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P1: FLV 2nd Revised Pages Qu: 00, 00, 00, 00 Encyclopedia of Physical Science and Technology EN002-95 May 19, 2001 20:57 Charged -Particle Optics P. W. Hawkes CNRS, Toulouse, France I. Introduction II. Geometric Optics III. Wave Optics IV. Concluding Remarks GLOSSARY Image processing Images can be improved in various ways by manipulation in a digital computer or by op- Aberration A perfect lens would produce an image that tical analog techniques; they may contain latent infor- was a scaled representation of the object; real lenses mation, which can similarly be extracted, or they may suffer from defects known as aberrations and measured be so complex that a computer is used to reduce the by aberration coefﬁcients. labor of analyzing them. Image processing is conve- Cardinal elements The focusing properties of optical niently divided into acquisition and coding; enhance- components such as lenses are characterized by a set ment; restoration; and analysis. of quantities known as cardinal elements; the most im- Optic axis In the optical as opposed to the ballistic study portant are the positions of the foci and of the principal of particle motion in electric and magnetic ﬁelds, the planes and the focal lengths. behavior of particles that remain in the neighborhood Conjugate Planes are said to be conjugate if a sharp im- of a central trajectory is studied. This central trajectory age is formed in one plane of an object situated in the is known as the optic axis. other. Corresponding points in such pairs of planes are Paraxial Remaining in the close vicinity of the optic axis. also called conjugates. In the paraxial approximation, all but the lowest order Electron lens A region of space containing a rotationally terms in the general equations of motion are neglected, symmetric electric or magnetic ﬁeld created by suit- and the distance from the optic axis and the gradient of ably shaped electrodes or coils and magnetic materials the trajectories are assumed to be very small. is known as a round (electrostatic or magnetic) lens. Scanning electron microscope (SEM) Instrument in Other types of lenses have lower symmetry; quadrupole which a small probe is scanned in a raster over the sur- lenses, for example, have planes of symmetry or face of a specimen and provokes one or several signals, antisymmetry. which are then used to create an image on a cathoderay Electron prism A region of space containing a ﬁeld in tube or monitor. These signals may be X-ray inten- which a plane but not a straight optic axis can be deﬁned sities or secondary electron or backscattered electron forms a prism. currents, and there are several other possibilities. 667

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P1: FLV 2nd Revised Pages Encyclopedia of Physical Science and Technology EN002-95 May 19, 2001 20:57 668 Charged -Particle Optics Scanning transmission electron microscope (STEM) are many devices in which charged particles travel in a As in the scanning electron microscope, a small probe restricted zone in the neighborhood of a curve, or axis, explores the specimen, but the specimen is thin and the which is frequently a straight line, and in the vast major- signals used to generate the images are detected down- ity of these devices, the electric or magnetic ﬁelds exhibit stream. The resolution is comparable with that of the some very simple symmetry. It is then possible to describe transmission electron microscope. the deviations of the particle motion by the ﬁelds in the Scattering When electrons strike a solid target or pass familiar language of optics. If the ﬁelds are rotationally through a thin object, they are deﬂected by the lo- symmetric about an axis, for example, their effects are cal ﬁeld. They are said to be scattered, elastically closely analogous to those of round glass lenses on light if the change of direction is affected with negligible rays. Focusing can be described by cardinal elements, and loss of energy, inelastically when the energy loss is the associated defects resemble the geometric and chro- appreciable. matic aberrations of the lenses used in light microscopes, Transmission electron microscope (TEM) Instrument telescopes, and other optical instruments. If the ﬁelds are closely resembling a light microscope in its general not rotationally symmetric but possess planes of symme- principles. A specimen area is suitably illuminated by try or antisymmetry that intersect along the optic axis, they means of condenser lenses. An objective close to the have an analog in toric lenses, for example the glass lenses specimen provides the ﬁrst stage of magniﬁcation, and in spectacles that correct astigmatism. The other important intermediate and projector lens magnify the image fur- ﬁeld conﬁguration is the analog of the glass prism; here ther. Unlike glass lenses, the lens strength can be varied the axis is no longer straight but a plane curve, typically at will, and the total magniﬁcation can hence be varied a circle, and such ﬁelds separate particles of different en- from a few hundred times to hundreds of thousands of ergy or wavelength