## Abstract

A photonic threshold comparator is presented. A step-like electrical-to-optical (E/O) response is obtained by employing Fourier series synthesis in which a set of sine-wave responses of different amplitudes and phases are superimposed according to the Fourier series representation of a square-wave. The proposed comparator does not rely on optical material non-linearity; rather it consists of multimode interference (MMI) couplers and phase shifters.

© 2013 OSA

## 1. Introduction

One of the challenges in realizing optical analog-to-digital conversion (OADC) is that of implementing high-speed optical threshold-comparators that can be incorporated into a fully monolithic device.

An optical analog-to-digital converter must perform three functions on a time-varying analog signal: (1) sample-and-hold the analog signal for a specified time, (2) quantize the sampled signal into a number of levels, and (3) assign a binary representation to each level.

Most of the previously published works on optical analog-to-digital converters focused on the first two functions. It is long recognized that short optical pulses can be used to make fast optoelectronic switches for electronic sampling. Optical pulses provide a very fast sampling technique with low pulse-to-pulse jitter, compared to an electronic sample and hold circuits. A large number of works were also published on optically assisted quantization, in which an electronic or optical signals were quantized by optical means. In the quantization process the incoming continuous analog signal is assigned to a finite number of distinguishable levels. The third step as mentioned in an analog-to-digital conversion is the encoding process, in which the quantized levels are assigned a digital code. Digital encoding requires a threshold comparator which senses an analog input level and according to it, outputs either binary 0 or 1.

Comparators based on Mach-Zehnder modulators were used for constructing OADC in [1–3]. A Mach-Zehnder modulator can be used as a comparator by setting its input around a bias value. Its response, however, is sinusoidal and therefore an electronic comparator has to be cascaded in order to provide two distinct levels at the output. Many reported schemes employed various nonlinear optical processes to implement a comparator [4–13].

There are several drawbacks in employing nonlinear optical processes for the purpose of realizing nonlinear responses: For standard materials implementing Integrated Optic devices, a relatively large optical power is required in order to manifest the nonlinear behavior, and significant nonlinear responses require special materials, their processing not being suitable for realizing all-optical ADC as a small monolithic device. Thus, most of the published results on OADCs rely on electronic comparators for providing the threshold required for delivering distinct binary 0 and 1 output levels.

Herein, an integrated optical device is presented that does not rely on optical material nonlinearity and still exhibits a sharp step-like response. This response is obtained via Fourier-series synthesis of a square-wave. The nonlinear response required for thresholding is provided here by the inherent mathematical nonlinear dependence between intensity and phase in multi-wave interference.

While all-optical comparators accept optical signal, the proposed comparator accepts an electronic signal as an input. The intensity at the output of the proposed comparator is of a binary nature, and hence by connecting a photodiode one obtains a digital (electronic) output. Therefore, the proposed comparator acts as an electrical-optical-electrical (E/O/E) device, which may be employed as a replacement for an all-electronic comparator.

The proposed device consists of multimode interference (MMI) couplers and phase shifters [14], both being basically linear, key components in optics. MMI couplers are based on the *self-imaging* effect in waveguides [15]. They are mostly used as efficient optical splitters and combiners having low loss and low crosstalk [16]. Moreover, MMIs were demonstrated to provide accurate splitting ratios with small footprint and good fabrication tolerances. They have been fabricated using various technologies, but mainly in Silicon and InP waveguide technologies. MMI couplers were also employed as building blocks in complex optical devices, e.g. generalized Mach-Zehnder interferometer switches [17].

Finally, optically-assisted thresholders have been investigated due to their potential speed advantage. More importantly, the periodic behavior inherent in the optical domain, which does not exists in the electronic domain, can provide extremely efficient Flash A/D architecture.

The rest of the paper is organized as follows. Section 2 provides relevant definitions and performance measures. Section 3 presents an optical threshold comparator by means of square-wave synthesis and discusses its design rules. In Section 4 we provide a design example of a 5-arm comparator, and in Section 5 we analyze its performance. In Section 6 we introduce a multi-threshold comparator and discuss its use in the context of optical analog-to-digital converter. Finally, Section 7 the conclusions are resumed.

