## Dispersion extraction with near-field measurements in periodic waveguides

Optics Express, Vol. 17, Issue 5, pp. 3716-3721 (2009)

http://dx.doi.org/10.1364/OE.17.003716

Acrobat PDF (804 KB)

### Abstract

We formulate and demonstrate experimentally the high-resolution spectral method based on Bloch-wave symmetry properties for extracting mode dispersion in periodic waveguides from measurements of near-field profiles. We characterize both the propagating and evanescent modes, and also determine the amplitudes of forward and backward waves in different waveguide configurations, with the estimated accuracy of several percent or less. Whereas the commonly employed spatial Fourier-transform (SFT) analysis provides the wavenumber resolution which is limited by the inverse length of the waveguide, we achieve precise dispersion extraction even for compact photonic structures.

© 2009 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,” Nature 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**, 123901–4 (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**, 073903–4 (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**, 125301–8 (2008). [CrossRef]

*k*≥ 2

*π*/

*L*, where Δ

*k*is the resolution of the wavenumber, and

*L*is the structure length. This severely restricts the potential to determine the mode dispersion.

5. 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]

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

7. 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]

## 2. Dispersion extraction based on the Bloch-wave symmetries

5. 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]

*M*) which primarily determine the system evolution. Let us consider how this methodology can be applied to extract dispersion for a periodic waveguide which supports

*M*modes in particular frequency range. Since each of the modes of a periodic waveguide satisfies the Bloch theorem [8], the complex electric field envelope of a waveguide mode with the index

*m*at the frequency

*ω*can be expressed as

*E*(

*x*,

*y*,

*z*;

*ω*) =

*ψ*(

_{m}*x*,

*y*,

*z*;

*ω*)exp(

*ik*/

_{m}z*d*). Here

*k*is the Bloch wavenumber (which real part defines the phase velocity and imaginary part can account for the evanescent field decay due to photonic band-gaps or the effect of losses),

_{m}*x*and

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

*z*is the direction of periodicity,

*d*is the waveguide period, and

*ψ*is the periodic Bloch-wave envelope function:

_{m}*ψ*(

_{m}*z*) =

*ψ*(

_{m}*z*+

*d*). Then, the total field inside the waveguide can be presented as a linear superposition of propagating modes and radiative waves:

*E*(

*x*,

*y*,

*z*;

*ω*) = ∑

^{M}

_{m=1}

*a*(

_{m}ψ_{m}*x*,

*y*,

*z*;

*ω*) exp(

*ik*/

_{m}z*d*) +

*w*(

*x*,

*y*,

*z*;

*ω*), where

*a*are the mode amplitudes and

_{m}*w*(

*x*,

*y*,

*z*;

*ω*) is the radiation field due to the excitation of non-guided waves.

5. 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]

*a*and wavenumbers

_{m}*k*which provide the best fitting for the experimental measurements of the electric field distribution. One approach is to use the periodicity property of Bloch-wave envelopes and represent them as infinite Fourier series,

_{m}*ψ*(

_{m}*x*,

*y*,

*z*;

*ω*) = ∑

^{+∞}

_{s=-∞}

*(*ψ ˜

_{m,s}*x*,

*y*;

*ω*)exp(

*i*2

*πsz*/

*d*). Then, by performing the measurements along a line with fixed transverse positions (

*x*

_{0},

*y*

_{0}), the problem is reduced to finding harmonics with the amplitudes

*(*ψ ˜

_{m,s}*x*

_{0},

*y*

_{0};

*ω*) and the corresponding wavenumbers

*k*= (

_{m,s}*k*+ 2

_{m}*πs*). In case of Bloch-waves with smooth profiles, terms with

*s*≠ 0 are small allowing efficient dispersion extraction. It appears that the latter condition was satisfied in the analysis of Ref [7

7. 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]

*s*, substantially complicating the spectral analysis.

*N*periodic locations:

*U*=

_{n}*E*(

*x*

_{0},

*y*

_{0},

*z*

_{0}+

*n*

*d*;

*ω*),

*n*= 1 :

*N*. The positions (

*x*

_{0},

*y*

_{0}) can be chosen at the locations with the maximum field amplitudes, reducing the effect of noise in measurements. Taking into account the periodicity property of Bloch-wave envelopes

*ψ*, we introduce new variables

_{m}*A*=

_{m}*a*(

_{m}ψ_{m}*x*

_{0},

*y*

_{0},

*z*

_{0};

*ω*) exp(

*ik*

_{m}*z*

_{0}/

*d*), and obtain a set of equations:

**205**, 109–152 (1991). [CrossRef]

*N*≥ 2

*M*.

*W*

_{min}will be larger than zero. Most importantly, this value can be used to estimate the accuracy of the dispersion extraction. Specifically, we define the confidence interval of wavenumbers as those corresponding to

*W*({

_{A}*k*}) < 2

_{m}*W*

_{min}.

