Abstract

A small-footprint nanobeam photonic crystal laser made of InGaAsP material is directly integrated on a SiO2/Si substrate without using adhesive material via transfer-printing processes (i.e., dry transfer-printing). The transferred nanobeam structure with a physical volume of ~6.6 × 0.58 × 0.28 µm3 (~10.5 (λ/n)3) shows single mode lasing near 1550 nm with continuous-wave (CW) operation at room-temperature, where effective lasing threshold power was as low as 9 µW. This CW operation was achieved mainly due to efficient heat dissipation provided by direct contact between the nanobeam and the substrate. This transfer-printed nanobeam laser could be a promising candidate for the next-generation light source with a feature of low-power consumption in ultracompact photonic integrated circuits.

© 2014 Optical Society of America

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2013 (3)

K.-Y. Jeong, Y.-S. No, Y. Hwang, K.-S. Kim, M.-K. Seo, H.-G. Park, Y.-H. Lee, “Electrically driven nanobeam laser,” Nat. Commun. 4, 2822 (2013).
[CrossRef] [PubMed]

T.-W. Lu, W.-C. Tsai, T.-Y. Wu, P.-T. Lee, “Laser emissions from one-dimensional photonic crystal rings on silicon-dioxide,” Appl. Phys. Lett. 102(5), 051103 (2013).
[CrossRef]

W. S. Fegadolli, S.-H. Kim, P. A. Postigo, A. Scherer, “Hybrid single quantum well InP/Si nanobeam lasers for silicon photonics,” Opt. Lett. 38(22), 4656–4658 (2013).
[CrossRef] [PubMed]

2012 (3)

S. Matsuo, K. Takeda, T. Sato, M. Notomi, A. Shinya, K. Nozaki, H. Taniyama, K. Hasebe, T. Kakitsuka, “Room-temperature continuous-wave operation of lateral current injection wavelength-scale embedded active-region photonic-crystal laser,” Opt. Express 20(4), 3773–3780 (2012).
[CrossRef] [PubMed]

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, W. Zhou, “Transfer-printed stacked nanomembrane lasers on silicon,” Nat. Photonics 6(9), 615–620 (2012).
[CrossRef]

B. Wang, T. Siahaan, M. A. Dundar, R. Notzel, S. He, R. W. van der Heijden, “Transfer printing and nanomanipulating luminescent photonic crystal membrane nanocavities,” J. Appl. Phys. 111(9), 093105 (2012).
[CrossRef]

2011 (3)

B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Haller, J. Vuckovic, “Ultralow-threshold electrically pumped quantum-dot photonic-crystal nanocavity laser,” Nat. Photonics 5(5), 297–300 (2011).
[CrossRef]

J. Huang, S.-H. Kim, J. Gardner, P. Regreny, C. Seassal, P. A. Postigo, A. Scherer, “Room temperature, continuous-wave coupled cavity InAsP/InP photonic crystal laser with enhanced far-field emission directionality,” Appl. Phys. Lett. 99(9), 091110 (2011).
[CrossRef]

Y. Halioua, A. Bazin, P. Monnier, T. J. Karle, G. Roelkens, I. Sagnes, R. Raj, F. Raineri, “Hybrid III-V semiconductor/silicon nanolaser,” Opt. Express 19(10), 9221–9231 (2011).
[CrossRef] [PubMed]

2010 (5)

Y. Halioua, A. Bazin, P. Monnier, T. J. Karle, I. Sagnes, G. Roelkens, D. Van Thourhout, F. Raineri, R. Raj, “III-V photonic crystal wire cavity laser on silicon wafer,” J. Opt. Soc. Am. B 27(10), 2146–2150 (2010).
[CrossRef]

Y. Zhang, M. Khan, Y. Huang, J. Ryou, P. Deotare, R. Dupuis, M. Lončar, “Photonic crystal nanobeam lasers,” Appl. Phys. Lett. 97(5), 051104 (2010).
[CrossRef]

Y. Gong, B. Ellis, G. Shambat, T. Sarmiento, J. S. Harris, J. Vuckovic, “Nanobeam photonic crystal cavity quantum dot laser,” Opt. Express 18(9), 8781–8789 (2010).
[CrossRef] [PubMed]

G. Roelkens, L. Liu, D. Liang, R. Jones, A. Fang, B. Koch, J. Bowers, “III-V/silicon photonics for on-chip and inter-chip optical interconnects,” Laser Photon. Rev. 4(6), 751–779 (2010).
[CrossRef]

D. Liang, J. E. Bowers, “Recent progress in lasers on silicon,” Nat. Photonics 4(8), 511–517 (2010).
[CrossRef]

2009 (2)

2008 (1)

2007 (4)

2006 (1)

M. A. Meitl, Z.-T. Zhu, V. Kumar, K. J. Lee, X. Feng, Y. Y. Huang, I. Adesida, R. G. Nuzzo, J. A. Rogers, “Transfer printing by kinetic control of adhesion to an elastomeric stamp,” Nat. Mater. 5(1), 33–38 (2006).
[CrossRef]

2005 (1)

B.-S. Song, S. Noda, T. Asano, Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005).
[CrossRef]

2003 (1)

K. Nozaki, A. Nakagawa, D. Sano, T. Baba, “Ultralow threshold and single-mode lasing in microgear lasers and its fusion with quasi-periodic photonic crystals,” IEEE J. Sel. Top. Quantum Electron. 9(5), 1355–1360 (2003).
[CrossRef]

2002 (1)

H.-G. Park, J.-K. Hwang, J. Huh, H.-Y. Ryu, S.-H. Kim, J.-S. Kim, Y.-H. Lee, “Characteristics of modified single-defect two-dimensional photonic crystal lasers,” IEEE J. Quantum Electron. 38(10), 1353–1365 (2002).
[CrossRef]

2000 (1)

H.-Y. Ryu, H.-G. Park, Y.-H. Lee, “Two-dimensional photonic crystal semiconductor lasers: computational design, fabrication, and characterization,” IEEE J. Sel. Top. Quantum Electron. 8(4), 891–908 (2000).

