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Two complementary types of SOI photonic wire based devices, the add/drop (A/D) filter using a racetrack resonator and the Mach-Zehnder interferometer with one arm consisting of an identical resonator in all-pass filter (APF) configuration, were fabricated and characterized in order to extract the optical properties of the resonators and predict the performance of the optical delay lines based on such resonators. We found that instead of well-known waveguide bending and propagation losses, mode conversion loss in the coupling region of such resonators dominates when the air gap between the racetrack resonator and access waveguide is smaller than 120nm. We also show that this additional loss significantly degrades the performance of the optical delay line containing cascaded resonators in APF configuration.
©2006 Optical Society of America
1. Introduction
Optical delay lines have a variety of applications in optical time division multiplexing (OTDM) communication systems, phased-array antennas, and optical buffers for optical interconnects, etc. For example, optical delay lines are used in OTDM systems for synchronization [1]. In a phased-array antenna, optical delay lines can provide true-time delays for beam steering purpose [2]. Madsen et al. recently demonstrated tunable optical delay lines with a tuning range of 0 to 2.56ns using all pass filter (APF) resonators in conjunction with spiral lines based on doped silica waveguides [3]. Owing to a relatively small index contrast of only 2%, the device size was around 1200mm2. Little Optics, Inc. also provides programmable optical delay lines with a tuning range of 0 to 2.047ns using spiral waveguides and crossbar switches in silicon oxynitride [4]. Due to a relatively large index contrast (up to 20%) achievable in silicon oxynitride material system [5], the device size is shrunk below 100mm2.
Nevertheless, optical delay line dimensions can be further scaled down aggressively if it is fabricated on SOI substrate due to the extremely large index contrast between Si and SiO2 (air). As it will be shown later in this paper, an optical delay line with a delay time up to 1ns can be realized on SOI substrate using cascaded racetrack resonators in APF configuration with an area less than 0.007mm2, or 4 to 5 orders of magnitudes smaller than the previously demonstrated delay lines. An additional advantage for the realization of general photonic devices on SOI substrate is the compatibility of the photonic device fabrication process to that of the CMOS circuitry. For example, electronic circuits can be integrated with optical delay lines monolithically on SOI substrate to realize programmable delay tuning functions. Currently, photonic devices on SOI substrates are being extensively investigated worldwide [6–8].
Generally in order to completely model the performance of the optical resonator coupled to a bus waveguide using a matrix approach [9], three important parameters have to be known: the self- and cross- coupling coefficients r and t which characterize the interaction between the access waveguide and the resonator, and the roundtrip loss in the resonator propagation region, a. Most of the previous work on optical resonators [8,10–11] in APF and A/D configurations was aimed at achieving very high quality factors (Q>50000) to explore enhanced nonlinear phenomena or to use them in cavity-QED experiments. In these high Q resonators, propagation and bending losses dominated light attenuation and therefore coupling losses in access waveguide-resonator interaction region were typically neglected. Therefore it is typically assumed that r 2+t 2=1. This assumption significantly simplifies the analysis of experimental data and allows us to extract all three major parameters, r, t, and loss per roundtrip a directly from the transmission spectrum [10].
However, for design of the optical delay line based on optical resonators, a high Q factor is undesirable since it can limit significantly the operational bandwidth of the delay line [12–13]. In order to enhance the interaction between the access waveguide and the resonator, and hence to achieve the desired moderate or low Q factor (<10,000 in this paper), the spacing between the access waveguide and the resonator is reduced. Here we found that, in resonators with an air gap width between the access waveguide and the resonator of 120 nm or smaller, mode conversion loss in interaction region is the dominant loss factor instead of the well-known waveguide bending and propagation losses. Therefore the usual assumption r 2+t 2=1 is no longer valid. This additional loss term can degrade the performance of an optical delay line based on cascaded multiple racetrack APFs.
Moreover, since this basic assumption is incorrect, direct extraction of three major parameters of the resonator from the experimental data is no longer possible. In order to accurately obtain optical properties of the resonators, two complementary types of structures were fabricated and measured-resonators in add/drop (A/D) configuration and Mach-Zehnder interferometers with the same resonator in the APF configuration in one of the arms. Combination of the data from two sets of structures allows us to extract all important parameters (r, t and a) without assuming that r 2+t 2=1 and without prior knowledge about propagation and bending losses (characterized by a) in the resonator.