## 3. Single-mode periodic waveguide

10. S. H. Fan, J. N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, “Guided and defect modes in periodic dielectric wave-guides,” J. Opt. Soc. Am. B **12**, 1267–1272 (1995). [CrossRef]

11. 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), http://www.opticsinfobase.org/abstract.cfm?URI=oe-8-3-173. [CrossRef] [PubMed]

12. D. A. Powell, I. V. Shadrivov, and Yu. S. Kivshar, “Multistability in nonlinear left-handed transmission lines,” Appl. Phys. Lett. **92**, 264104–3 (2008). [CrossRef]

*ω*(

*k*) =

*ω*(-

*k*) similar to dielectric waveguides, and therefore we use this symmetry in dispersion extraction. The line has 20 periods to ensure that end effects do not dominate the response, and the field distribution is registered at periodic locations along the line. We observe an excellent performance of extraction method for this periodic structure, as demonstrated by close agreement of dispersion curves based on experimental data and calculated numerically, see Fig. 2(b). We also determine the mode amplitudes [Fig. 2(c)] and find that the dominant propagating mode has negative wavenumber at lower frequencies, and positive wavenumber at higher frequencies above the stop-band. This result agrees with the nature of modes in the transmission line [12

12. D. A. Powell, I. V. Shadrivov, and Yu. S. Kivshar, “Multistability in nonlinear left-handed transmission lines,” Appl. Phys. Lett. **92**, 264104–3 (2008). [CrossRef]

## 4. Multi-mode waveguide

13. M. Born and E. Wolf, *Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light*, seventh ed. (Cambridge University Press, UK, 2002). [PubMed]

*ε*

_{eff}= 2.1. In Fig. 3(b) we show the numerically calculated dispersion for the first two modes with solid lines, presented for both forward- and backward-propagating waves with positive and negative wavenumbers, respectively.

*N*= 71 points with

*d*= 2mm spacing along the waveguide. The transverse antenna position is shifted away from the waveguide center by

*x*

_{0}= 1cm, as indicated by dashed line in Fig. 3(a). This position is chosen since at the waveguide center the amplitude of the second mode vanishes as its profile is anti-symmetric. Then, we process the data by running the retrieval algorithm for

*M*= 4 total number of modes, and we analyze the domain of real-valued wavenumbers

*k*in order to characterize the guided waves. The results of dispersion extraction are shown with dashed lines in Fig. 3(b), revealing excellent agreement with numerical calculations. Indeed, we observe that for all frequencies the relative error

_{j}*W*

_{min}remains below 8%.

## 5. Conclusion and outlook

## References and links

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

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 k space,” Phys. Rev. B |

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

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

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

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

9. | S.L. Marple, |

10. | S. H. Fan, J. N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, “Guided and defect modes in periodic dielectric wave-guides,” J. Opt. Soc. Am. B |

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

12. | D. A. Powell, I. V. Shadrivov, and Yu. S. Kivshar, “Multistability in nonlinear left-handed transmission lines,” Appl. Phys. Lett. |

13. | M. Born and E. Wolf, |

**OCIS Codes**

(230.7370) Optical devices : Waveguides

(250.5300) Optoelectronics : Photonic integrated circuits

(050.5298) Diffraction and gratings : Photonic crystals

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: November 30, 2008

Revised Manuscript: January 12, 2009

Manuscript Accepted: February 12, 2009

Published: February 25, 2009

**Citation**

Andrey A. Sukhorukov, Sangwoo Ha, Ilya V. Shadrivov, David A. Powell, and Yuri S. Kivshar, "Dispersion extraction with near-field measurements in periodic waveguides," Opt. Express **17**, 3716-3721 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-5-3716

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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," Nature Phys. 3, 401-405 (2007). [CrossRef]
- H. Gersen, T. J. Karle, R. J. P. Engelen,W. Bogaerts, J. P. Korterik, N. F. Hulst, van, T. F. Krauss, and L. Kuipers, "Direct observation of Bloch harmonics and negative phase velocity in photonic crystal waveguides," Phys. Rev. Lett. 94, 123901-4 (2005). [CrossRef] [PubMed]
- H. Gersen, T. J. Karle, R. J. P. Engelen,W. Bogaerts, J. P. Korterik, N. F. Hulst, van, T. F. Krauss, and L. Kuipers, "Real-space observation of ultraslow light in photonic crystal waveguides," Phys. Rev. Lett. 94, 073903-4 (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, 125301-8 (2008). [CrossRef]
- 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]
- 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]
- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).
- S. L. Marple, Digital Spectral Analysis with Applications (Prentice-Hall, Englewood Cliffs, NJ, 1987).
- S. H. Fan, J. N. Winn, A. Devenyi, J. C. Chen, R. D. Meade, and J. D. Joannopoulos, "Guided and defect modes in periodic dielectric wave-guides," J. Opt. Soc. Am. B 12, 1267-1272 (1995). [CrossRef]
- 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), http://www.opticsinfobase.org/abstract.cfm?URI=oe-8-3-173. [CrossRef] [PubMed]
- D. A. Powell, I. V. Shadrivov, and Yu. S. Kivshar, "Multistability in nonlinear left-handed transmission lines," Appl. Phys. Lett. 92, 264104-3 (2008). [CrossRef]
- M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, seventh ed. (Cambridge University Press, UK, 2002). [PubMed]

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