Abstract

A photonic crystal waveguide (PhC-WG) was reported to be usable as an optical sensor highly sensitive to various material parameters, which can be detected via changes in transmission through the PhC-WG caused by small changes of the refractive index of the medium filling its holes. To monitor these changes accurately, a precise optical model is required, for which the plane wave expansion (PWE) method is convenient. We here demonstrate the revision of the PWE method by employing the complex Fourier factorization approach, which enables the calculation of dispersion diagrams with fast convergence, i.e., with high precision in relatively short time. The PhC-WG is proposed as a line defect in a hexagonal array of cylindrical holes periodically arranged in bulk silicon, filled with a variable medium. The method of monitoring the refractive index changes is based on observing cutoff wavelengths in the PhC-WG dispersion diagrams. The PWE results are also compared with finite-difference time-domain calculations of transmittance carried out on a PhC-WG with finite dimensions.

© 2014 Optical Society of America

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2012 (2)

D. Threm, Y. Nazirizadeh, M. Gerken, “Photonic crystal biosensors towards on-chip integration,” J. Biophotonics 5, 601–616 (2012).
[CrossRef]

M. Piliarik, H. Sipova, P. Kvasnicka, N. Galler, J. R. Krenn, J. Homola, “High-resolution biosensor based on localized surface plasmons,” Opt. Express 20, 672–680 (2012).
[CrossRef] [PubMed]

2010 (4)

M. El Beheiry, V. Liu, S. Fan, O. Levi, “Sensitivity enhancement in photonic crystal slab biosensors,” Opt. Express 18, 22702–22714 (2010).
[CrossRef] [PubMed]

R. Antos, M. Veis, “Fourier factorization with complex polarization bases in the plane-wave expansion method applied to two-dimensional photonic crystals,” Opt. Express 18, 27511–27524 (2010).
[CrossRef]

D. Regatos, D. Farina, A. Calle, A. Cebollada, B. Sepulveda, G. Armelles, L. M. Lechuga, “Au/fe/au multilayer transducers for magneto-optic surface plasmon resonance sensing,” J. Appl. Phys. 108, 054502 (2010).
[CrossRef]

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

2009 (1)

2008 (2)

2007 (2)

2006 (1)

A. David, H. Benisty, C. Weisbuch, “Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape,” Phys. Rev. B 73, 075107 (2006).
[CrossRef]

2005 (3)

J. Jensen, P. Hoiby, G. Emiliyanov, O. Bang, L. Pedersen, A. Bjarklev, “Selective detection of antibodies in microstructured polymer optical fibers,” Opt. Express 13, 5883–5889 (2005).
[CrossRef] [PubMed]

N. Bonod, E. Popov, M. Neviere, “Light transmission through a subwavelength microstructured aperture: electromagnetic theory and applications,” Opt. Commun. 245, 355–361 (2005).
[CrossRef]

N. Bonod, E. Popov, M. Neviere, “Fourier factorization of nonlinear Maxwell equations in periodic media: application to the optical Kerr effect,” Opt. Commun. 244, 389–398 (2005).
[CrossRef]

2004 (1)

2003 (1)

L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A 5, 345–355 (2003).
[CrossRef]

2002 (4)

K. Watanabe, R. Petit, M. Neviere, “Differential theory of gratings made of anisotropic materials,” J. Opt. Soc. Am. A 19, 325–334 (2002).
[CrossRef]

K. Watanabe, “Numerical integration schemes used on the differential theory for anisotropic gratings,” J. Opt. Soc. Am. A 19, 2245–2252 (2002).
[CrossRef]

B. Cunningham, B. Lin, J. Qiu, P. Li, J. Pepper, B. Hugh, “A plastic colorimetric resonant optical biosensor for multiparallel detection of label-free biochemical interactions,” Sens. Actuators B Chem. 85, 219–226 (2002).
[CrossRef]

B. Cunningham, P. Li, B. Lin, J. Pepper, “Colorimetric resonant reflection as a direct biochemical assay technique,” Sens. Actuators B Chem. 81, 316–328 (2002).
[CrossRef]

2001 (1)

B. Chernov, M. Neviere, E. Popov, “Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings,” Opt. Commun. 194, 289–297 (2001).
[CrossRef]

2000 (1)

1999 (1)

J. Homola, S. S. Yee, G. Gauglitz, “Suraface plamson resonance sensors: review,” Sens. Actuators B 54, 3–15 (1999).
[CrossRef]

1998 (1)

L. Li, “Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials,” J. Mod. Opt. 45, 1313–1334 (1998).
[CrossRef]

1997 (3)

1996 (3)

1993 (1)

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

1992 (1)

H. S. Sozuer, J. W. Haus, R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B 45, 13962–13972 (1992).
[CrossRef]

1982 (1)

C. Nylander, B. Liedberg, T. Lind, “Gas-detection by means of surface-plasmon resonance,” Sens. Actuators 3, 79–88 (1982).
[CrossRef]

Alerhand, O. L.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

Antos, R.

