## Slow-light and evanescent modes at interfaces in photonic crystal waveguides: optimal extraction from experimental near-field measurements |

JOSA B, Vol. 28, Issue 4, pp. 955-963 (2011)

http://dx.doi.org/10.1364/JOSAB.28.000955

Acrobat PDF (1167 KB)

### Abstract

We develop a systematic approach for simultaneous extraction of the dispersion relations and profiles of multiple modes in periodic waveguides though a special global optimization procedure applied to near-field electric field measurements in the waveguide plane. We apply this method to perform in-depth analysis of experimental data on wave propagation close to an interface between waveguide sections with different dispersion characteristics, and we successfully identify several modes contributing to the experimentally measured fields. We find clear evidence that when the group velocity is reduced across the interface, evanescent modes that facilitate the excitation of propagating slow-light waves appear, confirming previous theoretical predictions.

© 2011 Optical Society of America

## 1. INTRODUCTION

1. R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, “Ultrafast evolution of photonic eigenstates in *k*-space,” Nat. Phys. **3**, 401–405 (2007). [CrossRef]

2. H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,” Phys. Rev. Lett. **94**, 123901 (2005). [CrossRef] [PubMed]

3. H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. **94**, 073903 (2005). [CrossRef] [PubMed]

4. N. Le Thomas, V. Zabelin, R. Houdre, M. V. Kotlyar, and T. F. Krauss, “Influence of residual disorder on the anticrossing of Bloch modes probed in *k* space,” Phys. Rev. B **78**, 125301 (2008). [CrossRef]

*L*is the structure length. Therefore, accurate dispersion results can only be obtained for long waveguides, extending over many periods of the underlying photonic structure. Another limitation of the SFT method is that it is difficult to extract information on the dispersion of evanescent waves, which may play an important role close to the structure boundaries or interfaces between different waveguides. For example, evanescent waves enable efficient excitation of slow-light modes in photonic crystal waveguides without a transition region [5

5. T. P. White, L. C. Botten, C. M. de Sterke, K. B. Dossou, and R. C. McPhedran, “Efficient slow-light coupling in a photonic crystal waveguide without transition region,” Opt. Lett. **33**, 2644–2646 (2008). [CrossRef] [PubMed]

6. S. H. Fan, I. Appelbaum, and J. D. Joannopoulos, “Near-field scanning optical microscopy as a simultaneous probe of fields and band structure of photonic crystals: a computational study,” Appl. Phys. Lett. **75**, 3461–3463 (1999). [CrossRef]

7. B. I. Popa and S. A. Cummer, “Determining the effective electromagnetic properties of negative-refractive-index metamaterials from internal fields,” Phys. Rev. B **72**, 165102 (2005). [CrossRef]

8. A. Andryieuski, R. Malureanu, and A. V. Lavrinenko, “Wave propagation retrieval method for metamaterials: unambiguous restoration of effective parameters,” Phys. Rev. B **80**, 193101 (2009). [CrossRef]

9. B. Dastmalchi, A. Mohtashami, K. Hingerl, and J. Zarbakhsh, “Method of calculating local dispersion in arbitrary photonic crystal waveguides,” Opt. Lett. **32**, 2915–2917 (2007). [CrossRef] [PubMed]

10. A. A. Sukhorukov, S. Ha, I. V. Shadrivov, D. A. Powell, and Yu. S. Kivshar, “Dispersion extraction with near-field measurements in periodic waveguides,” Opt. Express **17**, 3716–3721 (2009). [CrossRef] [PubMed]

11. R. Roy, B. G. Sumpter, G. A. Pfeffer, S. K. Gray, and D. W. Noid, “Novel methods for spectral analysis,” Phys. Rep. **205**, 109–152 (1991). [CrossRef]

12. V. A. Mandelshtam, “FDM: the filter diagonalization method for data processing in NMR experiments,” Prog. Nucl. Magn. Reson. Spectrosc. **38**, 159–196 (2001). [CrossRef]

13. S. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, C. M. de Sterke, and Yu. S. Kivshar, “Bloch-mode extraction from near-field data in periodic waveguides,” Opt. Lett. **34**, 3776–3778 (2009). [CrossRef] [PubMed]

