Abstract

The recently developed plasmonic and photonic metal-semiconductor nanolasers feature unique properties, such as ultra-small mode volume and footprint, high Purcell factor, and ultra-fast modulation. However, it is often difficult to recognize when the transition to lasing occurs, while the most important feature of laser radiation, i.e., coherence, is available only above the lasing threshold. Here we systematically study the second-order coherence properties of metal-semiconductor nanolasers at both low- and high-pump rates. We find the lasing threshold using a clear coherence definition and derive a simple expression for the threshold pump current (optical pump power), which can be applied to most thresholdless and non-thresholdless metal-semiconductor nanolasers.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2017 (2)

S. Kreinberg, W. W. Chow, J. Wolters, C. Schneider, C. Gies, F. Jahnke, S. Höfling, M. Kamp, and S. Reitzenstein, “Emission from quantum-dot high-β microcavities: transition from spontaneous emission to lasing and the effects of superradiant emitter coupling,” Light Sci. Appl. 6(8), e17030 (2017).
[Crossref] [PubMed]

D. I. Yakubovsky, D. Yu. Fedyanin, A. V. Arsenin, and V. S. Volkov, “Optical constant of thin gold films: Structural morphology determined optical response,” AIP Conf. Proc. 1874, 040057 (2017).
[Crossref]

2016 (4)

U. L. Andersen, T. Gehring, C. Marquardt, and G. Leuchs, “30 years of squeezed light generation,” Phys. Scr. 91(5), 053001 (2016).
[Crossref]

W. E. Hayenga, H. Garcia-Gracia, H. Hodaei, C. Reimer, R. Morandotti, P. LiKamWa, and M. Khajavikhan, “Second-order coherence properties of metallic nanolasers,” Optica 3(11), 1187 (2016).
[Crossref]

S. H. Pan, Q. Gu, A. El Amili, F. Vallini, and Y. Fainman, “Dynamic hysteresis in a coherent high-β nanolaser,” Optica 3(11), 1260 (2016).
[Crossref]

J.-H. Choi, Y.-S. No, J.-P. So, J. M. Lee, K.-H. Kim, M.-S. Hwang, S.-H. Kwon, and H.-G. Park, “A high-resolution strain-gauge nanolaser,” Nat. Commun. 7(1), 11569 (2016).
[Crossref] [PubMed]

2014 (6)

M. T. Hill and M. C. Gather, “Advances in small lasers,” Nat. Photonics 8(12), 908–918 (2014).
[Crossref]

Q. Zhang, G. Li, X. Liu, F. Qian, Y. Li, T. C. Sum, C. M. Lieber, and Q. Xiong, “A room temperature low-threshold ultraviolet plasmonic nanolaser,” Nat. Commun. 5(1), 4953 (2014).
[Crossref] [PubMed]

Y.-J. Lu, C.-Y. Wang, J. Kim, H.-Y. Chen, M.-Y. Lu, Y.-C. Chen, W.-H. Chang, L.-J. Chen, M. I. Stockman, C.-K. Shih, and S. Gwo, “All-color plasmonic nanolasers with ultralow thresholds: autotuning mechanism for single-mode lasing,” Nano Lett. 14(8), 4381–4388 (2014).
[Crossref] [PubMed]

Q. Gu, J. S. T. Smalley, M. P. Nezhad, A. Simic, J. H. Lee, M. Katz, O. Bondarenko, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Subwavelength semiconductor lasers for dense chip-scale integration,” Adv. Opt. Photonics 6(1), 1 (2014).
[Crossref]

Q. Gu, J. Shane, F. Vallini, B. Wingad, J. S. T. Smalley, N. C. Frateschi, and Y. Fainman, “Amorphous Al2O3 Shield for Thermal Management in Electrically Pumped Metallo-Dielectric Nanolasers,” IEEE J. Quantum Electron. 50(7), 499–509 (2014).
[Crossref]

W. W. Chow, F. Jahnke, and C. Gies, “Emission properties of nanolasers during the transition to lasing,” Light Sci. Appl. 3(8), 201 (2014).
[Crossref]

2013 (2)

2012 (5)

