Transverse emittance measurement

To measure the emittance fora space charge dominatedbeam the used technique isknow 1-D pepper-pot

222 xxxxThe emittance can bereconstructed from the secondmomentum of the distribution

C. Lejeune and J. Aubert, Adv. Electron. Electron Phys. Suppl. A 13, 159 (1980)

Emittance measurement with space charge

The beamlets must be emittance dominated

Assuming a round beam

d must be chosen to obtain <<1, in order to have a beam emittance dominated

12d

x

Design issues

The contribution of the slit width to the size of the beamlet profile should be negligible

The material thickness (usually tungsten) must be long enough to stop or heavily scatter beam at large angle

But the angular acceptance of the slit cannot be smaller of the expected angular divergence of the beam

12)(

22 d

L

12

dL

2dl

Design issues (2)

Examples

Phase space mapping

Phase space evolution

Transformation of

The inverse matrix transformation is

Inverse transformation

Matrix notation

Beam waist

The most used techniques for emittance measurements are quadrupole scan and multiple monitors

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200

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0

0

0

22

22

2

2

SCSCSSCSCSCC

SSCC

SCSC

ssM 21

Emittance measurement without space charge

Beam Matrix

2212

1211

12 22212

211 xxxx

TMM 01

Multiple screens

There are 3 unknown quantities i,11 is the rms beam size Ci and Si are the element of the transport matrix We need 3 measurements in 3 different positions

to evaluate the emittance

222

12112

11, 2 iiiii SCSC

Quadrupole scan

It is possible to measure in the same position changing the optical functions

The main difference with respect to the multi screen measurements is in the beam trajectory control and in the number of measurements

Schermo

222

12112

11 )()()(2)( kSkSkCkC

Condition to fulfill No interaction between particles The beam transport does not couple the various

two-dimensional projections of the six-dimensional phase space.

Higher moments than the second are not needed to characterize the density in phase space.

Usually the largest error is in the determination of the RMS beam size (Mini Workshop on "Characterization of High Brightness Beams“, Desy Zeuthen 2008, https://indico.desy.de/conferenceDisplay.py?confId=806)

Systematic error comes from the determination of the quadrupole strength, mainly for hysteresis. So a cycling procedure is required for accurate measurements

Thin lens model is not adequate Energy Large energy spread can gives chromatic effect Assumption: transverse phase space distribution fills an

ellipse

Source of errors

Phase space reconstruction

Tomography is related to the Radon theorem: a n-dimensional object can be reconstructed from a sufficient number of projection in (n-1) dimensional space

x

y

Tomography

A. C. Kak and Malcolm Slaney, Principles of Computerized Tomographic Imaging, IEEE Press, 1988.

D. Stratakis et al, “Tomography as a diagnostic tool for phase space mapping of intense particle beam”, Physical Review Special Topics – Accelerator and Beams 9, 112801 (2006)

Radon Transform

Fourier transform of the Radon transform

Fourier Slice theorem

Tomography measurements

C can be easily obtained from beam spatial distribution

s can be calculated from the beam line optics The accuracy of the result depends from the total

angle of the rotation and from the number of the projections

Scaling factor

Rotation angle

Emittance in a ring

Being ( an eigenvector

One complication

I cannot say it better

Slice properties

The space-charge force on a slice is determined by the density profile of the beam and the position of that slice within the beam.

As a result, different slices are exposed to different forces, and therefore expand at different rates and make different angles in phase space. As the slices separate in phase space angle, the rms emittance of the full beam increases

Slice emittance

Slice Emittance: A transverse cross section of the beam is called a slice, and the emittance of a slice is called slice emittance. Slice emittance consists of two parts, one part is the thermal emittance and the other part is due to nonlinear space charge force.

Correlated emittance: The growth of projected emittance is due in large part to the correlation between the phase space angle and the longitudinal position of slices. Normally, this part of projected emittance is called correlated emittance.