just as glass prisms redistribute white times. Either the object plane or the plane in which the light into a spectrum. diffraction pattern of the object is formed can be made In these remarks, we have been regarding charged par- conjugate to the image plane. ticles as classical particles, obeying Newton’s laws. The mention of wavelength reminds us that their behavior is also governed by Schro¨dinger’s equation, and the resulting OF THE MANY PROBES used to explore the structure description of the propagation of particle beams is needed of matter, charged particles are among the most versa- to discuss the resolution of electron-optical instruments, tile. At high energies they are the only tools available notably electron microscopes, and indeed any physical ef- to the nuclear physicist; at lower energies, electrons and fect involving charged particles in which the wavelength ions are used for high-resolution microscopy and many is not negligible. related tasks in the physical and life sciences. The behav- Charged-particle optics is still a young subject. The ior of the associated instruments can often be accurately ﬁrst experiments on electron diffraction were made in the described in the language of optics. When the wavelength 1920s, shortly after Louis de Broglie associated the notion associated with the particles is unimportant, geometric of wavelength with particles, and in the same decade Hans optics are applicable and the geometric optical proper- Busch showed that the effect of a rotationally symmet- ties of the principal optical components—round lenses, ric magnetic ﬁeld acting on a beam of electrons traveling quadrupoles, and prisms—are therefore discussed in de- close to the symmetry axis could be described in optical tail. Electron microscopes, however, are operated close terms. The ﬁrst approximate formula for the focal length to their theoretical limit of resolution, and to understand was given by Busch in 1926–1927. The fundamental equa- how the image is formed a knowledge of wave optics is tions and formulas of the subject were derived during the essential. The theory is presented and applied to the two 1930s, with Walter Glaser and Otto Scherzer contribut- families of high-resolution instruments. ing many original ideas, and by the end of the decade the German Siemens Company had put the ﬁrst commercial electron microscope with magnetic lenses on the market. I. INTRODUCTION The latter was a direct descendant of the prototypes built by Max Knoll, Ernst Ruska, and Bodo von Borries from Charged particles in motion are deﬂected by electric and 1932 onwards. Comparable work on the development of magnetic ﬁelds, and their behavior is described either by an electrostatic instrument was being done by the AEG the Lorentz equation, which is Newton’s equation of mo- Company. tion modiﬁed to include any relativistic effects, or by Subsequently, several commercial ventures were Schro¨dinger’s equation when spin is negligible. There launched, and French, British, Dutch, Japanese, Swiss,

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P1: FLV 2nd Revised Pages Encyclopedia of Physical Science and Technology EN002-95 May 19, 2001 20:57 Charged -Particle Optics 669 American, Czechoslovakian, and Russian electron micro- II. GEOMETRIC OPTICS scopes appeared on the market as well as the German instruments. These are not the only devices that depend A. Paraxial Equations on charged-particle optics, however. Particle accelerators Although it is, strictly speaking, true that any beam of also use electric and magnetic ﬁelds to guide the parti- charged particles that remains in the vicinity of an arbi- cles being accelerated, but in many cases these ﬁelds are trary curve in space can be described in optical language, not static but dynamic; frequently the current density in this is far too general a starting point for our present pur- the particle beam is very high. Although the traditional poses. Even for light, the optics of systems in which the optical concepts need not be completely abandoned, they axis is a skew curve in space, developed for the study of do not provide an adequate representation of all the prop- the eye by Allvar Gullstrand and pursued by Constantin erties of “heavy” beams, that is, beams in which the cur- Carathe´odory, are little known and rarely used. The same rent density is so high that interactions between individual is true of the corresponding theory for particles, devel- particles are important. The use of very high frequencies oped by G. A. Grinberg and Peter Sturrock. We shall in- likewise requires different methods and a new vocabulary stead consider the other extreme case, in which the axis that, although known as “dynamic electron optics,” is far is straight and any magnetic and electrostatic ﬁelds are removed from the optics of lenses and prisms. This ac- rotationally symmetric about this axis. count is conﬁned to the charged-particle optics of static ﬁelds or ﬁelds that vary so slowly that the static equations can be employed with negligible error (scanning devices); 1. Round Lenses it is likewise restricted to beams in which the current den- We introduce