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基于微波光子学的射频三角波生成方案如图 1所示。主要采用外调制方法,器件包括激光二极管(laser diode, LD)、偏振控制器(polarization controller, PC)、马赫-曾德尔调制器(MZM)、射频(radio frequency, RF)源、平衡光电二极管(balanced photodetector, BPD)、光耦合器(optical coupler, OC)和90°电桥。
激光器发出幅度为E0、中心频率为ω0的激光波束,经光耦合器,光束被等功率地分成两路。两路光信号分别经偏振控制器(PC)进入并联的马赫-曾德尔调制器(MZM1和MZM2)中,并被调制。加载在MZM1和MZM2上的微波信号是同一微波源经90°电桥分别生成,即两路微波信号相差90°,为方便推导,将加载在MZM1上的微波源信号定义为初相位为0°的余弦信号,激光波束经电光调制器后输出信号分别为:
$ \left\{ \begin{array}{l} {E_{\rm{u}}} = \frac{1}{{\sqrt 2 }}{E_0}{\rm{exp}}({\rm{j}}{\omega _0}t) \times \\ {\rm{cos}}\left[ {\frac{{{\rm{ \mathsf{ π} }}({V_{{\rm{RF}}}}{\rm{cos}}({\omega _{\rm{s}}}t) + {V_{{\rm{D}}{{\rm{C}}_{\rm{1}}}}})}}{{2{V_{\rm{ \mathsf{ π} }}}}}} \right]{\rm{exp}}\left( {{\rm{j}}\frac{{{\rm{ \mathsf{ π} }}{V_{{\rm{D}}{{\rm{C}}_{\rm{1}}}}}}}{{2{V_{\rm{ \mathsf{ π} }}}}}} \right)\\ {E_{\rm{l}}} = \frac{1}{{\sqrt 2 }}{E_0}{\rm{exp}}({\rm{j}}{\omega _0}t) \times \\ {\rm{cos}}\left[ {\frac{{{\rm{ \mathsf{ π} }}({V_{{\rm{RF}}}}{\rm{sin}}({\omega _{\rm{s}}}t) + {V_{{\rm{D}}{{\rm{C}}_2}}})}}{{2{V_{\rm{ \mathsf{ π} }}}}}} \right]{\rm{exp }}\left( {{\rm{j}}\frac{{{\rm{ \mathsf{ π} }}{V_{{\rm{D}}{{\rm{C}}_2}}}}}{{2{V_{\rm{ \mathsf{ π} }}}}}} \right) \end{array} \right. $
(1) 式中, VRF为微波源的幅度, ωs是微波源角频率,VDC1和VDC2是分别加载在两个MZM偏置电压, Vπ是调制器的半波电压, ω0表示激光器发生激光的中心频率,t表示时间。
需将MZM1和MZM2设置在最大偏置点或最小偏置点,需设置直流源的电压幅度,让直流源1(DC1)、直流源2(DC2)的电压分别为mVπ/2和nVπ/2,其中m, n为正偶数。
MZM1和MZM2的输出光谱理论上相同,如图 2所示。可以看出,光谱图中每两个相邻频率线之间相位相差π/2,相对于激光信号频率对称的两条频率线之间相位总是相差π的整数倍。
经微波源调制后,MZM1的输出端可表示为:
$ \begin{array}{l} {E_{\rm{u}}} = \frac{1}{{\sqrt 2 }}{E_0}{\rm{exp}}({\rm{j}}{\omega _0}t){\rm{cos}}\left[ {\frac{{{\rm{ \mathsf{ π} }}({V_{{\rm{RF}}}}{\rm{cos(}}{\omega _{\rm{s}}}t) + {V_{{\rm{D}}{{\rm{C}}_{\rm{1}}}}}){\rm{ }}}}{{2{V_{\rm{ \mathsf{ π} }}}}}} \right] \times \\ {\rm{exp}}\left( {{\rm{j}}\frac{{{\rm{ \mathsf{ π} }}{V_{{\rm{D}}{{\rm{C}}_{\rm{1}}}}}}}{{2{V_{\rm{ \mathsf{ π} }}}}}} \right) = \frac{1}{{\sqrt 2 }}{E_0}{\rm{exp}}({\rm{j}}{\omega _0}t){\rm{cos}}\left[ {m{\rm{ \mathsf{ π} }} + \frac{{{\rm{ \mathsf{ π} }}{V_{{\rm{RF}}}}}}{{2{V_{\rm{ \mathsf{ π} }}}}}{\rm{cos}}({\omega _{\rm{s}}}t)} \right]{\rm{ }} \times \\ {\rm{exp}}\left( {{\rm{j}}\frac{{\rm{ \mathsf{ π} }}}{4}m} \right){\rm{ }} = \frac{1}{{\sqrt 2 }}{E_0}{\rm{exp}}({\rm{j}}{\omega _0}t + {\rm{j}}m{\rm{ \mathsf{ π} }}) \times \\ {\rm{cos}}\left[ {\frac{{{\rm{ \mathsf{ π} }}{V_{{\rm{RF}}}}}}{{2{V_{\rm{ \mathsf{ π} }}}}}{\rm{cos}}({\omega _{\rm{s}}}t)} \right]{\rm{ }} \end{array} $
