Abstract

A theoretical study that follows the lines of previous photonic band structure investigations has been undertaken in search of a two-dimensional (2D) periodic dielectric structure that exhibits a certain type of photonic band structure for application as an external feedback mirror for laser diode arrays and broad-area devices. Instead of looking for an absolute band gap for a band of frequencies, we are particularly interested in structures that exhibit only stop bands, i.e., bands with a narrow extent in both frequency and angle. This type of reflecting structure, which can also be termed a 2D Bragg reflector, could provide improved mode control for both longitudinal and transverse characteristics of the modes for these devices. Our study, which employed photonic band structure calculational methods, showed that square lattice dielectric structures exhibit such stop bands for both TE and TM waves and could be used as photonic band structure mirrors to induce better mode control.

© 1993 Optical Society of America

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References

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  1. E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
    [Crossref] [PubMed]
  2. K. M. Leung and Y. F. Liu, Phys. Rev. Lett. 65, 2646 (1990).
    [Crossref] [PubMed]
  3. Z. Zhang and S. Satpathy, Phys. Rev. Lett. 65, 2650 (1990).
    [Crossref] [PubMed]
  4. K. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990).
    [Crossref] [PubMed]
  5. E. Yablonovitch, T. J. Gmitter, and K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991).
    [Crossref] [PubMed]
  6. S. L. McCall, P. M. Platzmann, R. Dalichaouch, D. Smith, and S. Schultz, Phys. Rev. Lett. 67, 2017 (1991).
    [Crossref] [PubMed]
  7. R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Phys. Rev. B 44, 10961 (1991).
    [Crossref]
  8. M. Plihal, A. Shambrook, A. A. Maradudin, and Ping Sheng, Opt. Commun. 80, 199 (1991).
    [Crossref]
  9. P. R. Villeneuve and M. Piché, J. Opt. Soc. Am. A 8, 1296 (1991).
    [Crossref]
  10. M. Plihal and A. A. Maradudin, Phys. Rev. B 44, 8565 (1991).
    [Crossref]
  11. R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Appl. Phys. Lett. 61, 495 (1992).
    [Crossref]
  12. A. Yariv and P. Yeh, Optical Waves in Crystals(Wiley, New York, 1984), p. 174.
  13. J. Dong, T. Ikeda, S. Arai, and K. Komori, IEEE Photon. Technol. Lett. 4, 491 (1992).
    [Crossref]
  14. K. M. Leung and Y. F. Liu, Phys. Rev. B 41, 10188 (1990).
    [Crossref]

1992 (2)

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Appl. Phys. Lett. 61, 495 (1992).
[Crossref]

J. Dong, T. Ikeda, S. Arai, and K. Komori, IEEE Photon. Technol. Lett. 4, 491 (1992).
[Crossref]

1991 (6)

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991).
[Crossref] [PubMed]

S. L. McCall, P. M. Platzmann, R. Dalichaouch, D. Smith, and S. Schultz, Phys. Rev. Lett. 67, 2017 (1991).
[Crossref] [PubMed]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Phys. Rev. B 44, 10961 (1991).
[Crossref]

M. Plihal, A. Shambrook, A. A. Maradudin, and Ping Sheng, Opt. Commun. 80, 199 (1991).
[Crossref]

M. Plihal and A. A. Maradudin, Phys. Rev. B 44, 8565 (1991).
[Crossref]

P. R. Villeneuve and M. Piché, J. Opt. Soc. Am. A 8, 1296 (1991).
[Crossref]

1990 (4)

K. M. Leung and Y. F. Liu, Phys. Rev. B 41, 10188 (1990).
[Crossref]

K. M. Leung and Y. F. Liu, Phys. Rev. Lett. 65, 2646 (1990).
[Crossref] [PubMed]

Z. Zhang and S. Satpathy, Phys. Rev. Lett. 65, 2650 (1990).
[Crossref] [PubMed]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990).
[Crossref] [PubMed]

1987 (1)

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[Crossref] [PubMed]

Arai, S.

J. Dong, T. Ikeda, S. Arai, and K. Komori, IEEE Photon. Technol. Lett. 4, 491 (1992).
[Crossref]

Brommer, K. D.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Appl. Phys. Lett. 61, 495 (1992).
[Crossref]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Phys. Rev. B 44, 10961 (1991).
[Crossref]

Chan, C. T.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990).
[Crossref] [PubMed]

Dalichaouch, R.

