## Abstract

In this paper, a photonic temporal integrator based on an active Fabry-Perot (F-P) cavity is proposed and theoretically investigated. The gain medium in the F-P cavity is a semiconductor optical amplifier (SOA) with high gain coefficient. Key feature of the proposed photonic integrator is that the length of integration time window is widely tunable and could be ideally extended to infinitely long when the injection current is approaching lasing condition. Based on an F-P cavity with practically feasible parameters, a photonic temporal integrator with an integration time window of 160 ns and an operation bandwidth of 180 GHz is achieved. The time-bandwidth product of this photonic temporal integrator is 28,800, which is about two-orders of magnitude higher than any previously reported results. Gain recovery effect has been also considered and analyzed for the impact on performance of the photonic integrator, followed by the simulation results of the impact of gain recovery.

© 2014 Optical Society of America

## 1. Introduction

In the past tens of years, information technology has witnessed a tremendous progress in optical communication systems, which are superior to electronic systems in many applications [1]. As a result of their severe speed limitation, traditional electronic systems cannot afford a high signal processing speed, and thus they are limited to offer a broad operation bandwidth. Fortunately, all-optical signal processing techniques could be used to overcome these limitations compared to their electronic counterparts. An all-optical processor inherently has the outstanding features of high signal processing speed, low power consumption and broad operation bandwidth.

A photonic temporal integrator is a basic building block to create more complex signal-processing and computing optical platforms. Such a device performs the temporal integral of an arbitrary input optical signal, which has many important applications such as optical dark soliton detection [2], pulse-shaping [3], optical memory [4] and photonic analog-to-digital conversion [5]. Several kinds of photonic temporal integrators have been proposed and experimentally demonstrated based on different optical components and systems, including fiber Bragg grating (FBG), ring resonator and a time-spectrum convolution system. N. Q. Ngo reported a photonic integrator based on a phase-shifted FBG working in transmission [6]. Y. Park *et al.* used a uniform FBG working in reflection to implement a photonic integrator which could integrate an optical pulse with a time width down to a few picoseconds (~6 ps) [3]. M. Ferrera *et al.* reported a photonic integrator using a microring resonator compatible with CMOS technology and experimentally proved that the temporal integration window could reach as long as 800ps with an operation bandwidth of 200 GHz [7]. In addition, a real-time and single-shot ultra-fast photonic time-intensity integrator for arbitrary temporal waveforms has been proposed and demonstrated based on a time-spectrum convolution system [8]. These reported photonic temporal integrators are implemented based on passive components, which necessarily translates into a limited and fixed integration time window. However, for an ideal integrator, the integration time window should be infinite. This feature is critical for many practical applications, particularly to create all-optical memory units with sufficiently long hold/storage times. Since a limited temporal integration window of a passive photonic integrator is fundamentally caused by the loss of the cavity, a loss-compensation mechanism should be introduced. To achieve a large temporal integration window, an optical filter with gain medium (e.g. an active resonator) is required. In this regard, an experimental demonstration of employing a gain medium to implement the optical integration has been realized by using an all-fiber active filter based on superimposed fiber Bragg gratings [9]. However, in practice, this design has proved to be limited to operation bandwidth in a few tens of GHz range [9].

In this paper, a photonic temporal integrator based on an integrated *active Fabry-Perot* (F-P) cavity is proposed and theoretically investigated. The gain medium in the F-P cavity is a semiconductor optical amplifier (SOA) with high gain coefficient. A key feature of the proposed photonic integrator is that the length of its integration time window can be widely tunable and could be ideally extended to be infinitely long by simply changing the injection current. In addition, SOAs can be easily integrated with other semiconductor devices, such as laser, modulator and photodetector in photonic integrated circuits, which is another critical feature for achieving integrated all-optical signal processing circuits with complicated functionality. Based on the use of an F-P cavity with practically feasible parameters, a photonic temporal integrator with an integration time window of 160 ns and an operation bandwidth of 180 GHz is successfully demonstrated through numerical simulations. The time-bandwidth product (TBP) of the simulated photonic temporal integrator is 28,800, which is about two-orders of magnitude higher than any previously reported results. A preliminary study of the technique has recently been presented by us [10], and a detailed analysis of the performance and limitations of the proposed integrator are also evaluated in this paper.

