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Analytic solution for a quartic electron mirror Jack C. Straton Portland State University, Department of Physics, PO Box 751, Portland, OR 97207, United States

art ic l e i nf o

a b s t r a c t

Article history: Received 22 April 2014 Received in revised form 29 August 2014 Accepted 8 September 2014 Available online 18 September 2014

A converging electron mirror can be used to compensate for spherical and chromatic aberrations in an electron microscope. This paper presents an analytical solution to a diode (two-electrode) electrostatic mirror including the next term beyond the known hyperbolic shape. The latter is a solution of the Laplace equation to second order in the variables perpendicular to and along the mirror's radius (z2 r 2 =2) to which we add a quartic term (kλz4 ). The analytical solution is found in terms of Jacobi cosine-amplitude functions. We ﬁnd that a mirror less concave than the hyperbolic proﬁle is more sensitive to changes in mirror voltages and the contrary holds for the mirror more concave than the hyperbolic proﬁle. & 2014 Elsevier B.V. All rights reserved.

Keywords: Electron mirror Electron microscope Microscopy Aberration correction Jacobi cosine-amplitude functions

1. Introduction

2. Theoretical model of a quartic diode mirror

In 1990 Gertrude F. Rempfer [1] built on the research of those such as Zworykin et al. [2] and Ramberg [3] to lay the theoretical foundations for the hyperbolic electron mirror as a means to counter the spherical and chromatic aberrations of electron lenses since aberrations in the former are of opposite sign to those in the latter. A number of other researchers have also employed such mirrors [4–8]. In order to gain more ﬂexibility in the ratio of these two aberrations Shao and Wu [9] designed a four-element mirror whose outer elements are not hyperbolic. Likewise, Fitzgerald, Word, and Könenkamp [10] extended Rempfer's diode hyperbolic electron mirror to include a third hyperbolic electrode that provides more ﬂexibility in the choice of potentials to better match spherical and chromatic aberrations and allow for different magniﬁcations. The present paper returns to the diode case to examine the utility of a mirror whose proﬁle deviates from a hyperbolic surface, either more concave or more convex, in order to better describe the equipotentials that result from the ﬁnite radii of real mirrors. As an added beneﬁt, we ﬁnd that mirrors that are more concave than the hyperbolic proﬁle can provide aberration corrections that are more stable against ﬂuctuations in the applied electric ﬁeld. Given that voltages inevitably ﬂuctuate, this robustness will likely improve the resolution of images.

Our approach to the more general class of mirror proﬁles closely tracks the case of the hyperbolic diode mirror [1]. At each step we reduce the extended results to the known hyperbolic proﬁle that Rempfer found. The one exception to that ﬂow is to bypass the derivation of solutions to Laplace's equation for the potential in cylindrical coordinates in terms of the axial potential V(z) near the axis that Rempfer [1] and Shao and Wu [9] use. Instead we simply write down the most general solution [11] in terms of the radial r, axial z, and rotational ϕ variables, as well as the coordinate-separation constants β and m. )( ) ( )( J m ðβ rÞ eimϕ eβz : ð1Þ Vðr; z; ϕÞ ¼ Y m ðβ rÞ e βz e imϕ (Here the stacked brackets are shorthand for the sum of eight possible products of three factors.) For rotationally symmetric solutions, m 0. Since we will place an electrode at some ﬁnite potential passing through z¼0, as in Fig. 1, we exclude the Bessel function of the second kind, Y 0 ðβ rÞ, that diverges there. Finally, for V nonzero at the origin (since we wish to reverse the path of electrons with a positively-charged electrode at the origin) we have the symmetric sum of exponentials, Vðr; zÞ ¼ V M coshðβ zÞJ 0 ðβrÞ:

ð2Þ

The functions cosh and J0 can each be replaced with a series representation, ( Vðr; zÞ V M ¼ V M 1 þ E-mail address: [email protected] http://dx.doi.org/10.1016/j.ultramic.2014.09.003 0304-3991/& 2014 Elsevier B.V. All rights reserved.

β 2 z2 β 4 z4 2

þ

4!

) ( þ⋯ 1

β2 r 2 22

þ

β4 r 4

)

⋯ VM ¼ 24 ð2!Þ2

J.C. Straton / Ultramicroscopy 148 (2015) 168–179

169

Fig. 1. Theoretical model of the quartic diode electron mirror. Dashed lines are the potential surfaces for the hyperbolic mirror proﬁle, having cylindrical symmetry about the z-axis with z ¼ 0 at the cone vertex. (a) Solid lines show the deviation of two quartic potential surfaces to a proﬁle more convex than the hyperbolic case, like the bell of a trumpet. (b) Solid lines show the deviation of the quartic diode mirror to a proﬁle more concave than the hyperbolic diode mirror, like a tulip blossom. The voltages VM and VA are on the mirror and terminating (here, grounded) electrodes, respectively. The distance from the vertex to the opening of the aperture is the mirror length ℓ, which we note has a larger value of z than in the hyperbolic diode mirror. A small aperture in the terminating electrode allows electron to enter and exit the mirror ﬁeld.

2 4 r2 k z r4 þ k z2 þ r 2 z2 þ ⋯; 2 2V M 3 23

2.1. Electron trajectories in a quartic ﬁeld, general solution ð3Þ

where we have subtracted the particular solution V 0 ¼ V M – the 2 potential at the origin – from both sides and where k ¼ ðβ =2ÞV M . The ﬁrst term in the last expression is the potential between the electrodes in a rotationally symmetric hyperboloid ﬁeld, the same arrived at via expansion of V in a conventional power series in r [12]. An analytic solution for the trajectory of an electron in this potential exists, from which the paraxial object/image distance and chromatic and spherical aberration coefﬁcients have been derived [1]. Since r is generally close to the axis, the next term in the last 2 expression, proportional to ðk =2V M Þz4 =3, is the next largest contributor. We here undertake to ﬁnd the analytical solution for electron trajectories in a potential containing this term, too. The present problem, then, is to ﬁnd the equations of motion for an electron in such a potential and to solve these equations for the position of the electron as a function of time. Fig. 1a shows a cross section of a pair of equipotential surfaces for the hyperbolic case (dashed lines) and the ﬂaring of such equipotentials outward from the horizontal axis with the introduction of the quartic term 2 ðk =2V M Þz4 =3 in the potential (solid lines). If, on the other hand, we change the sign of that quartic term, the equipotential surfaces will contract inward toward the horizontal axis as in Fig. 1b. Fig. 1b shows an equipotential labeled VA that is the physical conformation of a grounded electrode in a physical mirror, one containing a small aperture to let electrons pass through from the right. The equipotential labeled VM would also be the conformation of a physical electrode held at a negative voltage to stop the electron at equipotential VC (which is not a physical electrode) and reverse its course. One would suppose that this inward contraction seen in Fig. 1b would tend to more strongly focus the electron in its return trajectory, while an equivalent mirror based on the outward ﬂaring of Fig. 1a would tend to reduce the focusing of the mirror. Our goal in the next section is to turn these suppositions into precise analytical trajectories.

