1. Introduction

An electrooptic modulator is a key component of RF photonic links, where it is used to transfer an input RF signal into the optical domain for further processing using photonic means [1]. Integrated technology is attractive for implementing RF photonic circuits because of scalability to large circuit complexity, reduced packaging costs, improved reliability, reduced size and weight, and potentially higher energy efficiency and performance [2]. A proven solution for RF photonic links is to use Mach-Zehnder modulators, which have wide electrical and optical bandwidth and are relatively simple to control. A possible alternative to Mach-Zehnder modulators are microring resonator modulators [3, 4], which are significantly smaller in size and can provide higher modulation efficiency. The higher efficiency is achieved because light traveling around the resonator experiences the phase shift induced by the applied RF signal not once, but multiple times, in proportion to the photon lifetime in the resonator relative to the round trip travel time, where a long photon lifetime corresponds to high quality factor Q. However, the photon lifetime limits the speed with which the resonator responds to the applied RF field, with shorter lifetimes required for higher RF frequencies. However, shorter lifetimes brings the modulation efficiency down, so that there is a trade-off between the modulator bandwidth and efficiency. This work examines the maximum efficiency which can be achieved in a single-ring modulator with a given intrinsic quality factor as a function of RF frequency, and the modulator parameters need to be selected to achieve this maximum efficiency.

The RF response of a microring modulator considering the photon dynamics inside the resonator has been studied in multiple publications [5–12]. Analytic formulas for the frequency response has been derived in small-signal approximation [7–10, 12], and optimum conditions for the laser detuning from the microring resonance [7–9, 12] and the resonator coupling coefficient [8] have been obtained.

Similar to previous publications [7–10, 12], this work studies the small-signal modulator response considering the photon dynamics inside the resonator. However, this work optimizes the modulator for different applications, and the definition of modulation efficiency adopted in this work is different as well. Prior publications examined the modulation of the optical intensity at the output of the modulator, which is determined by the beating between the optical carrier and the sidebands produced by the modulator. On the other hand, this work considers the modulator as an RF mixer which upconverts the RF signal into one of the optical sidebands, and seeks to maximize the optical sideband amplitude independently from the carrier. Such a regime is of interest in coherent detection schemes, where the carrier is supplied by the local oscillator, as well as in RF receivers where the optical sidebands are detected without the carrier [13–15]. More details can be found in Sec. 2.1.

The expressions for efficiency of conversion into the optical sideband are derived in small-signal approximation and their physical significance is analyzed in Section 2. Optimization of modulator parameters – the detuning between the laser and the microring resonance frequency and the ring-to-bus coupling coefficient – is performed in Section 3 for narrowband, and in Section 4 for wideband RF signals. In both cases, analytic solutions for the optimum parameters and the obtained efficiency are given and analyzed.

2. Calculation of conversion efficiency

2.1. Problem Formulation

The single-ring resonant modulator studied in this work is illustrated in Fig. 1(a). The input laser light at frequency ω_l passes through a waveguide with a microring resonator modulator coupled to it. The resonant frequency of the resonator in absence of applied RF signal is ω_o. The transmission of the modulator depends on how close the laser frequency ω_l is to the resonance frequency ω_o, with the transmission minimum at ω_l = ω_o. The RF signal applied to the modulator changes the resonant frequency, modulating the transmission of laser light.

 

Fig. 1 (a) The microring resonator modulator studied in this work. (b) The spectra of the optical input s_in, applied RF signal v_RF, and the output optical signal s_out with multiple sidebands generated by the modulator. The modulator is considered as a mixer which upconverts the input RF signal at frequency Ω into the optical signal ω_l + Ω, which is the first sideband of the output signal s_out.

