Abstract

When a guided wave is impinging onto a Photonic Crystal (PC) mirror, a fraction of the light is not reflected back and is radiated into the claddings. We present a theoretical and numerical study of this radiation problem for several three-dimensional mirror geometries which are important for light confinement in micropillars, air-bridge microcavities and two-dimensional PC microcavities. The cause of the radiation is shown to be a mode-profile mismatch. Additionally, design tools for reducing this mismatch by tuning the mirror geometry are derived. These tools are validated by numerical results performed with a three-dimensional Fourier modal method. Several engineered mirror geometries which lower the radiation loss by several orders of magnitude are designed.

© 2005 Optical Society of America

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2004 (4)

P.  Lalanne, S.  Mias, J.P.  Hugonin, “Two physical mechanisms for boosting the quality factor to cavity volume ratio of photonic crystal microcavities,” Opt Express 12, 458–467 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-458
[CrossRef] [PubMed]

C.  Sauvan, P.  Lalanne, J.P.  Hugonin, “Tuning holes in Photonic Crystal nanocavities,” Nature 429, 1 (2004).
[CrossRef] [PubMed]

H.Y.  Ryu, M.  Notomi, E.  Kuramoti, T.  Segawa, “Large spontaneous emission factor (>0.1) in the photonic crystal monopole-mode laser,” Appl. Phys. Lett. 84, 1067–1069(2004).
[CrossRef]

P.  Lalanne, J.P.  Hugonin, J.M.  Gérard, “Electromagnetic study of the Q of pillar microcavities in the small limit diameter,” Appl. Phys. Lett. 84, 4726–4728 (2004).
[CrossRef]

2003 (5)

C.  Sauvan, P.  Lalanne, J.C.  Rodier, J.P.  Hugonin, A.  Talneau, “Accurate modeling of line-defect Photonic Crystal waveguides,” IEEE Photon. Technol. Lett. 15, 1243–1245 (2003).
[CrossRef]

Y.  Akahane, T.  Asano, B.S.  Song, S.  Noda, “High-Q photonic nanocavity in two-dimensional photonic crystal,” Nature 425, 944–947 (2003).
[CrossRef] [PubMed]

P.  Lalanne, J. P.  Hugonin, “Bloch-wave engineering for high Q’s, small V’s microcavities,” IEEE J. Quantum Electron. 39, 1430–1438 (2003).
[CrossRef]

K.J.  Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
[CrossRef] [PubMed]

Y.  Akahane, T.  Asano, B.S.  Song, S.  Noda, “Investigation of high-Q channel drop filters using donor-type defects in two-dimensional photonic crystal slabs,” Appl. Phys. Lett. 83, 1512–1514 (2003).
[CrossRef]

2002 (6)

K.  Srinivasan, O.  Painter, “Momentum space design of high-Q photonic crystal optical cavities,” Opt. Exp. 10, 670–684 (2002), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-15-670

J.  Vuckovic, M.  Loncar, H.  Mabuchi, A.  Scherer, “Optimization of the Q factor in Photonic Crystal microcavities,” IEEE J. Quantum Electron. 38, 850–856 (2002).
[CrossRef]

Q.  Cao, P.  Lalanne, J.P.  Hugonin, “Stable and efficient Bloch-mode computational method for one-dimensional grating waveguide,” J. Opt. Soc. Am. A 19, 335–338 (2002).
[CrossRef]

P.  Lalanne, “Electromagnetic analysis of photonic crystal waveguides operating above the light cone,” IEEE J. Quantum Electron. 38, 800–804 (2002).
[CrossRef]

J.  Ctyroky, S.  Helfert, R.  Pregla, P.  Bienstman, R.  Baets, R.  De Ridder, R.  Stoffer, G.  Klaasse, J.  Petracek, P.  Lalanne, J.P.  Hugonin, R.M.  De La Rue, “Bragg waveguide grating as a 1D photonic band gap structure: COST 268 modelling task,” Opt. Quant. Electron. 34, 455–470 (2002).
[CrossRef]

J.  Vuckovic, M.  Pelton, A.  Scherer, Y.  Yamamoto, “Optimization of three-dimensional micropost microcavities for cavity quantum electrodynamics,” Phys. Rev. A 66, #023808 (2002).
[CrossRef]

2001 (5)

