Abstract

The formation of localized states or modes at defects in waveguide arrays is investigated both, theoretically and experimentally. If the effective index or the coupling of the defect guide to its neighbors is varied the number and character of respective modes bound to the defect can be altered. Waveguide arrays may be considered as tailor-made or metamaterials with new and unexpected properties as e.g. guiding staggered modes bound to defects with reduced index. Although the symmetric defect waveguide becomes multimode for increased coupling it does not support antisymmetric modes. All theoretical predictions are confirmed in excellent agreement with experimental observations in polymer waveguide arrays.

© 2003 Optical Society of America

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References

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Appl. Phys. Lett.

G. von Freymanna, W. Koch, D. C. Meisel and M. Wegener, �??Diffraction properties of two-dimensional photonic crystals,�?? Appl. Phys. Lett. 83, 614-616 (2003).
[CrossRef]

S. Somekh, E. Garmire, A. Yariv, H. L. Garvin and R. G. Hunsperger, �??Channel optical waveguide directional couplers,�?? Appl. Phys. Lett. 22, 46-48 (1973).
[CrossRef]

U. Peschel, R. Morandotti, J. S. Aitchison, H. S. Eisenberg and Y. Silberberg, �??Nonlinearly Induced Escape from a Defect State,�?? Appl. Phys. Lett. 75, 1384-1386 (1999).
[CrossRef]

Electron. Lett.

J. C. Knight, T. A. Birks, R. F. Cregan, P. St. J. Russell, and J.-P. de Sandro, �??Large Mode Area Photonic Crystal Fiber,�?? Electron. Lett. 34, 1347-1348 (1998).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Mat.

U. Streppel, P. Dannberg, C. Wächter, A. Bräuer, L. Fröhlich, R. Houbertz and M. Popall, �??New wafer-scale fabrication method for stacked optical waveguide interconnects and 3D micro-optic structures using photo-responsive (inorganic-organic hybrid) polymers,�?? Opt. Mat. 21, 475-483 (2002).
[CrossRef]

Phys. Rev. B

M. Notomi, �??Theory of light propagation in strongly modulated photonic crystals: Refractionlike behavior in the vicinity of the photonic band gap,�?? Phys. Rev. B 62, 10696-10705 (2000).
[CrossRef]

Phys. Rev. Lett.

T. Pertsch, T. Zentgraf, U. Peschel, A. Bräuer and F. Lederer, �??Anomalous Refraction and Diffraction in Discrete Optical Systems,�?? Phys. Rev. Lett. 88, 093901-093904 (2002).
[CrossRef] [PubMed]

T. Pertsch, P. Dannberg, W. Elflein, A. Bräuer and F. Lederer, �??Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays,�?? Phys. Rev. Lett. 83, 4752-4755 (1999).
[CrossRef]

R. Morandotti, U. Peschel, J. S. Aitchison, H. S. Eisenberg and Y. Silberberg, �??Experimental Observation of Linear and Nonlinear Optical Bloch Oscillations,�?? Phys. Rev. Lett. 83, 4756-4759 (1999).
[CrossRef]

Science

P Russell, �??Photonic Crystal Fibers,�?? Science 299, 358-363 (2003).
[CrossRef] [PubMed]

Thin Solid Films

R. Houbertz, G. Domann, C. Cronauer, A. Schmitt, H. Martin, J.-U. Park, L. Fröhlich, R. Buestrich, M. Popall, U. Streppel, P. Dannberg, C. Wächter and A. Bräuer, �??Inorganic-organic hybrid materials for application in optical devices,�?? Thin Solid Films 422, 194-200 (2003)
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

Cross section of a polymer waveguide array with a defect. The defect is introduced by reducing the waveguide spacing compared with the homogenous array.

Fig. 2.
Fig. 2.

Dispersion relation of Bloch modes and the formation of defect modes (a) diffraction relation of Bloch-waves (longitudinal vs. transverse wavenumber) in a homogeneous waveguide array (propagation constant of isolated waveguides: β0, coupling constant C). In the shaded regions only evanescent waves exist. (b) Shift of the band structure and formation of a staggered mode (wavenumber βD) around a defect with a wavenumber reduced by δβ. (c) Expansion of the band structure and formation of staggered and unstaggered modes around a defect with increased coupling (C 1>C) (d) Compression of the band structure around a defect with reduced coupling (C 1<C).

Fig. 3.
Fig. 3.

Field and intensity of a staggered and an unstaggered mode for a dominant change of the propagation constant of the defect. Field (a) and intensity (b) distribution of an unstaggered defect mode for δβ/C=2.0 and C 1/C=0.8 (cross 1 in Fig. 4), solid line: theory, dots: experiment. Field (c) and intensity (d) distribution of a staggered defect mode for δβ/C=-1.4 and C 1/C=1.1 (cross 2 in fig. 4), solid line: theory, dots: experiment.

Fig. 4.
Fig. 4.

Regions of existence for symmetric staggered and unstaggered modes in the (C1/C)2-δβ-plane. The crosses mark the parameters for the experiments (Figs. 3, 5 and 6).

Fig. 5.
Fig. 5.

Interference pattern of a staggered and an unstaggered defect mode for dominant change of the coupling constant (δβ/C=-0.3 and C 1/C=1.4, cross 3 in Fig. 4) of the defect at a propagation distance of 59,95mm. Dots: experiment, lines: theory, dashed line: position of the excitation. (b) Intensity distribution for an excitation of the defect waveguide. (a) and (c) Intensity distribution for an excitation of the left and right nearest neighbor waveguide of the defect. Insets: schematic diagrams of the modal amplitude of the unstaggered and staggered mode, the superposition of both modal fields produces the actual interference pattern.

Fig. 6.
Fig. 6.

Diffraction pattern for an excitation of a repulsive defect ((a) theory, (b) experiment) with reduced coupling (C 1/C=0.5, δβ/C=0, cross 4 in Fig.4) and in a homogeneous array ((c) theory, (d) experiment).

Equations (8)

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( i Z + β 0 ) A n + C ( A n + 1 + A n 1 ) = 0 ,
β = β 0 + 2 C cos ( κ ) .
( i Z + β 0 ) A ± 1 + C 1 A 0 + C A ± 2 = 0 ,
A n ( Z ) = a n exp ( i β D Z )
a ± n = a ± 1 γ n 1 for n 2
γ < 1
β D = β 0 + C ( γ + 1 γ )
1 γ = δβ 2 C ± ( δβ 2 C ) 2 + 2 ( C 1 C ) 2 1 .

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