Analogous phenomena were previously discussed by Spillane et al. [14] in terms of the ideality factor and loss mechanisms between a tapered fiber and a microsphere. In this particular waveguide/resonator system, the dominant loss in the interaction region was due to the coupling of the fundamental resonator mode to the high order and radiation waveguide modes. However, in our single-mode access waveguide the loss is due to the mode conversion process in the interaction region. Such kind of loss mechanism is a sole characteristic of extremely high refractive index contrast waveguides, such as the SOI strip waveguides with submicron cross-section which are the subject of this study.
2. Add/drop filters based on racetrack resonators
2.1. Method
A schematic view of a racetrack resonator based add/drop filter is shown in Fig. 1. The interactions in the coupling regions can be described using the following matrices similar to those in Ref [9]:
where E8, E6, E4, E2, and E7, E5, E3, E1 are the output and input electric fields in the coupling regions, respectively, Lc is the total optical path length of the coupling region, r and t are the self- and cross- coupling coefficients which describe the interaction intensity, and k is the propagation constant. Compared to the matrix used in [9], there is an additional term in Eqs. (1) and (2), ejkLc, which is used to describe the phase delay due to the finite length of the coupling region. In this resonator the straight section of the racetrack is Ls, the radius of the semi-circle of the racetrack is R (measured from the center of the semi-circle to the center of the waveguide), the width of all the waveguides is w, and the gap between the access waveguide and the straight section of the racetrack is d.
We assume that when the spacing between the access waveguide and the racetrack is larger than dC (measured from the edges of both waveguides), the interaction between the access waveguide and resonator is negligible. Hence, given d, dC, LS, w, and R, the total optical path length of the coupling region, LC (see Fig. 1), can be determined using the following formula:
We also have:
where LP is the optical path length light travels inside the racetrack except for the coupling regions, and a is the optical field loss factor resulting from the light propagation over LP and which includes both waveguide bending and propagation losses.
If only a counter-clockwise propagating wave in the racetrack is considered and there is no input signal in the add port, we have
Combining Eqs. (1), (2) and (4)–(6), we have:
The accumulated phase in the racetrack, k(2LC+LP), can be approximated using the following equation:
where ng is the group index at the resonance wavelength, λ0, and k0(2LC+LP) equals 2mπ (m is an integer). From Eq. (8), we have:
Hence,
From Eq. (7), we have:
Combining Eqs. (10) and (12), we have:
By measuring the frequency response contrast at the drop port, r2a can be inferred from Eq. (11). On the other hand, (r2+t2)a can be obtained using Eq. (13) by measuring the frequency response contrast of both through and drop ports. If the coupling process between the access waveguide and the resonator is lossless (r2+t2=1) and the propagation and bending losses in the resonator are negligible (a=1), the combined frequency response contrast at drop and through ports will be infinitely large. Losses in the coupling region between the access waveguide and the resonator (r2+t2<1) and inside the resonator (a<1) will both reduce the combined frequency response contrast at drop and through ports, leading to a finite combined response contrast.
2.2. Experiment and modeling
Four groups of A/D filters with different air gap widths (all other parameters being identical) were fabricated on SOI Unibond 200mm wafers manufactured by SOITEC with 220nm lightly p-doped Si on a 1μm thick buried oxide (BOX) layer. Details of the fabrication process were described elsewhere [6, 15]. In this run instead of the previously used positive e-beam resist [6, 15], a negative e-beam resist was used to define all the sub-micron features. In order to achieve efficient and reproducible coupling from a single mode fiber to the Si single mode waveguide, spot-size converters based on inverted taper geometry were fabricated on both sides of the A/D filters as reported in [15].
Figure 2 shows a scanning electron micrograph of an A/D filter. The bend radius of the racetrack semi-circle, R, is 6.5μm and the straight section of the racetrack, LS, is 3.5μm. The single mode Si waveguide is 530nm wide by 220nm thick. In such highly confined SOI photonic wires designing polarization insensitive circuits is very difficult. We hence designed the cross-section of our waveguides in such a way that the TM mode has a cutoff at wavelength much shorter than that of TE, thus leaving the large bandwidth of 1500–1800nm free from the TM mode. Using the cutback method reported in [6], we measured a propagation loss of (9±1)dB/cm in the straight Si waveguide at a wavelength from 1540 to 1560nm, which is somewhat larger than what reported in [6]. The air gap widths in different groups of A/D filters were measured using scanning electron microscope (SEM). The measured values are (117±5) nm, (103±5) nm, (87±5) nm, and 0nm, respectively, for different groups of A/D filters.
