1 Introduction

Since their advent in the second half of the 1980s [1], semiconductor optical amplifiers (SOAs) have been attracting unceasing interest for optimizing and exploiting their technological features and capabilities in research and commercial level. Early efforts put forward in this context investigated the potential of SOAs not only for optical amplification of data but also for data encoding [2, 3]. This goal seemed normal to achieve, given that it should be possible to directly modulate the SOA bias current, which is indispensable for the SOA operation, by a data-carrying electrical signal. However, both experimental [2] and theoretical results [3] revealed that the direct modulation process could only be realized on information being as fast as the SOA modulation bandwidth would permit. This bandwidth is of the order of 1 GHz [4] because the SOA differential carrier lifetime, which depends on the launched optical power, SOA driving current and carrier recombination time [5], is finite. For this fundamental reason, the SOA can be directly modulated only up to the point dictated by the inverse of the specific critical parameter. Else the SOA suffers from pattern effects and their negative complications [5]. Although this limitation inevitably shifted the focus on other SOA functionalities [6], nevertheless during the last years applications like radio over fiber [7], slow and fast light [8], colorless passive optical networks [9], data erasing/rewriting [10] and signal transmission using advanced optical modulation formats [11] have emerged, which can benefit from using SOAs, instead of other technologies, like electroabsorption modulators (EAMs), as intensity modulators. In fact, the capability of a SOA modulator for simultaneous amplification of the encoded signal differentiates it from an EAM, which suffers from a significant insertion loss, thus actually requiring integration of a booster SOA to obtain high output power and extend the transmission reach [12]. In addition, owing to the large SOA gain bandwidth, a SOA-based modulator inherently exhibits small wavelength dependence than an EAM whose optical bandwidth is typically lower. This property renders SOAs ideal candidates as external modulators in applications where wavelength-independent, i.e., colorless, operation is required, as, for example, in passive optical networks (PONs) (to which we refer in more detail in Sect. 2.2). Furthermore, the nonlinear transfer function of an EAM makes it unsuitable for applications where modulation linearity is required, as it happens in subcarrier multiplexing optical systems [13]. In contrast, the modulation nonlinearity can be improved by using a SOA as external modulator, which hence is more suitable for this purpose.

The significant comparative advantages of SOAs as intensity modulators have renewed strong interest in this subject and spurred systematic attempts for overcoming the performance limitations imposed by the low-pass characteristic of the SOA electrical response. One popular approach followed for this purpose is based on the fact that the SOA current modulation induces, via active medium refractive index changes, components of varying instantaneous frequency deviation [13]. Thus, if these components are properly tailored, the waveform distortion of the encoded signal at the SOA output can be suppressed. The most straightforward way of doing this is by means of optical filtering. Among different existing technologies [14–17], we have recently employed a birefringent fiber loop (BFL), either alone [18] or assisted by an optical band-pass filter [19]. In this paper, we propose to use a double-stage BFL for resolving the target problem, which so far has not been addressed using this combination of BFLs [20–22]. By cascading two suitably detuned BFL modules, a comb filter is formed whose passband is more narrow, sharp and selective than that of the single-stage BFL [23, 24]. Owing to these features. the pattern-dependent impairments due to the SOA direct current modulation are canceled more efficiently. The feasibility of the scheme is demonstrated by experiment and simulation. From these results, it is deduced that the BFL pair can better restore the quality of the encoded optical signal than the solitary BFL at a given bit rate and extend further the SOA direct modulation speed with acceptable performance. Therefore, the proposed configuration constitutes an efficient solution for supporting the functional role of SOAs as intensity modulators and hence for extending the suite of applications of these active elements beyond usual optical signal amplification and processing [25].