## 2. Comparator overview

An ideal (non-inverting) comparator is a nonlinear device that converts all input values that are below a certain threshold into a single small value, and all signal values above that threshold into a single large value.

Figure 1 presents a comparator whose input and output are denoted by *V _{a}* and

*V*, respectively. The figure provides the static transfer characteristic of an ideal comparator (dashed line) and a realistic comparator (continuous line).

_{out}As shown, input values below *V _{IL}*, and above

*V*, generate well defined output levels, namely:

_{IH}*V*, representing logic-0; and

_{OL}*V*, representing logic-1, respectively. For input levels in the range [

_{OH}*V*,

_{IL}*V*] the comparator output state is undefined [18]. Since

_{IH}*V*is continuous in nature, it is not unlikely for the output to yield values in the transition region [

_{a}*V*,

_{OL}*V*] which may be erroneously identified by the following stage.

_{OH}Two important parameters that characterize the above behavior, and hence characterize the performance of a comparator, are the *gain* and *extinction ratio*. The *Gain* is defined as the slope of the output curve about the input value *V _{m}*[18]:

*V*is the input voltage for which the output voltage is

_{m}*V*= (

_{out}*V*+

_{OH}*V*)/2; this is actually the input threshold level. Note that the gain of the ideal comparator is infinite, while the gain is finite in a realistic comparator. High gain in the vicinity of the threshold point is a desirable property of a good comparator.

_{OL}The extinction ratio (*ER*) measures the distinction between the output high and low levels:

Finally, electronic comparators can be classified into two class: open-loop comparators and regenerative comparators. The main difference between the two types is whether a feed-back is applied. There are also configuration that employ a combination of the two. An open-loop comparators is an high gain amplifier that is designed to operate with its output saturated, close to the supply rails. The amplifier does not employ a feedback loop and therefore can be designed to achieve large bandwidth [19]. The proposed optical comparator falls in the category of open-loop comparators. As will be discussed later, this type of a comparator can achieve very high bandwidth.

## 3. Photonic threshold comparator using square-wave synthesis

The proposed device has a step-like electrical-to-optical (E/O) transfer characteristic: its input is an electrical voltage *v _{a}* and it outputs high optical power when

*v*exceeds a certain voltage level, and low optical power otherwise. This threshold behavior, which makes the device suitable as a comparator, is obtained here by using Fourier series synthesis whereby a set of sine-wave responses of appropriate amplitudes and phases are superimposed to generate a square-wave. Periodic function

_{a}*f*(

*x*) of period

*L*can be described by a complex Fourier series as:

*V*is:

_{π}*P*is the optical input power,

_{in}*P*is the optical output power,

_{out}*M*and |

*m*| are odd integers, and

*C*is a constant offset.

The proposed Fourier synthesis of a comparator shall be realized by means of photonic waveguided components consisting of MMI couplers and waveguide phase-shifters of both passive and active kind. Following is a description of the optical signal flow:

- A coherent optical signal splits into
*N*equal components, where*N*= 2*M*+ 1. E.g., if*N*= 5, then*M*= 3 and so*m*∈ {−3, −1, 1, 3} plus a DC component,*C*, are used. - The amplitude of each component is attenuated proportionally to the corresponding Fourier coefficient.
- The phase of each signal is adjusted accordingly. The individual phases have a fixed part and a variable part, the later being proportional to the electrical input signal
*v*._{a} - The signals are recombined into a single waveguide while preserving their phases and amplitudes.
- The outcoming optical signal is transduced into an electrical voltage
*V*by means of a square-law detector._{out}

As described, all the functions mentioned above are realized by standard linear optical components. The overall nonlinear behavior demanded by the function *V _{out}*(

*V*) is attained through the mathematical dependencies between between phases and amplitudes of interfering signals. A 5-arm design example for implementing Eq. (5) with

_{a}*N*= 5, is depicted in Figure 2. Optical power enters MMI-1 and equally splits into its 5 output arms. The arms serve as passive phase shifters (#1). These constant phases can be adjusted in such a way that when combined with MMI-2, produce a set of 5 optical signals whose amplitudes nearly obey the Fourier series formula. For these signals, the appropriate constant phase shifts are induced by the additional set of passive phase shifters (#2). The analog input voltage,

*v*, is simultaneously applied to the set of active phase shifters (#3) thus inducing the phases for the 5 signals as required by Eq. (5). The set of passive phase shifters (#4) together with the third coupler, MMI-3, serve as a coherent optical field combiner. In a practical design, the two passive shifters (#2 and #4), can be combined into a single one.

