Abstract

We examine the effects of disorder on propagation loss as a function of group velocity for W1 photonic crystal (PhC) waveguides. Disorder is deliberately and controllably introduced into the photonic crystal by pseudo-randomly displacing the holes of the photonic lattice. This allows us to clearly distinguish two types of loss. Away from the band-edge and for moderately slow light (group velocity c/20-c/30) loss scales sub-linearly with group velocity, whereas near the band-edge, reflection loss increases dramatically due to the random and local shift of the band-edge. The optical analysis also shows that the random fabrication errors of our structures, made on a standard e-beam lithography system, are below 1 nm root mean square.

© 2007 Optical Society of America

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  1. M. Soljačić, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen, and J. D. Joannopoulos, "Photonic-crystal slow-light enhancement of nonlinear phase sensitivity," J. Opt. Soc. Am. B 19, 2052 (2002).
    [CrossRef]
  2. T. F. Krauss, "Slow light in photonic crystal waveguides," J. App. Phys. D. 40, 2666-2670 (2007).
    [CrossRef]
  3. A. Melloni, F. Morichetti and M. Martinelli, "Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures," Opt. Quantum Electron. 35, 365 (2003).
    [CrossRef]
  4. Marin Soljavic and J. D. Joannopoulos, "Enhancement of nonlinear effects using photonic crystals," Nat. Mat. 3, 211 (2004).
    [CrossRef]
  5. J. B. Khurgin, "Optical buffers based on slow light in electromagnetically induced transparent media and coupled resonator structures: comparative analysis," J. Opt. Soc. Am. B 22, 1062 (2005).
    [CrossRef]
  6. S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, "Extrinsic Optical Scattering Loss in Photonic Crystal Waveguides: Role of Fabrication Disorder and Photon Group Velocity," Phys. Rev. Lett. 94, 033903 (2005).
    [CrossRef] [PubMed]
  7. YuriiA. Vlasov and Sharee J. McNab," Coupling into the slow light mode in slab-type photonic crystal waveguides," Opt. Lett. 31, 50, (2006).
    [CrossRef] [PubMed]
  8. <other>. However, non disordered devices have been modeled successfully in 3D, see S.Boscolo and M. Midrio, "3D Multiple-Scattering Technique for the Analysis of PhC Slabs" J. Lightwave Technol. 22, 2778 (2004), for an example.</other>
  9. A 2D model makes some significant simplifications when applied to this problem- it cannot account for out of plane scattering and coupling to substrate modes. Complex interactions between the loss mechanisms cannot be ruled out, however, the model shows good agreement with the experiment (2nm difference in shifts), suggesting that these interactions do not have a significant effect. The small discrepancy is probably due to absence of these effects in the simulation.
  10. D. Gerace and L. C. Andreani, "Disorder-induced losses in photonic crystal waveguides with line defects," Opt. Lett. 29, 1897 (2004)
    [CrossRef] [PubMed]
  11. R. Ferrini, D. Leuenberger, R. Houdré, H. Benisty, M. Kamp and A. Forchel, "Disorder-induced losses in planar photonic crystals," Opt. Lett. 31, 1426 (2006).
    [CrossRef] [PubMed]
  12. This analysis assumes that loss is disorder limited and that absorption (which would also scale with vg) is negligible. This is reasonable in the SOI system at this wavelength.
  13. This makes the non-trivial assumption that disorder does not change the group velocity. In order to verify this, we ran simulations of pulses propagating through the disorder. We found that up to the group indices where pulses break up due to dispersion (n>16), there was only minimal differences in group velocity between normal and disordered W1 waveguides.
  14. S.G. Johnson, M.L. Povinelli, M. Soukoulis, A. Karalis, S. Jacobs and J.D. Joannopoulos, "Roughness losses and volume-current methods in photonic-crystal waveguides," Appl. Phys. B 81, 283 (2005).
    [CrossRef]
  15. Eric Dulkeith, Sharee J. McNab, and Yurii A. Vlasov, "Mapping the optical properties of slab-type two-dimensional photonic crystal waveguides," Phys. Rev. B 72, 115102 (2005).
    [CrossRef]
  16. A. F. Koenderink, Ad Lagendijk and Willem L. Vos, "Optical extinction due to intrinsic structural variations of photonic crystals," Phys. Rev. B 72, 153102 (2005).
    [CrossRef]
  17. L. C. Andreani and D. Gerace, submitted Phys. Status Solidi B.

