Nuclear Instruments and Methods in Physics Research A 659 (2011) 537–542

Contents lists available at ScienceDirect

Nuclear Instruments and Methods inPhysics Research A

0168-90

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/nima

A new undulator scheme providing various polarization stateswith low on-axis power density

Takashi Tanaka �, Hideo Kitamura

RIKEN SPring-8 Center, Koto 1-1-1, Sayo, Hyogo 679-5148, Japan

a r t i c l e i n f o

Article history:

Received 6 June 2011

Received in revised form

26 July 2011

Accepted 3 August 2011Available online 12 August 2011

Keywords:

Undulator

Polarization

Heat load

02/$ - see front matter & 2011 Elsevier B.V. A

016/j.nima.2011.08.012

esponding author.

ail address: ztanaka@spring8.or.jp (T. Tanaka)

a b s t r a c t

A new undulator scheme is proposed to enable flexible utilization of polarization properties of

synchrotron radiation, which is based on the composite-period undulator originally proposed to

improve the wavelength tunability of free electron lasers. It has an advantage over the conventional

undulators that the on-axis power density is kept low even when a high K value is applied to lower the

photon energy. The basic principle and availability of various polarization states are presented together

with a practical example.

& 2011 Elsevier B.V. All rights reserved.

1. Introduction

Polarization is one of the most important properties of syn-chrotron radiation (SR) and thus a lot of undulator schemes havebeen proposed to make a variety of polarization states available toSR users. Among them, the most widely used is the APPLE-typedevice [1], in which two pairs of magnet arrays that are movablelongitudinally are placed below and above the electron path. Inprinciple, it can generate any polarization states: horizontal,vertical, inclined linear, left-hand circular and right-hand circular.

Thanks to the simplicity in structure and flexibility in polariza-tion, the APPLE-type device has now become a standard insertiondevice (ID) for polarization-dedicated SR experiments. It should benoted, however, that there is still a drawback especially for the long-wavelength applications with a high energy electron beam, in whichhigher K values are required. In such a case, operating the APPLE-type device in the linear polarization modes causes a serious heatproblem as explained in the followings.

When the electron moves along a sinusoidal orbit lying on acertain plane as in the linear polarization mode of the APPLE-typedevice, the intensity of high-harmonic radiation becomes moredominant when higher K values are applied. As a result, the on-axis power density increases drastically as the K value increases andthus the optical elements such as the monochromators and mirrorscan be seriously damaged or deformed so that the performance ofmonochromatization may be reduced significantly [2].

The ‘‘Figure-8 undulator’’ is an ID proposed to overcome thedifficulty explained above [3]. In addition to the vertical field, it

ll rights reserved.

.

generates a horizontal field with the period twice as long as thatof the vertical. As a result, the electron moves along a special orbitwhose projection on the transverse plane is a figure of eight, andgenerate horizontally polarized radiation at the fundamentalphoton energy. What is more important is that most of theradiation power is dispersed off axis without sacrificing signifi-cantly the fundamental flux. One major drawback of the Figure-8undulator is that it cannot be switched to the helical undulator asin the case of the APPLE-type device, because it contains twodifferent magnetic periods. Although several proposals have beenmade to make it possible to operate the Figure-8 undulator in acircular-polarization mode [4,5], they are not as advantageous asthe pure helical undulator in terms of the available flux andspectral purity.

The purpose of this paper is to propose a new undulatorscheme that makes it possible to switch from the helical undu-lator to the Figure-8 undulator and vice versa, in order to enablethe flexible utilization of polarization with low on-axis powerdensity. The idea is based on the composite-period undulator(CPU) scheme to generate magnetic fields composed of twodifferent periods, which has been recently proposed by theauthors for the purpose of improving the wavelength tunabilityof free electron lasers [6]. It is worth noting that other ID magnetconfigurations containing different periods have been proposed inRefs. [7,8] for special applications.

2. Principle

Let us consider an undulator magnet array as shown in Fig. 1(a),which is similar to that of the normal Halbach configuration [9]

λu

2λu

Fig. 1. (a) Halbach-type undulator magnet circuit with inclined easy axes to

generate a composite-period field. The magnetization vectors can be decomposed

into those for (b) the fundamental period and for (c) the double period. The

magnetic field profiles with the two different periods are shown for reference.

Fig. 2. Other possible configurations to generate the CPU field.

D

A B C

E F

A

D

B

E

C

F

Fig. 3. (a) Helical undulator composed of six magnet arrays and (b) its modifica-

tion to the Helical-8 undulator based on the CPU scheme. The magnet arrays B and

E generate the vertical field, while A, C, D and F generate the horizontal field.

