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Quadrupole Lenses
CHAPTER 19
Quadrupole Lenses Hitherto, we have been studying systems with an axis of rotational symmetry, which are by far the most common in electron optical devices in which the electron accelerating voltage does not exceed one or a few hundred kilovolts or, exceptionally, a few megavolts (in practice, 3 or 4 MV maximum). At higher voltages, focusing elements with lower symmetry are more commonly used, and, in particular, elements with planes of electrical symmetry forming quadrupoles. These possess the property of ‘strong focusing’, by which we mean that their fields exert a force directly on the electrons, towards or away from the axis, whereas in round magnetic lenses, the focusing force is more indirect, arising from the coupling between Bz and the azimuthal component of the electron velocity. Quadrupoles have also been very thoroughly studied as elements for use at conventional accelerating voltages for a quite different reason. We shall see in Section 24.3 that one of the most undesirable aberrations of round electron lenses cannot be eliminated in any straightforward manner but can in principle be cancelled by introducing quadrupoles and octopoles into the system. This has provided the incentive for exhaustive studies of quadrupole lens properties, comparable in thoroughness with those on round lenses. In this chapter, we derive the paraxial equations of quadrupole systems, and introduce the notion of an orthogonal system. A brief survey of the history of quadrupole studies is to be found at the beginning of Chapter 39 of Volume 2. Here we merely remark that although strong focusing at high energy and aberration correction have been the principal stimuli for research on quadrupoles, their properties were very fully explored long before either of these applications was known: the first study, which was thorough and meticulous, appeared as a Berlin dissertation in 1943 (Melkich, 1947). In the discussion of Section 7.2.3, the terms in 2ϕ in the potential expansion (7.37 or 7.59) and their magnetic counterparts (7.437.45) were described as quadrupole terms. Here we use the term in a slightly less restrictive sense: a magnetic or electrostatic quadrupole is characterized by the presence of two planes of symmetry in the potential or field (Fig. 19.1), and rotationally symmetric fields as well as octopoles (and higher order 2n-poles) are not excluded. From (7.36), we see that the term in p2(z) describes a potential for which the planes x 5 0 and y 5 0 are symmetry planes, Φ(x, y, z) 5 Φ(6x, 6y, z), and a typical electrostatic quadrupole thus has the form shown in Fig. 7.2 and Fig. 19.1A.
Principles of Electron Optics: Basic Geometrical Optics. DOI: http://dx.doi.org/10.1016/B978-0-08-102256-6.00019-5 © 2018 Elsevier Ltd. All rights reserved.
297
298 Chapter 19 y
(A)
(B)
–
S
y
N
a +
+
x
x
N
S
– (C)
Figure 19.1 Quadrupoles. (A) Electrostatic quadrupole characterized by p2(z), with equal and opposite voltages on the electrodes. (B) Magnetic quadrupole characterized by Q2(z). (C) Photograph of an electrostatic quadrupole.
The term in q2(z) simply corresponds to a similar quadrupole inclined at 45 to that of Fig. 19.1A. In the magnetic case, the planes x 5 0 and y 5 0 are planes of symmetry for P2(z) and planes of antisymmetry for Q2(z). For a field described by Q2 only, Bx(x, y, z) 5 Bx(2x, y, z) 5 2Bz(x, 2 y, z) 5 2Bz(2x, 2 y, z), with analogous relations for By (Fig. 19.1B). We shall learn that uncoupled equations can be obtained if the quadrupoles are orientated with respect to the coordinate axes in such a way that q2 5 0 and P2 5 0, that is, as shown in Fig. 19.1.
19.1 Paraxial Equations for Quadrupoles We set out from the general case in which rotationally symmetric magnetic and electrostatic fields may be present as well as the quadrupole fields themselves. The fields are thus characterized by six axial functions, φ(z) and B(z) for the round lens components, p2(z) and q2(z) for the electrostatic quadrupoles and Q2(z) and P2(z) for the magnetic quadrupoles.
Quadrupole Lenses 299 We substitute the field expansions (7.36) and (7.437.45) into the function M; M 5 M=ð2m0 eÞ1=2 (15.23) as in Section 15.2 and expand M as a power series in X, Y and their derivatives. We find M ð0Þ 5 φ^ M
ð2Þ
1=2
52 1
γφv 8φ^
1=2
X 1Y
γq2
2φ^
1=2
2
2
1
γp2 4φ^
1=2
1 2 ηQ2 X 2 2 Y 2 2
1 1=2 02 1 ηP2 XY 1 φ^ X 1 Y 02 2
(19.1)
1 2 ηBðXY 0 2 X 0 Y Þ 2 The term in XY0 X0 Y can be removed by introducing the rotating coordinate system employed in connection with round magnetic lenses (15.7, 15.9) whereupon M(2) becomes 0 1 2 2 η B C B γφv M ð2Þ 5 2@ 1=2 1 1=2 A x2 1 y2 8φ^ 8φ^ 80 9 1 0 1 > > = x2 2 y2 < γp B 2 C B γq2 C 1 @ 1=2 2 ηQ2 Acos 2θ 1 @ 1=2 1 ηP2 Asin 2θ > > ; 2 : 2φ^ 2φ^ (19.2) 8 0 9 1 1 0 > > < = C C B γp2 B γq2 1 2 @ 1=2 2 ηQ2 Asin 2θ 1 @ 1=2 1 ηP2 Acos 2θ xy > > : ; 2φ^ 2φ^ 1 1=2 02 1 φ^ x 1 y02 2 1=2 with θ0 5 ηB=2φ^ . The paraxial equations, @M(2)/@xi 5 d(@M(2)/dx0 i)/dz (x1 5 x, x2 5 y), still do not separate, however, unless
tan θðzÞ 5
1=2 γq2 =φ^ 1 2ηP2 1=2 γp2 =φ^ 1 2ηQ2
(19.3)
This is known as the orthogonality condition and has been known in various forms for many years (Melkich, 1947; Glaser, 1956, Section 37; Duˇsek, 1959). This condition can be satisfied in several ways but only one is a practical possibility. Most generally, θ(z) may vary with z, in which case electrodes and polepieces must be devised and constructed of
