Generalized Beta Functions in a General Coupled Lattice

Inspired by Chao’s solution by linear matrix (SLIM) formalism [21], we introduce the definition of the generalized beta functions in a 3D general coupled storage ring lattice as(1)where ^* means the complex conjugate, the sub or superscript k is the eigenmode index, Re() means the real component of a complex number or matrix, E_ki is the i-th component of the vector E_k which is the eigenvector of the 6×6 symplectic one-turn map M of storage ring with the eigenvalue (2)satisfying the following normalization condition(3)and(4)where i here means the imaginary unit, ^† means complex conjugate transpose, and(5)This normalization condition can be preserved around the ring owing to the symplecticity of the transfer matrix. Since the one-turn map is a real symplectic matrix, for a stable motion, we have(6)where νk is the eigen tune.

Similar to the real generalized beta function, here we define the imaginary generalized beta functions as(7)where Im() denotes the imaginary component of a complex number or matrix. Note that is a real value, similar to . The names ‘real’ and ‘imaginary’ here reflect that their definitions take the real and imaginary parts of , respectively. We use the symbol ^ on the top of these functions to indicate ‘imaginary’. Further we can define the real and imaginary generalized 6×6 Twiss matrices of a storage ring lattice corresponding to each eigen mode as(8)From their definitions we have(9)Therefore, we know that T_k is a real symmetric matrix, whereas is a real anti-symmetric matrix. Further, we can prove that(10)The generalized Twiss matrices at different places around the ring are related to each other according to(11)where is the symplectic transfer matrix of the state vector X from s₁ to s₂(12)and we have(13)where C₀ is the ring circumference.

With the help of the generalized Twiss matrices and eigen tunes, the one-turn map M can be parametrized as(14)where(15)and is the phase advance of the corresponding mode in one turn. For Mⁿ we only need to replace the above Φ_k with nΦ_k. We can write the matrix terms of the one-turn map more explicitly as(16)Applying , , we have(17)Using the generalized Twiss matrices, the actions or generalized Courant-Snyder invariants of a particle are defined according to(18)It is easy to prove that Jk are invariants of a particle when it travels around the ring, from Eqs. (11), (12), and the symplecticity of the transfer matrix(19)For a beam with N_p particles, the three beam invariants can then be defined according to(20)where Jk,i denotes the k-th mode invariant of the i-th particle. These invariants are the generalized root-mean-square (RMS) emittances of the beam in the ring.

The beam emittances defined above are based on the eigenmode motion of the particles in the storage ring. Another definition of emittance is based directly on the second moments matrix of a particle beam(21)where ⟨⟩ denotes the particle ensemble average. The beam second moment matrix at different places are related according to(22)From the symplecticity of R, we can prove that the eigenvalues of ∑(-iS) remain unchanged with beam transport in a linear symplectic lattice. The beam eigen emittances can thus be defined as the positive eigenvalues of ∑(-iS).

When the particle beam matches the storage ring lattice, which means the beam distribution at a given location repeats itself turn by turn, we have(23)It can be proven that for a matched beam the RMS emittances defined in Eq. (20) are the eigenvalues of ∑(-iS) with the eigenvector E_k, i.e.,(24)where we have used and(25)Using the generalized Twiss and second moments matrices, the eigen emittances for a matched beam can be calculated as(26)where Tr() denotes the trace of the matrix.

To ensure that the eigenvectors E_k are uniquely defined all around the ring once they are determined at a given location, we will let(27)Following this definition, we have(28)Using the generalized beta function, we can write the eigenvector component in an amplitude-phase form(29)And according to the definition we have(30)Using the generalized Courant-Snyder invariants and beta functions, we can express the phase space coordinate of a particle at s as(31)with determined by the initial condition of the particle state. The phase term at different locations are related according to(32)In particular, after n revolutions in the ring, we have(33)

Perturbations

After considering the parametrization of a general coupled lattice and the prescribed particle motion in it, let us now add perturbations, that from the lattice and also that from the beam. Assume there is a perturbation K to the one-turn map M, i.e.,(34)where I is the identity matrix, the subscripts ‘per’ and ‘unp’ denote ‘perturbed’ and ‘unperturbed’, respectively. When the perturbation is small, from cannonical perturbation theory, the tune shift of the k-th eigen mode can be calculated as(35)This tune shift formula can be used to calculate the real and imaginary tune shifts due to symplectic (e.g., lattice error) and non-symplectic (e.g., radiation damping) perturbations.

For example, given the radiation damping matrix D around the ring, the damping rate of each eigen mode per turn is(36)where denotes integration around the ring. Note that the damping rates here are those for the corresponding eigenvectors. The damping rates for particle actions or beam eigen-emittances are a factor of two larger, because they are quadratic with respect to the phase space coordinates.

Apart from the radiation damping, there are also various beam diffusion effects in the ring, such as quantum excitation and intra-beam scattering. Using Eq. (26), once we know the diffusion matrix N around the ring, the emittance growth per turn due to diffusion is(37)The equilibrium eigen emittance between a balance of diffusion and damping can be calculated as(38)It can be proven that the equilibrium beam distribution in the 6D phase space given by such a balance in a linear lattice is Gaussian [22], with . After getting the equilibrium eigen emittances, the second moments of beam can be written as(39)or in matrix form as(40)

Application to Electron Storage Rings

In an electron storage ring, intrinsic diffusion and damping are both caused by the emission of photons, namely, quantum excitation and radiation damping. For quantum excitation, we have all the other components of the diffusion matrix N zero except that(41)where is the number of photons emitted per unit time, u is the photon energy, , where r_e being the classical electron radius, ħ being the reduced Planck’s constant, m_e being the electron mass, γ is the Lorentz factor, c is the speed of light in free space, ρ is the bending radius. We take the convention that the sign of ρ is positive when the bending is inward and negative when the bending is outward. in the above formula is a result of Campbell’s theorem [23, 24].

For the damping effect, we have two sources: dipole magnets and RF cavity. For a horizontal dipole, we have all the other components of damping matrix D zero except that(42)where , is the transverse field gradient index. The physical origin of D₆₆ is the fact that a higher energy particle tends to radiate more photon energy in a given magnetic field, while D₆₁ is due to the fact that a transverse displacement of the particle will affect its path length in the dipole and when there is a transverse field gradient, the magnetic field strength observed, thus the radiation energy loss. For an RF cavity, we have all the other damping matrix terms of D zero except that(43)where e is the elementary charge assumed positive in this study, V_RF and ϕ_RF are the RF voltage and phase, respectively, δ(s) is Dirac’s delta function. Here, we assume that the RF cavity is zero length one. The physical origin of these damping terms is that the momentum boost of a particle in the RF cavity is along the longitudinal direction, whereas the transverse momenta of the particle are unchanged. Therefore, a damping effect was observed on the horizontal and vertical angles of the particles. We remind the readers that if we use the horizontal and vertical particle momentum px and py instead of x’ and y’, as the phase space coordinates, then the damping occurs only at bending magnets because the RF acceleration does not affect .

In an electron storage ring, RF acceleration compensates for the radiation energy loss of electrons. If there are N cavities in the ring, we have(44)where(45)is the radiation energy loss per particle per turn, with(46)If the ring consists of iso-bending magnets, then . From Eq. (42), we have(47)Similarly, based on Eqs. (43) and (44), we have(48)Combing with Eqs. (10) and (36), it is easy to show that for radiation damping, we have(49)This is the well-known Robinson’s sum rule [25].

The above formulation applies to a 3D general coupled lattice. For a ring without x-y coupling and when the RF cavity is placed at dispersion-free location, we can express the normalized eigenvectors using the classical Courant-Snyder functions [26] α, β, γ and dispersion D and dispersion angle as(50)where the subscripts x, y, z correspond to the horizontal, vertical, and longitudi, and are the phase factors of the eigenvectors. There is flexibility in choosing these phase factors as they do not affect the calculation of our defined Twiss matrices and physical quantities. However, note that once they are set at one location, their values around the ring are determined according to Eq. (27). In this case, the real and imaginary generalized Twiss matrices can be obtained explicitly(51)Correspondingly, the generalized Courant-Snyder invariants are given by(52)The equilibrium emittances determined by the balance of quantum excitation and radiation damping in an electron storage in this case reduce to the classical Sands radiation integrals formalism found in textbooks [27](53)with m for electrons, fm is the reduced Compton wavelength of electron, and the radiation integrals are given by(54)where(55)are the horizontal and vertical dispersion invariants, and is the longitudinal beta function [8]. The damping rate of three eigen modes are given by(56)where Jx,y,z are the damping partition numbers(57)The Robinson’s sum rule corresponds then to(58)The horizontal, vertical, longitudinal radiation damping times are(59)where T₀ is the particle revolution period in the ring.

After obtaining the equilibrium emittance, the beam second moments can be obtained by substituting the generalized Twiss matrices from Eq. (51) into Eq. (39). For example,(60)We remind the readers that the generalized Courant-Snyder formalism presented in this section has been briefly reported in Ref. [28].

