Abstract

Influence of Bloch wave decomposition on light coupling into planar photonic crystals is investigated using plane-wave-method and finite-difference time domain calculations. The contribution of each plane wave to the global Bloch wave energy is calculated to understand the relative power weights of fields excited in the patterned medium. It is shown that high coupling efficiency is obtained when plane-wave components excited through a direct parallel wave vector, i.e., without an additional lattice-induced translated parallel wave vector, bring an important amount of the overall Bloch propagating wave power.

© 2007 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  6. W. Jiang, R. T. Chen, and X. Lu, 'Theory of light refraction at the surface of a photonic crystal,' Phys. Rev. E 71, 245115-1-245115-12 (2005).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2006 (1)

2005 (3)

B. Lombardet, L. A. Dunbar, R. Ferrini, and R. Houdré, 'Fourier analysis of Bloch wave propagation in photonic crystals,' J. Opt. Soc. Am. B 22, 1179-1190 (2005).
[CrossRef]

W. Jiang, R. T. Chen, and X. Lu, 'Theory of light refraction at the surface of a photonic crystal,' Phys. Rev. E 71, 245115-1-245115-12 (2005).

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, 'Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,' Phys. Rev. Lett. 94, 123901.1-123901.4 (2005).
[CrossRef]

2004 (3)

A. Lupu, E. Cassan, S. Laval, L. El Melhaoui, P. Lyan, and J.-M. Fedeli, 'Experimental evidence for superprism phenomena in SOI photonic crystals,' Opt. Express 23, 5690-5696 (2004).
[CrossRef]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, 'Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,' Phys. Rev. E 70, 056606-1-056606-13 (2004).
[CrossRef]

T. P. White, L. C. Botten, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, 'Bloch mode scattering matrix methods for modeling extended photonic crystal structures. II. Applications,' Phys. Rev. E 70, 056607-1-056607-10 (2004).
[CrossRef]

2003 (2)

2002 (1)

L. Wu, M. Mazilu, T. Karle, and T. Krauss, 'Superprism phenomena in planar photonic crystals,' IEEE J. Quantum Electron. 38, 915-918 (2002).
[CrossRef]

2001 (1)

Adibi, A.

Asatryan, A. A.

T. P. White, L. C. Botten, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, 'Bloch mode scattering matrix methods for modeling extended photonic crystal structures. II. Applications,' Phys. Rev. E 70, 056607-1-056607-10 (2004).
[CrossRef]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, 'Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,' Phys. Rev. E 70, 056606-1-056606-13 (2004).
[CrossRef]

Askari, M.

Benisty, H.

J.-M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, and A. Tchelnokov, Photonic Crystals: Towards Nanoscale Photonic Devices (Springer, 2005).

Berger, V.

J.-M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, and A. Tchelnokov, Photonic Crystals: Towards Nanoscale Photonic Devices (Springer, 2005).

Bogaerts, W.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, 'Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,' Phys. Rev. Lett. 94, 123901.1-123901.4 (2005).
[CrossRef]

Botten, L. C.

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, 'Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,' Phys. Rev. E 70, 056606-1-056606-13 (2004).
[CrossRef]

T. P. White, L. C. Botten, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, 'Bloch mode scattering matrix methods for modeling extended photonic crystal structures. II. Applications,' Phys. Rev. E 70, 056607-1-056607-10 (2004).
[CrossRef]

Cassan, E.

A. Lupu, E. Cassan, S. Laval, L. El Melhaoui, P. Lyan, and J.-M. Fedeli, 'Experimental evidence for superprism phenomena in SOI photonic crystals,' Opt. Express 23, 5690-5696 (2004).
[CrossRef]

Chen, R. T.

W. Jiang, R. T. Chen, and X. Lu, 'Theory of light refraction at the surface of a photonic crystal,' Phys. Rev. E 71, 245115-1-245115-12 (2005).

de Sterke, C. M.

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, 'Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,' Phys. Rev. E 70, 056606-1-056606-13 (2004).
[CrossRef]

T. P. White, L. C. Botten, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, 'Bloch mode scattering matrix methods for modeling extended photonic crystal structures. II. Applications,' Phys. Rev. E 70, 056607-1-056607-10 (2004).
[CrossRef]

Dunbar, L. A.

El Melhaoui, L.

A. Lupu, E. Cassan, S. Laval, L. El Melhaoui, P. Lyan, and J.-M. Fedeli, 'Experimental evidence for superprism phenomena in SOI photonic crystals,' Opt. Express 23, 5690-5696 (2004).
[CrossRef]

Engelen, R. J. P.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, 'Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,' Phys. Rev. Lett. 94, 123901.1-123901.4 (2005).
[CrossRef]

Fedeli, J.-M.

