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低损耗Littman-Metcalf型光栅外腔半导体激光器结构如图1所示,在传统Littman-Metcalf结构的闪耀光栅G和反射镜M1之间新增一块反射镜M2,该反射镜垂直于光栅面放置。由半导体激光器出射的激光光束以
$ \alpha $ 角度入射闪耀光栅发生第一次衍射(如图1实线箭头所示),其衍射效应遵循光栅方程:图 1 低损耗Littman-Metcalf型光栅外腔半导体激光器结构图
Figure 1. Schematic of ECDL with the low loss Littman-Metcalf external-cavity configuration
$$ d(\sin\alpha + \sin\beta ) = m\lambda $$ (1) 式中:
$ d $ 为光栅常数;$ \lambda $ 为光波长;$ \alpha $ 为入射角;$\; \beta $ 为衍射角;$ m $ 为衍射级次。其中零级衍射光离开外腔直接输出,当$ m = 1 $ 时,满足$ {\lambda _1} = d\left( {\sin \alpha + \sin \gamma } \right) $ ($ \gamma $ 为一级衍射角)的一级衍射光则垂直入射调谐反射镜M1。经调谐反射镜M1反射的衍射光束会再次入射闪耀光栅,发生第二次衍射(如图1虚线箭头所示),此时大部分光束将以
$ \gamma $ 为入射角发生一级衍射返回LD本征腔。由于闪耀光栅无法实现理想的100%一级衍射效率,因此还有少量零级衍射光,其衍射角为$ - \gamma $ 。传统的Littman-Metcalf结构将此光束舍弃掉,为了进一步提高外腔的耦合效率,在闪耀光栅与反射镜M1之间再设置一个反射镜M2,使光束以$ \pi /2 - \gamma $ 角度入射M2,再次入射调谐反射镜M1并原路返回至半导体激光器有源腔,产生振荡。 -
如图2所示[16],在没有外腔反馈注入时,LD的两个解理面之间构成了F-P腔体,
$ {R_1} $ 、$ {R_2} $ 分别为激光器前后端面反射率。当有外腔反馈注入LD时,设$ l $ 、L分别为内腔和外腔长度,外腔反馈光的影响用一个反射系数为${R_{{\rm{out}}}}$ 的反射镜等效。这样一来,外腔反射损耗、衍射损耗等因素可以反映在系数${R_{{\rm{out}}}}$ 中。图 2 (a)外腔反馈的半导体激光器原理图;(b)引入等效反射系数后的等效腔
Figure 2. (a) Schematic diagram of semiconductor laser with external cavity feedback; (b) Equivalent resonator after introducing equivalent reflection coefficient
引入等效反射系数的概念,可以将
$ {R_2} - {R_{{\rm{out}}}} $ 平面等效为一个反射系数为$ {R_{{\rm{eff}}}} $ 的平面,如图2(b)所示,此时的等效反射率为:$$ {R_{{\rm{eff}}}} = \frac{{{R_2} + {R_{{\rm{out}}}} - 2{R_2}{R_{{\rm{out}}}}}}{{1 - {R_2}{R_{{\rm{out}}}}}} $$ (2) 当外腔模型为Littman-Metcalf结构时,根据光路可得此时的外腔反射率为:
$$ {R_{{\rm{out}}}} = R_d^2{R_3} $$ (3) 式中:
$ {R_d} $ 为闪耀光栅的一级衍射效率;$ {R_3} $ 为调谐反射镜的反射效率。当外腔模型为低损耗Littman-Metcalf结构时,根据光路可得此时的外腔反射率为:
$$ {R_{{\rm{out}}}} = R_d^2{R_3}(1 + (1 - {R_d})R_4^2{R_3}) $$ (4) 式中:
$ {R_4} $ 为新增反射镜反射率。光波在激光器内传播时,产生损耗的主要因素包括:自由载流子对光的吸收损耗
$ {\alpha _{fc}} $ ,衍射损耗${\alpha _{{\rm{dif}}}}$ 和端面输出引入的损耗[17],所以在有源区光波的损耗系数为:$$ \alpha = {\alpha _{fc}} + {\alpha _{{\rm{dif}}}} + \frac{1}{{2l}}{\rm{ln}}\frac{1}{{{R_1}{R_2}}} $$ (5) 由此可得Littman-Metcalf模型端面输出引入的损耗为:
$$ {L_\alpha } = \frac{1}{{2l}}{\rm{ln}}\frac{1}{{{R_1}{R_{eff}}}} = \frac{1}{{2l}}{\rm{ln}}\frac{1}{{{R_1}}} \cdot \frac{{1 - {R_2}R_d^2{R_3}}}{{{R_2} + R_d^2{R_3} - 2{R_2}R_d^2{R_3}}} $$ (6) 低损耗Littman-Metcalf模型端面输出引入的损耗为:
$$ \begin{gathered} {L_\alpha }^\prime = \frac{1}{{2l}}{\rm{ln}}\frac{1}{{{R_1}{R_{eff}}}} = \frac{1}{{2l}}{\rm{ln}}\frac{1}{{{R_1}}} \cdot \\ \frac{{1 - {R_2}R_d^2{R_3}(1 + (1 - {R_d})R_4^2{R_3})}}{{{R_2} + R_d^2{R_3}(1 + (1 - {R_d})R_4^2{R_3}) - 2{R_2}R_d^2{R_3}(1 + (1 - {R_d})R_4^2{R_3})}} \\ \end{gathered} $$ (7) 结合Littman-Metcalf模型阈值电流计算公式[18],改进后光栅外腔半导体激光器的阈值电流为:
$$ {I_{th}} = {I_0}\left( {1 - 2{f_d}{\tau _p}\frac{{R_d^2{R_3}\left( {1 + \left( {1 - {R_d}} \right)R_4^2{R_3}} \right)\left( {1 - R_2^2} \right)}}{{{R_2}}}\cos {w_0}\tau } \right) $$ (8) 式中:
$ {I_0} $ 为未加外腔反馈时半导体激光器的阈值电流;$ {\tau _p} $ 为光子寿命;$ {w_0} $ 为振荡频率。外腔半导体激光器输出线宽为[19]:
$$ \Delta v = \Delta {v_0}{\left[ {1 + \frac{\tau }{{{\tau _{in}}}}\left( {1 - \sqrt {\frac{{{R_2}}}{{{R_{{\rm{out}}}}}}} } \right)} \right]^{ - 2}} $$ (9) 式中:
$ \tau $ 为与谐振腔长度有关的光子寿命;$ \tau = 2 nl/c $ 。由公式(9)可以看出,外腔半导体激光器的输出线宽与谐振腔腔长、端面反射率以及外腔光强反馈效率有关,在外腔增加反射镜后等效于增大了激光器谐振器腔长,这样有利于进一步压窄输出线宽。在稳态条件下,增加外腔反馈的半导体激光器的输出功率为[19]:
$$ p = \nu \frac{\varOmega }{{{G_{th}}}}\left( {I - {I_{th}}} \right) $$ (10) 式中:
$ \nu $ 是半导体激光器的器件参数,表示为:$$ \nu = \frac{{{{{{\varGamma}}}} \varepsilon h}}{{4\pi {\tau _M}q}} $$ (11) 式中:
${\varGamma }$ 为制约因子;$ \varepsilon $ 为注入电流转换效率;$ h $ 为普朗克常量;$ {\tau _M} $ 为光子寿命;$ q $ 为电荷常量。$ \varOmega $ 和$ {G_{th}} $ 分别为振荡频率和阈值增益,表示为:$$ {G_{th}} = {\varGamma _0} + \frac{1}{{{\tau _d}}}{\rm ln}\left[ {\frac{{1 + 2{R_2}{R_{{\rm{out}}}}\cos (\varOmega {\tau _e}) + {R_2}{R_{{\rm{out}}}}}}{{1 + 2{R_{{\rm{out}}}}/{R_2}\left[ {\cos \left( {\varOmega {\tau _e}} \right)} \right] + {{\left( {{{{R_{{\rm{out}}}}} \mathord{\left/ {\vphantom {{{R_{{\rm{out}}}}} {{R_2}}}} \right. } {{R_2}}}} \right)}^2}}}} \right] $$ (12) $$ \begin{split} &\varOmega = {\omega _N} + \\ &\frac{1}{{{\tau _d}}}\arctan \left[ {\frac{{\left( {{R_2}{R_{{\rm{out}}}} - {R_{{\rm{out}}}}/{R_2}} \right)\sin \left( {\varOmega {\tau _e}} \right)}}{{1 + {R_{{\rm{out}}}}{R_{{\rm{out}}}} + \left( {{R_2}{R_{{\rm{out}}}} - {R_{{\rm{out}}}}/{R_2}} \right)\cos \left( {\varOmega {\tau _e}} \right)}}} \right] \end{split}$$ (13) 由上式可得,外腔半导体激光器的输出功率不仅与半导体激光器有源腔参数有关,还受外腔腔长、闪耀光栅衍射效率以及外腔反馈反射镜反射率的影响。
Optimal design for reducing diffraction loss of Littman-Metcalf grating external cavity semiconductor laser
