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受激喇曼散射效应是可以实现高效激光变频的一种3阶非线性光学效应，是扩展激光器光谱覆盖范围的一种实用且有效的方法。晶体喇曼介质有增益系数和损伤阈值高、热导性能好等优点，固体喇曼激光器已经成为近年来激光技术的研究热点[1-10]。喇曼晶体中通常存在多个不同强度的分子振动模，对应多个不同喇曼增益系数的喇曼频移，关于固体喇曼激光器的报道大部分集中在晶体喇曼增益系数最大的主要喇曼频移模。为了获得更丰富的激光波长，基于次级喇曼频移模的固体喇曼激光器受到越来越多的关注[8-10]。
双波长激光器在精密激光谱分析、激光干涉、差分激光雷达等领域有广泛应用[11-13]。利用喇曼晶体的主要喇曼频移模和次级喇曼频移模可以获得更多双波长喇曼激光[14-17]，从而拓展双波长激光器的应用范围。2015年，HUANG等人[14]分别利用磷酸钛氧钾(KTiOPO4，KTP)晶体的267 cm-1和694 cm-1频移，砷酸钛氧钾(KTiOAsO4，KTA)晶体的234 cm-1和671 cm-1频移获得了1478 nm/1503 nm和1474 nm/1480 nm的双波长喇曼激光器。2021年，XIE等人[15]利用KTA晶体234 cm-1和671 cm-1频移获得了1147 nm和1178 nm的双波长喇曼激光器。2020年，CHE等人[16]考虑谐振腔内光子数密度和激光介质的初始反转粒子数密度的空间分布，推导了由主要喇曼频移模和次级喇曼频移模产生1阶双波长喇曼光的主动调Q内腔抽运喇曼激光器的速率方程。同年，XU等人[17]推导了基于被动调Q的内腔式双波长喇曼激光器的速率方程。
外腔抽运的喇曼谐振腔可以显著降低受激喇曼散射的阈值, 同时大大提高喇曼光的转换效率，是实现喇曼激光器高效运转的方法之一[18-20]。本文作者从理论上分析了外腔抽运的1阶双波长喇曼激光器的特性，喇曼光1和喇曼光2分别由基频光经过主要频移和次级频移获得。通过光波的波动方程和喇曼晶体中振动波的阻尼谐振子波方程，得到了描述1阶双波长喇曼激光器中基频光、喇曼光1和喇曼光2相互作用的耦合波方程组。将空间坐标、时间坐标、喇曼增益系数、自发喇曼散射系数等参量归一化后，得到了归一化的耦合波方程组，并对该方程组进行了数值求解，分析了输出镜反射率和归一化参量对外腔抽运1阶双波长激光器性能的影响。该理论具有普遍适用性，可以用于指导外腔抽运1阶双波长喇曼激光器的设计和分析。
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外腔式双波长喇曼激光器耦合波方程的建立基于图 1所示的实验装置。喇曼晶体置于后腔镜M1和输出镜M2组成的谐振腔内，抽运光由M1入射，喇曼光1和喇曼光2在谐振腔内振荡并由M2输出。
图 1 外腔式双波长喇曼激光器的示意图
Figure 1. Schematics of extracavity dual-wavelength Raman lasers
喇曼晶体中的受激喇曼散射是光场与分子的喇曼振动模相互作用产生的，相互作用过程通过光场的波动方程和振动波的阻尼谐振子波方程表示为[21-23]：
$ \frac{\partial^{2} \varepsilon}{\partial z^{2}}-\frac{n^{2}}{c^{2}} \frac{\partial^{2} \varepsilon}{\partial t^{2}}=\frac{4 {\rm{ \mathsf{ π} }} N}{c^{2}} \frac{\partial^{2}}{\partial t^{2}}\left[\left(\frac{\partial \alpha}{\partial q_{1}} Q_{1}+\frac{\partial \alpha}{\partial q_{2}} Q_{2}\right) \varepsilon\right] $
(1) $ \left\{\begin{array}{l} \frac{\partial^{2} Q_{1}}{\partial t^{2}}+\frac{2}{T_{1}} \frac{\partial Q_{1}}{\partial t}+\omega_{\mathrm{v}, 1}{ }^{2} Q_{1}=\frac{1}{2 m_{0}} \frac{\partial \alpha}{\partial q_{1}} \varepsilon^{2} \\ \frac{\partial^{2} Q_{2}}{\partial t^{2}}+\frac{2}{T_{2}} \frac{\partial Q_{2}}{\partial t}+\omega_{\mathrm{v}, 2}{ }^{2} Q_{2}=\frac{1}{2 m_{0}} \frac{\partial \alpha}{\partial q_{2}} \varepsilon^{2} \end{array}\right. $
(2) 式中, $\varepsilon$为光辐射电场强度, $t$为时间, $n$为喇曼晶体的折射率, $N$为晶体的晶格密度, $m_{0}$为晶格的有效质量, $c$为真空中的光速, $Q_{l}(l=1, 2)$为产生喇曼光1和2的振动波的幅度, $\omega_{\mathrm{v}, l}$为振动波的频率, $T_{l}$为振动波振幅的退相时间, $\alpha$为分子极化率, $q_{l}$为分子振动的简正坐标, $\partial \alpha / \partial q_{l}$为分子极化率张量的简正模导数。
在平面波近似下，假设晶体中光场的传播方向沿z轴，(1)式的解为谐振腔内平面光波光矢量之和:
$ \begin{gathered} \varepsilon(z, t)=\frac{1}{2}\left\{\sum\limits_{j} E_{j}^{+} \exp \left[\mathrm{i}\left(\omega_{j} t-{k_{j}}^{+} z\right)\right]+\right. \\ \left.\sum\limits_{j} E_{j}^{-} \exp \left[\mathrm{i}\left(\omega_{j} t-k_{j}^{-} z\right)\right]\right\}+ \text { c. c. } \end{gathered} $
(3) 式中, $E_{j}{ }^{+}$和$E_{j}{ }^{-}$分别为正向和负向传播的基频光$(j=$ $0)$, 喇曼光1和喇曼光$2(j=1, 2)$的光矢量缓变复振幅, $\omega_{j}$为光波的频率, $k_{j}{ }^{+}$和$k_{j}{ }^{-}$分别为正向和负向传播光波的波数。
考虑完全喇曼共振的情况, 产生喇曼光1和喇曼光2的振动波幅度$Q_{l}(l=1, 2)$可表示为:
$ \begin{gathered} Q_{l}=\frac{1}{2}\left\{\rho_{l 1} \exp \left[\mathrm{i}\left(\omega_{\mathrm{v}, l} t-k_{l 1} z\right)\right]+\right. \\ \rho_{l 2} \exp \left[\mathrm{i}\left(\omega_{\mathrm{v}, l} t-k_{l 2} z\right)\right]+\rho_{l 3} \exp \left[\mathrm{i}\left(\omega_{\mathrm{v}, l} t-k_{l 3} z\right)\right]+ \\ \left.\rho_{l 4} \exp \left[\mathrm{i}\left(\omega_{\mathrm{v}, l} t-k_{l 4} z\right)\right]\right\}+ \text { c.c. } \end{gathered} $
(4) 式中, $\rho_{l m}$为$E_{0}{ }^{ \pm}$和$E_{l}{ }^{ \pm}$相互作用过程中4种振动波的缓变复振幅$(m=1, 2, 3, 4), \omega_{\mathrm{v}, l}=\omega_{0}-\omega_{l}$为振动波的频率, $k_{l m}$为振动波的波数:
$ \begin{aligned} & k_{l 1}=k_{0}{ }^{+}-k_{l}{ }^{+}, k_{l 2}=k_{0}{ }^{-}-k_{l}{ }^{+}, \\ & k_{l 3}=k_{0}{ }^{+}-k_{l}{ }^{-}, k_{l 4}=k_{0}{ }^{-}-k_{l}{ }^{-} \end{aligned} $
(5) 将(3)式~(5)式代入(1)式和(2)式中，同时引入基频光波长处主要喇曼频移模和次级喇曼频移模的稳态喇曼增益系数g0l[20]，单位变换后的光矢量振幅Ej′、振动波振幅ρlm′为：
$ \left\{\begin{array}{l} g_{0 l}=\frac{4 {\rm{ \mathsf{ π} }} ^{2} N \omega_{0} T_{l}}{c^{2} n_{l} m \omega_{\mathrm{v}, l}}\left(\frac{\partial \alpha}{\partial q_{l}}\right)^{2} \\ E_{j}{ }^{\prime}=\sqrt{c n_{j} /(8 {\rm{ \mathsf{ π} }})} E_{j} \\ \rho_{l m}{ }^{\prime}=\frac{\sqrt{n_{0} n_{l}} c m \omega_{\mathrm{v}, l}}{2 {\rm{ \mathsf{ π} }} T_{l}\left(\frac{\partial \alpha}{\partial q_{l}}\right)} \rho_{l m} \end{array}\right. $
(6) 式中，nj为各光波对应的喇曼晶体的折射率。在缓变振幅近似下，描述基频光、喇曼光1和喇曼光2之间相互作用的耦合波方程组表示为：
$ \left\{\begin{aligned} \frac{n_{0}}{c} \frac{\partial E_{0}{ }^{+}}{\partial t}+\frac{\partial E_{0}{ }^{+}}{\partial z}= & \frac{g_{01}}{2 \mathrm{i}}\left(E_{1}{ }^{+} \rho_{11}+E_{1}{ }^{-} \rho_{13}\right)+ \\ & \frac{g_{02}}{2 \mathrm{i}}\left(E_{2}{ }^{+} \rho_{21}+E_{2}{ }^{-} \rho_{23}\right) \\ \frac{n_{0}}{c} \frac{\partial E_{0}{ }^{-}}{\partial t}-\frac{\partial E_{0}{ }^{-}}{\partial z}= & \frac{g_{01}}{2 \mathrm{i}}\left(E_{1}{ }^{-} \rho_{14}+E_{1}{ }^{+} \rho_{12}\right)+ \\ & \frac{g_{02}}{2 \mathrm{i}}\left(E_{2}{ }^{-} \rho_{24}+E_{2}{ }^{+} \rho_{22}\right) \\ \frac{n_{l}}{c} \frac{\partial E_{l}{ }^{+}}{\partial t}+\frac{\partial E_{l}{ }^{+}}{\partial z}= & \frac{g_{0 l}}{2 \mathrm{i}} \frac{\omega}{l}_{\omega_{0}} E_{0}{ }^{+} \rho_{l 1}{ }^{*} \\ \frac{n_{l}}{c} \frac{\partial E_{l}{ }^{-}}{\partial t}-\frac{\partial E_{l}{ }^{-}}{\partial z}= & \frac{g_{0 l}}{2 \mathrm{i}} \frac{\omega_{l}}{\omega_{0}} E_{0}{ }^{-} \rho_{l 4}{ }^{*} \\ \partial \rho_{11} / \partial t+\rho_{l 1} / T_{l}= & E_{0}{ }^{+} E_{l}{ }^{+*} /\left(\mathrm{i} T_{l}\right) \\ \partial \rho_{l 2} / \partial t+\rho_{l 2} / T_{l}= & E_{0}{ }^{-} E_{l}{ }^{+*} /\left(\mathrm{i} T_{l}\right) \\ \partial \rho_{l 3} / \partial t+\rho_{l 3} / T_{l}= & E_{0}{ }^{+} E_{l}{ }^{-*} /\left(\mathrm{i} T_{l}\right) \\ \partial \rho_{14} / \partial t+\rho_{l 4} / T_{l}= & E_{0}{ }^{-} E_{l}{ }^{-*} /\left(\mathrm{i} T_{l}\right) \end{aligned}\right. $
(7) 式中，上标*表示共轭。
当基频光的脉冲宽度为纳秒量级时, 振动波振幅的退相时间$T_{l}$远远小于激光的脉冲宽度, 则有$\partial \rho_{l} /$ $\partial t \ll \rho_{l} / T_{l}, $ (7)式中的第5第8项改写为:
$ \begin{gathered} \rho_{l 1}=-\mathrm{i} E_{0}{ }^{+} E_{l}^{+*}, \rho_{l 2}=-\mathrm{i} E_{0}{ }^{-} E_{l}^{+*}, \\ \rho_{l 3}=-\mathrm{i} E_{0}{ }^{+} E_{l}{ }^{-*}, \rho_{l 4}=-\mathrm{i} E_{0}{ }^{-} E_{l}{ }^{-*} \end{gathered} $
