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We derive the frequency-dependent modulation transfer functions for microring resonator modulators, using a small-signal analysis. Modulation efficiencies are obtained in terms of familiar optical cavity parameters, which provide an intuitive understanding of device performance. The results are in good agreement with experimental data, and are used to compare operating characteristics between resonance-modulated and coupling-modulated microresonators.
© 2014 Optical Society of America
1. Introduction
Over the past decade, microring-based optical modulators have been researched extensively by many groups. The interest lies in the fact that these modulators can be energy efficient, high-speed, and densely integrated into systems requiring a large number of optical interconnections, as in data centers and high-performance computers. Furthermore, when combined with other photonic devices such as photodetectors and wavelength-division multiplexers, highly-integrated optical transceivers are becoming possible [1–3].
Intensity modulation is most commonly obtained from microring resonators in two different ways. In the first method, the refractive index within the ring is changed in order to shift the resonant wavelength, which in turn modulates the intensity transmission through the device [4, 5]. These devices will be referred to as resonance-modulated microring resonators (RM-MR). High modulation efficiency is achieved in RM-MR even for small shifts in the resonant wavelength because the resonators typically have a high quality factor Q, and therefore can have low power consumption. On the other hand, high Q resonators have a long photon lifetime τp which limits the rate at which optical power can be injected or extracted from the ring, which imposes a fundamental optical limitation on the modulation speed of the device. This may be mitigated by using lower Q resonators, but at the expense of modulation efficiency. A second method for using the microring as an intensity modulator is to change the coupling coefficient to the resonator while keeping the resonance fixed, which will be referred to as coupling-modulated microring resonators (CM-MR). This method has been investigated since resonance-enhanced operation is still obtained but the bandwidth is no longer limited by the photon lifetime and therefore there is the potential to be simultaneously low-power and high-speed [6, 7]. Therefore, it is important to quantify the difference in performance limitations between these microring modulators, in order to weigh the tradeoffs involved when designing photonic links.
In this paper, we provide a small-signal analysis of both types of modulators in order to evaluate the two types of devices in terms of bandwidth, modulation efficiency, footprint, and power consumption. We will not concern ourselves with the specific electrode structure of the device, which imposes its own electrical bandwidth limitation and contribution to the over all power consumption. Instead, we suppose a parameter of the resonator has been changed, such as index of refraction or coupling coefficient, and examine the response of the output power to this change. In this way, the fundamental optically-limited performance of the modulators is revealed. If one knows the relationship between drive voltage/current and index of refraction for a specific electrode structure, it is possible to combine that with the results provided in this paper to determine the overall speed and power consumption. We note that other authors have also analyzed the performance of these modulators either numerically or using small-signal methods [6, 8–10] and the results have provided significant insight, however simple expressions for the transfer functions and modulation efficiencies are not available to our knowledge. The results within are formulated in terms of the parameters of optical cavities, such as photon lifetime and quality factor Q. In our opinion these results offer a more lucid view of the limitations of microring modulators, in a familiar framework of cavity parameters and second-order transfer functions. In addition to descriptions of coupling controlled and index controlled microrings, we compare the results to the traditional Mach-Zehnder modulator (MZM) on the basis of speed, power consumption, and footprint in order to evaluate the merits of each device with respect to the well known and widely used MZM.
This paper is organized as follows. First, a static model is established in section 2 based on the coupling of modes in time (CMT). In section 3 a small-signal analysis is performed using the differential equations that govern the CMT resonator model, resulting in transfer functions and modulation efficiencies for both RM-MR and CM-MR resonators. In section 4 these results are compared with experimental data and in section 5 we compare the performance of the modulators and present power consumption estimates. The paper is concluded in section 6.
