Coherent X-ray source generation with off-resonance laser modulation

Ke Feng; Changhai Yu; Jiansheng Liu; Wentao Wang; Ye Tian; Zhijun Zhang; Rong Qi; Ming Fang; Jiaqi Liu; Zhiyong Qin; Ying Wu; Yu Chen; Lintong Ke; Cheng Wang; Ruxin Li

doi:10.1364/OE.26.019067

1. Introduction

Free-electron lasers (FELs) are capable of generating high-brilliance and short-wavelength coherent radiation and have become invaluable tools for a broad range of research [1,2]. A few facilities based on state-of-the-art linear accelerators have been operated successfully in the X-ray wavelength range [3–6]. These facilities are usually operated in the self-amplified spontaneous emission (SASE) mode, which causes a limited temporal coherence owing to the shot noise that seeds the amplification. An alternative method to improve the temporal coherence is based on the coherent harmonic generation (CHG) technique [7], which has been widely used in high-gain FELs [8, 9]. The standard CHG scheme consists of two short undulators separated by a small chicane. The e beam is energy-modulated by a seed laser in the first undulator (modulator) and then sent to the chicane to realize the conversion from energy modulation to density modulation (micro-bunching). The micro-bunched e beam is then sent to another undulator (radiator) to emit the harmonic radiation. Typically, generating the n^th harmonic of the seed laser requires the energy modulation amplitude to be approximately n times larger than the beam energy spread. The standard CHG scheme has a limitation of the harmonic number (3–5) for beams with a large energy spread (~0.1%) in the storage rings (SRs) [10, 11] and is not applicable for beams with an even larger energy spread (~1%) from laser wakefield accelerators (LWFAs) [12–17].

To avoid the large energy modulation and increase the harmonic number, the angular modulation-based CHG scheme has been proposed [18, 19]. An intense seed laser operating in the TEM01 mode is used to interact with the e beam in the undulator resonant to the laser frequency to introduce angular modulation to the beam instead of energy modulation. Then, a specially designed beam line is used to convert the angular modulation into density modulation. It has been verified that harmonic radiation in the soft X-ray regime can be generated via this technique. However, the required seed laser has a high peak power (~TW) and is operated in the TEM01 mode, increasing the complexity of the beamline.

In this paper, a promising method is discussed for introducing angular modulation into the e beam using an off-resonant seed laser. The off-resonant laser modulation was first proposed for beam-energy-spread cooling in generation of short-wavelength radiation [20]. Here, we use the off-resonant laser to modulate the angular distribution of the e beam. The seed laser in our scheme is operated in the fundamental mode, which differs from the aforementioned scheme and is easy to implement. “Off-resonant” means that the wavelength of the seed laser differs from the resonant wavelength of the modulator. Once the angular modulation is introduced, we can design a transport system to convert the angular modulation into density modulation, which contains high-order components of the seed laser. In this scheme, the difference between the longitudinal velocity of electrons and the speed of light must be considered, owing to the high intensity of the seed laser. The modulation of the transverse divergence of the e beam vanishes after the beam passes through one period of the undulator when the seed-laser wavelength equals the resonant wavelength. However, significant modulation can be retained by exploiting the off-resonant seed laser modulation (ORLM) scheme. The proposed scheme requires a beam with relatively low emittance (~0.1 μm) and avoids the large energy modulation, which is both suitable for the e beams from the LWFAs [21–23] and the SRs [19]. To describe our scheme more specific, we take the e beam from LWFAs as an example. With typical e-beam parameters, numerical simulations demonstrate the CHG in the soft X-ray regime with sub-gigawatts peak power by using an 800-nm seeding laser in the ORLM scheme.