The transverse electric field (TE) frequency response at the drop and through ports was measured using a tunable laser source with a resolution of 20pm. Polarized light was coupled into the polymer mode transformer using a polarization maintaining lensed fiber [6, 15]. Figures 3 and 4 show the typical measured/simulated (black/red) frequency response at the through and drop ports around 1550nm for A/D filters with an air gap, d, of (103±5) nm and 0nm, respectively.
Fig. 3. The transmission spectrum of a racetrack based A/D filter with an air gap width of (103±5) nm
To determine the group index of the strip waveguide and hence simulate the A/D filter spectrum response using Eqs. (7)–(9), we measured the TE frequency response of the drop port of an A/D filter with an air gap, d, of (103±5) nm from 1500nm to 1580nm using a broadband light emitting diode (LED) source as shown in Fig. 5.
Fig. 5. A broad band transmission spectrum of the drop port of an A/D filter with an air gap width of (103±5) nm
From the peak dropping wavelength λ1 to λ2, the accumulated phase in the racetrack, k(2LC+LP), changes by 2π. We then have:
From Eq. (14), we can deduce the group index as follows:
The measured group index is (4.25±0.02) from 1540 to 1560nm. More detailed discussions regarding the measurement of group indices will be presented in [16]. Effectively the measured group index is the average group index of the whole racetrack. Since the straight section of the race-track, LS, is much shorter than the total length of the semi-circle of the racetrack, the measured group index can be viewed as the group index of the curved waveguide. If we assume that the uncertainty in the total spectrum contrast measurement at the add/drop ports is ±1dB, then the measured (r2+t2)a is (0.981±0.002), (0.973±0.003), (0.963±0.004), and (0.92±0.01) for air gap widths of (117±5) nm, (103±5) nm, (87±5) nm, and 0nm, respectively. Using the measured ng, r2a, and (r2+t2)a, the frequency response of the A/D filters were simulated and also plotted in Figs. 3 and 4. In the simulations, the accumulated phase in the racetrack, k(2LC+LP), was approximated using Eq. (9), and the resonance wavelength, λ0, was obtained from the measurements. As shown in Figs. 3 and 4, the measured and the simulated frequency responses match very well, indicating accuracy in our measurements of r2a and (r2+t2)a.
2.3. Discussion
As the air gap width decreases, the interaction between the access waveguide and the resonator increases, leading to the reduction in the self-coupling coefficient (r). This effect is shown in Fig. 6. At the same time, the measured (r2+t2)a value also decreases monotonically as the air gap width decreases, as shown in Fig. 7. This indicates that the total loss (including the losses in the coupling and propagation regions) increases as the air gap width decreases.
In general the attenuation factor, a, and hence the propagation and the bending losses in the resonator should be independent of the air gap width. The following analysis confirms that this statement is still valid in our case. Previously, we reported that the bending loss per turn of a waveguide bend with a radius of 5μm is <0.005dB [6]. Since similar strip waveguides with a bend radius of 6.5μm are used in the current work, the bending loss is even smaller and hence negligible. Furthermore, if we follow the usual assumption that r2+t2=1 (in the lossless coupling region), the calculated propagation loss will be (35±4) dB/cm, (50±5) dB/cm, (68±7) dB/cm, and (151±20) dB/cm for racetrack resonators with air gap widths of (117±5) nm, (103±5) nm, (87±5) nm, and 0nm respectively. These numbers not only significantly exceed experimentally measured propagation losses of (9±1) dB/cm, but also exhibit an unphysical dependence on the gap width. This allows us to conclude that the assumption of lossless coupling, r2+t2=1 is no longer valid in our case. Therefore we assert that the increased total loss is due to the increased loss in the coupling region (r2+t2<1).
Unfortunately, in the A/D filter configuration, losses in the coupling region and in the propagation region contribute equally to the shape of the frequency response as shown by Eqs. (7) and (8), and we are not able to disentangle a from r and t. Although it is possible to estimate the a value by measuring linear propagation and bending losses [6], we introduce another structure,the Mach-Zenhder interferometer, to extract all three parameters (r, t and a) without prior knowledge about a.