1997 (1)

T. Baba, “Photonic crystals and microdisk cavities based on GaInAsP-InP system,” IEEE J. Sel. Top. Quantum Electron. 3(3), 808–830 (1997).
[CrossRef]

Adesida, I.

M. A. Meitl, Z.-T. Zhu, V. Kumar, K. J. Lee, X. Feng, Y. Y. Huang, I. Adesida, R. G. Nuzzo, J. A. Rogers, “Transfer printing by kinetic control of adhesion to an elastomeric stamp,” Nat. Mater. 5(1), 33–38 (2006).
[CrossRef]

Akahane, Y.

B.-S. Song, S. Noda, T. Asano, Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005).
[CrossRef]

Alén, B.

Andreani, L. C.

Asano, T.

B.-S. Song, S. Noda, T. Asano, Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005).
[CrossRef]

Baba, T.

K. Nozaki, S. Kita, T. Baba, “Room temperature continuous wave operation and controlled spontaneous emission in ultrasmall photonic crystal nanolaser,” Opt. Express 15(12), 7506–7514 (2007).
[CrossRef] [PubMed]

K. Nozaki, A. Nakagawa, D. Sano, T. Baba, “Ultralow threshold and single-mode lasing in microgear lasers and its fusion with quasi-periodic photonic crystals,” IEEE J. Sel. Top. Quantum Electron. 9(5), 1355–1360 (2003).
[CrossRef]

T. Baba, “Photonic crystals and microdisk cavities based on GaInAsP-InP system,” IEEE J. Sel. Top. Quantum Electron. 3(3), 808–830 (1997).
[CrossRef]

Baets, R.

G. Roelkens, J. Van Campenhout, J. Brouckaert, D. Van Thourhout, R. Baets, P. Rojo Romeo, P. Regreny, A. Kazmierczak, C. Seassal, X. Letartre, G. Hollinger, J. M. Fedeli, L. D. Cioccio, C. L. Blanchard, “III-V/Si photonics by die-to-wafer bonding,” Mater. Today 10(7–8), 36–43 (2007).
[CrossRef]

Bagheri, M.

Bazin, A.

Beaudoin, G.

Berggren, J.

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, W. Zhou, “Transfer-printed stacked nanomembrane lasers on silicon,” Nat. Photonics 6(9), 615–620 (2012).
[CrossRef]

Blanchard, C. L.

G. Roelkens, J. Van Campenhout, J. Brouckaert, D. Van Thourhout, R. Baets, P. Rojo Romeo, P. Regreny, A. Kazmierczak, C. Seassal, X. Letartre, G. Hollinger, J. M. Fedeli, L. D. Cioccio, C. L. Blanchard, “III-V/Si photonics by die-to-wafer bonding,” Mater. Today 10(7–8), 36–43 (2007).
[CrossRef]

Bouchoule, S.

Bowers, J.

G. Roelkens, L. Liu, D. Liang, R. Jones, A. Fang, B. Koch, J. Bowers, “III-V/silicon photonics for on-chip and inter-chip optical interconnects,” Laser Photon. Rev. 4(6), 751–779 (2010).
[CrossRef]

Bowers, J. E.

D. Liang, J. E. Bowers, “Recent progress in lasers on silicon,” Nat. Photonics 4(8), 511–517 (2010).
[CrossRef]

Braive, R.

Brouckaert, J.

G. Roelkens, J. Van Campenhout, J. Brouckaert, D. Van Thourhout, R. Baets, P. Rojo Romeo, P. Regreny, A. Kazmierczak, C. Seassal, X. Letartre, G. Hollinger, J. M. Fedeli, L. D. Cioccio, C. L. Blanchard, “III-V/Si photonics by die-to-wafer bonding,” Mater. Today 10(7–8), 36–43 (2007).
[CrossRef]

Choi, S.-J.

Chuwongin, S.

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, W. Zhou, “Transfer-printed stacked nanomembrane lasers on silicon,” Nat. Photonics 6(9), 615–620 (2012).
[CrossRef]

Cioccio, L. D.

G. Roelkens, J. Van Campenhout, J. Brouckaert, D. Van Thourhout, R. Baets, P. Rojo Romeo, P. Regreny, A. Kazmierczak, C. Seassal, X. Letartre, G. Hollinger, J. M. Fedeli, L. D. Cioccio, C. L. Blanchard, “III-V/Si photonics by die-to-wafer bonding,” Mater. Today 10(7–8), 36–43 (2007).
[CrossRef]

Dapkus, P. D.

Deotare, P.

Y. Zhang, M. Khan, Y. Huang, J. Ryou, P. Deotare, R. Dupuis, M. Lončar, “Photonic crystal nanobeam lasers,” Appl. Phys. Lett. 97(5), 051104 (2010).
[CrossRef]

Deotare, P. B.

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, M. Lončar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94(12), 121106 (2009).
[CrossRef]

Dundar, M. A.

B. Wang, T. Siahaan, M. A. Dundar, R. Notzel, S. He, R. W. van der Heijden, “Transfer printing and nanomanipulating luminescent photonic crystal membrane nanocavities,” J. Appl. Phys. 111(9), 093105 (2012).
[CrossRef]

Dupuis, R.

Y. Zhang, M. Khan, Y. Huang, J. Ryou, P. Deotare, R. Dupuis, M. Lončar, “Photonic crystal nanobeam lasers,” Appl. Phys. Lett. 97(5), 051104 (2010).
[CrossRef]

Ellis, B.

B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Haller, J. Vuckovic, “Ultralow-threshold electrically pumped quantum-dot photonic-crystal nanocavity laser,” Nat. Photonics 5(5), 297–300 (2011).
[CrossRef]

Y. Gong, B. Ellis, G. Shambat, T. Sarmiento, J. S. Harris, J. Vuckovic, “Nanobeam photonic crystal cavity quantum dot laser,” Opt. Express 18(9), 8781–8789 (2010).
[CrossRef] [PubMed]

Fang, A.