2007 (1)

T. F. Krauss, "Slow light in photonic crystal waveguides," J. App. Phys. D. 40, 2666-2670 (2007).
[CrossRef]

2006 (2)

2005 (5)

J. B. Khurgin, "Optical buffers based on slow light in electromagnetically induced transparent media and coupled resonator structures: comparative analysis," J. Opt. Soc. Am. B 22, 1062 (2005).
[CrossRef]

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, "Extrinsic Optical Scattering Loss in Photonic Crystal Waveguides: Role of Fabrication Disorder and Photon Group Velocity," Phys. Rev. Lett. 94, 033903 (2005).
[CrossRef] [PubMed]

S.G. Johnson, M.L. Povinelli, M. Soukoulis, A. Karalis, S. Jacobs and J.D. Joannopoulos, "Roughness losses and volume-current methods in photonic-crystal waveguides," Appl. Phys. B 81, 283 (2005).
[CrossRef]

Eric Dulkeith, Sharee J. McNab, and Yurii A. Vlasov, "Mapping the optical properties of slab-type two-dimensional photonic crystal waveguides," Phys. Rev. B 72, 115102 (2005).
[CrossRef]

A. F. Koenderink, Ad Lagendijk and Willem L. Vos, "Optical extinction due to intrinsic structural variations of photonic crystals," Phys. Rev. B 72, 153102 (2005).
[CrossRef]

2004 (2)

Marin Soljavic and J. D. Joannopoulos, "Enhancement of nonlinear effects using photonic crystals," Nat. Mat. 3, 211 (2004).
[CrossRef]

D. Gerace and L. C. Andreani, "Disorder-induced losses in photonic crystal waveguides with line defects," Opt. Lett. 29, 1897 (2004)
[CrossRef] [PubMed]

2003 (1)

A. Melloni, F. Morichetti and M. Martinelli, "Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures," Opt. Quantum Electron. 35, 365 (2003).
[CrossRef]

2002 (1)

Andreani, L. C.

Benisty, H.

Fan, S.

Ferrini, R.

Forchel, A.

Gerace, D.

Houdré, R.

Hughes, S.

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, "Extrinsic Optical Scattering Loss in Photonic Crystal Waveguides: Role of Fabrication Disorder and Photon Group Velocity," Phys. Rev. Lett. 94, 033903 (2005).
[CrossRef] [PubMed]

Ibanescu, M.

Ippen, E.

Jacobs, S.

S.G. Johnson, M.L. Povinelli, M. Soukoulis, A. Karalis, S. Jacobs and J.D. Joannopoulos, "Roughness losses and volume-current methods in photonic-crystal waveguides," Appl. Phys. B 81, 283 (2005).
[CrossRef]

Joannopoulos, J. D.

Joannopoulos, J.D.

S.G. Johnson, M.L. Povinelli, M. Soukoulis, A. Karalis, S. Jacobs and J.D. Joannopoulos, "Roughness losses and volume-current methods in photonic-crystal waveguides," Appl. Phys. B 81, 283 (2005).
[CrossRef]

Johnson, S. G.

Johnson, S.G.

S.G. Johnson, M.L. Povinelli, M. Soukoulis, A. Karalis, S. Jacobs and J.D. Joannopoulos, "Roughness losses and volume-current methods in photonic-crystal waveguides," Appl. Phys. B 81, 283 (2005).
[CrossRef]

Kamp, M.

Karalis, A.

S.G. Johnson, M.L. Povinelli, M. Soukoulis, A. Karalis, S. Jacobs and J.D. Joannopoulos, "Roughness losses and volume-current methods in photonic-crystal waveguides," Appl. Phys. B 81, 283 (2005).
[CrossRef]

Khurgin, J. B.