a Cartesian coordinate system in which the z sity is so low that interactions between individual parti- axis coincides with the symmetry axis, and we provision- cles can be neglected, except in a few local regions (the ally denote the transverse axes X and Y . The motion of a crossover of electron guns). charged particle of rest mass m0 and charge Q in an elec- New devices that exploit charged-particle optics are trostatic ﬁeld E and a magnetic ﬁeld B is then determined constantly being added to the family that began with the by the differential equation transmission electron microscope of Knoll and Ruska. Thus, in 1965, the Cambridge Instrument Co. launched (d/dt)(γ m 0v) = Q(E + v × B) the ﬁrst commercial scanning electron microscope after 2 2 −1/2 γ = (1 − v /c ) , (1) many years of development under Charles Oatley in the Cambridge University Engineering Department. Here, the which represents Newton’s second law modiﬁed for image is formed by generating a signal at the specimen by relativistic effects (Lorentz equation); v is the veloc- scanning a small electron probe over the latter in a regu- −19 ity. For electrons, we have e = −Q ≃ 1.6 × 10 C and lar pattern and using this signal to modulate the intensity e/m0 ≃ 176 C/µg. Since we are concerned with static of a cathode-ray tube. Shortly afterward, Albert Crewe of ﬁelds, the time of arrival of the particles is often of no the Argonne National Laboratory and the University of interest, and it is then preferable to differentiate not with Chicago developed the ﬁrst scanning transmission elec- respect to time but with respect to the axial coordinate z. tron microscope, which combines all the attractions of a A fairly lengthy calculation yields the trajectory equations scanning device with the very high resolution of a “con- ( ) 2 2 d X ρ ∂g ∂g ventional” electron microscope. More recently still, ﬁne ′ = − X 2 electron beams have been used for microlithography, for dz g ∂ X ∂z in the quest for microminiaturization of circuits, the wave- Qρ [ ] ′ ′ ′2 length of light set a lower limit on the dimensions attain- + Y (Bz + X BX ) − BY (1 + X ) g able. Finally, there are, many devices in which the charged ( ) 2 2 particles are ions of one or many species. Some of these d Y ρ ∂g ′ ∂g = − Y 2 operate on essentially the same principles as their electron dz g ∂Y ∂z counterparts; in others, such as mass spectrometers, the Qρ [ ] ′ ′ ′2 presence of several ion species is intrinsic. The laws that + −X (Bz + Y BY ) + BX (1 + Y ) (2) g govern the motion of all charged particles are essentially 2 ′2 ′2 the same, however, and we shall consider mainly electron in which ρ = 1 + X + Y and g = γ m0v. optics; the equations are applicable to any charged par- By specializing these equations to the various cases of ticle, provided that the appropriate mass and charge are interest, we obtain equations from which the optical prop- inserted. erties can be derived by the “trajectory method.” It is well

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P1: FLV 2nd Revised Pages Encyclopedia of Physical Science and Technology EN002-95 May 19, 2001 20:57 670 Charged -Particle Optics known that equations such as Eq. (1) are identical with the Euler–Lagrange equations of a variational principle of the form ∫ t1 W = L(r, v, t) dt = extremum (3) t0 provided that t0, t1, r(t0), and r(t1) are held constant. The Lagrangian L has the form FIGURE 1 Paraxial solutions demonstrating image formation. 2 2 2 1/2 L = m0c [1 − (1 − v /c ) ] + Q(v · A − ) (4) in which and A are the scalar and vector potentials we ﬁnd corresponding to E, E = −grad and to B, B = curl A. ′′ ′ ′ ′′ 2 2 ˆ ˆ For static systems with a straight axis, we can rewrite x + γφ x /2φ + [(γ φ + η B )/4φ]/x = 0 (10) Eq. (3) in the form ′′ ′ ′ ′′ 2 2 ˆ ˆ y + γφ y /2φ + [(γ φ + η B )/4φ]/y = 0. ∫ z1 ′ ′ S = M(x, y, z, x , y ) dz, (5) These differential equations are linear, homogeneous, z0 and second order. The general solution of either is a linear where combination of any two linearly independent solutions, and this fact is alone sufﬁcient to show that the corre- ′2 ′2 1/2 M = (1 + X + Y ) g(r) sponding ﬁelds B(z) and potentials φ(z) have an imaging ′ ′ +Q(X AX + Y AY + Az). (6) action, as we now show. Consider the particular solution h(z) of Eq. (10) that intersects the axis at z = z0 and z = zi The Euler–Lagrange equations, (Fig. 1). A pencil of rays that intersects the plane z = zo at ( ) ( ) d ∂M ∂M d ∂M ∂M some point Po(xo, yo) can be described by = ; = (7) ′ ′ dz ∂X ∂X dz ∂Y ∂Y x(z) = x og(z) + λh(z) (11) again deﬁne trajectory equations. A very powerful method y(z) = yog(z) + µh(z) of analyzing optical properties is based on a study of the in which g(z) is any solution of Eq. (10) that is linearly function M and its integral S; this is known as the method of characteristic functions, or eikonal method. independent of h(z) such that g(zo) = 1 andλ,µ are param- eters; each member of the pencil corresponds to a different We now consider the special case of rotationally sym- metric systems in the paraxial approximation; that is, we pair of values of λ, µ. In the plane z = zi, we ﬁnd examine the behavior of charged