High transmission of slow-light in a photonic crystal (PC) waveguide (WG) using a hetero group-velocity (Ht-Vg) PC-WG was proposed and experimentally investigated. The Ht-Vg WG, which comprises a low-group-velocity (L-Vg) PC-WG section between two identical high-group-velocity (H-Vg) PC-WGs, is designed to decrease the impedance mismatch of the L-Vg PC-WG. The increase in transmittance of a propagating pulse was confirmed in the Ht-Vg PC-WG even in the vicinity of the band-gap, whereas the homogeneous PC-WG showed a gradual decrease in transmittance with the pulse wavelength approaching the band-gap. The group index (ng) of the L-Vg region in the Ht-Vg PC-WG was measured by the cross-correlation method and attained a value above 20. On the other hand, the transmittance of the Ht-Vg structure recovered approximately 16dB compared to the homogeneous L-Vg WG having same ng, 17. This recovery is mainly dominated by the coupling improvement due to the Ht-Vg structure, around 12dB. These results indicate the effectiveness of the Ht-Vg structure to use slow light in a PC-WG, which leads to various applications in PC-based optical devices.
©2007 Optical Society of America
Slowing down the group velocity (Vg) of a light pulse in a photonic crystal (PC) waveguide (WG) can be realized by exploiting the unique dispersion of its mode in a two dimensional (2D) PC slab . Intensive efforts have been made to utilize pulsed light with low group velocity (L-Vg) in the PC-WG to enhance the optical nonlinearity (ONL) which finds various applications in PC-based optical devices.
So far, we have developed the PC-based all-optical switching devices utilizing the ONL of quantum dots (QDs) embedded in PC-WGs [2, 3]. These devices operate through the phase shift of the signal pulse induced by the ONL; refractive index changes due to the absorption saturation in the QDs by control pulse excitations. The QD’s efficient ONL provides various benefits for device performance such as reduced operation power. We investigated the utilization of slow light for ONL enhancement and predicted theoretically that the phase shift is inversely proportional to Vg .
However, in the slow light region, coupling into the PC-WGs becomes inefficient due to the large impedance mismatch at boundaries of the PC-WG resulting from the large difference of group index (ng). To solve this complex problem, several methods have been proposed: adiabatic changing of the ng [5, 6], adjusting the interface with insertion ports [7, 8], etc. However, from the practical point of view, these methods are not applicable to all optical integrated circuits of PC. For instance, the adiabatic changing requires a high precision fabrication technique and a longer WG. Concerning uses of the slow-light region in the integrated PC-WG circuits, e.g., our proposed all-optical switch: PC-SMZ [2, 3], simple and short change of PC at the slow light region is more useful for fabrications and provides much design flexibility.
In this paper, we thus propose the hetero group-velocity (Ht-Vg) WG  for the practical use of slow light in PC integrated circuits. We also confirmed the effectiveness of the Ht-Vg WG, both theoretically and experimentally, which is promising for its applications in PC based optical integrated circuits, especially for our proposed switch device, through an enhancement of the ONL.
2. Proposed and simulated Hetero-group velocity (Ht-Vg) waveguide
The proposed Ht-Vg WG, schematically shown in Fig. 1(a), has identical high-group-velocity (H-Vg) regions at both sides in addition to the conventional L-Vg WG. The added H-Vg regions play the role of a buffer, which matches the low-impedance interface to the input material such as air or semiconductor bulk WGs, and makes the recovery of the transmittance near the band-gap possible. Figure 1(b) explains the effectiveness of the Ht-Vg WG compared to the homogeneous (Homo)-Vg WG with a schematic of the spectrum of transmittance. In the case of the Homo-Vg WG, the transmittance (drawn in blue) decreases due to reflection at the boundaries where the group index increases for the propagating pulse. The spectrum of the Ht-Vg WG (drawn in red), however, is expected to show transmittance recovery in the slow light region, indicated by the red arrow.
