We report on the fabrication and evaluation of a directional coupler consisting of two different chirped photonic crystal (PC) waveguides that can generate wideband, low dispersion slow light. The directional coupler was fabricated from a silicon-on-insulator PC slab with airhole diameter chirping. For both waveguides, we observed a group index of 60–80 near the photonic band edge and opposite dispersion characteristics. In the directional coupler, we evaluated a group index of 30–40 in a wavelength bandwidth of 32 nm. The experimental results agree with the theoretical relation between the group index and bandwidth.
©2007 Optical Society of America
Slow light having an ultralow group velocity v g is of great interest because of its potential application to optical buffering and enhanced light-matter interaction. Photonic crystal (PC) waveguides [1–4] have been studied for use in generating slow light [5–11]. In general, slow light is constrained by the relation between the slowdown factor and operating spectral bandwidth. In addition, short optical pulses and high-bit-rate signals with a wide frequency spectrum are severely distorted due to large group velocity dispersion. Phase modulation of input signals by dynamic tuning of the material parameters has been discussed as a solution . However, this method operates like a one-bit optical memory rather than an optical buffer because it cannot store continuous signals without mutual interference. Previously, we proposed and theoretically demonstrated a chirped PC waveguide whose structural parameters are gradually changed along the waveguide . We also discussed combining the chirped structure with an ideal photonic band obtained by minute structural modifications to realize wideband, dispersion-compensated slow light [14,15]. In this case, the v g = 0 condition obtained by a flat photonic band is set at an inflection point of the band. When such an ideal band is shifted in a chirped structure, a light pulse is first affected by the dispersion but is recovered by the opposite dispersion, and is delayed by the v g = 0 condition. Such light propagation is obtained in a specified bandwidth by controlling the slope of the chirping. We showed theoretically that a directional coupler (DC) consisting of two different PC waveguides A and B with opposite dispersion characteristics equivalently realizes this operation .
In this paper, we experimentally demonstrate the operation of the DC. First, we briefly review the device structure and band profiles. Next, we explain the fabrication and evaluation methods. Then, we present the individual fabrication of A and B and evaluate the low v g and opposite dispersion characteristics. Finally, we demonstrate wideband, dispersion-compensated slow light in the DC and discuss its correspondence with theoretical results.
2. Device and photonic band
The structures of the waveguides A and B and the DC are shown with their band profiles in Fig. 1. A PC slab, consisting of a triangular lattice of airholes in a Si slab, is used as a platform. Waveguide A in Fig. 1(a) is a simple line defect waveguide. Its transmission band in the photonic bandgap below the air light line shows dispersion from the band edge to higher frequencies. Waveguide B in Fig. 1(b) is a line defect waveguide in which small airholes are arranged at the center with a half-period offset. It has two transmission bands, though we focus on the lower frequency band showing dispersion to lower frequencies. The DC in Fig. 1(c) consists of waveguides A and B with an inter-waveguide spacing of three lines of airholes. It has three bands originating from A and B. The lower two bands with opposite dispersion characteristics are used for dispersion compensation. When these bands are shifted by a chirped structure and the band edge frequencies of these bands are maintained to be equal, directional coupling from A to B occurs within a very short length due to the v g = 0 condition.
3. Experimental methods
The air-bridge PC slab was formed by e-beam lithography, SF6 inductively coupled plasma etching and HF wet etching of the silicon-on-insulator substrate with a top Si layer of 0.213 μm thickness. First, A and B were fabricated and evaluated individually. The lattice constant a of A and B was set at 0.48 μm and the airhole diameters were set to 2r 1 = 0.28 and 0.27 μm, respectively, so that the band edges were close to a wavelength of λ = 1.55 μm. For A, the line defect width defined by the distance between the centers of the innermost airholes was reduced to 85% of the simple line defect (W0.85) and the innermost airhole diameter of the waveguide was reduced by 5 nm. For B, W0.90 was used and the small airhole diameter 2r 2 was 0.21 μm.