G. Roelkens, L. Liu, D. Liang, R. Jones, A. Fang, B. Koch, J. Bowers, “III-V/silicon photonics for on-chip and inter-chip optical interconnects,” Laser Photon. Rev. 4(6), 751–779 (2010).
[CrossRef]

Farrell, S.

Fedeli, J. M.

G. Roelkens, J. Van Campenhout, J. Brouckaert, D. Van Thourhout, R. Baets, P. Rojo Romeo, P. Regreny, A. Kazmierczak, C. Seassal, X. Letartre, G. Hollinger, J. M. Fedeli, L. D. Cioccio, C. L. Blanchard, “III-V/Si photonics by die-to-wafer bonding,” Mater. Today 10(7–8), 36–43 (2007).
[CrossRef]

Fegadolli, W. S.

Feng, X.

M. A. Meitl, Z.-T. Zhu, V. Kumar, K. J. Lee, X. Feng, Y. Y. Huang, I. Adesida, R. G. Nuzzo, J. A. Rogers, “Transfer printing by kinetic control of adhesion to an elastomeric stamp,” Nat. Mater. 5(1), 33–38 (2006).
[CrossRef]

Frank, I. W.

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, M. Lončar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94(12), 121106 (2009).
[CrossRef]

Galisteo-López, J. F.

Galli, M.

Gardner, J.

J. Huang, S.-H. Kim, J. Gardner, P. Regreny, C. Seassal, P. A. Postigo, A. Scherer, “Room temperature, continuous-wave coupled cavity InAsP/InP photonic crystal laser with enhanced far-field emission directionality,” Appl. Phys. Lett. 99(9), 091110 (2011).
[CrossRef]

Gong, Y.

Guilet, S.

Halioua, Y.

Haller, E. E.

B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Haller, J. Vuckovic, “Ultralow-threshold electrically pumped quantum-dot photonic-crystal nanocavity laser,” Nat. Photonics 5(5), 297–300 (2011).
[CrossRef]

Hammar, M.

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, W. Zhou, “Transfer-printed stacked nanomembrane lasers on silicon,” Nat. Photonics 6(9), 615–620 (2012).
[CrossRef]

Harris, J.

B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Haller, J. Vuckovic, “Ultralow-threshold electrically pumped quantum-dot photonic-crystal nanocavity laser,” Nat. Photonics 5(5), 297–300 (2011).
[CrossRef]

Harris, J. S.

Hasebe, K.

He, S.

B. Wang, T. Siahaan, M. A. Dundar, R. Notzel, S. He, R. W. van der Heijden, “Transfer printing and nanomanipulating luminescent photonic crystal membrane nanocavities,” J. Appl. Phys. 111(9), 093105 (2012).
[CrossRef]

Hollinger, G.

G. Roelkens, J. Van Campenhout, J. Brouckaert, D. Van Thourhout, R. Baets, P. Rojo Romeo, P. Regreny, A. Kazmierczak, C. Seassal, X. Letartre, G. Hollinger, J. M. Fedeli, L. D. Cioccio, C. L. Blanchard, “III-V/Si photonics by die-to-wafer bonding,” Mater. Today 10(7–8), 36–43 (2007).
[CrossRef]

Huang, J.

J. Huang, S.-H. Kim, J. Gardner, P. Regreny, C. Seassal, P. A. Postigo, A. Scherer, “Room temperature, continuous-wave coupled cavity InAsP/InP photonic crystal laser with enhanced far-field emission directionality,” Appl. Phys. Lett. 99(9), 091110 (2011).
[CrossRef]

Huang, Y.

Y. Zhang, M. Khan, Y. Huang, J. Ryou, P. Deotare, R. Dupuis, M. Lončar, “Photonic crystal nanobeam lasers,” Appl. Phys. Lett. 97(5), 051104 (2010).
[CrossRef]

Huang, Y. Y.

M. A. Meitl, Z.-T. Zhu, V. Kumar, K. J. Lee, X. Feng, Y. Y. Huang, I. Adesida, R. G. Nuzzo, J. A. Rogers, “Transfer printing by kinetic control of adhesion to an elastomeric stamp,” Nat. Mater. 5(1), 33–38 (2006).
[CrossRef]

Huh, J.

H.-G. Park, J.-K. Hwang, J. Huh, H.-Y. Ryu, S.-H. Kim, J.-S. Kim, Y.-H. Lee, “Characteristics of modified single-defect two-dimensional photonic crystal lasers,” IEEE J. Quantum Electron. 38(10), 1353–1365 (2002).
[CrossRef]

Hwang, J.-K.

H.-G. Park, J.-K. Hwang, J. Huh, H.-Y. Ryu, S.-H. Kim, J.-S. Kim, Y.-H. Lee, “Characteristics of modified single-defect two-dimensional photonic crystal lasers,” IEEE J. Quantum Electron. 38(10), 1353–1365 (2002).
[CrossRef]

Hwang, Y.

K.-Y. Jeong, Y.-S. No, Y. Hwang, K.-S. Kim, M.-K. Seo, H.-G. Park, Y.-H. Lee, “Electrically driven nanobeam laser,” Nat. Commun. 4, 2822 (2013).
[CrossRef] [PubMed]

Jeong, K.-Y.

K.-Y. Jeong, Y.-S. No, Y. Hwang, K.-S. Kim, M.-K. Seo, H.-G. Park, Y.-H. Lee, “Electrically driven nanobeam laser,” Nat. Commun. 4, 2822 (2013).
[CrossRef] [PubMed]

Jones, R.

G. Roelkens, L. Liu, D. Liang, R. Jones, A. Fang, B. Koch, J. Bowers, “III-V/silicon photonics for on-chip and inter-chip optical interconnects,” Laser Photon. Rev. 4(6), 751–779 (2010).
[CrossRef]

Kakitsuka, T.

