## Abstract

In this work, we propose and demonstrate the performance of silicon-on-insulator (SOI) off-axis microring resonator (MRR) as electro-optic modulator (EOM). Adding an extra off-axis inner-ring in conventional microring structure provides control to compensate thermal effects on EOM. It is shown that dynamically controlled bias-voltage applied to the outer ring has the potency to quell the thermal effects over a wide range of temperature. Thus, besides the appositely biased conventional microring, off-axis inner microring with pre-emphasized electrical input message signal enables our proposed structure suitable for high data-rate dense wavelength division multiplexing scheme of optical communication within a very compact device size.

© 2014 Optical Society of America

## 1. Introduction

The reduction in feature size due to the advancement of semiconductor industry and integration of millions of transistors within the same chip area as prognosticated by Moore’s law have steered on-chip communications to the paradigm of multi-core processors which are fast enough to execute huge calculations within billionth of a second. However, drift velocity of electron limits the speed of electronic circuits. The delay at metallic interconnects remains the same which forces asynchronous data transfer between the processor and the other parts of a computational system and consequently creates an important bottleneck problem [1–3]. Also, with the increasing demand, communication bandwidth of a system is becoming a big issue day-by-day which has to be addressed very soon.

Photonic circuits have nascent potentiality to overcome almost all the drawbacks of present-day electronic technology [4, 5]. Low power consumption, low latencies, less interference, ultra compact size and wide bandwidth are the key advantages of optical circuits [6]. Such key benefits have engendered interest among scientists to find out an efficient design of electro-optic modulator (EOM), which is an indispensible active optical component of photonic circuits [7]. Recent advancements in silicon-on-insulator (SOI) technology [8, 9] has opened up a new window for the fabrication of micro optical devices with microelectronics on chip [10]. Nonetheless, to design a compact, high-speed EOM is still a big challenge due to the absence of Pockels effect, very low Franz-Keldysh and Kerr effect of pure crystalline Si [7]. Carrier plasma dispersion effect of Si is the only effective way to achieve a considerable amount of refractive index (RI) variation by applying moderate external electromagnetic field avoiding the breakdown of Si-device. Soref *et al*. have predicted the change in RI, Δ*n* and absorption coefficient, Δ*α* of Si due to carrier plasma dispersion effect [11]. Carrier dispersion and absorption [7] at *λ* = 1.55μm can be approximated by Eqs. (1) and (2),

^{−1}) are the changes in RI and absorption coefficient variation due to the electron concentration change Δ

*N*(cm

^{−}^{3}), $\Delta {n}_{h}$ and $\Delta {\alpha}_{h}$ (cm

^{−}^{1}) are the RI change and absorption coefficient variation due to the change in hole concentration Δ

*P*(cm

^{−}^{3}), respectively.

According to the previous results [7], a carrier depletion/injection of ~10^{18} cm^{−3} can yield a RI change Δ*n* ~2 × 10^{−3}. A modulating device typically of the order of few millimetres is capable to provide a high modulation with extinction ratio (ER) as high as 20dB with sufficient variation of RI [7] until the development of resonator based SOI EOM. MRR confines light within the ring cavity and can provide relatively longer path length without increasing device size. A small change in RI can detune a MRR and yields a very high modulation [7].One of the critical issues of Network on Chip (NoC) systems is to keep power budgets manageable while increasing performance. DWDM is one such solution which offers massive parallelism and is considered very important in order to realize high performance optical interconnects which can be achieved through MRRs.

