We explore slow light behavior of a specially designed optical waveguide by carrying out structural dispersion using numerical techniques. The structure proposed is composed of square-lattice photonic crystal waveguide integrated with side-coupled cavities. We report three orders of magnitude reduction in group velocity at around with strongly suppressed group velocity dispersion. The analysis is performed by using both plane-wave expansion and finite-difference time-domain methods. For the first time, we succeeded to show such a low group velocity in photonic structures. Slow light pulse propagation accompanied by light tunneling between each cavity is observed. These achievements show the feasibility of photonic devices to generate extremely large group index which in turn will eventually pave the way to new frontiers in nonlinear optics, optical buffers and low threshold lasers.
©2010 Optical Society of America
Slow light phenomenon has been pursued by various studies so far. While some groups study atomic resonances which rely on material dispersion of exotic media, some other researchers use micro-ring resonators and optical fibers [1–5]. Meanwhile, a large number of researchers show interest in the study of slow wave phenomena by employing periodic dielectric structures known as photonic crystals (PC) [6–16]. PCs sustain very special properties such as forbidden frequency interval (photonic band gap). Using this property, one can construct cavities, waveguides and coupled-resonator optical waveguide structures by introducing structural deformations either in terms of point wise or line like forms. For slow light implementations, photonic crystal waveguides (PCW) have already proven to possess exceptional importance. They can be obtained by inducing a line defect in pure PC structures, which in turn result in a waveguide mode within the PC’s forbidden band gap. Therefore, a line defect band is obtained in the band gap region, whose field distributions along transverse to propagation direction are strongly confined in the line defect. Dispersion diagrams of these PCW appear to have zero group velocities near the band edges which are high symmetry points. However, modes at these locations are very dispersive due to nonlinear relation between the radial frequency and wave vector. As a result, light pulses that carry data bits get heavily distorted while propagating along the transmission channel. This in turn necessitates tackling with group velocity dispersion (GVD) or finding a way of compensating it [6–16]. If there is a flat-band in the dispersion diagram then light pulse is expected to propagate without suffering from distortion.
The reported results in this study provide both ultra-low group velocity and very small GVD. We obtained a flat band which has a group velocity of. The time domain study yields the demonstration of a group velocity of the same mode at around with very small pulse distortion. To be best of authors’ knowledge, this is the first time that such a low group velocity is reported in PC structures. Later in the paper, we will provide propagation of a slow wave without any distortion. It is highly possible that the presented work may encourage researchers to pursue further studies of slow light targeting selected applications in areas such as nonlinear optics, optical buffers and low threshold lasers [17–22].
The rest of the paper consists of the following parts. We will show the designed structure and present frequency domain analysis using plane wave method (PWM) in the next section. The third section is devoted to time domain study. Finally, the last section concludes the paper with a discussion of the results and future prospects.
2. The designed slow light structure and frequency domain analysis
The proposed structure as shown in Fig. 1(a) is composed of a PCW and coupled cavities such that cavities are placed at the neighborhood of waveguide centerline with a periodicity of , where a is the lattice constant. Most of the slow light devices investigated so far have employed either waveguides or cavities [7–16]. There are even coupled waveguides structures . However, to the best of our knowledge, there are no studies proposing a structure which combines coupled cavities and waveguides in that fashion. We aimed to have good spectral properties of both types of defects (point and line) at the same time: the strong confinement and guiding capability of waveguides and the weak interactions of coupled cavities. By appropriately placing each defect along the transmission channel, the light speed will be reduced dramatically. The tuning of the slow light dispersion curve is achieved by altering the radii of the defects, R. No other tuning mechanisms such as position of defects were implemented in the present study.