Theoretical Minimum Horizontal Emittance

In Eq. (53), we can see that and βz, i.e., and as defined by us, at the bending magnets are of vital importance in determining the horizontal and longitudinal emittance, respectively. Therefore, it is necessary to understand how they evolve inside a bending magnet. The transfer matrix of the particle state vector for a sector bending magnet is(61)with ρ and α being the bending radius and angle of the bending magnet, respectively. In a planar uncoupled ring, the normalized eigenvectors of the one-turn map are expressed as Eq. (50). Assuming the Courant-Snyder functions, dispersion and dispersion angle at the dipole center where we set to be α=0 are given by . From Eqs. (1), (61) and (50) we have the evolution of in the dipole(62)Similarly, the evolution of βz in the dipole is given by(63)For simplicity, we have neglected the contribution of to R₅₆ of dipole in the above calculation of βz, since we are interested in the relativistic cases. However, we remind the readers that in a quasi-isochronous ring, the contribution of to the ring R₅₆ or phase slippage may not be negligible.

With the evolution of and βz known, we can now derive the theoretical minimum emittances. For simplicity, we assume the ring consists of iso-bending-magnets, with the bending angle induced by each bending magnet being θ, and the optical functions are identical in each bending magnets. Then from Eq. (53), we have(64)with(65)fx can be interpreted as the average value of in the dipoles. The lengthy explicit expression of is omitted here. It can be obtained straightforwardly by inserting Eq. (62) in Eq. (65). The mathematical problem we are trying to solve is to minimize fx by adjusting α_x0, β_x0, D_x0, . From Eq. (65) we have(66)We notice that the requirement of and leads to α_x0=0 and . Under the above conditions, then(67)The requirement of leads to(68)Under the above conditions, then(69)The requirement of leads to(70)Summarizing, the extreme value of fx is realized when(71)which means(72)and(73)One can verify that this is the minimum value of fx. Note that . Under these conditions we get the minimum horizontal emittance(74)These results are consistent with the classical results of Teng [29]. For practical use, and considering nominally , the above scaling can be written as(75)For example, if E₀=6 GeV and rad, we have pm.

Theoretical Minimum Longitudinal Emittance

We now analyze the theoretical minimum longitudinal emittance. Similar to the horizontal direction, we have(76)with(77)which can be interpreted as the average value of βz in dipoles. Then(78)where the lengthy explicit expression of is omitted here. Similar to the analysis of transverse minimum emittance, we notice again that the requirement of and leads to and . Under the above conditions, then(79)The requirement of leads to(80)Under the above conditions, then(81)The requirement of leads to(82)Summarizing, the extreme value of fz is realized when(83)and(84)One can verify that this is the minimum value of fz. Note that . Under these conditions we get the minimum longitudinal emittance(85)The above result was also reported in Ref. [7]. For practical use, and considering nominally Jz≈2, the above scaling can be written as(86)For example, if E₀=600 MeV and rad, we have pm.

Here, we remind the readers that in reality, it may not be easy to reach the optimal conditions Eq. (83) for all dipoles in a ring. This is based on the observation that when we realize Eq. (83) at the dipole center, the dipole as a whole will have a nonzero R₅₆ or more accurately, a nonzero phase slippage. Therefore, to make the longitudinal optics identical in different dipoles, there should be RF or laser modulator kicks between neighboring dipoles; otherwise, the required drift space between each pair of dipoles will be very long to compensate this nonzero phase slippage [7]. It may not be easy to apply too many RF cavities or laser modulators in a ring to manipulate the longitudinal optics, while in the transverse dimension it is straightforward to implement many quadrupoles to manipulate the transverse optics. Instead, we may choose a more practical strategy to realize a small longitudinal emittance, which is to let each half of the bending magnet be isochronous, and the longitudinal optics for each dipole can then be identical. This can be realized by requiring(87)In this case, we still have . Under such conditions, the minimum longitudinal emittance is [17](88)The emittance given in Eq. (88) is larger than the real theoretical minimum Eq. (86), but offers a more practical reference for the TMEz. For practical use, and considering nominally Jz≈2, the above scaling can be written as(89)Based on this emittance and β_z0, we can determine the bunch length at the dipole center contributed by the longitudinal emittance. We remind the readers that the bunch length can, in principle, be even smaller than this value, by pushing β_z0 to an even smaller value, although the longitudinal emittance will actually grow then because βz will diverge faster when moving away from the dipole center. If the ring works in a longitudinal weak focusing regime to be introduced in next section, there is actually a lower limit of bunch length in this process, which is a factor of smaller than the bunch length given by conditions of Eqs. (89) and (87). The energy spread diverges when the bunch length is pushed to this limit [30]. Putting in the numbers, we have this lower limit of bunch length in a longitudinal weak focusing ring [17](90)For example, if E₀=600 MeV, rad and ρ=1.5 m which corresponds to a bending field strength B₀=1.33 T, then nm.

Application of Transverse Gradient Bend

The previous analysis assumed that the transverse gradient of the bending field was zero. We now consider the application of transverse gradient bending (TGB) magnets to lower the horizontal and longitudinal emittances. For simplicity, we considered the case of a constant gradient. The transfer matrix of a sector bending magnet with a constant transverse gradient is(91)

Application of Longitudinal Gradient Bend

We can also apply longitudinal gradient bends (LGBs) to lower the transverse and longitudinal emittances. For simplicity, we studied the case of an LGB consisting of several sub-dipoles, each with a constant bending radius. Furthermore, we assume that each sub-dipole is a sector dipole. The analysis for the case of rectangular dipoles is similar, as long as the impact of the edge angles on the transfer matrix and damping partition is properly handled. We now investigate the case of sector sub-dipoles. For example, we may choose to let the LGB has a symmetric structure(99)The total bending angle of such a LGB is , and the total length is . Note that . We used this structure as an example for the analysis. However, the proposed method also applies to a more general setup.

3.4.1

Horizontal Emittance

Now, we calculate the theoretical minimum horizontal emittance by invoking LGBs with each structure given in Eq. (99). Still we assume all the LGB setup in the ring and optical functions in each LGB are identical, then(100)with(101)Note that the dimension of I_5x here is equivalent to given in the previous sections. The mathematical problem is then to minimize I_5x, by adjusting α_x0, β_x0, D_x0, . The calculation method of evolution in an LGB is the same as that given in Eq. (62), but note that here for different parts (sub-dipole) of the LGBs, we should apply the transfer matrix from the midpoint of the LGB to the corresponding location.

Following similar procedures, we find that to obtain the minimum emittance, we still need α_x0=0 and . Then we have , which means(102)We find a general analytical discussion of the combination of ρi and θi to be cumbersome. For simplicity and to obtain a concrete feeling, we first consider one specific case: . The physical consideration behind this choice is that is smaller in the central part of the bending magnet and larger at the entrance and exit. Therefore, we make the bending field in the center stronger and smaller at the entrance and exit regions, to minimize the quantum excitation of the horizontal emittance. For example, we may choose(103)The total length of such an LGB is . The minimum value of I_5x and horizontal emittance ϵx in this case is realized when(104)which means(105)and(106)which compared to Eq. (74) means that the theoretical minimum horizontal emittance can now become approximately one-third of the case with no longitudinal gradient. Therefore, the application of the LGB is quite effective in lowering the transverse emittance.

The next question is: what is the optimal combination of and ? This question is not straightforward to answer using analytical method. Here, we refer to the numerical method to perform the optimization directly. and are set to be zero in the optimization. The variables in the numerical optimization are . The two optimization goals are and the length of the LGB L_LGB, where is the theoretical minimum emittance of applying bending magnet without longitudinal gradient. We require . In the optimization, we kept the total bending angle of the LGB at a constant value.

The optimization result of a specific case where is presented in Fig. 2, from which we can see that by applying LGBs, we can lower the horizontal emittance by a factor of five with a reasonable length of the LGB. Note that in this optimization and the one in the following section, we assume that ρi>0. However, the formalism also applies to the case with anti-bends.

Fig. 2Full-screen View

Application of LGB to minimize horizontal (blue) and longitudinal (red) emittance, respectively. The subscripts “LGB" and “UB" represent the longitudinal gradient bend and uniform bend, respectively

Longitudinal Weak Focusing

We now begin the quantitative analysis. We begin with a longitudinal weak focusing (LWF) SSMB ring. In a LWF ring with a single laser modulator (LM) as shown in Fig. 3, the single-particle longitudinal dynamics turn by turn is modeled as(111)where the subscripts n, n + 1 means the number of revolutions, h is the energy chirp strength around the zero-crossing phase, is the wavenumber of the modulation laser and λ_L is the laser wavelength, C₀ is the ring circumference, and(112)is the phase slippage factor of the ring. Note that in the above model, we have assumed that the radiation energy loss in an SSMB storage ring will be compensated by other systems instead of the laser modulators; thus, microbunching will be formed around the laser zero-crossing phase. The laser can also be used for energy compensation; however, it is not a cost-effective choice and will also limit the output radiation power. Induction linacs or RF cavities can be used to supply the radiation energy loss. Linear stability of Eq. (111) around the zero-crossing phase requires that 0<hηC₀<4. To avoid strong chaotic dynamics which may destroy the regular longitudinal phase space structure, an empirical criterion is(113)In a LWF ring, when the synchrotron tune , the longitudinal beta function at the center of laser modulator is [17](114)Note that in this paper, we will use the subscript ‘M’ to represent modulator and ‘R’ to represent radiator. From Eq. (113) we then require(115)Now, we can use the previous analysis of the theoretical minimum longitudinal emittance, more specifically, Eq. (90), and the above result to evaluate a LWF ring. If E₀=600 MeV, the bending radius of dipoles in the ring ρ_ring=1.5 m, which corresponds to a bending field strength B_ring=1.33 T, and if our desired bunch length is nm ( for microbunches to be safely stored in the optical microbuckets for nm), then we may need nm to avoid significant energy widening when we reach the desired bunch length. From Eq. (90) we then need rad. Therefore, at least 30 bending magnets are required in the ring. Assuming that the length of each isochronous cell containing a bending magnet is 3 m, the arc section of such a storage ring has a length of approximately 90 m. Considering the straight section for beam injection/extraction, radiation energy loss compensation, and insertion device for radiation generation, the circumference of such a ring could be 100 m to 120 m.

