Abstract

A new class of planar optics has emerged using subwavelength gratings with a large refractive index contrast, herein referred to as high-contrast gratings (HCGs). This seemingly simple structure lends itself to extraordinary properties, which can be designed top-down based on intuitive guidelines. The HCG is a single layer of high-index material that can be as thin as 15% of one wavelength. It can be designed to reflect or transmit nearly completely and with specific optical phase over a broad spectral range and/or various incident beam angles. We present a simple theory providing an intuitive phase selection rule to explain the extraordinary features. Our analytical results agree well not only with numerical simulations but also experimental data. The HCG has made easy fabrication of surface-normal optical devices possible, including vertical-cavity surface-emitting lasers (VCSELs), tunable VCSELs, and tunable filters. HCGs can be designed to result in high-quality-factor (Q) resonators with surface-normal output, which is promising for wafer-scale lasers and optical sensors. Spatially chirped HCGs are shown to be excellent focusing reflectors and lenses with very high numerical apertures. This field has seen rapid advances in experimental demonstrations and theoretical results. We provide an overview of the underlying new physics and the latest results of devices.

© 2012 OSA

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  107. Yu. V. Troitski, “The energy conservation law for optical two-port devices,” Opt. Spectrosc. 92(4), 555–559 (2002).
  108. S. Fan, J. D. Joannopoulos, J. N. Winn, A. Devenyi, J. C. Chen, and R. D. Meade, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B 12(7), 1267–1272 (1995).

2012 (5)

C. Sciancalepore, B. B. Bakir, X. Letartre, J. Harduin, N. Olivier, C. Seassal, J. Fedeli, and P. Viktorovitch, “ CMOS-compatible ultra-compact 1.55-µm emitting VCSELs using double photonic crystal mirrors,” IEEE Photon. Technol. Lett. 24(6), 455–457 (2012).

W. Yang, J. Ferrara, K. Grutter, A. Yeh, C. Chase, Y. Yue, A. E. Willner, M. C. Wu, and C. J. Chang-Hasnain, ”Low loss hollow-core waveguide on a silicon substrate,” Nanophotonics 1(1), 23–29 (2012).

T. Sun, W. Yang, V. Karagodsky, W. Zhou, and C. Chang-Hasnain, “Low-loss slow light inside high contrast grating waveguide,” Proc. SPIE 8270, 82700A (2012).

L. Zhu, V. Karagodsky, and C. Chang-Hasnain, “Novel high efficiency vertical to in-plane optical coupler,” Proc. SPIE 8270, 82700L (2012).

V. Karagodsky and C. J. Chang-Hasnain, “Physics of near-wavelength high contrast gratings,” Opt. Express 20(10), 10888–10895 (2012).
[PubMed]

2011 (1)

2010 (9)

V. Karagodsky, F. G. Sedgwick, and C. J. Chang-Hasnain, “Theoretical analysis of subwavelength high contrast grating reflectors,” Opt. Express 18(16), 16973–16988 (2010).
[PubMed]

T. Stöferle, N. Moll, T. Wahlbrink, J. Bolten, T. Mollenhauer, U. Scherf, and R. F. Mahrt, “Ultracompact silicon/polymer laser with an absorption-insensitive nanophotonic resonator,” Nano Lett. 10(9), 3675–3678 (2010).
[PubMed]

F. Brückner, D. Friedrich, T. Clausnitzer, M. Britzger, O. Burmeister, K. Danzmann, E. B. Kley, A. Tünnermann, and R. Schnabel, “Realization of a monolithic high-reflectivity cavity mirror from a single silicon crystal,” Phys. Rev. Lett. 104(16), 163903 (2010).
[PubMed]

V. Karagodsky, B. Pesala, C. Chase, W. Hofmann, F. Koyama, and C. J. Chang-Hasnain, “Monolithically integrated multi-wavelength VCSEL arrays using high-contrast gratings,” Opt. Express 18(2), 694–699 (2010).
[PubMed]

F. Lu, F. G. Sedgwick, V. Karagodsky, C. Chase, and C. J. Chang-Hasnain, “Planar high-numerical-aperture low-loss focusing reflectors and lenses using subwavelength high contrast gratings,” Opt. Express 18(12), 12606–12614 (2010).
[PubMed]

D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. G. Beausolei, “Flat dielectric grating reflectors with focusing abilities,” Nat. Photonics 4(7), 466–470 (2010).

C. Chase, Y. Rao, W. Hofmann, and C. J. Chang-Hasnain, “1550 nm high contrast grating VCSEL,” Opt. Express 18(15), 15461–15466 (2010).
[PubMed]

W. Hofmann, C. Chase, M. Müller, Y. Rao, C. Grasse, G. Böhm, M.-C. Amann, and C. J. Chang-Hasnain, “Long-wavelength high-contrast grating vertical-cavity surface-emitting laser,” IEEE Photonics J. 2(3), 415–422 (2010).

P. Gilet, N. Olivier, P. Grosse, K. Gilbert, A. Chelnokov, I.-S. Chung, and J. Mørk, “High-index-contrast subwavelength grating VCSEL,” Proc. SPIE 7615, 76150J (2010).

2009 (4)

C. Chase, Y. Zhou, and C. J. Chang-Hasnain, “Size effect of high contrast gratings in VCSELs,” Opt. Express 17(26), 24002–24007 (2009).
[PubMed]

C. J. Chang-Hasnain, Y. Zhou, M. C. Y. Huang, and C. Chase, “High-contrast grating VCSELs,” IEEE J. Sel. Top. Quantum Electron. 15(3), 869–878 (2009).

Y. Zhou, V. Karagodsky, B. Pesala, F. G. Sedgwick, and C. J. Chang-Hasnain, “A novel ultra-low loss hollow-core waveguide using subwavelength high-contrast gratings,” Opt. Express 17(3), 1508–1517 (2009).
[PubMed]

S. Jatta, B. Kögel, M. Maute, K. Zogal, F. Riemenschneider, G. Böhm, M.-C. Amann, and P. Meißner, “Bulk-micromachined VCSEL at 1.55 µm with 76-nm single-mode continuous tuning range,” IEEE Photon. Technol. Lett. 21(24), 1822–1824 (2009).

2008 (6)

Y. Zhou, M. Moewe, J. Kern, M. C. Y. Huang, and C. J. Chang-Hasnain, “Surface-normal emission of a high-Q resonator using a subwavelength high-contrast grating,” Opt. Express 16(22), 17282–17287 (2008).
[PubMed]

I.-S. Chung, J. Mørk, P. Gilet, and A. Chelnokov, “Subwavelength grating-mirror VCSEL with a thin oxide gap,” IEEE Photon. Technol. Lett. 20(2), 105–107 (2008).

Y. Zhou, M. C. Y. Huang, and C. J. Chang-Hasnain, “Large fabrication tolerance for VCSELs using high contrast grating,” IEEE Photon. Technol. Lett. 20(6), 434–436 (2008).

M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A nanoelectromechanical tunable laser,” Nat. Photonics 2(3), 180–184 (2008).

R. Michalzik, J. M. Ostermann, and P. Debernardi, “Polarization-stable monolithic VCSELs,” Proc. SPIE,  6908, 69080A (2008).

W. Hofmann, E. Wong, G. Böhm, M. Ortsiefer, N. H. Zhu, and M. C. Amann, “1.55 µm VCSEL arrays for high-bandwidth WDM-PONs,” IEEE Photon. Technol. Lett. 20(4), 291–293 (2008).

2007 (6)

B. Kögel, H. Halbritter, S. Jatta, M. Maute, G. Böhm, M.-C. Amann, M. Lackner, M. Schwarzott, F. Winter, and P. Meissner, “Simultaneous spectroscopy of NH3 and CO using a >50 nm continuously tunable MEMS–VCSEL,” IEEE Sens. J. 7(11), 1483–1489 (2007).

H. Halbritter, C. Sydlo, B. Kögel, F. Riemenschneider, H. L. Hartnagel, and P. Meissner, “Impact of micromechanics on the linewidth and chirp performance of MEMS–VCSELs,” IEEE J. Sel. Top. Quantum Electron. 13(2), 367–373 (2007).

W. M. J. Green, M. J. Rooks, L. Sekaric, and Y. A. Vlasov, “Optical modulation using anti-crossing between paired amplitude and phase resonators,” Opt. Express 15(25), 17264–17272 (2007).
[PubMed]

S. Boutami, B. Ben Bakir, J.-L. Leclercq, and P. Viktorovitch, “Compact and polarization controlled 1.55 µm vertical-cavity surface emitting laser using single-layer photonic crystal mirror,” Appl. Phys. Lett. 91(7), 071105 (2007).

S. Boutami, B. Benbakir, X. Letartre, J. L. Leclercq, P. Regreny, and P. Viktorovitch, “Ultimate vertical checkerboard–Perot cavity based on single-layer photonic crystal mirrors,” Opt. Express 15(19), 12443–12449 (2007).
[PubMed]

M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A surface-emitting laser incorporating a high index-contrast subwavelength grating,” Nat. Photonics 1(2), 119–122 (2007).

2006 (7)

P. Lalanne, J. P. Hugonin, and P. Chavel, “Optical properties of deep lamellar gratings: a coupled Bloch-mode insight,” J. Lightwave Technol. 24(6), 2442–2449 (2006).

M. Lackner, M. Schwarzott, F. Winter, B. Kogel, S. Jatta, H. Halbritter, and P. Meissner, “CO and CO2 spectroscopy using a 60 nm broadband tunable MEMS–VCSEL at 1.55 µm,” Opt. Lett. 31(21), 3170–3172 (2006).
[PubMed]

J. M. Ostermann, P. Debernardi, and R. Michalzik, “Optimized integrated surface grating design for polarization-stable VCSELs,” IEEE J. Quantum Electron. 42(7), 690–698 (2006).

A. Haglund, J. S. Gustavsson, J. Bengtsson, P. Jedrasik, and A. Larsson, “Design and evaluation of fundamental-mode and polarization-stabilized VCSELs with a subwavelength surface grating,” IEEE J. Quantum Electron. 42(3), 231–240 (2006).

M. C. Y. Huang, K. B. Cheng, Y. Zhou, B. Pesala, C. J. Chang-Hasnain, and A. P. Pisano, “Demonstration of piezoelectric actuated GaAs-based MEMS tunable VCSEL,” IEEE Photon. Technol. Lett. 18(10), 1197–1199 (2006).

T. Asano, B.-S. Song, Y. Akahane, and S. Noda, “Ultrahigh-Q nanocavities in two-dimensional photonic crystal slabs,” IEEE J. Sel. Top. Quantum Electron. 12, 1121–1134 (2006).

W.-H. Chang, W.-Y. Chen, H.-S. Chang, T.-P. Hsieh, J.-I. Chyi, and T.-M. Hsu, “Efficient single-photon sources based on low-density quantum dots in photonic-crystal nanocavities,” Phys. Rev. Lett. 96(11), 117401 (2006).
[PubMed]

2005 (4)

A. Löffler, J. Reithmaier, G. Sek, C. Hofmann, S. Reitzenstein, M. Kamp, and A. Forchel, “Semiconductor quantum dot microcavity pillars with high-quality factors and enlarged dot dimensions,” Appl. Phys. Lett. 86(11), 111105 (2005).

P. J. Schuck, D. P. Fromm, A. Sundaramurthy, G. S. Kino, and W. E. Moerner, “Improving the mismatch between light and nanoscale objects with gold bowtie nanoantennas,” Phys. Rev. Lett. 94(1), 017402 (2005).
[PubMed]

N. Nishiyama, C. Caneau, B. Hall, G. Guryanov, M. Hu, X. Liu, M. Li, R. Bhat, and C. Zah, “Long-wavelength vertical-cavity surface-emitting lasers on InP with lattice matched AlGaInAs-InP DBR grown by MOCVD,” IEEE J. Sel. Top. Quantum Electron. 11(5), 990–998 (2005).

A. Haglund, S. J. Gustavsson, J. Vukusic, P. Jedrasik, and A. Larsson, “High-power fundamental-mode and polarisation stabilised VCSELs using sub-wavelength surface grating,” Electron. Lett. 41(14), 805–807 (2005).

2004 (10)

C. F. R. Mateus, M. C. Y. Huang, L. Chen, C. J. Chang-Hasnain, and Y. Suzuki, “Broad-band mirror (1.12–1.62 µm) using a subwavelength grating,” IEEE Photon. Technol. Lett. 16(7), 1676–1678 (2004).

C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultra-broadband mirror using low index cladded subwavelength grating,” IEEE Photon. Technol. Lett. 16(2), 518–520 (2004).

A. Syrbu, A. Mereuta, A. Mircea, A. Caliman, V. Iakovlev, C. Berseth, G. Suruceanu, A. Rudra, E. Deichsel, and E. Kapon, “1550 nm-band VCSEL 0.76 mW singlemode output power in 20–80°C temperature range,” Electron. Lett. 40(5), 306 (2004).

D. Sun, W. Fan, P. Kner, J. Boucart, T. Kageyama, D. Zhang, R. Pathak, R. F. Nabiev, and W. Yuen, “Long wavelength-tunable VCSELs with optimized MEMS bridge tuning structure,” IEEE Photon. Technol. Lett. 16(3), 714–716 (2004).

F. Riemenschneider, M. Maute, H. Halbritter, G. Boehm, M. C. Amann, and P. Meissner, “Continuously tunable long-wavelength MEMS–VCSEL with over 40-nm tuning range,” IEEE Photon. Technol. Lett. 16(10), 2212–2214 (2004).

J. P. Reithmaier, G. Sęk, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot-semiconductor microcavity system,” Nature 432(7014), 197–200 (2004).
[PubMed]

H. Takano, Y. Akahane, T. Asano, and S. Noda, “In-plane-type channel drop filter in a two-dimensional photonic crystal slab,” Appl. Phys. Lett. 84(13), 2226–2228 (2004).