(2) MZM2的输出端可表示为:
$ \begin{array}{l} {E_{\rm{l}}} = \frac{1}{{\sqrt 2 }}{E_0}{\rm{exp}}({\rm{j}}{\omega _0}t){\rm{cos}}\left\{ {\frac{{{\rm{ \mathsf{ π} }}[{V_{{\rm{RF}}}}{\rm{sin}}({\omega _{\rm{s}}}t) + {V_{{\rm{D}}{{\rm{C}}_{\rm{2}}}}}]}}{{2{V_{\rm{ \mathsf{ π} }}}}}} \right\} \times \\ {\rm{exp}}\left( {{\rm{ j}}\frac{{{\rm{ \mathsf{ π} }}{V_{{\rm{D}}{{\rm{C}}_{\rm{2}}}}}}}{{2{V_{\rm{ \mathsf{ π} }}}}}} \right) = \frac{1}{{\sqrt 2 }}{E_0}{\rm{exp}}({\rm{j}}{\omega _0}t) \times \\ {\rm{cos }}\left[ {\frac{{{\rm{ \mathsf{ π} }}{V_{{\rm{RF}}}}}}{{2{V_{\rm{ \mathsf{ π} }}}}}{\rm{sin}}({\omega _{\rm{s}}}t) + n{\rm{ \mathsf{ π} }}} \right]{\rm{exp}}({\rm{j}}n\pi ) = \\ \frac{1}{{\sqrt 2 }}{E_0}{\rm{exp}}({\rm{j}}{\omega _0}t + {\rm{j}}n{\rm{ \mathsf{ π} }}){\rm{cos }}\left[ {\frac{{{\rm{ \mathsf{ π} }}{V_{{\rm{RF}}}}}}{{2{V_{\rm{ \mathsf{ π} }}}}}{\rm{sin}}({\omega _{\rm{s}}}t)} \right] \end{array} $
(3) 两路光信号经光纤信道,进入平衡光电二极管中,经拍频处理后,输出的光电流可表示为:
$ \begin{array}{l} {I_{{\rm{out}}}} \propto {\rm{co}}{{\rm{s}}^2}\left[ {\frac{{{\rm{ \mathsf{ π} }}{V_{{\rm{RF}}}}}}{{2{V_{\rm{ \mathsf{ π} }}}}}{\rm{cos}}\left( {{\omega _{\rm{s}}}t} \right)} \right]{\rm{ }} - {\rm{co}}{{\rm{s}}^2}\left[ {\frac{{{\rm{ \mathsf{ π} }}{V_{{\rm{RF}}}}}}{{2{V_{\rm{ \mathsf{ π} }}}}}{\rm{sin}}\left( {{\omega _{\rm{s}}}t} \right)} \right]{\rm{ }} = \\ \frac{{1 - 2{\rm{cos}}\left[ {\alpha {\rm{cos}}({\omega _s}t)} \right]}}{2} - {\rm{ }}\frac{{1 - 2{\rm{cos}}\left[ {\alpha {\rm{sin}}({\omega _{\rm{s}}}t)} \right]}}{2}{\rm{ }} = \\ {\rm{cos}}[\alpha {\rm{sin}}({\omega _{\rm{s}}}t)\left] { - {\rm{cos}}} \right[\alpha {\rm{cos}}({\omega _{\rm{s}}}t)] \end{array} $
(4) 式中,α=πVRF/Vπ定义为调制系数。
通过贝塞尔展开式,对光电流信号进行Jacobi展开:
$ \begin{array}{l} {I_{{\rm{out}}}} \propto {{\rm{J}}_0}(\alpha ) + 2\sum\limits_{n = 1}^\infty {} {{\rm{J}}_{2n}}(\alpha ){\rm{cos}}(2n{\omega _{\rm{s}}}t) - \\ \left[ {{{\rm{J}}_0}(\alpha ) + 2\sum\limits_{n = 1}^\infty {} {\rm{ }}{{\left( { - 1} \right)}^n}{{\rm{J}}_{2n}}\left( \alpha \right){\rm{cos}}(2n{\omega _{\rm{s}}}t)} \right] = \\ 4\sum\limits_{n = 1, 3, 5}^\infty {} {{\rm{J}}_{2n}}(\alpha ){\rm{cos}}(2n{\omega _{\rm{s}}}t) \end{array} $
