The detuning effect of crystal monochromator in self-seeding and oscillator free electron laser

Chuan Yang; Cheng-Ying Tsai; Guanqun Zhou; Xiaofan Wang; Ye Hong; Emily Dong Krug; Alexandra Li; Haixiao Deng; Duohui He; Juhao Wu

doi:10.1364/OE.27.013229

1. Introduction

The successful operation of X-ray free electron lasers (FELs) [1–3] around the world has opened the route for a new era in science including materials, chemistry, and biology. To date, most of the FEL facilities operate in self-amplified spontaneous emission (SASE) [4,5] mode. However, the temporal coherence of SASE is poor due to starting from shot noise. In order to achieve fully coherent FEL [6], external laser-seed scheme was proposed in references [7–9]. Unfortunately, it is difficult to carry out external laser seeding down to X-ray regime. Therefore, self-seeding scheme was developed in soft X-ray regime [10]. In hard X-ray regime, a self-seeding scheme was proposed [11] by adopting four-bounce crystal monochromator. Afterwards, a more compact single-crystal self-seeding scheme was suggested in a reference [12], and has been successfully demonstrated at Linac Coherent Light Source (LCLS) in 2012 [13]. Single-crystal self-seeding FEL shown in Fig. 1 consists of three parts, including a SASE FEL operating in linear regime, a diamond crystal monochromator oriented in Bragg geometry, and a FEL amplifier.

Fig. 1 Schematic presentation of HXRSS FEL. The undulator U1 working in the linear regime is used to generate SASE radiation. The SASE pulse goes through the single-crystal monochromator, and the delayed monochromatic wake seed is produced. Then the monochromatic seed overlaps with the electron beam, and is amplified to saturation.

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For the purpose of high output efficiency of self-seeding FEL, several approaches can be adopted. Tapering the undulator is one of the approaches. Jiao et al. developed a model to extended the Kroll-Morton-Rosenbluth (KMR) theory [14] to include diffraction effect, wave guiding and particle trapping [15]. Meanwhile, a multi-dimensional optimization approach with multi-objective genetic algorithm is developed in a reference [16]. Recently, Tsai et al. have proposed a novel taper scheme for efficiency enhancement based on preservation of electron phase space area [17]. Different from the undulator tapering approaches, a new method is proposed to maximize the spectral flux by optimizing the electron bunch delay [18]. Although these schemes can dramatically increase the FEL efficiency, the cost is longer undulators or more complex software and hardware systems.

In this paper, a simple approach is proposed to enhance the seed power of HXRSS by tuning the incident angle off the Bragg condition corresponding to the central photon energy of the input X-ray pulse, which can further improve the output efficiency of HXRSS FEL. We first review the dynamical theory of X-ray diffraction to understand the diffraction geometry and crystal detuning effect. Then we focus on the discussion about crystal detuning effect both in Bragg and Laue geometries. Afterwards, we present the crystal detuning effect in HXRSS FEL. For a given diffraction geometry, the seed power can be maximized by tuning the incident angle. Furthermore, we also investigate the detuning effect in XFELO. We define the detuning ratio to evaluate whether the detuning effect will make an impact on FEL efficiency or not.

2. Dynamical theory of X-ray diffraction

In this section we briefly review the dynamical theory of X-ray diffraction [19–21]. We first establish the Maxwell’s equations inside the crystal in Sec. 2.1. Afterwards, we obtain the general solution of X-ray diffraction inside the crystal. Then the general solution is reduced to the two-beam diffraction solution in Sec. 2.2. Moreover, we give an interpretation of the dispersion surface in Sec. 2.3. Finally, the Bragg and Laue geometries are briefly introduced in Sec. 2.4.