Armelles, G.

D. Regatos, D. Farina, A. Calle, A. Cebollada, B. Sepulveda, G. Armelles, L. M. Lechuga, “Au/fe/au multilayer transducers for magneto-optic surface plasmon resonance sensing,” J. Appl. Phys. 108, 054502 (2010).
[CrossRef]

Bang, O.

Benisty, H.

A. David, H. Benisty, C. Weisbuch, “Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape,” Phys. Rev. B 73, 075107 (2006).
[CrossRef]

Bermel, P.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

Bjarklev, A.

Blum, L. J.

A. Sassolas, B. D. Leca-Bouvier, L. J. Blum, “Dna biosensors and microarrays,” Chem. Rev. 108, 109–139 (2008).
[CrossRef]

Bonod, N.

N. Bonod, E. Popov, M. Neviere, “Light transmission through a subwavelength microstructured aperture: electromagnetic theory and applications,” Opt. Commun. 245, 355–361 (2005).
[CrossRef]

N. Bonod, E. Popov, M. Neviere, “Fourier factorization of nonlinear Maxwell equations in periodic media: application to the optical Kerr effect,” Opt. Commun. 244, 389–398 (2005).
[CrossRef]

Borel, P. I.

Boyer, P.

Brommer, K. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

Calle, A.

D. Regatos, D. Farina, A. Calle, A. Cebollada, B. Sepulveda, G. Armelles, L. M. Lechuga, “Au/fe/au multilayer transducers for magneto-optic surface plasmon resonance sensing,” J. Appl. Phys. 108, 054502 (2010).
[CrossRef]

Cebollada, A.

D. Regatos, D. Farina, A. Calle, A. Cebollada, B. Sepulveda, G. Armelles, L. M. Lechuga, “Au/fe/au multilayer transducers for magneto-optic surface plasmon resonance sensing,” J. Appl. Phys. 108, 054502 (2010).
[CrossRef]

Chernov, B.

B. Chernov, M. Neviere, E. Popov, “Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings,” Opt. Commun. 194, 289–297 (2001).
[CrossRef]

Cunningham, B.

B. Cunningham, B. Lin, J. Qiu, P. Li, J. Pepper, B. Hugh, “A plastic colorimetric resonant optical biosensor for multiparallel detection of label-free biochemical interactions,” Sens. Actuators B Chem. 85, 219–226 (2002).
[CrossRef]

B. Cunningham, P. Li, B. Lin, J. Pepper, “Colorimetric resonant reflection as a direct biochemical assay technique,” Sens. Actuators B Chem. 81, 316–328 (2002).
[CrossRef]

David, A.

A. David, H. Benisty, C. Weisbuch, “Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape,” Phys. Rev. B 73, 075107 (2006).
[CrossRef]

El Beheiry, M.

Emiliyanov, G.

Fan, S.

Farina, D.

D. Regatos, D. Farina, A. Calle, A. Cebollada, B. Sepulveda, G. Armelles, L. M. Lechuga, “Au/fe/au multilayer transducers for magneto-optic surface plasmon resonance sensing,” J. Appl. Phys. 108, 054502 (2010).
[CrossRef]

Frandsen, L. H.

Frenner, K.

Galler, N.

Gauglitz, G.

J. Homola, S. S. Yee, G. Gauglitz, “Suraface plamson resonance sensors: review,” Sens. Actuators B 54, 3–15 (1999).
[CrossRef]

Gerken, M.

D. Threm, Y. Nazirizadeh, M. Gerken, “Photonic crystal biosensors towards on-chip integration,” J. Biophotonics 5, 601–616 (2012).
[CrossRef]

Gotz, P.

Granet, G.

Guizal, B.

Haus, J. W.

H. S. Sozuer, J. W. Haus, R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B 45, 13962–13972 (1992).
[CrossRef]

Hoiby, P.

Homola, J.