But, Si has high thermo-optic coefficient [12] (TOC = Δ*n*/Δ*T*), ~1.86 × 10^{˗4}/K. Small amount of change in RI results drastic change in transmission characteristics of SOI based MRR due to its large Q-factor and narrow bandwidth [13]. MRRs are highly susceptible to the thermal fluctuations. Heat generated from the surroundings, off-chip continuous-wave laser sources, local hot-spots etc. can cause shift in resonance wavelength. This instability can disturb precise channel allocations for DWDM systems. The shift in resonant wavelength can be represented as [13],

*λ*is the resonant wavelength at the room temperature,

_{o}*n*denotes the effective RI and

_{eff}*n*is the group index. The substrate expansion coefficient,

_{g}*α*of SiO

_{sub,}_{2}can be neglected in Eq. (3) due to its small value [13]. The Eq. (3) can be modified and written as,

In recent years, a few attempts were made to build athermal EOMs. A detailed comparative study on these approaches has been discussed in [13]. Methods to compensate the thermal effects can be broadly classified in two major categories (i) athermal solutions and (ii) control-based solutions. One of the most well-known control-based solution procedures to combat thermal degradation is the use of heat sink and integrated heater simultaneously [14–16] which later on has been proved highly power-inefficient [17]. Padmaraju *et al*. proposed an adaptive, innovative and dynamic athermal design exploiting the heating effect of current [17], which is more effective to deliver/extract heat using an external bias, albeit, this bias voltage can affect the message signal. This approach also fails to compensate wide range of temperature change. Moreover, it is not suitable for high data-rate due to its large response delay and inability to use pre-emphasized [18] input signal. Parallel efforts (athermal solution) to neutralize the thermal effect using negative temperature coefficient materials e.g. polymers [19–21] as the key component of the device cladding also exist. Mathematical representation of the effective TOC for a waveguide structure with Si-core and polymer-clad can be given by,

Where $\Gamma $*is the modal confinement factor. To overcome the thermal effects, one must produce a zero effective TOC. Since the TOC of Si is very high, according to Eq. (5) either we need a polymer with very high negative TOC or the mode confinement factor of the core must be deliberately reduced. Mode confinement can be reduced using narrow-ridged or slotted waveguides but these structures associate with high losses, greater radiation, lower Q-factor, small and corrupted FSR. The weak confinement of the mode also leads to higher device footprint. Furthermore, this method is not a successful commercial solution till date because of the CMOS-incompatibility of these polymers. CMOS compatible titanium oxide (TiO_{2}) with negative TOC [22] may be a possible future solution which allows moderate optical confinement in Si-core. Still the performance of these devices is not satisfactory due to low Q-factor and low FSR. Temperature independent operation over a small range has been shown employing MZI-coupled MRRs [23] and multiple cavity coupled devices [24]. However, relatively large footprints make them ineffectual for high-density on-chip applications. Adiabatic microdisk/microring modulators and cascaded MRRs [25,26] are capable of operating at as high data rate as 40Gbps and provide uncorrupted FSR but these devices are also very susceptible to the change in ambient temperature.*

*Recently, resonant mode-splitting method to negate the temperature effect has been reported [27]. Though, this method has potential to eliminate most of the previously mentioned drawbacks, it possesses a few serious disadvantages. Negative TOC CMOS-incompatible polymers have been used again to achieve satisfactory performance. Device size is large and it is impossible to implement this device in DWDM system due to its inability of using broad range of adjacent channels during the operation. It is also cumbersome to define and retain the material properties intact throughout the entire operational range.*

*The inefficacy of all the aforementioned mechanisms motivate us to propose a novel design of EOM using off-axis MRR. We have recently revealed many striking features of MRR with off-axis inner ring [28], where tunability aspects of conventional and extra resonant notches usher us the way to obtain a highly efficient and compact EOM. Two rings within the same structure provide us an extra degree of freedom of designing; we use one ring to deliver message and another to minimize the thermal effects. According to [13], our method falls in control-based solutions to avoid thermal degradation. Pre-emphasized input message signal can be applied to the conventional outer MRR to obtain high data-rate DWDM communication systems.*

*In this work, device parameters are extracted from commercial TCAD electro-optic device simulator based on finite difference time domain (FDTD)-Gummel solver. RI variations due to carrier injection or depletion are calculated from mode solver. Other necessary design parameters are calculated from transfer matrix method (TMM) and verified by FDTD method.*