S. L. McCall, P. M. Platzmann, R. Dalichaouch, D. Smith, and S. Schultz, Phys. Rev. Lett. 67, 2017 (1991).
[Crossref] [PubMed]

Dong, J.

J. Dong, T. Ikeda, S. Arai, and K. Komori, IEEE Photon. Technol. Lett. 4, 491 (1992).
[Crossref]

Gmitter, T. J.

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991).
[Crossref] [PubMed]

Ho, K. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990).
[Crossref] [PubMed]

Ikeda, T.

J. Dong, T. Ikeda, S. Arai, and K. Komori, IEEE Photon. Technol. Lett. 4, 491 (1992).
[Crossref]

Joannopoulos, J. D.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Appl. Phys. Lett. 61, 495 (1992).
[Crossref]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Phys. Rev. B 44, 10961 (1991).
[Crossref]

Komori, K.

J. Dong, T. Ikeda, S. Arai, and K. Komori, IEEE Photon. Technol. Lett. 4, 491 (1992).
[Crossref]

Leung, K. M.

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991).
[Crossref] [PubMed]

K. M. Leung and Y. F. Liu, Phys. Rev. Lett. 65, 2646 (1990).
[Crossref] [PubMed]

K. M. Leung and Y. F. Liu, Phys. Rev. B 41, 10188 (1990).
[Crossref]

Liu, Y. F.

K. M. Leung and Y. F. Liu, Phys. Rev. B 41, 10188 (1990).
[Crossref]

K. M. Leung and Y. F. Liu, Phys. Rev. Lett. 65, 2646 (1990).
[Crossref] [PubMed]

Maradudin, A. A.

M. Plihal, A. Shambrook, A. A. Maradudin, and Ping Sheng, Opt. Commun. 80, 199 (1991).
[Crossref]

M. Plihal and A. A. Maradudin, Phys. Rev. B 44, 8565 (1991).
[Crossref]

McCall, S. L.

S. L. McCall, P. M. Platzmann, R. Dalichaouch, D. Smith, and S. Schultz, Phys. Rev. Lett. 67, 2017 (1991).
[Crossref] [PubMed]

Meade, R. D.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Appl. Phys. Lett. 61, 495 (1992).
[Crossref]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Phys. Rev. B 44, 10961 (1991).
[Crossref]

Piché, M.

Platzmann, P. M.

S. L. McCall, P. M. Platzmann, R. Dalichaouch, D. Smith, and S. Schultz, Phys. Rev. Lett. 67, 2017 (1991).
[Crossref] [PubMed]

Plihal, M.

M. Plihal and A. A. Maradudin, Phys. Rev. B 44, 8565 (1991).
[Crossref]

M. Plihal, A. Shambrook, A. A. Maradudin, and Ping Sheng, Opt. Commun. 80, 199 (1991).
[Crossref]

Rappe, A. M.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Appl. Phys. Lett. 61, 495 (1992).
[Crossref]

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Phys. Rev. B 44, 10961 (1991).
[Crossref]

Satpathy, S.

Z. Zhang and S. Satpathy, Phys. Rev. Lett. 65, 2650 (1990).
[Crossref] [PubMed]

Schultz, S.

S. L. McCall, P. M. Platzmann, R. Dalichaouch, D. Smith, and S. Schultz, Phys. Rev. Lett. 67, 2017 (1991).
[Crossref] [PubMed]

Shambrook, A.

M. Plihal, A. Shambrook, A. A. Maradudin, and Ping Sheng, Opt. Commun. 80, 199 (1991).
[Crossref]

Sheng, Ping

M. Plihal, A. Shambrook, A. A. Maradudin, and Ping Sheng, Opt. Commun. 80, 199 (1991).
[Crossref]

Smith, D.

S. L. McCall, P. M. Platzmann, R. Dalichaouch, D. Smith, and S. Schultz, Phys. Rev. Lett. 67, 2017 (1991).
[Crossref] [PubMed]

Soukoulis, C. M.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990).
[Crossref] [PubMed]

Villeneuve, P. R.

Yablonovitch, E.

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991).
[Crossref] [PubMed]

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[Crossref] [PubMed]

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals(Wiley, New York, 1984), p. 174.

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals(Wiley, New York, 1984), p. 174.

Zhang, Z.

