Abstract
Neuromorphic photonic processors based on resonator weight banks are an emerging candidate technology for enabling modern artificial intelligence (AI) in high speed analog systems. These purposebuilt analog devices implement vector multiplications with the physics of resonator devices, offering efficiency, latency, and throughput advantages over equivalent electronic circuits. Along with these advantages, however, often come the difficult challenges of compensation for fabrication variations and environmental disturbances. In this paper, we review sources of variation and disturbances from our experiments, as well as mathematically define quantities that model them. Then, we introduce how the physics of resonators can be exploited to weight and sum multiwavelength signals. Finally, we outline automated design and control methodologies necessary to create practical, manufacturable, and high accuracy/precision resonator weight banks that can withstand operating conditions in the field. This represents a road map for unlocking the potential of resonator weight banks in practical deployment scenarios.
1 Introduction
Most of the recent successes in artificial intelligence (AI) are tied to the ability to perform an increasingly large amount of computation on a mathematical representation of information (i.e., data). In machine learning (ML) algorithms, data is first rearranged into mathematical objects called vectors living inside of a vector space. These vectors can then be manipulated by linear and nonlinear operations resulting in another vector that can be translated into a useful result. For example, in an image classification task the input image is typically encoded into a vector containing the contents of three 2dimensional arrays of numbers representing the intensity values of red, green, and blue for each pixel. A classifier then computes a function that maps the image vector into a classification vector, which is a 1dimensional array with each component representing a humaninterpretable label (e.g., cat, dog). In modern AI algorithms, especially deep neural networks, most of the hardware processing power in conventional processors is devoted to memory access for parallel linear operations on vectors [1], resulting in a limitation as models scaled up in size. Accordingly, specialized hardware architectures that efficiently perform linear operations have been introduced to accommodate the increasing demand for AI. These include GPUs [2], TPU [3] and neuromorphic processors [4, 5].
Inmemory computing and multiprocessing techniques are two innovations that have allowed specialized hardware such as multicore CPUs and GPUs to address the increasing demand for data processing. These techniques, however, were not a significant enough leap forward to close the gap between current computing capabilities and the needs of evolving AI tasks [4, 6], especially for highfrequency computation with low latency. Braininspired, or neuromorphic, computing schemes have emerged as a potential alternative for addressing these deficiencies in power efficiency and speed. Neuromorphic processor architectures are optimized on a hardware level to run neuralnetworkbased algorithms as fast and efficiently as possible [7]. Such architectures derive these benefits from the use of analog systems in which the computation is inherently tied to the physics of the device itself, rather than defined in software on a generalized digital architecture. Analog systems, however, must be engineered to control noise in such a way that the signals of interest are not corrupted by the processing itself, a task which is nearly trivial in digital electronics. It is also crucial to design systems which are usertransparent and are capable of interfacing with existing computational systems.
Most optical phenomena are linear, making multiplication and addition operations easy to implement using photonic devices. Photonic systems can, therefore, benefit from the body of knowledge in the field of ML, which has over the decades created algorithms that are heavily dominated by linear computation. These are directly utilized in photonic processors that operate using analog inputs and outputs. As an added benefit, information in different channels can be encoded onto noninteracting optical carriers using wavelengthdivision multiplexing (WDM) allowing for a high density of signals on a single waveguide. Utilizing WDM in turn allows for the use of resonator devices that produce controllable effects only around their specific resonant wavelength and do not interact with other optical carriers.
In neuromorphic photonics [8], the device responsible for linear computation is the weight bank, composed of a series of resonators with unique resonant wavelengths, which selectively weights (multiply) a series of incoming WDM signals [9] and optically sums all of the resulting light. Various implementations of resonator weights have been proposed, demonstrated, or leveraged by different groups over the last few years [9–18]. These are optimized specifically for realtime weighted summation of highbandwidth analog signals encoded onto multiwavelength lightwaves. Summation using photodetectors returns an analog electrical signal, which can either be read out directly or used to modulate another optical carrier. This multivariate processing ability has proven to be particularly useful in applications with analog inputs and outputs, such as microwave signal processing, ultrafast robotic control [19], and neuromorphic computing [20, 21]. It has also found uses in hardware accelerators for machine learning applications, even with the overhead of converting from digital to analog inputs and outputs [11], [12], [13], [14, 17].
One of the main engineering challenges facing weight banks is the ability to precisely and accurately control weight values, which is made difficult due to fabrication variations and environmental disturbances. In this paper, we aim to outline automated methodologies for designing and controlling resonator weights in order to enable practical WDM photonic processors. In Sections 2 and 3, we introduce the necessary background for understanding and engineering resonator weight banks. Next, in Section 4, we introduce a practical model of weight banks and a feedforward algorithm that can be used for calibration and control of real devices. Finally, in Section 5, we discuss options for designs that allow for compensation of fabrication variations and environmental disturbances as well as a feedback algorithm that can be used if a weight bank is manufactured with builtin weight sensing. The aim is to allow readers to design systems utilizing resonator weight banks that are robust enough to be brought beyond laboratory benches and into the field.
2 Weight bank overview
2.1 Matrix–vector multiplication
Tensor operations, which are the core of many machine learning algorithms, can be implemented via a collection of matrix–vector multiplications. Matrix–vector multiplications, in turn, are typically decomposed into a number of vector–vector dot products where the first vector corresponds to the rows of the matrix and the second vector (the vector of the matrix–vector operation) is identical for all such suboperations. Hardware units for performing matrix–vector units can be modularized in the same way, with the most basic units implementing vector–vector dot products. Vector–vector dot products can be written mathematically as
where
Since this operation is a collection of multiplication operations with a summation, it is common to refer to the performance of a system in terms of multiplyaccumulate (MAC) operations [22]. The number of MAC operations for a particular computation is hardwareagnostic, although some architectures, such as systolic arrays [3, 23], implement MACs directly while others, such as the WDM architecture analyzed in this paper, implement weights as individual units and perform all summation simultaneously.
