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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 10, Iss. 8 — Apr. 22, 2002
  • pp: 354–359
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Modal conversion with artificial materials for photonic-crystal waveguides

Ph. Lalanne and A. Talneau  »View Author Affiliations


Optics Express, Vol. 10, Issue 8, pp. 354-359 (2002)
http://dx.doi.org/10.1364/OE.10.000354


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Abstract

We study adiabatic mode transformations in photonic-crystal integrated circuits composed of a triangular lattice of holes etched into a planar waveguide. The taper relies on the manufacture of holes with progressively-varying dimensions. The variation synthesizes an artificial material with a gradient effective index. Calculations performed with a three-dimensional exact electromagnetic theory yield high transmission over a wide frequency range. To evidence the practical interest of the approach, a mode transformer with a length as small as λ/2 is shown to provide a spectral-averaged transmission efficiency of 92% for tapering between a ridge waveguide and a photonic crystal waveguide with a one-row defect.

© 2002 Optical Society of America

1. Introduction

2. Numerical study

Fig. 1. Tapering with artificial materials between two different PC waveguides. Gray zones correspond to a high refractive-index material. The specific case of a W1–W3 mode conversion is illustrated for a triangular lattice of air holes.
Fig. 2. Geometries considered for testing the taper principle. The lower ridge waveguides are assumed to be illuminated by the fundamental TE00 mode with a unit intensity. Ri and Ti denote the reflected and the transmitted intensities, respectively. (a) situation with tapers. To lower computational loads, only two different holes H1 and H2 are considered for the gradual variation. (b) situation without taper used as a reference. For both situations, the total length of the PC circuit is 18a.

We first estimate the accuracy of the present approach by providing numerical test convergence. Because as many as three PC periods are taken into account on each side of the PC waveguide in the unit cell, see Fig. 2, the Fourier expansion along the y-direction is more demanding in computer resource than that along the x-direction. Figure 4 shows the convergence performance of the method as a function of the truncation ranks for λ = 0.8 μm. We denote by My and Mx the truncation rank along the y- and x- directions. By truncation rank Mα, we mean that a total of 2Mα+1 Fouriers harmonics are retained for the computation along the α-direction. Circles and squares represent T1 data computed respectively as a function of My for Mx = 5, and as a function of Mx for My = 27. From Fig. 4, it is expected that the transmission T1 can be predicted with an accuracy of ≈ 0.01 for My > 25 and Mx > 5. In this specific case with a mirror symmetry along the x-direction, the CPU time on a PC computer equipped with a Pentium IV processor (1.8 Ghz clock frequency) is ≈ 1.3 hour for Mx = 5 and My = 35. A Matlab sofware is used for the calculation without compiling the codes.

Fig. 3. Cross section of a slice used for the computation. Perfect Matched Layers are used at the boundary of the unit cell. The example corresponds to a slice of a W1 waveguide.
Fig. 4. Convergence performance of the present method.
Fig. 5. Periodic waveguides associated to the taper segments. The horizontal line shows the z-plane location of the Bloch-mode profiles shown in Fig. 7.

3. Numerical results

For the following discussion, it is convenient to consider the periodic waveguides associated to the different taper segments. By WH1, we denote the periodic waveguides obtained by adding two rows of holes H1 on each side of a W3 waveguide, see Fig. 5. Similarly, WH2 is obtained by adding two rows of holes H2. In fact, an infinite set of periodic waveguides can be engineered by varying progressively the fillfactor of the added holes. The numerical transmission and reflection spectra for the diffraction problems of Fig. 2 are shown in Fig. 6. The horizontal axis covers the full band-gap spectral region, from 0.78 μm to 0.98 μm. Circles and squares are obtained for the non-tapered and tapered situations, respectively. Except for λ ≈ 0.87 μm, the transmission T2 is very high, the coupling between the ridge and the W3 waveguides is highly efficient. The low transmission at λ ≈ 0.87 μm does not result from any mode-mismatch problem. It is due to the hybridization of the fundamental W3-waveguide mode with a higher back-propagating order mode of W3. The hybridized mode acts as a lossy mirror. This phenomenon that corresponds to the opening of a mini-stop-band has been observed experimentally in Ref. [11

11. C.J.M. Smith, H. Benisty, S. Olivier, M. Rattier, C. Weisbuch, T.F. Krauss, R.M. De La Rue, R. Houdré, and U. Oesterle, “Low-loss channel waveguides with two-dimensional photonic crystal boundaries,” Appl. Phys. Lett. 77, 2813-15 (2000). [CrossRef]

].