To achieve the desired bunch length, according to Eq. (87) we need . In our present example case, if ρ=1.5 m and , then this value we need is μm. Then from Eq. (115) we have(116)If C₀=100 m, then we need a phase slippage factor , which is a quite small value. If we want a bunch length even smaller than 50 nm at a beam energy of 600 MeV, then the required phase slippage will be too demanding to be realized using the present technology. More details on the lattice design of an LWF SSMB ring that can store microbunches with a couple of 10 nm bunch lengths can be found in Ref. [16].

Longitudinal Strong Focusing

After discussing the LWF SSMB ring, we now begin the analysis of the LSF. First, we observe that the above analysis of the LWF SSMB considers the case with only a single LM. When there are multiple LMs, the longitudinal dynamics are similar to those of multiple quadrupoles in the transverse dimension, and the beam dynamics can have more possibilities. For example, a longitudinal strong focusing scheme can be invoked, similar to its transverse counterpart, which is the foundation of modern high-energy accelerators. Here, we use a setup with two LMs for the SSMB as an example to show the scheme of manipulating βz around the ring using the strong focusing regime. The schematic layout of the ring is illustrated in Fig. 4. The treatment of cases with more LMs is similar.

Fig. 4Full-screen View

(Color online) A schematic layout of a storage ring using two RF systems for longitudinal strong focusing and an example beam distribution evolution in the longitudinal phase space. (Figure from Ref. [8])

We divide the ring into five sections from the transfer matrix viewpoint, i.e., three longitudinal drifts (R₅₆) and two LM kicks (h), with the linear transfer matrices of the state vector in the longitudinal dimension given by(117)Then the one-turn map at the radiator center is(118)For the generation of coherent radiation, we usually want the bunch length to reach its minimum at the radiator, and then we need αz=0 at the radiator.

With the primary goal of presenting the principle, instead of a detailed design, here for simplicity, we focus on one special case: , . Denote(119)we then have(120)The linear stability requires which means 0<ζ₁ζ₂<1.

We remind the readers that the analysis above in this subsection about LSF has been presented in Ref. [17] and is presented here again for completeness of the paper. We now attempt to gain further insight. The longitudinal beta function at the radiator is(121)where is the synchrotron phase advance per turn. The longitudinal beta function at the opposite of the radiator, from where to LM2 has an , is(122)which can be obtained by switching ζ₁ and ζ₂ in the expression for β_zR in Eq. (121). So we have(123)The longitudinal beta function at the LM1 and LM2 (here we name them as modulators) is(124)For efficient bunch compression from the modulator to the radiator, we have . Then(125)from which we have the required energy chirp strength(126)This relationship is similar to the theorems presented in Sect. 5 about transverse-longitudinal coupling-based bunch compression schemes, which are the backbone of the GLSF scheme. The relation can also be casted as(127)with being the bunch length at the radiator.

We now investigate the required energy chirp strength and modulation laser power based on this analysis. We assume that in an LSF ring, the longitudinal emittance is dominated by quantum excitation in the ring dipoles. This assumption will be justified later to determine whether it is the case. Furthermore, we assume that the average longitudinal beta function at the dipoles equals that at the modulator, i.e., . Then the equilibrium longitudinal emittance given by the balance of quantum excitation and radiation damping in a ring consisting of iso-bending-magnets has the scaling(128)which combining with Eq. (127) gives(129)Here, σ_zR is determined by the desired radiation wavelength. Therefore, given the beam energy and desired bunch length, to lower the required energy chirp strength in the LSF, we should use as large ρ_ring as possible, which means as weak a bending magnet as possible. However, the total length of the bending magnets should be within a reasonable range. Also note that when the bending magnets in the ring are very weak, the assumption that the longitudinal emittance in a LSF ring is dominantly from them will fail, since the quantum excitation in the bending magnets will become weaker, while there are other contributions like quantum excitation at the laser modulators.

As mentioned in Sect. 1, a short bunch can generate coherent radiation. The parameter used to quantify the capability of beam for coherent radiation generation is called bunching factor, and in the 1D case is defined as(130)where ω is the radiation frequency, ψ(z) is the longitudinal charge density distribution satisfying the normalization . We will present a more in-depth discussion of the bunching factor in Sect. 6.1. The coherent radiation power of a beam with N_p particles at frequency ω is related to the radiation of a single particle according to(131)For a Gaussian bunch with an RMS bunch length of , we have For significant 13.5 nm-wavelength coherent EUV radiation generation, we may need nm which corresponds to . To increase the radiation power, we may need a radiator, which is assumed to be an undulator, with a large period number N_u (for example, ). To avoid significant bunch lengthening from the energy spread and the undulator , we then need , which then requires the energy spread at the radiator . So we need pm. Since in a LSF ring, as explained we cannot make the optimal conditions Eq. (83) be satisfied in all the bending magnets, a reasonable argument is that the real longitudinal emittance should be at least a factor of two larger than the true theoretical minimum, which then requires(132)Then according to Eq. (86), if E₀=600 MeV, we need , which means 59 bending magnets are needed. Assuming that the length of each isochronous cell containing a bending magnet is 3 m, the arc section of such a ring has a length of ~177 m. Considering the straight section for beam injection/extraction, radiation energy loss compensation, and longitudinal strong focusing section, the circumference of such a ring is ~200 m.

If ϵz=4 pm, to get nm, we then need(133)From Eq. (126), given β_zR, we should apply as large β_zM as possible to decrease the energy chirp strength h. Because with θ given, we will choose a reasonably large bending radius for the dipoles in the ring. If ρ=10 m, which corresponds to B₀=0.2 T and a total bending magnet length of 62.8 m, then the longitudinal beta function at the dipole center required to reach the practical theoretical minimum longitudinal emittance (see Eqs. (89) and (87)) is μm. Then we may let(134)Then from Eqs. (126), (133) and (134), the energy chirp strength required in such a LSF SSMB ring is(135)According to Eq. (221) to be presented later about the laser modulator induced energy chirp strength, if E₀=600 MeV, and for a modulator undulator with period λ_uM=8 cm (B_0M=1.13 T), length L_uM=1.6 m, and for a laser with wavelength λ_L=1064 m, to introduce the required energy chirp strength, we need a laser power GW. This is a large value, three orders of magnitude higher than the average stored laser power reachable in an optical enhancement cavity at the moment [31], which is at the level of one megawatt (MW). This causes the optical enhancement cavity to only work in a low duty cycle pulsed mode, thus limiting the filling factor of the microbunched electron beam in the ring and the average output EUV power.

We remind the readers that there is a subtle point in an LSF SSMB ring if we consider the nonlinear sinusoidal modulation waveform, since the dynamical system is then strongly chaotic and requires careful analysis to ensure a sufficiently large stable region for particle motion in the longitudinal phase space. More details on this respect can be found in Ref. [18].

For the completeness of discussion, let us evaluate the contribution of modulator undulators to the longitudinal emittance in our above example case, since there is also quantum excitation at the modulators. The quantum excitation contributions of two modulators to ϵz in a LSF ring are(136)where is the contribution of the two modulators to I_5z defined in Eq. (54), is the bending radius at the peak magnetic field of the modulator. Put in the numbers, and taking the approximation which means the radiation loss is mainly from dipoles in the ring, Jz≈2, we have(137)In our example case, B_ring=0.2 T, B_0M=1.13 T, β_zM=139 μm, L_uM=1.6 m, we have(138)which is even larger than the desired 4 pm longitudinal emittance and is therefore unacceptable.

The above evaluation of the longitudinal emittance contribution indicates that a weaker or shorter modulator should be used. Since our desired longitudinal emittance is pm, we need to control the contribution from the two modulators to be pm, since the ring dipoles will also contribute longitudinal emittance with a theoretical minimum of approximately 2 pm. For example, we may choose to weaken the modulator field by a factor of more than two. If λ_uM=0.15 m and B_0M=0.435 T, L_uM=1.5 m, then the contribution of two modulators to longitudinal emittance is(139)which should be acceptable for a target total longitudinal emittance of 4 pm. However, to introduce the desired energy chirp strength, we now need GW.