T. Tanaka, H. Kitamura / Nuclear Instruments and Methods in Physics Research A 659 (2011) 537–542538

except that the easy axis of each magnet block is inclined at acertain angle y according to a particular rule, which is explained inRef. [6] in detail. As a result, the magnetic field observed on theundulator axis has approximately the form

ByðzÞ ¼�B1y sinðkuzÞþB2y sinðkuz=2Þ ð1Þ

with

ku ¼ 2p=lu

being the undulator wave number. The above formula means thatthe magnetic field contains two different magnetic periods, lu and2lu. This is easily understood by decomposing the magnetic vectorsindicated in Fig. 1(a) into (b) and (c). The field amplitudes of the twocomponents, B1y and B2y, depend on the inclined angle y as well asthe remanent field of permanent magnet material and the dimen-sions of magnet blocks. It should be noted that we have a lot ofconfigurations that generate the magnetic field given by Eq. (1)other than that shown in Fig. 1(a). For example, three possibleconfigurations are schematically illustrated in Fig. 2(a)–(c). Inaddition, the most straightforward configuration is proposed inRef. [10]. The decision should be made according to the feasibilityand total cost of magnet fabrication, achievable magnetic perfor-mance, and expected optical characteristics.

By moving the top and bottom magnet arrays along theundulator axis to the opposite directions to introduce a relativelongitudinal distance of Dz, the magnetic field reduces to

ByðzÞ ¼�B1y sinðkuzÞ cos fþB2y sinðkuz=2Þ cosðf=2Þ

with

f¼ kuDz=2:

It is straightforward to show that by substituting Dz¼ lu orDz¼ lu=2, By(z) reduces to a simple sinusoidal field with theperiod of lu or 2lu. In other words, the magnetic period isselectable by adjusting the relative distance Dz. This is the basicprinciple of the CPU scheme.

Now let us apply the CPU scheme to the helical undulatorcomposed of six magnet arrays as illustrated in Fig. 3(a), whichhas been proposed and actually installed in SPring-8 [11]. Thecentral magnet arrays (B and E) generate the vertical field By,while the outer four arrays (A, C, D and F) generate the horizontalfield Bx. In other words, the horizontal and vertical magnetic fields

are generated independently unlike the APPLE-type device. Thehelicity can be flipped by shifting the central magnetic arrays byhalf a period. We assume that all the magnet arrays are modifiedto generate the CPU field as illustrated in Fig. 3(b), and are movedindependently to the longitudinal direction. To be specific, all themagnets are magnetized according to the arrows indicated inFig. 3(b) before assembling. It should be noted that the magne-tization vectors do not have to be changed after assembling.

The position of each magnet array is defined as zA,zB, . . . ,zF .Imposing the magnet array pairs (A, D) and (C, F) to movetogether, namely

zA ¼ zD ¼ zL, zC ¼ zF ¼ zR

the magnetic fields reduce to

Bx,yðzÞ ¼ �B1x,1y sinðkuz�f1x,1yÞ cos f2x,2y

þB2x,2y sinðkuz=2�f1x,1y=2Þ cosðf2x,2y=2Þ ð2Þ

with

f1x ¼ kuðzLþzRÞ=2, f2x ¼ kuðzL�zRÞ=2

T. Tanaka, H. Kitamura / Nuclear Instruments and Methods in Physics Research A 659 (2011) 537–542 539

f1y ¼ kuðzBþzEÞ=2, f2y ¼ kuðzB�zEÞ=2

being the phase parameters that can be adjusted to enable variousmodes of operation. Among them, we have three modes. Theelliptical mode is enabled by setting the phase parameters (f1x,f2x, f1y, f2y) to ð0,p,7p=2,pÞ. Substituting into Eq. (2), we havethe magnetic fields given as

BxðzÞ ¼ B1x sinðkuzÞ, ByðzÞ ¼ 8B1y cosðkuzÞ ð3Þ

which obviously agree with those of the elliptical undulator. If thedimensions of magnet blocks are designed to satisfy B1x ¼ B1y,they reduce to the helical undulator fields. As explicitly denotedabove, the helicity can be flipped by inverting the sign of thephase parameter f1y.