Fig. 6. Numerical results for the transmission (top) and reflection (bottom) spectra of the geometries shown in Fig. 2. The wavelength range covers the full band-gap spectral region. Circles : T2, squares : T1. The horizontal line indicate the spectral domain of interest.

Over this spectral region, the minimum and maximum values for T1 are 80% and 90%. On average, the transmission T1 is 85%. The amount of reflected light is rather weak, the maximum value for R1 being 1% with an average value of 0.4%. Thus, approximately 15% of the light is scattered into the substrate and in the air. Two different mechanisms contribute to the loss. For the low-index-contrast waveguide considered in this work, the W1 and W3 waveguides are leaky. The attenuation of the W1 waveguide is ≈10 dB/100μm and that of the W3 waveguide is two order of magnitude smaller [12

12. Ph. Lalanne, “Electromagnetic analysis of photonic crystal waveguides operating above the light cone,” to be published in IEEE Journal of Quantum Electronics.

]. Thus ≈3% of the light is lost by propagation in the 8-row-long W1 waveguide. The main channel for loss is the mode mismatch at the taper section. This mismatch mainly results in radiation into the claddings.

Fig. 7. Bloch-mode profiles for the different taper sections for λ = 0.9 μm. The effective indices (real parts) are given in parentheses. The horizontal dashed lines correspond to the stack interfaces cover/core/substrate. (a), (b), (c) and (d) correspond to the W3, WH1, WH2 and W1 waveguides. The profiles are obtained at longitudinal locations indicated by the horizontal solid lines in Fig. 5.

It is interesting to further consider the electromagnetic quantities that are involved into the tapering process. These quantities are clearly the fundamental TE00 mode supported by the ridge waveguides, and as shown in Ref. [4

4. M. Palamaru and Ph. Lalanne, “Photonic crystal waveguides: out-of-plane losses and adiabatic modal conversion,” Appl. Phys. Lett. 78, 1466-69 (2001). [CrossRef]

], the fundamental Bloch modes associated to the different sections of the taper. These sections are the W3, W1, WH1 and WH3 waveguides of Fig. 5. Figure 7 shows the different mode profiles (magnitude of the square of the x-component of the magnetic-field vector) for λ = 0.9 μm. The real part of the corresponding effective indices are given in parentheses. As shown by a comparison of the different profiles, the modes are progressively transformed. At the same time the effective indices vary monotonically from 3.04 for W1 to 3.33 for W3. The effective index of the ridge waveguide is 3.34. This monotonic variation confirms that the taper acts as a gradient-index thin film.

4. Conclusion

In this work, a new technique for tapering with subwavelength segments has been proposed and validated through 3D computational results. The approach, that exploits the analogy between subwavelength periodic structures and artificial homogeneous media, implements a gradient-effective-index structure through a progressive variation of the feature sizes. Because of the subwavelength periodicity, very short tapers can be implemented. The 3D computational results show that a taper length as short as of λ/2 (two PC periods) can achieve a 92% efficiency for coupling between a conventional ridge waveguide and a PC waveguides with a one-row defect. It should be emphasized that the concept used in this work for adiabatic transformation is general. The triangular lattice with holes may be replaced by other lattice and atom geometries and the approach can be easily generalized to engineered waveguide geometries with improved line defects.

Acknowledgments

The authors acknowledge Pierre Chavel for fruitful comments. Philippe Lalanne is also grateful to Jean Paul Hugonin for programming assistance and for numerous discussions.

References and links

1.

T. F. Krauss, R. M. De La Rue, and S. Brand, “Two-dimensional photonic bandgap structures operating at near-infrared wavelengths,” Nature 383, 699–702 (1996). [CrossRef]

2.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).

3.

Y. Xu, R. K. Lee, and A. Yariv, “Adiabatic coupling between conventional dielectric waveguides and waveguides with discrete translational symmetry,” Opt. Lett. 25, 755–757 (2000) [CrossRef]

4.

M. Palamaru and Ph. Lalanne, “Photonic crystal waveguides: out-of-plane losses and adiabatic modal conversion,” Appl. Phys. Lett. 78, 1466-69 (2001). [CrossRef]

5.

A. Mekis and J.D. Joannopoulos, “Tapered couplers for efficient interfacing between dielectric and photonic crystal waveguides,” J. Lightwave Technol. 19, 861–865 (2001). [CrossRef]

6.

