Abstract

We describe a novel scheme based on evanescent guided-wave coupling for optically interfacing between conventional fiber-optic and planar photonic crystal devices such as waveguides and resonant cavities. By considering the band structure of bulk photonic crystal slabs, we analyze the k space properties of a linear defect waveguide and establish a set of design rules to ensure efficient evanescent coupling with optical fiber tapers. These rules are used to design a waveguide in a square-lattice photonic crystal. The coupling efficiency is calculated with a coupled-mode theory incorporating the finite-difference time-domain-calculated uncoupled modes of the fiber taper and photonic crystal waveguide. On the basis of this coupled-mode theory, 95% power transfer from the fiber taper to the photonic crystal waveguide is possible over a coupling length of 80 lattice periods and with a bandwidth of 1.5% of the center wavelength. The integration of this waveguide with a photonic crystal defect resonant cavity is also presented, thus showing the usefulness of the combined fiber taper and photonic crystal waveguide system for efficient, optical fiber-based probing of optical elements based on planar photonic crystal technologies.

© 2003 Optical Society of America

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2003 (3)

P. E. Barclay, K. Srinivasan, M. Borselli, and O. Painter, “Experimental demonstration of evanescent coupling from optical fibre tapers to photonic crystal waveguides,” IEE Electron. Lett. 39, 842–844 (2003).
[CrossRef]

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

O. Painter, K. Srinivasan, and P. E. Barclay, “Wannier-like equation for the resonant cavity modes of locally perturbed photonic crystals,” Phys. Rev. B 68, 035214 (2003).
[CrossRef]

2002 (8)

H.-Y. Ryu, S.-H. Kim, H.-G. Park, J.-K. Hwang, Y.-H. Lee, and J.-S. Kim, “Square-lattice photonic band-gap single-cell laser operating in the lowest-order whispering gallery mode,” Appl. Phys. Lett. 80, 3883–3885 (2002).
[CrossRef]

K. Srinivasan and O. Painter, “Momentum space design of high-Q photonic crystal nanocavities in two-dimensional slab waveguides,” Opt. Exp. 10, 670–684 (2002), http://www.opticsexpress.org.
[CrossRef]

S. Kawakami, “Analytically solvable model of photonic crystal structures and novel phenomena,” J. Lightwave Technol. 20, 1644–1650 (2002).
[CrossRef]

W. Kuang, C. Kim, A. Stapleton, and J. O’Brien, “Grating-assisted coupling of optical fibers and photonic crystal waveguides,” Opt. Lett. 27, 1604–1606 (2002).
[CrossRef]

S. Mookherjea and A. Yariv, “Second-harmonic generation with pulses in a coupled-resonator optical waveguide,” Phys. Rev. E 65, 026607 (2002).
[CrossRef]

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. V. Daele, I. Moerman, S. Vertuyft, K. D. Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. 38, 949–955 (2002).
[CrossRef]

S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “The adiabatic theorem and a continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066608 (2002).
[CrossRef]

J. Vučković, M. Lončar, H. Mabuchi, and A. Scherer, “Design of photonic crystal microcavites for cavity QED,” Phys. Rev. E 65, 016608 (2002).
[CrossRef]

2001 (4)

E. Moreau, I. Robert, J. M. Gerard, I. Abram, L. Manin, and V. Thierry-Mieg, “Single-mode solid-state single photon source based on isolated quantum dots in pillar microcavities,” Appl. Phys. Lett. 79, 2865–2867 (2001).
[CrossRef]

S. Olivier, C. Smith, M. Rattier, H. Benisty, C. Weisbuch, T. Krauss, R. Houdré, and U. Oesterlé, “Miniband transmission in a photonic crystal coupled-resonator optical waveguide,” Opt. Lett. 26, 1019–1021 (2001).
[CrossRef]

A. Mekis and J. D. Joannopoulos, “Tapered couplers for efficient interfacing between dielectric and photonic crystal waveguides,” J. Lightwave Technol. 19, 861–865 (2001).
[CrossRef]

