## The effect of higher-order dispersion on slow light propagation in photonic crystal waveguides

Optics Express, Vol. 14, Issue 4, pp. 1658-1672 (2006)

http://dx.doi.org/10.1364/OE.14.001658

Acrobat PDF (1877 KB)

### Abstract

We have studied the dispersion of ultrafast pulses in a photonic crystal waveguide as a function of optical frequency, in both experiment and theory. With phase-sensitive and time-resolved near-field microscopy, the light was probed inside the waveguide in a non-invasive manner. The effect of dispersion on the shape of the pulses was determined. As the optical frequency decreased, the group velocity decreased. Simultaneously, the measured pulses were broadened during propagation, due to an increase in group velocity dispersion. On top of that, the pulses exhibited a strong asymmetric distortion as the propagation distance increased. The asymmetry increased as the group velocity decreased. The asymmetry of the pulses is caused by a strong increase of higher-order dispersion. As the group velocity was reduced to 0.116(9)·*c*, we found group velocity dispersion of -1.1(3)·10^{6} ps^{2}/km and third order dispersion of up to 1.1(4)·10^{5} ps^{3}/km. We have modelled our interferometric measurements and included the full dispersion of the photonic crystal waveguide. Our mathematical model and the experimental findings showed a good correspondence. Our findings show that if the most commonly used slow light regime in photonic crystals is to be exploited, great care has to be taken about higher-order dispersion.

© 2006 Optical Society of America

## 1. Introduction

2. S.G. Johnson, P.R. Villeneuve, S. Fan, and J.D. Joannopoulos, “Linear waveguides in photonic crystal slabs,” Phys. Rev. B **62**, 8212–8222 (2000) [CrossRef]

3. Y. Sugimoto, Y. Tanaka, N. Ikeda, Y. Nakamura, K. Asakawa, and K. Inoue, “Low propagation loss of 0.76 dB/mm in GaAs-based single-line-defect two-dimensional photonic crystal slab waveguides up to 1 cm in length,” Opt. Express **12**, 1090–1096 (2004) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1090 [CrossRef] [PubMed]

4. E. Chow, S.Y. Lin, J.R. Wendt, S.G. Johnson, and J.D. Joannopoulos, “Quantitative analysis of bending effiency in photonic crystal wavgeuide bends at *λ* = 1.55*μm* wavelengths,” Opt. Lett. **26**, 286–288 (2001) [CrossRef]

5. Y. Akahane, T. Asano, B.S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature **425**, 944–947 (2003) [CrossRef] [PubMed]

6. A.Y. Petrov and M. Eich, “Zero dispersion at small group velocities in photonic crystal waveguides,” Appl. Phys. Lett. **85**, 4866–4868 (2004) [CrossRef]

7. V.N. Astratov, R.M. Stevenson, I.S. Culshaw, D.M. Whittaker, M.S. Skolnick, T.F. Krauss, and R.M. de la Rue, “Heavy photon dispersions in photonic crystal waveguides,” Appl. Phys. Lett. **77**, 178–180 (2000) [CrossRef]

8. M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely Large Group-Velocity Dispersion of Line-Defect Waveguides in Photonic Crystal Slabs,” Phys. Rev. Lett. **87**, 253902 (2001) [CrossRef] [PubMed]

9. K. Inoue, N. Kawai, Y. Sugimoto, N. Ikeda, and K. Asakawa, “Observation of small group velocity in two-dimensional AlGaAs-based photonic crystal slabs,” Phys. Rev. B **65**, 121308 (2002) [CrossRef]

10. Y.A. Vlasov, M. O’Boyle, H.F. Hamann, and S.J. McNab, “Active control of slow light on a chip with photonic crystal waveguides” Nature **438**, 65–69 (2005) [CrossRef] [PubMed]

11.
for a review see
M. Soljacic and J.D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nature Mater. **3**211–219 (2004) [CrossRef]

12. M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, “Optical bistable switching action of Si high-Q photonic-crystal nanocavities,” Opt. Express **13**2678–2687 (2005) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-7-2678 [CrossRef] [PubMed]

13. A. Sugitatsu, T. Asano, and T, S. Noda, “Characterization of line-defect-waveguide lasers in two-dimensional photonic-crystal slabs,” Appl. Phys. Lett. **84**5395–5397 (2004) [CrossRef]

14. S.F. Mingaleev, Yu.S. Kivshar, and R.A. Sammut, “Longrange interaction and nonlinear localized modes in photonic crystal waveguides,” Phys. Rev. E **62**5777 (2000) [CrossRef]

15. M.D. Rahn, A.M. Fox, M.S. Skolnick, and T.F. Krauss, “Propagation of ultrashort nonlinear pulses through two-dimensional AlGaAs high-contrast photonic crystal waveguides,” J. Opt. Soc. Am. B **19**, 716–721 (2002) [CrossRef]

16. S. Yamada, Y. Watanabe, Y. Katayama, and J.B. Cole, “Simulation of optical pulse propagation in a two-dimensional photonic crystal waveguide using a high accuracy finite-difference time-domain algorithm,” J. Appl. Phys. **93**, 1859–1864 (2003) [CrossRef]