300 Chapter 19 such shapes that condition (19.3) is everywhere satisfied. More reasonably, θ(z) may be a 1=2 1=2 constant, not necessarily zero; this requires B(z) 5 0 and γq2 1 2ηP2 φ^ ~ γp2 2 2ηQ2 φ^ . Finally we may set θ(z) equal to zero, so that q2(z) 5 P2(z) 0, and at least one of p2(z) and Q2(z) is not zero everywhere. We retain only this final case. From now on, then, we consider only orthogonal systems, and we assume that the electrodes and polepieces are disposed so that only φ(z), p2(z) and Q2(z) are allowed to be nonzero. The paraxial equations then take the form 1=2 d ^ 1=2 0 γφv 2 2γp2 1 4ηQ2 φ^ x50 φ x 1 1=2 dz 4φ^
(19.4a)
d ^ 1=2 0 γφv 1 2γp2 2 4ηQ2 φ^ φ y 1 1=2 dz 4φ^
(19.4b)
1=2
y50
or nonrelativistically xv 1
φ0 0 φv 2 2p2 1 4ηQ2 φ1=2 x 1 x50 2φ 4φ
(19.5a)
yv 1
φ0 0 φv 1 2p2 2 4ηQ2 φ1=2 y 1 y50 2φ 4φ
(19.5b)
1=4 1=4 or in reduced form, ξx :¼ xφ^ ; ξy :¼ yφ^ , ( ) 1=2 2 3 φ0 4 ^ p2 2 2ηQ2 φ^ ξvx 1 1 1 εφ 2 ξx 5 0 16 φ^ 3 2φ^
) 2 1=2 3 φ0 4 ^ p2 2 2ηQ2 φ^ ξvy 1 1 1 εφ 1 ξy 5 0 16 φ^ 3 2φ^
(19.6a)
(
(19.6b)
Each paraxial equation is a linear, homogeneous, second-order differential equation and by any of the lines of reasoning set out in Chapter 16, Gaussian Optics of Rotationally Symmetric Systems: Asymptotic Image Formation, we may again establish the existence of cardinal elements. For quadrupoles, it is usually sufficient to list the asymptotic cardinal elements, though real (and osculating) elements can of course be defined if needed. Unlike round lenses, however, two sets of cardinal elements are needed, one to characterize the xz plane, the other the yz plane. In the absence of any rotationally symmetric lens field (φ 5 const), the lens action in one of these planes will be convergent while in the other it will be divergent. This is readily seen from (19.419.6).
Quadrupole Lenses 301 The action of a quadrupole lens on electron rays is most conveniently characterized by a pair of transfer matrices, similar to (16.12) except that the matrix describing the coordination between object and image space is different in the two planes. Before writing down these matrices, we must first introduce the notion of astigmatic objects and images. Quadrupoles are commonly employed as multiplets the quadruplet can have most attractive features and if the cardinal elements are different in the xz and yz planes, the image plane corresponding to a given object plane may clearly be different as well: the system will not produce a stigmatic image of a point object. If a further lens follows, this astigmatic image will be the object for the subsequent stage, and we must thus expect to have to deal with astigmatic objects. The most general transfer matrix relates position and slope in some plane in object space, z 5 z1, to the same quantities in a plane in image space, z 5 z2 (cf. 16.15): x2 x1 y2 y 5 Tx 0 ; 5 Ty 10 (19.7) 0 0 x2 x1 y2 y1 and as in (16.12), we write 0
ðxÞ B 2 z2 2 zFi B fxi B Tx 5 B B 1 @ 2 fxi
0
ðyÞ B 2 z2 2 zFi B fyi B Ty 5 B B 1 @ 2 fyi
z2 2 zðxÞ Fi
z1 2 zðxÞ Fo
1 fxo C C C C C A
fxi z1 2 zðxÞ Fo fxi
z2 2 zðyÞ Fi
z1 2 zðyÞ Fo
1
fyi z1 2 zðyÞ Fo fyi
(19.8a)
1 1 fyo C C C C C A
(19.8b)
The cardinal elements are most conveniently defined with the aid of the rays Gx(z), Gy(z), Gx ðzÞ and Gy ðzÞ, which satisfy conditions analogous to (16.1): lim
z- 2N
Gx ðzÞ 5 lim
z- 2N
Gy ðzÞ 5 1
lim Gx ðzÞ 5 lim Gy ðzÞ 5 1
z-N
z-N
(19.9)
The rays Gx(z) and Gx ðzÞ satisfy (19.4a) while Gy(z) and Gy ðzÞ satisfy (19.4b). The image foci are then the points of intersection of the image asymptotes to Gx and Gy with the axis; the rays Gx and Gy similarly define the object foci. The focal lengths are given by fxi 5 21=G0xi ; fyi 5 21=G0yi fxo 5 1=G0xo ; fyo 5 1=G0yo
(19.10)
Let us suppose that the planes z 5 zxo and z 5 zxi are conjugate, in the sense that (Tx)12 5 0, or
302 Chapter 19 ðxÞ zxi 2 zðxÞ 2 z z xo Fi Fo 5 2 fxi fxo
(19.11)
so that Tx 5
Mx 21=fxi
0 1=2
φ^ 0 =φ^ i
! =Mx
(19.12)
1=2 in which we have used the Wronskian of (19.4a) to show that fxo =fxi 5 φ^ 0 =φ^ i a similar relation is of course true for fyo/fyi. The magnitude Mx is the height of the image asymptote to Gx(z) in the plane z 5 zxi. In general, however, (Ty)12 6¼ 0 when (19.11) is satisfied and so although rays from a point P(xo, yo) all have the same x-coordinate xi 5 Mxxo in z 5 zxi, their y-coordinate is a function of both yo and the gradient y0 o: 1 0 ðyÞ ðyÞ ðyÞ z 2 z 2 z z xi xo F Fo z 2 z i Fi B 2 xi 1 fyo C C B f f yi yi C B (19.13) Ty 5 B C ðyÞ C B z 2 z 1 xo A @ Fo 2 fyi fyi A point P(x0, y0) is therefore imaged as a line in the plane z 5 zxi parallel to the y-axis. Likewise, if we consider a pair of planes z 5 zyo and z 5 zyi for which (Ty)12 5 0, we find that in general (Tx)12 does not vanish and again a line image is formed, now parallel to the x-axis. These line images are thus at right-angles to one another and separated by the astigmatic difference. One or both may be virtual (Fig. 19.2). If we move an axial point object along the axis, the line foci will move and there will always be real or virtual object positions for which the line foci coincide and the image is stigmatic. In general, however, the magnifications in the two planes will not be equal. The astigmatic difference can be expressed in terms of the cardinal elements and magnification. Using the quadrupole analogue of (16.25), zxi 2 zðxÞ Fi 5 2 fxi Mx zyi 2 zðyÞ Fi 5 2 fyi My
(19.14)
zxo 2 zðxÞ Fo 5 fxo =Mx zyo 2 zðyÞ Fo 5 fyo =My
(19.15)
and
we see that
N
y
x
Quadrupole Lenses 303
x z
y z
Figure 19.2 Formation of a line image in a magnetic quadrupole. The arrows show the directions of the currents in the windings. Courtesy D.F. Hardy
Λi :¼ zxi 2 zyi 5 ΛFi 2 fxi Mx 1 fyi My Λo :¼ zxo 2 zyo 5 ΛFo 1 fxo Mx 2 fyo My
(19.16)
ðyÞ ΛFi :¼ zðxÞ Fi 2 zFi 5 Λi Mx 5 My 5 0 ðyÞ ΛFo :¼ zðxÞ Fo 2 zFo 5 Λo Mx 5 My -N
