Group-index independent coupling to band engineered SOI photonic crystal waveguide with large slow-down factor

Group-index independent coupling to a silicon-on-insulator (SOI) based band-engineered photonic crystal (PCW) waveguide is presented. A single hole size is used for designing both the PCW coupler and the band-engineered PCW to improve fabrication yield. Efficiency of several types of PCW couplers is numerically investigated. An on-chip integrated Fourier transform spectral interferometry device is used to experimentally determine the group-index while excluding the effect of the couplers. A low-loss, low-dispersion slow light transmission over 18nm bandwidth under the silica light line with a group index of 26.5 is demonstrated, that corresponds to the largest slow-down factor of 0.31 ever demonstrated for a PCW with oxide bottom cladding. ©2011 Optical Society of America OCIS codes: (230.7390) Waveguides, planar; (130.5296) Photonic crystal waveguides. References and links 1. M. Soljačić, S. G. Johnson, S. H. Fan, M. Ibanescu, E. Ippen, and J. D. 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Introduction
The slow light effect in photonic crystal waveguides (PCW) provides a strong light-matter interaction, which enhances absorption, non-linearity and gain per unit length [1,2], with several applications ranging from low-power and compact optical modulation [3] to gas detection [4].However, the narrow bandwidth of PCW slabs due to their highly dispersive group velocity in the slow light regime, restricts their applications [5].As an example, a PCW Mach-Zehnder modulator operating with RF bandwidth as high as 10Gb/s was recently reported, but the optical bandwidth was only 0.7nm [6].Therefore, in order to cover over 20nm optical bandwidth in a typical integrated dense wavelength-division multiplexing (DWDM) system, several different designs will be necessary.In order to avoid having different PCW modulators for operation at each optical wavelength, a PCW design that can achieve slow light operation over a large bandwidth is required.
Band-engineered PCWs have been shown to provide low-dispersion slow-light suitable for applications such as optical buffers and delay lines [7].Different ways of achieving low group velocity dispersion in single PCWS have been presented including adjustments of a) the waveguide width [7][8][9], b) the air hole size of the first two innermost rows [10,11], c) the displacement of the first two innermost rows, and d) the displacement of the third innermost row [12,13].Most of the methods mentioned above have been tested for air-bridged structures [12][13][14][15].Slow down factor (SF), which is the normalized bandwidth -group index product, as high as 0.31 has been reported for air-bridged silicon-nanomembrane based PCWs [12].Realizing constant group velocity propagation over a wide bandwidth and with guided mode well-below the silica light line in such a SOI PCW is more challenging than in air-bridged PCWs.
Although air top and bottom claddings increase the PCW guided mode bandwidth, the SOI PCWs provides some engineering advantages compared to the air-bridged PCWs. 1) mature silicon CMOS fabrication technology tools can be used that enable PCW fabrication with sufficient accuracy [16], 2) the SOI structure facilitates fabrication of PCW devices that consist of separate silicon sections (please see Fig. 1 of [17]), 3) in general, eliminating any fabrication step (removal of the buried oxide layer) will improve the fabrication yield.
By perturbing holes adjacent to the waveguide core, the highest reported SF for a SOI PCW with bottom oxide cladding is 0.24 [10] corresponding to a group index of 34.Band engineering techniques that do not require multiple hole-sizes, such as changing the positions of the first two rows [12] or lattice-shifting [13,18], provide the advantage of higher yield and reproducibility.Among these techniques, lattice-shifting makes it easier to target a desired group velocity over a bandwidth of interest since these two parameters can be tuned relatively independently [13].The largest SF experimentally presented by this technique is 0.19 [13].
In this paper, we present a lattice-shifting based design for low-dispersion slow-light SOI PCW.Integration of several slow-light PCW based devices requires the problem of large insertion losses due to strong impedance mismatch between strip waveguides and slow light propagation in PCW to be addressed.Thus, we also discuss different band engineering techniques from coupling efficiency point of view.A step-coupler [19] is designed to provide group-index-independent coupling into and out of the band engineered PCW.A single holesize is used for both coupler and PCW.For an accurate measurement of group index, a complete on-chip device based on Fourier transform spectral interferometry [20] is designed and fabricated.A large SF of 0.31 and group-index-independent coupling over an 18nm bandwidth of low-dispersion slow light propagation is confirmed experimentally.Efficient optical coupling and slow-light operation over a large optical bandwidth provides a means for realization of larger optical bandwidth and high speed compact PCW based modulators.An example of such devices will be reported elsewhere.