Y.-J. Lu, J. Kim, H.-Y. Chen, C. Wu, N. Dabidian, C. E. Sanders, C.-Y. Wang, M.-Y. Lu, B.-H. Li, X. Qiu, W.-H. Chang, L.-J. Chen, G. Shvets, C.-K. Shih, and S. Gwo, “Plasmonic nanolaser using epitaxially grown silver film,” Science 337(6093), 450–453 (2012).
[Crossref] [PubMed]

K. Ding and C. Z. Ning, “Metallic subwavelength-cavity semiconductor nanolasers,” Light Sci. Appl. 1(7), 20 (2012).
[Crossref]

M. Khajavikhan, A. Simic, M. Katz, J. H. Lee, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Thresholdless nanoscale coaxial lasers,” Nature 482(7384), 204–207 (2012).
[Crossref] [PubMed]

D. Y. Fedyanin, “Toward an electrically pumped spaser,” Opt. Lett. 37(3), 404–406 (2012).
[Crossref] [PubMed]

D. Y. Fedyanin, A. V. Krasavin, A. V. Arsenin, and A. V. Zayats, “Surface plasmon polariton amplification upon electrical injection in highly integrated plasmonic circuits,” Nano Lett. 12(5), 2459–2463 (2012).
[Crossref] [PubMed]

2011 (2)

M. I. Stockman, “Nanoplasmonics: past, present, and glimpse into future,” Opt. Express 19(22), 22029–22106 (2011).
[Crossref] [PubMed]

R.-M. Ma, R. F. Oulton, V. J. Sorger, G. Bartal, and X. Zhang, “Room-temperature sub-diffraction-limited plasmon laser by total internal reflection,” Nat. Mater. 10(2), 110–113 (2011).
[Crossref] [PubMed]

2010 (2)

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4(6), 395–399 (2010).
[Crossref]

S. Ritter, P. Gartner, C. Gies, and F. Jahnke, “Emission properties and photon statistics of a single quantum dot laser,” Opt. Express 18(10), 9909–9921 (2010).
[Crossref] [PubMed]

2009 (4)

J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460(7252), 245–249 (2009).
[Crossref] [PubMed]

F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two-photon absorption in semiconductors,” Nat. Phys. 5(4), 267–270 (2009).
[Crossref]

M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y.-S. Oei, R. Nötzel, C.-Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17(13), 11107–11112 (2009).
[Crossref] [PubMed]

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[Crossref] [PubMed]

2007 (3)

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317(5839), 783–787 (2007).
[Crossref] [PubMed]

M. T. Hill, Y.-S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S.-H. Kwon, Y.-H. Lee, R. Nötzel, and M. K. Smit, “Lasing in metallic-coated nanocavities,” Nat. Photonics 1(10), 589–594 (2007).
[Crossref]

C. Gies, J. Wiersig, M. Lorke, and F. Jahnke, “Semiconductor model for quantum-dot-based microcavity lasers,” Phys. Rev. A 75(1), 013803 (2007).
[Crossref]

2004 (1)

F. Marquier, K. Joulain, J.-P. Mulet, R. Carminati, J.-J. Greffet, and Y. Chen, “Coherent spontaneous emission of light by thermal sources,” Phys. Rev. B Condens. Matter Mater. Phys. 69(15), 155412 (2004).
[Crossref]

2003 (1)

T. H. Gfroerer, L. P. Priestley, M. F. Fairley, and M. W. Wanlass, “Temperature dependence of nonradiative recombination in low-band gap InxGa1−xAs/InAsyP1−y double heterostructures grown on InP substrates,” J. Appl. Phys. 94(3), 1738–1743 (2003).
[Crossref]

2002 (1)

J.-J. Greffet, R. Carminati, K. Joulain, J.-P. Mulet, S. Mainguy, and Y. Chen, “Coherent emission of light by thermal sources,” Nature 416(6876), 61–64 (2002).
[Crossref] [PubMed]

1996 (1)

1994 (1)

P. R. Rice and H. J. Carmichael, “Photon statistics of a cavity-QED laser: A comment on the laser-phase-transition analogy,” Phys. Rev. A 50(5), 4318–4329 (1994).
[Crossref] [PubMed]

1989 (2)

M. Asada, “Intraband relaxation time in quantum-well lasers,” IEEE J. Quantum Electron. 25(9), 2019–2026 (1989).
[Crossref]