E. Chow, A. Grot, L. W. Mirkarimi, M. Sigalas, and G. Girolami, “Ultracompact biochemical sensor built with two-dimensional photonic crystal microcavity,” Opt. Lett. 29(10), 1093–1095 (2004).
[PubMed]

J. Niehusmann, A. Vörckel, P. H. Bolivar, T. Wahlbrink, W. Henschel, and H. Kurz, “Ultrahigh-quality-factor silicon-on-insulator microring resonator,” Opt. Lett. 29(24), 2861–2863 (2004).
[PubMed]

D. Ohnishi, T. Okano, M. Imada, and S. Noda, “Room temperature continuous wave operation of a surface-emitting two-dimensional photonic crystal diode laser,” Opt. Express 12(8), 1562–1568 (2004).
[PubMed]

2003 (2)

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003).
[PubMed]

S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the Fano resonance in optical resonators,” J. Opt. Soc. Am. A 20(3), 569–572 (2003).

2002 (2)

2001 (1)

S. Nakagawa, E. Hall, G. Almuneau, J. K. Kim, D. A. Buell, H. Kroemer, and L. A. Coldren, “88°C, continuous-wave operation of apertured, intracavity contacted, 1.55 µm vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 78(10), 1337 (2001).

2000 (4)

W. Yuen, G. S. Li, R. F. Nabiev, J. Boucart, P. Kner, R. J. Stone, D. Zhang, M. Beaudoin, T. Zheng, C. He, K. Yu, M. Jansen, D. P. Worland, and C. J. Chang-Hasnain, “High-performance 1.6 µm single-epitaxy top-emitting VCSEL,” Electron. Lett. 36(13), 1121–1123 (2000).

C. J. Chang-Hasnain, “Tunable VCSEL,” IEEE J. Sel. Top. Quantum Electron. 6(6), 978–987 (2000).

K. Iga, “Surface-emitting laser-its birth and generation of new optoelectronics field,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1201–1215 (2000).

M. Ortsiefer, R. Shau, G. Böhm, F. Köhler, and M. C. Amann, “Low-threshold index-guided 1.5 µm long-wavelength vertical-cavity surface-emitting laser with high efficiency,” Appl. Phys. Lett. 76(16), 2179 (2000).

1999 (1)

W. Shan, W. Walukiewicz, J. W. Ager III, E. E. Haller, J. F. Geisz, D. J. Friedman, J. M. Olson, and S. R. Kurtz, “Band anticrossing in GaInNAs alloys,” Phys. Rev. Lett. 82(6), 1221–1224 (1999).

1998 (3)

A. Mizutani, N. Hatori, N. Nishiyama, F. Koyama, and K. Iga, “InGaAs/GaAs vertical-cavity surface emitting laser on GaAs (311)B substrate using carbon auto-doping,” Jpn. J. Appl. Phys. 37(Part 1, No. 3B), 1408–1412 (1998).

S. Astilean, P. Lalanne, P. Chavel, E. Cambril, and H. Launois, “High-efficiency subwavelength diffractive element patterned in a high-refractive-index material for 633 nm,” Opt. Lett. 23(7), 552–554 (1998).
[PubMed]

S. Goeman, S. Boons, B. Dhoedt, K. Vandeputte, K. Caekebeke, P. Van Daele, and R. Baets, “First demonstration of highly reflective and highly polarization selective diffraction gratings (GIRO-gratings) for long-wavelength VCSELs,” IEEE Photon. Technol. Lett. 10(9), 1205–1207 (1998).

1997 (2)

D. Rosenblatt, A. Sharon, and A. A. Friesem, “Resonant grating waveguide structures,” IEEE J. Quantum Electron. 33(11), 2038–2059 (1997).

S. Nie and S. R. Emory, “Probing single molecules and single nanoparticles by surface-enhanced Raman scattering,” Science 275(5303), 1102–1106 (1997).
[PubMed]

1996 (3)

L. Zhuang, S. Schablitsky, R. C. Shi, and S. Y. Chou, “Fabrication and performance of thin amorphous Si subwavelength transmission grating for controlling vertical cavity surface emitting laser polarization,” J. Vac. Sci. Technol. B 14(6), 4055–4057 (1996).

S. J. Schablitsky, L. Zhuang, R. C. Shi, and S. Y. Chou, “Controlling polarization of vertical-cavity surface-emitting lasers using amorphous silicon subwavelength transmission gratings,” Appl. Phys. Lett. 69(1), 7–9 (1996).

M. S. Wu, E. C. Vail, G. S. Li, W. Yuen, and C. J. Chang-Hasnain, “Widely and continuously tunable micromachined resonant cavity detector with wavelength tracking,” IEEE Photon. Technol. Lett. 8(1), 98–100 (1996).

1995 (4)

S. Fan, J. D. Joannopoulos, J. N. Winn, A. Devenyi, J. C. Chen, and R. D. Meade, “Guided and defect modes in periodic dielectric waveguides,” J. Opt. Soc. Am. B 12(7), 1267–1272 (1995).

L. E. Eng, K. Bacher, W. Yuen, J. S. Harris, and C. J. Chang-Hasnain, “Multiple wavelength vertical cavity laser arrays on patterned substrates,” IEEE J. Quantum Electron. 1(2), 624–628 (1995).

F. Koyama, T. Mukaihara, Y. Hayashi, N. Ohnoki, N. Hatori, and K. Iga, “Wavelength control of vertical cavity surface-emitting lasers by using nonplanar MOCVD,” IEEE Photon. Technol. Lett. 7(1), 10–12 (1995).

T. Wipiejewski, M. Peters, E. Hegblom, and L. Coldren, “Vertical-cavity surface-emitting laser diodes with post-growth wavelength adjustment,” IEEE Photon. Technol. Lett. 7(7), 727–729 (1995).

1994 (1)

K. H. Hahn, M. R. Tan, and S. Y. Wang, “Intensity noise of large area vertical cavity surface emitting lasers in multimode optical fibre links,” Electron. Lett. 30(2), 139–140 (1994).

1993 (1)

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40(4), 553–573 (1993).

1992 (3)

R. Magnusson and S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61(9), 1022–1024 (1992).

L. A. Coldren, R. S. Geels, S. W. Corzine, and J. W. Scott, “Efficient vertical-cavity lasers,” Opt. Quantum Electron. 24(2), S105–S119 (1992).

C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, “Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity,” Phys. Rev. Lett. 69(23), 3314–3317 (1992).
[PubMed]

1991 (6)

M. Orenstein, A. Von Lehmen, C. J. Chang-Hasnain, N. G. Stoffel, J. P. Harbison, and L. T. Florez, “Matrix addressable vertical cavity surface emitting laser array,” Electron. Lett. 27(5), 437–438 (1991).

C. J. Chang-Hasnain, J. P. Harbison, C. E. Zah, M. W. Maeda, L. T. Florez, N. G. Stoffel, and T. P. Lee, “Multiple wavelength tunable surface emitting laser arrays,” IEEE J. Quantum Electron. 27(6), 1368–1376 (1991).

C. J. Chang-Hasnain, J. P. Harbison, G. Hasnain, A. Von Lehmen, L. T. Florez, and N. G. Stoffel, “Dynamic, polarization, and transverse mode characteristics of vertical cavity surface emitting lasers,” IEEE J. Quantum Electron. 27(6), 1402–1409 (1991).

M. W. Maeda, C. J. Chang-Hasnain, C. Lin, J. S. Patel, H. A. Johnson, and J. A. Walker, “Use of a multiwavelength surface-emitting laser array in a four-channel wavelength-division-multiplexed system experiment,” Photonics Technol. Lett. 3(3), 268–269 (1991).

H. A. Haus and Y. Lai, “Narrow-band distributed feedback reflector design,” J. Lightwave Technol. 9(6), 754–760 (1991).

K. Rastani, A. Marrakchi, S. F. Habiby, W. M. Hubbard, H. Gilchrist, and R. E. Nahory, “Binary phase Fresnel lenses for generation of two-dimensional beam arrays,” Appl. Opt. 30(11), 1347–1354 (1991).
[PubMed]

1990 (3)

1989 (3)

1987 (1)

F. Koyama, H. Uenohara, T. Sakaguchi, and K. Iga, “GaAlAs/GaAs MOCVD growth for surface emitting laser,” Jpn. J. Appl. Phys. Part 1 26(Part 1, No. 7), 1077–1081 (1987).

1982 (1)

1981 (1)

1979 (1)

D. C. Shaver and D. C. Flanders, “X-ray zone plates fabricated using electron-beam and x-ray lithography,” J. Vac. Sci. Technol. 16(6), 1626–1630 (1979).

Ager III, J. W.

W. Shan, W. Walukiewicz, J. W. Ager III, E. E. Haller, J. F. Geisz, D. J. Friedman, J. M. Olson, and S. R. Kurtz, “Band anticrossing in GaInNAs alloys,” Phys. Rev. Lett. 82(6), 1221–1224 (1999).

Akahane, Y.

T. Asano, B.-S. Song, Y. Akahane, and S. Noda, “Ultrahigh-Q nanocavities in two-dimensional photonic crystal slabs,” IEEE J. Sel. Top. Quantum Electron. 12, 1121–1134 (2006).

H. Takano, Y. Akahane, T. Asano, and S. Noda, “In-plane-type channel drop filter in a two-dimensional photonic crystal slab,” Appl. Phys. Lett. 84(13), 2226–2228 (2004).

Alexander, G. C.

P. C. Magnusson, G. C. Alexander, V. K. Tripathi, and A. Weisshaar, Transmission Lines and Wave Propagation, 4th ed. (CRC Press, 2001).

Almuneau, G.

S. Nakagawa, E. Hall, G. Almuneau, J. K. Kim, D. A. Buell, H. Kroemer, and L. A. Coldren, “88°C, continuous-wave operation of apertured, intracavity contacted, 1.55 µm vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 78(10), 1337 (2001).

Amann, M. C.

W. Hofmann, E. Wong, G. Böhm, M. Ortsiefer, N. H. Zhu, and M. C. Amann, “1.55 µm VCSEL arrays for high-bandwidth WDM-PONs,” IEEE Photon. Technol. Lett. 20(4), 291–293 (2008).

F. Riemenschneider, M. Maute, H. Halbritter, G. Boehm, M. C. Amann, and P. Meissner, “Continuously tunable long-wavelength MEMS–VCSEL with over 40-nm tuning range,” IEEE Photon. Technol. Lett. 16(10), 2212–2214 (2004).

M. Ortsiefer, R. Shau, G. Böhm, F. Köhler, and M. C. Amann, “Low-threshold index-guided 1.5 µm long-wavelength vertical-cavity surface-emitting laser with high efficiency,” Appl. Phys. Lett. 76(16), 2179 (2000).

Amann, M.-C.

W. Hofmann, C. Chase, M. Müller, Y. Rao, C. Grasse, G. Böhm, M.-C. Amann, and C. J. Chang-Hasnain, “Long-wavelength high-contrast grating vertical-cavity surface-emitting laser,” IEEE Photonics J. 2(3), 415–422 (2010).

S. Jatta, B. Kögel, M. Maute, K. Zogal, F. Riemenschneider, G. Böhm, M.-C. Amann, and P. Meißner, “Bulk-micromachined VCSEL at 1.55 µm with 76-nm single-mode continuous tuning range,” IEEE Photon. Technol. Lett. 21(24), 1822–1824 (2009).

B. Kögel, H. Halbritter, S. Jatta, M. Maute, G. Böhm, M.-C. Amann, M. Lackner, M. Schwarzott, F. Winter, and P. Meissner, “Simultaneous spectroscopy of NH3 and CO using a >50 nm continuously tunable MEMS–VCSEL,” IEEE Sens. J. 7(11), 1483–1489 (2007).

Arakawa, Y.

C. Weisbuch, M. Nishioka, A. Ishikawa, and Y. Arakawa, “Observation of the coupled exciton-photon mode splitting in a semiconductor quantum microcavity,” Phys. Rev. Lett. 69(23), 3314–3317 (1992).
[PubMed]

Armani, D. K.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(6926), 925–928 (2003).
[PubMed]

Asano, T.

T. Asano, B.-S. Song, Y. Akahane, and S. Noda, “Ultrahigh-Q nanocavities in two-dimensional photonic crystal slabs,” IEEE J. Sel. Top. Quantum Electron. 12, 1121–1134 (2006).

H. Takano, Y. Akahane, T. Asano, and S. Noda, “In-plane-type channel drop filter in a two-dimensional photonic crystal slab,” Appl. Phys. Lett. 84(13), 2226–2228 (2004).

Astilean, S.

Bacher, K.

L. E. Eng, K. Bacher, W. Yuen, J. S. Harris, and C. J. Chang-Hasnain, “Multiple wavelength vertical cavity laser arrays on patterned substrates,” IEEE J. Quantum Electron. 1(2), 624–628 (1995).

Baets, R.

S. Goeman, S. Boons, B. Dhoedt, K. Vandeputte, K. Caekebeke, P. Van Daele, and R. Baets, “First demonstration of highly reflective and highly polarization selective diffraction gratings (GIRO-gratings) for long-wavelength VCSELs,” IEEE Photon. Technol. Lett. 10(9), 1205–1207 (1998).

Bakir, B. B.

C. Sciancalepore, B. B. Bakir, X. Letartre, J. Harduin, N. Olivier, C. Seassal, J. Fedeli, and P. Viktorovitch, “ CMOS-compatible ultra-compact 1.55-µm emitting VCSELs using double photonic crystal mirrors,” IEEE Photon. Technol. Lett. 24(6), 455–457 (2012).