(8) 将(8)式代入(7)式中，忽略受激喇曼散射的级联效应和反斯托克斯散射效应，加入自发喇曼散射项和损耗项，描述基频光、喇曼光1和喇曼光2相互作用的耦合波方程为:
$ \left\{ \begin{gathered} \left[ { \pm \frac{\partial }{{\partial z}} + \frac{{{n_0}}}{c}\frac{\partial }{{\partial t}}} \right]{E_0}^ \pm = - \frac{{{g_{01}}}}{2}{E_0}^ \pm \left( {{{\left| {{E_1}^ \pm } \right|}^2} + {{\left| {E_1^ \mp } \right|}^2}} \right) - \hfill \\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \frac{{{g_{02}}}}{2}{E_0}^ \pm \left( {{{\left| {E_2^ \pm } \right|}^2} + {{\left| {E_2^ \mp } \right|}^2}} \right) - \hfill \\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; s\left( {{E_1}^ \pm + {E_1}^ \mp + {E_2}^ \pm + {E_2}^ \mp } \right) - \beta {E_0}^ \pm \hfill \\ \left[ { \pm \frac{\partial }{{\partial z}} + \frac{{{n_1}}}{c}\frac{\partial }{{\partial t}}} \right]{E_1}^ \pm = \frac{{{g_{01}}}}{2}\frac{{{\omega _1}}}{{{\omega _0}}}{E_1}^ \pm \left( {{{\left| {{E_0}^ \pm } \right|}^2} + {{\left| {{E_0}^ \mp } \right|}^2}} \right) + \hfill \\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; s\left( {{E_0}^ \pm + {E_0}^ \mp } \right) - \beta {E_1}^ \pm \hfill \\ \left[ { \pm \frac{\partial }{{\partial z}} + \frac{{{n_2}}}{c}\frac{\partial }{{\partial t}}} \right]{E_2}^ \pm = \frac{{{g_{02}}}}{2}\frac{{{\omega _2}}}{{{\omega _0}}}{E_2}^ \pm \left( {{{\left| {E_0^ \pm } \right|}^2} + {{\left| {E_0^ \mp } \right|}^2}} \right) + \hfill \\\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; s\left( {{E_0}^ \pm + {E_0}^ \mp } \right) - \beta {E_2}^ \pm \hfill \\ \end{gathered} \right. $
(9) 式中，s为晶体的自发喇曼散射系数，β为谐振腔内的损耗系数。
引人归一化空间坐标$\zeta$、归一化时间坐标$\tau$、归一化喇曼增益系数$G_{0 l}$、归一化光矢量振幅$\mathit{\Phi}_{j}$、归一化自发喇曼散射系数$S$和归一化损耗系数$L$ :
$ \begin{gathered} \zeta=\frac{z}{l_{\mathrm{c}}}, \tau=\frac{t}{t_{\mathrm{c}}}, G_{0 l}=g_{0 l}\left|E_{\mathrm{f}, \max }\right|^{2} l_{c}, \\ \mathit{\Phi}_{j}{ }^{ \pm}=\frac{E_{j}{ }^{ \pm}}{E_{\mathrm{f}, \max }}, S=s l_{\mathrm{c}}, L=\beta l_{\mathrm{c}} \end{gathered} $
(10) 式中, $l_{\mathrm{c}}$为喇曼谐振腔的光程, $t_{\mathrm{c}}=l_{\mathrm{c}} / c$为光波在谐振腔内单程通过所用的时间, $E_{\mathrm{f}, \text { max }}$为人射基频光脉冲的最大振幅。
将(10)式代入(9)式中，得到外腔式双波长喇曼激光器的归一化耦合波方程组：
$ \left\{\begin{aligned} {\left[ \pm \frac{\partial}{\partial \zeta}+n_{0} \frac{\partial}{\partial \tau}\right] \mathit{\Phi}_{0}{ }^{ \pm}=} & -\frac{G_{01}}{2} \mathit{\Phi}_{0}{ }^{ \pm}\left(\left|\mathit{\Phi}_{1}{ }^{ \pm}\right|^{2}+\left|\mathit{\Phi}_{1}{ }^{\mp}\right|^{2}\right)- \\ & \frac{G_{02}}{2} \mathit{\Phi}_{0}{ }^{ \pm}\left(\left|\mathit{\Phi}_{2}{ }^{ \pm}\right|^{2}+\left|\mathit{\Phi}_{2}{ }^{\mp}\right|^{2}\right)- \\ & S\left(\mathit{\Phi}_{1}{ }^{ \pm}+\mathit{\Phi}_{1}{ }^{\mp}+\mathit{\Phi}_{2}{ }^{ \pm}+\mathit{\Phi}_{2}{ }^{\mp}\right)-L \mathit{\Phi}_{0}{ }^{ \pm} \\ {\left[ \pm \frac{\partial}{\partial \zeta}+n_{1} \frac{\partial}{\partial \tau}\right] \mathit{\Phi}_{1}{ }^{ \pm}=} & \frac{G_{01}}{2} \frac{\omega_{1}}{\omega_{0}} \mathit{\Phi}_{1}^{ \pm}\left(\left|\mathit{\Phi}_{0}{ }^{ \pm}\right|^{2}+\left|\mathit{\Phi}_{0}{ }^{\mp}\right|^{2}\right)+ \\ & S\left(\mathit{\Phi}_{0}{ }^{ \pm}+\mathit{\Phi}_{0}{ }^{\mp}\right)-L \mathit{\Phi}_{1}^{ \pm} \\ {\left[ \pm \frac{\partial}{\partial \zeta}+n_{2} \frac{\partial}{\partial \tau}\right] \mathit{\Phi}_{2}{ }^{ \pm}=} & \frac{G_{02}}{2} \frac{\omega_{2}}{\omega_{0}} \mathit{\Phi}_{2}{ }^{ \pm}\left(\left|\mathit{\Phi}_{0}{ }^{ \pm}\right|^{2}+\left|\mathit{\Phi}_{0}{ }^{\mp}\right|^{2}\right)+ \\ & S\left(\mathit{\Phi}_{0}{ }^{ \pm}+\mathit{\Phi}_{0}{ }^{\mp}\right)-L \mathit{\Phi}_{2}{ }^{ \pm} \end{aligned}\right. $