M. Piliarik, H. Sipova, P. Kvasnicka, N. Galler, J. R. Krenn, J. Homola, “High-resolution biosensor based on localized surface plasmons,” Opt. Express 20, 672–680 (2012).
[CrossRef] [PubMed]

J. Homola, S. S. Yee, G. Gauglitz, “Suraface plamson resonance sensors: review,” Sens. Actuators B 54, 3–15 (1999).
[CrossRef]

A. D. Taylor, J. Ladd, J. Homola, S. Jiang, Surface Plasmon Resonance (SPR) Sensors for the Detection of Bacterial Pathogens (Springer, 2008), pp. 83–108.

Hugh, B.

B. Cunningham, B. Lin, J. Qiu, P. Li, J. Pepper, B. Hugh, “A plastic colorimetric resonant optical biosensor for multiparallel detection of label-free biochemical interactions,” Sens. Actuators B Chem. 85, 219–226 (2002).
[CrossRef]

Ibanescu, M.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

Inguva, R.

H. S. Sozuer, J. W. Haus, R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B 45, 13962–13972 (1992).
[CrossRef]

Jensen, J.

Jiang, S.

A. D. Taylor, J. Ladd, J. Homola, S. Jiang, Surface Plasmon Resonance (SPR) Sensors for the Detection of Bacterial Pathogens (Springer, 2008), pp. 83–108.

Joannopoulos, J. D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

Johnson, S. G.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

Kerwien, N.

Kjems, J.

Krenn, J. R.

Kristensen, M.

Kvasnicka, P.

Ladd, J.

A. D. Taylor, J. Ladd, J. Homola, S. Jiang, Surface Plasmon Resonance (SPR) Sensors for the Detection of Bacterial Pathogens (Springer, 2008), pp. 83–108.

Lalanne, P

Lalanne, P.

Leca-Bouvier, B. D.

A. Sassolas, B. D. Leca-Bouvier, L. J. Blum, “Dna biosensors and microarrays,” Chem. Rev. 108, 109–139 (2008).
[CrossRef]

Lechuga, L. M.

D. Regatos, D. Farina, A. Calle, A. Cebollada, B. Sepulveda, G. Armelles, L. M. Lechuga, “Au/fe/au multilayer transducers for magneto-optic surface plasmon resonance sensing,” J. Appl. Phys. 108, 054502 (2010).
[CrossRef]

Levi, O.

Li, L.

L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A 5, 345–355 (2003).
[CrossRef]

L. Li, “Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials,” J. Mod. Opt. 45, 1313–1334 (1998).
[CrossRef]

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

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

Li, P.

B. Cunningham, B. Lin, J. Qiu, P. Li, J. Pepper, B. Hugh, “A plastic colorimetric resonant optical biosensor for multiparallel detection of label-free biochemical interactions,” Sens. Actuators B Chem. 85, 219–226 (2002).
[CrossRef]

B. Cunningham, P. Li, B. Lin, J. Pepper, “Colorimetric resonant reflection as a direct biochemical assay technique,” Sens. Actuators B Chem. 81, 316–328 (2002).
[CrossRef]

Liedberg, B.

C. Nylander, B. Liedberg, T. Lind, “Gas-detection by means of surface-plasmon resonance,” Sens. Actuators 3, 79–88 (1982).
[CrossRef]

Lin, B.

B. Cunningham, P. Li, B. Lin, J. Pepper, “Colorimetric resonant reflection as a direct biochemical assay technique,” Sens. Actuators B Chem. 81, 316–328 (2002).
[CrossRef]

B. Cunningham, B. Lin, J. Qiu, P. Li, J. Pepper, B. Hugh, “A plastic colorimetric resonant optical biosensor for multiparallel detection of label-free biochemical interactions,” Sens. Actuators B Chem. 85, 219–226 (2002).
[CrossRef]

Lind, T.

C. Nylander, B. Liedberg, T. Lind, “Gas-detection by means of surface-plasmon resonance,” Sens. Actuators 3, 79–88 (1982).
[CrossRef]

Liu, V.

Meade, R. D.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

Morris, G.

Nazirizadeh, Y.

D. Threm, Y. Nazirizadeh, M. Gerken, “Photonic crystal biosensors towards on-chip integration,” J. Biophotonics 5, 601–616 (2012).
[CrossRef]

Neviere, M.