2. Static model
Before developing the small-signal model for microring modulators, we first review the static transmission characteristics. Figure 1(a) shows a basic microring resonator that is evanescently coupled to a straight bus waveguide. The optical input si travels along the upper straight waveguide and enters the coupling region (shown by dashed rectangle), which is modeled with the lossless coupling matrix [11]
where the amplitude coupling- and through-coefficients are given by κe and σe respectively, and |κe|2 + |σe|2 = 1. A portion of the input wave is then coupled into the resonator to form a travelling wave that circulates clockwise around the ring. During a single round-trip, the circulating wave is diminished, due to intrinsic optical losses, by an attenuation factor λ = exp(-αL/2) where α is the attenuation coefficient in the ring and L is the total circumference of the ring. This wave also undergoes a phase shift θ = ωT, where ω is the angular frequency of the input field si, and T = nL/c is the round-trip travel time of the ring, with effective refractive index n, ring circumference L, and vacuum light speed c. At the resonator output, the circulating field couples out of the ring and combines with the directly transmitted portion of si to form the output field st. The fields si and st are all normalized with the units (J/s)1/2 such that, for example, |si|2 = Pin has units of watts (J/s).The transmission spectra to the through port has been derived using either the coupling of modes in space (CMS) approach [11] or the coupling of modes in time (CMT) approach [12, 13]. Using CMS methods, the static transmission is given as [11, 14]
The interesting features of this transmission occur around the resonant frequencies of the ring, which are located at ωr = 2πmc/(nL) or λr = nL/m where m = 1,2,3,… is the mode number. Figure 1(b) shows an example transmission that has been plotted as a function of the detuning from resonance δ = ω-ωr. The parameters chosen for this example resonator were ωr = 1.22x1015 rad/s, L = 62.8 µm (a radius of 10 µm), n = 3, σe = 0.995, and α = 5 dB/cm. From Fig. 1(b) we see that |Ht|2 exhibits a notch at resonance (δ = 0), and manipulation of this notch, such as shifting its position by changing the resonant frequency or modifying its depth by changing κe, serves as the basis for using this device as an optical modulator.When the loss in the ring is small, and we are concerned only with operation near a single resonance, the resonator may be modeled using the CMT approach [12, 13]. This method represents the optical dynamics with a differential equation, which simplifies the small-signal analysis and is the reason we will adopt it. Here, we assume that within the ring a traveling wave A(t) is circulating with constant amplitude, such that |A(t)|2 is the total power flowing through any cross section of the ring. We can then view the resonator as a lumped oscillator with amplitude a(t), normalized so that |a(t)|2 is the total energy stored in the ring, and is related to the circulating power by |a(t)|2 = T·|A(t)|2. This oscillator has a resonant frequency ωr, and decay time constant τ. There are two sources of energy loss, each of which contributes to τ. The loss due to coupling to the straight waveguide is represented by time constant τe, and the intrinsic loss in the ring is accounted for with time constant τλ. The overall time constant is related to these components as τ −1 = τe−1 + τλ−1, and the photon lifetime of the cavity is τp = τ/2. The rate of change for a(t) can then be written as
and the output wave is given bywhere µe relates the coupling strength from the incoming wave si to the change in the energy amplitude a, and is related to κe and τe through μe2 = κe2/T = 2/τe [12]. For a purely harmonic input such that si = si0exp(jωt), we can solve Eq. (3) for a and then use Eq. (4) to obtainas the steady-state field transmission under the CMT formulation. The transmission spectrum |Ht(δ)|2 is shown in Fig. 1(b) by the dashed black line, which shows no noticeable disagreement between the CMS and CMT models, since the resonator loss is small. We note that the CMT method only models the resonator response around a single resonance, so that when δ becomes large enough, the two curves will vary dramatically since the CMS response is periodic in frequency with a free spectral range FSR = 1/T (where we have neglected dispersion for simplicity). Nevertheless, when operated near a single resonance and with low loss, the CMT model provides a good approximation and will be the basis for the dynamic analysis in the next section. Furthermore, a small-signal analysis of Eqs. (3) and (4) is found to be a straightforward process leading to transfer functions in terms of familiar cavity parameters. Small-signal analysis using CMS has also been carried out, although the analysis is complicated by the discrete-time nature of that formulation [6].The behavior of the microring strongly depends on the coupling strength between the ring and bus waveguides, relative to the loss in the ring. An important condition, known as critical coupling, occurs when the coupled power becomes equal to the roundtrip loss. When the ring is critically coupled to the bus waveguide, the notch depth at resonance shown in Fig. 1(b) is maximized, and no power exits the ring. Instead power builds up within the ring and is dissipated by the internal loss. Depending on whether the coupled power is greater or less than the round-trip power loss the ring is called overcoupled or undercoupled, respectively [7, 9]. Critical coupling occurs when τe−1 = τλ−1 or equivalently κe2 = αL. We now define a coupling ratio
that will be useful in the following analysis. For critical coupling we see that CR = 1, when CR>1 the ring is overcoupled, and when CR<1 the ring is undercoupled. Table 1 summarizes the resonator parameters that will be used in the following analysis.
