 2 I (x,) = |U,(Xl)| , (6.2.5) while in the case of bright-field imaging: 96 2 I(x() = |c| + 2Re{c exp(-2niax1)U(x1)l + fu-Uj)!'. (6.2.6) Usually, the contrast in a dark-field image is better than in its bright-field counterpart. Yet the signal-to-noise ratio (S/N) is in the same order of magnitude for both cases. We can prove this state- ment by the following calculations. The signal S in dark-field imaging is |U.(s.)|?, from which follows that the noise (Poisson dis- tribution) is proportional to v'S. Hence: S/N (dark-field) « |U.(x )|. (6,2.7) Let us consider semi-weak objects. The aberrations are chosen in such a way that the linear part in P(O and P (-£) is large with respect to the quadratic part. Choosing a=0 we notice from (6.2.6) that this implies that Re U. (x .)>Im U. (x ) and |l). (x ) \< S/N (bright-field) « 2Re 'u\(x ). (6.2.3) We may conclude now that the S/N »"tio for dark-field and bright- field imaging are of the same order. Furthermore, taking into account that the reconstruction of P(£) from (6.2,1) is less sensitive to noise than the reconstruction of P(Q from (6.2.3) (see Chapter V), we conclude that wave function reconstruction based solely on dark field imaging is rather senseless, as is evident when comparing the S/N ratios for bright-field and dark-field imaging. However, dark-field imaging in combination with bright-field imaging can be very useful under special circumstances. Such a situation arises when the inter- action between the incident wave and the object is strong, so that the object can no longer be considered to be semi-weak. In that case the algorithms of Chapter IV cannot be applied. By substracting the bright-field and the dark-field image, we obtain the terms in (6.2.1) which are linear in P(£|) and P (-£)> because the quadratic term in 97 of (6.2.1) is identical to f (C). [ 16 1. Unfortunately, the number of images has to be larger by a factor of 2. In addition to each bright- field image we need a dark-field image, which is unfavourable from the point of view of radiation damage and alignment of the micrographs [ 17 I. The use of special objective diaphragms can also be very useful. Suppose we take the direction of the illumination along the axis, such that the position of the unscnttered beam corresponds to J,=0 in the exit pupil. (We have semi-weak objects in mind). Consider now a diaphragm with an aperture given by Y^C^» (F.^0 or »•'()). Then by definition, we will obtain a dark-field image. If P(O has a single strong peak for £=p with y_p<>:, we are dealing with a special kind of dark-field imaging. In analogy with the calculations of Chapter IV, we now rewrite P(O as: Pg(O = c,a(C-p) + PS(O, Y<5 where the subscript reminds that we have chosen a special selective aperture y<(,^,<- Now we obtain for f (C): f?(O = c. Pc(p+?-Y)e,(pH-Y,fJ + c P*(p-r+Y)F;(p,O + J IS J 1 b J (6.2.10) with: (6.2.11) (j labels the various stages of defocusing of Che microscope). We now have obtained the same equations as have been derived in Chapter IV,' and the same kind of algorithms as in Chapter IV can be applied. The difference befw.en (6.2.1) and (6.2.10) is that the part linear in P(C) in (6.2.1) is obtained by interference with the unscattered wave, 98 while in (6.2.10) we have an interference with the peak at f.=p as follows from (6.2.10). P(£) can be computed, however, only in the interval Y'5<|C- The physical circumstances for which (6.2.10) is as- sumed to be valid, can be obtained e.g. for metallurgic objects by selecting the diaphragm such that it contains a single strong i. ak in the exit pupil (usually referred to a diffraction peak or Bragg-peak). In the case of partiallly coherent illumination we can obtain in stead of (6.2.10) equations similar to (6.1.1.5). Then the samu pro- cedures can be applied as have been introduced in the first sections of this chapter. References | 1 ] Beran, M.J. and Parrent, G.B., Theory of partial coherence, (Prentice-Hall), (1964). I 2] Hoenders, B.J., Thesis, State University at Groningen, (1972). I 3) We ignore inelastic scattering (for more details, see 6.1.4). Otherwise T(x -x') would be energy dependent. 14] Cowley, J.M., Appl. Phys. Let., Jj>, 58, (1969). 