2.1. Maxwell’s equations inside the crystal

Dynamical theory of X-ray diffraction was developed to find the solution for diffraction waves inside a perfect crystal. The Maxwell’s equations in a crystal can be described as

\nabla \times E = - \frac{\partial B}{\partial t}, \nabla \times H = \frac{\partial D}{\partial t}, \nabla \cdot D = 0, \nabla \cdot B = 0

D = ε_{0} E + P, P = ε_{0} χ_{e} E, B = μ_{0} (H + M), M = χ_{m} H

Where E, D, B, H, P, and M are the electric field, the electric displacement, the magnetic induction, the magnetic field, the electric polarization, and the magnetization, respectively. ε₀ is the dielectric constant, and μ₀ is the magnetic permeability in vacuum. χ_e and χ_m are the electric susceptibility and magnetic susceptibility, respectively. For a diamond crystal, the magnetization effect is negligible. Let us consider a Bravais lattice crystal with charges located at the lattice points. Under the excitation of electromagnetic wave, the charges oscillate in the same way. If the ρ(r) is the charge density, the polarization of the crystal is given by

P (r) = \frac{ρ (r) e^{2} E (r)}{m ({ω_{0}}^{2} - ω^{2}) - i ω α}

Where e and m are the charge and the mass of the electron. ω₀, ω and α are the resonant frequency of a dipole, the frequency of electromagnetic wave and the damping coefficient, respectively. For X-ray, ω₀ ≪ ω, the damping coefficient α is small. Let us substitute Eq. (2) into P = ε₀χ_eE, and the electric susceptibility of a three-dimensional periodic crystal can be expressed as

χ_{e} (r) = - Γ V_{c} ρ (r), Γ = \frac{r_{e} λ^{2}}{π V_{c}}, r_{e} = \frac{e^{2}}{4 π ε_{0} m c^{2}}

Where r_e and λ are the classical radius of the electron and the wavelength of incident wave, respectively. V_c is the volume of unit cell of crystal. Γ defines a dimensionless scattering amplitude per electron in an unit cell. Meanwhile, in a perfect crystal, the charge density ρ(r) is a periodic function of the three-dimensional space. Therefore, the electric susceptibility can be described in the reciprocal space by the Fourier series

χ_{e} (r) = \sum_{h} χ_{h} exp (i h \cdot r), χ_{h} = - Γ F_{h}, F_{h} = \int ρ (r) exp (- i h \cdot r) d r

Where h(|h| = 2π/d_h) is the reciprocal lattice vector, and d_h is the interplanar distance. χ_h is the Fourier coefficient of the electric susceptibility χ_e(r), and is related to the structure factor F_h. The scattered wave in the crystal can be given by

E (r) = exp (- i ω t) \sum_{h} E_{h} exp (i K_{h} \cdot r), H (r) = exp (- i ω t) \sum_{h} H_{h} exp (i K_{h} \cdot r)

Where K_h is the wave vector of the scattered wave inside the crystal. The scattered wave vector K_h is related with the incident wave vector K₀ by

K_{h} = K_{0} + h

Up to now we have already obtained the basic equations inside the crystal. We will solve these equations in the next subsection and get the two-beam diffraction solution.

2.2. Solution for X-ray diffraction inside the crystal

In this subsection, we will discuss the solution of the Maxwell’s equations inside the crystal. A general solution is first obtained. Then the solution of one-beam diffraction is discussed. Finally, the two-beam diffraction solution is analyzed.

From Eq. (1)–(6), we obtain the general solution relating the components of electric field E and wave vector K.

[k^{2} (1 - Γ F_{0}) - K_{h}^{2}] E_{h} - k^{2} Γ \sum_{h \neq p} F_{h - p} E_{p} = 0

Where k(|k| = 2π/λ) is the wave vector in vacuum. Equation (7) relates a given electric field E_h and other coherent scattered electric field E_p associated with all other reciprocal vectors p. We can imagine that the number of equations is huge. Fortunately, almost all of those equations can be neglected because the related components of electric field E_p are too small to have an effect.

