Adaptive High Linearity Intensity Modulator for Advanced Microwave Photonic Links

This chapter, first, presents the motivation behind the need for adaptive, highly linear electro-optic modulators and an overview of the different optical linearization approaches of electro-optic modulators. Then, the figures of merits in terms of linearity performance are described and analyzed. Next, the chapter focuses on one excellent linearization approach called interferometric modulator with phase-modulating and cavity-modulating components (IMPACC). Here, we model IMPACC by simulating each of the key building blocks separately before putting them together as IMPACC modulator. This adaptive IMPACC design is compared to typical Mach-Zehnder interferometer (MZI) based modulators, and ring-assisted Mach-Zenhder interferometer (RAMZI) modulators. Theoretical analysis and results show that the IMPACC provides both superior linearity performance and unique adaptive feature that can be used to compensate for manufacturing tolerances, thus, providing extra flexibility in terms of device manufacturability as well as system integration.


Technology trends in microwave photonic links (MPLs)
The current state of microwave photonic links (MPLs) is in the midst of transformation toward more dynamic and flexible MPLs, in response to three major on-going technical shifts, as depicted in Figure 1. The first technical shift is an outgrowth of the concept of software-defined The second technical shift is the rapid growth of the field integrated microwave photonics (IMWP) [21][22][23], which used to be a niche research area some three decades ago. Today, IMWP aims to miniaturize key photonic components, which are essential in the MPLs, by using the emerging photonic integrated circuits (PICs) [23][24][25][26] technology to the level that would be comparable to the performance of state-of-the-art microwave components and systems [21]. Clearly, the rapid developments and advances [22,[23][24][25][26] in fabrication technologies re-enforce more advanced IMWP devices and support the prospect of future software-defined MPLs.

Device programmability
The third technical shift is happening at the module level with the development of optical programmable devices and components. These programmable devices and components can have the traditional features of reconfigurability, tunability, or selectability. There are growing number of reports and demonstrations of so-called microwave photonics processors [27][28][29][30][31][32][33].
Another implementation of these devices is based on the concept of a photonic fieldprogrammable gate array (FPGA) or "Photonics FPGA". It is the analogous version of the electronic FPGA, which is an integrated circuit designed to be configured by a customer or a designer after manufacturing. "Photonic FPGA" is geared toward arbitrary function generation or function programmability in miniaturized form, which is essential in the full implementation of SDN. This technology is still in the early stages of development, but the growing numbers of demonstrations of on-chip optical processing have been reported and they look promising [27][28][29][30][31][32][33][34][35][36][37][38][39][40].

Linearity
However, to really support the march toward future IMWP-assisted, software-defined MPLs, it is imperative that these new photonic components not only have inherent programmable features that can be accessed by software control, but also superior linearity characteristics. Note that microwave photonic links, in its basic form, is essentially a linearization engineering challenge. Regardless of whether we are talking about current MPLs or the future SD-MPLs, the core requirement of the links is still the same-high linearity links! Superior device's linearity translates to high overall link performance. The combination of superior linearity and programmability features in key photonics devices would lead to adaptive, highly linearity SD-MPLs.

Intensity-modulated direct detection (IM-DD) link and figure of merits
One of the most dominant optical transport schemes in MPLs is the intensity modulated direct detection (IM-DD) because of its simplicity and low cost. IM-DD scheme is shown in the upper portion of Figure 2. In its basic form, it consists of a RF data signal, laser source (either in the form of (i) directly modulated laser or (ii) externally modulated laser, using a combination of CW laser transmitter and optical modulator), optical fibers, optical and electrical amplifiers, and photodetector.
For most low-bandwidth (<1 GHz) applications, the directly modulated laser is used as a laser source for economic reason. However, for applications that require higher linearity, larger link gain and wider bandwidth (e.g., 1-60 GHz), the use of an externally modulated laser is the most viable and common approach. It offers the best approach toward achieving higher link linearity compared with directly modulated lasers since CW lasers with very low RIN noise are commercially available. Furthermore, the high output power of a laser can act "as an indirect RF power booster" for the RF input signal to provide a positive gain rather than a negative gain (or equivalent loss) [41]. This important feature adds advantage to the link using an external modulator compared with a directly modulated laser. From this point forward of this book chapter, when we talk about the links or MPLs, we will specifically mean only externally modulated links.
The quality of the overall links performance is largely measured by four important figuresof-merit [42], namely: (1) its dynamic range in terms of the spurious-free-dynamic-range (SFDR), (2) link gain (G), (3) noise figure (NF), and (4) bandwidth. The link gain (G) describes the relationship of the RF input signal power to RF output signal power of the IM-DD link, whereas the noise figure (NF) defines the signal-to-noise ratio (SNR) degradation in the link. The SFDR depicts the RF signal power range that can be accommodated by the link, taking into account the effects of noise and nonlinear distortions. It represents the highest signal to noise ratio when the intermodulation signal power is equal to the noise floor. Thus, it combines noise and linearity performance of the link. More detailed technical discussions on these figures-of-merit are found in references [14][15][16]43].
The nonlinearities in any part of the links (such as laser, modulator, fiber, photodetector, amplifier, etc.) translate directly to signal noise and distortion. Nonlinear performance at the fiber occurs when the optical powers of the signals are high enough; hence, this is usually mitigated by proper engineering of the optical signal powers to avoid fiber-based nonlinearity. Similarly, detector nonlinearity is mitigated by proper optical power engineering so that the optical power impinging on the photodetector is within the linear response of the photodetector. The nonlinearity of the modulator affects the distortion and dynamic range of fiber optic links. From among these factors, the distortion produced by the external optical modulators has the most negative impact on the overall performance of the link. More specifically, the inherent nonlinear transfer function of the external intensity modulator is the dominant limitation in the performance of MPLs.

