Abstract

We demonstrate the spectral and spatial reconfigurability of photonic crystal double-heterostructure cavities in silicon by microfluidic infiltration of selected air holes. The lengths of the microfluidic cavities are changed by adjusting the region of infiltrated holes in steps of several microns. We systematically investigate the spectral signature of these cavities, showing high Q-factor resonances for a broad range of cavity lengths. The fluid can be removed by immersing the device in toluene, offering complete reconfigurability. Our cavity writing technique allows for tolerances in the infiltration process and provides flexibility as it can be employed at any time after photonic crystal fabrication.

© 2008 Optical Society of America

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2008 (6)

2007 (4)

F. B. Arango, M. B. Christiansen, M. Gersborg-Hansen, and A. Kristensen, "Optofluidic tuning of photonic crystal band edge lasers," Appl. Phys. Lett. 91, 223503 (2007).
[CrossRef]

C. Monat, P. Domachuk, and B. J. Eggleton, "Integrated optofluidics: A new river of light," Nat. Photon. 1, 106-114 (2007).
[CrossRef]

S. S. Xiao and N. A. Mortensen, "Proposal of highly sensitive optofluidic sensors based on dispersive photonic crystal waveguides," J. Opt. A: Pure Appl. Opt. 9, S463-S467 (2007).
[CrossRef]

C. L. C. Smith, D. K. C. Wu, M. W. Lee, C. Monat, S. Tomljenovic-Hanic, C. Grillet, B. J. Eggleton, D. Freeman, Y. Ruan, S. Madden, B. Luther-Davies, H. Giessen, and Y. H. Lee, "Microfluidic photonic crystal double heterostructures," Appl. Phys. Lett. 91, 121103 (2007).
[CrossRef]

2006 (6)

F. Intonti, S. Vignolini, V. Turck, M. Colocci, P. Bettotti, L. Pavesi, S. L. Schweizer, R. Wehrspohn, and D. Wiersma, "Rewritable photonic circuits," Appl. Phys. Lett. 89, 211117 (2006).
[CrossRef]

S. Balslev, A. M. Jorgensen, B. Bilenberg, K. B. Mogensen, D. Snakenborg, O. Geschke, J. P. Kutter, and A. Kristensen, "Lab-on-a-chip with integrated optical transducers," Lab on a Chip 6, 213-217 (2006).
[CrossRef] [PubMed]

D. Psaltis, S. R. Quake, and C. H. Yang, "Developing optofluidic technology through the fusion of microfluidics and optics," Nature 442, 381-386 (2006).
[CrossRef] [PubMed]

D. Erickson, T. Rockwood, T. Emery, A. Scherer, and D. Psaltis, "Nanofluidic tuning of photonic crystal circuits," Opt. Lett. 31, 59-61 (2006).
[CrossRef] [PubMed]

C. Grillet, C. Smith, D. Freeman, S. Madden, B. Luther-Davis, E. C. Mägi, D. J. Moss, and B. J. Eggleton, "Efficient coupling to chalcogenide glass photonic crystal waveguides via silica optical fiber nanowires," Opt. Express 14, 1070-1078 (2006).
[CrossRef] [PubMed]

S. Tomljenovic-Hanic, C. M. de Sterke, and M. J. Steel, "Design of high-Q cavities in photonic crystal slab heterostructures by air-holes infiltration," Opt. Express 14, 12451-12456 (2006).
[CrossRef] [PubMed]

2005 (4)

I. K. Hwang, S. K. Kim, J. K. Yang, S. H. Kim, S. H. Lee, and Y. H. Lee, "Curved-microfiber photon coupling for photonic crystal light emitter," Appl. Phys. Lett. 87, 131107 (2005).
[CrossRef]

M. L. Adams, M. Loncar, A. Scherer, and Y. M. Qiu, "Microfluidic integration of porous photonic crystal nanolasers for chemical sensing," IEEE J. Sel. Areas Commun. 23, 1348-1354 (2005).
[CrossRef]

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

J. C. Galas, J. Torres, M. Belotti, Q. Kou, and Y. Chen, "Microfluidic tunable dye laser with integrated mixer and ring resonator," Appl. Phys. Lett. 86, 264101 (2005).
[CrossRef]

2004 (3)

B. Maune, M. Loncar, J. Witzens, M. Hochberg, T. Baehr-Jones, D. Psaltis, A. Scherer, and Y. M. Qiu, "Liquid-crystal electric tuning of a photonic crystal laser," Appl. Phys. Lett. 85, 360-362 (2004).
[CrossRef]

P. E. Barclay, K. Srinivasan, M. Borselli, and O. Painter, "Probing the dispersive and spatial properties of photonic crystal waveguides via highly efficient coupling from fiber tapers," Appl. Phys. Lett. 85, 4-6 (2004).
[CrossRef]

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, 1093-1095 (2004).
[CrossRef] [PubMed]

2003 (6)

K. Srinivasan and O. Painter, "Fourier space design of high-Q cavities in standard and compressed hexagonal lattice photonic crystals," Opt. Express 11, 579-593 (2003).
[CrossRef] [PubMed]

C. Schuller, F. Klopf, J. P. Reithmaier, M. Kamp, and A. Forchel, "Tunable photonic crystals fabricated in III-V semiconductor slab waveguides using infiltrated liquid crystals," Appl. Phys. Lett. 82, 2767-2769 (2003).
[CrossRef]

T. F. Krauss, "Photonic crystals - Cavities without leaks," Nat. Mater. 2, 777-778 (2003).
[CrossRef] [PubMed]

K. J. Vahala, "Optical microcavities," Nature 424, 839-846 (2003).
[CrossRef] [PubMed]

B. S. Song, S. Noda, and T. Asano, "Photonic devices based on in-plane hetero photonic crystals," Science 300, 1537-1537 (2003).