particles, speciﬁcally x(zi) = xog(zi); y(zi) = yog(zi) (12) electrons, that remain very close to the axis. For such par- ticles, the trajectory equations collapse to a simpler form, for all λ and µ and hence for all rays passing through Po. namely, This is true for every point in the plane z = zo, and hence the latter will be stigmatically imaged in z = zi. ′ ′′ ′ γ φ γ φ ηB ηB ′′ ′ ′ Furthermore, both ratios and x(zi)/xo and y(zi)/yo are X + X + X + Y + Y = 0 ˆ ˆ ˆ1/2 ˆ1/2 2φ 4φ φ 2φ equal to the constant g(zi), which means that any pattern of (8) points in z = zo will be reproduced faithfully in the image ′ ′′ ′ γ φ γ φ ηB ηB ′′ ′ ′ plane, magniﬁed by this factor g(zi), which is hence known Y + Y + Y − X − X = 0 ˆ ˆ ˆ1/2 ˆ1/2 2φ 4φ φ 2φ as the (transverse) magniﬁcation and denoted by M. The form of the paraxial equations has numerous other in which φ(z) denotes the distribution of electrostatic consequences. We have seen that the coordinate frame x– ˆ potential on the optic axis, φ(z) = (0, 0, z); φ(z) = 2 y–z rotates relative to the ﬁxed frame X–Y –Z about the φ(z)[1 + eφ(z)/2m0c ]. Likewise, B(z) denotes the mag- optic axis, with the result that the image will be rotated netic ﬁeld distribution on the axis. These equations are with respect to the object if magnetic ﬁelds are used. In coupled, in the sense that X and Y occur in both, but this an instrument such as an electron microscope, the image can be remedied by introducing new coordinate axes x, therefore rotates as the magniﬁcation is altered, since the y, inclined to X and Y at an angle θ(z) that varies with z; latter is affected by altering the strength of the magnetic x = 0, y = 0 will therefore deﬁne not planes but surfaces. ﬁeld and Eq. (9) shows that the angle of rotation is a func- By choosing θ(z) such that tion of this quantity. Even more important is the fact that 1/2 1/2 ˆ dθ/dz = ηB/2φ ; η = (e/2m0) , (9) the coefﬁcient of the linear term is strictly positive in the

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P1: FLV 2nd Revised Pages Encyclopedia of Physical Science and Technology EN002-95 May 19, 2001 20:57 Charged -Particle Optics 671 case of magnetic ﬁelds. This implies that the curvature of any solution x(z) is opposite in sign to x(z), with the result that the ﬁeld always drives the electrons toward the axis; magnetic electron lenses always have a convergent action. The same is true of the overall effect of electrostatic lenses, although the reasoning is not quite so simple. A particular combination of any two linearly indepen- dent solutions of Eq. (10) forms the invariant known as the Wronskian. This quantity is deﬁned by 1/2 ′ ′ 1/2 ′ ′ ˆ ˆ φ (x1x 2 − x1x2); φ (y1y2 − y1y2) (13) Suppose that we select x1 = h and x2 = g, where h(zo) = h(zi) = 0 and g(zo) = 1 so that g(zi) = M. Then 1/2 ′ 1/2 ′ ˆ ˆ φ h = φ h M (14) o o i i ′ ′ The ratio h i/ho is the angular magniﬁcation MA and so 1/2 ˆ ˆ MMA = (φo/φi) (15) or MMA = 1 if the lens has no overall accelerating effect ˆ ˆ 1/2 and hence φo = φi. Identifying φ with the refractive in- dex, Eq. (15) is the particle analog of the Smith–Helmholtz formula of light optics. Analogs of all the other optical laws can be established; in particular, we ﬁnd that the lon- gitudinal magniﬁcation Ml is given by. 1/2 2 ˆ ˆ Ml = M/MA = (φi/φo) M (16) FIGURE 2 Typical electron lenses: (a–c) electrostatic lenses, of which (c) is an einzel lens; (d–e) magnetic lenses of traditional and that Abbe’s sine condition and Herschel’s condition design. take their familiar forms. We now show that image formation by electron lenses These are rays that arrive at or leave the lens parallel can be characterized with the aid of cardinal elements: to the axis (Fig. 3). As usual, the general solution is foci, focal lengths, and principal planes. First, however, ¯ x(z) = αG(z) + βG(z), where α and β are constants. We we must explain the novel notions of real and asymp- denote the emergent asymptote to G(z) thus: totic imaging. So far, we have simply spoken of rotation- ally symmetric ﬁelds without specifying their distribution lim G(z) = Gi(z − zFi) (18) z→∞ in space. Electron lenses are localized regions in which ¯ the magnetic or electrostatic ﬁeld is strong and outside of We denote the incident asymptote to G(z) thus: which the ﬁeld is weak but, in theory at least, does not ¯ ¯ ′ lim G(z) = G o(z − zFo) (19) vanish. Some typical lens geometries are shown in Fig. 2. z→−∞ If the object and image are far from the lens, in effec- tively ﬁeld-free space, or if the object is not a physical specimen but an intermediate image of the latter, the im- age formation can be analyzed in terms of the asymptotes to rays entering or emerging from the lens region. If, how- ever, the true object or image is immersed within the lens ﬁeld, as frequently occurs in the case of magnetic lenses, a different method of characterizing the lens properties must be adopted, and we shall speak of real cardinal elements. We consider the asymptotic case ﬁrst. It is convenient to introduce the solutions of Eq. (10) that satisfy the boundary conditions ¯ lim G(z) = 1; lim G(z) = 1 (17) z→−∞ z→∞ FIGURE 3 Rays G(z) and G(z).