Karle, T. J.

Kazmierczak, A.

G. Roelkens, J. Van Campenhout, J. Brouckaert, D. Van Thourhout, R. Baets, P. Rojo Romeo, P. Regreny, A. Kazmierczak, C. Seassal, X. Letartre, G. Hollinger, J. M. Fedeli, L. D. Cioccio, C. L. Blanchard, “III-V/Si photonics by die-to-wafer bonding,” Mater. Today 10(7–8), 36–43 (2007).
[CrossRef]

Khan, M.

Y. Zhang, M. Khan, Y. Huang, J. Ryou, P. Deotare, R. Dupuis, M. Lončar, “Photonic crystal nanobeam lasers,” Appl. Phys. Lett. 97(5), 051104 (2010).
[CrossRef]

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, M. Lončar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94(12), 121106 (2009).
[CrossRef]

Kim, J.-S.

H.-G. Park, J.-K. Hwang, J. Huh, H.-Y. Ryu, S.-H. Kim, J.-S. Kim, Y.-H. Lee, “Characteristics of modified single-defect two-dimensional photonic crystal lasers,” IEEE J. Quantum Electron. 38(10), 1353–1365 (2002).
[CrossRef]

Kim, K.-S.

K.-Y. Jeong, Y.-S. No, Y. Hwang, K.-S. Kim, M.-K. Seo, H.-G. Park, Y.-H. Lee, “Electrically driven nanobeam laser,” Nat. Commun. 4, 2822 (2013).
[CrossRef] [PubMed]

Kim, S.-H.

W. S. Fegadolli, S.-H. Kim, P. A. Postigo, A. Scherer, “Hybrid single quantum well InP/Si nanobeam lasers for silicon photonics,” Opt. Lett. 38(22), 4656–4658 (2013).
[CrossRef] [PubMed]

J. Huang, S.-H. Kim, J. Gardner, P. Regreny, C. Seassal, P. A. Postigo, A. Scherer, “Room temperature, continuous-wave coupled cavity InAsP/InP photonic crystal laser with enhanced far-field emission directionality,” Appl. Phys. Lett. 99(9), 091110 (2011).
[CrossRef]

H.-G. Park, J.-K. Hwang, J. Huh, H.-Y. Ryu, S.-H. Kim, J.-S. Kim, Y.-H. Lee, “Characteristics of modified single-defect two-dimensional photonic crystal lasers,” IEEE J. Quantum Electron. 38(10), 1353–1365 (2002).
[CrossRef]

Kita, S.

Koch, B.

G. Roelkens, L. Liu, D. Liang, R. Jones, A. Fang, B. Koch, J. Bowers, “III-V/silicon photonics for on-chip and inter-chip optical interconnects,” Laser Photon. Rev. 4(6), 751–779 (2010).
[CrossRef]

Kumar, V.

M. A. Meitl, Z.-T. Zhu, V. Kumar, K. J. Lee, X. Feng, Y. Y. Huang, I. Adesida, R. G. Nuzzo, J. A. Rogers, “Transfer printing by kinetic control of adhesion to an elastomeric stamp,” Nat. Mater. 5(1), 33–38 (2006).
[CrossRef]

Kuramochi, E.

Le Gratiet, L.

Lee, K. J.

M. A. Meitl, Z.-T. Zhu, V. Kumar, K. J. Lee, X. Feng, Y. Y. Huang, I. Adesida, R. G. Nuzzo, J. A. Rogers, “Transfer printing by kinetic control of adhesion to an elastomeric stamp,” Nat. Mater. 5(1), 33–38 (2006).
[CrossRef]

Lee, K.-H.

Lee, P.-T.

T.-W. Lu, W.-C. Tsai, T.-Y. Wu, P.-T. Lee, “Laser emissions from one-dimensional photonic crystal rings on silicon-dioxide,” Appl. Phys. Lett. 102(5), 051103 (2013).
[CrossRef]

Lee, Y.-H.

K.-Y. Jeong, Y.-S. No, Y. Hwang, K.-S. Kim, M.-K. Seo, H.-G. Park, Y.-H. Lee, “Electrically driven nanobeam laser,” Nat. Commun. 4, 2822 (2013).
[CrossRef] [PubMed]

H.-G. Park, J.-K. Hwang, J. Huh, H.-Y. Ryu, S.-H. Kim, J.-S. Kim, Y.-H. Lee, “Characteristics of modified single-defect two-dimensional photonic crystal lasers,” IEEE J. Quantum Electron. 38(10), 1353–1365 (2002).
[CrossRef]

H.-Y. Ryu, H.-G. Park, Y.-H. Lee, “Two-dimensional photonic crystal semiconductor lasers: computational design, fabrication, and characterization,” IEEE J. Sel. Top. Quantum Electron. 8(4), 891–908 (2000).

Letartre, X.

G. Roelkens, J. Van Campenhout, J. Brouckaert, D. Van Thourhout, R. Baets, P. Rojo Romeo, P. Regreny, A. Kazmierczak, C. Seassal, X. Letartre, G. Hollinger, J. M. Fedeli, L. D. Cioccio, C. L. Blanchard, “III-V/Si photonics by die-to-wafer bonding,” Mater. Today 10(7–8), 36–43 (2007).
[CrossRef]

Levenson, A.

Liang, D.

G. Roelkens, L. Liu, D. Liang, R. Jones, A. Fang, B. Koch, J. Bowers, “III-V/silicon photonics for on-chip and inter-chip optical interconnects,” Laser Photon. Rev. 4(6), 751–779 (2010).
[CrossRef]

D. Liang, J. E. Bowers, “Recent progress in lasers on silicon,” Nat. Photonics 4(8), 511–517 (2010).
[CrossRef]

Liu, L.