In the measurement, the group index n g = c/v g was evaluated by two methods. In the first method (frequency-domain method), the inter-peak spacing Δλ of the internal Fabry-Perot (FP) resonance is measured and n g is obtained from the relation ng = λ2/2LΔλ, where L is the waveguide length [16,5]. It is a very simple method that requires measuring the transmission spectrum using a tunable laser source. The wavelength resolution of the laser source restricts the upper limit of the evaluated n g and its accuracy is affected by disordering of the resonant spectrum. In the second method (modulation phase shift method), the shift of phase Φ between the input and output ends of the waveguide is measured for sinusoidally modulated light with frequency ωm [7,8,11,17,18]. Although this method requires a more complicated setup, it allows direct evaluation of the delay time τ from the relation ωmτ = ΔΦ. However, τ is increased or decreased when the internal FP resonance or antiresonance occurs, respectively. This effect can be eliminated and the n g of the waveguide can be evaluated through the following analysis. The electric field E o of the output light, which is a superposition of single pass and round-trip components of the FP resonance, is expressed as
where E i is the input electric field, r i, r o, ϕi, and ϕo are the amplitude and phase of the reflectivity at the input and output ends, respectively, α is the loss coefficient of the waveguide, k 0 = 2π/λ and c are the wavenumber and velocity of light in vacuum, respectively, and n eq is the equivalent index of the waveguide mode. In the absence of resonance, i.e. r i, r o = 0, the phase of E o is given by that of the numerator of Eq. (1), i.e. Φn = -ωm n g L/c. When resonance occurs, the phase is modulated by the following phase of the denominator of Eq. (1):
where the first plus and minus signs (and second minus and plus signs) correspond to the resonant and antiresonant conditions, respectively, which lead to over- and under-estimation of τ. The term r i r o e -αL represents attenuation of light in the waveguide and can be evaluated from the amplitude of resonant fringes (peak to bottom ratio of the resonant transmission spectrum) . Finally, n g is determined such that the measured phase Φ equals Φn-Φd. Figure 2 shows the normalized phase shift cΔΦ/ωm L calculated with the amplitude of fringes. For each n g assumed, larger and smaller phase shifts correspond to the resonance and antiresonance, respectively. In the absence of the resonance, the normalized phase shift becomes equivalent to n g. When the amplitude of the fringes is larger than e.g. 5 dB, the phase shift oscillates in the range of twice. By fitting a theoretical curve with maximum and minimum phase shifts measured, n g can be estimated.
Figure 3 shows measured transmission spectra and evaluated group index n g. The light blue curve shows n g obtained from the frequency-domain method. The red curve and circular data points show the normalized phase shift cΔΦ/ωm L and n g obtained by the modulation phase shift method. Two methods give almost similar values of n g. The band edge for A appears at the longer wavelength (lower frequency) side of the transmission band. As the wavelength approaches the band edge, n g increases to 50. The band edge for B appears at the shorter wavelength (higher frequency) side. In this waveguide, FP resonance is particularly strong due to its higher facet reflectivity. Accordingly, the red curve oscillates widely between about 20 and 50. Nevertheless, the circular points are also in good agreement with the light blue frequency-domain curve and increase to 80. These results show that both A and B generate slow light near the band edge and they have opposite dispersion characteristics.
