About Transport Envelope Fits

Envelope Fit The computer program Transport is very useful because it has some powerful fit capabilities, which are usually applied when designing new beam lines. For these cases the starting conditions (e.g. phase space, waist positions) are assumed being known. Thus the fields and gradients of different magnetic elements are varied in order to get the beam through the beam line by fulfilling the designer's boundary conditions.
Envelope Fit On an existing beam line with some profile monitors (eventually harps or beam scanners) Transport may also be used in a kind of opposite manner by helping to find out the initial beam conditions (which are usually unknown or changing from day to day) leading to the measured profile data. In practice this is done by using the measured beam half-widths as constraints and by freeing the appropriate vary codes of the initial beam. If there are 3 or more measured widths per direction, then also the correlation coefficients (r12 and r34) may be freed. The actual field values (conversion from hardware DAC value via look up table) for the gradients of the quadrupole lenses have also to be inserted at the appropriate places. At the Proton ACS of PSI, measuring the profile widths and insertion of all values is done since many years automatically with a special computer program (e.g. ENV.COM).
There is also the possibility to measure the beam profiles with several settings of quadrupoles on the same beam line with the same initial beam conditions and then performing a simultaneous multi envelope fit.

Why doing an envelope fit at all ? There are several reasons for it:

The better the fit, the more confidant you are to know the parameters of your beam line. A good fit is a kind of a consistency test. Eventually some wrong assumptions have to be corrected. (*

Inter- and extrapolation between the measured points are acceptable. You do not have to rely on manually drawn curves, which often are wrong.

It is the simplest way to measure the 87 % projected emittances of the beam without the interference of a slit system and without reducing the beam intensity. Another method to measure the emittance of a beam in a non-destructive manner is the Maximum Entropy Beam Tomography, which gives interesting possibilities to compare results.

If you know the initial beam conditions, you have the possibility to use these parameters in a following fit to modify quadrupole settings in order to adjust waist positions and beam sizes along the beam line. Usually, these newly computed settings could be tried out on-line with even the full powered beam staying on. If done so, a new envelope fit then shows, if the changed settings had the desired effect. If needed this process may be iterated several times.

Here is a Transport sample input file:

 /Sample Envelope Fit/
 0
 15. 1.0 /MM/ .1 ;
 15. 6.0 /PM/ .1 ;
 1. 1.5 8.5 1.0 17.0 0. 0. 1.17 /BEAM/ ;  (Guess of initial BEAM parameters)
 (12. 0.1 0. 0. 0. 0. -0.1 /CORR/ ;) (Correlations r12 and r34 => waist position)
 3. 5. ;
 5. 0.6 -5.4812 125. /Q1/ ;  (-5.4812 = inserted field value)
 3. .5;
 5. 0.6 4.9299 125. /Q2/ ;   (4.9299 = inserted field value)
 3. 4.5 ;
 -10. 1. 1. 40. .1 /X1/ ;  (40.0 = measured x-value)
 -10. 3. 3. 5. .1 /Y2/ ;   (5.0  = measured y-value)
 3. 5. ;
 -10. 1. 1. 20. .1 /X3/ ;  (20.0 = measured x-value)
 -10. 3. 3. 60. .1 /Y4/ ;  (60.0 = measured y-value)
 3. 2. ;
 SENTINEL
 /*PLOT*/
 -1
 1.111100 /BEAM/;    (vary x,x',y and y'of BEAM)
 (12.100001 /CORR/;) (vary r12 and r34 of CORR)
 10. /X1/ ;          (constraint)
 10. /Y2/ ;          (constraint)
 10. /X3/ ;          (constraint)
 10. /Y4/ ;          (constraint)
 SENTINEL
 SENTINEL

Envelope Fit In order to visualize these special fit constraints, you have to set the two parameters 'x-Measurements' and 'y-Measurements' in the 'Display Parameters' dialog box into the "on" position (MEASX=1.0 and MEASY=1.0 in FOR004.DAT). In the Graphic Transport display these constraints then appear as little "T"s. Very often the program does not find a solution to the problem and the fit is therefore diverging or sticking to the initial values. The reason for this behavior may be different from case to case and sometimes it is very time consuming to find a solution to the problem. Very often the guess of the initial parameters have to be changed for being successful.
Envelope Fit Be always aware that fitting with Transport is an art, which demands some understanding of the physics of the beam line under consideration and the functioning of the Transport code as well. If you feed the code with unphysical input, then it may crash with an arithmetic trap (division by zero etc.). Because of the complexity of the problems to be solved no special algorithms have been built into the code to prevent such events from happening.
Even with space charge effects being present, the Transport envelope fit may well be working. This is mainly due to the stochastic fit procedure, which does not require, that the derivatives of the transfer-matrices of all elements along the beam line have to be known. How well this fit may work has been presented in an article at the 16th International Conference of Cyclotrons and Their Applications 2001 (Figure 1).
Two more examples for demonstrating the usefulness of Transport envelope fits are given here: Simultaneous Envelope Fit of two 590 MeV Proton Beam Optics and TRANSPORT input file of envelope fit for isotope production.

Two hints for improving the quality of envelope fits:

For beam lines with large aperture quadrupoles the fringing field effects should be included in the transport input file. An example of how to accomplish this is given in the Compendium of Transport Enhancements (Chapter 1). Experience with PSI secondary beam lines has shown, that quadrupole settings extracted from model calculations including fringing field corrections are usually closer to settings found by experimental tuning for maximum beam transmission than those model-settings without fringing field corrections.
Experience at PSI has also shown that magnetic quadrupoles with mirror plates acting as field clamps give better envelope fit results, because the fringing fields of neighboring quadrupoles do practically not overlap. Therefore the effective lengths (computed from the field gradient along the axis of a single quadrupole) of the hard edge model used in transport remain independent of the neighboring quadrupole's field value settings.

*) There is even an additional and better way to check the correctness of all involved parameters by measuring horizontal and vertical displacements while deviating the beam off-axis with steering and bending magnets and performing a beam centroid shift fit with the acquired data (utilizing the charged particle beam as a "surveying tool").

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Last updated by Urs Rohrer on 27-Feb-2006