17. A. Imhof, W.L. Vos, R. Sprik, and A. Lagendijk, “Large dispersive effects near the band edges of photonic crystals,” Phys. Rev. Lett. **83**, 2942–2945 (1999) [CrossRef]

18. T. Asano, K. Kiyota, D. Kumamoto, B.S. Song, and S. Noda, “Time-domain measurement of picosecond light-pulse propagation in a two-dimensional photonic crystal-slab waveguide,” Appl. Phys. Lett. **84**, 4690–4692 (2004) [CrossRef]

19. 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]

*et al*. [18

18. T. Asano, K. Kiyota, D. Kumamoto, B.S. Song, and S. Noda, “Time-domain measurement of picosecond light-pulse propagation in a two-dimensional photonic crystal-slab waveguide,” Appl. Phys. Lett. **84**, 4690–4692 (2004) [CrossRef]

17. A. Imhof, W.L. Vos, R. Sprik, and A. Lagendijk, “Large dispersive effects near the band edges of photonic crystals,” Phys. Rev. Lett. **83**, 2942–2945 (1999) [CrossRef]

20. M. Miyagi and S. Nishida, “Pulse spreading in a single-mode fiber due to third-order dispersion,” Appl. Opt. **18**, 678–682 (1979) [CrossRef] [PubMed]

21. J. Khurgin, “Performance of nonlinear photonic crystal devices at high bit rates,” Opt. Lett. **30**, 643–645 (2005) [CrossRef] [PubMed]

22. M.L.M. Balistreri, H. Gersen, J.P. Korterik, L. Kuipers, and N.F. van Hulst, “Tracking femtosecond laser pulses in space and time,” Science **294**, 1080–1082 (2001) [CrossRef] [PubMed]

## 2. Experimental aspects and modelling

### 2.1. Sample and experimental setup

*μ*m-thick Al

_{0.6}Ga

_{0.4}As sacrificial layer on a GaAs substrate. A membrane-type photonic crystal structure was fabricated using high-resolution electron-beam lithography, dry etching, and selective wet-etching techniques. The 2D PhC thus consisted of a hexagonal array of air holes etched into a planar GaAs slab. The sacrificial layer was removed by an HF solution via the air holes. The lattice constant is 339 nm with air-holes of 204 nm in diameter. By leaving a single row of airholes unperforated, a so-called W1 waveguide was created. A section of the resulting structure is shown in Fig. 1.

23. 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.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173 [CrossRef] [PubMed]

3. Y. Sugimoto, Y. Tanaka, N. Ikeda, Y. Nakamura, K. Asakawa, and K. Inoue, “Low propagation loss of 0.76 dB/mm in GaAs-based single-line-defect two-dimensional photonic crystal slab waveguides up to 1 cm in length,” Opt. Express **12**, 1090–1096 (2004) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1090 [CrossRef] [PubMed]

*ω*= 0.254 the even mode bends towards

*k*

_{z}=

*π*/

*a*, corresponding to a strong reduction of the group velocity (

*ν*

_{g}), since the

*ν*

_{g}is determined by the inverse of the slope of

*k*

_{z}(

*ω*). Note that the 2D bandgap only exists for TE polarization. TM polarized light can either propagate through the crystal, or it can be confined to the waveguide by refractive index contrast [24

24. R.J.P. Engelen, T.J. Karle, H. Gersen, J.P. Korterik, T.F. Krauss, L. Kuipers, and N.F. van Hulst, “Local probing of Bloch mode dispersion in a photonic crystal waveguide,” Opt. Express **13**, 4457–4464 (2005) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-12-4457 [CrossRef] [PubMed]

*ν*

_{g}is found te be 0.23 ·

*c*. As the frequency is reduced, the group velocity decreases to become zero at

*ω*= 0.253. The group velocity dispersion (GVD) is defined by the second order derivative of the dispersion relation (

*β*

_{2}= d

^{2}

*k*/d

*ω*

^{2}). The frequency dependency of

*β*

_{2}is shown in Fig. 2(c). At high frequencies, the GVD is in the order of -10

^{4}ps

^{2}/km. As the frequency is reduced and the group velocity drops, the GVD increases by several orders of magnitude. The GVD is many orders of magnitude larger than in fiber optics. For comparison, the GVD in an optical fiber is between -28 and +8 ps

^{2}/km [25

25. Product information sheet, “Corning SMF-28e Optical Fiber, Product Information” (Corning Inc., 2005) http://corning.com/opticalfiber/products%5F%5Fapplications/products/smf%5F28e.aspx

22. M.L.M. Balistreri, H. Gersen, J.P. Korterik, L. Kuipers, and N.F. van Hulst, “Tracking femtosecond laser pulses in space and time,” Science **294**, 1080–1082 (2001) [CrossRef] [PubMed]

### 2.2. Local heterodyne detection of pulse propagation

26. P.St.J. Russell, “Optics of Floquet-Bloch Waves in Dielectric Gratings,” Appl. Phys. B **39**, 231–246 (1986) [CrossRef]

*k*

_{m}:

*a*is the period of the lattice. (For simplicity, a one-dimensional lattice is considered.) The wavevectors of the plane waves that comprise a single Bloch wave are spaced 2

*π*/

*a*apart in reciprocal space. The dispersion for all the higher-order Bloch harmonics, i.e.

