1. Introduction

Photon brightness is defined as the photon number within a fractional wavelength band (Δλ/λ), which emit into a unit angle from a unit area during a second. Roughly speaking, the total product of this emitting angle and area is the so-called emittance of a light source. High bright synchrotron radiation is useful for the scientific experiments. Continuous efforts have been made to increase the photon brightness with the method of reducing the electron beam emittance in the synchrotron light sources. The electron beam emittance is a statistical average of all the electrons’ action in the phase space of position and momentum. An equilibrium process between radiation damping and quantum excitation determines its value. The electrons radiate photons in their traveling direction. And then, with the decreases in both the transverse and longitudinal momentum, only the longitudinal one is compensated in the RF cavity. This consistently decreases the beam transverse emittance. At the same time, the electrons discontinuously radiate photons with different energy at different locations in the dipole, which disturbs the horizontal oscillations of the electrons due to the inevitable horizontal dispersion. The average effect of this disturbance always makes the beam emittance increase. At an eventual equilibrium, the beam emittance is given by the beam energy and the magnetic lattice functions [1]:

where C_q = 3.8319×10⁻¹³ m, γ is the relative energy, J_x is the horizontal damping partition number, and ρ(s) is the bending radius. The H function is calculated by Eq.(2) with the Courant-Snyder parameters (α, β, γ) [2], the dispersion and its divergence (η, η’).

An ideal lattice has no vertical dispersion, so the vertical oscillation is damped almost completely. The vertical emittance [3] in a real lattice is from betatron coupling and spurious vertical dispersion due to all kinds of imperfections, rather than the lattice design. In this paper, we discuss the horizontal emittance only.

Emittance reduction will be done by radiating the horizontal momentum more quickly or/and weakening the quantum excitation, without considering the beam energy. The methods having been proposed or implemented in the synchrotron light source community can be summarized as follows. Damping wiggler [4], installed in the dispersion-free straight section, radiates more beam energy (including the horizontal momentum), with negligible increase of the quantum excitation. Multi-Bend Achromatic (MBA) lattice [5, 6], i. e., more dipole in one cell, could apply tighter focusing to get very small average H function in all the dipoles, and thus get a very weak quantum excitation. This kind of lattice has been well known to dramatically reduce the beam emittance of the synchrotron light sources in the past two decades. It became much more practical in recent years [7], thanks to the great developments of the high gradient magnets, the small-aperture vacuum chamber and the precise alignment of magnets. Dipole with longitudinally varied field [8] reaches an emittance reduction condition of much/little frequent radiation while weak/strong quantum excitation. Anti-bend cell [9] allows independently adjusting the dispersion to reach a weak quantum excitation in the main dipoles.

Besides, a combined function dipole can adjust the equilibrium beam emittance. The horizontal betatron oscillation is changed by the energy radiation and horizontal dispersion. In this kind of dipole, the radiated energy is changed by the varying encountered bending field and the length change of the particle trajectory. This adjustment is made by the horizontal damping partition number J_x, which is calculated by Ref. [1]

where η_x is the horizontal dispersion, ρ is the bending radius, and k is the magnetic gradient. The D function has two parts. The first one refers to the path length change, which is very small in the medium- or large-size storage ring. The second one refers to changes in the bending field. A dipole of negative gradient reduces the beam emittance, as J_x is larger than 1.

The basic mechanism of emittance reduction in a Robinson wiggler is the same as in the combined function dipole. Several dipoles generate a chicane of the beam trajectory in the positive dispersion region. In order to get negative D function and thus increase J_x, there is inverse gradient with respect to the bending field in each dipole, shown as in Figs.1(a) and 1(b). The middle two dipoles can be replaced with a longer one (Fig.1c). Another option of the Robinson wiggler is multi-periodic (Fig.1d). K. W. Robinson introduced this kind of wiggler in 1958 [10]. It was applied on the Cambridge Electron Accelerator [11] and the PS ring at CERN [12], and the horizontal emittance reduction was indeed observed. In recent years, synchrotron light sources of SOLEIL, MLS, HLS etc. [13-15] have proposed to install the single-periodic or the multi-periodic Robinson wiggler in the straight section to get an emittance reduction by 30%–50%.