M. Loncar, A. Scherer, and Y. M. Qiu, "Photonic crystal laser sources for chemical detection," Appl. Phys. Lett. 82, 4648-4650 (2003).
[CrossRef]

2002 (1)

P. Mach, M. Dolinski, K. W. Baldwin, J. A. Rogers, C. Kerbage, R. S. Windeler, and B. J. Eggleton, "Tunable microfluidic optical fiber," Appl. Phys. Lett. 80, 4294-4296 (2002).
[CrossRef]

2001 (2)

X. Letartre, C. Seassal, C. Grillet, P. Rojo-Romeo, P. Viktorovitch, M. L. d'Yerville, D. Cassagne, and C. Jouanin, "Group velocity and propagation losses measurement in a single-line photonic-crystal waveguide on InP membranes," Appl. Phys. Lett. 79, 2312-2314 (2001).
[CrossRef]

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, "Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs," Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

2000 (1)

E. Centeno, and D. Felbacq, "Optical bistability in finite-size nonlinear bidimensional photonic crystals doped by a microcavity," Phys. Rev. B 62, R7683-R7686 (2000).
[CrossRef]

1999 (1)

K. Busch and S. John, "Liquid-crystal photonic-band-gap materials: The tunable electromagnetic vacuum," Phys. Rev. Lett. 83, 967-970 (1999).
[CrossRef]

1997 (1)

Adams, M. L.

M. L. Adams, M. Loncar, A. Scherer, and Y. M. Qiu, "Microfluidic integration of porous photonic crystal nanolasers for chemical sensing," IEEE J. Sel. Areas Commun. 23, 1348-1354 (2005).
[CrossRef]

Akahane, Y.

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

Arango, F. B.

F. B. Arango, M. B. Christiansen, M. Gersborg-Hansen, and A. Kristensen, "Optofluidic tuning of photonic crystal band edge lasers," Appl. Phys. Lett. 91, 223503 (2007).
[CrossRef]

Asano, T.

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

B. S. Song, S. Noda, and T. Asano, "Photonic devices based on in-plane hetero photonic crystals," Science 300, 1537-1537 (2003).

Baba, T.

Baehr-Jones, T.

B. Maune, M. Loncar, J. Witzens, M. Hochberg, T. Baehr-Jones, D. Psaltis, A. Scherer, and Y. M. Qiu, "Liquid-crystal electric tuning of a photonic crystal laser," Appl. Phys. Lett. 85, 360-362 (2004).
[CrossRef]

Baldwin, K. W.

P. Mach, M. Dolinski, K. W. Baldwin, J. A. Rogers, C. Kerbage, R. S. Windeler, and B. J. Eggleton, "Tunable microfluidic optical fiber," Appl. Phys. Lett. 80, 4294-4296 (2002).
[CrossRef]

Balet, L.

Balog, S.

Balslev, S.

S. Balslev, A. M. Jorgensen, B. Bilenberg, K. B. Mogensen, D. Snakenborg, O. Geschke, J. P. Kutter, and A. Kristensen, "Lab-on-a-chip with integrated optical transducers," Lab on a Chip 6, 213-217 (2006).
[CrossRef] [PubMed]

Barclay, P. E.

P. E. Barclay, K. Srinivasan, M. Borselli, and O. Painter, "Probing the dispersive and spatial properties of photonic crystal waveguides via highly efficient coupling from fiber tapers," Appl. Phys. Lett. 85, 4-6 (2004).
[CrossRef]

Belotti, M.

J. C. Galas, J. Torres, M. Belotti, Q. Kou, and Y. Chen, "Microfluidic tunable dye laser with integrated mixer and ring resonator," Appl. Phys. Lett. 86, 264101 (2005).
[CrossRef]

Bettotti, P.

F. Intonti, S. Vignolini, V. Turck, M. Colocci, P. Bettotti, L. Pavesi, S. L. Schweizer, R. Wehrspohn, and D. Wiersma, "Rewritable photonic circuits," Appl. Phys. Lett. 89, 211117 (2006).
[CrossRef]

Bilenberg, B.

S. Balslev, A. M. Jorgensen, B. Bilenberg, K. B. Mogensen, D. Snakenborg, O. Geschke, J. P. Kutter, and A. Kristensen, "Lab-on-a-chip with integrated optical transducers," Lab on a Chip 6, 213-217 (2006).
[CrossRef] [PubMed]

Birks, T. A.

Bog, U.

U. Bog, C. L. C. Smith, M. W. Lee, S. Tomljenovic-Hanic, C. Grillet, C. Monat, L. O'Faolain, C. Karnutsch, T. F. Krauss, R. McPhedran, and B. J. Eggleton, "High-Q microfluidic cavities in silicon-based 2D photonic crystal structures," Opt. Lett. 33 (2008).

Borselli, M.

P. E. Barclay, K. Srinivasan, M. Borselli, and O. Painter, "Probing the dispersive and spatial properties of photonic crystal waveguides via highly efficient coupling from fiber tapers," Appl. Phys. Lett. 85, 4-6 (2004).
[CrossRef]

Busch, K.

K. Busch and S. John, "Liquid-crystal photonic-band-gap materials: The tunable electromagnetic vacuum," Phys. Rev. Lett. 83, 967-970 (1999).
[CrossRef]

Campbell, K.

Cassagne, D.

X. Letartre, C. Seassal, C. Grillet, P. Rojo-Romeo, P. Viktorovitch, M. L. d'Yerville, D. Cassagne, and C. Jouanin, "Group velocity and propagation losses measurement in a single-line photonic-crystal waveguide on InP membranes," Appl. Phys. Lett. 79, 2312-2314 (2001).
[CrossRef]

Centeno, E.

E. Centeno, and D. Felbacq, "Optical bistability in finite-size nonlinear bidimensional photonic crystals doped by a microcavity," Phys. Rev. B 62, R7683-R7686 (2000).
[CrossRef]

Chen, Y.

J. C. Galas, J. Torres, M. Belotti, Q. Kou, and Y. Chen, "Microfluidic tunable dye laser with integrated mixer and ring resonator," Appl. Phys. Lett. 86, 264101 (2005).
[CrossRef]

Cheung, G.

Choi, J. H.

Chow, E.

Christiansen, M. B.