Adaptive linearized intensity modulator
As mentioned earlier, the major source of distortion in IMDD links is the nonlinearity of the intensity modulator. The quest to develop high linearity external modulator with wide bandwidth at low cost has been an on-going challenge for decades.
Besides linearity, other key and important attributes of an external intensity modulator include, (1) low half-wave (V π ) voltage, (2) low insertion loss (IL), (3) high bandwidth (BW) capability, (4) low-cost, and (5) low complexity design. The low V π voltage is related to the slope of the transfer function of the modulator. In general, the steeper the slope of the transfer function of the modulator, the smaller the V π voltage. The immediate impact is an increase of the link gain (G). A modulator with low V π voltage, as well as low insertion loss, is very attractive since it lowers the necessary input power level of the link and increases gain. Thus, low V π and low IL lead to higher dynamic range, and low noise figure (NF), while large bandwidth capability makes it better for high-end applications. Minimal complexity and low cost are interlinked and can open the use of high bandwidth MPLs applicable to more commercial applications.
Lastly, any linearized modulator must also have a high tolerance to changes in the RF power level, bias drifts, optical wavelength drifts, etc. This requirement calls for modulator with adaptive capability to provide better maintenance and stable operation of the overall MPLs.
Thus, the focus of this book chapter is two-fold. First, it provides a comprehensive overview of the different linearization approaches that have been proposed and implemented to obtain a highly linear external intensity modulator. Second, it reviews a special type of intensity modulator called interferometric modulator with phase-modulating and cavity-modulating components (IMPACC) that is not only highly linear but also adaptive. These combined features, as far as we know, are not available with any previously reported intensity modulators.

Objectives and organization
This chapter of the book is organized as follows. First, we give a brief outline of the different linearization technologies used for intensity modulator in Section 2. After a broad overview, we introduce the different optical approaches in linearizing the modulators. Then, we focus on one particular subgrouping known as MZI-based linearized modulator. In Section 3, we present a special type of MZI-based linearized modulator called IMPACC. IMPACC provides both superior linearity and unique adaptive feature. Here, we discuss its basic principle, configuration, and modeling. The comparison and discussion of the original IMPACC and ring-assisted Mach-Zenhder interferometer (RAMZI) modulator are also presented. In Section 4, we provide and analyze IMPACC and RAMZI's respective performances before giving our conclusion in Section 5.

Survey of linear intensity modulators
In this section, we discuss the different linear intensity modulators. Before presenting these modulators, we first discuss an important figure-of-merit called spurious free dynamic range (SFDR) which quantifies the linearity of a given link. In Section 2.1, we discuss its derivation using a 2-tone test and graphically describe its significance. In Section 2.2, we provide an overview of the different general techniques to linearize intensity modulators before focusing on one general approach called optical linearization in Section 2.3. Optical linearization has many implementation flavors depending on the particular optical structure used for the modulator. In Section 2.4, we narrow down to a subgroup known as MZI-based modulator because of its popularity and mature technology. Within this subgroup, there are different configurations that have been proposed and implemented, and we spotlight one configuration called IMPACC.

Measure of linearity: SFDR
A 2-tone frequency test is typically used to evaluate SFDR of the link. Figure 3(a) shows a typical IM-DD based MPL consisting of a laser, external modulator RF signal amplifier, optical fiber, photodetector, and RF output amplifier. Note that when the focus of the link is to establish the performance of the external modulator, the common practice is to consider only the intrinsic MPLs. Here, the calculation of dynamic range does not contain the effects of the optical fiber, electrical amplifiers, and optical amplifier, if present.
In Figure 3(a), RF input signal with two closely spaced RF frequencies, f 1 and f 2 , of equalpower levels, are simultaneously injected into a nonlinear external optical modulator. The nonlinear interaction between these two signals will create new output frequencies, as shown in the bottom portion of Figure 3(a). Thus, the detected RF output signal will contain two fundamental frequencies, (e.g., f 1 and f 2 ) together with the second-order IMD output frequencies, (e.g., f 1 + f 2 and f 2 Àf 1 ), second-order harmonics, (e.g., 2f 1 and 2f 2 ), and third-order IMD outputs (e.g., 2f 1 À f 2 and 2f 2 À f 1 ). Other higher order harmonics and IMD effects (e.g., fourth, fifth, etc.) can also be present, but their effects are generally small and can be filtered out. The second-order SFDR is only important for systems whose bandwidth is more than one octave because the second-order intermodulation products will fall outside the passband of a suboctave system. However, the most important and often most difficult to filter out are the third-order effects which occur very close to the fundamental frequencies within the system bandwidth of interest as shown in Figure 3(a).
Instead of giving the mathematical derivation of SFDR and other relevant parameters or merits, we present a graphical treatment in the discussion of SFDR [16]. For detailed mathematical derivation, we direct readers to relevant references [14][15][16]. Figure 3(b) depicts the output RF signal power versus the RF input power of the different frequency signals generated by the nonlinear interaction within the nonlinear external modulator. This single plot also demonstrates the relationships of the different parameters which we summarize below: • P ONF, BN, P ON, and P IN : P ONF refers to the output noise floor, which is the minimum discernable signal, whereas BN is the background noise or thermal noise limit and equal to À174 dBm/Hz at a temperature of 25 C. P in is the RF input signal power and P out is the RF output signal power.
• Link Gain, G: The MPL gain, in dB, is defined as the difference of the power at the output of the MPL, P out , to the available power of the input, P in [16]: We should note here that G is typically referred to as small signal gain, because only in the small-signal regime, the gain is independent of input signal power. That is, for any input power, the associated output power is G dB units higher than the input power. Hence, a straight line is obtained. However, as we continue to increase the modulation power (P in ), the straight gain line begins to saturate, until a maximum output power is reached and then starts curving downwards. Beyond this, any increase in the input power leads to lower output power. We should note that G is not necessarily positive. We use it here in its general form, where positive G means increase in the output power, whereas, negative G means reduction in the optical power or loss.
• Different Signal Slope Lines: The fundamental curve, third-order IMD curve, and fifthorder IMD curve have a slope of 1, slope of 3, and slope of 5 in the linear region, respectively. Note that oftentimes at the fifth-order IMD, the slope is not necessarily the same throughout the entire frequency band, but it can change from slope 3 to slope 5 [16,44]. Figure 3(b) shows the associated slope lines marked as A (fundamental curve with slope of 1), B (second-order harmonics and intermodulation terms with slope of 2), C (third-order intermodulation terms with slope of 3), and D (fifth-order intermodulation terms, with nonlinear slope and not the same slope in the entire frequency range, typical slope at the linear portion is 5. • Noise figure (NF): is defined as the 10 log(F), where F is the noise factor of the MPL and defined as the quotient of the signal-to-noise ratio (SNR) at the input to that at the output, taken for a thermal-noise-limited input at T s or -174 dBm. Typically this is described by the following equation [16,43]: • Spurious free dynamic range (SFDR): Graphically, the SFDR is the difference measured from the corresponding point on the fundamental curve to the associated point in the intermodulation distortion (IMD) curve, where IMD intersects or equals the P ONF . Physically, this means that this is the area for which any IMD terms are just below the noise floor and cannot be "seen" by the system. This approach determines the SFDR value, as discussed by Bridges et al. [45]. Mathematically, SFDR is defined as [45]: for a system limited by third-order IMD, where OIP3 is the output power at the thirdorder intercept point and B is the system bandwidth (with units Hz). It is simply twothirds the difference between the largest distortion-less signal that can be input into the system and the smallest detectable signal in the system. The lower end of this range (the minimum discernable signal) is the sum of the output noise floor (P ONF ) and the system bandwidth, B. The measurement bandwidth is assumed to be 1 Hz. In this case, the units of SFDR are dB/Hz 2/3 . However, in the case of more linear systems that are limited by fifth-order IMD, SFDR is mathematically defined as [45]: where OIP5 is the output power at the fifth-order intercept point. And, the units of SFDR for a system limited by fifth-order IMD are dB/Hz 4/5 . These respective SFDRs for the thirdorder and fifth-order are labeled as SFDR 3 and SFDR 5 , respectively in Figure 3(b).
• 1-dB Compression point: This is the point at which the small signal gain (e.g., straight extended line in Figure 3(b)) is 1-dB higher than the actual gain curve. This point signifies the end of the linear regime. A closer look of 1-dB Compression point is depicted in Figure 3(c).
• Compression dynamic range (CDR): is the difference between the output power at the 1-dB compression point (expressed in dBm) to the smallest such input power (e.g., noise floor).
• IP n -nth-order intercept point: If we extend the linear slope of the fundamental and the nth-IMD curve, then these lines intersect at a point. We called this point the nth-order intercept point (e.g., IP3 if the IMD is the third-order IMD curve, IP5 if the IMD is the fifthorder IMD curve). The point is defined by the input intercept power (IIP n ) and the output intercept power (OIP n ). These points are depicted in Figure 3(c).
Note that in some new modulator designs, the fifth-order IMD curves are not always straight lines with a constant slope, but have a slight deviation from a straight line just above the noise floor . This is clearly seen in Figure 3(b), in the circled region just below the output system noise level (P ONF -output noise floor), where the slope of an IMD curve increases beyond 5. Since the nonlinearity in the IMD curve is not taken into account in Eqs. (3) and (4), the graphical approach is best suited to be used in calculating the SFDR for the new modulator. In this approach, the SFDR is simply defined as the difference between the fundamental and intermodulation output powers at the input power location where the IMD curve intersects the output noise floor (P ONF ). Figure 4 summarizes the three general approaches used to linearize a generic external intensity modulator based on the physical mechanism used, namely: (1) electrical, (2) optical, and (3) digital (post detection). The electrical and digital linearization approaches can further be subdivided into smaller categories (as in the case of optical linearization) but we will not review this in much detail here, however more detailed information can be found in [13, 21-22, 41, 46-55].