Fig. 1.Full-screen View

Drafts of the Robinson wiggler.

In these proposals, the Robinson wiggler usually occupies a straight section, which is supposed to be installed by radiator. A local high-dispersion bump in this straight section is also desired, so as to make the modest bending field and gradient work well. There are two facts worth mentioning in the arc section. The first is the absolute need of quadrupoles to tune the beam optics, even a quadrupole array of defocusing-focusing-focusing-defocusing, which is the same array in the Robinson wiggler. The second is the intrinsically high dispersion in the arc section. This is necessary in the Robinson wiggler. So the arc section has a merit to be re-constructed to a single-periodic Robinson wiggler, by replaying the quadrupoles with short combined function dipoles. The storage ring of Shanghai Synchrotron Radiation Facility (SSRF) is taken as a test ring, to answer whether and how this scheme plays a role in emittance reduction. The results are presented in Sect. 2. In Sect. 3, nonlinearity of the ring is re-optimized, against the degraded longitudinal acceptance caused by decreased momentum compaction factor due to the inconsistent dispersion in this kind of Robinson wiggler. Conclusions are given in Sect. 4.

2. Application in the SSRF storage ring

SSRF, a third generation intermediate-energy light sources, has been serving the users since May 2009 [16, 17]. Its storage ring consists of 20 Double Bend Achromatic (DBA) cells, forming four super-periods. At present, a project is being implemented to install more new beamlines and insertion devices, which calls for upgrading the storage ring lattice with super bends [18-20]. However, this study is based on the current lattice, rather than the upgraded one. A DBA cell in the SSRF storage ring is shown schematically in Fig.2(a). It has two dipoles (yellow), ten quadrupoles (red for focusing, blue for defocusing), and seven sextupoles (magenta for focusing, green for defocusing). Fig.2(b) shows the structure after replacing four quadrupoles in the arc section with four short combined function dipoles, whose positions and lengths are the same as the replaced quadrupoles. This kind of DBA cell with Robinson wiggler in the arc section is abbreviated to RW-DBA in this paper.

Fig. 2.Full-screen View

One DBA (a) and RW-DBA (b) cell in the SSRF storage ring.

It is up to the bending field, gradient and dispersion to determine how much the Robinson wiggler reduces the beam emittance. D function is usually reduced to −1, and J_x is about 2. There are definite signs, while no any definite value for these parameters. Five independent gradient varibles are needed to match β_x, β_y and η_x at the straight section center and the horizontal and vertical phase advance in one DBA cell, as done as in the nominal lattice. Since it is so, the gradients of the Robinson wiggler in the RW-DBA cell are classed into two varibles (i.e. one is in the first and forth short dipole, another is in the second and third short dipole), and used to match the beam optics, in anditional to the three quadrupole gradient varibles. The gradient classifications are shown as k₁–k₅ in Fig.2(b). To obtain low emittance lattice, k₄ is almost negative, and k₅ is positive. So, the bending fields in the first and forth short dipole (B₁ in Fig.2b) are positive, and the ones in the other two short dipoles (B₂ in Fig.2b) are negative. The bending field and dispersion are scanned to obtain a good solution, while other optical functions are maintained as the nominal ones. The Robinson wiggler generates a chicane of the particle trajectory, and it means that there are equal absolute values of the kick angles in the four short dipoles. The first and forth short dipole are 0.276 m in length, while the other two short dipoles are 0.335 m in length, so B₁= − (0.335/0.276) B₂. There is only one independent varible for the bending fields, and B₂ will be presented in this section. Because of the inconsistent dispersion in the Robinson wiggler, the dispersion in the straight section, which is easily handled in the optics matching process, substitutes as the scanned parameter. In general, the lower the dispersion in the straight section, the higher the maximum dispersion in the arc section. The storage ring has a smaller H function in the dipole, hence a lower emittance, by allowing non-zero dispersion in the straight section, than the achromatic one. In this case, the energy spread will contribute a part of the beam size and the beam divergence.