G.S.  Solomon, M.  Pelton, Y.  Yamamoto, “Single-mode spontaneous emission from a single quantum dot in a three-dimensional microcavity,” Phys. Rev. Lett. 86, 3903–3906 (2001).
[CrossRef] [PubMed]

S.G.  Johnson, S.  Fan, A.  Mekis, J. D.  Joannopoulos, “Multipole-cancellation mechanism for high- Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78, 3388–3300 (2001).
[CrossRef]

E.  Silberstein, P.  Lalanne, J.P.  Hugonin, Q.  Cao, “On the use of grating theory in integrated optics,” J. Opt. Soc. Am. A. 18, 2865–28275 (2001).
[CrossRef]

M.  Palamaru, P.  Lalanne, “Photonic crystal waveguides: out-of-plane losses and adiabatic modal conversion,” Appl. Phys. Lett. 78, 1466–1469 (2001).
[CrossRef]

H.G.  Park, J.K.  Hwang, J.  Huh, H.Y.  Ryu, Y.H.  Lee, J.S.  Kim, “Nondegenerate monopole-mode two-dimensional photonic band gap laser,” Appl. Phys. Lett. 79, 3032–3034 (2001).
[CrossRef]

2000 (2)

A.  Chutinan, S.  Noda, “Waveguides and waveguide bends in two-dimensional photonic crystal slabs,” Phys. Rev. B 62, 4488–4492 (2000).
[CrossRef]

E.  Popov, M.  Nevière, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000).
[CrossRef]

1999 (1)

1998 (3)

B.E.  Little, H.A.  Haus, J.S.  Foresi, L.C.  Kimerling, E.P.  Ippen, D.J.  Ripin, “Wavelength switching and routing using absorption and resonance,” IEEE Phot. Technol. Lett. 10, 816–818 (1998).
[CrossRef]

T.F.  Krauss, O.  Painter, A.  Scherer, J.S.  Roberts, R.M.  De La Rue, “Photonic microstructures as laser mirrors,” Opt. Eng. 37, 1143–1148 (1998).
[CrossRef]

P.  Lalanne, “Effective properties and band structures of lamellar subwavelength crystals: plane-wave method revisited,” Phys. Rev B 58, 9801–9807 (1998).
[CrossRef]

1997 (2)

L.  Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J.Opt. Soc. Am. A 14, 2758–2767 (1997).
[CrossRef]

J.S.  Foresi, P.R.  Villeneuve, J.  Ferrera, E.R.  Thoen, G.  Steinmeyer, S.  Fan, J.D.  Joannopoulos, L.C.  Kimerling, H.I.  Smith, E.P.  Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390, 143–145 (1997).
[CrossRef]

1996 (4)

T.  Baba, M.  Hamasaki, N.  Watanabe, P.  Kaewplung, A.  Matsutani, T.  Mukaihara, F.  Koyama, K.  Iga, “A novel short-cavity laser with deep-grating distributed Bragg reflectors,” Jpn. J. Appl. Phys. 35, 1390–1394 (1996).
[CrossRef]

J.P.  Zhang, D.Y.  Chu, S.L.  Wu, W.G.  Bi, R.C.  Tiberio, R.M.  Joseph, A.  Taflove, C.W.  Tu, S.T.  Ho, “Nanofabrication of 1-D photonic bandgap structures along a photonic wire,” IEEE Photon. Technol. Lett. 8, 491–493 (1996).
[CrossRef]

L.  Li, “Use or Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
[CrossRef]

P.  Lalanne, G.M.  Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
[CrossRef]

1994 (2)

W.C.  Chew, W.H.  Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

N.  Château, J.P.  Hugonin, “Algorithm for the rigorous coupled-wave analysis of grating diffraction,” J. Opt. Soc. Am. A 11, 1321–1331 (1994).
[CrossRef]

Aarts, E.

E.  Aarts, J.  Korst, Simulated Annealing and Boltzmann Machine (John Wiley & Sons, New York 1989).

Akahane, Y.

Y.  Akahane, T.  Asano, B.S.  Song, S.  Noda, “Investigation of high-Q channel drop filters using donor-type defects in two-dimensional photonic crystal slabs,” Appl. Phys. Lett. 83, 1512–1514 (2003).
[CrossRef]

Y.  Akahane, T.  Asano, B.S.  Song, S.  Noda, “High-Q photonic nanocavity in two-dimensional photonic crystal,” Nature 425, 944–947 (2003).
[CrossRef] [PubMed]

Asano, T.