13. S. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, C. M. de Sterke, and Yu. S. Kivshar, “Bloch-mode extraction from near-field data in periodic waveguides,” Opt. Lett. **34**, 3776–3778 (2009). [CrossRef] [PubMed]

5. T. P. White, L. C. Botten, C. M. de Sterke, K. B. Dossou, and R. C. McPhedran, “Efficient slow-light coupling in a photonic crystal waveguide without transition region,” Opt. Lett. **33**, 2644–2646 (2008). [CrossRef] [PubMed]

## 2. DISPERSION EXTRACTION METHOD

### 2A. Symmetries of Modes in Periodic Waveguides

*M*). The value of

*M*can be established based on numerical modeling, taking into account both propagating and evanescent waves. Because each of the modes of a periodic waveguide satisfies the Bloch theorem [14], the complex electric field envelope of a waveguide mode with the index

*m*at the frequency

*ω*can be expressed as where

*x*and

*y*are the orthogonal directions transverse to the waveguide and

*z*is the direction of periodicity;

*d*is the waveguide period; and

*M*guided modes with amplitudes

### 2B. Locally Optimal Extraction at Individual Frequencies

13. S. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, C. M. de Sterke, and Yu. S. Kivshar, “Bloch-mode extraction from near-field data in periodic waveguides,” Opt. Lett. **34**, 3776–3778 (2009). [CrossRef] [PubMed]

*N*is the number of periods in the waveguide section. Let us denote with Then, taking into account the periodicity of Bloch-wave envelopes, Eq. (2) can be written as where

**r**belongs to the first unit cell.

**r**in the unit cell, it becomes mathematically equivalent to the problems considered in the spectral analysis of temporal series [11

11. R. Roy, B. G. Sumpter, G. A. Pfeffer, S. K. Gray, and D. W. Noid, “Novel methods for spectral analysis,” Phys. Rep. **205**, 109–152 (1991). [CrossRef]

12. V. A. Mandelshtam, “FDM: the filter diagonalization method for data processing in NMR experiments,” Prog. Nucl. Magn. Reson. Spectrosc. **38**, 159–196 (2001). [CrossRef]

10. A. A. Sukhorukov, S. Ha, I. V. Shadrivov, D. A. Powell, and Yu. S. Kivshar, “Dispersion extraction with near-field measurements in periodic waveguides,” Opt. Express **17**, 3716–3721 (2009). [CrossRef] [PubMed]

**r**in the unit cell. This allows us to determine the values of

**34**, 3776–3778 (2009). [CrossRef] [PubMed]

*ω*, we seek the values of

**r**in a unit cell, the optimal amplitudes satisfy the linear matrix equation where the components of vector

*C*are

**r**in a unit cell to determine

*W*from Eq. (7). The entire process can be combined into a single expression as We can numerically find the absolute minimum

### 2C. Globally Optimal Extraction across a Frequency Range

*ω*. However, in experimental data, the noise level can vary widely between different frequencies, especially if the group velocity has a strong frequency dependence, as is the case in the examples presented in Section 3 below. This can dramatically reduce the accuracy for determining group velocity and higher order dispersion characteristics, which are defined through the derivatives of the dispersion curves. In order to overcome this issue, we suggest here a globally optimal extraction procedure, which recovers dispersion properties simultaneously across a range of frequencies.

*m*can be approximated by Taylor expansion, i.e., Here the number of terms

*ω*(a particular branch will need to be chosen based on physical considerations). The significance of this transformation is that the modal dispersion

*same*set of parameters

*all values*of

*ω*in the data. Thus, instead of extracting

*ω*one at a time, we can perform a single global optimization and extract

*ω*: where

*ω*.