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摘要: 基于原有Littman-Metcalf型光栅外腔半导体激光器的工作原理,设计了一种可以降低衍射损耗的外腔结构。在Littman-Metcalf结构的基础上增加一个反射镜,将闪耀光栅二次衍射产生的零级衍射光反馈回半导体激光器本征腔。推导了新结构模型外腔损耗的表达式,通过等效腔的概念对两种结构激光器的外腔损耗、阈值电流、输出线宽以及输出功率进行了仿真分析。结果表明:将二次衍射产生的零级光反馈回有源区可有效降低Littman-Metcalf结构激光器的外腔损耗,提高了系统的耦合效率,从而降低阈值电流,提高了激光器的输出功率。同时,由于提高了外腔反射效率,该外腔结构进一步压窄激光器的输出线宽。对影响低损耗Littman-Metcalf外腔激光器输出线宽以及输出功率的因素(端面反射率、内外腔长、闪耀光栅衍射效率以及反射镜反射率等)也进行了仿真分析,为后期激光器制作提高了理论指导。Abstract: Based on the working principle of Littman-Metcalf type grating external cavity semiconductor laser, an external cavity structure which can reduce diffraction loss is designed. Based on the Littman-Metcalf structure, a reflector is added to feed back the zero-order diffraction light generated by the secondary diffraction of the shining grating to the intrinsic cavity of the semiconductor laser. The expression of the external cavity loss of the new structure model is derived, and the external cavity loss, threshold current, output line width and output power of the two laser structures are simulated by the concept of equivalent cavity. The results show that the zero-order light fed back to the active region can effectively reduce the external cavity loss of the Littman-Metcalf structure laser and improve the coupling efficiency of the system, thus reducing the threshold current and improving the output power of the laser. At the same time, the output linewidth of the laser is further narrowed by improving the reflection efficiency of the external cavity. The factors affecting the output linewidth and output power of low loss Littman-Metcalf external cavity laser (end reflectivity, internal and external cavity length, blazed grating diffraction efficiency and mirror reflectivity) are also simulated and analyzed. It improves the theoretical guidance for the later laser production.
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图 7 半导体激光器端面反射率(a)、内腔腔长(b)、外腔腔长(c)、反射镜M1反射率(d)、反射镜M2反射率(e)对低损耗Littman-Metcalf外腔半导体激光器输出线宽的影响
Figure 7. Effect of (a) the end-face reflectance of the semiconductor laser, (b) the length of internal cavity, (c) the length of external cavity, (d) the reflectance of M1, (e) the reflectance of M2 on the output linewidth of the low loss Littman-Metcalf external cavity semiconductor laser
图 8 半导体激光器前端面反射率(a)、半导体激光器后端面反射率(b)、外腔腔长(c)、反射镜M1反射率(d)、反射镜M2反射率、(e)对低损耗Littman-Metcalf外腔半导体激光器输出功率的影响
Figure 8. Effect of (a) the front-face reflectance of semiconductor laser, (b) the end-face reflectance of the semiconductor laser, (c) the length of external cavity, (d) the reflectance of M1, (e) the reflectance of M2 on the output power of the low-loss Littman-Metcalf external cavity semiconductor laser
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