Abstract

We investigate the use of two-dimensional photonic crystal slabs to improve the directionality of output coupling from planar waveguides and distributed-feedback lasers. We present the theory underlying the operation of such structures and design criteria for emission in desired directions. As an example, we demonstrate a vertical coupler that is integrated with an organic distributed-feedback laser, use computer simulations to find its coupling constant and efficiency, and then discuss its feasibility.

© 2000 Optical Society of America

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References

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  1. A. Katzir, A. C. Livanos, and A. Yariv, Appl. Phys. Lett. 30, 225 (1977).
    [CrossRef]
  2. E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997).
  3. D. Heitmann and R. V. Pole, Appl. Phys. Lett. 37, 585 (1980).
    [CrossRef]
  4. G. Hatakoshi, H. Fujima, and K. Goto, Appl. Opt. 23, 1749 (1984).
    [CrossRef]
  5. P-P. Borsboom and H. J. Frankena, J. Opt. Soc. Am. A 12, 1142 (1995).
    [CrossRef]
  6. M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, Appl. Phys. Lett. 75, 316 (1999).
    [CrossRef]
  7. M. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, Appl. Phys. Lett. 74, 7 (1999).
    [CrossRef]
  8. M. Meier, A. Dodabalapur, J. A. Rogers, R. E. Slusher, A. Timko, A. Mekis, C. A. Murray, R. Ruel, and O. Nalamasu, J. Appl. Phys. 86, 3502 (1999).
    [CrossRef]
  9. A. Dodabalapur, M. Berggren, R. E. Slusher, Z. Bao, A. Timko, P. Schiortino, E. Laskowski, H. E. Katz, and O. Nalamasu, IEEE J. Sel. Topics Quantum Electron. 4, 67 (1998).
    [CrossRef]
  10. H. Kogelnik, in Guided-Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1988), p. 56.
  11. See, for example, K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC, Boca Raton, Fla.1993).
  12. G. Mur, IEEE Trans. Electromag. Comput. EMC-23, 377 (1981).
    [CrossRef]
  13. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, San Diego, Calif., 1974).

1999 (3)

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, Appl. Phys. Lett. 75, 316 (1999).
[CrossRef]

M. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, Appl. Phys. Lett. 74, 7 (1999).
[CrossRef]

M. Meier, A. Dodabalapur, J. A. Rogers, R. E. Slusher, A. Timko, A. Mekis, C. A. Murray, R. Ruel, and O. Nalamasu, J. Appl. Phys. 86, 3502 (1999).
[CrossRef]

1998 (1)

A. Dodabalapur, M. Berggren, R. E. Slusher, Z. Bao, A. Timko, P. Schiortino, E. Laskowski, H. E. Katz, and O. Nalamasu, IEEE J. Sel. Topics Quantum Electron. 4, 67 (1998).
[CrossRef]

1995 (1)

P-P. Borsboom and H. J. Frankena, J. Opt. Soc. Am. A 12, 1142 (1995).
[CrossRef]

1984 (1)

G. Hatakoshi, H. Fujima, and K. Goto, Appl. Opt. 23, 1749 (1984).
[CrossRef]

1981 (1)

G. Mur, IEEE Trans. Electromag. Comput. EMC-23, 377 (1981).
[CrossRef]

1980 (1)

D. Heitmann and R. V. Pole, Appl. Phys. Lett. 37, 585 (1980).
[CrossRef]

1977 (1)

A. Katzir, A. C. Livanos, and A. Yariv, Appl. Phys. Lett. 30, 225 (1977).
[CrossRef]

Bao, Z.

A. Dodabalapur, M. Berggren, R. E. Slusher, Z. Bao, A. Timko, P. Schiortino, E. Laskowski, H. E. Katz, and O. Nalamasu, IEEE J. Sel. Topics Quantum Electron. 4, 67 (1998).
[CrossRef]

Berggren, M.

A. Dodabalapur, M. Berggren, R. E. Slusher, Z. Bao, A. Timko, P. Schiortino, E. Laskowski, H. E. Katz, and O. Nalamasu, IEEE J. Sel. Topics Quantum Electron. 4, 67 (1998).
[CrossRef]

Borsboom, P-P.

P-P. Borsboom and H. J. Frankena, J. Opt. Soc. Am. A 12, 1142 (1995).
[CrossRef]

Chutinan, A.

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, Appl. Phys. Lett. 75, 316 (1999).
[CrossRef]

Dodabalapur, A.

M. Meier, A. Dodabalapur, J. A. Rogers, R. E. Slusher, A. Timko, A. Mekis, C. A. Murray, R. Ruel, and O. Nalamasu, J. Appl. Phys. 86, 3502 (1999).
[CrossRef]

M. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, Appl. Phys. Lett. 74, 7 (1999).
[CrossRef]

A. Dodabalapur, M. Berggren, R. E. Slusher, Z. Bao, A. Timko, P. Schiortino, E. Laskowski, H. E. Katz, and O. Nalamasu, IEEE J. Sel. Topics Quantum Electron. 4, 67 (1998).
[CrossRef]

Frankena, H. J.