Y.  Akahane, T.  Asano, B.S.  Song, S.  Noda, “High-Q photonic nanocavity in two-dimensional photonic crystal,” Nature 425, 944–947 (2003).
[CrossRef] [PubMed]

Y.  Akahane, T.  Asano, B.S.  Song, S.  Noda, “Investigation of high-Q channel drop filters using donor-type defects in two-dimensional photonic crystal slabs,” Appl. Phys. Lett. 83, 1512–1514 (2003).
[CrossRef]

Baba, T.

T.  Baba, M.  Hamasaki, N.  Watanabe, P.  Kaewplung, A.  Matsutani, T.  Mukaihara, F.  Koyama, K.  Iga, “A novel short-cavity laser with deep-grating distributed Bragg reflectors,” Jpn. J. Appl. Phys. 35, 1390–1394 (1996).
[CrossRef]

Baets, R.

J.  Ctyroky, S.  Helfert, R.  Pregla, P.  Bienstman, R.  Baets, R.  De Ridder, R.  Stoffer, G.  Klaasse, J.  Petracek, P.  Lalanne, J.P.  Hugonin, R.M.  De La Rue, “Bragg waveguide grating as a 1D photonic band gap structure: COST 268 modelling task,” Opt. Quant. Electron. 34, 455–470 (2002).
[CrossRef]

Bi, W.G.

J.P.  Zhang, D.Y.  Chu, S.L.  Wu, W.G.  Bi, R.C.  Tiberio, R.M.  Joseph, A.  Taflove, C.W.  Tu, S.T.  Ho, “Nanofabrication of 1-D photonic bandgap structures along a photonic wire,” IEEE Photon. Technol. Lett. 8, 491–493 (1996).
[CrossRef]

Bienstman, P.

J.  Ctyroky, S.  Helfert, R.  Pregla, P.  Bienstman, R.  Baets, R.  De Ridder, R.  Stoffer, G.  Klaasse, J.  Petracek, P.  Lalanne, J.P.  Hugonin, R.M.  De La Rue, “Bragg waveguide grating as a 1D photonic band gap structure: COST 268 modelling task,” Opt. Quant. Electron. 34, 455–470 (2002).
[CrossRef]

Cao, Q.

Q.  Cao, P.  Lalanne, J.P.  Hugonin, “Stable and efficient Bloch-mode computational method for one-dimensional grating waveguide,” J. Opt. Soc. Am. A 19, 335–338 (2002).
[CrossRef]

E.  Silberstein, P.  Lalanne, J.P.  Hugonin, Q.  Cao, “On the use of grating theory in integrated optics,” J. Opt. Soc. Am. A. 18, 2865–28275 (2001).
[CrossRef]

Château, N.

Chew, W.C.

W.C.  Chew, W.H.  Weedon, “A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates,” Microwave Opt. Technol. Lett. 7, 599–604 (1994).
[CrossRef]

Chu, D.Y.

J.P.  Zhang, D.Y.  Chu, S.L.  Wu, W.G.  Bi, R.C.  Tiberio, R.M.  Joseph, A.  Taflove, C.W.  Tu, S.T.  Ho, “Nanofabrication of 1-D photonic bandgap structures along a photonic wire,” IEEE Photon. Technol. Lett. 8, 491–493 (1996).
[CrossRef]

Chutinan, A.

A.  Chutinan, S.  Noda, “Waveguides and waveguide bends in two-dimensional photonic crystal slabs,” Phys. Rev. B 62, 4488–4492 (2000).
[CrossRef]

Ctyroky, J.

J.  Ctyroky, S.  Helfert, R.  Pregla, P.  Bienstman, R.  Baets, R.  De Ridder, R.  Stoffer, G.  Klaasse, J.  Petracek, P.  Lalanne, J.P.  Hugonin, R.M.  De La Rue, “Bragg waveguide grating as a 1D photonic band gap structure: COST 268 modelling task,” Opt. Quant. Electron. 34, 455–470 (2002).
[CrossRef]

De La Rue, R.M.