(11) 谐振腔腔镜处的归一化边界条件为：
$ \left\{\begin{array}{l} \mathit{\Phi}_{j}^{+}(\tau, 0)=\sqrt{R_{\mathrm{p}, j}} \mathit{\Phi}_{j}^{-}(\tau, 0) \\ \mathit{\Phi}_{0}{ }^{+}(\tau, 0)=\sqrt{T_{0}} \mathit{\Phi}_{\mathrm{f}}(\tau) \\ \mathit{\Phi}_{j}^{-}(\tau, 1)=\sqrt{R_{\mathrm{o}, j}} \mathit{\Phi}_{j}^{+}(\tau, 1) \end{array}\right. $
(12) 式中, $\mathit{\Phi}_{\mathrm{f}}(\tau)$为人射基频光的归一化振幅, $T_{0}$为后腔镜对基频光光强的透过率, $R_{\mathrm{o}, j}$和$R_{\mathrm{p}, j}$分别为输出镜和后腔镜对各光波光强的反射率。假设人射基频光的脉冲光强在时间上为高斯分布, 归一化脉冲宽度$W_{\mathrm{f}}=w_{\mathrm{f}}$/$t_{\mathrm{c}}, w_{\mathrm{f}}$为入射基频光的脉冲宽度。
运用归一化边界条件对(11) 式进行数值求解, 出射喇曼光的归一化振幅为$\mathit{\Phi}_{l, \text { out }}(\tau)=\left(1-R_{\mathrm{o}, l}\right) \mathit{\Phi}_{l}(\tau$, $1)$, 喇曼光1和喇曼光2的转化效率为输出喇曼光的脉冲能量$e_{l}$与入射基频光脉冲能量$e_{\mathrm{f}}$的比值:
$ \frac{e_{l}}{e_{\mathrm{f}}}=\frac{A_{l} \int\left|\mathit{\Phi}_{l, \text { out }}(\tau)\right|^{2} \mathrm{~d} \tau}{A_{\mathrm{f}} \int\left|\mathit{\Phi}_{\mathrm{f}}(\tau)\right|^{2} \mathrm{~d} \tau}=\frac{A_{l}}{A_{f}} \eta_{l} $
(13) 式中, $A_{\mathrm{f}}$为人射基频光的光束面积, $A_{l}$为出射喇曼光的光束面积, $\eta_{l}$为喇曼分量的转化效率。定义$\mathit{\Psi}_{l, \text { max }}=$ $\left|\mathit{\Phi}_{l, \text { out }}(\tau)\right|_{\text {max }}^{2}$为出射喇曼光的归一化峰值光强, 出射喇曼光的归一化脉冲宽度$W_{l}$为$\mathit{\Psi}_{l, \text { max }} / 2 \sim \mathit{\Psi}_{l, \text { max }}$的上升时间和$\mathit{\Psi}_{l, \text { max }} \sim \mathit{\Psi}_{l, \text { max }} / 2$的下降时间之和。
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表 1中总结了几种常用喇曼晶体的喇曼频移模及其相对光谱强度和在$1064 \mathrm{~nm}$处的喇曼增益系数, 其中主要频移模的光谱强度设为1。设喇曼光1由基频光经主要频移产生, 喇曼光2由基频光经次级频移产生, 下面代人基频光脉冲的典型参数估算归一化参量的合理范围。假设人射基频光的单脉冲能量为$e_{\mathrm{f}}=$ $30 \mathrm{~mJ}$、脉冲宽度$w_{\mathrm{f}}=10 \mathrm{~ns}$、光束半径为$1 \mathrm{~mm}$, 谐振腔的光学长度$l_{\mathrm{c}}=100 \mathrm{~mm}$, 喇曼晶体的主要喇曼频移的增益系数$g_{01}=4 \mathrm{~cm} / \mathrm{GW}$, 由$G_{01}=g_{01}\left|E_{\mathrm{f}, m}\right|^{2} l_{\mathrm{c}}$可估算出$G_{01} \approx 3.8$, 由$W_{\mathrm{f}}=w_{\mathrm{f}} / t_{\mathrm{c}}$可估算出$W_{\mathrm{f}}=30$。考虑基频光、谐振腔和喇曼晶体的参量有一定的变化范围, 数值计算的部分参数由表 2给出。
表 1 常用喇曼晶体的喇曼频移模及其相对光谱强度和喇曼增益系数
Table 1. Raman modes with their relative spectra intensities and Raman gain coefficients for common Raman crystals
表 2 数值计算所用参数
Table 2. Parameters of numerical calculations
ωv, 1/cm-1 ωv, 2/cm-1 n Ro, 1 Ro, 2 S[21] L G01 G02 Wf 900 300 1.8 0.1~0.4 0.01~ 0.99 4×10-7 0.05 2~20 1~10 10~50 为提高基频光的利用率，后腔镜对基频光高透(T0=0.99)，同时输出镜对基频光高反，即基频光双程通过喇曼晶体。当基频光强度和喇曼晶体一定时，腔镜对喇曼光的反射率决定了喇曼光的转化效率。为了更直观地展示反射率对转化效率的影响，假设后腔镜对喇曼光1和喇曼光2的反射率相等且均等于0.99，输出镜对喇曼光1和喇曼光2部分反射。
图 2中给出了喇曼光1和2的转化效率$\eta_{l}$、归一化峰值光强$\mathit{\Psi}_{l, \text { max }}$和归一化脉冲宽度$W_{l}$随输出镜对喇曼光2反射率$R_{0, 2}$的变化。其中, $G_{01}=10 、G_{02}=5$、$W_{\mathrm{f}}=20$。从图 2中可以明显地看出喇曼光1和2之间的竞争关系, 由于喇曼光1的喇曼增益系数大于喇曼光2, 基频光更容易向喇曼光1转化, 输出镜对喇曼光1有较小的反馈就可以获得较大的转化效率, 当喇曼光1的反射率$R_{0, 1}>0.4$时, 喇曼光2几乎不能产生, 无法实现双波长输出。