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In this work, the modulator is considered as a mixer which upconverts the RF signal at frequency Ω to an optical signal at frequency ω_l + Ω. In general, when the modulator is driven by the RF frequency Ω, the output light has frequency components ω_l ± nΩ, where n = 0 corresponds to the optical carrier and n = ±1, ±2, … to the sideband frequencies, see Fig. 1(b). Usually, the two sidebands with n = ±1 have much larger magnitudes than the ones with |n| > 1. Out of these two sidebands, we pick the one with n = +1, and study conversion from RF frequency Ω to optical frequency ω_l + Ω, with a specific interest to maximize the efficiency of that single sideband’s generation. The conversion efficiency G is defined here as the fraction of the input optical power which is converted into that of the sideband at ω_l + Ω,

As mentioned in the introduction, this work analyzes the modulator operation in the regime which is different from the regime used in for regular data interconnects and conventional RF photonic links [5–12]. In particular, in RF photonics, the links are analyzed in terms of link gain which is defined as the RF power at frequency Ω at the output of the link relative to input RF power at the same frequency. The output RF signal at Ω is created at the photodetector due to the beating between the optical carrier at ω_l and the sidebands at ω_l ± Ω (the term S_out(Ω)S_out(ω_l + Ω)), so that the link gain depends not only on the magnitude of the sideband but also on the magnitude of the carrier. This means that maximizing magnitude of the sidebands alone, as it is done in this work, does not necessarily lead to the maximum gain in a conventional RF link. However, maximizing the sidebands is what needs to be done for example, for coherent detection, when the optical carrier is supplied by the local oscillator at the receiver so that the resulting RF power is maximized when the optical sidebands are maximized. The sideband needs to be maximized independently from the carrier also in the direct detection RF receivers [13–15]. Alternatively, one can envisage an integrated circuit where the optical carrier is controlled independently from the sidebands by analogy with the techniques used for conventional RF photonic links to improve the link noise performance and avoid photodetector saturation, such as low biasing of Mach-Zehnder modulators [16, 17] and carrier filtering [18, 19]. The carrier filtering technique in particular is well suited for implementation using integrated optics technology. The regime of operation studied in this work is also relevant for wavelength converters [20].

2.2. Small signal analysis

The optical sideband conversion efficiency G can be found analytically in small-signal approximation using the coupled mode theory in time (CMT) [21, 22]. According to CMT, the energy amplitude inside the ring resonator a evolves in time according to

The coefficients r_o and r_e are the decay rates of the energy amplitude a due to the optical losses inside the ring and the ring-to-waveguide external coupling, respectively. These decay rates are related to the loss-limited photon lifetime τ_p (or associated field decay time τ_o and the ring-to-waveguide power coupling coefficient κ² as [22] r_o = 1/τ_o = 1/2τ_p and $r_e=\frac{{\kappa }^2}{2}{\left(\frac{2\pi R}{v_g}\right)}^{-1}$ where R is the microring radius, and v_g is the group velocity. The loss quality factor is $Q_o={\omega }_o/2r_o={\omega }_o/\mathrm{\Delta }{\omega }_{3dB}^{\left(o\right)}$, the external quality factor is Q_e = ω_o/2r_e, and the total quality factor Q_total = ω_o/[2(r_e +r_o)] = ω_o/Δω_3dB, where $\mathrm{\Delta }{\omega }_{3dB}^{\left(o\right)}$ is the intrinsic linewidth and Δω_3dB is the full linewidth of the resonance.

The wave amplitude at the output of the modulator is

2.3. Analysis of the expression for conversion efficiency

The conversion efficiency G given by Eq. (6) can be expressed as

 

Fig. 2 (a) The Lorentzian profile Λ(ω) as defined by Eq. (9), describing resonant enhancement in the microring resonator; the red and the blue arrows indicate the input laser carrier at frequency ω_l and the generated sideband at frequency ω_l + Ω, respectively. In this example, laser detuning δω = ω_l − ω_o is assumed to be negative. (b) The conversion efficiency G as a function of applied RF frequency Ω.

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The following sections study optimization of the modulator design for maximum conversion efficiency. According to Eq. (6), the conversion efficiency G depends on the coupling rate r_e, intrinsic loss r_o, laser detuning δω, resonant frequency swing under drive δω_m, and the RF frequency Ω. The dependence on δω_m and r_o is simple: δω_m should be increased and r_o decreased for maximum conversion efficiency. Optimization with respect to the detuning δω and the coupling rate r_e is less straightforward and is carried out in the following sections.

3. Optimum design for a given RF driving frequency

Equation (6) can now be used to study how the modulator parameters and frequency detuning δω can be selected to maximize the efficiency. This section finds the parameters which maximize the conversion efficiency at a given RF frequency Ω_o, which is of interest for narrowband RF signals centered around Ω_o. The next section (Sec. 4) deals with wideband RF signals.