(5) 式中,J为Jacobi展开固定形式。理想三角波进行傅里叶级数展开:
$ {T_{{\rm{tr}}}}(t) = D + \sum\limits_{n = 1, 3, 5}^\infty {} \frac{1}{{{n^2}}}{\rm{cos}}(n\mathit{\Omega} t) $
(6) 考虑到高阶谐波衰减迅速,在实际生产生活中可以忽略,因此着重考虑低次谐波。可将理想三角波的信号分解进行简化:
$ {T_{{\rm{tr}}}}(t) = D + {\rm{cos}}(\mathit{\Omega} t) + \frac{1}{9}{\rm{cos}}(3\mathit{\Omega} t) $
(7) 式中, Ω是基频的角频率, D为直流信号幅度。从(5)式和(7)式对比可得,只需满足下式即可生成三角波:
$ {I_{{\rm{out}}}} \propto 4{{\rm{J}}_2}(\alpha ){\rm{cos}}(2{\omega _{\rm{s}}}t) + 4{{\rm{J}}_6}(\alpha ){\rm{cos}}(6{\omega _{\rm{s}}}t) $
(8) 通过调制系数α进行调节,使I2(α): I6(α)=9: 1。对调制系数进行计算,得到当α=3.895,可满足9: 1的比例,即可生成三角波, 如图 3所示。
基于微波光子学的倍频三角波生成方法
Triangular waveform generation with frequency doubling based on microwave photonics
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摘要: 为了克服传统任意波形生成方法中电子瓶颈问题,分析了基于微波光子学的射频任意波形的技术类型、特点和应用背景,采用一种基于并联马赫-曾德尔调制器的倍频三角波生成方法,引入均方根误差对输出信号和理想波形来评价,并进行了理论分析和仿真验证。结果表明,通过10GHz驱动信号生成了20GHz的三角波信号,均方根误差为0.038,即输出信号与理想信号吻合度较高; 与其它方法相比,该方法可生成倍频三角波,信号波形与理论波形吻合度良好。该研究对未来基于微波光子学的射频任意波生成有指导意义。Abstract: In order to overcome the problem of the electronic bottleneck in the traditional arbitrary waveform generation method, the technology type, characteristics and application background of the radio frequency arbitrary waveform based on microwave photonics were analyzed. A triangular waveform generation with frequency doubling based on parallel Mach-Zehnder modulator was adopted, and the theoretical analysis and simulation verification were carried out. The root mean square error was introduced to evaluate the output signal and the ideal waveform. The results show that, the triangle wave signal of 20GHz can be generated by 10GHz driving signal. The root mean square error is 0.038, which means the output signal is in good agreement with the ideal signal. Compared with other methods, the method can generate frequency doubling triangle wave, and the signal waveform agrees well with the theoretical waveform. It has guiding significance for the future generation of radio frequency arbitrary waves based on microwave photonics.
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