图 2 $G_{01}=10, G_{02}=5, W_{\mathrm{f}}=20, R_{\mathrm{o} 1}$取不同值时, $\eta_{l}, \mathit{\Psi}_{l, \text { max }}$和$W_{l}$随$R_{\mathrm{o}, 2}$的变化
Figure 2. $\eta_{l}, \mathit{\Psi}_{l, \text { max }}$ and $W_{l}$ versus $R_{0, 2}$ for different $R_{0, 1}$ with $G_{01}=10, G_{02}=$ $5, W_{\mathrm{f}}=20$
考虑双波长同时产生的情况$\left(R_{\mathrm{o}, 1} \leqslant 0.4\right)$, 当$R_{\mathrm{o}, 2}$较小时, 基频光仅向喇曼光1转化, $R_{\mathrm{o}, 2}$对$\eta_{1} 、\mathit{\Psi}_{1, \text { max }}$和$W_{1}$几乎没有影响。喇曼光2的产生存在阈值反射率$R_{\mathrm{o}, 2, \mathrm{th}}, R_{\mathrm{o}, 1}$越小, 腔内喇曼光1的增益越小, $R_{\mathrm{o}, 2, \mathrm{th}}$也越小; 在$R_{0, 2, \text { th }}$处, $\eta_{1} 、\mathit{\Psi}_{1, \text { max }}$和$W_{1}$有最大值。当$R_{\mathrm{o}, 2}>R_{\mathrm{o}, 2, \mathrm{th}}$时, 输出镜对喇曼光2的反馈增强, 随着$R_{0, 2}$的增大, 基频光向喇曼光2的转化增强, $\eta_{2} 、\mathit{\Psi}_{2, \text { max }}$和$W_{2}$随$R_{0, 2}$的增大而增大; 同时, 基频光向喇曼光1的转化减弱, $\eta_{1} 、\mathit{\Psi}_{1, \text { max }}$和$W_{1}$随$R_{\mathrm{o}, 2}$的增大而减小; $R_{0, 2}$一定时, $R_{0, 1}$越小, 出射的喇曼光2越强, $\eta_{2}$和$\mathit{\Psi}_{2, \text { max }}$越大, 喇曼光2在时间范围内消耗的基频光也越多, $W_{2}$也越大。$\eta_{2}$和$\mathit{\Psi}_{2, \max }$在某一个$R_{0, 2}$时增大至最大, 大于该$R_{0, 2}$值时, 由于输出镜对喇曼光2的透过率减小, $\eta_{2}$和$\mathit{\Psi}_{2, \text { max }}$随$R_{0, 2}$的增大逐渐降低。
从图 2a和图 2b中还可以看出, 选取某些$R_{\mathrm{o}, 1}$和$R_{\mathrm{o}, 2}$的值时, 可以使得$\eta_{1}=\eta_{2}$或$\mathit{\Psi}_{1, \text { max }}=\mathit{\Psi}_{2, \text { max }}$, 如: $R_{\mathrm{o}, 1}=0.1 、R_{\mathrm{o}, 2}=0.6$时, $\eta_{1}=\eta_{2}=0.35 ; R_{\mathrm{o}, 1}=0.1 、R_{\mathrm{o}, 2}=$ 0.66时, $\mathit{\Psi}_{1, \text { max }}=\mathit{\Psi}_{2, \text { max }}=0.66$。图 3a和图 3b分别为$R_{0, 1}=0.1 、R_{0, 2}=0.6$和$R_{0, 1}=0.1 、R_{0, 2}=0.66$时人射基频光、喇曼光1和喇曼光2的脉冲形状。从图中可以看出, 由于晶体介质主要喇曼频移模的增益系数大, 因此基频光首先向喇曼光1转化, 与此同时喇曼光2在腔内不断积累, 当达到喇曼光2的阈值后, 基频光开始向喇曼光2转化, 喇曼光1的光强迅速减小。
图 3 $G_{01}=10, G_{02}=5, W_{\mathrm{f}}=20$, 不同$R_{\mathrm{o}, 1}$和$R_{\mathrm{o}, 2}$时, 人射基频光、喇曼光1和喇曼光2的脉冲形状
Figure 3. Temporal profiles of the incident fundamental laser, Raman laser 1 and Raman laser 2 for different $R_{\mathrm{o}, 1}$ and $R_{\mathrm{o}, 2}$ with $G_{01}=10, G_{02}=5$, $W_{\mathrm{f}}=20$
由上所述, 喇曼光1的增益系数较大, 输出镜对喇曼光1稍有反射就可以形成激光脉冲。相反, 喇曼光2的增益系数较小, 要求输出镜对其有一定的反射率才会在谐振腔内有效积累, 喇曼光2产生阈值处输出镜的反射率$R_{0, 2, \mathrm{th}}$与$R_{\mathrm{o}, 1} 、G_{01}$和$G_{02}$均有关, 图 4中给出了$R_{0, 1}$和$G_{01}$取不同值时, $R_{\mathrm{o}, 2, \mathrm{th}}$随$G_{01}$与$G_{02}$比值$\left(G_{01} / G_{02}\right)$的变化关系, $R_{0, 1}$取$0.1 、0.2$和0.3, 可以得到不同$R_{\mathrm{o}, 1}$时双波长输出情况下的$R_{\mathrm{o}, 2}$的取值范围。当$G_{01} / G_{02}$一定时, $R_{0, 1}$越大, 输出镜对喇曼光1的反馈越大, 喇曼光2越难产生, 对应的$R_{\mathrm{o}, 2, \mathrm{th}}$也就越大。当$R_{\mathrm{o}, 1}$一定时, 有以下规律：(1) 对于相同的$G_{01} / G_{02}$, 喇曼增益系数较大时, 喇曼散射的阈值低, $R_{\mathrm{o}, 2, \mathrm{th}}$较小; (2) 对于相同的$G_{01}$, 随着$G_{01} / G_{02}$的增大, 喇曼光2的增益系数减小, 因此$R_{0, 2, \text { th }}$增大; $G_{01} / G_{02}$大于一定值时, 喇曼光2的喇曼增益系数过小, 基频光向喇曼光2的转化无法进行, 仅有喇曼光1产生。
图 4 $W_{\mathrm{f}}=20, R_{\mathrm{o}, 1}$和$G_{01}$取不同值时, $R_{\mathrm{o}, 2, \mathrm{th}}$随$G_{01} / G_{02}$的变化