3.1. Optimum detuning δω for a given modulator design

First, we consider a modulator with a fixed design (i.e. fixed ring-to-bus coupling rate, r_e), and find which frequency detuning δω = ω_l − ω_o maximizes the conversion efficiency at a given RF frequency Ω_o. The frequency detuning can be controlled by adjusting the laser frequency ω_l or tuning the resonant frequency of the modulator ω_o.

The optimum δω can be found by equating the first derivative of the conversion efficiency given by Eq. (6) to zero while making sure the second derivative is negative. This leads to the following solutions:

The optimum detuning δω is plotted in Fig. 3(a) as a function of RF frequency Ω_o, with the resulting frequency configurations illustrated in Fig. 3(b) at several representative points of this plot. If the RF frequency Ω_o is smaller than the 3dB linewidth of the resonance, Δω_3dB, the optimum solution is to set the laser at δω = −Ω_o/2. In this case the sideband is generated symmetrically at the opposite side of the resonance peak, at δω_s = +Ω_o/2, see point A in Fig. 3(b). If Ω_o exceeds the linewidth Δω_3dB, there are two optimum solutions for δω, see points B and C in Fig. 3(a). These two solutions are symmetric with respect to the resonance frequency in the sense that δω₁ = −δω_s2 and δω₂ = −δω_s1, where δω_s1 = δω₁ + Ω_o and δω_s2 = δω₂ + Ω_o are the sideband frequencies corresponding to the two values of the laser detuning, see Fig. 3(b). In the limiting case when Ω_o ≫ Δω_3dB, the first of these solutions corresponds to the laser being at the resonance frequency ω_o, see point D in Fig. 3, and the second solution corresponds to the sideband generated at the resonance frequency ω_o, see point E in Fig. 3.

 

Fig. 3 (a) The optimum frequency detuning δω as given by Eqs. (10), (11) which maximizes the conversion efficiency at frequency Ω_o shown along the x-axis. (b) Location of the laser frequency ω_l and the sideband frequency ω_l + Ω_o relative to the Lorentzian resonance of the microring centered at ω_o for several representative points from plot (a).

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3.2. Optimization of modulator design

The previous section determined the frequency detuning values which maximize the conversion efficiency for a modulator with fixed coupling coefficient r_e. This section goes a step further and finds the ring-to-bus coupling r_e which allows one to achieve the highest conversion efficiency, G_max, possible at RF frequency Ω_o, for given loss rate r_o (i.e. intrinsic quality factor Q_o) and modulation amplitude δω_m (i.e. the drive signal voltage and the electrooptic resonant frequency shift per volt).

The conversion efficiency G from Eq. (6) is plotted in Fig. 4 as a function of frequency detuning δω and coupling coefficient r_e. Figure 4(a) corresponds to the case when ${\mathrm{\Omega }}_o⩽\mathrm{\Delta }{\omega }_{3dB}^{\left(o\right)}$, where

 

Fig. 4 The conversion efficiency G as a function of coupling rate r_e (normalized by loss rate r_o) and frequency detuning δω (normalized by RF frequency Ω_o) as given by Eq. (6). Plot (a) represents the case when ${\mathrm{\Omega }}_o⩽\mathrm{\Delta }{\omega }_{3dB}^{\left(o\right)}$ (with this particular plot created for ${\mathrm{\Omega }}_o=\mathrm{\Delta }{\omega }_{3dB}^{\left(o\right)}/2)$, and (b) represents the case when ${\mathrm{\Omega }}_o>\mathrm{\Delta }{\omega }_{3dB}^{\left(o\right)}$ (with this particular plot created for ${\mathrm{\Omega }}_o=2\mathrm{\Delta }{\omega }_{3dB}^{\left(o\right)})$. Without loss of generality, $\delta {\omega }_m/\mathrm{\Delta }{\omega }_{3dB}^{\left(o\right)}=1$ is assumed.

