First Steps to the Optimization of Undulator Parameters for 125 - PowerPoint PPT Presentation

First Steps to the Optimization of Undulator Parameters for 125 GeV Drive Beam by Manuel Formela 04.09.2018 1

Overview • Introducing formulas for: • Power absorped by the undulator vessel in form of photons 𝑄 𝑤𝑓𝑡𝑡𝑓𝑚 • Number of produced 𝑓 + by 𝑓 − 𝑓 + - pair production in a Ti-6% Al-4%V target • Undulator scheme used in the RDR • Reproducing values for already calculated 𝑄 𝑤𝑓𝑡𝑡𝑓𝑚 for the RDR set-up • Calculations of 𝑂 𝑓 + for various parameter values for 𝐿, 𝜇, 𝑚 𝑣 , 𝑂 ℎ𝑑𝑓𝑚𝑚 • Dropping some parameter combinations due to restraints in 𝑂 𝑓 + and 𝑄 𝑤𝑓𝑡𝑡𝑓𝑚 • Outlook into possible future 04.09.2018 2

Radiated Synchrotron Energy Spectral Density per Solid Angle per Electron Formulas taken from: Kincaid, Brian M. "A short‐period helical wiggler as an improved source of synchrotron radiation." Journal of Applied Physics 48.7 (1977): 2684-2691. First approximations: Photon frequency - relativistic ( 𝛿 ≫ 1 ) 2 = 𝑒 2 𝑋(𝜕) 𝑓 2 𝜕 2 +∞ 𝑗𝜕 𝑢− ො 𝑜 Ԧ 𝑠 𝑢 𝑒𝐽(𝜕) - far field ( 𝑆 ≫ 𝜇 𝛿 ) 𝑜 × Ԧ 𝑑 = 14𝜌 3 𝜗 0 𝑑 න 𝑜 × ො ො 𝛾 𝑓 𝑒𝑢 𝑓 − → 0 ) 𝑒Ω 𝑒Ω𝑒𝜕 - pointlike charge ( 𝑊 −∞ For helical trajectory: Opening angle 𝜕 sin 2 𝑂 𝑣 𝜌 ∞ 𝜕 1 − 𝑜 2 e 2 𝜕 2 𝐿 2 𝑒𝐽(𝜕) ′2 𝑦 1 + 𝛿𝜄 𝐿 − 𝑜 2 𝑦 1 = 2 𝛿 2 ෍ 𝐾 𝑜 𝐾 𝑜 2 4𝜌 3 𝜗 0 𝑑𝜕 𝑣 𝑒Ω 𝑦 1 𝜕 𝜕 1 − 𝑜 𝑜=1 2nd approximations: - small (radiation) angle ( 𝜄 ≪ 1 ⇒ cos 𝜄 ≈ 1, sin 𝜄 ≈ 𝜄) ; this is reasonable, because the radiation cone has according to theory an Opening angle of 𝜄 ≈ 1/𝛿 - Many undulator periods ( 𝑂 𝑣 ≳ 100 ) - reasonably small undulator parameter ( 𝐿 ≲ 1 → 𝐿/𝛿 ≪ 1 ) 04.09.2018 3

ሶ 𝜕 sin 2 𝑂 𝑣 𝜌 ∞ 𝜕 1 − 𝑜 2 e 2 𝜕 2 𝐿 2 𝑒𝐽(𝜕) ′2 𝑦 1 + 𝛿𝜄 𝐿 − 𝑜 2 𝑦 1 = 2 𝛿 2 ෍ 𝐾 𝑜 𝐾 𝑜 2 4𝜌 3 𝜗 0 𝑑𝜕 𝑣 𝑒Ω 𝑦 1 𝜕 𝜕 1 − 𝑜 𝑜=1 Approximation sin 2 𝑂𝜌𝑧 /𝑧 2 → N𝜌𝜀(𝑧) : ∞ ∞ 𝑒𝐽(𝜕) 2 N u e 2 𝜕 𝑣 𝐿 2 8𝛿 4 𝑒𝑋 ′2 𝑦 𝑜 + 𝛿𝜄 𝐿 − 𝑜 Radiated energy 𝑜 2 𝐾 𝑜 2 𝑦 𝑜 1. 𝑒Ω = න 𝑒𝜕 ≈ 4𝜌𝜗 0 𝑑 1 + 𝐿 2 + 𝛿 2 𝜄 2 3 ෍ 𝐾 𝑜 per solid angle 𝑒Ω 𝑦 𝑜 0 𝑜=1 ∞ 2 𝑒Ω ≈ N u e 2 𝐿 2 𝑠 Radiated energy 2. 𝑒𝑋 𝑒𝜕 = න 𝑒𝐽(𝜕) ′2 𝑧 𝑜 + 𝛽 𝑜 𝐿 − 𝑜 𝑜 2 𝐾 𝑜 2 𝑧 𝑜 2 ) ෍ 𝐾 𝑜 𝐼(𝛽 𝑜 spectral density 𝑒Ω 𝜗 0 𝑑 𝑧 𝑜 𝑜=1 Numerical integration leads to: 𝜌 𝑂 𝑓 − න 𝑒𝑋 sin 𝜄 𝑒𝑋 𝑒Ω 𝑒Ω = 2𝜌 ሶ 𝑂 𝑓 − න 1. 𝑄 𝑤𝑓𝑡𝑡𝑓𝑚 = 𝑒𝜄 𝑒𝜄 Power deposited in the undulator vessel 𝜄 1 ∞ 1 𝑂 𝑓 + = 1 𝑒𝑋 𝑒𝜕 (1 − 𝑓 −𝑒𝜍𝜏(𝜕) )𝑒𝜕 ℏ න 2. Positron number produced by all photons 𝜕 0 Target thickness Cross section for 𝑓 − 𝑓 + -pair production by photon of target material Target density 04.09.2018 4