P-P. Borsboom and H. J. Frankena, J. Opt. Soc. Am. A 12, 1142 (1995).
[CrossRef]

Fujima, H.

G. Hatakoshi, H. Fujima, and K. Goto, Appl. Opt. 23, 1749 (1984).
[CrossRef]

Goto, K.

G. Hatakoshi, H. Fujima, and K. Goto, Appl. Opt. 23, 1749 (1984).
[CrossRef]

Hatakoshi, G.

G. Hatakoshi, H. Fujima, and K. Goto, Appl. Opt. 23, 1749 (1984).
[CrossRef]

Heitmann, D.

D. Heitmann and R. V. Pole, Appl. Phys. Lett. 37, 585 (1980).
[CrossRef]

Imada, M.

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, Appl. Phys. Lett. 75, 316 (1999).
[CrossRef]

Joannopoulos, J. D.

M. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, Appl. Phys. Lett. 74, 7 (1999).
[CrossRef]

Katz, H. E.

A. Dodabalapur, M. Berggren, R. E. Slusher, Z. Bao, A. Timko, P. Schiortino, E. Laskowski, H. E. Katz, and O. Nalamasu, IEEE J. Sel. Topics Quantum Electron. 4, 67 (1998).
[CrossRef]

Katzir, A.

A. Katzir, A. C. Livanos, and A. Yariv, Appl. Phys. Lett. 30, 225 (1977).
[CrossRef]

Kogelnik, H.

H. Kogelnik, in Guided-Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1988), p. 56.

Kunz, K. S.

See, for example, K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC, Boca Raton, Fla.1993).

Laskowski, E.

A. Dodabalapur, M. Berggren, R. E. Slusher, Z. Bao, A. Timko, P. Schiortino, E. Laskowski, H. E. Katz, and O. Nalamasu, IEEE J. Sel. Topics Quantum Electron. 4, 67 (1998).
[CrossRef]

Livanos, A. C.

A. Katzir, A. C. Livanos, and A. Yariv, Appl. Phys. Lett. 30, 225 (1977).
[CrossRef]

Loewen, E. G.

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997).

Luebbers, R. J.

See, for example, K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC, Boca Raton, Fla.1993).

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, San Diego, Calif., 1974).

Meier, M.

M. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, Appl. Phys. Lett. 74, 7 (1999).
[CrossRef]

M. Meier, A. Dodabalapur, J. A. Rogers, R. E. Slusher, A. Timko, A. Mekis, C. A. Murray, R. Ruel, and O. Nalamasu, J. Appl. Phys. 86, 3502 (1999).
[CrossRef]

Mekis, A.

M. Meier, A. Dodabalapur, J. A. Rogers, R. E. Slusher, A. Timko, A. Mekis, C. A. Murray, R. Ruel, and O. Nalamasu, J. Appl. Phys. 86, 3502 (1999).
[CrossRef]

M. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, Appl. Phys. Lett. 74, 7 (1999).
[CrossRef]

Mur, G.

G. Mur, IEEE Trans. Electromag. Comput. EMC-23, 377 (1981).
[CrossRef]

Murata, M.

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, Appl. Phys. Lett. 75, 316 (1999).
[CrossRef]

Murray, C. A.

M. Meier, A. Dodabalapur, J. A. Rogers, R. E. Slusher, A. Timko, A. Mekis, C. A. Murray, R. Ruel, and O. Nalamasu, J. Appl. Phys. 86, 3502 (1999).
[CrossRef]

Nalamasu, O.

M. Meier, A. Dodabalapur, J. A. Rogers, R. E. Slusher, A. Timko, A. Mekis, C. A. Murray, R. Ruel, and O. Nalamasu, J. Appl. Phys. 86, 3502 (1999).
[CrossRef]

M. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, Appl. Phys. Lett. 74, 7 (1999).
[CrossRef]

A. Dodabalapur, M. Berggren, R. E. Slusher, Z. Bao, A. Timko, P. Schiortino, E. Laskowski, H. E. Katz, and O. Nalamasu, IEEE J. Sel. Topics Quantum Electron. 4, 67 (1998).
[CrossRef]

Noda, S.

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, Appl. Phys. Lett. 75, 316 (1999).
[CrossRef]

Pole, R. V.

D. Heitmann and R. V. Pole, Appl. Phys. Lett. 37, 585 (1980).
[CrossRef]

Popov, E.

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997).

Rogers, J. A.

M. Meier, A. Dodabalapur, J. A. Rogers, R. E. Slusher, A. Timko, A. Mekis, C. A. Murray, R. Ruel, and O. Nalamasu, J. Appl. Phys. 86, 3502 (1999).
[CrossRef]

Ruel, R.

M. Meier, A. Dodabalapur, J. A. Rogers, R. E. Slusher, A. Timko, A. Mekis, C. A. Murray, R. Ruel, and O. Nalamasu, J. Appl. Phys. 86, 3502 (1999).
[CrossRef]

Sasaki, G.