The equations of motion we need to solve, for an electron in a quartic potential ﬁeld, are 2

d r dt

2

¼

e ∂V ek ¼ r ¼ ω2 r; m ∂r m

ð4aÞ

¼

e ∂V ¼ 2ω2 z þ 2ω4 μz3 ; m ∂z

ð4bÞ

2

d z dt

2

where e and m are the charge and mass of an electron and pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ ek=m. In the following μ ¼ m=ð3eVM Þ will be used as a quartic perturbation parameter that may be set to 0 to recover the hyperbolic case in the expressions below. Although Eq. (4b) is a nonlinear second-order differential equation, we suspected that it might be solved using Jacobi elliptic functions [13], since the Jacobi cosine-amplitude cnðujκ Þ, for instance, is a solution [14] to the differential equation

ω¼

2

d y dt

2

2

¼ ð2 k Þy 2y3 :

ð5Þ

Under that supposition the solution would be of the form pﬃﬃﬃ ð6Þ zðtÞ ¼ A cnðBð 2 t ω ψ ÞÞjκ Þ; where the parameters A, B, ψ, and κ will be determined to satisfy Eq. (4b). The derivative is pﬃﬃﬃ pﬃﬃﬃ z_ ðtÞ ¼ 2ABω dn B 2t ω ψ jκ pﬃﬃﬃ ð7Þ sn B 2t ω ψ jκ ; which is expressed in terms of Jacobi delta-amplitude dnðujκ Þ and sine-amplitude snðujκ Þ functions. We assert the boundary condition that the electron location at t¼0 is at the potential surface where z_ and r_ are both zero, the point of furthest penetration into the quartic ﬁeld (VC in Fig. 1b). At this point the incident and return electron trajectories are perpendicular to the reﬂecting potential surface. Since snð0jκ Þ ¼ 0, [15] then z_ ð0Þ ¼ 0 requires ψ 0.

170

J.C. Straton / Ultramicroscopy 148 (2015) 168–179

A second derivative of Eq. (6), for substitution into the lefthand side of Eq. (4b), leads to dnðujκ Þ2 and snðujκ Þ2 terms. These are transformed into cnðujκ Þ2 terms by [16] dnðujκ Þ2 ¼ 1 κ 2 snðujκ Þ2 ;

ð8aÞ

snðujκ Þ2 ¼ 1 cnðujκ Þ2 ;

ð8bÞ

so that the only Jacobi elliptic functions appearing on either leftor right-hand sides are cnðujκ Þ functions. The requirement that this equal the right-hand side of Eq. (4b) conﬁrms our supposition (6) and leads to the following four sets of values for the parameters: 0 1 1 1 þ 5 A2 μ ω2 C κ- B @ 1 þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃA 2 2 2 2 2 ð1 þ A μ ω Þ ð1 þ 5 A μ ω Þ ð9aÞ

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 þ A2 μ ω2 Þ ð1 þ5 A2 μ ω2 Þ 1 κ- þ 2 2 2 A2 μ ω2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ B- 7 i 1 þ A2 μ ω2

ð9bÞ

Inserting the ﬁrst of these into Eq. (6) gives

0 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ11 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ 5 z2C μ ω2 CC p ﬃﬃﬃ 1 B B z ¼ zC [email protected] 7 i 2 ω t 1 þ z2C μ ω2 @ 1 þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ AA; 2 1 þ z2C μ ω2 0

ð10Þ

after noting that zð0Þ zC ¼ A since cnð0jκ Þ ¼ 1 [15]. Suppose we let μ-0 in this expression, then pﬃﬃﬃ ð11Þ z⟶zC cosh 2 ω t ; μ-0

which is indeed the analytic result for the hyperbolic potential [1]. On the other hand, inserting (9b) into (6) under this limit gives zC times pﬃﬃﬃ ω2 t 2 3ω6 t 6 ⋯; ð12Þ cn 7 i 2 ωt 1 ¼ 1 þ ω2 t 2 2 10 so we exclude (9b) from further consideration as unphysical. Taking (10) as the ﬁnal form for z(t), then z_ is pﬃﬃﬃqﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ z_ ¼ iωzC 2 1 þz2C μ ω2 0 0 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ11 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ 5 z2C μ ω2 CC pﬃﬃﬃ 1 B B [email protected] 7i 2 ω t 1 þ z2C μ ω2 @ 1 þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ AA 2 1 þ z2C μ ω2 0 0 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ11 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ 5 z2C μ ω2 CC p ﬃﬃﬃ 1 B B [email protected] 7 i 2 ω t 1 þ z2C μ ω2 @ 1 þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ AA: 2 1 þ z2C μ ω2

reduces to the hyperbolic-potential pﬃﬃﬃ tanhð 2 ω tÞ if we let μ-0. The solution for r such that r_ ð0Þ ¼ 0, is [1]

3.1. Physical model and mathematical approach A physical model of the diode quartic mirror is presented in Fig. 1b. It consists of a mirror electrode at potential VM with a surface that is the solution to z2

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ B- 7 i 1 þ A2 μ ω2

This

we will ﬁnd where the electron exits the mirror into ﬁeld-free space and then where it crosses the optical axis. We consider the case of a symmetric mirror mode with the paraxial object and image at the same distance from the mirror (following the convention of Ref. [1]). In this scenario we need only calculate the aberration on the return (or incident) trajectory and multiply by a factor of 2.

value

ð13Þ pﬃﬃﬃ 2 ω zC

r2 1 4 r2 þk z z2 þ kλz4 ¼ 0; 6V M 2 2

ð15Þ

where we have introduced another simplifying parameter λ ω2 μ=ð2kÞ ¼ 1=ð6V M Þ that plays the same role as μ when we wish to seek the limit where the only hyperbolic term is present. When λ-0 (or μ-0), Eq. (15) describes a p cone ﬃﬃﬃ with vertex at the origin, with opening half-angle ϕ ¼ tan 1 2, the dashed lines in Fig. 1a and b.pﬃﬃﬃ When r ¼ 2 units, Eq. (15) requires that z¼ 1 unit if λ ¼ 0. But for λ 4 0 the value of z need not to be as large to give zero, so the effect of the quartic term on the equipotential surface is to ﬂare the outward from the horizontal axis like the bell of a trumpet, as in Fig. 1a, making it less converging. Of course, having obtained the solution including this term as dictated by the series expansion of coshðαzÞ, a mirror designer is nevertheless free to experiment with creating a more-converging mirror by reversing the sign of the quartic term, and/or reducing the coefﬁcient to small values to test the effect of slight perturbations of the hyperbolic design. We will retain the dictated coefﬁcient for the remainder of this derivation, knowing that it may be changed at a later step. Indeed we will demonstrate the beneﬁts of taking λ o 0 below. In such a case, for a given r, z must be larger to give zero in (15), so the effect of the negative quartic term in the potential is to contract the cone towards the horizontal axis, like a tulip blossom, as in Fig. 1b , making it more converging. Fig. 1b shows the full diode mirror in the more-converging case, with the right-hand electrode grounded. The reﬂecting electron starts with zero velocity at the quartic surface with the same voltage as the cathode V ¼ V C , after which it is focused by the quartic potential ﬁeld in the region 0 o z o ℓ. As it exits the initial region, it is deﬂected by the diverging thin lens ﬁeld created by the aperture in the terminating electrode at z ℓ. The full potential r2 2 Vðr; zÞ V M ¼ k z2 þ k λz 4 ; ð16Þ 2 at the point z ¼ ℓ becomes on axis V A V M ¼ kℓ2 þ k λℓ4 ; 2

ð17Þ

since it is an equipotential surface. We can solve this for kℓ 40: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ 1 þ 4V A λ 4V M λ 2 kℓ ¼ ð18aÞ 2λ 2

r ¼ r C cos ðωt Þ;

ð14aÞ

r_ ¼ ωr C sin ðωt Þ ¼ ωr tan ðωt Þ:

ð14bÞ

1

¼ ∑

n¼1

3. Finding the aberration coefﬁcients The coefﬁcient of chromatic aberration Cc is a measure of the change in the paraxial object/image distance in response to a change in the electron energy dE=E ¼ dV C =V C . Thus, in this section

1=2 n

22n 1 λ

n1

ðV A V M Þn ;

ð18bÞ

where in (18b) we have expanded the square root of (18a) in a series whose ﬁrst term is the λ-0 limit, the usual hyperbolic result, with ℓ identiﬁed as the semi-transverse axis of the hyperbola [17]. (For VA ¼ 0 and V M ¼ 1, the series gains one decimal place of accuracy for every three terms.) We discard the

J.C. Straton / Ultramicroscopy 148 (2015) 168–179

other root of the equation (a minus sign before the square root) in which the simple pole in λ does not cancel in an equivalent series expansion, giving an incorrect λ-0 limit.

