The Most General Transverse to Longitudinal Emittance
Exchanger
Ray Fliller
This is a rough draft to be perused, checked, corrected, scrutinized, and shot at by
experts and non-experts alike. I started this as an exercise to look for what one needs to
achieve in the beamline optics to have perfect emittance exchange (to first order).
The purpose of this paper is to answer the following question: What is needed of
the transport matricies of transverse to longitudinal emittance exchange beamline
sections to make a perfect emittance exchange? In particular, I assume that the
emittance exchange beamline is composed of three sections: a before the cavity section, a
deflecting mode cavity, and a after the cavity section with the following matricies.
a
b
0 



c
d
0  '

before the cavity matrix,
M bc  
c  a ' d  b ' 1  



0
0
0 1 

e
f
0 D



g
h
0 D' 

after the cavity matrix,
M ac  
gD  eD' hD  fD' 1  



0
0
0 1 

 1 0 0 0


 0 1 k 0
M cav  
cavity matrix from Ref[C&E].
0 0 1 0


 k 0 0 1


All of these matricies are formed to satisfy the symplectic condition, namely that
MTJM=J, where J is the symplectic matrix
 0 1 0 0


 1 0 0 0
J 
.
0 0 0 1


 0 0 1 0


In this notation, the matrix for the emittance exchange beamline is
d f (1 + k ) + b (e + kD - f ' ))
 c f (1 + k  ) + a (e + k ( D - f ' ))

d h (1 + k ) + b (g + k (D' - h ' ))
 c h (1 + k ) + a (g + k ( D ' - h ' ))
M   c (hD - fD ' ) + (1 + h kD - f kD ' ) (c - d (hD - f D' ) + (1 + h kD - f kD' ) (d 
a ' ) + a (gD - eD ' + k )
b ' ) + b (gD - eD' + k )

a
k
bk

fk
D + e  + k D  + f ( ' + k  ) 

hk
D' + g  + k D'  + h ( ' + k  ) 

gD  - e D'  + h D ' 
1  hkD - fkD'
f D'  '+  + h k D 

f k D'  +  (1 + k  )

0
1 + k

To simplify notation, I will refer to the emittance exchanger matrix with the notation of
Reference [C&E]
 A B

M  
C D
where A, B, C, D represent 2x2 blocks of the matrix.
Following from Equation 28 of C&E, we require that
det A  0
.
2  0
It is stated in Ref [C&E] that if all of the elements of the A block are zero, then 2  0
automatically. I have verified this and will include it in a subsequent draft. Nonetheless,
to design a beamline with perfect emittance exchange we need to produce a matrix with
the A block to be all zeros. As a consequence of this and the symplecticity of the matrix,
the following will automatically be true:
det B  1
det C  1 .
det D  0
What is not required from these considerations is that the D block be all zeros. However,
as I will show, setting the A block to all zeros will set the D block to all zeros
automatically. That this should happen at all is not obvious in the paper.
At this point the question becomes, what is needed of the elements of Mbc, Mcav, Mac,
to make the A block of the total matrix all zeros?? There are four equations, one for each
element of the A block. At first glance there are 15 unknowns (a, b, c, d, e, f, g, h, k,
', D, D’,However, the first 4 and the second 4 respectively are related by the
symplectic condition. The last 2 are determined by integrals of the dispersion function
for each line, this leaves 11 free parameters (a, b, c, e, f, h , k, ', D, D’.
One can solve the system of equations A11=A12=A21=A22=0, and one arrives at the
following relations between the free parameters:
k
1

D  e  f '
D' 
Dh
f
Plugging these into the matrix for the emittance exchange, one gets

0

0

M 

c


a

'

a



 a

0
 f
0
 h
d  b 'b 
 b
0
0
e  f ' f  


g  h 'h  

0


0

Now I will note the following interesting points of the solution. Firstly, it is general
involving only the properties of the cavity, symplectic transport lines, and perfect
emittance exchange. The second is that the solution of the cavity strength is only
determined by the dispersion in the cavity. It is not obvious that it should not depend on
other things, such as the slope of the dispersion through the cavity. Thirdly, the
requirements of the after cavity transport only depend on the dispersion and its slope in
the cavity. It does not depend on the other elements of the transport matrix prior to the
cavity.
And fourth, a beamline that returns the dispersion and its slope to zero in the
absence of the deflecting mode cavity (such as the chicane in the C&E paper with k=0),
cannot be a perfect emittance exchanger. The requirements to zero the dispersion and
slope in the absence of the cavity are
D  e  f '
D' 
Dh
f
.
Which cannot be satisfied with the conditions of perfect emittance exchange. However, a
beamline such as the double dog-leg proposed in SLAC-PUB-12038 (that is the paper
that Philippe is co-author) can. In fact if one uses
a  d  e  h 1
b f L
,
' 0
 
One can reproduce equations 25 and 26 of the paper.
Any questions?? Mistakes??