Bartelt, H.

Beaudoin, M.

W. Yuen, G. S. Li, R. F. Nabiev, J. Boucart, P. Kner, R. J. Stone, D. Zhang, M. Beaudoin, T. Zheng, C. He, K. Yu, M. Jansen, D. P. Worland, and C. J. Chang-Hasnain, “High-performance 1.6 µm single-epitaxy top-emitting VCSEL,” Electron. Lett. 36(13), 1121–1123 (2000).

Beausolei, R. G.

D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. G. Beausolei, “Flat dielectric grating reflectors with focusing abilities,” Nat. Photonics 4(7), 466–470 (2010).

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D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. G. Beausoleil, “A silicon lens for integrated free-space optics,” in Integrated Photonics Research, Silicon and Nanophotonics, OSA Technical Digest (CD) (Optical Society of America, 2011).

Ben Bakir, B.

S. Boutami, B. Ben Bakir, J.-L. Leclercq, and P. Viktorovitch, “Compact and polarization controlled 1.55 µm vertical-cavity surface emitting laser using single-layer photonic crystal mirror,” Appl. Phys. Lett. 91(7), 071105 (2007).

Benbakir, B.

Bengtsson, J.

A. Haglund, J. S. Gustavsson, J. Bengtsson, P. Jedrasik, and A. Larsson, “Design and evaluation of fundamental-mode and polarization-stabilized VCSELs with a subwavelength surface grating,” IEEE J. Quantum Electron. 42(3), 231–240 (2006).

Berseth, C.

A. Syrbu, A. Mereuta, A. Mircea, A. Caliman, V. Iakovlev, C. Berseth, G. Suruceanu, A. Rudra, E. Deichsel, and E. Kapon, “1550 nm-band VCSEL 0.76 mW singlemode output power in 20–80°C temperature range,” Electron. Lett. 40(5), 306 (2004).

Bhat, R.

N. Nishiyama, C. Caneau, B. Hall, G. Guryanov, M. Hu, X. Liu, M. Li, R. Bhat, and C. Zah, “Long-wavelength vertical-cavity surface-emitting lasers on InP with lattice matched AlGaInAs-InP DBR grown by MOCVD,” IEEE J. Sel. Top. Quantum Electron. 11(5), 990–998 (2005).

Boehm, G.

F. Riemenschneider, M. Maute, H. Halbritter, G. Boehm, M. C. Amann, and P. Meissner, “Continuously tunable long-wavelength MEMS–VCSEL with over 40-nm tuning range,” IEEE Photon. Technol. Lett. 16(10), 2212–2214 (2004).

Böhm, G.

W. Hofmann, C. Chase, M. Müller, Y. Rao, C. Grasse, G. Böhm, M.-C. Amann, and C. J. Chang-Hasnain, “Long-wavelength high-contrast grating vertical-cavity surface-emitting laser,” IEEE Photonics J. 2(3), 415–422 (2010).

S. Jatta, B. Kögel, M. Maute, K. Zogal, F. Riemenschneider, G. Böhm, M.-C. Amann, and P. Meißner, “Bulk-micromachined VCSEL at 1.55 µm with 76-nm single-mode continuous tuning range,” IEEE Photon. Technol. Lett. 21(24), 1822–1824 (2009).

W. Hofmann, E. Wong, G. Böhm, M. Ortsiefer, N. H. Zhu, and M. C. Amann, “1.55 µm VCSEL arrays for high-bandwidth WDM-PONs,” IEEE Photon. Technol. Lett. 20(4), 291–293 (2008).

B. Kögel, H. Halbritter, S. Jatta, M. Maute, G. Böhm, M.-C. Amann, M. Lackner, M. Schwarzott, F. Winter, and P. Meissner, “Simultaneous spectroscopy of NH3 and CO using a >50 nm continuously tunable MEMS–VCSEL,” IEEE Sens. J. 7(11), 1483–1489 (2007).

M. Ortsiefer, R. Shau, G. Böhm, F. Köhler, and M. C. Amann, “Low-threshold index-guided 1.5 µm long-wavelength vertical-cavity surface-emitting laser with high efficiency,” Appl. Phys. Lett. 76(16), 2179 (2000).

Bolivar, P. H.

Bolten, J.

T. Stöferle, N. Moll, T. Wahlbrink, J. Bolten, T. Mollenhauer, U. Scherf, and R. F. Mahrt, “Ultracompact silicon/polymer laser with an absorption-insensitive nanophotonic resonator,” Nano Lett. 10(9), 3675–3678 (2010).
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S. Goeman, S. Boons, B. Dhoedt, K. Vandeputte, K. Caekebeke, P. Van Daele, and R. Baets, “First demonstration of highly reflective and highly polarization selective diffraction gratings (GIRO-gratings) for long-wavelength VCSELs,” IEEE Photon. Technol. Lett. 10(9), 1205–1207 (1998).

Born, M.

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Boucart, J.

D. Sun, W. Fan, P. Kner, J. Boucart, T. Kageyama, D. Zhang, R. Pathak, R. F. Nabiev, and W. Yuen, “Long wavelength-tunable VCSELs with optimized MEMS bridge tuning structure,” IEEE Photon. Technol. Lett. 16(3), 714–716 (2004).

W. Yuen, G. S. Li, R. F. Nabiev, J. Boucart, P. Kner, R. J. Stone, D. Zhang, M. Beaudoin, T. Zheng, C. He, K. Yu, M. Jansen, D. P. Worland, and C. J. Chang-Hasnain, “High-performance 1.6 µm single-epitaxy top-emitting VCSEL,” Electron. Lett. 36(13), 1121–1123 (2000).

Boutami, S.

S. Boutami, B. Ben Bakir, J.-L. Leclercq, and P. Viktorovitch, “Compact and polarization controlled 1.55 µm vertical-cavity surface emitting laser using single-layer photonic crystal mirror,” Appl. Phys. Lett. 91(7), 071105 (2007).

S. Boutami, B. Benbakir, X. Letartre, J. L. Leclercq, P. Regreny, and P. Viktorovitch, “Ultimate vertical checkerboard–Perot cavity based on single-layer photonic crystal mirrors,” Opt. Express 15(19), 12443–12449 (2007).
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Britzger, M.

F. Brückner, D. Friedrich, T. Clausnitzer, M. Britzger, O. Burmeister, K. Danzmann, E. B. Kley, A. Tünnermann, and R. Schnabel, “Realization of a monolithic high-reflectivity cavity mirror from a single silicon crystal,” Phys. Rev. Lett. 104(16), 163903 (2010).
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Brückner, F.

F. Brückner, D. Friedrich, T. Clausnitzer, M. Britzger, O. Burmeister, K. Danzmann, E. B. Kley, A. Tünnermann, and R. Schnabel, “Realization of a monolithic high-reflectivity cavity mirror from a single silicon crystal,” Phys. Rev. Lett. 104(16), 163903 (2010).
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Burmeister, O.

F. Brückner, D. Friedrich, T. Clausnitzer, M. Britzger, O. Burmeister, K. Danzmann, E. B. Kley, A. Tünnermann, and R. Schnabel, “Realization of a monolithic high-reflectivity cavity mirror from a single silicon crystal,” Phys. Rev. Lett. 104(16), 163903 (2010).
[PubMed]

Caekebeke, K.

S. Goeman, S. Boons, B. Dhoedt, K. Vandeputte, K. Caekebeke, P. Van Daele, and R. Baets, “First demonstration of highly reflective and highly polarization selective diffraction gratings (GIRO-gratings) for long-wavelength VCSELs,” IEEE Photon. Technol. Lett. 10(9), 1205–1207 (1998).

Caliman, A.

A. Syrbu, A. Mereuta, A. Mircea, A. Caliman, V. Iakovlev, C. Berseth, G. Suruceanu, A. Rudra, E. Deichsel, and E. Kapon, “1550 nm-band VCSEL 0.76 mW singlemode output power in 20–80°C temperature range,” Electron. Lett. 40(5), 306 (2004).

Cambril, E.

Caneau, C.

N. Nishiyama, C. Caneau, B. Hall, G. Guryanov, M. Hu, X. Liu, M. Li, R. Bhat, and C. Zah, “Long-wavelength vertical-cavity surface-emitting lasers on InP with lattice matched AlGaInAs-InP DBR grown by MOCVD,” IEEE J. Sel. Top. Quantum Electron. 11(5), 990–998 (2005).

Chang, H.-S.

W.-H. Chang, W.-Y. Chen, H.-S. Chang, T.-P. Hsieh, J.-I. Chyi, and T.-M. Hsu, “Efficient single-photon sources based on low-density quantum dots in photonic-crystal nanocavities,” Phys. Rev. Lett. 96(11), 117401 (2006).
[PubMed]

Chang, W.-H.

W.-H. Chang, W.-Y. Chen, H.-S. Chang, T.-P. Hsieh, J.-I. Chyi, and T.-M. Hsu, “Efficient single-photon sources based on low-density quantum dots in photonic-crystal nanocavities,” Phys. Rev. Lett. 96(11), 117401 (2006).
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Chang-Hasnain, C.

T. Sun, W. Yang, V. Karagodsky, W. Zhou, and C. Chang-Hasnain, “Low-loss slow light inside high contrast grating waveguide,” Proc. SPIE 8270, 82700A (2012).

L. Zhu, V. Karagodsky, and C. Chang-Hasnain, “Novel high efficiency vertical to in-plane optical coupler,” Proc. SPIE 8270, 82700L (2012).

C. Chang-Hasnain, M. Maeda, N. Stoffel, J. Harbison, L. Florez, and J. Jewell, “Surface emitting laser arrays with uniformly separated wavelengths,” Electron. Lett. 26(13), 940–941 (1990).

Chang-Hasnain, C. J.

V. Karagodsky and C. J. Chang-Hasnain, “Physics of near-wavelength high contrast gratings,” Opt. Express 20(10), 10888–10895 (2012).
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W. Yang, J. Ferrara, K. Grutter, A. Yeh, C. Chase, Y. Yue, A. E. Willner, M. C. Wu, and C. J. Chang-Hasnain, ”Low loss hollow-core waveguide on a silicon substrate,” Nanophotonics 1(1), 23–29 (2012).

V. Karagodsky, C. Chase, and C. J. Chang-Hasnain, “Matrix Fabry–Perot resonance mechanism in high-contrast gratings,” Opt. Lett. 36(9), 1704–1706 (2011).
[PubMed]

V. Karagodsky, F. G. Sedgwick, and C. J. Chang-Hasnain, “Theoretical analysis of subwavelength high contrast grating reflectors,” Opt. Express 18(16), 16973–16988 (2010).
[PubMed]

V. Karagodsky, B. Pesala, C. Chase, W. Hofmann, F. Koyama, and C. J. Chang-Hasnain, “Monolithically integrated multi-wavelength VCSEL arrays using high-contrast gratings,” Opt. Express 18(2), 694–699 (2010).
[PubMed]

F. Lu, F. G. Sedgwick, V. Karagodsky, C. Chase, and C. J. Chang-Hasnain, “Planar high-numerical-aperture low-loss focusing reflectors and lenses using subwavelength high contrast gratings,” Opt. Express 18(12), 12606–12614 (2010).
[PubMed]

C. Chase, Y. Rao, W. Hofmann, and C. J. Chang-Hasnain, “1550 nm high contrast grating VCSEL,” Opt. Express 18(15), 15461–15466 (2010).
[PubMed]

W. Hofmann, C. Chase, M. Müller, Y. Rao, C. Grasse, G. Böhm, M.-C. Amann, and C. J. Chang-Hasnain, “Long-wavelength high-contrast grating vertical-cavity surface-emitting laser,” IEEE Photonics J. 2(3), 415–422 (2010).

C. Chase, Y. Zhou, and C. J. Chang-Hasnain, “Size effect of high contrast gratings in VCSELs,” Opt. Express 17(26), 24002–24007 (2009).
[PubMed]

C. J. Chang-Hasnain, Y. Zhou, M. C. Y. Huang, and C. Chase, “High-contrast grating VCSELs,” IEEE J. Sel. Top. Quantum Electron. 15(3), 869–878 (2009).

Y. Zhou, V. Karagodsky, B. Pesala, F. G. Sedgwick, and C. J. Chang-Hasnain, “A novel ultra-low loss hollow-core waveguide using subwavelength high-contrast gratings,” Opt. Express 17(3), 1508–1517 (2009).
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M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A nanoelectromechanical tunable laser,” Nat. Photonics 2(3), 180–184 (2008).

Y. Zhou, M. C. Y. Huang, and C. J. Chang-Hasnain, “Large fabrication tolerance for VCSELs using high contrast grating,” IEEE Photon. Technol. Lett. 20(6), 434–436 (2008).

Y. Zhou, M. Moewe, J. Kern, M. C. Y. Huang, and C. J. Chang-Hasnain, “Surface-normal emission of a high-Q resonator using a subwavelength high-contrast grating,” Opt. Express 16(22), 17282–17287 (2008).
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M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A surface-emitting laser incorporating a high index-contrast subwavelength grating,” Nat. Photonics 1(2), 119–122 (2007).

M. C. Y. Huang, K. B. Cheng, Y. Zhou, B. Pesala, C. J. Chang-Hasnain, and A. P. Pisano, “Demonstration of piezoelectric actuated GaAs-based MEMS tunable VCSEL,” IEEE Photon. Technol. Lett. 18(10), 1197–1199 (2006).

C. F. R. Mateus, M. C. Y. Huang, L. Chen, C. J. Chang-Hasnain, and Y. Suzuki, “Broad-band mirror (1.12–1.62 µm) using a subwavelength grating,” IEEE Photon. Technol. Lett. 16(7), 1676–1678 (2004).