Different linearization approaches
Briefly speaking, the electrical linearization approach employs different techniques [46][47][48][49] such as: (i) RF predistortion, (ii) RF Feed-forward, (iii) combined RF Feed-forward and predistortion, and (iv) others. These techniques generate circuits with either (a) inverse function (ArcSin) of the modulator to predistort the RF signal before it is injected into the modulator, or (b) split a portion of the RF signal, amplify, reverse its sign, and add it to delayed original RF signal. The overall goal is to cancel or eliminate the original distorted components in the RF signal. These techniques are generally simple and inexpensive but they are limited by the frequency bandwidth of the electrical circuits [51,53].
On the other hand, a digital linearization approach is based on the application of digital signal processing, DSP [50,52,54] techniques to post-detected output signal. The digital approach is very promising because of its flexibility and future growth of DSP. However, its practical usefulness will depend on the progress and development in analog-to-digital converter (ADC) and digital-to-analog converter (DAC) technologies. Unfortunately, ADC is limited by the sampling rate of <10 G/sample at this time [55]. Although there are ADC/DAC technology with >10 G/sample rates, they are usually for instrument purposes [55] and not for field deployment.
In contrast to electrical and digital linearization approaches, optical linearization is free from circuit frequency modulation limitation and does not utilize ADC/DAC technologies. For these reasons, we focus on the optical linearization approach in this chapter.

MZI-based linearized modulator
Technically speaking, the MZI-based modulator is the most well-studied type of modulator. A typical nonlinearized MZI modulator has a SFDR value of around 90-112 dB-Hz 2/3 . In the last decade, a wide range of linearization approaches has been proposed and implemented to improve the standard MZI modulator to have 120-130 dB-Hz 2/3 SFDR range using different configurations [74-76, 78-79, 81-84]. Recently, the goal is to push the SFDR further toward >130 dB-Hz 4/5 . Nevertheless, most of them include complex designs and low modulation bandwidth. Figure 5 shows these different types of MZI-based linearized electro-optic modulators, which can be divided into four groups depending on the actual implementations. The first group can be called dual-signal MZI-based modulator design [74][75][83][84]. It comprises of one modulator but with two optical signals injected into the device, as depicted in Figure 5(a). The two input signal amplitudes and phases must be properly matched with predetermined values of the RF amplitude phase signal. These two injected optical signals could be implemented using (i) two polarizations [74][75], (ii) two wavelengths [84], or (iii) bidirectional signals [83]. The major advantage of this family of modulators is the relative simplicity and low cost. Its disadvantages are design inflexibility, nonoptimum performance, and tight tolerance requirements. The second group can be referred to as cascaded modulator design which consists of two or more standard MZI modulators connected in series [76,[78][79][80][81][82] or in parallel [100,109] arrangements as illustrated in Figure 5(b). It is a generalization of the principle used in the first group. Its major drawbacks are the tight manufacturing tolerance requirements, higher optical loss, higher cost, due to use of multiple modulators, and complicated compensation arrangement.
The third group is generally designated as RAMZI modulators [87][88][97][98][99], which has received increased interest in recent years. Some implementations are shown in Figure 5(c). The RAMZI uses ring resonator(s) (RRs) instead of the standard phase modulator (PM) which is coupled in the arm(s) of the MZI. This design gives higher SFDR performance but often at higher manufacturing complexity, limited RF bandwidth range, and stricter transmission coefficient control requirement.
The fourth and last group of optically linearized MZI-based modulators was introduced by our group [89, 93-94, 96, 101-105, 107-108]. This modulator design is a family of modulators which we referred to as IMPACC. There are numerous configurations and variations within the family of IMPACC designs [93]. Two of these implementations are shown in Figure 5(d).
One of these specific structures of IMPACC is discussed in more details in Section 3 and its performance in Section 4.