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Figure 4(b) corresponds to ${\mathrm{\Omega }}_o>\mathrm{\Delta }{\omega }_{3dB}^{\left(o\right)}$. In this case, Ω_o > Δω_3dB for small values of r_e, and there are two optimum detuning values as given by Eq. (11). In this region the conversion efficiency plot of Fig. 4(b) has two ridges. As the coupling coefficient r_e increases, the linewidth Δω_3dB increases as well and at some point becomes larger than Ω_o. In this case, the optimum detuning is −Ω_o/2 as given by Eq. (10), and the conversion efficiency plot has only one ridge. Note that this agrees with Fig. 3(a), where there are two δω curves for frequencies above the linewidth Δω_3dB, and one δω curve for frequencies below the linewidth. The single-ridge region of Fig. 4(b) rises above the double-ridge region, which means that the maximum efficiency G_max is achieved for r_e high enough to fit Ω_o within the linewidth, i.e. Ω_o < Δω_3dB.

According to the analysis above, for both cases shown in Fig. 4 the maximum conversion efficiency is achieved when Ω_o < Δω_3dB. The optimum ring-to-bus coupling $r_e^{opt}$ can be found rigorously by setting the derivative of G from Eq. (10) over r_e to zero. This leads to the solution

Equation (13) suggests that as the frequency Ω_o increases, the coupling coefficient r_e needs to be increased as well to make sure the modulator is capable of operating at higher speeds. Another interpretation is that at slow RF frequencies, Ω_o ≪ r_o, r_e = r_o to be sure to get the sideband out of the cavity before it is lost due to r_o. For high Ω_o ≫ r_o, r_e ≈ Ω_o/2 to ensure the resonance is wide enough to span both the laser and the sideband. The corresponding maximum conversion efficiency G_max is found by substituting $r_e^{opt}$ into Eq. (10),

The maximum efficiency G_max is plotted in Fig. 5(a) as a function of the RF frequency normalized by the intrinsic linewidth $\mathrm{\Delta }{\omega }_{3dB}^{\left(o\right)}$ [solid red curve]. At each point of this plot, the modulator design is optimized as described above. Without loss of generality, the modulation amplitude is chosen to be such that $\delta {\omega }_m/\mathrm{\Delta }{\omega }_{3dB}^{\left(o\right)}=1$. The efficiencies for other values of δω_m can be found by multiplying the values shown in Fig. 5(a) by ${\left(\delta {\omega }_m/\mathrm{\Delta }{\omega }_{3dB}^{\left(o\right)}\right)}^2$ [as follows from Eq. (6)].

 

Fig. 5 (a) Solid red curve: the maximum conversion efficiency G_max(Ω_o) achievable in the modulator optimized for frequency Ω_o, with ${\mathrm{\Omega }}_o/\mathrm{\Delta }{\omega }_{3dB}^{\left(o\right)}$ plotted along the x-axis. Dashed curves: conversion efficiency profiles G(Ω) for modulators optimized for 3 different frequencies Ω_o, with the respective values of Ω_o indicated by circles and $\mathrm{\Omega }/\mathrm{\Delta }{\omega }_{3dB}^{\left(o\right)}$ plotted along the x-axis. The resonant frequency swing δω_m equal to the intrinsic linewidth $\mathrm{\Delta }{\omega }_{3dB}^{\left(o\right)}$ is assumed. (b) The maximum conversion efficiency G_max vs. non-normalized frequency Ω_o for 4 values of $\mathrm{\Delta }{\omega }_{3dB}^{\left(o\right)}$, with modulation frequency swing δω_m/2π = 1 GHz.

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Once the modulator parameters needed for maximum efficiency at the RF frequency Ω = Ω_o are determined from Eq. (13), the efficiency at all other frequencies Ω can be found from Eq. (6) or (7). The G(Ω) plots for several values of Ω_o are shown in Fig. 5(a) [dashed curves]. It is interesting to note that the response G(Ω) of a modulator optimized at frequency Ω_o peaks at Ω = Ω_o/2, rather than Ω_o which is the solid circle that lies on the negative slope of G(Ω). For example, for the modulator optimized for ${\mathrm{\Omega }}_o/\mathrm{\Delta }{\omega }_{3dB}^{\left(o\right)}=4$, the $\mathrm{\Omega }/\mathrm{\Delta }{\omega }_{3dB}^{\left(o\right)}=4$ point is on the negative slope of its response curve [the green dashed curve in Fig. 5(a)]. The modulator whose response peaks at $\mathrm{\Omega }/\mathrm{\Delta }{\omega }_{3dB}^{\left(o\right)}=4$ is the modulator optimized at ${\mathrm{\Omega }}_o/\mathrm{\Delta }{\omega }_{3dB}^{\left(o\right)}=8$ [the orange dashed curve]. However, the peak of the orange curve is below the $\mathrm{\Omega }/\mathrm{\Delta }{\omega }_{3dB}^{\left(o\right)}=4$ point of the green curve, justifying our optimum solution.