Before the experimental study, we simulated the properties of the Ht-Vg WG through 2D calculations. The model of the Ht-Vg WG shown in the upper portion of Fig. 2 was designed for a line defect consisting of one missing row (W1) in a 2D hexagonal air-hole PC. The H-Vg and L-Vg regions for a pulse of around 1300nm were fabricated in the WG by modulating the air-filling factor, r/a: the ratio of radius (r) to lattice constant (a) of the air-holes. The important parameters of the L-Vg region in Ht-Vg WG are the radius of air-holes (r) and the lattice constant (a). By varying these parameters independently, we obtained the optimized parameters for the highest transmittance in the Ht-Vg WG. The highest transmittance and lowest reflectivity was obtained by modulating r while holding the a constant. This result suggests that the r-modulation method is suitable for increasing the transmittance. A detailed discussion is given after the calculation results.
The calculated photonic band structures of line-defect-mode in H-Vg (r/a=0.30) and L-Vg (r/a=0.32) obtained via the 2D plane-wave-expansion method are shown in the lower part of Fig. 2. From the slope of the dispersion curves it is observed that a light pulse of certain wavelength is expected to propagate with Vg of 0.17c in the H-Vg regions and 0.05c in the L-Vg region respectively, where c is the speed of light in a vacuum. On the other hand, the time dependence of the pulse propagation was simulated with the 2D finite-difference time-domain method. Figure 3 shows the calculation models and results of the Homo-L-Vg and Ht-Vg WGs. A pulsed optical source of 2ps duration and wavelength produces the same Vgs as in Fig. 2 was introduced from the left air-edge interface of each WG, indicated by the red cross marks.
The pulse intensity was recorded using the ten monitors shown in each figure by the green lines and then normalized to the input power of the optical pulse . Using the duration of the monitored intensity peaks, the Vg in the WGs can be estimated. It was found that the Vg in the L-Vg regions of Ht-Vg WG and Homo-Vg WG are approximately 0.07c and 0.16c. These values closely correspond to the Vg deduced from the slope of the PC band calculations in Fig. 2, and the Vg in the Ht-Vg WG is confirmed to be modulated as intended.
Regarding the intensity of the pulse, it should be noted that the transmittance is higher in the Ht-Vg WG than in Homo-Vg WG. A significant decrease in the intensity of the pulse from 1 to 0.66 is observed at the first monitor of Homo-Vg WG, compared to 0.78 in the Ht-Vg WG; this indicates a larger impedance mismatch at the boundaries of the Homo-Vg WG due to large Vg difference. The monotonic decrease of the pulse intensity observed in the L-Vg regions originates from the in-plane leaks of electromagnetic field out of the PC. In the slow light regime, the electromagnetic field of the pulse expands out of the PC arrays and the pulse intensity, detected with the monitors set within the PC arrays, is observed to decrease. On the other hand, the intensity reduction at the interface between H-Vg and L-Vg regions in Ht-Vg WG is not significant. This implies that the low impedance mismatch is realized despite the abrupt change in the air hole radius at the interface. Considering that the impedance mismatch can be due to the Vg and the mode profile differences at the boundary, this could be explained as effects of lower Vg and mode profile differences at the boundary due to the correspondence of the lattice constant between L-Vg and H-Vg regions . For a complete interpretation, further investigation would be required.
The sample was prepared as a GaAs-based 2D air-bridge-type 500-μm-long straight PC-WG. First, a 250-nm-thick GaAs core layer on a 2-μm-thick Al0.55Ga0.45As cladding/sacrificial layer was grown by molecular beam epitaxy on a GaAs (001) substrate. Then, the PC-WG was fabricated by using electron-beam lithography, Cl2-based reactive ion-beam etching used for perforation of air holes. Finally, HF-solution-based wet-etching was performed for removal of the sacrificial layer and making an air-bridge slab . The periodic structure was constructed using an array of triangular-lattice air-holes with a period a=360nm, and a single missing row of air-holes in the Γ-K direction formed the WG. The air-filling factor, r/a, was set to 0.30 for H-Vg regions and 0.32 for L-Vg region. Fig. 4 shows scanning electron microscopy images of the sample. We prepared 4 different samples of Ht-Vg structure, varying the L-Vg region length (L) from 36μm to 144μm in 36μm increments. In addition, 500μm long Homo H-Vg and L-Vg WGs were prepared for comparison.
3.2 Transmission and group index measurements
To characterize the fabricated sample, we measured the transmission of continuous wave (CW) and pulsed light and determined the Vg through direct observations of the transmitted pulsed light using the time-of-flight (TOF) interference method.