Design and modeling of IMPACC
This section discusses the basic principle, configuration, and modeling of IMPACC. In Section 3.1, we present an overview of IMPACC configuration. For both educational as well as clarity purposes, we identify the different elements of IMPACC and discuss them separately in Section 3.2. Lastly, we bring all these elements together to come out with IMPACC modulator. While this section gives the basic principle of IMPACC, Section 4 presents its performance.

Configuration of IMPACC
The configuration of our modulator, known as IMPACC, is shown in Figure 6. IMPACC comprises of a MZI configuration, in which one of its arms (e.g., lower) has an active microring resonator (MRR) modulator, whereas the other arm (upper) has an active phase modulator (PM). It uses the standard PM as the phase modulating component and the MRR as the cavity modulating component within a MZI structure [89, 93-94, 96, 101-108]. The exact positions of the PM and MRR relative to one another offer some variations of the implementation, as shown in Figure 5(d).
Interestingly, the IMPACC configuration can be seen as a generalization of both MZI modulator and RAMZI. First, the normal combination of MZI structure and active PM in one of the arm of MZI leads to a typical single-electrode MZ modulator. Second, the typical combination of a MZI structure and an active MRR modulator in one of its arms results in a single-electrode RAMZI modulator [88,[91][92]98]. When MRR and PM are present in MZI's arms, then this configuration becomes IMPACC.
Another important aspect of IMPACC is that the RF power signal is split between the MRR and PM by the RF power splitter (PS) with a power split ratio of F:(1ÀF). This is a critical aspect in the operation of the IMPACC where PM and MRR are driven by the same frequency RF signal, but with different RF power, controlled by the power split ratio (F) using a variable or fixed RF power splitter (PS). Furthermore, the portions of the RF signals directed to MRR and PM electrodes are also delayed by an amount ξ MRR and ξ PM , respectively. These external control parameters from PS are not only instrumental in providing superior linear performance but also have key roles in delivering the unique adaptive mode of IMPACC. This adaptive feature is not available with any known modulators, as far as we know.
The input and output optical splitters (OS and OC) in the MZI configurations are generally assumed to have a (50:50) split so that the optical signal is divided equally into two portions. However, in this chapter of the book, we generalize this condition by taking OS and OC to have arbitrary optical power split ratios namely, P OS :1ÀP OS and P OC :1ÀP OC . This condition allows us to investigate the effect of unbalanced power split ratio of OS and OC on the performance of IMPACC.

Phase modulator (PM)
The basic principle of phase modulators is based on the Pockels or linear electro-optic effect [73]. Depending on the material used, the specific axes orientations and the relative alignment of the incoming polarization to the index ellipsoid of the material used, the net effect is the refractive index modulation based on the external applied field. There are two typical configurations, the longitudinal electro-optic modulation and the transverse electrooptic modulation. In the first case, the electric field of the modulating field is parallel to the direction of the optical beam propagation (except near the edges of the electrodes). In the second case, the electric field of the modulating field is perpendicular to the direction of the optical beam propagation. This second case offers refractive index changes induced by the external electric field E that are proportional to the field E. Hence, the electrically induced phase change (or retardation) for the optical beam through the material is proportional to the product EL, where L is the length of the optical beam and electric field interaction and E=V/d is the applied field, with V the applied voltage at the electrodes, which have a separation of d.
The net effect is that the ratio L/d can be engineered large enough to provide devices which require low driving voltages.
Referring to Figure 7, an incident beam propagating along the y-direction, with polarization along the z-axis (typical case for LiNbO 3 based electro-optic devices), will experience an electrically induced phase change, ϕ pm of [73]: where r 33 is the electro-optic coefficients implicated in this crystal case and orientation [73], λ is the wavelength of the optical beam, n e is the extraordinary index of refraction seen by the optical beam along this polarization orientation, L is the length of the crystal (or the interaction length of the optical and electric fields), and d is the separation of the electrodes. Note that the total phase change consists of two terms. The first one (e.g., φ 0 ) is the natural propagation phase change, whereas, the second term (Δφ) is the electrically induced phase change. Because of the natural birefringence term, a phase compensator is required to adjust the phase until the total phase retardation in the absence of the electric field is an odd multiple of π/2. The half wave voltage for this case is [73]: If the applied voltage V is sinusoidal V = V m sin(ω m t), where ω m is the modulation frequency and V m is the applied magnitude of the voltage, then the transmitted beam will be phase modulated which can be expressed as [73]: where A is the constant amplitude, and the modulation index δ is given as [73] δ We should note here that the phase modulator in a photonic integrated waveguide circuit is slightly different as it has to be modified to accommodate a substrate structure and the electrodes. A typical, but not the only, configuration is shown in Figure 7(b) and (c). The optical waveguide is placed at the center of the gap between the two electrodes. In the optical waveguide, the direction of the field of the applied electric field is parallel to the substrate surface along the z-axis. The induced refractive index change Δn z changes the phase in the way similar to Eq. 5.

Mach-Zehnder modulator (MZM)
A typical external modulator is usually in the form of an integrated optical modulator, which includes a waveguide MZI structure that is fabricated on a slab of lithium niobate (LiNbO 3 ). To ensure that light is channeled through the waveguide, the area is typically doped with impurities in order to increase the index of refraction. Figure 8 shows a typical MZI modulator configuration where the incident light is split by a Y-branch optical splitrer (OS) into the two arms of the interferometer structure. The light traversing the first arm is modulated by a RF signal. As described in the previous section, the applied electric signal induces a change in the index of refraction of the first waveguide and hence the optical signal experiences a phase shift as it propagates through the waveguide. The magnitude of the phase delay is proportional to the applied voltage and is given by [14]: where V is the voltage applied to the modulator and V π is the half-wave voltage that when applied, causes a phase shift of π. The second portion of the splitted optical signal propagates into the second arm without RF modulation. Then, the two optical signals are recombined at the output by a Y-branch optical combiner (OC). This recombination can be either constructive or destructive depending on the relative phase difference of the two components of the interfering optical signals. Since the phase of one arm can be controlled by the applied electric signal, the output signal can be modulated. The transfer function of the MZI modulator is given by [73]: where θ is the bias phase.