¼

3.2. Electron trajectories in the region 0 o z o ℓ

171

r 21 k r 21 λ ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ þℓ2 2 1þ 1þ4 VA λ4 VM λ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1 1 þ 1 þ 4 V A λ 4 V M λ þ 2k r 21 λ A pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ þ 1þ 1þ4 VA λ4 VM λ

An electron begins its return trip at time t ¼0 at the reﬂection point ðr C ; zC Þ, which is on the quartic equipotential surface V ¼ V C . Points on this quartic surface are described by

r 21 þℓ2 ð1 νA Þ 2

r2 V C V M zCð0Þ2 ¼ z2C C þkλz4C ; k 2

¼

ℓ2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ 1 þ 4 V A λ 4 V M λ (

ð19Þ

which is again less (more) converging than a hyperbolic equipotential surface when λ is positive (negative). Note that zð0Þ C is the axial coordinate of the vertex of the potential surface V ¼ V C . The solution to (19) is qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ 1 þ 4 V C λ 4 V M λ þ 2k r 2C λ z2C ¼ ð20aÞ 2 λk ¼

k λ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þ 1þ4 VA λ4 VM λ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1 1 þ 1 þ 4 V C λ 4 V M λ þ 2k r 2C λ A pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ þ 1 þ 1 þ4 V A λ 4 V M λ r 2C 2

þ ℓ2

r 2C

ð26cÞ

1þ 1þ4 VA λ4 VM λ þ

r 21 ð 1 þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ )1=2 ! 1 þ 4 V A λ 4 V M λÞ ℓ2

r2 ! 1 þℓ2 : λ-0 2

(Additional forms for this equation are given in Appendix A.) The angle at which the electron exits the initial region is r_ 1 : z_ 1

tan α1 ¼

r 2C þ ℓ2 ð1 νC Þ 2

ð20cÞ

3.3. Deﬂection by the terminating electrode aperture

r2 r2 V VC C þℓ2 ð1 νÞ: ! C þ ℓ2 1 A V A V M 2 λ-0 2

ð20dÞ

(For completeness, additional forms for this equation are given in Appendix A.) At time t1 the electron exits the initial region and enters the aperture in the terminating electrode with the coordinates given by r 1 ¼ r C cos θ1 ;

ð21Þ

r_ 1 ¼ ωr 1 tan θ1 ;

ð22Þ

0 0 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ11 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ5 z2C μ ω2 CC p ﬃﬃﬃ 1 B B 2 z1 ¼ zC [email protected] 7 i 2 θ1 1 þ zC μ ω2 @ 1 þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ AA; 2 1 þz2C μ ω2 ð23Þ pﬃﬃﬃqﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ z_ 1 ¼ iωz1 2 1 þz2C μ ω2 0 0 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ11 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ 5 z2C μ ω2 CC pﬃﬃﬃ 1 B B [email protected] 7 i 2 θ1 1 þ z2C μ ω2 @ 1 þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ AA 2 1 þ z2C μ ω2 0 0 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ11 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ 5 z2C μ ω2 CC p ﬃﬃﬃ 1 B B [email protected] 7i 2 θ1 1 þ z2C μ ω2 @ 1 þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ AA; 2 1 þ z2C μ ω2 ð24Þ where θ1 ¼ ωt 1 and scðujκ Þ ¼ snðujκ Þ=cnðujκ Þ. (An additional form for this equation is given in Appendix A.) The coordinates ðr 1 ; z1 Þ are on the quartic surface at the potential VA described by r2 VA VM ¼ ℓ2 þ λℓ4 ¼ z21 1 þkλz41 ; k 2 whose solution is qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ 1 þ4 V A λ 4 V M λ þ 2k r 21 λ z21 ¼ 2 λk

ð26dÞ

ð26eÞ

ð20bÞ

ð26bÞ

ð25Þ

ð26aÞ

ð27Þ

The aperture in the terminating electrode acts as a diverging thin lens. The focal length can be calculated using the Davisson– Calbick formula, f¼

4V beam : ð∂V=∂zÞ2 ð∂V=∂zÞ1

ð28Þ

where the electron is traveling from 1-2, which is good for weak lenses with relatively small openings compared to the axial dimensions [18]. Applying the formula to the potential distribution (Eq. (16)), and setting the potential gradient to zero outside the mirror (region 2), the focal length is fA ¼

4ðV A V C Þ 2ðV A V C Þ ¼ : 2 ð∂V=∂zÞ1 kz1 þ3k λz31

ð29Þ

(Note that we have the hyperbolic result in the λ-0 limit.) Here we treat the aperture lens as a deﬂection point, changing the trajectory of an electron passing through it by an angle 2 r 1 kz1 þ 3k λz31 tan δA ¼ r 1 =f A ¼ 2ðV A V C Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ r 1 1 þ 1 þ 4 V A λ 4 V M λ z1 ¼ 4 λðV A V C Þℓ2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2 3r 1 1 þ 1 þ 4 V A λ 4 V M λ z31 ð30aÞ 8 λðV A V C Þℓ4 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ r 1 1 þ 1 þ 4 V A λ 4 V M λ

¼ "

4 λðV A V C Þℓ2

ℓ2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þ 1þ4 VA λ4 VM λ 0 ( @ 1 þ 1 þ 4 V A λ 4 V M λ

þ

r 21 ð 1 þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ )1=2 1#1=2 1 þ 4 V A λ 4 V M λÞ A ℓ2

172

J.C. Straton / Ultramicroscopy 148 (2015) 168–179

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2 3r 1 1 þ 1 þ 4 V A λ 4 V M λ

"

terms of λ ¼ 1=ð6V M Þ as follows:

8 λðV A V C Þℓ4

zCð0Þ2 μ ω2 ð1 νC0 Þ

ℓ2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ 1 þ 4 V A λ 4 V M λ 0 ( @ 1þ 1þ4 VA λ4 VM λ

þ

r 21 ð 1 þ

¼ 2 λðV A V M Þð1 νÞ:

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ )1=2 1#3=2 1 þ 4 V A λ 4 V M λÞ A ℓ2

ð30bÞ

r 1 z1 ! 2 λ-0 2ℓ ν

ð30cÞ

at the point ðr 1 ; z1 Þ. After the deﬂection, the trajectory makes an angle with the axis of

αA ¼ α1 þ δA :

ð31Þ

Thus [19], 0 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1 1 þ5 zCð0Þ2 μ ω2 C 1B ﬃ A κ ¼ @ 1 þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 1 þzCð0Þ2 μ ω2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! 1 1 þ 10 λðV A V M Þð1 νÞ 1 þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 2 1 þ 2 λðV A V M Þð1 νÞ 2 λðV A V M Þð1 νÞ;

z0 ¼ z1 þ r 1 = tan αA :

3.4. The paraxial object/image distance The paraxial object/image distance z0 is computed from the onð0Þ 0 0 axis limits along the optical axis: zC -zð0Þ C ; z1 -z1 ; and z -z0 . If we set r 1 ¼ 0 in (26), we can write a zeroth-order version of (26) pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þ4 VA λ4 VM λ 2 λk pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ! 1þ 1þ4 VA λ4 VM λ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ ℓ2 : 1þ 1þ4 VA λ4 VM λ