2. Physical mechanism of ORLM

The physical scenario of angular modulation using the ORLM scheme is presented in Fig. 1. The 30-fs, 800-nm main laser beam with an on-target power of ~100 TW is focused onto the gas jet by an off-axis parabolic mirror. A fraction of the laser energy is split from the main beam as the seed laser to modulate the e beam. The seed laser, together with the produced e beam, is then sent into the modulator with a resonant wavelength different from the seed wavelength, where the angular modulation is introduced. The angular modulation can be converted into density modulation after the beam passes through a dispersion section, and the modulated e beam can efficiently emit the coherent high-order harmonics in the following undulator (radiator).

Fig. 1 Scheme for CHG using ORLM with the e beam from LWFAs.

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In the planar undulator, the magnetic field that the electron experiences is:

B_{y} (z) = B_{0} \cos (k_{u} z),

with

k_{u} = 2 π / λ_{u}

, where

λ_{u}

is the undulator period. The motion of the electron inside the undulator can be described as [18]:

\begin{array}{l} β_{x} (z) = - \frac{K_{0}}{γ} \sin (k_{u} z) \\ β_{z} (z) = \sqrt{1 - \frac{1}{γ^{2}} - β_{x}^{2}} \approx 1 - \frac{1}{2 γ^{2}} (1 + \frac{K_{0}^{2}}{2}) + \frac{K_{0}^{2}}{4 γ^{2}} \cos (2 k_{u} z), \end{array}

where

β_{x, z} = v_{x, z} / c

is the normalized velocity, and

K_{0} = e B_{0} / m c k_{u}

is the dimensionless undulator strength parameter.

Assuming that the seed laser is polarized in the horizontal plane, the electric field and associated magnetic field can be expressed as:

\begin{array}{l} E_{L x} (z, t) = E_{0} \sin (k_{L} (z - c t) + φ) \\ B_{L y} (z, t) = \frac{E_{0}}{c} \sin (k_{L} (z - c t) + φ), \end{array}

where

E_{0}

,

k_{L}

, and

φ

are the seed-laser amplitude, wave vector, and initial phase, respectively. The change of the electron horizontal momentum due to the interaction with the laser field in the undulator is obtained as:

\frac{d p_{x}}{d t} = e E_{L x} - e v_{z} (B_{y} + B_{L y}) = e (1 - β_{z}) E_{L x} + e c β_{z} B_{0} \cos (k_{u} z) .

The difference between the longitudinal velocity of electrons and the speed of light cannot be ignored, while the Coulomb field in the bunches can be neglected when the laser field is strong enough. We now explain the physical mechanism of ORLM using the slippage effects. The laser field and the current profile of the e beam at the entrance of the modulator are schematically illustrated in Fig. 2(a). The co-moving coordinate $s = z - c {\bar{β}}_{z} t$ denotes the longitudinal position of the electron in an electron bunch, with positive $s$ corresponding to the bunch head. We assume that the wavelength of the seed laser is twice the resonant wavelength of the one-period modulator. The electron moves backward by half of the seed wavelength relative to the seed laser after the one-period modulator. When the initial relative phase of the seed laser and the electron is as shown in Figs. 2(b) and 2(c), the electron obtains a maximum net kick in the horizontal direction because the electric field that the electron experiences stays positive or negative during the whole period. However, the kick is zero if the initial relative phase is as shown in Figs. 2(d) and 2(e). These kicks can be considerable owing to the strong seed laser which can be used to induce micro-bunching via angular modulation.

Fig. 2 Physical mechanism of angular ORLM. (a) Initial electric field of the seed laser (red) and the current profile of the e beam (blue); (b)-(e) Trajectory of the electron (blue) in the modulator and the laser field that the electron experiences (red) in different initial relative phases. The seed-laser wavelength is assumed to be twice the resonant wavelength of the one-period modulator.