Table 1. Summary of Resonator Parameters
3. Small-signal analysis
Optical modulation is achieved in a microring resonator by varying the parameters of its transfer characteristics, such as the resonant frequency or the coupling coefficient. For example, Fig. 2(a) shows that when the resonant frequency is shifted by Δfr, due to a Δn, then the transmission of the optical carrier is modified by Δ|Ht|2, which gives rise to intensity modulation. In this section we will find the transfer functions between small-signal perturbations of the resonator parameters and the small-signal output power. First, a general approach to finding the transfer functions is outlined, which is then applied to analyze two types of modulation. The framework for carrying out the small-signal analysis will be presented in terms of index modulation in the ring. However, the same procedure is followed for any type of microring modulation.
Fig. 2 Illustration of intensity modulation through resonance shift (a). Comparison between the simulated differential equations (grey curves) of Eq. (3) and (4), and the small-signal model (dashed black curves) given by Eq. (22) and (9) is shown in (b).
Consider varying the index sinusoidally so that n(t) = n + Δncos(ωmt) where n is the DC index, Δn is the modulation amplitude, and ωm is the angular modulation frequency. The sinusoidal index perturbation will impart the output wave st with upper and lower sidebands at ω0 + ωm and ω0-ωm, respectively, where ω0 is the (constant) carrier frequency of the input CW light source. The output wave can then be written as
where st1+ and st1- are complex amplitudes of the upper and lower sidebands, which are proportional to Δn, and st0 is the amplitude of the transmitted carrier. Note that the energy amplitude a(t) is similarly imparted with sidebands of amplitude a1+ and a1-. The output power of the modulator is the absolute value, squared, of Eq. (7) which consists of two beat signals generated between the carrier and each sideband, and the self-beating of the carrier. In the small-signal regime, where the modulation depth is much less than unity, we can neglect the second-order harmonic that arises from the beating of upper and lower sidebands, as well as the DC contribution from self-beating of the sidebands. With these adjustments the output power has the formIn general, the modulator output power can be written aswhere PDC = |st0|2 = |Ht|2·Pin is the DC output power, Δp is the small-signal output power amplitude, ϕ is the phase angle of the small-signal output power, and Hn(ωm) is the linearized transfer function for index modulation, which specifies the frequency dependent response of the output power to a change in index. Comparing Eqs. (8) and (9), the transfer function in terms of the complex amplitudes is found to beThe same procedure is followed to analyze the response of the modulator to changes in other cavity parameters such as Δκe or Δα. The focus of the next sections will be to find the sideband amplitudes st1+ and st1- through a small-signal analysis of Eqs. (3) and (4), which are then used with Eq. (10) to find the transfer functions for resonance modulation and coupling modulation.3.1 Resonance modulation
We have previously shown how a changing the resonant frequency of a microring produces optical intensity modulation in Fig. 2(a). Here, we will derive the transfer function between index change and output power. Substituting
andinto (3) yieldsExamining Eq. (11), we see that the equation can be separated into three new equations, by equating terms that have the same oscillation frequency. Neglecting the higher order oscillation terms, and cancelling the common exponential factors in each case, yields which can then be solved for a0, a1+, and a1- to obtain which are complex energy amplitudes at ω0, ω0 + ωm, and ω0-ωm. Using the fact that st1+ = -jμea1+,st1- = -jμea1-, and st0 = si0-jμea0, obtained from a similar small-signal analysis of Eq. (4), the carrier and sideband outputs are finally found to be The modulation transfer function is then calculated by substituting Eqs. (18)–(20) into (10), using Δωr = -ωrΔn/n, and carrying out some algebraic manipulation to ultimately findFinally, we see that response from index modulation to output power modulation is a second-order system that can be written in the standard normalized form aswhere ωn2 = δ2 + τ−2 is the undamped