Undulator set up (RDR, BCD) Taken from: Scott, Duncan J. "An Investigation into the Design of the Helical Undulator for the International Linear Collider Positron Source“ Undulator aperture = 5.85 mm 20 of such half-cell will be arranged in a row to form the full undulator (with 240 m of total magnetic length) 04.09.2018 5

Test: Power deposited in the undulator vessel - Dashed lines: single undulator piece - Solid lines: whole undulator scheme - Blue: RDR parameters - Red: BCD parameters Current calculations In good agreement Duncan J Scott‘s calculations 04.09.2018 6

Undulator mask (consisting of collimators with aperture of 4.4 mm) for preventing heating of the vessel due to photon absorption. Undulator aperture = 5.85 mm The limit of maximal absorped power is 1 Wm −1 (according to Duncan J Scott, who in turn names the source to be private communication with T Bradshaw) 04.09.2018 7

Current calculations In good agreement Duncan J Scott‘s calculations 04.09.2018 8

Current calculations Peridocity and peak values are in good agreement; Disagreement in dip values and local shape of the graph In good agreement Duncan J Scott‘s calculations 04.09.2018 9

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Examined parameter combination for the positron number 𝐿 = 0.65, 0.9, 1.15, 𝜇 𝑣 = 8.5, 10, 11.5 mm , 𝑚 𝑣 = 1.75, 2 m , 𝑂 ℎ𝑑𝑓𝑚𝑚 = 18, 20, 22 04.09.2018 11

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𝐿 = 1.15, 𝜇 = 8.5 mm, 𝑚 𝑣 = 2m, 𝑂 ℎ𝑑𝑓𝑚𝑚 = 22 ∼ 264 m magnetic length 04.09.2018 13

Possible future improvements • Drop a single or multiple approximations ( 𝛿 ≫ 1 , 𝜄 ≪ 1 , 𝑂 𝑣 ≳ 100 , 𝑓 − → 0 , sin 2 𝑂𝜌𝑧 /𝑧 2 → N𝜌𝜀(𝑧) , etc.) 𝐿 ≲ 1 , 𝑆 ≫ 𝜇 𝛿 , 𝑊 • Correcting possible flaws in the undulator mask considerations • For 𝑂 𝑓 + : Numerical integration over a solid angle, that only covers the target instead of the full 𝜄 = 0 − 𝜌 ∞ 2 𝑒Ω ≈ N u e 2 𝐿 2 𝑠 𝑒𝑋 𝑒𝜕 = න 𝑒𝐽(𝜕) ′2 𝑧 𝑜 + 𝛽 𝑜 𝐿 − 𝑜 𝑜 2 𝐾 𝑜 2 𝑧 𝑜 2 ) ෍ 𝐾 𝑜 𝐼(𝛽 𝑜 𝑒Ω 𝜗 0 𝑑 𝑧 𝑜 𝑜=1 • Examining more intermediate parameter values between the upper and lower limits • Adding more criteria for the optimazation besides lower limit for 𝑂 𝑓 + and upper limit for 𝑄 𝑤𝑓𝑡𝑡𝑓𝑚 04.09.2018 14

Thank you for your attention 04.09.2018 15