C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultra-broadband mirror using low index cladded subwavelength grating,” IEEE Photon. Technol. Lett. 16(2), 518–520 (2004).

C. J. Chang-Hasnain, “Tunable VCSEL,” IEEE J. Sel. Top. Quantum Electron. 6(6), 978–987 (2000).

W. Yuen, G. S. Li, R. F. Nabiev, J. Boucart, P. Kner, R. J. Stone, D. Zhang, M. Beaudoin, T. Zheng, C. He, K. Yu, M. Jansen, D. P. Worland, and C. J. Chang-Hasnain, “High-performance 1.6 µm single-epitaxy top-emitting VCSEL,” Electron. Lett. 36(13), 1121–1123 (2000).

M. S. Wu, E. C. Vail, G. S. Li, W. Yuen, and C. J. Chang-Hasnain, “Widely and continuously tunable micromachined resonant cavity detector with wavelength tracking,” IEEE Photon. Technol. Lett. 8(1), 98–100 (1996).

L. E. Eng, K. Bacher, W. Yuen, J. S. Harris, and C. J. Chang-Hasnain, “Multiple wavelength vertical cavity laser arrays on patterned substrates,” IEEE J. Quantum Electron. 1(2), 624–628 (1995).

M. W. Maeda, C. J. Chang-Hasnain, C. Lin, J. S. Patel, H. A. Johnson, and J. A. Walker, “Use of a multiwavelength surface-emitting laser array in a four-channel wavelength-division-multiplexed system experiment,” Photonics Technol. Lett. 3(3), 268–269 (1991).

M. Orenstein, A. Von Lehmen, C. J. Chang-Hasnain, N. G. Stoffel, J. P. Harbison, and L. T. Florez, “Matrix addressable vertical cavity surface emitting laser array,” Electron. Lett. 27(5), 437–438 (1991).

C. J. Chang-Hasnain, J. P. Harbison, C. E. Zah, M. W. Maeda, L. T. Florez, N. G. Stoffel, and T. P. Lee, “Multiple wavelength tunable surface emitting laser arrays,” IEEE J. Quantum Electron. 27(6), 1368–1376 (1991).

C. J. Chang-Hasnain, J. P. Harbison, G. Hasnain, A. Von Lehmen, L. T. Florez, and N. G. Stoffel, “Dynamic, polarization, and transverse mode characteristics of vertical cavity surface emitting lasers,” IEEE J. Quantum Electron. 27(6), 1402–1409 (1991).

C. J. Chang-Hasnain, “VCSEL for metro communications,” in Optical Fiber Communications, , I. Kaminow and T. Li, eds. (Academic, 2002), vol. IV A, pp. 666–698.

B. Pesala, V. Karagodsky, and C. J. Chang-Hasnain, “Ultra-compact optical coupler and splitter using high-contrast grating hollow-core waveguide,” in Silicon and Nanophotonics, OSA Technical Digest (CD) (Optical Society of America, 2010), paper IWH1.

V. Karagodsky, T. Tran, M. Wu, and C. J. Chang-Hasnain, “Double-resonant enhancement of surface enhanced Raman scattering using high contrast grating resonators,” in CLEO:2011—Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper CFN1.

Y. Rao, C. Chase, and C. J. Chang-Hasnain, “Multiwavelength HCG–VCSEL Array,” in 2010 Second IEEE International Semiconductor Laser Conference (ISLC) (IEEE, 2010), pp. 11–12.

Y. Rao, C. Chase, M. C. Y. Huang, S. Khaleghi, M. R. Chitgarha, M. Ziyadi, D. P. Worland, A. E. Willner, and C. J. Chang-Hasnain, “Continuous tunable 1550-nm high contrast grating VCSEL,” in CLEO: Applications and Technology, OSA Technical Digest (online) (Optical Society of America, 2012), paper CTh5C.3.

Chase, C.

W. Yang, J. Ferrara, K. Grutter, A. Yeh, C. Chase, Y. Yue, A. E. Willner, M. C. Wu, and C. J. Chang-Hasnain, ”Low loss hollow-core waveguide on a silicon substrate,” Nanophotonics 1(1), 23–29 (2012).

V. Karagodsky, C. Chase, and C. J. Chang-Hasnain, “Matrix Fabry–Perot resonance mechanism in high-contrast gratings,” Opt. Lett. 36(9), 1704–1706 (2011).
[PubMed]

F. Lu, F. G. Sedgwick, V. Karagodsky, C. Chase, and C. J. Chang-Hasnain, “Planar high-numerical-aperture low-loss focusing reflectors and lenses using subwavelength high contrast gratings,” Opt. Express 18(12), 12606–12614 (2010).
[PubMed]

V. Karagodsky, B. Pesala, C. Chase, W. Hofmann, F. Koyama, and C. J. Chang-Hasnain, “Monolithically integrated multi-wavelength VCSEL arrays using high-contrast gratings,” Opt. Express 18(2), 694–699 (2010).
[PubMed]

C. Chase, Y. Rao, W. Hofmann, and C. J. Chang-Hasnain, “1550 nm high contrast grating VCSEL,” Opt. Express 18(15), 15461–15466 (2010).
[PubMed]

W. Hofmann, C. Chase, M. Müller, Y. Rao, C. Grasse, G. Böhm, M.-C. Amann, and C. J. Chang-Hasnain, “Long-wavelength high-contrast grating vertical-cavity surface-emitting laser,” IEEE Photonics J. 2(3), 415–422 (2010).

C. J. Chang-Hasnain, Y. Zhou, M. C. Y. Huang, and C. Chase, “High-contrast grating VCSELs,” IEEE J. Sel. Top. Quantum Electron. 15(3), 869–878 (2009).

C. Chase, Y. Zhou, and C. J. Chang-Hasnain, “Size effect of high contrast gratings in VCSELs,” Opt. Express 17(26), 24002–24007 (2009).
[PubMed]

Y. Rao, C. Chase, M. C. Y. Huang, S. Khaleghi, M. R. Chitgarha, M. Ziyadi, D. P. Worland, A. E. Willner, and C. J. Chang-Hasnain, “Continuous tunable 1550-nm high contrast grating VCSEL,” in CLEO: Applications and Technology, OSA Technical Digest (online) (Optical Society of America, 2012), paper CTh5C.3.

Y. Rao, C. Chase, and C. J. Chang-Hasnain, “Multiwavelength HCG–VCSEL Array,” in 2010 Second IEEE International Semiconductor Laser Conference (ISLC) (IEEE, 2010), pp. 11–12.

Chavel, P.

Chelnokov, A.

P. Gilet, N. Olivier, P. Grosse, K. Gilbert, A. Chelnokov, I.-S. Chung, and J. Mørk, “High-index-contrast subwavelength grating VCSEL,” Proc. SPIE 7615, 76150J (2010).

I.-S. Chung, J. Mørk, P. Gilet, and A. Chelnokov, “Subwavelength grating-mirror VCSEL with a thin oxide gap,” IEEE Photon. Technol. Lett. 20(2), 105–107 (2008).

Chen, J. C.

Chen, L.

C. F. R. Mateus, M. C. Y. Huang, L. Chen, C. J. Chang-Hasnain, and Y. Suzuki, “Broad-band mirror (1.12–1.62 µm) using a subwavelength grating,” IEEE Photon. Technol. Lett. 16(7), 1676–1678 (2004).

Chen, W.-Y.

W.-H. Chang, W.-Y. Chen, H.-S. Chang, T.-P. Hsieh, J.-I. Chyi, and T.-M. Hsu, “Efficient single-photon sources based on low-density quantum dots in photonic-crystal nanocavities,” Phys. Rev. Lett. 96(11), 117401 (2006).
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Cheng, K. B.

M. C. Y. Huang, K. B. Cheng, Y. Zhou, B. Pesala, C. J. Chang-Hasnain, and A. P. Pisano, “Demonstration of piezoelectric actuated GaAs-based MEMS tunable VCSEL,” IEEE Photon. Technol. Lett. 18(10), 1197–1199 (2006).

Chitgarha, M. R.

Y. Rao, C. Chase, M. C. Y. Huang, S. Khaleghi, M. R. Chitgarha, M. Ziyadi, D. P. Worland, A. E. Willner, and C. J. Chang-Hasnain, “Continuous tunable 1550-nm high contrast grating VCSEL,” in CLEO: Applications and Technology, OSA Technical Digest (online) (Optical Society of America, 2012), paper CTh5C.3.

Chou, S. Y.

L. Zhuang, S. Schablitsky, R. C. Shi, and S. Y. Chou, “Fabrication and performance of thin amorphous Si subwavelength transmission grating for controlling vertical cavity surface emitting laser polarization,” J. Vac. Sci. Technol. B 14(6), 4055–4057 (1996).

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Chung, I.-S.

P. Gilet, N. Olivier, P. Grosse, K. Gilbert, A. Chelnokov, I.-S. Chung, and J. Mørk, “High-index-contrast subwavelength grating VCSEL,” Proc. SPIE 7615, 76150J (2010).

I.-S. Chung, J. Mørk, P. Gilet, and A. Chelnokov, “Subwavelength grating-mirror VCSEL with a thin oxide gap,” IEEE Photon. Technol. Lett. 20(2), 105–107 (2008).

Chyi, J.-I.

W.-H. Chang, W.-Y. Chen, H.-S. Chang, T.-P. Hsieh, J.-I. Chyi, and T.-M. Hsu, “Efficient single-photon sources based on low-density quantum dots in photonic-crystal nanocavities,” Phys. Rev. Lett. 96(11), 117401 (2006).
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Coldren, L. A.

S. Nakagawa, E. Hall, G. Almuneau, J. K. Kim, D. A. Buell, H. Kroemer, and L. A. Coldren, “88°C, continuous-wave operation of apertured, intracavity contacted, 1.55 µm vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 78(10), 1337 (2001).

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Danzmann, K.

F. Brückner, D. Friedrich, T. Clausnitzer, M. Britzger, O. Burmeister, K. Danzmann, E. B. Kley, A. Tünnermann, and R. Schnabel, “Realization of a monolithic high-reflectivity cavity mirror from a single silicon crystal,” Phys. Rev. Lett. 104(16), 163903 (2010).
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R. Michalzik, J. M. Ostermann, and P. Debernardi, “Polarization-stable monolithic VCSELs,” Proc. SPIE,  6908, 69080A (2008).

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Deichsel, E.

A. Syrbu, A. Mereuta, A. Mircea, A. Caliman, V. Iakovlev, C. Berseth, G. Suruceanu, A. Rudra, E. Deichsel, and E. Kapon, “1550 nm-band VCSEL 0.76 mW singlemode output power in 20–80°C temperature range,” Electron. Lett. 40(5), 306 (2004).

Deng, Y.

C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultra-broadband mirror using low index cladded subwavelength grating,” IEEE Photon. Technol. Lett. 16(2), 518–520 (2004).

Devenyi, A.

Dhoedt, B.

S. Goeman, S. Boons, B. Dhoedt, K. Vandeputte, K. Caekebeke, P. Van Daele, and R. Baets, “First demonstration of highly reflective and highly polarization selective diffraction gratings (GIRO-gratings) for long-wavelength VCSELs,” IEEE Photon. Technol. Lett. 10(9), 1205–1207 (1998).

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J. L. Jewell, S. L. McCall, Y. H. Lee, A. Scherer, A. C. Gossard, and J. H. English, “Lasing characteristics of GaAs microresonators,” Appl. Phys. Lett. 54(15), 1400–1402 (1989).

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Fan, W.

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C. J. Chang-Hasnain, J. P. Harbison, G. Hasnain, A. Von Lehmen, L. T. Florez, and N. G. Stoffel, “Dynamic, polarization, and transverse mode characteristics of vertical cavity surface emitting lasers,” IEEE J. Quantum Electron. 27(6), 1402–1409 (1991).

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W. Shan, W. Walukiewicz, J. W. Ager III, E. E. Haller, J. F. Geisz, D. J. Friedman, J. M. Olson, and S. R. Kurtz, “Band anticrossing in GaInNAs alloys,” Phys. Rev. Lett. 82(6), 1221–1224 (1999).

Gilbert, K.

P. Gilet, N. Olivier, P. Grosse, K. Gilbert, A. Chelnokov, I.-S. Chung, and J. Mørk, “High-index-contrast subwavelength grating VCSEL,” Proc. SPIE 7615, 76150J (2010).

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J. L. Jewell, S. L. McCall, Y. H. Lee, A. Scherer, A. C. Gossard, and J. H. English, “Lasing characteristics of GaAs microresonators,” Appl. Phys. Lett. 54(15), 1400–1402 (1989).

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W. Hofmann, C. Chase, M. Müller, Y. Rao, C. Grasse, G. Böhm, M.-C. Amann, and C. J. Chang-Hasnain, “Long-wavelength high-contrast grating vertical-cavity surface-emitting laser,” IEEE Photonics J. 2(3), 415–422 (2010).

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Grosse, P.

P. Gilet, N. Olivier, P. Grosse, K. Gilbert, A. Chelnokov, I.-S. Chung, and J. Mørk, “High-index-contrast subwavelength grating VCSEL,” Proc. SPIE 7615, 76150J (2010).

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Grutter, K.

W. Yang, J. Ferrara, K. Grutter, A. Yeh, C. Chase, Y. Yue, A. E. Willner, M. C. Wu, and C. J. Chang-Hasnain, ”Low loss hollow-core waveguide on a silicon substrate,” Nanophotonics 1(1), 23–29 (2012).

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N. Nishiyama, C. Caneau, B. Hall, G. Guryanov, M. Hu, X. Liu, M. Li, R. Bhat, and C. Zah, “Long-wavelength vertical-cavity surface-emitting lasers on InP with lattice matched AlGaInAs-InP DBR grown by MOCVD,” IEEE J. Sel. Top. Quantum Electron. 11(5), 990–998 (2005).

