The 13th Symposium on Accelerator Science and Technology, Suita, Osalm, Japan, October 2001

Focal Characteristic of Helical Quadrupole Focusing Channel
Akio Mrotia and Yoshihisa I washita
NSRF, ICR, Kyoto Univ, Japan
AbstractA helical quadrupole focusing channel has continuous fieldsymmetry and stronger focusing power compared with aconventional FODO focusing channel. The good field symmetry allowed us to construct the explicit transfer matrix
under the paraxial approximation. The paraxial analysis ofthe helical quadrupole focusing channel and the characteristic of the point-to-point focusing are reported.

0.15

1 INTRODUCTION

Figure 1: The pole surface of the helical quadrupole

A helical quadrupole focusing channel(HQFC) is a variation of a quadrupole focusing channel, which consists of along quadrupole field continuously twisted along the beamaxis. Its strong focus is adequate for a transport of a large
emittance beam such as secondary particles. Because focal strength of HQFC, unlike that of a solenoid channel,
depends on total momentum of a particle, it can be used
as a momentum filter for a beam from a point-like source.
In the conventional analysis, the HQFC has been treated asa series of rotating quadrupole blocks[1]. Accurate analysis, however, requires a treatment in a magnetic field distribution that satisfies the Maxwell equation. The analysis
and fabrication of a electrostatic helical quadrupole werereported in Ref. [2, 3, 4]. In the subsequent section, thefield distribution of the HQFC, the analytic solution of thetransfer matrix under the paraxial approximation and oneof the focal characteristic are presented. The comparison
with a FODO and a solenoid is described in Ref. [5].

2 HELICAL QUADRUPOLE FIELD
In a free space, the helical quadrupole field satisfying theMaxwell equation is expressed by the following form using
modified Bessel function.P(r, 0, z)
andB.Po.
kh(kr)sm(20-kz)(1)-Vel>,

(2)

and..

g
1:0

Figure 2: r dependence of radial part of both helical andpure quadrupole fields: Br,B9,Bz and Bquad
Figure 2 shows r dependences of the three components of
the helical quadrupole field together with Br componentof the pure quadrupole. Around the central axis, the helical quadrupole field can be approximated by a rotating
quadrupole with the angle of k2z radian. This paraxial approximation corresponds to the linear approximation. Inorder to examine the valid range of the linear approximation, we obtain following expansions by Taylor series:.Po(kr) 3.Po(kr) 3

Br = -4(kr + - 6 --<Poi~(kr)sin(20-kz),.Po

B9(r, 0, z) =-2cl>o~ cos (20- kz) (4)Bz(r, 0, z) =.Poh(kr) cos (20- kz). (5)52. (kr)4(6)(7)

6+ 12" + O(r )) cos (20- kz). (8)For a good linear approximation, the nonlinear terms in Br,
B9 and Bz components have to be negligible comparedwith the pure quadrupole component. If 10% deviation isallowed for either Br or B9 component, the maximum r
coordinate Tbore has to keep following condition:(3)h(kr).
+ O(r5 )) sm(20- kz) ,B9 = - 4 (kr + 12" + O(r )) cos (20- kz)
andBz = B((kr)

where k is 21r / L and L is the period of the magnetic field(the period of a pole is 2£). Figure 1 shows one of thepole surface of the helical quadrupole that is a equal potential surface of Eq. (1). The field component in cylindricalcoordinate system is given as follows:Br(r,O,z) =0.9 . - - - - - - , - - - . - - - - y - - , - - - - , - - ,B~ ___;_!··············' ··········+·······.,!
0.8
Bti, ------t-· ... ,........... L ............'·····/
0.7
B,!
.......
;..
,............, ......... V
0.6
0.5 .I!.'!.~t --~~·-·.·~~~~~-~~:1.·~~~
0.4
0.3
0.2
0.1
0-='--'""""-'---'-----'----'----''-----'
0.05 0.1 0.15 0.2 0.25 0.3
0r/L

Tbore ;S 0.12£.

(9)

Same limitation for Bz component is given by-328-

Tbore ;S 0.032£.