In the case of one-beam diffraction, there are no other scattered waves except for the forward scattered wave. If the incident wave is E₀, equation (7) can be reduced to

[k^{2} (1 - Γ F_{0}) - K_{0}^{2}] E_{0} = 0

and the solution of Eq. (8) is

K_{0} = k \sqrt{(1 - Γ F_{0})}

We can find that the wave vector inside the crystal is shorter than the wave vector in vacuum. Therefore the index of refraction can be expressed as

n = \sqrt{(1 - Γ F_{0})} \approx 1 - \frac{r_{e} λ^{2} F_{0}}{2 π V}

For the two-beam diffraction case, there are two strongly-scattered waves, including forward scattered wave (h = 0) and reflected wave with nonzero h. Therefore, there are two waves are coupled in the crystal, and Eq. (7) can be reduced to

[k^{2} (1 - Γ F_{0}) - K_{0}^{2}] E_{0} - k^{2} Γ {PF}_{\bar{h}} E_{h} = 0

- k^{2} Γ {PF}_{h} E_{0} + [k^{2} (1 - Γ F_{0}) - K_{h}^{2}] E_{h} = 0

Where P is the polarization factor (P = 1 for σ-polarization, P = cos 2θ for π-polarization). h and h̄ have opposite directions. Non-trivial solutions for E₀ and E_h are only possible if the determinant of Eq. (11) equals to zero

| \begin{matrix} [k^{2} (1 - Γ F_{0}) - K_{0}^{2}] & k^{2} Γ {PF}_{\bar{h}} \\ - k^{2} Γ {PF}_{h} & [k^{2} (1 - Γ F_{0}) - K_{h}^{2}] \end{matrix} | = 0

and let us define

2 k ξ_{0} = K_{0}^{2} - k^{2} (1 - Γ F_{0})

2 k ξ_{h} = K_{h}^{2} - k^{2} (1 - Γ F_{0})

Here, we make an approximation, K₀ ≈ K_h ≈ k. Thus, equation (13) can be rewritten as

ξ_{0} \approx K_{0} - k (1 - \frac{1}{2} Γ F_{0})

ξ_{h} \approx K_{h} - k (1 - \frac{1}{2} Γ F_{0})

and Eq. (12) becomes

ξ_{0} ξ_{h} = \frac{1}{4} k^{2} Γ^{2} P^{2} F_{h} F_{\bar{h}}

It is interesting to note that Eq. (15) is a parabolic relation between ξ₀ and ξ_h. The geometrical interpretation of ξ₀ and ξ_h will be discussed in the next subsection.

2.3. Dispersion surface in reciprocal space

In this subsection, we will make an interpretation of the solution of two-beam diffraction case by introducing the dispersion surface in the reciprocal space of crystal. According to Eq. (14), we figure out ξ₀ and ξ_h are the difference between K₀, K_h and nk. That means the scattered wave vectors slightly depart from the Bragg condition.

We first construct the schematic of dispersion surface shown in Fig. 2(a). L (Laue point) is the intersection of two spheres centered in origin O and H (hlk) with radii of k, while Q (Lorentz point) is the intersection of two spheres also centered in O and H with radii nk.

Fig. 2 (a) The solid purple curve is the dispersion surface; L is the Laue point, and Q is the lorentz point; $\bar{LO} = \bar{LH} = k$ ; $\bar{QO} = \bar{QH} = nk$ ; $\bar{AO}$ and $\bar{AH}$ are the coupled wave vectors. (b) l₀ and l_h are the tangents to the spheres of radius nk; l′₀ and l′_h are the tangents to the spheres of radius k; $Δ θ = \bar{LM} / k$ is the deviation from the Bragg angle of the incident wave.