The PC consists of dielectric rods with radii of and the modified rods that act as side-coupled cavities have larger radii. Additional corrugation of PCW occurs when cavities are placed at sideways. We proceed to scan the defect radius by coarse steps and found a band which is appropriate for slow light when R = . The location of the defects in the transverse direction is decided based on the fact that the slow light interaction is strong with the first rows of rods. As a result, the cavities are lined up along the first rows. The impact of the rows of rods gets stronger as they come close to the waveguide centerline. Separation between each defect is taken to be an intermediate value. A smaller distance will have strong interaction between each cavity which in turn degrades the slow light performance. An optimization procedure has to be taken to further enhance the performance of the declared values. The polarization is assumed to be TM (electric-field is along the rod axis) and high-refractive index contrast is obtained by taking.
The dispersion diagram of the structure is evaluated by freely available software which implements the plane wave expansion technique . The super-cell schematic is shown in Fig. 1(a) as a rectangle with dashed line. The result is depicted in Fig. 1(b). As it is shown in the dispersion diagram there are modes within the band gap region. The band gap of photonic crystal boundaries lie between and 0.416. Although all the bands possess very small slope, a close inspection of each of them reveals that the mode marked by circle has the smallest slope and very flat-band. It can be seen from Fig. 1(b) that it sustains resonant modes of different spectral widths (mode A: % 0.4, mode B: % 3, mode C: % 0.02). The frequency location of mode C is around 0.3258. The electric field plot of the corresponding mode is provided in Fig. 1(c) to show the spatial confinement of slow light.
The enlarged views of these bands are plotted in Figs. 2(a) –2(c). We selected the lowest mode and calculated its group index vs. frequency graph. Figure 2(d) shows a very large constant group index value at around 1200. The group index is defined as. The speed of light is represented by c and group velocity is defined as. The spectral region of interest is shaded in color in Fig. 2(c) whose inverse slope corresponds to approximately a constant value of 1200 as can be seen in Fig. 2(d). There are even larger group index region with a value of 4000 at the upper part of the band. However, due to the difficulty of addressing this portion with the finite-difference time-domain (FDTD) method, we focused on the lower shaded part. The larger group index part of the dispersion curve is even narrower in terms of bandwidth and it is multimode. Hence, pulse dispersion may occur. The difficulty arises while implementing FDTD because higher put stringent requirements in terms of computational resources. The extraction of time delay data necessitates performing simulations in a very long time. To focus exclusively on the shaded part which has low GVD, the center frequency and bandwidth of the modulated Gaussian pulse should be selected carefully. Because of the constant group index behavior of the flat-band, we expect that GVD will be considerably low. In the next section, we will explore further the slow light performance of the designed structure by extracting group delay information and showing slow light propagation.
3. Time-domain exploration of slow light structure
We constructed several FDTD [24,25] simulations for the structure shown in Fig. 1(a). We located the source inside the structure because input and output coupling issues are kept outside of the scope of the paper. We utilized perfectly matched layer (PML) as absorbing boundary condition .
To extract the group index value of the slow light structure, we stored arrival times of a modulated Gaussian pulse which is sent down the waveguide. The center frequency of the light pulse is adjusted atby investigating the transmission spectrum of pulse. The transmitted pulse is expected to have a strong peak at the propagating mode. The dispersion diagram informs us to use a center frequency value at 0.32586. There is a very small discrepancy between FDTD and PWM method. This difference is tolerable as we consider the limited discretization of FDTD and certain number of plane waves used in the super-cell calculation of PWM. The source has a bandwidth of . The sinusoidally modulated Gaussian pulse is sent to excite the resonance mode of the slow light structure. The temporal dependency of the source in terms of normalized power can be seen in Fig. 3(b) . The center frequency of the light pulse can be adjusted with an accuracy of 0.5nm if a is taken to be 500nm for operating slow light device in 1550nm region. If a broad pulse is sent to slow light structure, only the resonance modes can propagate and the other modes will be filtered out due to frequency selectivity of the structure. The compromise of sending broad pulse, however as a source is suffering from pulse distortion because band slopes are dispersive at band edges. The observation points are located inside the waveguide and separated at equal distance of 3a. Figure 3(a) presents the delay information of each point whose interpolated line slope gives us a group index value of 1360. This value is slightly larger than the PWM result. This discrepancy can be explained when we refer back to Fig. 2(d). As we can see, there are two different group index values at a very short frequency interval. Even though we targeted to excite the mode only at the shaded part of the band, due to slightly high bandwidth of the input pulse, it may be possible that light couples to the larger group index region. As a result, longer delay occurs for the light pulse. We also picked up the first and last detection points and presented the e-field forms at these locations in Fig. 3(b). Existence of very small distortion in pulse form is apparent in the intensity profiles of the light pulses.