natural frequency and ζ = (τωn)−1 is the damping ratio. This normalization allows the frequency dependent factor of Hn(ω) to be unity at DC. The first factor of Eq. (22) is the DC gain of the transfer function, or DC modulation efficiency, and will be referred to as ηn = |∂Pout/∂n| = |Hn(0)|.Figure 2(b) shows the modulator output power in response to sinusoidal modulation at fm = ωm/(2π) = 10 GHz, for two values of Δn. The grey curves are computed using Eqs. (3) and (4), while the dashed black curves are obtained from Eqs. (22) and (9). The resonator parameters used in these simulations were n = 3, L = 62.8 µm (a ring radius of R = 10 µm), κe = 0.15, α = 5 dB/cm, δ = 2 GHz, Pin = 1 mW, and m = 121 so that we are operating near a resonant wavelength of λr = 1.558 µm. Also, in this example we have fn = ωn/(2π) = 4.3 GHz and ζ = 0.868. Figure 2(b) shows that in the steady-state, after the initial transient response of the grey curves has settled (after about five times τp), the small-signal model agrees very well with the numerically simulated differential equation.
The frequency response (normalized) given by Eq. (22) is plotted in Fig. 3(a), with CR swept from strongly undercoupled (CR<1) to strongly overcoupled (CR>1). The same resonator parameters have been used that were given in the previous paragraph, except the coupling ratio is adjusted by holding α constant and varying κe. The normalized magnitude responses in Fig. 3(a) show that the 3-dB bandwidth of the modulator may be increased by increasing the coupling ratio. As the coupling ratio increases the photon lifetime decreases, thereby increasing ωn as well as the 3-dB bandwidth. Note that this bandwidth extension comes at the cost of reduced modulation efficiency, and therefore an increase in power consumption. Figure 3(b) shows that ηn is actually at a maximum near critical coupling, while in the heavily overcoupled case of CR = 10, ηn is reduced by almost a factor of 20.
Fig. 3 Normalized RM-MR modulation responses for various values of CR (a) and modulation efficiency versus CR (b).
We can find a simplified expression for ηn by setting the modulator to critical coupling (CR = 1), where Fig. 3(b) shows that the efficiency is near maximum. Setting ωm = 0 and τe−1 = τλ−1 in Eq. (22) and simplifying yields
for critically coupled microring modulators. Further simplification is obtained by specifying that the frequency detuning be equal to one half of the resonator linewidth (|δ| = Δν/2). This value of detuning results in an insertion loss of IL = 3 dB, which is the same IL of a Mach-Zhender modulator, biased at the linear operating point. Setting |δ| = Δν/2 in Eq. (23), the modulation efficiency reduces towhich explicitly shows that for index modulation in a microring, the efficiency is proportional to the photon lifetime, or quality factor, of the cavity. The efficiency is also proportional to Pin, which is true for all external intensity modulators. When CR = 1 and |δ| = Δν/2, it is also possible to find a simple expression for the 3-dB modulation bandwidth ω3dB. Taking the squared absolute value of the frequency-dependent portion in Eq. (22), setting the result equal to 1/2, and then solving for ωm = ω3dB yieldsand shows that the modulation cutoff frequency is proportional to the cavity linewidth, or inversely proportional to the photon lifetime. We note that this exact solution for ω3dB agrees well with the approximate result derived in [10]. Lastly, the DC frequency chirp parameter αH = −2Pout(∂θ/∂Pout) can also be determined for the critically coupled modulator [15, 16]. The phase angle of the output field is determined from Eq. (5) to be θ = tan−1[1/(τδ] which yieldsUsing Eqs. (23), (26), and (5), the chirp parameter simply becomesIn the following section the small-signal behavior of the CM-MR is analyzed, using the same procedure followed above.3.2 Coupling modulation
Coupling-modulated microrings are interesting because the modulation speed is not fundamentally limited by the photon lifetime of the cavity [6, 7]. Figure 4(a) shows a possible topology for a microring modulator, where coupling control is achieved using a 2-by-2 MZM operated in a push-pull configuration (shown in the dotted box). Figure 4(b) illustrates this type of modulation, where the depth of the notch is adjusted through a change in κe thereby modulating the output power of the microring.