N. Bonod, E. Popov, M. Neviere, “Fourier factorization of nonlinear Maxwell equations in periodic media: application to the optical Kerr effect,” Opt. Commun. 244, 389–398 (2005).
[CrossRef]

N. Bonod, E. Popov, M. Neviere, “Light transmission through a subwavelength microstructured aperture: electromagnetic theory and applications,” Opt. Commun. 245, 355–361 (2005).
[CrossRef]

P. Boyer, E. Popov, M. Neviere, G. Tayeb, “Diffraction theory in TM polarization: application of the fast Fourier factorization method to cylindrical devices with arbitrary cross section,” J. Opt. Soc. Am. A 21, 2146–2153 (2004).
[CrossRef]

K. Watanabe, R. Petit, M. Neviere, “Differential theory of gratings made of anisotropic materials,” J. Opt. Soc. Am. A 19, 325–334 (2002).
[CrossRef]

B. Chernov, M. Neviere, E. Popov, “Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings,” Opt. Commun. 194, 289–297 (2001).
[CrossRef]

E. Popov, M. Neviere, “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]

Nylander, C.

C. Nylander, B. Liedberg, T. Lind, “Gas-detection by means of surface-plasmon resonance,” Sens. Actuators 3, 79–88 (1982).
[CrossRef]

Oskooi, A. F.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

Osten, W.

Pedersen, L.

Pepper, J.

B. Cunningham, P. Li, B. Lin, J. Pepper, “Colorimetric resonant reflection as a direct biochemical assay technique,” Sens. Actuators B Chem. 81, 316–328 (2002).
[CrossRef]

B. Cunningham, B. Lin, J. Qiu, P. Li, J. Pepper, B. Hugh, “A plastic colorimetric resonant optical biosensor for multiparallel detection of label-free biochemical interactions,” Sens. Actuators B Chem. 85, 219–226 (2002).
[CrossRef]

Petit, R.

Piliarik, M.

Popov, E.

N. Bonod, E. Popov, M. Neviere, “Fourier factorization of nonlinear Maxwell equations in periodic media: application to the optical Kerr effect,” Opt. Commun. 244, 389–398 (2005).
[CrossRef]

N. Bonod, E. Popov, M. Neviere, “Light transmission through a subwavelength microstructured aperture: electromagnetic theory and applications,” Opt. Commun. 245, 355–361 (2005).
[CrossRef]

P. Boyer, E. Popov, M. Neviere, G. Tayeb, “Diffraction theory in TM polarization: application of the fast Fourier factorization method to cylindrical devices with arbitrary cross section,” J. Opt. Soc. Am. A 21, 2146–2153 (2004).
[CrossRef]

B. Chernov, M. Neviere, E. Popov, “Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings,” Opt. Commun. 194, 289–297 (2001).
[CrossRef]

E. Popov, M. Neviere, “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]

Qiu, J.

B. Cunningham, B. Lin, J. Qiu, P. Li, J. Pepper, B. Hugh, “A plastic colorimetric resonant optical biosensor for multiparallel detection of label-free biochemical interactions,” Sens. Actuators B Chem. 85, 219–226 (2002).
[CrossRef]

Rafler, S.

Rappe, A. M.

R. D. Meade, A. M. Rappe, K. D. Brommer, J. D. Joannopoulos, O. L. Alerhand, “Accurate theoretical analysis of photonic band-gap materials,” Phys. Rev. B 48, 8434–8437 (1993).
[CrossRef]

Regatos, D.

D. Regatos, D. Farina, A. Calle, A. Cebollada, B. Sepulveda, G. Armelles, L. M. Lechuga, “Au/fe/au multilayer transducers for magneto-optic surface plasmon resonance sensing,” J. Appl. Phys. 108, 054502 (2010).
[CrossRef]

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A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

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A. Sassolas, B. D. Leca-Bouvier, L. J. Blum, “Dna biosensors and microarrays,” Chem. Rev. 108, 109–139 (2008).
[CrossRef]

Schuster, T.

Sepulveda, B.

D. Regatos, D. Farina, A. Calle, A. Cebollada, B. Sepulveda, G. Armelles, L. M. Lechuga, “Au/fe/au multilayer transducers for magneto-optic surface plasmon resonance sensing,” J. Appl. Phys. 108, 054502 (2010).
[CrossRef]

Sipova, H.

Skivesen, N.

Sozuer, H. S.

H. S. Sozuer, J. W. Haus, R. Inguva, “Photonic bands: Convergence problems with the plane-wave method,” Phys. Rev. B 45, 13962–13972 (1992).
[CrossRef]

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Taylor, A. D.