Many applications involve multiplying a batch of vectors with the same matrix, such as running inference tasks with input data on a fixed (pretrained) neural network classifier. This approach optimizes inference tasks for high bandwidth signals and low latency results. In such cases, it can be advantageous to design analog MAC units where input signals x(t) can be modulated on fast time scales and are multiplied by “fixed” weights w that only change on relativelyslow time scales. In general, reconfiguring optoelectronic weights takes significantly longer than their optical signal bandwidth capacity. Optical processors thus need to be codesigned at a system level with an electronic circuitry that optimizes the decomposition of matrix–vector multiplications into smaller vector–vector products while reducing the number of redundant computations [24].
2.2 Role of individual resonators
Resonatorbased weights are specifically designed to take advantage of wavelengthdivision multiplexing (WDM), in which separate signals are encoded on optical carriers with nonoverlapping wavelengths. In optical communications, multigigahertz signals are modulated as amplitude or phase changes on a continuouswave (CW) carrier traveling through a bus waveguide. With WDM, hundreds of CW carriers with unique wavelengths can coexist in a single waveguide without interfering with one another, effectively creating many independent information channels within a single physical channel. An array of resonators, usually microring resonators, with resonances matching the optical carrier wavelengths is used to access each channel independently. Each resonator, when in resonance, transfers all of the optical energy traveling on one waveguide onto another waveguide. This property is utilized, for example, to enable compact optical interconnections on photonic chips for use in telecommunication systems [25]. If the resonances can be independently tuned, the amount of energy transferred between waveguides can be controlled for each channel, resulting in a precise weighting (multiplication) of the input WDM signal. Using this scheme for implementing matrix–vector multiplications is depicted in Figure 1.
Silicon photonics has emerged as a popular platform for creating photonic integrated circuits (PICs), owing to low loss and compatibility with commercial foundries [26]. Some process design kits (PDKs) are being generated [27, 28], but optical devices are very sensitive to temperature and fabrication variations. This poses unique challenges to the design of PICs on top of those usually associated with analog devices. In the case of microring resonator filters, this sensitivity can be counteracted via calibration and control techniques [24, 29]. Early demonstrations used integrated sensors to coarsely lock microring resonators on the transmission wavelength [30]. These involved applying a compensating electronic signal (either current or voltage) based on a measured property, mathematical state model, or both. This can be achieved easily with microcontroller circuits placed in a CMOS electronic circuit nearby or monolithically integrated on the same die [31]. Resonator weights, on the other hand, require more sophisticated control schemes, because the resonator not only needs to be stabilized near resonance, but also need to be finetuned with a high degree of accuracy.
3 Resonator weight physics
3.1 Resonator configurations
There are two common configurations for passive resonator weights: allpass and add–drop, consistent with the terminology used in optical filters. In the allpass configuration, there is only one waveguide bus coupled to the cavity, with one input port and one ‘thru’ port at the opposite end. On the other hand, in the add–drop configuration, an additional bus waveguide is coupled to the cavity, with one ‘add’ port and ‘drop’ opposite to add. In each configuration, the ports are defined relative to the input port. The ‘thru’ port receives most of the energy from the input port, except when the cavity is near resonance. Near resonance, a significant amount of the input energy enters the cavity and gets transferred to the ‘drop’ port. For passive, lossless cavities, all energy from the input port is divided between the ‘thru’ and ‘drop’ ports, in a ratio dependent on the input’s wavelength and the coupling between the bus waveguides and the cavity. In theory, the goal is to engineer the resonator so that when on resonance, 100% of the energy drops to the drop port, and when offresonance, 100% of the energy is transmitted to the ‘thru’ port. In a lossless resonator, this can be simply achieved by designing identical couplers between the allpass and add–drop waveguides, a condition called critical coupling.
In practice, cavities are not perfectly lossless. The main loss mechanisms in silicon photonics are scattering loss (due to roughness of the waveguide’s walls) and material absorption. This loss will perturb the critical coupling condition, causing the power transfer to be unbalanced and resulting in reduced extinction ratio [[32], Sec. 2]. This unbalance can be compensated by design if, and only if, the cavity loss and the coupling coefficients are well known prior to fabrication. These parameters are defined by fabrication and cannot be changed. In a completely passive allpass microring resonator, for example, this loss is so low that engineering the critical coupling condition is not possible due to fabrication variation.
We can use a simple technique to achieve nearcritical coupling, and therefore acceptable extinction ratios, by using a symmetric add–drop bus waveguide, even if we ignore its ports. Because the two couplers are spatially close to one another, the fabrication variation tends to be correlated, ensuring their coupling coefficients are similar. This technique only works if the coupling coefficient is much higher than the cavity loss. In this situation, the intrinsic loss of the microring, dominated by the coupling coefficient to the drop waveguide, automatically matches the coupling coefficient of the input waveguide, resulting in a critical coupling condition. However, as we show in Section 5, sometimes we intentionally modify the cavity’s waveguide via doping or evanescent coupling, increasing the cavity loss to a regime that warrants asymmetric coupling ratios.
The most common example of a photonic resonator is a microring resonator (refer Figure 2). MRRs are standard components in integrated photonics used for filtering, modulation given their simplistic design methodology. They are ring waveguides where the optical path length, i.e., ring circumference, determines the resonant wavelength. By varying the radius, a WDM link with MRRs with various resonances can be designed. Besides the resonance wavelength, the ring radius also affects the freespectral range (FSR) of the cavity, which is the spacing between adjacent resonator modes. FSR ultimately is the limiting factor to the channel count of the WDM link. Sophisticated measures have been employed to avoid this FSR limit, including gratingembedded MRR [33], etc. However, these schemes add to the design complexity and insertion loss. Another way to circumvent this issue would be using photonic crystal (PhC) cavities. Given the relatively strict resonance constraints of PhCs, resonance modes are fewer and can even be engineered to have a single resonance [34]. Eliminating FSR offers advantages in terms of channel count, which may be key for largescale networks. Nevertheless, the fabrication tolerances of PhC’s require more expensive fabrication techniques such as Ebeam lithography which poses a question to its scalable manufacturability.
This article was written with microring resonator cavities in mind, but all the discussions and modeling can be applied to any kind of cavity. We have designed a tool to help engineer MRRs based on known fabrication parameters and waveguide designs [35] which is also detailed in Section 7. Table 1 presents example microring resonator designs useful for weighting, and compares their predicted properties to measured properties. With no fit to experiment, the ring properties can be estimated.
Device  Geometry  Simulated (measured) parameters  Good for  