J.  Ctyroky, S.  Helfert, R.  Pregla, P.  Bienstman, R.  Baets, R.  De Ridder, R.  Stoffer, G.  Klaasse, J.  Petracek, P.  Lalanne, J.P.  Hugonin, R.M.  De La Rue, “Bragg waveguide grating as a 1D photonic band gap structure: COST 268 modelling task,” Opt. Quant. Electron. 34, 455–470 (2002).
[CrossRef]

T.F.  Krauss, O.  Painter, A.  Scherer, J.S.  Roberts, R.M.  De La Rue, “Photonic microstructures as laser mirrors,” Opt. Eng. 37, 1143–1148 (1998).
[CrossRef]

De Ridder, R.

J.  Ctyroky, S.  Helfert, R.  Pregla, P.  Bienstman, R.  Baets, R.  De Ridder, R.  Stoffer, G.  Klaasse, J.  Petracek, P.  Lalanne, J.P.  Hugonin, R.M.  De La Rue, “Bragg waveguide grating as a 1D photonic band gap structure: COST 268 modelling task,” Opt. Quant. Electron. 34, 455–470 (2002).
[CrossRef]

Fan, S.

S.G.  Johnson, S.  Fan, A.  Mekis, J. D.  Joannopoulos, “Multipole-cancellation mechanism for high- Q cavities in the absence of a complete photonic band gap,” Appl. Phys. Lett. 78, 3388–3300 (2001).
[CrossRef]

J.S.  Foresi, P.R.  Villeneuve, J.  Ferrera, E.R.  Thoen, G.  Steinmeyer, S.  Fan, J.D.  Joannopoulos, L.C.  Kimerling, H.I.  Smith, E.P.  Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390, 143–145 (1997).
[CrossRef]

Ferrera, J.

J.S.  Foresi, P.R.  Villeneuve, J.  Ferrera, E.R.  Thoen, G.  Steinmeyer, S.  Fan, J.D.  Joannopoulos, L.C.  Kimerling, H.I.  Smith, E.P.  Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390, 143–145 (1997).
[CrossRef]

Foresi, J.S.

B.E.  Little, H.A.  Haus, J.S.  Foresi, L.C.  Kimerling, E.P.  Ippen, D.J.  Ripin, “Wavelength switching and routing using absorption and resonance,” IEEE Phot. Technol. Lett. 10, 816–818 (1998).
[CrossRef]

J.S.  Foresi, P.R.  Villeneuve, J.  Ferrera, E.R.  Thoen, G.  Steinmeyer, S.  Fan, J.D.  Joannopoulos, L.C.  Kimerling, H.I.  Smith, E.P.  Ippen, “Photonic-bandgap microcavities in optical waveguides,” Nature 390, 143–145 (1997).
[CrossRef]

Gayral, B.

Gérard, J.M.

P.  Lalanne, J.P.  Hugonin, J.M.  Gérard, “Electromagnetic study of the Q of pillar microcavities in the small limit diameter,” Appl. Phys. Lett. 84, 4726–4728 (2004).
[CrossRef]

J.M.  Gérard, B.  Gayral, “Strong Purcell Effect for InAs quantum boxes in three-dimensional solid-state microcavities,” J. Lightwave Technol. 17, 2089–2095 (1999).
[CrossRef]

Hamasaki, M.

T.  Baba, M.  Hamasaki, N.  Watanabe, P.  Kaewplung, A.  Matsutani, T.  Mukaihara, F.  Koyama, K.  Iga, “A novel short-cavity laser with deep-grating distributed Bragg reflectors,” Jpn. J. Appl. Phys. 35, 1390–1394 (1996).
[CrossRef]

Haus, H.A.

B.E.  Little, H.A.  Haus, J.S.  Foresi, L.C.  Kimerling, E.P.  Ippen, D.J.  Ripin, “Wavelength switching and routing using absorption and resonance,” IEEE Phot. Technol. Lett. 10, 816–818 (1998).
[CrossRef]

Helfert, S.

J.  Ctyroky, S.  Helfert, R.  Pregla, P.  Bienstman, R.  Baets, R.  De Ridder, R.  Stoffer, G.  Klaasse, J.  Petracek, P.  Lalanne, J.P.  Hugonin, R.M.  De La Rue, “Bragg waveguide grating as a 1D photonic band gap structure: COST 268 modelling task,” Opt. Quant. Electron. 34, 455–470 (2002).
[CrossRef]

Ho, S.T.

J.P.  Zhang, D.Y.  Chu, S.L.  Wu, W.G.  Bi, R.C.  Tiberio, R.M.  Joseph, A.  Taflove, C.W.  Tu, S.T.  Ho, “Nanofabrication of 1-D photonic bandgap structures along a photonic wire,” IEEE Photon. Technol. Lett. 8, 491–493 (1996).
[CrossRef]

Hugonin, J. P.