First, we measured the CW light transmission to observe the band-width and band-edge wavelength of the samples. For the transmission measurements, a halogen lamp was used as a white light source and a monochromator attached InGaAs CCD was employed as a detector. The detailed setup for the halogen-lamp-based transmission measurement is described elsewhere . Next, for the pulse shape and group velocity measurements, the TOF interference method [11, 12] was employed. The shape of the propagating pulse through the sample can be observed by cross-correlation interference signal. The signal is produced by the sample propagating (probe) pulse and the original (reference) pulse detected collinearly as a function of the relative time delay. The Vg was derived from the time delay of the propagation of the pulse in the sample compared to in the air. The experimental setup for this method is depicted in Fig. 5. Pulses of around 830nm wavelength and 2 ps duration were generated with a mode-locked Ti-Sapphire laser at repetition rate of 80 MHz. For tuning the wavelength of the pulses around 1300nm, an optical parametric oscillator (OPO) was employed. The pulses were divided into two beams using a beam-splitter, one as a sample pulse and the other as a reference pulse. The pulse propagating in the sample is directly coupled through the cleaved edge-plane of the sample into the WG using an objective lens of numerical aperture (NA) 0.5. The output pulse from the WG is then collected by an identical objective lens, after which it interferes with the reference pulse at the secondary beam-splitter, and is then detected by a PIN photodiode. In order to vary the relative delay between the sample propagating pulse and the reference pulse, a delay system is constructed with a retroreflector mounted on a step-motor driven stage inserted into the optical path of the sample propagating pulse. Infrared vidicons were utilized to monitor the sample and transmitted pulse in order to optimize the optical alignment.
4. Results and Discussion
4.1 CW transmission
Figure 6 shows the CW transmission spectra of four of the straight WGs designed to have different Vgs: Homo H-Vg (r/a=0.30), Homo L-Vg (r/a=0.32) and the Ht-Vg WGs with L=36μm and 144μm. The transmittance was normalized to the intensity of light without the sample. The transmittance spectra of the Homo L-Vg and H-Vg WGs are shown using blue and black lines respectively. These are similar but shifted towards shorter wavelengths for increasing values of r/a. This behavior is consistent with the band calculation results shown in Fig. 2. In each spectrum, the transmittance decreased gradually as the wavelength increased close to their band edges. This gradual decrease in transmittance due to low Vg has been generally confirmed in conventional PC-WGs. On the other hand, in the spectra of the Ht-Vg WGs shown by the red and green lines, high transmittance is maintained in the low Vg region of the Ht-Vg WG, as indicated by the red arrow. Also, no significant change in the transmittance of the Ht-Vg WG due to the change in L was observed. These results clearly demonstrate one of the benefits of the Ht-Vg WG: High transmission near the slow light region. Using the results of CW transmission measurements, the wavelength near the band-gap can be determined; and with this, the Vg and pulse shape propagating in the PC-WGs near the band-edge were measured.
4.2 Pulse shape, Vg, and pulse transmittance measurements
First, we measured the pulse propagation through the homogeneous WG for comparison with the Ht-Vg WG. Figure 7(a) shows the cross correlation interference signals as a function of pulse wavelength from 1300nm to 1325nm. The origin of the horizontal axis is defined as the propagation time of a pulse through air without a sample. As can be seen in the graphs, the delay time of the pulse increased as its wavelength approached the PC band-edge. From the delay time at the maximum position of the signal, indicated by arrows in the Fig. 7(a), Vg of the pulse of each wavelength was estimated and the deduced group indices (ng=c/Vg) are plotted using black circles in Fig. 7(b). The maximum position of the signal was determined by using Gaussian-fitting, as shown by red lines in the Fig. 7(a). For shorter wavelength, such as 1300 and 1310nm, the Gaussian fitted well; the coefficient of determination was above 0.99. However, as the wavelength moved closer to the band-edge, the coefficient value decreased. We therefore determined the Vg up to the wavelength of 1325nm, at which the coefficient value is 0.91.
