## Activity Report

# Optics Design and Modeling of the PEEM3 Beam Separator

D. S. Robin, Y. Wu, R. Schlueter, E. Forest (KEK)

The Beam Separator is a center-piece for the next generation x-ray
Photo-emission electron microscopes (PEEMs). In x-ray PEEMs, the
chromatic aberrations are the main limiting factor for resolution
improvement. Unlike the current generation PEEM such as PEEM2 at the ALS
in which chromatic and spherical aberrations are uncorrected, the next
generation x-ray PEEM, such as PEEM3 at the ALS, depends on an elaborate
aberrations correction scheme to improve the resolution of the
microscope and increase the transmission at a given resolution.

The aberrations correction is performed in a normal incident angle
electron mirror which requires bending elements to bend electron
trajectory at the entrance and exit of the mirror. These bending
elements collectively form an electron beam separator for the
microscope. To reduce the resolution impact of the bending elements in
the PEEM3, the bending section should be designed to have minimal
aberrations. A highly symmetric and compact beam separator design with
two Pi-transformations in both transverse planes is desirable. As is
well understood in the accelerator physics, such an optical system is
free from all the second order geometric aberrations and second-order
chromatic aberrations with exception of the chromaticity terms. The
remaining aberrations are from third-order and higher terms. This basic
design has been proposed by a number researchers, and has been adopted
in a similar x-ray PEEM project , SMART, at BESSY II.

The focus of our work is to design such a highly symmetric beam
separator, evaluate its performance, and improve the design. During the
months that this activity report covers, we have performed the following
tasks for the ALS PEEM3 project:

-   developing/debugging a charged optics code for designing and
    modeling the beam separator (E. Forest, R. Schlueter );

-   developing an electron distribution model and a simple tracking code
    to analyze the impact of the Beam Separator aberrations on the
    e-beam distribution at various locations in the PEEM3 (Y. Wu, D.
    Robin);

-   designing and analyzing two specific Beam Separator designs: a 90 cm
    and a 28 cm design (E. Forest, D. Robin, Y. Wu).

### 

#### A. Beam Separator Design Code 

A beam separator optics design code has been developed by E. Forest from
KEK and his students using an automatic differential package TPSA (by E.
Forest). The code uses an analytic magnetic model of the separator
provided by R. Schlueter. Using this code, one can design beam
separators with various dimensions and compute the aberrations to
arbitrarily high orders. In addition, because the magnetic field can be
well modeled by an analytic expression, the aberrations can be computed
as a function of various separator parameters, such as the separator
current, alignment errors of the various grooves in the separator. Such
an analysis is particularly powerful to illustrate the tolerance impact
in different designs and to provide guidelines for optimizing the
separator design and for improving manufacturing techniques.

(A Flow Diagram to be inserted)

#### 

#### B. Electron Distribution and Beam Separator Analysis Code 

The full impact of an optical element in a PEEM can only be understood
with a detailed analysis of the electron distribution. In general, the
total aberration cannot be computed directly by adding up various
aberration terms of different nature while assuming a Gaussian
distribution for them. In fact, the total aberration presented in the
distribution not only depends on the field aberrations of optical
elements, but also depends on the initial distribution of the beam.

The initial distribution of the electron from the secondary electron
emission is assumed to be

> $N(\theta,E_{k}) = \text{Cos}(\theta)\frac{E_{k}}{(E + W)^{4}}$,

where W is the work function of the material under study, E~k~ is the
kinetic energy of the electron, θ is the azimuthal angle, and N(θ, E~k~)
is unnormalized electron distribution at E~k~ in a unit solid angle. We
use a macro-particle model to represent the initial electron
distribution on a uniform grid in (θ,ϕ) in the momentum space, the
weighting factor N(θ, E~k~) sin(θ) is recorded and used to normalize the
final distribution.

(A Sketch of the momentum vector to be inserted here)

In order to focus on the analysis of the beam separator, the front-end
elements, such as the acceleration gap, the objective lens, etc. are
modeled linearly. We assume that the acceleration gap is infinitely
small so that the macro-particle sees only a change of the transverse
momenta but no change in its position:

> $$\begin{array}{r}
> x_{2}^{'} \approx \frac{x_{1}^{'}}{\sqrt{1 + (x_{1}^{'})^{2} + (y_{1}^{'})^{2}}}\sqrt{\frac{E_{k1}}{E_{k2}}} \\
> y_{2}^{'} \approx \frac{y_{1}^{'}}{\sqrt{1 + (x_{1}^{'})^{2} + (y_{1}^{'})^{2}}}\sqrt{\frac{E_{k1}}{E_{k2}}}
> \end{array}$$

where $x_{1}^{'}$ and $y_{1}^{'}$ are the tangents of emission angles
and $x_{2}^{'}$ and $y_{2}^{'}$ are the new values after the
acceleration gap. The objective lens is represented by a linear matrix
of the following form:

> $M_{\text{obj}} = \begin{pmatrix}
> m & 0 & 0 & 0 \\
>  - \frac{1}{f} & \frac{1}{m} & 0 & 0 \\
> 0 & 0 & m & 0 \\
> 0 & 0 & - \frac{1}{f} & \frac{1}{m}
> \end{pmatrix}$,

where m is the magnification of the objective, and f is the effective
focal length.

A macro-particle is tracked from the sample to the entrance of the
separator using this linear model. The trajectory after the separator
calculated using the Taylor map produced by the separator design code.
Finally, all macro-particles are statically analyzed on the image plane.
We use a 68%-rule to define the width of the distribution in both
transverse planes. The full-width is defined as the area in which 68% of
the macro-particle is found. We then refer the resolution of the system
as the half width of the distribution. This definition is consistent
with those used for similar systems such as SMART.

A Fortran code has been developed for tracking macro-particles
trajectories. Typically, 27,000 macro-particles are tracked with 30
evenly or randomly distributed values of

θ, ϕ, and E~k~. To properly simulate the full distribution, we typically
select the value of θ to be between 0 and π, the value of ϕ between 0
and 2π, the value of E~k~ between 0 and 4W.

#### C. Analysis of 90 cm and 28 cm Beam Separator

#### D. Conclusion and Future Work