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We now provide the theoretical expression of the angular modulation in the ORLM scheme. After averaging over the undulator period on both sides of Eq. (4), we obtain:

〈 \frac{d x'}{d z} 〉 = \frac{e E_{0} (1 + K_{0}^{2} / 2)}{2 γ^{3} m c^{2}} 〈 \sin (k_{L} (z - c t) + φ) - \frac{ξ}{2} \sin (k_{L} (z - c t) + φ \pm 2 k_{u} z) 〉,

where

x' \approx p_{x} / γ m c

is the horizontal divergence angle. It is convenient to rewrite

t

in terms of the variable

z

[24]:

c t (z) = \int \frac{d z}{β_{z} (z)} \approx - \frac{s}{\bar{β_{z}}} + z + \frac{z}{2 γ^{2}} (1 + \frac{K_{0}^{2}}{2}) - \frac{K_{0}^{2}}{8 k_{u} γ^{2}} \sin (2 k_{u} z) .

The wavelength of the seed laser

λ_{L}

and the undulator resonant wavelength

λ_{r}

are assumed to have the relationship

λ_{r} = n_{1} λ_{L}

. By exploiting the resonant condition

λ_{r} = λ_{u} (1 + K_{0}^{2} / 2) / 2 γ_{r}^{2},

we obtain:

k_{L} (z - c t) = \frac{k_{L} s}{\bar{β_{z}}} - n_{1} k_{u} z \frac{γ_{r}^{2}}{γ^{2}} + \frac{n_{1}}{2} ξ \sin (2 k_{u} z),

where

γ_{r}

is the resonant energy of the electron and

ξ = K_{0}^{2} / (2 + K_{0}^{2})

. Assuming a small energy deviation of

(γ - γ_{r}) / γ_{r} ≪ 1

, the divergence change of the electron due to the angular kick at the end of the modulator is obtained, as details in Appendix A:

Δ x' = \frac{e E_{0} n_{1} λ_{L}}{2 γ m c^{2}} \frac{\sin (2 N υ π)}{\sin (υ π)} [J J] \sin (k_{L} s + ϕ),

where

θ = 2 k_{u} z, υ = n_{1} / 2, x = n_{1} ξ / 2

,

N

is the total period number of the modulator,

[J J] = [J_{υ} (x) - J_{υ - 1} (x) ξ /2 - J_{υ + 1} (x) ξ /2]

,

J

is the Anger’s function [25], and

ϕ = φ - (2 N - 1) υ π

is the phase of angular modulation. Equation (8) contains the singularities due to the zero value of

\sin (υ π)

in the denominator when

υ

is an arbitrary integer. Under this circumstance, the term

\sin (2 N υ π) / \sin (υ π)

reaches its maximum value 2N, while the term

[J J]

equals 0 according to the recursion relation of the Bessel function [25]. That is, the angular modulation vanishes when

υ

is an integer. As indicated by Eq. (8), the divergence of the e beam is modulated with the same spatial frequency but different phases of the seed laser. The amplitude of angular modulation behaves as the sine-function of the undulator period number, which means that the angular modulation can vanish if the undulator period number is chosen improperly. This expression is confirmed by three-dimensional (3D) simulation results in the Discussion section.

The phase space distribution that describes the state of electrons is generally characterized by $\vec{g} = {(x x' y y' z δ)}^{T}$ , where $x, y$ , and $z$ are the horizontal, vertical, and longitudinal coordinates, respectively; $x'$ and $y'$ are the horizontal and vertical divergences, respectively; and $δ$ is the particle energy deviation. After passing through a linear Hamiltonian system, the state of electron becomes ${\vec{g}}_{1} = R {\vec{g}}_{0}$ , where $R$ is a $6 \times 6$ transfer matrix that describes the beam dynamics associated with the system [24, 26]. Considering an e beam from the LWFA with an RMS horizontal beam size and divergence of $σ_{x}$ and $σ_{x'}$ , respectively, the initial distribution of the beam can be written as [19] $f_{0} (x, x') = N_{0} / (2 π σ_{x} σ_{x'}) \exp (- x^{2} / σ_{x}^{2}) \exp (- x'^{2} / σ_{x'}^{2})$ , where $N_{0}$ is the number of electrons per unit length. Assuming that the drift length between the LWFA and the modulator is $L_{1}$ , the horizontal position is changed to $x_{1} = x + L_{1} x'$ . After interaction with the seed laser in the modulator, the divergence and the horizontal position are changed to $x'_{1} = x' + A_{a} \sin (k_{L} s + ϕ)$ and $x_{2} = x_{1} + L_{m} x'$ , respectively, where $A_{a}$ is the amplitude of angular modulation and $L_{m}$ is the length of the modulator. The distribution of the e beam becomes