F. B. Arango, M. B. Christiansen, M. Gersborg-Hansen, and A. Kristensen, "Optofluidic tuning of photonic crystal band edge lasers," Appl. Phys. Lett. 91, 223503 (2007).
[CrossRef]

Colocci, M.

F. Intonti, S. Vignolini, V. Turck, M. Colocci, P. Bettotti, L. Pavesi, S. L. Schweizer, R. Wehrspohn, and D. Wiersma, "Rewritable photonic circuits," Appl. Phys. Lett. 89, 211117 (2006).
[CrossRef]

de Sterke, C. M.

Dolinski, M.

P. Mach, M. Dolinski, K. W. Baldwin, J. A. Rogers, C. Kerbage, R. S. Windeler, and B. J. Eggleton, "Tunable microfluidic optical fiber," Appl. Phys. Lett. 80, 4294-4296 (2002).
[CrossRef]

Domachuk, P.

C. Monat, P. Domachuk, and B. J. Eggleton, "Integrated optofluidics: A new river of light," Nat. Photon. 1, 106-114 (2007).
[CrossRef]

d'Yerville, M. L.

X. Letartre, C. Seassal, C. Grillet, P. Rojo-Romeo, P. Viktorovitch, M. L. d'Yerville, D. Cassagne, and C. Jouanin, "Group velocity and propagation losses measurement in a single-line photonic-crystal waveguide on InP membranes," Appl. Phys. Lett. 79, 2312-2314 (2001).
[CrossRef]

Eggleton, B. J.

U. Bog, C. L. C. Smith, M. W. Lee, S. Tomljenovic-Hanic, C. Grillet, C. Monat, L. O'Faolain, C. Karnutsch, T. F. Krauss, R. McPhedran, and B. J. Eggleton, "High-Q microfluidic cavities in silicon-based 2D photonic crystal structures," Opt. Lett. 33 (2008).

C. L. C. Smith, D. K. C. Wu, M. W. Lee, C. Monat, S. Tomljenovic-Hanic, C. Grillet, B. J. Eggleton, D. Freeman, Y. Ruan, S. Madden, B. Luther-Davies, H. Giessen, and Y. H. Lee, "Microfluidic photonic crystal double heterostructures," Appl. Phys. Lett. 91, 121103 (2007).
[CrossRef]

C. Monat, P. Domachuk, and B. J. Eggleton, "Integrated optofluidics: A new river of light," Nat. Photon. 1, 106-114 (2007).
[CrossRef]

C. Grillet, C. Smith, D. Freeman, S. Madden, B. Luther-Davis, E. C. Mägi, D. J. Moss, and B. J. Eggleton, "Efficient coupling to chalcogenide glass photonic crystal waveguides via silica optical fiber nanowires," Opt. Express 14, 1070-1078 (2006).
[CrossRef] [PubMed]

P. Mach, M. Dolinski, K. W. Baldwin, J. A. Rogers, C. Kerbage, R. S. Windeler, and B. J. Eggleton, "Tunable microfluidic optical fiber," Appl. Phys. Lett. 80, 4294-4296 (2002).
[CrossRef]

El-Kallassi, P.

Emery, T.

Erickson, D.

Fainman, Y.

Felbacq, D.

E. Centeno, and D. Felbacq, "Optical bistability in finite-size nonlinear bidimensional photonic crystals doped by a microcavity," Phys. Rev. B 62, R7683-R7686 (2000).
[CrossRef]

Ferrini, R.

Fiore, A.

Forchel, A.

C. Schuller, F. Klopf, J. P. Reithmaier, M. Kamp, and A. Forchel, "Tunable photonic crystals fabricated in III-V semiconductor slab waveguides using infiltrated liquid crystals," Appl. Phys. Lett. 82, 2767-2769 (2003).
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C. Grillet, C. Smith, D. Freeman, S. Madden, B. Luther-Davis, E. C. Mägi, D. J. Moss, and B. J. Eggleton, "Efficient coupling to chalcogenide glass photonic crystal waveguides via silica optical fiber nanowires," Opt. Express 14, 1070-1078 (2006).
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J. C. Galas, J. Torres, M. Belotti, Q. Kou, and Y. Chen, "Microfluidic tunable dye laser with integrated mixer and ring resonator," Appl. Phys. Lett. 86, 264101 (2005).
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Gersborg-Hansen, M.

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S. Balslev, A. M. Jorgensen, B. Bilenberg, K. B. Mogensen, D. Snakenborg, O. Geschke, J. P. Kutter, and A. Kristensen, "Lab-on-a-chip with integrated optical transducers," Lab on a Chip 6, 213-217 (2006).
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C. L. C. Smith, D. K. C. Wu, M. W. Lee, C. Monat, S. Tomljenovic-Hanic, C. Grillet, B. J. Eggleton, D. Freeman, Y. Ruan, S. Madden, B. Luther-Davies, H. Giessen, and Y. H. Lee, "Microfluidic photonic crystal double heterostructures," Appl. Phys. Lett. 91, 121103 (2007).
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C. L. C. Smith, D. K. C. Wu, M. W. Lee, C. Monat, S. Tomljenovic-Hanic, C. Grillet, B. J. Eggleton, D. Freeman, Y. Ruan, S. Madden, B. Luther-Davies, H. Giessen, and Y. H. Lee, "Microfluidic photonic crystal double heterostructures," Appl. Phys. Lett. 91, 121103 (2007).
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C. Grillet, C. Smith, D. Freeman, S. Madden, B. Luther-Davis, E. C. Mägi, D. J. Moss, and B. J. Eggleton, "Efficient coupling to chalcogenide glass photonic crystal waveguides via silica optical fiber nanowires," Opt. Express 14, 1070-1078 (2006).
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X. Letartre, C. Seassal, C. Grillet, P. Rojo-Romeo, P. Viktorovitch, M. L. d'Yerville, D. Cassagne, and C. Jouanin, "Group velocity and propagation losses measurement in a single-line photonic-crystal waveguide on InP membranes," Appl. Phys. Lett. 79, 2312-2314 (2001).
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X. Letartre, C. Seassal, C. Grillet, P. Rojo-Romeo, P. Viktorovitch, M. L. d'Yerville, D. Cassagne, and C. Jouanin, "Group velocity and propagation losses measurement in a single-line photonic-crystal waveguide on InP membranes," Appl. Phys. Lett. 79, 2312-2314 (2001).
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C. Schuller, F. Klopf, J. P. Reithmaier, M. Kamp, and A. Forchel, "Tunable photonic crystals fabricated in III-V semiconductor slab waveguides using infiltrated liquid crystals," Appl. Phys. Lett. 82, 2767-2769 (2003).
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U. Bog, C. L. C. Smith, M. W. Lee, S. Tomljenovic-Hanic, C. Grillet, C. Monat, L. O'Faolain, C. Karnutsch, T. F. Krauss, R. McPhedran, and B. J. Eggleton, "High-Q microfluidic cavities in silicon-based 2D photonic crystal structures," Opt. Lett. 33 (2008).