G. Roelkens, L. Liu, D. Liang, R. Jones, A. Fang, B. Koch, J. Bowers, “III-V/silicon photonics for on-chip and inter-chip optical interconnects,” Laser Photon. Rev. 4(6), 751–779 (2010).
[CrossRef]

Loncar, M.

Y. Zhang, M. Khan, Y. Huang, J. Ryou, P. Deotare, R. Dupuis, M. Lončar, “Photonic crystal nanobeam lasers,” Appl. Phys. Lett. 97(5), 051104 (2010).
[CrossRef]

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, M. Lončar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94(12), 121106 (2009).
[CrossRef]

Lu, T.-W.

T.-W. Lu, W.-C. Tsai, T.-Y. Wu, P.-T. Lee, “Laser emissions from one-dimensional photonic crystal rings on silicon-dioxide,” Appl. Phys. Lett. 102(5), 051103 (2013).
[CrossRef]

Ma, Z.

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, W. Zhou, “Transfer-printed stacked nanomembrane lasers on silicon,” Nat. Photonics 6(9), 615–620 (2012).
[CrossRef]

Martínez, L. J.

Matsuo, S.

Mayer, M. A.

B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Haller, J. Vuckovic, “Ultralow-threshold electrically pumped quantum-dot photonic-crystal nanocavity laser,” Nat. Photonics 5(5), 297–300 (2011).
[CrossRef]

McCutcheon, M. W.

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, M. Lončar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94(12), 121106 (2009).
[CrossRef]

Meitl, M. A.

M. A. Meitl, Z.-T. Zhu, V. Kumar, K. J. Lee, X. Feng, Y. Y. Huang, I. Adesida, R. G. Nuzzo, J. A. Rogers, “Transfer printing by kinetic control of adhesion to an elastomeric stamp,” Nat. Mater. 5(1), 33–38 (2006).
[CrossRef]

Mock, A.

Monnier, P.

Nakagawa, A.

K. Nozaki, A. Nakagawa, D. Sano, T. Baba, “Ultralow threshold and single-mode lasing in microgear lasers and its fusion with quasi-periodic photonic crystals,” IEEE J. Sel. Top. Quantum Electron. 9(5), 1355–1360 (2003).
[CrossRef]

No, Y.-S.

K.-Y. Jeong, Y.-S. No, Y. Hwang, K.-S. Kim, M.-K. Seo, H.-G. Park, Y.-H. Lee, “Electrically driven nanobeam laser,” Nat. Commun. 4, 2822 (2013).
[CrossRef] [PubMed]

Noda, S.

B.-S. Song, S. Noda, T. Asano, Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005).
[CrossRef]

Notomi, M.

Notzel, R.

B. Wang, T. Siahaan, M. A. Dundar, R. Notzel, S. He, R. W. van der Heijden, “Transfer printing and nanomanipulating luminescent photonic crystal membrane nanocavities,” J. Appl. Phys. 111(9), 093105 (2012).
[CrossRef]

Nozaki, K.

Nuzzo, R. G.

M. A. Meitl, Z.-T. Zhu, V. Kumar, K. J. Lee, X. Feng, Y. Y. Huang, I. Adesida, R. G. Nuzzo, J. A. Rogers, “Transfer printing by kinetic control of adhesion to an elastomeric stamp,” Nat. Mater. 5(1), 33–38 (2006).
[CrossRef]

O’Brien, J. D.

Park, H.-G.

K.-Y. Jeong, Y.-S. No, Y. Hwang, K.-S. Kim, M.-K. Seo, H.-G. Park, Y.-H. Lee, “Electrically driven nanobeam laser,” Nat. Commun. 4, 2822 (2013).
[CrossRef] [PubMed]

H.-G. Park, J.-K. Hwang, J. Huh, H.-Y. Ryu, S.-H. Kim, J.-S. Kim, Y.-H. Lee, “Characteristics of modified single-defect two-dimensional photonic crystal lasers,” IEEE J. Quantum Electron. 38(10), 1353–1365 (2002).
[CrossRef]

H.-Y. Ryu, H.-G. Park, Y.-H. Lee, “Two-dimensional photonic crystal semiconductor lasers: computational design, fabrication, and characterization,” IEEE J. Sel. Top. Quantum Electron. 8(4), 891–908 (2000).

Postigo, P. A.

Prieto, I.

Raineri, F.

Raj, R.

Regreny, P.

J. Huang, S.-H. Kim, J. Gardner, P. Regreny, C. Seassal, P. A. Postigo, A. Scherer, “Room temperature, continuous-wave coupled cavity InAsP/InP photonic crystal laser with enhanced far-field emission directionality,” Appl. Phys. Lett. 99(9), 091110 (2011).
[CrossRef]

G. Roelkens, J. Van Campenhout, J. Brouckaert, D. Van Thourhout, R. Baets, P. Rojo Romeo, P. Regreny, A. Kazmierczak, C. Seassal, X. Letartre, G. Hollinger, J. M. Fedeli, L. D. Cioccio, C. L. Blanchard, “III-V/Si photonics by die-to-wafer bonding,” Mater. Today 10(7–8), 36–43 (2007).
[CrossRef]

Roelkens, G.

Y. Halioua, A. Bazin, P. Monnier, T. J. Karle, G. Roelkens, I. Sagnes, R. Raj, F. Raineri, “Hybrid III-V semiconductor/silicon nanolaser,” Opt. Express 19(10), 9221–9231 (2011).
[CrossRef] [PubMed]

Y. Halioua, A. Bazin, P. Monnier, T. J. Karle, I. Sagnes, G. Roelkens, D. Van Thourhout, F. Raineri, R. Raj, “III-V photonic crystal wire cavity laser on silicon wafer,” J. Opt. Soc. Am. B 27(10), 2146–2150 (2010).
[CrossRef]

G. Roelkens, L. Liu, D. Liang, R. Jones, A. Fang, B. Koch, J. Bowers, “III-V/silicon photonics for on-chip and inter-chip optical interconnects,” Laser Photon. Rev. 4(6), 751–779 (2010).
[CrossRef]

G. Roelkens, J. Van Campenhout, J. Brouckaert, D. Van Thourhout, R. Baets, P. Rojo Romeo, P. Regreny, A. Kazmierczak, C. Seassal, X. Letartre, G. Hollinger, J. M. Fedeli, L. D. Cioccio, C. L. Blanchard, “III-V/Si photonics by die-to-wafer bonding,” Mater. Today 10(7–8), 36–43 (2007).
[CrossRef]

Rogers, J. A.