*2. Design of off-axis MRR as an EOM*

*In Fig. 1(a), it is depicted how a MRR with off-axis inner ring can be used as an EOM. One can apply a modulating signal, V_{m} as an input to the off-axis ring and another voltage source V_{B} to the outer MRR to nullify the thermal effect. Note that, V_{m} and V_{B} are interchangeable. Cross-sectional view of the waveguide is shown in Fig. 1(b). A rib structure has been grown on buried oxide layer. Waveguide with dimension of 450nm × 350nm is placed on a 50nm thick plateau which provides electrical connections to the optical device. The junction volume is 3.48 × 10^{˗12} cm^{3}.*

*We have created a p ^{+}n^{˗}n^{+} type structure. As the conductivity of pure Si is very low, n^{˗} epitaxial layer with nominal carrier concentration ~10^{15} cm^{˗3} has been sandwiched between p^{+} and n^{+} regions to keep the carrier flow uninterrupted. Moreover, this denuded layer reduces the number of recombination centres which further prevents the alteration of diode current (R-G current [29]) with the change in temperature. One must keep in mind that high doping concentration in n^{˗} region will obtain a better RI sensitivity (Δn/ΔV) for the waveguide with respect to applied bias but it will increase the device loss due to higher absorption coefficient [7]. We optimize our structure to limit the absorption loss due to extra carrier injection within 20 dB/cm. Doping concentration of p^{+} and n^{+} layers are ~10^{19} cm^{˗3}; metallic contacts are made through n^{++} and p^{++} with doping concentration of ~10^{20} cm^{˗3}. The doping concentration of n^{˗} layer is much lesser than the concentration of n^{+} or p^{+} layer. Basically, this layer acts as an intrinsic layer.*

*3. Theory, results and discussions*

*We have applied an 8-bit, 3Gbps pre-emphasized NRZ signal, V_{m} = ± 1V (V_{p-p} = 2V) to the inner off-axis ring with an off-set dc bias of 1V. The minority charge carriers diffuse from the junction to the terminal when a voltage is applied to the p^{+}n^{˗}n^{+} device (forward biasing). If ${n}_{{p}_{0}}$and ${p}_{{n}_{0}}$are the minority carrier concentrations at p and n sides at room temperature in thermal equilibrium, we can write the extra charge carrier density as the function of distance x [29,30] from the depletion layer as,*

*$$\Delta N\left(x\right)={n}_{{p}_{0}}\left({e}^{\frac{eV}{kT}}-1\right)\mathrm{exp}\left(\frac{{x}_{p}+x}{{L}_{p}}\right)$$*

*$$\Delta P\left(x\right)={p}_{{n}_{0}}\left({e}^{\frac{eV}{kT}}-1\right)\mathrm{exp}\left(\frac{{x}_{n}+x}{{L}_{n}}\right)$$where, *

*x*and_{n}*x*denote the depletion region widths at the n and p sides,_{p}*L*and_{n}*L*represent the diffusion lengths of electrons and holes, respectively. The_{p}*L*(~22μm) and_{p}*L*(~35μm) are greater than the depletion layer width (~640nm) which assures moderate current flow through the device._{n}Figure 2(a) shows the schematic cross-section of an off-axis micro ring resonator with three coupling regions dictated by dashed rectangular boxes where *R*_{1} and *R*_{2} are the radii of outer and inner rings, respectively. In order to understand the behavior of off-axis micro ring resonator, we have to first compute the effective index of the guided mode which can be either obtained by effective index method [31], frequently used in integrated optics by combining Eqs. (6), (7) and (1), (2) or through full-vectorial finite element method (FEM) solver. One can yield effective index of the guided mode *n _{eff}* for a particular mode and at a specific wavelength. Effective RI of a particular quasi-TE mode for outer ring (

*R*

_{1}= 3

*μ*m),

*n*and off-axis ring (

_{eff1}*R*

_{2}= 2

*μ*m),

*n*have been determined by changing applied bias and plotted in Fig. 2(b), whereas the spectral variation of the effective indices of the quasi-TE mode for different ring radii and different biasing voltages is depicted in Fig. 2(c). At wavelength 1.55

