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Abstract
We experimentally demonstrate the control of the hysteresis shape of a bistable optical signal at a location downstream of the nonlinear photonic resonator within which the bistability is generated; this shape control mechanism is performed without any signal sent upstream to the bistable system. The downstream shape-control element consists of a fiber-optic polarization controller and linear polarizer in series, and the nonlinear resonator generates an optical signal whose bistable branches exhibit different states of power and polarization. The diversity of demonstrated hysteresis shapes include the canonical counter-clockwise (CCW) shape (S-shape), the clockwise (CW) shape (inverted S-shape), and butterfly shapes; since all shapes originate from the same bistable signal, all shapes exhibit the same switching input powers. Moreover, the shape-selection process enhances the bistable switching contrast to 20 and 21 dB for the CCW and CW shapes, respectively. The shape-selection technique is demonstrated using a Fabry-Pérot semiconductor optical amplifier, is applicable to other nonlinear photonic resonators, and provides flexibility to future combinational and sequential all-optical signal processing applications.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical bistability has been studied widely as a means of achieving a range of sequential and combinational all-optical signal processing applications, including optical flip-flops [1–5], gate and packet switches [6,7], parity checkers [8], adders [9], and AND gates [10–12]. Many studies have been performed using a nonlinear photonic resonator exhibiting dispersive optical bistability, including distributed feedback (DFB) semiconductor optical amplifiers (SOAs) [2,3,13,14], vertical cavity SOAs (VC-SOAs) [4,11,15,16], and Fabry-Pérot SOAs (FP-SOAs) [5,10,13,17–21]. Previous studies of dispersive optical bistability have reported a variety of hysteresis shapes, including the canonical counter-clockwise shape (S-shape) [13–15,17–21], the clockwise shape (inverted S-shape) [13–16,20–25], and butterfly shapes [13,14,21,25].
Several of these bistable systems are capable of each producing more than one hysteresis shape. Counterclockwise (CCW), clockwise (CW), and butterfly shapes occur for reflective bistabtility from DFB-SOAs, VC-SOAs, FP-SOAs, with the shape controlled by either the SOA drive current (i.e., small-signal gain) [13,21,25] or the wavelength of the injected optical signal [14]; for these shape-control techniques, however, different hysteresis shapes exhibit different switching input powers. Both the CCW and CW shapes were also shown for dual-light injection into a FP-SOA, with the shape controlled by the wavelength of the non-dominant signal [20]. Control of the hysteresis shape has also been demonstrated using bistable systems other than a nonlinear photonic resonator, such as a coupled-laser, atomic-vapor system [26]; in this system, the hysteresis shape can be varied between a CW and CCW shape by tuning the wavelength of one of the lasers in the bistable system. In all of these cases, the hysteresis shape is controlled by a signal acting on the bistable system.
In this paper, we demonstrate a new control mechanism to produce a wide diversity of bistable hysteresis shapes. As depicted in Fig. 1, this new technique controls the hysteresis shape downstream of the nonlinear photonic resonator that generates the bistable signal; this shape control is performed without any signal sent upstream to the bistable system. The basic functional block are as follows: the Input Optical-Signal Generator produces the optical signal that is injected into the Nonlinear Resonator, which in turn generates the bistable optical signal. After a fiber distance $L$, the optical signal encounters the downstream Hysteresis-Shape Controller wherein the optical signal is controlled to produce a variety of hysteresis shapes. As will be shown, the CCW shape, the CW shape, and butterfly shapes are all selectable without changing any condition of the bistable system (such as drive current) or input optical signal (such as wavelength).
Fig. 1. Schematic of downstream control of the hysteresis shape; the bistable signal is generated within the nonlinear resonator and independent from (i.e., without any signal coming from) the downstream shape controller.
One advantage of this downstream technique is that all hysteresis shapes exhibit the same switching powers because these shapes originate from the same bistable signal generated by the upstream nonlinear resonator. Thus, logic gates based on the hysteresis shape may then be operated using the same control-pulse power irrespective of the particular shape. Previous demonstrations of shape diversity required different optical input powers to achieve different shapes [13,14,20,21,25], making the control-signal power of optical logic gates dependent on the selected hysteresis shape.