## 2. Operation principle

The integration of an input signal *f* (*t*) can be calculated by convolution between *f* (*t*) and the temporal impulse response of an ideal integrator, which is a step function *u*(*t*) (*u*(*t*) = 1, for *t*≥0 and *u*(*t*) = 0, for *t*<0) in the time domain. The output signal is expressed as

*ω*is the optical frequency,

*ω*

_{0}is the carrier frequency of the signal to be processed, $\mathbb{F}$is the Fourier transform operator, (

*ω-ω*

_{0}) is the baseband frequency variable,

*j*= $\sqrt{-1}$, and

*U*(

*ω*) is the Fourier transform of

*u*(

*t*) [9]. It can be seen from Eq. (2) that an optical device with a frequency response proportional to 1/

*j*(

*ω-ω*

_{0}), operates as a photonic integrator over the temporal complex envelope,

*f*(

*t*), of the incoming optical waveform.

Since the step function is fairly difficult to obtain in practice, researchers have employed several optical components [6,11] to simulate an ideal integrator and to implement photonic integrators experimentally [3,7,12]. Since an FP-cavity-based passive filter exhibits a similar temporal impulse response with an exponential decay of the stored light over time, mainly due to the facets leakage loss, it can be used as a photonic integrator but over a fixed and limited integration time window [7]. In this work, to realize an ideal photonic integrator with ultra-long and tunable integration time window, we employ a novel photonic temporal integrator based on an integrated *active F-P cavity*.

Figure 1(a) shows a concept diagram of an optical integrator, which helps to understand how it works. Figure 1(b) shows the schematic diagram of an active F-P cavity with proper reflectivity coatings on both facets. When an optical pulse is launched into the F-P cavity, it will be reflected by the two mirrors, in such a way that the output optical signal is the result of overlapping (with the same phase) of each transmitted pulse at the output end in the time domain. In a passive F-P cavity, the optical pulse in the cavity will be attenuated quickly due to the cavity loss and energy attenuation by partial reflection in the mirrors. Therefore, the integration time window is limited due to the overall energy loss in the optical cavity. Here, an SOA is inserted into the cavity. The pulse will be amplified by the gain medium to compensate the cavity energy loss, which in turn leads to the length extension of the integration time window. By tuning the gain coefficient in the cavity, the length of the integration time window can be changed. In principle, an optical pulse could be locked in the cavity to achieve an infinitely long integration time window when the loss is perfectly compensated.

Assuming that *E _{in}*(

*ω*) and

*E*(

_{out}*ω*) are the input and output optical signals, respectively. The output signal can be derived by superposition of infinite round-trips of the transmission part of

*E*(

_{oi}*ω*). We have

Based on Eq. (3), the transfer function of an active F-P cavity is given by [13,14]

*G*= exp[(

_{S}*Γg*-

_{m}*α*)

*L*] = exp(

*g*),

_{net}L*R*

_{1},

*R*

_{2}are the reflectivities of the input and output mirrors, respectively.

*Γ*is the optical confinement factor,

*g*denotes the material gain coefficient,

_{m}*α*is the total propagation loss in the cavity (SOA),

*τ*=

*nL/c*is the single-pass time delay,

*n*is the refractive index of SOA,

*g*is the net modal gain, and

_{net}*c*is the light speed in vacuum. It can be seen from Eq. (4) that when $\sqrt{{R}_{1}{R}_{2}}{G}_{S}$ approaches to 1,

*H*(

*ω*) can be approximated in the vicinity of the resonance frequency ω

_{0}:

_{0}is one of the resonant frequencies of the F-P cavity. Comparing Eq. (2) and Eq. (5), it can be concluded that the transfer function of an active F-P cavity matches well with the one of an ideal photonic temporal integrator at the resonant frequency when $\sqrt{{R}_{1}{R}_{2}}{G}_{S}\approx 1$. The integration time window of an optical integrator is defined by the decay time at (e.g.) 80% of its impulse response power. It is worth noting that when $\sqrt{{R}_{1}{R}_{2}}{G}_{S}$ approaches to 1, the energy loss in the F-P cavity is reduced thanks to the gain medium in the cavity, and accordingly the stored energy in the cavity (or the integration time window) is increased. In the other word, the decay time of an optical pulse propagating inside the cavity will be extended thanks to the energy loss compensation by SOA in each round trip. It is well known that quality factor (Q factor) is proportional to the ratio between stored energy (

*E*) in the cavity and energy loss

_{se}*E*due to the facet leakage, i.e.,

_{el}*E*. Therefore, the Q factor of a photonic temporal integrator determines the length of integration time window of a photonic temporal integrator.

_{se}/E_{el}## 3. Numerical analysis

To verify the theoretical analysis of the proposed photonic temporal integrator, numerical investigations are carried out as follows. The key to realize a photonic temporal integrator is to simulate an optical device with a transfer function approaching that of an ideal integrator as closely as possible. All the numerical analysis reported here is based on an F-P cavity with practically feasible parameters, as summarized in Table 1 [15,16,18].

Figure 2(a) shows the amplitude response of the proposed
photonic temporal integrator with a frequency spectral range (FSR) of 66.5 GHz for
*g _{net}* = 0.97 cm

^{−1}and other parameters presented in Table 1. The transmission peaks occur at the F-P resonant frequencies. Figure 2(b) shows that there is a pi phase shift at the resonance frequency which is identical to an ideal photonic integrator. In addition, it is worth noting that the quality factor of the transmission resonance is largely increased by increasing the gain coefficient in the cavity.

When the Q factor of an F-P cavity is tuned to 1 × 10^{9} (corresponding to a net
modal gain of 1.5047 cm^{−1}), an optical Gaussian pulse with a 3-dB time width
of 12.5 ps is introduced into the cavity as shown in Fig.
3(a). The output signal is a step-like waveform which agrees well with the output of an ideal
integrator. The length of the integration time window is about 68 ns which is at least one order
of magnitude higher than for any available passive photonic integrator [3,7]. To further investigate the
integration functionality of an active F-P cavity, as shown in Figs. 3(b) and 3(c), two Gaussian pulse pairs
in-phase and out-of-phase with a time spacing of about 820 ps and time width of 12.5 ps are
launched into the cavity. When the pulse pair are in-phase, the output signal is a result of
constructive superposition of two step-like waveform with an 820 ps time delay. However, when
there is a pi phase shift between the pulse pair, the output is a square-like waveform due to
the phase cancelation. This feature could be used for resetting purposes in an all-optical
memory unit [4]. The first input pulse functions as a
write signal (Set) and the second pulse with a pi phase shift to the first pulse functions as an
erase signal (Re-Set). Finally, a square waveform is successfully integrated into a triangular
waveform, as shown in Fig. 3(d).

In addition, as can be seen in Figs. 3(b) and 3(c), the pulse pair will result in different integration results, depending on their phase difference. Thus when the amplitude and phase of a reference pulse is available, the amplitude and phase of an unknown pulse can be measured according to their integration results. This is also a promising application for both amplitude and phase measurements [19] of the photonic integrator.

## 4. Discussions

To further evaluate and improve the performance of the proposed photonic integrator, some key parameters are investigated and discussed.

#### 4.1 The effect of Q factor on the length of temporal integration window

We first investigate the Q factor of the active F-P cavity. When the injection current approaches
the threshold current (lasing condition), the transmission peaks become narrower and a large Q
factor will be obtained, which is particularly desirable for a photonic integrator. The
relationship between the Q factor of an active F-P cavity and the injection current is shown in
Fig. 4(a). The x-axis is given by the ratio between absolute current and threshold current
I_{th}. Parameters used in the calculation are listed in Table 1.

