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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.
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Keywords:
- terahertz /
- quantum cascade laser /
- frequency comb /
- dispersion
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图 1 由于GVD引起的频率偏移Δ与模式数目m的关系
Figure 1. The relation between frequency offset Δ and mode numbers m.
图 2 复折射率随频率变化的关系
Figure 2. The relation between complex refractive index and frequency.
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Figure 3. (a) Simulated the relationship between waveguide loss
${\alpha _{\rm{W}}}$ and frequency; (b) the relation between the effective refractive index and frequency.
图 4 器件增益与频率的关系
Figure 4. The relation between the gain and frequency.
图 5 子带电子吸收随频率变化的关系
Figure 5. The relation between intersubband absorption and frequency.
图 6 材料折射率与频率的关系
Figure 6. The relation between the material refractive index and frequency.
图 7 器件的色散与频率的关系
Figure 7. The relation between GVD and frequency.
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
Figure 8. (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.
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[1] Udem T, Holzwarth R, Hansch T W 2002 Nature 416 233
Google Scholar
[2] Faist J, Capasso F, Sivco D L, Sirtori C, Hutchinson A L, Cho A Y 1994 Science 264 553
Google Scholar
[3] Williams B S 2007 Nature Photon. 1 517
Google Scholar
[4] Diddams S A 2010 J. Opt. Soc. Am. B-Opt. Phys. 27 B51
Google Scholar
[5] Siegel P H 2002 IEEE Trans. Microw. Theory Tech. 50 Pii s0018-9480(02)01958-0 910
[6] Villares G F F 2016 Ph. D. Dissertation (Zurich: Swiss Federal Institute of Technology Zurich)
[7] Hugi A, Villares G, Blaser S, Liu H C, Faist J 2012 Nature 492 229
Google Scholar
[8] Vitiello M S, Scalari G, Williams B, De Natale P 2015 Opt. Express 23 5167
Google Scholar
[9] Tzenov P, Burghoff D, Hu Q, Jirauschek C 2017 IEEE T. Thz. Sci. Techn. 7 351
Google Scholar
[10] Bachmann D, Rosch M, Scalari G, Beck M, Faist J, Unterrainer K, Darmo J 2016 Appl. Phys. Lett. 109 221107
Google Scholar
[11] Treacy E B 1969 IEEE J. Quantum Electron. QE 5 454
[12] Bonod N, Neauport J 2016 Adv. Opt. Photonics 8 156
Google Scholar
[13] Fork R L, Martinez O E, Gordon J P 1984 Opt. Lett. 9 150
Google Scholar
[14] Kane S, Squier J 1997 J. Opt. Soc. Am. B-Opt. Phys. 14 661
Google Scholar
[15] Matuschek N, Kartner F X, Keller U 1998 IEEE J. Sel. Top. Quantum Electron. 4 197
Google Scholar
[16] Tempea G, Krausz F, Spielmann C, Ferencz K 1998 IEEE J. Sel. Top. Quantum Electron. 4 193
Google Scholar
[17] Kartner F X, Morgner U, Ell R, Schibli T, Fujimoto J G, Ippen E P, Scheuer V, Angelow G, Tschudi T 2001 J. Opt. Soc. Am. B-Opt. Phys. 18 882
Google Scholar
[18] Rösch M, Scalari G, Villares G, Bosco L, Beck M, Faist J 2016 Appl. Phys. Lett. 108 171104
Google Scholar
[19] Faist J, Villares G, Scalari G, Rösch M, Bonzon C, Hugi A, Beck M 2016 Nanophotonics 5 272
[20] Wang F, Nong H, Fobbe T, Pistore V, Houver S, Markmann S, Jukam N, Amanti M, Sirtori C, Moumdji S, Colombelli R, Li L, Linfield E, Davies G, Mangeney J, Tignon J, Dhillon S 2017 Laser Photonics Rev. 11 1700013
Google Scholar
[21] Burghoff D, Kao T Y, Han N, Chan C W I, Cai X, Yang Y, Hayton D J, Gao J R, Reno J L, Hu Q 2014 Nature Photon. 8 462
Google Scholar
[22] Li H, Laffaille P, Gacemi D, Apfel M, Sirtori C, Leonardon J, Santarelli G, Rosch M, Scalari G, Beck M, Faist J, Hansel W, Holzwarth R, Barbieri S 2015 Opt. Express 23 33270
Google Scholar
[23] Wan W J, Li H, Zhou T, Cao J C 2017 Sci. Rep. 7 44109
Google Scholar
[24] Rösch M, Scalari G, Beck M, Faist J 2014 Nature Photon. 9 42
[25] Bidaux Y, Sergachev I, Wuester W, Maulini R, Gresch T, Bismuto A, Blaser S, Muller A, Faist J 2017 Opt. Lett. 42 1604
Google Scholar
[26] 朱永浩, 黎华, 万文坚, 周涛, 曹俊诚 2017 物理学报 66 099501
Google Scholar
Zhu Y H, Li H, Wan W J, Zhou T, Cao J C 2017 Acta Phys. Sin. 66 099501
Google Scholar
[27] Weber E R, Willardson R K, Liu H, Capasso F 1999 Intersubband Transitions in Quantum Wells: Physics and Device Applications (Vol. 62) (Beijing: Academic Press)
[28] Li H, Cao J C, Lue J T 2008 J. Appl. Phys. 103 103113
Google Scholar
[29] Adachi S 1994 GaAs and Related Materials: Bulk Semiconducting and Superlattice Properties (Beijing: World Scientific Press)
[30] Gires F, Tournois P 1964 Comptes Rendus Hebdomadaires Des Seances De L Academie Des Sciences 258 6112
[31] Lu Q Y, Manna S, Wu D H, Slivken S, Razeghi M 2018 Appl. Phys. Lett. 112 141104
Google Scholar
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