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P. Mach, M. Dolinski, K. W. Baldwin, J. A. Rogers, C. Kerbage, R. S. Windeler, and B. J. Eggleton, "Tunable microfluidic optical fiber," Appl. Phys. Lett. 80, 4294-4296 (2002).
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Kim, S. K.

I. K. Hwang, S. K. Kim, J. K. Yang, S. H. Kim, S. H. Lee, and Y. H. Lee, "Curved-microfiber photon coupling for photonic crystal light emitter," Appl. Phys. Lett. 87, 131107 (2005).
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Klopf, F.

C. Schuller, F. Klopf, J. P. Reithmaier, M. Kamp, and A. Forchel, "Tunable photonic crystals fabricated in III-V semiconductor slab waveguides using infiltrated liquid crystals," Appl. Phys. Lett. 82, 2767-2769 (2003).
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Kou, Q.

J. C. Galas, J. Torres, M. Belotti, Q. Kou, and Y. Chen, "Microfluidic tunable dye laser with integrated mixer and ring resonator," Appl. Phys. Lett. 86, 264101 (2005).
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J. Li, T. P. White, L. O'Faolain, A. Gomez-Iglesias, and T. F. Krauss, "Systematic design of flat band slow light in photonic crystal waveguides," Opt. Express 16, 6227-6232 (2008).
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F. B. Arango, M. B. Christiansen, M. Gersborg-Hansen, and A. Kristensen, "Optofluidic tuning of photonic crystal band edge lasers," Appl. Phys. Lett. 91, 223503 (2007).
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S. Balslev, A. M. Jorgensen, B. Bilenberg, K. B. Mogensen, D. Snakenborg, O. Geschke, J. P. Kutter, and A. Kristensen, "Lab-on-a-chip with integrated optical transducers," Lab on a Chip 6, 213-217 (2006).
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S. Balslev, A. M. Jorgensen, B. Bilenberg, K. B. Mogensen, D. Snakenborg, O. Geschke, J. P. Kutter, and A. Kristensen, "Lab-on-a-chip with integrated optical transducers," Lab on a Chip 6, 213-217 (2006).
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U. Bog, C. L. C. Smith, M. W. Lee, S. Tomljenovic-Hanic, C. Grillet, C. Monat, L. O'Faolain, C. Karnutsch, T. F. Krauss, R. McPhedran, and B. J. Eggleton, "High-Q microfluidic cavities in silicon-based 2D photonic crystal structures," Opt. Lett. 33 (2008).

C. L. C. Smith, D. K. C. Wu, M. W. Lee, C. Monat, S. Tomljenovic-Hanic, C. Grillet, B. J. Eggleton, D. Freeman, Y. Ruan, S. Madden, B. Luther-Davies, H. Giessen, and Y. H. Lee, "Microfluidic photonic crystal double heterostructures," Appl. Phys. Lett. 91, 121103 (2007).
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I. K. Hwang, S. K. Kim, J. K. Yang, S. H. Kim, S. H. Lee, and Y. H. Lee, "Curved-microfiber photon coupling for photonic crystal light emitter," Appl. Phys. Lett. 87, 131107 (2005).
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Lee, Y. H.

S. H. Kim, J. H. Choi, S. K. Lee, S. H. Kim, S. M. Yang, Y. H. Lee, C. Seassal, P. Regrency, and P. Viktorovitch, "Optofluidic integration of a photonic crystal nanolaser," Opt. Express 16, 6515-6527 (2008).
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X. Letartre, C. Seassal, C. Grillet, P. Rojo-Romeo, P. Viktorovitch, M. L. d'Yerville, D. Cassagne, and C. Jouanin, "Group velocity and propagation losses measurement in a single-line photonic-crystal waveguide on InP membranes," Appl. Phys. Lett. 79, 2312-2314 (2001).
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Li, J.

Li, L.

Loncar, M.

M. L. Adams, M. Loncar, A. Scherer, and Y. M. Qiu, "Microfluidic integration of porous photonic crystal nanolasers for chemical sensing," IEEE J. Sel. Areas Commun. 23, 1348-1354 (2005).
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B. Maune, M. Loncar, J. Witzens, M. Hochberg, T. Baehr-Jones, D. Psaltis, A. Scherer, and Y. M. Qiu, "Liquid-crystal electric tuning of a photonic crystal laser," Appl. Phys. Lett. 85, 360-362 (2004).
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M. Loncar, A. Scherer, and Y. M. Qiu, "Photonic crystal laser sources for chemical detection," Appl. Phys. Lett. 82, 4648-4650 (2003).
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C. L. C. Smith, D. K. C. Wu, M. W. Lee, C. Monat, S. Tomljenovic-Hanic, C. Grillet, B. J. Eggleton, D. Freeman, Y. Ruan, S. Madden, B. Luther-Davies, H. Giessen, and Y. H. Lee, "Microfluidic photonic crystal double heterostructures," Appl. Phys. Lett. 91, 121103 (2007).
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Mach, P.