H. Yokoyama and S. D. Brorson, “Rate equation analysis of microcavity lasers,” J. Appl. Phys. 66(10), 4801–4805 (1989).
[Crossref]

1987 (1)

S. Machida, Y. Yamamoto, and Y. Itaya, “Observation of amplitude squeezing in a constant-current-driven semiconductor laser,” Phys. Rev. Lett. 58(10), 1000–1003 (1987).
[Crossref] [PubMed]

1986 (1)

A. Sugimura, E. Patzak, and P. Meissner, “Homogeneous linewidth and linewidth enhancement factor for a GaAs semiconductor laser,” J. Phys. D Appl. Phys. 19(1), 7–16 (1986).
[Crossref]

Andersen, U. L.

U. L. Andersen, T. Gehring, C. Marquardt, and G. Leuchs, “30 years of squeezed light generation,” Phys. Scr. 91(5), 053001 (2016).
[Crossref]

Armani, A. M.

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317(5839), 783–787 (2007).
[Crossref] [PubMed]

Arsenin, A. V.

D. I. Yakubovsky, D. Yu. Fedyanin, A. V. Arsenin, and V. S. Volkov, “Optical constant of thin gold films: Structural morphology determined optical response,” AIP Conf. Proc. 1874, 040057 (2017).
[Crossref]

D. Y. Fedyanin, A. V. Krasavin, A. V. Arsenin, and A. V. Zayats, “Surface plasmon polariton amplification upon electrical injection in highly integrated plasmonic circuits,” Nano Lett. 12(5), 2459–2463 (2012).
[Crossref] [PubMed]

Asada, M.

M. Asada, “Intraband relaxation time in quantum-well lasers,” IEEE J. Quantum Electron. 25(9), 2019–2026 (1989).
[Crossref]

Assmann, M.

J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460(7252), 245–249 (2009).
[Crossref] [PubMed]

Bartal, G.

R.-M. Ma, R. F. Oulton, V. J. Sorger, G. Bartal, and X. Zhang, “Room-temperature sub-diffraction-limited plasmon laser by total internal reflection,” Nat. Mater. 10(2), 110–113 (2011).
[Crossref] [PubMed]

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461(7264), 629–632 (2009).
[Crossref] [PubMed]

Bayer, M.

J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460(7252), 245–249 (2009).
[Crossref] [PubMed]

Ben-Aryeh, Y.

Berstermann, T.

J. Wiersig, C. Gies, F. Jahnke, M. Assmann, T. Berstermann, M. Bayer, C. Kistner, S. Reitzenstein, C. Schneider, S. Höfling, A. Forchel, C. Kruse, J. Kalden, and D. Hommel, “Direct observation of correlations between individual photon emission events of a microcavity laser,” Nature 460(7252), 245–249 (2009).
[Crossref] [PubMed]

Boitier, F.

F. Boitier, A. Godard, E. Rosencher, and C. Fabre, “Measuring photon bunching at ultrashort timescale by two-photon absorption in semiconductors,” Nat. Phys. 5(4), 267–270 (2009).
[Crossref]

Bondarenko, O.

Q. Gu, J. S. T. Smalley, M. P. Nezhad, A. Simic, J. H. Lee, M. Katz, O. Bondarenko, B. Slutsky, A. Mizrahi, V. Lomakin, and Y. Fainman, “Subwavelength semiconductor lasers for dense chip-scale integration,” Adv. Opt. Photonics 6(1), 1 (2014).
[Crossref]

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, and Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4(6), 395–399 (2010).
[Crossref]

Brorson, S. D.

H. Yokoyama and S. D. Brorson, “Rate equation analysis of microcavity lasers,” J. Appl. Phys. 66(10), 4801–4805 (1989).
[Crossref]

Carmichael, H. J.