N. A. R. Bhat and J. E. Sipe, “Optical pulse propagation in nonlinear photonic crystals,” Phys. Rev. E 64, 056604 (2001).
[CrossRef]

2000 (7)

P. Paddon and J. F. Young, “Two-dimensional vector-coupled-mode theory for textured planar waveguides,” Phys. Rev. B 61, 2090–2101 (2000).
[CrossRef]

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. I. Glu, “A quantum dot single-photon turnstile device,” Science 290, 2282–2285 (2000).
[CrossRef] [PubMed]

M. Cai, O. Painter, and K. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[CrossRef] [PubMed]

S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature (London) 407, 608–610 (2000).
[CrossRef]

Y. Xu, R. K. Lee, and A. Yariv, “Adiabatic coupling between conventional dielectric waveguides and waveguides with discrete translational symmetry,” Opt. Lett. 25, 755–757 (2000).
[CrossRef]

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
[CrossRef]

M. Lončar, D. Nedeljković, T. Doll, J. Vučković, A. Scherer, and T. P. Pearsall, “Waveguiding in planar photonic crystals,” Appl. Phys. Lett. 77, 1937–1939 (2000).
[CrossRef]

1999 (5)

O. Painter, R. K. Lee, A. Yariv, A. Scherer, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1824 (1999).
[CrossRef] [PubMed]

A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. 24, 711–713 (1999).
[CrossRef]

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejaki, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Theoretical analysis of channel drop tunneling processes,” Phys. Rev. B 59, 15882–15892 (1999).
[CrossRef]

G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, “Nonlinear optics and normal-mode-coupling in semiconductor microcavities,” Rev. Mod. Phys. 71, 1591–1639 (1999).
[CrossRef]

1998 (1)

N. Stefanou and A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B 57, 12127–12133 (1998).
[CrossRef]

1997 (1)

M. Kanskar, P. Paddon, V. Pacradoui, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

1996 (2)

D. M. Atkin, P. S. J. Russell, T. A. Birks, and P. J. Roberts, “Photonic band structure of guided Bloch modes in high index films fully etched through with periodic microstructure,” J. Mod. Opt. 43, 1035–1053 (1996).
[CrossRef]

C. M. de Sterke, D. G. Salinas, and J. E. Sipe, “Coupled-mode theory for light propagation through deep nonlinear gratings,” Phys. Rev. E 54, 1969–1989 (1996).
[CrossRef]

1994 (2)

W.-P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11, 963–983 (1994).
[CrossRef]

O. Mitomi, K. Kasaya, and H. Miyazawa, “Design of a single-mode tapered waveguide for low-loss chip-to-fiber coupling,” IEEE J. Quantum Electron. 30, 1787–1793 (1994).
[CrossRef]

1992 (1)

T. A. Birks and Y. W. Li, “The shape of fiber tapers,” J. Lightwave Technol. 10, 432–438 (1992).
[CrossRef]

1991 (1)

Y. Yamamoto and S. Machida, “Microcavity semiconductor laser with enhanced spontaneous emission,” Phys. Rev. A 44, 657–668 (1991).
[CrossRef] [PubMed]

1990 (1)

S. John and K. Busch, “Quantum electrodynamics near a photonic band gap: photon bound states and dressed atoms,” Phys. Rev. Lett. 64, 2418–2421 (1990).
[CrossRef] [PubMed]

1989 (2)

H. A. Haus, W. P. Huang, and A. W. Snyder, “Coupled-mode formulations,” Opt. Lett. 14, 1222–1224 (1989).
[CrossRef] [PubMed]

W. P. Huang and H. A. Haus, “Power exchange in grating-assisted couplers,” J. Lightwave Technol. 7, 920–924 (1989).
[CrossRef]

1987 (1)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

1980 (1)

Abram, I.

E. Moreau, I. Robert, J. M. Gerard, I. Abram, L. Manin, and V. Thierry-Mieg, “Single-mode solid-state single photon source based on isolated quantum dots in pillar microcavities,” Appl. Phys. Lett. 79, 2865–2867 (2001).
[CrossRef]

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 (London) 421, 925–928 (2003).
[CrossRef]

Atkin, D. M.