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P1: FLV 2nd Revised Pages Encyclopedia of Physical Science and Technology EN002-95 May 19, 2001 20:57 672 Charged -Particle Optics ′ the emergent asymptote; x = dx/dz. The matrix that appears in this equation is widely used to study systems with many focusing elements; it is known as the (paraxial) transfer matrix and takes slightly different forms for the various elements in use, quadrupoles in particular. We denote the transfer matrix by T . If the planes z1 and z2 are conjugate, the point of arrival of a ray in z2 will vary with the position coordinates of its point of departure in z1 but will be independent of the gradient at that point. The transfer matrix element T12 must FIGURE 4 Focal and principal planes. therefore vanish, (zo − zFo)(zi − zFi) = − fo fi (25) Clearly, all rays incident parallel to the axis have emergent asymptotes that intersect at z = zFi; this point is known as in which we have replaced z1 and z2 by zo and zi to indicate the asymptotic image focus. It is not difﬁcult to show that that these are now conjugates (object and image). This the emergent asymptotes to any family of rays that are is the familiar lens equation in Newtonian form. Writing parallel to one another but not to the axis intersect at a point zFi = zPi + fi and zFo = zPo − fo, we obtain in the plane z = zFi. By applying a similar reasoning to ¯ G(z), we recognize that zFo is the asymptotic object focus. fo fi + = 1 (26) The incident and emergent asymptotes to G(z) intersect in zPo − zo zi − zPi a plane zPi, which is known as the image principal plane the thick-lens form of the regular lens equation. (Fig. 4). The distance between zFi and zPi is the asymptotic Between conjugates, the matrix T takes the form image focal length: ′ zFi − zPi = −1/G i = fi (20) M 0 We can likewise deﬁne zPo and fo: T = 1 fo 1 (27) − ¯ ′ fi fi M zPo − zFo = 1/G o = fo (21) ˆ1/2 ¯ ′ ′ ¯ in which M denotes the asymptotic magniﬁcation, the The Wronskian tells us that φ (GG − G G) is constant height of the image asymptote to G(z) in the image plane. and so If, however, the object is a real physical specimen and ˆ1/2 ¯ ′ ˆ1/2 ′ φ G = −φ G o o i i not a mere intermediate image, the asymptotic cardinal el- or ements cannot in general be used, because the object may / / 1/2 1/2 well be situated inside the ﬁeld region and only a part of ˆ ˆ fo φ o = fi φi (22) the ﬁeld will then contribute to the image formation. Fortu- In magnetic lenses and electrostatic lenses, that provide nately, objective lenses, in which this situation arises, are ˆ ˆ no overall acceleration, φo = φi and so fo = fi; we drop normally operated at high magniﬁcation with the speci- the subscript when no confusion can arise. men close to the real object focus, the point at which the The coupling between an object space and an image ¯ ray G(z) itself intersects the axis [whereas the asymptotic space is conveniently expressed in terms of zFo, zFi, fo, and ¯ object focus is the point at which the asymptote to G(z) in ¯ fi. From the general solution x = αG + βG, we see that object space intersects the optic axis]. The corresponding ¯ lim x(z) = α + β(z − zFo)/ fo real focal length is then deﬁned by the slope of G(z) at the z→−∞ ′ (23) object focus Fo: f = 1/G (Fo); see Fig. 5. lim x(z) = −α(z − zFi)/ fi + β z→∞ and likewise for y(z). Eliminating α and β, we ﬁnd [ ] z2 − zFi (z1 − zFo)(z2 − zFi) [ ] – fo + x2 fi fi x1 = x′ 1 zo − zFo x′ 2 – 1 fi fo (24) where x1 denotes x(z) in some plane z = z1 on the incident asymptote and x2 denotes x(z) in some plane z = z2 on FIGURE 5 Real focus and focal length.