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A. Haglund, J. S. Gustavsson, J. Bengtsson, P. Jedrasik, and A. Larsson, “Design and evaluation of fundamental-mode and polarization-stabilized VCSELs with a subwavelength surface grating,” IEEE J. Quantum Electron. 42(3), 231–240 (2006).

Gustavsson, S. J.

A. Haglund, S. J. Gustavsson, J. Vukusic, P. Jedrasik, and A. Larsson, “High-power fundamental-mode and polarisation stabilised VCSELs using sub-wavelength surface grating,” Electron. Lett. 41(14), 805–807 (2005).

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Haglund, A.

A. Haglund, J. S. Gustavsson, J. Bengtsson, P. Jedrasik, and A. Larsson, “Design and evaluation of fundamental-mode and polarization-stabilized VCSELs with a subwavelength surface grating,” IEEE J. Quantum Electron. 42(3), 231–240 (2006).

A. Haglund, S. J. Gustavsson, J. Vukusic, P. Jedrasik, and A. Larsson, “High-power fundamental-mode and polarisation stabilised VCSELs using sub-wavelength surface grating,” Electron. Lett. 41(14), 805–807 (2005).

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Halbritter, H.

B. Kögel, H. Halbritter, S. Jatta, M. Maute, G. Böhm, M.-C. Amann, M. Lackner, M. Schwarzott, F. Winter, and P. Meissner, “Simultaneous spectroscopy of NH3 and CO using a >50 nm continuously tunable MEMS–VCSEL,” IEEE Sens. J. 7(11), 1483–1489 (2007).

H. Halbritter, C. Sydlo, B. Kögel, F. Riemenschneider, H. L. Hartnagel, and P. Meissner, “Impact of micromechanics on the linewidth and chirp performance of MEMS–VCSELs,” IEEE J. Sel. Top. Quantum Electron. 13(2), 367–373 (2007).

M. Lackner, M. Schwarzott, F. Winter, B. Kogel, S. Jatta, H. Halbritter, and P. Meissner, “CO and CO2 spectroscopy using a 60 nm broadband tunable MEMS–VCSEL at 1.55 µm,” Opt. Lett. 31(21), 3170–3172 (2006).
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Hall, B.

N. Nishiyama, C. Caneau, B. Hall, G. Guryanov, M. Hu, X. Liu, M. Li, R. Bhat, and C. Zah, “Long-wavelength vertical-cavity surface-emitting lasers on InP with lattice matched AlGaInAs-InP DBR grown by MOCVD,” IEEE J. Sel. Top. Quantum Electron. 11(5), 990–998 (2005).

Hall, E.

S. Nakagawa, E. Hall, G. Almuneau, J. K. Kim, D. A. Buell, H. Kroemer, and L. A. Coldren, “88°C, continuous-wave operation of apertured, intracavity contacted, 1.55 µm vertical-cavity surface-emitting lasers,” Appl. Phys. Lett. 78(10), 1337 (2001).

Haller, E. E.

W. Shan, W. Walukiewicz, J. W. Ager III, E. E. Haller, J. F. Geisz, D. J. Friedman, J. M. Olson, and S. R. Kurtz, “Band anticrossing in GaInNAs alloys,” Phys. Rev. Lett. 82(6), 1221–1224 (1999).

Harbison, J.

C. Chang-Hasnain, M. Maeda, N. Stoffel, J. Harbison, L. Florez, and J. Jewell, “Surface emitting laser arrays with uniformly separated wavelengths,” Electron. Lett. 26(13), 940–941 (1990).

Harbison, J. P.

M. Orenstein, A. Von Lehmen, C. J. Chang-Hasnain, N. G. Stoffel, J. P. Harbison, and L. T. Florez, “Matrix addressable vertical cavity surface emitting laser array,” Electron. Lett. 27(5), 437–438 (1991).

C. J. Chang-Hasnain, J. P. Harbison, C. E. Zah, M. W. Maeda, L. T. Florez, N. G. Stoffel, and T. P. Lee, “Multiple wavelength tunable surface emitting laser arrays,” IEEE J. Quantum Electron. 27(6), 1368–1376 (1991).

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Harduin, J.

C. Sciancalepore, B. B. Bakir, X. Letartre, J. Harduin, N. Olivier, C. Seassal, J. Fedeli, and P. Viktorovitch, “ CMOS-compatible ultra-compact 1.55-µm emitting VCSELs using double photonic crystal mirrors,” IEEE Photon. Technol. Lett. 24(6), 455–457 (2012).

Harris, J. S.

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Hartnagel, H. L.

H. Halbritter, C. Sydlo, B. Kögel, F. Riemenschneider, H. L. Hartnagel, and P. Meissner, “Impact of micromechanics on the linewidth and chirp performance of MEMS–VCSELs,” IEEE J. Sel. Top. Quantum Electron. 13(2), 367–373 (2007).

Haruna, M.

Hasnain, G.

C. J. Chang-Hasnain, J. P. Harbison, G. Hasnain, A. Von Lehmen, L. T. Florez, and N. G. Stoffel, “Dynamic, polarization, and transverse mode characteristics of vertical cavity surface emitting lasers,” IEEE J. Quantum Electron. 27(6), 1402–1409 (1991).

Hatori, N.

A. Mizutani, N. Hatori, N. Nishiyama, F. Koyama, and K. Iga, “InGaAs/GaAs vertical-cavity surface emitting laser on GaAs (311)B substrate using carbon auto-doping,” Jpn. J. Appl. Phys. 37(Part 1, No. 3B), 1408–1412 (1998).

F. Koyama, T. Mukaihara, Y. Hayashi, N. Ohnoki, N. Hatori, and K. Iga, “Wavelength control of vertical cavity surface-emitting lasers by using nonplanar MOCVD,” IEEE Photon. Technol. Lett. 7(1), 10–12 (1995).

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F. Koyama, T. Mukaihara, Y. Hayashi, N. Ohnoki, N. Hatori, and K. Iga, “Wavelength control of vertical cavity surface-emitting lasers by using nonplanar MOCVD,” IEEE Photon. Technol. Lett. 7(1), 10–12 (1995).

He, C.

W. Yuen, G. S. Li, R. F. Nabiev, J. Boucart, P. Kner, R. J. Stone, D. Zhang, M. Beaudoin, T. Zheng, C. He, K. Yu, M. Jansen, D. P. Worland, and C. J. Chang-Hasnain, “High-performance 1.6 µm single-epitaxy top-emitting VCSEL,” Electron. Lett. 36(13), 1121–1123 (2000).

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Henschel, W.

Hofmann, C.

A. Löffler, J. Reithmaier, G. Sek, C. Hofmann, S. Reitzenstein, M. Kamp, and A. Forchel, “Semiconductor quantum dot microcavity pillars with high-quality factors and enlarged dot dimensions,” Appl. Phys. Lett. 86(11), 111105 (2005).

J. P. Reithmaier, G. Sęk, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot-semiconductor microcavity system,” Nature 432(7014), 197–200 (2004).
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Hofmann, W.

V. Karagodsky, B. Pesala, C. Chase, W. Hofmann, F. Koyama, and C. J. Chang-Hasnain, “Monolithically integrated multi-wavelength VCSEL arrays using high-contrast gratings,” Opt. Express 18(2), 694–699 (2010).
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C. Chase, Y. Rao, W. Hofmann, and C. J. Chang-Hasnain, “1550 nm high contrast grating VCSEL,” Opt. Express 18(15), 15461–15466 (2010).
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W. Hofmann, E. Wong, G. Böhm, M. Ortsiefer, N. H. Zhu, and M. C. Amann, “1.55 µm VCSEL arrays for high-bandwidth WDM-PONs,” IEEE Photon. Technol. Lett. 20(4), 291–293 (2008).

Hsieh, T.-P.

W.-H. Chang, W.-Y. Chen, H.-S. Chang, T.-P. Hsieh, J.-I. Chyi, and T.-M. Hsu, “Efficient single-photon sources based on low-density quantum dots in photonic-crystal nanocavities,” Phys. Rev. Lett. 96(11), 117401 (2006).
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N. Nishiyama, C. Caneau, B. Hall, G. Guryanov, M. Hu, X. Liu, M. Li, R. Bhat, and C. Zah, “Long-wavelength vertical-cavity surface-emitting lasers on InP with lattice matched AlGaInAs-InP DBR grown by MOCVD,” IEEE J. Sel. Top. Quantum Electron. 11(5), 990–998 (2005).

Huang, M. C. Y.

C. J. Chang-Hasnain, Y. Zhou, M. C. Y. Huang, and C. Chase, “High-contrast grating VCSELs,” IEEE J. Sel. Top. Quantum Electron. 15(3), 869–878 (2009).

Y. Zhou, M. C. Y. Huang, and C. J. Chang-Hasnain, “Large fabrication tolerance for VCSELs using high contrast grating,” IEEE Photon. Technol. Lett. 20(6), 434–436 (2008).

M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A nanoelectromechanical tunable laser,” Nat. Photonics 2(3), 180–184 (2008).

Y. Zhou, M. Moewe, J. Kern, M. C. Y. Huang, and C. J. Chang-Hasnain, “Surface-normal emission of a high-Q resonator using a subwavelength high-contrast grating,” Opt. Express 16(22), 17282–17287 (2008).
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M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A surface-emitting laser incorporating a high index-contrast subwavelength grating,” Nat. Photonics 1(2), 119–122 (2007).

M. C. Y. Huang, K. B. Cheng, Y. Zhou, B. Pesala, C. J. Chang-Hasnain, and A. P. Pisano, “Demonstration of piezoelectric actuated GaAs-based MEMS tunable VCSEL,” IEEE Photon. Technol. Lett. 18(10), 1197–1199 (2006).

C. F. R. Mateus, M. C. Y. Huang, L. Chen, C. J. Chang-Hasnain, and Y. Suzuki, “Broad-band mirror (1.12–1.62 µm) using a subwavelength grating,” IEEE Photon. Technol. Lett. 16(7), 1676–1678 (2004).

C. F. R. Mateus, M. C. Y. Huang, Y. Deng, A. R. Neureuther, and C. J. Chang-Hasnain, “Ultra-broadband mirror using low index cladded subwavelength grating,” IEEE Photon. Technol. Lett. 16(2), 518–520 (2004).

Y. Rao, C. Chase, M. C. Y. Huang, S. Khaleghi, M. R. Chitgarha, M. Ziyadi, D. P. Worland, A. E. Willner, and C. J. Chang-Hasnain, “Continuous tunable 1550-nm high contrast grating VCSEL,” in CLEO: Applications and Technology, OSA Technical Digest (online) (Optical Society of America, 2012), paper CTh5C.3.

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A. Syrbu, A. Mereuta, A. Mircea, A. Caliman, V. Iakovlev, C. Berseth, G. Suruceanu, A. Rudra, E. Deichsel, and E. Kapon, “1550 nm-band VCSEL 0.76 mW singlemode output power in 20–80°C temperature range,” Electron. Lett. 40(5), 306 (2004).

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K. Iga, “Surface-emitting laser-its birth and generation of new optoelectronics field,” IEEE J. Sel. Top. Quantum Electron. 6(6), 1201–1215 (2000).

A. Mizutani, N. Hatori, N. Nishiyama, F. Koyama, and K. Iga, “InGaAs/GaAs vertical-cavity surface emitting laser on GaAs (311)B substrate using carbon auto-doping,” Jpn. J. Appl. Phys. 37(Part 1, No. 3B), 1408–1412 (1998).

F. Koyama, T. Mukaihara, Y. Hayashi, N. Ohnoki, N. Hatori, and K. Iga, “Wavelength control of vertical cavity surface-emitting lasers by using nonplanar MOCVD,” IEEE Photon. Technol. Lett. 7(1), 10–12 (1995).

F. Koyama, H. Uenohara, T. Sakaguchi, and K. Iga, “GaAlAs/GaAs MOCVD growth for surface emitting laser,” Jpn. J. Appl. Phys. Part 1 26(Part 1, No. 7), 1077–1081 (1987).

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W. Yuen, G. S. Li, R. F. Nabiev, J. Boucart, P. Kner, R. J. Stone, D. Zhang, M. Beaudoin, T. Zheng, C. He, K. Yu, M. Jansen, D. P. Worland, and C. J. Chang-Hasnain, “High-performance 1.6 µm single-epitaxy top-emitting VCSEL,” Electron. Lett. 36(13), 1121–1123 (2000).

Jatta, S.

S. Jatta, B. Kögel, M. Maute, K. Zogal, F. Riemenschneider, G. Böhm, M.-C. Amann, and P. Meißner, “Bulk-micromachined VCSEL at 1.55 µm with 76-nm single-mode continuous tuning range,” IEEE Photon. Technol. Lett. 21(24), 1822–1824 (2009).

B. Kögel, H. Halbritter, S. Jatta, M. Maute, G. Böhm, M.-C. Amann, M. Lackner, M. Schwarzott, F. Winter, and P. Meissner, “Simultaneous spectroscopy of NH3 and CO using a >50 nm continuously tunable MEMS–VCSEL,” IEEE Sens. J. 7(11), 1483–1489 (2007).

M. Lackner, M. Schwarzott, F. Winter, B. Kogel, S. Jatta, H. Halbritter, and P. Meissner, “CO and CO2 spectroscopy using a 60 nm broadband tunable MEMS–VCSEL at 1.55 µm,” Opt. Lett. 31(21), 3170–3172 (2006).
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Jedrasik, P.

A. Haglund, J. S. Gustavsson, J. Bengtsson, P. Jedrasik, and A. Larsson, “Design and evaluation of fundamental-mode and polarization-stabilized VCSELs with a subwavelength surface grating,” IEEE J. Quantum Electron. 42(3), 231–240 (2006).