\begin{matrix} f_{1} (s, x_{2}, x_{1}') = \frac{N_{0}}{2 π σ_{x} σ_{x'}} \exp (- \frac{{(x_{2} - L_{T} x_{1}' + L_{T} A_{a} \sin (k_{L} s + ϕ))}^{2}}{2 σ_{x}^{2}}) \\ \exp (- \frac{{(x_{1}' - A_{a} \sin (k_{L} s + ϕ))}^{2}}{2 σ_{x'}^{2}}) . \end{matrix}

Where

L_{T} = L_{1} + L_{m}

is the total length of the drift section and the modulator. The e beam is then sent to a dispersion section that includes two focusing quadrupoles and a weak dipole. A brief description of the dispersion section is presented in Appendix B. The transfer matrix element

R_{56}

can be ignored owing to the weak dipole [26], which will be verified through simulation. Only the elements

R_{51}

and

R_{52}

, which represent the coupling between the transverse and longitudinal motion, are considered in our theoretical model. Because of the symplectic condition [26], one finds that the elements

R_{1j}

and

R_{2j}

remain for

j = 1, 2, 6

, which will be considered in the simulation results. Then, after the beam passes through the dispersion section, the particle longitudinal position is changed to

s_{1} = s + R_{51} x_{2} + R_{52} x_{1}'

and the electron distribution changes to

\begin{matrix} f_{1} (s_{1}, x_{2}, x_{1}') = \frac{N_{0}}{2 π σ_{x} σ_{x'}} \exp (- \frac{{(x_{2} - L_{T} x_{1}' + L_{T} A_{a} \sin ς)}^{2}}{2 σ_{x}^{2}}) \\ \exp (- \frac{{(x_{1}' - A_{a} \sin ς)}^{2}}{2 σ_{x'}^{2}}) . \end{matrix}

Where

ς = k_{L} (s_{1} - R_{51} x_{2} - R_{52} x_{1}') + ϕ

is the phase of the electron at the entrance of the radiator. After integrating over

x_{2}

and

x_{1}'

, the beam current distribution can be expanded to a Fourier series as

N (s) = N_{0} (1 + 2 \sum_{n_{2} = 1}^{\infty} b_{n_{2}} \sin (n_{2} k_{L} s + ϕ_{n_{2}}))

, where we define the bunching factor of the n₂^th harmonic as

b_{n_{2}} = J_{n_{2}} (- n_{2} k_{L} A_{a} R_{52}) \exp [- (\frac{1}{2} n_{2}^{2} k_{L}^{2} (R_{51}^{2} σ_{x}^{2} + {(R_{51} L_{T} + R_{52})}^{2} σ_{x'}^{2}))] .

Here, the first term is maximized when the argument of the Bessel function is equal to

n_{2} + 0.81 n_{2}^{1 / 3}

, so that

R_{52}

can be identified for a given value of

A_{a}

. The second term is an attenuation term that is maximized when

R_{51} = 0

or

R_{51} L_{T} + R_{52} =0

. For e beams from an LWFA, the initial beam size is small (~1 μm), which means that

R_{51} L_{T} + R_{52} =0

should be satisfied to obtain a large bunching factor.