T.D. Happ, M. Kamp, and A. Forchel, “Photonic crystal tapers for ultracompact mode conversion,” Opt. Lett. 26, 1102-04 (2001). [CrossRef]

7.

Z. Weissman and A. Hardy, “2-D mode tapering via tapered channel waveguide segmentation,” Electron. Lett. 28, 1514–1516 (1992). [CrossRef]

8.

E. Silberstein, Ph. Lalanne, J.P. Hugonin, and Q. Cao, “On the use of grating theory in integrated optics,” J. Opt. Soc. Am. A. 18, 2865 (2001). [CrossRef]

9.

N.P.K. Cotter, T.W. Preist, and J.R. Sambles, “Scattering-matrix approach to multilayer diffraction,” J. Opt. Soc. Am. A 12, 1097 (1995). [CrossRef]

10.

J.P Bérenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Computational. Physics , 114, 185–200 (1994). [CrossRef]

11.

C.J.M. Smith, H. Benisty, S. Olivier, M. Rattier, C. Weisbuch, T.F. Krauss, R.M. De La Rue, R. Houdré, and U. Oesterle, “Low-loss channel waveguides with two-dimensional photonic crystal boundaries,” Appl. Phys. Lett. 77, 2813-15 (2000). [CrossRef]

12.

Ph. Lalanne, “Electromagnetic analysis of photonic crystal waveguides operating above the light cone,” to be published in IEEE Journal of Quantum Electronics.

OCIS Codes
(130.3120) Integrated optics : Integrated optics devices
(230.3120) Optical devices : Integrated optics devices

ToC Category:
Research Papers

History
Original Manuscript: March 18, 2002
Revised Manuscript: April 8, 2002
Published: April 22, 2002

Citation
Philippe Lalanne and A. Talneau, "Modal conversion with artificial materials for photonic-crystal waveguides," Opt. Express 10, 354-359 (2002)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-8-354


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References

  1. T. F. Krauss, R. M. De La Rue and S. Brand, �Two-dimensional photonic bandgap structures operating at near-infrared wavelengths,� Nature 383, 699-702 (1996). [CrossRef]
  2. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1972).
  3. Y. Xu, R. K. Lee and A. Yariv, �Adiabatic coupling between conventional dielectric waveguides and waveguides with discrete translational symmetry,� Opt. Lett. 25, 755-757 (2000). [CrossRef]
  4. M. Palamaru and Ph. Lalanne, �Photonic crystal waveguides: out-of-plane losses and adiabatic modal conversion," Appl. Phys. Lett. 78, 1466-69 (2001). [CrossRef]
  5. A. Mekis and J. D. Joannopoulos, �Tapered couplers for efficient interfacing between dielectric and photonic crystal waveguides,� J. Lightwave Technol. 19, 861-865 (2001). [CrossRef]
  6. T. D. Happ, M. Kamp and A. Forchel, �Photonic crystal tapers for ultracompact mode conversion,� Opt. Lett. 26, 1102-04 (2001). [CrossRef]
  7. Z. Weissman and A. Hardy, "2-D mode tapering via tapered channel waveguide segmentation," Electron. Lett. 28, 1514-1516 (1992). [CrossRef]
  8. E. Silberstein, Ph. Lalanne, J. P. Hugonin and Q. Cao, �On the use of grating theory in integrated optics,� J. Opt. Soc. Am. A 18, 2865 (2001). [CrossRef]
  9. N. P. K. Cotter, T. W. Preist, and J. R. Sambles, �Scattering-matrix approach to multilayer diffraction,� J. Opt. Soc. Am. A 12, 1097 (1995). [CrossRef]
  10. J. P B�renger, "A perfectly matched layer for the absorption of electromagnetic waves," J. Computational. Phys. 114, 185-200 (1994). [CrossRef]
  11. C. J. M. Smith, H. Benisty, S. Olivier, M. Rattier, C. Weisbuch, T. F. Krauss, R. M. De La Rue, R. Houdr� and U. Oesterle, �Low-loss channel waveguides with two-dimensional photonic crystal boundaries,� Appl. Phys. Lett. 77, 2813-15 (2000). [CrossRef]
  12. Ph. Lalanne, �Electromagnetic analysis of photonic crystal waveguides operating above the light cone,� to be published in IEEE Journal of Quantum Electronics

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