D. M. Atkin, P. S. J. Russell, T. A. Birks, and P. J. Roberts, “Photonic band structure of guided Bloch modes in high index films fully etched through with periodic microstructure,” J. Mod. Opt. 43, 1035–1053 (1996).
[CrossRef]

Baets, R.

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. V. Daele, I. Moerman, S. Vertuyft, K. D. Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. 38, 949–955 (2002).
[CrossRef]

Barclay, P. E.

P. E. Barclay, K. Srinivasan, M. Borselli, and O. Painter, “Experimental demonstration of evanescent coupling from optical fibre tapers to photonic crystal waveguides,” IEE Electron. Lett. 39, 842–844 (2003).
[CrossRef]

O. Painter, K. Srinivasan, and P. E. Barclay, “Wannier-like equation for the resonant cavity modes of locally perturbed photonic crystals,” Phys. Rev. B 68, 035214 (2003).
[CrossRef]

Becher, C.

P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. I. Glu, “A quantum dot single-photon turnstile device,” Science 290, 2282–2285 (2000).
[CrossRef] [PubMed]

Benisty, H.

Bhat, N. A. R.

N. A. R. Bhat and J. E. Sipe, “Optical pulse propagation in nonlinear photonic crystals,” Phys. Rev. E 64, 056604 (2001).
[CrossRef]

Bienstman, P.

S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “The adiabatic theorem and a continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066608 (2002).
[CrossRef]

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. V. Daele, I. Moerman, S. Vertuyft, K. D. Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. 38, 949–955 (2002).
[CrossRef]

Birks, T. A.

D. M. Atkin, P. S. J. Russell, T. A. Birks, and P. J. Roberts, “Photonic band structure of guided Bloch modes in high index films fully etched through with periodic microstructure,” J. Mod. Opt. 43, 1035–1053 (1996).
[CrossRef]

T. A. Birks and Y. W. Li, “The shape of fiber tapers,” J. Lightwave Technol. 10, 432–438 (1992).
[CrossRef]

Bogaerts, W.

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. V. Daele, I. Moerman, S. Vertuyft, K. D. Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. 38, 949–955 (2002).
[CrossRef]

Borselli, M.

P. E. Barclay, K. Srinivasan, M. Borselli, and O. Painter, “Experimental demonstration of evanescent coupling from optical fibre tapers to photonic crystal waveguides,” IEE Electron. Lett. 39, 842–844 (2003).
[CrossRef]

Busch, A.

M. Kanskar, P. Paddon, V. Pacradoui, R. Morin, A. Busch, J. F. Young, S. R. Johnson, J. Mackenzie, and T. Tiedje, “Observation of leaky slab modes in an air-bridged semiconductor waveguide with two-dimensional photonic lattice,” Appl. Phys. Lett. 70, 1438–1440 (1997).
[CrossRef]

Busch, K.

S. John and K. Busch, “Quantum electrodynamics near a photonic band gap: photon bound states and dressed atoms,” Phys. Rev. Lett. 64, 2418–2421 (1990).
[CrossRef] [PubMed]

Cai, M.

M. Cai, O. Painter, and K. Vahala, “Observation of critical coupling in a fiber taper to a silica-microsphere whispering-gallery mode system,” Phys. Rev. Lett. 85, 74–77 (2000).
[CrossRef] [PubMed]

Chutinan, A.

S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature (London) 407, 608–610 (2000).
[CrossRef]

Daele, P. V.

D. Taillaert, W. Bogaerts, P. Bienstman, T. F. Krauss, P. V. Daele, I. Moerman, S. Vertuyft, K. D. Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” IEEE J. Quantum Electron. 38, 949–955 (2002).
[CrossRef]

Dapkus, P. D.

O. Painter, R. K. Lee, A. Yariv, A. Scherer, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1824 (1999).
[CrossRef] [PubMed]

de Sterke, C. M.