*m*≠ 0, is identical to that of

*k*

_{0}(

*ω*). Calculating the dispersion relation (such as in Fig. 2(a)) in the first Brillouin zone suffices to represent the dispersive properties of the entire Bloch wave. For the effect on either the pulse shape in time or in space on length scales larger than a single unit cell we therefore consider only

*E*˜

_{0}and

*k*

_{0}represent the amplitude of the electric field and the fundamental wavevec-tor of the Bloch wave, respectively [26

26. P.St.J. Russell, “Optics of Floquet-Bloch Waves in Dielectric Gratings,” Appl. Phys. B **39**, 231–246 (1986) [CrossRef]

27. B. Lombardet, L.A. Dunbar, R. Ferrini, and R. Houdre, “Fourier analysis of Bloch wave propagation in photonic crystals,” J. Opt. Soc. Am. B **22**, 1179–1190 (2005) [CrossRef]

*k*

_{0}(

*ω*) is approximated with a Taylor expansion:

*β*

_{i}represent the dispersive constants of a material. The wavevector equals

*β*

_{0},

*β*

_{1}is the inverse of the group velocity (

*ν*

_{g}),

*β*

_{2}and

*β*

_{3}represent the GVD and the third order dispersion (TOD), respectively. The evolution of the optical electric field in space and time is obtained by Fourier transforming Eq. 3:

*E*˜(

*ω*) is therefore required to describe a short pulse in time. ℱ

^{-1}denotes the inverse Fourier Transform. For a medium without TOD (

*β*

_{3}= 0), this is analytically solvable for a Gaussian pulse [28, 29

29. H. Gersen, J.P. Korterik, N.F. van Hulst, and L. Kuipers, “Tracking ultrashort pulses through dispersive media: Experiment and theory” Phys. Rev. E **68**, 026604 (2003) [CrossRef]

*τ*

_{p}. Clearly, when the electric field is considered at one time

*t*, a low group velocity (i.e. a high

*β*

_{1}) leads to a compression of the pulse in space. When the E-field at a specific point (

*z*) is evaluated, a

*β*

_{2}≠ 0 yields a broadened pulse. In addition, since the energy of the pulse is constant, the E-field amplitude is reduced. By evaluating the pulse broadening as a function of position (

*z*),

*β*

_{2}can be recovered.

*z*is

*zβ*

_{2}. When the quantity

*zβ*

_{2}is comparable to or larger than

*β*

_{2}does not change the shape of the pulse envelope in time, it only changes its duration.

20. M. Miyagi and S. Nishida, “Pulse spreading in a single-mode fiber due to third-order dispersion,” Appl. Opt. **18**, 678–682 (1979) [CrossRef] [PubMed]

*z*, the pulses are deformed asymmetrically in time. The temporal asymmetry of the pulses can be a measure for the TOD in the photonic material.

*k*(

*ω*)) is used as input. Hence the calculation is not limited by the accuracy of the Taylor expansion, since all dispersive orders are included, if present.

30. M.L.M. Balistreri, A. Driessen, J.P. Korterik, L. Kuipers, and N.F. van Hulst, “Quasi interference of perpendicularly polarized guided modes observed with a photon scanning tunneling microscope,” Opt. Lett. **25**, 637–639 (2000) [CrossRef]

*E*

_{ref}. Compared to

*E*

_{ref}, the E-field in the signal branch (

*E*

_{sig}) differed in spectrum and amplitude. With this in mind, we calculated the detection of pulses in our interferometric set-up. The pulses in the reference branch were assumed to be Fourier-limited, while the pulses in the signal branch could be distorted by the dispersion in the sample.

*E*

_{ref}is therefore not dependent on position

*z*, in contrast to

*E*

_{sig}. The light intensity on the detector can be described as:

*ω*. The delay time delay between the reference pulse signal pulse is denoted

*τ*. We can split Eq. (7) into a constant signal and an interference term:

*E*∣

^{2}terms are the constant detector signals, and the third term describes the interference. The real part of a function is denoted as “Re”. With our Lock-In Amplifier (LIA), we detected only the interference term as only this term varied with 40 kHz, given by Δ

*ω*. This signal was obtained in a characteristic time Δ

*t*. This characteristic time is given by the bandwidth of the detection system, and is therefore always much longer than the pulse duration (Δ

*t*≫

*τ*

^{p}), the repetition rate and is also kept much longer than 1 divided by the 40 kHz modulation. With this in mind, the output voltage of one of our LIA channels was proportional to:

*τ*between reference and signal pulse and was not dependent on the time

*t*. We can rewrite the above cross-correlation as a function multiplication in the spectral domain:

31. P. Sanchis, P. Bienstman, B. Luyssaert, R. Baets, and J. Marti, “Analysis of butt coupling in photonic crystals,” IEEE J. Quantum Electron. **40**, 541–550 (2004) [CrossRef]