## 3. DISPERSION AND MODE PROFILE EXTRACTION FROM NEAR-FIELD EXPERIMENTAL MEASUREMENTS

### 3A. Dispersion-Engineered Photonic Crystal Waveguides

15. J. Li, T. P. White, L. O’Faolain, A. Gomez Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express **16**, 6227–6232 (2008). [CrossRef] [PubMed]

16. S. Kubo, D. Mori, and T. Baba, “Low-group-velocity and low-dispersion slow light in photonic crystal waveguides,” Opt. Lett. **32**, 2981–2983 (2007). [CrossRef] [PubMed]

17. L. H. Frandsen, A. V. Lavrinenko, J. Fage Pedersen, and P. I. Borel, “Photonic crystal waveguides with semi-slow light and tailored dispersion properties,” Opt. Express **14**, 9444–9450 (2006). [CrossRef] [PubMed]

5. T. P. White, L. C. Botten, C. M. de Sterke, K. B. Dossou, and R. C. McPhedran, “Efficient slow-light coupling in a photonic crystal waveguide without transition region,” Opt. Lett. **33**, 2644–2646 (2008). [CrossRef] [PubMed]

**33**, 2644–2646 (2008). [CrossRef] [PubMed]

18. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**, 173–190 (2001). [CrossRef] [PubMed]

*p*” in Figs. 2a, 3a, 4a, where the slow-light regime is realized at wavenumbers around

**33**, 2644–2646 (2008). [CrossRef] [PubMed]

15. J. Li, T. P. White, L. O’Faolain, A. Gomez Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express **16**, 6227–6232 (2008). [CrossRef] [PubMed]

19. M. L. M. Balistreri, J. P. Korterik, L. Kuipers, and N. F. van Hulst, “Local observations of phase singularities in optical fields in waveguide structures,” Phys. Rev. Lett. **85**, 294–297 (2000). [CrossRef] [PubMed]

20. M. Burresi, R. J. P. Engelen, A. Opheij, D. Oosten, D. van Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. **102**, 033902 (2009). [CrossRef] [PubMed]

### 3B. Spatial Fourier Transform Spectra

### 3C. Application of Optimal Extraction Method

*ω*there are three solutions for

**33**, 2644–2646 (2008). [CrossRef] [PubMed]

21. C. M. de Sterke, K. B. Dossou, T. P. White, L. C. Botten, and R. C. McPhedran, “Efficient coupling into slow light photonic crystal waveguide without transition region: role of evanescent modes,” Opt. Express **17**, 17338–17343 (2009). [CrossRef]

### 3D. Results of Dispersion Extraction

*z*) should be very weak [5

**33**, 2644–2646 (2008). [CrossRef] [PubMed]

*p*” and “

*e*,” respectively) for different photonic structures are shown in Figs. 2a, 3a, 4a, and the corresponding mismatch values

22. M. Patterson, S. Hughes, S. Schulz, D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Disorder-induced incoherent scattering losses in photonic crystal waveguides: Bloch mode reshaping, multiple scattering, and breakdown of the Beer– Lambert law,” Phys. Rev. B **80**, 195305 (2009). [CrossRef]

22. M. Patterson, S. Hughes, S. Schulz, D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Disorder-induced incoherent scattering losses in photonic crystal waveguides: Bloch mode reshaping, multiple scattering, and breakdown of the Beer– Lambert law,” Phys. Rev. B **80**, 195305 (2009). [CrossRef]

**33**, 2644–2646 (2008). [CrossRef] [PubMed]

**34**, 3776–3778 (2009). [CrossRef] [PubMed]

21. C. M. de Sterke, K. B. Dossou, T. P. White, L. C. Botten, and R. C. McPhedran, “Efficient coupling into slow light photonic crystal waveguide without transition region: role of evanescent modes,” Opt. Express **17**, 17338–17343 (2009). [CrossRef]

### 3E. Comparison with Local Extraction Results

## 4. CONCLUSIONS

## ACKNOWLEDGMENTS

1. | R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, “Ultrafast evolution of photonic eigenstates in |

2. | H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,” Phys. Rev. Lett. |

3. | H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. |

4. | N. Le Thomas, V. Zabelin, R. Houdre, M. V. Kotlyar, and T. F. Krauss, “Influence of residual disorder on the anticrossing of Bloch modes probed in |

5. | T. P. White, L. C. Botten, C. M. de Sterke, K. B. Dossou, and R. C. McPhedran, “Efficient slow-light coupling in a photonic crystal waveguide without transition region,” Opt. Lett. |

6. | S. H. Fan, I. Appelbaum, and J. D. Joannopoulos, “Near-field scanning optical microscopy as a simultaneous probe of fields and band structure of photonic crystals: a computational study,” Appl. Phys. Lett. |

7. | B. I. Popa and S. A. Cummer, “Determining the effective electromagnetic properties of negative-refractive-index metamaterials from internal fields,” Phys. Rev. B |