15] Frank, J., Optik, 38, 519, (1973). 16] Burge, R.E., J. Micros. (G.B.), 98, 251, (1973); see also 171. 17] Pines, D. and Bohm, D. , Phys. Rev., 85, 338, (1952). [8] Wilska, A.P., Quant. El. Micr. (ed. Bahr, G.F. and Zeitler, E.H.), 325, (1964). 1 101 Unwin, P.M.T. and Henderson, R., J. Mol. Biol., £4, 425, (1975). (Ill We suppose that in STEM the number of electrons in the dark-field image can be neglected with respect to the electrons in the bright-field image (see Fig. VI.3). [ 12 ] If we suppose that the illimination in STEM is a plane wave (the source is a point source), does not mean that coherence is sup- posed, because the extent of the detector determines, as we will see, the degree of coherence. I 13] Dekkfrs, N.H. and de Lang, H., Optik, 4^ 452, (1974). ( 14] Huiscr, A.M.J., Thesis, State University at Groningen, (1979). [ 15] Glaser, W., Grundlagen der Elektronen Optik, (Springer), Chapter XII, (1952). 99 [ 161 Frank, J., Optik, 28, 582, (1973). I 17J A better way to obtain P(C), if the object is a strong, not weakly scattering object, is to increase the acceleration vol- tage. The phase shift is proportional to the wave length, (see Chapter I). I 18] Spatial stationary illumination means that l'(x ,x') depends on the difference of x and x^ (r(x ,x^) = f(x -x')); see also I 1 |. 100 Summary In this thesis we discuss the phase problem in electron microscopy, i.e. the determination of the unknown complex wave function in the image plane or in the exit pupil from the measured intensity distri- butions in both planes. The calculation of the wave function is the first problem to be solved for the determination of the object struc- ture from electron micrographs. We first assume -.imporal ly as well as spatially coherent illumina- tion. Relations arc derived between the unknown wave function in the exit pupil, P(C), and the Fourier transforms f(f.) of the image inten- sity distributions, obtained as exposures with various stages of the defocusing of the microscope, At first we restrict the discussions to a single lateral dimension. The resulting Volterra integral equations can be solved numerically by the application of a so-called direct method, which is developed for this purpose. Next the extension of this direct method to two lateral dimensions is made. The algorithm is verified by means of simulated objects. Analogously, a direct method is developed for the solution of P(£) from the intensity dis- tributions in the exit pupil and image plane. The results obtained with the latter direct method are in agreement with the theoretical pr. fictions. Both direct methods are compared with the algorithms proposed by Misell and Gerchberg/Saxton. It turns out that the direct methods give more accurate results than the other algorithms. In the case of semi-weak objects (and therefore bright-field ima- ging) an iterative algorithm is derived for solving the 'scattered' wave function P(O from defocused images. This algorithm is less sen- sitive to noise, because in the equation relating f(£) to "?(O and P*(-£) the terms linear in P(£) and P*(-£) are large with respect to the non-linear term. Since we get no accumulation of errors as in the direct method, the obtained solutions for P(O are less sensitive to the (quantum) noise, as has been confirmed by computer simulations. The best results are obtained, when the scattering percentage is so large, that the traditional weak phase/amplitude approximation is not valid, (but the object is still semi-weak). Using in addition the in- 101 tensity distribution in the exit pupil, a second iterative algorithm has been developed, which means a great improvement as compared to the results obtained with the first algorithm. The phase problem in the case of partially coherent illumination is also studied. Special cases for the solution fo the phase problem are considered. We show that the relations, which describe the ima- ging in STEM and CTEM, are analogous. We argue that CTEM, especially when filter lenses are applied, is to be prefered to STEM, because of the better signal to noise ratio in CTEM. Only in the case that more than one (and especially chosen) bright-field detectors are used in STEM, both operation modes of TEM are comparable. Next we study the advantages and disadvantages of bright-field and dark-field imaging. The signal to noise ratios in bright-field and dark-field imaging are in the same order. Taking also into account that in bright-field imaging the iterative algorithms can be used, while in dark-field imaging the non-linear Volterra integral equations have to be solved, then the application of dark-field imaging in stead of bright-field imaging is senseless. 