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D. Y. Fedyanin, A. V. Krasavin, A. V. Arsenin, and A. V. Zayats, “Surface plasmon polariton amplification upon electrical injection in highly integrated plasmonic circuits,” Nano Lett. 12(5), 2459–2463 (2012).
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Figures (5)

Fig. 1
Fig. 1 Schematic illustration of a typical electrically pumped metal-semiconductor nanolaser. It consists of the semiconductor layer stack etched into a pillar and covered by a thin insulator layer. The semiconductor-insulator structure is encapsulated in silver, gold or copper to form a metallic cavity, which usually has a quality factor in the range from 100 to 300 [1,3,6–9,15,28,32].
Fig. 2
Fig. 2 (a) L-I curve of the metal-semiconductor nanolaser based on the In0.53Ga0.47As bulk gain medium at T = 20 K. The operating wavelength is 1.55 μm, the quality factor of the nanolaser cavity is Q = 100, the confinement factor of the laser mode to the active region is Γ = 0.5 and the volume of the active region is V = 0.3 μm3, the β-factor is equal to 1. These parameters are similar to those of the experimentally measured metal-clad and plasmonic nanolasers [1,6,8,9,15,28]. The material gain is calculated in the parabolic band approximation, the material parameters of InGaAs are adopted from [29]. (b-d) g(2)-function at pump currents of 1 μA (panel (b)), 80 μA (panel (c)) and 1000 μA (panel (d)). These currents are shown in panel (a) by points A, B, and C, respectively. Note that the time scales are different for different panels. (e) g(2)(τ = 0) as a function of Ustim/Uspont. The red line is the result of direct calculations, while the blue dashed line is obtained using Eq. (11).
Fig. 3
Fig. 3 (a) L-I curve of the metal-semiconductor nanolaser for different β-factors. Other parameters are the same as in Fig. 2. (b) First-order derivatives of the L-I curves shown in panel (a). (c-d) g(2)(0) as a function of pump current (panel (c)) and Ustim/Uspont (panel (d)) calculated using Eq. (9). The color code in panel (d) is the same as in panel (c). The blue area indicates the region where g(2)(0) < 1.5. (e) Dependence of the threshold current defined by Eq. (12) on the quality factor of the laser cavity. Open dots in panel (a) indicate the lasing threshold obtained using Eqs. (9) and (14).
Fig. 4
Fig. 4 (a) L-I curve of the metal-semiconductor nanolaser for different Auger recombination constants, T = 300 K, other parameters are the same as in Fig. 2. (b) First-order derivative of the L-I curves shown in panel (a). (c-d) g(2)(0) as a function of pump current (panel (c)) and Ustim/Uspont (panel (d)) calculated using Eq. (9). The color code in panel (d) is the same as in panel (c). The blue area indicates the region where g(2)(0) < 1.5. Open dots in panel (a) indicate the lasing threshold obtained using Eqs. (9) and (14).
Fig. 5
Fig. 5 (a) L-I curve of the metal-semiconductor nanolaser at different temperatures. Open dots indicate the lasing threshold (g(2)(0) = 1.5), which is obtained using Eq. (9). (b) Temperature dependence of the threshold current obtained using three different definitions: green line – using the first-order derivative of the L-I curve in the log-log scale, red line – using the second-order coherence based on Eqs. (9) and (14), blue line – using the second-order coherence based on approximate Eq. (12).

Equations (14)

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{ d N e /dt= J pump /e U nr U spont /β U stim , dN/dt=(G1/ τ cav )N+ U spont .
n sp = 1 1exp( ω 0 ( F e F h ) k B T )
g (2) (τ)= N(t)N(t+τ) N 2
{ d dt δN=Aδ N e BδN+ F sp (t), d dt δ N e =Cδ N e DδN+ F rec (t).
A= ( U spont + U stim ) N e ,
B=1/ τ cav G,
C= ( U spont /β+ U stim + U nr ) N e ,
D=G.
g (2) (0)=1+ A 2 D rr +2AC D sr +(AD+BC+ C 2 ) D ss (B+C)(AD+BC) N 2 .
g (2) (0)=1+ U spont / τ cav AG/2 U spont / τ cav +AN/ τ cav .
g (2) (0)1 ( 1+ n sp dg/d n e | g= g th g th V ( U stim U spont ) 2 ) 1 = 1 1+ ( J pump J th ) 2 .
J th =e n sp n a V dg/dn | g= g th Γc τ cav 3 ,
P th = 2πc λ pump η n sp n a V dg/dn | g= g th Γc τ cav 3 ,
g (2) (0) | J= J th =1.5.