Figure 4. $\quad R_{\mathrm{o}, 2, \mathrm{th}}$ versus $G_{01} / G_{02}$ for different $R_{\mathrm{o}, 1}$ and $G_{01}$ with $W_{\mathrm{f}}=20$
$G_{01}$和$G_{02}$是影响双波长输出的重要参量, 由(10) 式可知, $G_{01}$和$G_{02}$与喇曼增益系数$g_{01}$和$g_{02}$以及基频光的峰值振幅和喇曼晶体的长度有关, 当喇曼晶体选定后, $G_{01}$和$G_{02}$随基频光的改变等比例变化。图 5中给出了$G_{01}=2 G_{02} 、W_{\mathrm{f}}=20$时, $\eta_{l} 、\mathit{\Psi}_{l, \text { max }}$和$W_{l}$随$G_{01}$的变化。
图 5 $G_{01}=2 G_{02}, W_{\mathrm{f}}=20, R_{0, 1}$和$R_{\mathrm{o}, 2}$取不同值时, $\eta_{l}, \mathit{\Psi}_{l, \text { max }}$和$W_{l}$随$G_{01}$的变化
Figure 5. $\eta_{l}, \mathit{\Psi}_{l, \text { max }}$ and $W_{l}$ versus $G_{01}$ for different $R_{\mathrm{o}, 1}$ and $R_{\mathrm{o}, 2}$ with $G_{01}=$ $2 G_{02}, W_{\mathrm{f}}=20$
从图 5a和图 5c可以看出，在不同$R_{\mathrm{o}, 1}$和$R_{\mathrm{o}, 2}$的组合下, $\eta_{1}$和$\eta_{2} 、W_{1}$和$W_{2}$随$G_{01}$的变化规律相似。开始阶段, $\eta_{1}$和$\eta_{2} 、W_{1}$和$W_{2}$随着$G_{01}$的增大迅速增大, 这是因为$G_{01}$和$G_{02}$较小时, 基频光没有被完全消耗, $G_{01}$越大, 基频光消耗得越多, 双波长的转化效率和脉冲宽度就越大。
图 6a为$G_{01}=2.5 、R_{0, 1}=0.1 、R_{0, 2}=0.6$时, 人射基频光、剩余基频光、喇曼光1和2的脉冲形状。$G_{01}>6$时, 基频光被喇曼转换几乎完全消耗, 因此随着$G_{01}$的增大, 喇曼光的转化效率和脉冲宽度增长缓慢趋于饱和, 这种情况下的脉冲形状如图 6b所示。$\eta_{l}$和$W_{l}$的饱和值与输出镜的反射率有关, $R_{0, 1}$一定, $R_{0, 2}$越大或$R_{\mathrm{o}, 2}$一定; $R_{\mathrm{o}, 1}$越小时, $\eta_{2}$和$W_{2}$的饱和值越大。反之, $\eta_{1}$和$W_{1}$的饱和值越大。
图 6 $G_{01}=2 G_{02}, W_{\mathrm{f}}=20$, 不同$G_{01}, R_{0, 1}$和$R_{\mathrm{o}, 2}$时, 人射基频光、剩余基频光、喇曼光1和喇曼光2的脉冲形状
Figure 6. Temporal profiles of the incident fundamental laser, depleted fundamental laser, Raman laser 1 and Raman laser 2 for different $G_{01}$, $R_{0, 1}$ and $R_{0, 2}$ with $G_{01}=2 G_{02}, W_{\mathrm{f}}=20$
如图 5b所示，$\mathit{\Psi}_{2, \text { max }}$的变化趋势与$\eta_{2}$相近, 首先随$G_{01}$的增大快速增大, $G_{01}>5$后增长缓慢。$\mathit{\Psi}_{1, \text { max }}$则与上述参量的变化明显不同, 在初始阶段, $\mathit{\Psi}_{1, \text { max }}$随$G_{01}$的增大而增大, 当大于一定$G_{01}$时, $\mathit{\Psi}_{1, \text { max }}$随$G_{01}$的变化与输出镜的反射率有关: $R_{0, 1}=0.2 、R_{0, 2}=0.6$和$R_{0, 1}=0.3 、R_{0, 2}=0.8$时, $\mathit{\Psi}_{1, \text { max }}$在$G_{01}>4$后基本保持不变; $R_{0, 1}=0.1 、R_{0, 2}=0.6 、G_{01}>4.4, R_{0, 1}=0.1 、R_{0, 2}=0.8$、$G_{01}>6.2$和$R_{0, 1}=0.2 、R_{0, 2}=0.8 、G_{01}>4.1$时, $\mathit{\Psi}_{1, \text { max }}$随$G_{01}$的增大而减小。前者产生的原因如图 6c所示, 在喇曼光1的脉冲峰值及之前, 喇曼光2的产生被喇曼光1抑制, 喇曼光2消耗的是喇曼光1脉冲后沿处的基频光, 峰值处的基频光只向喇曼光1转化, 当$G_{01}$比较大时, 不论$G_{01}$为何值, 喇曼光1都能完全消耗峰值处的基频光, 因此$\mathit{\Psi}_{1, \text { max }}$为一定值。后者则与之相反, 如图 6b所示, 喇曼光1和2均在产生于基频光脉冲的前沿, 喇曼光1的脉冲峰值也处于基频光的前沿, 喇曼光2产生于喇曼光1脉冲峰值处, 喇曼光2的阈值时间随着$G_{01}$的增大而提前, 则喇曼光1峰值对应的基频光强度减弱, 因此$\mathit{\Psi}_{1, \text { max }}$随$G_{01}$的增大逐渐减小。
脉冲宽度作为基频光脉冲的一个重要参量, 对双波长喇曼激光的输出有显著影响。图 7为$G_{01}=10$、$G_{02}=5$时, 不同输出镜反射率配置下, $\eta_{l} 、\mathit{\Psi}_{l, \text { max }}$和$W_{l}$随基频光归一化脉冲宽度$W_{\mathrm{f}}$的变化。$W_{\mathrm{f}}$加宽对喇曼光2的影响可以解释为: 基频光脉冲宽度较窄时, 由于喇曼光2的喇曼增益系数小, 输出镜对喇曼光2的反射率要相对较高才能使喇曼光2有效产生, 随着基频光脉冲宽度的加宽, 喇曼光在谐振腔内振荡的周期数增加, 输出镜对喇曼光2的高反射率使得喇曼光2在反射振荡的过程中放大效应增强, 喇曼光1的产生则受到抑制。因此, $\eta_{2} 、\mathit{\Psi}_{2, \text { max }}$和$W_{2}$随$W_{\mathrm{f}}$的加宽而增大。