3. Mach-Zenhder interferometer (MZI) containing a racetrack resonator
3.1. Method to extract loss factor a
To determine the attenuation factor, a, we also fabricated four groups of Mach-Zenhder interferometers with one arm containing a resonator in the APF configuration. Such MZI structure was previously reported in [17]. The resonator (including the air gap width between the access waveguide and the resonator) in each MZI group is identical to that in corresponding A/D filter group.
Fig. 8. A SEM micrograph of a Mach-Zehnder interferometer (MZI) with one arm consisting of a racetrack resonator-based APF.
Figure 8 shows a SEM image of such MZI. The interaction in the coupling region of the racetrack in Fig. 8 can also be described using the following formula:
where E6, E4 and E5, E3 are the output and input electric fields of the coupling region, and other parameters have already been defined in previous section. In our structure, if we assume a dc value of 0.4μm, we calculate a maximum Lc of 6.6μm from Eq. (3), a value much smaller than LP (34.64μm). Therefore the optical loss from E6 to E5 is about twice of that from E4 to E5 (or from E6 to E3) in the A/D filter configuration (see Fig. 1). We then have:
where a and LP have already been defined in Section II. Substituting Eq. (17) into (16), we have:
Hence, the relationship between the output and the input electric fields of the MZI, E2 and E1, can be described by the following formula:
where Ltotal is the total optical path length in the MZI reference arm. The intensity contrast at the output port is:
Given the measured output intensity contrast at the left side of Eq. (20) and r2a and (r2+t2)a obtained in Section II, a can then be deduced from Eq. (20) by solving a polynomial equation.
3.2. Experiment and discussion
The frequency response of the MZIs as shown in Fig. 8 was also measured using a similar method as described in Section II. Figure 9 shows the measured and the simulated spectra at the output port of the MZI with an air gap width in an APF resonator of (103±5) nm. Given the measured intensity contrast in Fig. 9 and r2a and (r2+t2)a measured in Section II, the loss factor in the resonator, a, was calculated using Eq. (20). The simulation was done using measured r, t and a through two sets of complementary devices (A/D filter and MZI) presented in Section II and this section. Excellent agreement between the measurement and simulation indicates the accuracy in determination of all three parameters (r, t and a).
Fig. 9. Transmission spectrum of a MZI with one arm consisting of a racetrack based all pass filter (APF) with an air gap width of (103±5) nm
Frequency response of MZIs in all four different groups were measured and the loss factor, a, was inferred using the method mentioned above. We measured an a value of (0.995±0.003), (0.994±0.004), (1.002±0.006), and (0.985±0.02) for resonators with an air gap width of (117±5) nm, (103±5) nm, (87±5) nm, and 0nm, respectively. The a value as a function of the air gap width, d, is plotted in Fig. 10. The dotted line in Fig. 10 shows the estimated a value using the measured propagation loss (9dB/cm). Since the bending loss in the bends with a radius of 5μm is already negligible according to previous measurements [6], we neglect the bending loss in devices here with bending radius of 6.5μm. The estimated a value using measured propagation loss is 0.995. As seen in Fig. 10, the a value determined by two sets of complementary devices (A/D filters and MZIs) is air gap width independent and close to the estimated value using the propagation loss only.
Fig. 10. The loss factor, a, deduced from measuring two sets of complementary devices (blue dots) and estimated from the propagation loss (red dashed line).
Given the loss factor, a, and the measured (r2+t2)a, single-pass optical intensity loss in the coupling region, 10log(r2+t2), can be calculated. The single-pass intensity loss in the coupling region is 0.06dB, 0.1dB, 0.14dB, and 0.34dB for resonators with an air gap width of (117±5) nm, (103±5) nm, (87±5) nm, and 0nm, respectively. While the single-pass propagation loss in the resonator, 10log(a2), is 0.04dB, the loss in the coupling region dominates the propagation loss as long as the air gap width in the resonator is ≤(117±5) nm, and increases as the gap width decreases.
