The evolution of the transmission spectrum of a photonic crystal waveguide under electro-optic tuning was studied in the band of an odd TE-like mode. The spectral signature of the interband scattering from the TM-like mode to the odd TE-like mode was characterized at various bias levels. The shift of the odd-mode band was determined based on a statistical approach to overcome the spectral noise. Simulations were performed to explain the spectral shift based on electro-optic and thermo-optic effects in the active photonic crystal structures. Potential impact of interband scattering on indirect interband-transition-based optical isolators is discussed and potential remedies are offered.
© 2013 OSA
Photonic crystal waveguides (PCWs) in silicon have potential for CMOS-compatible on-chip optical isolators that allow propagation of light in one direction but not the other. By comparison, magneto-optical isolators are not CMOS compatible. A non-magnetic PCW technique is to employ an indirect photonic transition between two modes (e.g. even and odd modes) in PCWs, following the principle that has been investigated for silicon rib waveguides with chip-scale interaction lengths of ~1cm [1, 2]. An RF-modulated even mode is converted to an odd mode through indirect interband transition after traveling forward in the PCW for a certain length, and that odd mode is filtered out at the end, blocking transmission; but in the reverse direction, the effective wave vector of the RF traveling-wave modulation does not match the wave vector difference between even and odd modes (for the reverse-propagation dispersion-relation branch). Therefore, the even-to-odd transition/conversion is weak and the even mode propagates with low loss in that direction. However, there exists another mechanism for mode conversion: inter-mode (interband) scattering due to structure imperfections. Such scattering is bi-directional and may possibly degrade the isolation. The present paper investigates interband scattering in electro-optically tuned PCW structures, and explores its impact on the performance of a PCW-based optical isolator. A potential remedy is described.
Photonic crystal waveguides have important applications in optical communications and signal processing due to their capabilities of modifying light propagation and dispersion characteristics. Particularly, light can be slowed down in a PCW. This enhances light-matter interaction and leads to significant reduction of interaction length for optical modulators and switches [3–6] and other electro-optic devices (including aforementioned isolators based on photonic transition). Most PCW research has been focused on an even mode with the TE-like polarization. Recently, odd modes in silicon-based conventional waveguides aroused interest for their potential important applications in CMOS-compatible one-way waveguides [1, 2]. A silicon-based PCW also has such an odd mode. While the passive properties of this odd mode have been experimentally studied in a few reports [7, 8], the transmission characteristics under electro-optic tuning have not been reported. Knowledge of such active properties is important for making one-way waveguides based on electro-optically induced interband indirect photonic transition. Here we report the evolution of the PCW transmission spectrum under electro-optic tuning due to interband scattering in the spectral range of the odd mode.
A decrease of TE transmission in the odd mode band has been observed in a passive photonic crystal waveguide . For a PCW fabricated under good conditions, small structural imperfections (e.g. small sidewall roughness of the air holes) still exist. The decrease of transmission can be attributed to structural imperfection-induced interband scattering from the even TE-like mode to the odd TE-like mode. According to the slow light scattering theory [9, 10], the loss coefficient arising from interband scattering is given by αi,f~Si,fDi(ω)Df(ω), where Si,f represents the scattering matrix element and Di(ω) and Df(ω) are the densities of states for the initial and final modes, respectively. Near the bandedge of the odd mode (the final mode), Df(ω) increases significantly due to slow light [Df(ω)~1/vg,f, where vg,f is the group velocity of the final mode] and Di(ω) is usually almost a constant. Thus αi,f increases and the transmission spectrum of the initial mode exhibits a notch at the odd-mode bandedge. Under carrier injection in an electro-optic device, the odd-mode bandedge shifts due to the change of refractive index, and the notch position will shift accordingly. Note that the above analysis is applicable to any initial mode, which can be the even TE-like mode as studied in Ref. . or the TM-like mode in the case of a PCW with asymmetric top/bottom cladding as discussed below. Note that understanding of the PCW TM-mode characteristics is of interest for optical isolators as it would be useful to have isolation for the TM mode as well.
