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Journals >Acta Photonica Sinica >Volume 49 >Issue 6 >Page 0630003 > Article

Acta Photonica Sinica
Vol. 49, Issue 6, 0630003 (2020)

Algorithm for Broadband Brillouin Spectra Analysis Based on Thomae's Function

Xu-sheng XIA^1、2, Xiang-long CAI¹, Zhong-hui LI¹, Chen-cheng SHEN^1、2, Wan-fa LIU¹, Yu-qi JIN¹, Feng-ting SANG¹, and Jing-wei GUO^1、*

Author Affiliations

¹Key Laboratory of Chemical Lasers, Dalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian, Liaoning 116023 China

²University of Chinese Academy of Sciences, Beijing 100049, China

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DOI: 10.3788/gzxb20204906.0630003 Cite this Article

Xu-sheng XIA, Xiang-long CAI, Zhong-hui LI, Chen-cheng SHEN, Wan-fa LIU, Yu-qi JIN, Feng-ting SANG, Jing-wei GUO. Algorithm for Broadband Brillouin Spectra Analysis Based on Thomae's Function[J]. Acta Photonica Sinica, 2020, 49(6): 0630003 Copy Citation Text

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Abstract

As the seed-injected lasers widely used in Brillouin Lidar area are too expensive and huge, the algorithm for a broadband Brillouin signal should be developed, which could accelerate the application of compact and economic diode lasers. A variation of Thomae's function was found during the signal processing of broadband Brillouin lidar, and the property of this function-reaching minimum value at the Brillouin frequency shift was used to recover the original signal spectrum and the corresponding frequency shift from a 1:1 superposition of pump light and Brillouin light spectra. Experiments using test data show nearly 100% accuracies for ideal cases; but for non-ideal cases, only when the noise level is less than -30 dB and the ratio of pump light to Brillouin light is less than 1.05 can this algorithm obtain accurate result.

Keywords

Algorithm Brillouin scattering Broadband Lidar Spectrum analysis Thomae''s function

$ g = - \frac{1}{4}({\beta _s} + \beta ){\rm{ }} + \frac{1}{4}\sqrt {{{({\beta _s} + \beta )}^2} - 4\left( {\beta {\beta _s} - \frac{{{\gamma ^2}{k_1}{k_s}{{\left| {{E_2}} \right|}^2}}}{{2\rho \upsilon _s^2\varepsilon }}} \right)} $

(1)

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$ E_1^*(q){\rm{ }} = E_1^*\left( 0 \right){{\rm{e}}^{gq}} $

(2)

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$ \begin{array}{*{20}{l}} {g(x){\rm{ }} = f(x){\rm{ }} + f(x + b)} \end{array} $

(3)

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$ \begin{array}{*{20}{l}} {g(x) = } \end{array}\sum\limits_{\begin{array}{*{20}{l}} {n \in \mathbb{Z}} \end{array}} {\begin{array}{*{20}{l}} {{F_n}{{\text{e}}^{ - {\text{i}}2\pi nx}} + } \end{array}} \sum\limits_{\begin{array}{*{20}{l}} {n \in \mathbb{Z}} \end{array}} {\begin{array}{*{20}{l}} {{F_n}{{\text{e}}^{ - {\text{i}}2\pi n\begin{array}{*{20}{l}} {(x + b)} \end{array}}}n = } \end{array}} \sum\limits_{\begin{array}{*{20}{l}} {n \in \mathbb{Z}} \end{array}} {\begin{array}{*{20}{l}} {\left( {{F_n} + {F_n}{{\text{e}}^{ - {\text{i}}2\pi n\begin{array}{*{20}{l}} b \end{array}}}} \right){{\text{e}}^{ - {\text{i}}2\pi nx}}} \end{array}} $

(4)

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$ {G_n} = {F_n} + {F_n}{{\rm{e}}^{ - {\rm{i}}2\pi n\begin{array}{*{20}{l}} b \end{array}}} \Rightarrow {F_n} = \frac{{{G_n}}}{{1 + {{\rm{e}}^{ - {\rm{i}}2\pi n\begin{array}{*{20}{l}} b \end{array}}}}} $