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Let us assume point A, nearby L and Q, from which the distances to the spheres centered in O and H with radii nk are ξ₀ and ξ_h. Obviously, A satisfies Eq. (15) (two-beam diffraction case) and can not lie on the spheres we mentioned above. It lies on the connection surface between the two spheres with radii of nk. Far from the intersection of the spheres, A lies on one of the two spheres (r = nk). That means there is one wave propagating in the crystal corresponding to Eq. (8). When A lies on the connection surface, there are two waves and both of the wave vectors slightly depart from nk. The connection surface has two branches. Therefore, we call it as dispersion surface. The point A connecting two waves is called tiepoint.

The dispersion surface is a surface of revolution around the axis $\bar{OH}$ . When close to Laue point L and Lorentz point Q, the relevant four spheres can be replaced by their tangential planes, and the dispersion surface can be approximate to a hyperboloid. The intersections between the four tangential planes and the diffraction plane are four straight lines (l₀, l_h, l′₀, l′_h), while the intersection between the dispersion plane and the diffraction plane is a hyperbola whose asymptotes are l₀ and l_h shown in Fig. 2(b).

We have understood the dispersion surface. The position of tiepoint A excited on the dispersion surface depends on the diffraction geometry which will be discussed in the next subsection.

2.4. Bragg and Laue diffraction geometries

In this subsection, we first introduce the excitation of the tiepoints. Then we discuss the diffraction geometry including Bragg and Laue geometries. Finally, the asymmetry ratio is defined to characterize the diffraction geometry.

The dispersion surface just illustrates the possible solutions of the Maxwell’s equations in the crystal. However, we still do not know which tiepoint on the dispersion surface is excited. The position of the tiepoint determines the propagation direction and the amplitudes of the coupled waves, and can be found by taking the boundary conditions into account. The boundary conditions include the conditions for the wave vectors and the amplitudes. Here we consider the boundary conditions for the wave vectors. According to the law of refraction (Snell’s law), we can know the difference between the incident wave vector and the refracted wave vector is parallel to the normal to the interface.

Let us draw a line parallel to the normal to the crystal surface from M. The intersections of this line and the dispersion surface are T₁ and T₂. $\bar{MO}$ and $\bar{T_{j} O}$ (j = 1, 2) are the incident and refracted wave vectors, respectively. According to the boundary conditions, T₁ and T₂ are the excited tiepoints. There are two different situations of the position of T₁ and T₂. When T₁ and T₂ lie on the same branch of the dispersion surface or are imaginary points, we call Bragg geometry shown in Fig. 3(a). In this case, the reflection beam is towards outside of the crystal. When T₁ and T₂ lie on different branches of the dispersion surface, we call Laue geometry shown in Fig. 3(b). In this case, the reflection beam is towards inside of the crystal.

Fig. 3 (a) Bragg diffraction geometry. (b) Laue diffraction geometry. The white lines are the atomic reflecting planes. θ_B is the Bragg angle. η is the asymmetry angle. n is the normal to the crystal surface. The positive angle is corresponding to the counterclockwise direction.

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The angle between the incident wave and the reflecting atomic plane is the Bragg angle θ_B, and the angle between the crystal surface and the reflecting atomic plane is the asymmetry angle η. The normal to the crystal surface n towards inside of the crystal. Ψ₀ and Ψ_h are the angles between the normal n and the incident direction, the reflection direction, respectively. The angular relations are in the following

Ψ_{0} = θ_{B} + η - \frac{π}{2}, Ψ_{h} = θ_{B} - η + \frac{π}{2}

The asymmetry angle η determines the diffraction geometry, and we have the angular relations

- θ_{B} < η < θ_{B} (Bragg), θ_{B} < η < π - θ_{B} (Laue)

Let us set γ₀ = cos Ψ₀, and γ_h = cos Ψ_h. Here, we define the asymmetry ratio

γ = \frac{γ_{h}}{γ_{0}} = \frac{cos Ψ_{h}}{cos Ψ_{0}}

The asymmetry ratio describes the diffraction geometry. Note that the asymmetry ratio is always negative for the Bragg geometry and is all positive for the Laue geometry. For symmetric Bragg geometry(η = 0), the asymmetry ratio γ = −1. For symmetric Laue geometry (η = π/2), the asymmetry ratio γ = 1. The diffraction geometry will affect the detuning effect, which will be discussed in the next section.