2. Structure design and simulation
Consider a PCW formed on a silicon-on-insulator (SOI) wafer with a 260-nm-thick top silicon layer, as shown in Fig. 1(a) . The line defect PCW was formed in a hexagonal lattice with a lattice constant of 400 nm and an air hole diameter of 240 nm. Using the plane wave expansion method , the photonic band diagram is calculated and shown as solid lines in Fig. 2(a) . The mode profiles in Fig. 2(a) show that the modes have imperfect vertical symmetry, due to the asymmetric air/SiO2 top/bottom cladding. Note that the TM-like mode also has odd x-symmetry in the odd-mode wavelength range. Thus the interband scattering from the TM-like mode to the odd TE-like mode is not prohibited by symmetry. The cutoff indicates the frequency at which a mode crosses the lightline. As the hole radius increases, both the cutoff frequency of the TM-like mode and the bandedge of the odd TE-like mode move up. However, the odd-mode bandedge moves up faster than the TM cutoff. For too large of an r (e.g. r = 0.33a), the TM cutoff will be substantially below the odd mode. As such, the TM mode transmission will be fairly low in the frequency range of the odd mode (as the TM mode is above the lightline), and the detection of interband scattering becomes difficult. For too small of an r (e.g. r = 0.27a), the odd-mode and TM-like mode will cross each other, which will create a mini stop gap due to anti-crossing . Because this mini-gap is usually very close to the odd mode bandedge, it complicates the observation of the bandedge shift. In this work, we have chosen r = 0.3a such that the TM cutoff is above the odd mode bandedge and the crossing point of these two modes is sufficiently above the lightline. Under this condition, the two modes will share a common frequency band below the lightline, yet they do not cross each other directly below the lightline. The TM-like mode can be scattered into the odd mode due to the structural imperfection. To model electro-optic tuning of the odd-mode bandedge, the carrier injection process in a silicon pin diode structure was simulated using Medici, an electronic device simulator. For p and n regions doped to Na = 1 × 1020cm−3 and Nd = 5 × 1019cm−3 separated by a 2μm wide i-region, the calculated i-region carrier concentrations at various bias levels are shown in Fig. 2(b). The refractive index change can be calculated according to the electro-optic coefficient of silicon . The band structure under carrier injection at 3V bias is then calculated readily [13–15] based on the changed refractive index and is plotted as dotted curves in Fig. 2(a). The odd-mode bandedge shift is on the order of 10nm for 3V forward bias, which should be relatively easily observed. Based on the bandedge shift (Δλ) in Fig. 2(a), the odd-mode overlap factor with Si [13–15] can be obtained σ = (Δλ/λ)/(Δn/n) ~0.9. Note that the TM-mode has a linear dispersion relation in the wavelength range of the odd mode. Hence its own transmission is featureless (i.e. flat) in this spectral range (and no wavelength shift would be visible) if interband scattering is absent.
3. Fabrication and characterization
The PCW structure was fabricated according to the parameters of the structure simulated in Fig. 2. The optical waveguide layer was patterned on a SOI wafer by electron-beam lithography and followed by inductively coupled plasma dry etching. P and N regions were defined using photolithography and were doped by implantations sequentially. Piranha cleaning was done after implantation. The contact windows were opened alongside the PCW by a third photolithography process, and titanium/aluminum alloy electrodes were deposited by e-beam evaporation, followed by a liftoff process and rapid thermal annealing. The contact resistances of the p and n pads were about 30Ω and 25Ω, respectively, which were obtained based on testing silicon resistors with same contact pads fabricated on the chip. The total device length including access silicon wire waveguides is about 3.5mm.
The spectral transmission characteristics of the fabricated sample were measured at various bias levels as shown in Fig. 3 . A broadband superluminescent LED with a useful spectral range of about 50nm was used as the light source. Polarization-maintaining lensed fibers were used to couple light into the TM polarization of a silicon wire waveguide. Then light is transported by the silicon wire waveguide to the 80μm-long PCW. Light exiting the PCW is delivered by another Si wire waveguide to the output lensed fiber. The transmission in Fig. 3 is normalized by the LED spectrum plus a fixed reference loss value of a reference Si wire waveguide. Compared to normalization by the spectrum of a reference wire waveguide, this approach avoids introducing extra spectral noise of the reference wire waveguide. This might make the peak insertion loss less precise—but we are primarily interested in the wavelength shift and lower noise here. As expected, a spectral valley appears in the odd-mode band due to interband scattering. As the bias increases, the spectral valley shifts towards shorter wavelengths. Due to noise, the spectrum is not smooth. Instead of a clear single minimum point on the spectrum, multiple local minima of similar depths may be present in the neighborhood of the spectral valley. It is difficult to assign a single spectral notch precisely. Here we employ a statistical approach to determining the spectral notch wavelength. For the spectrum at a given bias, the wavelengths of the n lowest local minima of the transmission spectrum were recorded; then the mean value and the variance of this set of wavelengths were calculated statistically. The results for n = 3, 4, and 5 are plotted in Fig. 4(a) for each bias level. Clearly, the results for various n values agree with each other fairly well and show a definite valley shift. The inset of Fig. 4 also shows the measured I-V relation of the pin diode. The differential resistance under large forward bias is estimated around 118Ω.