(5)

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$ {F_n} \approx \frac{{{G_n}}}{{1 + \left( {1 - \varepsilon } \right){{\rm{e}}^{ - {\rm{i}}2\pi n\begin{array}{*{20}{l}} b \end{array}}}}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \delta \ll 1 $

(6)

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$ {F_n}({b_{{\rm{trial}}}}){\rm{ }} = {G_n}{X_n}({b_{{\rm{trial}}}}) $

(7)

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$ {X_n}({b_{{\rm{trial}}}}) = \frac{1}{{1 + \left( {1 - \varepsilon } \right){{\rm{e}}^{ - {\rm{i}}2\pi n{b_{{\rm{trial}}}}}}}} $

(8)

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$ \left\{ \begin{array}{l} {b_{{\rm{trial}}}} = q/p{\rm{ }}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} p \in {\rm{even {\kern 1pt} {\kern 1pt}{\kern 1pt}{\kern 1pt} and }}{\kern 1pt}{\kern 1pt}{\kern 1pt}p, q{\kern 1pt} {\kern 1pt} {\kern 1pt} {\rm{are {\kern 1pt} {\kern 1pt} {\kern 1pt} coprime}}\\ n = kp/2{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} k \in {\rm{odd}} \end{array} \right. $

(9)

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$ \sum\limits_{\begin{array}{*{20}{l}} {n{\rm{ }} = {\rm{ }} - N/2} \end{array}}^{\begin{array}{*{20}{l}} {N/2} \end{array}} {{X_n}({b_{{\rm{trial}}}}) = 2} \left\lfloor {(N - p)/2p + 1} \right\rfloor \times \begin{array}{*{20}{l}} {(1/\varepsilon )} \end{array} $

(10)

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$ \Rightarrow \mathop {\lim }\limits_{\begin{array}{*{20}{c}} {N \to \infty } \\ {\varepsilon \to {0^ + }} \end{array}} \sum\limits_{\begin{array}{*{20}{l}} {n{\text{ }} = {\text{ }} - N/2} \end{array}}^{\begin{array}{*{20}{l}} {N/2} \end{array}} {{X_n}({b_{{\text{trial}}}})\varepsilon /N{\text{ }} = } \mathop {\lim }\limits_{N \to \infty } \frac{{\left\lfloor {(N - p)/2p + 1} \right\rfloor }}{N} = 1/p $

(11)

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$ \mathop {\lim }\limits_{\begin{array}{*{20}{c}} {N \to \infty } \\ {\varepsilon \to {0^ + }} \end{array}} \sum\limits_{\begin{array}{*{20}{l}} {n{\text{ }} = {\text{ }} - N/2} \end{array}}^{\begin{array}{*{20}{l}} {N/2} \end{array}} {{X_n}({b_{\text{t}}})\varepsilon /N{\text{ }} = } \left\{ \begin{gathered} {\kern 1pt} 1/p{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {b_{\text{t}}} = q/p,p{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{is}}{\kern 1pt} \;{\text{even}}\;{\text{and }}p,q{\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{are coprime}} \hfill \\ 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {b_{\text{t}}} = q/p,p{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{is}}{\kern 1pt} {\text{ odd and }}p,q{\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{are coprime}} \hfill \\ 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {b_{\text{t}}}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{is irrational}}{\kern 1pt} {\kern 1pt} {\kern 1pt} \hfill \\ \end{gathered} \right. $

(12)

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$ f(x){\text{ }} = \left\{ \begin{gathered} \frac{1}{p}{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{if}}{\kern 1pt} {\kern 1pt} {\kern 1pt} x = \frac{q}{p},{\text{with }}p \in {\text{ }}\mathbb{N}{\text{ and }}q \in {\text{ }}\mathbb{Z}{\text{ }}\mathbb{N}{\text{ coprime}} \hfill \\ 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{if}}{\kern 1pt} {\kern 1pt} {\kern 1pt} x{\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{is irrational}} \hfill \\ \end{gathered} \right. $

(13)

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$ {G_n} = (1 + {{\text{e}}^{ - {\text{i}}2\pi n{b_{{\text{real}}}}}}){F_{n, {\text{real}}}} $