3. Detuning effect of crystal monochromator

The crystal detuning effect is defined as the deviation from the Bragg’s angle of the middle of reflection domain. In practice, the incident wave vector corresponding to the middle of reflection domain is different for the Bragg condition (geometrical theory) and the dynamical theory. Because the former does not take the refraction effect into consideration. The angle of the two corresponding incident directions, θ_os, is defined as the deviation from the Bragg’s angle of the middle of reflection domain.

The gap between the two branches of the dispersion surface can be related with the bandwidth of the reflection domain. I is the intersection of the normal to the crystal surface drawn from the Lorentz point Q with the tangent l′₀ shown in Fig. 3. Therefore, $\bar{IO}$ is the incident direction of the middle of reflection domain for dynamical theory. The angle θ_os from the incidence $\bar{LO}$ of Bragg condition to $\bar{IO}$ can be written as

Δ θ_{os} = \bar{LI} / k

Let us define ζ_o and ζ_h as the distances from I to l_o and l_h. We know the distance between l_o and l′_o (and between l_h and l′_h) is −kχ₀/2. Now we can build the relevant equations

ζ_{o} = - k \frac{χ_{0}}{2} = γ_{0} \bar{QI}, ζ_{h} = - k \frac{χ_{0}}{2} - \bar{LI} sin 2 θ_{B} = γ_{h} \bar{QI}

Substituting Eq. (20) in Eq. (19), we have

Δ θ_{os} = - \frac{χ_{0} (1 - γ)}{2 sin 2 θ_{B}}

Note that Δθ_os is zero for symmetric Laue geometry (γ = 1). For the Bragg geometry, γ is negative, and Δθ_os ≠ 0. Therefore, there is always a deviation in Bragg geometry.

According to Bragg condition 2d sin θ_B = λ_B, the Bragg photon energy shift caused by the incident angle deviation can be expressed as

\frac{Δ ℰ}{ℰ_{B}} = - cot θ_{B} Δ θ,

Where ℰ_B, λ_B, θ_B, Δℰ, and Δθ are the Bragg photon energy, the Bragg wavelength, the Bragg angle, the Bragg photon energy shift, and the angle deviation, respectively. Substituting Eq. (21) into Eq. (22), we convert the angular detuning to the equivalent photon energy detuning Δℰ

Δ ℰ = - ℰ_{B} \frac{Δ θ_{os}}{tan θ_{B}} = ℰ_{B} \frac{χ_{0} (1 - γ)}{4 {sin}^{2} θ_{B}}

Equation (23) describes the fact, for a given incident Bragg angle, the photon energy corresponding to the middle of reflection domain has a blue shift relative to the Bragg photon energy in Bragg geometry.

Figure 4(a) shows the photon energy detuning (γ = −1) of diamond crystal in different atomic reflection planes, and we find that the detuning in the photon energy range of 3.25 keV∼ 15 keV is less than 1.5 eV which is much smaller than the typically SASE bandwidth (∼ 10 eV) of LCLS. However, it is impossible to orient each reflecting atomic plane to symmetric Bragg (Laue) geometry for a crystal. The detuning as a function of asymmetry ratio in Bragg and Laue geometries is shown in Fig. 4(b) and Fig. 4(c). For the C (111) case, the variation of detuning is 0.7 eV ∼ 27.2 eV in the asymmetry ratio range −80 ∼ 0 in Bragg geometry. For some diffraction geometries, the detuning is comparable with or larger than the FEL bandwidth. Therefore, we have to tune the incident angle to improve the seed efficiency.