(19.17)
where
From (19.13), it is readily seen that quadratic equations are obtained for Mx and My if we attempt to satisfy the stigmatic imaging condition, Λi 5 Λo 5 0. The discriminant δ is the same for Mx and My and we find
304 Chapter 19 Mx 5
ΛFo ΛFi 2 fxo fxi 1 fyo fyi 6 δ 2ΛFo fxi
2ΛFo ΛFi 2 fxo fxi 1 fyo fyi 6 δ My 5 2ΛFo fyi
(19.18a)
with 2 δ2 5 fxo fxi 2fyo fyi 1 Λ2Fo Λ2Fi 1 2 fxo fxi 1 fyo fyi ΛFo ΛFi 2 5 fxo fxi 1fyo fyi 1ΛFo ΛFi 2 4fxo fxi fyo fyi
(19.18b)
In the usual case in which the signs of fx and fy are different, δ2 is positive and there are two real roots and hence two pairs of stigmatic conjugates. The cardinal elements of multiplets are established most easily by multiplying the transfer matrices of the individual lenses; these must be separated by transfer matrices corresponding to the drift spaces, the spaces between the planes z 5 z2 for one lens and z 5 z1 for the next. We recall that these planes may be chosen in various ways; z1 5 z2 5 0, in which case incident position and gradient are related to emergent position and gradient in the same plane, conventionally the mid-plane of the lens, is one good choice, thoroughly explored by Duˇsek (1959). Here we have ! ! ! ðxÞ ðxÞ x2 x zðxÞ =f z z =f 1 f 1 xi xi xo Fi Fi Fo 5 x02 x01 21=fxi 2zðxÞ Fo =fxi (19.19) ! ! ! ðyÞ y2 y1 zðyÞ zðyÞ Fi =fyi Fi zFo =fyi 1 fyo 5 y02 y01 21=fyi 2zðyÞ Fo =fyi (We note that Duˇsek’s matrices are trivially different since he used the vectors (x0 x)T and (y0 y)T.) Another convenient choice involves using different pairs of planes for Tx and Ty, namely the focal planes, since the diagonal matrix elements then vanish: 0 fyo 0 fxo (19.20) Tx 5 ; Ty 5 21=fyi 0 21=fxi 0 This choice has been studied in great detail by Regenstreif (1966, 1967), who has established straightforward rules for writing down the transfer matrices of an arbitrary number of quadrupoles. His procedure can be applied to Duˇsek matrices (Hawkes, 1970), which we temporarily write as follows: ai bi Ti 5 (19.21) c i di where Ti denotes either Tx or Ty for the i-th quadrupole of a sequence and
Quadrupole Lenses 305 zFi zFo zFi ; bi :¼ 1 fo fi fi 1 zFo ci :¼ 2 ; di :¼ 2 fi fi ai :¼
(19.22)
The separation between the i-th and (i 1 1)-th quadrupoles is denoted by Li,i11 and we write 1 Li;i11 (19.23) T Li;i11 :¼ 0 1 Introducing the distances Xi,i11 between the image focus of the i-th quadrupole and the object focus of the (i 1 1)-th quadrupole, Xi;i11 :¼
ai di11 1 Li;i11 1 ci ci11
(19.24)
we can show (Regenstreif, 1966, 1967; Hawkes, 1970) that the transfer matrix of n quadrupoles separated by n1 drift spaces is given by aðnÞ bðnÞ ðnÞ T 5 cðnÞ dðnÞ n21
aðnÞ 5 an L ci αn ; n
i51
cðnÞ 5 L ci γ n ;
n
n
i52
1 γ γ αn 5 Xn21;n 2 2 n22 an cn n21 c2n21 1 δn22 δn21 2 2 β n 5 Xn21;n 2 an c n cn21 γ γ n 5 Xn21;n γ n21 2 2n22 cn21 δn22 δn 5 Xn21;n δn21 2 2 cn21
and γ0 5 δ0 5 0; γ2 5 X1;2 ;
(19.25)
i52
dðnÞ 5 d1 L ci δn
i51
in which
n21
bðnÞ 5 d1 an L ci β n
γ 1 5 δ1 5 1 1 δ2 5 X1;2 2 c1 d1
(19.26)
306 Chapter 19 Another expression for the elements of the transfer matrix between an arbitrary pair of planes, z 5 zn and z 5 z0, may be derived by using the transfer matrix between the principal planes. Writing ði21Þ 2#i#n21 DðiÞ :¼ zðiÞ Po 2 zPi ðnÞ Dð1Þ :¼ zð1Þ 2 z ; D :¼ zn 2 zðn21Þ 0 Po Pi
(19.27)
T ðz0 ; zn Þ 5 Dn Tn21 Dn21 . . .T2 D2 T1 D1
(19.28)
we form the matrix
The elements can be written as Gaussian brackets (Herzberger, 1943, 1958; Hawkes, 1967; Dymnikov, 1968), which are defined as follows: ½x1 ½x1 x2 ½x1 x2 x3 ½x1 x2 x3 . . .xn
5 5 5 5
x x1 x2 1 1 x1 x2 x3 1 x1 1 x3 ½x1 x2 x3 . . .xn22 1 xn ½x1 x2 x3 . . .xn21
(19.29)
We find T ðz0 ; zn Þ 5
½Dn cn21 . . .D2 c1 ½Dn cn21 . . .D2 c1 D1 ½cn21 Dn21 . . .c1 ½cn21 Dn21 . . .c1 D1
(19.30)
For a proof see Hawkes (1967). Gaussian brackets are also employed by Chechulin and Yavor (1969) in connection with prisms; renewed interest has been shown in them in light optics too (e.g., Tanaka, 1981, 1982, 1983, 1986). One common requirement for quadrupole multiplets is that their overall behaviour should be the same as that of a round lens; for this, the cardinal elements in the xz and yz planes must coincide and the focal lengths fx and fy must be equal. If we consider quadrupoles that may have a round electrostatic lens component, provided that the latter has no overall accelerating or retarding effect (φi 5 φo), we can see on symmetry grounds that one at least of these conditions is satisfied by imposing a certain symmetry on the system. In particular, we perceive that the focal lengths in the xz and yz planes are automatically equal in antisymmetric multiplets. The latter are defined as follows. If a multiplet consists of 2N quadrupoles (N $ 1) such that the central plane is a plane of geometrical symmetry and electrical antisymmetry, we say that the multiplet is antisymmetric. The case of N 5 2 was extensively studied by a group in Leningrad (Yavor, 1962; Dymnikov and Yavor, 1963; Dymnikov et al., 1963a,b, 1964a,b, 1965; Shpak and Yavor, 1964) and has come to be known as the Russian quadruplet (Fig. 19.3). Consider two rays, Gx(z) and Gy ðzÞ. Because of the electrical antisymmetry, the sequence of functions p2(z) or Q2(z) encountered by Gx(z) as it proceeds in the positive z-direction will be exactly the same as that traversed by Gy ðzÞ if we imagine it travelling in the negative z-direction. The gradients of
Quadrupole Lenses 307 x
C
D
C
D
z
D
C
D
C
y Antisymmetry plane
Figure 19.3 The antisymmetric or ‘Russian’ quadruplet. The centre plane is a plane of geometrical symmetry and electrical antisymmetry. C, D denote convergent and divergent action respectively.