A. Haglund, S. J. Gustavsson, J. Vukusic, P. Jedrasik, and A. Larsson, “High-power fundamental-mode and polarisation stabilised VCSELs using sub-wavelength surface grating,” Electron. Lett. 41(14), 805–807 (2005).

Jewell, J.

C. Chang-Hasnain, M. Maeda, N. Stoffel, J. Harbison, L. Florez, and J. Jewell, “Surface emitting laser arrays with uniformly separated wavelengths,” Electron. Lett. 26(13), 940–941 (1990).

Jewell, J. L.

J. L. Jewell, S. L. McCall, Y. H. Lee, A. Scherer, A. C. Gossard, and J. H. English, “Lasing characteristics of GaAs microresonators,” Appl. Phys. Lett. 54(15), 1400–1402 (1989).

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D. Sun, W. Fan, P. Kner, J. Boucart, T. Kageyama, D. Zhang, R. Pathak, R. F. Nabiev, and W. Yuen, “Long wavelength-tunable VCSELs with optimized MEMS bridge tuning structure,” IEEE Photon. Technol. Lett. 16(3), 714–716 (2004).

Kamp, M.

A. Löffler, J. Reithmaier, G. Sek, C. Hofmann, S. Reitzenstein, M. Kamp, and A. Forchel, “Semiconductor quantum dot microcavity pillars with high-quality factors and enlarged dot dimensions,” Appl. Phys. Lett. 86(11), 111105 (2005).

Kapon, E.

A. Syrbu, A. Mereuta, A. Mircea, A. Caliman, V. Iakovlev, C. Berseth, G. Suruceanu, A. Rudra, E. Deichsel, and E. Kapon, “1550 nm-band VCSEL 0.76 mW singlemode output power in 20–80°C temperature range,” Electron. Lett. 40(5), 306 (2004).

Karagodsky, V.

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F. Lu, F. G. Sedgwick, V. Karagodsky, C. Chase, and C. J. Chang-Hasnain, “Planar high-numerical-aperture low-loss focusing reflectors and lenses using subwavelength high contrast gratings,” Opt. Express 18(12), 12606–12614 (2010).
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D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. G. Beausolei, “Flat dielectric grating reflectors with focusing abilities,” Nat. Photonics 4(7), 466–470 (2010).

M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A surface-emitting laser incorporating a high index-contrast subwavelength grating,” Nat. Photonics 1(2), 119–122 (2007).

M. C. Y. Huang, Y. Zhou, and C. J. Chang-Hasnain, “A nanoelectromechanical tunable laser,” Nat. Photonics 2(3), 180–184 (2008).

Nature (2)

J. P. Reithmaier, G. Sęk, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot-semiconductor microcavity system,” Nature 432(7014), 197–200 (2004).
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Opt. Express (11)

D. Ohnishi, T. Okano, M. Imada, and S. Noda, “Room temperature continuous wave operation of a surface-emitting two-dimensional photonic crystal diode laser,” Opt. Express 12(8), 1562–1568 (2004).
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V. Karagodsky and C. J. Chang-Hasnain, “Physics of near-wavelength high contrast gratings,” Opt. Express 20(10), 10888–10895 (2012).
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W. M. J. Green, M. J. Rooks, L. Sekaric, and Y. A. Vlasov, “Optical modulation using anti-crossing between paired amplitude and phase resonators,” Opt. Express 15(25), 17264–17272 (2007).
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Y. Zhou, V. Karagodsky, B. Pesala, F. G. Sedgwick, and C. J. Chang-Hasnain, “A novel ultra-low loss hollow-core waveguide using subwavelength high-contrast gratings,” Opt. Express 17(3), 1508–1517 (2009).
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V. Karagodsky, F. G. Sedgwick, and C. J. Chang-Hasnain, “Theoretical analysis of subwavelength high contrast grating reflectors,” Opt. Express 18(16), 16973–16988 (2010).
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C. Chase, Y. Zhou, and C. J. Chang-Hasnain, “Size effect of high contrast gratings in VCSELs,” Opt. Express 17(26), 24002–24007 (2009).
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C. Chase, Y. Rao, W. Hofmann, and C. J. Chang-Hasnain, “1550 nm high contrast grating VCSEL,” Opt. Express 18(15), 15461–15466 (2010).
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Y. Zhou, M. Moewe, J. Kern, M. C. Y. Huang, and C. J. Chang-Hasnain, “Surface-normal emission of a high-Q resonator using a subwavelength high-contrast grating,” Opt. Express 16(22), 17282–17287 (2008).
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V. Karagodsky, B. Pesala, C. Chase, W. Hofmann, F. Koyama, and C. J. Chang-Hasnain, “Monolithically integrated multi-wavelength VCSEL arrays using high-contrast gratings,” Opt. Express 18(2), 694–699 (2010).
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F. Lu, F. G. Sedgwick, V. Karagodsky, C. Chase, and C. J. Chang-Hasnain, “Planar high-numerical-aperture low-loss focusing reflectors and lenses using subwavelength high contrast gratings,” Opt. Express 18(12), 12606–12614 (2010).
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R. Michalzik, J. M. Ostermann, and P. Debernardi, “Polarization-stable monolithic VCSELs,” Proc. SPIE,  6908, 69080A (2008).

B. Pesala, V. Karagodsky, and C. J. Chang-Hasnain, “Ultra-compact optical coupler and splitter using high-contrast grating hollow-core waveguide,” in Silicon and Nanophotonics, OSA Technical Digest (CD) (Optical Society of America, 2010), paper IWH1.

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V. Karagodsky, T. Tran, M. Wu, and C. J. Chang-Hasnain, “Double-resonant enhancement of surface enhanced Raman scattering using high contrast grating resonators,” in CLEO:2011—Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper CFN1.

Y. Rao, C. Chase, and C. J. Chang-Hasnain, “Multiwavelength HCG–VCSEL Array,” in 2010 Second IEEE International Semiconductor Laser Conference (ISLC) (IEEE, 2010), pp. 11–12.

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Figures (38)

Figure 1
Figure 1

(a) Generic HCG structure. The grating comprises simple dielectric bars with high refractive index n bar , surrounded by a low-index medium n o . A second low-index material n 2 is beneath the bars. The plane wave is incident from the top at an oblique angle. Λ, HCG period; s, bar width; a, air gap width; t g , HCG thickness. For surface-normal ( θ = 0 ) TE incidence, φ = 0 , and the electrical field is parallel to the grating bars; whereas for TM, φ = π / 2 , and the electrical field is perpendicular to the grating bars. (b) The HCG conditions that will be discussed in this paper are surface normal incidence, n o = 1 , n bar = 1 . 2 3 . 6 , n 2 = 1 .

Figure 2
Figure 2

Examples of three types of extraordinary reflectivity/transmission feature: (a) high-Q resonances (red) and (b) broadband high reflection (blue), and broadband high transmission (green). The high-Q resonances are characterized by very sharp transitions from 0 to ∼100% reflectivity and vice versa, e.g., 1.682 and 1.773 µm, as labeled by the black arrows. For the broadband high-reflection case, the 99% reflection bandwidth is 578 nm (from 1.344 to 1.922 µm); this corresponds to λ / λ 35 % . The dashed black line indicates the reflectivity equal to one. For the broadband high-transmission case, the transmission is larger than 99.68% over a broad spectrum, not limited to the 1–3 µm shown in the figure. The HCG parameters for the three different cases are high-Q resonances, Λ = 0 . 716 µ m , t g = 1 . 494 µ m , η = 0 . 70 , TE-polarized light; broadband high reflection, Λ = 0 . 77 µ m , t g = 0 . 455 µ m , η = 0 . 76 , TM-polarized light; and broadband high transmission, Λ = 0 . 8 µ m , t g = 0 . 6 µ m , η = 0 . 1 , TM polarized light. n bar = 3 . 48 for all three cases.

Figure 3
Figure 3

Nomenclature for Eqs. (1a)–(1e): The HCG input plane is z = 0 , and the output/exit plane is z = t g . In region II, k a and k s are the x wavenumbers in the air gaps and in the grating bars, respectively. The z wavenumber β is the same in both the air gaps and the bars. The x wavenumber outside the grating (regions I and III) is determined by the grating periodicity: 2 π n / Λ , where n = 0 , 1 , 2 , .

Figure 4
Figure 4

Dispersion curves of a single slab waveguide (dashed) and waveguide array modes (solid), for the same bar width s and index n bar , β being the z wavenumbers. Between the two light lines, the dispersion curves of the waveguide array modes are nearly identical to those of the single slab waveguide [48]. Below the air light line ( β < ω / c ) there is a discrete set of modes due to subwavelength grating periodicity. ω c 2 and ω c 4 are the cutoffs of the TE2/TM2 and the TE4/TM4 modes, respectively, and between them the grating operates at a dual-mode regime. For HCG with surface-normal incidence, we only need to consider the even modes. The TE condition is plotted in (a), and TM in (b). In this calculation, η = 0 . 6 , n bar = 3 . 48 .

Figure 5
Figure 5

Excellent agreement between analytical solutions and commercial numerical simulation on HCG reflectivity and field intensity profile. (a) Comparison of HCG reflectivity spectrum calculated by analytical solution (red) and RCWA (blue). HCG parameters are t g / Λ = 0 . 627 , η = 0 . 62 , n bar = 3 . 214 , and TM polarization. (b) Comparison of HCG field intensity profile ( | E y | 2 ) calculated by analytical solution and FDTD. HCG parameters are λ / Λ = 1 . 509 , t g / Λ = 0 . 2 , η = 0 . 4 , n bar = 3 . 48 , and TE polarization. The white boxes indicate the HCG bars.

Figure 6
Figure 6

(a) Reflectivity contour of an HCG as a function of wavelength and grating thickness. The incident wave has a TE-polarized, surface-normal incidence. The mode cutoffs ( λ c 2 , λ c 4 ) and the first-order diffraction cutoff line λ = Λ are marked to clearly illustrate the differences of the three wavelength regimes: deep-subwavelength, near-wavelength, and diffraction. (b) Analytical solutions of FP resonance conditions of the individual modes [Eq. (11)], shown by the white curves, superimposed on the reflectivity contour in (a). Excellent agreement is obtained between the analytic solutions and the simulation results. The insets show examples of an anticrossing and a crossing of the FP resonance lines (white curves). HCG parameters are η = 0 . 6 , n bar = 3 . 48 , TE polarization.

Figure 7
Figure 7

(a) Reflectivity contour of an HCG as a function of wavelength and grating thickness. The incident wave has a TM-polarized, surface-normal incidence. The mode cutoffs ( λ c 2 , λ c 4 ) and the first-order diffraction cutoff line λ = Λ are marked to clearly illustrate the differences of the three wavelength-regimes: deep-subwavelength, near-wavelength, and diffraction. (b) Analytical solutions of FP resonance conditions of the individual modes [Eq. (11)], shown by the white curves, superimposed on the reflectivity contour in (a). Excellent agreement is obtained between the analytic solutions and the simulation results. The inset (right) shows that at some regions, the two resonance curves pull in together. HCG parameters are η = 0 . 6 , n bar = 3 . 48 , TM polarization.

Figure 8
Figure 8

(a) Two-mode solution exhibiting perfect cancellation at the HCG output plane ( z = t g ) leading to 100% reflectivity for TE-polarized light. At the wavelengths of 100% reflectivity (marked by the two vertical dashed lines), both modes have the same magnitude of the “DC” lateral Fourier component ( | ( a 0 + b 0 ) Λ 1 0 Λ E y , 0 II ( x ) d x | = | ( a 2 + b 2 ) Λ 1 0 Λ E y , 2 II ( x ) d x | ) , but opposite phases: Δ ϕ = phase [ ( a 0 + b 0 ) Λ 1 0 Λ E y , 0 II ( x ) d x ] phase [ ( a 2 + b 2 ) Λ 1 0 Λ E y , 2 II ( x ) d x ] = π . This means that the overall DC Fourier component is zero, which leads to the cancellation of the zeroth transmissive diffraction order. When two perfect-cancellation points are located in close spectral vicinity of each other, a broad band of high reflectivity is achieved. HCG parameters are n bar = 3 . 48 , s / Λ = 0 . 4 , t g / Λ = 0 . 2 , and TE polarization of incidence. (b) Two-mode solution for the field profile at the HCG output plane ( z = t g ) in the case of perfect cancellation. The cancellation is shown to be only in terms of the DC Fourier component. The higher Fourier components do not need to be zero, since subwavelength gratings have no diffraction orders other than zeroth order. The left-hand plot shows the decomposition of the overall field profile into the two modes, whereby the DC components of these two modes cancel each other. HCG parameters are the same as (a), and λ / Λ = 1 . 563 ; this is the condition corresponding to the dashed line on the right-hand side in (a).

Figure 9
Figure 9

Reflectivity contour plot compared with the two modes’ “phase difference” at the HCG output plane (defined as Δ ϕ = phase [ ( a 0 + b 0 ) Λ 1 0 Λ E x , 0 II ( x ) dx ] phase [ ( a 2 + b 2 ) Λ 1 0 Λ E x , 2 II ( x ) dx ] ) and “magnitude difference” at the HCG output plane (defined as | ( a 0 + b 0 ) Λ 1 0 Λ E x , 0 II ( x ) dx | | ( a 2 + b 2 ) Λ 1 0 Λ E x , 2 II ( x ) dx | for TM polarization; for TE polarization, E x , m II should be changed to E y , m II ; the same applies to the definition of “magnitude difference”). The white curves overlaid onto the reflectivity contour plot indicate the HCG conditions where Eq. (15) is satisfied. At the two-modes region, except at the resonance curves, the input plane wave couples relatively equal to both modes. Thus their phase difference dominantly determines the reflectivity. (a) TM HCG, η = 0 . 75 , n bar = 3 . 48 . (b) TE HCG, η = 0 . 45 , n bar = 3 . 48 .