_{a}All three MMI couplers are of equal dimensions. Their length *L* is given by [20]:

*N*is the number of arms of the MMI and

*L*is its beat length defined by with

_{π}*β*

_{0}and

*β*

_{1}being the propagation constants of the fundamental and first-order modes supported by the multimode region of the MMI coupler, respectively.

Let us denote the transfer matrix of an *N* × *N* MMI device by *S _{MMI}*. The element in row

*i*and column

*k*of this matrix is given by

*a*is the field amplitude transfer coefficient from input

_{ik}*i*to output

*k*, and

*ϕ*is the associated phase. In a lossless MMI the field amplitude transfer coefficient is ${a}_{ik}=1/\sqrt{N}$[17].

_{ik}The phase difference between two given ports *i*, *k* is:

*M*

_{ps}_{1}is an

*N*×

*N*diagonal matrix whose non-zero entries are the phase shifts The optimum values for

*θ*are found by employing the method described by Lagali [20].

_{i}As stated earlier, the set of phase shifters #2 adjust the phases of **e**_{out}_{2} in accordance with the Fourier series formula. Its operation can be formulated by a *N* × *N* diagonal matrix *M _{ps}*

_{2}. For the 5-arm design, relative to the phase of the optical wave in the middle arm, the phases of the optical waves in the two upper arms are set to $\frac{\pi}{2}$.

The electrical input signal, *v _{a}*, is applied to the device through a number of electrodes of different lengths. The electrodes produce a set of phase shifts given mathematically by the diagonal matrix,

*M*

_{ps}_{3}, whose elements are given by:

*x*⌋ means rounding to nearest integer smaller than

*x*,

*n*indicates the relative electrode length, and

*V*is the half-wave voltage of the phase shifter. Note that no electrical signal is applied to the middle arm. For example, for

_{π}*N*= 5, the diagonal elements of

*M*

_{ps}_{3}are:

Finally, for MMI-3 to function as a coherent field combiner, the phases of the incoming optical waves must be adjusted in accordance with Equation (14) (with minus signs). The elements of the corresponding diagonal matrix *M _{ps}*

_{4}are

*k*= ⌊

*N*/2⌋.

The third coupler, MMI-3, serves as a coherent optical-field combiner. The field at the output port of MMI-3 is given by the following matrix equation:

**e**

_{out}_{3}| assumes a square-like shape as a function of the input voltage,

*v*. MMI-3 also induces passive phases that need to be cancelled by shifters #2, #4.

_{a}## 4. Design considerations

In choosing the materials with which to fabricate the comparator, one has to consider two critical issues: **1)** use of low-loss, high contrast materials for the fabrication of an efficient MMI coupler. Materials such as Silicon, InP, GaAs are suitable; **2)** it follows from Equation (5) that the active phase shifters must be linear so that the induced phase shift is linear with the applied voltage. Materials which posses the Pockels effect would be most suitable for this purpose.

In materials such as Ti-indiffused LiNbO3, which exhibit strong Pockels effect, the refractive index contrast is low and it is very difficult to utilize it as an efficient MMI coupler. On the other hand, Silicon does not posses a linear Pockels, but other modulation effects like carrier-depletion are intensively investigated [21], these effects have intrinsic non-linearity and mix amplitude and phase modulation and would require further consideration.

InP/InGaAsP technology [22] fulfills the technologies requirements and thus it is a good candidate for fabrication of the comparator. It posses a high refractive index contrast suitable for creating MMIs with it. Active phase shifters, utilizing the Pockels effect, fabricated on InP substrate, were demonstrated with a Mach-Zehnder modulator at bandwidth of 40GHz [23].