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P1: FLV 2nd Revised Pages Encyclopedia of Physical Science and Technology EN002-95 May 19, 2001 20:57 Charged -Particle Optics 673 2. Quadrupoles The paraxial equations are now different in the x–z and y–z planes: In the foregoing discussion, we have considered only ro- tationally symmetric ﬁelds and have needed only the axial ′′ ˆ1/2 distributions B(z) and φ(z). The other symmetry of great- d ˆ1/2 ′ γ φ − 2γ p2 + 4ηQ2φ (φ x ) + x = 0 est practical interest is that associated with electrostatic dz 4φˆ1/2 (28) and magnetic quadrupoles, widely used in particle accel- ′′ ˆ1/2 erators. Here, the symmetry is lower, the ﬁelds possessing d ˆ1/2 ′ γ φ + 2γ p2 − 4ηQ2φ (φ y ) + y = 0 planes of symmetry and antisymmetry only; these planes dz 4φˆ1/2 intersect in the optic axis, and we shall assume forthwith that electrostatic and magnetic quadrupoles are disposed in which we have retained the possible presence of a as shown in Fig. 6. The reason for this is simple: The round electrostatic lens ﬁeld φ(z). The functions p 2(z) and paraxial equations of motion for charged particles trav- Q 2(z) that also appear characterize the quadrupole ﬁelds; eling through quadrupoles separate into two uncoupled their meaning is easily seen from the ﬁeld expansions [for equations only if this choice is adopted. This is not merely B(z) = 0]: a question of mathematical convenience; if quadrupole ﬁelds overlap and the total system does not have the 1 2 2 ′′ (x, y, z) = φ(z) − (x + y )φ (z) symmetry indicated, the desired imaging will not be 4 achieved. 1 2 2 2 (4) + (x + y ) φ (z) 64 1 1 2 2 4 4 ′′ + (x − y )p2(z) − (x + y )p 2(z) 2 24 1 4 2 2 4 + p4(z)(x − 6x y + y ) + · · · (29) 24 1 1 2 ′′ 4 (4) (r, ψ, z) = φ(z) − r φ + r φ 4 64 1 1 2 ′′ 4 + p2r cos 2ψ − p 2r cos 2ψ 2 24 1 4 + p4r cos 4ψ + · · · 24 x 2 2 ′ Ax = − (x − 3y )Q 2(z) 12 y 2 2 ′ Ay = (y − 3x )Q 2(z) 12 (30) 1 1 2 2 4 4 ′′ Az = (x − y )Q2(z) − (x − y )Q 2(z) 2 24 1 4 2 2 4 + (x − 6x y + y )Q4(z) 24 The terms p4(z) and Q4(z) characterize octopole ﬁelds, and we shall refer to them brieﬂy in connection with the aberration correction below. It is now necessary to deﬁne separate transfer matrices for the x–z plane and for the y–z plane. These have ex- actly the same form as Eqs. (24) and (27), but we have to distinguish between two sets of cardinal elements. For FIGURE 6 (a) Magnetic and (b) electrostatic quadrupoles. arbitrary planes z1 and z2, we have

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P1: FLV 2nd Revised Pages Encyclopedia of Physical Science and Technology EN002-95 May 19, 2001 20:57 674 Charged -Particle Optics ( )( ) (x) (x) (x) z2 − z Fi z2 − zFi z2 − zFo − + f xi (x) fxi fxo T = (x) 1 z 1 − zFo − fxi fxi ( )( ) (y) (y) (y) z2 − z Fi z2 − zFi z1 − zFo − + fyi (y) fyi fyo T = · (y) 1 z 1 − zFo − fyi fyi (31) Suppose now that z = zxo and z = zxi and conjugate so that (x) (y) T = 0; in general, T ≠ 0 and so a point in the object 12 12 plane z = zxo will be imaged as a line parallel to the y axis. Similarly, if we consider a pair of conjugates z = zyo and z = zyi, we obtain a line parallel to the x axis. The imag- ing is hence astigmatic, and the astigmatic differences in FIGURE 7 Passage through a sector magnet. object and image space can be related to the magniﬁcation ∧i := zxi − zyi = ∧Fi − fxi Mx + fyi My (32) the quadrupole equations but do not have different signs, ∧i := zxo − zyo = ∧Fo + fxo/Mx − fyo/My, the particles will be focused in both directions but not in the 1 where same “image” plane unless kH = kv and hence n = . The 2 cases n = 0, for which the magnetic ﬁeld is homogeneous, (x) (y) ∧Fi := z Fi − zFi = ∧i(Mx = My = 0) and n = 1 have been extensively studied. Since prisms are (33) 2 (x) (y) ∧Fo := z Fo − zFo = ∧o(Mx = My → ∞). widely used to separate particles of different energy or mo- mentum, the dispersion is an important quantity, and the Solving the equations ∧i = ∧o = 0 for Mx and My, we ﬁnd transfer matrices are usually extended to include this infor- that there is a pair of object planes for which the image is mation. In practice, more complex end faces are employed stigmatic though not free of distortion. than the simple planes normal to the axis considered here, and the fringing ﬁelds cannot be completely neglected, as they are in the sharp cutoff approximation. 