Design and simulations
A schematic of the band engineered PCW is shown in Fig. 1.The lattice constant (a) is 392nm.The thickness of the silicon layer and the buried oxide layers are 250nm and 3µm, respectively.Band engineering can be done by tuning the hole sizes [10] or the positions of the rows adjacent to the line defect.However, lattice shifting is more desirable as it has proved to be a fabrication friendly approach [21], while the accurate control of several different hole sizes is a challenging task.Two different approaches of lattice shifting, longitudinal (parallel to the defect line) and lateral (perpendicular to the defect line), are shown in Fig. 1.A slow-down factor as high as 0.3 has been demonstrated for an air-bridged PCW band-engineered by lateral lattice shifting [12].Thus far, the highest experimentally determined SF for a PCW band-engineered by longitudinal lattice shifting has been 0.19 [13].Theoretically however, SF as large as 0.35 is estimated by longitudinal lattice shifting of the two innermost rows [18].
We perform simulations using RSoft 3D BandSolve.Figures 2(a) and 2(b) show the effect of longitudinal and lateral lattice shifting, respectively, on the band structure.An important observation is that longitudinal lattice shifting does not change the position of the fundamental PCW defect mode in the fast light region.In the case of lateral lattice shifting, for a large SF value, a large positive value of t 1 is necessary [12] that effectively increases the defect line width (dW) and thus causes red shift in the defect mode band.Therefore, for our design, we choose the longitudinal lattice shifting of the first three rows on the two sides of the defect line.As will be discussed later, this choice also facilitates the design of the input/output coupler.For design optimization process, we utilize sequential quadratic programming (SQP), a nonlinear programming technique that exploits the gradients of the objective and constraint functions at each iteration to accelerate convergence [22].The independent variables are s 1 , s 2 , s 3 and r, where s i (i = 1, 2, 3) represents the amount of lateral shifting for the i th row of air holes, and r is the hole radius.Similar to the results presented in [18], we find that a large longitudinal shift of the innermost rows results in a minimum in the defect mode dispersion curve below the silica light line and at normalized k<0.5 [Fig.2(a)].Although the resulting SF can be large, using FDTD we find that coupling into the desired section of the defect mode band in such PCWs is inefficient (typically, maximum normalized transmission <15%).In contrast, in the case of band engineered PCWs with their defect mode minimum at the band-edge (k = 0.5), normalized transmission over the low-dispersion bandwidth can reach 50% without use of any couplers [Fig.2(d)].Therefore, another constraint in the optimization process is to keep the defect mode minimum at the band-edge.Figure 2(c) shows the band structure of the designed band-engineered low-dispersion slow light PCW (r = 0.27a, s 1 = 0, s 2 = −0.05aand s 3 = 0.25a) with n g = 26.7 ± 10% over a bandwidth of 18nm corresponding to SF of 0.31.Note that the second and third rows are shifted in opposite directions.
An important consideration for practical applications is efficient coupling into the slowlight region from an access ridge waveguide.Coupling efficiency can be improved by inserting a group index taper [23][24][25] or a fast light PCW region between the access ridge waveguide and the PCW [12,19,26,27].In principle, the performance (transmission efficiency) of the group index tapers scale badly with increasing group index [21].The existence of evanescent modes at the interface of a fast light PCW and a slow light PCW helps satisfy the boundary conditions [19,28] thus improve coupling efficiency.Numerical and experimental investigations reveal that a step coupler, which is a special implementation for the fast light PCW region, outperforms group index tapers realized by the defect mode  Step coupler, Engineered step, Expanded lattice, Reduced hole-size couplers refer to (a = 392nm, r = 0.27a, dW = 0.15a, s1 = s2 = s3 = 0), (a = 392nm, r = 0.27a, dW = 0.15a, s1 = 0, s2 = −0.05a,s3 = 0.25a), (a = 420nm, r = 0.27a, dW = 0, s1 = s2 = s3 = 0), and (a = 392nm, r = 0.23a, dW = 0.15a, s1 = s2 = s3 = 0), respectively.In the case of the group index taper, the hole sizes of a W1 PCW (a = 392nm, r = 0.27a, dW = 0.0a, s1 = s2 = s3 = 0) at the interface with the slow light PCW are parabolically reduced to r = 0.23a at the interface with the ridge waveguide.In all cases, the coupler (or taper) length is 8 periods.
width modulation [19].In comparison to the slow light PCW, a fast light PCW coupler defect mode is red-shifted so that the steep section of its defect mode overlaps with the desired bandwidth of the slow light PCW.This can be realized in different ways, such as reducing the hole sizes while retaining the original lattice constant [27], expanding the hole spacing while retaining the same hole size [21], and increasing the defect mode width (also known as the step coupler) while retaining all the other parameters the same [19].
The step coupler presented in [19] was designed for efficient coupling into a W1 PCW.An increase in the defect line width red-shifts the defect mode band, and consequently, its low-group index bandwidth overlaps with the bandwidth of interest.Three-dimensional FDTD simulations show that increasing the defect line width more than about 0.15a-0.20aresults in diminishing returns [19].However, a question to be answered is whether the fast light PCW designed to couple light into band engineered PCW should itself be band engineered.In other words, how must the values of s 1 , s 2 and s 3 be chosen?Figure 2(c) shows the band structures of a non-engineered PCW step coupler (r = 0.27a, dW = 0.15a, s 1 = s 2 = s 3 = 0) and an engineered PCW step coupler (r = 0.27a, dW = 0.15a, s 1 = 0, s 2 = −0.05aand s 3 = 0.25a).One notices that the non-engineered coupler has a more consistent efficiency inside the low-dispersion bandwidth.Another important observation is that in the case of a step coupler, the usable part of the band lies slightly above the silica light line.However, both numerical and experimental results show that the silica bottom cladding does cause negligible radiation loss for a small number of periods (less than 16) [19].The same phenomenon was also observed in the case of the group index tapers [25].We believe that as long as the holes are not filled with silica, the effective light line lies between that of air and silica, and specifically, the radiation loss can be ignored for small number of periods.
In order to numerically investigate the efficiencies of different couplers, we simulate transmission through a structure consisting of a ridge silicon waveguide, 8 periods of a fast light region, and 13 periods of the designed band-engineered PCW (dW = 0, s 1 = 0, s 2 = −0.05aand s 3 = 0.25a) using RSoft 3D FDTD.Note that step couplers with longer than 8 periods do not further improve the transmission efficiency for group indices lower than 30 [19].The length of the slow light section is chosen to be 13 periods so that the transmission spectrum becomes independent from the number of periods.
For the sake of comparison, we also simulate a structure only consisting of a ridge silicon waveguide and 13 periods of the designed band-engineered PCW with no coupler.Figure 2(d) compares the normalized transmission of several different couplers against the case in which no coupler is used.Couplers (or fast light PCWs) include Step coupler (r = 0.27a, dW = 0.15a, s 1 = s 2 = s 3 = 0), Engineered Step coupler (r = 0.27a, dW = 0.15a, s 1 = 0, s 2 = −0.05aand s 3 = 0.25a), Group Index Taper (dW = 0, s 1 = s 2 = s 3 = 0), where the hole radii are gradually decreased (quadratically) from the interface of the engineered PCW with the fast light PCW to the interface of fast light PCW with the ridge waveguide by 15nm [25], Expanded Lattice coupler in which the hole spacing in the fast light region is increased by 30nm (see Fig. 5 of [21]), and finally, Reduced Hole-size coupler (r = 0.24a, dW = 0.15a, s 1 = s 2 = s 3 = 0), in which the holes sizes are about 10nm smaller in radius compared to the engineered PCW [27]. Only Step coupler and Expanded Lattice coupler result in complete designs with a single hole size.As demonstrated in Fig. 2(d), the Step coupler outperforms all the other types in the bandwidth of interest.Figure 2(d) also indicates that the Engineered Step coupler provides the sharpest transition near the band edge, and thus it is a good candidate for devices such as band-edge cut-off based modulators [29].
Here, we notice an advantage of longitudinal lattice shifting compared to lateral lattice shifting in designing the coupler region.When linearizing the defect mode band by lateral lattice shifting, the defect mode of the fast light PCW is red-shifted compared to that of the slow-light PCW.Since longitudinal lattice shifting does not red-shift the defect mode, designing the coupler requires less widening of the defect line and also the wider photonic band gap is expected from the experimental results.We find that using the step coupler design, the best coupling into the designed engineered PCW is at least 4% more efficient over the low-dispersion bandwidth compared to coupling into a laterally shifted lattice design (dW = 0, r = 0.295a, t 1 = 0.15a, and t 2 = 0) with nearly similar SF = 0.28 and n g = 25.5.Our final device consists of 250 periods of the designed band engineered PCW (r = 0.27a, dW = 0, s 1 = 0, s 2 = −0.05aand s 3 = 0.25a) and 8 periods of the Step coupler (r = 0.27a, dW = 0.15a, s 1 = s 2 = s 3 = 0) on each side.
In the case of SOI PCWs, the useful optical bandwidths of the step coupler, engineered step coupler, group index taper and the reduced hole-size coupler all lay near but above the Silica light line.However, based on the FDTD simulation results as long and the couplers are short [here all couplers are 8 periods (3.14µm) long], the propagation loss inside the couplers due to optical coupling to substrate radiation modes is negligible.Therefore, for the same designs we expect that the same conclusions hold true if the PCWs are air-bridged.
For an accurate measurement of group index, an integrated on chip device, which works based on Fourier transform spectral interferometry [20] is also designed and fabricated.
Compared to the other group-index measurement techniques, such as detecting Fabry-Perot interference inside the sample itself [30] or time of flight measurements [31], this method provides group index as a function of wavelength over the transmission band in a single-shot measurement and without uncertainty in choosing the fringes' maxima [20].The scheme presented herein integrates delay lines, beam splitters and beam combiners on the same chip as the PCW device and also eliminates post-sample polarization compared to the Mach-Zehnder interferometric setup presented in [20].

