Abstract

Mode coupling phenomena, manifested by transmission “mini-stopbands”, occur in two-dimensional photonic crystal channel waveguides. The huge difference in the group velocities of the coupled modes is a new feature with respect to the classical Bragg reflection occurring, e.g., in distributed feedback lasers. We show that an adequate ansatz of the classical coupled-mode theory remarkably well accounts for this new phenomenon. The fit of experimental transmission data from GaAs-based photonic crystal waveguides then leads to an accurate determination of the propagation losses of both fundamental and higher, low-group-velocity modes.

© 2003 Optical Society of America

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References

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  1. S. Olivier, M. Rattier, H. Benisty, C. Weisbuch, C. J. M. Smith, R. M. De La Rue, T. F. Krauss, U. Oesterle and R. Houdré, �??Mini stopbands of a one dimensional system: the channel waveguide in a two-dimensional photonic crystal,�?? Phys. Rev. B 63, 113311 (2001).
    [CrossRef]
  2. S. Olivier, H. Benisty, C. J. M. Smith, M. Rattier, C. Weisbuch, T. F. Krauss, R. Houdré and U. Oesterle, �??Transmission properties of two-dimensional photonic crystal channel waveguides,�?? Opt. Quantum Electron. 34, 171-181 (2002).
    [CrossRef]
  3. C. J. M. Smith, R. M. De La Rue, M. Rattier, S. Olivier, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdré and U. Oesterle, �??Coupled guide and cavity in a two-dimensional photonic crystal,�?? App. Phys. Lett. 78, 1487-1489 (2001).
    [CrossRef]
  4. H. Benisty, �??Modal analysis of optical guides with two-dimensional photonic band-gap boundaries,�?? J. Appl. Phys. 79, 7483-7492 (1996).
    [CrossRef]
  5. M. Qiu, �??Effective index method for heterostructure-slab-waveguide-based two-dimensional photonic crystals,�?? Appl. Phys. Lett. 81, 1163-1165 (2002).
    [CrossRef]
  6. H. Kogelnick and C.V. Shank, �??Coupled wave theory of distributed feedback lasers,�?? J. Appl. Phys. 43, 2328 (1972).
  7. See for example Tamir, Guided Wave Optoelectronics, Springer Verlag, Berlin, chaps 2,6 (1988).
  8. P. Ferrand, R. Romestain and J.C. Vial, �??Photonic band-gap properties of a porous silicon periodic planar waveguide,�?? Phys. Rev. B 63, 115106 (2001).
    [CrossRef]
  9. E. Peral and A. Yariv, �??Supermodes of grating-coupled multimode waveguides and application to mode conversion between copropagating modes mediated by backward Bragg scattering,�?? J. Lightwave Tech. 17, 942-947 (1999).
    [CrossRef]
  10. D. Labilloy, H. Benisty, C. Weisbuch, T.F. Krauss, R. Houdré and U. Oesterle, �??Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensionam photonic crystals,�?? Appl. Phys. Lett. 71, 738-740 (1997).
    [CrossRef]
  11. E. Schwoob, H. Benisty, S. Olivier, C. Weisbuch, C.J.M Smith, T.F. Krauss, R. Houdre and U. Oesterle, �??Two-mode fringes in planar photonic crystal waveguides with constrictions: a sensitive probe to propagation losses,�?? J. Opt. Soc. Am. B 19, 2403-2412 (2002).
    [CrossRef]
  12. M. Qiu, B. Jaskorzynska, M. Swillo and H. Benisty, �??Time-domain 2D modeling of slab-waveguide-based photonic-crystal devices in the presence of radiation losses,�?? Microwave and Opt. Tech. Lett. 34, 387-393 (2002).
    [CrossRef]

App. Phys. Lett. (1)

C. J. M. Smith, R. M. De La Rue, M. Rattier, S. Olivier, H. Benisty, C. Weisbuch, T. F. Krauss, R. Houdré and U. Oesterle, �??Coupled guide and cavity in a two-dimensional photonic crystal,�?? App. Phys. Lett. 78, 1487-1489 (2001).
[CrossRef]

Appl. Phys. Lett. (2)

M. Qiu, �??Effective index method for heterostructure-slab-waveguide-based two-dimensional photonic crystals,�?? Appl. Phys. Lett. 81, 1163-1165 (2002).
[CrossRef]