Type  Width (nm)  Radius (μm)  Gap (nm)  Tuning element  Thru E.R. (dB)  FWHM (nm)  Tuning efficiency  
Weight  500  8  200  ndoped heater  22 (25)  0.4 (0.4)  (0.2) nm/mW  Weighting 
Modulator  500  8  300  pn junction  13 (15)  0.1 (0.03)  (0.01) nm/V  Highspeed tuning 

Waveguide widths are singlevalued across bus and ring waveguide. Doping levels are ∼10^{17} cm^{−3} and overlapped with the central portion of the rings away from the couplers, leading to a propagation loss of roughly 5 dB/cm [36]. Parameters were simulated as described in the Appendix without experimental calibration. We compare them to parameters extracted from measurements taken from fabricated rings.
3.2 Context
Signals and Weights: In an analog matrix–vector multiplication, there are two kinds of vectors in a weight bank: the fast vector (x), which typically represents a signal; and the slow vector, or weight vector (w), which is understood to be much slower than or stationary compared to x. In digital hardware, this distinction might seem strange because both vectors are typically loaded from memory at the same rate. In analog hardware, however, it is much more energy efficient to use a slow signal to modulate the amplitude of another signal in a passive way. In photonics, for example, a voltage or currentcontrolled device can modulate the amplitude or phase of an optical signal, independent of its bandwidth. This is the first caveat for comparing performance between photonic matrix–vector multipliers and electronic ones, especially in terms of energy consumption or speed; in analog computing only one operand can be switched quickly, whereas in digital electronics both weight or signal are functionally equivalent.
One dimension in which optical weights offer a unique advantage is in the latency of the computation. Many applications, such as machine learning, signal processing, and control, require very high bandwidth input signals and low latency but can tolerate slowlyvarying weights. For these applications, implementing the MVM in the optical domain is ideal for two main reasons. First, passive optical weights support high bandwidth optical signals with constant (sometimes zero) power dissipation. Second, the latency between input and output is determined by the timeofflight of the lightwave carrier, or nanoseconds considering resonator and optoelectronic RClimited bandwidths [37].
Applications such as photonic accelerators, where the computation is offloaded from the CPU to a photonic tensor processor, would not benefit from this scheme since weights would need to be updated as fast as the incoming signals. Doing so imposes a significant energy and speed cost to the control hardware, and is challenging to do in a way that maintains the highbandwidth advantage of photonics. However, recent works have been able to marry the highly parallel efficient linearity for nonresonant weight elements, and custom digital electronic ASIC for accelerating deep neural networks [38]. To date, this approach has been favored by industryoriented startups.
Analog Summation: At the output of the weight bank there is a summing element, most commonly implemented with a photodetector or pair of balanced photodetectors (cf. Figure 1). Photodetectors generate a photocurrent proportional to the incident optical power. To first order, they are linear devices and a WDM system acts as a multilinear map, meaning that a linear change in each channel results in a linear change in the output. Realistic photodetectors, however, have nonlinearities at high optical powers, resulting in a saturation curve. In practice, this limits the optical power that can be used in a weight bank. They also typically have nonflat frequency response, so optical signals modulated at different frequencies but with same power generate different levels of photocurrent. Any nonlinear imperfection in the photodetector will contribute to precision and accuracy ‘errors’ in the weighting scheme.
Analog Subtraction: It is often desirable to implement negative weights (in addition to positive weights) in order to allow for the possibility of analog subtraction. Doing so in a WDM system requires the use of add–drop resonators along with balanced detectors, as depicted in Figure 1. In such a case, a channel’s weight value is zero when energy is evenly split between “thru” and “drop” ports.
Mach–Zehnder interferometer (MZI) Approaches: An alternative to WDMbased integrated photonic computational techniques are techniques which utilize Mach–Zehnder interferometers (MZIs) to perform arbitrary linear operations on a singlewavelength optical carrier [39]. Individual phase shifters found in MZIs are typically less sensitive than individual resonators, but process variations compound across large manyMZI PICs in a way that is not usually seen in large manyresonator PICs. Ultimately, compensating calibration and control techniques are required for practical deployment in either case. A discussion of the most common options for MZIbased PICs can be found in [39].
3.3 Definitions
Normalized Weights: For simplicity’s sake, we introduce a normalized weight
where w _{max} and w _{min} are the maximum and minimum weight values, respectively.
Bit Representation: Before defining error quantities such as accuracy, precision and resolution, sometimes it is useful to refer to them in units of ‘bits’. For example, if the relative error of a measurement is 0.125, or 12.5%, we can also refer to it as 3 bits, because it takes three digits to represent that number in binary representation. More generally, if the error is ɛ ∈ (0, 1], the bit representation can be computed as ɛ (bits) = log_{2}(1/ɛ) ∈ [0, ∞). For example, as the measurement error goes to 0, we say that it has infinite precision.
Accuracy: In the context of resonator weight banks, accuracy refers to the systematic error between the commanded weight (denoted
where
Precision: Precision is complementary to accuracy, and refers to random error caused by noise in the measurement system or control circuit. This is a challenge in any analog system. Here, we again make the distinction between individual precision and ensemble precision, and in general refer to ensemble precision whenever it is not specified. In this case, we chose to construct the ensemble precision so that it includes a contribution from individual accuracies in order to more closely reflect the user’s experience. This is intuitively understandable by examining each limiting case. On one extreme, if all individual accuracies are infinite (zero error) then random spread of the ensemble is just the average of individual precisions. On the other extreme, if all individual precisions are infinite (no random spread) then one still expects to see spread in commanded weight values over the entire ensemble due to finite accuracy. Here, we associate ensemble precision to an error bound users would expect from a randomly chosen weight vector. Mathematically, individual and ensemble precision are defined as:
with the same notation as in the accuracy definition.
Resolution: Resolution ordinarily relates to the discretization error of the signals x _{ i }(t) or y(t), which is not to be confused with weight accuracy and precision. For analog continuous signals, the equivalent terminology is signaltonoise ratio, or SNR. In this article, we use resolution and SNR interchangeably, often expressed in bits. Because of Eq. (1), the finite precision error of w causes a degradation in the resolution of y(t). In practice, weight banks are designed such that the resolution is lower than the ensemble precision (e.g., 5bit precision for a 4bit resolution), otherwise ensemble precision will determine the resolution of y(t).
Actuation: Actuation refers to the physical process of changing a weight. One common actuation mechanism is the application of current to a microheater near a resonator. Applied current generates heat, which changes the temperature around the resonator and as a result changes the weight. The actuation mechanism often sets an upper limit on achievable accuracy. For example, consider a resonator controlled by a typical 16bit DAC within a 0–20 mA range. If the resonator needs 2 mA of current to lock to resonance and 1.5 mA to change the effective weight between −1 and 1, a single LSB of the DAC corresponds to 0.3 µA and −log_{2}(0.3 µA/1.5 mA) = 12.3 bits is the maximum achievable accuracy. In practice, DAC nonlinearities, parasitic resistances, and onchip thermal crosstalk tend to decrease system accuracy below this upper limit.
Sensing: Sensing is the ability to directly measure each effective weight independently from the actuation mechanism. This plays an important role in simplifying the calibration and control of the weight bank as well as improving accuracy and reconfiguration speed. The ideal sensor would be a spectral sensor that would detect the transmission of each resonator as a function of wavelength and map it to a voltage with high accuracy (0–10 V would be compatible with popular DAC/ADC components currently available from chip manufacturers). More practical sensors measure indirect quantities correlated with the weight, such as local temperature or circulating optical intensity. In this case, the relationship must be incorporated into the calibration and control.
Trimming: Trimming refers to the process of adjusting a physical property of a device to a desired value with high accuracy. This is necessary because common microfabrication processes based on photolithography or ebeam lithography are fundamentally incapable of producing predictable, identical resonator cavities (thus resonance wavelengths) by design. For example, a typical requirement in EDM telecomm systems is a fixed operating wavelength; say 1545.32 nm (ITU Cband channel 40). A resonator designed to operate on this channel needs a resonance wavelength accuracy of 0.01 nm, which is currently unattainable by geometric design alone. There are two main ways of achieving trimming postfabrication. Active trimming involves measuring each resonator’s wavelength in advance and using a control circuit to apply a voltage or current bias that compensates for fabrication variations. This approach is referred to as active because it requires the use of external powered circuitry to apply the correction. In contrast, passive trimming involves permanently adjusting the device to the target wavelength via modifications to its physical properties (e.g., phasechange materials or ion implantation). Passive approaches are preferred because they reduce extra control logic and power requirements. Passive trimming technologies, however, currently require specific postprocessing steps that may not be economically viable in silicon photonic foundries.
It is worth noting that active trimming and weighting occur via the same physical mechanisms. The difference between the two, therefore, exists solely in the software layer (i.e., how the user interacts with each). An end user should only need to control weighting, and should not need to worry about active trimming. In other words, the same weight command should produce the same results in two separate chips regardless of manufacturing imperfections. Trimming, therefore, must be completely automatic and free from user interaction. This separation is absolutely critical to enable use of photonic computing hardware by nonexperts, and consequently for the proliferation of the technology. In order to realize this, there should be two parallel circuits that are independent and are characterized by independent metrics: trimming, which makes sure that any chip with the same design has the same baseline behavior; and weight control, which translates weight command to effective weight.
4 Control algorithms
One of the biggest advantages of using analog photonic weights is that highbandwidth optical signals can be weighted by an optoelectronic device that consumes constant power (or static power). It is worth noting, however, that as bandwidth is increased, more static power is required at the optical source to compensate for the added noise captured by the summing photodiodes. In the mechanisms presented in the previous section, the analog weight is defined by the electrical voltage or current applied to the microresonator. The accuracy and precision of each weight is a function of the completeness of the model along with the effects of noise and disturbances. In this section, we outline a simplified weight bank model and discuss practical examples of noise and disturbances.
4.1 Simplified microresonator weight bank model
To illustrate the ramifications of control algorithms, we present a simplified model of a resonator weight bank (Table 2). Each weight (e.g., a microring resonator) can be tuned via silicon’s thermooptic effect by controlling the local temperature. By sourcing current through a resistor placed nearby, this temperature can be adjusted directly and efficiently. This phenomenon is referred to as Joule heating. We assume, for simplicity, that the properties of the resistor do not change with its temperature or with the circulating optical power in the resonator cavity. Due to the phasematching condition on resonators, there is a direct relationship between resonant wavelength and refractive index, namely λ/n = constant. In a simple model, the transmission of the microresonator is a function of only the “detuning,” or the difference between the fixed wavelength (λ
_{0}) and the tunable resonance frequency (λ). In high Qfactor resonators, this function can be approximated by a Lorentzian line shape. In Table 2, this is denoted by
Step  Physical mechanism 