*z*or

*τ*in Eq. (11) was fixed, while the other variable was varied. In this way, two measurement approaches were possible: a time-resolved measurement of the interference or a space-resolved measurement, experimentally corresponding to scanning either probe or delay time. In the first measurement scheme, the fiber probe was scanned over the sample. This resulted in the distribution of the interference as a function of position. We will refer to this measurement mode as the spatial interference distribution (SID) measurement and it contains a combination of dispersion and pulse shape information. With this technique, spatial properties can be derived on the propagation of light, for example wavevectors and mode profiles [32

32. H. Gersen, E.M.P.H. van Dijk, J.P. Korterik, N.F. van Hulst, and L. Kuipers, “Phase mapping of ultrashort pulses in bimodal photonic structures: A window on local group velocity dispersion,” Phys. Rev. E **70**, 066609 (2004) [CrossRef]

17. A. Imhof, W.L. Vos, R. Sprik, and A. Lagendijk, “Large dispersive effects near the band edges of photonic crystals,” Phys. Rev. Lett. **83**, 2942–2945 (1999) [CrossRef]

## 3. Results

### 3.1. Near-field experiments

*τ*was increased going from Fig. 4a to Fig. 4c. The pulses are found to be well confined to the waveguide in the lateral direction. At each frequency, the spatial distribution along the propagation direction is different. At the highest frequency (

*ω*=0.2664), the pulse in the structure was least distorted by the dispersion and we find a smooth and short envelope of 25(2)

*μ*m (FWHM) in width. As the frequency was reduced the measured pattern was elongated along the propagation direction. At

*ω*=0.2635 the elongation was still moderate, but the measurement at

*ω*=0.2603 shows significantly broader pulses (48(3)

*μ*m FWHM) than at higher optical frequencies. A clear beating pattern is visible in the measurement at

*ω*=0.2635. Most probably, an unwanted TM polarized mode was generated while coupling light to the structure, which “quasi-interfers” with the TE mode, resulting in the modulated amplitude pattern. Note that due to the mixing of polarizations in the near-field probe, quasi-interference can occur between orthogonally polarized modes [30

30. M.L.M. Balistreri, A. Driessen, J.P. Korterik, L. Kuipers, and N.F. van Hulst, “Quasi interference of perpendicularly polarized guided modes observed with a photon scanning tunneling microscope,” Opt. Lett. **25**, 637–639 (2000) [CrossRef]

24. R.J.P. Engelen, T.J. Karle, H. Gersen, J.P. Korterik, T.F. Krauss, L. Kuipers, and N.F. van Hulst, “Local probing of Bloch mode dispersion in a photonic crystal waveguide,” Opt. Express **13**, 4457–4464 (2005) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-12-4457 [CrossRef] [PubMed]

*μ*m and 116

*μ*m distance from the waveguide input facet, respectively. The position of the fiber probe at each of these locations is highlighted with crosses in Fig. 4. The interferograms resulting from the PXC measurements are shown in Fig. 5(a-c). For clarity only the amplitude of the interferograms is presented, while all the underlaying fringes are also measured. The underlaying fringes of the measurement at position “F” in Fig. 5(a) are shown in Figs. 5(e) and 5(f). The interference fringes are well resolved. By Fourier transformation, the optical carrier frequency is recovered. We find frequencies of

*ω*=0.2664(19), 0.2635(12) and 0.2603(8). The error in these values is given by the FWHM of the corresponding Fourier transforms.

*ω*=0.2664 are shown in Fig. 5(a). The interferograms are found at different delay times, corresponding to the travelling time of the pulse. We analyzed both the center of mass (CoM) and the maxima of the interferograms as a function of probe position. We found a linear dependence of the found delay time

*τ*as a function of probe position

*z*. The slope represents the group velocity. Due to the asymmetric pulse dispersion (especially at low

*ω*), the found velocity is slightly different, depending on with method was used. We found that, the CoM-approach results in a too low

*ν*

_{g}, whereas analyzing the maxima results in a too high group velocity. Therefore, we used both approaches to determine the group velocity, were the results represent the upper and lower error margin. In this way, we find a group velocity of

*c*· 0.188(5) for

*ω*= 0.2664. Similarly, the group velocities of the other measurements are also calculated. The group velocities for the measurements at

*ω*=0.2635 and 0.2603 are c · 0.151(5) and

*c*· 0.116(9), respectively.

*ω*=0.2664, the interferogram retains its initial shape, but as the pulses travel further in the PhCW, the interferogram is broadened up to 0.57(5) ps at position “F”. This broadening is still moderate at this high frequency, but as the frequency is reduced to

*ω*=0.2603 the pulses experience a much stronger broadening. At the same spatial position (“F”), we measured a FWHM of the interferogram of 2.2(2) ps. Interestingly, as the pulse has propagated through 106

*μ*m, the shape of the pulse has become asymmetric for all frequencies. The largest asymmetry is found in the measurements at

*ω*=0.2603.

*ω*=0.2664. These modes are visible as smaller side-lobes on the main pulse. We could discriminate these modes by their group velocities found in Fig. 5 and their spatial profile in the measurements of Fig. 4. Particularly at the measurement position at 10

*μ*m from the facet, the air-guided light is very strong. This is not surprising, since we couple light into the structure via an objective, that focusses light onto the facet. Since a diffraction limited spot at these wavelength is larger in diameter (3

*μ*m) than the membrane thickness, such coupling results in some light skimming over the surface of the PhCW.