Note that in this updated parameter choice, there is still one issue that needs to be addressed. In the evaluation of the quantum excitation contribution of modulators to the longitudinal emittance, we implicitly assumed that the longitudinal beta function did not change inside it. This is not strictly true. The undulator itself has an , with λ₀ being the fundamental resonance wavelength of the undulator, which in our case is the modulation laser wavelength. The criterion for whether the thin-lens approximation applies is to evaluate whether or not , where R₅₆ is that of the undulator. Here in this updated example, we have for the modulator, which means that the thin-lens kick approximation does not apply here. Therefore, more accurately we should use the thick-lens map of the modulator [17] to calculate the evolution of longitudinal beta function in the modulator, and then evaluate the contribution to longitudinal emittance.

With all the subtle points carefully handled, the LSF, as analyzed above, can realize the desired nm bunch length and thus generate coherent EUV radiation. The main issue with such an EUV source is that the required modulation laser power (GW level) is too high, and the optical enhancement cavity can only work in low-duty-cycle pulsed mode, thus limiting the average EUV output power.

Generalized Longitudinal Strong Focusing

The previous analysis of the LWF and LSF led us to consider the generalized longitudinal strong focusing (GLSF) scheme [9]. The basic idea of the GLSF is to take advantage of the ultrasmall natural vertical emittance in a planar electron storage ring. More specifically, we will apply a partial transverse-longitudinal emittance exchange in the optical laser wavelength range to achieve efficient microbunching generation. As shown in Fig. 3, the schematic setup of a GLSF ring is very similar to that of an LSF ring. However, as stressed before, the energy chirp strength required in GLSF is much smaller than that in the LSF scheme, which means that the required modulation laser power can also be smaller. A sharp reader may also notice that in Fig. 3, the longitudinal phase space area of the beam is not conserved in the bunch compression or harmonic generation section of a GLSF ring. Fundamental physical laws, such as Liouville’s theorem, cannot be violated in a symplectic system. The reason for this apparent “contradiction" is that GLSF invokes 4D or 6D phase space dynamics, as summarized in Table 1, and what is conserved are the eigen emittances, instead of the projected emittances. It should be noted that in the plot, the phase space rotation direction in the GLSF scheme is reversed after the radiator compared to that before the radiator, whereas this is not the case in the LSF scheme. In other words, in the GLSF, we choose to make the upstream and downstream modulations cancel each other. In this sense, this setup is a special case of the reversible seeding scheme of the SSMB [32]. The reason for this is that we want to make the system transverse-longitudinal coupled only in a limited local region in the ring, the so-called GLSF section, such that we can maintain being at the majority places of the ring to minimize the quantum excitation and IBS contribution to vertical emittance, thus keeping the small vertical emittance of a planar uncoupled ring. Furthermore, the cancellation of the nonlinear sinusoidal modulation waveforms makes the nonlinear dynamics of the ring easier to handle. To ensure that the modulations perfectly cancel, the lattice between the upstream and downstream modulators must be an isochronous achromat. This reversible seeding setup makes the decoupling of the system straightforward. All we need to do is make the GLSF section an achromat, as the section from the upstream modulator to the downstream modulator is transparent to the longitudinal dynamics. Another advantage of this reversible seeding setup is that it makes the bunch length at the modulator more flexible. It can be a short microbunned beam, as shown in Fig. 3. It can also be a conventional RF-bunched or coasting beam. A coasting beam in this context means the beam is not pre-microbunched, and its length is much longer than the modulation laser wavelength, or even longer than that of an RF bunch. In fact, in our present design, which will be presented in Sect. 8, we use an RF-bunched beam in the ring. Therefore, the third laser modulator of such a GLSF ring is an RF system. Having explained the reason why we choose this reversible seeding setup for the GLSF, we remind the readers that this is not the only possible way to realize the GLSF scheme [9]. For example, a symmetric lattice setup with respect to the radiator is also possible, although the nonlinear dynamics may be challenging.

After this general introduction of the GLSF scheme, we now appreciate in a more physical way why GLSF could be favored compared to LSF, in lowering the required modulation laser power. The key is that the LSF has a contribution of ϵz from both the LSF section and the ring dipoles, while the GLSF has only a contribution of ϵy from the GLSF section, because outside the GLSF section is zero, as explained above. This is the key physical argument from the single-particle dynamics perspective of why the GLSF may require a smaller energy modulation than the LSF to realize the same desired bunch length at the radiator. If the longitudinal emittance in the LSF is only from the quantum excitation of LSF modulators, and the vertical emittance in the GLSF is only from the quantum excitation of GLSF modulators, then GLSF and LSF are equivalent in essence (only a factor of two difference in damping rates) concerning the requirement on energy modulation strength from the single-particle dynamics perspective.

Now, we explain the above argument more clearly using formulas. As we will show in the following section, more specifically Eq. (150), in GLSF at best case we have(140)where is defined in Sect. 2 and quantifies the contribution of the vertical emittance to the bunch length. Note that we have used , which means that the bunch length at the radiator in the GLSF scheme is solely determined by the beam vertical emittance. One can appreciate the similarity between the above formula and Eq. (127) for the LSF case. Therefore, GLSF will be advantageous to LSF in lowering the required energy modulation strength if(141)We now compare the two schemes in a more quantitative manner. We assume that the two schemes operate at the same beam energy. As we will show in Sect. 8.3, in a GLSF SSMB ring and if we consider only single-particle dynamics, the dominant contribution of vertical emittance is from the quantum excitation of two modulators in the GLSF section, and we have(142)In the LSF, we assume that the longitudinal emittance is mainly from the quantum excitation in dipoles of the ring, and the average longitudinal beta function around the ring dipoles is the same as that at the modulators .

Taking the approximation which means in both rings the radiation loss is mainly from the dipoles in the ring, Jz≈2, Jx≈1, and combining with Eq. (128), Eq. (141) then corresponds to(143)The above condition should be straightforward to fulfill in practice. For example, if the bending magnet strengths are the same in both schemes, i.e., if , the above relation corresponds to(144)which is easily satisfy in practice. Therefore, GLSF can be favored compared to LSF in lowering the required modulation laser power.

Short Summary

From the analysis in this section, our tentative conclusion is that an LWF SSMB ring can be used to generate bunches with a couple of 10 nm bunch lengths, thus generating coherent visible and infrared radiation. If we want to push the bunch length to an even shorter range, the required phase slippage factor of the ring will be too small from an engineering perspective. An LSF SSMB ring can create bunches with a bunch length at the nanometer level, thus generating coherent EUV radiation. However, the required modulation laser power is at the GW level, and the optical enhancement cavity can only work at a low duty cycle pulsed mode, thus limiting the average output EUV radiation power. At present, a GLSF SSMB ring is the most promising among these three schemes to realize nm bunch length with a smaller modulation laser power compared to LSF SSMB, thus allowing a higher average power for EUV radiation generation.

Problem Definition

First, let us define the problem that we are trying to solve. We assume ϵy is the small eigen-emittance we want to exploit. The case of using ϵx is similar. The schematic layout of the TLC-based bunch compression section is shown in Fig. 5. Suppose the beam at the entrance of the bunch compression section is x-y-z decoupled, with its second moments matrix given by(145)where α, β and γ are the Courant-Snyder functions, the subscript i indicates the initial state, and ϵx, ϵy and ϵz are the eigen-emittances of the beam corresponding to the horizontal, vertical, and longitudinal modes, respectively. The eigen-emittances are beam invariants with respect to linear symplectic transport. For the application of TLC for bunch compression, it means that the final bunch length at the exit or radiator σ_zR depends only on the vertical emittance ϵy, and neither on the horizontal one ϵx and nor on the longitudinal one ϵz.

Fig. 5Full-screen View

(Color online) Schematic layout of applying TLC dynamics for bunch compression. (Figure adapted from Ref. [33])

We divide such a bunch compression section into three parts, with their symplectic transfer matrices given by(146)with M₁ representing “from entrance to modulator", M₂ representing “modulation kick" and M₃ representing “modulator to radiator". Note that M₁ and M₃ are in their general thick-lens form and do not necessarily need to be x-y decoupled. The transfer matrix from the entrance to the radiator is then(147)From the problem definition, for σ_zR to be independent of ϵx and ϵz, we need(148)

Theorems and Proof

Dragt’s Minimum Emittance Theorem

Theorem one in Eq. (150) can also be expressed as(163)Note that in the above formula, is the bunch length at the modulator contributed by the vertical emittance ϵy. Therefore, given a fixed ϵy and desired σ_zR, a smaller h, i.e., a smaller RF acceleration gradient or modulation laser power (), means a larger , thus a larger , is needed. As quantifies the energy spread introduced by the modulation kick, we thus also have(164)Similarly for Theorem 2 and 3, we have(165)and(166)respectively, and Eq. (164). Note that in the above formulas, the vertical beam size or divergence at the modulator contains only the vertical betatron part, i.e., that from the vertical emittance ϵy.

Equation (164) is actually a manifestation of the classical uncertainty principle [34], which states that(167)where is the minimum of the three eigen-emittances . In our bunch compression case, we assume that ϵy is smaller than ϵz.