M. A. Meitl, Z.-T. Zhu, V. Kumar, K. J. Lee, X. Feng, Y. Y. Huang, I. Adesida, R. G. Nuzzo, J. A. Rogers, “Transfer printing by kinetic control of adhesion to an elastomeric stamp,” Nat. Mater. 5(1), 33–38 (2006).
[CrossRef]

Rojo Romeo, P.

G. Roelkens, J. Van Campenhout, J. Brouckaert, D. Van Thourhout, R. Baets, P. Rojo Romeo, P. Regreny, A. Kazmierczak, C. Seassal, X. Letartre, G. Hollinger, J. M. Fedeli, L. D. Cioccio, C. L. Blanchard, “III-V/Si photonics by die-to-wafer bonding,” Mater. Today 10(7–8), 36–43 (2007).
[CrossRef]

Ryou, J.

Y. Zhang, M. Khan, Y. Huang, J. Ryou, P. Deotare, R. Dupuis, M. Lončar, “Photonic crystal nanobeam lasers,” Appl. Phys. Lett. 97(5), 051104 (2010).
[CrossRef]

Ryu, H.-Y.

H.-G. Park, J.-K. Hwang, J. Huh, H.-Y. Ryu, S.-H. Kim, J.-S. Kim, Y.-H. Lee, “Characteristics of modified single-defect two-dimensional photonic crystal lasers,” IEEE J. Quantum Electron. 38(10), 1353–1365 (2002).
[CrossRef]

H.-Y. Ryu, H.-G. Park, Y.-H. Lee, “Two-dimensional photonic crystal semiconductor lasers: computational design, fabrication, and characterization,” IEEE J. Sel. Top. Quantum Electron. 8(4), 891–908 (2000).

Sagnes, I.

Sano, D.

K. Nozaki, A. Nakagawa, D. Sano, T. Baba, “Ultralow threshold and single-mode lasing in microgear lasers and its fusion with quasi-periodic photonic crystals,” IEEE J. Sel. Top. Quantum Electron. 9(5), 1355–1360 (2003).
[CrossRef]

Sarmiento, T.

B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Haller, J. Vuckovic, “Ultralow-threshold electrically pumped quantum-dot photonic-crystal nanocavity laser,” Nat. Photonics 5(5), 297–300 (2011).
[CrossRef]

Y. Gong, B. Ellis, G. Shambat, T. Sarmiento, J. S. Harris, J. Vuckovic, “Nanobeam photonic crystal cavity quantum dot laser,” Opt. Express 18(9), 8781–8789 (2010).
[CrossRef] [PubMed]

Sato, T.

Scherer, A.

W. S. Fegadolli, S.-H. Kim, P. A. Postigo, A. Scherer, “Hybrid single quantum well InP/Si nanobeam lasers for silicon photonics,” Opt. Lett. 38(22), 4656–4658 (2013).
[CrossRef] [PubMed]

J. Huang, S.-H. Kim, J. Gardner, P. Regreny, C. Seassal, P. A. Postigo, A. Scherer, “Room temperature, continuous-wave coupled cavity InAsP/InP photonic crystal laser with enhanced far-field emission directionality,” Appl. Phys. Lett. 99(9), 091110 (2011).
[CrossRef]

Seassal, C.

J. Huang, S.-H. Kim, J. Gardner, P. Regreny, C. Seassal, P. A. Postigo, A. Scherer, “Room temperature, continuous-wave coupled cavity InAsP/InP photonic crystal laser with enhanced far-field emission directionality,” Appl. Phys. Lett. 99(9), 091110 (2011).
[CrossRef]

L. J. Martínez, B. Alén, I. Prieto, J. F. Galisteo-López, M. Galli, L. C. Andreani, C. Seassal, P. Viktorovitch, P. A. Postigo, “Two-dimensional surface emitting photonic crystal laser with hybrid triangular-graphite structure,” Opt. Express 17(17), 15043–15051 (2009).
[CrossRef] [PubMed]

G. Roelkens, J. Van Campenhout, J. Brouckaert, D. Van Thourhout, R. Baets, P. Rojo Romeo, P. Regreny, A. Kazmierczak, C. Seassal, X. Letartre, G. Hollinger, J. M. Fedeli, L. D. Cioccio, C. L. Blanchard, “III-V/Si photonics by die-to-wafer bonding,” Mater. Today 10(7–8), 36–43 (2007).
[CrossRef]

Seo, J.-H.

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, W. Zhou, “Transfer-printed stacked nanomembrane lasers on silicon,” Nat. Photonics 6(9), 615–620 (2012).
[CrossRef]

Seo, M.-K.

K.-Y. Jeong, Y.-S. No, Y. Hwang, K.-S. Kim, M.-K. Seo, H.-G. Park, Y.-H. Lee, “Electrically driven nanobeam laser,” Nat. Commun. 4, 2822 (2013).
[CrossRef] [PubMed]

Shambat, G.

B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Haller, J. Vuckovic, “Ultralow-threshold electrically pumped quantum-dot photonic-crystal nanocavity laser,” Nat. Photonics 5(5), 297–300 (2011).
[CrossRef]

Y. Gong, B. Ellis, G. Shambat, T. Sarmiento, J. S. Harris, J. Vuckovic, “Nanobeam photonic crystal cavity quantum dot laser,” Opt. Express 18(9), 8781–8789 (2010).
[CrossRef] [PubMed]

Shih, M. H.