the emergent asymptotes will hence be equal but opposite in sign, and the focal lengths fx and fy will hence be equal (since φi 5 φo, we already have fxi :¼ fxo ¼: fx and fyi :¼ fyo ¼: fy). If we wish to arrange that an antisymmetric multiplet has the same image-forming properties as a round lens, therefore, we have only to ensure that the foci (or principal points) in the xz and yz planes coincide. For a given geometry, we need only vary the relative excitation until the condition is satisfied. We therefore obtain a set of pairs of excitations, which is known as the load characteristic of the quadruplet. Some examples are given in Chapter 39 of Volume 2.
19.2 Transaxial Lenses The foregoing discussion has been confined to the optics of quadrupoles in general and we have said little about the shapes of the electrodes and polepieces and hence about the effects of any additional symmetries. Two particular additional symmetry properties are of interest, however; one leads to cylindrical lenses, at least in the electrostatic case, as we see in Chapter 20, Cylindrical Lenses. A different symmetry characterizes transaxial lenses, in which the field or potential is rotationally symmetric but the optic axis is no longer the same as the symmetry axis but is perpendicular to it. The electron beam now passes between rotationally symmetric, typically plane electrodes and is focused by the fields in any gaps. Fig. 19.4 shows such a system. The electrodes are ideally circular or annular but are in practice reduced to sectors since the electron beam occupies so little of the space available. It can be seen by comparison with Chapter 20, Cylindrical Lenses, that such structures bear some resemblance to cylindrical lenses but differ from the latter in that the potential is not independent of one of the transverse cartesian coordinates but is the same for all azimuthal
308 Chapter 19
x
φ1 φc
ψ
φ0
z (Optic axis)
Figure 19.4 A transaxial lens. The electrodes shown lie in some plane y 5 const and an identical set lies in the plane y 5 2const.
angles ϕ at a given radial distance from the axis of rotational symmetry. The optical behaviour of such systems was first investigated formally by Strashkevich (1962), to whom we owe the name ‘transaxial lenses’; the theory was set out in some detail in his book of 1966. In the early 1970s, it was realized (by V.M. Kel’man and colleagues in Alma-Ata) that certain features of these structures rendered them attractive for use in the collimator of a prism spectrometer, and their properties were investigated in some detail (Glikman et al., 1971; Brodskii and Yavor, 1970, 1971; Karetskaya et al., 1970, 1971a,b). This work is presented in full in Kel’man et al. (1979), one of the three chapters of which is devoted wholly to these lenses. The symmetry conditions are now such that Φðx; y; zÞ 5 Φðx; 2y; zÞ 1=2 Φð0; y; zÞ 5 Φ x; y; z2 2x2
(19.31)
For small values of x and y, therefore, ! φ0 2 1 φ0 4 x Φðx; y; zÞ 5 φðzÞ 1 x 1 2 φv 2 z 2z z φ0 1 φ2 y 1 2 x 2 y 2 1 φ4 y 4 2z 2
with
(19.32)
Quadrupole Lenses 309 1 φ0 φ2 :¼ 2 φv 1 z 2
! !
(19.33)
p2 ðzÞ 5 φ0 ðzÞ=z 1 φv=2
(19.34)
1 φ0 φ4 :¼ 2 φv2 1 2 12 z Comparing Eq. (19.32) with (7.36), we see that
The paraxial equations have the form xv 1
φ0 0 φ0 x 2 x50 2φ 2zφ
(19.35a)
yv 1
φ0 0 φ2 y 2 y50 2φ φ
(19.35b)
All the theory for quadrupoles can hence be employed without further discussion. The form of (19.35a) is, however, such that simple expressions can be obtained for the focal length and foci. Thus on writing ξ :¼ x=z
(19.36)
Eq. (19.35a) becomes ξv 1
2 1 φ0 z=2φ 0 ξ 50 z
(19.37)
and after some trivial calculation, we find ð 1=2 z dζ xo z 0 x5 1 x o z o 2 x o φo z 1=2 2 zo ζ zo φ ðz 0 1 z φ dζ 5 x0o z xo 2 x0o zo φ1=2 1 o 1=2 2 zo φ3=2 ζ φ giving the transfer matrix 1=2 x φo =φ 2 z=fi 5 x0 21=fi with
1=2 xo z 2 zo φo =φ 1 zzo =fi x0o 1 1 zo =fi
(19.38)
(19.39)
310 Chapter 19 1 φ1=2 52 o fi 2
ðN
φ0 dz
2N φ
3=2
(19.40)
z
in which we have extended the limits of integration to infinity in fi since it is asymptotic imagery that will be of interest. The planes zo and zi will be conjugate if 1=2 φo zo zi 1 50 (19.41) zi 2 zo φi fi or 1 φ1=2 o zo
2
1 1=2 φi zi
52
and the transfer matrix becomes zi =zo zi =zo 0 5 21=fi 1 1 zo =fi 21=fi
1
(19.42)
fi φ1=2 o
0 1=2 zo φ1=2 o =zi φi
(19.43)
In the converging or yz plane, there is no such simple solution of the paraxial equation.