_{eff2}*μ*m, the dispersion coefficient

*D*$\left(=\delta {n}_{eff}/\delta \lambda \right)$for outer and inner rings are obtained as ˗0.98965

*μ*m

^{−1}and ˗1.0135

*μ*m

^{−1}, respectively, when no voltage is applied and ˗0.9897

*μ*m

^{−1}and ˗1.01295

*μ*m

^{−1}, respectively when an external bias of 1V is applied across the p

^{+}n

^{-}n

^{+}junction of outer and inner rings, respectively. Since, we intend to implement the structure for several DWDM channels within a broad operating wavelength range,

*n*and

_{eff1}*n*will decrease with increasing wavelength due to dispersion which must be taken into account in TMM method. We use the dispersion coefficients to calculate accurate

_{eff2}*n*values over this broad range of operating wavelengths. Both the effective RI and dispersion coefficient of off-axis MRR are greater than the effective RI and dispersion coefficient of outer ring since the bending radius of the inner MRR is smaller than the outer one. Both the values of effective RIs and dispersion coefficients depend upon external voltage bias.

_{eff}In Fig. 2(d), we have plotted total losses (absorption, bending, radiation etc.) as a function of bias voltage *V _{m}*. Note that,

*V*

_{m}and

*V*are interchangeable. As discussed earlier, absorption losses have been restricted within 20dB/cm. It is noticed that both absorption loss and effective RI change rapidly after

_{B}*V*>0.7V (built-in potential of Si p-n junction). In this work, we have achieved Δ

_{m}*n*= −1.163 × 10

^{−3}and Δ

*n*= −1.179 × 10

^{−3}for outer and inner rings with an external bias voltage of 1V when applied individually on outer and inner rings, respectively at

*λ*= 1.55μm.

Off-axis inner ring as shown in Fig. 2(a) gives rise to the extra resonant notches in the transmission spectrum [28]. Off-axis MRR follows extra resonant conditions which results in the appearance of extra resonant notches in the transmission characteristics of off-axis MRR. These extra resonant notches are impeccably discernible beside the conventional notches due to their high Q-factor. The shift in resonant notches (conventional and extra) with respect to applied bias, *V _{m}* and

*V*can be calculated through propagation matrix ${P}^{\prime}$. For the off-axis MRR, propagation matrix is calculated [28] as,

_{B}where${\tau}_{2}$ and ${\kappa}_{2}$ are the transmission and coupling coefficients between off-axis inner and outer rings, respectively. The change in ambient temperature of MRR affects the effective RI (*n _{eff1}* and

*n*) of outer and inner rings, modifying the transmission characteristics. Voltages applied at the individual rings change the RI of respective rings only. Propagation matrix ${P}^{\prime}$ is a function of both ${\varphi}_{1}\left({n}_{eff1}\right)$ and ${\varphi}_{2}\left({n}_{eff2}\right)$. A positive voltage to the outer ring modifies ${P}^{\prime}$and blue shifts the conventional resonant notches as well as extra resonant notches. It is found that, in that case the shift in conventional resonant notches is larger in comparison to the extra notches. If we apply a positive voltage at the inner ring, blue-shift in extra notches will be greater than the blue-shift in conventional notches. It has been observed that the shift of extra resonant notches due to the change in

_{eff2}*n*is more than the shift of conventional resonant notches due to the change in

_{eff1}*n*. These facts have been theoretically established and validated by FDTD method [28,32]. Mathematically, the phase matching conditions (which cause the conventional notches) for the outer ring can be given by [28],

_{eff2}*m*= 1, 2, 3…

Both *ϕ*_{1} and *ϕ*_{2} depend on *n _{eff1}* and

*n*

_{eff2}_{,}respectively. From Eq. (13) it is viable that change in

*n*(due to the voltage applied to outer MRR) is capable for a slight shift in the extra resonant notches. On contrary, from Eq. (12), it is conspicuous that change in

_{eff1}*n*(due to the voltage applied to the inner MRR) has ideally no effect on the conventional resonant notches. The details are tabulated in Table 1.