P.  Lalanne, J. P.  Hugonin, “Bloch-wave engineering for high Q’s, small V’s microcavities,” IEEE J. Quantum Electron. 39, 1430–1438 (2003).
[CrossRef]

Hugonin, J.P.

P.  Lalanne, S.  Mias, J.P.  Hugonin, “Two physical mechanisms for boosting the quality factor to cavity volume ratio of photonic crystal microcavities,” Opt Express 12, 458–467 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-3-458
[CrossRef] [PubMed]

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Y.  Akahane, T.  Asano, B.S.  Song, S.  Noda, “Investigation of high-Q channel drop filters using donor-type defects in two-dimensional photonic crystal slabs,” Appl. Phys. Lett. 83, 1512–1514 (2003).
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Figures (7)

Fig. 1.
Fig. 1.

Optical microcavities considered in this work. (a) Air-bridge microcavity. (b) Micropillar. (c) Single defect PC microcavity in a semiconductor membrane (top view).

Fig. 2.
Fig. 2.

Modal reflectivity and transverse mode-profile mismatch. (a) Modal reflectivity spectrum for the air-bridge mirror (a=420 nm and d=230 nm). Solid red curve: computational data using exact electromagnetic theory. Blue circles: square of the overlap integral η2. The vertical dashed lines indicate the band edges. (b)–(f) Comparison between the y-component of the transverse magnetic field of the fundamental air-bridge mode H 1 (bottom) and that of the half-Bloch wave H T (top) for several wavelengths covering the whole bandgap. White solid lines indicate the semiconductor-air boundaries of the air-bridge. The transverse H T field is calculated in a symmetry plane shown as vertical dashed lines in the left-hand side of Fig. 3.

Fig. 3.
Fig. 3.

Illustration of geometry tuning for tapering. Two segments of length a’ and a” are inserted between the PC mirror and the air-bridge.

Fig. 4.
Fig. 4.

Relevant quantities for the evanescent Bloch mode associated to three segments with different hole diameters, d=230, 170 and 100 nm, for λ=1.5 µm. (a) 1-η as a function of the segment period. (b) Real part of the effective index neff of the Bloch mode of the segments. The curves for d=230 and 170 nm are down shifted by 0.07 and 0.03, respectively, for the sake of clarity [otherwise, Real(neff)=λ/(2a)]. The horizontal dashed line represents the effective index of the fundamental air-bridge guided mode. The vertical arrow labelled A indicates the location associated to the mirror with a 420-nm period. The vertical arrows labelled B and C indicate the location associated to the segments used in Section 4.2 to reduce the losses.

Fig. 5.
Fig. 5.

Effect of hole shifting on the modal reflectivity of 1D and 2D PC mirrors. (a) 2D PC configuration. (b) Related air-bridge configuration. (c) and (d) Modal reflectivities R1 and R2 for λ=1.54 µm as a function of the normalized hole shift s/a. (e) and (f) Corresponding modal reflectivity spectra. The solid and dashed curves are obtained for s=0 and s=0.18a, respectively.

Fig. 6.
Fig. 6.

Single-segment tapers for micropillar Bragg reflectors. (a) Reflector geometry (a=d 1+d 2=228 nm). Dark and light regions correspond to GaAs and AlOx materials. (b) Modal reflectivity as a function of the normalized thicknesses x1/a and x2/a of the first GaAs and AlOx layers for λ=0.95 µm. Points A and B correspond to geometries with periodic and optimized mirrors, respectively.

Fig. 7.
Fig. 7.

Radiation loss spectra L=1-R for air-bridge mirrors with two-segment tapers. Red bold curve: hand-driven design for segments defined by (a’, d)=(320, 170) nm, and (a”, d)=(280, 100) nm. Blue thin curve: optimized design for segments defined by (a’, d)=(240, 210) nm, and (a”, d)=(414, 100) nm. The dashed curve corresponds to the modal reflectivity of the periodic mirror and the vertical dotted lines indicate the band edge.

Equations (1)

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η = ℜe { [ dxdy ( E 1 × H T * ) e z dxdy ( E T × H 1 * ) e z ] [ dxdy ( E T × H T * ) e z ] } ℜe { dxdy ( E 1 × H 1 * ) e z }

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