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It is shown that two mutually uncoupled microresonators in series can adequately cover the entire I–Q space and render the realization of QAM signals possible. This approach is based on the independent optimization of each microresonator for amplitude and phase modulation respectively. Generation of 16 quadrature amplitude modulation is demonstrated by means of simulation.
© 2013 Optical Society of America
1. Introduction
Advanced modulation schemes, such as M-ary quadrature amplitude modulation (QAM) can provide solutions to high speed transmission systems. QAM is a modulation scheme that conveys data by means of modulating both the amplitude and the phase of a sinusoid carrier, thus providing spectral efficiencies exceeding 2 bits/symbol.
Many reported QAM modulators were realized with LiNbO3 Mach-Zehnder modulators and phase shifters (for example see [1–3]). However, such structures are large and unsuitable for chip-scale optical interconnects. Devices based on microring resonators are attractive as building blocks for chip-scale interconnects systems. Microring resonators were demonstrated for generating OOK [4], DPSK [5, 6] and QPSK [7] signals. Recently, microring resonators were proposed for the generation of QAM signals: A high-Q microring QAM modulator incorporating a dual 2x2 Mach-Zehnder interferometers was proposed [8].
In a recent work, dual parallel-coupled racetrack microring resonators were proposed for generating QAM signals [9] and the optical structure was fabricated [10]. Therein, it was stated that two mutually uncoupled racetrack resonators in series cannot provide adequate coverage for generating QAM signals, and that mutual coupling is indispensable.
Herein, we present a method for generating QAM signals by way of two, serially connected and mutually uncoupled, microring resonators. Besides reducing the number of parameters needed for design the QAM modulator, our method has the advantage of allowing the independent optimization of each microresonator for amplitude and phase modulation respectively.
2. Generation of complex signals using a microring resonators
A schematic layout of the proposed device is depicted in Fig. 1. The design of each microring-resonator is based on a previously demonstrated physical layout [4]. The current design is extended to include two mutually uncoupled ring resonators in series, aiming to allow the modulation of the amplitude and the phase of an incoming optical wave in a quasi-independent manner.
Fig. 1 Optical M-ary modulator based on a two microring resonators. As an input, the modulator accepts two voltages V1 and V2.
The two rings have the same radius R, and by applying input voltages V1 and V2, the resonance frequency of each ring changes, respectively. The first ring modulates mostly the amplitude and the second ring modulates the phase of the incoming optical wave. Such a device is capable of covering the entire phase space, thus generating serially optical amplitude and phase modulated signals for advanced coherent communication. Considering a single microring resonator, its steady-state transfer function [11] is:
where t and a are the straight waveguide transmission factor and the ring internal loss factor, respectively. The accumulated phase shift over one round-trip is ϕ. This phase is controlled by tuning the effective index of the ring waveguide. The resonance wavelength is then modified and induces a strong modulation of the transmitted signal. In a Silcon based device, the effective index of the ring can be modulated electrically by injecting electrons and holes using a p-i-n junction embedded in the ring resonator [4].Defining the set of parameters t1, a1 to characterize Ring #1 and t2, a2 to characterize Ring #2, the design process involves setting the values of the transmission and loss factors t1 and a1 of Ring #1, in such a way the total transmission of the modulator is nearly linear with changes in the phase ϕ1 (concurrent phase modulation is unavoidable). In the second ring ϕ2 modulates mostly the phase, compensating for the phase variations induced by Ring #1, and determining the total desired output phase changes. Both phases ϕ1 and ϕ2 are assumed proportional to the soperating voltages V1 and V2 respectively. These voltages can be realized by employing N-bit digital-to-analog converters (DAC). Each N-bit DAC then outputs 2N equally-spaced voltage levels.
3. Modulator design
The design process involves the setting of the straight waveguide transmission factors, t1 and t2, the internal loss factors for both rings, a1 and a2 and for the phase shifts ϕ1, ϕ2. The serial disposition of the two rings that are mutually uncoupled, allows the consideration of the two transmission functions in an independent way. Our approach is to optimize each of them for amplitude and phase modulation respectively. It is assumed that the input voltages V1 and V2 that induce the phase shift ϕ1 and ϕ2, respectively, are fed using N-bit digital-to-analog converters (DAC). An N-bit DAC outputs 2N equally-spaced voltage levels.