Single microring resonator (MRR)
The basic formulation for the physics of microring resonators (MRR) has been extensively discussed [85][86]90]. We follow this analysis, in which an MRR has a total circumference length equal to d MRR is coupled to a bus waveguide as shown in Figure 9(a). The MRR round trip phase is given as θ MRR = ω·n 0 ·d MRR /c, where n 0 is the index of refraction of the material, α is the loss factor of the MRR waveguide, and c is the velocity of the light.
In Figure 9(a), the electric fields E 1 , E' 1 , E 2 , and E' 2 are related by the matrix equation which describes their interactions as [91]: where τ is the transmission coupling coefficient of the coupler between the bus waveguide and the MRR while κ is the associated cross coupling coefficient of the coupler. We assume that only a single mode of the MRR is excited, and the coupler is lossless, so that τ 2 + κ 2 = 1.
The output electric field E 2 , after one round trip, can be expressed as [91]: Here, α is the RR waveguide loss factor (α = 1 indicates that the RR is lossless).
After two roundtrips, we have [91]: Repeating this circulation around the MRR n times, output electric field E 2 leads to [91]: And finally, after infinite roundtrips, we have got the output electric field E 2 as [91]: Thus, its corresponding normalized transmission transfer function T MRR (θ) under steady-state condition is given by [91]: Accordingly, the corresponding phase response, Θ MRR and group-delay (GD) [95] response of MRR can be derived as [110,111]: Eqs. (14) and (16) will be handy in the analysis of the new modulator that we discuss in Section 4. The relationship between the magnitude of the bus waveguide-to-MRR transmission coupling coefficient (τ) and the RR waveguide loss factor (α) defines the three different coupling conditions of MRR [73,85,91,[110][111][112]. Figure 10(a) shows the typical MRR while Figure 10(b) depicts its three different operating coupling conditions known as (1) critical-coupled (CC) when α = τ, (2) over-coupled (OC) when α>τ, and (3) under-coupled (UC) when α<τ. Their respective intensity and phase responses are shown in Figures 10(b) and (c), respectively. One notable observation in Figure 10(b) is that the intensity output becomes zero at resonance frequency (normalized frequency Ω = 0) under CC condition. In comparison, the respective dip intensity outputs of OC MRR and UC MRR are always higher than zero. The second significant observation is found in Figure 10(c) where the phase response of UC MRR is "opposite in direction" compared with phase responses of both OC MRR and CC MRR. This is significant and is used in many group-delay/chromatic dispersion mitigation approaches. Recall that if waveguide lossfactor α = 1, then MRR is ideally lossless and becomes an all-pass filter or phase only filter.

(B). Dynamic operation
If a voltage V, is applied to an electro-optic (EO) MRR, a corresponding round-trip phase shift θ MRR induced in the ring resonator is then given as [73]: where θ 0 = ωn d MRR /c is the static MRR phase response, and The parameter Δβ is responsible for the dynamic phase change due to applied voltage. Here, r is the electro-optic coefficient of the waveguide, λ is the incident light wavelength, g is the electrode gap and Γ is the overlap integral.
For a sinusoidal voltage, V=V 0 sin ω m t applied to the MRR, then using the same approach that was used to derive Eq. (14), the output amplitude, E out (t) is given by [77] : where φ = ω m /FSR with ω m being the modulation frequency and FSR is the free spectral range of MRR, n is the number of round trip around the ring, and δ n is the modulation index. The modulation index depends on the electrode structure. For a lumped electrode, the modulation index is [91][92]: For a traveling wave electrode, the modulation index is given by [91][92]: Where t r = n 0 L/c is the optical round-trip time and ψ = ω m ·Δn·d MRR /c is the velocity matching factor. Here, Δn = n mn 0 , where n m is the microwave effective index and n 0 is the optical index. Figure 11(a) illustrates a RAMZI configuration where MRR is coupled to a MZI in one of its arms. We assumed that the OS and OC of the MZI have balanced (50:50) power split ratios. Then, the transfer function of RAMZI is given as [66-67, 91-92, 98]:

RAMZI
where α(θ MRR ) is the complex amplitude transfer function of the MRR under steady-state (SS) condition, which is given by Eq. (14) or (15), the phase response of MRR, Θ MRR = Arg[α(θ MRR )] is given by Eq. (17), and φ MZI is the DC phase bias between the two arms of the MZI. In Eq. (24), we assumed that the nominal MZI-arm lengths L 1 and L 2 are equal, except for the length due to DC phase bias. If the MRR is lossless (α = 1), τ ¼ 2 À ffiffi ffi 3 p and φ MZI = π/2, then RAMZI becomes linearized around θ = π, which is off-resonance for the MRR [66-67, 91-92, 98]. Figure 11(b) shows the typical linearized transfer function of RAMZI in comparison to standard MZI output intensity profile.
On the other hand, under dynamic state (DS) where a modulating sinusoidal RF voltage V is applied to MRR electrode, the corresponding transfer function of RAMZI is given as [91]: where α(θ MRR ) is given by Eq. (21), θ bias is the phase bias of MZI and [91] for a lumped electrode case. The dynamic performance of RAMZI will be shown in Section 4 together with IMPACC performance side-by-side for clear comparison.

IMPACC
As described briefly in Section 3.1, IMPACC is a generalization of standard MZI modulator and RAMZI. Figure 12(a) shows the IMPACC configuration (a) together with its linearized transfer function (b) under its steady-state condition. Here, we assume that both OS and OC have 50:50 optical power split ratios, although later we will relax this restriction and consider arbitrary optical power split ratios namely, P OS :(1ÀP OS ) and P OC :(1ÀP OC ).
Structurally speaking, IMPACC is very much like the RAMZI configuration except for two big differences. First, the passive path-difference, ΔL of the MZI in the RAMZI modulator is now replaced by an active phase modulator (PM) element. Second, the total RF signal driving the electrode in MRR for RAMZI is now split into two portions, by a ratio of F:(1ÀF), and drives both the electrodes of MRR and PM elements of IMPACC. The two split RF signals are also delayed by an amount ξ MRR and ξ PM , respectively. The external control of the RF power split ratio F:(1ÀF) adds a degree of freedom in the design and serves to drive both the PM and MRR under specific power ratios. The corresponding linearized transfer function of IMPACC under its steady-state condition is shown in Figure 12(b). Here, we assume that both OS and OC have 50:50 optical power split ratios although later in Section 3.3.1, we will relax this restriction by taking optical power split ratios namely, P OC :(1ÀP OC ) to have 50:50 split while OS has arbitrary split of P OS :(1ÀP OS ).