Quadrupole Lenses Hitherto, we have been studying systems with an axis of rotational symmetry, which are by far the most common in electron optical devices in which the electron accelerating voltage does not exceed one or a few hundred kilovolts or, exceptionally, a few megavolts (in practice, 3 or 4 MV maximum). At higher voltages, focusing elements with lower symmetry are more commonly used, and, in particular, elements with planes of electrical symmetry forming quadrupoles. These possess the property of ‘strong focusing’, by which we mean that their fields exert a force directly on the electrons, towards or away from the axis, whereas in round magnetic lenses, the focusing force is more indirect, arising from the coupling between Bz and the azimuthal component of the electron velocity. Quadrupoles have also been very thoroughly studied as elements for use at conventional accelerating voltages for a quite different reason. We shall see in Section 24.3 that one of the most undesirable aberrations of round electron lenses cannot be eliminated in any straightforward manner but can in principle be cancelled by introducing quadrupoles and octopoles into the system. This has provided the incentive for exhaustive studies of quadrupole lens properties, comparable in thoroughness with those on round lenses. In this chapter, we derive the paraxial equations of quadrupole systems, and introduce the notion of an orthogonal system. A brief survey of the history of quadrupole studies is to be found at the beginning of Chapter 39 of Volume 2. Here we merely remark that although strong focusing at high energy and aberration correction have been the principal stimuli for research on quadrupoles, their properties were very fully explored long before either of these applications was known: the first study, which was thorough and meticulous, appeared as a Berlin dissertation in 1943 (Melkich, 1947). In the discussion of Section 7.2.3, the terms in 2ϕ in the potential expansion (7.37 or 7.59) and their magnetic counterparts (7.437.45) were described as quadrupole terms. Here we use the term in a slightly less restrictive sense: a magnetic or electrostatic quadrupole is characterized by the presence of two planes of symmetry in the potential or field (Fig. 19.1), and rotationally symmetric fields as well as octopoles (and higher order 2n-poles) are not excluded. From (7.36), we see that the term in p2(z) describes a potential for which the planes x 5 0 and y 5 0 are symmetry planes, Φ(x, y, z) 5 Φ(6x, 6y, z), and a typical electrostatic quadrupole thus has the form shown in Fig. 7.2 and Fig. 19.1A.
Principles of Electron Optics: Basic Geometrical Optics. DOI: http://dx.doi.org/10.1016/B978-0-08-102256-6.00019-5 © 2018 Elsevier Ltd. All rights reserved.
297
298 Chapter 19 y
(A)
(B)
–
S
y
N
a +
+
x
x
N
S
– (C)
Figure 19.1 Quadrupoles. (A) Electrostatic quadrupole characterized by p2(z), with equal and opposite voltages on the electrodes. (B) Magnetic quadrupole characterized by Q2(z). (C) Photograph of an electrostatic quadrupole.
The term in q2(z) simply corresponds to a similar quadrupole inclined at 45 to that of Fig. 19.1A. In the magnetic case, the planes x 5 0 and y 5 0 are planes of symmetry for P2(z) and planes of antisymmetry for Q2(z). For a field described by Q2 only, Bx(x, y, z) 5 Bx(2x, y, z) 5 2Bz(x, 2 y, z) 5 2Bz(2x, 2 y, z), with analogous relations for By (Fig. 19.1B). We shall learn that uncoupled equations can be obtained if the quadrupoles are orientated with respect to the coordinate axes in such a way that q2 5 0 and P2 5 0, that is, as shown in Fig. 19.1.
19.1 Paraxial Equations for Quadrupoles We set out from the general case in which rotationally symmetric magnetic and electrostatic fields may be present as well as the quadrupole fields themselves. The fields are thus characterized by six axial functions, φ(z) and B(z) for the round lens components, p2(z) and q2(z) for the electrostatic quadrupoles and Q2(z) and P2(z) for the magnetic quadrupoles.
Quadrupole Lenses 299 We substitute the field expansions (7.36) and (7.437.45) into the function M; M 5 M=ð2m0 eÞ1=2 (15.23) as in Section 15.2 and expand M as a power series in X, Y and their derivatives. We find M ð0Þ 5 φ^ M
ð2Þ
1=2
52 1
γφv 8φ^
1=2
X 1Y
γq2
2φ^
1=2
2
2
1
γp2 4φ^
1=2
1 2 ηQ2 X 2 2 Y 2 2
1 1=2 02 1 ηP2 XY 1 φ^ X 1 Y 02 2
(19.1)
1 2 ηBðXY 0 2 X 0 Y Þ 2 The term in XY0 X0 Y can be removed by introducing the rotating coordinate system employed in connection with round magnetic lenses (15.7, 15.9) whereupon M(2) becomes 0 1 2 2 η B C B γφv M ð2Þ 5 2@ 1=2 1 1=2 A x2 1 y2 8φ^ 8φ^ 80 9 1 0 1 > > = x2 2 y2 < γp B 2 C B γq2 C 1 @ 1=2 2 ηQ2 Acos 2θ 1 @ 1=2 1 ηP2 Asin 2θ > > ; 2 : 2φ^ 2φ^ (19.2) 8 0 9 1 1 0 > > < = C C B γp2 B γq2 1 2 @ 1=2 2 ηQ2 Asin 2θ 1 @ 1=2 1 ηP2 Acos 2θ xy > > : ; 2φ^ 2φ^ 1 1=2 02 1 φ^ x 1 y02 2 1=2 with θ0 5 ηB=2φ^ . The paraxial equations, @M(2)/@xi 5 d(@M(2)/dx0 i)/dz (x1 5 x, x2 5 y), still do not separate, however, unless
tan θðzÞ 5
1=2 γq2 =φ^ 1 2ηP2 1=2 γp2 =φ^ 1 2ηQ2
(19.3)
This is known as the orthogonality condition and has been known in various forms for many years (Melkich, 1947; Glaser, 1956, Section 37; Duˇsek, 1959). This condition can be satisfied in several ways but only one is a practical possibility. Most generally, θ(z) may vary with z, in which case electrodes and polepieces must be devised and constructed of