P. Mach, M. Dolinski, K. W. Baldwin, J. A. Rogers, C. Kerbage, R. S. Windeler, and B. J. Eggleton, "Tunable microfluidic optical fiber," Appl. Phys. Lett. 80, 4294-4296 (2002).
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C. L. C. Smith, D. K. C. Wu, M. W. Lee, C. Monat, S. Tomljenovic-Hanic, C. Grillet, B. J. Eggleton, D. Freeman, Y. Ruan, S. Madden, B. Luther-Davies, H. Giessen, and Y. H. Lee, "Microfluidic photonic crystal double heterostructures," Appl. Phys. Lett. 91, 121103 (2007).
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C. Grillet, C. Smith, D. Freeman, S. Madden, B. Luther-Davis, E. C. Mägi, D. J. Moss, and B. J. Eggleton, "Efficient coupling to chalcogenide glass photonic crystal waveguides via silica optical fiber nanowires," Opt. Express 14, 1070-1078 (2006).
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Maune, B.

B. Maune, M. Loncar, J. Witzens, M. Hochberg, T. Baehr-Jones, D. Psaltis, A. Scherer, and Y. M. Qiu, "Liquid-crystal electric tuning of a photonic crystal laser," Appl. Phys. Lett. 85, 360-362 (2004).
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U. Bog, C. L. C. Smith, M. W. Lee, S. Tomljenovic-Hanic, C. Grillet, C. Monat, L. O'Faolain, C. Karnutsch, T. F. Krauss, R. McPhedran, and B. J. Eggleton, "High-Q microfluidic cavities in silicon-based 2D photonic crystal structures," Opt. Lett. 33 (2008).

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Mogensen, K. B.

S. Balslev, A. M. Jorgensen, B. Bilenberg, K. B. Mogensen, D. Snakenborg, O. Geschke, J. P. Kutter, and A. Kristensen, "Lab-on-a-chip with integrated optical transducers," Lab on a Chip 6, 213-217 (2006).
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U. Bog, C. L. C. Smith, M. W. Lee, S. Tomljenovic-Hanic, C. Grillet, C. Monat, L. O'Faolain, C. Karnutsch, T. F. Krauss, R. McPhedran, and B. J. Eggleton, "High-Q microfluidic cavities in silicon-based 2D photonic crystal structures," Opt. Lett. 33 (2008).

C. L. C. Smith, D. K. C. Wu, M. W. Lee, C. Monat, S. Tomljenovic-Hanic, C. Grillet, B. J. Eggleton, D. Freeman, Y. Ruan, S. Madden, B. Luther-Davies, H. Giessen, and Y. H. Lee, "Microfluidic photonic crystal double heterostructures," Appl. Phys. Lett. 91, 121103 (2007).
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O'Faolain, L.

J. Li, T. P. White, L. O'Faolain, A. Gomez-Iglesias, and T. F. Krauss, "Systematic design of flat band slow light in photonic crystal waveguides," Opt. Express 16, 6227-6232 (2008).
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U. Bog, C. L. C. Smith, M. W. Lee, S. Tomljenovic-Hanic, C. Grillet, C. Monat, L. O'Faolain, C. Karnutsch, T. F. Krauss, R. McPhedran, and B. J. Eggleton, "High-Q microfluidic cavities in silicon-based 2D photonic crystal structures," Opt. Lett. 33 (2008).

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P. E. Barclay, K. Srinivasan, M. Borselli, and O. Painter, "Probing the dispersive and spatial properties of photonic crystal waveguides via highly efficient coupling from fiber tapers," Appl. Phys. Lett. 85, 4-6 (2004).
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F. Intonti, S. Vignolini, V. Turck, M. Colocci, P. Bettotti, L. Pavesi, S. L. Schweizer, R. Wehrspohn, and D. Wiersma, "Rewritable photonic circuits," Appl. Phys. Lett. 89, 211117 (2006).
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D. Erickson, T. Rockwood, T. Emery, A. Scherer, and D. Psaltis, "Nanofluidic tuning of photonic crystal circuits," Opt. Lett. 31, 59-61 (2006).
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D. Psaltis, S. R. Quake, and C. H. Yang, "Developing optofluidic technology through the fusion of microfluidics and optics," Nature 442, 381-386 (2006).
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B. Maune, M. Loncar, J. Witzens, M. Hochberg, T. Baehr-Jones, D. Psaltis, A. Scherer, and Y. M. Qiu, "Liquid-crystal electric tuning of a photonic crystal laser," Appl. Phys. Lett. 85, 360-362 (2004).
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Qiu, Y. M.

M. L. Adams, M. Loncar, A. Scherer, and Y. M. Qiu, "Microfluidic integration of porous photonic crystal nanolasers for chemical sensing," IEEE J. Sel. Areas Commun. 23, 1348-1354 (2005).
[CrossRef]

B. Maune, M. Loncar, J. Witzens, M. Hochberg, T. Baehr-Jones, D. Psaltis, A. Scherer, and Y. M. Qiu, "Liquid-crystal electric tuning of a photonic crystal laser," Appl. Phys. Lett. 85, 360-362 (2004).
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M. Loncar, A. Scherer, and Y. M. Qiu, "Photonic crystal laser sources for chemical detection," Appl. Phys. Lett. 82, 4648-4650 (2003).
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D. Psaltis, S. R. Quake, and C. H. Yang, "Developing optofluidic technology through the fusion of microfluidics and optics," Nature 442, 381-386 (2006).
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Regrency, P.

Reithmaier, J. P.

C. Schuller, F. Klopf, J. P. Reithmaier, M. Kamp, and A. Forchel, "Tunable photonic crystals fabricated in III-V semiconductor slab waveguides using infiltrated liquid crystals," Appl. Phys. Lett. 82, 2767-2769 (2003).
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Rogers, J. A.