A. D. Taylor, J. Ladd, J. Homola, S. Jiang, Surface Plasmon Resonance (SPR) Sensors for the Detection of Bacterial Pathogens (Springer, 2008), pp. 83–108.

Tetu, A.

Threm, D.

D. Threm, Y. Nazirizadeh, M. Gerken, “Photonic crystal biosensors towards on-chip integration,” J. Biophotonics 5, 601–616 (2012).
[CrossRef]

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Watanabe, K.

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A. David, H. Benisty, C. Weisbuch, “Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape,” Phys. Rev. B 73, 075107 (2006).
[CrossRef]

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J. Homola, S. S. Yee, G. Gauglitz, “Suraface plamson resonance sensors: review,” Sens. Actuators B 54, 3–15 (1999).
[CrossRef]

Chem. Rev. (1)

A. Sassolas, B. D. Leca-Bouvier, L. J. Blum, “Dna biosensors and microarrays,” Chem. Rev. 108, 109–139 (2008).
[CrossRef]

Comput. Phys. Commun. (1)

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181, 687–702 (2010).
[CrossRef]

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D. Regatos, D. Farina, A. Calle, A. Cebollada, B. Sepulveda, G. Armelles, L. M. Lechuga, “Au/fe/au multilayer transducers for magneto-optic surface plasmon resonance sensing,” J. Appl. Phys. 108, 054502 (2010).
[CrossRef]

J. Biophotonics (1)

D. Threm, Y. Nazirizadeh, M. Gerken, “Photonic crystal biosensors towards on-chip integration,” J. Biophotonics 5, 601–616 (2012).
[CrossRef]

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A. D. Taylor, J. Ladd, J. Homola, S. Jiang, Surface Plasmon Resonance (SPR) Sensors for the Detection of Bacterial Pathogens (Springer, 2008), pp. 83–108.

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Figures (10)

Fig. 1
Fig. 1

Geometry of the hexagonal PhC (a) and PhC-WG made by omitting one row (b).

Fig. 2
Fig. 2

Space distribution of the permittivity function (a) and of the vector û of the polarization basis, its azimuth (b) and ellipticity (c) in degrees, plotted within the PhC-WG supercell. The white circles denote the permittivity discontinuities (the hole edges). Note that û is always linear at and perpendicular to the hole edges. The azimuth distribution corresponding the normal vector method (d) is similar to the azimuth in (b).

Fig. 3
Fig. 3

Convergence properties of the standard factorization (Ho) method (using Eq. (10), the dashed curve with squares), the normal vector method (using Eq. (27) and real û ), the dash-dotted curve with triangles), and the complex Fourier factorization method (also using Eq. (27), but with complex û , the solid curve with circles). The 9th eigenfrequency at the X-point (the cutoff frequency) is displayed according to the number of the retained Fourier harmonics M (in our PhC-WG M corresponds to N = 7M).

Fig. 4
Fig. 4

(a) Dispersion diagram of the PhC-WG. Normalized eigenfrequencies are plotted according to the normalized kx (in the units of 2π x ). (b) Detail of the dispersion diagram showing the PhC-WG’s forbidden band and the position of the cutoff frequency.

Fig. 5
Fig. 5

Calculated cutoff wavelengths according to the RI of the medium inside holes in the PhC-WG.

Fig. 6
Fig. 6

Transmission spectrum of a finite PhC-WG with vacuum inside holes. The inset displays the PhC-WG geometry.

Fig. 7
Fig. 7

Details of the transmission spectra for various RIs of media inside holes.

Fig. 8
Fig. 8

Space distribution of the functions gx (a), gy (b), g′x (c), e′x (d), e′y (e), ûx (f), and ûy (g). The distributions (a)–(c) are same for the CFF method and normal vector method. The distributions (d)–(g) correspond only to the normal vector method.

Fig. 9
Fig. 9

Space distribution of the functions Re(g″x) (a), Im(g″x) (b), Re(g″y) (c), and Im(g″y) (d), all corresponding the CFF method.

Fig. 10
Fig. 10

Space distribution of the functions Re(ûx) (a), Im(ûx) (b), Re(ûy) (c), and Im(ûy) (d), all corresponding the CFF method.

