Abstract

We suggest a new type of grating reflector denoted hybrid grating (HG) which shows large reflectivity in a broad wavelength range and has a structure suitable for realizing a vertical cavity laser with ultra-small modal volume. The properties of the grating reflector are investigated numerically and explained. The HG consists of an un-patterned III–V layer and a Si grating. The III–V layer has a thickness comparable to the grating layer, introduces more guided mode resonances and significantly increases the bandwidth of the reflector compared to the well-known high-index-contrast grating (HCG). By using an active III–V layer, a laser can be realized where the gain region is integrated into the mirror itself.

© 2014 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  4. S. Boutami, B. Benbakir, 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, 071105 (2007).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2012 (3)

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).
[CrossRef]

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

T. Estruch, F. Pardo, B. Portier, J. Jaeck, S. Derelle, and R. Haidar, “Masons rule and Signal Flow Graphs applied to subwavelength resonant structures,” Opt. Express 20(24), 27155–27162 (2012).
[CrossRef] [PubMed]

2011 (1)

2010 (3)

I.-S. Chung and J. Mørk, “Silicon-photonics light source realized by III–V/Si-grating-mirror laser,” Appl. Phys. Lett. 97(15), 151–153 (2010).
[CrossRef]

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

I.-S. Chung, V. Iakovlev, A. Sirbu, A. Mereuta, A. Caliman, E. Kapon, and J. Mørk, “Broadband MEMS-tunable high-index-contrast subwavelength grating long-wavelength VCSEL,” IEEE J. Quantum Electron. 46(9), 1245–1253 (2010).
[CrossRef]

2008 (2)

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).
[CrossRef]

R. Magnusson and M. Shokooh-Saremi, “Physical basis for wideband resonant reflectors,” Opt. Express 16(5), 3456–3462 (2008).
[CrossRef] [PubMed]

2007 (3)

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).
[CrossRef]

S. Boutami, B. Benbakir, 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, 071105 (2007).
[CrossRef]

M. C.Y. Huang, Y. Zhou, and Connie J. Chang-Hasnain, “Nano electro-mechanical optoelectronic tunable VCSEL,” Opt. Express 15, 1222 (2007).
[CrossRef] [PubMed]

2006 (1)

2004 (3)

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

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

Y. Ding and R. Magnusson, “Resonant leaky-mode spectral-band engineering and device applications,” Opt. Express 12(23), 5661–5674 (2004).
[CrossRef] [PubMed]

2001 (1)

1997 (1)

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

1996 (2)

1995 (1)

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).
[CrossRef]

Beausoleil, R. G.

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

Benbakir, B.

S. Boutami, B. Benbakir, 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, 071105 (2007).
[CrossRef]

Boutami, S.

S. Boutami, B. Benbakir, 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, 071105 (2007).
[CrossRef]

Caliman, A.

I.-S. Chung, V. Iakovlev, A. Sirbu, A. Mereuta, A. Caliman, E. Kapon, and J. Mørk, “Broadband MEMS-tunable high-index-contrast subwavelength grating long-wavelength VCSEL,” IEEE J. Quantum Electron. 46(9), 1245–1253 (2010).
[CrossRef]

Cao, Q.

Carletti, L.

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).
[CrossRef] [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).
[CrossRef]

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

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

Chang-Hasnain, Connie J.

Chavel, P.

Chelnokov, A.

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).
[CrossRef]

Chen, L.

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

Chung, I.-S.

L. Carletti, R. Malureanu, J. Mørk, and I.-S. Chung, “High-index-contrast grating reflector with beam steering ability for the transmitted beam,” Opt. Express 19(23), 23567 (2011).
[CrossRef] [PubMed]

I.-S. Chung and J. Mørk, “Silicon-photonics light source realized by III–V/Si-grating-mirror laser,” Appl. Phys. Lett. 97(15), 151–153 (2010).
[CrossRef]

I.-S. Chung, V. Iakovlev, A. Sirbu, A. Mereuta, A. Caliman, E. Kapon, and J. Mørk, “Broadband MEMS-tunable high-index-contrast subwavelength grating long-wavelength VCSEL,” IEEE J. Quantum Electron. 46(9), 1245–1253 (2010).
[CrossRef]

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).
[CrossRef]

Deng, Y.