*ω*=0.2635, a TM-polarized mode is very strong and causes a quasi-interference with the dominant TE-polarized mode. Specifically, a TM-polarized crystal mode travels at roughly the same group velocity as the TE defect mode. This causes the irregular pulse shapes in the interferograms in the measurement at

*ω*=0.2635. The TM-crystal mode is particularly strong in the measurements at this frequency. In the other measurements, it’s influence is negligible. From a Fourier transform of the complex fields in Fig. 4, we know that multiple

*k*-vectors are present in the measurement [24

24. R.J.P. Engelen, T.J. Karle, H. Gersen, J.P. Korterik, T.F. Krauss, L. Kuipers, and N.F. van Hulst, “Local probing of Bloch mode dispersion in a photonic crystal waveguide,” Opt. Express **13**, 4457–4464 (2005) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-12-4457 [CrossRef] [PubMed]

32. H. Gersen, E.M.P.H. van Dijk, J.P. Korterik, N.F. van Hulst, and L. Kuipers, “Phase mapping of ultrashort pulses in bimodal photonic structures: A window on local group velocity dispersion,” Phys. Rev. E **70**, 066609 (2004) [CrossRef]

19. 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]

**13**, 4457–4464 (2005) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-12-4457 [CrossRef] [PubMed]

*ω*and

*k*

_{z}known, a portion of the dispersion relation can be drawn. In Fig. 6(a), the measured dispersion relation is slightly offset (0.5% of

*ω*) with respect to the theoretical dispersion relation. This can be attributed to a slight offset of the theoretical dispersion relation of Δ

*ω*≈ 0.006, compared tot the found results in the measurements. This small difference can easily be explained by assuming that the actual geometric and optical properties of the sample are slightly different from the parameters used in the simulations.

*c*down to 0.108(9) ·

*c*. The qualitative agreement between experimental data and the theoretic dispersion relation is clear. Again, the measured points are offset in frequency with respect to the theoretic curve. The discrepancy in Fig. 6(b) (2% of

*ω*) is larger than in Fig. 6(a). This suggests that the actual dispersion relation differs from the simulated curve in two aspects. First, the actual curve is offset in frequency as is observed in Fig. 6(a). Second, the actual curve runs steeper than the simulated curve, due to the difference in group velocity.

### 3.2. The effect of higher-order dispersion

*ω*=0.2620, 0.2585 and 0.2565 correspond to the experimental frequencies

*ω*=0.2664, 0.2535 and 0.2503. We have calculated six interferograms at positions from 10

*μ*m up to 116

*μ*m and the results are shown in Fig. 7. Again, only the amplitude of the interfero-grams are shown for clarity.

*ω*=0.2620. At this frequency, only a slight symmetric broadening is visible after 116

*μ*m of propagation. This in contrast to the results at

*ω*=0.2585. Here, the interferograms are clearly broadened when propagating through the waveguide. This effect is present much stronger when calculating the envelopes at

*ω*=0.2565. Now, the dispersion is very strong and after 116

*μ*m, the pulse are approximately 3 times longer (FWHM) than initially. At this frequency, the asymmetry of the interferogram becomes clear. Since the spectrum of the pulses is symmetric along the carrier wavelength, the asymmetry in temporal pulse shape (and also in the interferogram) can only be caused by the enhanced higher-order dispersion in the waveguide. By comparing Figs. 5 and 7, we can conclude that the overall interference envelope change is similar in the measurements and the calculation.

*ω*= 0.2603, we calculated the magnitude of the dispersive terms. We found the following GVD:

*β*

_{2}= -1.1(3) · 10

^{6}ps

^{2}/km. For TOD we find

*β*

_{3}= 1.1 (4) · 10

^{5}ps

^{3}/km and for the fourth order dispersion we found

*β*

_{4}= -8(4) · 10

^{3}ps

^{4}/km.

*β*

_{>2}= 0 in Eq. (4) and using this dispersion, the interferogram was recalculated. In steps, the higher-order dispersive terms were added to the calculation. First only GVD and then TOD was added to the calculation. The results are shown in Fig. 8, where they are compared to the calculation using the full dispersion relation, with all dispersive orders included. Clearly, if we compare the dispersion-free propagating pulse with the full calculation, it is clear that the pulse broadens significantly and we obtain only 25% overlap between the two curves. This broadening is also visible if the GVD is included in the calculation. Now the overlap increased to 64%. However, the shape of the interferogram is still symmetric. We obtain an asymmetric interferogram if TOD is considered. This approximates the actual interferogram quite well with 88% overlap. If also the fourth order dispersion is included (not shown), a slightly better overlap is found up to 89%. Note however, that the group velocity in this calculation is still quite high (

*ν*

_{g}= 0.116 ·

*c*). If the group velocity is further decreased, higher-order dispersive effect play an increasingly important role and then, using a Taylor expansion as in Eq. (4) is an inaccurate approximation to the dispersion relation if only lower order dispersive terms are considered.

