The DIMAD second order matrices for the cryomodule to wiggler and wiggler to reinjection point transports are presented in the following tables. Aberrations are limited in size and good behavior is expected. Of principle concern are the T336, T346, T436, and T446 chromatic aberrations for transport from wiggler to reinjection point. These couple to steering errors at the wiggler to produce dispersive-like effects at reinjection, and can lead to beam spot growth. Design effort was expended to limit their values to order 100 (m/(m-rad) for T336, m/rad2 for T346,...) or smaller; this, coupled with stringent steering requirements at the FEL (~30 microns/30 microradians) will limit spot growth at reinjection to order 1 mm or less during laser operation.
LINEAR MATRIX
-.2433893E-01 -.1576606E+01 -.5704803E-16 -.2173345E-15 .0000000E+00 -.9784089E-16
.6027802E+00 -.2040069E+01 -.9956494E-16 -.7123737E-15 .0000000E+00 -.3322702E-15
-.9721565E-16 -.2315058E-15 .2408158E+00 -.1670497E+01 .0000000E+00 .7223294E-15
.1697674E-16 -.1550203E-15 .3365967E+00 .1817639E+01 .0000000E+00 .3816392E-16
.1110223E-15 .2220446E-15 .2125036E-15 .9992007E-15 .1000000E+01 -.2888239E+00
.0000000E+00 .0000000E+00 .0000000E+00 .0000000E+00 .0000000E+00 .1000000E+01
SECOND ORDER TERMS
-.3814156E-01 -.1472129E+00 .6755982E-04 .3648265E-03 .0000000E+00 -.1888529E+01
-.2400861E+00 -.5646827E-04 -.3049315E-03 .0000000E+00 .6221657E+01
.1792152E+00 .7255648E+00 .0000000E+00 .4027762E-15
.3381167E+00 .0000000E+00 .2578181E-14
.0000000E+00 .0000000E+00
.1058543E-14
-.8056675E-01 -.3978873E+00 .5075830E-04 .2740974E-03 .0000000E+00 -.4231192E+00
-.7713685E+00 -.9763334E-03 -.5272251E-02 .0000000E+00 -.2319967E+02
.4822751E+00 .2571348E+01 .0000000E+00 -.8419149E-15
.2245231E+01 .0000000E+00 -.3287684E-14
.0000000E+00 .0000000E+00
-.1557638E-14
.6232850E-04 -.1717214E-03 .5203990E+00 .9594861E+00 .0000000E+00 .9681608E-16
-.2115431E-02 .1938579E+01 .5676789E+01 .0000000E+00 .1193511E-14
.1153396E-15 -.1279179E-14 .0000000E+00 -.3546576E+01
-.1215899E-13 .0000000E+00 -.1575583E+02
.0000000E+00 .0000000E+00
-.1511581E-15
.9107298E-17 .9020562E-16 -.2670441E+00 -.7343428E+00 .0000000E+00 -.3634002E-15
.1457168E-15 -.5680505E+00 -.2756969E+01 .0000000E+00 -.9643265E-15
.4259830E-15 -.3368833E-14 .0000000E+00 .1254032E+01
-.4024905E-13 .0000000E+00 -.3952480E+01
.0000000E+00 .0000000E+00
.2567391E-15
.5669975E+00 -.4135984E+01 -.4470151E-15 -.1475329E-14 .0000000E+00 -.7267533E-15
.2624287E+02 .1197158E-14 .8538915E-14 .0000000E+00 .5476520E-14
.7883958E+00 .3789087E+01 .0000000E+00 .6661440E-15
.1610114E+02 .0000000E+00 -.3372601E-14
.0000000E+00 .0000000E+00
.4600039E+00
LINEAR MATRIX
.3149358E+01 -.9740546E+00 -.2520805E-15 -.1356412E-15 .0000000E+00 -.1423875E-14
.3492349E+00 .2095113E+00 .8150145E-16 .5835819E-16 .0000000E+00 .9714451E-15
-.3259238E-15 .1586386E-16 -.8731791E+00 -.1531041E+01 .0000000E+00 .1390114E-15
-.1800362E-15 .1998037E-16 .5753147E+00 -.1364778E+00 .0000000E+00 .4130536E-15
-.1942890E-14 -.7771561E-15 -.4406450E-15 .7258713E-15 .1000000E+01 .1900000E+00
.0000000E+00 .0000000E+00 .0000000E+00 .0000000E+00 .0000000E+00 .1000000E+01
SECOND ORDER TERMS
-.3124403E+01 .1988059E+02 .8914805E-14 .4323082E-03 .0000000E+00 -.4234840E+01
-.2858985E+01 -.9146040E-14 -.1309814E-02 .0000000E+00 .2337674E+02
.2839433E+02 .1269786E+02 .0000000E+00 .5334722E-13
-.2319591E+02 .0000000E+00 .1294763E-13
.0000000E+00 .0000000E+00
.1093241E-15
.2576255E+01 .1026675E+01 -.1270772E-14 -.4095468E-03 .0000000E+00 -.1004639E+01
-.1137082E+01 -.2427895E-14 -.4733824E-04 .0000000E+00 .3184710E+01
-.2862298E+01 -.9167513E+01 .0000000E+00 .1047124E-13
-.2258212E+01 .0000000E+00 -.9902863E-14
.0000000E+00 .0000000E+00
.1354472E-13
-.6290325E-03 .2692426E-03 .2888689E+02 .5272429E+02 .0000000E+00 -.1349767E-13
.1399404E-03 .1515271E+02 .2725806E+01 .0000000E+00 .3178429E-13