(10)

The 13th Symposium on Accelerator Science and Technology, Suita, Osaka, Japan, October 2001

3

TRANSFER MATRIX OF HELICALQUADRUPOLE FOCUSING CHANNEL

Linearized transfer matrix of the HQFC will be derived inthis chapter. The stability condition for the matrix is deduced by Eigenvalue analysis.

space(x, x', y, y') are given byR dO = (),(cos dO

0- sin d(}0sin dO0cos dO
00cos d(}0sind(}( -Kdsand MQ(dsiO) =0

Cell Matrix

Considering the rotating characteristics of the HQFC's
magnetic scalar potential, a transfer matrix from location
Sa to Sb MHQ(sblsa) can be written as

where matrix R is a rotation matrix. Using the translation operator Eq. (11), a transfer matrix of the HQFC
MHQ(Soutlsin) can be divided into products of small cellmatrices:s ) , (16)

0Kds

1

where K, q and p are the field gradient of the quadrupole
field ( -qk if!o/ 4p), the charge and the momentum of theparticle, respectively. Because the sign of the parameter K
can be chosen without loss of generality, we will assume
that K is positive hereafter.

3.2

Eigenvalue Analysis and Stability Condition

Let us derive the Eigenvalue and the Eigenvectors of theinfinity product in Eq. (14) from those of the cell matrix.
The Eigenvalue problem of the cell matrix indexed by s j Nis defined as

N-1

MHQ(SoutiSin) =(15)~ 1

100d8 )

cos d(}ds1

3.1-si;

IJ MHQ(Si+llsi)
i=Oks>-.i(s/N)xi(s/N) = R(- 2N)MQ(sfNIO)X;(sjN). (17)A

Using the Eigensolutions of Eq. (17), the Eigesolutionsof the infinity product Mcore are described as follows:
Xi= limN-.oo

where { Si li = 0, ... , N, so = Sin, SN = S0 ut} is a division of the region [sin, Soutl· Applying the equal weight
division and a limit operator to Eq. (12), we obtain

Mcore(s) = lim (N->oo= lim (N->oo)
11 R(- 2kNs )MHQ( ~IO)
N)..i = lim 5.i(s/N)N.
N->oo(18)After some mathematical operations, we can obtain theEigensolution of the matrix Mcore as follows:(>-. 1 , ).. 2 , )..3 , )..4 ) = (e-iWHS l e+iWHS l e-iWLS l e+iWLS) l ( 19)
where WH and W£ are y'k 2 /4+ K and y'k 2 /4- K, respectively. The set of the Eigenvectors isX = [x1, x2, x3, X4] =·2WH-i2WH
2Jt(13)where 6.s is the length of slice: 6.s = (sout - Sin)/N.
Considering that a thin helical quadrupole is equivalent to athin quadrupole under the linear approximation, the transfer matrix MHQ(6.sl0) in Eq. (13) can be replaced by atransfer matrix of a thin quadrupole MQ(Ll.siO). Let us
introduce the matrix Mcore to denote the infinity matrix
product in Eq. (13) as follows:xi(s/N),

(\kTT

10

10k2Kk2K

00-iS]t-~2K

11}

(20)

where the vector set X forms the basis of the Eigenspace.
When both WH and WL are real, the solutions are periodic
and the corresponding motion of a particle is stabilized.
Thus the stability condition is given byk22O<K<-=!!._p·- 4-(21)The transfer matrix MHQ can be constructed by )..i and X
as follows:

N-1
i=Oksin-),)M HQ (Sout ISin ) = R (-ksout
22 - Mcore(Sout-Sin)R(--))MQ( ~IO) . (14)11 R( -~N2N

N-1
i=O

The infinite matrix product Mcore is the products of thecell matrix R( -ksj2N)MQ(s/NIO). The explicit representations of R(dO) and MQ (dsiO) in the transversal phase

M,M,(s) =X (-329->., O A,A, OM)

x- 1 .