2 Experiment

2.1 Setup

Figure 1 depicts the experimental setup, which is an extension of that built in [18, 19]. A laser source provided at \(\lambda _{\mathrm{seed}}=1540\hbox { nm}\) center wavelength a continuous wave (CW) signal whose power and polarization were adjusted using a variable optical attenuator (VOA) and polarization controller (PC), respectively. This signal was launched at \(-5\) dBm into an SOA (CIP model XN-OEC-1550 of active region length 2 mm), which exhibited an input saturation power (defined at 3 dB gain compression from gain at \(-30\) dBm input power) of \(P_{\mathrm{in,sat}}\approx -17\,\hbox {dBm}\) and a noise figure of \({\mathrm{NF}}\approx 5\,{\mathrm{dB}}\) when biased at 250 mA. The SOA was directly modulated by a pulse pattern generator (PPG) with a non-return-to-zero (NRZ) data signal which incurred on the \(5\,\Omega\) SOA impedance a current variation of \(\pm 50\hbox { mA}\). The SOA was followed by two BFLs connected in such way that the output of the first one, OUT1, drove the input of the second one, IN2. Each BFL comprised a 50/50 dB coupler, a PC, which produced a rotation of \(90^\circ\) to light coming from both of its sides, and a piece of polarization-maintaining fiber (PMF) of birefringence \(B=3.3\times 10^{-4}\) and length \(L=8.5\, {\mathrm{m}}\). An erbium-doped fiber amplifier (EDFA 1) was placed between the two BFLs to compensate for the 8.5 dB insertion loss of the first BFL and restore the initial signal power level so that the second BFL was operated under the same working conditions as its other counterpart and the two BFLs could be combined in the most favorable way. Similarly, an optical isolator (ISO) between the BFL stages prevented backward reflections from causing undesirable oscillations that could distort the shape of the combined transfer function [20]. The role of these as well as of the other passive components, such as optical band-pass filters (OBPFs), that were involved in the setup was as described in [18, 19]. Finally, since the BFL was sensitive to environmental perturbations due to the manually controlled PC and the temperature dependence of the PMF [26], each BFL was packaged in a box. This enclosure, which has also been used for similar reasons in other types of comb filters [27], enhanced the operation stability of each BFL. This fact permitted to run the experiment over a longer time window than [18, 19] and subsequently to take measurements, which were important for verifying the capability of the scheme to enhance the SOA direct modulation potential, such as those for the error performance that are reported in the next section.

Fig. 1
figure 1

Experimental setup

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2.2 Practical implementation matters

The implementation of the double-stage BFL-assisted SOA-based modulator setup involves the use of bulky OBPFs and EDFAs. However, this fact does not necessarily compromise its practicality, since these components may anyway be indispensable in real applications of SOA-based modulators. One such characteristic application is that of PONs, where a SOA is primarily intended to be exploited as external modulator [9, 10]. In this case, the OBPFs emulate [14] array waveguide gratings, which are located at the remote and central nodes and are necessary elements for the interconnection of these nodes [28], while the EDFAs emulate the respective active modules, which compensate for transmission and distribution losses incurred along the bidirectional link [29]. Accordingly, the degree of practicality of the double-stage BFL configuration depends on the applications that the directly modulated SOA is intended to serve. Thus, in applications where there is a higher degree of freedom regarding the placement of the OBPFs and EDFAs, as in those which involve bidirectional signal transmission at some flexible distance [9, 10, 28], the use of the specific filtering scheme is more practical. If, on the other hand, the applications are such that the margins for separately placing these components are more tight [7, 8, 11], then the practicality is more limited, but nevertheless this limitation can be compensated by the performance benefits offered by the scheme.

On the other hand, enabling the direct modulation of the SOA with the double-stage BFL comes at the expense of higher total insertion losses, which are inevitable when this active module is followed by multiple optical filters [30]. Still after these losses are compensated by means of booster EDFAs, which as explained above are not necessarily extra modules, the remaining net gain of the SOA–BFL combination, i.e., 5.5 dB [18], is sufficient for the aforementioned transmission applications of the directly modulated SOA. Furthermore, by dynamically adjusting the SOA gain through fast control of the SOA injection current, the desired gain level offered in addition to signal modulation can be preserved [31].