Equations (47)

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× ( × E ) = ω 2 c 2 ε ( x , y ) E ,
E j ( x , y , z ) = e i k r m , n = + E j , m n e i ( m p x + n q y ) = m , n = + E j , m n e i ( p m x + q n y ) ,
ε ( x , y ) = m , n = + ε m n e i ( m p x + n q y ) ,
η ( x , y ) = m , n = + η m n e i ( m p x + n q y ) .
( x 2 + y 2 ) E z = ω 2 c 2 ε E z
[ y 2 x y x y x 2 ] [ E x E y ] = ω 2 c 2 ε [ E x E y ]
( x η x + y η y ) H z = ω 2 c 2 H z
( p 2 + q 2 ) [ E z ] = ω 2 c 2 [ [ ε ] ] [ E z ] ,
[ q 2 p q p q p 2 ] [ [ E x ] [ E y ] ] = ω 2 c 2 [ [ ε ] ] [ [ E x ] [ E y ] ] ,
( p [ [ ε ] ] 1 p + q [ [ ε ] ] 1 q ) [ H z ] = ω 2 c 2 [ H z ] ,
D ˜ = ε E ,
[ D ˜ z ] = [ [ ε ] ] [ E z ] ,
[ D ˜ x D ˜ y ] = ε [ E x E y ] ,
u ^ ( x , y ) = F 11 ( x , y ) x ^ + F 21 ( x , y ) y ^ ,
v ^ ( x , y ) = F 12 ( x , y ) x ^ + F 22 ( x , y ) y ^ ,
[ E x E y ] = F [ E u E v ] ,
u ^ = [ F 11 F 21 ] = [ ξ ζ ] ,
v ^ = [ F 12 F 22 ] = [ ζ * ξ * ] ,
[ D ˜ u D ˜ v ] = ε [ E u E v ] ,
[ D ˜ u ] = [ [ η ] ] 1 [ E u ] ,
[ D ˜ v ] = [ [ ε ] ] [ E v ] ,
[ [ D ˜ x ] [ D ˜ y ] ] = ε ˜ [ [ E x ] [ E y ] ]
[ [ E x ] [ E y ] ] = η ˜ [ [ D ˜ x ] [ D ˜ y ] ]
ε ˜ = [ [ [ η ] ] 1 [ [ ξ ξ * ] ] + [ [ ε ] ] [ [ ζ ζ * ] ] , [ [ η ] ] 1 [ [ ξ ζ * ] ] [ [ ε ] ] [ [ ξ ζ * ] ] [ [ η ] ] 1 [ [ ξ * ζ ] ] [ [ ε ] ] [ [ ξ * ζ ] ] , [ [ η ] ] 1 [ [ ζ ζ * ] ] + [ [ ε ] ] [ [ ξ ξ * ] ] ] ,
η ˜ = [ [ [ η ] ] [ [ ξ ξ * ] ] + [ [ ε ] ] 1 [ [ ζ ζ * ] ] , [ [ η ] ] [ [ ξ ζ * ] ] [ [ ε ] ] 1 [ [ ξ ζ * ] ] [ [ η ] ] [ [ ξ * ζ ] ] [ [ ε ] ] 1 [ [ ξ * ζ ] ] , [ [ η ] ] [ [ ζ ζ * ] ] + [ [ ε ] ] 1 [ [ ξ ξ * ] ] ] .
[ q 2 p q p q p 2 ] [ [ E x ] [ E y ] ] = ω 2 c 2 ε ˜ [ [ E x ] [ E y ] ] ,
( p η ˜ y y p p η ˜ y x q q η ˜ x y p + q η ˜ x x q ) [ H z ] = ω 2 c 2 [ H z ] ,
d ( x , y ) = ε ( x , y ) f ( x , y ) ,
g x ( x , y ) = x f ( x , y ) ,
g y ( x , y ) = y f ( x , y )
d m n = k , l = + ε m k , n l f k l ,
g x , m n = i p m f m n ,
g y , m n = i q n f m n .
α ( m , n ) = m + M + 1 + ( n + N ) ( 2 M + 1 ) ,
n ( α ) = ( α 1 ) div ( 2 M + 1 ) N ,
m ( α ) = ( α 1 ) mod ( 2 M + 1 ) M ,
[ d ] = [ [ ε ] ] [ f ] ,
[ g x ] = i p [ f ] ,
[ g y ] = i q [ f ] ,
[ [ ε ] ] α β = ε m ( α ) m ( β ) , n ( α ) n ( β ) ,
p α β = p m ( α ) δ α β ,
q α β = q n ( α ) δ α β ,
d = ε f ,
[ d ] = [ [ ε ] ] [ f ]
f = 1 ε d ,
[ f ] = [ [ 1 ε ] ] [ d ] .
[ d ] = [ [ 1 ε ] ] 1 [ f ] ,

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