Transfer function
The dynamic transfer function of IMPACC, T DS IMPACC (θ) is given as [104]: where and The parameters p MRR and p PM are the input optical power split ratio coefficients of the input optical splitter (OS) for the MRR and the PM arms, respectively. Note that p MRR = 1 À P OS , p PM = P OS, where P OS was defined in Section 3.3. Also, here we assumed that OC has 50:50 power coupling ratio. The parameters θ pm and arg[a(θ MRR )] are the phase responses of the PM and MRR respectively, while the parameters d PM and d MRR are the lengths of the PM and MRR, respectively with n being the number of times the beam propagates inside the MRR, ω m is the modulation frequency, φ = ω m /FSR, ϕ B is MZI optical bias phase, ξ MRR and ξ PM , are the RF signal phase biases injected to the MRR and PM electrodes, respectively, and Δβ is given by Eq. (20). We assume negligible microwave loss and no velocity mismatch.
Unlike RAMZI, IMPACC has an external control of either the RF power split ratio or the RF bias. As we will show in the next section, while both of these modulator designs can result in high SFDR performance, IMPACC has an additional and unique property-an adaptive, inherent compensation capability that can be used to maintain its high SFDR under three nonideal parameter conditions namely: (1) increasing MRR waveguide loss factor, (2) unfavorable MRR coupling operating conditions, and (3) unbalanced input power split ratios of either the optical splitter, OS or optical combiner, OC. This adaptive property provides an additional level of design flexibility and can greatly simplify the overall operation of high-performing modulator.

Performance of IMPACC
This section provides details of the linearity performance of IMPACC in terms of SFDR, and compares it with RAMZI and standard MZI modulators. Unlike RAMZI and MZI modulators, IMPACC modulator has two different operational modes, namely: (i) the typical basic mode, and (ii) the unique adaptive mode. As far as we know, IMPACC is the only intensity modulator that has an adaptive mode. In Section 4.1, we focus on its basic operating mode, whereas in Section 4.2 we discuss the adaptive mode.

Basic operating mode of IMPACC
In the basic mode, IMPACC operates under fixed, predetermined parameter conditions aimed to optimize its performance. This mode of operation is typical in all intensity modulators. Here, we investigate the effects of the different parameter variations (from these optimum conditions) on IMPACC's SFDR performance, as well as its bandwidth capability, to assess its performance limits. These parameters include (i) MRR loss factor α, (ii) τ and α that define the CC, UC, and OC conditions, and (iii) unbalanced power split ratio (e.g., P os :(1ÀP os )) of optical splitters (OS). We also compare IMPACC performance with RAMZI and standard MZI modulators.

Effect of ring resonator's loss on SFDR
Any MRR waveguide has an intrinsic loss, (expressed as loss factor, α in Section 3). Here, we assess its effect on both IMPACC and RAMZI's SFDR performance [104]. Figure 13(a) shows the respective SFDRs of both IMPACC and RAMZI under two conditions; (a) an ideal lossless MRR waveguide (e.g., α = 1.0), and (b) lossy case (e.g., α = 0.8). Note that parameter τ is chosen for both IMPACC and RAMZI to get their respective highest SFDR values. We choose to use τ = 0.6355 for IMPACC and the ideal value of τ = 0.2679 for RAMZI. Furthermore, the modulation frequency, ω m is set to 1 Hz, while the power split ratios of OS and OC are assumed both balanced (50:50).
Here, we see that IMPACC has an SFDR value of 132.9 dB under lossless condition (labeled as A) that reduces drastically to 113.3 dB (labeled as C) when loss is introduced. A similar pattern is observed for the RAMZI case, where its SFDR value deteriorates from 129.4 dB for lossless case (labeled as B) to 109.0 dB for lossy case (labeled as D). Figure 13(b) summarizes the full effect of MRR loss factor α on the SFDR of both IMPACC and RAMZI when the value of α is allowed to vary from 1 to 0.8. We have the following three observations. First, the respective SFDR performances of IMPACC and RAMZI are affected drastically and negatively by the decreasing value of the MRR loss. It is clear from Figure 13(b) that the MRR loss value must be minimized within a < 0.99 to obtain the higher than 126 dB-Hz SFDR performance. Second, IMPACC always outperforms RAMZI. It exhibits a consistently higher SFDR value (e.g., more than 1 dB) compared than RAMZI for the range a = 1 to a = 0.85. Third, at the range of a = 0.85 -0.80, the SFDR performance of RAMZI degrades more than three times compared with IMPACC. The comparative SDFR difference between IMPACC and RAMZI increases from 1.2 dB when a = 0.85 -4.3 dB with a = 0.8. This implies that IMPACC has a higher parameter tolerance to MRR loss variation especially at higher loss condition.