2.3 Results

Following our previous relevant work [18, 19], the SOA current was directly modulated by an RF NRZ data signal of repetition rate \(B_{\mathrm{rep}}=5\hbox { Gb/s}\). However, the equivalent bandwidth of this excitation, \(B_{\mathrm{rep}}/2\) [32], exceeds the measured SOA modulation bandwidth, \(B_{\mathrm{mod,SOA}}=950\hbox { MHz}\). The negative by-product of this mismatch is that the signal inscribed on the CW input and transferred at the SOA exit is distorted. Nevertheless, this distortion can be sufficiently mitigated by means of the double-stage BFL configuration. More concretely, according to its principle of operation [33] each BFL has a periodic comb-like spectral response shown in Fig. 2. This response was obtained by connecting a broadband light source, first to each single-stage BFL and then to their combination, and displaying the output in an optical spectrum analyzer (OSA). The response consists of alternating maxima that are spaced apart by the free spectral range, \({\mathrm{FSR}}_{\mathrm{BFL}1}={\mathrm{FSR}}_{\mathrm{BFL}2}=\lambda ^2/(\mathrm{BL})=0.85\hbox { nm}\), as well as of notches that are situated in between at 18.5 dB lower intensity level. By adjusting the PC of each BFL so that the transmission minimum of the second BFL coincided with the transmission maximum of the first BFL, the instantaneous frequency deviation components could be efficiently filtered [34]. This allowed the SOA to be directly modulated at extended data speed than that permitted by its limited modulation bandwidth, and with better performance than when using only one BFL. The filtering efficiency is critically determined by the detuning [34], which in this work was defined as the difference between the CW seeding wavelength, \(\lambda _{\mathrm{seed}}\), and the wavelength of the nearest notch, \(\lambda _{\mathrm{notch}}\), i.e., \({\Delta }\lambda =\lambda _{\mathrm{seed}}-\lambda _{\mathrm{notch}}\), as schematically shown in Fig. 2. During the experiment, the detuning was achieved by fine-tuning of the CW laser wavelength relative to the notch position, which was kept fixed after being set through proper handling of the transfer function of each BFL [35]. In this manner, we could precisely control the detuning and subsequently find which was appropriate for the performance of the scheme. As illustrated in Fig. 2, the detuning is negative if the laser wavelength is located at the left-hand side of the reference notch, while it changes sign if this wavelength lies on the opposite side. Since the ultimate goal of filtering is to suppress the longer-wavelength spectral components that have been induced on the encoded signal due to the SOA direct modulation [18, 19], the detuning was negative so that the filter transmission decreased as the wavelength increased [34].

Fig. 2
figure 2

BFL(s) measured power responses. Solid line first BFL. Dashed line second BFL. Dotted line Double-stage BFL. FSR: Free spectral range equal to wavelength spacing between consecutive maxima. \({\Delta }\lambda\): Detuning between CW seeding wavelength, \(\lambda _{\mathrm{seed}}\), and wavelength of the nearest notch \(\lambda _{\mathrm{notch}}\)

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Fig. 3
figure 3

Pulse waveforms (upper row) and eye diagrams (lower row) at double-stage BFL output for different detunings: a \({\Delta }\lambda =-0.25\hbox { nm}\), b \({\Delta }\lambda =-0.15\hbox { nm}\), c \({\Delta }\lambda =-0.1\hbox { nm}\), d \({\Delta }\lambda =-0.05\hbox { nm}\). The dashed circle in d demonstrates the overshoot (see relevant discussion in Sect. 3.2)