8. | A. Andryieuski, R. Malureanu, and A. V. Lavrinenko, “Wave propagation retrieval method for metamaterials: unambiguous restoration of effective parameters,” Phys. Rev. B |

9. | B. Dastmalchi, A. Mohtashami, K. Hingerl, and J. Zarbakhsh, “Method of calculating local dispersion in arbitrary photonic crystal waveguides,” Opt. Lett. |

10. | A. A. Sukhorukov, S. Ha, I. V. Shadrivov, D. A. Powell, and Yu. S. Kivshar, “Dispersion extraction with near-field measurements in periodic waveguides,” Opt. Express |

11. | R. Roy, B. G. Sumpter, G. A. Pfeffer, S. K. Gray, and D. W. Noid, “Novel methods for spectral analysis,” Phys. Rep. |

12. | V. A. Mandelshtam, “FDM: the filter diagonalization method for data processing in NMR experiments,” Prog. Nucl. Magn. Reson. Spectrosc. |

13. | S. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, C. M. de Sterke, and Yu. S. Kivshar, “Bloch-mode extraction from near-field data in periodic waveguides,” Opt. Lett. |

14. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, |

15. | J. Li, T. P. White, L. O’Faolain, A. Gomez Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express |

16. | S. Kubo, D. Mori, and T. Baba, “Low-group-velocity and low-dispersion slow light in photonic crystal waveguides,” Opt. Lett. |

17. | L. H. Frandsen, A. V. Lavrinenko, J. Fage Pedersen, and P. I. Borel, “Photonic crystal waveguides with semi-slow light and tailored dispersion properties,” Opt. Express |

18. | S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express |

19. | M. L. M. Balistreri, J. P. Korterik, L. Kuipers, and N. F. van Hulst, “Local observations of phase singularities in optical fields in waveguide structures,” Phys. Rev. Lett. |

20. | M. Burresi, R. J. P. Engelen, A. Opheij, D. Oosten, D. van Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. |

21. | C. M. de Sterke, K. B. Dossou, T. P. White, L. C. Botten, and R. C. McPhedran, “Efficient coupling into slow light photonic crystal waveguide without transition region: role of evanescent modes,” Opt. Express |

22. | M. Patterson, S. Hughes, S. Schulz, D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Disorder-induced incoherent scattering losses in photonic crystal waveguides: Bloch mode reshaping, multiple scattering, and breakdown of the Beer– Lambert law,” Phys. Rev. B |

**OCIS Codes**

(230.7370) Optical devices : Waveguides

(250.5300) Optoelectronics : Photonic integrated circuits

(050.5298) Diffraction and gratings : Photonic crystals

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: December 20, 2010

Manuscript Accepted: February 8, 2011

Published: March 31, 2011

**Virtual Issues**

April 21, 2011 *Spotlight on Optics*

**Citation**

Sangwoo Ha, Marko Spasenović, Andrey A. Sukhorukov, Thomas P. White, C. Martijn de Sterke, L. (Kobus) Kuipers, Thomas F. Krauss, and Yuri S. Kivshar, "Slow-light and evanescent modes at interfaces in photonic crystal waveguides: optimal extraction from experimental near-field measurements," J. Opt. Soc. Am. B **28**, 955-963 (2011)