The observed pulse broadened and weakened at increasing wavelengths particularly near the band-edge. The pattern of measured interference signals were utilized for definition of the transmittance. The transmittance of the interference signal, the square of its amplitude normalized to the pulse without the sample is also plotted in Fig. 7(b) with blue squares. This summarized result shown in Fig. 7(b) clearly exhibits the properties of the Homo-Vg WG; gradual decrease of transmittance with increase in group index as the wavelength approaches the band-edge. Thus the TOF interference method is confirmed to be effective for measuring the Vg and transmittance of the propagating pulse.
Next, we moved on the pulse measurements with the Ht-Vg WG. Although the length of L-Vg region (L) in the WG varies from 36μm to 144μm as mentioned earlier, results on the interference measurement of the WG for L=36μm are shown in Fig. 8 as an example. Similar to the case of the Homo-Vg WG, the delay time of the interference signal was increased for longer wavelengths. It should be noted that these delay times include the delay in the L-Vg and H-Vg regions, and hence Vg cannot be determined directly from this result. To deduce the Vg for each wavelength, we used delay times measured for various WGs with different L as shown in Fig. 9. The fitted lines of the plots for each wavelength indicate a linear variation of delay time with L. The extrapolated values at L=0 determine the delay time of homogeneous H-Vg WG of 500μm length for each wavelength. Thus, the delay time within the L-Vg region can be derived by subtracting the delay time in H-Vg regions from total delay time observed for each sample. The deduced Vgs are summarized in a table shown in Fig. 9(b). Let us discuss the accuracy of the Vg, derived from the Gaussian-fitting for each signal drawn in red lines in Fig. 8. According to the coefficients of determination for the Gaussian-fitting, the Vg at λ=1327nm is less reliable than the other Vgs; the coefficient is 0.85, as opposed to 0.99–0.97 for the others. This is likely to be due to the pulse reshaping at slow light regime.
We also measured the transmittance of the Ht-Vg WG from the intensities of the transmitted pulse. Figure 10 summarizes the transmittance and ng of the propagating pulse through the Ht-Vg WG, with L=36μm, and the Homo-L-Vg WG as a function of pulse wavelength. As is clearly seen in Fig. 10, the transmittance of Ht-Vg WG is higher than that of the Homo-L-Vg WG, although the values of ng are fairly close. For instance, the transmittance and ng of the Ht-Vg WG for a 1323-nm-wavelength pulse were -24dB and 17.3 respectively, whereas their corresponding values for the Homo-L-Vg WG for a 1325-nm-wavelength pulse were -40dB and 16.5. Thus nearly a 16dB recovery in transmittance is realized when ng is around 17. Regarding this recovery value, it should be noted that the value possibly includes a reduction of propagating loss due to the L-Vg length (L) difference. As known in general, the propagating loss increases in the slow light regime. Thus, the transmittance difference between the 500-μm-length Homo L-Vg WG and the Ht-Vg WG with 36-μm-L could include the propagation loss except the coupling improvement by the hetero structure. To distinguish the propagation loss value, we verified the L dependence of the transmittance in Ht-Vg WG by referring the transmittance difference between samples with minimum L, 36μm and maximum L, 144μm, as shown in Fig. 10. Assuming the propagation loss at high Vg regime should be negligible, the transmittance difference value obtained at λ=1310nm, 1.1dB, can be considered as an error value. Therefore, we can estimate the propagation loss to be 0.9dB at λ=1323nm, resulting in around 8dB/mm at ng = 17. Referring our previous estimation of the propagation loss in GaAs-core 2D PC slab WG at ng = 17, 4.5dB/mm , this estimated loss value can be reasonable. As a result, approximately 12dB recovery is expected as the coupling improvement by using the Ht-Vg WG.
Finally, we shall discuss group velocity dispersion (GVD). In case of using an ultra-short pulse, of sub-ps order, GVD problems will be inevitable for a PC-WG. However, in the case of our proposed PC-based switch, the required pulse of approximately 2ps time width and 2.5nm line width is not seriously affected by GVD so long as the ng is less than 20. In addition, recent papers [6, 14, 15] suggest that minimization of GVD is possible via slight modifications, such as variation of the size of air holes adjacent to the WG or WG width. Making these adjustments to the Ht-Vg WG would further increase its effective utilization of slow light.
We proposed and investigated a Hetero-group-velocity WG for utilizing slow light with high transmittance. From the experimental results, a 12dB coupling improvement against the homogeneous WG was confirmed while ng around 17 was realized. This effect shows great promise in improving the efficiency of our proposed ONL-induced, PC-based all-optical switch.