A. Lupu, E. Cassan, S. Laval, L. El Melhaoui, P. Lyan, and J.-M. Fedeli, 'Experimental evidence for superprism phenomena in SOI photonic crystals,' Opt. Express 23, 5690-5696 (2004).
[CrossRef]

Ferrini, R.

Gérard, J.-M.

J.-M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, and A. Tchelnokov, Photonic Crystals: Towards Nanoscale Photonic Devices (Springer, 2005).

Gersen, H.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, 'Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,' Phys. Rev. Lett. 94, 123901.1-123901.4 (2005).
[CrossRef]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, 2005).

Houdré, R.

Huang, J.

Jiang, W.

W. Jiang, R. T. Chen, and X. Lu, 'Theory of light refraction at the surface of a photonic crystal,' Phys. Rev. E 71, 245115-1-245115-12 (2005).

Joannopoulos, J. D.

Johnson, S. G.

Karle, T.

L. Wu, M. Mazilu, T. Karle, and T. Krauss, 'Superprism phenomena in planar photonic crystals,' IEEE J. Quantum Electron. 38, 915-918 (2002).
[CrossRef]

Karle, T. J.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, 'Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,' Phys. Rev. Lett. 94, 123901.1-123901.4 (2005).
[CrossRef]

Korterik, J. P.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, 'Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,' Phys. Rev. Lett. 94, 123901.1-123901.4 (2005).
[CrossRef]

Krauss, T.

L. Wu, M. Mazilu, T. Karle, and T. Krauss, 'Superprism phenomena in planar photonic crystals,' IEEE J. Quantum Electron. 38, 915-918 (2002).
[CrossRef]

Krauss, T. F.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, 'Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,' Phys. Rev. Lett. 94, 123901.1-123901.4 (2005).
[CrossRef]

T. F. Krauss, 'Planar photonic crystal waveguide devices for integrated optics,' Phys. Status Solidi A 197, 688-702 (2003).
[CrossRef]

Kuipers, L.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, 'Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,' Phys. Rev. Lett. 94, 123901.1-123901.4 (2005).
[CrossRef]

Langtry, T. N.

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, 'Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,' Phys. Rev. E 70, 056606-1-056606-13 (2004).
[CrossRef]

T. P. White, L. C. Botten, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, 'Bloch mode scattering matrix methods for modeling extended photonic crystal structures. II. Applications,' Phys. Rev. E 70, 056607-1-056607-10 (2004).
[CrossRef]

Laval, S.

A. Lupu, E. Cassan, S. Laval, L. El Melhaoui, P. Lyan, and J.-M. Fedeli, 'Experimental evidence for superprism phenomena in SOI photonic crystals,' Opt. Express 23, 5690-5696 (2004).
[CrossRef]

Lombardet, B.

Lourtioz, J.-M.

J.-M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, and A. Tchelnokov, Photonic Crystals: Towards Nanoscale Photonic Devices (Springer, 2005).

Lu, X.

W. Jiang, R. T. Chen, and X. Lu, 'Theory of light refraction at the surface of a photonic crystal,' Phys. Rev. E 71, 245115-1-245115-12 (2005).

Lupu, A.

A. Lupu, E. Cassan, S. Laval, L. El Melhaoui, P. Lyan, and J.-M. Fedeli, 'Experimental evidence for superprism phenomena in SOI photonic crystals,' Opt. Express 23, 5690-5696 (2004).
[CrossRef]

Lyan, P.

A. Lupu, E. Cassan, S. Laval, L. El Melhaoui, P. Lyan, and J.-M. Fedeli, 'Experimental evidence for superprism phenomena in SOI photonic crystals,' Opt. Express 23, 5690-5696 (2004).
[CrossRef]

Maystre, D.

J.-M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, and A. Tchelnokov, Photonic Crystals: Towards Nanoscale Photonic Devices (Springer, 2005).

Mazilu, M.

L. Wu, M. Mazilu, T. Karle, and T. Krauss, 'Superprism phenomena in planar photonic crystals,' IEEE J. Quantum Electron. 38, 915-918 (2002).
[CrossRef]

McNab, S. J.

McPhedran, R. C.

T. P. White, L. C. Botten, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, 'Bloch mode scattering matrix methods for modeling extended photonic crystal structures. II. Applications,' Phys. Rev. E 70, 056607-1-056607-10 (2004).
[CrossRef]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, 'Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,' Phys. Rev. E 70, 056606-1-056606-13 (2004).
[CrossRef]

Mohammadi, S.