While Fig. 5(a) plots the efficiency versus frequency normalized by intrinsic linewidth $\mathrm{\Delta }{\omega }_{3dB}^{\left(o\right)}$, Fig. 5(b) plots the maximum efficiency G_max as a function of absolute frequency Ω_o for several values of $\mathrm{\Delta }{\omega }_{3dB}^{\left(o\right)}$, for modulation frequency swing δω_m/2π = 1 GHz. At high RF frequencies ${\mathrm{\Omega }}_o\gg \mathrm{\Delta }{\omega }_{3dB}^{\left(o\right)}$, the maximum efficiency becomes independent of $\mathrm{\Delta }{\omega }_{3dB}^{\left(o\right)}$,

At low frequencies ${\mathrm{\Omega }}_o\gg \mathrm{\Delta }{\omega }_{3dB}^{\left(o\right)}$, the maximum efficiency is determined by the frequency shift δω_m relative to loss r_o (and intrinsic linewidth $\mathrm{\Delta }{\omega }_{3dB}^{\left(o\right)}=2r_o$),

3.3. Obtained RF bandwidth

The modulator design optimization so far has been performed for best performance at a single RF frequency Ω_o. This section examines how well a modulator optimized at Ω_o performs if the RF signal has non-zero spectral bandwidth around Ω_o. Figure 6(a) illustrates the frequency response G(Ω) of the modulator optimized for frequency Ω_o. The efficiency goes up below Ω_o and goes down above Ω_o; from Eq. (6), the frequency at which the response drops by 3dB relative to G(Ω_o) is

 

Fig. 6 (a) An example of frequency response G(Ω) of the modulator optimized for maximum efficiency at Ω_o. 3dB bandwidth BW is defined as the bandwidth around Ω_o within which the response goes down by no more than 3dB with respect to the response at the center frequency G(Ω_o). (b) The bandwidth BW normalized by center frequency Ω_o in the modulator optimized for Ω_o as a function of Ω_o, as given by Eq. (15).

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From Eq. (13), δω = −Ω_o/2 and Ω_3dB becomes

Therefore, if the RF signal has bandwidth BW centered around Ω_o, see Fig. 6(a),

The bandwidth BW normalized by Ω_o is plotted in Fig. 6(b) as a function of ${\mathrm{\Omega }}_o/\mathrm{\Delta }{\omega }_{3dB}^{\left(o\right)}$, with the modulator optimized at each point according to Eq. (13). The relative bandwidth decreases with Ω_o and in the high frequency limit becomes

This means that the optimization strategy presented in this section works well for band-limited RF signals with relatively high RF bandwidths relative to their RF center frequency. Modulator optimization for wideband RF signals with frequencies going down to DC is carried out in the following section.

4. Optimization for a wideband RF signal

The results of the previous sections can conveniently be used for signals with bandlimited spectra centered around some center frequency. However, many applications require even baseband operation of the modulator, when the RF spectrum has components that can go down to zero frequency. This section optimizes the modulator design for performance with such RF signals. Similar to the analysis above, the optimization is performed in two steps: optimization of frequency detuning δω for fixed modulator design (i.e. fixed r_e), and optimization of both δω and r_e.

4.1. Formulation of the optimization problem

A representative profile of the conversion efficiency G(Ω) given by Eq. (6) is shown in Fig. 7 with logarithmic scale used for the x-axis. The response profile can be described by the following parameters:

 

Fig. 7 A representative frequency response G(Ω) of a resonant modulator, indicating the response metrics used in modulator design optimization for baseband RF signals. The x-axis is the frequency Ω in logarithmic scale. Plots of G(Ω) with x axis in linear scale can be found in Fig. 2(b) and Fig. 6(a).