当$W_{\mathrm{f}}$较大时, $\eta_{2}$和$\mathit{\Psi}_{2, \text { max }}$随$W_{\mathrm{f}}$的变化还有以下特点: (1) 如图 7a所示, 当$R_{\mathrm{o}, 1}=0.1 、R_{\mathrm{o}, 2}=0.8 、W_{\mathrm{f}}>24, $$ R_{\mathrm{o}, 1}=0.1 、R_{\mathrm{o}, 2}=0.6 、W_{\mathrm{f}}>38$和$R_{\mathrm{o}, 1}=0.2 、R_{\mathrm{o}, 2}=$ $0.8 、W_{\mathrm{f}}>41$时, 喇曼光1的转化效率$\eta_{1}<0.05$, 基频光几乎完全消耗并转化为喇曼光$2, \eta_{2}$不再快速增长, 而是趋于饱和状态; 这时, 在$R_{0, 1}=0.1$的情况下, 对喇曼光2透过率高的输出镜$\left(R_{0, 2}=0.6\right)$得到的$\eta_{2}$更大; (2) 如图 7b所示, $\mathit{\Psi}_{2, \max }$随$W_{\mathrm{f}}$的变化规律与$\eta_{2}$相似, 首先随$W_{\mathrm{f}}$的增大快速增大, 大于一定值时, 基频光的峰值光强完全转化为喇曼光2的峰值光强, $\mathit{\Psi}_{2, \max }$成为一稳定值。此时, 对于不同的反射率组合, 由于腔内喇曼光2的峰值光强相等, 相同的透过率对应的稳定值相等, 且$R_{0, 1}$相等时, 透过率越大, 稳定值越大。
图 7 $G_{01}=10, G_{02}=5, R_{\mathrm{o}, 1}$和$R_{\mathrm{o}, 2}$取不同值时, $\eta_{l}, \mathit{\Psi}_{l, \text { max }}$和$W_{l}$随$W_{\mathrm{f}}$的变化
Figure 7. $\eta_{l}, \mathit{\Psi}_{l, \max }$ and $W_{1}$ versus $W_{\mathrm{f}}$ for different $R_{\mathrm{o}, 1}$ and $R_{\mathrm{o}, 2}$ with $G_{01}=$ $10, G_{02}=5$
图 7中喇曼光1的归一化参量随$W_{\mathrm{f}}$的变化可以做如下解释。图 7b中, $\mathit{\Psi}_{1, \max }$随$W_{\mathrm{f}}$的加宽首先缓慢下降, 进而在拐点后迅速减小; 图 7c中, $W_{1}$先随$W_{\mathrm{f}}$的增大而增大, 在拐点后减小, 且相同反射率组合时, $\mathit{\Psi}_{1, \text { max }}$和$W_{1}$的拐点值基本相同。这是因为在拐点前双波长激光的强弱很大程度上由喇曼增益系数决定，大部分基频转化为喇曼光1, 喇曼光2在基频光的后沿产生, 因此, $\mathit{\Psi}_{1, \max }$随$W_{\mathrm{f}}$的增大变化缓慢, $W_{1}$随$W_{\mathrm{f}}$的增大而增大。拐点的位置由反射率决定, $R_{0, 1}$一定时, $R_{\mathrm{o}, 2}$越小, 拐点的$W_{\mathrm{f}}$越大; $R_{\mathrm{o}, 2}$一定时, $R_{\mathrm{o}, 1}$越大, 拐点的$W_{\mathrm{f}}$越大。$W_{\mathrm{f}}$大于拐点后, 喇曼光2在不断振荡中放大增强, 喇曼光2的产生时间逐渐提前至基频光的前沿, 且$W_{\mathrm{f}}$越大, 喇曼光2阈值的归一化时间$\tau$越小, 因此, $\mathit{\Psi}_{1, \max }$和$W_{1}$随$W_{\mathrm{f}}$的增大而减小。基于以上的分析, 图 7a中, $\eta_{1}$随$W_{\mathrm{f}}$的变化同样存在拐点, 拐点之前$\eta_{1}$变化不大, 拐点之后$\eta_{1}$迅速减小。
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通过光场的波动方程和振动波的阻尼谐振子波方程推导出了外腔抽运喇曼谐振腔产生两束1阶喇曼光的耦合波方程。波动方程是光场传播的基本方程，振动波的阻尼谐振子波方程在本质上描述了光场与喇曼介质的相互作用，因此，本文中的耦合波理论能够准确分析外腔式双波长喇曼激光器的运转性能。为了使该理论具有普适性，引入归一化参量对方程组进行了归一化处理，数值求解方程组分析了双波长喇曼激光器的运转规律。通过分析结果可以看到，实现双波长喇曼光运转的前提是对喇曼增益较大的喇曼光1进行抑制，从而提高喇曼光2的有效输出。平衡两束喇曼光输出的主要途径有：(1)选择对喇曼光1反射率小(Ro, 1 < 0.5)同时对喇曼光2反射率大(Ro, 2>0.5)的输出镜；(2)选择喇曼增益系数大的喇曼晶体，同时两个喇曼振动模的增益系数不能相差太大，当喇曼光2的喇曼增益系数过小时，喇曼光2将被完全抑制；(3)适当提高入射基频光的脉冲宽度可以提高喇曼光2的转化效率，但喇曼光2的脉冲宽度也会随之增大。
本文中的理论推导和数值计算可以作为设计和分析外腔式双波长喇曼激光器的理论工具，能够为该类激光器的实验研究提供参考。
外腔式双波长喇曼激光器的耦合波理论
Coupled wave theory of extracavity pumped dual-wavelength Raman lasers
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摘要: 为了对基于喇曼晶体的主要喇曼频移模和次级喇曼频移模的外腔式双波长喇曼激光器进行理论分析, 采用光场的波动方程和喇曼晶体中振动波的阻尼谐振子波方程推导出了描述基频光、喇曼光1和喇曼光2的耦合波方程组, 通过引入归一化参量对耦合波方程组进行了归一化, 并数值分析了输出镜反射率和归一化参量对外腔式双波长喇曼激光器性能的影响。结果表明, 选择对主要频移模反射率小于0. 5且对次级频移模反射率大于0. 5的输出镜、两个喇曼振动模的增益系数相差不大的喇曼晶体, 适当提高入射基频光的脉冲宽度可以提高次级喇曼频移模的转化效率, 可实现有效双波长运转。所提出的归一化耦合波理论可以作为分析外腔式双波长喇曼激光器的辅助工具。