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In Fig. 3, the experimental results, which have been obtained after PIN photodetection and monitoring in a digital communications analyzer (DCA) at 5 Gb/s SOA direct modulation, indicate indeed that the condition \({\Delta }\lambda <0\) must be satisfied in order to improve the quality of the encoded optical signal. Specifically, Fig. 3a shows that initially when the detuning was near the peak of the double-stage BFL transfer function, at \({\Delta }\lambda =-0.25\hbox { nm}\), there was no improvement on the profile of the temporal waveform. However, as the detuning was moved away from this point (Fig. 3b, c), the peak amplitude fluctuations between marks and between spaces, as well as the level of the spaces, were all gradually reduced. As the detuning tended closer to the reference notch, this reduction became more noticeable and eventually reached an equilibrium when \({\Delta }\lambda =-0.05\hbox { nm}\) (Fig. 3d). In fact, the encoded data stream became quite uniform and resembled that of the electrically applied signal. This signal reshaping was further supported by the form and histogram of the corresponding eye diagrams shown in the bottom row of Fig. 3. These results were obtained using a \(2^{31}-1\) long pseudorandom binary sequence (PRBS) from the PPG that directly modulated the SOA current. As the detuning was varied in the same range as before, the eye diagrams became more symmetrical and open. Also, the high- and low-bit levels were less scattered, as observed in the probability distributions of the marks and spaces. These improvements were notably more pronounced than those realized with one BFL only, which is evident when comparing against the pairs of results obtained for \({\Delta }\lambda =-0.05\hbox { nm}\), i.e., Fig. 3d versus Fig. 4, at a SOA direct modulation data rate of 5 Gb/s.

Fig. 4
figure 4

Pulse waveform (top) and eye diagram (bottom) at single-stage BFL output for \({\Delta }\lambda =-0.05\hbox { nm}\)

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Fig. 5
figure 5

Measured error probability (EP) versus detuning (\({\Delta }\lambda\)) at different data rates for a single- and b double-stage BFL

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Figure 5 shows that the applied detuning also influences the error probability (EP) that is attained at different SOA direct modulation data speeds. This holds both for single- (a) and double-stage BFL (b). For better depiction of the EP curves, the horizontal axis spans up to 0.1 nm, although, as explained, the appropriate detuning should be negative. As outlined in [36], the error probability can be estimated through the Q-factor. For this purpose, the knowledge of the mean and standard deviation of the peak power of the encoded logical bits is required. This information is contained in the histogram that is constructed from the photodetected voltage samples, which are recorded and stored by the DCA. After the Q-factor has been calculated, it can be linked directly to the error probability, based on the assumption that the signal and noise statistics follow the Gaussian distribution [32]. Then from Fig. 5a, b the following observations can be made: (a) For SOA direct modulation speed up to 2 Gb/s, the EP is smaller for the single- than for the double-stage BFL. This happens because the instantaneous frequency components incurred due to the SOA direct modulation are not yet so spectrally broadened and hence sufficiently discriminated. Consequently, their elimination by the steeper slope of the double-stage BFL response also cuts useful information carried by the encoded optical signal, which impairs its correct reception. Moreover, at \({\Delta }\lambda =-0.25\hbox { nm}\) the EP attains a minimum both for the single- and double-stage BFL, while the opposite occurs at the other extreme, \({\Delta }\lambda =-0.05\hbox { nm}\), where the EP is comparatively higher. (b) For SOA direct modulation speed over 2 Gb/s, the situation with regard to the EP is reversed. More specifically, as the data rate is increased above this borderline, the EP is shifted and becomes minimum at \({\Delta }\lambda =-0.05\hbox { nm}\), where the instantaneous frequency components, which have been strongly and distinctly manifested, can be suppressed more efficiently. Furthermore, the EP curves for the single- and double-stage BFL change slope and the local peak at 2 Gb/s in Fig. 5a is replaced by a dip. The double-stage BFL scheme achieves a better EP, which for \({\Delta }\lambda =-0.05\hbox { nm}\) is at least an order of magnitude lower than that of the single-stage BFL. Although the EP is increased as the bit rate gets higher, still even at 6 Gb/s the cascade of the two BFLs allows to keep the EP around \(10^{-3}\), which is the forward error correction (FEC) limit [15, 37], when this is not possible with the single-stage BFL. Thus, the proposed scheme extends the SOA direct modulation speed, which is beneficial for the target applications it is destined to serve.