3. Prisms Electrostatic prisms can be analyzed in a similar way There is an important class of devices in which the optic and will not be discussed separately. axis is not straight but a simple curve, almost invariably lying in a plane. The particles remain in the vicinity of this curve, but they experience different focusing forces in the B. Aberrations plane and perpendicular to it. In many cases, the axis is 1. Traditional Method a circular arc terminated by straight lines. We consider the situation in which charged particles travel through a The paraxial approximation describes the dominant fo- magnetic sector ﬁeld (Fig. 7); for simplicity, we assume cusing in the principal electron-optical devices, but this that the ﬁeld falls abruptly to zero at entrance and exit is inevitably perturbed by higher order effects, or aberra- planes (rather than curved surfaces) and that the latter are tions. There are several kinds of aberrations. By retaining normal to the optic axis, which is circular. We regard the higher order terms in the ﬁeld or potential expansions, we plane containing the axis as horizontal. The vertical ﬁeld at obtain the family of geometric aberrations. By considering −n the axis is denoted by Bo, and off the axis, B = Bo(r/R) small changes in particle energy and lens strength, we ob- in the horizontal plane. It can then be shown, with the tain the chromatic aberrations. Finally, by examining the notation of Fig. 7, that paraxial trajectory equations of the effect of small departures from the assumed symmetry of form the ﬁeld, we obtain the parasitic aberrations. ′′ 2 ′′ 2 All these types of aberrations are conveniently studied x + k x = 0; y + k y = 0 (34) v H by means of perturbation theory. Suppose that we have ob- 2 2 describe the particle motion, with k = (1 − n)/R and tained the paraxial equations as the Euler–Lagrange equa- H 2 2 k = n/R . Since these are identical in appearance with tions of the paraxial form of M [Eq. (6)], which we denote v

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P1: FLV 2nd Revised Pages Encyclopedia of Physical Science and Technology EN002-95 May 19, 2001 20:57 Charged -Particle Optics 675 ∫ (P) (A) zi (4) M . Apart from a trivial change of scale, we have where S denotes M dz, with a similar expression oi zo (A) (P) ˆ1/2 ′′ 2 2 2 2 for y . The quantities with superscript (A) indicate the M = −(1/8φ )(γ φ + η B )(x + y ) departure from the paraxial approximation, and we write 1 1/2 ′2 ′2 ˆ + φ (x + y ) (35) (A) (A) 2 xi = x /M = −(1/W) ∂Soi /∂xa (40) Suppose now that M(P) is perturbed to M(P) + M(A). (A) (A) yi = y /M = −(1/W) ∂S oi /∂y(a) (A) The second term M may represent additional terms, The remainder of the calculation is lengthy but straight- neglected in the paraxial approximation, and will then (4) forward. Into M , the paraxial solutions are substituted enable us to calculate the geometric aberrations; alterna- (A) (P) and the resulting terms are grouped according to their de- tively, M may measure the change in M when particle (A) energy and lens strength ﬂuctuate, in which case it tells pendence on xo, yo, xa, and ya. We ﬁnd that S can be written us the chromatic aberration. Other ﬁeld terms yield the parasitic aberration. We illustrate the use of perturbation 1 1 1 (A) 4 4 2 2 −S /W = Er + Cr + A(V − v ) o a theory by considering the geometric aberrations of round 4 4 2 lenses. Here, we have 1 2 2 2 2 + Fr r + Dr V + Kr V o a o a 1 (A) (4) 2 2 2 2 M = M = − L1(x + y ) ( ) 4 2 2 + v dr + kr + aV (41) o a 1 2 2 ′2 ′2 − L2(x + y )(x + y ) with 2 2 2 2 2 2 2 1 r = x + y ; r = x + y ′2 ′2 2 o o o a a a − L3(x + y ) (42) 4 V = xoxa + yoya; v = xoya − xayo ′ ′ 2 − R(xy − x y) and 1/2 2 2 ′ ′ ˆ − Pφ (x + y )(xy − x y) [ ] 2 2 xi = xa Cr a + 2K V + 2kv + (F − A)ro 1/2 ′2 ′2 ′ ′ ˆ − Qφ (x + y )(xy − x y) (36) ( ) 2 2 + xo Kr a + 2AV + av + Dro with ( ) ( 2 2 1 φ′′2 2γ φ′′η2 B2 − yo kra + aV + dro (43) (4) L1 = − γφ + [ ] ˆ1/2 ˆ ˆ 2 2 32φ φ φ y i = ya Cra + 2K V + 2kv + (F − A)ro 4 4 ) ( ) η B 2 2 + − 4η2 B B′′ + xo kra + aV + dro ˆ φ Each coefﬁcient A, C, . . . , d, k represents a differ- 1 ′′ 2 2 ent type of geometric aberration. Although all lenses L2 = (γ φ + η B ) 8φˆ1/2 suffer from every aberration, with the exception of the ( ′′ 2 2 ) anisotropic aberrations described by k, a, and d, which are 1 η γφ B η B 1/2 ′′ L3 = φˆ ; P = + − B peculiar to magnetic lenses, the various aberrations are of 2 16φˆ1/2 φˆ φˆ very unequal importance when lenses are used for differ- 2 2 ηB η B ent purposes. In microscope objectives, for example, the Q = ; R = (37) ˆ1/2 ˆ1/2 incident electrons are scattered within the specimen and 4φ 8φ ∫ emerge at relatively steep angles to the optic axis (sev- (A) z (4) and with S = M dz, we can show that z0 eral milliradians or tens of milliradians). Here, it is the spherical (or