_{eff2}This makes us confident to negate the thermal effects on MRR using an extra biasing voltage *V _{B}* at outer ring. The position of

*V*and

_{m}*V*can be swapped at the cost of overall performances. For example, if we apply a message signal at the outer ring and an extra bias voltage to the inner ring, it will take higher voltage and consume more power to compensate the thermal effect as the effect of external bias in the inner microring (or, change in

_{B}*n*) is negligible on conventional notches. Moreover, due to high

_{eff2}*Q*-factor and narrow bandwidth of the extra resonant notches, only a small message signal applied to inner ring would be sufficient to achieve acceptable ER. Thereof, it is always preferable to apply the message signal at the inner ring. The normalized power transmission of off-axis MRR within optical communication wavelength range is depicted in Fig. 3(a).

It is conspicuous from inset of Fig. 3(a) that *V _{m}* = + 1V results into ~380pm blue-shift of extra resonant notches (negligible blue-shift in conventional notches) which is sufficient to achieve almost 100% (>20 dB ER) modulation due to an ultra-high

*Q*-factor of extra resonant notches of off-axis MRR [28]. Figure 3(b) shows the effect of

*V*on conventional and extra resonant notches. Applying ~1V at conventional ring (i.e. outer ring) causes a blue shift of ~250-300pm in the conventional resonant notches while ~50-100pm blue shift of extra resonant notches. It is found that per Kelvin temperature rise can incur ~40-50pm and ~50-70pm red shifts for conventional and extra notches, respectively. Even 5K change in temperature may result into 20-30% change in modulation as

_{B}*Q*-factor is very high. Temperature effect, a nonlinear phenomenon in silicon waveguides, can be compensated through proper biasing voltage

*V*at the outer ring. In Fig. 3(c), it is shown that

_{B}*V*= 3.4V has nullified the effect of 20K rise in temperature. The solid blue curve shows an extra resonant notch at

_{B}*T*= 300K with no voltage bias applied at outer ring. When 1V modulating signal is applied to the inner ring the notch is blue-shifted towards the left-hand side as shown by the solid violet curve. At 1V, the off-axis MRR renders an extinction ratio of ~38dB. The solid red curve shows the transmission of the MRR with 20K rise in temperature without modulating signal while the solid green curve depicts the transmission of the MRR with 20K rise in temperature with an applied message signal of 1V to the inner ring. The solid black curve represents the transmission characteristics of MRR with a temperature compensating extra bias

*V*= 3.4V applied to the outer ring. This bias voltage can counterbalance the thermal effect, which is clearly visible from the inset where the solid black curve overlaps the solid violet curve. Therefore, for the Fig. 3(c) one can write,

_{B}*represent the shift in resonant wavelength due to*

*V*= 1V, Δ_{m}*T*= 20K and*V*= 3.4V, respectively._{B}The athermal behavior has been further validated and corroborated through eye diagrams. Figure 4(a) shows the waveform of user-defined pre-emphasized message signal. NRZ signal is passed through a low noise amplifier-differentiator block and the output is added with the original NRZ signal itself to provide pre-emphasized NRZ signal. Using pre-emphasized input, rise time, *t _{r}* and fall-time,

*t*can be reduced significantly (as small as 100ps and 40ps respectively) [18]. Figure 4(b) shows the eye-diagram for the optical transmission in room temperature.