The authors would like to thank Prof. K. Inoue, Dr. Y. Tanaka, Dr. H. Kawashima, and Dr. K. Kanamoto for fruitful discussion and support in the pulse propagation measurement. This work was partly supported by the New Energy and Industrial Technology Development Organization (NEDO) projects and a Grant-in-Aid from the Ministry of Education, Culture, Sports, Science and Technology, Japan.
References and links
1. 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]
2. H. Nakamura, Y. Sugimoto, K. Kanamoto, N. Ikeda, Y. Tanaka, Y. Nakamura, S. Ohkouchi, Y. Watanabe, K. Inoue, H. Ishikawa, and K. Asakawa, “Ultra-Fast Photonic Crystal/Quantum Dot All-Optical Switch for Future Photonic Network,” Opt. Express 12, 6606–6614 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-26-6606 [CrossRef] [PubMed]
3. K. Asakawa, Y. Sugimoto, Y. Watanabe, N. Ozaki, A. Mizutani, Y. Takata, Y. Kitagawa, H. Ishikawa, N. Ikeda, K. Awazu, X. Wang, A. Watanabe, S. Nakamura, S. Ohkouchi, K. Inoue, M. Kristensen, O. Sigmund, P. I. Borel, and R. Baets, “Photonic crystal and quantum dot technologies for all-optical switch and logic device,” New J. Phys. 8, 208 (2006). [CrossRef]
4. Y. Watanabe, N. Yamamoto, K. Komori, H. Nakamura, Y. Sugimoto, Y. Tanaka, N. Ikeda, and K. Asakawa, “Simulation of group-velocity-dependent phase shift induced by refractive-index change in an air-bridge-type AlGaAs two-dimensional photonic crystal slab waveguide,” J. Opt. Soc. Am. B 21, 1833–1838 (2004). [CrossRef]
5. S. G. Johnson, P. Bienstman, M. A. Skorobogatiy, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, “Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals,” Phys. Rev. E 66, 066608 (2002). [CrossRef]
6. D. Mori and T. Baba, “Wideband and low dispersion slow light by chirped photonic crystal coupled waveguide,” Opt. Express 13, 9398–9408 (2005),http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-23-9398 [CrossRef] [PubMed]
7. E. Miyai and S. Noda, “Structural dependence of coupling between a two-dimensional photonic crystal waveguide and a wire waveguide,” J. Opt. Soc. Am. B 21, 67–72 (2004). [CrossRef]
9. H. Nakamura, K. Kanamoto, Y. Sugimoto, N. Ikeda, Y. Tanaka, Y. Nakamura, S. Ohkouchi, and K. Asakawa, “High-efficiency coupling to photonic crystal waveguide with low group velocity by hetero photonic crystal technique,” presented at the Fifth Int. Symp. Photonic and Electromagnetic Crystal Structures (PECS-V), Kyoto, Japan, 7-11 March 2004.
10. Y. Sugimoto, N. Ikeda, N. Carlsson, K. Asakawa, N. Kawai, and K. Inoue, “Fabrication and characterization of different types of two-dimensional AlGaAs photonic crystal slab,” J. Appl. Phys. 91, 922–929 (2002). [CrossRef]
11. K. Inoue, in Photonic Crystals, edited by K. Inoue and K. Ohtaka (Springer-Verlag, Berlin Heidelberg, 2004).
13. Y. Tanaka, Y. Sugimoto, N. Ikeda, H. Nakamura, K. Asakawa, K. Inoue, and S. G. Johnson, “Group velocity dependence of propagation losses in single-line-defect photonic crystal waveguides on GaAs membranes,” Electron. Lett. 40, 174–176 (2004). [CrossRef]
14. A. Yu. Petrov and M. Eich, “Zero dispersion at small group velocities in photonic crystal waveguides,” Appl. Phys. Lett. 85, 4866–4868 (2004). [CrossRef]
15. Lars H. Frandsen, Andrei V. Lavrinenko, Jacob Fage-Pedersen, and Peter I. Borel, “Photonic crystal waveguides with semi-slow light and tailored dispersion properties,” Opt. Express 14, 9444–9450 (2006). http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-20-9444 [CrossRef] [PubMed]