Effect of OC, UC, and CC operating conditions on SFDR
The SFDR performance of any MRR-based devices, including IMPACC and RAMZI, depends highly on the relative ratio between the MRR coupling coefficient strength (τ) and intrinsic loss factor (a) As mentioned in Section 3, there are three MRR regions of operation namely; (i) overcoupling (OC) condition, when a > τ, (ii) critical coupling (CC) when a = τ, and (iii) undercoupling (UC) when a < τ.
Here, we selected the three regions with the following parameters; (i) OC condition (τ = 0.6364, α = 0.7364), (ii) CC condition (τ = 0.6364 and α = 0.6364), and (iii) UC condition (τ = 0.6364 and α = 0.5364). Note again that we choose τ = 0.6364 for IMPACC and the ideal value of τ = 0.2679 for RAMZI. Furthermore in order to simplify the analysis and gain better understanding of the different parameter effects, we assumed an ideal, balanced 50/50 power split ratio for both the input optical splitters (OS) and output optical combiner (OC). The effect on nonideal 50:50 split ratio of OS will be discussed in the next section. Moreover, we assumed that IMPACC and RAMZI are driven at one of its resonant modulation frequency of 23 GHz.
Briefly speaking, the SFDR performance of both IMPACC and RAMZI under OC condition is shown in Figure 14(a). IMPACC has superior linearity with SFDR =130 dB which is far better than RAMZI (e.g., 110 dB). This superior SFDR value is maintained (e.g., 129.7 dB) under CC condition as shown in Figure 14(b). Lastly, the SFDR performance of both IMPACC and RAMZI under UC condition is also shown in Figure 14(c). Under UC condition, both IMPACC and RAMZI are no better than the typical nonlinearized MZI modulator having SFDR value in the range of 110-113 dB. However, we can state that IMPACC's SFDR value (e.g., 113 dB) is still 3 dB higher when compared with RAMZI (e.g., 110 dB). Thus, the takeaway from Figure 14 is that the huge SFDR advantage of IMPACC to RAMZI exists only when MRR is operated either in OC or CC condition.

Effect of unbalanced optical coupler on SFDR
This section studies the effect on nonideal, unbalanced split ratios of the optical splitter (OS) or optical combiner (OC) on the SFDR performance of IMPACC for three different MRR operating conditions (OC, UC, CC) [110]. To simplify the analysis, we limit our study to unbalanced OS by maintaining OC to have 50:50 power split ratio. Note that we use the same parameters as found in Section 4.1.2 with τ = 0.6364 for IMPACC and the ideal value of τ = 0.2679 for RAMZI.  various split ratios. It is plotted as a function of the offset from the ideal optical splitter balanced condition (0 deviation corresponds to an ideal power split ratio of 50:50) to a very unbalanced condition (e.g., 0.20 corresponds to power split ratio of 70:30 and À0.20 corresponds to power split ratio of 30:70). One conclusion from Figure 15(b) is that in order to maintain a SFDR value of above 124 dB-Hz, we need to ensure that the OS power split ratio deviation should be no greater than AE 0.08 or no more than 58:42 or 42:58 power split ratio. Note that the results of Figure 15(b) were obtained without changing any external IMPACC parameters (e.g., RF power split ratio-F or the RF phase bias-ξ PM ) except the coupling ratio OS. Figure 16 shows IMPACC's SFDR performance under (a) CC and (b) UC conditions for balanced (i.e., 50:50) and unbalanced (e.g., 55:45) cases. We see that the SFDR value of the OC condition degrades by 11 dB from balanced case of 129 dB-Hz to 118 dB-Hz for unbalanced case. Compared with the OC case ( Figure 15a) where the SFDR drops only by 6 dB, the CC condition implies higher sensitivity to OS split ratio deviation. On the other hand, Figure 16(b) illustrates that OS split ratio deviation has no effect in the balanced or unbalanced cases for UC condition. However, IMPACC has only low SFDR value of 113dB-Hz.

SFDR versus modulation bandwidth response of IMPACC
We know that the modulation bandwidth of all ring resonator based modulators, such as RAMZI and IMPACC is typically limited by the MRR's free-spectral range (FSR) [91,102,[104][105]. Here, we describe the SFDR-vs-modulation bandwidth capability of both IMPACC and RAMZI, and assess their respective performance limits. We evaluate the SFDR-vs-modulation bandwidth under lossless case (A), and under the three MRR operating conditions (B).

MRR lossless case (OC Condition)
Here, we assume that the loss factor a in the MRR waveguide, for both the IMPACC and the RAMZI, is lossless or equal to 1. Although, this is not a realistic case, it is done to focus on the bandwidth capability and understand its performance limits. Later in the next subsection, we consider the effect of MRR loss. Here, we use τ = 0.208 for IMPACC, τ = 0.268 (or τ = 00.2679) for RAMZI [90-91, 95, 104], MRR RF phase bias ξ MRR = 0 , and a PM RF bias control sensitivity ξ PM of 0.2 [105]. The two τ values have been selected such that they provide maximum SFDR value for each case.
First, Figure 17(a) shows the SFDR-vs-Modulation frequency of IMPACC (blue line), RAMZI (red line), and MZI modulator (green line) for modulation frequencies up to 75 GHz. IMPACC has an increasingly higher SFDR, (e.g., 3.7 dB to 13.7 dB) compared with RAMZI and standard MZI modulators for these increasing resonant frequencies. It is important to note that the resonance enhancement in IMPACC is accomplished without resorting to a smaller ring size. Smaller ring size not only makes the fabrication more challenging, but would also introduce larger resonator waveguide losses.
Next, we identify two frequency regions of operation: (i) the resonance region at around~23 GHz (shown in Figure 17(b) and~70GHz (shown in Figure 17(c)), and (ii) the non-resonance region at around 56 GHz (shown in Figure 17(d)) [105]. The resonance region is defined as the region where the SFDR is greater than that obtained from an ideal MZI, which has a relative flat response as a function of frequency. These resonant regions occur at multiples of the FSR of the modulator (e.g., 23.3 GHz, 69.9 GHz).
As shown in Figure 17(a), IMPACC provides an increasingly higher SFDR, (e.g., 3.7 dB to 13.7 dB) compared with RAMZI and MZI at these resonant frequency regions. Figures 17(b) and (c) depict the respective IMPACC's SFDR performance when the central RF modulation frequencies are set to 23 GHz and 69.9 GHz, respectively. We note that the modulation linewidth around the central frequency 23 GHz is wider compared with case of central frequency of 69.9 GHz. On the other hand, in the non-resonance region, (defined as the region between the resonance regions), the SFDR is typically less than that obtained from the ideal MZI as shown in Figure 17(d). However, the IMPACC still shows an improved SFDR performance by 0.5 to 2.0 dB compared to RAMZI. Figure 18 depicts the SFDR performance of both IMPACC and RAMZI under over-coupled (OC), critical-coupled (CC), and under-coupled (UC) conditions. Figure 18(a) shows the OC IMPACC (τ = 0.6364, α = 0.7364) and OC RAMZI (τ = 0.2679, α = 1) as a function of the RF modulation frequency. As we have described earlier, in the regions close to the resonance peak (e.g.,~23 GHz), IMPACC clearly outperforms RAMZI, which operates under ideal condition by 10-20 dB.