The HF8 and VF8 modes are enabled by setting the phaseparameters (f1x, f2x, f1y, f2y) to ð0,3p=2,0,pÞ and ð0,p,0,p=2Þ,respectively. The magnetic fields in the HF8 mode are given as

BxðzÞ ¼ �ðB2x=ffiffiffi2pÞ sinðkuz=2Þ, ByðzÞ ¼ B1y sinðkuzÞ ð4Þ

while those in the VF8 mode are given as

BxðzÞ ¼ B1x sinðkuzÞ, ByðzÞ ¼ ðB2y=ffiffiffi2pÞ sinðkuz=2Þ: ð5Þ

Based on the analysis in Ref. [3], it is found that the radiation emittedfrom an electron moving in the magnetic fields given in Eq. (4) or (5)is horizontally or vertically polarized at the fundamental photonenergy. What should be stressed more is that the on-axis powerdensity is much lower than that of the conventional planar undulator,in which the electron moves along a trajectory lying on a plane.

Summarizing the discussions above, the helical undulator withsix magnet arrays combined with the CPU scheme works not onlyas the simple helical undulator but also as the HF8 and VF8undulators by properly adjusting the phase parameters. Theundulator based on this concept is hereinafter referred to as the‘‘Helical-8 undulator’’ for simplicity.

Fig. 4. Field quality parameter d as a function of a, showing the feasibility of the

Figure-8 undulator providing inclined linear polarization.

3. Feasibility of inclined linear polarization

It has been shown in the previous section that the Helical-8undulator can be switched from the HF8 undulator to the VF8undulator. In other words, the direction of linear polarization canbe switched from horizontal to vertical. Now let us consider theavailability of linear polarization inclined at an arbitrary angle.For this purpose, we define a parameter a, with which the phaseparameters are given as

ðf1x,f2x,f1y,f2yÞ ¼ ð0,3p=2�a,0,p�aÞ: ð6Þ

Substituting into Eq. (2), we have

BxðzÞ ¼ B1 sinðkuzÞ sin aþB2 sinðkuz=2Þ sinða=2�p=4Þ

ByðzÞ ¼ B1 sinðkuzÞ cos aþB2 sinðkuz=2Þ sinða=2Þ

where we have assumed B1x ¼ B1y ¼ B1 and B2x ¼ B2y ¼ B2 forsimplicity. As explained in the previous section, the magneticfields described above reduce to those of the HF8 and VF8undulators when a is set to 0 and p=2, respectively. For othervalues of a, the magnetic fields do not necessarily satisfy thecondition of the Figure-8 undulator. In order to investigate theimpact of the parameter a on the field quality as the Figure-8undulator, let us introduce a new coordinate defined by twovariables (v,u), which is obtained by rotating the original (x,y)coordinate by an angle �a, namely:

ev ¼ ex cos a�ey sin a, eu ¼ ex sin aþey cos a

where ex is the unit vector in the x direction, and so on. Then themagnetic vector is written in the form:

B¼ euB1 sinðkuzÞþB2ðeubuþevbvÞ sinðkuz=2Þ

with

bu ¼ sin a sinða=2�p=4Þþcos a sinða=2Þ

bv ¼ cos a sinða=2�p=4Þ�sin a sinða=2Þ:

Considering the fact that bu vanishes when a is set to 0 or p=2 andthe magnetic vector reduces to that of the Figure-8 undulator, it isnatural to define the parameter d¼ bv=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2

uþb2v

pas a figure of

merit to denote the field quality. To be specific, d¼ 1 means thatthe magnetic fields are purely those of the Figure-8 undulator,while d¼ 0 means that the magnetic vector lies on a planeperpendicular to ev and thus the on-axis power density can belarge upon high K-value application, which should be avoided.

Fig. 4 shows the field quality parameter d calculated as afunction of a, where we find that d is larger than 0.99 when a iswithin the region between 01 and 901, while it rapidly decreaseswith distance from this region. Thus we impose the parameter ato be between 01 and 901 to avoid the heat load problem.

Next, we investigate the polarization characteristics for possi-ble values of a. In order to do so, we need to calculate the complexamplitude of radiation at the photon energy of o, which is givenin the form

Fo ¼ocR

ZbeiotðzÞ dz

with

tðzÞ ¼1

2cg2zþ

Z z

g2b2ðz0Þ dz0

� �

where c is the speed of light, R is the distance from the undulatorto the observer, g and b are the Lorentz factor and relative velocityof the electron moving in the undulator field, respectively. Aftersome mathematical operations, we find that the radiation spec-trum has a sharp peak at the photon energy:

o¼ n4pcg2

2lu

1

1þK21=2þK2

2 PðaÞ=2¼

n

2o1

with

K1 ¼eB1lu

2pmc, K2 ¼

eB2lu

pmc

PðaÞ ¼ 1�sin aþcos a

2

where e and m are the electron charge and rest mass, and n is aninteger. The parameters K1 and K2 are obviously the K values

Table 1Magnet design parameters and configurations for the arrays from A to F. The

undulator period and remanent field of the permanent magnet are assumed to be

100 mm and 1.3 T, respectively.

