We report on time-of-flight experimental measurements and numerical calculations of the group-index dispersion in a photonic crystal waveguide realized in silicon-on-insulator material. Experimentally group indices higher than 230 has been observed. Numerical 2D and 3D time-domain simulations show excellent agreement with the measured data.
©2005 Optical Society of America
It is well-known that various classical nonlinear effects can be enhanced by a small group velocity of light, since the light-matter interaction increases with decreasing group velocity vg (GV) [1–7]. For the traditional ridge or slab waveguides usually considered in integrated optics, a nonlinear mode dispersion, which leads to a small value of the group velocity, is a highly exotic phenomenon. However, photonic crystal waveguides (PhCWs) deliver very large group-index dispersion combined with extremely low values of the group velocity at the cutoff for the defect mode located in a photonic band gap (PBG). Other prominent features may occur in a PBG of a photonic crystal without defects, e.g., superluminal propagation with GVs exceeding that of light in vacuum [8–10]. Therefore, photonic crystals are considered to be interesting new aspirants for future applications in a large variety of all-optical photonic circuit devices.
A key question when investigating slow light in PhCWs is how the group velocity in the proximity of a PBG can be determined in a reliable way. Various different experimental and/or theoretical approaches have previously been presented in the literature on the determination of the GV or, equivalently, the group index ng, which is connected to the GV by the relation vg = c/ng.
Presumably the most straightforward way is to obtain the GV directly from its definition:
When the dispersion relations have been obtained experimentally or numerically, it is straightforward to perform a numerical differentiation of the data to obtain the GV. Numerous examples of this approach exist in the literature [4,11–15]. Disadvantages of this method include that the limited accuracy of the numerical differentiation impedes the precise determination of the GV, and that only a limited number of points can be inferred for the GV.
Another method concerns the determination of the phase of the transmitted impulse [9,10,16]. Such phase calculations can be performed with very high precision. For example, the calculations can be used to obtain the overall phase shift when passing through the photonic crystal or to determine the phase delay of an arriving pulse relative to a reference arm. A closely related method is to extract the GV from the Fabry-Perot (F-P) fringes in a transmission spectrum [1,17]. In this case the GV can be found by
where ∆λ is the wavelength spacing between adjacent the F-P peaks and L is the length of the PhCW, which forms the F-P cavity. As before a drawback of this method is that only a limited number of points may be deducted for the GV behavior.
Finally, the GV can be obtained utilizing the time-of-flight (ToF) method, which exists in several varieties [6,12–14,18–22]. Some of the experimental methods deal with the direct observation of the time delay of a pulse when it traverses through a PhCW or with the visualization of the pulse propagation employing near-field scanning optical microscopy measurements [14,20]. In the latter case, it has been shown that the field of a very slow mode appears to be frozen at a certain place in a W3 waveguide, which is defined by three rows of missing holes in the photonic crystal. An estimate of the GV hints a value at roughly c/1000 . However, in this case the understanding of the mode propagation has not yet been clarified, so these quoted data has assessing character only. One possible limitation of the ToF method is caused by the heavy distortion of a pulse in the PBG region. In the case of a complete corrugation of the pulse shape a large uncertainty will unavoidably arise in the determination of the centre of mass of the pulse. Furthermore, in numerical calculations the abrupt decrease in the intensity of the pulse in the PBG region may give numerical distortions in form of spurious reflections from the absorbing boundaries that truncate the numerical space. Despite these potential limitations, the ToF method provides values for the GV in a straightforward manner. Therefore, this method may be considered fundamental and universal in character. While most of the abovementioned examples permit experimental realization of the ToF method, a theoretical realization is considered to be challenging .
In this paper, the ToF method is utilized in modeling as well as in experimental characterization of light propagation in PhCWs. In the experiment the group index is extracted by measuring the difference in ToF whereas it is determined by direct pulse tracking in the calculations. The study is based on silicon based planar photonic crystal structures that are encapsulated between glass claddings. The periodic structure constitutes of holes having diameter d arranged in a triangular lattice with pitch Λ. The W1 PhCWs are formed by carving out 1-row line-defects in the Γ-K direction. Description of the experimental procedure and measurement results are given in Section 2. Details on the ToF modeling and discussions follow in Section 3. The conclusion is found in Section 4.