1 þ

¼ ℓ2

ð33aÞ ð33bÞ

We also need the zeroth-order version of (20) – which is simply the axial coordinate of the vertex of the VC surface in the deﬁnition (19): zCð0Þ2 ¼

V C V M 2 λℓ ðV C V M Þ ﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ k 1þ 1þ4 VA λ4 VM λ 2

"

¼ℓ

2

2 λðV C V M Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 1 1 1þ 1þ4 VA λ4 VM λ

ð34aÞ #! ℓ2 ð1 νC0 Þ

λ-0

ð34cÞ

(Additional forms for this equation are given in Appendix A.) This is not (20) with rC ¼0, as is shown in Appendix B, Eqs. (B.1) and (B.2). The zeroth-order expressions for the Jacobi cosine-amplitude function parameters κ and B involve the product of ½ℓ2 zCð0Þ2 μ ω2 that can be expressed, using (34b) and the ﬁrst term in (18b), in

ð36bÞ

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ zCð0Þ2 μ ω2 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 1 þ 2 λðV A V M Þð1 νÞ

ð37aÞ

1 þ λðV A V M Þð1 νÞ:

ð37bÞ

The hyperbolic aberration calculations are greatly simpliﬁed by utilizing the zeroth order relation pﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð0Þ cosh 2θ1 ¼ ℓ=zCð0Þ ¼ 1= 1 ν: ð38Þ to give pﬃﬃﬃ pﬃﬃﬃ ð0Þ tanh 2θ1 ¼ ν:

ð39Þ

for use in the λ-0 version of the denominator of (27), that in turn appears in the denominator in the last term of (32). In the quartic case one has an equivalent relation to (38), z1ð0Þ zCð0Þ

ð1 νA Þ1=2

¼

ð1 νC0 Þ1=2

ð40aÞ

0 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ11 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p ﬃﬃﬃ 1 þ 5 zCð0Þ2 μ ω2 CC 1 B ð0Þ B 1 þ zCð0Þ2 μ ω2 @ 1 þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ AA ¼ [email protected] 7i 2 θ1 2 1 þ zCð0Þ2 μ ω2 0

ð40bÞ 0 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p ﬃﬃﬃ ð0Þ B ¼ [email protected] 7i 2 θ1 1 þ 2 λðV A V M Þð1 νÞ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!! 1 1 þ 10 λðV A V M Þð1 νÞ 1 þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 1 þ2 λðV A V M Þð1 νÞ

ð40cÞ

1 p ﬃﬃﬃ

ð0Þ B C [email protected] 7i 2 θ1 1 þ λðV A V M Þð1 νÞ 2 λðV A V M Þð1 νÞA: 0

ð40dÞ

ð34bÞ !ℓ2 ð1 νÞ:

ð36aÞ

8 iB ¼

ð32Þ

As an aside, if one contemplates the extension of the present work to a triode mirror in parallel to Fitzgerald, Word, and Könenkamp's [10] electron mirror including three hyperbolic electrodes, one must revisit the use of the Davisson–Calbick formula for the intermediate electrode. There seems to be merit in this approximation for the terminating electrode where the electron is passing into a ﬁeld-free region. The corresponding point transformation, however, does not seem to be generalizable for passage into a region of nonzero ﬁeld.

ð35Þ

and

At small angles, this rotation will only have signiﬁcant impact on the r_ 1 component of the trajectory. The electron exits the mirror into ﬁeld-free space, crossing the optical axis at

z1ð0Þ2 ¼

m ekℓ2 3eV M m

The zeroth-order hyperbolic angle may be found from inverting (40c) 1 pﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 7 i 2 1 þ 2 λðV A V M Þð1 νÞ 0sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!1 ð1 νA Þ 1 1 þ 10 λðV A V M Þð1 νÞ @ 1 þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ A arccn ð1 νC0 Þ2 1 þ 2 λðV A V M Þð1 νÞ

θð0Þ 1 ¼

ð41aÞ

J.C. Straton / Ultramicroscopy 148 (2015) 168–179

0sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 1 2 λðV A V M Þð1 νÞA [email protected] ð1 νC0 Þ pﬃﬃﬃ

7 i 2 1 þ λðV A V M Þð1 νÞ 1 1 ! pﬃﬃﬃ arccosh pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : λ-0 2 1ν

0

tan θ1 pﬃﬃﬃ

z1 2 1 þ λðV A V M Þð1 νÞ

2 1 2 λðV A V M Þð1 νÞ

1=2 1 þð1 νA Þ=ð1 νC0 Þ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 νA νC0 νA !

tan θ1 tan θ1 pﬃﬃﬃ : pﬃﬃﬃpﬃﬃﬃ ¼ pﬃﬃﬃ 2 ν z1 2 tanh 2θð0Þ 1

λ-0 z1

tan θ1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

ℓ 2νC0 1 þ λðV A V M Þð1 νÞ

2 1 2 λðV A V M Þð1 νÞ !1=2

1 þ 1=ð1 νC0 Þ

ð45bÞ

tan θ1 ! pﬃﬃﬃpﬃﬃﬃ: λ-0 ℓ 2 ν

ð45cÞ

Applying these same approximations to (30) gives pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð0Þ 1 þ 1 þ4 V A λ 4 V M λ tan δA ¼ r1 4 λðV A V M Þν ℓ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2 3 1þ 1þ4 VA λ4 VM λ 8 λðV A V M Þν ℓ ¼

1 þ 2ðV A V M Þ λ 2ℓν

! λ-0

ð46aÞ ð46bÞ

1 : 2ℓν

ð46cÞ

3.5. The on-axis and small-angle limit The on-axis and small-angle limit of Eq. (32) is z0 ¼ lim z0 ¼ zð0Þ 1 þ lim r C -0

r C -0

¼ ℓ þ q0 ℓ; ð43Þ

r C -0

r1

α1 þ δA

ð47Þ

!1=2

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2νC0 1 þ 1 þ 4 V A λ 4 V M λ ν 4 λðV A V M Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ2 1 2νC0 3 1 þ 1 þ 4 V A λ 4 V M λ C A ν 8 λðV A V M Þ

ð48aÞ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð0Þ 2νC0 tan θ1 2νC0 1 þ λðV A V M Þð1 νÞ

2 1 2 λðV A V M Þð1 νÞ

1=2 1 þ1=ð1 νC0 Þ ! 1

ð44cÞ

¼ zð0Þ 1 þ lim

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð0Þ 2νC0 tan θ1 1 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ q0 2νC0 1 þ 2 λðV A V M Þð1 νÞ 0 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!2 1 1 þ 10 λðV A V M Þð1 νÞ @ 1 þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 4 1 þ2 λðV A V M Þð1 νÞ

ð44aÞ

ð44bÞ

r1

αA

where

1 þ 1=ð1 νC0 Þ

We set z1 ℓ and νA ¼ 0, in consonance with (34), to get the zeroth-order version of (44): tan α1ð0Þ tan θ1 ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ r1 ℓ 2νC0 1 þ 2 λðV A V M Þð1 νÞ

ð45aÞ

ð41cÞ

and (35) in (24) so that (27) becomes tan α1 ω tan θ1 ¼ pﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ r1 iωz1 2 1 þ 2 λðV A V M Þð1 νÞ 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 κ 2 κ 2 ð1 νA Þ=ð1 νC0 Þ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 νA νA νC0

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!2 1 1 þ 10 λðV A V M Þð1 νÞ @1 1 þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 4 1 þ 2 λðV A V M Þð1 νÞ !1=2