_{f}The parameters ${\mu}_{1}$ and ${\mu}_{2}$ refer to the eye amplitude levels, the standard deviations (SDs) *σ*_{1} and *σ*_{2} measure the noises at the levels ${\mu}_{1}$ and ${\mu}_{2}$, respectively. The eye-opening is defined as ${\mu}_{2}-{\mu}_{1}$ which varies from 0 to 1 without thermal effects. The eye signal-to-noise ratio (SNR) can be defined as:

From TMM, it has been found numerically that the modulator amplitude varies from ~0.42 to 1 (3.82dB) at 5K rise in temperature. Consequently, the eye squints, which is demonstrated in Fig. 4(c). Figure 4(d) establishes that an extra bias of 2.2V at the outer ring can nullify the thermal effect. It is clearly observed that$\left({\mu}_{2}-{\mu}_{1}\right)>\left({\mu}_{2}^{5}-{\mu}_{1}^{5}\right)$, ${\sigma}_{2}^{5}>{\sigma}_{2}$ and ${\sigma}_{1}^{5}>{\sigma}_{1}$ due to the thermal effects, but at the same time$\left({\mu}_{2}-{\mu}_{1}\right)\approx \left({\mu}_{2}^{{5}^{\prime}}-{\mu}_{1}^{{5}^{\prime}}\right)$, where the superscript 5 has been used to explicate the amplitude levels at Δ*T* = 5K and a prime is used in superscript to indicate the thermally compensated eye-levels. One should also note carefully that even after the recompense, ${\sigma}_{2}^{{5}^{\prime}}\approx {\sigma}_{2}$ and ${\sigma}_{1}^{{5}^{\prime}}\approx {\sigma}_{1}$ as the thermal noise in Si-waveguide remains intact and cannot be eliminated through this procedure. If temperature is raised to 10K the eye opening further decreases and the transmission amplitude varies from 0.65 to 1 (swing 1.86 dB). The closed eye can be reopened by counteracting the thermal effect using a bias voltage *V _{B}* of ~2.82V. These two cases have been delineated in Figs. 4(e) and 4(f). Further, an additional voltage of only 0.58V can negate the temperature effect of 20K.

Padamraju *et al*., has clearly mentioned that thermal compensation with control-based solution consumes more power than athermal based solutions, which is the main disadvantage of the control-based solutions. Our process can be conceived as control-based solution. Besides the fabrication complexity, low fabrication tolerance and the requirement of an extra power source are the challenges in the off-axis MRR. While the thermal noise cannot be eliminated, however this does not affect the performance of the proposed device. Moreover, the RI change with respect to the bias voltage is nonlinear due to the exponential I-V characteristics [30] of the PIN diode. Until the device breaks down, this characteristic enables the device worthwhile to control the temperature effects more easily when the bias voltage is already very high. Note that, the extra bias voltage (Δ*V*_{B}) applied to the outer ring in order to compensate the thermal effect will be lesser for the same amount of temperature rise (Δ*T*) while the temperature is already high in comparison to the required feedback voltage for low initial temperature value, for example, if Δ*V _{B}*

_{1}and Δ

*V*

_{B}_{2}are the bias voltages to compensate the effect of same amount of temperature rise Δ

*T*at two different room temperatures

*T*and

_{1}*T*(

_{2}*T*≤

_{1}*T*), respectively, then within the break-down limit of the SOI EOM one can write mathematically,

_{2}*R*(i.e. Δ

_{T}*R*/Δ

_{T}*T*) (Fig. 1a) by choosing suitable material and doping concentration. An ultra-sensitive on-chip optical MRR sensor, temperature monitoring integrated photo-detector or fibre Bragg grating sensor can be used to detect the ambient temperature. One can also use pre-calibrated external feedback mechanism to control and maintain the required value of

*V*dynamically to compensate thermal effect. Other possibilities are still under investigations.

_{B}## 4. Conclusion

To conclude, off-axis MRR EOM can obviate the thermal effects for wide range of temperatures very effectively through an extra ring within the same MRR structure which not only creates highly sensitive and sharp resonant notches (with high Q-factor) but also offers another degree of freedom of designing. This renders us an opportunity to build up an electro-optic modulator untarnished by thermal effects within the same compact dimension of a single MRR. The eye diagrams of the proposed off-axis MRR show the potential to overcome the thermal detuning for DWDM applications.

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