-.8443050E-14 -.2676589E-13 .0000000E+00 -.1328226E+03
-.1084230E-13 .0000000E+00 .2425963E+02
.0000000E+00 .0000000E+00
-.8324817E-13
.4144529E-03 -.1773969E-03 .2432808E+02 .3403446E+01 .0000000E+00 .1420188E-13
-.9220303E-04 -.2745497E+01 -.8978311E+01 .0000000E+00 -.2294271E-14
-.6730989E-14 -.9164386E-14 .0000000E+00 -.1696746E+02
.2368960E-14 .0000000E+00 -.2497480E+02
.0000000E+00 .0000000E+00
-.6542549E-14
-.2925741E+00 .1525646E+01 .4581897E-14 -.1665575E-14 .0000000E+00 -.3553450E-13
-.4101924E+01 -.1242043E-13 .8368605E-14 .0000000E+00 .5096308E-14
.4536357E+02 .6968952E+01 .0000000E+00 .5403262E-13
.2087710E+02 .0000000E+00 .1038665E-13
.0000000E+00 .0000000E+00
-.1051748E-12
Higher order aberration analysis was performed using various DIMAD numerical tools; results are described on subsequent pages. Because it is a second order code, it was deemed desirable for these large phase space volumes to benchmark DIMAD against a higher order, more accurate model. Comparison was made to simulations based on the code TLIE, and results for the two codes were observed to be generally consistent. Nonlinear effects out to fifth order were found to be significant and to be modeled similarly by both codes.
This transport system manifests an uncommon violation of symplecticity
in the second order transform from wiggler to reinjection point. Simulation
by Li using TLIE exhibited an unphysical phase space growth when a second
order Taylor series representation of the wiggler to reinjection point
transfer map was used. This growth was absent when the fifth order
representation was used. Cross-checks were performed with DIMAD;
use of a single second order transform from wiggler to reinjection
produced results very similar to second order TLIE; use of element
to element tracking (the usual DIMAD integration mode) behaved
similarly to the fifth order TLIE tracking. It was hypothesized
that truncation of the map to second order eliminated higher order
chromatic terms (that were retained, for example, in TLIE fifth order
and DIMAD element-to-element tracking) necessary for the transform
to remain symplectic.
To test this idea, DIMAD symplectic ray-tracing was activated. Changes in
element to element tracking were very small, but tracking results using the
single wiggler to reinjection point second order transform changed dramatically.
The phase space no longer grew, but was, however, quite distorted. We
conclude that this transport system is inherently nonlinear, particularly
in its chromatic behavior. Higher order aberration analysis is therefore
required, and is discussed in the
next link. These observations provide an interesting counterpoint to
the usual concerns about symplecticity in optics codes. Typically, code
users are concerned that small violations of symplecticity in single
element maps can accumulate over many turns of a storage ring simulation
to produce spurious phase space growth or damping. In this case, it is
violations of the symplectic condition produced by truncation of multi-element
cross terms which is of concern. Second order transforms model individual
elements quite well, as demonstrated by the similarity in results from
TLIE, nonsymplectic DIMAD and symplectic DIMAD. It is the interaction
amongst the elements which produces chromatic nonlinearities of importance.
| Project Overview | |
| System Design Process | |
| Application of Process to High Power IR FEL | |
| Description of Solution | |
| System Performance | |
| A. Linear Optics | |
| **you are here! ** | B. Aberration Analysis |
| **the next link is | C. Chromatic Performance |
| D. Geometric Performance | |
| E. Simulation of Energy Recovery | |
| Error Studies | |
| Upgrade Scenarios | |
| | |
| Go to The FODOmat's FEL Page | |
Last modified: 10 March 1997
http://www.jlab.org/~douglas/
is maintained by: douglas@jlab.org