FODO Cell Introduction to OptiM

Transcription

1 FODO Cell Introduction to OptiM S. Alex Bogacz Jefferson Lab 1

2 FODO Optics cell Most accelerator lattices are designed in modular ways Design and operational clarity, separation of functions One of the most common modules is a FODO module Alternating focusing and defocusing strong quadrupoles Spaces between are combinations of drifts and dipoles Strong quadrupoles dominate the focusing Periodicity is one FODO cell so we ll investigate that motion Horizontal beam size largest at centers of focusing quads Vertical beam size largest at centers of defocusing quads 2

3 Periodic FODO Cell Phase Advance cell Select periodicity between centers of focusing quads only has real solutions (stability) if 3

4 FODO Cell Beta Max/Min cell What is the maximum beta function,? Follow a similar strategy reversing F/D quadrupoles to find the minimum b(s) within a FODO cell (center of D quad) 4

5 FODO Betas vs Phase Advance Generally one wants Strong focusing Smaller magnets Less expensive accelerator /L small [deg] 5

6 Stability Diagrams cell Designers often want or need to change the focusing of the two transverse planes in a FODO structure What happens if the focusing/defocusing strengths differ? Recalculate the M matrix and use dimensionless quantities Then take the trace for stability conditions to find 6

7 Stability Diagrams or using For stability, we must have Using, stability limits are where These translate to an a necktie stability diagram for FODO 7

8 What Does OptiM Compute? OptiM is aimed at assisting with the linear optics design of particle accelerators, including linear space charge effects. The program can also be used for particle tracking, in which case non-linear elements are accounted for. Outline of the Program Features and Capabilities : All linear optics calculations are based on 6 6 transfer matrices. Dispersion and betatron functions (uncoupled or coupled (X/Y) particle motion) Beam sizes Betatron phase advances Results can be plotted or printed along a reference trajectory, or at the dowstream end(s) of a selection of element(s). Parameters of accelerator elements can be iteratively adjusted to fit prescribed optical functions (matching). The matching operation can be performed either for a transport line, where initial Twiss parameters are fixed or for a ring, where Twiss parameters are periodic. 8

9 What Does OptiM Compute? All calculated values are nominally referenced to a moving coordinate frame attached to the central (design) beam orbit. If desired, the program can calculate and output the position of this reference frame in global coordinates. A wide variety of elements is provided to support the design of linear accelerators, recirculators or circular synchrotron accelerators. A simple input syntax and a well-developed set of menus minimize the amount of time needed to learn the program. Not limited to small machines; all necessary measures have been taken to support design and optics studies of very large machines consisting of thousands of elements. Interactive: In complex situations where a designer cannot formulate a full set of requirements in the early stage of a design, working in interactive mode can significantly improve productivity. 9

10 What Does OptiM Compute? Particle tracking in presence of non-linear magnetic fields in accelerator magnets Beam space charge in the KV-distribution (linear space charge field) approximation. Computations can be performed not only on the design orbit but also on any reference orbit defined by machine errors (e.g quad offsets or errors in dipole bend strength), correctors or energy offset. In this case, a new "reference" orbit is established and maps for machine elements obtained from an expansion about this new orbit (see Optics Calculations at Reference Orbit and Control). Linear optics computations (tunes, beta-functions, etc.) and non-linear tracking can be performed relative to a reference orbit. 10

11 Fitting Menu This command performs fitting (matching) of beta-functions, dispersion and momentum compaction. If the input (lattice) file contains no fitting block, invoking "Fitting Betas" command will prompt the user and append a template to the lattice description in the lattice editor window. The command may be invoked again once the required parameter entries have been filled-in. While the calculations proceed, messages from the non-linear optimizer will appear into a new dedicated text window. Iterations may be stopped by pressing the "Stop" button on the top button bar or invoking the "Fitting Stop" menu action. While it is safe to examine the contents of other windows while fitting is in progress, one should abstain from invoking other commands as this will likely produce unexpected results or may even cause the program to crash. 11

12 Fitting Menu Nonlinear optimization is performed using the method of steepest descent with automatic step size adjustment. The initial step sizes used for length, magnetic field and field gradient are prescribed in the fitting block. Each parameter is assigned an accuracy; a zero or negative accuracy implies that the corresponding parameter is not participating in the fit. BetaFitBlock dl[cm]=0.01 db[kg]=0.01 dg[kg/cm]=0.001 #Required parameters and their accuracy listed below(dparm<=0. - no fitting) #Maximum Betas[cm] and MomentumCompaction are on the next line BtXmax=5000. dbtxmax=0. BtYmax=5000. dbtymax=0. Alfa=0. dalfa=0. #Fitting parameters at the end of the lattice Beta_X[cm]=100. dbeta_x[cm]=0.1 Alfa_X=0. dalfa_x= Beta_Y[cm]=100. dbeta_y[cm]=0.1 Alfa_Y=0. dalfa_y= Disp_X[cm]=0. ddisp_x[cm]=0.01 D_prime_X=0. dd_prime_x= Disp_Y[cm]=0. ddisp_y[cm]=0.01 D_prime_Y=0. dd_prime_y= Qx=0. dqx=0. Qy=0. dqy=0. #Fit at Element with number =2 #To create a Fitting at intermediate Element: uncomment the line above, # write the correct Element number and insert six lines describing the # fit parameters. You can use up to 4 intermediate points #Each point has to be specified as described above # #Insert groups of Elements below. Each group has to be located on one line. #Start from the letter describing the type of changeable parameter such as: L:, B:, G: G: q1 G: q2 q3 EndBetaFitBlock 12

13 Fitting Menu As demonstrated in this example, elements may be assigned to groups. Parameters of elements belonging to a given group are varied proportionally. To compute the objective function, the program makes use of the accuracy parameters The relative magnitudes of the latter act as relative weighting factors If the iterations fail to converge or if convergence is too slow, modifying the relative magnitudes of the accuracy parameters may help. 13

14 Elements - Description Drift Quad Sector Bend Edge Focusing 14

15 Rectangular Bend Path length s a L q - a q q s L 2 sin q cos( q -a )

16 FODO Lattice Again.in OptiM 16