Shinya, A.

Shuai, Y.

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, W. Zhou, “Transfer-printed stacked nanomembrane lasers on silicon,” Nat. Photonics 6(9), 615–620 (2012).
[CrossRef]

Siahaan, T.

B. Wang, T. Siahaan, M. A. Dundar, R. Notzel, S. He, R. W. van der Heijden, “Transfer printing and nanomanipulating luminescent photonic crystal membrane nanocavities,” J. Appl. Phys. 111(9), 093105 (2012).
[CrossRef]

Song, B.-S.

B.-S. Song, S. Noda, T. Asano, Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005).
[CrossRef]

Suh, N.-K.

Takeda, K.

Taneau, A.

Taniyama, H.

Tsai, W.-C.

T.-W. Lu, W.-C. Tsai, T.-Y. Wu, P.-T. Lee, “Laser emissions from one-dimensional photonic crystal rings on silicon-dioxide,” Appl. Phys. Lett. 102(5), 051103 (2013).
[CrossRef]

Van Campenhout, J.

G. Roelkens, J. Van Campenhout, J. Brouckaert, D. Van Thourhout, R. Baets, P. Rojo Romeo, P. Regreny, A. Kazmierczak, C. Seassal, X. Letartre, G. Hollinger, J. M. Fedeli, L. D. Cioccio, C. L. Blanchard, “III-V/Si photonics by die-to-wafer bonding,” Mater. Today 10(7–8), 36–43 (2007).
[CrossRef]

van der Heijden, R. W.

B. Wang, T. Siahaan, M. A. Dundar, R. Notzel, S. He, R. W. van der Heijden, “Transfer printing and nanomanipulating luminescent photonic crystal membrane nanocavities,” J. Appl. Phys. 111(9), 093105 (2012).
[CrossRef]

Van Thourhout, D.

Y. Halioua, A. Bazin, P. Monnier, T. J. Karle, I. Sagnes, G. Roelkens, D. Van Thourhout, F. Raineri, R. Raj, “III-V photonic crystal wire cavity laser on silicon wafer,” J. Opt. Soc. Am. B 27(10), 2146–2150 (2010).
[CrossRef]

G. Roelkens, J. Van Campenhout, J. Brouckaert, D. Van Thourhout, R. Baets, P. Rojo Romeo, P. Regreny, A. Kazmierczak, C. Seassal, X. Letartre, G. Hollinger, J. M. Fedeli, L. D. Cioccio, C. L. Blanchard, “III-V/Si photonics by die-to-wafer bonding,” Mater. Today 10(7–8), 36–43 (2007).
[CrossRef]

Vecchi, G.

Viktorovitch, P.

Vuckovic, J.

B. Ellis, M. A. Mayer, G. Shambat, T. Sarmiento, J. Harris, E. E. Haller, J. Vuckovic, “Ultralow-threshold electrically pumped quantum-dot photonic-crystal nanocavity laser,” Nat. Photonics 5(5), 297–300 (2011).
[CrossRef]

Y. Gong, B. Ellis, G. Shambat, T. Sarmiento, J. S. Harris, J. Vuckovic, “Nanobeam photonic crystal cavity quantum dot laser,” Opt. Express 18(9), 8781–8789 (2010).
[CrossRef] [PubMed]

Wang, B.

B. Wang, T. Siahaan, M. A. Dundar, R. Notzel, S. He, R. W. van der Heijden, “Transfer printing and nanomanipulating luminescent photonic crystal membrane nanocavities,” J. Appl. Phys. 111(9), 093105 (2012).
[CrossRef]

Wu, T.-Y.

T.-W. Lu, W.-C. Tsai, T.-Y. Wu, P.-T. Lee, “Laser emissions from one-dimensional photonic crystal rings on silicon-dioxide,” Appl. Phys. Lett. 102(5), 051103 (2013).
[CrossRef]

Yacomotti, A.

Yang, H.

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, W. Zhou, “Transfer-printed stacked nanomembrane lasers on silicon,” Nat. Photonics 6(9), 615–620 (2012).
[CrossRef]

Yang, W.

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, W. Zhou, “Transfer-printed stacked nanomembrane lasers on silicon,” Nat. Photonics 6(9), 615–620 (2012).
[CrossRef]

Zhang, Y.

Y. Zhang, M. Khan, Y. Huang, J. Ryou, P. Deotare, R. Dupuis, M. Lončar, “Photonic crystal nanobeam lasers,” Appl. Phys. Lett. 97(5), 051104 (2010).
[CrossRef]

Zhao, D.

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, W. Zhou, “Transfer-printed stacked nanomembrane lasers on silicon,” Nat. Photonics 6(9), 615–620 (2012).
[CrossRef]

Zhou, W.

H. Yang, D. Zhao, S. Chuwongin, J.-H. Seo, W. Yang, Y. Shuai, J. Berggren, M. Hammar, Z. Ma, W. Zhou, “Transfer-printed stacked nanomembrane lasers on silicon,” Nat. Photonics 6(9), 615–620 (2012).
[CrossRef]

Zhu, Z.-T.

