Coherent free-electron light sources

Dongdong Zhang; Yushan Zeng; Ye Tian; Ruxin Li

doi:10.3788/PI.2023.R07

Photonics Insights, Volume. 2, Issue 3, R07(2023)

Coherent free-electron light sources Story Video , On the Cover

Dongdong Zhang¹, Yushan Zeng¹, Ye Tian^1、*, and Ruxin Li^1,2、*

¹State Key Laboratory of High Field Laser Physics and CAS Center for Excellence in Ultra-intense Laser Science, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai, China

²School of Physical Science and Technology, ShanghaiTech University, Shanghai, China

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Equations(46)

E^{2} - p^{2} c^{2} = m_{e}^{2} c^{4} .

(1)

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ω^{2} - k^{2} c^{2} = 0,

(2)

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ω^{2} - k^{2} c^{2} = 0,

(3)

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P_{f} - P_{i} = P_{p h} + m ℏ︀ k_{u},

(4)

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E_{i} + ℏ︀ ω_{1} = E_{f} + ℏ︀ ω_{2},

(5)

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P_{i} - ℏ︀ ω_{1} / c = P_{f} + ℏ︀ ω_{2} / c .

(6)

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ρ (r, t) = - e δ (r - r_{e} (t)),

(7)

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J (r, t) = - e v_{e} δ (r - r_{e} (t)),

(8)

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ρ (r, ω) = - e δ (r_{⊥}) \exp (i ω r_{∥} / v),

(9)

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J (r, ω) = - e \hat{r_{∥}} δ (r_{⊥}) \exp (i ω r_{∥} / v) .

(10)

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E (r, t) = \frac{e}{4 π ɛ_{0}} (\frac{(1 - β^{2}) (n - β)}{k^{3} (r - r_{e} (t))} + \frac{n \times (n - β) \times \dot{β}}{c k^{3} (r - r_{e} (t))}),

(11)

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B (r, t) = \frac{n}{c} \times E (r, t) .

(12)

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P = \frac{2}{3} \frac{e^{2}}{m_{e} c^{3}} (\frac{d p}{d t} \cdot \frac{d p}{d t}),

(13)

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P = - \frac{2}{3} \frac{e^{2}}{m_{e} c^{3}} (\frac{d p_{μ}}{d τ} \cdot \frac{d p^{μ}}{d τ}),

(14)

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- \frac{d p_{μ}}{d τ} \cdot \frac{d p^{μ}}{d τ} = {(\frac{d p}{d τ})}^{2} - \frac{1}{c^{2}} {(\frac{d E}{d τ})}^{2} = {(\frac{d p}{d τ})}^{2} - β^{2} {(\frac{d p}{d τ})}^{2} .

(15)

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P = \frac{2}{3} \frac{e^{2}}{c} γ^{6} ({(\dot{β})}^{2} - {(β \times \dot{β})}^{2}),

(16)

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\frac{d^{2} U}{d ω d Ω} = \frac{e^{2}}{4 π^{2} c} \times {| \int_{- \infty}^{+ \infty} d t e^{i ω (t - n \cdot r (t) / c)} \frac{n \times [(n - β) \times \dot{β}]}{{(1 - β \cdot n)}^{2}} |}^{2} .

(17)

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\frac{\partial^{2} N}{\partial x \partial λ} = \frac{2 π α}{λ^{2}} (1 - 1 / β^{2} n^{2}) .

(18)

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\frac{d E}{d ω d z} = \frac{z^{2} e^{2}}{c^{2}} ω (1 - \frac{1}{β^{2} ε (ω)}) H (z) (β n - 1),

(19)

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\frac{d U}{d ω d Ω} = \frac{μ_{0} e^{2} β^{2} c}{4 π^{3}} \frac{\sin^{2} θ}{{(1 - β \cos θ)}^{2}} .

(20)

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\frac{d^{2} U}{d Ω d ω} = \frac{e^{2}}{16 π^{3} ε_{0} c} γ^{2} {(\frac{ω}{ω_{c}})}^{2} {(1 + γ^{2} θ^{2})}^{2} (K_{3 / 2}^{2} (ξ) + \frac{γ^{2} θ^{2}}{1 + γ^{2} θ^{2}} K_{1 / 3}^{2} (ξ)) .

(21)

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x (z) = x_{0} \sin (k_{u} z) = \frac{ψ}{k_{u}} \sin (k_{u} z) = \frac{K}{γ k_{u}} \sin (k_{u} z),

(22)

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β_{z} ≃ β (1 - \frac{K^{2}}{2 γ^{2}} \cos^{2} (k_{u} z)),

(23)

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{\bar{β}}_{z} ≃ β (1 - \frac{K^{2}}{4 γ^{2}}) = 1 - \frac{1}{2 γ^{2}} (1 + \frac{K^{2}}{2}) .