Actually there is a stronger inequality compared to the classical uncertainty principle, i.e., the minimum emittance theorem [34], which states that the projected emittance cannot be smaller than the minimum one among the three eigen emittances,(168)

Theorems Cast in Another Form

As another way to appreciate the result, here we cast the theorems in a form using the generalized beta functions, as introduced in Sect. 2. According to definition, we have(169)

Theorem 1: If M₂ is as shown in Eq. (149), then(170)where M_2,65 is the ₆₅ matrix term of M₂, i.e., h. The superscript 2 of means the square of M_2,65. For better visualization, in this subsection, we use brackets to denote the location, with Ent, Mod, and Rad meaning entrance, modulator, and radiator, respectively.

Theorem 2: If M₂ is as shown in Eq. (151), then(171)

Theorem 3: If M₂ is as shown in Eq. (153), then(172)At the entrance, the generalized Twiss matrix corresponding to eigen mode I is(173)and similar expressions for , with x replaced by y, z and the location of the 2×2 matrix shifted in the diagonal direction. Then(174)(175)(176)(177)(178)(179)For σ_zR to be independent of ϵx and ϵz, we need and , which then lead to Eq. (148). The following proof procedures are the same as those shown in the Sect. 5.2.2.

Form Function and Bunching Factor

Modulation Strength

After deriving the bunching factor, we derived the formula for the modulation strength, given the laser, electron, and undulator parameters. This is necessary for quantitative analysis and comparison.

6.2.1

A Normally Incident Laser

The most common method of imprinting energy modulation on an electron beam at the laser wavelength is to use a TEM₀₀ mode laser that resonates with the electrons in an undulator. In this study, we used a planar undulator as the modulator. A helical undulator can also be applied for energy modulation; however, because we want to preserve the ultrasmall vertical emittance, we need to avoid x-y coupling as much as possible. Thus, a planar undulator might be preferred. The electromagnetic field of a TEM₀₀ mode Gaussian laser polarized in the horizontal plane is [36](207)where and are the horizontal, vertical and longitudinal electric and magnetic field, respectively, s is the longitudinal global path length variable, t is the time variable, c is the speed of light in free space, , is the Rayleigh length, w₀ the beam waist radius, and(208)with i here being the imaginary unit. We remind the readers that it has been implicitly assumed that all the fields will take the real part of their complex expressions.

The relationship between the peak electric field E_x0 and the laser peak power P_L is given by(209)where is the impedance of free space. The prescribed wiggling motion of electron in a planar undulator is(210)with γ being the Lorentz factor,(211)being the dimensionless undulator parameter, where B₀ is the peak magnetic field, λ_u is the undulator period, and being the undulator wavenumber. The resonant condition of laser-electron interaction inside a planar undulator is(212)From the prescribed motion we can calculate the electron horizontal and longitudinal velocity(213)with , and(214)the average longitudinal velocity of electrons in the undulator. Therefore, we have the longitudinal path length of electron as a function of time given by(215)Then(216)where(217)Note that because the longitudinal coordinate of the electron will affect the observed laser phase, we need to calculate its precision to the order of , while for the horizontal coordinate x, we only need to calculate it to the order of . In the following, we will adopt the approximation because we are interested in relativistic cases.

Given the electron prescribed motion and laser electric field, the laser and electron exchange energy according to(218)where is the work done on the electron by the laser, or the energy transfer from the laser to the electron. Note that in this manuscript e represents the elementary charge and is assumed to be positive. Assuming that the laser beam waist is in the middle of the undulator, whose length is L_u, and when the electron transverse coordinates are much smaller than the laser beam waist , which is true in most of the cases under consideration, we drop the factor in the laser electric field. Furthermore, when the transverse displacement of the electron is much smaller than the Rayleigh length which is also usually the case, we can also drop the contribution from Ez on the energy modulation. Assuming the relative phase of laser field to the electron horizontal velocity vx at the undulator center is ϕ₀, the integrated modulation voltage induced by the laser on the electron beam in a planar undulator is then(219)where Re() denotes the real component of the complex number. When , which means the undulator period number , and when , in the above integration, only the term with n=0 and n=1 will give a notable non-vanishing value. Denote , we then have(220)We want the energy modulation strength to be as large as possible; therefore, we choose ϕ₀=0. Put in the expression of peak electric field from Eq. (209), the linear energy chirp strength around the zero-crossing phase is therefore(221)Once the modulator length is given, we can optimize the laser Rayleigh length to maximize energy modulation. Figure 6 is a plot of as a function of x. The maximum value of f(x) is 0.8034 and is realized when x=1.392. So when , the energy modulation reaches the maximum value. Note that when Z_R is within a small range close to the optimal value, the impact of Rayleigh length on the energy modulation strength is not very sensitive. Therefore, for easy of remembering, the optimal condition can be expressed as(222)From Eq. (221), we can see that under the optimal condition, we have . Using Eq. (221), we can also perform some example calculations to obtain a more concrete feeling. If E₀=600 MeV, λ_L=1064 nm, λ_u=8 cm (B₀=1.13 T), L_u=0.8 m (N_u=10), , then the induced energy chirp strength with P_L=1 MW is h=955 m^-1.

Fig. 6Full-screen View

vs. x. This curve describes the laser induced energy modulation strength as a function of the laser Rayleigh length, with the laser power and modulator length given

6.2.2

Dual-Tilted-Laser for Energy Modulation

Now with a hope to increase the energy modulation strength with a given laser power, we can use a configuration of crossing two lasers for energy modulation. The basic idea is that if two crossing lasers can double the energy modulation strength of a single laser, then the effect is similar to that induced by a single laser with a laser power four times larger. Our calculation shows that indeed dual-tilted-laser (DTL) can induce a larger energy modulation compared to that of a single laser, but the issue is that the required crossing angle (less than 2 mrad) is too small from an engineering viewpoint. We remind the readers that we can boost the bunching factor and thus the radiation power by adding one or more high-harmonic modulation in addition to the fundamental-frequency modulation, as discussed in Ref. [33]. Here in this paper we focus on the case of using a fundamental-frequency modulation only.

The analysis is presented in the following sections. First, we consider the case of two lasers crossing in the y-z plane. The laser field of an oblique TEM₀₀ laser is given by first replacing the physical coordinate with the rotated coordinates(223)with θ in this section and below being the tilted angle of the incident laser instead of the dipole bending angle in Sect. 3. We hope their difference is clear from context. The resulting field expression according to Eq. (207) is(224)However, in the above expression, x, y, s of the electromagnetic fields are defined according to the oblique laser propagation direction. To get the expression back in the original coordinate system, i.e., undulator axis as the z axis, we need to rotate the laser field as,(225)and for the electric field the result is(226)Assuming that the two crossing lasers are in phase and have the same amplitude. In addition, we assume that , and that the two lasers have the same Rayleigh length. Then the superimposed field is(227)As before, we focus on the impact of Ex on the laser-electron interaction and ignore the contribution from Ez. When θ is very small, the superimposed Ex can be approximated as(228)Note that in the final approximated expression, we have retained in the laser phase term. This is because the laser phase is of key importance in laser-electron interactions, and the accuracy requirement is high. In addition, we have also kept the in the intensity decay term, because may not be small compared to the laser beam wait radius w₀. The expression for Ex in the case of crossing in x–z plane is similar. Therefore, the difference in crossing in x–z plane and y–z plane is not significant in inducing energy modulation.

For effective laser-electron interaction, the off-axis resonance condition now is(229)or(230)Then(231)Note that χ now depends on θ, more specifically,(232)Assume that the laser beam waists are in the middle of the undulator, whose length is L_u, and denote(233)the integrated modulation voltage induced by the DTL in a planar undulator is then(234)To maximize the energy modulation, we chose ϕ₀=0. Put in the expression of E_x0 from Eq. (209), the linear energy chirp strength around the zero-crossing phase is therefore(235)where(236)A flat contour plot of is shown in Fig. 7.

Fig. 7Full-screen View

(Color online) Contour plot of given by Eq. (236)

Now, we can use the derived formula to calculate the energy chirp strength induced by the DTL. First we consider the case of keeping λ_u fixed when changing θ, then the off-axis resonant condition leads to the undulator parameter as a function of θ given by(237)With the increase of θ, decreases. Note that in this case, , so the magnetic field strength also decreases with the increase of θ. An example calculation of versus θ is shown in Fig. 8. The corresponding contour plot of the energy chirp strength normalized by the largest energy chirp induced by a single normally incident laser, versus θ and Z_R/L_u, is given in Fig. 9.

Fig. 8Full-screen View

Undulator parameter vs. θ with λ_u fixed. Parameters used: E₀=600 MeV, λ_L=1064 nm, λ_u=8 cm

Fig. 9Full-screen View

(Color online) Energy chirp strength h normalized by the largest energy chirp induced by a single normally incident laser, vs. θ and Z_R/L_u. Keep λ_u fixed when changing θ. Parameters used: E₀=600 MeV, λ_L=1064 nm, λ_u=0.08 m, N_u=10, L_u=0.8 m

The previous calculation assumes λ_u remains unchanged when we adjust the incident angle θ. This will result in a limited region of θ to fulfill the resonant condition, as shown in Fig. 8. We now conduct the calculation by assuming that the peak magnetic field B₀ remains unchanged when we adjust θ. Put the expression of undulator parameter Eq. (211) in the off-axis resonant condition Eq. (230), we have(238)from which we get(239)with(240)

For example, if λ_L=1064 nm and B₀=1.2 T, then λ_u as a function of θ for the case of E₀=600 MeV is shown in Fig. 10. Note that in this case, which depends on θ, so the undulator parameter also decreases with the increase of θ. However, the decrease was not as rapid as that presented in Fig. 8. The corresponding contour plot of the energy chirp strength induced by the DTL, normalized by the largest energy chirp induced by a single normally incident laser, vs. θ and Z_R/L_u, is shown in Fig. 11.