1 þ 1=ð1 νC0 Þ

ð41bÞ

We have found numerical instabilities in implementing this closed-form of the inverse cosine-amplitude (some sources use the notation cn 1 for this) in a program such as Mathematica. So we substitute a series expansion in the parameter κ, (36) which is small for the full range of mirror voltage ratios 0:65 o ν o 0:99. Although the deﬁnition V m λ 1=6 arises from the separation constant in (3), we can contemplate independently varying it to achieve a desired mirror design. For λ ¼ 10 6 , κ o 0:007 even for V m ¼ 104 and κ -0:617 as jV m j 4109 . If we set V A 4 0 this simply decreases the value for which Vm approaches its asymptote. Taking jλj ¼ 1=6 simply hastens that asymptotic approach. In fact, we ﬁnd that just three terms in the series [20] are sufﬁcient in (41): qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ arccnðz; κ Þ ¼ arccosðzÞ þ 1=4κ 2 ½ z ð1 z2 Þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ þ arccosðzÞ þ 3=64κ 4 ½ 5z ð1 z2 Þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ þ 2z3 ð1 z2 Þ þ 3 arccosðzÞ þ⋯: ð42Þ Then one substitutes [22] sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ snðujκ Þ2 dnðujκ Þ scðujκ Þ ¼ dnðujκ Þ2 cnðujκ Þ2 vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u u 1 cnðujκ Þ2 t 2 ¼ 1 κ 2 κ 2 cnðujκ Þ cnðujκ Þ2 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 νC0 Þ ð1 νA Þ ¼ ð1 νA Þ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 νC0 Þ κ 2 ð1 νC0 Þ κ 2 ð1 νA Þ ð1 νC0 Þ

173

νC0 ð1 þ 2ðV A V M Þ λÞ ν

ð48bÞ

pﬃﬃﬃﬃﬃﬃ ð0Þ 2ν tan θ1 1 : 2 ν μ;λ-0

ð48cÞ

!

174

J.C. Straton / Ultramicroscopy 148 (2015) 168–179

Quartic Mirror

Quartic Mirror

8

150

6

Cc

zq0

100

4

50

2

0

0

0.4

0.5

0.6

0.7 V A VC

0.8

0.9

0.6

0.7 VA VC

V A VM

Fig. 2. The paraxial object/image distance z0 (relative to the mirror length ℓ) for the quartic mirror in the symmetric mode as a function of the ratio of accelerating voltage to mirror voltage, ν ¼ ðV A V C Þ=ðV A V M Þ. The solid line is the hyperbolic limit, which matches Rempfer's Fig. 4(a), though her abscissa has higher values on the left rather than the right [1]. As ν decreases, z0 increases continuously in the range shown, diverging as ν approaches 0.607. This divergence point shifts to higher voltages as the quartic mirror proﬁle becomes less concave than a hyperbola – the highest shown (dashed) curve is for λ ¼ þ 3=18 – and to lower voltages as the mirror proﬁle becomes more concave. For λ ¼ 3=18, the lowest (dot-dash) curve, the divergence point moves down to about 0.4. The intermediate curves are for λ ¼ 7 2=18 and λ ¼ 7 1=18. For values of ν very close to 1.0, the incident or returning electron crosses the axis more than once.

Fig. 2 shows the effects on the paraxial object/image distance z0 of deforming the hyperbolic mirror (the solid line) by adding quartic mirror terms, as a function of the ratio of accelerating voltage to mirror voltage ν ¼ ðV A V C Þ=ðV A V M Þ. The lowest (dotdash) curve is for a quartic mirror proﬁle that is more concave than a hyperbola, with λ ¼ 3=18, which gives a much larger range of voltages before the divergent image behavior is reached, ν ¼ 0:4 vs. 0:6. On the other hand, as the quartic mirror proﬁle becomes less concave, the highest (dashed) curve for λ ¼ þ3=18, the mirror becomes more diverging. The inﬂection point near ν ¼ 0:92 is caused by the incident or returning electron crossing the axis more than once. The present work, thus, provides a tool for gauging the correctness of the proﬁle of a hyperbolic mirror. If one tests a manufactured electron mirror whose divergence point shifts higher or lower than ν ¼ 0:607 that may provide a diagnostic of an imperfectly hyperbolic proﬁle. Seen from another perspective, analytic results such as those in Refs. [1] and [10], as well as the present work, hold for mirrors of inﬁnite radial extent, but manufactured mirrors are necessarily of ﬁnite radius. One typically adds a bead on the concave side of the mirror to create a ﬁeld approximating an inﬁnite one. In some sense such a bead provides a somewhat quartic proﬁle for the electric ﬁeld.

length of its hyperbolic limit [1] 3q2 ν C c ! 1 þ q0 þ 02 : 4ν λ-0 ð1 νÞ

ð49Þ

It is measured in units of length (cm). The result for the quartic mirror is too long to display in this paper since it is 79 times the

0.9

1.0

ð50Þ

We do, however, show it graphically in Fig. 3 for a series of values of λ between 7 3/18 as a function of ν ¼ ðV A V C Þ= ðV A V M Þ. The negative value of the chromatic aberration denotes overcorrection, which can be used to compensate for undercorrected electron lenses. The solid line is the hyperbolic limit. The lowest (dot-dash) curve is for a quartic mirror proﬁle that is more concave than a hyperbola, with λ ¼ 3=18, for which there is a much larger range of voltages ν in which the chromatic aberration does not change. The opposite is true as the quartic mirror proﬁle becomes less concave, the highest (dashed) curve for λ ¼ þ 3=18, and the ﬂat range of responses to voltage is much diminished. (The intermediate curves are for λ ¼ 7 2=18 and λ ¼ 71=18.) 3.7. Coefﬁcient of spherical aberration We can likewise now calculate the coefﬁcient of spherical aberration, which is a measure of the change in the distance at which the electron crosses the optical axis in response to a change in the angle of incidence (squared), according the linear equation [1], 2ðz0 z0 Þ 2Δz ¼ C s α2A

ð51aÞ

d d 2 Δz ¼ C s 2 α A αA dr C dr C

ð51bÞ

d We now have all we need to calculate the coefﬁcient of chromatic aberration Cc, which is a measure of the change in the paraxial object/image distance in response to a change in the electron energy dE=E ¼ dV C =V C or [1]

0.8 V A VM

Fig. 3. The chromatic aberration coefﬁcient Cc of the quartic mirror for symmetric rays as a function of ν ¼ ðV A V C Þ=ðV A V M Þ. The aberration coefﬁcient is plotted relative to ℓ, the length of the mirror, and its negative value denotes overcorrection. The solid line is the hyperbolic limit, which matches one of the three curves in Rempfer's Fig. 4(b), though her abscissa has higher values on the left rather than on the right [1]. The lowest (dot-dash) curve is for a quartic mirror proﬁle that is more concave than a hyperbola, with λ ¼ 3=18. One sees a much larger range of voltages ν in which the chromatic aberration does not change. As the quartic mirror proﬁle becomes less concave, the highest (dashed) curve for λ ¼ þ 3=18, this ﬂat range of responses to voltage is diminished. The intermediate curves are for λ ¼ 7 2=18 and λ ¼ 7 1=18.

