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Chinese Optics Letters
Vol. 16, Issue 4, 040701 (2018)

Extracting cavity and pulse phases from limited data for coherent pulse stacking

Yilun Xu^1、2、3, Russell Wilcox¹, John Byrd¹, Lawrence Doolittle¹, Qiang Du¹, Gang Huang¹, Yawei Yang¹, Tong Zhou¹, Lixin Yan^2、3、*, Wenhui Huang^2、3, and Chuanxiang Tang^2、3

Author Affiliations

¹Accelerator Technology and Applied Physics Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA

²Department of Engineering Physics, Tsinghua University, Beijing 100084, China

³Key Laboratory of Particle and Radiation Imaging, Ministry of Education, Tsinghua University, Beijing 100084, China

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DOI: 10.3788/COL201816.040701 Cite this Article Set citation alerts

Yilun Xu, Russell Wilcox, John Byrd, Lawrence Doolittle, Qiang Du, Gang Huang, Yawei Yang, Tong Zhou, Lixin Yan, Wenhui Huang, Chuanxiang Tang. Extracting cavity and pulse phases from limited data for coherent pulse stacking[J]. Chinese Optics Letters, 2018, 16(4): 040701 Copy Citation Text

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Abstract

Coherent pulse stacking (CPS) is a new time-domain coherent addition technique that stacks several optical pulses into a single output pulse, enabling high pulse energy and high average power. A

Z

-domain model targeting the pulsed laser is assembled to describe the optical interference process. An algorithm, extracting the cavity phase and pulse phases from limited data, where only the pulse intensity is available, is developed to diagnose optical cavity resonators. We also implement the algorithm on the cascaded system of multiple optical cavities, achieving phase errors less than 1.0° (root mean square), which could ensure the stability of CPS.

{\tilde{O}}_{4} = r {\tilde{W}}_{1} + i t {\tilde{W}}_{2},

(1a)

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{\tilde{O}}_{3} = i t {\tilde{W}}_{1} + r {\tilde{W}}_{2},

(1b)

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{\tilde{W}}_{2} = z^{- 1} α e^{i φ} {\tilde{O}}_{3},

(1c)

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H (z) = \frac{Y (z)}{X (z)} = \frac{r - α e^{i φ} z^{- 1}}{1 - r α e^{i φ} z^{- 1}},

(2)

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H_{cascaded} (z) = \prod_{m} H_{m} (z) .

(3)

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X (z) = {\sum_{k = 1}^{n}}^{in} A_{k} \cdot e^{i^{in} ψ_{k}} \cdot z^{- (k - 1)},

(4)

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Y (z) = {\sum_{k = 1}^{n}}^{out} A_{k} \cdot e^{i^{out} ψ_{k}} \cdot z^{- (k - 1)} .

(5)

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(r - α e^{i φ} z^{- 1}) {\sum_{k = 1}^{n}}^{in} A_{k} \cdot e^{i^{in} ψ_{k}} \cdot z^{- (k - 1)} = (1 - r α e^{i φ} z^{- 1}) {\sum_{k = 1}^{n}}^{out} A_{k} \cdot e^{i^{out} ψ_{k}} \cdot z^{- (k - 1)} .

(6)

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{\begin{matrix} r \cdot {}^{in}A_{1} \cdot e^{i^{in} ψ_{1}} = {}^{out}A_{1} \cdot e^{i^{out} ψ_{1}} \\ - α e^{i φ} \cdot {}^{in}A_{1} \cdot e^{i^{in} ψ_{1}} + r \cdot {}^{in}A_{2} \cdot e^{i^{in} ψ_{2}} = {}^{out}A_{2} \cdot e^{i^{out} ψ_{2}} - α e^{i φ} \cdot r \cdot {}^{out}A_{1} \cdot e^{i^{out} ψ_{1}} \\ ⋮ \\ - α e^{i φ} \cdot {}^{in}A_{k - 1} \cdot e^{i^{in} ψ_{k - 1}} + r \cdot {}^{in}A_{k} \cdot e^{i^{in} ψ_{k}} = {}^{out}A_{k} \cdot e^{i^{out} ψ_{k}} - α e^{i φ} \cdot r \cdot {}^{out}A_{k - 1} \cdot e^{i^{out} ψ_{k - 1}} \\ ⋮ \\ - α e^{i φ} \cdot {}^{in}A_{n - 1} \cdot e^{i^{in} ψ_{n - 1}} + r \cdot {}^{in}A_{n} \cdot e^{i^{in} ψ_{n}} = {}^{out}A_{n} \cdot e^{i^{out} ψ_{n}} - α e^{i φ} \cdot r \cdot {}^{out}A_{n - 1} \cdot e^{i^{out} ψ_{n - 1}} \\ {}^{in}A_{n} \cdot e^{i^{in} ψ_{n}} = r \cdot {}^{out}A_{n} \cdot e^{i^{out} ψ_{n}} . \end{matrix}

(7)

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α e^{i φ} (r \cdot {}^{out} A_{k - 1} \cdot e^{i^{out} ψ_{k - 1}} - {}^{in}A_{k - 1} \cdot e^{i^{in} ψ_{k - 1}}) = {}^{out}A_{k} \cdot e^{i^{out} ψ_{k}} - r \cdot {}^{in}A_{k} \cdot e^{i^{in} ψ_{k}} .

(8)

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{}^{out}ψ_{k} = {}^{in}ψ_{k} \pm \arccos [\frac{r^{2} \cdot {}^{in}A_{k}^{2} + {}^{out}A_{k}^{2} - α^{2} \cdot {}^{in}A_{k - 1}^{2} - α^{2} \cdot r^{2} \cdot {}^{out}A_{k - 1}^{2} + 2 α^{2} r \cdot {}^{in}A_{k - 1} \cdot {}^{out}A_{k - 1} \cdot \cos ({}^{in}ψ_{k - 1} - {}^{out}ψ_{k - 1})}{2 r \cdot {}^{in}A_{k} \cdot {}^{out}A_{k}}] .

(9)

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φ = \arg [\frac{{}^{out}A_{k} \cdot e^{i^{out} ψ_{k}} - r \cdot {}^{in}A_{k} \cdot e^{i^{in} ψ_{k}}}{α (r \cdot {}^{out}A_{k - 1} \cdot e^{i^{out} ψ_{k - 1}} - {}^{in}A_{k - 1} \cdot e^{i^{in} ψ_{k - 1}})}],

(10)

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δ_{ka, lb} = \min (| φ_{ka} - φ_{lb} |, 2 π - | φ_{ka} - φ_{lb} |),

(11)

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δ = δ_{ka, lb} + δ_{ka, pc} + δ_{ka, qd} + δ_{lb, pc} + δ_{lb, qd} + δ_{pc, qd},

(12)

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References (16)

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Yilun Xu, Russell Wilcox, John Byrd, Lawrence Doolittle, Qiang Du, Gang Huang, Yawei Yang, Tong Zhou, Lixin Yan, Wenhui Huang, Chuanxiang Tang. Extracting cavity and pulse phases from limited data for coherent pulse stacking[J]. Chinese Optics Letters, 2018, 16(4): 040701

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