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The ripple can be found from Eq. (6),

The modulator optimization problem in the broadband case can be formulated as follows: for given loss r_o, find such values of frequency detuning δω and coupling rate r_e that

4.2. Optimum detuning δω for a given modulator design

First, the optimization is performed for a modulator with fixed coupling rate r_e. The optimization problem formulated above can be solved analytically using the conversion efficiency given by Eq. (6); the obtained solutions are given in the Appendix. The results are analyzed below.

Figure 8 plots the conversion efficiency G_DC as a function of the normalized coupling rate r_e/r_o and minimum bandwidth ${\mathrm{\Omega }}_{3dB}^{min}/\mathrm{\Delta }{\omega }_{3dB}^{\left(o\right)}$ for $\delta {\omega }_m/\mathrm{\Delta }{\omega }_{3dB}^{\left(o\right)}=1$. At each point, the detuning δω is selected in such a way that the ripple α [which is directly related to δω through Eq. (17)] has the minimum possible value which ensures that ${\mathrm{\Omega }}_{3dB}⩾{\mathrm{\Omega }}_{3dB}^{min}$; according to Eq. (16), such ripple also maximizes the efficiency G_DC and therefore corresponds to the solution of the modulator optimization problem. The obtained ripple values are shown with the white contour lines in Fig. 8 for few values of the ripple. This plot can be used in the following way: if the required maximum ripple α^max for the modulator with given r_e and the required ${\mathrm{\Omega }}_{3dB}^{min}$ is higher than the ripple α shown in the plot, the value of α from the plot [and δω related to it through Eq. (17)] should be used for the modulator design. If the required α^max is lower than shown in the plot, such an optimization problem has no solutions.

 

Fig. 8 Conversion efficiency G_DC (in dB) as a function of the required minimum 3 dB bandwidth ${\mathrm{\Omega }}_{3dB}^{min}$ and coupling rate r_e normalized by loss rate r_o. The white lines are the contour lines of the obtained ripple values for the obtained ripple value α = 0, 1, 3, and 6 dB. Red circles indicate the designs with the best efficiency for several fixed values of ${\mathrm{\Omega }}_{3dB}^{min}$, assuming the ripple α is not constrained. The efficiencies are calculated for $\delta {\omega }_m/\mathrm{\Delta }{\omega }_{3dB}^{\left(o\right)}=1$. The formulas used to create this plot can be found in the Appendix.

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Note that the points below the 0 dB ripple line all have 0 dB ripple and correspond to δω = 0 with Ω_3dB = r_e + r_o = Δω_3dB/2. When the required minimum bandwidth ${\mathrm{\Omega }}_{3dB}^{min}$ exceeds Δω_3dB/2, the value of δω has to be decreased below 0 to increase Ω_3dB, which leads to higher ripple α and lower efficiency G_DC.

4.3. Optimization of modulator design

Figure 9 shows the conversion efficiency G_DC for the case when the coupling rate r_e is optimized in addition to δω. This plot can be understood by going back to Fig. 8 which has r_e as its x-axis. Let’s assume first that there no restrictions on the ripple, so that the optimum value of r_e is the one which maximizes G_DC for the required ${\mathrm{\Omega }}_{3dB}^{min}$, regardless of the ripple. Such points for several values of ${\mathrm{\Omega }}_{3dB}^{min}$ are shown in Fig. 8 with red circles. Looking at the location of the red circles with respect to the ripple contour lines in Fig. 8, one can conclude that for each required bandwidth ${\mathrm{\Omega }}_{3dB}^{min}$ there exists an optimum ripple value, i.e. the ripple value which corresponds to the maximum G_DC. These optimum ripple values are plotted in Fig. 9 as the “optimum ripple” line; note that the red points of Fig. 8 lie on this line. Now, let us consider the case when the maximum ripple α^max is constrained. If the required α^max value is above the optimum line, the optimum modulator design is the one with the optimum ripple; this is the reason why the efficiency contour lines in Fig. 9 are vertical above the optimum line. If the required ripple α^max is below the optimum line, one needs to go back to Fig. 8 and select a larger value of r_e which corresponds to this α^max; the design with such r_e will have smaller efficiency G_DC.