(22)

The 13th Symposium on Accelerator Science and Technology, Suita, Osaka, Japan, October 2001

Explicit representation of Mcore ( s) is given byg~ (gpSp- ~)

§.m.
g.,.

g~ (Cm- Cp)_JS..JLS

Cp

0Sp
gp

_§z

g~ (Cp- Cm)

Cm

Cp
L gp

M=•(s) = (Pgp_§.m.

0g~ (~ -gmSm)

7rJLS

Lgmgm

Cmm}

CpCm
SpSmcosgps
cosgmssingpssingms=1rSr;-;-:~
8 = L' 9p = y 1 + g, 9m = y 1- g and

4 POINT-TO-POINT FOCUSING
Considering the transportation of a secondary beam, thepoint-to-point focusing characteristic is useful. Let us discuss about the condition of the point-to-point focusing.
The transfer matrix in Eq. (22) is described by the matrix Mcore except a coordinate rotation. These coordinate
rotation matrix is unrelated to the property of the point focusing. The core matrix Mcore has two different frequencyof the oscillation in the Eigenspace and these two oscillation is not isolated in the real phase space. Thus in orderto focus by point-to-point, two oscillation phases WHS andWLS have to be multiples of 1r and the difference of twophases (wH - WL)s should be a multiple of 21r:

3s,p, q, nWHS

(23)

of the transformation matrix from the theory of the linear
algebra. Thus the cross section S(s, G) is written down as:. v'1r 2 - Gs . V1r 2 + Gs
27r4 - G 2SID)
LL
21r2J1r4 _ Q2-;:;--:;::::::;:=~ SID•(27)

The peaks of the logarithm of the S(s, G)- 1 in Eq. (27)

= p7r, WLS = Q'Tr, (WH- WL)S = 27rns E R,p,q,n EN(24)

In order to obtain the minimum period of the point-to-point
focusing, the rational number p / q should be irreducible rational. Thus the dimensionless focusing parameter G andthe minimum period so is obtained as:

4K
k2

10~

p2-q2
p2+q2'

so =Ly~gcd(p, q) = 1,

p > q,

p, q E Odd

For example, the minimum period so and the focusing parameter G, which corresponds to the minimum period so,
is 0.81r 2 "" 7.9 and ../5L, respectively.
Let us estimate the cross section of the beam from a pointsource. Using the notations in the transfer matrices in Eqs.
(22),(23), a beam boundary that began from a point sourceis given by:(g S (YX) = Lr'g1r
Cp- Cm
pp

§.m.
g.,.

Cm- Cp ) (cosO)§z- gmSm
gpsinO

Figure 3: The cross section of the point source beam

(25)

'(26)

where r' and 0 are the divergence of the initial beam and theazimuthal angle of initial beam, respectively. From the estimation of the capture range against the initial divergence
under the linear approximation [5], the upper limit of theproduct Lr' is about 0.8 times of the bore radius of theHQFC.
Because the transformation in Eq. (26) is linear, the scale
factor of the cross section is obtained as the detenninant

shown in Fig. ·3 are the points on which the beam crosssection shrinks. Thus the HQFC might be usable as a momentunl discriminator by combining with a collimator.

5REFERENCES[1] H. 0. Funsten, "Properties of an Infinite Focusing Channel
with Successive Quadrupole Rotations Not Equal to 90°",
NIM 44 (1966) 301[2] F. S. Chute, F. E. Vermeulen and E. A. Youssef, "A 1\visted
Electrostatic Quadrupole for Guiding Heavy Charged Particles",NIM 82 (1970) 86[3] E. A. Youssef, F. S. Chute and F. E. Vermeulen, "A Note on
Particle Trajectories in a Helical Electrostatic Quadrupole",
NIM 93 (1971) 181[4] R. G. McPherson, F. E. Vermeulen and F. S. Chute, "Use ofLine Charge System for a Helical Electrostatic Quadrupole",
NIM 148 (1978) 5[5] A. Morita and Y. !washita, "Analysis of Helical Quadrupole
Focusing Channel(II)", Proc. of the 26th Linear Accelerator
Meeting in Japan, Tsukuba, (2001) 329-330-