3 Simulation

3.1 Model

Given the central role of the detuning on the performance of the double-stage BFL scheme, we further investigated its impact in order to quantify how it must be chosen. For this purpose, we conducted numerical simulation to provide more information for the fine adjustment of this critical parameter. This was done by considering and combining the responses of the SOA and BFLs. More specifically, the SOA amplitude response is obtained by numerically solving the first-order differential equation [38], which has been adapted so as to account for current transients [39]

$$\begin{aligned} \frac{\mathrm{d}h(t)}{\mathrm{d}t}=\frac{{\varGamma } gN_{o}\Bigg [\dfrac{I(t)}{I_{o}}-1\Bigg ]L_{\mathrm{SOA}}-h(t)}{T_{\mathrm{car}}}-\frac{P_{\mathrm{CW}}}{P_{\mathrm{sat}}T_{\mathrm{car}}}\Big \{\exp \big [h(t)\big ]-1\Big \} \end{aligned}$$
(1)

where h(t) is the power gain integrated over the SOA length, \(L_{\mathrm{SOA}}=2\, {\mathrm{m}}\)m, \({\varGamma }=0.25\) is the SOA confinement factor, \(g=3.3\times 10^{-20}\, {\mathrm{m}}^2\) is the SOA differential gain, \(N_{o}=0.15\times 10^{24}\, {\mathrm{m}}^{-3}\) is the SOA carrier density required for transparency, \(I_{o}=75\, {\mathrm{mA}}\) is the SOA current required for transparency, \(T_{\mathrm{car}}=312\hbox { ps}\) is the SOA carrier lifetime, \(P_{\mathrm{sat}}=10\,\hbox {dBm}\) is the SOA saturation power and \(P_{\mathrm{CW}}=-5\,\hbox {dBm}\) is the power of the CW optical signal. The time-varying bias current, I(t), is given by

$$\begin{aligned} I(t)=I_{\mathrm{dc}}-I_{\mathrm{m}}+ \sum _{k=-\infty }^{\infty } A_{k}I_{\mathrm{p}}(t-kT) \end{aligned}$$
(2)

where \(I_{\mathrm{dc}}=250\, {\mathrm{mA}}\) is the fixed bias level, \(A_{k}=\hbox {`1'}\) or ‘0’ depending on whether a mark or space, respectively, is contained in the k-th bit slot of the driving RF NRZ PRBS of repetition interval T, and \(I_{\mathrm{p}}(t)\) represents the stream of applied current pulses whose temporal shape is theoretically described by [40]

$$\begin{aligned} I_{\mathrm{p}}(t)= {\left\{ \begin{array}{ll} 2I_{\mathrm{m}}\left[ 1-\exp (-t^2/t_{\mathrm{r}}^2)\right] &{} 0\le t<T \\ 2I_{\mathrm{m}}\exp \left[ -(t-T)^2/t_{\mathrm{r}}^2)\right] &{} t>T \end{array}\right. } \end{aligned}$$
(3)

where \(I_{\mathrm{m}}=50\, {\mathrm{mA}}\) is the peak modulation current. The above expression allows to properly account for the experimental pulse modulation characteristics, namely for the pulse finite rise and fall times, which are both equal to \(t_{\mathrm{r}}\) and occupy approximately 17 % of the current bit period, as well as for the maximum and minimum amplitudes of the modulation current, which according to (2) are given by \(I_{\mathrm{dc}}+I_{\mathrm{m}}\) and \(I_{\mathrm{dc}}-I_{\mathrm{m}}\), respectively. The above values assigned to the SOA operating parameters ensure that there is computational consistency throughout the model, which is necessary in order to be able to obtain valid simulation results. Thus, the unsaturated single-pass amplifier gain, \(G_{o}=\exp \{{\varGamma } gN_{o}[(I_{\mathrm{dc}}/I_{o})-1]L_{\mathrm{SOA}}\}\), equals \(G_{\mathrm{CW}}/\exp [-(G_{\mathrm{CW}}-1)P_{\mathrm{CW}}/P_{\mathrm{sat}}]\) when substituting the measured SOA CW gain, \(G_{\mathrm{CW}}=14.5\,{\mathrm{dB}}\), as required according to [38]. Similarly, the SOA differential carrier lifetime value, 167 ps, is the same either if numerically calculated from \(\tau _{\mathrm{{d}}}=T_{\mathrm{car}}/(1+P_{\mathrm{CW}}G_{\mathrm{CW}}/P_{\mathrm{sat}})\), or experimentally extracted from \(\tau _{\mathrm{{d}}}=1/(2\pi B_{\mathrm{mod,SOA}})\) [5].