(24)

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m λ = \frac{λ_{u}}{\bar{β_{z}}} - λ_{u} \cos θ ≃ \frac{λ_{u}}{2 γ^{2}} (1 + \frac{K^{2}}{2} + γ^{2} θ^{2}) .

(25)

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λ \approx \frac{λ_{u}}{2 m γ^{2}} (1 + \frac{K^{2}}{2}),

(26)

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B_{6 D} = \frac{I}{ε_{n x} ε_{n y} 0.1 % \frac{Δ W}{W}},

(27)

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ω = ω_{q} \frac{β n (ω_{q}) \cos θ_{q} - 1}{1 - β \cos θ} .

(28)

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ω = ω_{q} \frac{β n (ω_{q}) \cos θ_{q} - 1}{1 - β \cos θ} \approx 2 ω_{q} γ^{2} (β n (ω_{q}) \cos θ_{q} - 1),

(29)

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E = m_{e} ω_{p} c \sqrt{a_{0}} / e,

(30)

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r_{b} = (2 / k_{p}) \sqrt{a_{0}},

(31)

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λ_{u} = \sqrt{2 γ} λ_{p},

(32)

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K = γ_{β} k_{p} \sqrt{γ / 2} .

(33)

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B = \frac{photons / second}{{(mrad)}^{2} ({mm}^{2} source area) (0.1 % band width)} .

(34)

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E_{H} = 2 γ_{e}^{2} f (a_{0}) \frac{1 + \cos θ_{L}}{1 + (γ_{e} θ_{γ}) + a_{0}^{2} + \frac{4 γ_{e} E_{L}}{m_{e} c^{2}}} E_{L},

(35)

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λ_{γ} ≃ \frac{λ_{L}}{4 γ_{e}^{2}} \frac{1}{f (a_{0})} (1 + \frac{a_{0}^{2}}{2}) .

(36)

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λ_{u} = \frac{β λ_{L}}{1 + β} \approx \frac{λ_{L}}{2} .

(37)

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B_{u} = B + \frac{E}{β c} = \frac{(1 + β) B}{β} \approx 2 B .

(38)

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\frac{λ_{γ}}{λ_{u}} = \frac{k_{u}}{k_{γ}} = \frac{1 + (1 + β) K^{2} / 4 + β (1 + β) γ^{2} θ_{L}^{2} / 2}{β (1 + β) γ^{2}} .

(39)

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λ_{u} = \frac{λ_{L} / 2}{2 γ^{2}} \frac{1}{f (a_{0})} (1 + \frac{a_{0}^{2}}{2}),

(40)

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P = {| \sum_{j = 1}^{N_{e}} A (r, t) \exp (i ω t + ϕ_{j}) |}^{2} = {| A (r, t) |}^{2} (\sum_{j = 1}^{N_{e}} N_{e} + {| \sum_{j = 1}^{N_{e}} \sum_{k = 1}^{N_{e}} \exp (i (ω t_{j} + ω t_{k} + ϕ_{j} - ϕ_{k})) |}^{2}),

(41)

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2 σ_{t b} < T = 2 π / ω,

(42)

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P = ɛ_{0} E^{2} Σ v_{e} = \frac{1}{2} η^{2} I_{p}^{2} Z_{0} \frac{f^{2}}{β} \frac{z^{2}}{Σ},

(43)

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Δ W = \frac{e λ}{\sqrt{2 m_{e}} π c} \sqrt{W_{0}} E_{\max},

(44)

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F_{P} = - \nabla U_{p} = \frac{e^{2}}{4 m_{e} ω^{2}} \nabla E_{L} {(r, t)}^{2},

(45)

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U_{P} ≅ 2 E_{L} (t, t_{0}, z, z_{0}) \frac{e^{2}}{m_{e} {(ω_{1} + ω_{2})}^{2}} \cdot \cos ((ω_{1} - ω_{2}) (t - t_{0}) - (ω_{1} \cos α - ω_{2} \cos β) \frac{z - z_{0}}{c} + (ϕ_{1} - ϕ_{2})),

(46)

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Dongdong Zhang, Yushan Zeng, Ye Tian, Ruxin Li. Coherent free-electron light sources[J]. Photonics Insights, 2023, 2(3): R07

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Paper Information

Category: Review Articles

Received: Jul. 11, 2023

Accepted: Aug. 25, 2023

Published Online: Sep. 28, 2023

The Author Email: Tian Ye (tianye@siom.ac.cn), Li Ruxin (ruxinli@mail.siom.ac.cn)

DOI:10.3788/PI.2023.R07

Topics

laser devices and laser physics

Lasers and Laser Optics

Laser physics

laser manufacturing

Instrumentation, Measurement and Metrology