(14)

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$ \sum\limits_{\begin{array}{*{20}{l}} {n{\text{ }} = {\text{ }} - N/2} \end{array}}^{\begin{array}{*{20}{l}} {N/2} \end{array}} {\left| {{F_n}({b_{{\text{trial}}}}){\text{ }}} \right| = \sum\limits_{\begin{array}{*{20}{l}} {n{\text{ }} = {\text{ }} - N/2} \end{array}}^{\begin{array}{*{20}{l}} {N/2} \end{array}} {\left| {{G_n}{X_n}({b_{{\text{trial}}}})} \right|} } = \sum\limits_{\begin{array}{*{20}{l}} {n{\text{ }} = {\text{ }} - N/2} \end{array}}^{\begin{array}{*{20}{l}} {N/2} \end{array}} {\left| {\frac{{1 + {{\text{e}}^{ - {\text{i}}2\pi n{b_{{\text{real}}}}}}}}{{1 + \left( {1 - \varepsilon } \right){{\text{e}}^{ - {\text{i}}2\pi n{b_{{\text{trial}}}}}}}}{F_{n, {\text{real}}}}} \right|} $

(15)

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$ \sum\limits_{\begin{array}{*{20}{l}} {{\text{ }} - N/2} \end{array}}^{\begin{array}{*{20}{l}} {N/2} \end{array}} {\left| {{F_n}({b_{{\text{trial}}}}){\text{ }}} \right| = {\kern 1pt} {\kern 1pt} } \sum\limits_{\begin{array}{*{20}{l}} {{\text{ }} - N/2} \end{array}}^{\begin{array}{*{20}{l}} {N/2} \end{array}} {\left| {\frac{{{G_n}}}{{1 + \left( {1 - \varepsilon } \right){\text{exp}}\left( { - {\text{i}}2\pi n{b_{{\text{trial}}}}} \right)}}{\text{ }}} \right|{\kern 1pt} {\kern 1pt} } $

(16)

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$ \begin{array}{l} \sum\limits_{\begin{array}{*{20}{l}} {{\rm{ }} - N/2} \end{array}}^{\begin{array}{*{20}{l}} {N/2} \end{array}} {\frac{1}{{1 + \left( {1 - \varepsilon } \right){\rm{ex}}{{\rm{p}}^{ - {\rm{i}}2\pi n{b_{{\rm{trial}}}}}}}}{\kern 1pt} {\kern 1pt} } = \sum\limits_{\begin{array}{*{20}{l}} {{\rm{ }} - N/2} \end{array}}^{\begin{array}{*{20}{l}} {N/2} \end{array}} {\frac{1}{{1 + \left( {1 - \varepsilon } \right){\rm{ex}}{{\rm{p}}^{ - {\rm{i}}2\pi \left( { - n} \right)\left( { - {b_{{\rm{trial}}}}} \right)}}}}{\kern 1pt} {\kern 1pt} } = \sum\limits_{\begin{array}{*{20}{l}} {{\rm{ }}N/2} \end{array}}^{\begin{array}{*{20}{l}} { - N/2} \end{array}} {\frac{1}{{1 + \left( {1 - \varepsilon } \right){\rm{ex}}{{\rm{p}}^{ - {\rm{i}}2\pi \left( n \right)\left( {1 - {b_{{\rm{trial}}}}} \right)}}}}{\kern 1pt} {\kern 1pt} } \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \Rightarrow {X_n}({b_{{\rm{trial}}}}) = {X_n}(1 - {b_{{\rm{trial}}}}) \end{array} $

(17)

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$ {F_n}({b_{{\text{trial}}}}) = \left( {{G_n} + {N_n}} \right){{\text{X}}_n}({b_{{\text{trial}}}}) $

(18)

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Xu-sheng XIA, Xiang-long CAI, Zhong-hui LI, Chen-cheng SHEN, Wan-fa LIU, Yu-qi JIN, Feng-ting SANG, Jing-wei GUO. Algorithm for Broadband Brillouin Spectra Analysis Based on Thomae's Function[J]. Acta Photonica Sinica, 2020, 49(6): 0630003

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