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, Appl. Phys. Lett. 75, 316 (1999).
[CrossRef]

Schiortino, P.

A. Dodabalapur, M. Berggren, R. E. Slusher, Z. Bao, A. Timko, P. Schiortino, E. Laskowski, H. E. Katz, and O. Nalamasu, IEEE J. Sel. Topics Quantum Electron. 4, 67 (1998).
[CrossRef]

Slusher, R. E.

M. Meier, A. Dodabalapur, J. A. Rogers, R. E. Slusher, A. Timko, A. Mekis, C. A. Murray, R. Ruel, and O. Nalamasu, J. Appl. Phys. 86, 3502 (1999).
[CrossRef]

M. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, Appl. Phys. Lett. 74, 7 (1999).
[CrossRef]

A. Dodabalapur, M. Berggren, R. E. Slusher, Z. Bao, A. Timko, P. Schiortino, E. Laskowski, H. E. Katz, and O. Nalamasu, IEEE J. Sel. Topics Quantum Electron. 4, 67 (1998).
[CrossRef]

Timko, A.

M. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, Appl. Phys. Lett. 74, 7 (1999).
[CrossRef]

M. Meier, A. Dodabalapur, J. A. Rogers, R. E. Slusher, A. Timko, A. Mekis, C. A. Murray, R. Ruel, and O. Nalamasu, J. Appl. Phys. 86, 3502 (1999).
[CrossRef]

A. Dodabalapur, M. Berggren, R. E. Slusher, Z. Bao, A. Timko, P. Schiortino, E. Laskowski, H. E. Katz, and O. Nalamasu, IEEE J. Sel. Topics Quantum Electron. 4, 67 (1998).
[CrossRef]

Tokuda, T.

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, Appl. Phys. Lett. 75, 316 (1999).
[CrossRef]

Yariv, A.

A. Katzir, A. C. Livanos, and A. Yariv, Appl. Phys. Lett. 30, 225 (1977).
[CrossRef]

Appl. Opt. (1)

G. Hatakoshi, H. Fujima, and K. Goto, Appl. Opt. 23, 1749 (1984).
[CrossRef]

Appl. Phys. Lett. (4)

A. Katzir, A. C. Livanos, and A. Yariv, Appl. Phys. Lett. 30, 225 (1977).
[CrossRef]

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, Appl. Phys. Lett. 75, 316 (1999).
[CrossRef]

M. Meier, A. Mekis, A. Dodabalapur, A. Timko, R. E. Slusher, J. D. Joannopoulos, and O. Nalamasu, Appl. Phys. Lett. 74, 7 (1999).
[CrossRef]

D. Heitmann and R. V. Pole, Appl. Phys. Lett. 37, 585 (1980).
[CrossRef]

IEEE J. Sel. Topics Quantum Electron. (1)

A. Dodabalapur, M. Berggren, R. E. Slusher, Z. Bao, A. Timko, P. Schiortino, E. Laskowski, H. E. Katz, and O. Nalamasu, IEEE J. Sel. Topics Quantum Electron. 4, 67 (1998).
[CrossRef]

IEEE Trans. Electromag. Comput. (1)

G. Mur, IEEE Trans. Electromag. Comput. EMC-23, 377 (1981).
[CrossRef]

J. Appl. Phys. (1)

M. Meier, A. Dodabalapur, J. A. Rogers, R. E. Slusher, A. Timko, A. Mekis, C. A. Murray, R. Ruel, and O. Nalamasu, J. Appl. Phys. 86, 3502 (1999).
[CrossRef]

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

P-P. Borsboom and H. J. Frankena, J. Opt. Soc. Am. A 12, 1142 (1995).
[CrossRef]

Other (4)

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997).

H. Kogelnik, in Guided-Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1988), p. 56.

See, for example, K. S. Kunz and R. J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC, Boca Raton, Fla.1993).

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, San Diego, Calif., 1974).

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Figures (4)

Fig. 1
Fig. 1

Schematic of light output from three types of grating coupler. The arrows pointing to the right the guides show the guided-mode propagation direction, and the arrows out of plane indicate the light-output directions. Only a small section of the grating is shown. (a) One-dimensional grating coupler. (b) Grating lens. (c) Two-dimensional PC coupler.

Fig. 2
Fig. 2

Reciprocal lattice for a two-dimensional grating. Shown are the guided-mode wave vector k, two possible reciprocal lattice vectors, G and G, that satisfy the phase-matching condition, and the in-plane components of the two corresponding radiation-mode wave vectors K and K.

Fig. 3
Fig. 3

(a) Horizontal cross section of the unit cell of the grating in the grating region. (b) Vertical view of the unit cell.

Fig. 4
Fig. 4

Solid curve, coupling constant for the structure in Fig. 3, with lattice constant a=400 nm. Dashed curve, ratio of power coupled into air to total power coupled out for the same structure. The circles are calculated values, and the curves are interpolated curves.

Equations (4)

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

k=K+G,
sin θ=λk-G2πn,
P=vgE/Lx,
α=-12PdPdx=-12PPsLx=12vgdln Edt.

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