2. Experimental results
In the experimental part of our work we investigated a 20-μm long W1 PhCW, which was realized in a planar photonic crystal with glass filled holes (d = 282±5 nm) arranged in a triangular lattice having period Λ=435 nm. The waveguide was fabricated in a core silicon layer (thickness 216±5 nm) that was encapsulated between glass claddings of approximately 1 μm thicknesses. The group velocity of the quasi-even PBG mode decreases dramatically when approaching the edge of the Brillouin zone. The mode cutoff is located in the c-band at 1562.4 nm for the fabricated sample.
2.1 The experimental setup.
The basic idea is to measure the difference in ToF and thereby to determine the change in the group index versus wavelength, ∆ng(λ). This is done by measuring the phase delay of a transmitted signal as function of wavelength. The transmitted signal is an envelope function for the amplitude of the light and, therefore, the envelope phase velocity corresponds to the group velocity of the light. Light that is amplitude-modulated is sent through the PhCW, and the envelope phase is recorded as function of the wavelength. The envelope phase is then used to calculate the group index of the light. The envelope phase is measured using a network analyzer, which modulates the light and detects it after passing the PhCW. The experimental setup is sketched in Fig. 1. Our experimental setup has some similarities with the one mentioned in Ref. . In the latter case, however, the envelope phase was detected using an electronic comparator instead of a network analyzer. As a consequence, we have improved the accuracy of the measured time delay roughly by a factor of 5.
In order to measure ∆ng(λ) in the PhCW, light emitted from a continuous-wave (cw) laser is utilized. The laser light is small-signal amplitude-modulated in a LiNbO3 Mach-Zehnder modulator at 19 GHz. The frequency is chosen as high as possible to obtain the best temporal resolution (i.e., the highest possible number of oscillations per second). The tradeoff of this choice manifests itself in the spectral resolution, as the high-speed modulation of the light introduces sidebands as seen in Fig. 2. The sidebands are only detectable in the spectrum when the modulation amplitude is large. The modulation signal is carried in the sidebands. Hence, the probe pulse can be considered to be contained within a spectral width of less than 0.4 nm whether it is modulated with small or large amplitude.
2.2 Experimental method and results
The modulated wave is transmitted through the sample under test into a photodiode, and the envelope phase is recorded as function of wavelength utilizing the network analyzer. The measured envelope phase gives the dispersion of the whole measurement system. In order to obtain the phase change due to the PhCW alone, a calibration measurement is made using a buried ridge waveguide that contains no photonic crystal region. The envelope phase as function of wavelength is approximated with a third order polynomial and this calibration curve is subtracted from the envelope phase measurement for the PhCW. The calibration curve and the phase change due to the PhCW alone are displayed in Figs. 3 and 4, respectively.
The phase delay of the modulated signal due to the PhCW part, ∆ϕelectric (λ), can be used to determine the change in the group index ∆ng(λ) of the light in the PhCW from the equation
Here c is the speed of light, lPhc is the length of the PhCW, and felectric is the frequency of the envelope function. The negative sign is due to the fact that a larger phase delay corresponds to a larger group index. The change in the group index determined by this method is partly influenced by two F-P cavities in the sample, namely between the two end facets of the sample and the two internal facets between the buried ridge waveguides and the PhCW. The two buried ridge waveguides are used to route light to and from the PhCW. The lengths of the two buried ridge waveguides are 4334±7μm and 786±15μm, respectively. To determine the F-P spectral width, a high-resolution spectrum was made and it revealed two F-P periods of 0.07 nm and 0.35 nm as shown in Fig. 5. These two F-P periods correspond to an effective index of ~4, in reasonable agreement with the expected effective index in multimode buried ridge waveguides made in silicon. A relatively high propagation loss for light transmitted through the PhCW ensures that the F-P contribution from light bouncing back and forth inside the PhCW may be ignored.