Based on the above formulation and using the numerical method outlined in [41], Eq. (1) was solved for h(t) to find the electric field of the encoded signal at the SOA output from [38]

$$\begin{aligned} E_{\mathrm{SOA}}(t)=\sqrt{P_{\mathrm{CW}}}\exp \Bigg [\Big (1/2\big (1-j\alpha _{\mathrm{LEF}}\big )h(t)\Big )\Bigg ] \end{aligned}$$
(4)

where \(\alpha _{\mathrm{LEF}}\) is the SOA linewidth enhancement factor. The value of this parameter is set using the fact that it relates the modification of the gain, \({\Delta } G\), that is induced by the upper and lower extremes of the SOA modulation current, to the variation per time increment, \({\Delta } t\), of the accompanying phase change, which defines the chirp, \({\Delta }\nu =\alpha \ln ({\Delta } G)/2{\Delta } t\) [42]. The involved quantities in turn are experimentally obtained from the static characterization of the SOA gain against bias current and from the dynamic measurement of the chirp at the SOA output [18]. Thus, taking and replacing \({\Delta } G=8.8\,{\mathrm{dB}}\), max\(({\Delta }\nu )=50\hbox { GHz}\), where the operator ‘max’ denotes the maximum value of the measured chirp, and \({\Delta } t=200\hbox { ps}\) gives \(\alpha _{\mathrm{LEF}}=10\), which lies within the typical range of values for SOAs operating around the 1550 nm spectral window.

On the other hand, the interconnection of the two BFLs in series is modeled through the product of their individual field transfer functions [20–22]

$$\begin{aligned} T_{\mathrm{BFL}1+{\mathrm{BFL}}2}(\lambda )=T_{\mathrm{BFL}1}(\lambda )T_{\mathrm{BFL}2}(\lambda ) \end{aligned}$$
(5)

where [26] \(T_{\mathrm{BFL}1,2}(\lambda )=e^{-j{\varGamma }(\lambda )/2}+j\sin [{\varGamma }(\lambda )/2]\), \({\varGamma }(\lambda )=(2\pi {\mathrm{BL}})/\lambda\), while, in direct accordance with the way the intensity transmission of BFL2 is adjusted against that of BFL1, \(T_{\mathrm{BFL}2}^2(\lambda )=1-T_{\mathrm{BFL}1}^2(\lambda )\). The parameter values of the BFLs are the same as those in the experiment. The squared modulus of (5) is plotted in Fig. 6, which clearly shows that compared to the single-stage BFL the FSR and slope of the concatenated BFLs response become halved and steeper, respectively. The detuning of BFL1 is set at the transmission maximum, and the BFL2 detuning is varied at the left-hand side of the wavelength notch so that filtering is realized as in the experiment.

Fig. 6
figure 6

Spectral response of single- (dashed line) and double-stage (solid line) BFL

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Using (4) and (5), the optical signal power at the output of the BFLs cascade is obtained from [26]

$$\begin{aligned} P_{\mathrm{BFL}1+{\mathrm{BFL}}2}(t)&=|E_{\mathrm{BFL}1+{\mathrm{BFL}}2}(t)|^2,\nonumber \\ E_{\mathrm{BFL}1+{\mathrm{BFL}}2}(t)&=F^{-1}\Big \{F\big [E_{\mathrm{SOA}}(t)\big ]T_{\mathrm{BFL}1+{\mathrm{BFL}}2}(\lambda )\Big \} \end{aligned}$$
(6)

where F[.] and \(F^{-1}\{.\}\) are the Fourier transform and its inverse, respectively. These functions are executed in MATLAB software. In this manner, the mean peak power of the encoded pulses, \(\bar{P_{1}}\) for marks and \(\bar{P_{0}}\) for spaces, can be found and replaced in the Q-factor. This metric is defined as [32]