http://www.opticsinfobase.org/josab/abstract.cfm?URI=josab-28-4-955

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### References

- R. J. P. Engelen, Y. Sugimoto, H. Gersen, N. Ikeda, K. Asakawa, and L. Kuipers, “Ultrafast evolution of photonic eigenstates in k-space,” Nat. Phys. 3, 401–405 (2007). [CrossRef]
- H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides,” Phys. Rev. Lett. 94, 123901(2005). [CrossRef] [PubMed]
- H. Gersen, T. J. Karle, R. J. P. Engelen, W. Bogaerts, J. P. Korterik, N. F. van Hulst, T. F. Krauss, and L. Kuipers, “Real-space observation of ultraslow light in photonic crystal waveguides,” Phys. Rev. Lett. 94, 073903 (2005). [CrossRef] [PubMed]
- N. Le Thomas, V. Zabelin, R. Houdre, M. V. Kotlyar, and T. F. Krauss, “Influence of residual disorder on the anticrossing of Bloch modes probed in k space,” Phys. Rev. B 78, 125301 (2008). [CrossRef]
- T. P. White, L. C. Botten, C. M. de Sterke, K. B. Dossou, and R. C. McPhedran, “Efficient slow-light coupling in a photonic crystal waveguide without transition region,” Opt. Lett. 33, 2644–2646(2008). [CrossRef] [PubMed]
- S. H. Fan, I. Appelbaum, and J. D. Joannopoulos, “Near-field scanning optical microscopy as a simultaneous probe of fields and band structure of photonic crystals: a computational study,” Appl. Phys. Lett. 75, 3461–3463 (1999). [CrossRef]
- B. I. Popa and S. A. Cummer, “Determining the effective electromagnetic properties of negative-refractive-index metamaterials from internal fields,” Phys. Rev. B 72, 165102 (2005). [CrossRef]
- A. Andryieuski, R. Malureanu, and A. V. Lavrinenko, “Wave propagation retrieval method for metamaterials: unambiguous restoration of effective parameters,” Phys. Rev. B 80, 193101 (2009). [CrossRef]
- B. Dastmalchi, A. Mohtashami, K. Hingerl, and J. Zarbakhsh, “Method of calculating local dispersion in arbitrary photonic crystal waveguides,” Opt. Lett. 32, 2915–2917 (2007). [CrossRef] [PubMed]
- A. A. Sukhorukov, S. Ha, I. V. Shadrivov, D. A. Powell, and Yu. S. Kivshar, “Dispersion extraction with near-field measurements in periodic waveguides,” Opt. Express 17, 3716–3721(2009). [CrossRef] [PubMed]
- R. Roy, B. G. Sumpter, G. A. Pfeffer, S. K. Gray, and D. W. Noid, “Novel methods for spectral analysis,” Phys. Rep. 205, 109–152(1991). [CrossRef]
- V. A. Mandelshtam, “FDM: the filter diagonalization method for data processing in NMR experiments,” Prog. Nucl. Magn. Reson. Spectrosc. 38, 159–196 (2001). [CrossRef]
- S. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, C. M. de Sterke, and Yu. S. Kivshar, “Bloch-mode extraction from near-field data in periodic waveguides,” Opt. Lett. 34, 3776–3778 (2009). [CrossRef] [PubMed]
- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, 1995).
- J. Li, T. P. White, L. O’Faolain, A. Gomez Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express 16, 6227–6232 (2008). [CrossRef] [PubMed]
- S. Kubo, D. Mori, and T. Baba, “Low-group-velocity and low-dispersion slow light in photonic crystal waveguides,” Opt. Lett. 32, 2981–2983 (2007). [CrossRef] [PubMed]
- L. H. Frandsen, A. V. Lavrinenko, J. Fage Pedersen, and P. I. Borel, “Photonic crystal waveguides with semi-slow light and tailored dispersion properties,” Opt. Express 14, 9444–9450(2006). [CrossRef] [PubMed]
- S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001). [CrossRef] [PubMed]
- M. L. M. Balistreri, J. P. Korterik, L. Kuipers, and N. F. van Hulst, “Local observations of phase singularities in optical fields in waveguide structures,” Phys. Rev. Lett. 85, 294–297 (2000). [CrossRef] [PubMed]
- M. Burresi, R. J. P. Engelen, A. Opheij, D. Oosten, D. van Mori, T. Baba, and L. Kuipers, “Observation of polarization singularities at the nanoscale,” Phys. Rev. Lett. 102, 033902 (2009). [CrossRef] [PubMed]
- C. M. de Sterke, K. B. Dossou, T. P. White, L. C. Botten, and R. C. McPhedran, “Efficient coupling into slow light photonic crystal waveguide without transition region: role of evanescent modes,” Opt. Express 17, 17338–17343 (2009). [CrossRef]
- M. Patterson, S. Hughes, S. Schulz, D. M. Beggs, T. P. White, L. O’Faolain, and T. F. Krauss, “Disorder-induced incoherent scattering losses in photonic crystal waveguides: Bloch mode reshaping, multiple scattering, and breakdown of the Beer–Lambert law,” Phys. Rev. B 80, 195305 (2009). [CrossRef]

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