The F-P contribution to the change in the group index can be accounted for (and hence removed) by averaging over the corresponding F-P period, thereby giving an average effective optical path length that does not change with wavelength. Since the graph for ∆ng(λ) displays interesting spectral features smaller than 0.35 nm, it was decided to remove only the contribution from the physically long F-P cavity by averaging. To arrive at absolute values for the group index of the PhCW mode from the relative change in Eq. (3), the vertical offset of the experimental curve is adjusted to match the calculated results, and the experimental result for ng(λ) is given in Fig. 6. Arguably the most distinct feature of these measurement results is the direct experimental observation of an extremely high group index exceeding 230 near the cutoff. Another interesting observation is that even though the transmission in general is decreasing when approaching the cutoff then we clearly measure local maxima for the transmission in the wavelength ranges where the group index displays local maxima. This observation does not agree with the behavior proposed in Ref. , where the loss is predicted to grow with increasing group index. Also shown in Fig. 6 is the result of the numerical 2D FDTD calculation for the group index. The numerical and experimental data are seen to be in excellent agreement. No fitting procedures have been applied except for a scaling of the wavelength axis of the numerical data. It is noted that there is a large dip in the experimental data at the very edge of the transmission band that is not covered by the 2D FDTD data points.
The details of the numerical calculations are given in the next section, where also data from the 3D FDTD simulations are presented and discussed.
3. Numerical results
In the numerical investigations of the ToF method calculations have been made by utilizing 2D and 3D finite-difference time-domain (FDTD) methods to track the pulse propagation in the PhCW. These methods are particularly suitable when exploring the dynamics of optical pulses in various devices including the photonic crystal based ones. In the following the 2D FDTD calculations are discussed and succeeded by a treatment of the 3D case.
Before initiating the ToF calculations the position of the mode cutoff needs to be determined. Therefore, band diagrams were calculated using a 2D plane wave expansion (PWE) method . From these calculations it is found that the fundamental PhCW mode cuts off at the normalized frequency 0.215 (left plot in Fig. 7). The transmission spectrum was calculated by the 2D FDTD method and is shown next to the band diagram (right plot in Fig. 7). In the 2D FDTD calculations the length of the PhCW was 30 μm. The other PhCW parameters used in the modeling were chosen to match as exactly as possible the experimental parameters given in the previous section.
It should be emphasized that no fitting procedures have been applied to the plots shown in Fig. 7. The figure displays an unambiguous frequency correlation between the PWE and 2D FDTD data. All the distinctive features of the dispersion behavior of the PhCW modes are reflected in the transmission spectrum. For example, the stop band around f= 0.17, the cutoff for the guided PBG mode at f= 0.215, and the transmission dip due to interactions between even and odd modes near f= 0.245 are all clearly mimicked in the 2D FDTD plot.
The ToF method used in our modeling is carried out as follows. A TE-polarized impulse is launched from a ridge waveguide into the PhCW. The tracing of the impulse is performed by using two detector plates that are placed at the input and output ports of the PhCW. The dynamics of the impulse propagation through the detectors are recorded by storing the field values at the detector plates for all time steps. By determining the time when the top of the impulse passes the detectors we can calculate the time delay ∆t, which together with the known distance ∆L between the detectors, is used to compute the group index
Our first set of calculations was performed with a 2.7-ps long impulse (time interval of nonzero emission from the exciter). This impulse had a spectral full-width-half-maximum (FWHM) of approximately 6.5 nm. In these calculations it was impossible to obtain group indices higher than 50. The reason for this is due to the fact that the result of the calculations depends on the spectral width of the impulse. If the impulse is too broad it may span over the whole nonlinear part of the dispersion band and, partially, cover the linear dispersion zone. Hence, it is obvious that impulses with longer lifetime are needed when it is taken into account that the extent of the wavelength region with highly nonlinear growth of the group index is around 5-6 nm.