3. Simulation results

To investigate the feasibility of the proposed scheme, 3D simulations are performed with typical parameters of the e beam and the undulator, as listed in Table 1. The e-beam parameters chosen in our simulation are based on a previous experimental work [21]. The e-beam size $σ_{x}$ and duration at the exit of the LWFA were not measured but rather assumed to be 1 μm and 3 fs, respectively, according to the reported values [27–29]. The minimal e-beam divergence of 0.1 mrad was obtained [21], which results in a normalized transverse emittance of 0.1 μm. In the simulation, the drift length between the LWFA and the modulator is 1 m. The waist radius of the seed laser is 200 μm. The e beam, together with the seed laser, is sent into a short modulator with only one period and a period length of 4 cm. The peak power of the seed laser overlapped with the e beam is assumed to be 1 TW. The resonant wavelength of the modulator is 400 nm, while the seed laser wavelength is 800 nm, which can be regarded as the 1/2 harmonic of the modulator. The e beam moves half a seed wavelength backward relative to the seed laser after one period of the modulator. The interaction between the e beam and the seed laser is simulated using a 3D algorithm [30] based on fundamental electrodynamics.

Table 1. Electron beam and undulator parameters used in our scheme for CHG.

View Table

After passing through the modulator, the e beam is sent to a dispersion section, which includes two focusing quadrupoles and one dipole magnet. The dipole magnet has a length of 0.1m and a bending angle of 0.0021 rad. The distance between the modulator and the first quadrupole is 0.2 m, which is the same as that between the second quadrupole and the dipole, and that between the two quadrupoles is 1 m. The strengths of the quadrupoles are properly adjusted to achieve the optimized condition of R₅₂ = 0.0029 m and R₅₁ = −0.0029. The corresponding $R_{56}$ for the dispersion section is 7.49 × 10⁻⁸ m. The e beam after leaving the modulator was tracked using the ELEGANT code [31], considering second-order transport effects.

With the aforementioned parameters, the z – x’ and z – p phase space distributions of the e beam at the exit of the modulator are shown in Figs. 3(a) and 3(b), respectively, where $p = (γ - γ_{0}) / γ_{0}$ is the dimensionless energy deviation of a particle. The amplitude of the angular modulation induced by the off-resonant seed laser is 46 μrad, which is comparable to the initial value of the e beam. The e-beam energy spread is 1.56% at the exit of the modulator, which avoids the increase of the bandwidth due to large energy modulation. Despite the weak angular modulation, we note that the transverse coordinate is closely related to the transverse divergence because of the long drift section. This correlation can be corrected by introducing a weak dispersion [32]. Strong coherent micro-bunching at the exit of the dispersion section is shown in Fig. 3(c), and the corresponding bunching factor for various harmonic numbers is shown in Fig. 3(d). The simulation results agree well with the theoretical predictions for a small harmonic number. However, the deviation becomes prominent when the harmonic number exceeds 15, which is mainly caused by the element $R_{56}$ of the dispersion section. For generating temporally coherent radiation, the bunching factor must be larger than 1% to effectively suppress the shot noise [33]. In our scheme, the bunching factor for the 20th harmonic is approximately 8.5%, which is sufficient for generating intense coherent harmonic radiation.

Fig. 3 (a) $z - x'$ and (b) $z - p$ phase space distribution of the e beam at the exit of the modulator. The seed wavelength is 800 nm and the resonant wavelength of the modulator is 400 nm. (c) $z - x'$ phase space distribution of the e beam and (d) the corresponding bunching factor at the entrance of the radiator.

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The modulated e beam is then sent into a radiator with period length of 2.5 cm and a total length of 1.5 m. The corresponding resonant wavelength is 40 nm, which is the 20th harmonic of the seed laser. The radiation process was simulated using the 3D time-dependent code GENESIS [34]. Figures 4(a) and 4(b) show the results for both the CHG scheme and the SASE scheme. The FEL performance is significantly enhanced in the proposed scheme owing to the pre-bunching of the e beam. The radiation exceeds 160MW in peak power with a 0.7% bandwidth, which has a significant improvement compared with the SASE scheme.