Ring #1 modulates the amplitude. The power transfer function of Ring #1 is:
The design of Ring #1 involves the setting of three parameters: t1, a1 and the range of operating voltages, V1. For the entire phase span of ϕ1, which is induced by the input voltages V1, the optical amplitude should assume values between 0 and 1.
The transmission drops to zero when the internal losses are equal to the coupling losses, t1 = a1 and the resonator is said to be critically coupled. The maximum transmission for Eq. (2) is achieved for ϕ1 = π. In this case the maximum transmission is . and for a critically coupled resonator with t1 = a1, the maximum transmission reduces to . As t1 (or a1) decreases, the transmission decreases. For maximum transmission, the highest voltage level applied on Ring #1 should induce a phase shift of ϕ1 = π.
The driving voltage can be lowered on the expense of reducing the maximum output optical power. If fraction r is sufficient as maximum output power, then the maximum induced phase that is needed:
Figure 2 shows the power and phase transmissions for Ring #1 when t1 = a1 = 0.99. The output power is between 0 and 0.91, where the input power is 1. Ring #1 is driven by a range of voltages that induces induces small extra phase shifts in the range: 0 ≤ ϕ1 ≤ 0.04π.
Fig. 2 Power and phase transmissions for Ring #1 when t1 = a1 = 0.99; ring is driven by input voltage range that induces 0 ≤ ϕ1 ≤ 0.04π.
Ring #2 modulates the phase. The phase transfer function of Ring #2 is:
For the entire phase span 0 ≤ ϕ2 ≤ 2π, which is induced by the input voltages V2, the induced phase span inside Ring #2 should cover 0 ≤ Φ2 ≤ 2π. The design goal is to find the best match parameters t2 and a2 for Ring #2 such that the phase transmission will follow a straight line with minimum power loss. Ring #2 acts as lossless linear phase shifter by setting practical values of t2 = 0.1 and a2 = 0.99 in Eq. (4). At this situation light comes out almost entirely after one round and practically does not interfere with residual uncoupled. Figure 3 shows the power and phase transmissions for Ring #2.
Fig. 3 Power and phase transmissions for Ring #2 with t2 = 0.1 and a2 = 0.99. The ring modulator acts as a phase shifter; light exits after one round and does not interfere at all.
4. Generation of optical 16QAM constellation
The role of an M-ary modulator is to generate a predefined set of M distinct points in the complex plain. Such a constellation of points can be generally formulated as follows:
Simulation results for t1 = a1 = 0.99 and t2 = 0.1, a2 = 0.99 are provided in Fig. 4. The figure shows the desired constellation provided as model example, namely, an ideal 16-QAM, along with the signal pool generated using two mutually uncoupled microring resonators. In the simulation, each ring was controlled by a 6-bit DAC, i.e. 26 different analog voltage levels were applied to each ring in order to generate the signal pool. It is clearly seen that the signal pool resides inside a filled circle rim. From this pool of available points, 16 points (highlighted in the figure) are drawn so as to match the ideal 16QAM. Qualitatively, good match between the ideal 16QAM and the 16 points drawn from the pool can be observed. The matching can be further improved by employing higher resolution DACs. This, in turn, would generate a denser pool of signal points from which a desirable constellation can be drawn with greater accuracy.Fig. 4 Simulation results with t1 = a1 = 0.99 and t2 = 0.1, a2 = 0.99. Ideal 16QAM (∇); available signal pool generated by way of two microring modulators (×); obtained 16QAM constellation (Δ) (i.e. 16 points selected from the pool).
To quantify the modulator performance, one can employ the so-called normalized error vector magnitude (EVM), which measures the deviation of the chosen set of points from the ideal constellation [12]. EVM is defined mathematically as
The EVM for the generated constellation presented in Fig. 4 is −32dB. As mentioned earlier, the EVM can be further improved to −44dB by increasing the DAC resolution to 8 bits.16QAM requires 32 distinct control voltage levels; 16 voltage levels are required for driving each microring modulator. The required number of control voltages increases as the number of levels in the M-ary modulation scheme is increased. For example, 64QAM modulator may require as many as 128 distinct control voltage levels.