1  Joule heating: 


T _{ i }: local temperature; T _{0}: room temperature; K: effective thermooptic coefficient matrix (diagonal in the absence of thermal crosstalk); R _{ i }: electrical heater resistance.  
2  Thermooptic effect: 


λ _{ i } (λ _{0,i }): resonance frequency of filter with applied heat (at room temperature), respectively; β _{eff}: thermooptic coefficient; n _{eff}(λ _{0,i }): effective waveguide index; ΔT: temperature offset.  
3  Lorentzian transfer function: 


Δλ: halfwidth halfmaximum of Lorentzian transfer function modeling a single resonator weight.  
4  Weighted sum: 


μ ∈ [0, 1]: weighted sum (photocurrent normalized by PD’s responsivity); P _{ i }: optical intensity of input lightwave at wavelength λ _{ l,i }; P _{max}: maximum optical intensity. 

Changed to table formatting in response to reviewer 1.
After weighting, the output of the weight bank is fed to a photodetector. In our simplified model, this photodetector is assumed to have a constant responsivity independent of input power. The electrical properties of the photodetector (parasitic capacitance, parasitic resistance, or frequency response) can be ignored at this point and instead captured later by the model of the electrical receiver (in neuromorphic photonics, this would be the E/O conversion of the neuron). In practice, the photodetector’s responsivity can have a range of values depending on fabrication variation and operating wavelength, and this should be captured by the calibration model via a linear correction. Once the optical power input is too high, however, the responsivity drops due to spacecharge screening effects [40]. This nonlinear relationship between photocurrent and optical intensity cannot be calibrated away, and results in reduced precision.
4.2 Compensating for noise and disturbances
The simplified model presented in Section 4.1 is effective at modeling the weighted sum μ as a function of the optical intensities at each wavelength P _{ i }. In this scheme, the signal is modulated onto each carrier wavelength. To simplify, we assume a perfect modulation scheme where the instantaneous intensity is described as follows:
where x _{ i }(t) is the same as in Eq. (1) and P _{ i,input}.
In this analog representation, we can identify μ(t)/P _{0} as the equivalent of the weighted sum y(t) scalar, and the weight as:
where P _{0} is a normalization constant.
As revealed by Eq. (8), however, noise and temporal disturbances to the values of P _{ i,input} can negatively affect the weight’s precision or the resolution of the sum y. These effects cannot be captured by any calibration model, since by definition a calibration model must be based on static parameters. Disturbances must nonetheless be corrected independently from the transmission control, since the user will reasonably expect the chip to have a constant average weight vector.
In practice, there are two relevant sources of disturbances to the input intensity. One is ambient temperature, and the other is polarization drift accumulated in the input fibers. To show the impact these sources have on the operation of a weight bank, we have setup a microring resonator weight bank with four rings, placed on top of a temperature controlled base (Figure 3). These effects can be observed, for instance, in the optical spectrum of a microring resonator weight bank (Figure 4). We observe that the temperature drift essentially maintains the shape of the transmission curve for each resonator, as expected for microring resonators. As a result, the microcontroller can compensate for that by either dissipating extra heat uniformly across all devices, or by using a thermal bath that holds the average temperature of the entire chip constant, such as the thermoelectric module sketched in Figure 3. This invariance property is unique to microrings and cannot be generalized to other kinds of resonator structures. Keeping temperature constant, however, will stabilize any temperaturedependent device such as resonators and optical couplers. A polarization drift in the fibers outside the chip, however, alters the power spectrum irregularly, and can only be remedied by recalibrating the circuit. We have found that using polarizationmaintaining fibers and optical adhesives can stabilize the polarization spectrum indefinitely in laboratory settings. More studies must be completed in order to evaluate this effect in field conditions, where significant contributions are expected from mechanical stress and vibrations.
4.3 Measuring effective weights
The summing element is an integral part of the photonic circuit, so we typically only have access to output photocurrent while operating the weight bank. This creates the challenge of independently verifying that the applied weight was, in fact, the correct one. Mathematically speaking, in the linear approximation of the summing element, for a set of signals x
_{
i
}(t), and a measurement of y(t), we need to recover the weights w
_{
i
} in Eq. (1). One approach could be to ensure that only one channel has a positive value at any given time, for example x
_{
i
}(t) = 1 for time slot
4.4 Calibration and feedforward control
Regardless of the weighting mechanism, the control strategy of a resonator involves a calibration stage and a control stage. The calibration is required to adjust the weight bank model parameters caused by fabrication variation or uncertain initial conditions. The control stage runs continuously thereafter.
Calibration and control strategies must take into consideration an array of resonators in the weight bank, because coupled resonators suffer from both optical and thermal/electrical crosstalk. There is no noticeable electrical crosstalk in this device, since all heaters are surrounded by silicon dioxide, a highly insulating material. As shown in Figure 5, applying Joule heating to the vicinity of one resonator directly changes its transmission function, which according to the model introduced in Section 4.1 results in changing the weight for that wavelength. Notice, however, that the neighboring resonator also undergoes a small change in its transmission function due to optical crosstalk. Though barely visible, this second order effect, if uncompensated, reduces the accuracy of the weight bank. For a 8bit accuracy level, the transmission on the other rings must be kept within 0.4% of their target. In our model, we can capture this crosstalk in the matrix K, shown in Table 2.