C. M. de Sterke, D. G. Salinas, and J. E. Sipe, “Coupled-mode theory for light propagation through deep nonlinear gratings,” Phys. Rev. E 54, 1969–1989 (1996).
[CrossRef]

Doll, T.

M. Lončar, D. Nedeljković, T. Doll, J. Vučković, A. Scherer, and T. P. Pearsall, “Waveguiding in planar photonic crystals,” Appl. Phys. Lett. 77, 1937–1939 (2000).
[CrossRef]

Fan, S.

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B 62, 8212–8222 (2000).
[CrossRef]

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejaki, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751–5758 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “Theoretical analysis of channel drop tunneling processes,” Phys. Rev. B 59, 15882–15892 (1999).
[CrossRef]

Gerard, J. M.

E. Moreau, I. Robert, J. M. Gerard, I. Abram, L. Manin, and V. Thierry-Mieg, “Single-mode solid-state single photon source based on isolated quantum dots in pillar microcavities,” Appl. Phys. Lett. 79, 2865–2867 (2001).
[CrossRef]

Gibbs, H. M.

G. Khitrova, H. M. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, “Nonlinear optics and normal-mode-coupling in semiconductor microcavities,” Rev. Mod. Phys. 71, 1591–1639 (1999).
[CrossRef]

Glu, A. I.

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P. Michler, A. Kiraz, C. Becher, W. V. Schoenfeld, P. M. Petroff, L. Zhang, E. Hu, and A. I. Glu, “A quantum dot single-photon turnstile device,” Science 290, 2282–2285 (2000).
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S. Mookherjea and A. Yariv, “Second-harmonic generation with pulses in a coupled-resonator optical waveguide,” Phys. Rev. E 65, 026607 (2002).
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In addition to the weak coupling between this mode and the fiber taper, higher-order odd slab modes may make coupling in this region of k space impractical. Nonetheless, the calculations shown here demonstrate the importance of a mode’s transverse Fourier components.

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Figures (13)

Fig. 1
Fig. 1

(a) Schematic of the coupling scheme showing the four-mode basis used in the coupled-mode theory. (b) Coupling geometry. In the case considered here the coupling is contradirectional. WG, waveguide. (c) Grading of the hole radius used to form the waveguide and a top view of the graded-defect compressed-lattice (Λx/Λz=0.8) waveguide unit cell.

Fig. 2
Fig. 2

Approximate band structure of fundamental even (TE-like) modes for a square-lattice PC of air holes with radius r/Λ=0.35 in a slab of thickness d=0.75Λ and dielectric constant =11.56, calculated with an effective index of nTEeff=2.64 that corresponds to the propagation constant of the fundamental TE mode of the untextured slab. The inset shows the first Brillouin zone of a rectangular lattice.

Fig. 3
Fig. 3

Projection of the square-lattice band structure onto the first Brillouin zone of a line defect with the same periodicity of the lattice and oriented in the X1Γ direction. Band edges whose modes have dominant wave numbers in the X1Γ direction (i.e., k=kzˆ) are shown by the dashed curves. Band edges whose modes have dominant wave numbers in the X2M direction (i.e., k=kzzˆ+π/Λxxˆ) are shown by the dot dashed curves.

Fig. 4
Fig. 4

Approximate projected band structure for a (a) donor-type and (b) acceptor-type compressed square-lattice waveguide. Possible defect modes and the fundamental fiber taper mode are indicated by the dashed curves.

Fig. 5
Fig. 5

3-D FDTD calculated band structure for the waveguide shown in Fig. 1(c). The shaded regions indicate continuums of unbound modes. The dashed lines are the dispersion of fiber tapers with radius r=0.8Λz=1Λx (upper dashed line) and r=1.5Λz=1.875Λx (lower dashed line). The solid lines are the air (upper line) and fiber (lower line) light lines. The energies and wave numbers of modes A and B are ω˜Λz/2π=0.304 and 0.373 at βΛz/2π=0.350 and 0.438, respectively.