Fig. 10Full-screen View

λ_u vs. θ, with B₀ fixed. Parameters used: E₀=600 MeV, λ_L=1064 nm, B₀=1.2 T

Fig. 11Full-screen View

(Color online) Energy chirp strength h normalized by the largest energy chirp induced by a single normally incident laser, vs. θ and Z_R/L_u. Keep B₀ fixed when changing θ. Parameters used: E₀=600 MeV, λ_L=1064 nm, B₀=1.2 T, L_u=0.8 m

As shown in the calculation results in Figs. 9 and 11, DTL indeed can induce a larger energy modulation compared to a single normally incident laser. However, the required crossing angle (less than 2 mrad) is too small for engineering. Therefore, the usual setup of a single normally incident TEM₀₀ mode laser remains the preferred choice in practical applications.

Realization Examples

After the derivation of the bunching factor and laser-induced modulation strengths, finally in this section we provide some realization examples of microcbunching schemes that belong to what we have analyzed. FEL seeding technique like phase-merging enhanced harmonic generation (PEHG) [37, 38], and angular dispersion-induced microbunching (ADM) [39] can be viewed as specific examples of our general definition of TLC-based microbunching schemes in Theorem One. A more detailed discussion in this respect has been presented in Ref. [33].

Here, we briefly comment on the relation between FEL seeding techniques, such as HGHG, PEHG, and ADM, and the storage ring schemes, such as LSF and GLSF, discussed in this paper. One is single-pass, and the other is multipass. One invokes matrix multiplication or nonlinear transfer map once, and the other invokes eigenanalysis or normal form analysis of the one-turn map. They share the bunch compression or harmonic generation mechanism. The relationship between HGHG and LSF was similar to that between PEHG/ADM and GLSF.

We also remind the readers that the GLSF SSMB scheme analyzed in this paper bears similarity to the approach discussed in Ref. [40], in the transverse-longitudinal-coupling-based microbunching and modulation-demodulation processes. The difference is that here we aim for a true steady state in the context of storage ring dynamics and the electron beam pass the radiator turn by turn, while in Ref. [40] the radiator is placed at a bypass line and the ring is used to prepare the electron beam which is sent to the bypass line every multiple revolutions in the ring.

Bunching Factor

After investigating the energy modulation-based TLC microbunching schemes, in this section, we discuss angular modulation-based schemes [42-48]. The problem definition is similar to that in energy modulation-based schemes, with the exception of replacing energy modulation with angular modulation. We use y’ modulation as an example because we will take advantage of the ultrasmall vertical emittance in a planar ring, as explained before. The lumped laser-induced angular modulation is modeled as:(241)Note that we must add the second equation above to make the modulation symplectic. Such an angular modulation process also satisfies the Panofsky-Wenzel theorem [42](242)Following the derivations in the section of energy modulation-based schemes, the final form function in this case is(243)The integration of the above results is given by the generalized hypergeometric function. If and R₅₆=0, then , and the initial beam distribution is Gaussian as given in Eq. (190), then the 1D bunching factor is(244)with M_p given by Eq. (188).

To appreciate the physical principle, we use a specific case as an example instead of a general mathematical analysis. If R₅₁=0, R₅₂=0, R₅₅=1, R₅₆=0, and the initial beam is transverse-longitudinal decoupled and has an upright distribution in the longitudinal phase space, then(245)with being the Courant-Snyder functions before the modulation, and σ_δ0 being the initial RMS energy spread. If the initial bunch length is much longer than the laser wavelength, the above exponential terms will be non-zero only when , which means that there is only bunching at the laser harmonics. In this case, we have the bunching factor at the n-th laser harmonic to be(246)where . One can appreciate the similarity of the above result with the bunching factor of the energy modulation-based TLC microbunching schemes, i.e., Eq. (206). The R₅₄ here plays the role of R₅₆ in the previous equation. If further R₅₃=0, then Eq. (246) reduces to(247)where σy’ is the initial RMS beam angular divergence. Therefore, R₅₄ and σy’ in this scheme play the role of R₅₆ and σ_δ in HGHG, as shown in Eq. (195). In a planar uncoupled ring, the natural vertical emittance is quite small, as is σy’. Therefore, using this scheme, we can realize high-harmonic bunching in a storage ring to generate ultrashort soft X-ray pulse.

As can be seen from our analysis, both the energy modulation-based and angular modulation-based TLC microbunching schemes share the same spirit, i.e., to take advantage of the small transverse emittance, the vertical emittance in our case, to generate microbunching with a shallow modulation strength. These TLC-based microbunching schemes can be viewed as partial transverse-longitudinal emittance exchanges in the optical laser wavelength range. They do not necessarily need to be complete emittance exchanges because for microbunching, the most important coordinate is z, and δ is relatively less important. As we will show soon, although the spirit is the same, given the same level of modulation laser power, the physical realization of energy modulation-based TLC microbunching schemes is more effective for our SSMB application compared to the angular modulation-based schemes.

Modulation Strength

7.2.2

Dual-Titled-Laser-Induced Angular Modulation

Another way to imprint angular modulation on the electron beam is to use a titled incident TEM₀₀ mode laser to modulate the beam in an undulator. To further lower the required laser power, a dual-tilted-laser (DTL) with a crossing configuration can be applied [44, 45]. In this study, we focused on an angular modulation scheme based on a DTL setup. Note that if we want to use a DTL for energy modulation, the two lasers should be in phase to ensure that the two laser-induced energy modulations add. For angular modulation, they should be π-phase shifted with respect to each other. This is because, for angular modulation, the particle on the reference orbit should receive a zero-energy kick. Only when the particle transverse coordinate is nonzero will it receive an energy kick. So the energy modulations induced by the two lasers should cancel on axis.

To induce vertical angular modulation, we let the two lasers cross in y-z plane and be polarized in the horizontal plane. The laser field of a normal incident TEM₀₀ laser is given by Eq. (207). Assuming that the two lasers are π-phase-shifted with respect to each other and have the same amplitude. In addition, we assume that , and the two lasers have the same Rayleigh length. Then the superimposed field is(254)As before, we focus on the impact of Ex on the laser-electron interaction and ignore the contribution from Ez. Note that if y=0, then Ex=0. We want to know when y is close to zero. For this purpose, we do Taylor expansion of the above horizontal electric field with respect to θ when θ is small,(255)When , , we have(256)However, note that when is comparable to w₀, the term should be kept in the exponential term. In addition, for the laser phase, we should use a more accurate . These arguments have been explained before when we analyzed the application of a DTL for energy modulation. So the more correct approximate expression of Ex is(257)Again taking the same notation as given in Eq. (233), the integrated modulation voltage induced by a DTL (two lasers π-phase-shifted with respect to each other) in a planar horizontal undulator can be calculated to be(258)We choose to maximize V_L.

Therefore, we get the maximum linear angular chirp strength as(259)where(260)A contour plot of is presented in Fig. 12.

Fig. 12Full-screen View

(Color online) Contour plot of given by Eq. (260)

Then the angular chirp strength introduced by a DTL compared to the energy chirp strength introduced by a single TEM₀₀ laser modulator, i.e., that given in Eq. (221), with the same laser parameters and undulator length can be expressed as(261)with(262)Note that in the numerator is a function of θ according to the off-axis resonance condition given in Eq. (230). Since , and θ is usually in the mrad level, given the same laser power, the DTL-induced angular chirp strength will be much smaller than the energy modulation strength induced by a TEM₀₀ mode laser. This observation was supported by a more quantitative calculation of the angular chirp strength induced by a DTL, as shown in Figs. 13 and 14. As before, we considered the case of keeping λu or B₀ unchanged when we change the crossing angle θ. In both cases, the maximal angular chirp strength induced with PL=1 MW was approximately m^-1. From the evaluation in Sect. 6.2.1, at the same power level, a TEM₀₀ mode laser modulation can induce an energy chirp strength of . This is because θ is only 1 to 2 mrad.

Fig. 13Full-screen View

(Color online) Angular chirp strength g vs. θ and Z_R/L_u, keeping λu fixed when changing θ. Parameters used: E₀=600 MeV, λ_L=1064 nm, P_L=1 MW, λ_u=0.08 m, N_u=10, L_u=0.8 m

Fig. 14Full-screen View

(Color online) Angular chirp strength g vs. θ and Z_R/L_u, keeping B₀ fixed when changing θ. Parameters used: E₀=600 MeV, λ_L=1064 nm, P_L=1 MW, B₀=1.2 T, L_u=0.8 m

Therefore, we can see that the DTL-induced angular chirp strength, although a factor of three larger than that induced by a single normally incident TEM₀₁ mode laser with the same laser power, is still generally quite small. There are two reasons why a DTL is not effective in imprinting angular modulation:

The crossing angle between the laser and the electron propagating directions results in that they have a rather limited effective interaction region. For example, if the crossing angle is θ=5 mrad and the undulator length is L_u=0.8 m. Then, the center of the electron beam and the center of the laser beam at the undulator entrance and exit depart from each other with a distance of mm, which is a large value compared to the laser beam waist μm and results in a very weak laser electric field felt by the electron there.