Array B,E A,C,D,F

Configuration Fig. 1(a) Fig. 2(c)

Inclined angle (deg.) 27 23

Magnet width (mm) 27 100

Magnet height (mm) 100 100

T. Tanaka, H. Kitamura / Nuclear Instruments and Methods in Physics Research A 659 (2011) 537–542540

corresponding to the fundamental and double period field com-ponents. Note that we have defined the fundamental energy o1

and the harmonic number n=2� k to be consistent with thosedefined in Ref. [3]. The horizontal and vertical complex ampli-tudes at the k-th harmonic are then calculated as

Fkx ¼ real const:�

Z p

�p½K1 cos a cosð2ZÞþK2 sinða=2Þ cosZ�e2ikcðZÞ

and

Fky ¼ real const:�

Z p

�p½K1 sin a cosð2ZÞþK2 sinða=2�p=4Þ cos Z�e2ikcðZÞ

with

cðZÞ ¼ Zþ 1

1þK21=2þK2

2 PðaÞ=2

�K2

1

8sinð4ZÞþ

K22 PðaÞ

4sinð2ZÞþK1K2Q ðaÞ sinð3ZÞ

3þsin Z

� �� �

Q ðaÞ ¼ cos a sinða=2Þþsin a sinða=2�p=4Þ:

Because the function cðZÞ is an odd function of Z, we find thatFkx and Fky are real numbers. This means that the radiation islinearly polarized at any photon energy as long as the phaseparameters satisfy the condition (6). The polarization anglemeasured from the horizontal axis is given as w¼ tan�1ðFky=FkxÞ

and obtained by the numerical integration of the above formulae.The results at the 1=2-th harmonic (k¼ 1=2) and fundamentalenergy (k¼1) are plotted in Fig. 5 as functions of a together withthe difference between the two. The K values are assumed to beK1 ¼ K2 ¼ 4, but it has been verified that the results in Fig. 5 are soinsensitive to the K values as to be considered to be universal, aslong as the conditions B1x � B1y and B2x � B2y are satisfied.

We find that the polarization angle at the fundamental energyvaries from �901 to 01, while that at the 0.5-th harmonic variesfrom 901 to 01, and the difference between them keeps almostconstant at 901. This is the basic property of radiation from theFigure-8 undulator. In terms of the availability of polarizationangle, a single quadrant (in this case from 01 to �901) is availablewith the fundamental radiation alone. If two quadrants (i.e., from

Fig. 5. Linear polarization angle w as a function of a, the parameter directly related

to the translational shift of magnet array (see text for details). Dashed and solid

lines show the results at the 0.5-th harmonic and fundamental photon energy,

respectively, while the dotted line shows the difference between the two.

�901 to 901) are required, we have to utilize the 0.5-th harmonicas well as the fundamental, but at the expense of the photon flux.It should be noted that the available angular region with thefundamental radiation depends on the polarity of each magnetand it cannot be changed after assembling the magnet blocks.

4. Examples

Now let us calculate the magnetic and optical performances ofthe Helical-8 undulator. In general, the magnet configuration anddimension should be optimized to meet the requirement from theSR users. As an example, the magnetic period lu is assumed to be100 mm and the magnet design is optimized so that the horizontal

Fig. 6. Magnetic field distributions in the four operation modes: (a) helical,

(b) HF8, (c) 451-F8 and (d) VF8 modes. The undulator parameters are referred

from Table 1 and its caption.

T. Tanaka, H. Kitamura / Nuclear Instruments and Methods in Physics Research A 659 (2011) 537–542 541

and vertical K values are comparable with each other in anyoperation mode. This is the condition not only to realize the helicalorbit in the helical mode but also to validate the discussions on theutilization of linear polarization in the previous section. The opti-mization has been carried out based on the magnetic analysis withRADIA [12], the results of which are summarized in Table 1. Notethat the remanent field of magnet material is 1.3 T and a 1-mmspace is inserted between the magnet arrays for the horizontal (B, E)and vertical (A, C, D, F) fields to allow the longitudinal motion.

Fig. 6(a)–(d) shows the magnetic field distributions over twoperiods (7100 mm) in the helical, HF8, 451-F8 and VF8 modes,respectively, where the 451-F8 mode means that the parameter ais set to 451 to generate linearly polarized radiation with thepolarization angle of �451 (refer to the discussions in the

Fig. 7. Projected electron beam orbits calculated with the field distributions

shown in Fig. 6(a)–(d).