Fig. 4 A coupling-modulated microring topology that uses a 2-by-2 MZM tunable coupler (a). Illustration of a change in optical transmission for two values of κe (b).
The transfer function for coupling modulation is obtained in this section by applying the same procedure used previously for resonance modulation. For the sake of brevity we will omit the derivations for the sideband amplitudes. Note, however, that they are obtained in the same manner as in the previous section, by inserting
andinto Eq. (3). The result of the small-signal analysis yieldsandwhile st0 is given by Eq. (18). Using the fact that Δμe = Δκe/T1/2 and substituting Eqs. (28), (29), and (18) into Eq. (10), with Δn replaced by Δκe, the transfer function is found to bewhere G0 = 2κePin/(Tωn2) andThis is a general result that is valid for any coupling ratio and frequency detuning. Typically however, this type of modulator would be operated with the input wavelength on resonance of the cavity, in order to maximize the modulation efficiency. In this case, when δ = 0, Eq. (30) reduces to a much simpler form ofwhere Ht0 = Ht(0) is given by Eq. (5) and both ζ and ωn are the same parameters used in Eq. (22). Looking at this result, it is interesting to notice that the response consists of two terms, one which is frequency independent. This means that the output power of the modulator can change instantaneously in response to a change in κe. This is the reason that coupling modulated microrings have been recently studied, because the modulation rate is not fundamentally limited by the photon lifetime of the resonator [6, 7, 17].The result given by Eq. (32) is the modulation response in terms of changes in the coupling coefficient κe. In practice, a 2-by-2 Mach Zehnder modulator or a tunable directional coupler may be employed to implement the coupling control [7, 11, 15]. In both types of couplers, the coupling strength is controlled by inducing refractive index changes in the device. Also, if operated in the push-pull configuration, the CM-MR that utilizes the MZM coupler is ideally chirp-free, because the resonance of the microring remains fixed [7]. We will opt to use the 2-by-2 MZM as shown in Fig. 4(a), and find the change in coupling coefficient Δκ, in response to index change Δn in the arms of the interferometer. For the 2-by-2 MZM operated in the push-pull configuration, the coupling coefficient is given by κe = sin(ϕd/2) where ϕd = ωLκn/c is the phase difference in the MZI arms, ω is the optical frequency, Lκ is the length of the MZI arms, and n is the index difference in the arms [17]. Differentiating κe with respect to n yields Δκe = ωLκ(1-κe2)1/2/(2c)·Δn as the small-signal relation between index change and the coupling coefficient. Now we can use this result with Eq. (32) to write the response in terms of index change as
where we notice that, as was the case for index modulation, the response is proportional to the quality factor of the cavity. The added subscript in Eq. (33) indicates that the transfer function now represents changes in output power due to changes in the refractive index of the MZM coupler. Unlike index modulation, the response is also proportional to κe which is small when Q is large, and therefore will act to weaken the response of the CM-MR. The response is also proportional to the ratio Lκ/L, indicating that the coupling region should occupy as much of the ring circumference as possible. Of course there is a limit to the maximum obtainable value of Lκ/L, so we will consider the maximum practical value of Lκ to be one quarter of the total ring length L, for the remainder of this paper.Figure 5(a) shows several frequency responses for the CM-MR, with varying coupling ratios. The optical carrier has been placed on resonance (δ = 0), the ring length is L = 800 μm, Lκ = 200 µm, and α = 5 dB/cm. This ring length matches that of the fabricated modulator reported in [7]. The most important feature to notice is that, due to the frequency independent term in (33), the response has no high frequency roll off. Unlike the RM-MR, the speed of the CM-MR is not fundamentally limited by the photon lifetime of the resonator, since the output power can change instantaneously in response to a change in coupling strength. Also seen, for undercoupling, the response steps down beyond the natural frequency, with the magnitude of the step increasing for smaller values of CR. For overcoupling, the response increases beyond the natural frequency, with a larger increase for larger values of CR. The response flattens near critical coupling, but as shown in Fig. 5(b), the modulation efficiency ηκ = |Hκ(0)| is significantly reduced near critical coupling and the modulation vanishes completely at CR = 1 since for critical coupling Ht0 = 0. Figure 5(b) also suggests that a CM-MR should be operated in the undercoupled region, in order to obtain the largest possible ηκ.