 

Fig. 9 The conversion efficiency G_DC (in dB) for modulators optimized to have the 3 dB bandwidth of at least the ${\mathrm{\Omega }}_{3dB}^{min}$ value shown along the x axis, and the ripple of at most α^max shown along the y axis. Both detuning δω and coupling rate r_e are optimized at each point; the analytic solutions are given in the Appendix. The white line labeled the “optimum ripple” indicates the optimum ripple α which maximizes the efficiency for given ${\mathrm{\Omega }}_{3dB}/\delta {\omega }_{3dB}^{\left(o\right)}$. Without loss of generality $\delta {\omega }_m/\delta {\omega }_{3dB}^{\left(o\right)}=1$ is assumed.

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The conversion efficiency of optimized modulators is shown in Fig. 10 in a different way. The efficiency is plotted versus the minimum required bandwidth ${\mathrm{\Omega }}_{3dB}^{min}$ which is not normalized by $\mathrm{\Delta }{\omega }_{3dB}^{\left(o\right)}$ for several values of the ripple α^max and the intrinsic linewidth $\mathrm{\Delta }{\omega }_{3dB}^{\left(o\right)}$, for modulation frequency swing δω_m/2π = 1 GHz. One can see that for low bandwidths, the efficiency of the optimized modulator is limited mostly by the loss while for high bandwidths, the efficiency is almost independent of the loss, which is consistent with the behavior of the modulators optimized for narrowband operation (see Fig. 5). For high bandwidths, a modest efficiency improvement (up to about 3dB) can be achieved by allowing some ripple in the response.

 

Fig. 10 The conversion efficiency G_DC for modulators optimized for given minimum 3 dB bandwidth ${\mathrm{\Omega }}_{3dB}^{min}$ for several values of the required maximum ripple α^max. Three sets of curves are for different loss rates r_o. Modulation frequency swing δω_m/2π = 1 GHz.

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5. Summary and conclusions

This work solves the modulator design problem for a single-mode resonant modulator and provides analytic solutions for modulator parameters needed to maximize the conversion efficiency for narrowband and wideband RF signals driving a microring modulator as an RF mixer which upconverts the RF signal into the optical domain. It finds the maximum conversion efficiency possible using a single-ring modulator design, establishing a baseline that other, more complex modulator designs such as [15, 20, 23] can be benchmarked against.

As expected, there is a trade-off between the modulation speed and efficiency, with the efficiency going down at higher frequencies. The fact that the modulation efficiency goes down at higher frequencies shows the fundamental limitation of the selected modulator architecture, and suggests that different architectures might need to be considered when high efficiency at high frequencies is required. Note that the efficiency reduction happens even in the idealized case considered in this work when the modulation frequency swing δω_m is not a function of frequency. In reality, δω_m can go down with frequency as well because of the RC constant of the modulator, which further reduces the efficiency.

It should be noted that the results presented in this work are obtained in small-signal approximation, when the resonant frequency modulation δω_m is small enough so that the conversion of optical power from the carrier to the sideband does not deplete the carrier. It remains to be studied how the presented results are to be adjusted when the small-signal approximation no longer applies.

While this work establishes the fundamental limits of modulation efficiency which are possible to achieve with a single-ring modulator, the performance in any real-life scenario will be affected by fabrication tolerances. For instance, the frequency detuning δω will be affected by temperature fluctuations on the chip, requiring utilization of resonant frequency stabilization techniques [24]. The coupling between the microring and the bus waveguide in a fabricated device will deviate from the design target provided in this work, which can be mitigated by using a tunable Mach-Zehnder interferometer-based couplers.

This work optimizes the modulator operation for applications in coherent optical links and direct detection RF receivers, and the conversion efficiency used in this work is defined as the fraction of input optical power which is converted into the optical sideband, see Eq. (1). As explained in Sec. 2.1, this definition is different from the link gain usually used to describe RF links, which differentiates this work from prior art [7–10, 12]. The efficiency definition used here is useful for RF links where the magnitude of the carrier is controlled independently from the sidebands, such as in coherent links where the carrier is supplied by the local oscillator, direct detection links [13–15], or optical wavelength converters [20].