Abstract

We show how the forbidden bands of a Bragg reflector may shrink to points and how some classes of the guided modes of a planar Bragg waveguide may disappear altogether by shrinking with the forbidden bands. We derive the general conditions to determine the missing modes and explain these conditions with examples. It is possible, for example, to design a Bragg waveguide that rejects all antisymmetric modes and supports only symmetric modes for the TE polarization. We also highlight the effect of Brewster incidence on the interpretation of the missing modes for the TM polarization.

© 2007 Optical Society of America

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  1. P. Yeh, A. Yariv, and C. S. Hong, J. Opt. Soc. Am. A 67, 423 (1977).
    [CrossRef]
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    [CrossRef] [PubMed]
  3. N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, Opt. Lett. 27, 1592 (2002).
    [CrossRef]
  4. B. Nistad, M. W. Haakestad, and J. Skaar, Opt. Commun. 265, 153 (2006).
    [CrossRef]
  5. E. Simova and I. Golub, Opt. Express 11, 3425 (2003).
    [CrossRef] [PubMed]
  6. A. L. Y. Low, Y. S. Yong, A. H. You, S. F. Chien, C. F. Teo, IEEE Photon. Technol. Lett. 16, 1673 (2004).
    [CrossRef]
  7. J. M. Heaton, R. M. Jenkins, D. R. Wight, J. T. Parker, J. C. H. Birbeck, and K. P. Hilton, Appl. Phys. Lett. 61, 1754 (1992).
    [CrossRef]
  8. H. Taniyama, M. Notomi, and Y. Yoshikuni, Phys. Rev. B 71, 153103 (2005).
    [CrossRef]
  9. P. Yeh, A. Yariv, and E. Marom, J. Opt. Soc. Am. A 68, 1196 (1978).
    [CrossRef]
  10. J. C. Knight, Nature 424, 847 (2003).
    [CrossRef] [PubMed]

2006 (1)

B. Nistad, M. W. Haakestad, and J. Skaar, Opt. Commun. 265, 153 (2006).
[CrossRef]

2005 (1)

H. Taniyama, M. Notomi, and Y. Yoshikuni, Phys. Rev. B 71, 153103 (2005).
[CrossRef]

2004 (1)

A. L. Y. Low, Y. S. Yong, A. H. You, S. F. Chien, C. F. Teo, IEEE Photon. Technol. Lett. 16, 1673 (2004).
[CrossRef]

2003 (2)

2002 (1)

1998 (1)

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef] [PubMed]

1992 (1)

J. M. Heaton, R. M. Jenkins, D. R. Wight, J. T. Parker, J. C. H. Birbeck, and K. P. Hilton, Appl. Phys. Lett. 61, 1754 (1992).
[CrossRef]

1978 (1)

P. Yeh, A. Yariv, and E. Marom, J. Opt. Soc. Am. A 68, 1196 (1978).
[CrossRef]

1977 (1)

P. Yeh, A. Yariv, and C. S. Hong, J. Opt. Soc. Am. A 67, 423 (1977).
[CrossRef]

Abeeluck, A. K.

Birbeck, J. C. H.

J. M. Heaton, R. M. Jenkins, D. R. Wight, J. T. Parker, J. C. H. Birbeck, and K. P. Hilton, Appl. Phys. Lett. 61, 1754 (1992).
[CrossRef]

Chen, C.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef] [PubMed]

Chien, S. F.

A. L. Y. Low, Y. S. Yong, A. H. You, S. F. Chien, C. F. Teo, IEEE Photon. Technol. Lett. 16, 1673 (2004).
[CrossRef]

Eggleton, B. J.

Fan, S.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef] [PubMed]

Fink, Y.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef] [PubMed]

Golub, I.

Haakestad, M. W.

B. Nistad, M. W. Haakestad, and J. Skaar, Opt. Commun. 265, 153 (2006).
[CrossRef]

Headley, C.

Heaton, J. M.

J. M. Heaton, R. M. Jenkins, D. R. Wight, J. T. Parker, J. C. H. Birbeck, and K. P. Hilton, Appl. Phys. Lett. 61, 1754 (1992).
[CrossRef]

Hilton, K. P.

J. M. Heaton, R. M. Jenkins, D. R. Wight, J. T. Parker, J. C. H. Birbeck, and K. P. Hilton, Appl. Phys. Lett. 61, 1754 (1992).
[CrossRef]

Hong, C. S.

P. Yeh, A. Yariv, and C. S. Hong, J. Opt. Soc. Am. A 67, 423 (1977).
[CrossRef]

Jenkins, R. M.

J. M. Heaton, R. M. Jenkins, D. R. Wight, J. T. Parker, J. C. H. Birbeck, and K. P. Hilton, Appl. Phys. Lett. 61, 1754 (1992).
[CrossRef]

Joannopoulos, J. D.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef] [PubMed]

Knight, J. C.

J. C. Knight, Nature 424, 847 (2003).
[CrossRef] [PubMed]

Litchinitser, N. M.

Low, A. L. Y.

A. L. Y. Low, Y. S. Yong, A. H. You, S. F. Chien, C. F. Teo, IEEE Photon. Technol. Lett. 16, 1673 (2004).
[CrossRef]

Marom, E.

P. Yeh, A. Yariv, and E. Marom, J. Opt. Soc. Am. A 68, 1196 (1978).
[CrossRef]

Michel, J.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef] [PubMed]

Nistad, B.

B. Nistad, M. W. Haakestad, and J. Skaar, Opt. Commun. 265, 153 (2006).
[CrossRef]

Notomi, M.