This paper is organized as follows. Section 2 describes the experimental set-up. Section 3 discusses the process of achieving optical bistability and polarization-mode-resolved hysteresis shapes. Section 4 demonstrates the diversity of hysteresis shapes.
2. Experimental set-up
Downstream hysteresis-shape control is demonstrated in the laboratory using the experimental set-up shown in Fig. 2. The Input Optical-Signal Generator, introduced in Fig. 1, consists of a tunable laser source (TLS), a Mach-Zehnder modulator (MZM), a signal generator (SG), and two polarization controllers, PC1 and PC2. TLS generates a continuous wave optical signal that is modulated by signal-generator-driven MZM in a sinusoidal fashion at 0.5 MHz. The polarization controller PC1 aligns the continuous-wave signal state of polarization (SOP) for optimal passage through an MZM having an insertion loss of 4 dB.
Fig. 2. (a) Experimental set-up of downstream hysteresis-shape control using a Fabry-Pérot semiconductor optical amplifier (FP-SOA) as the nonlinear resonator: TLS = tunable laser source; PC = polarization controller; SG = RF signal generator; MZM = Mach-Zehnder modulator; TE = transverse electric; TM = transverse magnetic; PLZR = polarizer; EDFA = erbium-doped fiber amplifier; OTF = optical tunable filter; OSA = optical spectrum analyzer; PLR = polarimeter; PD = photodiode.
The polarization controller PC2 orients the modulated-signal SOP to the desired SOP for injection into the FP-SOA, the nonlinear resonator used for this study. For all hysteresis data reported in this paper, the injected SOP is linearly polarized with 33% of the optical power injected into the transverse-magnetic (TM) mode of the FP-SOA. The FP-SOA itself is driven with an injection current of 62.5 mA to bias it at 97% of lasing threshold. The peak optical power entering the FP-SOA is −7.8 dBm.
After a distance of $L = 9$ m, the optical signal enters the Downstream Hysteresis-Shape Controller introduced in Fig. 1. This downstream controller is comprised of two components — a fiber-coupled polarization control PC4 followed by a linear polarizer (PLZR). PC4 alters the power passing through the polarizer by varying the SOP closer to or farther from the block-state of the polarizer. As discussed in Section 3, this SOP-dependent power passage is a key to produce the diversity of hysteresis shapes.
Input and output signals are measured by a suite of diagnostics. The input signal is measured from the monitor port of a 50/50 fiber-optic splitter located upstream of the FP-SOA. The input and output temporal powers are measured using 23-GHz photodiodes and a 1-GHz oscilloscope. The output signals are boosted with an Erbium-doped fiber amplifier (EDFA) that provides 16.2-dB gain for the optical-signal wavelength of 1607.09 nm used in our demonstrations. The 1.2-nm-bandwidth optical tunable filter (OTF) following the EDFA is centered on the signal wavelength to remove out-of-signal amplified spontaneous emission (ASE). The optical spectrum analyzer (OSA), polarimeter (PLR), and polarization controller PC3 are used to configure the optical signal as discussed in Section 3.
3. Polarization-mode-resolved hysteresis curves
Bistable hysteresis behavior at the output of the FP-SOA is readily obtainable. The optical-signal wavelength is tuned to the long-wavelength side of a Fabry-Pérot resonance, as shown in Fig. 3(a); four Fabry-Pérot resonances in the figure are labeled with "R" and are evident due to the ASE generated by the SOA. The hysteresis shape of the optical power exiting the FP-SOA is shown in Fig. 3(b); optical power in this plot (and in all hysteresis plots in this paper) is given in units of mV, which is proportional to optical power and is provided directly by the two photodiode-oscilloscope sets shown in Fig. 2. The formation of a CCW shape is common for dispersive optical bistability [14]. When the input power increases, the optical signal saturates the carrier density and thus increases the refractive index. Consequently, the interferometric resonance shifts towards the optical-signal wavelength, which leads to a general increase in output power. As the resonance snaps onto the signal wavelength, the output power switches to its higher branch. The signal remains on the upper branch of the hysteresis curve until the input power is decreased. As the input power falls, the optical signal allows the carrier density to recover and thus decreases the refractive index. The resonance then shifts away from the optical-signal wavelength, and as it snaps off of the optical signal, the output power switches to its lower hysteresis branch. (To observe spectral and temporal data directly after the FP-SOA, data shown in Figs. 3(a) and 3(b) are measured with the downstream hysteresis-shape controller components temporarily removed from the experimental set-up.)