It can be seen in Fig. 4(a) that when the injection current is smaller than the threshold current, the Q factor of the active F-P cavity monotonically increases as the injection current is increased. In particular, when the injection current becomes extremely close to the threshold current, the Q factor experiences a sharp increase and will reach an infinitely large value at the threshold current value. It can be concluded that the Q factor of the active F-P cavity is tunable and can achieve an extremely high value, ideally approaching infinity, by tuning the injection current.

The influence of Q factor on the length of temporal integration window is then investigated, as shown in Fig. 4(b). When the Q factor is increased from 1 × 10^{9} to 3 × 10^{9} (corresponding to a net modal gain of 1.505 cm^{−1} with other parameters listed in Table 1), the length of the integration time window is extended from 68 ns to 160 ns, while keeping the nominal operation bandwidth of 66.5 GHz. When the Q factor is 3 × 10^{9}, the time-bandwidth product of the photonic temporal integrator is then estimated to 28,800, which is about two-orders of magnitude higher than in any previously reported photonic integrator (Here TBP is defined as the multiplication of the integration time window and the broadest operation bandwidth (FSR of the cavity) of the input signal, which is equivalent to the definition in [7] only that the fastest signal that can be accurately processed is featured by its frequency bandwidth). These results verify that the proposed active photonic integrator based on an active F-P cavity is able to extend the temporal integration window well beyond the capabilities of a passive integrator by simply tuning the injection current.

#### 4.2 Operation bandwidth

Another key parameter of a photonic temporal integrator is its operation bandwidth, which is
mainly determined by the FSR of the optical resonator [7]. FSR can be calculated by *FSR = c/2nL*, where *n* is
the refractive index and *L* denotes the length of the cavity. By tuning the
refractive index and cavity length, FSR can be changed. Variation of injection current would
lead to the change of the refractive index of the active region. Given a typical threshold
carrier density of 2 × 10^{24} m^{−3} [15,16], the changes of FSR versus the
carrier density over a wide range is shown in Fig. 5(a).
It can be seen that the FSR cannot be largely tuned by simply changing the carrier
density.

The other way to realize a large FSR is to reduce the cavity length [7], but this would simultaneously lead to a reduced gain. This problem can be solved by selecting a SOA with a large modal gain. Three kinds of SOAs (i.e., quantum wells, quantum dashes and quantum dots) are investigated and compared as the gain medium. Quantum wells based SOA has the largest modal gain. On the other hand, quantum dots and dash based SOA has a larger material gain but a smaller modal gain due to a poor optical confinement factor [17,18,20]. In practice, a net modal gain as high as 105 cm^{−1} can be easily achieved by tuning the injection current based on Quantum wells structures [20]. The relation between gain and cavity length is given by *g _{0}* = exp(

*g*), where

_{net}L*g*is gain,

_{0}*g*=

_{net}*Γg*-

_{m}*α*is the net modal gain and

*L*is the cavity length. When the integration time window is kept as constant (68 ns), i.e. a fixed Q factor, the relation between operation bandwidth and net modal gain is shown in Fig. 5(b). Obviously, a higher net modal gain corresponds to a larger operation bandwidth. With an available SOA reported in [18], when the net modal gain is 4.1 cm

^{−1}, an operation bandwidth of about 180 GHz could be achieved.

#### 4.3 The effects of gain recovery time of SOA on F-P active integrator

In the above analysis, we have assumed that the gain of SOA is tuned just by the injection current and it keeps constant for each round trip of the optical pulse. The gain recovery effect of SOA induced by the input optical pulse has not been taken into consideration. In fact, when an optical signal pulse is launched into the SOA, it will consume the carriers and induce a decrease in the gain and then an increase in the refractive index of the active region of the SOA. If a light pulse train with binary formats passes through the SOA, the “1” bit, which corresponds to a high light power, will consume part of the carrier and the gain then decreases; during the “0” bit that corresponds to a low light power, the carriers and gain would recover to its steady state or initial state. The gain recovery time ranges from a few picoseconds to several hundred picoseconds, depending on the structures, injection current of the SOA and characteristics of the input light signal. This nonlinear feature (known as self/cross gain modulation and self/cross phase modulation) makes SOA a widely used device for application of wavelength conversions [21–23]. But for our proposed F-P active integrator, this feature is adverse as it may decrease the Q factor at the peak power of the input pulse and affects the frequency response of the SOA.