Emittance compensation: Correlated emittance evolves as the beam propagates in the beamline, and can be eliminated at one or more specific points through selecting proper parameters of the beamline. This process of eliminating correlated emittance is called emittance compensation

No panic! Note that even though the emittance changes in this

process, Liouville’s theorem is not violated. Actually, Liouville’s theorem does not strictly apply to the (x,px) phase plane since the transverse space-charge forces on a slice are coupled (through the longitudinal charge distribution) to that slice’s position in z

Slice emittance definition

When the beam is strongly dominated by space charge the correlation term is the leading one.

In the case of only two slices

Again the space charge field

r

z

The field is linear in radius, but depends on the radial extent of the slice, a, which is a function of z

Electric field

Laminar flow imposes the condition that particle trajectories do not cross.

This means that the number of particles within a set fraction of the beam radius, r/a, is constant

Motion equation

Constants

classical electron radius

transverse plasma wavenumber (spatial plasma frequency),

Equation and solution

Trace space

The projected emittance — that is, the rms emittance of the full beam —has increased even through the emittance of any given slice has not (the slice emittance is zero in this model).

Growth of projected emittance

Put a thin lens in zl

Now z is the distance after the lens!

Some steps…

After the lens

Waist and cross-over Physically, the slice trajectory is initially converging,

and will evolve into one that is diverging. This can happen in two different ways

Avoid bifurcation

The way to avoid the bifurcation of the beam’s phase space is to minimize the range of current values between slices

But

the phase space angles of the slices is equal and independent of kp0, and the emittance vanishes if

Graphics

After a thin lens

Closing the fan

Double minimum

The chromaticity of the solenoid produces a double minimum structure

Drift inside a solenoid

The stationary solution is called “Brillouin flow” This solution represent the matching conditions for

which the external focusing completely balances the internal space charge force. Unfortunately since kext has a slice independent constant value, the Brillouinmatching condition cannot be achieved at the same time for all the bunch slices

Small perturbation Considering a small perturbation from the equilibrium

in the form

3

Plasma oscillation

The slice envelopes oscillate altogether around the equilibrium solution with the same frequency for all slices, often called plasma frequency) dependent only on the external focusing forces. This solution represents a collective behavior of the bunch.

Where 0=σso −σsB is the amplitude of the initial slice mismatch that we assume for convenience the same for all slices

Slice oscillations Perturbed trajectories oscillate around the equilibrium

with the same frequency but with different amplitudes

Emittance oscillation

Emittance Oscillations are driven by space charge differential defocusing in core and tails of the beam

Envelope oscillations of the mismatched slices induce correlated emittance oscillations which periodically goes back to zero, showing the reversible nature of the correlated emittance growth

x

px

Oscillations

The size is smaller than equilibrium for all portions of the beam

Some below, one above

The angle of the low current slice is of opposite sign from the other two

Acceleration

After a focusing kick is applied by means of an external solenoid, the fan slices distribution tends to close in the following drift space until a minimum phase space area is reached, corresponding to a partial re-alignment of bunch slices

In order to avoid additional space charge emittance growth in the subsequent beam line, the emittance minimum has to be reached at high beam energy so that space charge forces are sufficiently damped. The beam has to be properly matched to the following accelerating sections in order to keep under control emittance oscillations and obtain the required emittance minimum at the injector exit.

Invariant envelope

It has been demonstrated that in the space charge-dominated regime, i.e. when the space charge collective force is largely dominant over the emittance pressure, mismatches between the space charge correlated forces and the external focusing gradient produce slice envelope oscillations that cause normalized emittance oscillations. It has been shown that to also damp these emittance oscillations, the beam has to be propagated close to a the so-called invariant envelope

This solution represents a beam equilibrium mode that turns out to be the transport mode for achieving minimum emittance at the end of the emittance correction process

Fundamental property of the invariant

Constant phase space angle

Real word Serafini and and Rosenzweig provide a recipe for

emittance compensation, which works for simple cases (e.g. matching beam into long focusing channel / linac in the injector)

For other more complicated scenarios one can try solving envelope equation for slices (or write a code to do that)

Particle tracking is indispensable for analysis and design of the injector where the assumptions made are invalid or theory is too complicated to be useful.