H. Taniyama, M. Notomi, and Y. Yoshikuni, Phys. Rev. B 71, 153103 (2005).
[CrossRef]

Parker, J. T.

J. M. Heaton, R. M. Jenkins, D. R. Wight, J. T. Parker, J. C. H. Birbeck, and K. P. Hilton, Appl. Phys. Lett. 61, 1754 (1992).
[CrossRef]

Simova, E.

Skaar, J.

B. Nistad, M. W. Haakestad, and J. Skaar, Opt. Commun. 265, 153 (2006).
[CrossRef]

Taniyama, H.

H. Taniyama, M. Notomi, and Y. Yoshikuni, Phys. Rev. B 71, 153103 (2005).
[CrossRef]

Teo, C. F.

A. L. Y. Low, Y. S. Yong, A. H. You, S. F. Chien, C. F. Teo, IEEE Photon. Technol. Lett. 16, 1673 (2004).
[CrossRef]

Thomas, E. L.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef] [PubMed]

Wight, D. R.

J. M. Heaton, R. M. Jenkins, D. R. Wight, J. T. Parker, J. C. H. Birbeck, and K. P. Hilton, Appl. Phys. Lett. 61, 1754 (1992).
[CrossRef]

Winn, J. N.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef] [PubMed]

Yariv, A.

P. Yeh, A. Yariv, and E. Marom, J. Opt. Soc. Am. A 68, 1196 (1978).
[CrossRef]

P. Yeh, A. Yariv, and C. S. Hong, J. Opt. Soc. Am. A 67, 423 (1977).
[CrossRef]

Yeh, P.

P. Yeh, A. Yariv, and E. Marom, J. Opt. Soc. Am. A 68, 1196 (1978).
[CrossRef]

P. Yeh, A. Yariv, and C. S. Hong, J. Opt. Soc. Am. A 67, 423 (1977).
[CrossRef]

Yong, Y. S.

A. L. Y. Low, Y. S. Yong, A. H. You, S. F. Chien, C. F. Teo, IEEE Photon. Technol. Lett. 16, 1673 (2004).
[CrossRef]

Yoshikuni, Y.

H. Taniyama, M. Notomi, and Y. Yoshikuni, Phys. Rev. B 71, 153103 (2005).
[CrossRef]

You, A. H.

A. L. Y. Low, Y. S. Yong, A. H. You, S. F. Chien, C. F. Teo, IEEE Photon. Technol. Lett. 16, 1673 (2004).
[CrossRef]

Appl. Phys. Lett. (1)

J. M. Heaton, R. M. Jenkins, D. R. Wight, J. T. Parker, J. C. H. Birbeck, and K. P. Hilton, Appl. Phys. Lett. 61, 1754 (1992).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

A. L. Y. Low, Y. S. Yong, A. H. You, S. F. Chien, C. F. Teo, IEEE Photon. Technol. Lett. 16, 1673 (2004).
[CrossRef]

J. Opt. Soc. Am. A (2)

P. Yeh, A. Yariv, and C. S. Hong, J. Opt. Soc. Am. A 67, 423 (1977).
[CrossRef]

P. Yeh, A. Yariv, and E. Marom, J. Opt. Soc. Am. A 68, 1196 (1978).
[CrossRef]

Nature (1)

J. C. Knight, Nature 424, 847 (2003).
[CrossRef] [PubMed]

Opt. Commun. (1)

B. Nistad, M. W. Haakestad, and J. Skaar, Opt. Commun. 265, 153 (2006).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. B (1)

H. Taniyama, M. Notomi, and Y. Yoshikuni, Phys. Rev. B 71, 153103 (2005).
[CrossRef]

Science (1)

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, Science 282, 1679 (1998).
[CrossRef] [PubMed]

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Figures (5)

Fig. 1
Fig. 1

Refractive-index profile of a symmetric planar Bragg waveguide.

Fig. 2
Fig. 2

Variation of χ with the normalized frequency V for n 1 n 2 = 1.5 , d 1 Λ = 0.5 , and b = 0.333 .

Fig. 3
Fig. 3

Dispersion curves for the TE modes of the waveguides with n g = n 2 , n 1 n 2 = 1.5 , and d 1 Λ = 0.5 for three values of d g d 2 : (a) 2.9, (b) 3.0, and (c) 3.1. The modes with m = 3 m 2 1 or m = 2 , 5 , 8 , disappear in (b).

Fig. 4
Fig. 4

Electric-field profiles in the x direction calculated at points (a) 1, (b) 2, and (c) 3. The guiding layer is marked by two dashed vertical lines.

Fig. 5
Fig. 5

Dispersion curves for the TE modes of the waveguide with n g = n 2 , n 1 n 2 = 1.5 , d 1 Λ = 0.5 , and d g d 2 = 2.0 , where only symmetric modes are present and all antisymmetric modes disappear.

Equations (5)

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i k 2 G = { k g tan ( k g d g 2 ) , for symmetric modes k g cot ( k g d g 2 ) , for antisymmetric modes , }
G = ( e i K Λ T 11 T 12 ) ( e i K Λ T 11 + T 12 ) ,
χ = 1 2 [ 1 + 1 2 ( k 1 k 2 + k 2 k 1 ) ] cos ( k 1 d 1 + k 2 d 2 ) + 1 2 [ 1 1 2 ( k 1 k 2 + k 2 k 1 ) ] cos ( k 1 d 1 k 2 d 2 ) .
2 k g d g + 2 Φ = 2 m π , m = integers ,
k 1 d 1 = m 1 π , k 2 d 2 = m 2 π , k g d g = m 3 π .

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