The effective emittance, taking the energy spread into account, is a more practical optimal-objective than the natural emittance, to increase the photon brightness. It is expressed as [21]:

where δ_E is the energy spread, which can be calculated by

J_s is the longitudinal damping partition number.

The scanning process is carried out with the bending field from 0 to 1.4 T and the dispersion from 0 to 0.14 m. Other beam optical functions, such as β functions and working points, are kept as the same as the nominal ones. Then, the resulted beam parameters are recorded (Fig. 3), including the effective emittance in the straight section, the natural emittance, the horizontal damping partition number, the energy spread, and the momentum compaction factor. In the dispersion-free case, the beam emittance of the nominal lattice of SSRF is 11 nm·rad. It is reduced to 4.3 nm·rad in the RW-DBA lattice, with the bending field of 0.7 T and J_x of about 2.5. The gradients of the Robinson wiggler are −0.65 and 1.20 m⁻² (7.58 and 14.0 T/m). At bending field of 0.4 T, the J_x is about 2, the beam emittance becomes about 5 nm·rad, with the gradients of −0.85 and 1.26 m⁻² (9.92 and 14.7 T/m). In the non-zero dispersion case, a minimum effective emittance of 3.2 nm·rad is reached, at bending field of 0.5 T, the dispersion is 0.07 m, the gradients are −0.94 and 1.29 m⁻² (11.0 and 15.1 T/m), and the resulted J_x is about 1.9. The effective emittance is significantly reduced from the nominal one of 5.2 nm·rad. The minimum natural emittance of 1.6 nm·rad is reached, at bending field of 0.5 T, the dispersion is 0.12 m, the gradients are −1.03 and 1.32 m⁻² (12.0 and 15.4 T/m), and the J_x is 1.7.

Fig. 3.Full-screen View

Contour maps of the beam parameters of effective emittance (a), natural emittance (b), horizontal damping partition number (c), energy spread (d), and momentum compaction factor (e), as functions of the bending field in the Robinson wiggler and the dispersion in the straight section.

Fig.3(e) plots the momentum compaction factor as a function of the bending field and the dispersion. At bending field of 0.5 T and dispersion of 0.07 m, the momentum compaction factor is about 0.0001, reduced by a factor of 4 from the nominal one, and it even reaches about zero at 0.5 T and 0.12 m. This tendency to isochronous prevents from obtaining the minimum beam emittance. Rearrangement of the magnets in the arc section will reduce the difference of dispersion in the Robinson wiggler. It can improve the lattice performance from this bad situation. Nevertheless, replacement of several DBA cells with the RW-DBA cells can also reduce the beam emittance of SSRF.

Eight DBA cells (i. e. the second and forth cells in each super-period) are replaced with the RW-DBA cells in the SSRF storage ring. The bending field in the Robinson wiggler is 0.5 T, and the dispersion in the straight section is 0.1 m. The gradients in the Robinson wiggler are −1.01 and 1.31 m⁻² (11.7 and 15.3 T/m), respectively. Fig. 4 plots the linear optics of this upgraded lattice, and the beam parameters are summerized in Table 1. The beam optics is very close to the nominal one. Variations of the beam parameters are mainly contributed by the Robinson wigglers. J_x increases to 1.312, and the natural emittance decreases by about 31%. The energy spread increases by about 8.2%, conflicting with the natural emittance reduction. It makes the effective emittance in the straight section reduced by about 21% (from 5.2 nm.rad to 4.1 nm.rad). Fig. 5 shows the effective emittance in a super-period of the SSRF storage ring, which is reduced all along the ring. The energy loss per turn increases by 3%, which is insignificant to the RF cavity. The synchrotron tune and the bunch length decrease a little because of the lower momentum compaction factor.

Fig. 4.Full-screen View

Linear optics in one super-period of SSRF with Robinson wiggler in the arc section. The dotted lines are the nominal ones.

Fig. 5.Full-screen View

The effective emittance in one super-period of the SSRF storage ring.