Koenderink, A. F.

A. F. Koenderink, Ad Lagendijk and Willem L. Vos, "Optical extinction due to intrinsic structural variations of photonic crystals," Phys. Rev. B 72, 153102 (2005).
[CrossRef]

Krauss, T. F.

T. F. Krauss, "Slow light in photonic crystal waveguides," J. App. Phys. D. 40, 2666-2670 (2007).
[CrossRef]

Leuenberger, D.

Martinelli, M.

A. Melloni, F. Morichetti and M. Martinelli, "Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures," Opt. Quantum Electron. 35, 365 (2003).
[CrossRef]

Melloni, A.

A. Melloni, F. Morichetti and M. Martinelli, "Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures," Opt. Quantum Electron. 35, 365 (2003).
[CrossRef]

Morichetti, F.

A. Melloni, F. Morichetti and M. Martinelli, "Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures," Opt. Quantum Electron. 35, 365 (2003).
[CrossRef]

Povinelli, M.L.

S.G. Johnson, M.L. Povinelli, M. Soukoulis, A. Karalis, S. Jacobs and J.D. Joannopoulos, "Roughness losses and volume-current methods in photonic-crystal waveguides," Appl. Phys. B 81, 283 (2005).
[CrossRef]

Ramunno, L.

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, "Extrinsic Optical Scattering Loss in Photonic Crystal Waveguides: Role of Fabrication Disorder and Photon Group Velocity," Phys. Rev. Lett. 94, 033903 (2005).
[CrossRef] [PubMed]

Sipe, J. E.

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, "Extrinsic Optical Scattering Loss in Photonic Crystal Waveguides: Role of Fabrication Disorder and Photon Group Velocity," Phys. Rev. Lett. 94, 033903 (2005).
[CrossRef] [PubMed]

Soljacic, M.

Soukoulis, M.

S.G. Johnson, M.L. Povinelli, M. Soukoulis, A. Karalis, S. Jacobs and J.D. Joannopoulos, "Roughness losses and volume-current methods in photonic-crystal waveguides," Appl. Phys. B 81, 283 (2005).
[CrossRef]

Young, J. F.

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, "Extrinsic Optical Scattering Loss in Photonic Crystal Waveguides: Role of Fabrication Disorder and Photon Group Velocity," Phys. Rev. Lett. 94, 033903 (2005).
[CrossRef] [PubMed]

Yurii,

Appl. Phys. B (1)

S.G. Johnson, M.L. Povinelli, M. Soukoulis, A. Karalis, S. Jacobs and J.D. Joannopoulos, "Roughness losses and volume-current methods in photonic-crystal waveguides," Appl. Phys. B 81, 283 (2005).
[CrossRef]

J. App. Phys. D. (1)

T. F. Krauss, "Slow light in photonic crystal waveguides," J. App. Phys. D. 40, 2666-2670 (2007).
[CrossRef]

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

Nat. Mat. (1)

Marin Soljavic and J. D. Joannopoulos, "Enhancement of nonlinear effects using photonic crystals," Nat. Mat. 3, 211 (2004).
[CrossRef]

Opt. Lett. (3)

Opt. Quantum Electron. (1)

A. Melloni, F. Morichetti and M. Martinelli, "Linear and nonlinear pulse propagation in coupled resonator slow-wave optical structures," Opt. Quantum Electron. 35, 365 (2003).
[CrossRef]

Phys. Rev. B (2)

Eric Dulkeith, Sharee J. McNab, and Yurii A. Vlasov, "Mapping the optical properties of slab-type two-dimensional photonic crystal waveguides," Phys. Rev. B 72, 115102 (2005).
[CrossRef]

A. F. Koenderink, Ad Lagendijk and Willem L. Vos, "Optical extinction due to intrinsic structural variations of photonic crystals," Phys. Rev. B 72, 153102 (2005).
[CrossRef]

Phys. Rev. Lett. (1)