Fig. 3. Bistable temporal powers and corresponding spectra, showing gain anisotropy, birefringence, and polarization-resolved hysteresis curves. (a) and (b) show the spectral and temporal power exiting the nonlinear resonator (FP-SOA), respectively; these pre-polarizer measurements show the total power and do not reveal polarization dynamics. (c) and (e) show the spectrum after the polarizer for the cases of injecting 67% and 33% of the optical power into the TE and TM modes of the FP-SOA, respectively. The Fabry-Pérot resonances are denoted by "R"; comparing the TM and TE resonances reveals gain anisotropy and birefringence. (d) and (f) show the polarization-resolved hysteresis curves, indicating a change in SOP during bistable action.
Recently, it was demonstrated that dispersive optical bistability can be accompanied by the simultaneous occurrence of nonlinear polarization rotation (NPR) [27]. In this case, there are two sets of interferometric resonances, one set belonging to the TE polarization mode of the nonlinear resonator and one set belonging to the TM polarization mode. Birefringence and gain anisotropy cause these sets of resonances to exhibit different strengths and different spectral locations with respect to each other. By the launching the optical signal into both polarization modes, its orthogonal components experience different variations in optical power and phase due to the anisotropic resonances, yielding the demonstrated polarization rotation during dispersive optical bistability [27].; as such, the state of polarization (SOP) of the lower stable branch curve [such as that shown in Fig. 3(b)] is different than that of the upper stable branch. Launching 67% and 33% of the optical signal into the TE and TM modes, respectively, was found to achieve a 30% orthogonality in SOP between the two stable states [27]. This optimal launch condition is used in the present work, as discussed below.
To support this polarization-selective optical injection, the downstream polarization controller PC4, polarizer, and OSA are first used to identify the FP-SOA polarization modes. By rotating PC4, it is readily found that the FP-SOA ASE can be polarization-resolved into two unique sets of resonances, one set for each FP-SOA polarization mode [28]. In this manner, the TM spectrum is identified as shown in Fig. 3(c) with four of the resonances labeled with $R_{TM}$. This figure also shows the unmodulated optical signal sent through the FP-SOA with PC2 set to provide a linear SOP with 33% of its power injected into the TM mode. In all cases shown in Sections 3 and 4, the small-signal optical signal wavelength is detuned 0.112 nm to the long-wavelength side of a TM resonance. (Detuning from a TE resonance is different because of birefringence, as discussed below.)
The hysteresis shown in Fig. 3(d) is the TM-resolved portion of the hysteresis curve of total power shown in Fig. 3(b). Like the latter, it exhibits the canonical CCW shape, indicating that the TM resonance shifts onto the optical-signal wavelength in the manner discussed in the first paragraph of this section. However, its switching contrast of 5.1 dB is noticeably better than that of the initial-hysteresis contrast of 2.0 dB shown in Fig. 3(b).
The TE spectrum is found by rotating PC4 to reveal the other set of FP-SOA spectra, highlighted in Fig. 3(e) using four resonances labeled with $R_{TE}$; Fig. 3(e) also shows the unmodulated optical signal sent through the FP-SOA with PC2 set to provide a linear SOP with 67% of its power injected into the TE mode. The high sensitivity of the OSA is essential for finding the weak resonance peaks of the TE spectrum. The different ASE power levels of the TM and TE resonances shown in Figs. 3(c) and 3(e) indicate gain anisotropy; specifically, the TM-mode gain is stronger than that of the TE mode [27]. Moreover, their relative displacement along the wavelength axis indicates birefringence; specifically, the spectral difference between peaks of the TE and TM resonances is 0.054 nm; (which corresponds to a 30% offset in terms of the 0.18-nm free-spectral range). For this amount of birefringence, the small-signal optical signal wavelength is detuned 0.058 nm to the long-wavelength side of a TE resonance.
