Some Considerations Regarding Pulsed Correction of Chromatic Aberrations in Final Focusing Systems Page: 2 of 10
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Some Considerations Regarding Pulsed Correction of Chromatic
Aberrations in Final Focusing Systems
31 March 2010
Nearly all designs of accelerators for heavy ion fusion rely on a velocity (energy) ramp to
compress the beam longitudinally from its length in the accelerator to the length required
at the target. The size of the velocity ramp is constrained by the longitudinal emittance of
the beam. For example, if the longitudinal emittance is 0.05 eV-s and we wish to
produce a pulse having a width of 2.5 ns at the target, we must supply an energy tilt
such that the energy spread at the target is at least 0.05 eV-s / 2.5 ns = 2 x107 eV. The
minimal value of energy spread occurs when the beam has propagated to the point where
there is no correlation between the time and energy variables of the beam particles. (In
the simple approximation where the boundary of the longitudinal phase space containing
the particles is an ellipse, the ellipse is erect at this point, i.e., not tilted with respect to the
axes.) In any case, the energy spread can affect focusing. If, for example, the beam
kinetic energy is of the order of 5 GeV, a tilt of 2 x107 eV corresponds to a fractional
energy spread of 0.004 and it may be possible to focus the beam to the required spot size
without using an achromatic optical system. Nevertheless, an optical system that allows
larger longitudinal emittance should lead to a less expensive accelerator since the
tolerances on acceleration waveforms could be relaxed. Moreover, at lower kinetic
energies the problem becomes more serious. If the kinetic energy of our example beam
were 1 GeV rather than 5 GeV, the fractional energy spread would be 0.02. This much
energy spread would likely produce serious chromatic aberrations leading to an unwanted
increase in focal spot size. It is interesting to note that the lower limit on energy spread at
the target does not depend on whether the beam is neutralized as it approaches the target.
If the beam is not neutralized, it will require a larger initial velocity tilt to overcome
longitudinal space-charge forces; but these forces will remove part of the tilt as the beam
Al Maschke suggested that it is possible to reduce the chromatic aberrations by applying
a time-dependent transverse focusing correction to the beam upstream of the final
focusing lenses . At this point, because of the energy tilt, there is a correlation
between longitudinal position in the beam and particle energy. In other words, the
average beam energy at the tail of the beam is larger than the average beam energy at the
head of the beam. If the beam is completely neutralized as it drifts toward the final
focusing lenses, the kinetic energies of the individual particles will remain nearly
unchanged during compression. In this case, it is possible, in principle, to apply some
"pre-focusing" to the higher energy particles (those nearer to the tail of the beam) to
compensate for their weaker focusing in the final lenses. Although kinetic energies of
individual particles are not conserved if the beam is not neutralized, one still expects a
positive correlation between the particle energies at the beginning of compression and at
the end of compression so correction is still assumed to be possible. It is important that
the pulse duration is larger upstream than it is at the final focusing lenses. Larger pulse
duration makes it easier, from an engineering standpoint, to supply the power needed to
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Bangerter, Roger. Some Considerations Regarding Pulsed Correction of Chromatic Aberrations in Final Focusing Systems, report, March 31, 2010; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc1015375/m1/2/: accessed February 17, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.