Fig. 5 Normalized CM-MR modulation responses for various values of CR (a) and modulation efficiency versus CR (b).
An expression for the DC modulation efficiency is obtained from Eq. (33) by setting ωm = 0 and rewriting Ht0 as Ht0 = (1-CR)/(1 + CR), which provides
as the result. In order to compare this value to that of a MZM and the result from Eq. (24), we again want to configure the modulator to have a 3-dB insertion loss. Depending on whether the modulator is overcoupled or undercoupled, there are two cases where the modulator has IL = 3 dB. Setting |Ht0|2 = 1/2 these appropriate conditions are found to be CR = 3 + 2·21/2 and CR = 3-2·21/2, for overcoupling and undercoupling, respectively. With these two values of CR determined, the modulation efficiency at IL = 3 dB becomesfor the overcoupled case, andfor the undercoupled case. Since κe is generally small in the undercoupled case, this result can be simplified using the approximation (1-κe2)1/2≈1.4. Comparison to experiment
In a recent report, microring modulators were demonstrated for both coupling and resonance modulation. The devices were fabricated using the IBM Silicon CMOS Integrated Nanophotonics process, and used PN junction phase shifters to control the refractive index via the injection of free carriers [7]. A microscope image of one of the devices in shown in Fig. 6(a). Each modulator contained a phase shifter within the ring as well as a 2-by-2 MZM for coupling control, and therefore the devices could operate as resonance-modulated rings as well as coupling-modulated rings. Also the devices contained thermal tuners, so that the DC resonance frequency could be adjusted via the thermooptic effect. The MZM couplers were implemented with the same PN junction phase shifters and were driven in push-pull mode. In order to isolate the optically limited response from the influence of the device electrodes, reference MZMs with an identical structure as used for coupling control were tested, and the obtained frequency response was then used to calibrate out that contribution to the overall response. Figure 6(b) shows the experimental data for both types of modulation. Two values of detuning are shown for index modulation, while for coupling modulation the carrier was aligned to the microring resonance. Also shown, by the dashed curves, are the small-signal frequency responses calculated from Eqs. (22) and (32) using the parameters provided in [7]. It is seen that the results derived above fit well with the experimental results, accurately capturing the optically limited dynamics of the index modulated ring as well as the reduced damping that results from larger detuning frequency. The coupling-modulated responses also agree well, and are relatively flat since the ring is near critical coupling, a characteristic which was illustrated in Fig. 5(a).
Fig. 6 Comparison of transfer functions given by (22) and (33) to experimental data reported (a) and microscope image (b) of the fabricated CM-MR [7].
5. Discussion
Using the results of section 3 we can now compare the merits of both types of microring modulators in terms of efficiency, bandwidth, and footprint. In order to make fair comparisons, we will set the insertion loss of each device to be equal to IL = 3 dB since this is the insertion loss of a MZM operating at the linear bias point, where the modulation efficiency is maximum. Thus, the widely deployed and understood MZM performance becomes a benchmark for comparison with microring modulators. Note that this is the insertion loss introduced due to the operating point of the modulator, additional insertion loss is introduced in the input and output waveguides of the device, and will reduce the modulation efficiency due to the reduction of Pin.