The hysteresis curve shown in Fig. 3(f) is the TE-resolved portion of the hysteresis curve shown in Fig. 3(b). It exhibits a CW shape, an inverted form as compared to the TM-resolved hysteresis. The inverted nature of this shape indicates that a TE resonance shifts away from the optical-signal wavelength as the TM resonance shifts onto the signal; this shifting-away process ultimately drives the optical power of the TE signal downward. Thus, the TM resonance is the dominate contribution to realization of dispersive optical bistability.
The difference in behavior between the TE- and TM-resolved hysteresis curves clearly demonstrates that NPR occurs during optical bistability. Indeed, the difference of the optical power in the TE and TM modes, which varies significantly upon switching between hysteresis branches, defines the $s_1$ Stokes parameter; thus, the SOP is shown to vary in a bistable way.
To precisely achieve the 33/67 split into the TM/TE modes stated above, the input polarimeter (PLR) is calibrated using PC2 to maximize the signal power in the TM spectrum (not depicted in Fig. 3); doing so ensures that the SOP injected into the FP-SOA is aligned to the TM mode [27]. With the optical signal SOP aligned to the TM mode of the FP-SOA, the polarimeter (PLR) in the input-signal diagnostic arm is calibrated by varying PC3 to make the measured SOP equal the normalized Stokes parameter $s_1 = -1$. The 33/67 split was found to yield the highest polarization rotation between the low and high stable branches of the hysteresis curve [27]. This simultaneous occurrence of nonlinear polarization rotation and optical bistability [and shown in Figs. 3(d) and 3(f)] is leveraged in the next section to realize a variety of hysteresis shapes.
4. Hysteresis-shape diversity
A diversity of hysteresis shapes is achieved using the same downstream PC (PC4) and polarizer used in Section 3 to produce the polarization-mode-resolved hystereses. Four distinct hysteresis shapes are shown in Fig. 4 — a CCW shape, a CW shape, a downward-switching butterfly shape, and an upward-switching butterfly shape. The variety of these shapes is based on the NPR that can be made to accompany dispersive optical bistability [27], as discussed in the previous section. Since each stable branch of the bistable signal exhibits different SOPs, PC4 controls the relative power passing through the polarizer for the upper and lower branches; this relative power control is a key for realizing shape diversity as discussed below.
Fig. 4. Demonstration of hysteresis-shape control via the downstream shape controller depicted in Figs. 1 and 2. All shapes occur for the same bistable action occurring within the nonlinear resonator. The (a) high-contrast (20-dB) counter-clockwise shape is obtained by blocking the SOP at the center of the bottom bistable branch. The (b) high-contrast (21-dB) clockwise shape is obtained by blocking the SOP at the center of the upper bistable branch. The (c) downward-switching and (d) upward-switching butterfly shapes are obtained by blocking the two branches in a manner that matches the optical power near the middle of the input switching-power range.
The CCW hysteresis shape shown in Fig. 4(a) is similar in shape to the total-power and TM-resolved hysteresis shapes shown in Figs. 3(b) and 3(d), respectively; the CCW shape is also referred to as a S-shape because time-independent modeling of the hysteresis curve traces out an S shape that includes a dynamically unstable, reverse-slope middle branch [14]. The CCW shape is traversed first by upward switching from the low-output-power branch followed by downward switching from the high-output-power branch. However, the CCW shapes shown in Figs. 3(b) and 3(d) suffer from a low output-power contrast of 2.0 and 5.1 dB, respectively. This contrast is improved by using PC4 to orient the SOP at the middle of the lower branch of the hysteresis curve exiting the FP-SOA to the block state of the polarizer; the resulting contrast is measured to be 20.0 dB, an improvement of nearly 15 dB over that of the TM-resolved hysteresis curve.
The CW hysteresis shape shown in Fig. 4(b) is similar in shape to the TE-resolved hysteresis shape shown in Fig. 3(f); the CW shape is also referred to as a inverted-S-shape because time-independent modeling of the hysteresis curve traces out an upside-down S shape that includes a dynamically unstable, reverse-slope middle branch [14]. The CW shape is traversed first by downward switching from the high-output-power branch followed by upward switching from the low-output-power branch. However, the CW shape shown in Fig. 3(f) suffers from a low output-power contrast of 2.9 dB. This contrast is improved by using PC4 to orient the SOP at the middle of the upper branch of the hysteresis curve exiting the FP-SOA to the block state of the polarizer; the resulting contrast is measured to be 21.4 dB, and improvement of 18.5 dB over that of the TE-resolved hysteresis curve. Moreover, it is likely that the PD and oscilloscope diagnostic system artificially limits the measurement of the low-power branch power and therefore that of the contrast.