3. Nonlinear re-optimization

The momentum compaction factor is defined as the path length change rate of the particle to the energy deviation, as

where C is the path length, δ is the energy deviation, α_1,2,3 are different orders, and O is the ignored higher order term. The momentum compaction factor, mentioned in Sections 1 and 2, is usually the first order one (α₁), which is calculated by

The high positive dispersion and the negative bending radius in the two inner dipoles of the Robinson wiggler make α₁ always decrease. Then, effects of the high order terms of the momentum compaction factor become significant. It modulates RF-bucket to be α-bucket [22, 23], and thus degrades the phase acceptance of the storage ring (Fig. 6).

Fig. 6.Full-screen View

The longitudinal phase space of the SSRF storage ring with eight RW-DBA cells. The blue lines are the acceptances.

So, nonlinear re-optimization of the lattice is necessary, so as to obtain good transverse and longitudinal beam dynamics simultaneously. There are 140 sextupoles with eight power supplies (eight families) in the SSRF storage ring. Two sextupole families, installed in the arc sections, are applied to correct the linear chromaticities, which are always kept to be 1 in both transverse planes in the optimization process. Six harmonic sextupole families are used to cancel the high order aberrations in order to get sufficient dynamic aperture and energy acceptance. The maximum gradients of all the sextupoles are 460 T/m². The nonlinear optimization is under these constraints. Since the electron beam is horizontally injected in the SSRF storage ring, the horizontal dynamic aperture at the injection point is one of the optimal-objectives. Another optimal-objective is width of the longitudinal phase space, because it is necessary to re-shape the bucket of the storage ring with RW-DBA cells. In general, it is difficult to reach the best values of the two objectives simultaneously. Multi-objective genetic algorithm [24] is applied to find the Pareto optimal front, which is defined as an assemblage of all the solutions that are non-dominated with respect to each other. In the special case here, the Pareto optimal front records the horizontal dynamic aperture and width of the longitudinal phase space, resulted from a sextupole solution. It takes dozens of iterations.

A good solution is found. Accelerator Toolbox [25-26] is used for particle tracking. The results are shown in Figs. 6, 7 and 8. Fig.6 plots the longitudinal phase space, and the re-optimized results. The bucket shape is modified to close to the RF-bucket, and the phase acceptance increases. Fig. 7 plots the energy acceptance along the ring, which is determined by the RF voltage, the nonlinear dynamics, the dispersion, and the size of vacuum chamber (half height 16 mm; half width 32 mm). Tracking turn is 1000. The minimum energy acceptance is ±2.5%, while the mean values are about −3.5% and 2.7%. It even reaches about −6% somewhere, and this asymmetry is related to the effect of the high order α. The dynamic aperture and the frequency maps [27], resulted from the particle tracking of 1000 turns, are shown in Fig. 8. The color map indicates the diffusion rate of the oscillation frequency. The blue points show slow diffusion, and the red points show fast diffusion that is made by the nonlinear resonances. The horizontal dynamic aperture reaches ±20 mm, which is sufficient to the off-axis injection. A nonlinear resonance disturbs the horizontal dynamic aperture. The particle loss is absent, while it will be sensitive to special multiple magnetic error. A little shift of the tune will play a good role in avoiding this resonance.

Fig. 7.Full-screen View

The energy acceptance along the ring of SSRF with eight RW-DBA cells.

Fig. 8.Full-screen View

The dynamic aperture and the frequency maps at the injection point of the SSRF storage ring with eight RW-DBA cells.

4. Conclusion

The single-periodic Robinson wiggler is re-constructed in the arc section. It reduces the beam emittance, without occupying any straight section. The SSRF storage ring with eight RW-DBA cells obtains an effective emittance of 4.1 nm·rad and a natural emittance of 2.65 nm·rad, which are reduced by 21% and 31% from the nominal ones, respectively. The bending field and gradients in the Robinson wiggler are modest (0.5T, 11.7 and 15.3 T/m) due to the intrinsic high dispersion in the arc section. Nonlinearity of the lattice is re-optimized with the current sextupoles. Sufficient dynamic aperture and energy acceptance are obtained simultaneously, in spite of lower momentum compaction factor. So, there is no insurmountable problem in the beam dynamics or hardware for operating SSRF with this kind of lattice. This scheme is also useful in TBA and MBA lattice with high dispersion bump in the arc section.