# Presentation: 676969

## Optimization of adiabatic buncher and phase rotator

***A.Poklonskiy (SPbSU, MSU), D.Neuffer (FNAL)***
***C.Johnstone (FNAL), M.Berz (MSU), K.Makino (MSU)***

## Key Parameters

**Drift**
**Length ****L****D**
**Buncher****  **
**Length ****L****B**
**RF Gradients ****E****B**
**Final RF frequency ********RF**** (****L****D****, ****L****B, ********RF****: (L****D**** + L****B****) ********(1/********) = ********RF****)**
**Phase Rotator**
**Length ****L********R**
**Vernier offset, spacing ****N********R****, ********V**
**RF gradients ****E********R**

## Equations

## Central Kinetic Energies

**From****	****	****	****	**** buncher synchronism condition we derive following relation for kinetic energies of central particles: **

- Puts limits on n_min and n_max => n_bunches!

## Final Central Kinetic Energies

**From the rotator concept one could derive amount of energy gained by n-th central particle in each RF (kept const in ROTATOR)**
**                       ****or, more generally**
**So for final energy n-th central particle has after the ROTATOR with m RFs we have**

## Energies Shape after Buncher and Amount of Energy Gain they Get in Rotator

## Evolution of Central Energies Shape

**Evolution of Central Energies Shape**

## Evolution of Central Energies Shape (projecton)

## Objective Functions

**One of the purposes of the structure is to reduce overall beam energy spread and to put particles energies close to some central energy. It seems natural to use objective function:**

## Objective function 1

**First, we can set               and get**

## Different optimized paremters (n vs T_fin), COSY built-in optimizer

## Objective Function 2

**As     we can use particle’s energies distribution in a beam**

**	**** ****n   energy   particles       %**
**----------------------------------------------**
** ****-12   963.96     1023      17.050000**
** ****-11   510.85      692       11.533333**
** ****-10   374.64      537         8.950000**
**   ****-9   302.98      412         6.866667**
**                 ****	****...**

## Objective Function 2

## Modeled Optimization (OBJ1), whole domain search

- Fixed params:
-    Desired central kinetic energy (T_c) =   125.0000000000000
-    T_0 in buncher (T_0) =   125.0000000000000
-    Drift+Buncher length (L_buncher) =   90.00000000000000
-    Final frequency (final_freq) =   200000000.0000000
- Varied params:
-    1st lever particle (n1) : -4.000000000000000 ==>   1.000000000000000
-    2nd lever particle (n2) : -3.000000000000000 ==>   27.00000000000000
-    Vernier parameter (vernier) : 0.1000000000000000 ==>  0.3000000000000000
-    RF gradient (V_RF) :  1.000000000000000 ==>   10.00000000000000
-    Number of RFs in rotator (m) :  1.000000000000000 ==>   10.00000000000000
- Objective functions:
- !!      188624.7593430086 ==>   7434.672341694457 =   181190.0870013142
-         21918840.88332907 ==>   1643615.384258914 =   20275225.49907015
-         215.4599659295154 ==>   238.7961711271633 =  -23.33620519764796
-         31921138.21770231 ==>   15260635.43728683 =   16660502.78041548

## Modeled Optimization (OBJ2), whole domain search

- Fixed params:
-    Desired central kinetic energy (T_c) =   125.0000000000000
-    T_0 in buncher (T_0) =   125.0000000000000
-    Drift+Buncher length (L_buncher) =   90.00000000000000
-    Final frequency (final_freq) =   200000000.0000000
- Varied params:
-    1st lever particle (n1) : -7.000000000000000 ==>   1.000000000000000
-    2nd lever particle (n2) : -6.000000000000000 ==>   14.00000000000000
-    Vernier parameter (vernier) : 0.1000000000000000 ==>  0.4000000000000000
-    RF gradient (V_RF) :  1.000000000000000 ==>   10.00000000000000
-    Number of RFs in rotator (m) :  1.000000000000000 ==>   10.00000000000000
- Objective functions:
-         552558.8685890788 ==>   331194.3462835634 =  -221364.5223055154
- !!      1120051685.347023 ==>   740898336.4248258 =  -379153348.9221967
-         790.5115736637570 ==>   714.6396495910632 =  -75.87192407269379
-         1614049125.098601 ==>   1105871829.498042 =  -508177295.6005592

## Other possible objective functions

**We may try to incorporate information about other particles (?)**
**We may combine this objective function with any of the first two with any weight coefficients (?)**