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

Derelle, S.

Ding, Y.

Estruch, T.

Fattal, D.

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

Fedeli, 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).
[CrossRef]

Fiorentino, M.

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

Friesem, A. A.

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

Gaylord, T. K.

Gilet, P.

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).
[CrossRef]

Granet, G.

Grann, E. B.

Guizal, B.

Haidar, R.

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).
[CrossRef]

Huang, M. C. Y.

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

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

Huang, M. C.Y.

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).
[CrossRef]

M. C.Y. Huang, Y. Zhou, and Connie J. Chang-Hasnain, “Nano electro-mechanical optoelectronic tunable VCSEL,” Opt. Express 15, 1222 (2007).
[CrossRef] [PubMed]

Hugonin, J. P.

Iakovlev, V.

I.-S. Chung, V. Iakovlev, A. Sirbu, A. Mereuta, A. Caliman, E. Kapon, and J. Mørk, “Broadband MEMS-tunable high-index-contrast subwavelength grating long-wavelength VCSEL,” IEEE J. Quantum Electron. 46(9), 1245–1253 (2010).
[CrossRef]

Jaeck, J.

Kapon, E.

I.-S. Chung, V. Iakovlev, A. Sirbu, A. Mereuta, A. Caliman, E. Kapon, and J. Mørk, “Broadband MEMS-tunable high-index-contrast subwavelength grating long-wavelength VCSEL,” IEEE J. Quantum Electron. 46(9), 1245–1253 (2010).
[CrossRef]

Karagodsky, V.

Lalanne, P.

Leclercq, J. L.

S. Boutami, B. Benbakir, 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, 071105 (2007).
[CrossRef]

Letartre, X.

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).
[CrossRef]

Li, J.

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

Li, L.

Magnusson, R.

Malureanu, R.

Mateus, C. F. R.

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

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

Mereuta, A.

I.-S. Chung, V. Iakovlev, A. Sirbu, A. Mereuta, A. Caliman, E. Kapon, and J. Mørk, “Broadband MEMS-tunable high-index-contrast subwavelength grating long-wavelength VCSEL,” IEEE J. Quantum Electron. 46(9), 1245–1253 (2010).
[CrossRef]

Moharam, M. G.

Mørk, J.

L. Carletti, R. Malureanu, J. Mørk, and I.-S. Chung, “High-index-contrast grating reflector with beam steering ability for the transmitted beam,” Opt. Express 19(23), 23567 (2011).
[CrossRef] [PubMed]

I.-S. Chung and J. Mørk, “Silicon-photonics light source realized by III–V/Si-grating-mirror laser,” Appl. Phys. Lett. 97(15), 151–153 (2010).
[CrossRef]

I.-S. Chung, V. Iakovlev, A. Sirbu, A. Mereuta, A. Caliman, E. Kapon, and J. Mørk, “Broadband MEMS-tunable high-index-contrast subwavelength grating long-wavelength VCSEL,” IEEE J. Quantum Electron. 46(9), 1245–1253 (2010).
[CrossRef]

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).
[CrossRef]

Neureuther, A. R.

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

Olivier, N.

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).
[CrossRef]

Pardo, F.

Peng, Z.

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

Pommet, D. A.

Portier, B.

Rosenblatt, D.

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

Sciancalepore, C.

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).
[CrossRef]

Seassal, C.

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).
[CrossRef]

Sharon, A.

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

Shokooh-Saremi, M.

Silberstein, E.

Sirbu, A.

I.-S. Chung, V. Iakovlev, A. Sirbu, A. Mereuta, A. Caliman, E. Kapon, and J. Mørk, “Broadband MEMS-tunable high-index-contrast subwavelength grating long-wavelength VCSEL,” IEEE J. Quantum Electron. 46(9), 1245–1253 (2010).
[CrossRef]

Viktorovitch, P.

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).
[CrossRef]

S. Boutami, B. Benbakir, 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, 071105 (2007).
[CrossRef]

Zhou, Y.