3.6. Coefﬁcient of chromatic aberration

∂ z0 =ℓ ∂q ¼ 2ν 0 : C c ¼ 2ν ∂ν ∂ν

0.5

1.0

2

2 dr C

2 Δz ¼ C s 2

d αA drC

2 ð51cÞ

where the factor of 2 comes from the symmetric path and negative sign is the convention for overcorrection. The second and third forms are from L'Hospital's rule. Like Cc, it is measured in units of length (cm). Positive values indicate that the image distance decreases from the paraxial value as the trajectory angle increases. As in the hyperbolic case, we seek an iterative solution for z and α. For the narrow beams used in electron microscopy, a ﬁrst-order off-axis approximation for z0 -z0ð1Þ and αA -αAð1Þ is sufﬁcient. We

J.C. Straton / Ultramicroscopy 148 (2015) 168–179

or 1

θð1Þ 1 ¼ pﬃﬃﬃarccosh

qp ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ 4V A λ 4V M λ 1

100

50

0

2 1 þ 4V C λ 4V M λ

0.65

0.70

0.75

0.80 VA VC

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 1=2 1 þ4V A λ 4V M λ 1 þ r 2C 1 2 3 vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u 1 1 6u 7 ! pﬃﬃﬃarccosh4tqﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ5: λ-0 2 1 ν þ r 2C =2

ð53aÞ

ð53bÞ

Finally, the second-order approximation for the time function

ð1Þ in Eq. (26d) and using this θð2Þ 1 is found by setting r 1 ¼ r C cos θ 1

in the equivalent of Eq. (52). The effect is to complexify the argument of the internal square root in the equivalent of the ﬁrst line of Eq. (53a) to read 1 þ 4V A λ 4V M λ-1 þ 4V A λ 4V M λ h i2 ð1Þ þ r C cos θ1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ 4V A λ 4V M λ 1

0.85

0.90

0.95

V A VM

Fig. 4. The spherical aberration coefﬁcient Cs of the hyperbolic mirror for symmetric rays as a function of ν ¼ ðV A V C Þ=ðV A V M Þ. The aberration coefﬁcient is plotted relative to ℓ, the length of the mirror, and its negative value denotes overcorrection. The solid line is the hyperbolic limit, which matches a second of the three curves in Rempfer's Fig. 4(b), though her abscissa has higher values on the left rather than on the right [1]. The lowest (dot-dash) curve is for a quartic mirror proﬁle that is more concave than a hyperbola, with λ ¼ 3=18. As with chromatic aberration, one sees a much larger range of voltages ν in which the spherical aberration does not change appreciable. On the other hand, as the quartic mirror proﬁle becomes less concave, the highest (dashed) curve for λ ¼ þ 3=18, this ﬂat range of responses to voltage is diminished. The intermediate curves are for λ ¼ 7 2=18 and λ ¼ 7 1=18.

þ sin ðzÞ þ 8 sin ð3zÞ þ 8 sin ð5zÞ þ sin ð7zÞ 40z cos ðzÞ 24z cos ð3zÞ 8z cos ð5zÞ þ κ 2 ½16 sin ð3zÞ þ 16 sin ð5zÞ 128z cos ðzÞÞ þ 128 sin ðzÞ þ 128 sin ð3zÞÞ

ð54Þ ð1Þ

and then to insert the expression for θ1 in the second line of Eq. (54). In the hyperbolic limit, the effect is to complexify the numerator of the outer square root in Eq. (53b) to read 1-1 þr 2C =2 2 0

0 1132 vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u 1 1 6 B Bu CC7 4 cos @pﬃﬃﬃ[email protected]ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃAA5 : 2 1 ν þ r 2C =2

ð55Þ

Finally, Eqs. (31) and (32) for αA and z0 can be expanded in a power series in rC around r C ¼ 0. The coefﬁcient of spherical aberration can be computed using the ﬁrst non-zero terms of these expansions. The second and third forms of Eq. (51) are used in ﬁnding the limit as r C -0 via L'Hospital's rule since the ﬁrst version, straight division of 2Δz2 =α2A , gives a pole in r2C from α2A. For the hyperbolic mirror, the ﬁrst form may be used if one sets 10 3 ≳r C ≳10 4 . But the quartic mirror has no easy-to-identify plateau value of rC for which results are stable. Cc was found for the quartic mirror using an approximation for the Jacobi cosine amplitude to eliminate numerical instabilities. For Cs the Jacobi delta amplitude function (dn) and sin-amplitude/ cosine-amplitude ratio function (sc) in Eq. (24b), which appears in Eq. (27), will easily exhaust the memory capacity of a computer running an algebra program such as Mathematica. Thus we introduce a series approximation at the beginning of the computer algebra processes rather than later at the numerical evaluation step. We found a trigonometric representation of the product dnðzjκ Þ scðzjκ Þ

Quartic Mirror 150

Cs

will, however need the hyperbolic angle θ to second order in this iteration. Given that the zeroth-order version of z21 , Eq. (26), is z1ð0Þ2 ¼ ℓ2 , Eq. (33), the next order z1ð1Þ2 is found by setting r 1 ¼ 0 in Eq. (26c). To obtain zCð1Þ2 , we use the full third expression for z2C, Eq. (20c), but with νC -νCðrC ¼ 0Þ from (B.2a). (That is, we set r C ¼ 0 only in the term in parentheses in Eq. (20b). Then the ﬁrst-order approximation for the time function is pﬃﬃﬃ ð1Þ ð1Þ cosh 2θ1 ¼ zð1Þ ð52Þ 1 =zC

175

1 sec3 ðzÞ κ 4 32z2 sin ðzÞ 512

ð56Þ

that is valid for the small values of κ that arise from any voltage on the mirror. Even the hyperbolic limit of the spherical aberration coefﬁcient is 32 times the length of the hyperbolic limit of the chromatic aberration coefﬁcient (Eq. (50)), due to the need to use the secondð2Þ order time function θ1 , so we do not display it here, nor did Rempfer [1]. The result for the quartic mirror is vastly longer. We do, however, plot the quartic result in Fig. 4 for a series of values of λ between 7 3=18 as a function of ν ¼ ðV A V C Þ= ðV A V M Þ. The negative value of the spherical aberration denotes overcorrection, which can be used to compensate for undercorrected electron lenses. The solid line is the hyperbolic limit. The highest (dashed) curve is for a quartic mirror proﬁle that is less concave than a hyperbola, with λ ¼ þ3=18. As with the chromatic aberration coefﬁcient, there is a much greater sensitivity to electrode voltages ν that a diode mirror designer may use to correct for issues such as changes in magniﬁcation between samples, for which Cc and Cs scale differently, and electron energy differences that affect the aberrations in the transfer lens differently for Cc and Cs. The lowest (dot-dash) curve is for a quartic mirror proﬁle that is more concave than a hyperbola, with λ ¼ 3=18. As with the chromatic aberration coefﬁcient, there is a much larger range of voltages ν for which the spherical aberration does not change much. Given that voltages inevitably ﬂuctuate somewhat and one has a region as λ becomes more negative in which such ﬂuctuations would not appreciably affect mirror performance, there are applications for which this, too, can be an advantage for a mirror designer. Users of diode mirrors ﬁnd that instabilities in power supplies, sample charging and quality, external magnetic ﬁelds, mechanical vibrations, astigmatism, beam alignment, etc., are the major limits to best resolution. From this point of view, a more stable mirror gives the microscopist one reliable variable in a sea of problems.