In Fig. 4(b), the band shift Δλ evidently saturates at large bias. This is because the band shift is not solely determined by the electro-optic effect, but also by the thermo-optic effect accompanying the joule heating in the pin diode. Also, the electro-optic effect and the thermo-optic effect change the refractive index in opposite directions. The thermo-optic effect was simulated using the approach described in Ref. . The total temperature rise in the PCW core includes contributions from the heating in the i, p and n regions. The heat generated by the contact resistances had little effect on the temperature rise in the PCW core because the contacts are far from the core (~20μm, more than three times the thermal spreading length). At low voltages (V<1.5V), the electro-optic effect dominates. As the current rises at high forward biases, the thermo-optic effect due to joule heating becomes substantial. Together, the electro-optic and thermo-optic effects shift the odd-mode band as plotted in Fig. 4(b). The simulation result provides a reasonable trend of the spectral shift of this odd mode.
Note that in the odd TE-like band, the even TE-like mode is far above the lightline and has substantially higher loss and worse noise. This obscures the observation of the spectral features and therefore the TM-like mode is suitable for input in the present structure instead of the TE-like mode. In addition to the coupling via structural imperfection-induced interband scattering throughout the PCW length, the TM-like mode may also couple to the odd TE-like mode at the PCW interface. Due to the asymmetric top/bottom cladding, the TM-like mode and odd TE-like mode are not perfectly orthogonal (these two modes have the same x-symmetry and approximately opposite z-symmetry, as shown in Fig. 2a), which causes this coupling at the PCW interface. However, such coupling is extremely weak because the two modes have approximately opposite z-symmetry and a large group velocity mismatch (ng<5 for the TM-like mode of a Si wire waveguide and ng≥15 for the odd TE-like mode). FDTD simulations show that such interface coupling causes a very small loss <0.3dB per PCW interface over the entire odd-mode band. Such a weak effect was easily buried in the noise (~0.5dB in Fig. 3) of the spectrum. Note that at PCW interface, the coupling ratio between the TM-like mode and even TE-like mode is zero because of their opposite x-symmetry (for comparison, the interface coupling ratio between the TM-like and odd TE-like mode is <7% from FDTD simulations). However, the interband scattering between two modes of different polarizations is not prohibited/restrained by symmetry in this structure due to asymmetric top/bottom cladding (also note each mode is not purely TE or TM). This is different from the air-bridge structures [17, 18]. The even TE-like mode transmission below the light line is normal: it shows a bandedge around 1570nm but no interband scattering valley because the TM bandedge is at λ>1570nm. To launch an odd TE-like mode needs a carefully fabricated Mach-Zehnder coupler , which is not included in the present device. For the odd TE-like mode spectrum, it is expected that while the interband scattering from the odd TE mode to the TM mode is enhanced by 1/vg,odd at the odd mode bandedge, this signal will be buried by the backscattering (more strongly enhanced by 1/vg,odd2 [9, 10]) of the launched mode (odd mode). Note that backscattering and out-of-plane scattering (to radiation modes) are significant only at the bandedge of the initial mode (the launched mode) [9, 10]. In this work, we have chosen a wavelength range near the bandedge of the final mode (odd mode) and far away from the bandedge of the initial mode (TM-like mode). Thus, only the interband scattering is enhanced, and backscattering and out-of-plane scattering are not enhanced and are much weaker than interband scattering. Despite the spectral noise, the spectral notch position due to the interband scattering can be determined statistically within 1σ of all data sets for n = 3~5. This indicates that our method is independent of the number of spectral minima.
