Abstract

We derive that, in all-pass (unity amplitude transmittance or reflectance) one-dimensional layered structures, the tangent of the group-propagation angle, the group delay, and the stored energy are approximately proportional. Thus photonic crystal superprisms and spatial dispersion in nonperiodic photonic nanostructures are generally related to wavelength-dependent stored energy in the stack.

© 2005 Optical Society of America

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  1. R. Zengerle, J. Mod. Opt. 34, 1589 (1987).
    [CrossRef]
  2. B. E. Nelson, M. Gerken, D. A.B. Miller, R. Piestun, C.-C. Lin, and J. S. Harris, Opt. Lett. 25, 1502 (2000).
    [CrossRef]
  3. T. Baba and M. Nakamura, IEEE J. Quantum Electron. 38, 909 (2002).
    [CrossRef]
  4. L. Wu, N. Mazilu, T. Karle, and T. F. Krauss, IEEE J. Quantum Electron. 38, 915 (2002).
    [CrossRef]
  5. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sata, and S. Kawakami, Phys. Rev. B 58, R10096 (1998).
    [CrossRef]
  6. M. Gerken and D. A.B. Miller, Appl. Opt. 42, 1330 (2003).
    [CrossRef] [PubMed]
  7. M. Gerken and D. A.B. Miller, papers presented at the Materials Research Society’s Spring Meeting, San Francisco, Calif. (April 21–25, 2003).
  8. A. Yariv and P. Yeh, J. Opt. Soc. Am. 67, 438 (1977).
    [CrossRef]
  9. M. Centini, C. Sibilia, M. Scalora, G. D’Aguannol, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, Phys. Rev. E 60, 4891 (1999).
    [CrossRef]
  10. M. Gerken, “Wavelength multiplexing by spatial beam shifting in multilayer thin-film structures,” Ph.D. dissertation (Stanford University, 2003).
  11. G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, Phys. Rev. E 63, 036610 (2001).
    [CrossRef]
  12. P. Penfield, R. Spence, and S. Duinker, Tellegen’s Theorem and Electrical Networks (MIT Press, 1970).
  13. C. Ernst, V. Postoyalko, and N. G. Khan, IEEE Trans. Microwave Theory Tech. 49, 192 (2001).
    [CrossRef]
  14. H. G. Winful, Phys. Rev. E 68, 016615 (2003).
    [CrossRef]
  15. G. D’Aguanno, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, Phys. Rev. E 70, 016612 (2004).
    [CrossRef]

2004 (1)

G. D’Aguanno, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, Phys. Rev. E 70, 016612 (2004).
[CrossRef]

2003 (2)

2002 (2)

T. Baba and M. Nakamura, IEEE J. Quantum Electron. 38, 909 (2002).
[CrossRef]

L. Wu, N. Mazilu, T. Karle, and T. F. Krauss, IEEE J. Quantum Electron. 38, 915 (2002).
[CrossRef]

2001 (2)

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, Phys. Rev. E 63, 036610 (2001).
[CrossRef]

C. Ernst, V. Postoyalko, and N. G. Khan, IEEE Trans. Microwave Theory Tech. 49, 192 (2001).
[CrossRef]

2000 (1)

1999 (1)

M. Centini, C. Sibilia, M. Scalora, G. D’Aguannol, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, Phys. Rev. E 60, 4891 (1999).
[CrossRef]

1998 (1)

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sata, and S. Kawakami, Phys. Rev. B 58, R10096 (1998).
[CrossRef]

1987 (1)

R. Zengerle, J. Mod. Opt. 34, 1589 (1987).
[CrossRef]

1977 (1)

Baba, T.

T. Baba and M. Nakamura, IEEE J. Quantum Electron. 38, 909 (2002).
[CrossRef]

Bertolotti, M.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, Phys. Rev. E 63, 036610 (2001).
[CrossRef]

M. Centini, C. Sibilia, M. Scalora, G. D’Aguannol, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, Phys. Rev. E 60, 4891 (1999).
[CrossRef]

Bloemer, M. J.

G. D’Aguanno, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, Phys. Rev. E 70, 016612 (2004).
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, Phys. Rev. E 63, 036610 (2001).
[CrossRef]

M. Centini, C. Sibilia, M. Scalora, G. D’Aguannol, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, Phys. Rev. E 60, 4891 (1999).
[CrossRef]

Bowden, C. M.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, Phys. Rev. E 63, 036610 (2001).
[CrossRef]

M. Centini, C. Sibilia, M. Scalora, G. D’Aguannol, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, Phys. Rev. E 60, 4891 (1999).
[CrossRef]

Centini, M.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, Phys. Rev. E 63, 036610 (2001).
[CrossRef]

M. Centini, C. Sibilia, M. Scalora, G. D’Aguannol, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, Phys. Rev. E 60, 4891 (1999).
[CrossRef]

D’Aguanno, G.

