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非厄米的描述对于开放系统有重要意义, 满足parity-time对称性的哈密顿量, 其参数在一定范围内可以使能量具有实的本征值. 本文通过模拟, 研究了损耗大小以及结构对称性对条形波导中的parity-time对称性的影响, 并通过实验发现了电注入条件下由parity-time对称破缺导致的脊条波导模式间隔加倍、模式数减半的现象.
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关键词:
- parity-time对称性 /
- 电注入 /
- 半导体激光器 /
- 模式控制
The non-Hermitian description is of great significance for open systems, and the Hamiltonian which satisfies parity-time symmetry can make the energy have real eigenvalue within a certain range. The properties of parity-time symmetry have bright application prospects in optical systems. For semiconductor lasers, the parity-time symmetry can be constructed by adjusting the level of electrical injection to help achieve better mode control. Electric injection is easier to realize than optical pump when the device size is small and the structure is complex. Therefore, we hope to analyze the characteristics of the laser that satisfies the parity-time symmetry condition under the condition of electric injection. In this paper, we simulate the effects of different set loss values on parity-time symmetry. It is found that with the increase of set loss value, the imaginary part of the refractive index of the gain cavity corresponding to the parity-time symmetry breaking point so-called exceptional point will decrease, and the imaginary part of the characteristic frequency corresponding to the exceptional point will also decrease. We also simulate the effect of structural size ratio of gain region and loss region on parity-time symmetry. On condition that the total cavity length and the imaginary part of the refractive index of the loss region remain unchanged, as the gain cavity becomes longer and the loss cavity becomes shorter, the imaginary part of the refractive index of the gain cavity corresponding to the exceptional point will increase, and the imaginary part of the characteristic frequency corresponding to the exceptional point will also increase. And we qualitatively explain the above phenomenon through the coupled mode equations. Through experiments, metal organic chemical vapor deposition (MOCVD) and standard lithography techniques are used to fabricate asymmetric ridge lasers. Under thermoelectric cooler (TEC) refrigeration and by controlling the injection level of the gain area, the doubled mode spacing and halved mode number of ridge waveguide are found for the first time due to the parity-time symmetry breaking under the condition of electric injection. We believe that the study of parity-time symmetry in ridge laser under the condition of electric injection will be of great help in implementing the mode control.-
Keywords:
- parity-time symmetry /
- electric injection /
- semiconductor laser /
- mode control
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图 2 器件结构图, 其中黄色部分为增益区, 蓝色部分为损耗区
Fig. 2. Device structure diagram, the yellow part is the gain region and the blue part is the loss region.
图 4 折射率虚部为
${n_{{\rm{Ir}}}} = - 0.01$ 和${n_{{\rm{Ir}}}} = - 0.05$ 时, 波长与${n_{{\rm{Il}}}}$ 的关系图Fig. 4. Relationship between wavelength and
${n_{{\rm{Il}}}}$ when${n_{{\rm{Ir}}}} = - 0.01$ and${n_{{\rm{Ir}}}} = - 0.05$ .
图 5 折射率虚部为nIr = –0.01以及nIr = –0.05时, 特征频率虚部与
${n_{{\rm{Il}}}}$ 的关系图Fig. 5. Relationship between the imaginary part of the characteristic frequency and
${n_{{\rm{Il}}}}$ when${n_{{\rm{Ir}}}} = - 0.01$ and${n_{{\rm{Ir}}}} = - 0.05$ .
图 6 增益区和损耗区长度比为5∶5, 7∶3以及8∶2时的结构图
Fig. 6. Simulation structure of ridged waveguide when length ratio of gain region and loss region is 5∶5, 7∶3, and 8∶2.
图 8 长度比为5∶5, 7∶3以及8∶2时, 特征频率虚部与
${n_{{\rm{Il}}}}$ 的关系图Fig. 8. Relationship between the imaginary part of the characteristic frequency and
${n_{{\rm{Il}}}}$ when the length ratio is 5∶5, 7∶3, and 8∶2.
图 7 长度比为5∶5, 7∶3以及8∶2时, 波长与
${n_{{\rm{Il}}}}$ 的关系图Fig. 7. Relationship between wavelength and
${n_{{\rm{Il}}}}$ when the length ratio is 5∶5, 7∶3, and 8∶2.