These results were significantly refined by extending the temporal impulse to 58.3 ps (requiring 1 million time steps in the FDTD calculations) resulting in a FWHM of 0.3 nm, which is comparable to the experimental resolution of 0.4 nm. 106 time steps for each point in the spectrum prove to be a heavy burden in terms of processor time, especially in the case of the 3D calculations and hence the PhCW length was limited to 8.64 μm for the 3D FDTD calculations. However, the benefit was that well-defined impulses traversing the second detector were observed and that higher values of the group index were obtained. This is clearly seen in Fig. 8, which shows the ToF of the electric field recorded at the input and output detectors.
The results from the 2D modeling have already been shown in Fig. 6. The group index reaches its highest value ng = 438 at a wavelength near the cutoff. It was not possible to obtain values for longer wavelengths beyond the cutoff, since the impulses in this case were strongly suppressed in intensity and severely distorted in shape. Hence, it was no longer valid to define the impulse for tracking the ToF by the peak of the amplitude when the shape of the outgoing impulses was distorted in such a degree that the pulse no longer was symmetrically shaped. This typically occurred when the normalized transmission was below 0.1 %.
In the case of the 3D FDTD calculations the parameters were also chosen to match the set of the experimental ones. However, due to high memory requirements both the heights of the SiO2 cladding layers and the waveguide length L were truncated. Furthermore, all the parameters were rounded to the nearest integer multiple of the space grid resolution δ = 27 nm. Only 5 rows of holes were left on both sides of the W1 waveguide channel, thereby making the photonic crystal surrounding the PhCW reasonably good but not perfect. In the 3D ToF calculations, the central wavelength of the exciter was rearranged to fit with the 3D cutoff wavelength, which was found to be 1605 nm. The 3D FDTD cutoff deviates less than 3% from the experimental observed cutoff at 1562.4 nm. For comparison the cutoff was located at 2020 nm in the 2D case.
The calculated group indices in the 3D case are found to be similar to the ones computed in the 2D case, except that they do not reach the record values obtained by the 2D model. However, the 3D calculated group index of 166 is still the highest value obtained in a direct numerical experiment.
Figure 9 shows the 3D calculated group indices compared to the measured ones from previous section. Again an excellent agreement between measurements and modeling results are found. It is noted that contrary to the 2D case, the numerical 3D FDTD calculated group indices mimic the large dip in the experimental data at the very edge of the transmission band.
We have reported on measurements and numerical simulations of group indices of light propagating in a photonic crystal waveguide (PhCW) near the cutoff edge. The group-index change versus wavelength was measured by transmitting a 19 GHz small-modulated signal through the PhCW and detecting the envelope phase of the transmitted signal using a network analyzer. The dispersion in the rest of the measurement setup was determined by a calibration measurement on an ordinary waveguide containing no PhCW. The measurement determines the change in time-of-flight (ToF) through the PhCW versus wavelength and therefore reveals ∆ng(λ). Results for group index exceeding 230 were measured. The experimental curves fit very well with ToF modeling. The numerical method was realized by tracking the top of the impulse in time domain, either in 2D or in 3D space. A group index of 438 was found in 2D FDTD simulations and 166 when performing 3D calculations. These group index values for a photonic crystal waveguide are to the best of our knowledge the largest numbers reported so far by direct tracking of pulse propagation.
Direct comparison was made between the transmission spectrum and the group index versus wavelength. The measurement shows a one to one correlation between high local group index and a high local transmission in the investigated spectral window. This appears to contradict an earlier proposed connection between high group index and low transmission. However, deeper insight on this phenomenon is required.
This work was supported in parts by the Danish Technical Research Council via the PIPE (Planar Integrated PBG Elements) project and by the New Energy and Industrial Technology Development Organization (NEDO) via the Industrial Technology Research Area. R. S. Jacobsen acknowledges the NKT academy for financial support.
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