Fabrication and test setup
The chip is fabricated on commercially available SOI wafers from SOITEC, which have a 250nm thick silicon device layer, 3μm thick buried oxide (BOX) layer and a 500μm thick silicon handle substrate layer.After electron beam lithography (JEOL JBX-6000) and developing, the resist pattern is transferred to the silicon layer using an HBr/Cl 2 RIE etch.A microscope image of the fabricated chip is shown in Fig. 3.A scanning electron microscope (SEM) picture of the fabricated PCW device is shown in Fig. 4. The red dashed line shown in Fig. 4(a) demarcates the high n g PCW from the specially engineered PCW step coupler.The purpose of the step coupler, as discussed in Section 2, is to provide efficient light coupling from a silicon strip waveguide into the high n g PCW device.A zoomed-in SEM image of the PCW air holes is shown in Fig. 4(b).
In order to test the device, light from a broadband amplified spontaneous emission (ASE) source, covering 1520-1620 nm, was Transverse Electric (TE)-polarized with an extinction ratio of over 30dB and butt coupled through a polarization maintaining (PM) fiber into each of the two devices shown in Fig. 3.In the top device, the input power is then uniformly divided into two arms of a Mach-Zehnder structure using a 1x2 multimode interference coupler (MMI) shown in Fig. 4(c).One arm (Signal) in the MZ is down tapered and connected to the band-engineered PCW with input/output step couplers.The other arm (Reference) in the MZ is down tapered and connected to 16 periods of the step coupler (combining the input 8 period and output 8 period couplers with no band-engineered PCW).Following the step coupler in the reference arm is a long delay waveguide with length > 5mm, which is needed for phase calculations.The reason for including the PCW couplers is to enhance transmission though the device and to suppress Fabry-Perot oscillations between highly reflective interfaces.Both the Signal and Reference arms are connected to 1x2 MMIs to provide 2 samples of each signal (S and R).One output from each of these MMIs is combined to provide the Interference signal (I).The bottom device is similar to the top one with this difference that the engineered PCW is replaced with a ridge silicon waveguide.The bottom device is required for normalize the test results and explained in Section 4.