Crosstalk can be significantly, but not totally, reduced by carefully engineering the geometry of each device. To reduce the thermal crosstalk, MRRs can be spaced further apart, taking advantage of the low thermal conductivity of silicon dioxide cladding [29]. Optical crosstalk can be reduced by increasing the wavelength spacing between resonators, at the expense of spectral capacity [9].
As the number of weights N in a system increase, so does the number of independent variables in the model. Mitigating the thermal and optical crosstalk between resonators in a weight bank requires measuring crosstalk terms and fitting a multidimensional model with
A series of approaches have been developed to tackle the weight bank control problem. The earliest method presented in the literature is called the feedforward scheme [41], shown in Figure 6. In this scheme, the controller would possess an electrical model of the resonators, including an approximation of the thermal and optical crosstalk effect. The model maps a vector of electrical current values to a vector of weight values, and vice versa. Based on a desired weight, the model converts to an actuation electrical signal for each resonator. But once signals are set, there is no way to verify or validate that the effective weight is correct.
The advantage of this approach is the simplicity of the control algorithm – it is similar to a lookup table in electronics. This calibration algorithm scales with
To resolve these fundamental issues with the feedforward scheme, the microresonator device can be designed with an embedded sensor, capable of measuring the applied weight in real time. With a sensor located near each resonator in the weight bank, measuring, e.g., the local temperature, we can directly feed an electric signal back to the controller to adjust for model deviations. This is called a feedback scheme, and will be revisited in Section 5.3, after we delve into index modulation, actuation and sensing physics in photonic waveguides in Section 5.
5 Design for manufacturability
Passive optical devices are sensitive to the effective refractive index of waveguides, which determines the optical path length along the path of propagation. In integrated photonics platforms, the effective refractive index varies with the height and width of the waveguide [42]. The width of integrated waveguides is lowerbounded by a photolithography process with a limited resolution. In the case of silicon photonic chips, deep ultraviolet photolithography is used, with a wavelength of 193 nm, limiting the minimum lateral feature size of devices to 65 nm [26]. Wider waveguides are typically used, however, since increasing width leads to tighter optical mode confinement and lower losses. This, in turn, is upperbounded by the width at which the waveguide supports a second lateral mode at the wavelength of operation. As a result waveguide width is usually chosen to be the maximum singlemode width minus some engineering margin. For similar reasons, the height of the waveguide is often chosen to be the maximum that only admits a single infiniteslab mode minus some engineering margin. A de facto industry standard height is 220 nm with a standard deviation of 2 nm [42].
The combination of lateral and vertical waveguide manufacturing imperfections results in fixed, random phase shifts in each waveguide both across the same chip and between chips. That means that identicallydesigned interferometric devices will behave differently once fabricated. For resonator structures, random phase offsets above 2π are sufficient to render any prediction of a postfabrication resonance wavelength impossible. However, within the same chips the phase offsets are spatially correlated [43], meaning that resonators in close proximity have resonance offsets similar in sign and magnitude.
There are several postfabrication options available to compensate for undesired random phase shifts. These techniques are referred to as trimming, where a section of an optical waveguide material is physically or chemically altered until the desired phase is achieved. The range of phase shifts that can be compensated is typically limited, and as a result designlevel compensation is often required. Design for manufacturability tackles this challenge by predicting the amount of compensation required via Monte Carlo simulations of fabrication variability [44]. A successful design for manufacturability is able to reduce the range of fabrication variation that will need to be compensated by trimming, and can result in considerable cost saving overall.
5.1 Index tuning mechanisms
Many refractive index tuning mechanisms have been developed for silicon photonics, and Table 3 displays some stateoftheart devices from the literature. In silicon, the strongest effects are the thermooptic effect, freecarrier absorption and freecarrier dispersion (also known as plasma dispersion). Exploiting the thermooptic effect for tuning with metal filament microheaters is the easiest and most popular way to effect large index changes, but it is slow and power inefficient [45]. Thermal tuning with waveguideembedded heaters is similar in efficiency [46], but provides a potential feedback signal for weight control [47]. To exploit freecarrier effects, we can directly manipulate carrier concentrations by selectively p and ndoping the waveguide in a lateral junction [48]. Freecarrier absorption dominates if the junction is forward biased, while freecarrier dispersion dominates if the junction is reverse biased.
Modulation effect  Speed  Efficiency  Ref. 