For the DC, the parameters were set to a = 0.48 μm and 2r 2 = 0.21 μm. For simple fabrication, chirping was introduced into 2r 1 by controlling the e-beam exposure time. In the chirping, a linear change of 2r 1 from 0.27 to 0.29 μm was obtained over a device length L of 120 μm with a local fluctuation of ±5 nm. In the device, the precise matching of two band-edges is crucial for high efficiency light transmission. We reduced the difference between the band-edge wavelengths of A and B to less than 5 nm by fine tuning the line defect width of A around W0.85 and the innermost airhole diameter. In the measurement, light was incident on A. When the wavelength was too short, the light was directly output from A even in the chirped structure. When the wavelength was in the target range, light reached the band-edge position and was output from B through the directional coupling. Figure 4 shows transmission spectra observed at the output ends of A and B. The light output was switched from A to B at λ = 1.525 μm and directional coupling occurred for a wavelength bandwidth of 32 nm at longer wavelengths. The maximum output intensity from B was 3 dB lower than that from A, and the spectrum for B exhibits large fringes of 5–15 dB. These might be caused by a small mismatch of the band edges and insufficient smoothness of the chirping, which lead to nonuniform reflection and localization of light at the directional coupling points. The fringes, which look to be arising from simple FP resonances, have amplitudes of 5–10 dB. Therefore, larger amplitudes of some spectral dips might be caused by the above imperfections. To investigate this effect, we calculated the transmission spectrum in the device using finite-difference time-domain (FDTD) method. Due to the limitation of computer resources, we did not directly model the device with a fluctuation in airhole diameter of a few nanometers, but assumed a small and random fluctuation in slab index, which corresponds to the airhole diameter fluctuation. In this calculation, we found that the spectral fringes can be almost eliminated if the airhole diameter fluctuation is reduced to ±2 nm (<1% of the airhole diameter). The solid curve in Fig. 5 shows the normalized phase shift cΔΦ/ωm L measured by the modulation phase shift method. Similarly to the case of Fig. 3, it oscillates according to the spectral fringes. However, the minimum line is larger than 25. This means that the average group index ñ g in the chirped structure is larger than this value over the transmission bandwidth. We evaluated ñ g for some FP-like fringes using the experimental data analysis explained in Section 3. As shown by the circular data points in Fig. 5, ñ g lies in the range of 30–40.
5. Comparison with calculation results
As discussed in Ref. , ñ g is constrained by the bandwidth even in chirped structures. The balance between ñ g and the bandwidth is changed by the slope and range of the chirping (change of structural parameters between the start and end points of the chirping against the chirped length); a smaller slope gives a larger ñ g and a narrower bandwidth. This relationship was confirmed by a summary of ñ g for sample devices shown in Fig. 6. The circles and error-bars indicate the average and fluctuation of ñ g over the bandwidth, respectively. The solid line shows a theoretical value obtained by band calculations. The experimental plots for Fig. 5 agree well with the theoretical value. Here, the theoretical value saturates to nearly 100 at the narrow bandwidth limit. This saturation is caused by distortion of coupled bands near the band edge, as shown in Fig. 1(c), which reduces the useful bandwidth. The band distortion can be eliminated by expanding the inter-waveguide spacing of A and B and reducing the coupling strength. Under these conditions, an ideal increase in ñ g, as shown by the dashed line in Fig. 6, is expected; i.e. ñ g = 550 for a normalized bandwidth Δω/ω = 2.06×10-4(40 GHz at λ = 1.55 μm).
It should be noted, however, that there is a limit to the increase in ñ g actually because the narrow bandwidth operation is disturbed by the fluctuation in airhole diameter. For example, the fluctuation of ±5 nm in our fabrication corresponds to half of the total chirping range of 2r 1 = 0.27 – 0.29 μm and the operating bandwidth is 32 nm (Δω/ω = 2.06 × 10-2). This means that half of this bandwidth, i.e. 16 nm is the limit to be realized. In fact, device samples with narrower chirping ranges were fabricated and the narrowing of the operating bandwidth was observed up to this value. However, the increase in ñ g was almost saturated, as shown by experimental plots in Fig. 6. The fluctuation can be reduced to ±0.5 nm by improving the e-beam lithography; ±1 nm was achieved by the present technology , and further improvement is expected in future. Then, ñ g = 200 for the bandwidth of 3 nm (Δω/ω = 2.06×10-3) will be available.
In this paper, we fabricated two kinds of PC waveguides with opposite dispersion characteristics, and demonstrated wideband, dispersion-compensated slow light in a chirped PC directional coupler consisting of these waveguides. In the device, the average group index ñ g was 30–40 in a wide wavelength bandwidth of 32 nm. The relation between ñ g and bandwidth agrees with the theoretical calculation. Structural optimizations can enhance ñ g to, at least, 200 by narrowing the operating bandwidth to 3 nm at λ = 1.55 μm. Completely flat transmission and group index spectra are crucial for more precise evaluation of the group index, narrowing the bandwidth, and practical use of this device. For this purpose, more precise matching of band edges, smoothing of the chirping, and the use of anti-reflection structure at the input and output facets are important future issues.
This work was partly supported by The CREST Project of JST, The IT program and The 21st Century COE program of MEXT, and The Grant-In-Aid and The Research Fellowship of JSPS.
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