D. Labilloy, H. Benisty, C. Weisbuch, T.F. Krauss, R. Houdré and U. Oesterle, �??Use of guided spontaneous emission of a semiconductor to probe the optical properties of two-dimensionam photonic crystals,�?? Appl. Phys. Lett. 71, 738-740 (1997).
[CrossRef]

J. Appl. Phys (1)

H. Kogelnick and C.V. Shank, �??Coupled wave theory of distributed feedback lasers,�?? J. Appl. Phys. 43, 2328 (1972).

J. Appl. Phys. (1)

H. Benisty, �??Modal analysis of optical guides with two-dimensional photonic band-gap boundaries,�?? J. Appl. Phys. 79, 7483-7492 (1996).
[CrossRef]

J. Lightwave Tech. (1)

E. Peral and A. Yariv, �??Supermodes of grating-coupled multimode waveguides and application to mode conversion between copropagating modes mediated by backward Bragg scattering,�?? J. Lightwave Tech. 17, 942-947 (1999).
[CrossRef]

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

Microwave and Opt. Tech. Lett. (1)

M. Qiu, B. Jaskorzynska, M. Swillo and H. Benisty, �??Time-domain 2D modeling of slab-waveguide-based photonic-crystal devices in the presence of radiation losses,�?? Microwave and Opt. Tech. Lett. 34, 387-393 (2002).
[CrossRef]

Opt. Quantum (1)

S. Olivier, H. Benisty, C. J. M. Smith, M. Rattier, C. Weisbuch, T. F. Krauss, R. Houdré and U. Oesterle, �??Transmission properties of two-dimensional photonic crystal channel waveguides,�?? Opt. Quantum Electron. 34, 171-181 (2002).
[CrossRef]

Phys. Rev. B (2)

S. Olivier, M. Rattier, H. Benisty, C. Weisbuch, C. J. M. Smith, R. M. De La Rue, T. F. Krauss, U. Oesterle and R. Houdré, �??Mini stopbands of a one dimensional system: the channel waveguide in a two-dimensional photonic crystal,�?? Phys. Rev. B 63, 113311 (2001).
[CrossRef]

P. Ferrand, R. Romestain and J.C. Vial, �??Photonic band-gap properties of a porous silicon periodic planar waveguide,�?? Phys. Rev. B 63, 115106 (2001).
[CrossRef]

Other (1)

See for example Tamir, Guided Wave Optoelectronics, Springer Verlag, Berlin, chaps 2,6 (1988).

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Figures (4)

Fig. 1.
Fig. 1.

(a) Two-dimensional photonic crystal etched through a vertically monomode semiconductor heterostructure; (b) Micrograph of a three-missing-row PCCW (top view) with period a=260 nm in a GaAs heterostructure.

Fig. 2.
Fig. 2.

(a) Dispersion relations of the W3 waveguide folded into the first Brillouin zone. The waveguide supports up to 6 modes; (b) Local picture of the mini-gap associated to the coupling of the index-guided fundamental mode and the Bragg-guided 5th order mode; (c)–(d) magnetic field patterns associated to the fundamental mode and to the 5th order mode.

Fig. 3.
Fig. 3.

Influence of the propagation losses on the spectral shapes of the transmission spectrum of mode a at the output of the guide and of the reflection spectrum of mode b at the input of the guide, showing the disappearance of secondary lobes.

Fig. 4.
Fig. 4.

(a) Experimental set-up for the selective measurement of the transmission of the fundamental guided mode; (b) Experimental transmission spectra for W3 waveguides of lengths indocated on the graph (coloured curves) and fit using the coupled-mode theory (black curves) with the propagation losses as the only adjustable parameters.

Equations (4)

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d dy ( A ( y , u ) B ( y , u ) ) = ( i δ a ( u ) i κ ab i κ ab i δ b ( u ) ) ( A ( y , u ) B ( y , u ) )
δ a ( u ) = β a 0 β a ( u ) = 2 π a n ga ( u 0 u )
δ b ( u ) = β b 0 β b ( u ) = 2 π a n gb ( u 0 u )
Δ u ab = a 2 π 4 κ ab n ga + n gb

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