102 Samenvatting In dit proefschrift wordt het fase probleem in de electronen rai- croscopie besproken. Het fase probleem is gedefinieerd als het bepa- len van de golffunctie in het beeldvlak dan wel de golffum-tie in de uittreepupil uitgaande van gemeten intensiteitsverdelingen in beide vlakken. De berekening van de golffunctie is het eerste probleem dat opgelost dient te worden, indien we liet voorwerp willen reconstrueren uit electronon microscopische opnamen. We veronderstellen eerst dat de belichting ruimtelijk coherent is en coherent in de tijd. Relaties tussen de onbekende golffunctie P(C) in de uittreepupil en de Fourier getransformeerden f(C) van de intensiteitsverdelingen in het beeldvlak worden afgeleid. De betref- fende intensiteitsverdelingen zijn verkregen door middel van verschil- lende gedefocuseerde opnamen. De hieruit volgende Volterra integraal vergelijkingen kunnen numeriek opgelost worden door middel van de di- recte methode, die in detail besproken wordt. Tevens wordt de directe methode beschreven voor twee dwars dimensies. Het algorithme voor één dimensie is getoest met behulp van gesimuleerde voorwerpen. Er van uitgaande dat de intensiteitsverdelingen in de uittreepupil, g(5), en in het beeldvlak bekend zijn, is een directe methode afgeleid, die analoog is aan de eerste methode. De resultaten die verkregen zijn met de tweede directe methode, zijn in overeenstemming met de theorie. Tevens zijn de resultaten uit de directe methoden vergeleken met de resultaten uit de algorithtnen van Misell en Gerchberg/Saxton. Hier- uit volgt dat de directe methoden beter voldoen dan de laatst genoem- de algorithmen. Voor het geval van semi-zwakke voorwerpen (helder veld belichting) is er een iteratief algorithme afgeleid voor het oplossen van P(C), waarbij P(5) de verstrooiing door het voorwerp voorstelt. Er wordt weer gebruik gemaakt van gedefocuseerden beelden. Het voordeel van dit iteratieve algorithme ten opzichte van de directe methode is, dat er een, in hoofdzaak, lineair verband bestaat tussen f(5) enerzijds en P(5) en P*(-£) anderzijds. Dit betekent dat het oplossen van P/S) minder gevoelig voor (quantum) ruis is dan wanneer P(5) opgelost dient 103 te worden uit de niet-lineaire Volterra integraal vergelijkingen. Verreweg de beste resultaten kunnen worden verkregen, indien de ver- strooiing zo groot is, dat de traditionele zwak fase/amplitude bena- dering niet meer voldoet. Indien we tevens g(O kennen, kunnen we ge- bruik maken van een tweede iteratief algorithms, die een aanzienlijke verbetering geeft vergeleken met het eerste algorithme. Ook voor partieel coherente belichtingen is het fase probleem be- studeerd. Het oplossen van het fase probleem wordt beschreven voor speciale gevallen. Afgeleid wordt ook, dat de relaties die de af- beelding beschrijven in CTEM en STEM, analoog zijn. Echter, indien we rekening houden met betere signaal ruis verhouding in CTKM ten opzichte van STEM, verdient het aanbeveling CTEM toe te passen in plaats van STEM, met name indien filter lensen worden toegepast. Alleen in het geval dat een speciale combinatie van helder-veld detec- toren wordt toegepast in STEM, zijn beide afbeeldings systemen van STEM vergelijkbaar. Verder bestuderen we de voor- en nadelen van helder-veld en don- ker-veld microscopie. Voor helder-veld belichting kunnen we de itera- tieve algorithmen toe passen, terwijl voor donker-veld belichting niet lineaire Volterra integraal vergelijkingen opgelost dienen te worden. Indien we er tevens rekening mee houden, dat de signaal ruis verhoudingen voor donker-veld en helder-veld belichting in de zelfde orde van grootte liggen, is de toepossing van donker-veld belichting in plaats van helder-veld belichting zinloos. 104