## 4. Conclusion

*c*and 0.108(9) ·

*c*. Though these group velocities are moderately low, the found values for GVD and TOD are already substantial. We have quantified the GVD from our measurements, and found anomalous dispersion up to

*β*

_{2}= -1.1(3) · 10

_{6}ps

^{2}/km. Similarly, the TOD is found to be very large: values up to

*β*

_{3}= 1.1(4) · 10

^{5}ps

^{3}/km are found. These parameters are sufficient to simulate the dispersive effects of a femtosecond pulse travelling through our photonic crystal waveguide at a moderate speed of 0.116 ·

*c*. As the group velocity reduces further, even higher-order dispersive terms will start to play a role. Especially at these lower optical frequencies the common approach to approximate the dispersive properties by a Taylor expansion, is only valid if many orders of the expansion are included. We have found that the effect of higher-order dispersion in a typical photonic crystal (waveguide) strongly increases when the group velocity decreases.

*c*/10). However, if one exploits a lower group velocity, for example

*c*/50 to achieve the same delay, the elongation would be up to 4 times larger (approximately 0.5 ns) and the pulse shape would become asymmetric, due to the increased higher-order dispersion. In this respect, the large higher-order dispersion in the studied waveguide makes the simple W1 geometry an unlikely candidate for slow-light applications. However, in photonic crystals, one has the freedom to selectively alter the dispersive properties by changing the geometry of the lattice, or even by using combinations of lattices. This freedom will need to be used to create large-bandwidth slow-light applications that do not suffer from higher-order dispersion.

## Acknowledgements

## References and links

1. |
See for example, |

2. | S.G. Johnson, P.R. Villeneuve, S. Fan, and J.D. Joannopoulos, “Linear waveguides in photonic crystal slabs,” Phys. Rev. B |

3. | Y. Sugimoto, Y. Tanaka, N. Ikeda, Y. Nakamura, K. Asakawa, and K. Inoue, “Low propagation loss of 0.76 dB/mm in GaAs-based single-line-defect two-dimensional photonic crystal slab waveguides up to 1 cm in length,” Opt. Express |

4. | E. Chow, S.Y. Lin, J.R. Wendt, S.G. Johnson, and J.D. Joannopoulos, “Quantitative analysis of bending effiency in photonic crystal wavgeuide bends at |

5. | Y. Akahane, T. Asano, B.S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature |

6. | A.Y. Petrov and M. Eich, “Zero dispersion at small group velocities in photonic crystal waveguides,” Appl. Phys. Lett. |

7. | V.N. Astratov, R.M. Stevenson, I.S. Culshaw, D.M. Whittaker, M.S. Skolnick, T.F. Krauss, and R.M. de la Rue, “Heavy photon dispersions in photonic crystal waveguides,” Appl. Phys. Lett. |

8. | M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, “Extremely Large Group-Velocity Dispersion of Line-Defect Waveguides in Photonic Crystal Slabs,” Phys. Rev. Lett. |

9. | K. Inoue, N. Kawai, Y. Sugimoto, N. Ikeda, and K. Asakawa, “Observation of small group velocity in two-dimensional AlGaAs-based photonic crystal slabs,” Phys. Rev. B |

10. | Y.A. Vlasov, M. O’Boyle, H.F. Hamann, and S.J. McNab, “Active control of slow light on a chip with photonic crystal waveguides” Nature |

11. |
for a review see
M. Soljacic and J.D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nature Mater. |

12. | M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, and T. Tanabe, “Optical bistable switching action of Si high-Q photonic-crystal nanocavities,” Opt. Express |

13. | A. Sugitatsu, T. Asano, and T, S. Noda, “Characterization of line-defect-waveguide lasers in two-dimensional photonic-crystal slabs,” Appl. Phys. Lett. |

14. | S.F. Mingaleev, Yu.S. Kivshar, and R.A. Sammut, “Longrange interaction and nonlinear localized modes in photonic crystal waveguides,” Phys. Rev. E |

15. | M.D. Rahn, A.M. Fox, M.S. Skolnick, and T.F. Krauss, “Propagation of ultrashort nonlinear pulses through two-dimensional AlGaAs high-contrast photonic crystal waveguides,” J. Opt. Soc. Am. B |

16. | S. Yamada, Y. Watanabe, Y. Katayama, and J.B. Cole, “Simulation of optical pulse propagation in a two-dimensional photonic crystal waveguide using a high accuracy finite-difference time-domain algorithm,” J. Appl. Phys. |

17. | A. Imhof, W.L. Vos, R. Sprik, and A. Lagendijk, “Large dispersive effects near the band edges of photonic crystals,” Phys. Rev. Lett. |

18. | T. Asano, K. Kiyota, D. Kumamoto, B.S. Song, and S. Noda, “Time-domain measurement of picosecond light-pulse propagation in a two-dimensional photonic crystal-slab waveguide,” Appl. Phys. Lett. |

19. | 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. |

20. | M. Miyagi and S. Nishida, “Pulse spreading in a single-mode fiber due to third-order dispersion,” Appl. Opt. |

21. | J. Khurgin, “Performance of nonlinear photonic crystal devices at high bit rates,” Opt. Lett. |