Many theoretical models have been built to investigate the gain recovery effects [17,24–26]. In these models the following carrier density rate equation [27] is used:

*N*is the carrier density,

*V*is the volume of the active region of SOA,

*q*is the electron charge,

*I*is the injection current,

*A*is the surface and defect recombination coefficient,

*B*is the bimolecular recombination coefficient,

*C*is the Auger recombination coefficient,

*h*is the planck constant,

*ν*is the input light frequency, and

*P*denotes the average light power in the SOA. From Eq. (6), it can be seen that the light signal launched into the SOA would cause a reduction of carrier density and hence a decrease of gain. By using a similar model and simplifications in [28], we have calculated the gain recovery time of the SOA with parameters listed in Table 2 through a pump-probe method.

_{av}The light signal pulse launched into the SOA is Gaussian shape with a 3-dB time width of ~12.5 ps (FWHM) and a peak power of 10 mW. The numerical simulation of normalized gain variation versus time is plotted in Fig. 6 (a). From Fig. 6(a), it can be seen that the injection of the pump signal causes a gain compression from its steady state value due to a depletion of carrier density. When the pulse exits the SOA, the gain recovery process starts immediately. This complex gain recovery process includes current injection, carrier density recovery, intraband and interband effects etc [29]. After a few tens of picoseconds the gain recovers to 90% of its initial value and totally recovers after a longer time. Specific analysis of gain recovery process can be found in [29]. The transient gain profile of the SOA is presented in Fig. 6(b). From this figure, it can be seen that for the first trip of the input optical signal, the carrier density does not need the recovery process and the input signal experiences a flat gain. However, from the second trip on, the input optical pulse would experience a different gain at different point along the SOA due to the carrier density depletion and recovery process. It can be seen that the gain at different point along the SOA recovers to a different value, depending on the gain recovery time.

The nonuniform profile of the gain will lead to a modification of the frequency response of the F-P SOA. In Fig. 1, the optical pulse experiences a flat and identical gain for each round trip, but when taking the gain recovery process into consideration, the optical pulse would experience a different gain for each trip in the SOA. By assuming that the optical pulse would experience an identically nonuniform gain variation from the second trip on, as shown in Fig. 6(b), for simplisity, we can easily obtain and simplify the frequency response of the F-P SOA as:

*g*(

*l*) is the net modal gain profile. Comparing to Eq. (4), the gain expression of

*G*in Eq. (6) is different from

_{nu}*G*and is smaller than

_{S}*G*. This deviation is caused by the gain recovery process and gain nonuniform. In this situation, we have calculated and compared the integration curves presented in Fig. 7, for different gain recovery times with parameters listed in Table 1 and 2.

_{S}Figure 7(a) shows the gain dynamics of the SOA after launching an optical pulse with different carrier lifetime. Using this model, we have found that the gain recovery time is closely related to the carrier lifetime as a small variation of the carrier lifetime will lead to an obvious change of gain recovery time. This can also be seen from the two different curves in Fig. 7(a). When the carrier lifetime is 20ps and 10ps, the gain recovery time is 116ps and 55ps, respectively. These are also consistent with results in [28]. Figure 7(b) presents the comparisons of integration results, from which we can evaluate the influence of the gain nonlinear effect on the temporal integration window. In Fig. 7(b), a gain recovery time of 55ps corresponds to a temporal integration window of 4.5ns and it is obvious that a longer gain recovery time leads to a decrease of the temporal integration window due to a reduction of the gain and the Q factor.