Analysis and discussions of test results
The mathematical approach to map the group index of the dispersion profile is given in [20].For the sake of completeness, we summarize the technique as follows.The interference (I) spectral intensity from any of the two arms is given as where S(ω) and R(ω) are the spectral intensity outputs of the Signal (S) and Reference (R) arm, respectively, and τ is the time delay from the delay line.φ(ω) is the phase difference between S and I, and therefore, contains the group index data.In order to calculate the group index, the complex interference term is extracted from I by numerically filtering the signal peak with a peck at τ in the time domain.The result is then Fourier transformed back into frequency domain from which the phase data, ( ) ) is obtained.Finally, the group index from the PCW is calculated from the difference between the calculated group delays from the two devices, given as where L is the engineered PCW without the couplers and n RW (from RSoft FEMSim simulations) is effective index of the ridge waveguide replacing the engineered PCW in the bottom device.In order to increase the accuracy of the measurement we also include the input/output PCW couplers in Mach-Zehnder setup.However, their effect on the group index measurement is cancelled out because the PCW couplers are included in both arms of both devices.
The measured I, S (the same as PCW transmission), and R signals for our device are shown in Fig. 5(a), where the measured spectrums are vertically shifted for better visibility.Note that our choice for the length of reference arm is made so that the slowing of light in the PCW arm decreases the difference in group delay between the two arms, thus leading to an increase in fringe spacing.We note that backscattering loss (induced by side wall roughness), which is the dominant factor for n g >10, scales with n g 2 in PCWs [32].The relatively constant measured transmission not only over the low-dispersion slow light bandwidth (1539nm-1556nm), but also at shorter wavelength (λ<1539nm) that corresponds to smaller group indices, is due to small length of the PCW compared to the ridge waveguide.The TE propagation loss in a fabricated 500nm x 230nm waveguide is estimated using a cutback technique and is determined to be 7.3dB/cm at λ = 1550.Similar observation was made before in [12], where for a band-engineered PCW designed for n g ~30 over the low-dispersion bandwidth (similar to our PCW), group-index independent transmission was observed over the low-dispersion band width and the fast light region (n g <<30).However, for the band-engineered PCWs designed for n g >35 over the lowdispersion bandwidth, a clear group-index dependent transmission was observed.
In order to estimate the device insertion loss we measure transmission through two waveguides D and N. Waveguide D includes 1) 1mm long width taper [2.5µm (matched to the lensed fiber mode size) to 0.5µm wide], about 4mm long single mode waveguide (0.5µm wide), 0.5mm long width taper [0.5µm to 0.738µm (matched to the PCW step coupler width)], 2.17µm long step PCW input step coupler, 100µm band engineered PCW, 2.17µm long step PCW output coupler, 0.5mm long width taper (0.738µm to 0.5µm), 4mm long straight single mode waveguide (0.5µm wide), and 1mm long (0.5µm to 2.5µm wide].Waveguide N is the same as waveguide D, but the PCW step couplers and the band engineered PCW waveguides are entirely replaced by a 0.738µm wide straight ridge waveguide.All the waveguides are 230nm high.We normalized the transmission through waveguide D with regard to waveguide N to cancel out 1) input/output fiber-ridge waveguide coupling loss, 2) propagation loss in the silicon ridge waveguide, 3) taper losses.Our results indicate that the insertion loss over the low dispersion bandwidth (1539nm-1557nm) is 3.8 ± 1.1dB, which only includes input/output ridge waveguide-PCW coupling loss and propagation loss inside the PCW.