Fig. 4 (a) FEL peak power as a function of the undulator distance. (b) Single-shot spectra for the CHG scheme (red solid line) and SASE scheme (blue dashed line).

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4. Discussion

For a given electric field $E_{0}$ , the amplitude of the angular modulation scales with $λ_{L} / γ$ according to Eq. (8), while for a TEM01 mode, the scaling follows $ω_{0} / γ^{2}$ [18]. The efficacy of the angular modulation in our proposed scheme is comparable to that in the scheme employing TEM01 mode for highly relativistic e beams (GeV) and an 800-nm laser with 100-μm spot size.

The performance of the FEL radiation in our proposed scheme relies strongly on the period number of the modulator and the ratio of the wavelength of the modulator to the seed wavelength according to the aforementioned theoretical analysis. We define the relative angular-modulation amplitude as $A_{a} / σ_{x'}$ , which acts as a sinusoidal curve with the increasing period number, as shown in Fig. 5(a). Figure 5(a) indicates that the angular-modulation amplitude is maximized when the distance that the electron moves backward relative to the seed laser is an odd multiple of the half-wavelength of the seed laser. However, the angular modulation vanishes when the distance is a multiple of the wavelength of the seed laser. Figure 5(b) shows the amplitude of angular modulation as a function of the resonant wavelength of the one-period modulator. The angular modulation vanishes only when the laser frequency equals the harmonics of the resonant frequency of the modulator. These simulation results exhibit remarkable agreement with the theoretical expectations in Eq. (8). Figures 5(c) and 5(d) show the bunching factors for the 10th, 15th, and 20th harmonics with different initial beam sizes and beam divergence angles, respectively. The bunching factor decreases as the initial beam size increases when the divergence angle remains constant, as predicted by Eq. (11). The divergence angle of the e beam must be optimized to improve the up-conversion efficiency. Our proposed scheme provides an alternative route for optimizing the property of the e beam to realize the LWFA-based FEL.

Fig. 5 (a) Normalized angular-modulation amplitude as a function of the period number of the modulator with resonant wavelengths of 400, 200 and 100 nm, respectively. (b) Normalized angular-modulation amplitude as a function of the resonant wavelength of the one-period modulator. The bunching factors for the 10th, 15th and 20th harmonics with different (c) initial beam sizes and (d) initial divergence angles of the e beam from the LWFA. The 800-nm, 30-fs seed laser has a peak power of 1 TW.

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5. Conclusion

We proposed a new scheme to generate short-wavelength coherent radiation using ORLM. A theoretical analysis and simulation of ORLM were performed. It is found that the maximum bunching factor can be achieved by properly choosing the period number and resonant wavelength of the modulator. As an example, simulations demonstrate an ORLM-based FEL operating at 40 nm with an output power of 160 MW employing an 800-nm seeding laser. In addition to the LWFA, the proposed scheme can be used with the e beams from SRs having a small geometric vertical emittance [19]. To reduce the wavelength, one may increase the seed laser power or decrease the emittance of the e beams. A cascade ORLM on angular can also be used to improve the harmonic up-conversion efficiency. Further investigations into driving a short wavelength and LWFA-based FELs are ongoing.

Appendix A

The change of the electron horizontal divergence in the modulator is described by Equation (5). By inserting Equation (7) into Equation (5) and integrating over the variable $z$ , we obtain