S. Hughes, L. Ramunno, J. F. Young, and J. E. Sipe, "Extrinsic Optical Scattering Loss in Photonic Crystal Waveguides: Role of Fabrication Disorder and Photon Group Velocity," Phys. Rev. Lett. 94, 033903 (2005).
[CrossRef] [PubMed]

Other (5)

L. C. Andreani and D. Gerace, submitted Phys. Status Solidi B.

<other>. However, non disordered devices have been modeled successfully in 3D, see S.Boscolo and M. Midrio, "3D Multiple-Scattering Technique for the Analysis of PhC Slabs" J. Lightwave Technol. 22, 2778 (2004), for an example.</other>

A 2D model makes some significant simplifications when applied to this problem- it cannot account for out of plane scattering and coupling to substrate modes. Complex interactions between the loss mechanisms cannot be ruled out, however, the model shows good agreement with the experiment (2nm difference in shifts), suggesting that these interactions do not have a significant effect. The small discrepancy is probably due to absence of these effects in the simulation.

This analysis assumes that loss is disorder limited and that absorption (which would also scale with vg) is negligible. This is reasonable in the SOI system at this wavelength.

This makes the non-trivial assumption that disorder does not change the group velocity. In order to verify this, we ran simulations of pulses propagating through the disorder. We found that up to the group indices where pulses break up due to dispersion (n>16), there was only minimal differences in group velocity between normal and disordered W1 waveguides.

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Figures (9)

Fig. 1.
Fig. 1.

An SEM image of a deliberately disordered photonic crystal- the ideal positions are shown in white. The root mean square value (RMS) of the deliberate disorder was 20nm. This device was made for demonstration purposes only- near zero transmission would be seen for such a level of disorder, while the disorder of a real device (up to 4.5 nm RMS) would not be discernible in such a picture.

Fig. 2.
Fig. 2.

The transmission and reflection characteristics (power) of a normal and disordered W1 waveguide. A significant increase in reflection can be seen for the disordered waveguide approaching the band-edge. No such increase is present in the normal W1. The calculated reflection is the back reflection into the guided mode of the access photonic wire and does not include off-axis scattering at the interface into free space (and thus the sum of the reflection and transmission does not equal one).

Fig. 3.
Fig. 3.

Transmission through varied length w1 photonic crystal waveguides. The inset shows the etching quality corresponding to this measurement.

Fig. 4.
Fig. 4.

The passband of the W1 waveguide (no deliberate disorder). A low power LED source was used to obtain the pink curve and a high power ASE source to obtain high resolution over a smaller range (the LED spectrum was scaled to match). The inset shows a close up of the high resolution fringes. The group velocity was determined from a 3D bandstructure (MPB) with r/a=0.28. The inset shows a close-up of the mode cut-off. While a change in fringe spacing is apparent, it is not possible to extract the group velocity from this data.

Fig. 5.
Fig. 5.

Transmission through a 200μm W1 waveguide as a function of increasing disorder. The inset gives the RMS value of the deliberate positional variations.

Fig. 6.
Fig. 6.

W1 mode cut-off frequency with disorder. As the disorder increases the position of the W1 mode cut-off moves to higher frequencies.

Fig. 7.
Fig. 7.

Normalized transmission as a function of disorder at different spectral positions. (The data was averaged to smooth out the Fabry-Perot ripple). The inset gives the normalized frequency for each data set. The rise at the end (e.g. circles at 4.5 nm) is somewhat of an artifact- past cut-off, transmission slowly increases, particularly in the case of strong disorder. The intercept (parameter A in equation (1)) is determined by the coupling coefficient into the PhC waveguide, which is generally lower in the slow light regime. The lines correspond to the fits made using equation 1.

Fig. 8.
Fig. 8.

The experimental dependence of the disorder parameter on the group velocity, plotted on logarithmic scales. A 1/vg1/2 dependence of parameter group velocity is the best fit (R2 value of 0.972). The group velocity is derived from the bandstructure (Fig. 4).

Fig. 9.
Fig. 9.

|Hz|2 field plots for light propagating in W1 at group velocities of c/5 and c/25. The expansion of the mode at low group velocity is evident even at c/25.

Equations (1)

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T = A B ( x + C ) 2

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