M. A. Meitl, Z.-T. Zhu, V. Kumar, K. J. Lee, X. Feng, Y. Y. Huang, I. Adesida, R. G. Nuzzo, J. A. Rogers, “Transfer printing by kinetic control of adhesion to an elastomeric stamp,” Nat. Mater. 5(1), 33–38 (2006).
[CrossRef]

Appl. Phys. Lett. (4)

Y. Zhang, M. Khan, Y. Huang, J. Ryou, P. Deotare, R. Dupuis, M. Lončar, “Photonic crystal nanobeam lasers,” Appl. Phys. Lett. 97(5), 051104 (2010).
[CrossRef]

J. Huang, S.-H. Kim, J. Gardner, P. Regreny, C. Seassal, P. A. Postigo, A. Scherer, “Room temperature, continuous-wave coupled cavity InAsP/InP photonic crystal laser with enhanced far-field emission directionality,” Appl. Phys. Lett. 99(9), 091110 (2011).
[CrossRef]

T.-W. Lu, W.-C. Tsai, T.-Y. Wu, P.-T. Lee, “Laser emissions from one-dimensional photonic crystal rings on silicon-dioxide,” Appl. Phys. Lett. 102(5), 051103 (2013).
[CrossRef]

P. B. Deotare, M. W. McCutcheon, I. W. Frank, M. Khan, M. Lončar, “High quality factor photonic crystal nanobeam cavities,” Appl. Phys. Lett. 94(12), 121106 (2009).
[CrossRef]

IEEE J. Quantum Electron. (1)

H.-G. Park, J.-K. Hwang, J. Huh, H.-Y. Ryu, S.-H. Kim, J.-S. Kim, Y.-H. Lee, “Characteristics of modified single-defect two-dimensional photonic crystal lasers,” IEEE J. Quantum Electron. 38(10), 1353–1365 (2002).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (3)

K. Nozaki, A. Nakagawa, D. Sano, T. Baba, “Ultralow threshold and single-mode lasing in microgear lasers and its fusion with quasi-periodic photonic crystals,” IEEE J. Sel. Top. Quantum Electron. 9(5), 1355–1360 (2003).
[CrossRef]

H.-Y. Ryu, H.-G. Park, Y.-H. Lee, “Two-dimensional photonic crystal semiconductor lasers: computational design, fabrication, and characterization,” IEEE J. Sel. Top. Quantum Electron. 8(4), 891–908 (2000).

T. Baba, “Photonic crystals and microdisk cavities based on GaInAsP-InP system,” IEEE J. Sel. Top. Quantum Electron. 3(3), 808–830 (1997).
[CrossRef]

J. Appl. Phys. (1)

B. Wang, T. Siahaan, M. A. Dundar, R. Notzel, S. He, R. W. van der Heijden, “Transfer printing and nanomanipulating luminescent photonic crystal membrane nanocavities,” J. Appl. Phys. 111(9), 093105 (2012).
[CrossRef]

J. Opt. Soc. Am. B (1)

Laser Photon. Rev. (1)

G. Roelkens, L. Liu, D. Liang, R. Jones, A. Fang, B. Koch, J. Bowers, “III-V/silicon photonics for on-chip and inter-chip optical interconnects,” Laser Photon. Rev. 4(6), 751–779 (2010).
[CrossRef]

Mater. Today (1)

G. Roelkens, J. Van Campenhout, J. Brouckaert, D. Van Thourhout, R. Baets, P. Rojo Romeo, P. Regreny, A. Kazmierczak, C. Seassal, X. Letartre, G. Hollinger, J. M. Fedeli, L. D. Cioccio, C. L. Blanchard, “III-V/Si photonics by die-to-wafer bonding,” Mater. Today 10(7–8), 36–43 (2007).
[CrossRef]

Nat. Commun. (1)

K.-Y. Jeong, Y.-S. No, Y. Hwang, K.-S. Kim, M.-K. Seo, H.-G. Park, Y.-H. Lee, “Electrically driven nanobeam laser,” Nat. Commun. 4, 2822 (2013).
[CrossRef] [PubMed]

Nat. Mater. (2)

B.-S. Song, S. Noda, T. Asano, Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. 4(3), 207–210 (2005).
[CrossRef]

M. A. Meitl, Z.-T. Zhu, V. Kumar, K. J. Lee, X. Feng, Y. Y. Huang, I. Adesida, R. G. Nuzzo, J. A. Rogers, “Transfer printing by kinetic control of adhesion to an elastomeric stamp,” Nat. Mater. 5(1), 33–38 (2006).
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Nat. Photonics (3)

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Opt. Express (8)

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Opt. Lett. (1)

Other (2)

S.-L. Chuang, Physics of Photonic Devices (Wiley, 2009).

M. Shearn, X. Sun, M. D. Henry, A. Yariv, and A. Scherer, “Advanced plasma processing: etching, deposition, and wafer bonding techniques for semiconductor applications,” in Semiconductor Technologies, J. Grym, ed. (InTech, 2010), pp. 80–104.

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Figures (3)

Fig. 1
Fig. 1

(a) Schematic of nanobeam on a SiO2/Si substrate. (b) Formation of a photonic well. (c) Ey field profile of 0th order and 1st order modes of the nanobeam in (a). (d) Q-factor of the nanobeam as a function of total number of air holes (N).

Fig. 2
Fig. 2

(a) SEM images of the fabricated sample i) top and ii) tilted view. (b) Schematic of the transfer-printing processes i) micron-sized, rectangular PDMS stamp (15 μm × 15 μm × 40 μm) was precisely aligned and placed on the top of the nanobeam. A small impulse force was applied on the stamp in order to break two small tethers at the end of the nanobeam. ii) After breaking the tethers, the nanobeam will be attached to the stamp due to non-specific, general van der Walls force. iii) PDMS stamp inked with the nanobeam is put in contact with the receiver substrate. iv) The stamp is retracted slowly in order to separate it from nanobeam by kinetically controlled viscoelastic effect of PDMS.

Fig. 3
Fig. 3

(a) SEM images of the printed nanobeam on a SiO2/Si substrate taken at top view (top panel) and side view (bottom panel): the nanobeam was conformably printed on a SiO2/Si substrate. (b) Photoluminescence spectrum of the nanobeam printed on a SiO2/Si substrate. Insets show the lasing-mode image captured by a CCD camera and the polarization dependence of the laser emission. (c) L-L curve and spectral linewidth as a function of input pump power of the 0thmode. (d) Rate equation fitted L-L measurement on log-log scales.

Tables (1)

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Table 1 Parameters Used in the Rate Equations

Equations (2)

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dN dt =η L in ω p V a ( A a V a v s N+B N 2 +C N 3 )ΓG(N)P
dP dt =ΓG(N)P P τ p +βB N 2

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