Volatile mechanisms  
Thermooptic TiN (a) 

P _{ π } L = 6.8 mW mm  [45] 
Thermooptic N^{+}/N/N^{+} Si (b) 

P _{ π } L = 0.8 mW mm  [46] 
Reversebiased PN (c)  41 GHz  V _{ π } L = 46 V mm  [48] 
Graphene SLG (g)  30 GHz^{1}  V _{ π } L = 28 V mm  [49] 
LiNbO_{3}/Si hybrid (f)  70 GHz  V _{ π } L = 22 V mm  [50] 
III–V MQW/Si hybrid (d)  27 GHz  V _{ π } L = 2.4 V mm  [51] 
III–V/Si MOS (e)  2.2 GHz  V _{ π } L = 0.9 V mm  [52] 
ITO MOS (e) 

V _{ π } L = 0.52 V mm  [53] 
Forwardbiased PIN (c)  0.5 GHz  V _{ π } L = 0.36 V mm  [54] 
Nonvolatile mechanisms  
PCM Ge_{2}Sb_{2}Te_{5} (h)  0.8 GHz^{3}  E _{ π } = 400 pJ  [10] 
PCM Sb_{2}Se_{3} (h)  1.25 MHz  E _{ π } = 176 nJ  [55] 
Ion implantation  0  0  [56] 

Options for phase modulation of silicon waveguides. (a) Thermal tuning with TiN filament; (b) thermal tuning with embedded photoconductive heater; (c) PN/PIN junction across the waveguide for injection and/or depletion modulation; (d) III–V/Si hybrid waveguide; (e) metaloxidesemiconductor (MOS), where the ‘metal’ is actually an active semiconductor; (f) lithium niobate cladding adds a strong electrooptic effect; (g) 2 singlelayergraphene (SLG) ‘capacitor’; (h) nonvolatile phase change material.