P. Mach, M. Dolinski, K. W. Baldwin, J. A. Rogers, C. Kerbage, R. S. Windeler, and B. J. Eggleton, "Tunable microfluidic optical fiber," Appl. Phys. Lett. 80, 4294-4296 (2002).
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X. Letartre, C. Seassal, C. Grillet, P. Rojo-Romeo, P. Viktorovitch, M. L. d'Yerville, D. Cassagne, and C. Jouanin, "Group velocity and propagation losses measurement in a single-line photonic-crystal waveguide on InP membranes," Appl. Phys. Lett. 79, 2312-2314 (2001).
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Ruan, Y.

C. L. C. Smith, D. K. C. Wu, M. W. Lee, C. Monat, S. Tomljenovic-Hanic, C. Grillet, B. J. Eggleton, D. Freeman, Y. Ruan, S. Madden, B. Luther-Davies, H. Giessen, and Y. H. Lee, "Microfluidic photonic crystal double heterostructures," Appl. Phys. Lett. 91, 121103 (2007).
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Scherer, A.

D. Erickson, T. Rockwood, T. Emery, A. Scherer, and D. Psaltis, "Nanofluidic tuning of photonic crystal circuits," Opt. Lett. 31, 59-61 (2006).
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B. Maune, M. Loncar, J. Witzens, M. Hochberg, T. Baehr-Jones, D. Psaltis, A. Scherer, and Y. M. Qiu, "Liquid-crystal electric tuning of a photonic crystal laser," Appl. Phys. Lett. 85, 360-362 (2004).
[CrossRef]

M. Loncar, A. Scherer, and Y. M. Qiu, "Photonic crystal laser sources for chemical detection," Appl. Phys. Lett. 82, 4648-4650 (2003).
[CrossRef]

Schuller, C.

C. Schuller, F. Klopf, J. P. Reithmaier, M. Kamp, and A. Forchel, "Tunable photonic crystals fabricated in III-V semiconductor slab waveguides using infiltrated liquid crystals," Appl. Phys. Lett. 82, 2767-2769 (2003).
[CrossRef]

Schweizer, S. L.

F. Intonti, S. Vignolini, V. Turck, M. Colocci, P. Bettotti, L. Pavesi, S. L. Schweizer, R. Wehrspohn, and D. Wiersma, "Rewritable photonic circuits," Appl. Phys. Lett. 89, 211117 (2006).
[CrossRef]

Seassal, C.

S. H. Kim, J. H. Choi, S. K. Lee, S. H. Kim, S. M. Yang, Y. H. Lee, C. Seassal, P. Regrency, and P. Viktorovitch, "Optofluidic integration of a photonic crystal nanolaser," Opt. Express 16, 6515-6527 (2008).
[CrossRef] [PubMed]

X. Letartre, C. Seassal, C. Grillet, P. Rojo-Romeo, P. Viktorovitch, M. L. d'Yerville, D. Cassagne, and C. Jouanin, "Group velocity and propagation losses measurement in a single-line photonic-crystal waveguide on InP membranes," Appl. Phys. Lett. 79, 2312-2314 (2001).
[CrossRef]

Shinya, A.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, "Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs," Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Sigalas, M.

Smith, C.

Smith, C. L. C.

U. Bog, C. L. C. Smith, M. W. Lee, S. Tomljenovic-Hanic, C. Grillet, C. Monat, L. O'Faolain, C. Karnutsch, T. F. Krauss, R. McPhedran, and B. J. Eggleton, "High-Q microfluidic cavities in silicon-based 2D photonic crystal structures," Opt. Lett. 33 (2008).

C. L. C. Smith, D. K. C. Wu, M. W. Lee, C. Monat, S. Tomljenovic-Hanic, C. Grillet, B. J. Eggleton, D. Freeman, Y. Ruan, S. Madden, B. Luther-Davies, H. Giessen, and Y. H. Lee, "Microfluidic photonic crystal double heterostructures," Appl. Phys. Lett. 91, 121103 (2007).
[CrossRef]

Snakenborg, D.

S. Balslev, A. M. Jorgensen, B. Bilenberg, K. B. Mogensen, D. Snakenborg, O. Geschke, J. P. Kutter, and A. Kristensen, "Lab-on-a-chip with integrated optical transducers," Lab on a Chip 6, 213-217 (2006).
[CrossRef] [PubMed]

Song, B. S.

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

B. S. Song, S. Noda, and T. Asano, "Photonic devices based on in-plane hetero photonic crystals," Science 300, 1537-1537 (2003).

Srinivasan, K.

P. E. Barclay, K. Srinivasan, M. Borselli, and O. Painter, "Probing the dispersive and spatial properties of photonic crystal waveguides via highly efficient coupling from fiber tapers," Appl. Phys. Lett. 85, 4-6 (2004).
[CrossRef]

K. Srinivasan and O. Painter, "Fourier space design of high-Q cavities in standard and compressed hexagonal lattice photonic crystals," Opt. Express 11, 579-593 (2003).
[CrossRef] [PubMed]

Steel, M. J.

Takahashi, C.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, "Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs," Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Takahashi, J.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, "Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs," Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Tomljenovic-Hanic, S.

U. Bog, C. L. C. Smith, M. W. Lee, S. Tomljenovic-Hanic, C. Grillet, C. Monat, L. O'Faolain, C. Karnutsch, T. F. Krauss, R. McPhedran, and B. J. Eggleton, "High-Q microfluidic cavities in silicon-based 2D photonic crystal structures," Opt. Lett. 33 (2008).

C. L. C. Smith, D. K. C. Wu, M. W. Lee, C. Monat, S. Tomljenovic-Hanic, C. Grillet, B. J. Eggleton, D. Freeman, Y. Ruan, S. Madden, B. Luther-Davies, H. Giessen, and Y. H. Lee, "Microfluidic photonic crystal double heterostructures," Appl. Phys. Lett. 91, 121103 (2007).
[CrossRef]

S. Tomljenovic-Hanic, C. M. de Sterke, and M. J. Steel, "Design of high-Q cavities in photonic crystal slab heterostructures by air-holes infiltration," Opt. Express 14, 12451-12456 (2006).
[CrossRef] [PubMed]

Torres, J.