Since the temporal integration window is strongly dependent on Q factor, which is closely related to gain that has been demonstrated in section 2, a decrease of gain would directly lead to a low Q factor and hence a short temporal integration window. As result, a fast gain recovery speed and short gain recovery time is required for a good performance of this proposed photonic integrator based on F-P SOA. Both theoretical investigations and experimental results have shown that a larger injection current, a higher differential gain of the active region, a short carrier lifetime and increasing the length of active region [25,26] will promote the gain recovery process. Besides, some other authors found that by introducing a carrier storage region near the active region, the gain recovery time can be shortened [30]. In this approach, when a pulse passes through the active region of SOA, it depletes part of the carrier and thus a carrier density gradient is formed between the active region and the carrier-storing region, resulting in carrier diffusion from the latter region to the former region and this carrier diffusion lead to an acceleration of carrier band-filling process. Using this carrier compensation mechanism, a 100 ps gain recovery time has been obtained [31]. In this regard, we suggest that the amplifiers should be implemented with optimized structures that have built-in carrier storage region near the active region for a fast gain recovery speed. Moreover, with a lower input pulse energy, a short gain recovery time can also be achieved as the light pulse signal consumes only a small part of the carrier, leading to a small variation of the gain of the SOA and thus the Q factor of the F-P SOA almost keeps constant, which means that the gain recovery has only a slight impact on the performance of the photonic integrator based on SOA.

#### 4.4 The effects of the noise figure on the integration results

No matter whether it is used for signal amplification or signal processing, the noise figure of the gain medium is always of great importance. The noise typically comes from the amplified spontaneous emission (ASE). In [32], it is pointed out that both the gain and the internal loss of the gain medium play important roles in the noise figure. A large internal loss corresponds to a large value of noise figure and thus a reduced single-pass gain, which will decrease the integration time window. Furthermore, the noise floor of the output result may also deteriorate due to the ASE noise.

#### 4.5 Practical challenges for implementing an optical integrator

When fabricating such an SOA, the practical considerations mainly include the uniformity of the material gain coefficient, facet reflectivity and the chip length. In the simulations, we have assumed that the gain profile is uniform. However, in practical cases, the gain may show some fluctuations. And we should first test and select a chip with relatively uniform gain profile before the implementations. Furthermore, the cavity length should be compromised, since a short cavity length on one hand corresponds to a large FSR (integration bandwidth), but on the other hand it will decrease the single-pass gain (*G _{S}* = exp[(

*Γg*-

_{m}*α*)

*L*] = exp(

*g*)). To solve this problem, materials with high gain coefficient should be preferred in order to simultaneously obtain a large FSR and a relatively large single-pass gain.

_{net}LFurthermore, the stability is a key problem both for passive and active cavity. For the F-P SOA, the main factor leads to the cavity instability is the temperature fluctuation of the gain medium. A solution to this problem is that a temperature-controlling system, such as a heatsink and a thermoelectric cooler, should be attached to the gain medium to reduce the cavity instability.

In the simulations, we have pointed out that in order to obtain a large integration time window (a large Q factor), the injection current should be as close to the lasing threshold current as possible, which is a crucial requirement for the practical implementations. In fact, the question that how close the injection current can approach the lasing condition mainly depends on the accuracy of the current source and the gain stability of SOA. To obtain a higher Q-factor, i. e, longer integration time window, a high precision current source and a SOA with high stable gain efficiency are required in practical implementations.

## 5. Conclusion

In this paper, a photonic temporal integrator based on an active F-P cavity was proposed and theoretically investigated. A key feature of the proposed photonic integrator is that the length of integration time window is widely tunable and could be ideally extended to be infinitely long by changing the injection current. Based on an F-P cavity with practically feasible parameters, a photonic temporal integrator with an integration time window of 160 ns and an operation bandwidth of 180 GHz is achieved. Besides, the gain recovery effects is also investigated and analyzed and methods to mitigate this effect are presented, such as improving the differential gain, a built-in carrier storage region. Practical considerations for implementing an optical integrator are also evaluated. We are convinced that these entire analysis pose a practical guide towards the realization of the SOA based photonic temporal integrator.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China under 61377002, 61321063, and 61090391. Ming Li was supported in part by the “Thousand Young Talent” program.

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