\begin{matrix} Δ x' = \frac{e E_{0} (1 + K_{0}^{2} /2)}{2 γ^{3} m c^{2}} \int [\sin (k_{L} (z - c t) + φ) - \frac{ξ}{2} \sin (k_{L} (z - c t) + φ \pm 2 k_{u} z)] d z \\ = \frac{e E_{0}}{γ m c^{2}} \frac{n_{1} λ_{L}}{λ_{u}} [\begin{array}{l} \int \sin (\frac{k_{L} s}{{\bar{β}}_{z}} - 2 k_{u} z \frac{n_{1}}{2} + n_{1} \frac{ξ}{2} \sin (2 k_{u} z) + φ) d z \\ - \frac{ξ}{2} \sin (\frac{k_{L} s}{{\bar{β}}_{z}} - 2 k_{u} z (\frac{n_{1}}{2} \mp 1) + n_{1} \frac{ξ}{2} \sin (2 k_{u} z) + φ) d z \end{array}] \\ = \frac{e E_{0}}{γ m c^{2}} \frac{n_{1} λ_{L}}{λ_{u}} \frac{1}{2 k_{u}} [\begin{array}{l} \sin (\frac{k_{L} s}{{\bar{β}}_{z}} + φ) \int_{0}^{4 π N} \cos (x \sin (θ) - υ θ) d θ \\ + \cos (\frac{k_{L} s}{{\bar{β}}_{z}} + φ) \int_{0}^{4 π N} \sin (x \sin (θ) - υ θ) d θ \\ - \frac{ξ}{2} \sin (\frac{k_{L} s}{{\bar{β}}_{z}} + φ) \int_{0}^{4 π N} \cos (x \sin (θ) - (υ \mp 1) θ) d z \\ - \frac{ξ}{2} \cos (\frac{k_{L} s}{{\bar{β}}_{z}} + φ) \int_{0}^{4 π N} \sin (x \sin (θ) - (υ \mp 1) θ) d z \end{array}], \end{matrix}

where

θ = 2 k_{u} z, υ = n_{1} / 2, x = n_{1} ξ / 2

, and

N

is the period number of the modulator. To simplify our theoretical model, we introduce two basic functions:

J_{υ} (x) = \frac{1}{π} \int_{0}^{π} \cos (x \sin (θ) - υ θ) d θ,

E_{υ} (x) = \frac{1}{π} \int_{0}^{π} \sin (x \sin (θ) - υ θ) d θ,

where

J_{υ} (x)

and

E_{υ} (x)

are Anger’s function and Weber’s function, respectively [25]. By substituting these expressions into Eq. (12) and assuming that

υ

is not an integer, we obtain

\begin{array}{l} \int_{0}^{4 π N} sin (x sin (θ) - υ θ) d θ \\ = π \frac{sin (2 N υ π)}{sin (υ π)} {\begin{cases} E_{υ} (x) \cos ((2 N - 1) υ π) - E_{- υ} (x) \cos (2 N υ π) \\ - J_{υ} (x) sin ((2 N - 1) υ π) - J_{- υ} (x) sin (2 N υ π) \end{cases}}, \end{array}

and

\begin{array}{l} \int_{0}^{4 π N} \cos (x \sin (θ) - υ θ) d θ \\ = π \frac{\sin (2 N υ π)}{\sin (υ π)} {\begin{cases} J_{υ} (x) \cos ((2 N - 1) υ π) + J_{- υ} (x) \cos (2 N υ π) \\ + E_{υ} (x) \sin ((2 N - 1) υ π) - E_{- υ} (x) \sin (2 N υ π) \end{cases}} . \end{array}

When Eqs. (15) and (16) are substituted into (12), the first two terms on the right-hand side change to

\frac{e E_{0} n_{1} λ_{L}}{2 γ m c^{2}} \frac{\sin (2 N υ π)}{\sin (υ π)} J_{υ} (x) \sin (k_{L} s + ϕ)

. The last two terms in Eq. (12) can be similarly derived. The total angular modulation can then be determined:

Δ x' = \frac{e E_{0} n_{1} λ_{L}}{2 γ m c^{2}} \frac{\sin (2 N υ π)}{\sin (υ π)} [J_{υ} (x) - \frac{ξ}{2} J_{υ - 1} (x) - \frac{ξ}{2} J_{υ + 1} (x)] \sin (k_{L} s + ϕ) .

where

ϕ = φ - (2 N - 1) υ π

is the phase of the angular modulation. When

υ

is an arbitrary integer, the Anger function equals the corresponding Bessel function with the same order, and the Eq. (17) changes to

Δ x' = \frac{e E_{0} n_{1} λ_{L}}{2 γ m c^{2}} 2 N [J_{υ} (x) - \frac{ξ}{2} J_{υ - 1} (x) - \frac{ξ}{2} J_{υ + 1} (x)] \sin (k_{L} s + ϕ) .