J. C. Galas, J. Torres, M. Belotti, Q. Kou, and Y. Chen, "Microfluidic tunable dye laser with integrated mixer and ring resonator," Appl. Phys. Lett. 86, 264101 (2005).
[CrossRef]

Turck, V.

F. Intonti, S. Vignolini, V. Turck, M. Colocci, P. Bettotti, L. Pavesi, S. L. Schweizer, R. Wehrspohn, and D. Wiersma, "Rewritable photonic circuits," Appl. Phys. Lett. 89, 211117 (2006).
[CrossRef]

Vahala, K. J.

K. J. Vahala, "Optical microcavities," Nature 424, 839-846 (2003).
[CrossRef] [PubMed]

Vignolini, S.

F. Intonti, S. Vignolini, V. Turck, M. Colocci, P. Bettotti, L. Pavesi, S. L. Schweizer, R. Wehrspohn, and D. Wiersma, "Rewritable photonic circuits," Appl. Phys. Lett. 89, 211117 (2006).
[CrossRef]

Viktorovitch, P.

S. H. Kim, J. H. Choi, S. K. Lee, S. H. Kim, S. M. Yang, Y. H. Lee, C. Seassal, P. Regrency, and P. Viktorovitch, "Optofluidic integration of a photonic crystal nanolaser," Opt. Express 16, 6515-6527 (2008).
[CrossRef] [PubMed]

X. Letartre, C. Seassal, C. Grillet, P. Rojo-Romeo, P. Viktorovitch, M. L. d'Yerville, D. Cassagne, and C. Jouanin, "Group velocity and propagation losses measurement in a single-line photonic-crystal waveguide on InP membranes," Appl. Phys. Lett. 79, 2312-2314 (2001).
[CrossRef]

Wehrspohn, R.

F. Intonti, S. Vignolini, V. Turck, M. Colocci, P. Bettotti, L. Pavesi, S. L. Schweizer, R. Wehrspohn, and D. Wiersma, "Rewritable photonic circuits," Appl. Phys. Lett. 89, 211117 (2006).
[CrossRef]

White, T. P.

Wiersma, D.

F. Intonti, S. Vignolini, V. Turck, M. Colocci, P. Bettotti, L. Pavesi, S. L. Schweizer, R. Wehrspohn, and D. Wiersma, "Rewritable photonic circuits," Appl. Phys. Lett. 89, 211117 (2006).
[CrossRef]

Windeler, R. S.

P. Mach, M. Dolinski, K. W. Baldwin, J. A. Rogers, C. Kerbage, R. S. Windeler, and B. J. Eggleton, "Tunable microfluidic optical fiber," Appl. Phys. Lett. 80, 4294-4296 (2002).
[CrossRef]

Witzens, J.

B. Maune, M. Loncar, J. Witzens, M. Hochberg, T. Baehr-Jones, D. Psaltis, A. Scherer, and Y. M. Qiu, "Liquid-crystal electric tuning of a photonic crystal laser," Appl. Phys. Lett. 85, 360-362 (2004).
[CrossRef]

Wu, D. K. C.

C. L. C. Smith, D. K. C. Wu, M. W. Lee, C. Monat, S. Tomljenovic-Hanic, C. Grillet, B. J. Eggleton, D. Freeman, Y. Ruan, S. Madden, B. Luther-Davies, H. Giessen, and Y. H. Lee, "Microfluidic photonic crystal double heterostructures," Appl. Phys. Lett. 91, 121103 (2007).
[CrossRef]

Xiao, S. S.

S. S. Xiao and N. A. Mortensen, "Proposal of highly sensitive optofluidic sensors based on dispersive photonic crystal waveguides," J. Opt. A: Pure Appl. Opt. 9, S463-S467 (2007).
[CrossRef]

Yamada, K.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, "Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs," Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Yang, C. H.

D. Psaltis, S. R. Quake, and C. H. Yang, "Developing optofluidic technology through the fusion of microfluidics and optics," Nature 442, 381-386 (2006).
[CrossRef] [PubMed]

Yang, J. K.

I. K. Hwang, S. K. Kim, J. K. Yang, S. H. Kim, S. H. Lee, and Y. H. Lee, "Curved-microfiber photon coupling for photonic crystal light emitter," Appl. Phys. Lett. 87, 131107 (2005).
[CrossRef]

Yang, S. M.

Yokohama, I.

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, "Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs," Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Zamek, S.

Zuppiroli, L.

Appl. Phys. Lett. (11)

M. Loncar, A. Scherer, and Y. M. Qiu, "Photonic crystal laser sources for chemical detection," Appl. Phys. Lett. 82, 4648-4650 (2003).
[CrossRef]

P. Mach, M. Dolinski, K. W. Baldwin, J. A. Rogers, C. Kerbage, R. S. Windeler, and B. J. Eggleton, "Tunable microfluidic optical fiber," Appl. Phys. Lett. 80, 4294-4296 (2002).
[CrossRef]

B. Maune, M. Loncar, J. Witzens, M. Hochberg, T. Baehr-Jones, D. Psaltis, A. Scherer, and Y. M. Qiu, "Liquid-crystal electric tuning of a photonic crystal laser," Appl. Phys. Lett. 85, 360-362 (2004).
[CrossRef]

J. C. Galas, J. Torres, M. Belotti, Q. Kou, and Y. Chen, "Microfluidic tunable dye laser with integrated mixer and ring resonator," Appl. Phys. Lett. 86, 264101 (2005).
[CrossRef]

C. L. C. Smith, D. K. C. Wu, M. W. Lee, C. Monat, S. Tomljenovic-Hanic, C. Grillet, B. J. Eggleton, D. Freeman, Y. Ruan, S. Madden, B. Luther-Davies, H. Giessen, and Y. H. Lee, "Microfluidic photonic crystal double heterostructures," Appl. Phys. Lett. 91, 121103 (2007).
[CrossRef]

F. Intonti, S. Vignolini, V. Turck, M. Colocci, P. Bettotti, L. Pavesi, S. L. Schweizer, R. Wehrspohn, and D. Wiersma, "Rewritable photonic circuits," Appl. Phys. Lett. 89, 211117 (2006).
[CrossRef]