Moll, N.

Momeni, B.

Rakhshandehroo, M.

Soltani, M.

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, 2005).

Tchelnokov, A.

J.-M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, and A. Tchelnokov, Photonic Crystals: Towards Nanoscale Photonic Devices (Springer, 2005).

van Hulst, N. F.

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, 'Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,' Phys. Rev. Lett. 94, 123901.1-123901.4 (2005).
[CrossRef]

Vlasov, Y.

White, T. P.

T. P. White, L. C. Botten, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, 'Bloch mode scattering matrix methods for modeling extended photonic crystal structures. II. Applications,' Phys. Rev. E 70, 056607-1-056607-10 (2004).
[CrossRef]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, 'Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,' Phys. Rev. E 70, 056606-1-056606-13 (2004).
[CrossRef]

Wu, L.

L. Wu, M. Mazilu, T. Karle, and T. Krauss, 'Superprism phenomena in planar photonic crystals,' IEEE J. Quantum Electron. 38, 915-918 (2002).
[CrossRef]

IEEE J. Quantum Electron. (1)

L. Wu, M. Mazilu, T. Karle, and T. Krauss, 'Superprism phenomena in planar photonic crystals,' IEEE J. Quantum Electron. 38, 915-918 (2002).
[CrossRef]

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

Opt. Express (4)

Phys. Rev. E (3)

W. Jiang, R. T. Chen, and X. Lu, 'Theory of light refraction at the surface of a photonic crystal,' Phys. Rev. E 71, 245115-1-245115-12 (2005).

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, 'Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory,' Phys. Rev. E 70, 056606-1-056606-13 (2004).
[CrossRef]

T. P. White, L. C. Botten, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, 'Bloch mode scattering matrix methods for modeling extended photonic crystal structures. II. Applications,' Phys. Rev. E 70, 056607-1-056607-10 (2004).
[CrossRef]

Phys. Rev. Lett. (1)

H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, 'Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,' Phys. Rev. Lett. 94, 123901.1-123901.4 (2005).
[CrossRef]

Phys. Status Solidi A (1)

T. F. Krauss, 'Planar photonic crystal waveguide devices for integrated optics,' Phys. Status Solidi A 197, 688-702 (2003).
[CrossRef]

Other (3)

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, 2005).

Crystalwave software, http://www.photond.com/products/crystalwave.htm.

J.-M. Lourtioz, H. Benisty, V. Berger, J.-M. Gérard, D. Maystre, and A. Tchelnokov, Photonic Crystals: Towards Nanoscale Photonic Devices (Springer, 2005).

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Figures (6)

Fig. 1
Fig. 1

Schematic picture of the studied refraction problem.

Fig. 2
Fig. 2

Frequency band diagram of the 2D triangular lattice PC configuration studied in the paper.

Fig. 3
Fig. 3

EFS obtained at λ = 1320 nm ( a λ = 0.3485 ) for bands 2 and 3. The wave vector conservation condition is shown for a 30 ° input angle incident wavevector (zeroth and first order are shown). The EFS of the homogeneous medium is also shown.

Fig. 4
Fig. 4

Refraction angles of Bloch waves excited in the PC area as a function of plane wave light input angle at λ = 1320 nm .

Fig. 5
Fig. 5

Study of light refraction for θ i = 30 ° at λ = 1320 nm : (a) FDTD steady-state E-field map (the arrow indicates the direction of light group velocity calculated using PWM), (b) decomposition of Bloch waves into a series of electromagnetic plane waves; relative power weights ( h n , m ( k ) 2 ) are indicated in percents.

Fig. 6
Fig. 6

(Color online) Study of light refraction for θ i = 20 ° at λ = 1320 nm : (a) FDTD steady-state E-field map (arrows indicate the direction of light group velocity calculated using PWM), (b) decomposition of Bloch waves into a series of electromagnetic plane waves; relative power weights ( h n , m ( k ) 2 ) are indicated in percents for bands 2 and 3.

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

H k ( r ) n , m h n , m ( k ) H 0 exp ( j k n , m r ) .
k n , m = k + G n , m = k + n b 1 + m b 2 .
η n , m ( k ) = h n , m ( k ) 2 .
v G = n , m h n , m ( k ) 2 v n , m ,
v n , m = ω k n , m 2 k n , m being the phase velocity of plane wave ( n , m ) .
n , m κ n n , m m × k n , m k n , m × h n , m ( k ) = ( ω c ) 2 h n , m ( k ) .

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