## Other Optimization (OBJ1) (short structure), whole domain search

- Fixed params:
-    Desired central kinetic energy (T_c) =   125.0000000000000
-    T_0 in buncher (T_0) =   125.0000000000000
-    Drift+Buncher length (L_buncher) =   75.00000000000000
-    Final frequency (final_freq) =   100000000.0000000
- Varied params:
-    1st lever particle (n1) : -3.000000000000000 ==>   1.000000000000000
-    2nd lever particle (n2) : -2.000000000000000 ==>   4.000000000000000
-    Vernier parameter (vernier) : 0.1000000000000000 ==>  0.2000000000000000
-    RF gradient (V_RF) :  1.000000000000000 ==>   10.00000000000000
-    Number of RFs in rotator (m) :  1.000000000000000 ==>   10.00000000000000
- Objective functions:
- !!      920802.1870498012 ==>   716144.2078318644 =  -204657.9792179369
-         2906605812.448390 ==>   2289340374.907225 =  -617265437.5411658
-        -1154.766872358291 ==>  -1012.150761595708 =   142.6161107625823
-         2905272325.918894 ==>   2288315925.743026 =  -616956400.1758685

## Other Optimization (OBJ2) (short structure), whole domain search

- Fixed params:
-    Desired central kinetic energy (T_c) =   125.0000000000000
-    T_0 in buncher (T_0) =   125.0000000000000
-    Drift+Buncher length (L_buncher) =   75.00000000000000
-    Final frequency (final_freq) =   100000000.0000000
- Varied params:
-    1st lever particle (n1) : -3.000000000000000 ==>   0.000000000000000
-    2nd lever particle (n2) : -2.000000000000000 ==>   5.000000000000000
-    Vernier parameter (vernier) : 0.1000000000000000 ==>  0.4000000000000000
-    RF gradient (V_RF) :  1.000000000000000 ==>   10.00000000000000
-    Number of RFs in rotator (m) :  1.000000000000000 ==>   10.00000000000000
- Objective functions:
-         920802.1870498012 ==>   716325.0523539253 =  -204477.1346958759
- !!      2906605812.448390 ==>   2288276641.662843 =  -618329170.7855473
-        -1154.766872358291 ==>  -1015.263140233114 =   139.5037321251762
-         2905272325.918894 ==>   2287245882.418927 =  -618026443.4999671

## Study 2a params (OBJ1)

**Study 2a params (OBJ1)**

- Fixed params:
-    Desired central kinetic energy (T_c) =   200.0000000000000
-    T_0 in buncher (T_0) =   200.0000000000000
-    Drift+Buncher length (L_buncher) =   150.0000000000000
-    Final frequency (final_freq) =   200000000.0000000
- Varied params:
-    1st lever particle (n1) :
- 	0 ==>   3.000000000000000
-    2nd lever particle (n2) :
-          18 ==>   6.000000000000000
-    Vernier parameter (vernier) :
-          0.032 ==>  0.08
-    RF gradient (V_RF) :
- 	8 ==>   9.000000000000000
-    Number of RFs in rotator (m) :
- 	10 ==>   10.00000000000000
- Objective functions:
-         619593.7642709546 ==>   522561.7532899606 =  -97032.01098099403
- !!      750907264.4334378 ==>   615875434.3135488 =  -135031830.1198890
-        -1066.685941047459 ==>  -869.7924497231580 =   196.8934913243012
-         749769445.5366095 ==>   615118895.4079534 =  -134650550.1286561

## Study 2a params (OBJ2)

**Study 2a params (OBJ2)**

- Fixed params:
-    Desired central kinetic energy (T_c) =   200.0000000000000
-    T_0 in buncher (T_0) =   200.0000000000000
-    Drift+Buncher length (L_buncher) =   150.0000000000000
-    Final frequency (final_freq) =   200000000.0000000
- Varied params:
-    1st lever particle (n1) :
- 	0 ==>   3.000000000000000
-    2nd lever particle (n2) :
-          18 ==>   6.000000000000000
-    Vernier parameter (vernier) :
-          0.032 ==>  0.07
-    RF gradient (V_RF) :
- 	8 ==>   9.000000000000000
-    Number of RFs in rotator (m) :
- 	10 ==>   10.00000000000000
- Objective functions:
-         619593.7642709546 ==>   522561.7532899606 =  -97032.01098099403
- !!      750907264.4334378 ==>   615875434.3135488 =  -135031830.1198890
-        -1066.685941047459 ==>  -869.7924497231580 =   196.8934913243012
-         749769445.5366095 ==>   615118895.4079534 =  -134650550.1286561

## Summary

**Model of central energies shape optimization for buncher and phase rotator is proposed.**
** ****Code is written and debugged (hope so), It allows to perform optimization on any set of supported parameters (length of the buncher and rotator, final frequency, central energy, E field gradient, phases). **
**Some example results are presented**

## To do

**Check results in (t,E) space as more important (problem: energy is changing ********              )**
**Different RF field waveform?**
**Check optimized parameters for the whole beam distribution (COSY, ICOOL?) Is it really better?**
**Switch to 3D-motion simulation and optimization**

## Overall RFs Effect