Fig. 8. Flux and power densities in the conventional planar and helical undulators,

and in the various operation modes of the Helical-8 undulator. All the undulators

are assumed to be installed in SPring-8 with the beam energy of 8 GeV. The

undulator parameters are referred from Table 2.

Table 2Undulator parameters used to compare the optical

performances between the conventional and

Helical-8 undulators.

Period length 100 mm

Number of periods 44

Magnet length 4.4 m

Fundamental energy 500 eV

previous section). The electron orbits projected on the transverseplane obtained by integrating the field distributions twice areplotted in Fig. 7(a)–(d), where we find the typical orbits specific tothe helical and Figure-8 undulators.

Fig. 8 shows the comparison between the conventional andHelical-8 undulators in terms of the angular flux density at thefundamental energy and the angular power density, which havebeen computed with SPECTRA [13]. We assumed that the undu-lators have the parameters summarized in Table 2 and areinstalled in SPring-8.

The angular flux density, which directly relates to the photonflux actually available to the SR users, is of the order of1017 photon/s/mrad2/0.1%b.w., and is more or less the same inany case. On the other hand, the power density, which results in theheat load on optical elements, depends largely on the type of thedevice. In the case of the conventional planar undulator, it reachesnearly 100 kW/mrad2 and thus is larger than others approximatelyby two orders of magnitude. What should be stressed more is thatthe radiation power from the conventional planar undulator in thisexample mainly comes from the spectral region higher than 10 keVas opposed to the other three cases, as shown in Fig. 9(a)–(d). Thephotons in such a high energy region are not actually used for SRexperiments but cause the heat load on optical elements, whichshould be avoided as much as possible. This is the reason why theconventional planar undulator (and thus the APPLE-type device) isnot recommended in the SR facility using a high energy electronbeam when high K values are required. It is now clear that this kindof heat load problem can be avoided by adopting the Helical-8undulator and operating it in a proper operation mode.

5. Discussion

We have proposed a new undulator scheme, the Helical-8undulator, to overcome the difficulty in providing various

Fig. 9. Comparison of spectra between the (a) conventional planar undulator,

(b) conventional helical undulator, (c) Helical-8 undulator (helical mode) and

(d) Helical-8 undulator (HF8 mode). All the undulators are assumed to be installed

in SPring-8 with the beam energy of 8 GeV. The undulator parameters are referred

from Table 2.

T. Tanaka, H. Kitamura / Nuclear Instruments and Methods in Physics Research A 659 (2011) 537–542542

polarization states with ordinary undulators. The idea is based onthe CPU scheme to generate the magnetic field composed of twodifferent periods. Although its basic principle and fundamentalperformances have been successfully demonstrated in this paper,we have several points to be discussed for more practical andefficient application.

First, we have to explore further the possible magnet config-uration for the CPU fields besides those in Figs. 1(a) and 2(a)–(d)to improve not only the magnetic but also the optical perfor-mances. For example, the calculation results presented in theprevious section show that both the flux and power densitiessomewhat depend on the polarization angle, or the parameter a,which is not the case for the APPLE-type device, and may not be

desirable in some cases. This is attributable to the fact that themagnetic field distributions in the Figure-8 operation mode,which are shown in Fig. 6(c) and (d), are more or less far fromthe sinusoidal ones. It is thus important to look for the magnetconfiguration to generate the CPU field as sinusoidal as possible inany mode of operation.

Second, we have to look for the best combination of the phaseparameters ðf1x,f2x,f1y,f2yÞ to utilize the inclined linear polar-ization. In Section 3, we have investigated its availability underthe condition given in Eq. (6). In principle, however, we have aninfinite number of combinations, and not all of them are exam-ined. More detailed analysis is therefore important to take fulladvantage of the flexibility of the Helical-8 undulator.

Finally, we have to be very careful about the quality of thepermanent magnet with an inclined easy axis. Using the state-of-the-art manufacturing technology of permanent magnets, it is nottoo challenging to produce magnets with an arbitrary inclinedangle [14]. It should be note, however, that the deviation ofmagnetization angle from the mean value can be somewhat largerthan the orthogonally magnetized magnets. In such a case, thefield errors, such as the first and second field integrals andintegrated multipoles, can be too large to be corrected using anordinary undulator field correction scheme such as shimming andflipping and thus we need to implement a special scheme tocorrect them. For example, the so-called magic finger [15] ordynamic multipole shimming technique [16] can be applied toreduce the errors related to the inclined magnetization.

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