F. B. Arango, M. B. Christiansen, M. Gersborg-Hansen, and A. Kristensen, "Optofluidic tuning of photonic crystal band edge lasers," Appl. Phys. Lett. 91, 223503 (2007).
[CrossRef]

P. E. Barclay, K. Srinivasan, M. Borselli, and O. Painter, "Probing the dispersive and spatial properties of photonic crystal waveguides via highly efficient coupling from fiber tapers," Appl. Phys. Lett. 85, 4-6 (2004).
[CrossRef]

I. K. Hwang, S. K. Kim, J. K. Yang, S. H. Kim, S. H. Lee, and Y. H. Lee, "Curved-microfiber photon coupling for photonic crystal light emitter," Appl. Phys. Lett. 87, 131107 (2005).
[CrossRef]

X. Letartre, C. Seassal, C. Grillet, P. Rojo-Romeo, P. Viktorovitch, M. L. d'Yerville, D. Cassagne, and C. Jouanin, "Group velocity and propagation losses measurement in a single-line photonic-crystal waveguide on InP membranes," Appl. Phys. Lett. 79, 2312-2314 (2001).
[CrossRef]

C. Schuller, F. Klopf, J. P. Reithmaier, M. Kamp, and A. Forchel, "Tunable photonic crystals fabricated in III-V semiconductor slab waveguides using infiltrated liquid crystals," Appl. Phys. Lett. 82, 2767-2769 (2003).
[CrossRef]

IEEE J. Sel. Areas Commun. (1)

M. L. Adams, M. Loncar, A. Scherer, and Y. M. Qiu, "Microfluidic integration of porous photonic crystal nanolasers for chemical sensing," IEEE J. Sel. Areas Commun. 23, 1348-1354 (2005).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

S. S. Xiao and N. A. Mortensen, "Proposal of highly sensitive optofluidic sensors based on dispersive photonic crystal waveguides," J. Opt. A: Pure Appl. Opt. 9, S463-S467 (2007).
[CrossRef]

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

Lab on a Chip (1)

S. Balslev, A. M. Jorgensen, B. Bilenberg, K. B. Mogensen, D. Snakenborg, O. Geschke, J. P. Kutter, and A. Kristensen, "Lab-on-a-chip with integrated optical transducers," Lab on a Chip 6, 213-217 (2006).
[CrossRef] [PubMed]

Nat. Mater. (2)

T. F. Krauss, "Photonic crystals - Cavities without leaks," Nat. Mater. 2, 777-778 (2003).
[CrossRef] [PubMed]

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

Nat. Photon. (1)

C. Monat, P. Domachuk, and B. J. Eggleton, "Integrated optofluidics: A new river of light," Nat. Photon. 1, 106-114 (2007).
[CrossRef]

Nature (2)

D. Psaltis, S. R. Quake, and C. H. Yang, "Developing optofluidic technology through the fusion of microfluidics and optics," Nature 442, 381-386 (2006).
[CrossRef] [PubMed]

K. J. Vahala, "Optical microcavities," Nature 424, 839-846 (2003).
[CrossRef] [PubMed]

Opt. Express (7)

Opt. Lett. (4)

Phys. Rev. B (1)

E. Centeno, and D. Felbacq, "Optical bistability in finite-size nonlinear bidimensional photonic crystals doped by a microcavity," Phys. Rev. B 62, R7683-R7686 (2000).
[CrossRef]

Phys. Rev. Lett. (2)

K. Busch and S. John, "Liquid-crystal photonic-band-gap materials: The tunable electromagnetic vacuum," Phys. Rev. Lett. 83, 967-970 (1999).
[CrossRef]

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, "Extremely large group-velocity dispersion of line-defect waveguides in photonic crystal slabs," Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Science (1)

B. S. Song, S. Noda, and T. Asano, "Photonic devices based on in-plane hetero photonic crystals," Science 300, 1537-1537 (2003).

Other (1)

"Datasheet TYPIOB, Immersion Oil Type B," (Cargille Laboratories, 2002).

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Figures (7)

Fig. 1.
Fig. 1.

(a) Schematic of fluid-filled PhC. (b) Calculated dispersion relation of the W0.9 PhC: blue triangles represent the PhC with air holes and red circles represent the PhC with fluid-filled holes. The green (solid) curve represents the evanescent coupling taper. The gray (dashed) curve represents the light-line. (c) Dispersion relation of a regular W1 waveguide with otherwise the same parameters as (b).

Fig. 2.
Fig. 2.

Infiltration schematic: a glass micro-tip is used to draw the infiltration fluid across the PhC section, forming the DH stripe.

Fig. 3.
Fig. 3.

Evanescent coupling schematic: polarization controller (PC) and polarizer select TE-like light from the edge-emitting light-emitting diode (EELED) source. The evanescent fiber (EF) couples light to the PhC sample and is connected to an optical spectrum analyzer (OSA) for monitoring.

Fig. 4.
Fig. 4.

(a) Spectrum of the case before infiltration. (b) Spectrum for varying infiltrated cavity lengths. (c) Spectrum of the case after cleaning. Insets are 150× microscope images of the corresponding waveguide/cavities.

Fig. 5.
Fig. 5.

Comparison of measured and calculated dispersive behavior for the fluid-filled PhCs. Cavity lengths of 8.2 µm (black squares), 16.8 µm (red circles) and 20.1 µm (blue triangles) are plotted against numerical data (black solid).

Fig. 6.
Fig. 6.

Transmission and Q-factor versus frequency. (a) and (b) represent spectra from 5.3 µm and 16.8 µm cavity lengths with highest Q-factors shown inset. (c) and (d) are the corresponding Q-factors of the spectra to (a) and (b).

Fig. 7.
Fig. 7.

Change of Q-factor against cavity length at an arbitrarily fixed frequency of 0.291 a/λ. The entire PhC structure is 25 µm.

Equations (1)

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2 k x L = 2 π p ,

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