aperture) aberration C that dominates, and (A) ∂ S (A) (A) 1/2 ′ = p t(z) − x φˆ t (z) since this aberration does not vanish on the optic axis, x ∂xa (38) being independent of ro, it has an extremely important (A) ∂ S effect on image quality. Of the geometric aberrations, it (A) (A) 1/2 ′ ˆ = p s(z) − x φ s (z) x is this spherical aberration that determines the resolving ∂ya power of the electron microscope. In the subsequent lenses where s(z) and t(z) are the solutions of Eq. (10) for of such instruments, the image is progressively enlarged which s(z0) = t(za) = 1, s(za) = t(z0) = 0, and z = za de- until the ﬁnal magniﬁcation, which may reach 100,000× notes some aperture plane. Thus, in the image plane, or 1,000,000×, is attained. Since angular magniﬁcation (A) (A) x = −(M/W)∂S oi /∂xa (39) is inversely proportional to transverse magniﬁcation, the

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P1: FLV 2nd Revised Pages Encyclopedia of Physical Science and Technology EN002-95 May 19, 2001 20:57 676 Charged -Particle Optics angular spread of the beam in these projector lenses will be tiny, whereas the off-axis distance becomes large. Here, therefore, the distortions D and d are dominant. A characteristic aberration ﬁgure is associated with each aberration. This ﬁgure is the pattern in the image plane formed by rays from some object point that cross the aper- ture plane around a circle. For the spherical aberration, this ﬁgure is itself a circle, irrespective of the object position, and the effect of this aberration is therefore to blur the im- age uniformly, each Gaussian image point being replaced 3 by a disk of radius MCr . The next most important aber- a ration for objective lenses is the coma, characterized by K and k, which generates the comet-shaped streak from which it takes its name. The coefﬁcients A and F describe Seidel astigmatism and ﬁeld curvature, respectively; the astigmatism replaces stigmatic imagery by line imagery, two line foci being formed on either side of the Gaussian image plane, while the ﬁeld curvature causes the image to be formed not on a plane but on a curved image surface. The distortions are more graphically understood by con- sidering their effect on a square grid in the object plane. Such a grid is swollen or shrunk by the isotropic distortion D and warped by the anisotropic distortion d; the latter has been evocatively styled as a pocket handkerchief dis- tortion. Figure 8 illustrates these various aberrations. Each aberration has a large literature, and we conﬁne this account to the spherical aberration, an eternal pre- occupation of microscope lens designers. In practice, it is more convenient to deﬁne this in terms of angle at the specimen, and recalling that x(z) = xos(z) + xat(z), we see ′ ′ ′ that x o = xos (zo) + xat (zo) Hence, ( ) C ( ) 2 2 ′ ′2 ′2 xi = Cxa x a + ya = ′3 xo xo + yo + · · · (44) t o ′3 and we therefore write Cs = c/t o so that ( ) ( ) ′ ′2 ′2 ′ ′2 ′2 xi = Csx o xo + yo ; yi = Csyo xo + yo (45) It is this coefﬁcient Cs that is conventionally quoted and tabulated. A very important and disappointing property of Cs is that it is intrinsically positive: The formula for it can be cast into positive-deﬁnite form, which means that we FIGURE 8 Aberration patterns: (a) spherical aberration; (b) coma; (c–e) distortions. cannot hope to design a round lens free of this aberration by skillful choice of geometry and excitation. This result is known as Scherzer’s theorem. An interesting attempt to lished the lower limit for Cs as a function of the practical upset the theorem was made by Glaser, who tried setting constraints imposed by electrical breakdown, magnetic the integrand that occurs in the formula for Cs, and that saturation, and geometry. can be written as the sum of several squared terms, equal Like the cardinal elements, the aberrations of objective to zero and solving the resulting differential equation for lenses require a slightly different treatment from those of the ﬁeld (in the magnetic case). Alas, the ﬁeld distribution condenser lenses and projector lenses. The reason is eas- that emerged was not suitable for image formation, thus ily understood: In magnetic objective lenses (and probe- conﬁrming the truth of the theorem, but it has been found forming lenses), the specimen (or target) is commonly useful in β-ray spectroscopy. The full implications of the immersed deep inside the ﬁeld and only the ﬁeld re- theorem were established by Werner Tretner, who estab- gion downstream contributes to the image formation. The

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