The decrease of undulator parameter K with the increase of θ to meet the off-axis resonance condition, as can be seen in Figs. 8 and 10.

Because the angular chirp strength is small, according to Theorem Two, the required vertical beta function at the modulator β_yM will be large. For example, if ϵy=4 pm, nm, and , then we need β_yM=2.5×10⁵ m, which is too large to be used in practice in a storage ring. To lower β_yM, a higher modulation laser power is required. This is why we tend to use energy modulation-based coupling schemes for bunch compression or microbunching generation in the GLSF SSMB. Our analysis and conclusion here is consistent with what reported in Ref. [49] when comparing energy and angular modulation microbunching schemes for laser plasma accelerator based light source.

Generally, when we compare the energy modulation-based and DTL-induced angular modulation-based bunch compression schemes, from the Theorem One and Two, i.e., Eqs. (150) and (152), for the same modulation laser wavelength λ_L and power P_L, vertical emittance ϵy and target linear bunch length at the radiator , we have(263)

Realization Examples

Similar to the section on energy modulation-based schemes, we introduce some realization examples of angular modulation-based microbunching. To the best of our knowledge, the first proposal of applying an angular modulated beam for harmonic generation was from Ref. [43]. Later, an emittance-exchange-based harmonic generation scheme was proposed in Ref. [46]. These two schemes apply a TEM₀₁ mode laser to induce angular modulation. Following these developments, there have been proposals to realize angular modulation using TEM₀₀ mode lasers, with Ref. [47] using an off-resonance laser, and Ref. [48] using a tilted incident laser. Later, a dual-tilted-laser (DTL) modulation scheme was applied in emittance exchange in the optical laser wavelength range [44]. Recently, the DTL scheme was proposed to compress the bunch length in SSMB and lower the requirement on the laser power by a factor of four compared to a single-tilted laser scheme [45]. Note that for these angular modulation-based harmonic or bunch compression schemes, we have the inequality given by Eq. (152), i.e., our Theorem Two.

A Solution of 1 kW EUV Source

Based on what we have studied in the previous sections and the various important physical effects to be discussed in this section, we present an example parameter set of a 1 kW-average-power EUV light source based on GLSF SSMB, as shown in Table 2. All parameter lists should be feasible from an engineering perspective. Such a table summarizes the investigations presented in this study.

Some Basic Considerations

Now, we present the detailed considerations and calculations to support our solution. First, we discuss some basic considerations on the choice of parameters. As explained at the beginning of Sect. 4, we used a beam energy of E₀=600 MeV and a modulation laser wavelength of λ_L=1064 nm. A small vertical emittance is crucial in the GLSF scheme. We assume that the vertical emittance used to accomplish our goal stated above is ϵy=40 pm. It should be noted that the vertical emittance used is not extremely small. This conservative choice is mainly a reflection of the consideration for intra-beam scattering (IBS) to be introduced soon. To realize significant coherent EUV generation, and considering that we may generate microbunching based on a coasting beam or RF bunched beam in the ring, which is much longer than the modulation laser wavelength, we may need to compress the linear bunch length to be as short as 2 nm. The bunching factor at 13.5 nm according to Eq. (206) is then 0.0675.

With ϵy=40 pm, to get the desired linear bunch length 2 nm at the radiator, we need μm. If βy at the radiator is around 1 m, then the required control precision of dispersion and dispersion angle at the radiator is at the level of 0.3 mm and 0.3 mrad. Such precise control of is challenging but realizable using present technology. We remind the readers that the dispersion function is not well defined when the system is transverse-longitudinal coupled, and here should be replaced by the defined generalized beta function . However, here we still use the classical definition of in obtaining the numbers for the above Dx and Dx’ control precision to give the readers a more concrete feeling. Following a similar line of thought, we also need a precision control of or at the radiator, since the horizontal emittance is even larger than the vertical one. Generally, we require precision control of both Dx,y and . Besides, we also need to ensure the coupling of horizontal emittance ϵx to the vertical plane to be less than 1% since our applied ϵx=2 nm is about two orders of magnitude larger than ϵy=40 pm.

Another important parameter is the beam current, including the average and peak currents. First, we observe that given the same average beam current, the average radiation power will be higher with a decreasing beam filling factor f_e. This is because the peak power of the coherent radiation is proportional to the peak current squared , where I_P and I_A are the peak and average currents, respectively, and fe is the electron beam filling factor. The average radiation power is then . Therefore, we tend to choose a high average current and a small filling factor as much as possible, as long as collective effects, such as IBS and coherent synchrotron radiation, do not degrade the beam properties. We applied a 40 A peak current and a 200 mA average current in our solution. Note that in the calculation of the filling factor, for simplicity, we assumed that the beam current was similar to square waves. The real beam distribution is more like a smooth curve, for example, a Gaussian profile if there is only a single RF cavity in the ring. However, this observation only results in some numerical factor adjustment and does not affect the core part of our analysis. So in Table 2, we take this simplification of a square wave current distribution.

Quantum Excitation Contribution to Vertical Emittance

After the general considerations, let us now take a closer look at the critical parameter ϵy. The first contribution of the vertical emittance is the quantum excitation in the GLSF section itself, since there is nonzero. More clearly, according to our Theorem One, i.e., Eq. (150), the modulator should be placed at a dispersive location if energy modulation-based coupling is used for bunch compression or harmonic generation. Therefore, the quantum excitation of all bend-like elements, such as dipoles, modulators, and radiators in the GLSF section, will contribute to the vertical emittance.

Like the calculation in Eq. (136), the quantum excitation contribution of a radiator to ϵy is(264)where L_uR is the radiator undulator length, is the bending radius corresponding to the peak magnetic field of the radiator B_0R. Assuming that the radiation energy loss in the GLSF section is much less than that induced by the bending magnets in the ring, then we have . Taking the the approximation Jy≈1, in an easy-to-use form we have(265)For the parameters listed in Table 2, B_ring=1.33 T, λ_uR=1.8 cm (B_0R=0.867 T), μm, L_uR=5.69 m (N_uR=4×79), then Δϵ_yR=2.5 fm, which is much less than ϵy. Generally, because at the radiator is quite small, the contribution of the radiator to the vertical emittance is not dominant, compared to that from modulators and dipoles to be introduced.

Similar to the analysis for radiator, the contribution of two GLSF modulators to ϵy is(266)where B_0M and L_uM are the peak magnetic field and length of the modulator undulators, respectively. For the parameters listed in Table 2, B_ring=1.33 T, B_0M=0.806 T, m, L_uM=1.5 m, then Δϵ_yM=592 pm, which is a quite large value compared to that contributed from the radiator. This is mainly due to the fact that . As will be introduced soon, we can apply a horizontal planar damping wiggler to control this contribution by increasing the radiation damping rate.

In the above analysis, we have assumed is a constant value throughout an undulator. This is not strictly true, since intrinsic dispersion is generated inside an undulator. We can refer to the transfer matrix of an undulator or laser modulator to study the evolution of in an undulator radiator and modulator to obtain a more accurate evaluation of their quantum excitation contribution, as shown in Sect. 8.4.2. Our calculation shows that this will result in a significant difference in the radiator’s contribution to ϵy, but not much difference for the modulators. However, because the radiator’s contribution to ϵy is negligible compared to that of the modulators, we still use the simplified formula in Eq. (266) in our following discussion.

Vertical bending magnets are also present in the GLSF section for optical manipulation to fulfill the bunch compression or harmonic generation condition. However, in principle, we can use weak dipoles to minimize their quantum excitation contribution to vertical emittance and satisfy the symplectic optics requirement simultaneously. The total length of these dipoles should not be excessively long. Therefore, in our present evaluation, we will assume that the quantum excitation contribution from the two modulators is the dominant source of ϵy if we consider single-particle dynamics alone.

Application of Damping Wigglers

Intra-beam Scattering

We mentioned that our conservative choice of ϵy=40 pm is mainly due to the consideration of IBS. This can be understood with more quantitative calculations. We will see that IBS is the most fundamental obstacle in obtaining ultrasmall vertical emittance in GLSF SSMB. This is partially because our choice of beam energy was not too high. In addition, to realize high EUV power, we need a high peak current, which means a high charge density in the phase space.

We use Bane’s high-energy approximation [52] to calculate the IBS diffusion rate(298)where m is the classical electron radius and is the Coulomb Log factor, with b_max and b_min being the maximum and minimal impact factor for the scattering process, respectively. Typically L_c is in the range of 10 to 20. N is the number of electrons in the bunch. For a coasting beam, we need to replace where L is the bunch length. Note that according to our definition, where I_P is the peak current.