Note that optical isolators based on an indirect photonic transition can, in principle, achieve very high isolation ; up to 20dB difference for forward-to-backward transmission has been demonstrated in fiber-optic devices . In silicon-based waveguide devices, the isolation can be degraded by a variety of factors including RF impedance mismatch of the driving circuitry (which reduces the modulation/transition efficiency), the thermo-optic effect (which counteracts the electro-optic modulation and is not unidirectional), and interband scattering. Most of these secondary effects have been studied in optical modulators, therefore there is a knowledge base available for further improvement. However, the interband scattering effect is seldom studied in electro-optic devices, particularly for slow-light PCWs under electro-optic tuning. Here the notch depth due to interband scattering in Fig. 3 is found to be relatively small (about 2.5~3.5dB for all applied voltages) compared to ~10dB modulation depth achievable in PCW modulators with a similar interaction length [4, 20]. For the one-way waveguide (isolator) based upon an electro-optic modulation-induced photonic transition , this is a positive indication that the effect of electro-optic modulation would likely be substantially stronger than the undesired interband scattering.
For further analysis, consider an isolator composed of a modulated waveguide with a mode filter at each end . For the forward path (ideal case) in Fig. 5(a) , the launched mode (e.g. TM mode) is fully converted to a final mode (e.g. odd mode) by interband transition, and the odd mode is blocked by the mode filter. By proper design, the indirect interband transition can occur from the initial mode to the final mode (i→f) in the forward path only (not f→i nor any type in the backward path) . When interband scattering is included in Fig. 5(b), the back-and-forth scattering between the TM and odd modes results in some residual TM mode at the end. Based on the scattering coefficient formula [9, 10], one can readily see that the interband scattering loss coefficients between any two guided modes are symmetric, αi,f = αf,i. Assuming other losses are negligible, after a sufficiently long propagation length y = LB, the interband transition and two-way scattering reach a balance: tIi(y) + αi,fIi(y)−αf,iIf(y) = 0, where t is the interband transition coefficient per unit length (in dB/cm), Ii(y) and If(y) are the intensity for the initial launched mode (TM) and converted mode (odd), respectively. Under the lossless assumption, the total intensity at y = LB satisfies Ii(LB) + If(LB) = Ii(0). Thus we obtainFigure 5(b) also indicates that Ii(y) and If(y) remain constant for y>LB (after the balance between the interband transition and scattering is reached). This suggests rather than having an isolator of 2LB length, two cascaded isolators of LB each can possibly improve the isolation by two times. Here excellent mode filters are assumed in each isolator section, otherwise the isolation improvement will be less. Note that mode filters (or passive mode converters) with >20dB mode-selectivity  are achievable. The primary effect of the interband scattering for the backward path, shown in Fig. 5(d), is some excess loss (a few dB) for the isolator. Note that free carrier absorption caused overall decrease of transmission in Fig. 3. The absorption affects both forward and backward propagation. For the backward propagation, it increases the device insertion loss. Also note that under modulated bias, the time average of the scattering loss spectra at varying bias levels may result in a broad valley roughly covering all spectral notches from 0V to 3V.
This work is focused on investigating the effect of interband scattering on the new isolator based on the indirect interband transition. Of course, to further demonstrate a PCW based isolator, the dispersion relations of two waveguide modes and the doping of silicon need to be judiciously tailored  to translate the full potential of electro-optic modulation depth into high isolation and better impedance matching. Prudent junction design and higher electric driving power can also help. Detailed discussion of these techniques is beyond the scope of this work. Besides the cascading approach (one remedy) discussed above, the interband scattering effect can also be minimized by reducing structure imperfection (e.g. through tight process control in industrial-grade cleanroom facilities). An additional way to reduce interband scattering is to increase the wavelength of operation into the mid infrared range  by scaling up the waveguide dimensions. In that case, the structure imperfections of the present paper will have a much smaller effect.
In summary, the evolution of the PCW transmission spectrum in the odd mode band is studied under electro-optic tuning. The shift of the odd-mode band under carrier injection has been observed through interband scattering. The band shift is determined statistically from the measured spectrum with noise. The spectral shift was accounted for by electro-optic and thermo-optic effects. Potential impact of interband scattering for the isolator application is analyzed and potential remedies are discussed.
This work is supported by AFOSR Grant No. FA9550-08-1-0394 (G. Pomrenke) and by the DARPA Young Faculty Award (Grant No. N66001-12-1-4246). Part of the fabrication was carried out at the Microelectronics Research Center of UT-Austin (supported by NSF NNIN Grant No. ECS-0335765), Princeton PRISM, and at the CFN of Brookhaven National Lab. We are grateful to Marylene Palard and Mike Gaevski for fabrication assistance.
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