Conclusion
In conclusion, we presented the design of a low-loss, low dispersion and high group index photonic crystal waveguide with large slow down factor of 0.31.In order to improve coupling to the PCW, a step coupler with just one hole-size was numerically investigated and found to be more efficient over other different types of couplers.For enabling accurate on-chip group index measurement of the PCW, a scaled-down Fourier transform spectral interferometer on

Fig. 1 .
Fig. 1.Schematic of band engineered PCW and PCW taper.Longitudinal (parallel to the defect line) and lateral (perpendicular to the defect line) lattice shifting are depicted by red (solid) and yellow (dashed) arrows, respectively.The directions of the arrows indicate the positive direction assumed in this paper.

Fig. 3 .
Fig. 3.A microscope image of the fabricated on-chip group index measurement device.The top device contains the PCW and PCW couplers in one arm of the Mach-Zehnder structure and a delay line and PCW couplers in the other arm.In the bottom device, the PCW is replaced with a ridge waveguide.

Fig. 4 .
Fig. 4. (a) SEM pictures of the fabricated PCW device showing the PCW coupler and high ng region, (b) close up of the PCW air holes, (c) 1x2 MMI device employed for splitting and merging the signal coming from two arms of MZI device.
ω ωτ − , is extracted.Based on the phase data, the group delay of the top device containing the device under test (PCW), Applying the same technique on the signals measured from the bottom device, the group delay ( RW T ∆

Figure 5 (
b) shows the calculated group index as a function of wavelength based on the experimental data.Our results indicate a low-dispersion (with less than ±10% fluctuations in group index) transmission over 18 nm bandwidth with an average group index of 26.5.

Fig. 5 .
Fig. 5. (a) Output spectrum of PCW device (red curve), ridge waveguide (blue curve) and their interference (black curve).(b) Output spectrum of PCW device (red curve) and the calculated group index based on FT method (blue curve).The PCW transmission curve is also provided as a reference.