$$\begin{aligned} Q=\frac{\bar{P_{1}}-\bar{P_{0}}}{\sigma _{1, {\mathrm{pe}}}+\sigma _{0, {\mathrm{pe}}}} \end{aligned}$$
(7)

where \(\sigma _{1, {\mathrm{pe}}}^2/\sigma _{0, {\mathrm{pe}}}^2\) is the variance of the peak powers of the marks/spaces due to the pattern effect caused by the SOA direct modulation. Then, the error probability (EP) can be extracted from the Q-factor according to \({\mathrm{EP}}=\frac{1}{2}{\mathrm{erfc}}\bigg (\dfrac{Q}{\sqrt{2}}\bigg )\), where erfc(.) is the complementary error function.

3.2 Results

Initially, we validated the model against experiment by comparing the results obtained at the SOA output. This validation was done at 5 Gb/s to ensure continuity with our previous relevant work on the addressed research topic [18, 19], but also because, as it will be shown in the following, this data rate is the border which distinguishes the double- from the single-stage BFL in terms of the capability to allow SOA direct modulation with enhanced performance. From Fig. 7, it can be seen that the form of the simulated modulated data frame is identical to the experimental one. Moreover, the pseudo-eye diagram (PED), which is generated as explained in [43], captures the distortion characteristics observed in the real measurement. Therefore, there is a good matching between the respective pairs of results, which supports the suitability of the model to make correct and reasonable predictions.

Fig. 7
figure 7

a Modulated data pattern and b eye diagram at SOA output. Left column Experimental results. Right column Simulation results

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Next, the scope of specifying the detuning was extended beyond demonstrating the proof of principle of the double-stage BFL scheme, by providing further insight into the influence of this parameter on the EP. The results are shown in Fig. 8 and have been obtained by scanning the detuning with a precision of 0.01 nm, which was not practically possible during the experiment. Because the \(2^{31-1}\) bit-long PRBS used for the EP measurements is prohibitive for conducting simulations, we employed a 127 bit-long PRBS, which is sufficient for acquiring the EP at the examined data rates in a physically correct and computationally efficient manner [44]. For this reason, the simulated SOA direct modulation-induced pattern effect and its compensation are less intense than in the experiment. Thus, compared to Fig. 5 the EP curves are much deeper and deviate more distinctly from each other at 4 and 6 Gb/s, while at 2 Gb/s, where the pattern effect is not as pronounced as at 4 and 6 Gb/s, the curve drops so much that there is no essential difference between the single- and double-stage BFL. Furthermore, the finer resolution of the detuning allows to observe that as the data rate gets faster the detuning must be shifted closer to the notch in order to minimize the EP. We also confirm the experimentally drawn conclusions that (a) the EP scales inversely with the SOA direct modulation data rate, (b) at 4 and 6 Gb/s, the EP minimum is lower for the double- than for the single-stage BFL, (c) at 6 Gb/s, the minimum EP achieved with the double-stage BFL differs by an order of magnitude against that attained with the single-stage BFL. Especially at 6 Gb/s, we can verify that, similar to the experiment, this difference can result in the EP being acceptable with the double-stage but not with the single-stage BFL. For this purpose, we compensated for the reduced PRBS size and intentionally worsened the simulated pattern effect and EP by appropriately intervening on the SOA response. This was efficiently done by adopting the approach of numerically increasing the SOA carrier recombination time [45]. This action affects analogously the SOA differential carrier lifetime, \(\tau _{\mathrm{{d}}}\), and consequently reduces the SOA modulation bandwidth, \(B_{\mathrm{mod,SOA}}\). Thus, by running the model using \(T_{\mathrm{car}}=750\hbox { ps}\), which by the way is a physical possibility, we can see in Fig. 9 that the EP is brought close to, but still below, the FEC limit, thanks to the double-stage BFL, while this is not possible with its single-stage counterpart. Moreover, the minimum EP occurs at \({\Delta }\lambda =-0.05\hbox { nm}\), which agrees with, and hence supports, the experimental evidence. These trends hold similarly for 4 Gb/s provided that T car is further increased by 80 %, i.e. more than 6/4 times, from the numerical value employed at 6 Gb/s, else the corresponding EP curve is just shifted upward from its position in Fig. 8.