A complete 5-arm comparator was designed and simulated. A single-mode rib waveguide of width 2 *μm* is formed with a 1 *μm* thickness layer of InGaAsP on InP substrate. The refractive index of the InP is *n _{InP}* = 3.167 and that of the InGaAsP is

*n*= 3.22. The waveguide is surrounded by air cladding. The widths of the MMI couplers are

_{InGaAsP}*W*= 40

_{MMI}*μm*, and the corresponding lengths are

*L*= 2527

_{MMI}*μm*.

## 5. Performance

A quasi-static analysis was conducted in order to investigate the performance of the proposed comparator. Figure 3 presents the calculated E/O response for the 5-arm comparator for an input voltage range of 0 to 2.1*V* where *P _{in}* = 10

*mW*.

The x-axis represents the applied voltage *v _{a}* while the y-axis is the resulting optical signal intensity,

*P*, at the device output. The applied voltages,

_{out}*v*, in the figure are DC reversed biased. The horizontal bar labeled

_{a}*P*= 3.5

_{OH}*mW*, is the minimum output power-level representing logic-1, while the horizontal bar labeled

*P*= 0.35

_{OL}*mW*, is the maximum power level representing logic-0. The output levels,

*P*and

_{OH}*P*, were chosen as explained in Section 2. In a practical system, the comparator typically drives a logical stage. Therefore, one has to make sure that these output levels suitably match the specified input logic levels of the following digital stage.(for E/E response one may employ a photodetector at the device output to convert the output optical power to a corresponding voltage level.)

_{OL}#### 5.1. Input dynamic range

In practice, the voltage range allowed at the comparator input is limited by the power supply voltages. Thus, *input dynamic range* (DR) will be defined as |*V _{IE}* −

*V*|, where

_{IS}*V*and

_{IS}*V*are the lowest and highest voltage values allowed at the input for proper operation of the comparator. In Figure 3, it can be seen that an input voltage below

_{IE}*V*= 0.45

_{IL}*V*will output low level (not greater than

_{π}*P*) and input above

_{OL}*V*= 0.55

_{IH}*V*, will output high level (not smaller than

_{π}*P*). This, in turn, means that input voltage levels in the range

_{OH}*V*<

_{IS}*v*<

_{a}*V*produce low output power, while input voltage levels

_{IL}*V*<

_{IH}*v*<

_{a}*V*produce high output power.

_{IE}Input voltage levels in the range [*V _{IL}*,

*V*] produce output power levels of undefined logic interpretation [18]. The uncertainty in this transition region may lead to signal jittering or phase noise in designs where a comparator directly precedes a digital circuit. Since the input voltage

_{IH}*v*is analog, of a continuous nature, the region [

_{a}*V*,

_{IL}*V*] should clearly be minimized. It follows that a well designed comparator must present a steep transition slope, which one may refer to as gain.

_{IH}#### 5.2. Comparator gain

Typically, the, so-called, *Gain* is measured between the input and the output of a device. Since the proposed comparator is an electro-optical device in which the input is electrical and the output is optical, the gain may be defined as the following ratio:

*V*is the input voltage for which the output power is

_{m}*P*= (

_{out}*P*+

_{OH}*P*)/2.

_{OL}At first glance the gain in the above definition appears as a ”mismatched” ratio between power and voltage. Nevertheless, a photodetector can be employed to convert the output optical power into electrical voltage. The conversion expression is given by *V _{pd}* =

*P*·

_{out}*r*·

*R*, where

*r*is the

*responsivity*of the detector in units of

*A/W*and

*R*is the output resistance, usually 50Ω. It follows that

*V*is proportional to the output optical power, and consequently the gain given by Equation (16) is suitable for device evaluation purposes.

_{pd}#### 5.3. Design examples

Figures 4(a) and 4(b) present the calculated output for the proposed comparator using *N* = 5 and 9 arms. Table 1 summarizes the input power, MMI dimensions and static performance for these designs. To provide baseline for comparison, the comparators were all designed to achieve high digital output level of *P _{OH}* = 2

*mW*.