Figure 10
Figure 10

Magnitude (a) and phase (b) spectrum of the two grating modes’ lateral average m = 0 , 2 ( a m + b m ) Λ 1 0 Λ E x , m II ( x ) d x at the HCG output plane, and m = 0 , 2 [ a m e + j β m t g + b m e j β m t g ] Λ 1 0 Λ E x , m II ( x ) d x at the input plane, compared with the reflection spectrum. At the 0% reflection points (i.e., 100% transmission points), marked by the three vertical dashed lines, Eqs. (18) and (19) are satisfied. The input light is TM-polarized, HCG t g / Λ = 0 . 84 , η = 0 . 55 , n bar = 3 . 48 .

Figure 11
Figure 11

(a) Resonance lines on t g λ diagram. Depending on | ψ | , the intersecting resonance curves either crosses or anticrosses. HCG parameters: η = 0 . 70 , n bar = 3 . 48 , TE polarization light. (b) Intensity profile inside the grating for an anticrossing, showing 1 0 7 -fold resonant energy buildup. The input light is TE polarized, λ / Λ = 2 . 32912867 , HCG t g / Λ = 0 . 8415 , η = 0 . 70 , n bar = 3 . 48 . (c) Intensity profile for a crossing, showing only weak energy buildup. The input light is TE polarized, λ / Λ = 2 . 02 , HCG t g / Λ = 0 . 32 , η = 0 . 70 , n bar = 3 . 48 . The HCG conditions for (b) and (c) are labeled in the t g λ diagram in (a). (d)–(i) Field profile [real( E y / E incident )] of different orders of HCG resonance at anticrossings: (d), (g) first order; (e), (h) second order; (f), (i) third order. The assigned index for each resonance is labeled in the figures. The HCG parameters for (d)–(i) are λ / Λ = 2 . 32912867 , t g / Λ = 0 . 8415 ; λ / Λ = 2 . 04704662 ; t g / Λ = 1 . 0267 ; λ / Λ = 1 . 80528405 , t g / Λ = 1 . 1555 ; λ / Λ = 2 . 32912867 , t g / Λ = 2 . 452394271 ; λ / Λ = 2 . 313806663 , t g / Λ = 1 . 8975 ; λ / Λ = 2 . 164125301 , t g / Λ = 2 . 0857 . For all conditions, the input light is TE polarized, HCG η = 0 . 70 , n bar = 3 . 48 . [Note that (b) and (d) are the same HCG resonance.]

Figure 12
Figure 12

(a) Validation of the HCG resonance condition det [ I ( ρ φ ) 2 ] = 0 . The absolute value of the determinant in Eq. (20) is plotted against the reflectivity spectrum of the HCG, containing five resonances (one strong and four weaker). The figure shows that not only do the dips of the determinant predict the exact resonant wavelengths, but also the closer the determinant to zero, the stronger the resonance. The bottom plot validates the HCG Q factor formulation in Eqs. (21) and (22). The overall Q factor of the HCG is the maximal among Q j . The spikes of Q are shown to predict precisely the wavelengths of the resonances in the reflectivity spectrum. The Q factor is in good agreement with a known method of inferring Q factors by curve fitting to Fano-resonant line shapes [55]. (b) Example of Fano-shape spectral fitting. The HCG parameter is η = 0 . 7 , t g / Λ = 2 . 086 , and n bar = 3 . 48 . The polarization of the incident light is TE [45].

Figure 13
Figure 13

Schematic of a typical 850 nm design, consisting of an HCG-based top mirror, a 5/4-λ air gap, a λ-cavity containing an active region, and a conventional semiconductor-based bottom n-DBR mirror. Here M = 2 or 4 [14].

Figure 14
Figure 14

Typical near-field intensity distribution, showing a Gaussian fundamental mode determined by the VCSEL cavity, despite the square shape of the HCG [18].

Figure 15
Figure 15

Schematic of a 1550 nm VCSEL with a suspended TE HCG in place of a typical top DBR. Current confinement is provided through the use of a proton-implant-defined aperture [19].

Figure 16
Figure 16

Scanning electron microscope images of (a) a completed 1550 nm HCG VCSEL. (b) Enlarged image of the HCG, which is just 195 nm thick [19].

Figure 17
Figure 17

Light– (solid lines) and voltage–current (dashed lines) characteristics of an HCG VCSEL with a 13 µm proton implant aperture at various heat sink temperatures. Devices show over 1.1 mW output power at room temperature and operate CW to >60°C. (b) Spectrum of the same device under various heat sink temperatures. A wavelength shift of 0.12 nm/K is extracted [19].

Figure 18
Figure 18

(a) Polarization-resolved light–current characteristics of a 1550 nm HCG VCSEL. A polarization suppression ratio of >20 dB is achieved, with the measurement limited by the polarizer [19]. (b) Near-field intensity profile of the device at 2 . 5 × I t h (where I t h is threshold current). A FWHM of ∼6.5 µm is obtained with a VCSEL with a proton implant aperture size of 15 µm [19]. (c) Frequency response (S21 characteristics) of a directly modulated 1550-nm HCG VCSEL under room temperature CW operation, biased at various current levels [26].

Figure 19
Figure 19

Schematic of tunable VCSEL, consisting of an n-doped HCG top mirror, a sacrificial layer, two pairs of p-doped DBRs, an AlGaAs oxidation layer, a cavity layer containing the active region, and a bottom standard n-doped DBR mirror. Electrical current injection is conducted through the middle laser p-contact and back side n-contact. An aluminum oxide aperture is formed on the AlGaAs layer just above the active region to provide current and optical confinement. The HCG is freely suspended above a variable air gap and supported via a nanomechanical structure. The tuning contact is fabricated on the top n-doped HCG layer [16].

Figure 20
Figure 20

Measured wavelength tuning spectra of tunable VCSEL using a bridge nanomechanical actuator. The laser is biased at ∼1.2 times the threshold current and actuated under various applied voltages across the tuning contact [16].

Figure 21
Figure 21

Mechanical response for a 3 µm × 3 µm HCG with 3 µm long membrane bridges. The 3 dB frequency is as high as 27 MHz [17].

Figure 22
Figure 22

Schematic of tunable proton-implanted VCSEL [26].

Figure 23
Figure 23

(a) Mechanical tuning range of a tunable HCG VCSEL. The laser lases over a range of 16.5 nm. (b) The maximum output power (blue) and threshold current (red) as a function of the wavelength. The device outputs over 1.5 mW over most of the mechanical tuning range [26].

Figure 24
Figure 24

Simulation of HCG reflectivity by RCWA. The design tolerance of the HCG is calculated at a fixed wavelength of 1550 nm, and the grating air gap variation could be > ± 100 nm  [25].

Figure 25
Figure 25

(a) Light–current curve and (b) spectrum of multiwavelength HCG VCSELs under CW operation at room temperature. All the devices in (b) are biased at 10 mA, and the wavelength range is from 1540 to 1591 nm [25].

Figure 26
Figure 26

Schematic of multiwavelength VCSEL array. The lasers have the same epitaxy, including HCG thickness. The VCSEL wavelength is changed by varying the HCG period and duty cycle [32].

Figure 27
Figure 27

(a) Magnitude and (b) phase of reflectivity spectra of HCGs with different periods and duty cycles in (c), keeping the same HCG thickness.

Figure 28
Figure 28

Emission spectrum of HCG resonator (in red) and emission spectrum in the area without grating (in blue). The inset shows an enlarged picture of the HCG emission spectrum. The FWHM of the HCG resonator emission peak is ∼0.07 nm, and Q is ∼14000 [30].

Figure 29
Figure 29

The TE HCG t g λ reflectivity contour plot with the locations of resonances used in the SERS enhancement structure (Fig. 30). The horizontal dashed line t g / Λ = 2 . 70 cuts across various resonance points. Point B is M70 resonance, used to enhance the Raman signal, whereas point A is M82, using for the pump laser. The silicon HCG is half-buried in the oxide substrate. Liquid containing the sensed molecules is placed on top of the grating structure. Its refractive index is ∼1.33. η = 0 . 6 for the HCG.

Figure 30
Figure 30

Proposed structure for SERS application. The HCG is half-buried in the oxide substrate in order to benefit from its odd-order resonances, all of which facilitate field enhancement at the center plane of the HCG thickness. The nanoanennae will then be lithographically defined at the grating gaps [31].

Figure 31
Figure 31

Intensity profile, calculated by using FDTD, for (a) M82 HCG resonance (third-order resonance) designed for the pump, point A in Fig. 29, and (b) M70 HCG resonances (first-order resonance) designed for the Raman signal, point B in Fig. 29. The intensity peaks are at the HCG center plane in both cases. We chose the dimensions so that the intensity peaks are at the gaps, rather than inside the semiconductor bars, which will make it easy for the molecules to have access to the hot spots. For the third- and first-order resonance, the peak enhancements are 1 0 4 and 1 0 2 , respectively. HCG parameters are Λ = 372 nm , t g = 1003 nm , and η = 0 . 6 . Wavelength is 785 nm for the pump. Light is TE polarized. The Raman shift is 1197 cm−1. Liquid containing the sensed molecules is placed on top of the grating structure. Its refractive index is ∼1.33 [31].

Figure 32
Figure 32

Intensity enhancement spectrum of HCG overlaid on the Raman spectrum of the BPE molecule, measured under a 785 nm pump. The M82 resonance (third-order resonance) is high Q, and it enhances the pump by 4 orders of magnitude. The M70 resonance (first-order resonance) is broader, in order to engulf the Raman peak [31].

Figure 33
Figure 33

FDTD simulation of intensity enhancement of (a) the entire structure (HCG + nanoantenna), (b) nanoantenna only, and (c) HCG only. The intensity enhancement of the entire structure is > 1 0 5 at the Raman wavelengths, agreeing well with simple multiplicative dependence on the individual enhancements of the HCG and nanoantenna. Nanoantenna dimensions are as follows: dipole gap 11 nm, thickness 32 nm, width 32 nm, and length 210 nm [31].

Figure 34
Figure 34

(a) Phase distribution ϕ ( x ) for a lens, with each sawtooth corresponding to a different 2 π window. A chirped HCG with different bar widths and gaps can provide a discrete phase distribution to approximate the ideal continuous distribution (green dots). (b) The reflectivity and phase calculated as a function of HCG dimensions with 1.2 µm thickness. The actual dimensions chosen for an HCG reflector design are shown on this map as a series of circles, colored according to their corresponding 2 π window in (a) [33].

Figure 35
Figure 35

(a) H-field intensity distribution (normalized by incident field intensity) on both the reflection side and the transmission side of an HCG focusing reflector. HCG bars are denoted by yellow boxes (b) H-field intensity distribution (normalized by incident field intensity) at the reflection focal plane. This field distribution is plotted after the incident wave is subtracted [33].

Figure 36
Figure 36

(a) Top view of a 2D HCG lens designed by varying bar width and air gap. (b) A 3D FDTD simulated E-field intensity distribution of the output at the focal plane with a Gaussian beam as input source. The Gaussian beam is focused from 3.5 µm (waist radius) down to 0.89 µm, a 15X reduction in area [33].

Figure 37
Figure 37

(a) Reflectivity contour of an HCG as a function of wavelength and grating thickness. The incident wave is TE polarized, with an incident angle θ = 50 ° . The mode cutoffs ( λ c 1 , λ c 2 , λ c 3 ) as well as the first-order diffraction cutoff lines are marked to clearly illustrate the three wavelength regimes: deep-subwavelength, near-wavelength, and diffraction. (b) Analytical solutions of FP resonance conditions of the individual modes [Eq. (11), but for oblique incident angle], shown by the blue curves, superimposed on the reflectivity contour in (a). Excellent agreement is obtained between the analytical solutions and the simulation results. The insets show examples of an anticrossing and a crossing of the FP resonance lines (blue curves). HCG parameters are η = 0 . 6 , n bar = 3 . 48 , TE polarization, and θ = 50 ° . In Subsection 6.2, the HCG band diagram will be discussed. Figure 37(a) is used to illustrate how to extract the band diagram. A horizontal line t g = 0 . 5 Λ is plotted for the chosen thickness, which cuts across various resonance curves of different modes. The crossing point regions are labeled A–D. In region A, t g = 0 . 5 Λ crosses first- and zeroth-order resonance; in region B, first-order resonance; in region C, second- and zeroth-order resonance; in region D, first-, second-, and third-order resonance. The incident angle is then varied from 0° to 90°, and these crossing points can be traced along the wavelength, forming a photonic band, as shown in Fig. 38(a). See Subsection 6.2 for more discussion.

Figure 38
Figure 38

Band diagram analysis of HCG and 1D photonic crystal. (a) HCG band diagram calculated with HCG analytical solution, for t g = 0 . 5 Λ . The photonic bands are signified with lines across which sharp reflectivity change happens or lines with full transmission. A dashed curve indicating θ = 50 ° is plotted and crosses the band. The crossing point regions are labeled A–D, corresponding to those in Fig. 37(a). (b) Full band diagram simulated with FDTD. The bands are symmetric along k x = 0 . 5 ( 2 π / Λ ) owing to Brillouin zone folding, and the light line is indicated by the dotted lines. In comparison with (a), one can associate different bands with different orders of supermodes in HCG. Because of the low quality factor of the zeroth-order supermodes above the light line, they do not show up clearly above the light line in (b). (c) The FDTD simulated band diagram for a pure 1D photonic crystal, where t g = . (d) Schematic showing the different operation regime for HCG and photonic crystal (PhC), separated by the light line. The bands are extracted from (a) and (b). The diffraction line of Eq. (26) is basically the folded light line due to Brillouin zone folding, above which lies the diffraction regime. The HCG operates above the light line but below the diffraction line, whereas the photonic crystal typically operates below the light line [108]. We also show the dual-mode regime for HCG with cutoff frequencies for HCG first-, second-, third-, and fourth-order supermodes, labeled ω c 1 , ω c 2 , ω c 3 , and ω c 4 , respectively. HCG parameters, n bar = 3 . 48 , η = 0 . 6 , TE incident polarized light.