M. C.Y. Huang, Y. Zhou, and Connie J. Chang-Hasnain, “Nano electro-mechanical optoelectronic tunable VCSEL,” Opt. Express 15, 1222 (2007).
[CrossRef] [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).
[CrossRef]

Appl. Phys. Lett. (2)

S. Boutami, B. Benbakir, 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, 071105 (2007).
[CrossRef]

I.-S. Chung and J. Mørk, “Silicon-photonics light source realized by III–V/Si-grating-mirror laser,” Appl. Phys. Lett. 97(15), 151–153 (2010).
[CrossRef]

IEEE J. Quantum Electron. (2)

I.-S. Chung, V. Iakovlev, A. Sirbu, A. Mereuta, A. Caliman, E. Kapon, and J. Mørk, “Broadband MEMS-tunable high-index-contrast subwavelength grating long-wavelength VCSEL,” IEEE J. Quantum Electron. 46(9), 1245–1253 (2010).
[CrossRef]

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

IEEE Photon. Technol. Lett. (4)

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).
[CrossRef]

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).
[CrossRef]

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

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

J. Lightwave Technol. (1)

J. Opt. Soc. Am. A (4)

Nat. Photonics (2)

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).
[CrossRef]

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

Opt. Express (6)

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Figures (7)

Fig. 1
Fig. 1

(a) and (b) show schematic cross-sectional views of an HCG and an HG, respectively. In both cases a plane wave is incident from the top in the surface normal direction and similar results can be obtained for incident light from bottom. ni and ti represent refractive index and thickness of each part, respectively. Λ and f denote the periodicity and duty cycle of the grating. (c) and (d) are dispersions of leaky modes (blue curves) and reflectivity spectra (green curves) for the HCG and the HG, respectively. k0 = 2π/λ0, and β is the real part of propagation constant of a leaky mode. The red curves show the appeared resonance due to individual GMRs which will overlap to make the total reflectivity spectra (green curves)

Fig. 2
Fig. 2

Reflectivity and transmissivity spectra of (a) conventional HCG and (b) HG. Grating parameters common for HCG and HG are Λ=720 nm, tg=497 nm, f =45 %, nh=3.48, nl=1.0, n1=1.0, and n2=1.0. Cap layer parameters for HG are tc=370 nm and nc=3.166. Parameter symbols are defined in Fig. 1. (c) Incident angle dependence of reflectivity at 1550-nm wavelength for HCG and HG. (d) Reflectivity phase spectra for HCG and HG.

Fig. 3
Fig. 3

Fabrication tolerance analysis with 100 samples. Histogram distributions of (a) cap layer thickness, (b) grating thickness, and (c) grating bar width. (d) Monte Carlo analysis result for reflectivity.

Fig. 4
Fig. 4

Contour plot of transmissivity in dB as a function of wavelength, grating thickness, tg, and cap layer thickness, tc. In the upper graph, tg is increased from 0.497 μm to 1.997 μm while tc is zero. In the lower graph, tc is increased from 0 to 1.5 μm while tg is kept to 0.497 μm. The white dashed line shows the cap layer thickness of the HG structure considered in Fig. 2 and the circles indicate the wavelengths of the four GMRs.

Fig. 5
Fig. 5

Magnetic field amplitude profiles in the HG, decomposed into 0-th (red), 1-st(blue), and 2-nd(magenta) harmonics at different GMR wavelengths, λ: (a) λ =1340 nm. (b) λ =1541 nm. (c) λ =1605 nm. (d) λ =1759 nm. Incident light is assumed to be from the left.

Fig. 6
Fig. 6

SFGs of the propagating modes in each layer for (a) HCG and (b) HG structure. Black dots represent propagating modes at each layer. Red arrows denote the interactions between the modes which take place at interfaces. Circular arrows mean self coupling (reflectivity of a mode to itself) at interfaces.

Fig. 7
Fig. 7

(a) Schematic cross-sectional view of the HG with considered modes in each layer. (b) The equivalent reflectivity from the two interface of the cap layer and grating. (c) Calculated reflectivity spectrum of the structure shown in Fig. 2 using full RCWA (red solid line) and with discarding non-propagating modes (blue dashed line). It shows a very good agreement especially in the wavelength region with high reflectivity.

Equations (1)

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r eq = r 1 + r 2 exp ( j 2 β d ) 1 + r 1 r 2 exp ( j 2 β d )

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