22. | M.L.M. Balistreri, H. Gersen, J.P. Korterik, L. Kuipers, and N.F. van Hulst, “Tracking femtosecond laser pulses in space and time,” Science |

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

24. | R.J.P. Engelen, T.J. Karle, H. Gersen, J.P. Korterik, T.F. Krauss, L. Kuipers, and N.F. van Hulst, “Local probing of Bloch mode dispersion in a photonic crystal waveguide,” Opt. Express |

25. | Product information sheet, “Corning SMF-28e Optical Fiber, Product Information” (Corning Inc., 2005) http://corning.com/opticalfiber/products%5F%5Fapplications/products/smf%5F28e.aspx |

26. | P.St.J. Russell, “Optics of Floquet-Bloch Waves in Dielectric Gratings,” Appl. Phys. B |

27. | B. Lombardet, L.A. Dunbar, R. Ferrini, and R. Houdre, “Fourier analysis of Bloch wave propagation in photonic crystals,” J. Opt. Soc. Am. B |

28. | G.P. Agrawal, |

29. | H. Gersen, J.P. Korterik, N.F. van Hulst, and L. Kuipers, “Tracking ultrashort pulses through dispersive media: Experiment and theory” Phys. Rev. E |

30. | M.L.M. Balistreri, A. Driessen, J.P. Korterik, L. Kuipers, and N.F. van Hulst, “Quasi interference of perpendicularly polarized guided modes observed with a photon scanning tunneling microscope,” Opt. Lett. |

31. | P. Sanchis, P. Bienstman, B. Luyssaert, R. Baets, and J. Marti, “Analysis of butt coupling in photonic crystals,” IEEE J. Quantum Electron. |

32. | H. Gersen, E.M.P.H. van Dijk, J.P. Korterik, N.F. van Hulst, and L. Kuipers, “Phase mapping of ultrashort pulses in bimodal photonic structures: A window on local group velocity dispersion,” Phys. Rev. E |

**OCIS Codes**

(160.3130) Materials : Integrated optics materials

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

(230.7370) Optical devices : Waveguides

(260.2030) Physical optics : Dispersion

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: December 13, 2005

Revised Manuscript: February 13, 2006

Manuscript Accepted: February 13, 2006

Published: February 20, 2006

**Citation**

R.J.P. Engelen, Y. Sugimoto, Y. Watanabe, J.P. Korterik, N. Ikeda, N.F. van Hulst, K. Asakawa, and L. Kuipers, "The effect of higher-order dispersion on slow light propagation in photonic crystal waveguides," Opt. Express **14**, 1658-1672 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-4-1658