G. D’Aguanno, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, Phys. Rev. E 70, 016612 (2004).
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, Phys. Rev. E 63, 036610 (2001).
[CrossRef]

D’Aguannol, G.

M. Centini, C. Sibilia, M. Scalora, G. D’Aguannol, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, Phys. Rev. E 60, 4891 (1999).
[CrossRef]

Duinker, S.

P. Penfield, R. Spence, and S. Duinker, Tellegen’s Theorem and Electrical Networks (MIT Press, 1970).

Ernst, C.

C. Ernst, V. Postoyalko, and N. G. Khan, IEEE Trans. Microwave Theory Tech. 49, 192 (2001).
[CrossRef]

Gerken, M.

M. Gerken and D. A.B. Miller, Appl. Opt. 42, 1330 (2003).
[CrossRef] [PubMed]

B. E. Nelson, M. Gerken, D. A.B. Miller, R. Piestun, C.-C. Lin, and J. S. Harris, Opt. Lett. 25, 1502 (2000).
[CrossRef]

M. Gerken, “Wavelength multiplexing by spatial beam shifting in multilayer thin-film structures,” Ph.D. dissertation (Stanford University, 2003).

M. Gerken and D. A.B. Miller, papers presented at the Materials Research Society’s Spring Meeting, San Francisco, Calif. (April 21–25, 2003).

Harris, J. S.

Haus, J. W.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, Phys. Rev. E 63, 036610 (2001).
[CrossRef]

Karle, T.

L. Wu, N. Mazilu, T. Karle, and T. F. Krauss, IEEE J. Quantum Electron. 38, 915 (2002).
[CrossRef]

Kawakami, S.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sata, and S. Kawakami, Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Kawashima, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sata, and S. Kawakami, Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Khan, N. G.

C. Ernst, V. Postoyalko, and N. G. Khan, IEEE Trans. Microwave Theory Tech. 49, 192 (2001).
[CrossRef]

Kosaka, H.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sata, and S. Kawakami, Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Krauss, T. F.

L. Wu, N. Mazilu, T. Karle, and T. F. Krauss, IEEE J. Quantum Electron. 38, 915 (2002).
[CrossRef]

Lin, C.-C.

Mattiucci, N.

G. D’Aguanno, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, Phys. Rev. E 70, 016612 (2004).
[CrossRef]

Mazilu, N.

L. Wu, N. Mazilu, T. Karle, and T. F. Krauss, IEEE J. Quantum Electron. 38, 915 (2002).
[CrossRef]

Miller, D. A.B.

M. Gerken and D. A.B. Miller, Appl. Opt. 42, 1330 (2003).
[CrossRef] [PubMed]

B. E. Nelson, M. Gerken, D. A.B. Miller, R. Piestun, C.-C. Lin, and J. S. Harris, Opt. Lett. 25, 1502 (2000).
[CrossRef]

M. Gerken and D. A.B. Miller, papers presented at the Materials Research Society’s Spring Meeting, San Francisco, Calif. (April 21–25, 2003).

Nakamura, M.

T. Baba and M. Nakamura, IEEE J. Quantum Electron. 38, 909 (2002).
[CrossRef]

Nefedov, I.

M. Centini, C. Sibilia, M. Scalora, G. D’Aguannol, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, Phys. Rev. E 60, 4891 (1999).
[CrossRef]

Nelson, B. E.

Notomi, M.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sata, and S. Kawakami, Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Penfield, P.

P. Penfield, R. Spence, and S. Duinker, Tellegen’s Theorem and Electrical Networks (MIT Press, 1970).

Piestun, R.

Postoyalko, V.

C. Ernst, V. Postoyalko, and N. G. Khan, IEEE Trans. Microwave Theory Tech. 49, 192 (2001).
[CrossRef]

Sata, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sata, and S. Kawakami, Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Scalora, M.

G. D’Aguanno, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, Phys. Rev. E 70, 016612 (2004).
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, Phys. Rev. E 63, 036610 (2001).
[CrossRef]

M. Centini, C. Sibilia, M. Scalora, G. D’Aguannol, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, Phys. Rev. E 60, 4891 (1999).
[CrossRef]

Sibilia, C.