The ideal MZM output power is given by
where ϕd = ωLn/c is the phase difference between the MZM arms, n is the difference in effective index between the arms, L is the modulator length, ω0 is the frequency of the input CW light, ϕ0 is the bias point phase difference between the arms, and α is the attenuation coefficient in the MZM waveguides. Setting the MZM at the linear bias point, ϕ0 = π/2, and differentiating (37) with respect to n yieldsas the MZM modulation efficiency when ϕd is small, i.e. small-signal modulation.Figure 7 shows the modulation efficiency versus α, using Eqs. (24), (35), (36), and (38), for RM-MR, CM-MR, and the MZM. The loss coefficient α depends on which material system the modulators are built in as well as the fabrication quality of the waveguide sidewalls. First, it is seen that the resonance-modulated microring (dotted curve) has the highest efficiency for all values of α, and also has the smallest footprint of ~20x20 µm2. The 3-dB bandwidth of the RM-MR, given by Eq. (25), is shown on the right-hand side of Fig. 7 and is increasing with α as the linewidth also increases, at the expense of efficiency. One interesting property of this type of modulator is that, for critical coupling, the ηn vs. α curve is independent of L, which can be seen by rewriting (24) as
and therefore all critically-coupled and half linewidth-detuned RM-MR modulators have the same efficiency for a given waveguide loss, resonant frequency, and optical input power. RM-MR modulators can therefore be scaled down in size without reducing modulation efficiency.Fig. 7 Comparison of slope efficiencies as a function of α. For the RM-MR, the 3-dB bandwidth is shown on the right y-axis.
The CM-MR modulation efficiencies are shown in Fig. 7 by the dashed curves for two values of L, and IL = 3 dB. We first note that the 3-dB modulation bandwidth for the CM-MR and MZM is not plotted because, when the electrical response of the modulator electrode is neglected, the frequency response for these modulators does not roll off at high frequencies. We also note that for the overcoupled situation, the efficiency is the weakest due to the low Q obtained when CR = 3 + 2·21/2 and will not be considered further. The undercoupled CM-MR, however has a much stronger response due to the higher Q when CR = 3-2·21/2, and is shown for L = 800 µm and L = 5 mm where we have assumed Lκ/L = 4. For comparison, MZM efficiencies are plotted (solid curves) with the same two values of L. When L = 800 µm, the CM-MR is more efficient the MZM across all values of α. When the length is increased to L = 5 mm however, the efficiencies are approximately equal when the loss is greater than α = 3 dB/cm. If the length were increased further, the MZM would become more efficient since ηMZML, as seen in Eq. (38), while for the CM-MR ηκ,nL1/2. The latter relation can be seen by rearranging Eq. (36), with Lκ/L = 4, to show
which also reveals that the CM-MR is less sensitive to variations in α than the RM-MR since ηκ,n1/α1/2 while ηκ,n1/α. The footprints of the CM-MR and MZM, for equal L, are similar but with different aspect ratios. For example, in [7] and [18], CM-MRs were demonstrated with an elongated racetrack topology, such that the coupling region was approximately one half of the total ring length, i.e. Lκ/L≈1/2. In these situations the CM-MR is about half the length of an MZM with the same L, but also about twice as wide.In previous sections we have derived the modulation efficiencies in terms of changes in refractive index, so that we did not have to be concerned with specific electrode structures, or physical mechanisms responsible for the index control and these contributions to the modulator operation. This allowed us to analyze the optically-limited modulation response, and although the results suggest which modulators would be most power efficient for certain configurations, the power dissipation was not quantified. We can extend the analysis to provide estimates for power consumption, however, if the properties of a certain device configuration are known, i.e. the voltage-to-index relationship ηe = |∂n/∂V| which will be referred to as the electrode efficiency. This parameter encapsulates the physical mechanism for the index change, as well as the confinement factor between the active waveguide region and optical mode. Knowing a specific ηe, the small-signal output power of a given modulator can be written as