Tables (1)

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Table 1. Differences between TM and TE Polarizations of Incidence

Equations (76)

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H y I ( x , z ) = e j k 0 z n = 0 r n H y , n I ( x ) e + j γ n z , z < 0 ,
E x I ( x , z ) = μ 0 ε 0 e j k 0 z + n = 0 r n E x , n I ( x ) e + j γ n z , z < 0 ,
H y , n I ( x ) = cos [ ( 2 n π / Λ ) ( x a / 2 ) ] , E x , n I ( x ) = ( γ n / k 0 ) μ 0 ε 0 H y , n I ( x ) ,
k 0 = 2 π λ , γ n 2 = k 0 2 ( 2 n π / Λ ) 2 .
H y III ( x , z ) = n = 0 τ n H y , n III ( x ) e j γ n ( z t g ) , z > t g ,
E x III ( x , z ) = n = 0 τ n E x , n III ( x ) e j γ n ( z t g ) , z > t g .
H y , n III ( x ) = H y , n I ( x ) and E x , n III ( x ) = E x , n I ( x ) .
r ( r 0 r 1 r 2 ) T R ( 1 0 0 ) T ,
τ ( τ 0 τ 1 τ 2 ) T T ( 1 0 0 ) T .
H y II ( x , z ) = m = 0 H y , m II ( x ) [ a m e j β m ( z t g ) b m e + j β m ( z t g ) ] , 0 < z < t g ,
E x II ( x , z ) = m = 0 E x , m II ( x ) [ a m e j β m ( z t g ) + b m e + j β m ( z t g ) ] , 0 < z < t g ,
H y , m II ( x ) = A cos [ k a , m ( x a / 2 ) ] + B sin [ k a , m ( x a / 2 ) ] , 0 < x < a ,
E x , m II ( x ) = ( β m / k 0 ) μ 0 ε 0 H y , m II ( x ) , 0 < x < a ,
H y , m II ( x ) = C cos { k s , m [ x ( a + Λ ) / 2 ] } + D sin { k s , m [ x ( a + Λ ) / 2 ] } , a < x < Λ ,
E x , m II ( x ) = n bar 2 ( β m / k 0 ) μ 0 ε 0 H y , m II ( x ) , a < x < Λ .
H y , m II ( x ) = H y , m II ( x + l Λ ) , l  is an integer,
E x , m II ( x ) = E x , m II ( x + l Λ ) , l  is an integer,
A = cos ( k s , m s / 2 ) ,
C = cos ( k a , m a / 2 ) ,
B = 0 ,
D = 0 ,
H y , m II ( x ) = E x , m II ( x ) = 0 when m = 1 , 3 , 5 , .
β m 2 = ( 2 π / λ ) 2 k a , m 2 = ( 2 π n bar / λ ) 2 k s , m 2 .
k s , m 2 k a , m 2 = ( 2 π / λ ) 2 ( n bar 2 1 ) .
n bar 2 k s , m tan ( k s , m s / 2 ) = k a , m tan ( k a , m a / 2 ) .
a ( a 0 a 1 a 2 ) T , b ( b 0 b 1 b 2 ) T .
b ρ a .
φ n , m = { e j β m t g for n = m 0 for n m .
n = 0 τ n H y , n III ( x ) = m = 0 H y , m II ( x ) ( a m b m ) ,
n = 0 τ n E x , n III ( x ) = m = 0 E x , m II ( x ) ( a m + b m ) .
τ n Λ 1 0 Λ [ H y , n III ( x ) ] 2 d x = m = 0 ( a m b m ) Λ 1 0 Λ H y , m II ( x ) H y , n III ( x ) d x ,
τ n Λ 1 0 Λ [ E x , n III ( x ) ] 2 d x = m = 0 ( a m + b m ) Λ 1 0 Λ E x , m II ( x ) E x , n III ( x ) d x .
H n , m = Λ 1 ( 2 δ n , 0 ) 0 Λ H y , m II ( x ) H y , n III ( x ) d x ,
E n , m = Λ 1 ( 2 δ n , 0 ) ( μ 0 ε 0 γ n / k 0 ) 2 0 Λ E x , m II ( x ) E x , n III ( x ) d x ,
τ = H ( a b ) = E ( a + b ) .
τ = H ( I ρ ) a = E ( I + ρ ) a .
H ( I ρ ) = E ( I + ρ )   ρ = ( I + H 1 E ) 1 ( I H 1 E ) .
n = 0 ( δ n , 0 r n ) H y , n I ( x ) = m = 0 H y , m II ( x ) [ a m e + j β m t g b m e j β m t g ] ,
n = 0 ( δ n , 0 + r n ) E x , n I ( x ) = m = 0 E x , m II ( x ) [ a m e + j β m t g + b m e j β m t g ] .
( I R ) 1 H ( I φ ρ φ ) = ( I + R ) 1 E ( I + φ ρ φ ) .
E ( I + φ ρ φ ) ( I φ ρ φ ) 1 H 1 = ( I + R ) ( I R ) 1 Z in .
R = ( Z in + I ) 1 ( Z in I ) , where Z in = E ( I + φ ρ φ ) ( I φ ρ φ ) 1 H 1 .
T = 2 E ( I + ρ ) φ [ ( Z in 1 + I ) E ( I + φ ρ φ ) ] 1 .
| R 00 | 2 + | T 00 | 2 1 .
ψ m phase   ( m th   eigenvalue   of   ρ φ ) .
ψ m = n π , where  n = 0 , 1 , 2 , .
ρ φ = [ 0 . 5534 e i 0 . 2042 π 0 . 1855 e i 0 . 0569 π 0 . 4743 e i 0 . 3554 π 0 . 8757 e i 0 . 0193 π ] ,
V 1 ( ρ φ ) V = [ 0 . 4988 e i 0 . 2798 π 0 0 0 . 9993 e i 0 . 0000 π ] .
| Δ ψ | = | ψ 2 ψ 0 | = l π , where  l = 0 , 1 , 2 , .
| R 00 | = 100 % τ 0 = [ E ( a + b ) ] 0 = m E 0 m ( a m + b m ) = 0 .
τ 0 = m E 0 m ( a m + b m ) = ( μ 0 ε 0 γ 0 / k 0 ) 2 m ( a m + b m ) Λ 1 0 Λ E x , m II ( x ) E x , 0 III ( x ) d x = ( μ 0 ε 0 γ 0 / k 0 ) 1 m ( a m + b m ) Λ 1 0 Λ E x , m II ( x ) d x = 0 .
| R 00 | = 100 % ( a 0 + b 0 ) Λ 1 0 Λ E x , 0 II ( x ) d x lateral average of the 0th-order mode ( forward + backward )  at  z = t g + ( a 2 + b 2 ) Λ 1 0 Λ E x , 2 II ( x ) d x lateral average of the 2nd-order mode ( forward + backward )  at  z = t g “destructive interference” (cancellation) between the 0th-order and the 2nd-order modes at  z = t g = 0 .
| T 00 | = 100 % | τ 0 | = | [ E ( a + b ) ] 0 | = | m E 0 m ( a m + b m ) | = 1 .
| τ 0 | = | m E 0 m ( a m + b m ) | = | ( μ 0 ε 0 γ 0 / k 0 ) 2 m ( a m + b m ) Λ 1 0 Λ E x , m II ( x ) E x , 0 III ( x ) d x | = | ( μ 0 ε 0 γ 0 / k 0 ) 1 m ( a m + b m ) Λ 1 0 Λ E x , m II ( x ) d x | = 1 .
| m ( a m + b m ) Λ 1 0 Λ E x , m II ( x ) d x | = μ 0 ε 0 .
m [ a m e + j β m t g + b m e j β m t g ] Λ 1 0 Λ E x , m II ( x ) d x = μ 0 ε 0 .
det [ I ( ρ φ ) 2 ] = 0 .
Q j = 2 π t g ( n g r / λ ) | r j / ( 1 r j 2 ) | .
Q HCG = max Q j .
ϕ ( x ) = 2 π λ ( f + ϕ max 2 π λ x 2 + f 2 ) ,
ϕ R ϕ T = π 2 + m π , m = 1 , 2 , ,
H y , m II ( x + l Λ ) = H y , m II ( x ) e i k x 0 l Λ , l  is an integer,
E x , m II ( x + l Λ ) = E x , m II ( x ) e i k x 0 l Λ , l  is an integer
k x 0 = k 0 sin θ = 2 π λ sin θ .
ω D / ( 2 π c Λ ) 1 k x / ( 2 π Λ ) .
H n , m = Λ 1 ( 2 δ n , 0 ) 0 Λ H y , m II ( x ) H y , n III ( x ) d x ,
H n , m = Λ 1 ( 2 δ n , 0 ) [ 0 a H y , m II ( x ) H y , n III ( x ) d x + a Λ H y , m II ( x ) H y , n III ( x ) d x ] ,
H n , m = Λ 1 ( 2 δ n , 0 ) { { 2 Λ cos ( k s , m s / 2 ) [ 2 n π cos ( k a , m a / 2 ) sin ( a n π / Λ ) + k a , m Λ cos ( a n π / Λ ) sin ( k a , m a / 2 ) ] } / ( 4 n 2 π 2 + Λ 2 k a , m 2 ) + { Λ cos ( k a , m a / 2 ) [ 2 n π cos ( k s , m s / 2 ) [ sin ( a n π / Λ ) + sin ( n π ( a 2 Λ ) / Λ ) ] k s , m Λ [ cos ( a n π / Λ ) + cos ( n π ( a 2 Λ ) / Λ ) ] sin ( k s , m s / 2 ) ] } / ( 4 n 2 π 2 Λ 2 k s , m 2 ) } .
E n , m = Λ 1 ( 2 δ n , 0 ) ( μ 0 ε 0 γ n / k 0 ) 2 0 Λ E x , m II ( x ) E x , n III ( x ) d x ,
E n , m = Λ 1 ( 2 δ n , 0 ) ( μ 0 ε 0 γ n / k 0 ) 2 [ 0 a E x , m II ( x ) E x , n III ( x ) d x + a Λ E x , m II ( x ) E x , n III ( x ) d x ] ,
E n , m = Λ 1 ( 2 δ n , 0 ) ( β m / γ n ) [ 0 a H y , m II ( x ) H y , n III ( x ) d x + n bar 2 a Λ H y , m II ( x ) H y , n III ( x ) d x ] ,
E n , m = Λ 1 ( 2 δ n , 0 ) ( β m / γ n ) { { 2 Λ cos ( k s , m s / 2 ) [ 2 n π cos ( k a , m a / 2 ) sin ( a n π / Λ ) + k a , m Λ cos ( a n π / Λ ) sin ( k a , m a / 2 ) ] } / ( 4 n 2 π 2 + Λ 2 k a , m 2 ) + { Λ cos ( k a , m a / 2 ) [ 2 n π cos ( k s , m s / 2 ) [ sin ( a n π / Λ ) + sin ( n π ( a 2 Λ ) / Λ ) ] k s , m Λ [ cos ( a n π / Λ ) + cos ( n π ( a 2 Λ ) / Λ ) ] sin ( k s , m s / 2 ) ] } / [ n bar 2 ( 4 n 2 π 2 Λ 2 k s , m 2 ) ] } .
H n , m = Λ 1 ( 2 δ n , 0 ) 0 Λ H x , m II ( x ) H x , n III ( x ) d x ,
H n , m = Λ 1 ( 2 δ n , 0 ) { { 2 Λ cos ( k s , m s / 2 ) [ 2 n π cos ( k a , m a / 2 ) sin ( a n π / Λ ) + k a , m Λ cos ( a n π / Λ ) sin ( k a , m a / 2 ) ] } / ( 4 n 2 π 2 + Λ 2 k a , m 2 ) + { Λ cos ( k a , m a / 2 ) [ 2 n π cos ( k s , m s / 2 ) [ sin ( a n π / Λ ) + sin ( n π ( a 2 Λ ) / Λ ) ] k s , m Λ [ cos ( a n π / Λ ) + cos ( n π ( a 2 Λ ) / Λ ) ] sin ( k s , m s / 2 ) ] } / ( 4 n 2 π 2 Λ 2 k s , m 2 ) } .
E n , m = Λ 1 ( 2 δ n , 0 ) ( μ 0 ε 0 ) 2 ( γ n / k 0 ) 2 0 Λ E y , m II ( x ) E y , n III ( x ) d x ,
E n , m = Λ 1 ( 2 δ n , 0 ) ( γ n / β m ) { { 2 Λ cos ( k s , m s / 2 ) [ 2 n π cos ( k a , m a / 2 ) sin ( a n π / Λ ) + k a , m Λ cos ( a n π / Λ ) sin ( k a , m a / 2 ) ] } / ( 4 n 2 π 2 + Λ 2 k a , m 2 ) + { Λ cos ( k a , m a / 2 ) [ 2 n π cos ( k s , m s / 2 ) [ sin ( a n π / Λ ) + sin ( n π ( a 2 Λ ) / Λ ) ] k s , m Λ [ cos ( a n π / Λ ) + cos ( n π ( a 2 Λ ) / Λ ) ] sin ( k s , m s / 2 ) ] } / ( 4 n 2 π 2 Λ 2 k s , m 2 ) } .

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