300 Chapter 19 such shapes that condition (19.3) is everywhere satisfied. More reasonably, θ(z) may be a 1=2 1=2 constant, not necessarily zero; this requires B(z) 5 0 and γq2 1 2ηP2 φ^ ~ γp2 2 2ηQ2 φ^ . Finally we may set θ(z) equal to zero, so that q2(z) 5 P2(z) 0, and at least one of p2(z) and Q2(z) is not zero everywhere. We retain only this final case. From now on, then, we consider only orthogonal systems, and we assume that the electrodes and polepieces are disposed so that only φ(z), p2(z) and Q2(z) are allowed to be nonzero. The paraxial equations then take the form 1=2 d ^ 1=2 0 γφv 2 2γp2 1 4ηQ2 φ^ x50 φ x 1 1=2 dz 4φ^
(19.4a)
d ^ 1=2 0 γφv 1 2γp2 2 4ηQ2 φ^ φ y 1 1=2 dz 4φ^
(19.4b)
1=2
y50
or nonrelativistically xv 1
φ0 0 φv 2 2p2 1 4ηQ2 φ1=2 x 1 x50 2φ 4φ
(19.5a)
yv 1
φ0 0 φv 1 2p2 2 4ηQ2 φ1=2 y 1 y50 2φ 4φ
(19.5b)
1=4 1=4 or in reduced form, ξx :¼ xφ^ ; ξy :¼ yφ^ , ( ) 1=2 2 3 φ0 4 ^ p2 2 2ηQ2 φ^ ξvx 1 1 1 εφ 2 ξx 5 0 16 φ^ 3 2φ^
) 2 1=2 3 φ0 4 ^ p2 2 2ηQ2 φ^ ξvy 1 1 1 εφ 1 ξy 5 0 16 φ^ 3 2φ^
(19.6a)
(
(19.6b)
Each paraxial equation is a linear, homogeneous, second-order differential equation and by any of the lines of reasoning set out in Chapter 16, Gaussian Optics of Rotationally Symmetric Systems: Asymptotic Image Formation, we may again establish the existence of cardinal elements. For quadrupoles, it is usually sufficient to list the asymptotic cardinal elements, though real (and osculating) elements can of course be defined if needed. Unlike round lenses, however, two sets of cardinal elements are needed, one to characterize the xz plane, the other the yz plane. In the absence of any rotationally symmetric lens field (φ 5 const), the lens action in one of these planes will be convergent while in the other it will be divergent. This is readily seen from (19.419.6).
Quadrupole Lenses 301 The action of a quadrupole lens on electron rays is most conveniently characterized by a pair of transfer matrices, similar to (16.12) except that the matrix describing the coordination between object and image space is different in the two planes. Before writing down these matrices, we must first introduce the notion of astigmatic objects and images. Quadrupoles are commonly employed as multiplets the quadruplet can have most attractive features and if the cardinal elements are different in the xz and yz planes, the image plane corresponding to a given object plane may clearly be different as well: the system will not produce a stigmatic image of a point object. If a further lens follows, this astigmatic image will be the object for the subsequent stage, and we must thus expect to have to deal with astigmatic objects. The most general transfer matrix relates position and slope in some plane in object space, z 5 z1, to the same quantities in a plane in image space, z 5 z2 (cf. 16.15): x2 x1 y2 y 5 Tx 0 ; 5 Ty 10 (19.7) 0 0 x2 x1 y2 y1 and as in (16.12), we write 0
ðxÞ B 2 z2 2 zFi B fxi B Tx 5 B B 1 @ 2 fxi
0
ðyÞ B 2 z2 2 zFi B fyi B Ty 5 B B 1 @ 2 fyi
z2 2 zðxÞ Fi
z1 2 zðxÞ Fo
1 fxo C C C C C A
fxi z1 2 zðxÞ Fo fxi
z2 2 zðyÞ Fi
z1 2 zðyÞ Fo
1
fyi z1 2 zðyÞ Fo fyi
(19.8a)
1 1 fyo C C C C C A
(19.8b)
The cardinal elements are most conveniently defined with the aid of the rays Gx(z), Gy(z), Gx ðzÞ and Gy ðzÞ, which satisfy conditions analogous to (16.1): lim
z- 2N
Gx ðzÞ 5 lim
z- 2N
Gy ðzÞ 5 1
lim Gx ðzÞ 5 lim Gy ðzÞ 5 1
z-N
z-N
(19.9)
The rays Gx(z) and Gx ðzÞ satisfy (19.4a) while Gy(z) and Gy ðzÞ satisfy (19.4b). The image foci are then the points of intersection of the image asymptotes to Gx and Gy with the axis; the rays Gx and Gy similarly define the object foci. The focal lengths are given by fxi 5 21=G0xi ; fyi 5 21=G0yi fxo 5 1=G0xo ; fyo 5 1=G0yo
(19.10)
Let us suppose that the planes z 5 zxo and z 5 zxi are conjugate, in the sense that (Tx)12 5 0, or
302 Chapter 19 ðxÞ zxi 2 zðxÞ 2 z z xo Fi Fo 5 2 fxi fxo
(19.11)
so that Tx 5
Mx 21=fxi
0 1=2
φ^ 0 =φ^ i
! =Mx
(19.12)
1=2 in which we have used the Wronskian of (19.4a) to show that fxo =fxi 5 φ^ 0 =φ^ i a similar relation is of course true for fyo/fyi. The magnitude Mx is the height of the image asymptote to Gx(z) in the plane z 5 zxi. In general, however, (Ty)12 6¼ 0 when (19.11) is satisfied and so although rays from a point P(xo, yo) all have the same x-coordinate xi 5 Mxxo in z 5 zxi, their y-coordinate is a function of both yo and the gradient y0 o: 1 0 ðyÞ ðyÞ ðyÞ z 2 z 2 z z xi xo F Fo z 2 z i Fi B 2 xi 1 fyo C C B f f yi yi C B (19.13) Ty 5 B C ðyÞ C B z 2 z 1 xo A @ Fo 2 fyi fyi A point P(x0, y0) is therefore imaged as a line in the plane z 5 zxi parallel to the y-axis. Likewise, if we consider a pair of planes z 5 zyo and z 5 zyi for which (Ty)12 5 0, we find that in general (Tx)12 does not vanish and again a line image is formed, now parallel to the x-axis. These line images are thus at right-angles to one another and separated by the astigmatic difference. One or both may be virtual (Fig. 19.2). If we move an axial point object along the axis, the line foci will move and there will always be real or virtual object positions for which the line foci coincide and the image is stigmatic. In general, however, the magnifications in the two planes will not be equal. The astigmatic difference can be expressed in terms of the cardinal elements and magnification. Using the quadrupole analogue of (16.25), zxi 2 zðxÞ Fi 5 2 fxi Mx zyi 2 zðyÞ Fi 5 2 fyi My
(19.14)
zxo 2 zðxÞ Fo 5 fxo =Mx zyo 2 zðyÞ Fo 5 fyo =My
(19.15)
and
we see that
N
y
x
Quadrupole Lenses 303
x z
y z
Figure 19.2 Formation of a line image in a magnetic quadrupole. The arrows show the directions of the currents in the windings. Courtesy D.F. Hardy
Λi :¼ zxi 2 zyi 5 ΛFi 2 fxi Mx 1 fyi My Λo :¼ zxo 2 zyo 5 ΛFo 1 fxo Mx 2 fyo My
(19.16)
ðyÞ ΛFi :¼ zðxÞ Fi 2 zFi 5 Λi Mx 5 My 5 0 ðyÞ ΛFo :¼ zðxÞ Fo 2 zFo 5 Λo Mx 5 My -N
(19.17)
where