4. Mode conversion loss in resonator coupling region
The optical field in the resonator coupling region experiences mode conversion and power is transferred partially from one waveguide to another. The loss mechanism in the coupling region which results in (r2+t2)<1 can be explained by considering the mode conversion process using a coupled local mode theory [18]. The SEM micrograph of a resonator coupling region with an air gap width, d, of (103±5) nm is shown in Fig. 11.
Fig. 11. A SEM micrograph of a racetrack resonator coupling region with an air gap width of (103±5) nm
Before the incident light reaches the cross section A, at which the waveguide spacing is dC≈0.4μm, the interaction between the waveguides can be ignored since the power coupling length (defined as the length needed for the power to be transferred from one waveguide to another) at 0.4μm waveguide spacing is as large as 250μm. The super eigen-modes at the cross section A are shown in Table 1. The modes are calculated using a full-vector eigen-mode solver, FIMMWAVE (by Photon Design www.photond.com), and the modes shown in Table 1 are the TE components (Ex) of the modes as a function of the x position at the central height of the Si strip waveguide. As can be seen from Table 1, at the cross section A, the spacing between the waveguides is large enough so that the super eigen-modes can be approximated accurately using the modes of the individual waveguide [19], and the power in the incoming light will be coupled into both modes evenly. As the waveguides spacing decreases from cross section A to B, the super even and odd modes undergo transitions in a relatively small distance (1.59μm from A to B in our structure). If the final waveguide spacing, d, is big enough so that the super modes at the cross section B do not differ significantly from those at the cross section A, the mode transition losses will be small, and hence we can assume r2+t2=1. However, if the final waveguide spacing, d, is small, the super eigen-modes at the cross section B differ significantly from those at the cross section A. While the distance between A and B is not long enough to allow for lossless adiabatic mode transformation, some mode conversion loss will occur and consequently r2+t2<1.
As shown in Table 1, the shapes of both even and odd eigen-modes at the cross section B (with a waveguide spacing of around 100nm) significantly differ from those at the cross section A (with a waveguide spacing of 400nm). Actually, the even (fundamental) mode profile at the cross section B is very similar to the mode in the slot waveguides reported in Refs. [20] and [21], where a significant amount of light is concentrated in the low index slot between two high index waveguides. When the air gap between the waveguides decreases, the mode shape difference between sections A and B increases, and hence the mode conversion loss increases. This phenomenon is clearly shown in Fig. 7 as (r2+t2)a drops monotonically when the air gap width decreases.
Table 1. Normalized eigen-mode profiles at cross sections A and B of a racetrack resonator coupling region with the air gap width of (103±5) nm
5. Performance degradation of optical delay lines using resonators in an APF configuration
For the optical delay lines based on racetrack resonators, the mode conversion loss can have a significant impact on the performance. For example, in our current racetrack resonators with the smallest mode conversion loss (with air gap width of (117+5) nm), the measured quantity (r2+t2)a is 0.981+0.002. Assuming an a value of 0.995, parameters r2 and t2 are around 0.931 and 0.055, respectively. Even if we can optimize our fabrication processes so that the propagation loss is completely removed (a=1), for a single racetrack resonator APF-based delay component, the calculated group delay and the optical loss at resonance are 51ps and 4.4dB, respectively. Hence, we need around 20 racetracks to achieve a delay of ~1ns. Using the device parameters mentioned in this paper, the total delay line footprint will be ~15×420μm2 (0.0063mm2), or 4 to 5 orders of magnitudes smaller than the previous demonstrations [3, 4]. However, the peak loss at the resonance introduced solely by poor mode conversion will be as large as 88dB which would make this design impractical.
6. Conclusion
By measuring two complementary types of SOI photonic wire based devices - the add/drop (A/D) filter using a racetrack resonator and the Mach-Zehnder interferometer with one arm consisting of an identical resonator in an all-pass filter configuration, we found that mode conversion losses in the coupling region of the racetrack resonators are significantly larger than the bending and the propagation losses in the resonator propagation region when the air gap between the access waveguide and the racetrack is ≤(117±5) nm. This loss mechanism hinders the realization of practical optical delay lines based on APF racetrack resonators.
Acknowledgment
Partial financial support from DARPA/ONR (J. Lowell, DSO), grant N00014-04-C-0455, is gratefully acknowledged. We thank W. Green and E. Dulkeith (IBM T. J. Watson Research Center) for helpful technical discussions.
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