图 9 腔模强度与波长关系 (a)注入电流为100 mA; (b)注入电流分别为310 mA(黑线)、320 mA(红线)以及330 mA(蓝线)
Fig. 9. Relationship between the normalized intensity and wavelength of cavity modes: (a) The injection current is 100 mA; (b) the injection current is 310 mA (black line), 320 mA (red line) and 330 mA (blue line), respectively.
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[1] Gamow G 1928 Z. Für Phys. 51 204
Google Scholar
[2] Dirac P A M, F R S 1942 Proceedings A 180 1
[3] Feshbach H, Weisskopf V F, Porter C E 1954 Nucl React 96 227
[4] Brower R C, Furman M A, Moshe M 1978 Phys. Lett. B 76 213
Google Scholar
[5] Denham S A, Harms B C, Jones S T 1981 Nucl. Phys. B 188 155
Google Scholar
[6] Andrianov A A 1982 Ann. Phys.-New York 140 82
Google Scholar
[7] Hollowood T J 1992 Nucl. Phys. B 384 523
Google Scholar
[8] Scholtz F G, Geyer H B, Hahne F J W 1992 Ann. Phys.-New York 213 74
Google Scholar
[9] Caliceti E, Graffi S, Maioli M 1980 Commun. Math. Phys. 75 51
Google Scholar
[10] Bender C M, Boettcher S 1998 Phys. Rev. Lett. 80 5243
Google Scholar
[11] Bender C M, Dunne G V, Meisinger P N 1998 Phys. Lett. A 252 272
[12] Dorey P, Dunning C, Roberto T 2001 J. Phys. A: Gen. Phys. 34 L391
Google Scholar
[13] Ruschhaupt A, Delgado F, Muga J G 2017 J. Phys. A: Gen. Phys. 38 L171
[14] Brandstetter M, Liertzer M, Deutsch C, Klang P, Schöberl J, Türeci H E, Strasser G, Unterrainer K, Rotter S 2014 Nat. Commun. 5 4034
Google Scholar
[15] Gao Z, Fryslie S T M, Thompson B J, Carney P S, Choquette K D 2017 Optica 4 323
Google Scholar
[16] Guo A, Salamo G J, Duchesne D, Morandotti R, Volatierravat M, Aimez V, Siviloglou G A, Christodoulides D N 2009 Phys. Rev. Lett. 103 093902
Google Scholar
[17] Makris K G, El-Ganainy R, Christodoulides D N, Musslimani Z H 2008 Phys. Rev. Lett. 100 103904
Google Scholar
[18] Zhen B, Hsu C W, Igarashi Y, Lu L, Kaminer I, Pick A, Chua S L, Joannopoulos J D, Marin S 2015 Nature 525 354
Google Scholar
[19] Doppler J, Mailybaev A A, Böhm J, Kuhl U, Girschik A, Libisch F, Milburn T J, Rabl P, Moiseyev N, Rotter S 2016 Nature 537 76
Google Scholar
[20] Xu H, Mason D, Jiang L, Harris J G E 2016 Nature 537 80
Google Scholar
[21] Gu Z, Zhang N, Lyu Q, Li M, Xiao S, Song Q 2016 Laser Photonics Rev. 10 588
Google Scholar
[22] Zhang N, Gu Z, Wang K, Li M, Ge L, Xiao S, Song Q 2017 Laser Photonics Rev. 11 1700052
Google Scholar
[23] Zhao S, Qi A, Wang M, Qu H, Lin Y, Dong F, Zheng W 2018 Opt. Express 26 3518
Google Scholar
[24] Coldren L A 2012 Diode Lasers and Photonic Integrated Circuits (America: John Wiley & Sons) pp335–391
[25] Rüter C E, Makris K G, El-Ganainy R, Christodoulides D N, Segev M, Kip D 2010 Nat. Phys. 6 192
Google Scholar
[26] Hodaei H, Hassan A U, Wittek S, Garcia-Gracia H, El-Ganainy R, Christodoulides D N, Khajavikhan M 2017 Nature 548 187
Google Scholar
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