The butterfly shapes in Figs. 4(c) and 4(d) have a crossing point in the center of the switching region; at this point, the output power of either stable branch is the same. This crossing point is achieved by leveraging the different SOP exhibited by the two stable branches of the FP-SOA hysteresis curve shown in Fig. 3(b). Specifically, PC4 is oriented to block the SOP of the upper branch more than the SOP of the lower branch. This mechanism of achieving a butterfly shape is very different than that used for previous studies of reflective bistability from nonlinear resonators [13,14,21,25], which is based on the flipping of reflective resonances upside down during gain saturation.
The butterfly shapes in Figs. 4(c) and 4(d) differ in their switching dynamics. Whereas the shape in Fig. 4(c) is traversed first by downward switching and followed again by downward switching, the shape in Fig. 4(d) is traversed first by upward switching and followed again by upward switching. The switching dynamics are determined by the manner in which the SOP changes along a stable hysteresis branch. Downward switching occurs for a signal whose SOP initially varies away from the block-state of the polarizer, followed by a switching event that moves the SOP towards the block-state of the polarizer. Conversely, upward switching occurs for a signal whose SOP initially varies towards the block-state of the polarizer, followed by a switching event that moves the SOP away from the block-state of the polarizer. All four possible cases of switching dynamics are represented in Fig. 4 — 4(a) upward-downward, 4(b) downward-upward, 4(c) downward-downward, and 4(d) upward-upward switching
5. Conclusion
A diversity of bistable hysteresis shapes has been experimentally demonstrated by using a mechanism occurring downstream from the nonlinear resonator that gives rise to optical bistability; this shape control is performed without any signal sent upstream to the bistable system. The hysteresis shape can be controlled to realize the canonical CCW shape (S-shape), a CW shape (inverted S-shape), and butterfly shapes exhibiting either downward or upward switching. The selectability of these shapes allows for re-configurable combinational & sequential all-optical processing applications; for example, selecting between the CCW and CW hysteresis shapes would change the optical gate between AND and XOR functionality.
Several features demonstrated here are particularly useful for all-optical signal processing applications. First, all hysteresis shapes exhibit the same switching input powers, whereas previous techniques of realizing different hysteresis shapes required different switching input powers for different shapes [13,14,20,21,25]. This means that AND and XOR gates, for example, can be driven using the same-power control pulses. Second, the CCW and CW shapes are produced with switching contrasts at or exceeding 20 dB, which are substantially higher contrast values than shown by previous demonstrations (less than 10 dB) [14–25] . Third, shape diversity is available using the transmission port of the nonlinear photonic resonator, unlike most previous geometries based on dispersive optical bistability that exhibit shape diversity only in the reflection port [13–15,21,25]; the transmission port may lend itself to a more straightforward concatenation of photonic-logic gates.
In this work, a distance of $L = 9$ m was used between the nonlinear resonator and downstream shape controller. This length may be set by the specific application, and future work may study the impact of extending this length. Furthermore, a single bistable signal can be multicasted to several downstream shape controllers, supporting different gates at different locations. This multicasting ability leverages the downstream selectability capability of the demonstrated technique.
Our work is based on a FP-SOA that was not designed for the bistable polarization rotation at the heart of the demonstrated shape-control technique; further improvement in the switching contrast and hysteresis shapes may be possible through the development of a mathematical model encompassing both the bistable FP-SOA and the downstream control element. Also, although our investigation uses a Fabry-Pérot cavity as the resonator, the downstream shape control can be investigated using other resonator structures such as ring resonators, vertical cavities, and distributed feedback structures. Likewise, although our investigation used the linewidth-enhancement factor of an SOA as the nonlinearity, other nonlinear effects, such as the Kerr nonlinearity, can be used to realize shape diversity.
Disclosures
The authors declare no conflicts of interest.
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