Abstract: The extracavity pumped dual-wavelength Raman laser based on the main and the secondary Raman shifts of Raman crystal was theoretically analyzed. From the wave equation of the light field and the damped harmonic oscillator wave equation for the vibrational wave in Raman crystal, a group of coupled wave equations were deduced to describe the fundamental laser, Raman laser 1 and Raman laser 2 were then derived. The coupled wave equations were normalized by introducing several normalized parameters. The effects of the normalized Raman gain coefficient, normalized fundamental pulse width and output mirror reflectivities on the performance of extracavity dual-wavelength Raman lasers were numerically analyzed. It is found that selecting the output mirror with reflectivity less than 0.5 for the main frequency shift and larger than 0.5 for the secondary frequency shift, and the Raman crystal with small difference gain coefficients of the two Raman modes, and properly increasing the pulse width of the incident fundamental frequency light can improve the conversion efficiency of the secondary Raman frequency shift and achieve effective dual wavelength operation. The theoretical derivation and numerical calculation of this paper can be used as a theoretical tool for the design and analysis of extracavity dual-wavelength Raman lasers, and can provide a reference for the experimental research of this kind of lasers.
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图 2 $G_{01}=10, G_{02}=5, W_{\mathrm{f}}=20, R_{\mathrm{o} 1}$取不同值时, $\eta_{l}, \mathit{\Psi}_{l, \text { max }}$和$W_{l}$随$R_{\mathrm{o}, 2}$的变化
Figure 2. $\eta_{l}, \mathit{\Psi}_{l, \text { max }}$ and $W_{l}$ versus $R_{0, 2}$ for different $R_{0, 1}$ with $G_{01}=10, G_{02}=$ $5, W_{\mathrm{f}}=20$
图 3 $G_{01}=10, G_{02}=5, W_{\mathrm{f}}=20$, 不同$R_{\mathrm{o}, 1}$和$R_{\mathrm{o}, 2}$时, 人射基频光、喇曼光1和喇曼光2的脉冲形状
Figure 3. Temporal profiles of the incident fundamental laser, Raman laser 1 and Raman laser 2 for different $R_{\mathrm{o}, 1}$ and $R_{\mathrm{o}, 2}$ with $G_{01}=10, G_{02}=5$, $W_{\mathrm{f}}=20$
图 4 $W_{\mathrm{f}}=20, R_{\mathrm{o}, 1}$和$G_{01}$取不同值时, $R_{\mathrm{o}, 2, \mathrm{th}}$随$G_{01} / G_{02}$的变化
Figure 4. $\quad R_{\mathrm{o}, 2, \mathrm{th}}$ versus $G_{01} / G_{02}$ for different $R_{\mathrm{o}, 1}$ and $G_{01}$ with $W_{\mathrm{f}}=20$
图 5 $G_{01}=2 G_{02}, W_{\mathrm{f}}=20, R_{0, 1}$和$R_{\mathrm{o}, 2}$取不同值时, $\eta_{l}, \mathit{\Psi}_{l, \text { max }}$和$W_{l}$随$G_{01}$的变化
Figure 5. $\eta_{l}, \mathit{\Psi}_{l, \text { max }}$ and $W_{l}$ versus $G_{01}$ for different $R_{\mathrm{o}, 1}$ and $R_{\mathrm{o}, 2}$ with $G_{01}=$ $2 G_{02}, W_{\mathrm{f}}=20$
图 6 $G_{01}=2 G_{02}, W_{\mathrm{f}}=20$, 不同$G_{01}, R_{0, 1}$和$R_{\mathrm{o}, 2}$时, 人射基频光、剩余基频光、喇曼光1和喇曼光2的脉冲形状
Figure 6. Temporal profiles of the incident fundamental laser, depleted fundamental laser, Raman laser 1 and Raman laser 2 for different $G_{01}$, $R_{0, 1}$ and $R_{0, 2}$ with $G_{01}=2 G_{02}, W_{\mathrm{f}}=20$
图 7 $G_{01}=10, G_{02}=5, R_{\mathrm{o}, 1}$和$R_{\mathrm{o}, 2}$取不同值时, $\eta_{l}, \mathit{\Psi}_{l, \text { max }}$和$W_{l}$随$W_{\mathrm{f}}$的变化
Figure 7. $\eta_{l}, \mathit{\Psi}_{l, \max }$ and $W_{1}$ versus $W_{\mathrm{f}}$ for different $R_{\mathrm{o}, 1}$ and $R_{\mathrm{o}, 2}$ with $G_{01}=$ $10, G_{02}=5$
表 1 常用喇曼晶体的喇曼频移模及其相对光谱强度和喇曼增益系数
Table 1. Raman modes with their relative spectra intensities and Raman gain coefficients for common Raman crystals
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