Now, we input some example numbers to estimate the IBS diffusition rate in a GLSF EUV SSMB ring:(299)Note that in Table 2 we have m at the two modulators, whose length are both 1.6 m. In evaluating we only considered the contribution from the two modulators, where reaches its maximum value. This is a simplification but should provide a correct order of magnitude estimation. Putting in the example numbers, we then have(300)Compared with the radiation damping times given in Eq. (272), we can see that even for the vertical dimension, the IBS diffusion is more than three times slower than the radiation damping. Therefore, the IBS diffusion can now be controlled by the strong damping induced by damping wigglers. This calculation also justifies the necessity or benefit of applying the damping wigglers.

Microwave Instability

Now, we want to evaluate whether the peak current of 40 A we applied in the above example is feasible. One of the main limitations of the peak current is the microwave instability induced by coherent synchrotron radiation (CSR). According to Ref. [53], the CSR-induced microwave instability threshold is(301)with(302)and 2g is the separation between the two plates. So the threshold peak current is(303)with kA being the Alfven current. The is that of the entire ring. is the linear energy chirp strength around the synchronous RF phase ϕ_s. Putting in some typical parameters for the EUV SSMB: E₀=600 MeV, B₀=1.33 T, ρ=1.5 m, |R₅₆|=1 m, σ_δ0=8.5×10^-4, g=4 cm, h_RF=0.01^-1, then(304)Therefore, our application of a peak current of 40 A should be safe from microwave instability.

The astute reader may notice that one of the main reasons we have a large threshold current here is the large phase slippage or R₅₆ we applied for the ring. To avoid confusion, first we need to clarify that in this example of a GLSF SSMB EUV source, the electron bunch in the ring can be a coasting beam or an RF-bunched beam, and microbunching appears only at the radiator, owing to the phase space manipulation of the electron beam in the GLSF section. In our setup, we used an RF-bunched beam in the ring. Therefore, the phase slippage factor of the ring does not need to be small, whereas this is required in an LWF SSMB ring. Then, the question becomes whether the required large phase slippage is feasible and what beam dynamics effects it may have. As a reference, the Metrology Light Source storage ring [54] in standard user mode has an m, which means a phase slippage factor of 3.3×10^-2 given a circumference of 48 m. Therefore, we believe that our application of |R₅₆|=1 m is realizable. However, we recognize that such a large R₅₆ requires a large horizontal dispersion Dx at the dipoles, since where ⟨ ⟩_ρ means average around the dipoles in the ring. A large Dx may result in a large , and the quantum excitation of dipoles to the horizontal emittance ϵx should be carefully evaluated. A large ϵx will degrade the coherent EUV radiation [17, 55]. What we need is m, and at the same time nm, and some optimization may be required to realize such a goal.

Now, we check if the required RF system in the above example calculation is feasible. The longitudinal beta function at the RF cavity is m, and the RMS bunch length is mm. To obtain a beam filling factor of 0.5%, we require an RF wavelength of approximately 1.7 m, which corresponds to an RF frequency of 176.5 MHz. Then, the required energy chirp strength of h_RF=0.01^-1 means that the required RF voltage is 1.62 MV. Such an RF voltage should be achievable in this frequency range. Multiple cavities can be invoked if it is too demanding for a single cavity to reach the desired voltage.

We remind the readers that in this evaluation, we assumed that the beam was RF bunched. In our evaluation of the IBS, we assumed that the beam was a coasting beam. These evaluations mainly serve as an order of magnitude estimation that supports the general feasibility of our parameter choice. A more detailed analysis of the collective effects will be necessary in the future development of such a GLSF SSMB light source.

Energy Compensation System

The large radiation loss of the electron beam in the ring, particularly that induced by the strong damping wigglers, must be compensated. Here, we present some preliminary analysis on the requirements of the energy compensation system of such a GLSF SSMB EUV light source. From Table 2, the total radiation loss per particle per turn is keV, where the three terms on the right hand side represent the radiation loss in dipoles, damping wigglers and radiator, respectively. Therefore, the synchronous acceleration phase corresponds to an acceleration voltage of V_acc=341.3 kV. Such a high voltage is not easy to realize using an induction linac, considering that the required repetition rate is at the MHz level. Therefore, we use conventional RF cavities to supply radiation loss. Following the discussion in the last section, and assuming that we have used an RF frequency of 166.6 MHz (RF wavelength 1.8 m), we need an RF voltage of 1.72 MV to realize an energy chirp strength of h_RF=0.01^-1. Under this parameter set, and consider , the RF bucket half-height is [17](305)which is with the energy spread σ_δ0=8.5×10^-4. Therefore, the bucket half-height should be sufficiently large to ensure the beam quantum and Touschek lifetime.

To relieve the burden on the RF cavities, we may use three RF cavities to achieve the total RF voltage, with each cavity having a voltage of 573 kV. To minimize the power disspated on the cavity, we need a large shunt impedance of each cavity which here we assume to be , then the total power dissipated on the three cavity walls is(306)The power delivered to the 200 mA-average current beam is(307)Therefore, the power dissipated on the cavity walls was at the same level as that delivered to the beam. We recognize that a large shunt impedance may require effort to realize in practice. If the shunt impedance is lower than the assumed value, the power consumption on the wall will be correspondingly higher. Superconducting RF cavities can be used to reduce the power dissipation on the wall.

Generally, the total power consumption of the RF system of an SSMB ring is at 100–200 kW level. Together with the power consumption of other systems, such as electromagnets, superconducting damping wigglers, and vacuum and water cooling systems, the overall power consumption of such an SSMB storage ring is at the level of several hundred kWs. The output EUV power per radiator is approximately 1 kW. In principle, an SSMB ring can accommodate multiple GLSF insertions and, therefore, multiple radiators; however, we consider the case of only one radiator in the ring. Therefore, for such a GLSF SSMB storage ring, it takes a couple of 100 kWs of electricity to generate 1 kW of EUV light. Such a large power consumption may raise the question of the advantage of such an SSMB-EUV source compared to the superconducting RF-based high-repetition rate FEL-EUV source, particularly the energy recovery linac-based FEL-EUV source. For comparison, according to Ref. [56], it takes 7 MW overall power consumption to generate 10 kW EUV light in an ERL-FEL EUV source, which means 700 kW electricity power per 1 kW EUV light. Therefore, the overall power efficiency from electricity to EUV light for the SSMB-EUV and ERL-FEL EUV source is comparable. However, we remind readers that these numbers are only rough estimates, and more in-depth studies are required to reach a concrete conclusion. Another side comment is that the radiation emitted by the damping wigglers may also be useful.

Modulation Laser Power

We now evaluate the modulation laser power required. Given the laser wavelength, modulator undulator parameters and the required energy chirp strength, we can use Eq. (221) to calculate the required laser power(308)Using Theorem one, i.e., Eq. (150), and let the equality in the relation holds, we have(309)where is the fine structure constant and is given by Eq. (268). In the above derivation, we used the electron momentum and the approximation . We now attempt to derive more useful scaling laws to offer guidance in our parameter choice for a GLSF SSMB storage ring. As shown in Fig. 6, to maximize the energy modulation, we need . When , we approximate the resonance condition as(310)and . Then we have(311)Putting in the numbers for the constants, we obtain the quantitative expressions of the above scalings for practical use(312)The above scaling laws are accurate when , and it should be noted that the calculated power refers to the peak power of the laser. For completeness, here we give also the modulator length scaling(313)Therefore, to lower the required modulation laser power, we can apply a large , which indicates strong damping wigglers. A low beam energy and short laser wavelength are also preferred. However, their choices should consider more factors, such as IBS and engineering experience of the optical enhancement cavity, as explained in Sect. 4. A strong bending magnetic field in the ring is also desired. A weaker modulator field strength is favored to lower the required laser power. However, note that the required modulator length may be longer if we keep the quantum excitation contribution of modulators to vertical emittance Δϵ_yM unchanged in this process.

Radiation Power

After microbunching is formed, now comes the radiation generation. We will use a planar undulator as a radiator. Coherent undulator radiation power at the odd-H-th harmonic from a transversely-round electron beam is [17, 55](314)where N_u is the number of undulator periods,(315)with , and the transverse form factor is(316)where , is the RMS transverse electron beam size, is the bunching factor at the H-th harmonic determined by the longitudinal current distribution, and I_P is the peak current.

The above formula was derived by assuming that the longitdudinal and transverse distributions of the electron beam do not change significantly in the radiator. The energy spread of the electron beam can lead to a current distribution change inside the undulator, considering that the undulator has an , where λ₀ is the fundamental on-axis resonant wavelength of the undulator. If we consider the impact of the energy spread on coherent radiation, and assuming that the microbunching length reach its minimum at the radiator center, there will be a correction or reduction factor multiplied to the radiation power given by Eq. (314)(317)Here, we remind the readers that in our case of the GLSF SSMB, the beam is not transversely round at the radiator. Therefore, we used an effective round beam size to evaluate the radiation power using the above formula. This effective transverse beam size is between σx and σy. For a more accurate calculation of the radiation power, we can use the real beam distribution and invoke a numerical method [17].

With all the beam physics issues properly handled, we have finally obtained a solution for a 1 kW EUV source based on GLSF SSMB, as shown in Table 2.