where M = Δp/PDC is the desired modulation depth and ηx is one of the modulation efficiencies we have previously derived, depending on what type of modulator is to be analyzed. Solving for ΔV in (41) gives ΔV = MPDC/(ηxηn) as the required peak drive voltage required to achieve a certain modulation depth. The average dynamic power dissipation, when the electrode is driven with a DC-balanced pseudo random bit stream (PRBS) [19], can be expressed as Pdyn = (1/4)CVpp2BR with Vpp being the peak-to-peak amplitude of driving voltage signal (Vpp = 2ΔV), C is the capacitance of the electrode structure, and BR is the bitrate of the input PRBS. This expression then becomesas the dynamic power dissipation if a certain C and ηe is known. For example in silicon-on-insulator (SOI) platforms, modulators based on the metal-oxide-semiconductor (MOS) structure or lateral/vertical p-n junctions have been well represented in the literature [20]. Both structures rely on the plasma dispersion effect, which alters the refractive index in the structure by manipulating the free carrier concentration in the waveguides either through field-effect in the MOS devices, or carrier injection/depletion in the p-n junction devices. In [21], MOS electrode structures are investigated and the authors report an electrode efficiency of ηe = 2.32 x 10−5 V−1 and a capacitance of C = 26.4 pF for a 3.45 mm long electrode, to give a capacitance per unit length of C’ = 7.65 fF/µm. Using this information with Fig. 7 and Eq. (42), we can estimate the dynamic power dissipation, per Gbps of data modulation, for the modulators described in Fig. 7, if they were implemented with such a MOS structure. For example, when α = 5 dB/cm the RM-MR consumes 258 μW/(Gbps). When L = 800 µm, the CM-MR consumes 49.7 mW/(Gbps) and the MZM consumes 1.7 W/(Gbps). When L = 5 mm, the dynamic power dissipation is also 49.7 mW/(Gbps) for the CM-MR and for the MZM it is 196 mW/(Gbps). Note that even though in this case ηκ,n≈ηMZM, the CM-MR consumes less power since Lκ = L/4 is used to calculate the capacitance while L is used for the MZM. It is also interesting that the power dissipated by the CM-RM does not change with increased L. This is because Pdyn is proportional to C = C’L while ηκ,n is proportional to L1/2, and therefore the power consumption of the CM-MR is independent of L. The required voltage swing however, will decrease with increasing L. This is important since the voltage swings available to the modulator are often determined by the drive circuitry, which for CMOS circuits, would typically be limited to Vpp = 1-2 V.To verify the validity of Eq. (42), we compare the predicted Pdyn to the quoted value for a depletion-mode, PN junction based RM-MR presented in [5]. In that work, the authors report a wavelength shift of |∂λr/∂V| = 10 pm/V. Using the relation ∂λr/∂n = λr/n·La/L, where La is the length of the active waveguide in the ring, the wavelength shift is converted to the electrode efficiency through ηe = |∂n/∂V| = |∂λr/∂V·∂n/∂λr|. Using n = 4, typical for SOI waveguides [22], and La/L≈0.7 we obtain ηe = 4.31e-5 V−1. The reported RM-MR quality factor is 8300, and the input optical power was approximately Pin = 50.2 μW which yields ηn = 0.104 W, given by (24). The approximate capacitance of the electrode is C = 125 fF, the DC optical output power is PDC = 12.6 μW, and an extinction ratio ER = 3 dB (M = 1/3). The estimate therefore, using the small-signal analysis above, is found by substituting these parameters into Eq. (42) to obtain Pdyn = 109 μW/Gbps which agrees reasonably well with the reported value of 120 μW/Gbps.
In addition to the dynamic power dissipation, there will be power consumed by biasing and stabilizing the modulators. For both types of microring modulators, it is crucial that the resonance be correctly aligned to the incoming laser wavelength, or modulation will be impaired or lost completely. The resonant frequency in microrings is often tuned by placing heater pads within the ring, which tune the resonance through the thermooptic effect, and also add to the total power consumption of the device. In addition, since resonant modulators are very sensitive to index variation, these modulators will require stabilization circuitry to maintain the alignment of the input wavelength to the resonance due to environmental temperature fluctuations [23]. In general these contributions to the total power dissipation are often neglected in the literature, as well as the laser power, but should be accounted for when comparing modulators and calculating power budgets for optical transmitters.
6. Conclusion
We have derived the small-signal transfer functions for both RM-MR and CM-MR modulators on the basis of CMT. From these results the modulation efficiencies were obtained along with the 3-dB modulation bandwidth. To validate the developed small-signal model, the transfer functions were compared with experimental data, and found to agree well. Modulation efficiencies for the RM-MR and CM-MR modulators were compared with each other and with the common MZM. Finally, an expression for the dynamic power dissipation was presented and found to provide good agreement to data reported in the literature.
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