The term

[J_{υ} (x) - \frac{ξ}{2} J_{υ - 1} (x) - \frac{ξ}{2} J_{υ + 1} (x)]

is equal to zero, according to the recursion relation of the Bessel function [25], and the angular modulation vanishes.

Appendix B

The purpose of this part is to provide a brief description of the dispersion section using the transfer-matrix method. An electron located in the six-dimensional phase space co-moving with the electron beam is characterized by the vector $\vec{g} = {(x x' y y' z δ)}^{T}$ , where $x, y$ , and $z$ are the horizontal, vertical, and the longitudinal coordinates, respectively; $x'$ and $y'$ are the horizontal and vertical divergences, respectively; and $δ$ is the particle energy deviation. When the electron moves through a linear system, the phase space of each electron is transformed from the initial state to the final state as follows [24]:

{\vec{g}}_{1} = R {\vec{g}}_{0},

where

R

is a

6 \times 6

transfer matrix associated with the system.

In our proposed scheme, the dispersion section includes a pair of quadrupoles and a dipole, whose transfer matrix is assumed to be $M_{Q 1}, M_{Q 2}$ , and $M_{B}$ [26]. There is a drift section between any two magnet devices, and the corresponding transfer matrices of the drift section are $M_{D 1}, M_{D 2}, M_{D 3}$ , and $M_{D 4}$ [26]. Thus, the total transfer matrix of the dispersion section is

R = M_{D 4} \cdot M_{B} \cdot M_{D 3} \cdot M_{Q 2} \cdot M_{D 2} \cdot M_{Q 1} \cdot M_{D 1},

with the transfer elements

R_{i j} (i, j = 1, 2 \dots 6)

. The essential transfer matrix elements used to convert the angular modulation into the density modulation are

R_{51}

and

R_{52}

, which represent the coupling between the longitudinal and transverse motion. The quadrupole and dipole strength are properly adjusted to achieve

R_{51} L_{1} + R_{52} =0

for the maximum bunching factor.

Funding

National Natural Science Foundation of China (Grants No. 11127901, No. 11425418, No. 61521093, and No.11505263); the Shanghai Sailing Program (No. 18YF1426000); the Strategic Priority Research Program (B) (Grant No. XDB16); the Youth Innovation Promotion Association CAS; and the State Key Laboratory Program of the Chinese Ministry of Science and Technology.

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Electron beam energy $γ_{0}$	500 MeV
Relative energy spread $σ_{δ}$	1%
RMS transverse beam size $σ_{x} / σ_{y}$	1/1 μm
Normalized emittance $γ_{0} ε_{x} / γ_{0} ε_{y}$	0.1/0.1 μm
RMS bunch duration $T$	3 fs
Peak current $I_{p}$	10 kA
Wavelength for seed laser $λ_{L}$	800 nm
Laser waist size in modulator $σ_{L}$	200 μm
$N_{p} \times λ_{u}$ for the modulator	1 × 4 cm
Resonant wavelength for modulator $λ_{1}$	400 nm
$N_{p} \times λ_{u}$ for the radiator	60 × 2.5 cm
Resonant wavelength for radiator $λ_{2}$	40 nm

Coherent X-ray source generation with off-resonance laser modulation

Abstract

1. Introduction

2. Physical mechanism of ORLM

3. Simulation results

4. Discussion

5. Conclusion

Appendix A

Appendix B

Funding

References

Cited By

Figures (5)

Tables (1)

Equations (20)

Optics Express