Fig. 6
Fig. 6

Mode A field profiles calculated by FDTD. Dominant magnetic field component (a) |By(x, y=0, z)|, (b) |By(x, y,z=0)|. (c) Dominant electric field component transverse Fourier transform |E˜x(kx, y=0, z)|. Note that the dominant transverse Fourier components are near kx=0.

Fig. 7
Fig. 7

Mode B field profiles calculated by FDTD. Dominant magnetic field component (a) |By(x, y=0, z)|, (b) |By(x, y,z=0)|. (c) Dominant electric field component transverse Fourier transform |E˜x(kx, y=0, z)|. Note that the dominant transverse Fourier components are near kx=±π/Λx.

Fig. 8
Fig. 8

(a) Power coupled to PC mode A from a tapered fiber with radius r=1.15Λx placed with a d=Λx gap above the PC as a function of detuning from phase matching and coupler length. (b) Power coupled at ω=ω0 to the forward- and backward-propagating PC and fiber modes as a function of coupler length.

Fig. 9
Fig. 9

Power coupled to PC mode B from a tapered fiber with radius r=1.55Λx placed with a d=Λx gap above the PC as a function of detuning from phase matching and coupler length.

Fig. 10
Fig. 10

(a) FDTD-calculated band structure of the full fiber–PC system. The fiber taper has a radius r=1.17Λz=1.46Λx placed at d=Λz=1.25Λx above the PC waveguide. (b) Band structure of the boxed region in (a). Mode A and fiberlike dispersion are identified, and the symmetric and antisymmetric superpositions of these modes at the anticrossing are labeled by the plus and minus signs. (c) The By(x, y, 0) component of the lower-frequency anticrossing edge supermode (symmetric). (d) The By(x, y, 0) component of the higher-frequency anticrossing edge supermode (antisymmetric).

Fig. 11
Fig. 11

High-Q defect cavity mode of Ref. 43. Plots of the magnetic field pattern are shown in (a) the xz plane [|By(x,y=0, z)|] and (b) the xy plane [|By(x, y, z=0)|]. In (c) the Fourier transform of the dominant electric field component is plotted [|E˜x(kx, 0, kz)|].

Fig. 12
Fig. 12

(a) Waveguide to high-Q cavity coupling scheme. (b) Schematic mode frequency diagram. The dotted line represents the cavity mode frequency. The solid black curve represents the local laterally guided-mode frequency and is a function of the filling fraction and lattice compression. The shaded region represents the finite bandwidth of the waveguide mode, which is a function of its dispersion (a flatter band has a smaller bandwidth and vice versa).

Fig. 13
Fig. 13

Coupling from the defect cavity to the PC waveguide for varying waveguide lattice compressions at instances in time when the cavity magnetic field is a minimum (left) and a maximum (right). The envelope modulating the waveguide field is a standing wave caused by interference with reflections from the boundary of the computational domain. The diagonal radiation pattern of the cavity is due to coupling to the square-lattice M points and is sufficiently small to ensure a cavity Q of 105. |B|: (a) ΛxWG/ΛzWG=20/20, (b) ΛxWG/ΛzWG=20/25 (ratio used in Section 4), (c) ΛxWG/ΛzWG=20/29.

Equations (14)

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××Eνμ(r)=ω˜2μ(r)Eνμ(r),
E(r)=μνCνμ(z)Eνμ(r),
Hβνeβνμ(r)=ω˜2μ(r)eβνμ(r),
Hβν=[-βν2zˆ×zˆ+iβν(zˆ×+×zˆ)+×]×,
zz(E1×H2*+E2*×H1)zˆdxdy
=iω˜zE1E2*(1-2*)dxdy,
E1=jCj(z)Ej(r),
E2=Ei,
1=,
2=i,
PijdCjdz=iω˜KijCj,
Pij(z)=z(Ei*×Hj+Ej×Hi*)zˆdxdy,
Kij(z)=zEi*Ej(-j)dxdy,
B=yˆ[cos(kX1 r)]Γ(r),

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