From (19.13), it is readily seen that quadratic equations are obtained for Mx and My if we attempt to satisfy the stigmatic imaging condition, Λi 5 Λo 5 0. The discriminant δ is the same for Mx and My and we find
304 Chapter 19 Mx 5
ΛFo ΛFi 2 fxo fxi 1 fyo fyi 6 δ 2ΛFo fxi
2ΛFo ΛFi 2 fxo fxi 1 fyo fyi 6 δ My 5 2ΛFo fyi
(19.18a)
with 2 δ2 5 fxo fxi 2fyo fyi 1 Λ2Fo Λ2Fi 1 2 fxo fxi 1 fyo fyi ΛFo ΛFi 2 5 fxo fxi 1fyo fyi 1ΛFo ΛFi 2 4fxo fxi fyo fyi
(19.18b)
In the usual case in which the signs of fx and fy are different, δ2 is positive and there are two real roots and hence two pairs of stigmatic conjugates. The cardinal elements of multiplets are established most easily by multiplying the transfer matrices of the individual lenses; these must be separated by transfer matrices corresponding to the drift spaces, the spaces between the planes z 5 z2 for one lens and z 5 z1 for the next. We recall that these planes may be chosen in various ways; z1 5 z2 5 0, in which case incident position and gradient are related to emergent position and gradient in the same plane, conventionally the mid-plane of the lens, is one good choice, thoroughly explored by Duˇsek (1959). Here we have ! ! ! ðxÞ ðxÞ x2 x zðxÞ =f z z =f 1 f 1 xi xi xo Fi Fi Fo 5 x02 x01 21=fxi 2zðxÞ Fo =fxi (19.19) ! ! ! ðyÞ y2 y1 zðyÞ zðyÞ Fi =fyi Fi zFo =fyi 1 fyo 5 y02 y01 21=fyi 2zðyÞ Fo =fyi (We note that Duˇsek’s matrices are trivially different since he used the vectors (x0 x)T and (y0 y)T.) Another convenient choice involves using different pairs of planes for Tx and Ty, namely the focal planes, since the diagonal matrix elements then vanish: 0 fyo 0 fxo (19.20) Tx 5 ; Ty 5 21=fyi 0 21=fxi 0 This choice has been studied in great detail by Regenstreif (1966, 1967), who has established straightforward rules for writing down the transfer matrices of an arbitrary number of quadrupoles. His procedure can be applied to Duˇsek matrices (Hawkes, 1970), which we temporarily write as follows: ai bi Ti 5 (19.21) c i di where Ti denotes either Tx or Ty for the i-th quadrupole of a sequence and
Quadrupole Lenses 305 zFi zFo zFi ; bi :¼ 1 fo fi fi 1 zFo ci :¼ 2 ; di :¼ 2 fi fi ai :¼
(19.22)
The separation between the i-th and (i 1 1)-th quadrupoles is denoted by Li,i11 and we write 1 Li;i11 (19.23) T Li;i11 :¼ 0 1 Introducing the distances Xi,i11 between the image focus of the i-th quadrupole and the object focus of the (i 1 1)-th quadrupole, Xi;i11 :¼
ai di11 1 Li;i11 1 ci ci11
(19.24)
we can show (Regenstreif, 1966, 1967; Hawkes, 1970) that the transfer matrix of n quadrupoles separated by n1 drift spaces is given by aðnÞ bðnÞ ðnÞ T 5 cðnÞ dðnÞ n21
aðnÞ 5 an L ci αn ; n
i51
cðnÞ 5 L ci γ n ;
n
n
i52
1 γ γ αn 5 Xn21;n 2 2 n22 an cn n21 c2n21 1 δn22 δn21 2 2 β n 5 Xn21;n 2 an c n cn21 γ γ n 5 Xn21;n γ n21 2 2n22 cn21 δn22 δn 5 Xn21;n δn21 2 2 cn21
and γ0 5 δ0 5 0; γ2 5 X1;2 ;
(19.25)
i52
dðnÞ 5 d1 L ci δn
i51
in which
n21
bðnÞ 5 d1 an L ci β n
γ 1 5 δ1 5 1 1 δ2 5 X1;2 2 c1 d1
(19.26)
306 Chapter 19 Another expression for the elements of the transfer matrix between an arbitrary pair of planes, z 5 zn and z 5 z0, may be derived by using the transfer matrix between the principal planes. Writing ði21Þ 2#i#n21 DðiÞ :¼ zðiÞ Po 2 zPi ðnÞ Dð1Þ :¼ zð1Þ 2 z ; D :¼ zn 2 zðn21Þ 0 Po Pi
(19.27)
T ðz0 ; zn Þ 5 Dn Tn21 Dn21 . . .T2 D2 T1 D1
(19.28)
we form the matrix
The elements can be written as Gaussian brackets (Herzberger, 1943, 1958; Hawkes, 1967; Dymnikov, 1968), which are defined as follows: ½x1 ½x1 x2 ½x1 x2 x3 ½x1 x2 x3 . . .xn
5 5 5 5
x x1 x2 1 1 x1 x2 x3 1 x1 1 x3 ½x1 x2 x3 . . .xn22 1 xn ½x1 x2 x3 . . .xn21
(19.29)
We find T ðz0 ; zn Þ 5
½Dn cn21 . . .D2 c1 ½Dn cn21 . . .D2 c1 D1 ½cn21 Dn21 . . .c1 ½cn21 Dn21 . . .c1 D1
(19.30)
For a proof see Hawkes (1967). Gaussian brackets are also employed by Chechulin and Yavor (1969) in connection with prisms; renewed interest has been shown in them in light optics too (e.g., Tanaka, 1981, 1982, 1983, 1986). One common requirement for quadrupole multiplets is that their overall behaviour should be the same as that of a round lens; for this, the cardinal elements in the xz and yz planes must coincide and the focal lengths fx and fy must be equal. If we consider quadrupoles that may have a round electrostatic lens component, provided that the latter has no overall accelerating or retarding effect (φi 5 φo), we can see on symmetry grounds that one at least of these conditions is satisfied by imposing a certain symmetry on the system. In particular, we perceive that the focal lengths in the xz and yz planes are automatically equal in antisymmetric multiplets. The latter are defined as follows. If a multiplet consists of 2N quadrupoles (N $ 1) such that the central plane is a plane of geometrical symmetry and electrical antisymmetry, we say that the multiplet is antisymmetric. The case of N 5 2 was extensively studied by a group in Leningrad (Yavor, 1962; Dymnikov and Yavor, 1963; Dymnikov et al., 1963a,b, 1964a,b, 1965; Shpak and Yavor, 1964) and has come to be known as the Russian quadruplet (Fig. 19.3). Consider two rays, Gx(z) and Gy ðzÞ. Because of the electrical antisymmetry, the sequence of functions p2(z) or Q2(z) encountered by Gx(z) as it proceeds in the positive z-direction will be exactly the same as that traversed by Gy ðzÞ if we imagine it travelling in the negative z-direction. The gradients of
Quadrupole Lenses 307 x
C
D
C
D
z
D
C
D
C
y Antisymmetry plane
Figure 19.3 The antisymmetric or ‘Russian’ quadruplet. The centre plane is a plane of geometrical symmetry and electrical antisymmetry. C, D denote convergent and divergent action respectively.
