# Presentation: 096010

## Transverse-Longitudinal Phase Space Manipulations

**Kwang-Je Kim**
- Argonne National Laboratory
- and The University of Chicago
- B&A Seminar
- January 20, 2006
- Advanced Photon Source

## To Improve the Performance of X-Ray FELs

***To Improve the Performance of X-Ray FELs***

## SASE FEL  for 30 keV

- LCLS reference parameters:
-  = 8 keV, u = 3 cm, K = 3.7, Ip = 3.5 kA, Ee = 15 GeV,
- E/E = 10-4 , n = 1.2 mm-mrad,   Lsat = 100 m
- Vary K, n, and Ee                                            (Z.R. Huang)
***It pays to strive for an ultralow emittance e-beam***

| K | Ee(GeV) | n(mm-mrad) | L sat(m) |
| --- | --- | --- | --- |
| 	3.7 | 	30	 | 	1.2 | 	300 |
| 	3.7 | 	30 | 	0.5 | 	130 |
| 	3.7 | 	30 | 	0.1 | 	40 |
| 	1 | 	12 | 	0.1 | 	60 |

**shorter undulator**

**    ****shorter undulator**
**    ****and shorter linac**

## Need for Phase Space Manipulation

- RF Photocathode Gun
- Transverse emittance   x, y ~1 × 10-6 m
- Energy spread very small
- 		E ~ 1.5 keV
- FEL requires z/E < 10-4
- Can we do the transformation:

- ~

## Contents

**Some properties of Hamiltonian Transport**
**Transverse-longitudinal *****exchange***** for x-ray FELs**
**Implementation**
**Space charge effect**
**Flat beam technique**
**Optical system for exchange**

## Phase Space Manipulation

**Accelerators are for manipulation of beams in 6D phase space**
**Phase space manipulation in accelerators has been mainly in one 2D subspaces (z, ********) or (x, x’) and in Hamiltonian system**
**Beam cooling is an advanced phase space manipulation in weakly *****non-Hamiltonian***** system**
**Manipulation involving 6-D phase space of *****Hamiltonian***** system can greatly enhance accelerator performance**

## Beam Transport and Manipulation

**6D phase space:  (x, x**** ́, y, y ́, z, ********)**
**We will work mostly in 4D**
-  **X**
**Beam matrix:**
**Transfer matrix: M**
- 	X = M Xo,  Σ = M Σo

## Hamiltonian Transport

**Unit symplectic matrix**
**M is symplectic:**
***Det M = 1***
**Conserved quantities**

- Projected emittances
- Iff uncoupled; Σc = 0

## If Coupled, Projected Emittances

**Projected emittances are ****not**** conserved**
**Some properties:** (K.L. Brown & R.V. Servanckx, SLAC-PUB 4679 (1989) )
- Start from uncoupled emittances *xo** *and* **yo*
- If *xo** = **yo* *** **x** = *y* *for all s
- 			*x* ≥ *xo*
- If *xo** **≠ *y*o*
- 			*x *+ *y* ≥ *xo *+ *yo *
**Useful applications appear difficult, but ...**

## Emittance Exchange Theorem               (E. Courant, “Perspectives in Modern Physics, Essays in 	Honor of H.A. Bethe,” Interscience Pub., 1966)

**For transport from an uncoupled to another uncoupled system, *********a***** and *********b***** are uniquely determined up to switching**.
- (*a*, *b*)  → (*a*, *b*) or (*b*, *a*)
**Proof****:**
** ****Can be generalized to higher dimensions**

## An Emittance Switching Scheme for Improved X-Ray FEL Performance

-                                  (P. Emma, Z. Huang, P. Piot, and KJK, under preparation)
**Flat beam technique (units in m-rad)**
**Use short electron beam  *********z***** = 33******
-                                          Q = 33 pC, I = 100 A
**Exchange ****(x ****↔ z)**
**Final bunch length**

## Power gain length LG of an x-ray FEL at 0.4 Å versus the undulator parameter K for (a) a beam with a normalized transverse emittance 1 × 10-6 m-r and a peak current 3.5 kA and (b) a beam with a normalized transverse emittance 1 × 10-7 m-r and a peak current 1 kA. The relative rms energy spread in both cases is 1 × 10-4 (courtesy of Z. Huang).

- Power gain length *LG* of an x-ray FEL at 0.4 Å versus the undulator parameter *K* for (a) a beam with a normalized transverse emittance 1 × 10-6 m-r and a peak current 3.5 kA and (b) a beam with a normalized transverse emittance 1 × 10-7 m-r and a peak current 1 kA. The relative rms energy spread in both cases is 1 × 10-4 (courtesy of Z. Huang).

## Is (10-6, 10-6, 10-7)m consistent with the space charge degradation?

- Use spot size x = 0.5 mm, z = 33 m, A ~ 18
- 	Eo = 120 MV/m, I = 100 A, /4 sin o ~ 1
- → (10-6, 10-6, 10-7) !

## Generating a Flat Beam with Angular Mom. Dominated Beam

***Generating a Flat Beam with Angular Mom. Dominated Beam***
***   ******(Y. Derbenev), (R. Brinkmann, Y. Derbenev, K. Fl******öttmann), (D. Edwards ...), (Y.-e Sun)***

- experiment

- simulation

## Electron Emission into Axial Magnetic Field

- Xth

- B

- Xo

## Does Flat Beam Technique Violate the EET?

- (x, y):  (10-6, 10-6) → (10-5,10-7)
- Thermal distribution before emission
- Distribution after emission
- Xth → Xo non-symplectic
- o: coupled

## Optical System Producing the x-z Exchange

- (x, z) = (10-5, 10-7) → (10-7, 10-5)
- Dipole mode cavity:
- Dog leg

## An Approximate Scheme for (x,z) Exchange 	M. Cornacchia and P. Emma, PRSTAB, 5, 084001 (2002)

- M = MD (-η,ξ,L) MC(k) MD (η,ξ,L)
- Choose ηk = 1    then
- Works if                                                      η cannot be reduced arbitrarily
*	**	**	**	**            *small due to second order* *aberration

## An Exact Scheme for Emittance Exchange (KJK)

- Reverse second dog leg
- Choose kη = -1
- Then

## Transverse phase space (left two plots) and longitudinal phase space (right two plots) before (top) and after (bottom) emittance exchange.

- 				(Courtesy of P. Emma)

## Technical Detail

- High aspect ratio is difficult to maintain
- Phillipe Piot will solve the problem