Z. Zhang and S. Satpathy, Phys. Rev. Lett. 65, 2650 (1990).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Appl. Phys. Lett. 61, 495 (1992).
[Crossref]

IEEE Photon. Technol. Lett. (1)

J. Dong, T. Ikeda, S. Arai, and K. Komori, IEEE Photon. Technol. Lett. 4, 491 (1992).
[Crossref]

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

Opt. Commun. (1)

M. Plihal, A. Shambrook, A. A. Maradudin, and Ping Sheng, Opt. Commun. 80, 199 (1991).
[Crossref]

Phys. Rev. B (3)

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, Phys. Rev. B 44, 10961 (1991).
[Crossref]

M. Plihal and A. A. Maradudin, Phys. Rev. B 44, 8565 (1991).
[Crossref]

K. M. Leung and Y. F. Liu, Phys. Rev. B 41, 10188 (1990).
[Crossref]

Phys. Rev. Lett. (6)

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[Crossref] [PubMed]

K. M. Leung and Y. F. Liu, Phys. Rev. Lett. 65, 2646 (1990).
[Crossref] [PubMed]

Z. Zhang and S. Satpathy, Phys. Rev. Lett. 65, 2650 (1990).
[Crossref] [PubMed]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 (1990).
[Crossref] [PubMed]

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991).
[Crossref] [PubMed]

S. L. McCall, P. M. Platzmann, R. Dalichaouch, D. Smith, and S. Schultz, Phys. Rev. Lett. 67, 2017 (1991).
[Crossref] [PubMed]

Other (1)

A. Yariv and P. Yeh, Optical Waves in Crystals(Wiley, New York, 1984), p. 174.

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Figures (9)

Fig. 1
Fig. 1

Schematic showing (a) in-phase mode reflected by photonic band structure mirror integrated monolithically with the laser diode array and (b) far-field intensity profile for the in-phase mode. The mode could be outcoupled from a standard mirror (facet) located at the opposite end of the array from the photonic band structure mirror.

Fig. 2
Fig. 2

Schematic showing (a) out-of-phase mode transmitted by photonic band structure mirror integrated monolithically with the laser diode array and (b) far-field intensity profile for the out-of-phase mode.

Fig. 3
Fig. 3

Schematic of 2D rectangular photonic crystal lattice. The lattice constant is c, and the potential width is a. The indices of refraction are na for the center region and nb for the surrounding region. θ is the Fourier component angle of k in the first Brillouin zone, i.e., tan θ = ky/kx.

Fig. 4
Fig. 4

Cross section of approximation (solid curve) to the square-well index-of-refraction variation (dotted curve) obtained by using 21 × 21 reciprocal lattice vectors.

Fig. 5
Fig. 5

Cross section of approximation (solid curve) to the square-well index-of-refraction variation (dotted curve) obtained by using 11 × 11 reciprocal lattice vectors. λmin is the minimum wavelength in the medium for which the approximation has been shown to be valid (i.e., λmin/nb = c/2.5).

Fig. 6
Fig. 6

Photonic band structure, showing the TE-wave stop bands for a square lattice as a function of Fourier component angle, θ, and frequency parameter, nbc/λ, where λ is the free-space wavelength, nb is the host index of refraction, and c is the lattice period. The index contrast ratio is 1.38, and the lattice ratio a/c = 0.922.

Fig. 7
Fig. 7

Photonic band structure, showing TE-wave stop bands; the case is the same as in Fig. 6 except that the index contrast ratio is 1.03 and a/c = 0.493.

Fig. 8
Fig. 8

Stop bands for TE waves in a 1D Bragg reflector, equivalent to the 2D case in Fig. 7.

Fig. 9
Fig. 9

Photonic band structure, showing TM-wave stop bands; the case is the same as Fig. 7. Note the absence of the stop band at the Brewster angle near 45° for first-order Bragg reflection.

Equations (10)

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( 2 x 2 + 2 y 2 ) E z ( x , y ) + k 0 2 n 2 ( x , y ) E z ( x , y ) = 0 ,
f k = d 2 r exp ( - i k · r ) E z ( x , y ) ,
U G = 1 A cell d 2 r exp ( - i G · r ) n 2 ( x , y ) ,
k - G 2 f k - G - k b 2 G U G - G f k - G = 0 ,
k 0 [ n a a + n b ( c - a ) ] cos θ = π ,             0 θ π / 4 ,
n b c λ = 1 2 cos θ [ a c ( n a n b - 1 ) + 1 ] - 1 .
n b c λ = 0.37 cos θ .
k b 2 h k - G = G ( k - G ) · ( k - G ) U G - G h k - G ,
h k = d 2 r exp ( - i k · r ) H ( x , y ) ,
U G = 1 A cell d 2 r exp ( - i G · r ) n - 2 ( x , y ) ,

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