Fig. 8
figure 8

Simulated error probability (EP) versus detuning (\({\Delta }\lambda\)) at different data rates for a single- and b double-stage BFL. The exponent ‘n’ indicates the EP order when this metric is extremely small [32], i.e., \(\hbox {n}>15\) for \(\hbox {EP}<10^{-15}\)

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Fig. 9
figure 9

Simulated error probability (EP) versus detuning (\({\Delta }\lambda\)) at 6 Gb/s for a single- and b double-stage BFL when the SOA carrier recombination time is numerically increased to 750 ps. The horizontal dashed line denotes the forward error correction (FEC) limit

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Furthermore, we accounted for the waveform distortion on intervals of consecutive encoded pulses. Due to the finite SOA differential carrier lifetime, the peak amplitude at the leading pulse edge is not the same as in the subsequent time slots. This temporal overshoot (OVS) may affect the operation of the target applications employing a SOA as external modulator unless it is kept below 25 % when defined as in [46]. For this reason, we investigated its effect on the performance of the double-stage BFL by plotting in Fig. 10 (with T car set back to 312 ps) its variation at 2, 4 and 6 Gb/s over a wide detuning range. From this figure, it can be seen that for all data rates up to 6 Gb/s the overshoot is initially zero but from \({\Delta }\lambda =-0.15\hbox { nm}\) it is gradually increased, until \({\Delta }\lambda =-0.08\hbox { nm}\) where it inclines sharply and becomes unacceptable. The existence of OVS when the double-stage BFL is detuned at \(-0.05\) nm is experimentally confirmed at 2 and 4 Gb/s (see insets in Fig. 10), while above 5 Gb/s the OVS is hardly visible (for this reason, it is indicated with the help of the dashed circle in Fig. 3d-top) due to the limited DCA bandwidth and the associated negative implications on the OVS [47]. However, from this data rate onwards the OVS exceeds 25 %, as it can be seen in the simulated pattern and PED profiles (Fig. 11a), which have been obtained at 5 Gb/s for the same reasons explained at the beginning of this section. This means that a trade-off is necessary for selecting the detuning so that it is the best for both EP and OVS metrics. Thus, by choosing \({\Delta }\lambda =-0.08\hbox { nm}\), the OVS is dropped to the tolerable level of 16 % and the signal quality is significantly improved, as shown in Fig. 11b by the reduction in the transient spikes at the beginning of repetitive marks, which become more equalized, and the rectification of the trailing edge of the PED so that its form becomes more rectangular. Concurrently, the experimental and theoretical curves indicate that the EP is acceptable at 2 and 4 Gb/s and quasi-acceptable at 6 Gb/s. Therefore, when detuned in this manner the double-stage BFL enables fast SOA direct modulation with overall enhanced performance.

Fig. 10
figure 10

Simulated overshoot versus detuning (\({\Delta }\lambda\)) at different data rates. The horizontal dashed-dotted line denotes the overshoot (OVS) limit. Insets Encoded data patterns at 2 and 4 Gb/s

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Fig. 11
figure 11

Double-stage BFL output for different detunings: a \({\Delta }\lambda =-0.05\hbox { nm}\), b \({\Delta }\lambda =-0.08\hbox { nm}\). Upper row Output data. Lower row Pseudo-eye diagrams with overshoot (OVS) definition

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4 Conclusion

In conclusion, we have demonstrated that an optical notch filter comprising of two cascaded birefringent fiber loops can be employed to overcome the electrical bandwidth limitations of a directly modulated SOA. The obtained experimental and simulation results confirm that if properly detuned then the proposed scheme can realize a manifold increase in the SOA direct modulation speed with better performance with regard to encoded signal reshaping and error probability than its single-stage BFL counterpart. This proven potential, which can be enhanced if the SOA electrical modulation capability is inherently assisted by the structure, fabrication, packaging and impedance matching characteristics of the SOA device itself [7, 48–50], renders the double-stage BFL a notable configuration for allowing SOAs to be exploited as intensity modulators in diverse applications.