1.Key Laboratory of Terahertz Solid State Technology, Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai 200050, China 2.Center of Materials Science and Optoelectronics Engineering, University of Chinese Academy of Sciences, Beijing 100049, China
Fund Project:Project supported by the “Hundred-Talent" Program of Chinese Academy of Sciences, the National Natural Science Foundation of China (Grant Nos. 61875220, 61575214, 61404150, 61405233, 61704181), the National Key R&D Program of China (Grant Nos. 2017YFF0106302, 2017YFA0701005), and Shanghai Municipal Commission of Science and Technology, China (Grant No. 17YF1430000).
Received Date:19 February 2019
Accepted Date:12 March 2019
Available Online:01 May 2019
Published Online:20 May 2019
Abstract:The frequency comb which is characterized by equally-spaced frequency lines with high mode coherence has received much attention since its first demonstration in near-infrared and optical frequency range. In the terahertz frequency range, the electrically-pumped terahertz quantum cascade laser (THz QCL) based on semiconductors is an ideal candidate for achieving frequency comb operation in a frequency range between 1 THz and 5 THz. The group velocity dispersion (GVD) is a key factor for the frequency comb. A higher GVD can pull the frequencies from their equidistant values and limit the comb bandwidth. Therefore the laser dispersion needs to be compensated for in order to make the total GVD sufficiently low and flat, such as using a Gires-Tournois interferometer (GTI) or the double chirped mirror (DCM). However, a successful design still depends on the knowledge of the total GVD in the laser. In this paper, we show how to calculate the GVD in metal-metal waveguide THz QCLs by taking into account the dispersions from the GaAs material, the waveguide, and the laser gain, which conduces to the understanding of the frequency comb behavior. The waveguide loss is modelled by the finite element method. The loss due to intersubband absorption is calculated by Fermi's gold rule. All the losses, i.e., waveguide loss, mirror loss, and intersubband absorption loss, are summed up to calculate the clamped gain. The material loss can be calculated by using the reststrahlen band model. Because of these losses and gain, the refractive index needs to be replaced by a complex refractive index. The real part of the complex refractive index is the refractive index, which can be calculated from the Kramers-Kronig relationship that connects the loss or gain with the refractive index. Then the GVD introduced by the material loss, waveguide loss, and clamped gain can be finally calculated. The results show that the total GVD of THz QCL is approximately –8 × 105~8 × 105 fs2/mm which is strongly determined by the clamped gain. Finally, the developed numerical model is employed to study the dispersion compensation effect of a GTI mirror which is coupled into a QCL gain cavity. The design of the THz QCL based on GTI structure is more flexible and feasible than that of the DCM. The result shows that by carefully designing the geometry of GTI, the dispersion of a THz QCL can be compensated for, thus achieving the broadband terahertz frequency combs. Keywords:terahertz/ quantum cascade laser/ frequency comb/ dispersion
以有源区为例, 图2给出了有源区复折射率的实部(图2红色实线)和虚部(图2蓝色虚线)随频率的关系. 图 2 复折射率随频率变化的关系 Figure2. The relation between complex refractive index and frequency.
根据(8)式计算得到器件各个部分的复折射率与频率的关系后, 利用有限元法便可以计算得到系统的损耗和有效折射率, 如图3所示, 其中图3(a)为计算得到的波导损耗${\alpha _{\rm{W}}}\left( \omega \right)$随频率变化的关系, 图3(b)为计算得到的等效折射率与频率的关系. 图 3 (a)计算得到的波导损耗${\alpha _{\rm{W}}}$与频率的关系; (b)等效折射率与频率的关系 Figure3. (a) Simulated the relationship between waveguide loss ${\alpha _{\rm{W}}}$ and frequency; (b) the relation between the effective refractive index and frequency.
由(17)式可以发现, GTI结构引起的色散补偿的最大值与其长度的平方成正比, 但是过长的GTI会使其GDD的周期过小, 无法对THz QCL进行色散补偿. 我们基于参考文献[31]设计了一种可直接集成到THz QCL上的GTI结构进行色散补偿, 其三维结构示意图如图8(a)所示, 通过在THz QCL上刻蚀出一个空气间隙, 从而在器件的末端形成一个GTI结构, 其中GTI结构的前端面反射系数${r_1}$由空气间隙的长度以及GTI腔长决定, 后端面反射系数${r_2}$由器件的波导结构决定(当GTI腔长远小于器件的腔长时). 该结构相比于在波导上设计DCM或者是在器件端面生长材料形成GTI等方式, 在工艺以及设计上更加简便灵活, 仅通过控制空气间隙以及GTI长度便可以设计出具备不同色散的GTI结构. 在这里, 我们计算了不同前端面反射系数${r_1}$下的GTI引起的GDD变化, 其中THz QCL结构为双面金属波导结构(${r_2} = $0.83), GTI的腔长$d$ = 58 μm, 结果如图8(b)所示. 由图8(b)可以发现, 较大或者较小的前端面反射系数${r_1}$均不能提供足够强、平滑的色散补偿, 因此, 一个优秀的色散补偿结构设计需要基于准确的GVD计算得到. 图 8 (a)基于GTI结构THz QCL色散补偿的三维示意图; (b)不同前端面反射系数下的群延迟色散与频率的关系 Figure8. (a) Three-dimensional schematic of the THz QCL based on GTI structure for dispersion compensations; (b) calculated group delay dispersions as a function of frequency for different reflection coefficients.