Fig. 4 (a) The photon energy detuning of diamond crystal for different reflection planes in symmetric Bragg geometry(γ = −1). (b) Photon energy detuning varies as a function of γ in Bragg geometry (ℰ_B =8300 eV). (c) Photon energy detuning varies as a function of γ in Laue geometry (ℰ_B =8300 eV). Here, the lines with different colors are corresponding to the atomic reflection planes with different Miller indices.

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In this section, we have discussed the crystal detuning effect. The further discussion of the detuning effect in the application of self-seeding FEL and XFELO will be analyzed in the following sections.

4. Crystal detuning in hard X-ray self-seeding FEL

In this section, we will investigate the seed energy enhancement by tuning the incident angle off the Bragg condition where the Bragg photon energy is corresponding to the central photon energy of the input SASE spectrum. The normalized bandwidth of SASE is Δω/ω₀ ∼ ρ, where ρ is the FEL Pierce parameter. We can obtain the FEL spectrum before the crystal with the help of FEL simulation code GENESIS [22]. In our study, the SASE part consists of 10 undulator segments. The seed can be produced by the interaction between X-ray pulse and the diamond crystal [23,24]. The relevant FEL simulation parameters are summarized in Table 1.

Table 1. Parameters of electrum beam and undulator.

View Table

The SASE spectrum is shown in Fig. 5(a), and the central frequency is 8300 eV. The average seed energy as a function of Bragg photon energy and asymmetry ratio in Bragg and Laue geometries is shown in Fig. 5(b). For the cases of γ = −1, γ = −5.4, γ = −11.1 and γ = −16.3 shown in Fig. 5(c), the detuning are 0.67 eV, 2.15 eV, 4.05 eV and 5.81 eV, respectively. With the increase of |γ|, the seed energy decreases, because the transmissivity drops.

Fig. 5 (a) The SASE spectrum before the diamond crystal. The black curve refers to the average over 100 simulation runs, and the purple curves refer to single shot realizations. (b),(c) The average seed energy as the function of Bragg photon energy and asymmetry ratio (Bragg geometry). Here, the crystal thickness is 110μm, and C (111) is used.

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In Section 3, we figure out that the photon energy corresponding to the middle of reflection domain has a blue shift (detuning effect) relative to the Bragg condition. Therefore, for the purpose of obtaining the maximal seed energy efficiency, we need to adjust the Bragg photon energy down to the central photon energy (8300 eV) of SASE spectrum. In the cases of γ = −5.4 and γ = −11.1, the maximal seed power efficiency is obtained by adjusting the Bragg photon energy at 8297.85 eV and 8295.95 eV, respectively. Compared with the case of setting the Bragg photon energy as 8300 eV, the seed power efficiency of the two cases is about 1.5 and 2 times enhanced shown in Fig. 5(c), respectively.

In short, for a given crystal thickness, a given reflecting atomic plane and a given asymmetry ratio, we can always find an optimal detuning to get the maximal seed energy efficiency which is corresponding to the central photon energy of SASE spectrum. This discussion also can give us some inspirations in the optimization of diffraction geometry, and in the selection of reflecting atomic plane and the bandwidth of the monochromatic seed in the application of HXRSS FEL.

5. Crystal detuning effect in XFELO

In this section, we discuss the detuning effect in XFELO [25–27]. The input bandwidth of XFELO can be regarded as the Darwin width

θ_{D} = \frac{2 P \sqrt{| γ | χ_{h} χ_{\bar{h}}}}{sin 2 θ_{B}}

Where χ_h, χ_h̄ are the Fourier coefficients of the electric susceptibility. P is the polarization factor. The Darwin width in units of the photon energy can be converted by substituting Eq. (24) into Eq. (21)

Δ ℰ_{D} = ℰ_{B} \frac{Δ θ_{D}}{tan θ_{B}} = ℰ_{B} \frac{P \sqrt{| γ | χ_{h} χ_{\bar{h}}}}{{sin}^{2} θ_{B}}

Figure 6(a) and 6(b) show the Darwin width as a function of asymmetry ratio in Bragg geometry and Laue geometry. However, it is still hard to directly identify whether the detuning effect makes an impact or not. Here, we define a parameter κ which is called detuning ratio of XFELO.