J.C. Straton / Ultramicroscopy 148 (2015) 168–179

Speciﬁcally, in higher resolution multi-photon nP-PEEM studies of plasmonics and photonics, very low emission currents available as well as the broader range of electron emission energies of nPPEEM vs. UV PEEM mean that achieving the theoretical limit to resolution is an even more difﬁcult challenge than in UV-PEEM. Practical magniﬁcation limits (i.e., exposure times) generally limit nP-PEEM resolution to that of the recording device (CCD). However, any gains in intensity and contrast due to even just spherical aberration correction are still a beneﬁt, as exposure times of many minutes are often necessary. Gains that reduce image exposure times lead to real image improvements, especially in combating image drift or other time-sensitive sources of image defects. Improvements in brightness generally mean that larger magniﬁcations can be used for the same exposure time, leading to higher resolution where the CCD is the limit, or simply by allowing the operator to improve corrections to astigmatism and focus. Finally, some UV PEEMs simply do not have large electron energy variations. In such circumstances, a mirror with stable spherical aberration correction may be the best choice. Looking to the future, the insights gained with a more stable diode mirror design might lead to better triode mirror designs. Fitzgerald, Word, and Könenkamp's [10] analytic extension of Rempfer's diode hyperbolic electron mirror to include a third hyperbolic electrode indeed provides more ﬂexibility in the choice of potentials to better match spherical and chromatic aberrations and allow for different magniﬁcations. In this triode conﬁguration, the expected ﬂat range of responses to voltage afforded by a quartic mirror proﬁle that is more concave than a hyperbola, with λ ¼ 3=18, might be a win–win solution that a designer could choose. So from either perspective, the ﬂexibility afforded by the addition of the quartic term may well be an advantage. (The intermediate curves in Fig. 4 are for λ ¼ 7 2=18 and λ ¼ 7 1=18.)

4. Aberration compensation using a diode mirror Since all rotationally symmetric lenses have positive coefﬁcients of aberration, a mirror is one of the few ways to cancel spherical and chromatic aberration in an electron microscope [1,4,23,24]. An aberration-free image is obtained by pairing mirror's negative aberration coefﬁcients with the lenses that have positive aberration coefﬁcients. The details of how this is done are given in Rempfer's 1990 paper [1] so only a brief outline of the process is given here. Since aberration coefﬁcients scale with the length of the mirror ℓ, it is important to ﬁrst match the scale-independent ratio C c =C s of the overcorrected aberrations of the mirror to the undercorrected aberrations of a lens or lens system. This ﬁxes ν and z0 =ℓ and, for a given beam energy VC, this also ﬁxes VM if the terminating electrode is grounded V A ¼ 0. The length of the mirror ℓ is then adjusted so that its negative values of Cs and Cc match the positive values of the lens system. This also sets the paraxial object/image distance of the mirror z0. The ratio of the chromatic and spherical aberration coefﬁcients is shown in Fig. 5. We see that as the quartic mirror proﬁle becomes less concave, the bottom-most (dashed) curve for λ ¼ þ 3=18, the aberration ratio does not vary much from the hyperbolic ratio (the solid line) until below 0.74 where the diverging nature of the former causes spikes. The (dot-dash) curve for a quartic mirror proﬁle that is more concave than a hyperbola, with λ ¼ 3=18, gives the highest C c =C s ratio for the largest range voltages, with an inﬂection point at about 0.92. Since each aberration is made ﬂatter by the inclusion of this negative-valued quartic term, their ratio is likewise ﬂatter for λ ¼ 3=18 than for the hyperbolic mirror. The intermediate curves are for λ ¼ 72=18 and λ ¼ 7 1=18.

Quartic Mirror

3.0 2.5 2.0 Cc Cs

176

1.5 1.0 0.5 0.0

0.65

0.70

0.75

0.80 VA VC

0.85

0.90

0.95

V A VM

Fig. 5. The ratio of the chromatic aberration coefﬁcient to the spherical aberration coefﬁcient C c =C s of the quatric mirror for symmetric rays as a function of ν ¼ ðV A V C Þ=ðV A V M Þ. The solid line is the hyperbolic limit, which matches the third of the three curves in Rempfer's Fig. 4(b), though her abscissa has higher values on the left rather than on the right [1]. The (dot-dash) curve for a quartic mirror proﬁle that is more concave than a hyperbola, with λ ¼ 3=18, has moved to the higher position for the largest range voltages, with an inﬂection point at about 0.92. On the other hand, as the quartic mirror proﬁle becomes less concave, the (dashed) curve for λ ¼ þ 3=18, the aberration ratio does not vary much from the hyperbolic ratio until below 0.74 where the diverging nature of the ﬂared mirror proﬁle causes spikes. The intermediate curves are for λ ¼ 7 2=18 and λ ¼ 7 1=18.

Thus, the present work adds an additional parameter λ that may be used to expand the range of possible mirror/lens conﬁgurations. Although one would presume that the parameter λ would be ﬁxed in the design stage, working well with one particular magniﬁcation in the microscope, it is conceivable that λ could be varied somewhat to achieve different magniﬁcations. There is a long history of the use of actuators to adjust the proﬁle of optical telescope mirrors to compensate for atmospheric distortion [25], as well as subdividing such mirrors into 10 or more independently moveable hexagonal mirror segments [26]. Though wholesale reconﬁguration of the electron mirror proﬁle is a daunting proposition, the small size of the mirror would allow one to use aluminized materials having some plasticity that would not stand up to the gravitational stresses of a meter-scale optical mirror. The small scale might also allow for the use of a turret such those found on Omega D-5 photographic enlargers [27] that would rotate into the beam mirrors whose ﬁxed proﬁles would be matched to a series of increasing instrument magniﬁcations. This would require a vacuum housing of the order of 24 cm in diameter and the ability to maintain precision alignment as each mirror came into place.

5. Possible extensions of this work The present work sought to derive the electron trajectory and aberrations for a diode electron mirror departing from the hyperbolic proﬁle Rempfer [1] derived by inclusion of z4 terms. Because of the limitations imposed by having only two electrodes with which to tune out two aberrations (spherical and chromatic), it would be desirable to extend the present work to a quartic triode or tetrode mirror. Fitzgerald, Word, and Könenkamp's [10] analytic extension of Rempfer's diode hyperbolic electron mirror to include a third hyperbolic electrode indeed provides more ﬂexibility in the choice of potentials to better match spherical and chromatic aberrations and allow for different magniﬁcations. With sufﬁcient care for the increasing complexity of the analytic trajectory, and the appropriate application of approximations within the calculation of spherical

J.C. Straton / Ultramicroscopy 148 (2015) 168–179

177

3 0.15

2 0.10

1 0.05

0

0.00

1

0.05

2

0.10

3 3

2

1

0

1

2

3

0.15 0.60

0.65

0.70

0.75

0.80

0.85

0.90

Fig. 6. An electron mirror with a spherical proﬁle of radius 1.5 units (shifted rightwards 1.828 units) may be ﬁtted near axis with a quartic proﬁle given by z2 r 2 =2 þ z4 =4 ¼ 0:4, shown near-axis in (b).

3 0.15 2 0.10 1

0.05

0

0.00

1

0.05

2

0.10

3 3

2

1

0

1

2

3

0.15 0.60

0.65

0.70

0.75

0.80

0.85

0.90

Fig. 7. An electron mirror with a spherical proﬁle of radius 1.0 units (shifted rightwards 1.67 units) has been ﬁtted near axis with a quartic contour given by z2 r 2 =2 z4 =4 ¼ 0:4, shown near-axis in (b).

aberration in particular, a quartic triode mirror is certainly within reach, and perhaps a quartic tetrode mirror. We noted above that the lowest (dot-dash) curves for a quartic mirror proﬁle that is more concave than a hyperbola, with λ ¼ 3=18, created a much larger range of voltages ν in which the spherical aberration does not change appreciable. While this may impose some limitations and some beneﬁts on the diode mirror, the added freedoms of a triode mirror might fully use this ﬂat range in which responses to voltage ﬂuctuations are diminished to achieve better resolution. A second possible use of the present work is in the analysis of low energy electron microscopy (LEEM) [28] and photoelectron emission microscopy (PEEM) [29] tetrode mirrors such as SMART [30], IBM/SPECS [31], PEEM-III [32], and Elmitec [33]. These use

spherical mirrors, optimized via numerical techniques, rather than hyperbolic mirrors that can be optimized via analytical as well as numerical techniques. While the numerical modeling of these systems has already produced mirrors with very good performance, there has been no systematic optimization of the shape of the mirrors. Additional optimization including a parallel, analytical component might reveal, say, a global minimum missed in a purely numerical approach. Though beyond the scope of the present paper, we wish to sketch how this might be done in an approximate fashion. The present analytical result, having both z2 and z4 terms, has more ﬂexibility for approximating the curvature of spherical mirrors near the axis than the hyperbolic proﬁle alone. In Fig. 6