the emergent asymptotes will hence be equal but opposite in sign, and the focal lengths fx and fy will hence be equal (since φi 5 φo, we already have fxi :¼ fxo ¼: fx and fyi :¼ fyo ¼: fy). If we wish to arrange that an antisymmetric multiplet has the same image-forming properties as a round lens, therefore, we have only to ensure that the foci (or principal points) in the xz and yz planes coincide. For a given geometry, we need only vary the relative excitation until the condition is satisfied. We therefore obtain a set of pairs of excitations, which is known as the load characteristic of the quadruplet. Some examples are given in Chapter 39 of Volume 2.
19.2 Transaxial Lenses The foregoing discussion has been confined to the optics of quadrupoles in general and we have said little about the shapes of the electrodes and polepieces and hence about the effects of any additional symmetries. Two particular additional symmetry properties are of interest, however; one leads to cylindrical lenses, at least in the electrostatic case, as we see in Chapter 20, Cylindrical Lenses. A different symmetry characterizes transaxial lenses, in which the field or potential is rotationally symmetric but the optic axis is no longer the same as the symmetry axis but is perpendicular to it. The electron beam now passes between rotationally symmetric, typically plane electrodes and is focused by the fields in any gaps. Fig. 19.4 shows such a system. The electrodes are ideally circular or annular but are in practice reduced to sectors since the electron beam occupies so little of the space available. It can be seen by comparison with Chapter 20, Cylindrical Lenses, that such structures bear some resemblance to cylindrical lenses but differ from the latter in that the potential is not independent of one of the transverse cartesian coordinates but is the same for all azimuthal
308 Chapter 19
x
φ1 φc
ψ
φ0
z (Optic axis)
Figure 19.4 A transaxial lens. The electrodes shown lie in some plane y 5 const and an identical set lies in the plane y 5 2const.
angles ϕ at a given radial distance from the axis of rotational symmetry. The optical behaviour of such systems was first investigated formally by Strashkevich (1962), to whom we owe the name ‘transaxial lenses’; the theory was set out in some detail in his book of 1966. In the early 1970s, it was realized (by V.M. Kel’man and colleagues in Alma-Ata) that certain features of these structures rendered them attractive for use in the collimator of a prism spectrometer, and their properties were investigated in some detail (Glikman et al., 1971; Brodskii and Yavor, 1970, 1971; Karetskaya et al., 1970, 1971a,b). This work is presented in full in Kel’man et al. (1979), one of the three chapters of which is devoted wholly to these lenses. The symmetry conditions are now such that Φðx; y; zÞ 5 Φðx; 2y; zÞ 1=2 Φð0; y; zÞ 5 Φ x; y; z2 2x2
(19.31)
For small values of x and y, therefore, ! φ0 2 1 φ0 4 x Φðx; y; zÞ 5 φðzÞ 1 x 1 2 φv 2 z 2z z φ0 1 φ2 y 1 2 x 2 y 2 1 φ4 y 4 2z 2
with
(19.32)
Quadrupole Lenses 309 1 φ0 φ2 :¼ 2 φv 1 z 2
! !
(19.33)
p2 ðzÞ 5 φ0 ðzÞ=z 1 φv=2
(19.34)
1 φ0 φ4 :¼ 2 φv2 1 2 12 z Comparing Eq. (19.32) with (7.36), we see that
The paraxial equations have the form xv 1
φ0 0 φ0 x 2 x50 2φ 2zφ
(19.35a)
yv 1
φ0 0 φ2 y 2 y50 2φ φ
(19.35b)
All the theory for quadrupoles can hence be employed without further discussion. The form of (19.35a) is, however, such that simple expressions can be obtained for the focal length and foci. Thus on writing ξ :¼ x=z
(19.36)
Eq. (19.35a) becomes ξv 1
2 1 φ0 z=2φ 0 ξ 50 z
(19.37)
and after some trivial calculation, we find ð 1=2 z dζ xo z 0 x5 1 x o z o 2 x o φo z 1=2 2 zo ζ zo φ ðz 0 1 z φ dζ 5 x0o z xo 2 x0o zo φ1=2 1 o 1=2 2 zo φ3=2 ζ φ giving the transfer matrix 1=2 x φo =φ 2 z=fi 5 x0 21=fi with
1=2 xo z 2 zo φo =φ 1 zzo =fi x0o 1 1 zo =fi
(19.38)
(19.39)
310 Chapter 19 1 φ1=2 52 o fi 2
ðN
φ0 dz
2N φ
3=2
(19.40)
z
in which we have extended the limits of integration to infinity in fi since it is asymptotic imagery that will be of interest. The planes zo and zi will be conjugate if 1=2 φo zo zi 1 50 (19.41) zi 2 zo φi fi or 1 φ1=2 o zo
2
1 1=2 φi zi
52
and the transfer matrix becomes zi =zo zi =zo 0 5 21=fi 1 1 zo =fi 21=fi
1
(19.42)
fi φ1=2 o
0 1=2 zo φ1=2 o =zi φi
(19.43)
In the converging or yz plane, there is no such simple solution of the paraxial equation.