^{1}This bandwidth was not yet shown experimentally. A big challenge is to reduce the contact resistance with Graphene, reducing RCloading effect. ^{2}Not experimentally shown at highspeed. ^{3}Demonstrated up to 20 MHz. We broke down the mechanisms into volatile and nonvolatile based on the reviewer recommendations.
Moving beyond pure silicon approaches, one can make hybrid waveguides consisting of a silicon core and additional materials with favorable index modulation properties that are placed close enough to the core that interact with the evanescent field. Some examples include III–V semiconductors [51], lithium niobate [50], and graphene [49]. These mechanisms are faster and require much less power compared to heaters, but typically provide smaller tuning range before the onset of electrical damage.
Nonvolatile mechanisms: Tuning methods based on phase change materials (PCMs) allow weights to retain their values without active power consumption after being set. This is called nonvolatile weight (or memory) and provides the most energy efficient weight setting method: note that in Table 3 the efficiency is measured in Joules rather than Watts. This method involves depositing chalcogenide films atop silicon waveguides, e.g., Ge_{2}Sb_{2}Te_{5} (GST) [10, 57], Ge_{2}Sb_{2}Se_{4}Te_{1} (GSST) [58], Sb_{2}Se_{3} [55, 59]. Microring resonator nonvolatile weights with low loss have recently been demonstrated with Sb_{2}Se_{3} PCMs [55].
Combining volatile and nonvolatile methods: Another important variation source is ambient temperature. Automotiveclass devices must have an operating temperature range of −40 °C to 125 °C. Due to silicon’s large thermooptic coefficient, this temperature range results in a large resonance drift, albeit no worse than the original fabrication variation (Figure 7). Depending on the final application of the circuit, these two sources of variation can be addressed with different index modulation options from Table 3.
We propose two practical options for addressing variations, including the ones present in automotive temperature ranges. The first is to use postfabrication trimming to lock resonators to their desired resonance wavelengths at a set operating temperature, which can be chosen, for example, to be above room temperature to avoid condensation. A thermoelectric controller can then be used to stabilize the chip’s resting temperature and finetuning can be performed via either microheaters or PN junctions, depending on the required operation speed and actuation range. This greatly simplifies the photonic integrated circuit design, but the use of temperature control adds thermal engineering complexity to the packaging. Another option is to rely on active trimming, in which onchip microheaters are used to compensate for wide operating temperature ranges. Accompanying this, finetuning can be performed via microheaters or PN junctions. This approach requires more control circuitry, but eases the overall burden on packaging [30, 31]. Both approaches can be visualized in Figure 7.
5.2 Electrical actuation and sensing
As reviewed above, there are multiple physical phenomena that are available for electrically adjusting the refractive index of a waveguide, and consequently the resonance wavelength of an integrated resonator. Since the weight is determined by the distance between a lightwave’s wavelength and a resonator’s resonance wavelength (Table 2), we would ideally have a sensor that can directly determine resonance wavelength (λ _{0}). With such a sensor coupled with resonance actuation, we would only need to model the mapping between resonance wavelength and weight (i.e., a Lorentzian function for microring resonators) in order to precisely perform weighting. In reality, it is impractical to measure λ _{0} directly because it would require a precision spectrometer.
One practical alternative is to use a temperature sensor [61] surrounding the resonator. This allows absolute temperature stabilization independent from the environment’s temperature. Although this provides an efficient mechanism for ‘fixing’ a weight, it usually requires sourcing very precise currents and reading tiny voltage signals with high dynamic range. So far, it has only been successfully employed to stabilize microring resonators around maximum transmission, which is useful for optical switching but not smoothlyvariable weighting.
Another practical alternative is to measure the optical power circulating within a resonator with a photosensitive element, shown in Figure 8. This method only works if the weight value is proportional to the circulating power relative to input power, and if input power is otherwise known. A simple method for measuring circulating power is by lightly doping the resonator’s waveguide, which results in a photoconductive resistor. This resistor can be used for both heating and sensing the circulating power, and was successfully employed in both resonancelocking circuits [46] and weight control [47].
This sensing modality will suffer if the nonheater parasitic resistances^{[1]} are too large (e.g., more than 10 kOhm). Fortunately, when chips are packaged with lowimpedance wirebonds and properly designed printed circuit boards (PCBs) all such parasitic resistances become insignificant compared to the heater resistance [62]. We have found that a heater resistance range of 1.5–2.5 kOhm works well. This is because in a typical silicon waveguide, a π phase shift is achieved with the application of
5.3 Feedback control
Equipped with an appropriate sensing element, it is then possible to build a closedloop control circuit that dynamically responds to external disturbances to a resonator. Similarly, with a sensor for each resonator in a weight bank it becomes possible to correct for disturbances to each resonator individually. This modality of control based on sensed signals is called a feedback (or closedloop) control scheme. Compared to the feedforward scheme introduced in Section 4, the feedback scheme offers superior accuracy and precision as well as reduced control complexity.
A typical control circuit involves a currentbased actuator, voltagebased sensors, and a model that maps the sensing signal to the desired control quantity (Figure 9). Having a sensor for every weight element simplifies the feedback control law from one large model for the weight bank to a collection of independent models for each resonator. Unlike feedforward control, it is no longer necessary to take into account the thermal crosstalk between resonators or the ambient temperature because the closedloop circuit corrects for any deviation from the target.
A photoconductive sensor (PCS) provides a voltage signal that is linearly dependent to the transmission function. The transmission function can be translated in wavelength space with an actuator, leading to smoothlycontrollable weight values. Rather than modeling how transmission is affected by actuation and other environmental variables, the controller can sense the voltage directly and simply adjust the actuation accordingly.
Despite using a closedloop/feedback control circuit, each resonator may vary slightly due to fabrication imperfections. In order to compensate for this, all imperfections must be parameterized and calibrated out on a chipbychip basis. Fortunately, this model only has to be built once per device. Therefore, it can be done at the manufacturing facility and stored in a readonly memory copackaged with the chip.
6 Conclusions
Practical neuromorphic photonic processors employing resonatorbased weight banks must be able to achieve high precision and accuracy while being readily manufacturable and deployable in variable/dynamic environments. This is made particularly challenging by the fact that resonators are highly sensitive to both manufacturing variations and environmental disturbances. In order to compensate for this, individual resonator weights must be specifically designed with calibration and control in mind.
In this paper, we have outlined index tuning and sensing mechanisms that can be used for actuation and sensing of weights, as well as feedforward and feedback calibration/control methodologies for use with weight banks. With multiple index tuning mechanisms with varying speed, efficiency, and dynamic range available, system designers can choose which mechanism or combination of mechanisms are suitable based on application needs.
As an example, we outline a fabrication design strategy that can compensate for worstcase fabrication variation (Δϕ = ±2π) and employment in automotiveclass operating temperature ranges (−40 °C–125 °C). Germanium ion implantation is used for postfabrication trimming, a phasechange material (PCM) is used to compensate for large temperature variations, an Ndoped heater is used for lowerbandwidth weight tuning, and a PN junction is used for fast modulation. Complementing this, photoconductive sensors are used on each resonator in order to enable feedback calibration/control procedures.
Funding source: Universal Display Corporation and the Office of Naval Research
Award Identifier / Grant number: (Grants N000141812297 and N000142012664)

Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

Research funding: The authors acknowledge Universal Display Corporation and the Office of Naval Research (Grants N000141812297 and N000142012664) for funding support. S. Bilodeau acknowledges funding from the Fonds de recherche du Québec  Nature et technologies.

Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
Appendix: Microring resonator weight design methodology
In this section, we outline our design methodology for symmetric add–drop microring resonators. We follow the nomenclature from Ref. [32]. The ring’s add and drop transmission coefficients can be computed analytically using transfer matrices with only four parameters: selfcoupling coefficient r (thru port amplitude), roundtrip loss coefficient a → ra (accounting for the second coupler’s coupling loss), cavity length L, and effective index of refraction n
_{eff} as a function of wavelength. Each parameter depends on the geometry of the resonator such as waveguide width, the gap between the bus waveguide and the ring, the ring’s radius R, and the doping level of the waveguide. r can be computed through finitedifference time domain (FDTD) calculations of the coupler section. a is related to power attenuation coefficient α via
The quantities of interest for weighting can be extracted from the parameters above. The free spectral range is calculated as
The finesse determines roughly how many WDM channels (or resonators) can share the same bus waveguide
The Qfactor approximates the optical rise and fall times of the resonator:
Finally, the thru and drop extinction ratios allow calculation of the weighting dynamic range (assuming the index tuning mechanism can be used to tune the resonator on and offresonance):
We used Lumerical FDTD to perform the finitedifference time domain coupler simulations to extract r(λ) as a function of geometry, and Lumerical MODE for the curved waveguide eigensolver simulations to obtain n _{eff}(λ). A Python script implements the above analysis pipeline, and the results can be visualized in a Jupyter Notebook. The Python script and the results of the Lumerical simulations are made accessible online [35].
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