κ = \frac{Δ ℰ}{Δ ℰ_{D}} = \frac{χ_{0} (1 - γ)}{2 P \sqrt{| γ | χ_{h} χ_{\bar{h}}}}

The detuning ratio equals to the photon energy detuning over the bandwidth of X-ray pulse before the crystal monochromator. If κ ≪ 1, the detuning effect can be neglected. However, if κ ≈ 1 or κ ≫ 1, the detuning effect can not be ignored, because the total reflection domain of Bragg diffraction is out of or at the falling edge of the input spectrum where there are very few photons, which results in low efficiency of output, and we need to tune the incident angle.

For the case of 8300 eV, we note the detuning ratio of XFELO is larger than 1 shown in Fig. 6(c). Therefore, we have to take the detuning effect into account. In hard X-ray regime, the normalized bandwidth Δω/ω₀ of SASE is typically in the order of 10⁻³, while the normalized input bandwidth of XFELO is typically in the order of 10⁻⁴ ∼ 10⁻⁵ which is one or two order of magnitude smaller than the bandwidth of SASE (the input of HXRSS). This is the reason why XFELO is more sensitive than HXRSS. The detuning ratio also can be applied to HXRSS FEL.

Fig. 6 (a) Darwin width changes as a function of γ in Bragg geometry. (b) Darwin width changes as a function of γ in Laue geometry. (c) The detuning ratio varies as a function of γ in Bragg geometry. The Bragg photon energy is 8300 eV.

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6. Conclusion

In this paper we have introduced the crystal detuning effect and analyzed the application in HXRSS FEL and XFELO. The detuning effect caused by the refraction effect is studied both in Bragg and Laue geometries. In the investigation of HXRSS FEL, the seed power efficiency always can be enhanced for a given reflecting atomic plane by tuning the incident angle off the Bragg condition where the Bragg photon energy is corresponding to the central photon energy of the input SASE spectrum. For XFELO, the output is very sensitive to the detuning effect, because the bandwidth before the crystal monochromator is usually smaller than the photon energy detuning. The detuning ratio κ is defined to describe how the detuning affects the output efficiency by comparing with 1. If κ ≪ 1, the detuning effect has little impact on the output or seed efficiency. Otherwise, we have to tune the incident angle to obtain the optimal output. The detuning effect investigated in this paper can give a guidance to reach a high-efficiency and more stable output of HXRSS FEL and XFELO.

Funding

The US Department of Energy (DOE) (DE-AC02-76SF00515); The US DOE Office of Science Early Career Research Program grant (FWP-2013-SLAC-100164).

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Parameter	Value	Unit	Parameter	Value	Unit

Electron beam energy	9.5064	GeV	Average beta function	15	m
Energy spread	1.5	MeV	Bunch length (rms)	1.0	μm
Peak current	3000	A	Undulator parameter K	2.4385	-
Normalized emittance	0.4	mm-mrad	Undulator period	2.6	cm

The detuning effect of crystal monochromator in self-seeding and oscillator free electron laser

Abstract

1. Introduction

2. Dynamical theory of X-ray diffraction

2.1. Maxwell’s equations inside the crystal

2.2. Solution for X-ray diffraction inside the crystal

2.3. Dispersion surface in reciprocal space

2.4. Bragg and Laue diffraction geometries

3. Detuning effect of crystal monochromator

4. Crystal detuning in hard X-ray self-seeding FEL

5. Crystal detuning effect in XFELO

6. Conclusion

Funding

References

Cited By

Figures (6)

Tables (1)

Equations (30)

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