The first row in Table 1 lists the input powers required by the two designs to generate a high level output of 2 *mW*. It follows from the discussion in previous sections that increasing the arm count leads to reduction of the peak output power and consequently *P _{OH}*. This occurs because for a larger arm count, the input optical power is distributed among a larger number of ports, and the higher Fourier components are of decreasing amplitudes. So, to compensate for that loss, and achieve the required

*P*, the input power

_{OH}*P*must be increased. Thus, for

_{in}*N*= 5,

*P*= 5.9

_{in}*mW*, while increasing the arm count to

*N*= 9 the required input power is 10.3

*mW*.

The second row in the table provides the gain of the two devices: the *N* = 5 design achieves gain of 10.6 *dB*, while increasing the arm count to 9 monotonically increases the gain by additional 2.6 *dB*. For the two designs, the achieved low level output and extinction ratios, are *P _{OL}* = 0.2

*mW*and

*ER*= −10

*dB*, respectively.

The normalized transition region, defined as *TR* = (*V _{IH}* −

*V*)/

_{IL}*DR*, provides a good measure for the quality of the comparator. A wider transition region implies greater error probably (undefined logic state) at the output of the comparator. Obviously, the higher is the gain, the better is the TR value. The transition region improves as

*N*increases.

Fabrication deviations may possibly degrade the performance of the comparator. The impact of deviations in length or width of the MMI couplers, on the coupler performance, has already been shown to be quite minor [17]. Fabrication sensitivity analysis we conducted for the complete device validated this result also for the comparator. Rather, we found that the main impact on the device performance comes from phase-shift inaccuracies originating in the arms connecting the MMI couplers. These phase inaccuracies can be quite easily compensated via biasing. A wavelength sensitivity analysis reveals that that the optical bandwidth is ±20*nm*. Greater deviations from the nominal wavelength results with a distorted output.

## 6. Multi-threshold comparator for optical analog-to-digital conversion

Increasing the input voltage span reveals a periodic output. The output of the device may then be treated as if the comparator has multiple threshold levels. Figure 5 depicts the output of a device with *N* = 5 when the input voltage spans from 0*V* to 10*V*. The response of the (single-threshold) comparator of Figure 4(a) is duplicated 4 times.

In a previous publication [14], a similar comparator was presented based on silicon technology. In silicon photonics the Pockels effect is very weak and hence plasma dispersion effect is widely used. Unfortunately, this effect also exhibits a voltage-dependent loss, which, in turn, prevents the generation of a (pure periodic) multi-threshold response. InP technology which posses the Pockels effect, is suitable for obtaining a multi-threshold response with the comparator. A multi-threshold comparator is an extremely valuable building block in the construction of an efficient Flash A/D. An OADC scheme on that base was presented back in 1979 by Taylor [1]. The main idea of that scheme was to use a set of modulators whose lengths double with each modulator and where the output of each modulator is one bit. Each modulator exhibits a different response curve based on its (binary) weight. Several variations on Taylor’s scheme have been reported since (see [12]). The problem, however, with the original scheme as well as with the follow-up variations, is that electronic threshold-comparators have to be used in order to convert the analog intensity at the output of each modulator to a two-level digital signal.

The output periodicity of the multi-threshold comparator can be determined by lengthening or shorting the electrodes of the active phase shifters. It allows one to make multiple devices with different threshold levels for parallel multi-bit quantization of a shared electronic input. Employing the multi-threshold comparator proposed herein makes an electronic comparator superfluous and allows the output of the photodiodes to be wired directly to a digital processing system. Thus, associated delays can be avoided and overall system complexity is reduced. An advantage of using InP in making the optical comparators, is that a monolithic device can be built by integrating a laser diode, comparator, and photodiodes.

## 7. Conclusions

An optical device that has a step-like E/O response was presented. Its sharp transition response is achieved by employing a set of superimposed sine-wave responses differing in amplitude and phase according to a square-wave Fourier Series formula. A detailed design based on InP technology for the comparator was given. The calculated performances for a design based on InP technology predicts a sharp response and good distinction between digital low and high levels.

Increasing the arm count improves the approximation of the output to an ideal square wave and the output slope increases. The improvement comes at the expense of increased input optical power. It was demonstrated that a comparator designed with N = 5 arms is suitable to be used as a stand-alone comparator. No electronic comparator needs to follow the proposed optical comparator, to encode the data into two distinct levels.

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