178

J.C. Straton / Ultramicroscopy 148 (2015) 168–179

a circle of radius 1.5 units (shifted rightwards 1.828 units) has been ﬁtted near axis with a quartic contour given by z2 r 2 =2 þ z4 =4 ¼ 0:4, shown near-axis in (b). The ﬁtting possibilities become better as the radius of curvature of the spherical mirror electrode becomes larger. Perhaps more promising for ﬁtting circles (particularly those of small radius) than the open proﬁle given by a hyperbolic contour or a quartic contour with positive sign on the z4 term is to use a quartic contour having a negative sign to create a contour that, while not circular, is at least closed. Such is shown in Fig. 7 where a circle of radius 1.0 units (shifted rightwards 1.67 units) has been ﬁtted near axis with a quartic contour given by z2 r 2 =2 z4 =4 ¼ 0:4, shown near-axis in (b). Once ﬁtted, the results of the present approach might be applied as an analytical supplement to purely numerical modeling to systematically optimize the proﬁles of a tetrode mirror having spherical surfaces.

¼

1þ

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ 4 k zCð0Þ2 λ þ 2k r 2C λ

ðA:1cÞ

2 λk

¼ k zCð0Þ2 λ þ ℓ2 ð1 νC Þ ¼

1 1 ∑ kn¼1

ðA:1dÞ

1=2 n1 2 ðk zCð0Þ2 þ kr C =2Þn : 22n 1 λ n

ðA:1eÞ

In addition to Eq. (24) z_ 1 may be expressed as pﬃﬃﬃqﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ z_ 1 ¼ iωzC 2 1 þz2C μ ω2 0 0 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ11 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ 5 z2C μ ω2 CC p ﬃﬃﬃ 1 B B 2 [email protected] 7 i 2 θ1 1 þ zC μ ω2 @ 1 þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ AA 2 1 þ z2C μ ω2 0 0 q ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ11 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ 5 z2C μ ω2 CC p ﬃﬃﬃ 1 B B [email protected] 7 i 2 θ1 1 þ z2C μ ω2 @ 1 þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ AA; 2 1 þ z2C μ ω2 ðA:2aÞ

6. Conclusion An analytical solution in terms of Jacobi cosine-amplitude functions has been found for the electron mirror whose curvature extends the well-known hyperbolic proﬁle – a solution of the Laplace equation to second order in the variables perpendicular to and along the mirror's radius (z2 r 2 =2) – to one somewhat less concave or more concave with the addition of a quartic term (kλz4 ). The paraxial object/image distance z0 (relative to the mirror length ℓ) is found, and from that the chromatic and spherical aberration coefﬁcients, Cc and Cs. With this quartic term that can be shifted from positive to negative values, a mirror designer has additional ﬂexibility in setting the response of these aberrations to mirror voltage ratios ν. Finally, the possible extension of this work to a triode and tetrode quartic mirror is discussed, as is the possibility for using these analytical results to approximately model spherical tetrode mirrors close to axis to supplement a purely numerical optimization of their proﬁles.

Acknowledgments

As with the substitution of Eq. (43) in Eq. (24), in the above one may substitute [21] d dnðujκ Þ snðujκ Þ ¼ cnðuκ Þ du rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 1 cnðujκ Þ2 1 κ 2 κ 2 cnðujκ Þ2 sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 νC0 Þ ð1 νA Þ ¼ ð1 νC0 Þ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1 νC0 Þ κ 2 ð1 νC0 Þ κ 2 ð1 νA Þ : ð1 νC0 Þ ¼

In addition to the forms listed in Eq. (26) one also has 1=2 1 1 n1 2 ðV A V M þkr 1 =2Þn z21 ¼ ∑ 22n 1 λ kn¼1 n ¼ ¼

I would like to thank Joe Fitzgerald for providing me with his Mathematica notebooks for calculating aberrations in the hyperbolic mirror and, along with Rolf Könenkamp and Rob Word, for helpful discussions about this problem.

1 þ

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ 4 k ðℓ2 þ λℓ4 Þλ þ2k r 21 λ

1=2 n

22n 1 λ

n1

ðA:4aÞ

ðA:4bÞ

2 λk 1 1 ∑ kn¼1

ðA:3Þ

ðk ðℓ2 þ λℓ4 Þ þkr 1 =2Þn : 2

ðA:4cÞ

Finally, one may write 1

zð0Þ2 ¼ ℓ2 ð1 νÞ 1 þ ðVa VmÞλ ∑ C

ðVa VmÞj þ 1 λ

j¼1

jþ1

j!

∏ij ¼20 ði 2jÞ

! :

ðA:5Þ Appendix A

in addition to the forms listed in Eq. (34).

Here we present, for completeness, forms of several equations in the body of the paper other than those explicitly referenced that nevertheless may be helpful to other researchers in this ﬁeld. In addition to the forms listed in Eq. (20) ℓ2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ z2C ¼ 1 þ 1 þ 4 V A λ 4 V M λ 8 0 < @ 1þ 1þ4 VC λ4 VM λ :

¼

1 1 ∑ kn¼1

1=2 n

22n 1 λ

n1

To see why Eq. (34) is not Eq. (20) with rC ¼ 0, consider the explicit form for νC after inserting the expression for k, Eq. (18), in Eq. (20) (and anticipating a desire to replace VC by an expression involving the dimensionless hyperbolic parameter ν),

ðA:1aÞ

r 2C 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2ℓ2 1 þ 1 þ 4 V A λ 4 V M λ 0 ( @ 1þ 1þ4 VC λ4 VM λ

ðA:1bÞ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ91=2 1 r 2C 1 þ 1 þ 4 V A λ 4 V M λ = A þ ; ℓ2

νC ¼ 1 þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ91=2 1 r 2C 1 þ 1 þ 4 V A λ 4 V M λ = A þ ; ℓ2

Appendix B

2

ðV C V M þ kr C =2Þn

ðB:1aÞ

J.C. Straton / Ultramicroscopy 148 (2015) 168–179

r 2C 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2ℓ2 1 þ 1 þ 4 V A λ 4 V M λ 0 (

¼ 1þ

@ 1 þ 1 þ 4ð1 νÞV A λ þ 4 νV M λ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ91=2 1 r 2C 1 þ 1 þ 4 V A λ 4 V M λ = A: þ ; ℓ2

ðB:1bÞ

We can form an approximate expression by setting rC ¼ 0 in this to get pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þ 1þ4 VC λ4 VM λ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðB:2aÞ νðrC C ¼ 0Þ ¼ 1 1 þ 1 þ 4 V A λ 4 V M λ ¼ 1

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ 4ð1 νÞV A λ þ 4 νV M λ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þ 1þ4 VA λ4 VM λ

1þ

a νC0 ¼ 1

¼ 1

ðB:2bÞ ðB:2cÞ

2 λðV C V M Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ 1þ4 VA λ4 VM λ

ðB:2dÞ

2 λðV A V M Þð1 νÞ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þ4 VA λ4 VM λ

ðB:2eÞ

1þ

1þ

ν λðV A V M Þð1 νÞ; although both expressions go to

ðB:2f Þ

ν as λ-0.

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