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, Phys. Rev. E 63, 036610 (2001).
[CrossRef]

M. Centini, C. Sibilia, M. Scalora, G. D’Aguannol, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, Phys. Rev. E 60, 4891 (1999).
[CrossRef]

Spence, R.

P. Penfield, R. Spence, and S. Duinker, Tellegen’s Theorem and Electrical Networks (MIT Press, 1970).

Tamamura, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sata, and S. Kawakami, Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Tomita, A.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sata, and S. Kawakami, Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Winful, H. G.

H. G. Winful, Phys. Rev. E 68, 016615 (2003).
[CrossRef]

Wu, L.

L. Wu, N. Mazilu, T. Karle, and T. F. Krauss, IEEE J. Quantum Electron. 38, 915 (2002).
[CrossRef]

Yariv, A.

Yeh, P.

Zengerle, R.

R. Zengerle, J. Mod. Opt. 34, 1589 (1987).
[CrossRef]

Zheltikov, A. M.

G. D’Aguanno, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, Phys. Rev. E 70, 016612 (2004).
[CrossRef]

Appl. Opt. (1)

IEEE J. Quantum Electron. (2)

T. Baba and M. Nakamura, IEEE J. Quantum Electron. 38, 909 (2002).
[CrossRef]

L. Wu, N. Mazilu, T. Karle, and T. F. Krauss, IEEE J. Quantum Electron. 38, 915 (2002).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

C. Ernst, V. Postoyalko, and N. G. Khan, IEEE Trans. Microwave Theory Tech. 49, 192 (2001).
[CrossRef]

J. Mod. Opt. (1)

R. Zengerle, J. Mod. Opt. 34, 1589 (1987).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Lett. (1)

Phys. Rev. B (1)

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sata, and S. Kawakami, Phys. Rev. B 58, R10096 (1998).
[CrossRef]

Phys. Rev. E (4)

M. Centini, C. Sibilia, M. Scalora, G. D’Aguannol, M. Bertolotti, M. J. Bloemer, C. M. Bowden, and I. Nefedov, Phys. Rev. E 60, 4891 (1999).
[CrossRef]

H. G. Winful, Phys. Rev. E 68, 016615 (2003).
[CrossRef]

G. D’Aguanno, N. Mattiucci, M. Scalora, M. J. Bloemer, and A. M. Zheltikov, Phys. Rev. E 70, 016612 (2004).
[CrossRef]

G. D’Aguanno, M. Centini, M. Scalora, C. Sibilia, M. J. Bloemer, C. M. Bowden, J. W. Haus, and M. Bertolotti, Phys. Rev. E 63, 036610 (2001).
[CrossRef]

Other (3)

P. Penfield, R. Spence, and S. Duinker, Tellegen’s Theorem and Electrical Networks (MIT Press, 1970).

M. Gerken, “Wavelength multiplexing by spatial beam shifting in multilayer thin-film structures,” Ph.D. dissertation (Stanford University, 2003).

M. Gerken and D. A.B. Miller, papers presented at the Materials Research Society’s Spring Meeting, San Francisco, Calif. (April 21–25, 2003).

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Figures (5)

Fig. 1
Fig. 1

Superprism effect in a 1-D photonic crystal.

Fig. 2
Fig. 2

Group-propagation angle as a function of wavelength for an infinite 1-D photonic crystal consisting of alternating 167 nm SiO 2 ( n = 1.46 at 830 nm) and 110 nm Ta 2 O 5 ( n = 2.06 ) layers at θ in = 45 ° and of p polarization calculated by Bloch theory.

Fig. 3
Fig. 3

Group velocities v gx and v gz in units of the speed of light, c, as a function of wavelength. Dotted curves were calculated by Bloch theory for the infinite stack in Fig. 2. The solid v gx curve represents a constant-value approximation. The solid v gz curve was calculated from the stored energy in the periodic region of the stack in Fig. 4.

Fig. 4
Fig. 4

(a) Layer thickness as a function of position in the stack for a stack with 120 periods of alternating 167 nm SiO 2 and 110 nm Ta 2 O 5 layers in the center and 40 period impedance-matching regions on either side. The bulk material on either side is assumed to be SiO 2 . (b) Transmittance calculated using transfer matrices.

Fig. 5
Fig. 5

E-field amplitude squared in the x direction of the forward (solid curves) and backward- (dotted curves) propagating waves as a function of the position in the stack from Fig. 4 for three wavelengths.

Equations (3)

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

θ gr = tan 1 ( v gx v gz ) .
τ group = L v gz = L tan ( θ gr ) v gx = s x v gx ,
τ group = W e + W m P = W tot P .

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