MRR lossy cases (OC, CC and UC Conditions)
The case of CC IMPACC is shown in Figure 18(b). IMPACC also outperforms RAMZI within or in the resonant peak regions. Outside the resonant regions, both IMPACC and RAMZI have nearly the same performance. Lastly, Figure 18(c) shows IMPACC performance versus RAMZI in the case of UC condition at the same resonance region as above (e.g., modulation frequencỹ 23 GHz). In regions close to the resonance peak, the IMPACC still outperforms the ideal overcoupled RAMZI, but now the maximum difference is much smaller (e.g.,~4 dB). We note that in all these situations, we did not optimize the performance of IMPACC by changing its external parameters (e.g., F or ξ PM ).

Adaptive operating mode of IMPACC
IMPACC's adaptive operating mode is all about using dynamically the two built-in, adjustable, and externally controllable two parameters to maintain IMPACC's 130 dB SFDR performance under unfavorable parameter conditions such as: (i) intrinsic RR loss, (ii) parameter variation of τ value, and (iii) unbalanced OS power split ratio. The unavoidable parameter variations can be due to (a) environmental temperature change, (b) device aging, and (c) fabrication tolerance and errors. The externally controllable parameters used for mitigation purposes are the RF power split ratio F, and the RF phase shift, ξ PM . Here, we set the other RF phase shift, ξ MRR equal to 0, for simplicity. Using again these two external parameters, we show that IMPACC can also extend its SFDR-vs-modulation bandwidth capability beyond that of RAMZI [91,[106][107][108]110]. Overall, the adaptive mode of IMPACC allows the link design to be flexible with excellent results.

Extending parameter tolerance to ring resonator loss
The ability to adjust the control parameters F and ξ PM can be exploited to compensate for the negative effect of loss factorα of MRR. In Section 4.1.1, we showed that the SFDR performance of both IMPACC and RAMZI can deteriorate by as much as~20 dB when the RR loss factor (e.g., α< 1) increases. This effect is reproduced in Figure 19(a) for IMPACC and RAMZI with fixed F values. Figure 19(a) also shows the SFDR performance for IMPACC with dynamic or adaptive F parameter value. The two cases with fixed F parameter value are similar to the one shown in Figure 13(b) and discussed in Section 4.1.1.
The result of this adaptive mode is to maintain the SFDR value to higher than 130 dB by dynamically changing the required F value as a function of the MRR loss factorα as shown in Figure 19(b). It shows the range of required changes in the value of F to achieve the SFDR performance shown in Figure 19(a). This unique ability of IMPACC overcomes the negative effect of the intrinsic insertion loss due to MRR. This feature is not feasible in RAMZI or any In this section, we highlight IMPACC's ability to compensate the negative effect due to unbalanced optical splitter performance [110,113]. This is emphasized in Figure 20(a) where the original drop of 6 dB in the SFDR performance, when the optical splitter is nonideal (55:45), can be compensated to return to SFDR value equal to 129 dB by adjusting the external parameters, F and ξ PM to 0.2003 and 1.307π, respectively.  Hence, clearly, IMPACC's inherent compensation capability is quite effective and able to overcome this limitation. Figure 20(b) depicts the SFDR performance of IMPACC at a power balanced for the case of CC condition (i.e., 50:50), unbalanced condition (e.g., 55:45) without compensation and with compensation for the case of CC condition. In the unbalanced condition, the SFDR drops 11 dB from 129 to 118 dB. Nevertheless, IMPACC's inherent compensation capability enables one to increase the SFDR back up to 129 dB for the unbalanced (55:45) case under CC by adjusting the F and ξ PM parameters.

Under-coupled (UC) IMPACC
In UC condition, the inherent compensation capability of IMPACC is severely limited and cannot increase the SFDR of IMPACC for the unbalanced case. Adjusting the external parameters,F or ξ PM do not improve the SFDR value. Similar to previous Section 4.1.3, we observed that the IMPACC SFDR performance (e.g., 111 dB) is still higher than that of RAMZI (e.g., 108 dB) but the difference is noticeably smaller compared with the over-and critical-coupled conditions with SFDR of around 129-130 dB.

Extending SFDR versus modulation bandwidth response
As discussed earlier in Section 4.1.3, all ring resonator based modulators (e.g., RAMZI and IMPACC) are limited by the FSR of the ring resonator [88, 91-92, 94,102,104-108]. This is graphically represented by the narrow Lorentzian shape of its SDFR profile as depicted in Figures 17 and 18. This limitation is troublesome, especially for wideband RF link applications. Fortunately, the IMPACC has significant advantage compared with RAMZI and other linear modulators that relaxes this built-in RF modulation bandwidth limitation due to MRR.
IMPACC can broaden the SFDR profile at its peak value by dynamically adjusting both F and ξ PM for any given modulation frequency band. Figure 21 shows this remarkable feature for a frequency of operation from 16 GHz to 20 GHz, where the IMPACC can maintain a SFDR of >130 dB. For the results presented in Figure 21, we have used λ = 1550 nm and τ = 0.208, and both the RF phase, ξ PM and power split, F parameters were changed in other to maintain the high SFDR. We are not aware of any MRR-based modulator designs that can accomplish such a dynamically adjustable wide frequency response using externally controlled parameter.

Conclusion
In summary, electro-optic modulators are critical part of optical communications. They will continue to play an important role in fiber optic links. However, an essential requirement is that they have linear response with a high dynamic range of input RF powers of multiple tones. We have shown that IMPACC can operate with highly linear response with peak SFDR value of 132 dB.
Another important requirement in the link design is the inherent versatility of the device or link to compensate for unexpected changes or unavoidable parameter variations due to environmental temperature change, device aging, and fabrication errors. In this regard, IMPACC has a unique place among previously reported linear intensity modulators since it is the only linear modulator design, as far as we know, that has an inherent compensation capability. We demonstrated this inherent feature by compensating various nonideal and often detrimental effects in the modulator from RR waveguide loss and unbalanced optical inputs to various coupling conditions that could lower manufacturing tolerance and degrade linearity performance if not mitigated. The ability to maintain high SFDR (e.g., >130 dB) under these conditions makes IMPACC a viable candidate for many high-bandwidth RF FO-link applications, and well positioned as ideal linear intensity modulator for software-defined MPLs