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### References

- See for example, Photonic Crystals and Light Localization in the 21st Century, in NATO Science Series, C.M. Soukoulis, ed. (Kluwer Academic, Dordrecht, The Netherlands, 2001)
- S.G. Johnson, P.R. Villeneuve, S. Fan, J.D. Joannopoulos, "Linear waveguides in photonic crystal slabs," Phys. Rev. B 62, 8212-8222 (2000) [CrossRef]
- Y. Sugimoto, Y. Tanaka, N. Ikeda, Y. Nakamura, K. Asakawa, K. Inoue, "Low propagation loss of 0.76 dB/mm in GaAs-based single-line-defect two-dimensional photonic crystal slab waveguides up to 1 cm in length," Opt. Express 12, 1090-1096 (2004) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1090 [CrossRef] [PubMed]
- E. Chow, S.Y. Lin, J.R. Wendt, S.G. Johnson, J.D. Joannopoulos, "Quantitative analysis of bending effiency in photonic crystal wavgeuide bends at ⌊= 1.55 m wavelengths," Opt. Lett. 26, 286-288 (2001) [CrossRef]
- Y. Akahane, T. Asano, B.S. Song, S. Noda, "High-Q photonic nanocavity in a two-dimensional photonic crystal," Nature 425, 944-947 (2003) [CrossRef] [PubMed]
- A.Y. Petrov, M. Eich, "Zero dispersion at small group velocities in photonic crystal waveguides," Appl. Phys. Lett. 85, 4866-4868 (2004) [CrossRef]
- V.N. Astratov, R.M. Stevenson, I.S. Culshaw, D.M. Whittaker, M.S. Skolnick, T.F. Krauss, R.M. de la Rue, "Heavy photon dispersions in photonic crystal waveguides," Appl. Phys. Lett. 77, 178-180 (2000) [CrossRef]
- M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, I. Yokohama, "Extremely Large Group-Velocity Dispersion of Line-Defect Waveguides in Photonic Crystal Slabs," Phys. Rev. Lett. 87, 253902 (2001) [CrossRef] [PubMed]
- K. Inoue, N. Kawai, Y. Sugimoto, N. Ikeda, K. Asakawa, "Observation of small group velocity in twodimensional AlGaAs-based photonic crystal slabs," Phys. Rev. B 65, 121308 (2002) [CrossRef]
- Y.A. Vlasov, M. O’Boyle, H.F. Hamann, S.J. McNab, "Active control of slow light on a chip with photonic crystal waveguides" Nature 438, 65-69 (2005) [CrossRef] [PubMed]
- for a review see M. Soljacic, J.D. Joannopoulos, "Enhancement of nonlinear effects using photonic crystals," Nature Mater. 3211-219 (2004) [CrossRef]
- M. Notomi, A. Shinya, S. Mitsugi, G. Kira, E. Kuramochi, T. Tanabe, "Optical bistable switching action of Si high-Q photonic-crystal nanocavities," Opt. Express 132678-2687 (2005) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-7-2678 [CrossRef] [PubMed]
- A. Sugitatsu, T. Asano T, S. Noda, "Characterization of line-defect-waveguide lasers in two-dimensional photonic-crystal slabs," Appl. Phys. Lett. 845395-5397 (2004) [CrossRef]
- S.F. Mingaleev, Yu.S. Kivshar, R.A. Sammut, "Longrange interaction and nonlinear localized modes in photonic crystal waveguides," Phys. Rev. E 625777 (2000) [CrossRef]
- M.D. Rahn, A.M. Fox, M.S. Skolnick, T.F. Krauss, "Propagation of ultrashort nonlinear pulses through twodimensional AlGaAs high-contrast photonic crystal waveguides," J. Opt. Soc. Am. B 19, 716-721 (2002) [CrossRef]
- S. Yamada, Y. Watanabe, Y. Katayama, J.B. Cole, "Simulation of optical pulse propagation in a two-dimensional photonic crystal waveguide using a high accuracy finite-difference time-domain algorithm," J. Appl. Phys. 93, 1859-1864 (2003) [CrossRef]
- A. Imhof, W.L. Vos, R. Sprik, A. Lagendijk, "Large dispersive effects near the band edges of photonic crystals," Phys. Rev. Lett. 83, 2942-2945 (1999) [CrossRef]
- T. Asano, K. Kiyota, D. Kumamoto, B.S. Song, S. Noda, "Time-domain measurement of picosecond light-pulse propagation in a two-dimensional photonic crystal-slab waveguide," Appl. Phys. Lett. 84, 4690-4692 (2004) [CrossRef]
- H. Gersen, T.J. Karle, R.J.P. Engelen, W. Bogaerts, J.P. Korterik, N.F. van Hulst, T.F. Krauss, L. Kuipers, "Real space observation of ultraslow light in photonic crystal waveguides," Phys. Rev. Lett. 94, 073903 (2005) [CrossRef] [PubMed]
- M. Miyagi, S. Nishida, "Pulse spreading in a single-mode fiber due to third-order dispersion," Appl. Opt. 18, 678-682 (1979) [CrossRef] [PubMed]
- J. Khurgin, "Performance of nonlinear photonic crystal devices at high bit rates," Opt. Lett. 30, 643-645 (2005) [CrossRef] [PubMed]
- M.L.M. Balistreri, H. Gersen, J.P. Korterik, L. Kuipers, N.F. van Hulst, "Tracking femtosecond laser pulses in space and time," Science 294, 1080-1082 (2001) [CrossRef] [PubMed]
- S.G. Johnson, J.D. Joannopoulos, "Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis," Opt. Express 8173-190 (2001) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173 [CrossRef] [PubMed]
- R.J.P Engelen, T.J. Karle, H. Gersen, J.P. Korterik, T.F. Krauss, L. Kuipers, N.F. van Hulst, "Local probing of Bloch mode dispersion in a photonic crystal waveguide," Opt. Express 13, 4457-4464 (2005) http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-12-4457 [CrossRef] [PubMed]
- Product information sheet, "Corning SMF-28e Optical Fiber, Product Information" (Corning Inc., 2005) http://corning.com/opticalfiber/products%5F%5Fapplications/products/smf%5F28e.aspx
- P.St.J. Russell, "Optics of Floquet-Bloch Waves in Dielectric Gratings," Appl. Phys. B 39, 231-246 (1986) [CrossRef]
- B. Lombardet, L.A. Dunbar, R. Ferrini, R. Houdre, "Fourier analysis of Bloch wave propagation in photonic crystals," J. Opt. Soc. Am. B 22, 1179-1190 (2005) [CrossRef]
- G.P. Agrawal, Nonlinear Fiber Optics (Academic Press, San Diego, Calif., 2001)
- H. Gersen, J.P. Korterik, N.F. van Hulst, L. Kuipers, "Tracking ultrashort pulses through dispersive media: Experiment and theory" Phys. Rev. E 68, 026604 (2003) [CrossRef]
- M.L.M. Balistreri, A. Driessen, J.P. Korterik, L. Kuipers, N.F. van Hulst, "Quasi interference of perpendicularly polarized guided modes observed with a photon scanning tunneling microscope," Opt. Lett. 25, 637-639 (2000) [CrossRef]
- P. Sanchis, P. Bienstman, B. Luyssaert, R. Baets, J. Marti, "Analysis of butt coupling in photonic crystals," IEEE J. Quantum Electron. 40, 541-550 (2004) [CrossRef]
- H. Gersen, E.M.P.H. van Dijk, J.P. Korterik, N.F. van Hulst, L. Kuipers, "Phase mapping of ultrashort pulses in bimodal photonic structures: A window on local group velocity dispersion," Phys. Rev. E 70, 066609 (2004) [CrossRef]

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