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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,*|Show fewer author(s)

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}{l} g (x) = f (x) + f (x + b) \end{array}

$

(3)

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$

\begin{array}{l} g (x) = \end{array}

\sum\limits_{

\begin{array}{l} n \in Z \end{array}

} {

\begin{array}{l} F_{n} e^{- i 2 π n x} + \end{array}

} \sum\limits_{

\begin{array}{l} n \in Z \end{array}

} {

\begin{array}{l} F_{n} e^{- i 2 π n \begin{matrix} (x + b) \end{matrix}} n = \end{array}

} \sum\limits_{

\begin{array}{l} n \in Z \end{array}

} {

\begin{array}{l} (F_{n} + F_{n} e^{- i 2 π n \begin{matrix} b \end{matrix}}) e^{- i 2 π n x} \end{array}

} $

(4)

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$ {G_n} = {F_n} + {F_n}{{\rm{e}}^{ - {\rm{i}}2\pi n

\begin{array}{l} b \end{array}

}} \Rightarrow {F_n} = \frac{{{G_n}}}{{1 + {{\rm{e}}^{ - {\rm{i}}2\pi n

\begin{array}{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}{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_{t r i a l} = q / p p \in e v e n a n d p, q a r e c o p r i m e \\ n = k p / 2 k \in o d d \end{array}

\right. $

(9)

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$ \sum\limits_{

\begin{array}{l} n = - N / 2 \end{array}

}^{

\begin{array}{l} N / 2 \end{array}

} {{X_n}({b_{{\rm{trial}}}}) = 2} \left\lfloor {(N - p)/2p + 1} \right\rfloor \times

\begin{array}{l} (1 / ε) \end{array}

$

(10)

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$ \Rightarrow \mathop {\lim }\limits_{

\begin{matrix} N \to \infty \\ ε \to 0^{+} \end{matrix}

} \sum\limits_{

\begin{array}{l} n = - N / 2 \end{array}

}^{

\begin{array}{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{matrix} N \to \infty \\ ε \to 0^{+} \end{matrix}

} \sum\limits_{

\begin{array}{l} n = - N / 2 \end{array}

}^{

\begin{array}{l} N / 2 \end{array}

} {{X_n}({b_{\text{t}}})\varepsilon /N{\text{ }} = } \left\{

\begin{matrix} 1 / p b_{t} = q / p, p is even and p, q are coprime \\ 0 b_{t} = q / p, p is odd and p, q are coprime \\ 0 b_{t} is irrational \end{matrix}

\right. $

(12)

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$ f(x){\text{ }} = \left\{

\begin{matrix} \frac{1}{p} if x = \frac{q}{p}, with p \in N and q \in Z N coprime \\ 0 if x is irrational \end{matrix}

\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}{l} n = - N / 2 \end{array}

}^{

\begin{array}{l} N / 2 \end{array}

} {\left| {{F_n}({b_{{\text{trial}}}}){\text{ }}} \right| = \sum\limits_{

\begin{array}{l} n = - N / 2 \end{array}

}^{

\begin{array}{l} N / 2 \end{array}

} {\left| {{G_n}{X_n}({b_{{\text{trial}}}})} \right|} } = \sum\limits_{

\begin{array}{l} n = - N / 2 \end{array}

}^{

\begin{array}{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}{l} - N / 2 \end{array}

}^{

\begin{array}{l} N / 2 \end{array}

} {\left| {{F_n}({b_{{\text{trial}}}}){\text{ }}} \right| = {\kern 1pt} {\kern 1pt} } \sum\limits_{

\begin{array}{l} - N / 2 \end{array}

}^{

\begin{array}{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_{\begin{matrix} - N / 2 \end{matrix}}^{\begin{matrix} N / 2 \end{matrix}} \frac{1}{1 + (1 - ε) e x p^{- i 2 π n b_{t r i a l}}} = \sum_{\begin{matrix} - N / 2 \end{matrix}}^{\begin{matrix} N / 2 \end{matrix}} \frac{1}{1 + (1 - ε) e x p^{- i 2 π (- n) (- b_{t r i a l})}} = \sum_{\begin{matrix} N / 2 \end{matrix}}^{\begin{matrix} - N / 2 \end{matrix}} \frac{1}{1 + (1 - ε) e x p^{- i 2 π (n) (1 - b_{t r i a l})}} \\ \Rightarrow X_{n} (b_{t r i a l}) = X_{n} (1 - b_{t r i a l}) \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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