Finally, we plan to show the features of slow light propagation in the designed structure. A comparison of light pulses traveling between a standard PCW and our structure will help us to distinguish the physical mechanism which is responsible for reduction of light speed. Considering these aspect, we prepared Fig. 4 . The e-field profile of the pulse propagation in a regular PCW is indicated in Fig. 4(a) while the slow light counterpart is shown in Fig. 4(b). Periodic oscillations of the field profile are apparent for the fast light. A similar oscillation behavior is also present for the slow light with one major difference. The side located cavities dramatically modifies light pulse propagation. They spatially confine wave inside each cavity and allow it to slowly hop from one cavity to another. The balance between PCW guiding capability and PC cavity confinement ability ensures slow wave propagation. Such a field localization and confinement properties will induce field enhancement due to the group velocity reduction. This is very crucial for nonlinear optics which needs very strong light-matter interactions. In the figure, we present time snap-shots of both reference and slow wave cases. The minimum and maximum ranges of the e-filed plots are indicated as blue and red colors. The weak interaction of light pulse with the periodic structure is the cause of fast light. The opposite direction where the speed of light is reduced dramatically is made possible by increasing light interaction with the periphery cells.
A figure of merit can be defined as normalized delay bandwidth product (DBP) i.e., x. The PWM result produces a normalized DBP value of around 0.25. The ultra-slow light propagation comes at the expense of bandwidth limitation. We recently investigated a trade-off relation between group index and bandwidth . The scalability of photonic crystal structures comes to rescue in this problem. One may envisage a parallel version of many of the proposed photonic structure which is tuned to operate at slightly different frequencies. This can be achieved by appropriately altering for example the lattice constant of the structure. We restrict the scope of the paper to the proof of a large group index achievement due to the first investigation of such a designed slow light structure. The robustness of the structure under the presence of losses will be studied deeply in another work. It is known that light propagation is very sensitive to structural disorder in the slow light regime. However, to estimate the performance of designed slow light structure under the small disorder that may occur during fabrication stage, we performed additional simulations and provided Table 1 . As it can seen, the group index value is on the same order for R<0.45a but decreases in the other region. By looking at this table, it can be argued that R = 0.44a value is more resistive to fabrication error. The findings of the study with R = 0.45a are valid for R = 0.44a and 0.43a values except minor variations in group index and bandwidth values.
The designed structure performs ultra slow light under the mentioned structural parameters in 2D analysis. One may question the performance of the designed structure in 3D. In this case, lower refractive index as an effective index should be used instead of higher refractive index so that 3D analysis can be replaced with 2D analysis. The group index value for R = 0.45a degrades when refractive index is taken to be 3.12. However, a parametric scan can be performed for searching the optimum R and r values which would provide small group velocity with low GVD.
The ultra-large group index value at around 1200 with very small GVD is achieved by incorporating coupled-cavities and photonic crystal waveguide. Consistency of the results between PWM and FDTD proves the fact that waveguides merged with coupled-cavities are structures that diminish the speed of light by more than three orders of magnitude. We show that structural modifications will have prominent role in the slow light research and the proposed type of structure will preserve its importance as an active research direction.
While reducing the speed of light, a number of important issues arise such as input and output coupling losses and disorder and material absorption induced losses. These issues have been recently targeted in a number of studies [27–33]. Coupling to the presented structure is important research topic and will be a subject of the subsequent study.
The authors gratefully acknowledge the financial support of the Scientific and Technological Research Council of Turkey (TUBITAK), Project no: 108T717. HK acknowledges support from the Turkish Academy of Sciences Distinguished Young Scientist Award (TUBA GEBIP).
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