5. Sensitivity analysis
In Section 4 we provided the design parameters used in simulating the modulator. Deviation from the design parameters may degrade its performance. Since the two rings are mutually decoupled, one can analyze the sensitivity of the proposed design with respect to each of the rings independently.
5.1. Ring #1
It is important to design Ring #1 fulfilling the equality t1 = a1 as close as possible in order to achieve critical coupling. Otherwise, the output power will not drop to zero, and consequently a circular void will be shown to appear in the center of the signal pool.
The coupling coefficient κ is defined as the fraction of power coupled from a ring to an adjacent waveguide at the coupling section. In the above equations, the straight waveguide transmission factor t was employed. According to a previously published work [13] a lateral offset of the waveguide (with respect to the ring) of 50nm corresponds to about 2% decrease in the coupling coefficient κ. Since the two factors are related by (owing to power conservation), a 2% reduction in κ translates to less than 0.05% reduction in t, assuming the fabrication conditions of [13] are given.
As an example, we shall consider a worst case error of 0.5% in t. Figure 5 shows the signal pool generated with two microring resonators with t1 = 0.985, a1 = 0.99 and t2 = 0.1, a2 = 0.99, along with 16 points drawn from the pool. A void has been created in the center of the pool because the minimum power is 0.04. The EVM grows to −31.5dB, which indicates that the match to 16QAM is less good.
Fig. 5 Signal pool and 16 selected points. Pool is generated by two microring resonators with t1 = 0.985, a1 = 0.99 and t2 = 0.1, a2 = 0.99.
5.2. Ring #2
Ring #2 acts as a phase shifter having minimum power loss. The ring is controlled by a range of input voltages that induce phase shifts between 0 and 2π.
The coupling through parameter, t2, should be small in order to couple as much of the incoming optical power into the ring. Otherwise, the light will pass through the waveguide and interfere with the light exiting the ring. Consequently, the phase at the output of Ring #2 will not be linear with respect to the input voltage V2, nor will the ring be lossless, as long as a1 ≠ 1. For example, a deviation of t2 by 0.5% to a value of 0.105 will increase the EVM by 2dB to −30dB. A deviation of 0.5% in t1 and t2 will result in EVM of −31.4dB for 16QAM. For 64QAM, when using 7-bit DACs, this error will degrade the EVM from about −36.3dB to −31.5dB. It follows from the above examples that today’s fabrication capabilities are sufficient for obtaining good QAM modulators based on two uncoupled micro rings.
The fabrication of two microring modulators placed side-by-side with different parameters may not be easy. However, in our case the sensitivity analysis carried out above revealed that the fabrication is feasible with the state-of-art technology. Notably, Xu et. al. [14] fabricated a structure with several micro-rings placed side-by-side, where each microring has different geometrical parameters.
5.3. Further tolerance and sensitivity considerations
As seen in Figs. 2 and 3, the power and phase dependencies on operating voltages are not perfectly linear. Nevertheless, the QAM constellation is well fitted by adjusting the voltages to their best-fit values. This procedure can be applied to cope also with other non-linear physical effects not taken into-account like carrier-voltage control and carrier-induced absorption. The method will be limited only in the case that the IQ plane coverage leaves substantial voids and constellation points are not satisfactorily approximated.
6. Conclusions
A compact modulator architecture is presented for generating M-ary signal constellations using two mutually uncoupled microring resonators. Each ring has a specific task and the operation of the modulator can therefore be optimized by independently controlling each of the rings. Thus the modulator can generate arbitrary optical signals.
The analysis presented assumes quasi-static regime which holds as long as the photon lifetime inside the microring is substantially SHORTER than the inverse of the signal bandwidth. For micro-rings of 8 micron radii at the presented parameter set this lifetime is of the order of a few psecs, meaning bandwidth limit bellow 100Ghz. Larger signal bandwidths will require a detailed time-domain analysis.
A detailed example for a practical device capable of generating 16-QAM constellations achieving EVM of −32dB was presented. The EVM can be further improved by feeding the rings with control voltages coming from higher resolution DACs. The proposed modulator can be used for optical interconnects to generate arbitrary M-ary signals.
References and links
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